A Mathematical Journal
Universidad de La Frontera
Vol. 7, No 3, December 2005.
SUMMARY
Fuzzy Taylor Formulae
....
1
....
15
....
27
....
39
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49
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65
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75
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87
George A. Anastassiou
Some special classes of neutral functional
differential equations
Constantin Corduneanu
Sufficiency of the maximum principle for time
optimality
H. O. Fattorini
The exact solution of the Potts models with
external magnetic field on the Cayley tree
Nasir Ganikhodjaev
The ergodic measures related with nonautonomous
hamiltonian systems and their homology structure.
Part 1
Denis L .Blackmore
Yarema A .Prykarpatsky
Anatoly M. Samoilenko
Anatoly K. Prykarpatsky
Conjectures in Inverse Boundary Value Problems for
Quasilinear Elliptic Equations
Ziqi Sun
Relations of al Functions over Subvarieties in a
Hyperelliptic Jacobian
Shigeki Matsutani
Convergence rates in regularization for ill-posed
variational inequalities
Nguyen Buong
A Mathematical Journal
Universidad de La Frontera
Vol. 7, No 3, December 2005.
CONTENTS
George A. Anastassiou Fuzzy Taylor Formulae
1
Constantin Corduneanu Some special classes of neutral functional
differential equations
H. O. Fattorini Sufficiency of the maximum principle for time optimality
15
27
Nasir Ganikhodjaev The exact solution of the Potts models with external
magnetic field on the Cayley tree
39
Denis L .Blackmore, Yarema A .Prykarpatsky, Anatoly M.
Samoilenko and Anatoly K. Prykarpatsky The ergodic measures related
with nonautonomous hamiltonian systems and their homology structure. Part 1
49
Ziqi Sun Conjectures in Inverse Boundary Value Problems for Quasilinear
Elliptic Equations
65
Shigeki Matsutani Relations of al Functions over Subvarieties in a
Hyperelliptic Jacobian
75
Nguyen Buong Convergence rates in regularization for ill-posed variational
inequalities
87
A Mathematical Journal
Vol. 7, No 3, (1 - 13). December 2005.
Fuzzy Taylor Formulae
George A. Anastassiou
Department of Mathematical Sciences University of Memphis
Memphis, TN 38152 U.S.A.
ganastss@memphis.edu
ABSTRACT
We produce Fuzzy Taylor formulae with integral remainder in the univariate
and multivariate cases, analogs of the real setting.
RESUMEN
Se presentan versiones Fuzzy análogas a las reales de fórmulas de Taylor con
resto integral en el caso univariado y multivariado.
Key words and phrases:
2000 AMS Subj. Class.:
1
Fuzzy Taylor formula, Fuzzy–Riemann integral
remainder, H-fuzzy derivative, fuzzy real analysis.
26E50.
Background
We need the following
Definition A (see [10]). Let µ : R → [0, 1] with the following properties.
(i) is normal, i.e., ∃x0 ∈ R; µ(x0 ) = 1.
2
George A. Anastassiou
7, 3(2005)
(ii) µ(λx + (1 − λ)y) ≥ min{µ(x), µ(y)}, ∀x, y ∈ R, ∀λ ∈ [0, 1] (µ is called a convex
fuzzy subset).
(iii) µ is upper semicontinuous on R, i.e., ∀x0 ∈ R and ∀ε > 0, ∃ neighborhood
V (x0 ): µ(x) ≤ µ(x0 ) + ε, ∀x ∈ V (x0 ).
(iv) The set supp(µ) is compact in R (where supp(µ) := {x ∈ R; µ(x) > 0}).
We call µ a fuzzy real number. Denote the set of all µ with RF .
E.g., X{x0 } ∈ RF , for any x0 ∈ R, where X{x0 } is the characteristic function at x0 .
For 0 < r ≤ 1 and µ ∈ RF define [µ]r := {x ∈ R: µ(x) ≥ r} and
[µ]0 := {x ∈ R : µ(x) > 0}.
Then it is well known that for each r ∈ [0, 1], [µ]r is a closed and bounded interval of
R. For u, v ∈ RF and λ ∈ R, we define uniquely the sum u ⊕ v and the product λ u
by
[u ⊕ v]r = [u]r + [v]r , [λ u]r = λ[u]r ,
∀r ∈ [0, 1],
where [u]r + [v]r means the usual addition of two intervals (as subsets of R) and λ[u]r
means the usual product between a scalar and a subset of R (see, e.g., [10]). Notice
1 u = u and it holds u ⊕ v = v ⊕ u, λ u = u λ. If 0 ≤ r1 ≤ r2 ≤ 1 then
(r)
(r)
(r)
(r)
(r)
(r)
[u]r2 ⊆ [u]r1 . Actually [u]r = [u− , u+ ], where u− ≤ u+ , u− , u+ ∈ R, ∀r ∈ [0, 1].
(r)
(r)
For λ > 0 one has λu± = (λ u)± , respectively.
Define
D : R F × RF → R+
by
(r)
(r)
(r)
(r)
D(u, v) := sup max{|u− − v− |, |u+ − v+ |},
r∈[0,1]
(r)
(r)
where [v]r = [v− , v+ ]; u, v ∈ RF . We have that D is a metric on RF . Then (RF , D)
is a complete metric space, see [10], with the properties
D(u ⊕ w, v ⊕ w) = D(u, v), ∀u, v, w ∈ RF ,
D(k u, k v) = |k|D(u, v), ∀u, v ∈ RF , ∀k ∈ R,
D(u ⊕ v, w ⊕ e) ≤ D(u, w) + D(v, e), ∀u, v, w, e ∈ RF .
Let f, g : R → RF be fuzzy number valued functions. The distance between f, g is
defined by
D∗ (f, g) := sup D(f (x), g(x)).
x∈R
(r)
(r)
On RF we define a partial order by “≤”: u, v ∈ RF , u ≤ v iff u− ≤ v− and
(r)
(r)
u+ ≤ v+ , ∀r ∈ [0, 1].
We mention
Fuzzy Taylor Formulae
7, 3(2005)
Lemma 2.2 ([5]). For any a, b ∈ R : a, b ≥ 0 and any u ∈ RF we have
D(a u, b u) ≤ |a − b| · D(u, õ),
where õ ∈ RF is defined by õ := X{0} .
Lemma 4.1 ([5]).
(i) If we denote õ := X{0} , then õ ∈ RF is the neutral element with respect to ⊕,
i.e., u ⊕ õ = õ ⊕ u = u, ∀u ∈ RF .
(ii) With respect to õ, none of u ∈ RF , u 6= õ has opposite in RF .
(iii) Let a, b ∈ R : a · b ≥ 0, and any u ∈ RF , we have (a + b) u = a u ⊕ b u.
For general a, b ∈ R, the above property is fale.
(iv) For any λ ∈ R and any u, v ∈ RF , we have λ (u ⊕ v) = λ u ⊕ λ v.
(v) For any λ, µ ∈ R and u ∈ RF , we have λ (µ u) = (λ · µ) u.
(vi) If we denote kukF := D(u, õ), ∀u ∈ RF , then k · kF has the properties of a usual
norm on RF , i.e.,
kukF
ku ⊕ vkF
= 0 iff u = õ, kλ ukF = |λ| · kukF ,
≤ kukF + kvkF , kukF − kvkF ≤ D(u, v).
Notice that (RF , ⊕, ) is not a linear space over R, and consequently (RF , k · kF )
is not a normed space.
We need
Definition B (see [10]). Let x, y ∈ RF . If there exists a z ∈ RF such that x = y + z,
then we call z the H-difference of x and y, denoted by z := x − y.
Definition 3.3 ([10]). Let T := [x0 , x0 + β] ⊂ R, with β > 0. A function f : T → RF
is H-differentiable at x ∈ T if there exists a f 0 (x) ∈ RF such that the limits (with
respect to metric D)
lim
h→0+
f (x + h) − f (x)
,
h
lim
h→0+
f (x) − f (x − h)
h
exist and are equal to f 0 (x). We call f 0 the derivative or H-derivative of f at x. If
f is H-differentiable at any x ∈ T , we call f differentiable or H-differentiable and it
has H-derivative over T the function f 0 .
The last definition was given first by M. Puri and D. Ralescu [9].
We need also a particular case of the Fuzzy Henstock integral (δ(x) = 2δ ) introduced
in [10], Definition 2.1.
That is,
3
4
George A. Anastassiou
7, 3(2005)
Definition 13.14 ([6], p. 644). Let f : [a, b] → RF . We say that f is Fuzzy-Riemann
integrable to I ∈ RF if for any ε > 0, there exists δ > 0 such that for any division
P = {[u, v]; ξ} of [a, b] with the norms ∆(P ) < δ, we have
!
X
∗
D
(v − u) f (ξ), I < ε,
P
where
P∗
denotes the fuzzy summation. We choose to write
Z b
I := (F R)
f (x)dx.
a
We also call an f as above (F R)-integrable.
We mention
Lemma 1 ([3]). If f, g : [a, b] ⊆ R → RF are fuzzy continuous functions, then the
function F : [a, b] → R+ defined by F (x) := D(f (x), g(x)) is continuous on [a, b], and
! Z
Z
Z
b
D (F R)
b
f (x)dx, (F R)
a
b
≤
g(x)dx
a
D(f (x), g(x))dx.
a
Lemma 2 ([3]). Let f : [a, b] → RF fuzzy continuous (with respect to metric D), then
D(f (x), õ) ≤ M , ∀x ∈ [a, b], M > 0, that is f is fuzzy bounded. Equivalently we get
χ−M ≤ f (x) ≤ χM , ∀x ∈ [a, b].
Lemma 3 ([3]). Let f : [a, b] ⊆ R → RF be fuzzy continuous. Then
Z x
(F R)
f (t)dt is a fuzzy continuous function in x ∈ [a, b].
a
Lemma 5 ([4]). Let f : [a, b] → RF have an existing H-fuzzy derivative f 0 at c ∈ [a, b].
Then f is fuzzy continuous at c.
We need
Rb
Theorem 3.2 ([7]). Let f : [a, b] → RF be fuzzy continuous. Then (F R) a f (x)dx
exists and belongs to RF , furthermore it holds
#
"
# r "Z
Z b
Z b
b
(r)
(r)
(F R)
f (x)dx =
(f )− (x)dx,
(f )+ (x)dx , ∀r ∈ [0, 1].
(1)
a
a
a
(r)
Clearly f± : [a, b] → R are continuous functions.
We also need
Theorem 5.2 ([8]). Let f : [a, b] ⊆ R → RF be H-fuzzy differentiable. Let t ∈ [a, b],
0 ≤ r ≤ 1. (Clearly
(r)
(r) [f (t)]r = (f (t))− , (f (t))+ ⊆ R.)
(2)
Fuzzy Taylor Formulae
7, 3(2005)
5
(r)
Then (f (t))± are differentiable and
(r)
(r) [f 0 (t)]r = ((f (t))− )0 , ((f (t))+ )0 .
(3)
The last can be used to find f 0 .
Here C n ([a, b], RF ), n ≥ 1 denotes the space of n-times fuzzy continuously Hdifferentiable functions from [a, b] ⊆ R into RF . By above Theorem 5.2 of [8] for
f ∈ C n ([a, b], RF ) we obtain
(r)
(r)
[f (i) (t)]r = ((f (t))− )(i) , ((f (t))+ )(i) ,
(4)
for i = 0, 1, 2, . . . , n and in particular we have
(i)
(r)
(f± )(r) = (f± )(i) ,
∀r ∈ [0, 1].
(5)
Definition 1. Let a1 , a2 , b1 , b2 ∈ R such that a1 ≤ b1 and a2 ≤ b2 . Then we define
[a1 , b1 ] + [a2 , b2 ] = [a1 + a2 , b1 + b2 ].
(6)
Let a, b ∈ R such that a ≤ b and k ∈ R, then we define,
if k ≥ 0,
if k < 0,
k[a, b] = [ka, kb],
k[a, b] = [kb, ka].
(7)
Here we use
Lemma 1. Let f : [a, b] → RF be fuzzy continuous and let g : [a, b] → R+ be continuous. Then f (x) g(x) is fuzzy continuous function ∀x ∈ [a, b].
Proof. The same as of Lemma 2 ([1]), using Lemma 2 of [3].
2
Main Results
We present the following fuzzy Taylor theorem in one dimension.
Theorem 1. Let f ∈ C n ([a, b], RF ), n ≥ 1, [α, β] ⊆ [a, b] ⊆ R. Then
f (β) = f (α) ⊕ f 0 (α) (β − α) ⊕ · · · ⊕ f (n−1) (α) ⊕
1
(F R)
(n − 1)!
Z
(β − α)n−1
(n − 1)!
β
(β − t)n−1 f (n) (t) dt.
(8)
α
The integral remainder is a fuzzy continuous function in β.
(r)
(r)
Proof. Let r ∈ [0, 1]. We have here [f (β)]r = [f− (β), f+ (β)], and by Theorem 5.2
(r)
([8]) f± is n-times continuously differentiable on [a, b]. By (5) we get
(i)
(r)
(f± (α))(r) = (f± (α))(i) ,
all i = 0, 1, . . . , n,
(9)
6
George A. Anastassiou
7, 3(2005)
and
(r)
(r)
[f (i) (α)]r = (f− (α))(i) , (f+ (α))(i) .
Thus by Taylor’s theorem we obtain
(r)
f± (β)
(r)
(r)
= f± (α) + (f± (α))0 (β − α)
(r)
+ · · · + (f± (α))(n−1)
(β − α)n−1
1
+
(n − 1)!
(n − 1)!
β
Z
(r)
(β − t)n−1 (f± )(n) (t)dt.
α
Furthermore by (9) we have
(r)
f± (β)
(r)
0
= f± (α) + (f±
(α))(r) (β − α)
(n−1)
+ · · · + (f±
(α)(r)
(β − α)n−1
1
+
(n − 1)!
(n − 1)!
Z
β
(n)
(β − t)n−1 (f± )(r) (t)dt.
α
Here it holds β − α ≥ 0, β − t ≥ 0 for t ∈ [α, β], and
(i)
(i)
(f− (t))(r) ≤ (f+ (t))(r) ,
∀t ∈ [a, b]
all i = 0, 1, . . . , n, and any r ∈ [0, 1].
We see that
(β − α)n−1
(n−1)
(r)
(r)
(r)
0
(α))(r)
(α))(r) (β − α) + · · · + (f−
f− (β), f+ (β)] = [f− (α) + (f−
(n − 1)!
Z β
1
(r)
(n)
+
(β − t)n−1 (f− )(r) (t)dt, , f+ (α)
(n − 1)! α
(β − α)n−1
(n−1)
0
(α))(r)
+ (f+
(α))(r) (β − α) + · · · + (f+
(n − 1)!
Z β
1
(n)
+
(β − t)n−1 (f+ )(r) (t) dt .
(n − 1)! α
To split the above closed interval into a sum of smaller closed intervals is where we
use β − α ≥ 0. So we get
[f (β)r ] =
(r)
(r)
(r)
(n−1)
+ · · · + [(f−
1
+ (n−1)!
=
(r)
0
0
[f− (β), f+ (β)] = [f− (α), f+ (α)] + [(f−
(α))(r) , (f+
(α))(r) ](β − α)
hR
β
(β
α
(n−1)
(α))(r) , (f+
n−1
(α))(r) ] (β−α)
(n−1)!
(n)
− t)n−1 (f− )(r) (t)dt,
Rβ
(n)
(β − t)n−1 (f+ )(r) (t)dt
α
i
n−1
[f (α)]r + [f 0 (α)]r (β − α) + · · · + [f (n−1) (α)]r (β−α)
(n−1)!
1
+ (n−1)!
hR
β
((β
α
(r)
− t)n−1 f (n) (t))− dt,
Rβ
α
i
(r)
((β − t)n−1 f (n) (t))+ dt .
Fuzzy Taylor Formulae
7, 3(2005)
7
By Theorem 3.2 ([7]) we next get
[f (β)]r
(β − α)n−1
= [f (α)]r + [f 0 (α)]r (β − α) + · · · + [f (n−1) (α)]r
(n − 1)!
#r
"
Z β
1
+
(F R)
(β − t)n−1 f (n) (t)dt .
(n − 1)!
α
Finally we obtain
[f (β)]r
=
(β − α)n−1
f (α) ⊕ f 0 (α) (β − α) ⊕ · · · ⊕ f (n−1) (α) (n − 1)!
r
Z β
1
(β − t)n−1 f (n) (t)dt , all r ∈ [0, 1].
⊕
(F R)
(n − 1)!
α
By Theorem 3.2 of [7] and Lemma 1 we get that the remainder of (8) is in RF , and by
Lemma 3 ([3]) is a fuzzy continuous function in β.
The theorem has
been proved.
Next we present a multivariate fuzzy Taylor theorem.
We need the following multivariate fuzzy chain rule. Here the H-fuzzy partial
derivatives are defined according to the Definition 3.3 of [10], see Section 1, and the
analogous way to the real case.
Theorem 3 ([2]). Let φi : [a, b] ⊆ R → φi ([a, b]) := Ii ⊆ R, i = 1, . . . , n, n ∈ N, are
strictly increasing and differentiable functions. Denote xi := xi (t) := φi (t), t ∈ [a, b],
i = 1, . . . , n. Consider U an open subset of Rn such that ×ni=1 Ii ⊆ U . Consider
f : U → RF a fuzzy continuous function. Assume that fxi : U → RF , i = 1, . . . , n,
the H-fuzzy partial derivatives of f , exist and are fuzzy continuous. Call z := z(t) :=
f (x1 , . . . , xn ). Then dz
dt exists and
n
dz X∗ dz
dxi
=
,
dt
dx
dt
i
i=1
∀t ∈ [a, b]
(10)
dz
where dz
dt , dxi , i = 1, . . . , n are the H-fuzzy derivatives of f with respect to t, xi ,
respectively.
The interchange of the order of H-fuzzy differentiation is needed too.
Theorem 4 ([2]). Let U be an open subset of Rn , n ∈ N, and f : U → RF be a
fuzzy continuous function. Assume that all H-fuzzy partial derivatives of f up to
order m ∈ N exist and are fuzzy continuous. Let x := (x1 , . . . , xn ) ∈ U . Then the
H-fuzzy mixed partial derivative of order k, Dx`1 ,...,x`k f (x) is unchanged when the
indices `1 , . . . , `k are permuted. Each `i is a positive integer ≤ n. Here some or all
of `i ’s can be equal. Also k = 2, . . . , m and there are nk partials of order k.
We give
8
George A. Anastassiou
7, 3(2005)
Theorem 2. Let U be an open convex subset of Rn , n ∈ N and f : U → RF be a
fuzzy continuous function. Assume that all H-fuzzy partial derivatives of f up to order
m ∈ N exist and are fuzzy continuous. Let z := (z1 , . . . , zn ), x0 := (x01 , . . . , x0n ) ∈ U
such that xi ≥ x0i , i = 1, . . . , n. Let 0 ≤ t ≤ 1, we define xi := x0i + t(zi − z0i ),
i = 1, 2, . . . , n and gz (t) := f (x0 + t(z − x0 )). (Clearly x0 + t(z − x0 ) ∈ U .) Then for
N = 1, . . . , m we obtain

!N 
n
X
∗
∂
f  (x1 , x2 , . . . , xn ).
(11)
gz(N ) (t) = 
(zi − x0i ) ∂x
i
i=1
Furthermore it holds the following fuzzy multivariate Taylor formula
f (z) = f (x0 ) ⊕
m−1 (N )
X
∗ gz (0)
⊕ Rm (0, 1),
N!
(12)
N =1
where
Rm (0, 1) :=
1
(F R)
(m − 1)!
Z
1
(1 − s)m−1 gz(m) (s)ds.
(13)
0
Comment. (Explaining formula (11)). When N = n = 2 we have (zi ≥ x0i , i = 1, 2)
gz (t) = f (x01 + t(z1 − x01 ), x02 + t(z2 − x02 )),
0 ≤ t ≤ 1.
We apply Theorems 3 and 4 of [2] repeatedly, etc. Thus we find
gz0 (t) = (z1 − x01 ) ∂f
∂f
(x1 , x2 ) ⊕ (z2 − x02 ) (x1 , x2 ).
∂x1
∂x2
Furthermore it holds
gz00 (t)
=
∂2f
(x1 , x2 ) ⊕ 2(z1 − x01 ) · (z2 − x02 )
∂x21
∂ 2 f (x1 , x2 )
∂2f
⊕ (z2 − x02 )2 (x1 , x2 ).
∂x1 ∂x2
∂x22
(z1 − x01 )2 (14)
When n = 2 and N = 3 we obtain
gz000 (t)
=
∂3f
(x1 , x2 ) ⊕ 3(z1 − x01 )2 (z2 − x02 )
∂x31
∂ 3 f (x1 , x2 )
∂ 3 f (x1 , x2 )
⊕ 3(z1 − x01 )(z2 − x02 )2 ·
2
∂x1 ∂x2
∂x1 ∂x22
∂3f
⊕ (z2 − x02 )3 (x1 , x2 ).
∂x32
(z1 − x01 )3 (15)
Fuzzy Taylor Formulae
7, 3(2005)
9
When n = 3 and N = 2 we get (zi ≥ x0i , i = 1, 2, 3)
gz00 (t)
∂2f
∂2f
(x1 , x2 , x3 ) ⊕ (z2 − x02 )2 (x1 , x2 , x3 )
2
∂x1
∂x22
∂2f
(x1 , x2 , x3 ) ⊕ 2(z1 − x01 )(z2 − x02 )
⊕ (z3 − x03 )2 ∂x23
∂ 2 f (x1 , x2 , x3 )
⊕ 2(z2 − x02 )(z3 − x03 )
∂x1 ∂x2
∂ 2 f (x1 , x2 , x3 )
∂2f
⊕ 2(z3 − x03 )(z1 − x01 ) (x1 , x2 , x3 ), (16)
∂x2 ∂x3
∂x3 ∂x1
= (z1 − x01 )2 etc.
Proof of Theorem 2. Let z := (z1 , . . . , zn ), x0 := (x01 , . . . , x0n ) ∈ U , n ∈ N, such
that zi > x0i , i = 1, 2, . . . , n. We define
xi := φi (t) := x0i + t(zi − x0i ),
Thus
dxi
dt
0 ≤ t ≤ 1;
i = 1, 2, . . . , n.
= zi − x0i > 0. Consider
Z := gz (t) := f (x0 + t(z − x0 ))
= f (x01 + t(z1 − x01 ), . . . , x0n + t(zn − x0n ))
= f (φ1 (t), . . . , φn (t)).
Since by assumptions f : U → RF is fuzzy continuous, also fxi exist and are fuzzy
continuous, by Theorem 3 (10) of [2] we get
dZ(x1 , . . . , xn )
dt
n
X
∗ ∂Z(x1 , . . . , xn )
dxi
=
∂x
dt
i
i=1
=
Thus
gz0 (t) =
n
X
∗ ∂f (x1 , . . . , xn )
(zi − x0i ).
∂xi
i=1
n
X
∗ ∂f (x1 , . . . , xn )
(zi − x0i ).
∂xi
i=1
Next we observe that
d2 Z
dt2
!
n
X
∗ ∂f (x1 , . . . , xn )
=
(zi − x0i )
∂xi
i=1
n
X
∗
d ∂f (x1 , . . . , xn )
=
(zi − x0i ) dt
∂xi
i=1


n
n
X
X
∗
∗ ∂ 2 f (x1 , . . . , xn )
=
(zi − x0i ) 
(zj − x0j )
∂xj ∂xi
i=1
j=1
gz00 (t)
=
d
=
dt
n
n
X
∗ X∗ ∂ 2 f (x1 , . . . , xn )
(zi − x0i ) · (zj − x0j ).
∂xj ∂xi
i=1 j=1
10
George A. Anastassiou
That is
gz00 (t) =
7, 3(2005)
n
n
X
∗ X∗ ∂ 2 f (x1 , . . . , xn )
(zi − x0i ) · (zj − x0j ).
∂xj ∂xi
i=1 j=1
The last is true by Theorem 3 (10) of [2] under the additional assumptions that fxi ;
∂2f
∂xj ∂xi , i, j = 1, 2, . . . , n exist and are fuzzy continuous.
Working the same way we find


n
n
2
3
X
X
∗
∗
∂ f (x1 , . . . , xn )
d 
d Z
= gz000 (t) =
(zi − x0i ) · (zj − x0j )
dt3
dt i=1 j=1
∂xj ∂xi
n
n
X
∗ X∗
d ∂ 2 f (x1 , . . . , xn )
(zi − x0i ) · (zj − x0j )
=
dt
∂xj ∂xi
i=1 j=1
"
#
n
n
n
X
X
∗ X∗
∗ ∂ 3 f (x1 , . . . , xn )
=
(zi − x0i ) · (zj − x0j )
(zk − x0k )
∂xk ∂xj ∂xi
i=1 j=1
k=1
n
n
n
X
∗ X∗ X∗ ∂ 3 f (x1 , . . . , xn )
=
(zi − x0i ) · (zj − x0j ) · (zk − x0k ).
∂xk ∂xj ∂xi
i=1 j=1
k=1
Therefore,
gz000 (t) =
n
n
n
X
∗ X∗ X∗ ∂ 3 f (x1 , . . . , xn )
(zi − x0i ) · (zj − x0j ) · (zk − x0k ).
∂xk ∂xj ∂xi
i=1 j=1
k=1
That last is true by Theorem 3 (10) of [2] under the additional assumptions that
∂ 3 f (x1 , . . . , xn )
,
∂xk ∂xj ∂xi
do exist and are fuzzy continuous.
N = 1, . . . , m ∈ N,
gz(N ) (t) =
n
n
X
∗ X∗
i1 =1 i2 =1
···
i, j, k = 1, . . . , n
Etc.
In general one obtains that for
n
X
∗
iN
N
Y
∂ N f (x1 , . . . , xn )
(zi − x0ir ),
∂xiN ∂xiN −1 · · · ∂xi1 r=1 r
=1
which by Theorem 4 of [2] is the same as (11) for the case zi > x0i , see also (14),
(15), and (16). The last is true by Theorem 3 (10) of [2] under the assumptions that
all H-partial derivatives of f up to order m exist and they are all fuzzy continuous
including f itself.
Next let tm̃ → t̃, as m̃ → +∞, tm̃ , t̃ ∈ [0, 1]. Consider
xim̃ := x0i + tm̃ (zi − x0i )
and
x̃i := x0i + t̃(zi − x0i ),
i = 1, 2, . . . , n.
Fuzzy Taylor Formulae
7, 3(2005)
11
That is
xm̃ = (x1m̃ , x2m̃ , . . . , xnm̃ ) and x̃ = (x̃1 , . . . , x̃n ) in U .
Then xm̃ → x̃, as m̃ → +∞. Clearly using the properties of D-metric and under the
theorem’s assumptions, we obtain that
gz(N ) (t) is fuzzy continuous for N = 0, 1, . . . , m.
Then by Theorem 1, from the univariate fuzzy Taylor formula (8), we find
(m−1)
gz (1) = gz (0) ⊕ gz0 (0) ⊕
gz
(0)
gz00 (0)
⊕ ··· ⊕
⊕ Rm (0, 1),
2!
(m − 1)!
where Rm (0, 1) comes from (13).
By Theorem 3.2 of [7] and Lemma 1 we get that Rm (0, 1) ∈ RF . That is we get
the multivariate fuzzy Taylor formula
(m−1)
f (z) = f (x0 ) ⊕ gz0 (0) ⊕
(0)
gz00 (0)
gz
⊕ ··· ⊕
⊕ Rm (0, 1),
2!
(m − 1)!
when zi > x0i , i = 1, 2, . . . , n.
Finally we would like to take care of the case that some x0i = zi . Without loss of
generality we may assume that x01 = z1 , and zi > x0i , i = 2, . . . , n. In this case we
define
Z̃ := g̃z (t) := f (x01 , x02 + t(z2 − x02 ), . . . , x0n + t(zn − x0n )).
Therefore one has
g̃z0 (t) =
n
X
∗ ∂f (x01 , x2 , . . . , xn )
(zi − x0i ),
∂xi
i=2
and in general we find
g̃z(N ) (t) =
n
X
∗
i2 =2,...,iN
N
∂ N f (x01 , x2 , . . . , xn ) Y
(zir − x0ir ),
∂xiN ∂xN −1 · · · ∂xi2
r=2
=2
(N )
for N = 1, . . . , m ∈ N. Notice that all g̃z
g̃z (0) = f (x01 , x02 , . . . , x0n ),
, N = 0, 1, . . . , m are fuzzy continuous and
g̃z (1) = f (x01 , z2 , z3 , . . . , zn ).
(N )
Then one can write down a fuzzy Taylor formula, as above, for g̃z . But g̃z (t)
(N )
coincides with gz (t) formula at z1 = x01 = x1 . That is both Taylor formulae in
that case coincide.
At last we remark that if z = x0 , then we define Z ∗ := gz∗ (t) := f (x0 ) =: c ∈ RF a
constant. Since c = c + õ, that is c − c = õ, we obtain the H-fuzzy derivative (c)0 = õ.
Consequently we have that
gz∗(N ) (t) = õ,
N = 1, . . . , m.
12
George A. Anastassiou
7, 3(2005)
(N )
The last coincide with the gz
formula, established earlier, if we apply there
z = x0 . And, of course, the fuzzy Taylor formula now can be applied trivially for
gz∗ . Furthermore in that case it coincides with the Taylor formula proved earlier for
gz . We have established a multivariate fuzzy Taylor formula for the case of zi ≥ x0i ,
i = 1, 2, . . . , n. That is (11)–(13) are true.
Note. Theorem 2 is still valid when U is a compact convex subset of Rn such that
U ⊆ W , where W is an open subset of Rn . Now f : W → RF and it has all the
properties of f as in Theorem 2. Clearly here we take x0 , z ∈ U .
Received:
March 2003.
Revised:
July 2003.
References
[1]
George A. Anastassiou, Fuzzy wavelet type operators, submitted.
[2]
George A. Anastassiou, On H-fuzzy differentiation, Mathematica Balkanica,
New Series, Vol. 16 Volumen Fasc. 1-4 (2002), 153-193.
[3]
George A. Anastassiou, Rate of convergence of fuzzy neural network operators, univariate case, Journal of Fuzzy Mathematics, 10, No. 3 (2002),
755–780.
[4]
George A. Anastassiou, Univariate fuzzy-random neural network approximation operators, submitted.
[5]
George A. Anastassiou and Sorin Gal, On a fuzzy trigonometric approximation theorem of Weierstrass-type, Journal of Fuzzy Mathematics, 9, No. 3
(2001), 701–708.
[6]
S. Gal, Approximation theory in fuzzy setting. Chapter 13 , Handbook of
Analytic Computational Methods in Applied Mathematics (edited by G.
Anastassiou), Chapman & Hall CRC Press, Boca Raton, New York, 2000,
pp. 617–666.
[7]
R. Goetschel , Jr. and W. Voxman, Elementary fuzzy calculus, Fuzzy Sets
and Systems, 18 (1986), 31–43.
[8]
O. Kaleva,Fuzzy differential equations, Fuzzy Sets and Systems, 24 (1987),
301–317.
[9]
M. L. Puri and D. A. Ralescu, Differentials of fuzzy functions, J. of Math.
Analysis & Appl., 91 (1983), 552–558.
7, 3(2005)
Fuzzy Taylor Formulae
13
[10]
Congxin Wu and Zengtai Gong, On Henstock integral of fuzzy number valued
functions (I), Fuzzy Sets and Systems, 120, No. 3, 2001, 523–532.
[11]
L. A. Zadeh, Fuzzy sets, Information and Control, 8, 1965, 338–353.
A Mathematical Journal
Vol. 7, No 3, (15 - 26). December 2005.
Some special classes of neutral
functional differential equations
Constantin Corduneanu
The University of Texas at Arlington
Box 19408, UTA, Arlington TX 76019
concord@uta.edu
ABSTRACT
This paper is dedicated to the investigation of existence, mainly local, of solutions of two classes of neutral functional differential equations. A reduction
method and fixed point methods are emphasized.
RESUMEN
Este artı́culo está dedicado al estudio de existencia, principalmente local, de
soluciones de dos clases de ecuaciones diferenciales funcionales neutrales. Un
método de reducción y de punto fijo son puestos con algún énfasis.
Key words and phrases:
Math. Subj. Class.:
1
functional equation, neutral equation, existence
of solution
34K40
Introduction
In several recent papers of the author [3], [4], [5], as well as in some joint papers
with M. Mahdavi [7], [8], certain types of neutral functional equations (including
functional–differential ones) have been investigated in regard to the existence of so-
16
Constantin Corduneanu
7, 3(2005)
lutions. Such equation, in a rather general form, can be written as
(U x)(t) = (V x)(t), t ∈ [0, T ],
(1)
or, in functional differential form,
d
(U x)(t) = (V x)(t), t ∈ [0, T ].
dt
(2)
In (1) and (2), U and V stand for some operators acting on convenient function
spaces, whose elements are defined on [0, T ]. The case [0, T ), T ≤ ∞, has been also
discussed, while the solution has been sought in various function spaces (usually, C,
Lp , 1 ≤ p < ∞).
Let us notice that equation (2), by integrating both sides on [0, t], t ≤ T , takes
the form (1):
Z t
(U x)(t) = C +
(V x)(s)ds, t ∈ [0, T ].
(3)
0
Moreover, when V is a causal operator, the right hand side of (3) is also causal. The
case of causal operators has been dealt with in the author’s recent book [6].
The basic method of investigating the existence of solutions to equations(1) and
(2) consisted in reducing such equations to the simpler form
x(t) = (W x)(t), t ∈ [0, T ],
(4)
and then applying existing results available for (4). In our book [2] we provided existence results for (4), based on the existing literature, while in [6] we have illustrated
how this method works for neutral equations like (1) or (2).
The aim of this paper is to investigate some classes of neutral equations encountered in the existing literature, by using mainly the above described method. In other
words, to reduce such equations to the form (4), and then to apply known results.
We are not necessarily intended to reobtain results already known in the literature,
but to see what kind of results one obtains by means of the above described method.
We shall particularly refer to the papers of T.A. Burton [1] and Loris Faina [9], in
which various classes of functional differntial equations, of neutral type, are investigated.
2
Reduction of neutral equations to the form (4)
Let us start with the neutral equation
x0 (t) = f (t, x(t), x0 (t)), t ∈ [0, T ],
(5)
investigated by Loris Faina [9] and other authors. This is the simplest form, in which
x and f take scalar values, or values belonging to IRn .
The initial value condition attached to (5) will be
x(0) = x0 ∈ IRn ,
(6)
7, 3(2005)
Some special classes of neutral functional differential equations
17
if one deals with the vector case.
Formally, let us denote
x0 (t) = y(t),
which implies under rather general assumptions (see below)
Z t
0
x(t) = x +
y(s)ds.
(7)
(8)
0
The neutral equation (5) can be now written as
Z t
0
y(s)ds, y(t) .
y(t) = f t, x +
(9)
0
Obviously, the right hand side of (9) engages only the values of y on the interval
0 ≤ s ≤ t (t ≤ T ).
This means that equation (9) is an equation of the form (4), in which the right
hand side is a causal operator (in y). There is, therefore, an equivalence between
the initial value problem (5), (6), and the problem (8), (9). This equivalence will be
further discussed when we precise the underlying function spaces. Since equation (9)
contains only the unknown y(t), we shall be able to investigate it in various function
spaces (continuous or measurable functions), by using known results.
In L. Faina [9] there are more general neutral functional differential equations than
(5). For instance, in (5) one assumes that f is a map from R × C(R) × L1 (R) into
IRn , while the initial condition (6) is replaced by x(t) = ϕ(t), t ∈ (−∞, t0 ], t0 ∈ R
fixed. In other terms, the infinite delay is dealt with. This case can be also covered
by the scheme described above, though the procedure is more intricate.
In T.A. Burton [1], the following neutral functional differential equation is studied
x0 (t) = f (t, x(t), x0 (t − h(t)) + g(t, x(t), x(t − h(t)),
where 0 ≤ h(t) ≤ h0 , h0 > 0 being fixed. To the above equation one attaches the
typical initial condition for delay equations, namely x(t) = ϕ(t), t ∈ [−h0 , 0].
We shall rewrite Burton’s equation in the form
x0 (t) = f (t, x(t), x0 (α(t))) + g(t, x(t), x(α(t))),
(10)
where α(t) is such that 0 ≤ α(t) ≤ t on some interval [0, T ]. The right hand side in
(10) can be regarded as a causal operator in x. Hence, (10) is also of the form (4).
Moreover, one can use the initial condition (6) for determining a (unique) solution to
(10), (6).
The literature on neutral functional equations is very rich, and a good amount of
references can be found in our book [6]. In many more cases than those illustrated
above, the method of reduction to equations of the form (1), with causal operator in
the right hand side, can be successfully applied.
In what follows, we shall dwell on the equations (5) and (9), trying to apply the
reduction procedure in these cases, as well as other methods.
18
Constantin Corduneanu
3
7, 3(2005)
The equation (9) in the space C([0, T ], IRn )
We shall consider in this section the equation (9) in the space C([0, T ], IRn ). The
assumption to be made are of such a nature that the right hand side of this equation
represent a compact operator on C([0, T1 ], IRn ), with T1 ≤ T. Obviously, the ArzelàAscoli criterion of compactness in C([0, T ], IRn ) will be used.
The compactness of the operator
y(t) −→ f
t, x0 +
t
Z
y(s)ds, y(t) ,
(11)
0
on C([0, T ], IRn ) can be achieved under various sets of hypotheses. We shall describe
such a set of hypeotheses, which will imply the existence of a local solution to the
equation (9). Such a solution will generate a continuously differentiable solution to
the problem (5), (6).
Before we proceed with the statement of the hypotheses, it is instructive to look
at a simple example for the equation (5). Namely, we will choose f = 2x(t)x0 (t) + 1.
This leads to the integral x(t) = x2 (t) + t + c, from which we derive x(t) =
√
1
1 + 1 − 4c − 4t . Choosing the initial value x0 = 1/2, we get c = 1/4, which
2
√ 1
means the solution is x(t) =
1 + 2 −t . This shows that we have no solutions of
2
(5) on any [0, T ], T > 0, for x0 = 1/2.
Therefore, a problem of the form (5), (6) may be deprived of (local) solutions,
even though the right hand side of (5) is quite a usual function.
We shall return now to the general problem (5), (6), and provide some conditions
which assure the existence of local solutions. But before getting into details, we
shall modify somewhat the equation (5), in order to encompass a larger category of
situations. Namely, let us rewrite (9) in the form
y(t) = f
t, x0 +
Z
t
y(s)ds; y ,
(9)0
0
and assume the right hand side in (9)0 is defined on the set [0, T ]×IRn ×C([0, T ], IRn ).
We shall assume continuity of the map y −→ f, but further hypotheses will be formulated. Compared to (9), the equation (9)0 involves now an operator in the right
hand side, defined on the space C([0, T ], IRn ).
The following hypotheses will be made, in view of obtaining the existence of solutions to the equation (9)0 :
H1 The map
y −→ f
t, x0 +
Z
t
y(s)ds; y
0
is continuous from [0, T ] × IRn × C([0, T ], IRn ) into IRn , and causal.
(12)
Some special classes of neutral functional differential equations
7, 3(2005)
19
H2 For each γ > 0, there exist two functions ω1 (r) and ω2 (r), continuous on [0, ∞),
ω1 (0) = ω2 (0) = 0, and positive for r > 0, such that
Z u
Z t
0
f t, x0 +
y(s)ds; y ≤
y(s)ds; y − f u, x +
(13)
0
0
≤ ω1 (|t − u|) + ω2 (γ|t − u|),
for arbitrary t, u ∈ [0, T ], and all y ∈ C, with |y|C ≤ γ.
Let us notice the fact that choosing ω1 (r) = αr, ω2 (r) = βr, α, β > 0, the
condition (13) becomes a Lipschitz type continuity condition.
H3 The map (12) takes bounded sets in C([0, T ], IRn ), into bounded sets of IRn .
We can now prove the following (local) existence theorem for the equation (9)0 .
Theorem 1. Consider the functional equation (9)0 in the space C([0, T ], IRn ). Assume that the map (12) satisfies the hypotheses H1 , H2 and H3 . Then equation (9)0
has a local solution (i.e. defined on some interval [0, T1 ], T1 ≤ T ), provided f (0, x0 ; y)
is independent of y.
Moreover, the equation
x0 (t) = f (t, x(t); x0 ),
(5)0
under initial condition (6), has a local solution, which is continuously differentiable.
Remark 1. The localization is possible due to the causality of the operator (12)
(hypothesis H1 ).
Remark 2. In case f does not depend on the last argument, the equation (5)0
becomes x0 (t) = f (t, x(t)), while hypotheses H2 and H3 are automatically satisfied in
case of continuity. The result of Theorem 1 reduces to the classical Peano’s existence
theorem. The existence of the functions ω1 (r) and ω2 (r) is a simple consequence of
the uniform continuity of f (t, x) on a set of the form [0, T ]×B, with B compact in IRn .
The
of hypothesis
H2 is motivated by the fact that the last argument in
imposition
Z
t
f
t, x0 +
y(s)ds; y
belongs to an infinite dimensional space, i.e. to C([0, T ], IRn ).
0
Proof of Theorem 1. The equation (9)0 is, according to our hypotheses, a functional
equation with causal operaor of the form (4). The hypotheses H1 and H2 assure the
continuity of the map (12) from C([0, T ], IRn ) into itself. Based on hypothesis H3 , the
map (12) from C([0, T ], IRn ) into C([0, T ], IRn ) is also compact. Indeed, according to
the hypothesis H3 , the image of the ball |y|C ≤ γ is a bounded set in C([0, T ], IRn ).
The inequality (13) in hypothesis H2 tells us that the image of the ball |x|C ≤ γ
consists of equicontinuous functions on [0, T ], with values in IRn . Since γ > 0 is an
arbitrary number, we conclude that the operator (12) is compact (takes bounded sets
into relatively compact sets). Hence Theorem 3.1 in [6] applies directly, keeping also
20
Constantin Corduneanu
7, 3(2005)
in mind that the operator (12) enjoys the property of fixed initial value. Consequently,
(9)0 has a local solution in some space C([0, T1 ], IRn ), with T1 ≤ T.
This result leads immediately to the existence of a local solution for the problem
(5)0 , 6. This ends the proof of Theorem 1.
Remark 3. The case of measurable solutions to the equation (9)0 , when the corresponding solutions to (5)0 will be absolutely continuous functions, can be treated
in the same manner as in the continuous case. One has to use Theorem 3.3 in [6],
instead of Theorem 3.1. We shall leave to the reader the task of formulating existence
results.
4
Existence of solutions to equation (10)
If we denote again x0 (t) = y(t), and take into account the initial condition (6), then
equation (10) becomes
Z t
0
y(t) = f t, x +
y(s)ds, y(α(t)) +
0
(14)
Z t
+g t, x0 +
y(s)ds, y(α(t)) ,
0
which is precisely of the form (4), with causal operator in the right hand side. This
is due to the assumption on α(t), namely 0 ≤ α(t) ≤ t, t ∈ [0, T ].
We shall consider now a particular case of equation (14), as far as the function
g is concerned. Instead of the term f , we shall consider another operator-like term.
More precisely, we shall deal with the functional equation
Z t
(W y)(s)ds,
(15)
y(t) + g(y(α(t))) = C +
0
under the following assumptions:
1) g : C([0, T ], IRn ) −→ C([0, T ], IRn ) is a contraction map on this space:
|g(x) − g(y)|C ≤ λ|x − y|C , λ ∈ [0, 1);
2) W : C([0, T ], IRn ) −→ C([0, T ], IRn ) is a continuous causal operator, taking
bounded sets of C([0, T ], IRn ) into bounded sets;
3) α : [0, T ] −→ [0, ∞) is continuous, and such that α(0) = 0 and 0 ≤ α(t) ≤ t for
t ∈ [0, T ].
Remark 4. The vector C ∈ IRn is arbitrary, but it can be chosen in such a way to
satisfy some kind of initial condition. For instance, if we assign to y the initial value
y 0 , and assume (without loss of generality) that g(θ) = θ ∈ IRn , then one obtains
C = y0 .
Some special classes of neutral functional differential equations
7, 3(2005)
21
In regard to the equation (15), the following existence result can be stated:
Theorem 2. Consider the functional equation (15), under conditions 1), 2), 3) stated
above.
Then, there exists a solution y = y(t), defined on some interval
[0, T1 ] ⊂ [0, T ], for each C ∈ IRn . This solution is such that y(t) + g(y(α(t))) is
continuously differentiable.
Proof. The hypotheses accepted are of such a nature that allow the application of
Theorem 6.1 in [6], which yields the existence result.
The idea of proof is based on the fact that the functional equation y(t)+g(y(α(t))) =
f (t) is uniquely solvable in C([0, T ], IRn ). Moreover, y(t) depends continuously of f (t),
which allows to deal with (15) by contraction mapping principle, or by another fixed
point methods.
Details of this approach can be found in our paper [5], where further existence
results are obtained.
5
Further considerations on equation (10)
The idea of proof mentioned above can be adapted to other functional equations. For
an illustration we will consider the equation (10), as well as the auxiliary equation
x0 (t) = g(t, x(t), x(α(t))) + f (t),
(16)
with f ∈ C([0, T ], IRn ). The attached initial condition is (6). The functional integral
equation equivalent to (16), (6) is
x(t) = x0 +
Z
t
Z
f (s)ds +
0
t
g(s, x(s), x(α(s)))ds.
(17)
0
Let us assume the following conditions on the data in equation (17):
1) g : [0, T ] × IRn × IRn −→ IRn is continuous, and satisfies the Lipschitz condition
|g(t, x, y) − g(t, x̄, ȳ)| ≤ L(|x − x̄| + |y − ȳ|),
with L > 0;
2) f ∈ C([0, T ], IRn );
3) α(t) is continuous on [0, T ], and α(0) = 0, 0 ≤ α(t) ≤ t.
It is easy to see that the usual process of iteration leads to the following relationship:
x(k+1) (t) − x(k) (t) =
Z th i
=
g s, x(k) (s), x(k) (α(s)) − g s, x(k−1) (s), x(k−1) (α(s)) ds, k ≥ 1,
0
22
Constantin Corduneanu
with
x(0) (t) = x0 +
7, 3(2005)
t
Z
f (s)ds.
0
We further derive on behalf of condition 1)
|x(k−1) (t) − x(k) (t)| ≤
≤L
Z th
i
|x(k) (s) − x(k−1) (s)| + |x(k) (α(s)) − x(k−1) (α(s))| ds,
0
which leads to
sup |x(k+1) (s) − x(k) (s)| ≤ 2L
0≤s≤t
Z
t
sup |x(k) (u) − x(k−1) (u)|ds,
(18)
0 0≤u≤s
if we keep in mind that 0 ≤ α(t) ≤ t.
The inequality (18) can be processed in the usual manner, and one finds that
lim x(k) (t) = x(t) as k −→ ∞, uniformly on [0, T ], with x(t) satisfying (16). The
uniqueness can be also proven by the standard method, as well as the continuous
dependence of the solution with respect to f ∈ C([0, T ], IRn ).
The auxiliary result established above enables us to make some progress in regard
to the equation (10). We rewrite it for the reader’s convenience,
x0 (t) = g(t, x(t), x(α(t))) + f (t, x(t), x0 (α(t))),
and regard it as a (nonlinear) perturbed equation associated to (16). Of course, we
preserve the initial condition (6).
The following fixed point scheme can be attached to the equation (19): for each
continuously differntiable u(t) on [0, T ], with values inIRn , we shall attach the unique
solution x(t) of the equation like (16)
x0 (t) = g(t, x(t), x(α(t))) + f (t, u(t), u0 (α(t)))
(19)
with the initial condition (6). The existence and uniqueness of x(t), under the above
scheme, is guaranteed under the conditions 1), 2) and 3) specified above. Consequently, in the space C([0, T ], IRn ), or rather in the space C (1) ([0, T ], IRn ), we have
defined an opeator u −→ x, where u and x are related by the equation (19), with x
satisfying also (6).
Let us denote by V the operator defined above, i.e.
x(t) = (V u)(t), t ∈ [0, T ],
(20)
with u and x as described above. The operator V appears as a compound operator:
first, u −→ f (t, u(t), u0 (α(t))), and second f −→ x, with x the solution of (19), (6).
As noticed earlier in this section, the second operator is continuous on C([0, T ], IRn ).
The first operator involved, u −→ f (t, u(t), u0 (α(t))) can be made continuous, under
adequate hypotheses on the function f . It is obviously continuous from C (1) ([0, T ], IRn )
into C([0, T ], IRn ) when f (t, u, v) is continuous on [0, T ] × IRn × IRn .
7, 3(2005)
Some special classes of neutral functional differential equations
23
Instead of pursuing the above scheme, which can certainly lead to results of existence for (10), we shall attempt to apply the contraction mapping principle to the
equation (10), but modifying somewhat the scheme presented above. Namely, we will
consider the scheme described by the following equation, attached to (10):
x0 (t) = g(t, x(t), x(α(t))) + f (t, x(t), u0 (α(t))).
(21)
By means of (21) and (6), we shall define the operator on C (1) ([0, T ], IRn ), say x(t) =
(W u)(t), in the following manner. Given u ∈ C (1) ([0, T ], IRn ), the equation (21)
can be solved in x under rather mild assumptions (as seen above, under Lipschitz
condition). The unique solution of (21), (6) will be denoted by x(t) = (W u)(t).
From (2) we derive the following relationship between x = W u and y = W v, where
u, v ∈ C (1) ([0, T ], IRn ):
x0 (t) − y 0 (t) = g(t, x(t), x(α(t))) − g(t, y(t), y(α(t)))+
+f (t, x(t), u0 (α(t))) − f (t, y(t), v 0 (α(t))).
(22)
Assuming also a Lipschitz condition on f (t, x, y), as we did already on g(t, x, y), we
obtain
|x0 (t) = y 0 (t)| ≤ L(|x(t) − y(t)| + |x(α(t)) − y(α(t))|)+
+M |x(t) − y(t)| + m|u0 (α(t)) − v 0 (α(t))|,
for any t ∈ [0, T ], where L, M and m are positive numbers. The above inequality
yields
sup |x0 (t) − y 0 (t(| ≤ (2L + M ) sup |x(t) − y(t)|+
(23)
+m sup |u0 (α(t)) − v 0 (α(t))|,
with sup taken on [0, T ], or on any [0, T1 ], T1 ≤ T.
But
Z t
x(t) − y(t) =
[x0 (s) − y 0 (s)]ds,
(24)
0
because x(0) = y(0) = x0 , according to (6). From (23) we derive
sup |x(t) − y(t)| ≤ T sup |x0 (s) − y 0 (s)|,
(25)
with sup taken on [0, T ]. Taking into account (23), (24) and (25) we obtain
sup |x0 (t) − y 0 (t)| ≤ (2L + M )T sup |x0 (t) − y 0 (t)|+
+m sup |u0 (t) − v 0 (t)|.
(26)
Since we want (26) to be a relation showing the fact that the operator W is a contraction on C (1) ([0, T ], IRn ), we see from (26) that a first condition to be imposed
is
(2L + M )T < 1.
(27)
If we admit (27), then (26) allows us to write
sup |x0 (t) − y 0 (t)| ≤ m[1 − (2L + M )T ]−1 sup |u0 (t) − v 0 (t)|,
24
Constantin Corduneanu
7, 3(2005)
which really represents a contraction condition for W , as soon as
λ = m[1 − (2L + M )T ]−1 < 1.
(28)
It is appropriate to notice the fact that the norm in C (1) ([0, T ], IRn ) is (by our choice)
|x0 | + sup |x0 (t)|.
(29)
Accordingly, the norm for u − v should be |u0 − v 0 | + sup |u0 (t) − v 0 (t)|, which is in
advantage of the contraction inequality
|W u − W v|C (1) ≤ λ|u − v|C (1) ,
(30)
as derived from above.
Therefore, we can now state the following (global) existence result for the problem
(10), (6):
Theorem 3. Consider the problem (10), (6), and assume the following conditions
are verified by the functions f and g:
1) f, g : [0, T ] × IRn × IRn −→ IRn are continuous maps;
2) f and g satisfy the Lipschitz type conditions
|f (t, x, y) − f (t, x̄, ȳ)| ≤ L(|x − x̄| + |y − ȳ|),
|g(t, x, y) − g(t, x̄, ȳ)| ≤ M |x − x̄| + m|y − ȳ|,
with positive constants L, M and m;
3) the inequalities (27) and (28) are satisfied.
Then, there exists a unique solution x(t) ∈ C (1) ([0, T ], IRn ), which can be approximated by the scheme described by the equation (21).
The proof of Theorem 3 has been carried out above, before its statement.
Remark 5. It is obvious from the inequalities (27) and (28) that severe restrictions
must be imposed to the constants L, M, m and T .
First, if L, M are fixed, it is obvious that the inequality (27) can be satisfied
provided we choose T small enough: T < (2L + M )−1 . This restriction suggests that
we need to confine our investigation, possibly, to a smaller interval than the original
interval [0, T ]. But this kind of restriction is in accordance with the fact we are looking
for local solutions to our problem.
Second, once we choose T such that (27) takes place, there remains the inequality
(28) to be satisfied. If the constants L, M and T are fixed, then the only way to
satisfy (28) is to choose m small enough.
In conclusion, local existence for the problem (10), (6) is always assured by choosing the constant m sufficiently small.
7, 3(2005)
Some special classes of neutral functional differential equations
25
Other neutral equations can be investigated in regard to the existence of their
solutions, using approaches described above. We suggest to the reader to try such
procedures on equations of the form
x0 (t) = f (t, x(t), x(t − h)) + g(t, x(t), x0 (t − h)),
under an initial condition of the form x(t) = ϕ(t), t ∈ [−h, 0]. Also, similar to the
equation (15), is
d
[x(t) + g(x(t − h))] = (W x)(t),
dt
with initial datum x(t) = ϕ(t), t ∈ [−h, 0].
See our paper [4] for details.
Received:
July 2003.
Revised:
December 2003.
References
[1] T.A. Burton, An existence theorem for neutral equations. Nonlinear Studies
5 (1998), 1-6.
[2] C. Corduneanu, Integral Equations and Applications. Cambridge Univ.
Press, 1991.
[3] C. Corduneanu, Neutral functional equations with abstract Volterra operators. In ”Advances in Nonlinear Dynamics”, Gordon & Breach, (1997), 229235.
[4] C. Corduneanu, Neutral functional equations of Volterra type. Functional
Differential Equations (Israel), (1997), 265-270.
[5] C. Corduneanu, Existence of solutions to neutral functional differential equations. J. Diff. Equations, 168 (2000), 93-101.
[6] C. Corduneanu, Functional Equations with Causal Operators, Taylor & Francis, 2002.
[7] C. Corduneanu, M. Mahdavi, On neutral functional differential equations
with causal operators. Proc. Third Workshop of the Int. Inst. General System
Science, Tianjin (China), 1998, 43-48.
[8] C. Corduneanu, M. Mahdavi, On neutral functional differential equations
with causal operators, II. In ”Integral Methods in Science and Engineering”,
Chapman & Hall CRC Press, Research Notes #418 (2000), 102-106.
26
Constantin Corduneanu
7, 3(2005)
[9] Loris Faina, Existence and continuous dependence for a class of neutral functional differential equations. Annales Polonici Mathematici, LXIV.3 (1996),
215-226.
A Mathematical Journal
Vol. 7, No 3, (27 - 37). December 2005.
Sufficiency of the maximum principle for
time optimality
H. O. Fattorini
Department of Mathematics, University of California
Los Angeles, California 90095-1555
hof@math.ucla.edu
ABSTRACT
For infinite dimensional linear systems, Pontryagin’s maximum principle is
shown to be sufficient for time optimality with conditions on the initial condition
and on the target. These conditions cannot be given up and are shown to be best
possible by means of counterexamples.
RESUMEN
Para sistemas lineales en dimensión infinita, el principio del máximo de Pontryagin es suficiente para alcanzar optimalidad en el tiempo con condiciones en
el valor inicial y el final. Estas condiciones no se pueden relajar y se muestra que
son las mejores posibles, por medio de contraejemplos.
Key words and phrases:
Math. Subj. Class.:
1
linear control systems in Banach spaces, time
optimal problem.
93E20, 93E25.
Introduction.
Consider the time optimal problem of driving the solution y(t) of
y 0 (t) = Ay(t) + u(t) ,
y(0) = ζ
(1.1)
28
H. O. Fattorini
7, 3(2005)
from the initial point ζ to a point target,
y(T ) = ȳ
(1.2)
with maximum-norm bound
ku(t)k ≤ 1
a. e. in 0 ≤ t ≤ T
(1.3)
in minimum time T ; A is the infinitesimal generator of a strongly continuous semigroup S(t) in a Banach space E and the controls u(t) are strongly measurable (so
that, in view of (1.3), belong to the unit ball of L∞ (0, T ; E)). Solutions or trajectories
Z t
y(t) = S(t)ζ +
S(t − σ)u(σ)dσ
0
of (1.1) are named y(t) = y(t, ζ, u) and controls satisfying (1.3) are called admissible.
Let Z be an arbitrary linear space with E ∗ ⊆ Z. We say that Z is a multiplier space
if (i) S(t)∗ is defined in Z,(ii) S(t)∗ Z ⊆ E ∗ for t > 0. A control ū(t) in the interval
0 ≤ t ≤ T satisfies the weak maximum principle if there exists z in a multiplier space
Z such that S(t)∗ z is not identically zero in 0 < t ≤ T and
hS(T − t)∗ z, ū(t)i = max hS(T − t)∗ z, ui a. e. in 0 ≤ t ≤ T ,
kuk≤1
(1.4)
where h· , ·i is the duality of the space E and the dual E ∗ . The control satisfies the
strong maximum principle if (1.4) holds and
Z T
kS(t)∗ zkE ∗ dt < ∞ .
(1.5)
0
The space Z(T ) consists of all multipliers that satisfy
is equivalent to
S(T − t)∗ z
ū(t) =
kS(T − t)∗ zk
1
(1.5). In Hilbert space, (1.4)
(1.6)
whenever the denominator is not zero.It is known [3] that if the control ū(t) drives
ζ ∈ E to ȳ = y(T, ζ, ū) ∈ D(A) then the strong maximum principle (1.4)-(1.5) is a
necessary condition for time optimality.
It is also known [4] that(1.4)-(1.5) is a sufficient condition if ζ = 0 or ȳ =
y(T, ζ, ū) = 0; then ū(t) drives ζ to ȳ time optimally.2 We prove below (Theorem 1.2)
that these conditions on the initial and final point of the trajectory can be relaxed to
one of the two assumptions
ζ ∈ D(A), kAζk ≤ 1
or ȳ ∈ D(A), kAȳk ≤ 1 ,
(1.7)
1 The semigroup S(t)∗ may not be strongly continuous, but in all cases the norm kS(t)∗ k is lower
semicontinuous, thus theintegral (1.5) makes sense. The spaces Z(T ) are the same for all T > 0;
condition (1.5) only bears on the behavior of kS(t)∗ k near zero.
2 The weak maximum principle(1.4) is not a sufficient condition for time optimality; for a counterexample, see[6]
Sufficiency of the maximum principle for time optimality
7, 3(2005)
29
(the first with an additional condition on the adjoint semigroup). That restrictions
on the initial condition ζ or the target ȳ cannot be completely given up is illustrated
with several examples, two of which show that conditions (1.7) are the best possible
of their kind. We also see (in Example 4.2) that restrictions on kζk, kȳk (rather
than on kAζk, kAȳk) do not guarantee sufficiency of the maximum principle for time
optimality.
Remark 1.1. If S(t) is a group or, more generally, if S(T )E = E (t > 0) then
the condition ȳ ∈ D(A) is not required to show that the maximum principle is a
necessary condition for time optimality; moreover, Z(T ) = E ∗ . Sufficiency of the
maximum principle, however, requires the same conditions as those in the general
case.
2
Sufficiency of the maximum principle.
Let R∞ (T ) ⊆ E be the space of all elements of the form
Z
T
S(T − σ)u(σ)dσ ,
y = y(T, 0, u) =
u(·) ∈ L∞ (0, T ; E) .
(2.1)
0
The norm kykR∞ (T ) is the infimum of ku(·)kL∞ (0,T ;E) for all u(·) that satisfy (2.1);
in other words, R∞ (T ) is the quotient of L∞ (0, T ; E) by the closed subspace characterized by y(T, 0, u) = 0. An element z ∈ Z(T ) defines a bounded linear functional ξz
in R∞ (T ) through the formula
Z
hhξz , yii =
T
hS(T − σ)∗ z, u(σ)idσ
(2.2)
0
where y and u(·) are related by (2.1) and hh·, ·ii indicates the duality of the space
R∞ (T ) and its dual R∞ (T )? ; the norm of ξz satisfies
Z
kξz kR∞ (T )? =
T
kS(t)∗ zkE ∗ dt .
(2.3)
0
Theorem 2.1. Assume that ū(t) satisfies (1.4) - (1.5) and that either
(a) ȳ = y(T, ζ, ū) ∈ D(A), kAȳk < 1 , or
(2.4)
(b)
ζ ∈ D(A), kAζk < 1 , S(t)∗ z 6= 0 in 0 ≤ t < T .
Then ū(t) drives ζ to ȳ time optimally in 0 ≤ t ≤ T.
Proof of case (a). Assume ū(·) does not drive ζ to ȳ time optimally. Then there
exists δ > 0 and a control ũ(·) ∈ L∞ (0, T − δ; E), kũ(·)kL∞ (0,T −δ;E) ≤ 1, that drives
ζ to ȳ in time T − δ. The control
30
H. O. Fattorini
(0 ≤ σ < T − δ)
ũ(σ)
(
7, 3(2005)
v(σ) =
(2.5)
−Aȳ
(T − δ ≤ σ ≤ T )
satisfies kv(·)kL∞ (0,T ;E) ≤ 1 . We have
ȳ − S(t − (T − δ))ȳ
=−
R t−(T −δ)
=−
R t−(T −δ)
=
0
0
Rt
T −δ
S(σ)Aȳ dσ
S(t − (T − δ) − σ)Aȳ dσ
S(t − σ)(−Aȳ) dσ
(2.6)
(T − δ ≤ t ≤ T )
hence the trajectory y(t, ζ, v) starts at ζ, reaches ȳ at time T − δ and stays at ȳ for
T − δ ≤ t ≤ T;
y(t, ζ, v) = ȳ (T − δ ≤ t ≤ T ) .
(2.7)
This can be also be seen noting that if y(t) = ȳ then we have y 0 (t) − Ay(t) = −Aȳ in
T − δ ≤ t ≤ T.
The maximum principle (1.4) is equivalent to
RT
0
RT
hS(T − σ)∗ z, u(σ)idσ ≤
0
∞
(u(·) ∈ L (0, T ; E),
hS(T − σ)∗ z, ū(σ)idσ
(2.8)
ku(·)kL∞ (0,T ;E) ≤ 1) .
In terms of the linear functional ξz in (2.2), this is
RT
RT
ξz , 0 S(T − σ)u(σ)dσ ≤ ξz , 0 S(T − σ)ū(σ)dσ
(2.9)
(u(·) ∈ L∞ (0, T ; E),
ku(·)kL∞ (0,T ;E) ≤ 1) .
We have y(T, ζ, v) = y(T, ζ, ū), thus
Z
T
Z
S(T − σ)v(σ)dσ =
0
T
S(T − σ)ū(σ)dσ ,
(2.10)
0
and it follows from (2.9) that
RT
RT
ξz , 0 S(T − σ)u(σ)dσ ≤ ξz , 0 S(T − σ)v(σ)dσ
(2.11)
(u(·) ∈ L∞ (0, T ; E),
ku(·)kL∞ (0,T ;E) ≤ 1) ,
which, being equivalent to (1.4), gives
hS(T − t)∗ z, v(t)i = max hS(T − t)∗ z, ui a. e. in 0 ≤ t ≤ T .
kuk≤1
(2.12)
Sufficiency of the maximum principle for time optimality
7, 3(2005)
31
We have S(T − σ)∗ z 6= 0 near3 T, hence (2.12) implies kv(σ)k = 1 near T. This is a
contradiction, since by hypothesis kv(σ)k = kAȳk < 1.
Proof of case (b). This time we define
−Aζ
(0 ≤ σ ≤ δ)
v(σ) =
(2.13)
ũ(σ − δ)
(δ ≤ σ ≤ T ) .
As in (2.6) we have
Z
t
ζ − S(t)ζ =
S(t − σ)(−Aζ)dσ ,
0
hence the trajectory y(t, ζ, v) stays at ζ for 0 ≤ t ≤ δ,
y(t, ζ, v) = ȳ
(0 ≤ t ≤ δ) ,
and then starts for the target ȳ, which hits at time T. The proof ends in the same
way as that of (a) noting that S(T − σ)∗ z 6= 0 in 0 ≤ σ ≤ T (in particular, in
0 ≤ σ ≤ δ) hence (2.12) implies kv(σ)k = 1 in 0 ≤ σ ≤ δ, in contradiction to the fact
that kv(σ)k = kAζk < 1 in 0 ≤ σ ≤ δ.
3
Counterexamples, I.
To see that (2.4) cannot be relaxed, we have
Example 3.1.Consider the one dimensional system
y 0 (t) = −ay(t) + u(t),
y(0) = ζ
(3.1)
with a > 0. We have S(t) = e−at = S(t)∗ , thus controls satisfying (1.4) with z 6= 0
are of one of the two forms
(
1 if z > 0 ,
ū(t) =
(3.2)
−1 if z < 0 .
For the initial condition and target ζ = 1/a, ȳ = 1/a we have
Z
T
Z
S(T − σ) · 1 dσ =
0
0
T
e−a(T −σ) dσ =
1 − e−aT
= ȳ − S(T )ζ
a
so that the first control in (3.2) drives ζ to ȳ in any time T ≥ 0; in other words,
y(T, ζ, ū) = ȳ for all T ≥ 0. None of these drives is time optimal except forthe one
where T = 0.
3 The semigroup equation for the adjoint semigroup S(t)∗ implies: if S(T − t)∗ z = 0, then
S(T − σ)∗ z = S(t − σ)∗ S(T − t)∗ z = 0 for σ ≤ t. Accordingly, unless S(T − t)∗ z 6= 0 in an
interval (ρ, T ), ρ < T,S(T − t)∗ z will be identically zero in 0 < t ≤ T.
32
H. O. Fattorini
7, 3(2005)
Example 3.2. For another counterexample (or, rather, family of counterexamples)
we use an arbitrary unitary group S(t) in Hilbert space. Here we have Z(T ) = E
(see Remark 1.1), S(t)∗ = S(−t) = S(t)−1 , kS(t)yk = kyk. Controls satisfying the
maximum principle are given by (1.6) (the denominator satisfies kS(T − t)∗ zk = kzk).
Assuming (as we may) that kzk = 1 we have
Z T
Z T
S(T − σ)∗ z
dσ
=
S(T − σ)S(T − σ)∗ z dσ = T z
(3.3)
S(T − σ)
kS(T − σ)∗ zk
0
0
so that the control (1.6) drives ζ to ȳ in time T if and only if T is a solution of the
equation
T z = ȳ − S(T )ζ .
(3.4)
This equation implies the scalar equation
T = kS(T )ζ − ȳk
(3.5)
and, conversely, if T > 0 is a solution of (3.5) it is clear that (3.4) will holdwith
z=
ȳ − S(T )ζ
.
kȳ − S(T )ζk
(3.6)
Theorem 3.3. Assume (3.5) has only one nonnegative solution T. Then the control
(1.6) drives ζ to ȳ in optimal time T. If (3.5) has multiple solutions, only the control
(1.6) corresponding to the smallest T drives ζ to ȳ time optimally.
Proof. Let T ≥ 0 be the smallest solution of (3.5). If T = 0 we don’t need to drive
at all so that T is the optimal time. If T > 0 there exists an admissible control
driving from ζ to ȳ, hence the standard existence theorem [2, Theorem 1.2] provides
a control ū(t) driving from ζ to ȳ in optimal time T . Since S(t) is a group (Remark
1.1) this control must satisfy the maximum principle (1.4) with a nonzero multiplier
z ∈ Z(T ) = E ∗ = E. We are in a Hilbert space, which means this control must of the
form (1.6) (with kzk = 1),
ū(t) =
S(T − t)∗ z
= S(T − t)∗ z
kS(T − t)∗ zk
(the denominator cannot be zero since kS(T − t)∗ zk = kzk). As in (3.3) we then have
Z
T
S(T − σ)ū(σ)dσ = T z = ȳ − S(T )ζ
0
hence T is a solution of (3.5) and, as T is the optimal time, we must have T = T.
Corollary 3.4. Assume that, either (a) ζ ∈ D(A), kAζk > 1, or (b) ζ ∈
/ D(A). Then
there exists a control of theform (1.6) that drives ζ to ζ in time T > 0, thus is not
time optimal.
Proof. We write (3.5) for ζ = ȳ as
kS(t)ζ − ζk
= 1.
t
(3.7)
Sufficiency of the maximum principle for time optimality
7, 3(2005)
33
In case (a) we have
S(t)ζ − ζ kS(t)ζ − ζk
= lim → kAζk > 1 ,
t→0+
t→0+
t
t
lim
and we deduce that (3.7) has a positive solution, since the left side tends to 0 as
t → ∞.
In case (b),
lim inf
t→0+
S(t)ζ − ζ kS(t)ζ − ζk
= lim inf =∞
t→0+
t
t
since a finite lim inf implies that ζ ∈ D(A) ([1, Theorem 2.1.2. (c), p.88]).
Remark 3.5. Corollary 3.4 has an interesting application. The equivalence
T
Z
Z
S(T − σ)u(σ)dσ = ȳ
S(T )ζ +
0
⇐⇒
T
S(T − σ)u(σ)dσ = ȳ − S(T )ζ
0
says that u(t) drives ζ to ȳ in time T ⇐⇒ u(t) drives 0 to ȳ − S(T )ζ in time T. If
“drives” is changed to “drives optimally”, the implication =⇒ remains. In fact, if ū(·)
does not drive 0 to ȳ − S(T )ζ time optimally then there exists δ > 0 and acontrol u(·)
with ku(·)kL∞ (0,T −δ;E) ≤ 1 and
Z
T −δ
S(T − δ − σ)u(σ)dσ = ȳ − S(T )ζ .
0
Then, if we define
(
0
(0 ≤ σ < δ)
u(σ − δ) (δ ≤ σ ≤ T )
v(σ) =
we have
Z
T
Z
S(T − σ)v(σ)dσ =
0
T −δ
S(T − δ − σ)u(σ)dσ = ȳ − S(T )ζ ,
0
thus v(t) drives from ζ to ȳ in time T. If this drive were time optimal,the “bangbang” Theorem 2.2 in [2] would say that kv(σ)k = 1 a. e., which is not the case since
v(σ) = 0 in 0 ≤ σ ≤ δ. Accordingly, the optimal driving time from ζ to ȳ is < T.
The implication ⇐= is not true; in the setting of unitary semigroups in Hilbert
spaces it suffices to take ȳ = ζ ∈ D(A) with kAζk > 1, and, applying Corollary 3.4
construct a control ū(·) satisfying (1.6) and driving ζ to ζ in time T > 0. The same
control drives 0 to ζ − S(T )ζ, but this drive is optimal since the initial condition
satisfies (b) in Theorem 2.1.
34
4
H. O. Fattorini
7, 3(2005)
Counterexamples, II.
The next example belongs to the family in Example 3.2.
Example 4.1. Consider E = IR2 ,
A=
0
−1
1
0
.
(4.1)
The semigroup generated by A,
At
S(t) = e
=
cos t
− sin t
sin t
cos t
(4.2)
is unitary. In polar coordinates, ζ = (r cos θ, r sin θ),ȳ = (s cos ϕ, s sin ϕ), and
2
r cos θ
s cos ϕ S(t)
−
kS(t)ζ − ȳk2 = r sin θ
s sin ϕ = (r cos t cos θ + r sin t sin θ − s cos ϕ)2
+(−r cos θ sin t + r cos t sin θ − s sin ϕ)2
= r2 + s2 − 2rs cos(t − θ + ϕ) .
We have kAyk = kyk. For ζ = ȳ = (1.1, 0) (so that kAζk = kAȳk = 1.1) we have
r = s = 1.1,θ = ϕ = 0. Equation (3.5) (Figure 1) has a positive solution
T = 1.49797
(4.3)
thus we can drive from ζ = (1.1, 0) back to ζ in time T with a control satisfying (1.6),
ū(σ) = S(T − σ)z ,
(4.4)
with z given by (3.6),
z = (0.68090, 0.73238) .
Figure 2 below shows the drive (moving clockwise) whichis obviously not time optimal.
7, 3(2005)
Sufficiency of the maximum principle for time optimality
35
For ζ = (4, 0), ȳ = (−4, 0) we have r = s = 4, θ = 0, ϕ = π. Equation (3.5) (Figure
3) has three solutions,
T0 = 2.50471 ,
T1 = 4.26666 ,
T2 = 7.19061 .
(4.5)
thus we can drive from (4, 0) to (−4, 0) with three different controls that satisfy (1.6),
ūj (σ) = S(Tj − σ)∗ zj
j = 0,1,2,
(4.6)
where the zj are given by (3.6) for each Tj ,
z0 = (−0.31308, 0.94972) , z1 = (−0.53333, −0.84590) , z2 = (−0.89882, 0.43830) .
Figure 2 shows the three trajectories, each plotted for 0 ≤ t ≤ Tj ; only the first
(thicker curve) is time optimal.
Remark 4.2. The strong maximum principle (1.4)-(1.5) is a sufficient condition for
norm optimality [4] with no conditions on ζ or ȳ so that each of the controls ūj (t),
36
H. O. Fattorini
7, 3(2005)
j = 0, 1, 2 in (4.6) is norm optimal in its own interval; this means, if ζ = (4, 0) can be
drivento ȳ = (−4, 0) in the interval 0 ≤ t ≤ Tj by means of a control u(t) then
ku(·)kL∞ (0,Tj ;E) ≥ 1 = kūj (·)kL∞ (0,Tj ;E) .
The same observation applies to the control (4.4); it drives ζ back to ζ norm optimally
in the interval 0 ≤ t ≤ T.
Example 4.3. Example 3.1 can be manipulated into evidence that restrictions on
kζk or kȳk such as kζk ≤ or ȳ ≤ don’t guarantee sufficiency of the maximum
principle for time optimality. To thisPend we consider thespace E = `2 of all sequences
y = {y1 , y2 , . . . } such that kyk2 =
|yk |2 < ∞, equipped with the norm k · k. The
operator is
Ay = A{yk } = {−nyk }
with maximal domain. It generates the semigroup
S(t){yk } = {e−kt yk } = S(t)∗ .
Let
1
{δnk } = ȳn ,
n
the Kronecker delta). We have
ζn =
(δnk
kζn k = kȳn k =
1
,
n
zn = {δnk }
kAζn k = kAȳn k = 1 .
If ūn (·) satisfies (1.6) in an interval 0 ≤ t ≤ T with z = zn then ūn (σ) = {δnk } and
Z
T
S(T − σ)ū(σ)dσ =
0
1 − e−nT
δnk = ȳn − S(T )ζn
n
for any T > 0. Accordingly, ūn (σ) drives ζn to yn in an arbitrary interval 0 ≤ t ≤ T.
The driveis not optimal unless T = 0.
Received:
April 2004.
Revised:
May 2004.
References
[1] P. L. Butzer, H. Berens, Semi-Groups of Operators and Approximation,
Springer, Berlin 1967.
[2] H. O. Fattorini, Time-optimal control of solutions of operational differential
equations, SIAM J. Control 2 (1964) 54-59.
7, 3(2005)
Sufficiency of the maximum principle for time optimality
37
[3] H. O. Fattorini, The maximum principle in infinite dimension,Discrete &
Continuous Dynamical Systems 6 (2000) 557-574.
[4] H. O. Fattorini, Existence of singular extremals and singular functionals in
reachable spaces, Jour. Evolution Equations 1 (2001)325-347.
[5] H. O. Fattorini, A survey of the time optimal problem and the norm optimal
problem in infinite dimension, Cubo Mat. Educacional 3 (2001) 147-169.
[6] H. O. Fattorini, Time optimality and the maximum principle in infinite dimension, Optimization 50 (2001) 361-385.
A Mathematical Journal
Vol. 7, No 3, (39 - 48). December 2005.
The exact solution of the Potts models
with external magnetic field on the Cayley
tree
Nasir Ganikhodjaev 1
Centre for Computational and Theoretical Sciences,Faculty of Science,
International Islamic University Malaysia,53100 Kuala Lumpur,Malaysia and
Department of Mechanics and Mathematics, National University of
Uzbekistan,Vuzgorodok
700095,Tashkent ,Uzbekistan
nasirgani@hotmail.com
Seyit Temir
Harran University, Department of Mathematics Sanliurfa-Turkey
temirseyit@harran.edu.tr
Hasan Akin
Harran University, Department of Mathematics Sanliurfa-Turkey
akinhasan@harran.edu.tr
ABSTRACT
The exact solution is found for the problem of phase transition in the Potts
model and the Potts model with competing ternary and binary interactions with
external magnetic field.
1 This research was supported in part by the grant Uz.R.FTM F-2.1.56. The first named author
(N.G.) thanks NATO-TUBITAK for providing financial support and Harran University for kind
hospitality and providing all facilities.
40
Nasir Ganikhodjaev, Seyit Temir and Hasan Akin
7, 3(2005)
RESUMEN
Se encuentran soluciones exactas para los problemas de transisión de fases en
el modelo de Pott y también para el modelo de Pott con interacciones binarias y
ternarias en un campo magnético externo.
Key words and phrases:
Math. Subj. Class.:
1
Cayley tree, Potts model, Competing interactions,
External magnetic field.
82B20 Secondary 82B26
Introduction
The Potts model was introduced as a generalization of the Ising model. The idea
came from the representation of the Ising model as interacting spins which can be
either parallel or antiparallel. An obvious generalization was to extend the number
of directions of the spins. Such a model was proposed by C.Domb as a PhD thesis for
his student R.Potts in 1952. At present the Potts model encompasses a number of
problems in statistical physics and lattice theory. It has been a subject of incresing
intense research interest in recent years. It includes the ice-rule vertex and bond
percolation models as special cases.
We consider a semi-infinite Cayley tree J k for order k ≥ 2, i.e., a graph having no
cycles, from each vertex of which, except on vertex x0 , emanates exactly k + 1 edges
and from vertex x0 , which is the root of the tree, emanates k edges.
The vertices x and y are called nearest neighbors, which is denoted by < x, y >,
if there exists an edge connecting them. The vertices x,y and z are called a triple
of neighbors ,which is denoted by < x, y, z >,if < x, y > and < y, z > are nearest
neighbors and x 6= z
Let V be the set of vertices in J k . We set
Wn = {x ∈ V |d(x, x0 ) = n},
Vn = ∪nm=0 Wm = {x ∈ V |d(x, x0 ) ≤ n}.
where the distance d(x, y), x, y, ∈ V is given by the formula,
d(x, y) = min{d|x = x0 , x1 , x2 , ..., xd−1 , xd = y ∈ V
such that the pairs < x0 , x1 >, ..., < xd−1 , xd > are nearest neighbors.
The set Wn is called n-th level of J k and the set Vn is called n-storeyed home with
root x0 .
We consider models where the spin takes values in the set Φ = {0, 1, 2, ..., q}, q ≥ 2
and assigned to the vertices of the tree.A configuration σ on V is then defined as a
7, 3(2005)
The exact solution of the Potts models ...
41
function x ∈ V → σ(x) ∈ Φ;the set of all configurations coincides with Ω = ΦV . The
Potts model on the Cayley tree is defined by the Hamiltonian
H(σ) = −J
X
δσ(x)σ(y) − h
<x,y>
X
δ0σ(x)
(1)
x∈V
where the first sum is taken over all nearest neighbors, δ in the first and second sums
is the Kroneker’s symbol,J, h ∈ R are coupling constants and σ ∈ Ω.
Along with this model, we will consider the Potts model with competing interactions on the Cayley tree which is defined by the Hamiltonian below
H(σ) = −J1
X
δσ(x)σ(y)σ(z) − J2
<x,y,z>
X
δσ(x)σ(y) − h
<x,y>
X
δ0σ(x)
(2)
x∈V
where the first sum is taken over all neighbors tripples, and δ in this sum is the
generalized Kroneker’s symbol (see [1]-[4] for models with competing interactions).
Such model was investigated in [3], where for the neighbors tripple < x, y, z >the
generalized Kroneker’s symbol δ had a form
1
if σ(x) = σ(y) = σ(z),
δσ(x)σ(y)σ(z) =
0
else.
For the neighbors tripple < x, y, z >, we assume

if σ(x) = σ(y) = σ(z),
 1
1
if
σ(x) = σ(y) 6= σ(z) or σ(x) 6= σ(y) = σ(z);
δσ(x)σ(y)σ(z) =
 2
0
else.
(3)
where x, z ∈ Wn for some n and y ∈ Wn−1 . This definition is well coordinated with
the theory of quadratic stochastic operators, where the quadratic stochastic operator
corresponding to the generalized Kroneker’s symbol (4) is the identity transformation
[5].
Let
S m−1 = {x = (x1 , · · · , xm ) ∈ Rm :
m
X
xi = 1} xi ≥ 0 ∀i = 1, · · · , m}
i=1
be the (m − 1)-dimensional simplex in Rm . The transformation V : S m−1 → S m−1 is
called quadratic stochastic operator , if
(V x)k =
m
X
pij,k xi xj
i,j=1
Pm
where pij,k ≥ 0 , pij,k = pji,k and k=1 pij,k = 1 for arbitrary i, j, k ∈ {1, · · · , m} .
Such operator have applications in mathematical biology, namely theory of heredity,
42
Nasir Ganikhodjaev, Seyit Temir and Hasan Akin
7, 3(2005)
where the coefficients pij,k are interpreted as coefficients of heredity. Assume pij,k =
δijk ,where the generalized Kroneker’s symbol δ has a form

if i = j = k,
 1
1
if
i = k 6= j or i 6= j = k;
δijk =
(4)
 2
0
else.
Then it is easy to show that the corresponding quadratic stochastic operator is the
identity transformation .
2
Recurrent Equations for partition function
There are several approaches to derive the equation or a system of equations describing
the limiting Gibbs measures for lattice models on a Cayley tree. One approach is
based on the properties of the Markov random fields on a Cayley tree [6, 7]. Another
approach is based on recurrent equations for partition functions(see for example [8]).
Naturally both approaches lead to the same equation(for example [9]). The second
approach, however, is more suitable for models with competing interactions.
Let Λ be a finite subset of V. Assume σ(Λ) and σ(V \ Λ) are the restriction of σ
to Λ and V \ Λ respectively. Let σ(V \ Λ) be a fixed boundary configuration. The
total energy of configuration σ(Λ) under condition σ(V \ Λ) is defined as
H(σ(Λ)|σ(V \ Λ))
X
X
X
= −J δσ(x)σ(y) − J
δσ(x)σ(y) − h
δ0σ(x) .
< x, y >
x, y ∈ Λ
x∈Λ
< x, y >
x ∈ Λ, y ∈
/ Λ
in the first case and
H(σ(Λ)|σ(V \ Λ))
= −J1
X
X
X
δ0σ(x)
δσ(x)σ(y)σ(z) − J2 δσ(x)σ(y) − h
−J1
X
x∈Λ
< x, y >
x, y ∈ Λ
< x, y, z >
x, y, z ∈ Λ
δσ(x)σ(y)σ(z) − J2
< x, y, z >
x ∈
/ Λ, y ∈ Λ, z ∈
/ Λ
X
δσ(x)σ(y) .
< x, y >
x ∈ Λ, y ∈
/ Λ
for the second Hamiltonian respectively.
The partition function ZΛ (σ(V \ Λ)) in volume Λ under boundary condition
σ(V \ Λ) is defined as
X
ZΛ =
exp(−βH(σ(Λ))|σ(V \ Λ)),
(5)
σ(Λ)∈Ω(Λ)
1
is the inverse temperature.
T
We consider the configurations σ(Vn ), the partition functions ZVn in the volume Vn
and for brevity we denote it as σn , Z (n) respectively.
where Ω(Λ) is the set of all configuration on Λ, and β =
The exact solution of the Potts models ...
7, 3(2005)
43
Let us first consider the model (1).We decompose the partition function Z (n) into
the following summands
q
P
Z (n) =
i=1
(n)
Zi ,
where
(n)
Zi
X
=
exp(−βHn (σn )).
(6)
σn ∈Ω(Vn ):σn (x0 )=i
Let θ = exp (βJ), θ3 = exp (βh) . From (5) and (6), the following system of
recurrent equations can be easily derived
(n)
Z0
(n)
Zi
h
ik
(n−1)
(n−1)
(n−1)
(n−1)
= θ3 θZ0
+ Z1
+ Z2
+ ... + Zq
h
ik
(n−1)
(n−1)
(n−1)
(n−1)
(n−1)
= Z0
+ ... + Zi−1 + θZi
+ Zi+1 ...Zq
(7)
(n−1)
for i=1,2,...,q , where Zi
is a partition function in (n − 1)-storeyed home with
root located a vertex x ∈ W1 for which σ(x) = i.
(n)
After replacing ui
(n)
=
Zi
(n)
Z0
, we have the following system of recurrent equations
(n−1)
(n)
ui
1
= (
θ3
1 + (θ − 1)ui
θ+
+
q
P
j=1
q
P
j=1
(n−1)
uj
)k ;
(8)
(n−1)
uj
for
i=1,2,...,q and n=2,3,...
We describe the fixed points of this system recurrent equation (5). For this, it
suffices to solve the system of equations
1
ui = (
θ3
1 + (θ − 1)ui +
q
P
j=1
θ+
q
P
uj
)k ;
i = 1, 2, ..., q.
(9)
uj
j=1
Before we begin to solve this system of equations, we turn to the model (2). Here we
consider a slight modification of the Hamiltonian (2)
Definition 1 A triple of neighbours < x, y, z > is said to be two-level and is denoted
by < x, ¯y, z > if the vertices x and z belong to Wn for some n, i.e. they are located
on the same level and y ∈ Wn−1 .
We consider the Hamiltonian
X
X
X
H(σ) = −J1
δσ(x)σ(y)σ(z) − J2
δσ(x)σ(y) − h
δ0σ(x)
¯
<x,y,z>
<x,y>
x∈V
(10)
44
Nasir Ganikhodjaev, Seyit Temir and Hasan Akin
7, 3(2005)
where J1 6= 0 and in contrast to (2), the first sum inlcudes only the two-level triples
of neighbours. Such a model is called a two-level model (see [3], [4] and the references
there for the physical motivation underlying the study of these model).
It is not hard to derive, in this case, the system of recurrent equations is as the
following
(n)
Z0
(n)
Zi
h
ik
(n−1)
(n−1)
(n−1)
(n−1)
= θ3 θ1 θ2 Z0
+ Z1
+ Z2
+ ... + Zq
h
ik
(n−1)
(n−1)
(n−1)
(n−1)
(n−1)
= Z0
+ ... + Zi−1 + θ1 θ2 Zi
+ Zi+1 ...Zq
for i = 1, 2, ..., q
where θ1 = exp(βJ1 ), θ2 = exp(βJ2 ) and θ3 = exp(βh).
Thus both models (1) and (10) are described by the same system of recurrent
equations.
3
The proof of existence of phase transitions for
zero external field
In this section, we let J > 0 for model (1), that is we consider model (1) as a
ferromagnetic Potts model and J1 + J2 > 0 for model (10). Then θ > 1 in the first
case and θ1 θ2 > 1 for second case. We consider the system of equations (9).Assume
ui = exp hi , i = 1, 2, ..., q Then
h0i = k ln
1
(
θ3
1 + (θ − 1)hj +
q
P
exp(hj )
j=1
θ+
q
P
);
i = 1, 2, ..., q
(11)
exp hj
j=1
is the transformation Rq into Rq . Evidently the line l0 : h1 = h2 = ... = hq in Rq is
invarinat with respect to transformation (11) and the restriction of (11) on the line
l0 has the following form
h0 = k ln
1 (θ + q − 1) exp(h) + 1
(
)
θ3
q exp(h) + θ
where h ∈ R. Again, after renaming u = exp(h), we have
u=
1 (θ + q − 1)u + 1 k
(
) .
θ3
qu + θ
The following Lemma is a generalization of the Proposition 10.7 from [8].
Lemma 1 The equation
θ3 u = (
(θ + q − 1)u + 1 k
)
qu + θ
(12)
7, 3(2005)
The exact solution of the Potts models ...
45
(with u > 0, k ≥ 2, q ≥ 2) has a single solution if
p
−(k − 1)(q − 1) + (k − 1)2 (q + 1)2 + 8q(k − 1)
1 < θ < θcr =
2(k + 1)
If θ > θcr then there are numbers η1 (θ, q, k), η2 (θ, q, k) with 0 < η1 (θ, q, k) <
η2 (θ, q, k) such that equation (12) has three roots, when 0 < η1 (θ, q, k) < θ3 <
η2 (θ, q, k) and it has two roots if either θ3 = η1 (θ, q, k) or θ3 = η2 (θ, q, k) or θ = θcr .
The numbers ηi , i = 1, 2 are defined from the formula
ηi (θ, q, k) =
1 (θ + q − 1)ui + 1 k
(
)
ui
qui + θ
(13)
where u1 and u2 are the solution of the equation
(θ + q − 1)qu2 − [k(θ − 1)(θ + q) − θ(θ + q − 1) − q]u + θ = 0
(14)
)k . It is easy to check that equation (12) has more
Proof. Assume f (u) = ( (θ+q−1)u+1
qu+θ
than one root if and only if the equation uf 0 = f (u) has more than one solution. The
equation uf 0 = f (u) is no other than just equation (14).
Although there are three solutions for the system of equations (9) for θ > θcr ,
one cannot claim that there is a phase transition. Among these solutions, only one of
them is a stable solution. It is necessary to find other stable solutions. This problem
is rather complete for arbitrary k and q when θ 6= 1. The case with θ 6= 1 will be
considerd separately when k = 2 and q = 2. We shall now solve this problem for
θ3 = 1, that is, h = 0. Then, the system of equation (9) has the following form
1 + (θ − 1)ui +
q
P
uj
j=1
ui =
θ+
q
P
k
i = 1, 2, ..., q
(15)
uj
j=1
and the transformation Rq into Rq (11) has the following form
1 + (θ − 1) exp hj +
q
P
j=1
h0i = k ln(
θ+
q
P
exp(hj )
);
i = 1, 2, ..., q.
(16)
exp hj
j=1
Then, apart from the invariant line l0 we can find other q invariant lines, namely the
line lj : h1 = ... = hj−1 = hj+1 = ... = hq = 0, j = 1, 2, ..., q. The transformation (11)
reduces to the following transformation of R:
h0 = k ln(
on each invariant line lj , j = 1, 2, ..., q.
θ exp h + q
)
exp h + θ + q − 1
46
Nasir Ganikhodjaev, Seyit Temir and Hasan Akin
7, 3(2005)
Now we will solve this simpler equation
u=(
θu + q
)k
u+θ+q−1
(17)
θu + q
)k . With the help of the
u+θ+q−1
Lemma, it is not hard to show that the equation (17) has three solutions when
k + 2q − 1
∗
θ > θcr
=
. In this case only one of these roots is stable, namely, largest of
k−1
them. For equation (12), when θ3 = 1, we showed above that it has three solutions
∗
when θ > θcr
(see Lemma) and only one of them is stable. It is easy to check that
∗
θcr > θcr .
(x0 =j)
k+2q−1
∗
As uj = PP (x
0 =0) for some limiting Gibbs measure P with θ > θcr =
k−1 , we
have q + 1 differences translated invariant limiting Gibbs measures. The same way as
in [9], it is possible to prove that all of them are extremal.
Let us consider the function φ(u) = (
Theorem 1 For Potts model (1) with null external field, a phase transition occurs
when,
k + 2q − 1
.
θ>
k−1
Similar assertation is also valid for the two-level Potts model with competing ternary
and binary potentials with null external field.
Theorem 2 For the two-level Potts model (10) with competing ternaty and binary
potentials with null external magnetic field, a phase transition occurs when θ1 θ2 >
k+2q−1
k−1 .
4
The case of non-zero external magnetic field when
k=q=2
Here we consider Potts models both (1) and (10) with external magnetic field h 6= 0,
when k = q = 2 and θ > 1 for model (1) and θ1 θ2 > 1 for model (10) respectively
(The case h = 0 was considered in [9]for model (1) and for model (10) in [10] ). Then
the system of equations (9) reduces to the following
1 θx + y + 1 2
(
)
θ3 x + y + θ
1 x + θy + 1 2
y= (
)
θ3 x + y + θ
x=
where x = u1 , y = u2 for brevity. As
x−y =
1 (θ1 )(x − y)[(θ + 1)(x + y) + 1]
,
θ3
(x + y + θ2 )
(18)
The exact solution of the Potts models ...
7, 3(2005)
47
then some solutions of (18) can be found from equation
u=
1 (θ + 1)u + 1 2
(
)
θ3
2u + θ
(19)
where x = y = u and other solutions can be found from equation
θ3 z 2 − (θ2 − 2θ3 θ − 1)z + θ3 θ2 − 2θ + 2 = 0
(20)
where z = x + y. First of all, let us consider equation (19).Then the equation (13)(see
Lemma)has the following form
2(θ + 1)u2 − (θ2 + θ − 6)u + θ = 0.
(21)
√
This equation has two roots u1 , u2 if θ > 73−1
. Then by Lemma, equation (19)have
2
√
73−1
i +2 2
three roots if θ > 2 and η1 (θ) < θ3 < η2 (θ), where ηi = u1i ( (θ+1)u
3ui +θ ) , i = 1, 2.
Now we consider the equation (20). Again with the help of elementary analysis
it is not hard to show that the equation has two solutions for θ3 > 21 with θ >
√
p
3
. By virtue of
2θ3 − 1 + 2 θ3 (θ3 + 1) and for 0 < θ3 < 12 , with θ > 1+ θ1−2θ
3
symmetry of equations (18) we have two stable solutions.
√
73 − 1
Assume A = {(θ3 , θ) : η1 (θ) < θ3 < η2 (θ) ; θ >
} where η1 (θ) and η2 (θ)
√2
1+ 1−2θ3
1
as above and B = {(θ3 , θ): 0 < θ3 < 2 ; θ >
} ∪ {(θ3 , θ): θ3 > 12 ;
θ3
p
θ > 2θ3 − 1 + 2 θ3 (θ3 + 1) }. Then for arbitrary (θ3 , θ) ∈ A ∩ B there are three
stable solutions of the equations (18). We have thus proved the following theorems.
Theorem 3 For Potts model (1) with q = k = 2 and non-zero external magnetic
field, a phase transition occurs when (θ3 , θ) ∈ A ∩ B
A similar result is valid for model (10).
Theorem 4 For the two-level Potts model (10) with competing ternary and binary
potentials q = k = 2 and non-zero external magnetic field a phase transition occurs
when (θ3 , θ1 θ2 ) ∈ A ∩ B.
Received:
July 2005.
Revised:
August 2005.
References
[1] M. Mariz ,C.Tsalis, A.L.Albuquerque, Phase diagram of the Ising model on
a Cayley tree in the presence of competing interactions and magnetic field,
J. Stat. Phys., 40,(1985), 577-592.
48
Nasir Ganikhodjaev, Seyit Temir and Hasan Akin
7, 3(2005)
[2] C.R. da Silca, S.Coutinho, Ising model on the Bethe lattice with competing
interactions up to the third-nearest-neighbor generation, Phys. Review B,
34, (1986), 7975-7985.
[3] J.L. Monroe, Phase diagrams of Ising models on Husimi trees II. Pure multisite interactions systems, J. Stat. Phys. 67, (1992), 1185-2000.
[4] J.L. Monroe, A new criterion for the location of phase transitions for spin
systems on a recursive lattices, Phys. Lett. A 188, (1994), 80-84.
[5] S.N. Bernstein, The solution of a mathematical problem concerning the theory of heredity, Uchnye Zapiski Naucho-Issled. Kaf. Ukr. Otd. Mat. 1, (1924),
83-115.(Russian)
[6] F. Spitzer, Markov random field on infinite tree, Ann. Prob., 3, (1975), 387398.
[7] K. Preston, Gibbs States on Countable Sets, Cambridge, London (1974).
[8] R. Kindermann and J.L. Snell, Markov Random Fields and their Applications, Contemporary Mathematics 1, (1980).
[9] N.N.Ganikhodjaev (Ganikhodzhaev), On Pure Phases of the Ferromagnetic
Potts Model with three states on the Bethe Lattice order two, Theor. Math.
Phys. 82(2), (1990), 163-175.
[10] N.N. Ganikhodjaev, S. Temir, H. Akin, The exact solution of the three-state
Potts model with competing interactions on the Cayley tree, Uzbek Math.
Journal 3-4, (2002), 37-40.
A Mathematical Journal
Vol. 7, No 3, (49 - 63). December 2005.
The ergodic measures related with
nonautonomous hamiltonian systems and
their homology structure. Part 1 1
Denis L.Blackmore
Dept. of Mathematical Sciences at the NJIT, Newark, NJ 07102, USA
deblac@m.njit.edu
Yarema A.Prykarpatsky
The AGH University of Science and Technology, Department of Applied
Mathematics, Krakow 30059 Poland,
and Brookhaven Nat. Lab., CDIC, Upton, NY, 11973 USA
yarchyk@imath.kiev.ua, yarpry@bnl.gov
Anatoliy M.Samoilenko
The Institute of Mathematics, NAS, Kyiv 01601, Ukraine
Anatoliy K.Prykarpatsky 2
Department of Applied Mathematics, The AGH University of Science and
Technology
Applied Mathematics, Krakow 30059 Poland
pryk.anat@ua.fm, prykanat@cybergal.com
ABSTRACT
There is developed an approach to studying ergodic properties of time-dependent
periodic Hamiltonian flows on symplectic metric manifolds having applications
in mechanics and mathematical physics. Based both on J. Mather’s [9] results
about homology of probability invariant measures minimizing some Lagrangian
1 The authors are cordially indebted to Profs. Anthoni Rosato (NJIT, NJ,USA) and Alexander
S. Mishchenko (Moscow State University, Russia) for useful comments on the article. They are also
thankful to participants of the Seminar ” Nonlinear Analysis” at the Dept. of Applied Mathematics
of the AGH University of Science and Technology of Krakow for valuable discussions.
2 The fourth author was supported in part by a local AGH grant.
50
D.L.Blackmore, Y.A.Prykarpatsky, A.M.Samoilenko & A.K.Prykarpatsky
7, 3(2005)
functionals and on the symplectic field theory devised by A. Floer and others
[3-8,12,15] for investigating symplectic actions and Lagrangian submanifold intersections, an analog of Mather’s β-function is constructed subject to a Hamiltonian
flow reduced invariantly upon some compact neighborhood of a Lagrangian submanifold. Some results on stable and unstable manifolds to hyperbolic periodic
orbits having applications in the theory of adiabatic invariants of slowly perturbed
integrable Hamiltonian systems are stated within the Gromov-Salamon-Zehnder
[3,5,12] elliptic techniques in symplectic geometry.
RESUMEN
Un método para estudiar propiedades ergódicas de flujos Hamiltonianos que
dependen del tiempo sobre variedades simplécticas es desarrollado. Basados tanto
en un trabajo de J. Mather [9] sobre homologı́a de medidas invariantes de probabilidad que minimizan algunos funcionales lagrangianos, como en la teorı́a de
campos simplécticos, desarrollada por A. Floer y otros [3-8,12,15] para investigar
acciones simplécticas e intersecciones de subvariedades lagrangianas, se construye
un análogo de la función β de Mather sujeto a un flujo hamiltoniano reducido
invariantemente sobre una vecindad compacta de una subvariedad Lagrangiana.
Se plantean algunos resultados sobre variedades estables e intestables de órbitas
hiperbólicas periódicas. Estas tienen aplicaciones en la teorı́a de sistemas hamiltonianos integrables con perturbaciones lentas, en el marco de las técnicas elı́pticas
de Gromov-Salamon-Zehnder [3,5,12] en geometrı́a simpléctica.
Key words:
Math. Subj. Class.:
Ergodic measures, Holonomy groups, Dynamical systems,
Quasi-complex structures, Symplectic field theory
37A05, 37B35, 37C40, 37C60, 37J10, 37J40, 37J45
Introduction
The past years have given rise to several exciting developments in the field of symplectic geometry and dynamical systems [3-12], which introduced new mathematical
tools and concepts suitable for solving many before too hard problems. When studying periodic solutions to non-autonomous Hamiltonian systems Salamon & Zehnder
[3] developed a proper Morse theory for infinite dimensional loop manifolds based on
previous results on symplectic geometry of Lagrangian submanifolds of Floer [4, 6].
Investigating at the same time ergodic measures related with Lagrangian dynamical
systems on tangent spaces to configuration manifolds, Mather [9] devised a new approach to studying the correspondingly related invariant probabilistic measures based
on a so called β-function. The latter made it possible to describe effectively the so
called homology of these invariant probabilistic measures minimizing the correspond-
7, 3(2005)
The ergodic measures related with nonautonomous hamiltonian ...
51
ing Lagrangian action functional.
As one can easily see, the Mather approach doesn’t allow any its direct application to the problem of describing the ergodic measures related naturally with a given
periodic non-autonomous Hamiltonian system on a closed symplectic space. Thereby,
to overcome constraints to this task we suggest in the present work some new way to
imbedding the non-autonomous Hamiltonian case into the Mather β-function theory
picture, making use of the mentioned above Salamon & Zehneder and Floer [3, 4, 6]
loop space homology structures. Based further on the Gromov elliptic techniques
in symplectic geometry, the latters make it possible to construct the invariant submanifolds of our Hamiltonian system, naturally related with corresponding compact
Lagrangian submanifolds, and the related on them a β-function analog.
1
Symplectic and analytic problem setting
Let (M 2n , ω (2) ) be a closed symplectic manifold of dimension 2n with a symplectic structure ω (2) ∈ Λ(M 2n ) being weakly exact, that is ω (2) (π2 (M 2n )) = 0. Every
smooth enough time-dependent 2π-periodic function H : M 2n × S1 → R gives rise to
the non-autonomous vector field XH : M 2n × S1 → T (M 2n ) defined by the equality
iXH ω (2) = −dH,
(1)
where as usually [1], the operation ” iXH ” denotes the intrinsic derivation of the
Grassmann algebra Λ(M 2n ) along the vector field XH . The corresponding flow on
M 2n × S1 takes the form:
du/ds = XH (u; t),
dt/ds = 1,
(2)
where u : R →M 2n is an orbit, t ∈ R/2πZ ' S1 and s ∈ R is an evolution parameter.
We shall assume that solutions to (2) are complete and determine a one-parametric
ψ-flow of diffeomorphisms ψ s : M 2n × S1 → M 2n × S1 for all s ∈ R which are due
to (1) evidently symplectic, that is ψts∗
ω (2) = ω (2) where ψts0 := ψ s |M 2n at any fixed
0
t0 ∈ R/2πZ ' S1 . Take now an (n + 1)-dimensional submanifold Ln+1 ⊂ M 2n × R,
such that for any closed contractible curve γ with γ ⊂ Ln+1 the following integral
equality
I
(α(1) − H(t)dt) = 0
(3)
γ
holds,
where α(1) ∈ Λ1 (M 2n ) is such a 1-form on M 2n which satisfies the condition
R
(2)
(ω − dα(1) ) = 0 for any compact two-dimensional disk D2 ⊂ M 2n due to the
D2
weak exactness of the symplectic structure ω (2) ∈ Λ2 (M 2n ) and existing globally on
Ln+1 due to Floer results [4, 6]. Assume now also that for the flow of symplectomorphisms ψts0 : M 2n → M 2n , s ∈ R, the condition
{(ψts0 Lnt0 , t0 + s) : s ∈ R} ⊂ Ln+1
(4)
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D.L.Blackmore, Y.A.Prykarpatsky, A.M.Samoilenko & A.K.Prykarpatsky
7, 3(2005)
holds for some compact Lagrangian submanifold Lnt0 ⊂ M 2n upon which ω (2) Ln = 0.
t0
The condition (4) in particular means [2] that the following expression
α(1) − H(t)dt = dA(t),
(5)
t = t0 + s(mod2π) ∈ R/2πZ, holds in some vicinity of the Lagrangian submanifold
Lnt0 ⊂ M 2n , where a mapping A : R/2πZ → R is the so called [1, 2] generating
function for the defined above continuous set of diffeomorphisms ψts0 ∈Diff(M 2n ),
s ∈ R. The expression (5) makes it possible to define naturally the following PoincareCartan type functional on a set of almost everywhere differentiable curves γ : [0, τ ] →
M 2n × S1
Z
1
(τ )
(α(1) − H(t)dt),
At0 (γ) :=
(6)
τ γ
n
n
with end points { γ(τ ) = ψ τ (γ(0)) }, supp γ ⊂ U(Lt0 ) × S1 for all τ ∈ R and U(Lt0 )
is some compact neighborhood of the Lagrangian submanifold Lnt0 ⊂ M 2n satisfying
n
n
the condition ψts0 U(Lt0 ) ⊂ U(Lt0 ) for all s ∈ R.
n
Let us denote by Σt0 (H) the subset of curves γ with support in U(Lt0 ) × S1 and
fixed end-points as before minimizing the functional (6). If the infimum is realized,
one easily shows that any such curve γ ∈ Σt0 (H) solves the system (2). For the above
set of curves Σt0 (H) to be specified more suitably, choose, following Floer’s ideas
[3-8,12], an almost complex structure J : M 2n → End(T (M 2n )) on the symplectic
manifold M 2n , where by definition J 2 = −I, compatible with the symplectic structure
ω (2) ∈ Λ2 (M 2n ). Then the expexpression
< ξ, η >:= ω (2) (ξ, Jη),
(7)
where ξ, η ∈ T (M 2n ), naturally defines a Riemannian metric on M 2n . Subject to the
metric (7) our Hamiltonian vector field XH : M 2n × S1 → T (M 2n ) is now represented
as XH = J∇H, where ∇ : D(M 2n ) → T (M 2n ) denotes the usual gradient mapping
with respect to this metric.
Consider now the space Ω := Ω(M 2n × S1 ) of all continuous curves in M 2n × S1
with fixed end-points. Then one can similarly define the gradient mapping grad
(τ )
At0 : Ω → T (Ω) as follows:
(τ )
(grad At0 (γ), ξ) :=
1
τ
Z
τ
ds < J(γt0 )γ̇t0 (s) + ∇H(γt0 ; s + t0 ), ξ >,
(8)
0
where γ = {(γt0 (s); t0 + s( mod 2π)) : s ∈ [0, τ ]} ∈ Ω as before, and ξ ∈ T (Ω).
Since all critical curves γ ∈ Σt0 (H) minimizing the functional (6) solve (2), this fact
motivates a way of construction of an invariant subset ΩH ⊂ Ω, such that ΩH :=
n
Ω(U(Lt0 ) × S1 ). Namely, define a curve γ ∈ ΩH (γ (−) ) ⊂ ΩH as satisfying [3] the
following gradient flow in U(Lnt0 ) × S1 :
(τ )
∂ut0 /∂z = −grad At0 (u),
∂t/∂z = 0
(9)
7, 3(2005)
The ergodic measures related with nonautonomous hamiltonian ...
53
for all z ∈ R and any τ ∈ R under the asymptotic conditions
(−)
lim ut0 (s; z) = γt0 (s),
z→−∞
lim γt0 (s; z) = γt0 (s)
z→∞
(10)
(−)
with the corresponding curves γt0 , γt0 : R →M 2n satisfying the system (2), and
(−)
moreover, with the curve γt0 : R →M 2n being taken to be hyperbolic [1, 2] with supp
(−)
(−)
γt0 ⊂ Lnt0 . Now we can construct a so called [1] unstable manifold W u (γt0 ) to this
(−)
hyperbolic curve γt0 defined for all τ ∈ R. Thus due to the above construction, the
(−)
functional manifold W u (γt0 ) when compact can be imbedded as a point submanifold
into M 2n thereby interpreting supports of all curves solving (9) and (10) where supp
(−)
γt0 ⊂ Lnt0 , as a compact neighborhood Lt0 (H) ⊂ U(Lnt0 ) of the compact Lagrangian
n
2n
submanifold Lt0 ⊂ M
looked for above.
The same construction can be done evidently for the case when the conditions
(10) are changed either by
(+)
lim γt0 (s; z) = γt0 (s),
z→+∞
lim γt0 (s; z) = γt0 (s),
z→−∞
(10a)
or by
(−)
lim γt0 (s; z) = γt0 (s),
z→−∞
(−)
(+)
lim γt0 (s; z) = γt0 (s),
z→∞
(10b)
(+)
where γt0 : R →M 2n and γt0 : R →M 2n are some strictly different hyperbolic
(±)
curves on M 2n with supp γt0 ⊂ Lnt0 and solving (2). Based on (10a) one constructs
(+)
(+)
similarly the stable manifold W s (γt0 (s)) to a hyperbolic curve γt0 and further the
(+)
corresponding compact neighborhood Lt0 (H) ⊂ U(Lnt0 ) of the compact Lagrangian
2n
n
submanifold Lt0 ⊂ M which is of crucial importance when studying intersection
(+)
(−)
properties of stable W s (γt0 ) and unstable W u (γt0 ) manifolds. Based similarly on
(10b), one constructs the neighborhood Lt0 (H) ⊂ U(Lnt0 ) of the compact Lagrangian
submanifold Lnt0 ⊂ M 2n being of interest when investigating so called adiabatic
perturbations of integrable autonomous Hamiltonian flows on the symplectic manifold
M 2n .
Now we make use of some statements [3, 5, 12] about the properties of the set ΩH
constructed above. For a generic choice of the Hamiltonian function H : M 2n ×S1 → R
the functional space of curves ΩH is proved to be finite-dimensional what gives rise
(−)
right away to hereditary finite-dimensionality of the neighborhood Lt0 (H) with the
compact manifold structure. To see this linearize equation (9) in the direction of a
vector field ξ ∈ T (ΩH ). This leads to the linearized first-order differential operator:
Ft0 (u)ξ := ∇z ξ + J(u)∇s ξ + ∇ξ J(u)∂u/∂s + ∇ξ ∇H(u; t0 + s),
(11)
where u ∈ ΩH satisfies the following equation stemming from (9) :
∂u/∂z + J(u)∂u/∂s + ∇H(u; s + t0 ) = 0
(12)
54
D.L.Blackmore, Y.A.Prykarpatsky, A.M.Samoilenko & A.K.Prykarpatsky
7, 3(2005)
and ∇z , ∇s and ∇ξ denote here the corresponding covariant derivatives with
respect to the metric (7) on M 2n . If u ∈ ΩH satisfies (12), the curve γt0 in M 2n
(−)
has supp γt0 ⊂ Lnt0 and a curve γt0 in Lnt0 is hyperbolic and nondegenerate [3],
then the operator Ft0 (u) : T (ΩH ) → T (ΩH ) defined by (11) is a Fredholm operator
[12] between appropriate Sobolev spaces. The corresponding pair (H, J) with J :
M 2n → End(T (M 2n )) satisfying (7) is called regular [3] if every hyperbolic solution
to (2) is nondegenerate [1, 3] and the operator Ft0 (u) is onto for u ∈ ΩH . In general
one can prove that the space (H, J )reg ⊂ (H, J ) of regular pairs (H, J) ∈ (H, J ) is
dense with respect to the C ∞ -topology. Thus, for the regular pairs it follows from
(−)
an implicit function theorem [1] that the space ΩH (γt0 ) is indeed for any curve γt0
n
with supp γt0 ⊂ Lt0 a finite-dimensional compact functional submanifold whose local
(−)
dimension near u ∈ ΩH (γt0 ) is exactly the Fredholm index of the operator Ft0 (u).
(−)
As a simple inference from the finite-dimensionality of the set ΩH (γt0 ) and its com(−)
pactness one gets that the corresponding point set Lt0 (H) is finite-dimensional and
2n
compact submanifold smoothly imbedded into M . The same is evidently true for
(+)
the point manifolds Lt0 (H) and Lt0 (H) supplying us with compact neighborhoods
of the compact Lagrangian submanifold Lnt0 ⊂ M 2n . Let us specify the structure of
(−)
the manifold Lt0 (H) more exactly making use of the Floer type analytical results
[3, 8, 12] about the space of solutions to the problem (9) and (10). One has that for
any two curves γ (−) , γ : [0, τ ] → Lnt0 × S1 satisfying the system (2), the following
functional
Z
Z
1 τ
(τ )
2
2
ds dz(|∂u/∂z| + |∂u/∂s − XH (u; s + t0 )| )
(13)
Φt0 (u) :=
τ 0
R
if bounded satisfies the characteristic equality
(τ )
(τ )
(τ )
Φt0 (u) = At0 (γ (−) ) − At0 (γ)
(14)
for any τ ∈ R. Thereby, in the case when the right hand side of (14) doesn’t vanish, the
(+)
functional space ΩH (γ (−) ) will be a priori nontrivial. Similarly, for any u ∈ Lt0 (H)
one finds that
(τ )
(τ )
(τ )
Φt0 (u) = At0 (γ) − At0 (γ (+) ),
(14a)
(+)
where the corresponding curve γt0 : [0, τ ] → M 2n satisfies the system (2), is hy(+)
perbolic having supp γt0 ⊂ Lnt0 , and the curve γt0 : [0, τ ] → M 2n also satisfies the
system (2) having supp γt0 ⊂ Lnt0 , and at last, for u ∈ Lt0 (H)
(τ )
(τ )
(τ )
Φt0 (u) = At0 (γ (−) ) − At0 (γ (+) ),
(14b)
where γ (±) : [0, τ ] → M 2n × S1 , τ ∈ R, are taken to be strictly different, hyperbolic
(+)
(−)
and having supp γ (±) ⊂ Lnt0 . The case when γt0 = γt0 needs some modification
of the construction presented above on which we shall not dwell here. Thus we
(±)
have constructed the corresponding neighborhoods Lt0 (H) and Lt0 (H) of the compact Lagrangian submanifold Lnt0 ⊂ M 2n consisting of all bounded solutions to the
7, 3(2005)
The ergodic measures related with nonautonomous hamiltonian ...
55
corresponding equations (9), (10) and (10a,b). Based now on this fact and the analytical expressions (14) and (14a,b) one derives the following important lemma.
(±)
Lemma 1.1. All neighborhoods Lt0 (H) and Lt0 (H) constructed via the scheme
presented above are compact and invariant with respect to the Hamiltonian flow of
diffeomorphisms ψ s ∈Diff(M 2n ) × S1 , s ∈ R.
Let us consider below the case of the neighborhood Lt0 (H) ⊂ M 2n . The preceding
characterization of the space of curves ΩH leads us following Mather’s approach [9] to
another important for applications description of the compact neighborhood Lt0 (H)
by means of the space of normalized probability measures Mt0 (H) := M(T (Lt0 (H))×
S) with compact support and invariant with respect to our Hamiltonian ψ-flow of
diffeomorphisms ψ s ∈Diff(M 2n ) × S1 , s ∈ R, naturally extended on T (Lt0 (H)) × S.
The Hamiltonian ψ-flow due to Lemma 1.1 can be reduced invariantly upon the
compact submanifold Lt0 (H) × S ⊂ M 2n × S. For the behavior of this reduced ψflow upon Lt0 (H) × S to be studied in more detail let us assume that our extended
(τ )
Hamiltonian ψ∗ -flow on T (Lt0 (H)) × S is ergodic, that is the limτ →∞ At0 (γ) doesn’t
depend on initial points (u0 , u̇0 ; t0 ) ∈ T (Lt0 (H)) × S.
Recall now that the basic result [13] in functional analysis (the Riesz representation
theorem) states that the set of Borel probability measures on a compact metric space
X is a subset of the dual space C(X)∗ of the Banach space C(X) of continuous
functions on X. It is obviously a convex set and it is well known [13] to be metrizable
and compact with respect to the weak topology on C(X)∗ defined by C(X), also
called the weak (∗)-topology. The restriction of this topology to the set of Borel
measures is frequently called the vague topology on measures [9]. Since the space
Pt0 := T (Lt0 (H))×S is metrizable and can be as well compactified, it follows that the
set of Borel probability measures on Pt0 is a metrizable, compact and convex subset
of the dual to the Banach space of continuous functions on Pt0 . The corresponding
set Mt0 (H) is then evidently a compact, convex subset of this set. The well known
result of the Kryloff and Bogoliuboff [14] states that any ψ-flow on a compact metric
space X has an invariant probability measure. This result one can suitably adapt [9]
to our metric compactified space Pt0 := T (Lt0 (H)) × S as follows. Take a trajectory
γ ∈ ΩH of the extended ψ∗ -flow on Pt0 with supp γ ⊂ Lt0 (H) × S defined on a time
interval [0, τ ] ⊂ R and let a measure µτ on T (Lt0 (H)) × S be evenly distributed along
the orbit γ. Then evidently ||ψ∗s µτ − µτ || ≤ 2s/τ for s ∈ [0, τ ]. Denote by µ a point of
accumulation of the set {µτ : τ ∈ R+ } as τ → ∞ with respect to the before mentioned
vague topology. For any continuous
function fR ∈ C(Pt0 ), any s ∈ R and any τ0 , ε > 0
R
there exists τ > τ0 such that | Pt f ◦ ψ∗s̄ dµ − Pt f ◦ ψ∗s̄ dµτ | < ε for s̄ ∈ {0, s}. Then
0
0
it follows from the above estimations
Z
Z
Z
s
|
f ◦ ψ∗ dµ −
f dµ| ≤ |
f ◦ ψ∗s dµ −
Pt0
Z
f◦
Pt0
Z
Pt0
Pt0
ψ∗s dµτ |
Pt0
Z
+|
f◦
Pt0
ψ∗s dµτ
Z
−
Z
f dµτ | + |
Pt0
f dµτ −
Pt0
f dµ| ≤ 2ε + ||f || ||ψ∗s µτ − µτ || ≤ 2ε + 2s||f ||/τ,
56
D.L.Blackmore, Y.A.Prykarpatsky, A.M.Samoilenko & A.K.Prykarpatsky
7, 3(2005)
R
R
that is | Pt f ◦ ψ∗s dµ − Pt f dµ| = 0 since ε > 0 can be taken arbitrarily small and
0
0
τ0 > 0 arbitrarily large. Thereby one sees that the constructed measure µ ∈ Mt0 (H),
that is it is normalized and invariant with respect to the extended Hamiltonian ψ∗ -flow
on Pt0 .
Thus, in the case of ergodicity of the ψ∗ -flow on T (Lt0 (H)) × S the mentioned
above limit
Z
(τ )
lim At0 (γ) =
(α(1) − H)dµ,
(15)
τ →∞
Pt0
with 1-form α(1) ∈ Λ1 (M 2n ) being considered above as a function α(1) : Pt0 → R,
since the submanifold Lt0 (H) by construction is compact and invariantly imbedded
into M 2n due to Lemma 1.1. So, it is natural to study properties of the functional
Z
At0 (µ) :=
(α(1) − H)dµ
(16)
Pt0
on the space Mt0 (H), where we omitted for brevity the natural pullback of the 1-form
α(1) ∈ Λ1 (M 2n ) upon the invariant compact submanifold Lt0 (H) ⊂ M 2n . Being interested namely in ergodic properties of ψ∗ -orbits on T (Lt0 (H))×S), we shall develop
below an analog of the J. Mather Lagrangian measure homology technique [9, 10] to a
more general and complicated case of the reduced Hamiltonian ψ-flow on the invariant
compact submanifold Lt0 (H) ⊂ M 2n . In particular, we shall construct an analog of
the so called Mather β-function [9] on the homology group H1 (Lt0 (H); R) whose linear
domains generate exactly ergodic components of a measure µ ∈ Mt0 (H) minimizing
the functional (16), being of great importance for studying regularity properties of
ψ∗ -orbits on T (Lt0 (H)) × S. The results can be extended further to adiabatically perturbed integrable Hamiltonian systems depending on a small parameter ε ↓ 0 via the
continuous dependence H(t) := H̃(εt), where H̃(τ + 2π) = H̃(τ ) for all τ ∈ [0, 2π].
It makes also possible to state the existence of so called adiabatic invariants with
compact supports in Lt0 (H) having many applications in mathematical physics and
mechanics. Some of the results can be also applied to investigating the problem of
transversal intersections of corresponding stable and unstable manifolds to hyperbolic
curves or singular points, related closely with existence of highly irregular motions in
a periodic time-dependent Hamiltonian dynamical system under regard.
2
Invariant measures and mather’s type β-function
Before studying the average functional (16) on the measure space Mt0 (H), let us first
analyze properties of the functional
I
a(1) :=≺ a(1) , σ (17)
σ
on H 1 (Lt0 (H); R) at a fixed σ ∈ H1 (Lt0 (H); R). Since the 1-form a(1) ∈ H 1 (Lt0 (H); R)
in (17) can be considered as a function a(1) : Pt0 → R, in virtue of the Riesz theorem
7, 3(2005)
The ergodic measures related with nonautonomous hamiltonian ...
57
[13] there exists a Borel measure µ : Pt0 → R+ (still not necessary ψ-invariant), such
that
Z
≺ a(1) , σ =
a(1) dµ.
(18)
Pt0
The following lemma characterizing the right hand side of (18) holds.
Lemma 2.1. Let a 1-form a(1) = dλ(0) ∈ Λ1 (Lt0 (H)) be exact, that is the cohomology
class [dλ(0) ] = 0 ∈ H 1 (Lt0 (H); R). Then for any µ ∈ Mt0 (H)
I
a(1) = 0.
(19)
σ
C Really, for a(1) = dλ(0) , where λ(0) : Lt0 (H) → R is an absolutely continuous
mapping, the following holds due to The Fubini theorem for any τ ∈ R+ :
Rτ R
R
| Pt dλ(0) dµ.| = | τ1 0 ds Pt dλ(0) (ψ∗s dµ)| =
0
0
R
R
τ
| τ1 Pt dµ 0 dsd(λ(0) ◦ ψ∗s )/ds|
(20)
0
R
= | τ1 Pt dµ[λ(0) ◦ ψ∗τ − λ(0) ◦ ψ∗0 ]| ≤ 2||λ(0) ||/τ.
0
The latter inequality as τ → ∞ gives rise to the wanted equality (19), that proves
the lemma.B
Thus, the right hand side of (18) defines a true functional
Z
1
(1)
H (Lt0 (H); R) 3 a →
a(1) dµ ∈ R
(21)
Pt0
on the cohomology space H 1 (Lt0 (H); R). All the above can be formulated as the
following theorem.
Theorem 2.2. Let an element σ ∈ H1 (Lt0 (H); R) be fixed. Then there exists a ψinvariant probability measure (not unique) µ ∈ Mt0 (H), such that the representation
(18) holds and vice versa, for any measure µ ∈ Mt0 (H) there exists the homology
class σ := ρt0 (µ) ∈ H1 (Lt0 (H); R), such that
Z
≺ a(1) , ρt0 (µ) =
a(1) dµ
(22)
Pt0
for all a(1) ∈ H 1 (Lt0 (H); R).
Definition 2.3. ([10]) For any measure µ ∈ Mt0 (H) the homology class ρt0 (µ) ∈
H1 (Lt0 (H); R) is called its homology.
Corollary 2.4. The homology mapping ρt0 : Mt0 (H) → H1 (Lt0 (H); R) defined
within Theorem 2.2 is surjective.
C Sketch of a proof of Theorem 2.2. The fact that for each µ ∈ Mt0 (H)
there exists the unique homology class σ := ρt0 (µ) ∈ H1 (Lt0 (H); R) is based on
the well known Poincare duality theorem [1]. The inverse statement is about the
surjectivity of the mapping ρt0 : Mt0 (H) → H1 (Lt0 (H); R). For it to be stated, consider following [8-10] a covering space Lt0 (H) over Lt0 (H) defined by the condition
58
D.L.Blackmore, Y.A.Prykarpatsky, A.M.Samoilenko & A.K.Prykarpatsky
7, 3(2005)
that π1 (Lt0 (H)) = ker ht0 , where ht0 : π1 (Lt0 (H)) → H1 (Lt0 (H); R) denotes the
Hurewicz homomorphism [10]. Since in reality the functional (22) is defined on the
covering space Lt0 (H), it is necessary to lift all curves γ ∈ ΩH on Lt0 (H)×S to curves
γ̃ ∈∈ Ω̃H on Lt0 (H) × S. In the case when the homotopy group π1 (Lt0 (H)) is abelian,
the covering space L̃t0 (H) becomes universal, but in general it is obtained as some
universal covering of L̃t0 (H) quotioned further with respect to the action of the kernel
of the corresponding Hurewicz homomorphism ht0 : π1 (Lt0 (H)) → H1 (Lt0 (H); R).
Take now any element σ ∈ H1 (Lt0 (H); R) and construct a set of approximating
it so called Deck transformations τ −1 στ ∈ im ht0 ⊂ H1 (Lt0 (H); R), τ ∈ R+ , such
that weakly limτ →∞ τ −1 στ = σ holds. Put further x̃τ := στ ◦ x̃0 ∈ Lt0 (H) × S,
τ ∈ R+ , where x̃0 ∈ Lt0 (H) × S is taken arbitrary and consider such a curve γ̃ :
[0, τ ] → Lt0 (H) × S with end-points γ̃(0) = x̃0 , γ̃(τ ) = x̃τ whose projection on
Lt0 (H)×S is the curve γ ∈ Σt0 (H), minimizing the functional (6). Consider also a set
{µτ : τ ∈ R+ } of probability measures on Pt0 evenly distributed along corresponding
curves γ ∈ Σt0 (H) for each τ ∈ R+ and denote by µ a point of its accumulation
as τ → ∞. Due to the uniform distribution of measures µτ , τ ∈ R+ , along curves
γ ∈ Σt0 (H) having the end-points agreed with chosen above Deck transformations
στ ∈ H1 (Lt0 (H); R), τ ∈ R+ , one gets right away from the Birkhoff-Khinchin ergodic
theorem [1, 2] that
Z
a(1) dµτ =≺ a(1) , τ −1 στ ) (23)
Pt0
for any a(1) ∈ H 1 (Lt0 (H); R). Passing now to the limit in (23) as τ → ∞ and taking
into account that weakly limτ →∞ τ −1 στ = σ, one gets right away that the equality
(22) holds for some measure µ ∈ Mt0 (H), such that ρt0 (µ) = σ ∈ H1 (Lt0 (H); R),
thereby giving rise to the surjectivity of the mapping ρt0 : Mt0 (H) → H1 (Lt0 (H); R)
and proving the theorem. B
Return now to treating the average functional (16) subject to the space of all
invariant measures Mt0 (H). Namely, consider the following β-function
βt0 : H1 (Lt0 (H); R) → R defined as
βt0 (σ) := inf {At0 (µ) : ρt0 (µ) = σ ∈ H1 (Lt0 (H); R)}
µ
(24)
It will be further called a Mather type β-function due to its analogy to the definition
given in [9,10]. The following lemma holds.
Lemma 2.5. Let a 1-form a(1) ∈ H 1 (Lt0 (H); R) be taken arbitrary. Then the
Mather type β-function
(a)
(a)
βt0 (σ) := inf {At0 (µ) : ρt0 (µ) = σ ∈ H1 (Lt0 (H); R)},
µ
(25)
where by definition
(a)
At0 (µ)
Z
:=
(α(1) + a(1) − H)dµ,
(26)
Pt0
satisfies the following equation:
(a)
βt0 (σ) = βt0 (σ)+ ≺ a(1) , σ) .
(27)
The ergodic measures related with nonautonomous hamiltonian ...
7, 3(2005)
59
C The proof easily stems from the definition (25) and the equality (22). B
Assume now that the infimum in (24) is attained at a measure µ(σ) ∈ Mt0 (H).
Then evidently, ρt0 (µ(σ)) = σ for any homology class σ ∈ H1 (Lt0 (H); R). Denote by
(σ)
Mt0 (H) the set of all minimizing the functional (24) measures of Mt0 (H). In the
next chapter we shall proceed on study its ergodic and homology properties.
3
Ergodic measures and their homologies
(a)
Consider the introduced above Mather type β-function βt0 : H1 (Lt0 (H); R) → R for
any a(1) ∈ H 1 (Lt0 (H); R). It is evidently a convex function on H1 (Lt0 (H); R), that
is for any λ1 , λ2 ∈ [0, 1], λ1 + λ2 = 1, and σ1 , σ2 ∈ H1 (Lt0 (H); R) there holds the
inequality
(a)
(a)
(a)
βt0 (λ1 σ1 + λ2 σ2 ) ≤ λ1 βt0 (σ1 ) + λ2 βt0 (σ2 ).
(28)
As usually dealing with convex functions, one says that an element σ ∈ H1 (Lt0 (H); R)
(a)
(a)
(a)
is extremal point [13] if βt0 (λ1 σ1 + λ2 σ2 ) < λ1 βt0 (σ1 ) + λ2 βt0 (σ2 ) for all λ1 , λ2 ∈
(0, 1), λ1 + λ2 = 1, and σ = λ1 σ1 + λ2 σ2 . Correspondingly, we shall call a convex set
Zt0 (H) ⊂ H1 (Lt0 (H); R) by a linear domain of the Mather type function (25) if
(a)
(a)
(a)
βt0 (λ1 σ1 + λ2 σ2 ) = λ1 βt0 (σ1 ) + λ2 βt0 (σ2 )
(29)
for any σ1 , σ2 ∈ Zt0 (H) and λ1 , λ2 ∈ R. It is easy to see now that if σ ∈ H1 (Lt0 (H); R)
(σ)
is extremal, then the set Mt0 (H) contains [15] ergodic minimizing measure components. Namely, following [9, 10] one states that if Zt0 (H) is a linear domain and
(σ)
(σ)
Pt0 ⊂ Pt0 is the closure of the union of the supports of measures µ(σ) ∈ Mt0 (H)
(σ)
with σ ∈ Zt0 (H), then the set Pt0 is compact and the inverse mapping ( pt0 |P (σ) )−1 :
t0
(σ)
(σ)
pt0 (Pt0 ) → Pt0 is Lipschitzian, where pt0 : Pt0 → Lt0 (H) × S is the standard pro(σ)
jection, being injective upon Pt0 . Moreover, one can show [9] that if a measure
(σ)
(σ)
µ ∈ Mt0 (H) is minimizing the functional (26), then its support supp µ ⊂ Pt0 and
all its ergodic components {µ̄} are minimizing this functional too, and the convex
(σ)
hull of the corresponding homologies conv{ρt0 (µ̄)} is a linear domain Zt0 (H) of the
Mather type β-function (25). These results are of very interest concerning many applications in dynamics. Especially, the ergodic measures, as is well known, possess
the crucial property that every invariant Borel set has measure either 0 or 1, giving
rise to the following important equality:
(τ )
lim At0 (γ) = At0 (µ̄))
τ →∞
(30)
uniformly on (γt0 , (0), γ̇t0 (0); t0 ) ∈ Pt0 ∩ supp µ̄, where γ ∈ Σt0 (H). All of the properties formulated above are inferred from the following theorem modeling the similar
one in [10].
Theorem 3.1. Let a measure µ ∈ Mt0 (H) be minimizing the functional (26)
(a)
satisfying the condition βt0 (ρt0 (µ)) = At0 (µ). Then supp µ ⊂ Σt0 (H) and the convex
60
D.L.Blackmore, Y.A.Prykarpatsky, A.M.Samoilenko & A.K.Prykarpatsky
7, 3(2005)
hull of the set of homologies ρt0 (µ̄) ∈ H1 (Lt0 (H); R), where {µ̄} ⊂ Mt0 (H) are the
corresponding ergodic components of the measure µ ∈ Mt0 (H), is a linear domain
Zt0 (H) of the Mather type β-function (25).
C Sketch of a proof. Let ht0 : π1 (Lt0 (H)) → H1 (Lt0 (H); R) be the corresponding
Hurewicz homomorphism and take some basis σk ∈ im ht0 ⊂ H1 (Lt0 (H); R), k = 1, r,
(1)
where r = dim im ht0 , being its dual basis aj ∈ H 1 (Lt0 (H); R), j = 1, r. Then for
any points x̃, ỹ ∈ Lt0 (H) × S one can define an element ξ (τ ) (x̃, ỹ|γ̃) ∈ H1 (Lt0 (H); R)
as the sum
Z τ
r
1X
(1)
(τ )
σj
ãj (γ̃),
(31)
ξ (x̃, ỹ|γ̃) :=
τ j=1
0
where γ : [0, τ ] → Lt0 (H) × S is any continuous arc joining these two chosen points
(1)
x̃, ỹ ∈ Lt0 (H) × S, and ãj ∈ H 1 (Lt0 (H); R) are the corresponding liftings to Lt0 (H)
(1)
of 1-forms aj ∈ H 1 (Lt0 (H); R), j = 1, r. One can show then that if µ ∈ Mt0 (H)
is ergodic and supp µ ⊂ Σt0 (H), then the measure µ is minimizing the functional
(26). Put σ := ρt0 (µ) and let a set Zt0 (H) ⊂ H1 (Lt0 (H); R) be a supporting
domain containing this homology class σ ∈ H1 (Lt0 (H); R). Thus, one can see that the
extremal points of the convex set Zt0 (H) are extremal points also of the Mather type
β-function (25). Next expand the homology class σ = ρt0 (µ) as a convex combination
of extremal points σ̄j ∈ Zt0 (H), j = 1, m, for some m ∈ Z+ . Then, since elements
(σ )
σ̄j ∈ Zt0 (H), j = 1, m, are extremal, there exist ergodic measures µ̄j ∈ Mt0 j (H),
(σ)
j = 1, m, such that ρt0 (µ̄j ) = σ̄j , j = 1, m. Moreover, since Zt0 (H) is a linear
domain, one easily brings about that
(a)
βt0 (σ) =
m
X
(a)
cj βt0 (σ̄j ) =
j=1
m
X
(a)
cj At0 (µ̄j ),
(32)
j=1
Pm
where σ =
j=1 cj σ̄j with some real coefficients cj ∈ R, j = 1, m. Due to the
ergodicity of the measure µ ∈ Mt0 (H) from the Birkhoff-Khinchin ergodic theorem
[1] one derives that there exists an orbit γ̃ : [0, τ } → Lt0 (H) × S with the supp γ ⊂
supp µ, such that the property (30) together with the equality
σ := ρt0 (µ) = lim ξ (τ ) (x̃, ỹ|γ̃)
τ →∞
(33)
hold. Further, there exist curves γ̃j ∈ Σt0 (H), supp γj ⊂ supp µ̄j , j = 1, m, such the
expressions
σ̄j := ρt0 (µ̄j ) = lim ξ (τ ) (x̃, ỹ|γ̃j )
(34)
τ →∞
(a)
βt0 (σ̄j )
(a)
At0 (µ̄j )
(τ )
as well as
=
= limτ →∞ At0 (γ̃j ) hold for every j = 1, m. Under the
conditions (14b) involved on the invariant neighborhood Lt0 (H) one shows that for
(a)
(a)
any measure µ ∈ Mt0 (H) such that ρt0 (µ) = σ, the inequality At0 (µ) ≤ βt0 (ρt0 (µ))
holds thereby proving its minimality. Suppose now that the measure µ ∈ Mt0 (H)
has all its ergodic components with supports contained in Σt0 (H) and the convex
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The ergodic measures related with nonautonomous hamiltonian ...
61
hull of its homologies is a linear domain of the Mather type function (25). One can
approximate (in the
Pmweak topology) a measure µ ∈ Mt0 (H) by means of a convex
combination µ̂ := j=1 ĉj µ̄j , where ĉj ∈ R and µ̄j ∈ Mt0 (H), j = 1, m, are ergodic
components of the measure µ ∈ Mt0 (H). Then supp µ̄j ⊂ Σt0 (H) implying that all
µ̄j ∈ Mt0 (H), j = 1, m, are minimizing (26), that is are minimal. Therefore, since
the convex hull of homologies {ρt0 (µ̄j ) ∈ H1 (Lt0 (H); R) : j = 1, m} is a linear domain
due to its minimality, one gets that
Pm
Pm
(a)
(a)
(a)
At0 (µ̂) = j=1 ĉj At0 (µ̄j ) = j=1 ĉj βt0 (ρt0 (µ̄j ))
(35)
P
(a)
(a)
m
= βt0 (ρt0 ( j=1 ĉj µ̄j )) = βt0 (ρt0 (µ),
meaning evidently that the measure µ̂ ∈ Mt0 (H) is minimal too. Making use now
of the fact that limits of minimizing measures are minimizing too, one obtains finally
that the measure µ ∈ Mt0 (H) is minimizing the functional (26), thereby proving the
theorem. B
Consider some properties of a so called [10] supporting domain
(a)
(a)
(a)
Zt0 (H) := {σ ∈ H1 (Lt0 (H); R) : βt0 (σ) =≺ a(1) , σ +ct0 }
(36)
(a)
for the Mather type β-function (25) at some fixed a(1) ∈ H 1 (Lt0 (H); R) with ct0 ∈
(a)
R properly defined by (27). Define also by Pt0 := ∪σ∈Z (a) (H) supp µ(σ), where
t0
(a)
µ(σ) ∈ Mt0 (H) and ρt0 (µ(σ)) = σ ∈ Zt0 (H). Present now a supporting domain
(a)
Zt0 (H) ⊂ H1 (Lt0 (H); R) due to the expression (27) as follows:
(a)
(0)
(a)
Zt0 (H) = {σ ∈ H1 (Lt0 (H); R) : βt0 (σ) = ct0 },
(0)
(37)
where the function βt0 : H1 (Lt0 (H); R) being bounded from below is chosen in such
(0)
(a)
a way that βt0 (σ) ≥ ct0 for all σ ∈ H1 (Lt0 (H); R). Take now a measure µ ∈ Mt0 (H)
(0)
(a)
and suppose that supp µ ⊂ Σt0 (H). Since βt0 (σ) ≥ ct0 for all σ ∈ H1 (Lt0 (H); R)
(a)
(0)
(a)
and due to (37) Zt0 (H) = (βt0 )−1 {ct0 } at some fixed a(1) ∈ H 1 (Lt0 (H); R), this
evidently implies that the measure µ ∈ Mt0 (H) is minimizing the functional (26) and
(a)
ρt0 (µ) ∈ Zt0 (H). Thereby the following theorem is stated.
(a)
Theorem 3.2. Suppose that Zt0 (H) ⊂ H1 (Lt0 (H); R) is a supporting domain of
the Mather type function (27) and a measure µ ∈ Mt0 (H) satisfies the condition supp
(a)
µ ⊂ Σt0 (H). Then this measure µ ∈ Mt0 (H) is minimizing and ρt0 (µ) ∈ Zt0 (H).
The following corollaries from the Theorem 3.2 as in [10] hold.
Corollary 3.3. The minimizing measure µ ⊂ Mt0 (H) with supp µ ⊂ Σt0 (H)
(0)
(a)
satisfies the condition At0 (µ) = ct0 . By means of choosing the element a(1) ∈
(a)
(a)
H 1 (Lt0 (H); R) one can make the value ct0 be zero, that is one can put ct0 = 0.
Corollary 3.4. For any strictly extremal closed curve σ ∈ H1 (Lt0 (H); R) the
following properties take place:
i) there exists an ergodic measure µ̄(σ) ∈ Mt0 (H) whose support is a minimal
set and ρt0 (µ̄(σ)) = σ;
62
D.L.Blackmore, Y.A.Prykarpatsky, A.M.Samoilenko & A.K.Prykarpatsky
7, 3(2005)
ii) for every closed 1-form a(1) ∈ H 1 (Lt0 (H); R) the equality ≺ a(1) , σ =
R t +τ
limτ →∞ τ1 t00 a(1) (γ̇)ds holds uniformly for all (γt0 (0) , γ̇t0 (0) ; t0 ) ∈ Pt0 ∩ supp µ̄(σ),
ρt0 (µ̄(σ)) = σ and γ ∈ Σt0 (H);
iii) if (γt0 (0) , γ̇t0 (0) ; t0 ) ∈ Pt0 ∩ supp µ̄(σ), ρt0 (µ̄(σ)) = σ and γ ∈ Σt0 (H) is the
(a)
(τ )
corresponding orbit in Lt0 (H) × S, then βt0 (σ) = limτ →∞ At0 (γ) uniformly.
The statements formulated above can be effectively used for studying dynamics
of many perturbed integrable Hamiltonian flows and their regularity properties. As
it is well known, they are strongly based on the intersection theory of stable and
unstable manifolds related with hyperbolic either closed orbits or singular points of
a Hamiltonian system under regard. These aspects of our study of ergodic measure
and homology properties of such Hamiltonian flows are supposed to be treated in a
proceeding article under preparation.
Received:
June 2004.
Revised:
January 2005.
References
[1] Abraham R. and Marsden J., Foundations of Mechanics, Cummings, NY,
1978.
[2] Arnold V.I., Mathematical methods of classical mechanic, Springer, 1978.
[3] Salamon D. and Zehnder E., Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math. 1992,45,
1303-1360.
[4] Floer A., A relative Morse index for the symplectic action, Comm. Pure
Appl. Math., 1988, 41, 393-407.
[5] Aebischer B., Borer M. and others, Symplectic geometry:
course, Birkhauses Verlag, Basel, 1992, 79-165.
Introductory
[6] Floer A., Morse theory for Lagrangian intersections, J. Diff. Geom., 1988,
28, 513-547.
[7] Hofer H., Lusternik-Schnirelman-theory for Lagrangian intersections, Ann.
Ins. H. Poincare, 1988, 5,N5, 465-499.
[8] Eliashberg Y., Givental A. and Hofer H., Introduction to symplectic field
theory,. //arXive: math.SG/0010059 6 Oct 2000, 1-102.
[9] Mather J. N., Action minimizing invariant measures for positive definite
Lagrangian systems, Math. Zeitschr., 1991, 2017, 169-207.
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[10] Mane R., On the minimizing measures of Lagrangian dynamical systems,
Nonlinearity, 1992, 5, 623-638.
[11] Prykarpatsky A.K., On invariant measure structure of a class of ergodic
discrete dynamical systems, Nonlin. Oscillations, 2000, 3, N1, 78-83.
[12] McDuff D., Elliptic methods in symplectic geometry, Bull. AMS, 1990, 23,
311-358.
[13] Edwards R.E., Functional analysis, Holt, Rinehart and Winston Publ., New
York, 1965.
[14] Kryloff N.M. and Bogoliubov N.N., La theorie generale de la mesure et son
application a l’etude des systemes dynamiques de la mechanique nonlineaire,
Ann. Math.,II, 1937, 38, 65-113.
[15] Niemycki V.V. and Stepanov V.V., Qualitive theory of differential equations,
Princeton, Univ. Press, 1960.
A Mathematical Journal
Vol. 7, No 3, (65 - 73). December 2005.
Conjectures in Inverse Boundary Value
Problems for Quasilinear Elliptic Equations
Ziqi Sun
Department of Mathematics and Statistics Wichita State university
Wichita, KS 67226, USA
ziqi.sun@wichita.edu
ABSTRACT
Inverse boundary value problems originated in early 80’s, from the contribution of A.P. Calderon on the inverse conductivity problem [C], in which one
attempts to recover the electrical conductivity of a body by means of boundary measurements on the voltage and current. Since then, the area of inverse
boundary value problems for linear elliptic equations has undergone a great deal
of development [U]. The theoretical growth of this area contributes to many areas of applications ranging from medical imaging to various detection techniques
[B-B][Che-Is].
In this paper we discuss several conjectures in the inverse boundary value
problems for quasilinear elliptic equations and their recent progress. These problems concern anisotropic quasilinear elliptic equations in connection with nonlinear materials and the nonlinear elasticity system.
RESUMEN
Problemas inversos a valores en la frontera se desarrollaron a comienzos de
la década de los 80, a partir de contribuciones de A.P. Calderon en el problema
de conductividad inversa [C], en el cual se intenta recuperar las conductividad
eléctrica de un cuerpo mediante mediciones de voltaje y corriente en la frontera.
Desde entonces, el área de problemas a valores en la forntera inversos para ecuaciones lineales elı́pticas ha sido objeto de mucho desarrollo [U]. El crecimiento de
la teorı́a en esta área tiene aplicaciones en muchas aplicaciones, las que varı́an
desde imagenologı́a médica, hasta diversos métodos de detección [BB], [Che-Is].
En este artı́culo, discutimos varias conjeturas en problemas inversos de valores en
66
Ziqi Sun
7, 3(2005)
la frontera para ecuaciones elı́pticas quasi-lineales y sus progresos recientes. Estos problemas dicen relación con ecuaciones elı́pticas quasilineales anisotrópicas
en conexión con materiales nolineales y sistemas de elasticidad no lineal.
Key words and phrases:
Math. Subj. Class.:
1
Inverse boundary value problem. Dirichlet to
Neumann map
35R30
Anisotropic Quasilinear Conductivity Equations
Consider the quasilinear elliptic equation
LA u =
n
X
(aij (x, u)uxi )xj = 0, u|Γ = f ∈ C 2,α (Γ)
(1)
i,j=1
on a bounded domain Ω ⊂ Rn , n ≥ 2, with smooth boundary Γ. Here A(x, t) =
(aij (x, t))n×n is the quasilinear coefficient matrix which is assumed to be in the C 1,α
class with 0 < α < 1. The nonlinear Dirichlet to Neumann map
ΛA : f → ν · A(x, f )∇u|Γ
is an operator from C 2,α (Γ) to C 1,α (Γ), which carries essentially all information about
the solution u which can be measured on the boundary. Here we denote ν to be the
unit outer normal of Ω. The inverse problem under discussion is to recover information
about the quasilinear coefficient matrix A from the knowledge of ΛA .
This problem was raised by R. Kohn and M. Vogelius [KV] in mid 80’s as a nonlinear analogue of the well known inverse conductivity problem posed by A.P. Calderon
[C]. Physically, the problem is connected to Electrical Impedance Tomography in
nonlinear media.
It has been shown in [Su1] that, in the isotropic case of the problem, i.e., when A
is a scalar matrix, the Dirichlet to Neumann map ΛA gives full information about A.
In other words, ΛA determines A uniquely as a function on Ω × R. This generalizes
to the quasilinear case the well known uniqueness theorems of the linear case (i.e.,
when A is scalar and is indenpendent on t)[SU1,2][SuU2] [N].
In the anisotropic case, however, one only expects to recover A module the group
G = {all C 3,α diffeomorphisms Φ: Ω̄ → Ω̄ with Φ∂Ω = identity}.
In fact, ΛA is invariant under G: For any A and Φ ∈ G, ΛA = ΛHΦ A . Here HΦ A is
the pull back of A under Φ:
HΦ A(x, t) = (|detDΦ|−1 (DΦ)T A(x, t)(DΦ)) ◦ Φ−1
(2)
7, 3(2005)
Conjectures in Inverse Boundary Value Problems for ...
67
where DΦ is the Jacobian matrix of Φ. One should observe that (2) holds only when
Φ is independent on t. Thus, the following conjecture is natural:
Conjecture 1: Assume that ΛA1 = ΛA2 . Then there exists a unique diffeomorphism
Φ ∈ G so that A2 = HΦ A1 .
In [SuU1] we have verified this conjecture in the C 2,α category for dimension n = 2
and in the real analytic category for dimension n ≥ 3. These results extend all known
results regarding this conjecture in the case of linear coefficient matrices (i.e. when
A is independent of t), obtained earlier in the works of Sylvester [S], Nachman [N]
and Lee-Uhlmann [LU]. We mention that in the two dimensional case the unique
diffeomorphism Φ in the result belongs to the C 3,α class, which is one order smoother
than A1 and A2 and in the case n ≥ 3, Φ is in the real analytic category. Assuming
Holder smoothness for the coefficient seems quite essential to assure that Φ is one
order smoother than the coefficient matrices. As explained in [SuU1], this is closely
related to the elliptic regularity theory.
The proof is based on a well known linearization technique introduced in [I1] and
further developed in [I2][IS][IN][Su1,3] which reduces the nonlinear problem to a linear
one. Let t ∈ R and g ∈ C 2,α (Γ). From ΛA one determines two linear operators:
(1)
KA,t : g → d/dsΛA (t + sg)|s=0
(2)
KA,t : g → d2 /ds2 (s−1 ΛA (t + sg))|s=0
(3)
(1)
One observes that KA,t = ΛAt , the Dirichlet to Neumann map corresponding to the
linear coefficient matrix At (x) = A(x, t) for a fixed t. So, if ΛA1 = ΛA2 for two
quasilinear coefficient matrices A1 and A2 , then ΛAt1 = ΛAt2 , ∀t ∈ R, and since the
conjecture is true in the linear case, one obtains a family of diffeomorphisms Φt ∈ G,
depending on the parameter t, so that
HΦt At1 = At2 , ∀t ∈ R.
(4)
The mathematical difficulty is to show that Φt is actually independent on t, which
would imply the result. It has been verified in [SuU1] that Φt is smooth in t. For
dimension n ≥ 3, this was achieved by studying a related geometrical problem in
which Φt becomes a family of isometries between two families of Riemannian metrics
|Ati |1/(n−2) (Ati )−1 on Ω̄, i = 1, 2. For n = 2, One can transform it to a similar problem
where Φt becomes a family of conformal diffeomorphisms between Riemannian metrics
(Ati )−1 , i = 1, 2. In the latter case, the smoothness is verified via the standard theory
of the Beltrami equation [AB].
So, the task is to show that Φ̇|t=0 , where dot means differentiation in t variable.
We only give a very brief description of the proof. One only needs to show
Φ̇0 = Φ̇t |t=0 = 0
(5)
68
Ziqi Sun
7, 3(2005)
since the same argument works for t 6= 0. By a transformation one may assume that
Φ0 = identity map. The proof of (5) is then based on the information obtained from
(3):
(2)
(2)
KA1 ,t = KA2 ,t .
(6)
(2)
A crucial step of the proof is to show that one can recover from KA,t information
about ∂A/∂t(x, 0). So (6) implies
∂
∂
A1 (x, 0) = A2 (x, 0), ∀x ∈ Ω.
∂t
∂t
(7)
One views (7) as a certain control over the flows At1 and At2 at t = 0. Actually,
the assumption Φ0 =Id. together with (7) give A01 = A02 and Ȧ01 = Ȧ02 . Consider
now the solution flows uti,f for the linear equations LAti (uti,f ) = 0 with uti,f |Γ = f ,
i = 1, 2. One observes that the control over the flows of coefficient matrices translates
to a control over the solution flows. In fact, for every f , u01,f = u02,f and u̇01,f = u̇02,f .
Since the transformation in (4) links ut1,f to ut2,f via the relation u̇t1,f = u̇t2,f ◦ Φt , one
differentiates it in t at t = 0 to get Φ̇0 ·∇u01,f = 0 for all boundary value f , from which
(5) follows by an argument based on Runge approximation. See [SuU1] for details.
The above result obtained in [SuU1] covers the two dimensional case and the real
analytic case in dimension three or higher. However, the remaining case in dimension
n ≥ 3 is essentially open even when the equation (1) is linear. An interesting problem
for further study in this direction is whether one can reduce the conjecture in the
quasilinear case directly to the conjecture in the linear case. In other words, one
would like to verify Conjecture 1 under the assumption that Conjecture 1 holds in
the linear case. Such a full reduction has already been obtained in the scalar case
(where A is a scalar matrix) [Su1]. It is possible that the same reduction also hold
in the anisotropic case. One possible approach to attack this problem is to further
study the relation between (6) and (7) in the general case, which is the heart of proof
in [SuU1]. The main issue is how to avoid the use of the property of completeness
of products of solutions which is currently available only in the two dimensional case
and the case of real analytic coefficient matrices.
2
Quasilinear Equations in Connection with Nonlinear Elastic Materials
Consider the quasilinear elliptic equation
∇ · A(x, ∇u) = 0, u|Γ = f ∈ C 3,α (Γ),
(8)
on a bounded domain Ω ⊂ Rn , n ≥ 2, with smooth boundary Γ. Here A(x, p) =
(a1 (x, p), a2 (x, p), ..., an (x, p)) is the quasilinear coefficient vector. We assume that
A and Ap (which is assumed to be symmetric) are both in C 2,α (Ω̄×R) with 0 < α < 1,
A(x, 0) = 0 and the structure conditions which guarantee the unique solvability in
the C 3,α class [HSu].
7, 3(2005)
Conjectures in Inverse Boundary Value Problems for ...
69
The nonlinear Dirichlet to Neumann map
ΛA : f → ν · A(x, ∇u)|Γ ,
(9)
is an operator from C 3,α (Γ) to C 2,α (Γ), which carries essentially all information about
the solution u observable on the boundary. One verifies that ΛA is invariant under
the group G: ΛA = ΛHΦ A for all Φ ∈ G. Here the transformation HΦ is defined as
HΦ A(x, p) = (|detDΦ|−1 (DΦ)T A(x, (DΦ)p)) ◦ Φ−1 .
The main problem is whether the converse is true.
Conjecture 2: Assume that ΛA1 = ΛA2 . Then there exists a unique diffeomorphism
Φ ∈ G so that A2 = HΦ A1 .
The equation (8) can be considered as a simple scalar model of the nonlinear
elasticity system, which takes the form
∇{σ(x, E) + (∇u)σ(x, E)} = 0,
(10)
where u is the displacement vector function resulting from a deformation of an elastic
body and the matrix function σ is the constitutive relation with the strain tensor
E=
1
(∇uT + ∇u + ∇uT ∇u).
2
In [HSu], we developed a mathematical framework towards proving this conjecture
in the case of two dimensions. In the discussion below, we assume ΛA1 = ΛA2 for two
quasilinear coefficient vectors A1 and A2 in dimension two. By linearizing (9) one
obtains, as in the case of Conjecture 1, a family of diffeomorphisms {Φf }⊂ G which
transforms A1,p (x, ∇u1,f ) to A2,p (x, ∇u2,f ):
A2,p (x, ∇u2,f ) = HΦf A1,p (x, ∇u1,f ),
and the main problem is to show that Φf is independent on f . Here we denote by
ui,f solution of (11) with A replaced by Ai , i = 1, 2.
One notices that {Φf , f ∈ C 2,α (Γ)} is an infinite dimensional family rather than
an one dimensional family in the case of Conjecture 1. Also, contrary to (3), any
further linearization on (9) would not provide any new information about Φf . So,
technically, the task in this case is much harder to accomplish.
For a f ∈ C 3,α (Γ), let gi,f be the Riemannian metric (on Ω̄) generated by the
metrix A−1
i,p (x, ∇ui,f ), i = 1, 2. One verifies that Φf is a family of conformal diffeomorphisms sending (Ω̄, g1,f ) to (Ω̄, g2,f ). If one uses Φ∗f g to denote the pullback of a
tensor g under Φf , then (15) can be rewritten as
Φ∗f g2,f = |DΦf |g1,f .
Given f , h ∈ C 3,α (Γ), Let’s denote by ġi,f,h the Frechet derivative of gi,f at the
reference point f in the direction h, i = 1, 2. Once again, one can show that Φf
70
Ziqi Sun
7, 3(2005)
is smooth in f (parallel to those in Conjecture 1) and we denote by X = Φ̇f,h the
corresponding derivative of Φf in the direction h (viewed as a vector field). For a
fixed f , we may once again assume that Φf = identity and set g1,f = g2,f =: gf and
u1,f = u2,f =: uf .
In order to prove the conjecture by showing
X = Φ̇f,h = 0, ∀h ∈ C 3,α (Γ),
(11)
Let us take a deep look at the relation Φ∗f g2,f =| DΦf | g1,f by differentiating it in f
with a direction h ∈ C 3,α (Γ). We get
ġ1,f,h − ġ2,f,h = LX gf − (eσ ∇gf · (e−σ X))gf .
(12)
p
where LX gf stands for Lie derivative of gf under the vector field X and σ = log det(g).
Equation (12) implies that X is connected to the inhomogeneous conformal Killing
field equation (with respect to the metric gf ) with the boundary condition X |Γ = 0.
However, this equation has no real consequence if one just considers one direction.
The main observation made in [HSu] is that if one considers a pair of directions, then
one can use the theory of conformal Killing field to obtain useful consequences leading
to (11). Indeed, when one is given a pair of directions h1 , h2 ∈ C 2,α (Γ), one can show
that the following symmetric relation
ġf,h1 lf,h2 = ġf,h2 lf,h1
holds for ġf,h1 = ġ1,f,h1 or ġ2,f,h1 and lf,h = ∇gf u̇f,h = gf−1 ∇u̇f,h . This is proven in
[HSu] using the special structure of the linearized coefficient matrix. Combining this
symmetric relation together with (12) one gets
lf,h2 c(LX1 gf − (eσ ∇gf · (e−σ X1 ))gf ) = lf,h1 c(LX2 gf − (eσ ∇gf · (e−σ X2 ))gf ), (13)
where Xi = Φ̇f,hi , i = 1, 2. Equation (13) implies that both Xi , i = 1, 2, satisfy the
inhomogeneous conformal Killing field equation of the type
lc(LX (g) − (eσ ∇ · (e−σ X))g) = F
(14)
with the same inhomogeneous term F , which is a 1-form. The equation (14) is the
crucial equation for the proof. We have proven
that if
X and l satisfy the equation (14)
with X |Γ = 0, then both inner products l, X g and l⊥ , X g are uniquely determined
by F, where l⊥ stands for the unique vector perpendicular to l with l⊥ = l
in the counterclockwise direction under the metric g [Su2], Base on this result, one
concludes from (13) that the vector fields Xi and lf,hi must satisfy the following system
of equations:
( X1 ,lf,h
= X2 , lf,h1 g
2 g
f
f
,
(15)
⊥
⊥
X1 ,lf,h
=
X
,
l
2 f,h1 g
2 g
f
f
7, 3(2005)
Conjectures in Inverse Boundary Value Problems for ...
71
To understand (15) better, consider now a two-parameter family of conformal
diffeomorphisms Φf +η1 h1 +η2 h2 ⊂ G with parameters η1 and η2 in R. For a fixed
point x ∈ Ω, define
ω(η1 , η2 ) = Φf +η1 h1 +η2 h2 (x) : R2 → Ω̄
as a function from (η1 , η2 ) to the image of x under Φf +η1 h1 +η2 h2 . One checks that
ωη1 = Φ̇f +η1 h1 +η2 h2 ,h1 (x), ωη2 = Φ̇f +η1 h1 +η2 h2 ,h2 (x).
By Replacing f by f + η1 h1 + η2 h2 one can shows from (15) that the function ω
satisfies the following first order system:
( ω η , l 2 = ω η , l 1 1
g
2
g
(16)
,
ωη1 , l2⊥ g = ωη2 , l1⊥ g
where
lj = lf +η1 h1 +η2 h2 ,hj ◦ Φf +η1 h1 +η2 h2 , j = 1, 2.
Here the additional term Φf +η1 h1 +η2 h2 is needed once one removes the assumption
Φf = identity.
System (16) can be viewed as a generalized Cauchy-Riemann system under the
vector fields l1 and l2 . The proof of (11) with h = h1 and h2 is now reduced to
showing that System (16) admits no bounded nonconstant solution ω. Note that ω is
always bounded. In order to do that, one way is to apply Liouville’s type theorems
to the system (16). However, one must choose the directions h1 and h2 in a way that
the gradients of the solution l1 and l2 are uniformly independent. Once (11) is proven
with two independent directions, one can show that (11) holds for all directions. This
is proven in [HSu] using the geometric argument developed in [Su2].
In [HSu] the above framework has been successfully to two important special
cases: The case in which A(x, p) is independent of x and the case in which Ap (x, p)
is independent of p. In both cases one is allowed to construct the needed independent
directions h1 and h2 . See [HSu] for details.
To verify the conjecture completely, the main difficulty is the construction of
special directions. The construction of special directions in the known cases has
been completed by using techniques of exponentially growing solutions, which is not
available in the general case. One possible way to overcome this difficulty is to replace
the two-parameter family of conformal diffeomorphisms Φf +η1 h1 +η2 h2 by ΦF (η1 ,η2 ) ,
where F (η1 , η2 ) is a two dimensional nonlinear variety in C 3,α (Γ) passing through f .
The nonlinearity of F (η1 , η2 ) should correspond to the quasilinear nature of A(x, p).
Once one identifies the correct form of F (η1 , η2 ), the rest of the argument can be
modified to cover the general case.
Received:
April 2004.
Revised:
May 2004.
72
Ziqi Sun
7, 3(2005)
References
[AB]
L. Ahlfors and L. Bers, Riemann’s mapping theorem for variable metrics,
Ann. of Math. 72 (1960), 385-404.
[BB]
D. C. Barber and B. H. Brown Applied potential tomography J. Phys. E. 17
(1984), 723-733.
[C]
A.P.Calderon, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasileira de
Matematica, Rio de Janeiro, (1980), 65-73.
[CheIs]
M. Cheney and D. Isaacson, An overview of inversion algorithm for impedance imaging, Contemporary Math. 122 (1991), 29-39.
[HSu]
. Hervas and Z. Sun, An inverse boundary value problem for quasilinear
elliptic equations, Comm. in PDE 27 (2002), 2449-2490.
[I1]
V. Isakov, On uniqueness in inverse problems for semilinear parabolic equations, Arch. Rat. Mech.Anal. 124 (1993), 1-12.
[I2]
V. Isakov, Uniqueness of recovery of some systems of semilinear partial differential equations, Inverse Problems 17 (2001) 607-618.
[IN]
V. Isakov and A. Nachman, Global uniqueness for a two-dimensional semilinear elliptic inverse problem, Trans, of AMS 347 (1995), 3375-3390
[IS]
V. Isakov and J. Sylvester, Global uniqueness for a semilinear elliptic inverse
problem, Comm. Pure Appl. Math. 47 (1994), 1403-1410.
[KV]
R. Kohn and M. Vogelius, Identification of an unknown conductivity by
means of measurements II, Inverse Problems, D. W. McLaughlin, ed., SIAMAMS Proc. 14 (1984), 113-123.
[LU]
J. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivity
by boundary measurements, Comm. Pure Appl. Math, 42 (1989), 1097-1112.
[N]
A. Nachman, Global uniqueness for a two-dimensional inverse boundary
value problem, Ann. of Math, 143 (1996), 71-96.
[S]
J. Sylvester, An anisotropic inverse boundary value problem, Comm. Pure
Appl. Math. 43 (1990), 201-232.
[SU1]
J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse
boundary value problem, Ann. of Math. 125 (1987), 153-169.
[SU2]
J. Sylvester and G. Uhlmann, Inverse problems in anisotropic media, Contemporary Math. 122 (1991), 105-117.
[Su1]
Z. Sun, On a quasilinear inverse boundary value problem, Math. Z. 221
(1996), 293-305.
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[Su2]
Z. Sun, An inverse problem for inhomogeneous conformal Killing field equations, Proc. Amer. Math. Soc. 131 (2003), 1583-1590.
[Su3]
Z. Sun, Inverse boundary value problems for a class of semilinear elliptic
equations, to appear in Advances in Applied Math.
[SuU1]
Z. Sun and G. Uhlmann, Inverse problems in quasilinear anisotropic media,
Amer. J. of Math. 119 (1997), 771-797.
[SuU2]
Z. Sun and G. Uhlmann, Anisotropic inverse problems in two dimensions,
Inverse Problems 19 (2003), 1-10.
[U]
G. Uhlmann, Developments in inverse problems since Calderon’s foundational paper, Harmonic Analysis and Pde, University of Chicago Press, 1999.
A Mathematical Journal
Vol. 7, No 3, (75 - 85). December 2005.
Relations of al Functions over Subvarieties
in a Hyperelliptic Jacobian
Shigeki Matsutani
8-21-1 Higashi-Linkan, Sagamihara, 228-0811, JAPAN
rxb01142@nifty.com
ABSTRACT
The sine-Gordon equation has hyperelliptic al function solutions over a hyperelliptic Jacobian for y 2 = f (x) of arbitrary genus g. This article gives an
extension of the sine-Gordon equation to that over subvarieties of the hyperelliptic Jacobian. We also obtain the condition that the sine-Gordon equation in a
proper subvariety of the Jacobian is defined.
RESUMEN
La ecuación de sine-Gordon tiene soluciones funciones hiperelı́pticas sobre un
Jacobiano hiperelı́ptico para y 2 = f (x) de género arbitrario g. En este artı́culo
damos una extensión de la ecuación de Sine-Gordon sobre subvariedades de Jacobiano hiperelı́ptico. También obtenemos la condición para que la ecuación de
sine-Gordon esté definida en una subvariedad propia del Jacobiano.
Key words and phrases:
Math. Subj. Class.:
sine-Gordon equation, nonlinear integrable
differential equation, hyperelliptic functions,
a subvariety in a Jacobian
Primary 14H05, 14K12; Secondary 14H51, 14H70
76
1
Shigeki Matsutani
7, 3(2005)
Introduction
Q2g+1
For a hyperelliptic curve Cg given by an affine curve y 2 = i=1 (x − bi ), where bi ’s
are complex numbers, we have a Jacobian Jg as a complex torus Cg /Λ by the Abel
map ω [Mu]. Due to the Abelian theorem, we have a natural morphism from the
symmetrical product Symg (Cg ) to the Jacobian Jg ≈ ω[Symg (Cg )]/Λ. As zeros of an
appropriate shifted Riemann theta function over Jg , the theta divisor is defined as
Θ := ω[Symg−1 (Cg )]/Λ
which is a subvariety of Jg . Similarly, it is natural to introduce a subvariety
Θk := ω[Symk (Cg )]/Λ
and a sequence,
Θ0 ⊂ Θ1 ⊂ Θ2 ⊂ · · · ⊂ Θg−1 ⊂ Θg ≡ Jg
Vanhaecke studied the structure of these subvarieties as stratifications of the Jacobian
Jg using the strategies developed in the studies of the infinite dimensional integrable
system [V1]. He showed that these stratifications of the Jacobian are connected with
stratifications of the Sato Grassmannian. Further Vanhaecke investigated Lie-Poisson
structures in the Jacobian in [V2]. He showed that invariant manifolds associated with
Poisson brackets can be identified with these strata; it implies that the strata are characterized by the Lie-Poisson structures. He also showed that the Poisson brackets are
connected with a finite-dimensional integrable system, Henon-Heiles system. Following the study, Abenda and Fedorov [AF] investigated these strata and their relations
to Henon-Heiles system and Neumann systems.
On the other hand, functions over the embedded hyperelliptic curve Θ1 in a hyperelliptic Jacobian Jg were also studied from viewpoint of number theory in [C, G, Ô].
In [Ô], Ônishi also investigated the sequence of the subvarieties, and explicitly studied behaviors of functions over them in order to obtain higher genus analog of the
Frobenius-Stickelberger relations for genus one case. Though Vanhaecke, Abenda and
Fedorov found some relations of functions over these subvarieties explicitly using the
infinite universal grassmannians and so-called Mumford’s U V W expressions [Mu],
Ônishi gave more explicit relations on some functions over the subvarieties using the
theory of hyperelliptic functions in the nineteenth century fashion [Ba1, Ba2, Ba3].
In this article, we will also investigate some relations of functions over the subvarieties based upon the studies of the hyperelliptic function theory developed in the
nineteenth century [Ba2, Ba3, W]. Especially this article deals with the “sine-Gordon
equation” over there.
Modern expressions of the sine-Gordon equation in terms of Riemann theta functions were given in [[Mu] 3.241],
∂ ∂
log([2P − 2Q]) = A([2P − 2Q] − [2Q − 2P]),
∂tP ∂tQ
(1.1)
where P and Q are ramified points of Cg , A is a constant number, [D] is a meromorphic
function over Symg (Cg ) with a divisor D for each Cg and tP0 is a coordinate in the
7, 3(2005)
Relations of al Functions over Subvarieties in a Hyperelliptic Jacobian
77
Jacobi variety such that it is identified with a local parameter at a ramified point P0
up to constant.
In the previous work [Ma], we also studied (1.1) using the fashion
of the nineteenth
p
century. In [W] Weierstrass defined al function by alr := γr Fg (br ) and Fg (z) :=
(x1 − z) · · · (xg − z) over Jg with a constant factor γr . Let γr = 1 in this article.
As Weierstrass implicitly seemed to deal with it, (1.1) is naturally described by alfunctions as [Ma],
∂2
(g)
(g)
∂v1 ∂v2
1
al
log r =
als
(br − bs )
f 0 (bs )
alr
als
2
+ f 0 (br )
als
alr
2 !
.
(1.2)
Here f 0 (x) := df (x)/dx and v (g) ’s are defined in (2.4). ((1.2) was obtained in the
previous article [Ma] by more direct computations and will be shown as Corollary 3.3
in this article). We call (1.2) Weierstrass relation in this article.
In this article,
p we will introduce an “al” function over the subvariety in the Jacobian, al(m)
:=
Fm (br ) and Fm (z) := (x1 − z) · · · (xm − z) for a point ((x1 , y1 ), · · · ,
r
(xm , ym )) in the symmetric product of the m curves Symm Cg (m = 1, · · · , g − 1). In
[Mu], Mumford dealt with Fm function (he denoted it by U ) for 1 ≤ m < g and studied the properties. Further Abenda and Fedorov also studied some properties of the
and Fm functions in [AF] though they did not mention about Weierstrass’s paal(m)
r
per nor the relation (1.2). We will consider a variant of the Weierstrass relation (1.2)
over subvariety in non-degenerated and degenerated hyperelliptic Jacobian.
to al(m)
r
As in our main theorem 3.1, even on the subvarieties, we have a similar relation
to (1.1),
∂
(m)
∂vr
∂
(m)
∂vs
log
al(m)
r
al(m)
s

0
1
 f (br )
=
2
(br − bs ) (Q(2)
m (br ))
al(m)
s
al(m)
r
!2
+
f 0 (bs )
al(m)
r
(2)
al(m)
s
(Qm (bs ))2
!2 

+ reminder terms.
(1.3)
(2)
Here Qm is defined in (2.2). We regard (1.3) or (3.1) as a subvariety version of the
Weierstrass relation (1.2). In fact, (1.3) contains the same form as (1.1) up to the
(2)
factors (Qm (bt ))2 (t = r, s) and the reminder terms. Thus (1.3) or (3.1) should be
regarded as an extension of the sine-Gordon equation (1.2) over the Jacobian to that
over the subvariety of the Jacobian.
Further a certain degenerate curve, the remainders in (1.3) vanishes. Then we
have a relations over subvarieties in the Jacobian, which is formally the same as
(2)
the Weierstrass relations (1.2) up to the factors (Qm (bt ))2 (t = r, s), which means
that we can find solutions of sine-Gordon equation over subvarieties in hyperelliptic
Jacobian. We expect that our results shed a light on the new field of a relation
between “integrability” and a subvariety in the Jacobian, which was brought off by
[V1, V2, AF].
78
Shigeki Matsutani
7, 3(2005)
The author is grateful to the referee for directing his attensions to the references
[AF] and [V2].
2
Differentials of a Hyperelliptic Curve
In this section, we will give our conventions of hyperelliptic functions of a hyperelliptic
curve Cg of genus g (g > 0) given by an affine equation,
y 2 = f (x) = (x − b1 )(x − b2 ) · · · (x − b2g )(x − b2g+1 )
= Q(x)P (x),
(2.1)
(1)
(2)
where bj ’s are complex numbers. Here we use the expressions Q(x) := Qm (x)Qm (x),
Q(1)
m (x) := (x − a1 )(x − a2 ) · · · (x − am ),
Q(2)
m (x) := (x − am+1 )(x − am+2 ) · · · (x − ag ),
P (x) := (x − c1 )(x − c2 ) · · · (x − cg )(x − c),
(2.2)
where ak ≡ bk , ck ≡ bg+k , (k = 1, · · · , g) c ≡ b2g+1 .
Definition 2.1 [Ba1, Ba2] For a point (xi , yi ) ∈ Cg , we define the following quantities.
1. The unnormalized differentials of the first kind are defined by,
(g,i)
dvk
:=
Q(xi )dxi
,
2(xi − ak )Q0 (ak )yi
(k = 1, · · · , g)
(2.3)
2. The Abel map for g-th symmetric product of the curve Cg is defined by,
(g)
v (g) ≡ (v1 , · · · , vg(g) ) : Symg (Cg ) −→ Cg ,
(g)
vk ((x1 , y1 ), · · ·
, (xg , yg )) :=
g Z
X
(xi ,yi )
!
(g,i)
dvk
.
(2.4)
Λ.
Ξm := v (g) (Symm (Cg ) × (am+1 , 0) × · · · × (ag , 0))/Λ
(2.5)
i=1
∞
3. For v (g) ∈ Cg , we define the subspace,
Here C is a complex field and Λ is a g-dimensional lattice generated by the
related periods or the hyperelliptic integrals of the first kind.
7, 3(2005)
Relations of al Functions over Subvarieties in a Hyperelliptic Jacobian
79
The Jacobi variety Jg are defined as complex torus as Jg := Ξg . As Ξm (m < g)
is embedded in Jg whose complex dimension as subvariety is m, the differential forms
(g)
(dvk )k=1,··· ,g are not linearly independent. We select linearly independent bases
(m)
(g)
such as vk := vk ((x1 , y1 ), · · · , (xm , ym ), (am+1 , 0), · · · , (ag , 0)), (k = 1, · · · , m) at
Ξm .
Ξ0 ⊂ Ξ1 ⊂ Ξ2 ⊂ · · · ⊂ Ξg−1 ⊂ Ξg ≡ Jg
For (x1 , · · · , xm ) ∈ Symm (Cg ), we introduce
Fm (x) := (x − x1 ) · · · (x − xm ),
(2.6)
and in terms of Fm (x), a hyperelliptic al-function over (v (m) ) ∈ Ξm , [Ba2 p.340, W],
p
(m)
(2.7)
) = Fm (br ).
al(m)
r (v
Further we introduce m × m-matrices,

1
1
 x1 − a1
x2 − a1

1
1


x
−
a
x
−
a2

1
2
2
Mm := 
..
..

.
.


1
1
x1 − am x2 − am
 s
Qm






=






···
..
1
xm − a1
1
xm − a2
..
.
1
xm − am





,





Q(x2 )
P (x2 )
..


Am = 

.
···
Q(x1 )
P (x1 )
s

Lemma 2.2
···
.






,




s

Q(xm ) 
P (xm )
Q0 (a1 )

Q0 (a2 )
..
.
Q0 (am )


.

1.
det Mm =
(−1)m(m−1)/2 P (x1 , · · · , xm )P (a1 , · · · , am )
Q
,
k,l (xk − al )
80
Shigeki Matsutani
7, 3(2005)
where
P (z1 , · · · , zm ) :=
Y
(zi − zj ).
i<j
2.

(1)
Fm (aj )Qm (xi )

M−1
m =
(1)0
0 (x )Q
Fm
m (aj )(aj − xi )
i
(1)0
! 
,
i,j
(1)
0
where Fm
(x) := dFm (x)/dx and Qm (x) = dQm (x)/dx.
3.

(MQ)−1 A = 
2yi Fm (aj )
(2)
0 (x )Q
Fm
m (xi )(aj − xi )
i
! 
.
(2.8)
i,j
Proof. (1) is a well-known result [T]. The zero and singularity in the left hand side
give the right hand side as
Y
CP(x1 , · · · , xm )P(a1 , · · · , am )/ (xk − al ),
k,l
Q
for a certain constant C. In order to determine C, we multiply k,l (xk − al ) both
sides and let x1 = a1 , x2 = a2 , · · · , and xm = am . Then C is determined as above. (2)
is obtained by the Laplace formula using the minor determinant
pfor the inverse matrix.
(1)
(1) (2)
(2)
On (3) we note that Qm Qm = Q(x) in (2.2) and thus Qm (x) P (x)/Q(x) = y/Qm .
Then we obtain (3).
(r)
(r)
Corollary 2.3 Let ∂vi := ∂/∂vi , and ∂xi := ∂/∂xi .

v1
 ∂v2

 ..
 .
∂vm
3


x1

 ∂x2


 = 2(MQm )−1 Am  ..

 .
∂xm



.

(2.9)
Weierstrass relation on Ξm
The hyperelliptic solution of the sine-Gordon equation over Jg related to ramified
points (a1 , 0) and (a2 , 0) is obtained as (1.1) by Mumford [Mu], whose expression in
an old fashion is the Weierstrass relation (1.2). Let us consider an extension of the
Weierstrass relation (1.2) over Ξm as our main theorem. We will give the theorem as
follows.
7, 3(2005)
Relations of al Functions over Subvarieties in a Hyperelliptic Jacobian
81
Theorem 3.1 al(m)
and al(m)
(r, s ∈ {1, 2, · · · , m}) over Ξm in (2.5) obey the relar
s
tion,
∂
∂
(m)
∂vr
(m)
log
(m)
al(m)
)
r (v
(m) )
al(m)
s (v

0
1
 f (ar )
=
2
(ar − as ) (Q(2)
m (ar ))
∂vs
+
(m)
al(m)
)
s (v
!2
(m) )
al(m)
r (v
+
f 0 (as )
(m)
al(m)
)
r (v
(2)
(m) )
al(m)
s (v
(Qm (as ))2
!2 

(m) 2
(m) 2
f 0 (am+1 )(al(m)
)) (al(m)
)) (ar − as )
r (v
s (v
(m)
(2)0
(am+1 − ar )(am+1 − as )(alm+1 (v (m) ))4 (Qm (am+1 ))2
+ ······
+
(m) 2
(m) 2
f 0 (ag )(al(m)
)) (al(m)
)) (ar − as )
r (v
s (v
(2)
0 2
(m) ))4 (Q
(ag − ar )(ag − as )(al(m)
m (ag ) )
g (v
.
(3.1)
Proof. From (2.7), we will consider the following formula instead of (3.1) without
loss of generality,
∂
(m)
∂v1
∂
(m)
log
∂v2
Fm (a1 )
Fm (a1 )Fm (a2 ) f 0 (a1 )
=2
(2)
Fm (a2 )
(a1 − a2 )
Fm (a1 )2 (Qm (a1 ))2
+
+
f 0 (a2 )
(2)
Fm (a2 )2 (Qm (a1 ))2
f 0 (am+1 )(a1 − a2 )2
(am+1 − a1 )(am+1 −
+ ···
+
(3.2)
(2)0
a2 )Fm (am+1 )2 (Qm (am+1 ))2
f 0 (ag )(a1 − a2 )2
(2)0
(ag − a1 )(ag − a2 )Fm (ag )2 (Qm (ag ))2
.
Before we start the proof, we will comment on our strategy, which is essentially the
same as [Ba3]. First we translate the words of the Jacobian into those of the curves;
(m)
we rewrite the differentials v(r) ’s in terms of the differentials over curves as in (3.3).
We count the residue of an integration and use a combinatorial trick as in Lemma
3.2. Then we will obtain (3.2).
From (2.8) and (2.9), the derivative v’s over Ξm in (2.5) are expressed by the affine
coordinate xi ’s,
∂
(m)
∂vi
= Fm (ai )Q(2)
m (ai )
m
X
2yj
(2)
0
j=1 Fm (xj )Qm (xj )(xj
∂
.
∂x
− ai ) j
(3.3)
82
Shigeki Matsutani
7, 3(2005)
The right hand side of (3.2) becomes,
∂2
Fm (a1 )
log
= Fm (a1 )Q(2)
m (a1 )
∂v1 ∂v2
Fm (a2 )
m
(2)
X
2yi Fm (a2 )Qm (a1 )
2yj
∂
(xi −
i,j=1
0 (x )Q(2) (x )
a1 )Fm
m
j
j
(a1 − a2 )
.
0 (x )Q(2) (x )(x − a ) (xi − a1 )(xi − a2 )
∂xj Fm
m
i
i
i
2
The right hand side is
m
X
Fm (a1 )Fm (a2 )
i=1
−
"
∂
1
0
Fm (xi ) ∂x
!#
f (x)(a2 − a1 )
(2)
0 (x)
(x − a1 )2 (x − a2 )2 (Qm (x))2 Fm
x=xi
X
2yk yl (a2 − a1 )
k,l,k6=l
0 (x )F 0 (x )(x − a )(x − a )Q
Fm
m (xl )(xk − a1 )(xk − a2 )Qm (xk )(xl − xk )
k
l
1
l
2
m l
(2)
(2)
.
Then the proof of Theorem 3.1 is completely done due to next lemma.
Lemma 3.2 1)
m
X
i=1
=
+
"
1
∂
0 (x ) ∂x
Fm
i
2
!#
f (x)
(2)
0 (x)
(x − a1 )2 (x − a2 )2 (Qm (x)2 Fm
x=xi
f 0 (a2 )
f 0 (a1 )
+
(2)
2
(a1 − a2 )2 Fm (a1 )2 (Q(2)
Fm (a2 )2 (Qm (a1 ))2
m (a1 ))
f 0 (am+1 )(a1 − a2 )2
(2)0
(am+1 − a1 )(am+1 − a2 )Fm (am+1 )2 (Qm (am+1 ))2
+ ···
f 0 (ag )(a1 − a2 )2
+
.
(2)0
(ag − a1 )(ag − a2 )Fm (ag )2 (Qm (ag ))2
X
2yk yl (a2 − a1 )
k,l,k6=l
0 (x )F 0 (x )(x − a )(x − a )Q
Fm
m (xl )(xk − a1 )(xk − a2 )Qm (xk )(xl − xk )
k
l
1
l
2
m l
(2)
(2)
Proof. : (1) will be proved by the following residual computations: Let ∂Cgo be the
boundary of a polygon representation Cgo of Cg ,
I
f (x)
dx = 0.
(3.4)
(2)
∂Cgo (x − a1 )2 (x − a2 )2 Fm (x)2 (Qm (x))2
The divisor of the integrand of (3.4) is
2g+1
X
i=1
(bi , 0) − 4
X
i=1,2,m+1,m+2,··· ,g
(ai , 0) − 2
m
X
i=1
(xi , yi ) − 2
m
X
i=1
(xi , −yi ) + 3∞
= 0.
7, 3(2005)
Relations of al Functions over Subvarieties in a Hyperelliptic Jacobian
83
We check these poles: First we consider the contribution around ∞ point.
res(xk ,±yk )
f (x)
dx
(2)
(x − a1 )2 (x − a2 )2 Fm (x)2 (Qm (x))2
"
!#
∂
f (x)
1
= 0
2 0
Fm (xk ) ∂x (x − a1 )2 (x − a2 )2 (Q(2)
m (x)) Fm (x)
.
x=xk
At the point (a1 , 0), noting that the local parameter t is given by t =
there, we have
res(a1 ,0)
f (x)
(2)
(x − a1 )2 (x − a2 )2 Fm (x)2 (Qm (x))2
dx =
p
(x − a1 )
2f 0 (a1 )
(2)
(a1 − a2 )2 Fm (a1 )2 (Qm (a1 ))2
.
The residue at (a2 , 0) is similarly obtained. For the points (ak , 0) (g ≥ k > m), we
have
res(ak ,0)
f (x)
(2)
(x − a1 )2 (x − a2 )2 Fm (x)2 (Qm (x))2
=
dx
2f 0 (ak )
(2)0
(ak − a1 )2 (ak − a2 )2 Fm (a2 )2 (Qm (ak ))2
.
By arranging them, we obtain (1). (2) is obvious.
As a corollary, we have Weierstrass relation (1.2) which was proved in [Ma]:
Corollary 3.3 For m = g case, we have the Weierstrass relation for a general curve
Cg ,

!2
!2 
(m)
(m)
∂
∂
al(g)
1
al
al
r
s
f 0 (ar )
 . (3.5)
log r(g) =
+ f 0 (as )
(g)
(g)
(m)
(ar − as )
als
al(m)
al
∂vr ∂vs
r
s
Now we will give our final proposition as corollary.
Corollary 3.4 For a curve satisfying the relations cj = aj for j = m + 1, · · · , g,
al(m)
and al(m)
(r, s ∈ {1, 2, · · · , m}) over Ξm in (2.5) obey the relation,
r
s
∂
(m)
∂vr
∂
(m)
log
al(m)
r
al(m)
s

0
1
 f (ar )
=
2
(ar − as ) (Q(2)
m (ar ))
∂vs
al(m)
s
al(m)
r
!2
+
f 0 (as )
al(m)
r
(2)
al(m)
s
(Qm (as ))2
!2 
.
(3.6)
84
Shigeki Matsutani
7, 3(2005)
Proof. Since the condition cj = aj for j = m + 1, · · · , g means f 0 (aj ) = 0 for
j = m + 1, · · · , g, Theorem 3.1 reduces to this one.
p
2 f 0 (ar )f 0 (as )
Under the same assumption of Corollary 3.4, letting A =
,
(ar − as )Qm (ar )Qm (as )
and
s
f 0 (ar ) Qm (ar ) Fm (ar )
1
log
,
φ(r,s)
m (u) := √
f 0 (as Qm (as ) Fm (as )
−1
(r,s)
defined over Ξm , φm
obeys the sin-Gordon equation,
∂
(m)
∂vr
Received:
∂
(m)
∂vs
φ(r,s)
= A cos(φ(r,s)
m
m ).
September 2004.
Revised:
(3.7)
November 2004.
References
[AF]
S. Abenda, Yu. Fedorov, On the Weak Kowalevski-Painleve Property for
Hyperelliptically Separable Systems, Acta Appl. Math., 60 (2000) 137-178.
[Ba1]
H. F. Baker, Abelian functions – Abel’s theorem and the allied theory
including the theory of the theta functions –, Cambridge Univ. Press, (1897)
republication 1995.
[Ba2]
H. F. Baker, On the hyperelliptic sigma functions, Amer. J. of Math., XX
(1898) 301-384.
[Ba3]
H. F. Baker, On a system of differential equations leading to periodic functions, Acta Math., 27 (1903) 135-156.
[C]
D.G. Cantor, On the analogue of the division polynomials for hyperelliptic
curves, J. reine angew. Math., 447 (1994) 91-145.
[G]
D. Grant, A Generalization of a Formula of Eisenstein, 62 (1991) 121–132
Proc. London Math. Soc., .
[Ma]
S. Matsutani, On relations of Hyperelliptic Weierstrass al-Functions, Int.
J. Appl. Math., (2002) 11 295-307.
[Mu]
D. Mumford, Tata Lectures on Theta, vol II, Birkhäuser, (1984) Boston,
.
7, 3(2005)
Relations of al Functions over Subvarieties in a Hyperelliptic Jacobian
85
[Ô]
Y. Ônishi, Determinant Expressions for Hyperelliptic Functions (with an
Appendix by Shigeki Matsutani), preprint math.NT/0105189, to appear in
Proc. Edinburgh Math. Soc., , (2004) .
[V1]
P. Vanhaecke, Stratifications of hyperelliptic Jacobians and the Sato Grassmannian, Acta. Appl. Math., 40 (1995) 143-172.
[V2]
Pol Vanhaecke, Integrable systems and symmetric products of curves, Math.
Z., 227 (1998) 93-127.
[T]
T. Takagi, Daisuu-Gaku-Kougi (Lecture of Algebra), Kyouritsu, Tokyo,
(1930) japanese.
[W]
K. Weierstrass,
Zur Theorie der Abel’schen Functionen,
Aus dem
Crelle’schen Journal, 47 (1854) in Mathematische Werke I, Mayer und
Müller, Berlin, (1894) .
A Mathematical Journal
Vol. 7, No 3, (87 - 94). December 2005.
Convergence rates in regularization for
ill-posed variational inequalities
Nguyen Buong 1
Vietnamse Academy of Science and Technology, Institute of Information Technology
18, Hoang Quoc Viet, q. Cau Giay, Ha Noi, Vietnam
nbuong@ioit.ncst.ac.vn
ABSTRACT
In this paper the convergence rates for ill-posed inverse-strongly monotone
variational inequalities in Banach spaces are obtained on the base of choosing
the regularization parameter by the generalized discrepancy principle.
RESUMEN
En este artı́culo se obtienen tasas de convergencia para desigualdades variacionales en problemas inversos mal puestos fuertemente monótonos en espacios
de Banach, sobre la base de la elección del parámetro de regularización por medio
del principio de discrepancia generalizada.
Key words and phrases:
Math. Subj. Class.:
Monotone operators, hemi-continuous, strictly convex
Banach space, Frechet differentiable and Tikhonov
regularization.
47H17; CR: G1.8.
1 The author would like to express his thanks to the referees for their valuable remarks. This work
was supported by the National Fundamental Research Program in Natural Sciences.
88
1
Nguyen Buong
7, 3(2005)
Introduction.
Let X be a real reflexive Banach space having the E-property and X ∗ , the dual space
of X, be strictly convex. For the sake of simplicity, the norms of X and X ∗ will be
denoted by the symbol k.k. We write hx∗ , xi instead of x∗ (x) for x∗ ∈ X ∗ and
x ∈ X. Let A be a hemi-continuous and monotone operator from X into X ∗ , and
K be a closed convex subset of X.
For a given f ∈ X ∗ , consider the variational inequality: find an element x0 ∈ K
such that
hA(x0 ) − f, x − x0 i ≥ 0, ∀x ∈ K.
(1.1)
Variational inequalities and their approximations have been extensively studied in the
last two decates. Existence and approximations of solutions of variational inequalities
for various classes of operators in Hilbert and Banach spaces have been considered in
[1]-[5], [7], [8], [10], [11] and [13]. We mention, in particular, the paper [3], [11], where
the operator method or iterative method of regularization are considered. Further, in
[7] the convergence rates of the operator method of regularization is investigated under
the inverse-strongly monotone A in Hilbert space when the parameter of regularization
α is chosen a priory.
In the Banach space X, the operator method of regularization is the following
variational inequality
hAh (xτα ) + αU (xτα − x0 ) − fδ , x − xτα i ≥ 0,
xτα ∈ K,
∀x ∈ K,
(1.2)
where Ah are also monotone operators from X into X ∗ and approximate A in the
sense
kAh (x) − A(x)k ≤ hg(kxk)
(1.3)
with a nonegative continuous and bounded (image of bounded set is bounded) function
g(t), U is the normalized duality mapping of X, i.e., U is the mapping from X onto
X ∗ satisfying the condition (see [14])
hU (x), xi = kxk2 ,
kU (x)k = kxk,
fδ are the approximations of f : kfδ − f k ≤ δ, τ = (h, δ), and x0 is some element
in X playing the role of a criterion selection. By the choice of x0 , we can influence
which solution we want to approximate.
In [11], it is showed the existence and uniqueness of the solution xτα for every α > 0
and for arbitrary Ah , fδ . And, the regularized solution xτα converges to x0 ∈ S0 , the
set of solutions of (1.1) which is assumed to be nonempty, with
kx0 − x0 k = min kx − x0 k,
x∈S0
if (h+δ)/α, α → 0. Moreover, for each fixed τ = (δ, h) the papameter of regularization
α can be chosen by the discrepancy principle
τ
k)hp ,
ρ(α) = (k − 1)(δ + h)p + δ p + g(kxα
0 < p < 1, k > 1,
7, 3(2005)
Convergence rates in regularization for ill-posed variational inequalities
89
where ρ(α) = αkxτα − x0 k, under the conditions: x0 ∈ int K and
kAh (x0 ) − fδ k > (k − 1)(δ + h)p + δ p + g(kx0 k)hp
for 0 < δ < δ < 1, 0 < h < h < 1. The case x0 ∈ ∂K also is considered when xτα ∈
int K.
In this paper, under the condition x0 ∈ K\S0 without the restriction xτα ∈ int K
we shall show that the parameter of regularization α = α(δ, h) can be chosen by the
generalized discrepancy principle
ρ(α) = (δ + h)p α−q ,
p, q > 0,
(1.4)
for arbitrary monotone operator A, and on the base of the result we can estimate the
convergence rates when A is an inverse-strongly monotone operator, i.e., A possesses
the property
hA(x) − A(y), x − yi ≥
1
kA(x) − A(y)k2 ,
β
∀x, y ∈ X,
(1.5)
where β is some positive constant. In facts, variational inequalities with inversestrongly monotone operator belong to a class of nonlinear ill-posed problems (see
[7]).
Note that the generalized discrepancy principle for parameter choice is presented
first in [6] for the ill-posed operator equation
A(x) = f
(1.6)
when A is a linear and bounded operator in Hilbert space. Recently, it is considered
and applied in estimating convergence rates of the regularized solution for equation
(1.6) involving an m-accretive (in general nonlinear) operator (see [9]).
Later, the symbols * and → denote weak convergence and convergence in norm,
respectively, and the notation a ∼ b is meant that a = O(b) and b = O(a).
2. Main result
To obtain the result on the convergence rate for {xτα(δ,h) } as in [6] we need the
following lemmas.
Lemma 1. For each p, q, δ, h > 0, there exists at least a value α such that (1.4) holds.
Proof. It follows from [11] that ρ(α) is a continuous and nondecreasing function on
[α0 , +∞), α0 > 0. Moreover, ρ(α) > 0 ∀ α 6= 0. Indeed, if α1 6= 0 with ρ(α1τ ) = 0,
then xτα1 = x0 and from (1.2) it follows
hAh (x0 ) − fδ , x − x0 i ≥ 0,
∀x ∈ K.
After passing δ and h to zero in this inequality we see x0 ∈ S0 . This contradicts the
assumption x0 ∈ K\S0 . Therefore, αq ρ(α) → +∞, as α → +∞. On the other hand,
since
0 ≤ ρ(α) = αkxτα − x0 k
≤ δ + hg(kx0 kk) + 2αkx0 − x0 k
90
Nguyen Buong
7, 3(2005)
(see also [11]), we have αq ρ(α) → 0, as α → +0. Hence, there exists a value α such
that (1.4) holds.
Lemma 2. limδ,h→0 α(δ, h) = 0.
Proof. Let δn , hn → 0, and αn = α(δn , hn ) → ∞ as n → ∞. From (1.3),
hAhn (xταnn ) + αn U (xταnn − x0 ) − fδn , x − xταnn i ≥ 0,
∀x ∈ K,
(2.1)
the monotone property of Ahn and x0 ∈ K it follows
kxταnn − x0 k ≤ kAhn (x0 ) − fδn k/αn → 0,
as n → ∞. Therefore, xταnn → x0 , as n → ∞. On the other hand, by using the
monotone property of Ahn and the property of U we can write (2.1) in the form
hAhn (x) − fδn , x − xταnn i ≥ −αn hU (xταnn − x0 ), x − xταnn i
≥ −αn kxταnn − x0 kkx − xταnn k
≥ −ρ(αn )kx − xταnn k
≥ −(δn + hn )p αn−q kx − xταnn k → 0,
as n → ∞. It means that
hA(x0 ) − f, x − x0 i ≥ 0,
∀x ∈ K,
i.e., x0 is a solution of (1.1). It contradicts x0 ∈
/ S0 .
Thus, α(δ, h) remains bounded as δ, h → 0. Let δn , hn → 0 as n → ∞, and
meantime αn → c > 0. Since αn1+q kxταnn − x0 k = (δn + hn )p , we have kxταnn − x0 k → 0,
as n → ∞. Again, x0 ∈ S0 . Hence, limδ,h→0 α(δ, h) = 0.
Lemma 3. If 0 < p < q, then limδ,h→0 (δ + h)/α(δ, h) = 0.
Proof. It is easy to see that
δ+h
α(δ, h)
p
[(δ + h)p α(δ, h)−q ]α(δ, h)q−p
= ρ(α(δ, h))α(δ, h)q−p = α(δ, h)kxτα(δ,h) − x0 kα(δ, h)q−p
≤ δ + hg(kx0 k) + 2α(δ, h)kx0 − x0 k α(δ, h)q−p → 0
as δ, h → 0. Therefore,
lim
δ,h→0
The lemma is proved.
δ+h
α(δ, h)
p
= 0.
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Convergence rates in regularization for ill-posed variational inequalities
91
Lemma 4. Let 0 < p < q. Then, there exist constants C1 , C2 > 0 such that, for
sufficiently small δ, h > 0, the relation
C1 ≤ (δ + h)p α(δ, h)−1−q ≤ C2
holds.
Proof. From
(δ + h)p α(δ, h)−1−q = α(δ, h)−1 ρ(α(δ, h)) = kxτα(δ,h) − x0 k
≤
δ
h
+
g(kx0 k) + 2kx0 − x0 k
α(δ, h) α(δ, h)
and lemma 3, it implies the existence of a positive constant C2 in the lemma.
On the other hand, as X is reflexive and {xτα(δ,h) } is bounded, there exists a
subsequence of the sequence {xτα(δ,h) } that converges weakly to some element x̃0 in
K such that
kx̃0 − x0 k ≤ lim inf kxτα(δ,h) − x0 k.
We can conclude that x̃0 6= x0 . Indeed, if x̃0 = x0 , then from the monotone hemicontinuous property of Ah and (1.2) it follows
hAh (x) + αU (x − x0 ) − fδ , x − xτα i ≥ 0,
∀x ∈ K.
After passing δ and h in the last inequality to zero we obtain
hA(x) − f, x − x̃0 i ≥ 0,
∀x ∈ K
which is equivalent to (1.1). It is meant that x̃0 ∈ S0 . It contradicts x0 ∈
/ S0 .
Therefore, there exists a constant C1 in the lemma.
To estimate the convergence rates for {xτα(δ,h) } we assume that
hU (x) − U (y), x − yi ≥ mU kx − yks ,
mU > 0, s ≥ 2,
∀x, y ∈ X.
(2.2)
It is well-known that when X ≡ H, the Hilbert space, mU = 1, s = 2, and when
X = Lp or Wp , mU = p − 1, s = 2 for the case 1 < p < 2. In the case p > 2 we have
to use the duality mapping U s satisfying the condition
hU s (x), xi = kxks ,
kU s (x)k = kxks−1 ,
s≥2
instead of U . Then, mU s = 22−p /p and s = p in (2.2) (see [12]).
Theorem 1. Assume that the following conditions hold:
(i) A is an inverse-strongly-monotone operator in X with
kA(x) − A(x0 ) − A0 (x0 )(x − x0 )k ≤ τ̃ kA(x) − A(x0 )k,
where τ̃ is some positive constant;
∀x ∈ X,
92
Nguyen Buong
7, 3(2005)
(ii) There exists an element z ∈ X such that A0 (x0 )∗ z = U (x0 − x0 );
(iii) The parameter α is chosen by (1.4) with p < q.
Then, we have
p
.
kxτα(δ,h) − x0 k = O((δ + h)θ ), θ =
(1 + q)(2s − 1)
Proof. From (1.1) - (1.3) it follows
hA(xτα(δ,h) ) − A(x0 ), xτα(δ,h) − x0 i + α(δ, h)
× hU (xτα(δ,h) − x0 ) − U (x0 − x0 ), xτα(δ,h) − x0 i
≤ (δ + hg(kxα(δ,h) k))kxα(δ,h) − x0 k
+α(δ, h)hU (x0 − x0 ), x0 − xτα(δ,h) i.
(2.3)
Thus, by using (1.5) and the monotone property of U we obtain
p
kA(xτα(δ,h) ) − A(x0 )k ≤ O( δ + h + α(δ, h))kxτα(δ,h) − x0 k1/2 .
On the other hand, from (2.2), (2.3) and the monotone property of A which is followed
from (1.5) we have
mU kxτα(δ,h) − x0 ks ≤ hU (xτα(δ,h) − x0 ) − U (x0 − x0 ), xτα(δ,h) − x0 i
≤
δ + C̃0 h τ
kx
− x0 )k + hz, A0 (x0 )(x0 − xτα(δ,h) )i
α(δ, h) α(δ,h)
where C̃0 is some positive constant, and
τ
hz, A0 (x0 )(x0 −xτ
α(δ,h) )i ≤ kzk(τ̃ + 1)kA(xα(δ,h) ) − A(x0 )k
p
≤ kzk(τ̃ + 1)O( δ + h + α(δ, h))kxτα(δ,h) ) − x0 k1/2 .
Now, from lemma 4 it implies that
−1/(1+q)
α(δ, h) ≤ C1
(δ + h)p/(1+q) .
and
δ+h
≤ C2 (δ + h)1−p α(δ, h)q
α(δ, h)
−q/(1+q)
(δ + h)1−p (δ + h)pq/(1+q)
−q/(1+q)
(δ + h)1−p/(1+q) .
≤ C2 C1
≤ C2 C1
In final, we have
−q/(1+q)
mU kxτα(δ,h) −x0 ks−1/2 ≤ max{1, C̃0 }C2 C1
(δ + h)1−p/(1+q)
p
× kxτα(δ,h) − x0 k1/2 + O( δ + h + α(δ, h))
≤ O((δ + h)1−p/(1+q) kxτα(δ,h) − x0 k1/2 + O((δ + h)p/2(1+q) ).
7, 3(2005)
Convergence rates in regularization for ill-posed variational inequalities
93
Using the implication
a, b, c ≥ 0, s > t, as ≤ bat + c =⇒ as = O(bs/(s−t) + c)
we obtain
kxτα(δ,h) − x0 )k = O((δ + h)θ ).
Received:
July 2004.
Revised:
August 2004.
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