Electronic Journal of Differential Equations, Vol. 2002(2002), No. 46, pp. 1–15.
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
ftp ejde.math.swt.edu (login: ftp)
Positive solutions for a nonlocal boundary-value
problem with vector-valued response ∗
Andrzej Nowakowski & Aleksandra Orpel
Abstract
Using variational methods, we study the existence of positive solutions
for a nonlocal boundary-value problem with vector-valued response. We
develop duality and variational principles for this problem and present
a numerical version which enables the approximation of solutions and
gives a measure of a duality gap between primal and dual functional for
approximate solutions for this problem.
1
Introduction
The aim of this paper is to establish the conditions for which the differential
equation
(k(t)|x0 (t)|q−2 x0 (t))0 + Vx (t, x(t)) = 0, a.e. in [0, T ]
(1.1)
possesses a positive solution x : [0, T ] → Rn such that x(0) = 0, and satisfies
the non-local boundary condition
|x0 (T )|q−2 x0 (T ) =
Z
T
|x0 (s)|q−2 x0 (s)dg(s),
(1.2)
t0
pPn
2
where |z| =
i=1 zi and the integral is understood in the Riemann-Stieltjes
sense. The general assumptions for this article are as follows:
(H) The number q is even and greater than zero, T is an arbitrary positive
number, t0 is a real number in the open interval (0, T ), g = (g1 , . . . , gn ) :
[0, T ] → Rn where gi increases and g(t0 ) = 0, V : [0, T ] × Rn → R is
Gateaux differentiable in the second variable and measurable in t, k :
[0, T ] → R+ , and k(T ) = 1.
Of course, when k is an absolutely continuous, the solution of (1.1) belongs
to C 1,+ ([0, T ], Rn ), the space of continuously differentiable functions whose first
∗ Mathematics Subject Classifications: 34B18.
Key words: Nonlocal boundary-value problems, positive solutions, duality method,
variational method.
c
2002
Southwest Texas State University.
Submitted February 27, 2002. Published May 28, 2002.
1
2
Positive solutions for a nonlocal boundary-value problem
EJDE–2002/46
derivative is absolutely continuous. Note that we do not assume that Vx is
superlinear or sublinear.
In this paper we shall apply some variational methods and consider (1.1) as
the Euler-Lagrange equation to the functional
J(x) =
Z
T
(−V (t, x(t)) +
0
k(t) 0 q
|x (t)| )dt
q
(1.3)
defined on the space A0 of absolutely continuous functions x : [0, T ] → Rn ,
1/q
RT
x(0) = 0 with x0 in Lq ([0, T ], Rn ) and the norm kxkA0 = 0 |x0 (t)|q dt
. We
shall denote by A0b the subset of A0 consisting of functions satisfying (1.2) .
This problem appears in mathematical models of physical phenomena and it
is associated with the principle of minimal action, which holds true universally
in nature. Problems like (1.1), (1.2) have been studied by many authors, mainly
in one-dimensional case (n = 1) with q = 2. This problem is discussed also in
[7], where a topological approach is presented and the methods are based on
the fixed point theorem in cones by Krasnosielski [9]. In [7], it is assumed that
g(t0 +) > 0, that V has the special form Vx (t, x) = q(t)f (x) for some continuous
functions q : [0, 1] → R, f : R → R, f is nonnegative for x > 0, and V quiet at
infinity, and
sup f (x) ≤ θv
(1.4)
x∈[0,v]
for some v > 0 and θ > 0. It appears that weaker assumptions (n ≥ 1, t and
x not separated in the left-hand side of the equation, Vx (·, x) only measurable,
and Vx (t, ·) not necessarily quiet at infinity) are still sufficient to conclude the
existence of solutions for (1.1). We consider the general case when V satisfies
hypothesis (H) which are not as strong as in [7]. We are also able to omit the
condition g(t0 +) > 0.
In this paper we study (1.1), (1.2) by duality methods analogous to the
methods developed for (1.1) in sublinear cases [12, 13]. Functional (1.3) is, in
general, unbounded in A0 (especially in the superlinear case), so that we must
look for critical points of (1.3) of ”minmax” type, or find subsets X and X d on
which the action functional J or its dual JD is bounded. We shall apply the
second approach; i.e., choose the special sets over which we calculate minimum
of J and JD and then link this value with critical points of J. Of course, we have
the Morse theory and its generalization, the saddle points theorems, and the
mountain pass theorems [12, 14, 15]. However, because of the boundary condition (1.2) they cannot be applied directly to find critical points of J. Moreover,
our assumptions are not strong enough to use, for example, the Mountain Pass
Theorem: V is not sufficiently smooth, Vx and V do not have growth conditions. In consequence, J is not necessary of C 1 class and it does not satisfy, in
general, the PS-condition. We shall develop duality, and because of this theory
we are able to omit in our proof of the existence of critical points, the deformation lemmas, the Ekeland variational principle, and the PS type conditions.
Our approach also enables the numerical characterization of solutions for this
EJDE–2002/46
Andrzej Nowakowski & Aleksandra Orpel
3
problem. It seems to us that this is the first publication that applies variational
methods to problem (1.1), (1.2).
Let the positive cone in Rn be denoted by
P = x = (x1 , . . . , xn ) ∈ Rn : xi > 0, i = 1, . . . , n
and let P̄ = {x ∈ Rn : xi ≥ 0, i = 1, . . . , n}. We say that x ≥ y for x, y ∈ Rn
if x − y ∈ P̄ . If b, c ∈ Rn by bc we always mean a vector [bi ci ]i=1,...,n and by
1
√
√
1
q−1
b = b q−1 a vector [ q−1 bi ]i=1,...,n = [biq−1 ]i=1,...,n . We denote 1 = (1, . . . 1)
the vector in Rn .
e for x ∈ A0b , x(t) ∈ P , t ∈ [0, T ].
Let us investigate the operator A,
Z T
Z T
Z T
1
1
1
e
Ax(t)
=α
Vx (s, x(s))dsdg(r) +
Vx (s, x(s))ds.
k(t) t0 k(r) r
k(t) t
where α := [αi ]i=1,...,n = [(1 − ai )−1 ]i=1,...,n and
"Z
#
Z T
T
1
1
a :=
dg(t) =
dgi (t)
t0 k(t)
t0 k(t)
.
i=1,...,n
For the functions that appear in this paper we assume the following:
(H1) The function k is continuous and positive, V (t, ·) is convex in P , Vx (t, ·)
RT
is continuous and nonnegative in P , t ∈ [0, T ], | 0 V (t, 0)dt| < ∞, and
α > 0.
(H2) There exist a function u ∈ L∞ ([0, T ], Rn ), u(t) ∈ P , for a.e. t ∈ (0, T ),
and constants c, e ∈ P , such that for
Z t
q−2
(αa + 1)c + e
b(t) :=
, v(t) =
|b(s)|− q−1 u(s)ds,
k(t)
0
Z t
q−2
− q−1
z(t) =
|u(s)|
b(s)ds
0
we have v(t) < z(t), t ∈ (0, T ) and
Z T
e
Vx (t, z(t))dt ≤ c, Av(t)
≥ u(t), t ∈ (0, T ).
t0
We see that our hypotheses on V concern only convexity of V (t, ·) in P̄ and
that this function is rather of general type.
To construct the set X first we put
n
0
X = x ∈ A0b : p(t) = k(t)|x0 (t)|q−2 x0 (t), t ∈ [0, T ] belongs to Aq ,
Z T
o
x(t) ≥ v(t), t ∈ (0, T ) and
Vx (t, x(t))dt ≤ c
t0
0
where Aq is the set of absolutely continuous functions x : [0, T ] → Rn with x0
0
in Lq ([0, T ], Rn ) and t0 is given above.
4
Positive solutions for a nonlocal boundary-value problem
EJDE–2002/46
Remark 1.1 Assumptions (H1) and (H2) imply that X 6= ∅. Indeed; each
Rt
q−2
element x ∈ A0b such that v(t) = 0 |b(s)|− q−1 u(s)ds < x(t) for all t ∈ (0, T )
and x(t) < z(t) for all t ∈ [t0 , T ] belongs to X.
We reduce the space X to the set
X = {x ∈ X : x ∈ P̄ , x0 (t) ≥ 0, t ∈ [0, T ]}.
It is clear that X depends strongly on the type of nonlinearity of V . We easily
see that X is not in general a closed set in A. Assume additionally that
(H3) For e ∈ P given in (H2)
Z t0
Vx (t, x(t))dt ≤ e for all x ∈ X
0
and there exists d ∈ R such that
Z t0
V (t, x(t))dt ≤ d for all x ∈ X.
0
Let an operator A : X → P̄ be given by
Z t
q−2
− q−1
e
e
Ax(t) :=
Ax(s)|
Ax(s)|
ds.
0
e
Since Ax(t)
> 0 for all t ∈ [0, T ], so the operator A is well-defined. We easily see
that solving the boundary problem (1.1)-(1.2) is equivalent to solve the operator
equation x = Ax in A0b .
Now we give some auxiliary results
Lemma 1.2 The set X satisfies AX ⊂ X; i.e. for each x ∈ X there exists
w ∈ X such that w = Ax.
Proof.
It is clear that if x ∈ X then Ax(t) ≥ 0 and
q−2
− q−1
e
e
(Ax(t))0 = Ax(t)|
Ax(t)|
≥ 0.
e
First we will prove that Ax(t)
≤ b(t), t ∈ (0, T ] for x ∈ X. Indeed; fix x ∈ X,
then using definition of X we have for all t ∈ (0, T ]
1
e
Ax(t)
=α
k(t)
Z
T
t0
Z T
1
k(r)
Z
T
Z
1
Vx (s, x(s))ds +
k(t)
t0
Z
r
1
1
≤α
dg(r)
k(t) t0 k(r)
(αa + 1)c + e
≤
= b(t)
k(t)
Z
T
1
k(t)
Vx (s, x(s))dsdg(r) +
T
Vx (s, x(s))ds
t
0
T
Vx (s, x(s))ds
EJDE–2002/46
and
Andrzej Nowakowski & Aleksandra Orpel
5
q−2
− q−1
− q−2
q−2
e
e
Ax(t)
q−1 ≤ |u(t)|− q−1 .
≤ Av(t)
Combining these results we obtain
Z t
Z t
q−2
q−2
− q−1
e
e
Ax(t) =
Ax(s)|Ax(s)|
ds ≤
b(s)|u(s)|− q−1 ds = z(t).
0
0
Thus, assumptions (H1) and (H2) imply
Z T
Z T
Vx (s, Ax(s))ds ≤
Vx (s, z(s))ds ≤ c.
t0
t0
On the other hand we have for all t ∈ (0, T )
Z t
− q−2
e
e
q−1 ds
Ax(t) =
Ax(s)|
Ax(s)|
0
Z t
Z t
q−2
q−2
e
e
≥
|b(s)|− q−1 Ax(s)ds
≥
|b(s)|− q−1 Av(s)ds
0
0
Z t
q−2
≥
|b(s)|− q−1 u(s)ds = v(t).
0
Summarizing: Ax ∈ X. As the dual set to X we shall consider the set
n
0
X d = p ∈ Aq : there exists x ∈ X such that
o
p(t) = k(t)|x0 (t)|q−2 x0 (t), t ∈ [0, T ] .
Remark 1.3 From the definition of A and X, we derive that for each x ∈ X
there exists p ∈ X d such that p0 (·) = −Vx (·, x(·)) and therefore
Z T
Z T
Z T
h−p0 (t), x(t)idt −
V ∗ (t, −p0 (t))dt =
V (t, x(t))dt.
0
0
0
Indeed; fixing x ∈ X, Lemma 1.2 leads to the existence of x
e ∈ X such that
x
e = Ax. So that we have
q−2
q−2
− q−1 q−2 e
− q−1
e
e
e
|e
x0 (t)|q−2 x
e0 (t) =[|Ax(t)||
Ax(t)|
] Ax(t)|Ax(t)|
q−2
1
− q−1
e
e
e
e
q−1 ]q−2 Ax(t)|
=[|Ax(t)|
Ax(t)|
= Ax(t).
Set p(t) = k(t)|e
x0 (t)|q−2 x
e0 (t), t ∈ [0, T ]. Since x
e ∈ X , p ∈ X d we get what
follows
(p(t))0 =(k(t)|e
x0 (t)|q−2 x
e0 (t))0
0
e
=(k(t)Ax(t))
Z
Z T
Z T
T
0
1
= α
Vx (s, x(s))dsdg(r) +
Vx (s, x(s))ds
t0 k(r) r
t
6
Positive solutions for a nonlocal boundary-value problem
EJDE–2002/46
= − Vx (t, x(t)).
Taking into account the properties of the subdifferential, we can infer that the
required relation is satisfied.
Now we study the action functional
J(x) =
T
Z
0
1
(−V (t, x(t)) + k(t)|x0 (t)|q )dt − hx(T ), |x0 (T )|q−2 x0 (T )i
q
on the set X. To show that x̄ ∈ X realizing “min” is a critical point of J we
develop a duality theory between J and dual to it JD .
2
Duality results
In this section we shall develop the duality, which describes the relationship
between the critical value of J and the infimum of the dual functional JD :
X d → R,
JD (p) = −
Z
0
T
0
0
1
[k(t)]1−q |p(t)|q +
0
q
Z
T
V ∗ (t, −p0 (t))dt,
(2.1)
0
where q 0 := q/(q − 1).
Now we need a kind of perturbation of J and the convexity of a function
considered on a whole space. Therefore, for each x ∈ X the perturbation
Jx : Lq ([0, T ], Rn ) × Rn → R for the functional J is defined as
Jx (y, a) =
Z
=
Z
T
(V̆ (t, x(t) + y(t)) −
0
k(t) 0 q
|x (t)| )dt + hx(T ) − a, |x0 (T )|q−2 x0 (T )i
q
T
k(t) 0 q
|x (t)| )dt
q
0
+ hx(T ), |x0 (T )|q−2 x0 (T )i − ha, |x0 (T )|q−2 x0 (T )i,
(V̆ (t, x(t) + y(t)) −
with
(
V (t, x) if x ∈ P̄ , t ∈ [0, T ]
V̆ (t, x) =
∞
if x ∈
/ P̄ , t ∈ [0, T ].
We use this notation only for the purpose of duality and we will not change a
notation for the functional J containing V or V̆ . It is associated with the fact
that our all investigation reduce to the set X on which V̆ (t, x) = V (t, x) for all
t ∈ [0, T ]. Let Jx# : X d → R, where x ∈ X, be defined as a type of conjugate of
EJDE–2002/46
Andrzej Nowakowski & Aleksandra Orpel
7
Jx :
Jx# (p)
=
sup
y∈Lq ([0,T ],Rn ),a∈Rn
=
sup
y∈Lq ([0,T ],Rn )
nZ
0
T
o
hy(t), p0 (t)idt + hp(T ), ai − Jx (y, a)
T
nZ
0
hy(t), p (t)idt −
V̆ (t, x(t) + y(t))dt
0
0
0
o
(2.2)
q−2 0
+ sup ha, p(T )i + ha, |x (T )|
a∈Rn
Z T
T
Z
x (T )i
k(t) 0 q
|x (t)| dt − hx(T ), |x0 (T )|q−2 x0 (T )i.
q
0
(
0,
for b = 0
n
Put l : R → {0, +∞}, l(b) =
We see at once that
+∞, for b 6= 0.
+
Z
Z T
1 T
k(t)|x0 (t)|q dt +
V ∗ (t, p0 (t))dt
q
0
0
0
− hx(T ), |x0 (T )|q−2 x0 (T )i + l(|x0 (T )|q−2 x0 (T ) + p(T ))
Z T
(2.3)
=
hx0 (t), p(t)idt − hx(T ), p(T )i
Jx# (p) = −
Z
T
hx(t), p0 (t)idt +
0
Z
Z T
1 T
k(t)|x0 (t)|q dt +
V ∗ (t, p0 (t))dt
q 0
0
− hx(T ), |x0 (T )|q−2 x0 (T )i + l(|x0 (T )|q−2 x0 (T ) + p(T )).
+
Theorem 2.1 For functionals Jand JD , we have the duality relation
inf J(x) ≤ inf JD (p).
x∈X
p∈X d
Our proof starts with the observation that for all p ∈ X d
Proof.
inf Jx# (−p) = JD (p)
(2.4)
inf Jx# (−p) ≥ J(x).
(2.5)
x∈X
and for all x ∈ X
p∈X d
Because X is not a linear space we need some trick to avoid calculation of the
conjugate with respect to a nonlinear space. To this effect we use the special
structure of the sets X d and X. Indeed; fix p ∈ X d . The definition of X d
implies that there exists xp ∈ X satisfying the equality
p(t) = k(t)|x0p (t)|q−2 x0p (t)
and, in consequence
Z T
Z
0
hxp (t), p(t)idt −
0
0
T
for all t ∈ [0, T ]
k(t) 0
|xp (t)|q dt =
q
Z
0
T
0
0
1
[k(t)]1−q |p(t)|q dt.
q0
8
Positive solutions for a nonlocal boundary-value problem
EJDE–2002/46
An easy calculation yields
T
Z
hx0p (t), p(t)idt −
0
≤
T
Z
0
sup
k(t) 0
|xp (t)|q dt
q
Z
nZ T
0
hx (t), p(t)idt −
x∈{z∈X,p(T )=|z 0 (T )|q−2 z 0 (T )}
T
nZ
Z
0
0
T
T
o
k(t)
|x(t)0 |q dt
q
k(t)
|x(t)0 |q dt
q
x∈X
0
0
Z T
nZ T
k(t) 0 q o
≤
sup
hx0 (t), p(t)idt −
|x (t)| dt
q
x0 ∈Lq ([0,T ],Rn )
0
0
Z T
0
0
1
=
[k(t)]1−q |p(t)|q dt .
0
0 q
≤ sup
hx0 (t), p(t)idt −
o
Actually all inequalities above are equalities. Finally for p ∈ X d , we obtain
− inf Jx# (−p) = sup (−Jx# (−p))
x∈X
x∈X
= sup
x∈X
Z
nZ
T
hx0 (t), p(t)idt −
0
1
q
Z
T
k(t)|x0 (t)|q dt
0
T
V ∗ (t, −p0 (t))dt − hx(T ), p(T )i + hx(T ), |x0 (T )|q−2 x0 (T )i
0
o
− l(|x0 (T )|q−2 x0 (T ) − p(T ))
Z T
Z T
0
0
1
[k(t)]1−q |p(t)|q dt = −JD (p),
=−
V ∗ (t, −p0 (t))dt +
0
0
0 q
−
which proves our claim. To show the other assertion, fix x ∈ X. Remark 1.3
leads to the existence of p̄ ∈ X d such that
T
Z
T
Z
h−p̄0 (t), x(t)idt −
0
V ∗ (t, −p̄0 (t))dt =
T
Z
0
V̆ (t, x(t))dt.
0
Moreover,
Z
T
h−p̄0 (t), x(t)idt −
0
T
Z
V ∗ (t, −p̄0 (t))dt
0
Z
≤ sup {
p∈X d
≤
=
T
0
h−p (t), x(t)idt −
0
sup
p0 ∈Lq0 ([0,T ],Rn )
Z
0
Z
T
V ∗ (t, −p0 (t))dt}
0
Z
{
0
T
V̆ (t, x(t))dt.
T
h−p0 (t), x(t)idt −
Z
0
T
V ∗ (t, −p0 (t))dt}
EJDE–2002/46
Andrzej Nowakowski & Aleksandra Orpel
9
Combining both results we infer
sup (−Jx# (−p)) = sup
p∈X d
p∈X d
nZ
T
T
Z
hx(t), −p0 (t)idt −
0
V ∗ (t, −p0 (t))dt
0
o
− l(|x0 (T )|q−2 x0 (T ) − p(T ))
Z
1 T
k(t)|x0 (t)|q dt
+ hx(T ), |x0 (T )|q−2 x0 (T )i −
q 0
≤ sup − l(|x0 (T )|q−2 x0 (T ) − p(T ))
p∈X d
+ sup
p∈X d
nZ
T
hx(t), −p0 (t)idt −
0
V ∗ (t, −p0 (t))dt
0
1
q
+ hx(T ), |x0 (T )|q−2 x0 (T )i −
Z
T
Z
o
T
Z
k(t)|x0 (t)|q dt
0
T
k(t) 0 q
=−
(−V̆ (t, x(t)) +
|x (t)| )dt + hx(T ), |x0 (T )|q−2 x0 (T )i
q
0
= − J(x);
so that inf p∈X d Jx# (−p) ≥ J(x). Combining (2.4) and (2.5) we obtain the chain
of relations
inf J(x) ≤ inf inf Jx# (−p) = inf inf Jx# (−p) = inf JD (p).
x∈X
p∈X d x∈X
x∈X p∈X d
p∈X d
Denote by ∂Jx (y) the subdifferential of Jx . Calculating ∂Jx at 0 we obtain
T
Z
n
0
q0
n
∂Jx (0) = p ∈ L ([0, T ], R ) :
V ∗ (t, p0 (t))dt
0
+
Z
T
V̆ (t, x(t))dt =
0
Z
(2.6)
T
0
0
hp (t), x(t)idt
o
Our task is now to prove a variational principle for “min” arguments.
Theorem 2.2 Let x̄ ∈ X be a minimizer of J : X → R, J(x̄) = inf x∈X J(x).
Then there exists p̄ ∈ X d with −p̄0 ∈ ∂Jx̄ (0), such that p̄ satisfies
JD (p̄) = inf JD (p).
p∈X d
Furthermore,
Jx (0) + Jx̄# (−p̄) = 0
(2.7)
JD (p̄) − Jx̄# (−p̄) = 0.
(2.8)
10
Positive solutions for a nonlocal boundary-value problem EJDE–2002/46
Proof. From Theorem 2.1, J(x̄) ≤ inf p∈X d JD (p), so to prove the first assertion we need to show only the existence of p̄ ∈ X d such that J(x̄) ≥ JD (p̄). To
this effect we use Remark 1.3 which implies the existence of p̄ ∈ X d such that
p̄0 (t) = −Vx (t, x(t)) a.e. on [0, T ]
and further
Z T
h−p̄0 (t), x(t)idt −
0
Z
T
V ∗ (t, −p̄0 (t))dt =
0
Z
T
V̆ (t, x(t))dt.
(2.9)
0
Combining (2.9) and (2.6) we get the inclusion −p̄0 ∈ ∂Jx̄ (0). On the other
hand an easy computation gives the equality Jx̄∗ (−p̄0 , −p(T )) = Jx̄# (−p̄) (where
Jx̄∗ (−p̄0 , −p(T )) denotes the Fenchel transform of Jx̄ at (−p̄0 , −p(T )) ). Indeed;
from the definitions of Jx̄∗ and of Jx̄# we have
Jx̄∗ (−p̄0 , −p(T ))
=
sup
y∈Lq ([0,T ],Rn ),a∈Rn
=
sup
y∈Lq ([0,T ],Rn )
nZ
nZ
0
T
o
hy(t), −p0 (t)idt + ha, −p(T )i − Jx̄ (y, a)
T
hy(t), −p0 (t)idt −
0
Z
T
V̆ (t, x̄(t) + y(t))dt
0
o
+ sup ha, −p(T )i + ha, |x̄0 (T )|q−2 x̄0 (T )i
a∈Rn
Z T
k(t) 0 q
|x̄ (t)| dt − hx̄(T ), |x̄0 (T )|q−2 x̄0 (T )i
q
0
Z T
Z
1 T
=−
hx̄(t), −p0 (t)idt +
k(t)|x̄0 (t)|q dt
q 0
0
Z T
+
V ∗ (t, −p0 (t))dt − hx̄(T ), |x̄0 (T )|q−2 x̄0 (T )i + l(|x̄0 (T )|q−2 x̄0 (T ) − p(T ))
+
0
=Jx̄# (−p).
Therefore, we obtain (2.7). Finally
J(x̄) = Jx̄# (−p̄) ≥ inf Jx# (−p̄) = JD (p̄),
x∈X
where the last equality is due to (2.4). Hence J(x̄) ≥ JD (p̄). Now Theorem 2.1
leads to J(x̄) = JD (p̄) = inf p∈X d JD (p). (2.8) follows from (2.7) and the chain
of equalities Jx̄ (0) = −J(x̄) = −JD (p).
Corollary 2.3 Let x̄ ∈ X be such that J(x̄) = inf x∈X J(x). Then there exists
p̄ ∈ X d such that the pair (x̄, p̄) satisfies the relations
JD (p̄) = inf JD (p) = inf J(x) = J(x̄)
(2.10)
−(k(t)|x̄0 (t)|q−2 x̄0 (t))0 = Vx (t, x̄(t)),
(2.11)
p∈X d
x∈X
and
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Andrzej Nowakowski & Aleksandra Orpel
11
Proof. From Theorems 2.1 and 2.2, we obtain immediately (2.10). To show
(2.11) we use (2.7) and (2.8) obtaining the two equalities:
Z T
Z T
Z T
V (t, x̄(t))dt +
V ∗ (t, −p̄0 (t))dt −
hx̄(t), −p̄0 (t)idt = 0,
0
Z
0
T
0
0
0
1
[k(t)]1−q |p̄(t)|q dt +
0
q
0
Z
0
T
k(t) 0 q
|x̄ (t)| dt −
q
Z
T
hx̄0 (t), p̄(t)idt = 0
0
and further
−p̄0 (t) = Vx (t, x̄(t)) and p̄(t) = k(t)|x̄0 (t)|q−2 x̄0 (t)
which gives (2.11).
3
Variational principles and a duality gap for
minimizing sequences
In this section we prove that a statement analogous to Theorem 2.2 is true for
a minimizing sequence of J. It is worth noting that as a consequence of our
duality we obtain for the first time in the superlinear case a measure of a duality
gap between primal and dual functional for approximate solutions to (1.1) (for
the sublinear case see [13]).
Theorem 3.1 Let {xj }, xj ∈ X, j = 1, 2, . . . , be a minimizing sequence for J.
Then there exist pj ∈ X d with −p0j ∈ ∂Jxj (0) such that {pj } is a minimizing
sequence for JD i.e.,
inf J(x) = inf J(xj ) = inf JD (pj ) = inf JD (p).
x∈X
j∈N
j∈N
p∈X d
(3.1)
Furthermore,
Jxj (0) + Jx#j (−pj ) = 0,
and for all ε > 0 there exists j0 ∈ N such that for all j ≥ j0 ,
JD (pj ) − Jx#j (−pj ) ≤ ε,
(3.2)
0 ≤ J(xj ) − JD (pj ) ≤ ε.
(3.3)
Proof. We first show the boundedness of J on X. By the assumptions made
RT
on V and the definition of X we get for all x ∈ X, 0 ≤ 0 Vx (r, x(r))dr ≤ c + e
and, in consequence,
Z T
kxk
Vx (r, x(r))dr ≤ kxk(c + e),
0
where kxk = max{|x(t)| : t ∈ [0, T ]}. Therefore,
Z T
Vx (r, x(r))x(r)dr ≤ kxk(c + e).
0
12
Positive solutions for a nonlocal boundary-value problem EJDE–2002/46
Hence, by the property of the subdifferential
Z T
Z T
Z T
V (r, 0)dr −
V (r, x(r))dr ≥
Vx (r, x(r))[0 − x(r)]dr ≥ −kxk(c + e)
0
0
0
and further
Z T
k(t) 0 q
|x (t)| dt −
V (t, 0)dt
J(x) ≥
q
0
0
− kxk(c + e) − hx(T ), |x0 (T )|q−2 x0 (T )i
Z T
Z T
k(t) 0 q
≥
|x (t)| dt −
V (t, 0)dt
q
0
0
− T kx0 kLq ([0,T ],Rn ) (c + e) − |x(T )||x0 (T )|q−1 .
Z
T
(3.4)
It is clear that the definition of J and (3.4) give the estimate
∞ > inf J(xj ) = a > −∞.
j∈N
Fix ε > 0. From the above there exists j0 such that J(xj ) − a < ε, for all j ≥ j0 .
We can now proceed analogously to the proof of Theorem 2.2. First we observe
that for each j ∈ N there exists pj ∈ X d such that p0j (t) = −Vx (t, xj (t)) a.e. on
[0, T ]. This implies
−p0j ∈ ∂Jxj (0) for all j ∈ N,
which gives Jxj (0) + Jx#j (−pj ) = 0 for all j ∈ N. As in the proof of Theorem
2.2, taking into account (2.4), this assertion yields
J(xj ) = Jx#j (−pj ) ≥ inf Jx# (−pj ) = JD (pj ).
x∈X
Thus, by Theorem 2.1, we deduce that (3.1) holds. Then (3.2) and (3.3) follow
from (3.1) and two facts:
JD (pj ) + ε ≥ J(xj ) for j ≥ j0 ,
−J(xj ) = Jxj (0) = −Jx#j (−pj ) for each j ∈ N.
Theorem 3.1 gives the below corollary
Corollary 3.2 Let {xj }, xj ∈ X, j = 1, 2, . . . , be a minimizing sequence for
J. If −p0j (t) = Vx (t, xj (t)), then
Z T
pj (t) = pj (T ) −
p0j (s)ds, pj (T ) = |x0j (T )|q−2 x0j (T )
t
d
which belongs to X and {pj } is a minimizing sequence for JD ; i.e.,
inf J(x) = inf J(xj ) = inf JD (pj ) = inf JD (p).
x∈X
j∈N
j∈N
p∈X d
Furthermore for a given ε > 0 and sufficiently large j,
JD (pj ) − Jx#j (−pj ) ≤ ε, 0 ≤ J(xj ) − JD (pj ) ≤ ε.
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4
Andrzej Nowakowski & Aleksandra Orpel
13
The existence result
In this section we prove the existence of a minimizer of x̄ ∈ X of J on X;
i.e., J(x̄) = minx∈X J(x) being a solution of (1.1) with the non-local boundary
condition (1.2).
Theorem 4.1 Under hypotheses (H) and (H1)-(H3) there exists x̄ ∈ X such
that J(x̄) = minx∈X J(x) and x̄ satisfies (1.1)-(1.2).
Proof. Let Sz = {x ∈ X, J(x) ≤ z}, where z ∈ R. It follows from (3.4),
the assumptions made on V , and the definition of X that Sz is nonempty for
sufficiently large z and bounded with respect to the norm kx0 kLq ([0,T ],Rn ) . This
statement implies that Sz , z ∈ R, are relatively weakly compact in A0b . It
is a well known fact that the functional J is weakly lower semicontinuous in
A0 and thus also in X. Therefore, there exists a sequence {xn }, xn ∈ X,
such that xn * x̄ weakly in A0b with x̄ ∈ A0b (we use the fact that {xn }
is uniformly convergent to x̄) and lim inf n→∞ J(xn ) ≥ J(x̄). Moreover, the
uniform convergence of {xn } to x̄, implies that x̄ ∈ X.
Our task is now to show that x̄ ∈ X. For this purpose, we recall from
Corollary 3.2 that for
p0n (t) = −Vx (t, xn (t)),
t ∈ [0, T ]
(4.1)
RT
pn (t) = pn (T ) − t p0n (s)ds, where pn (T ) = |x0n (T )|q−2 x0n (T ), belongs to X d .
Then {pn } is a minimizing sequence for JD . We easily check that {pn (T )} is a
bounded sequence and therefore we may assume (up to a subsequence) that it
is convergent. From (4.1) we infer that {p0n } is a bounded sequence in L1 norm
and that it is pointwise convergent to
p̄0 (t) = −Vx (t, x̄(t)).
(4.2)
RT
Thus {pn } is uniformly convergent to p̄ where p̄(t) = p̄(T )− t p̄0 (s)ds. By
Corollary 3.1 we also have that for εn → 0 (n → ∞)
Z T
Z T
0
0
1
k(t) 0
0≤
( 0 [k(t)]1−q |pn (t)|q +
|xn (t)|q )dt −
hx0n (t), pn (t)idt ≤ εn
q
q
0
0
where 1/q + 1/q 0 = 1, and so, taking the limit
Z T
Z T
Z T
1
k(t) 0
1−q 0
q0
q
0=
[k(t)]
|p̄(t)|
dt
+
lim
|x
(t)|
dt
−
hx̄0 (t), p̄(t)idt.
n
0
n→∞
q
q
0
0
0
Therefore, in view of the property of Fenchel inequality,
Z T
Z T
Z T
1
k(t) 0 q
1−q 0
q0
0=
[k(t)]
|p̄(t)|
dt
+
|x̄
(t)|
dt
−
hx̄0 (t), p̄(t)idt
0
q
0 q
0
0
and further
p̄(t) = k(t)|x̄0 (t)|q−2 x̄0 (t).
(4.3)
Thus x̄ ∈ X. Substituting (4.3) into (4.2) we can assert that the minimizer
x̄ ∈ X is the solution of problem (1.1)-(1.2).
14
Positive solutions for a nonlocal boundary-value problem EJDE–2002/46
Acknowledgments The authors are in debt to anonymous referee whose suggestions led to the improvement of the presentation and the corrections of a few
mistakes in this paper.
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Andrzej Nowakowski & Aleksandra Orpel
15
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Andrzej Nowakowski (e-mail: annowako@math.uni.lodz.pl)
Aleksandra Orpel (e-mail: orpela@math.uni.lodz.pl)
Faculty of Mathematics, University of Lodz,
Banacha 22, 90-238 Lodz, Poland