I nternat. J. Math. & Mah. Sci.
(1982) 31-40
Vol. 5 No.
31
SEMIGROUP STRUCTURE UNDERLYING EVOLUTIONS
G. EDGAR PARKER
Department of Mathematics
Pan American University
78539 U.S.A.
Edinburg, Texas
(Received December 4, 1979)
ABSTRACT.
A member of a class of evolution systems is defined by averaBing a one-
parameter family of invertible transformations
G
with a semigroup
G(t)T(t-s)G(s)
resulting evolution system, U(t,s)
strong continuity, and in case
G
-1,
T
The
preserves continuity and
is a linear family, may have an identifiable
generator and resolvent both of which are constructed from
T.
Occurrences of the
class of evolutions are given to show possible applications.
KEY WORDS AND PHRASES.
Evolon system, semigroup
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
i.
of transformations,
resolvent.
Primary 47H15, Secondary 34G05.
INTRODUCTION.
Given a subset
C
system of continuous transformations on
U(t,s):.
i.
ii.
t "_; s
_>
U(t,s)
U(s,s)
0
H, a strongly continuous evolution
of a Banach space
so that if t
_>
s
_>
is a family of functions
C
r
_>
0
then
is a continuous function from C into C,
I, the identity on C,
(DI
iii.
U(t,s)U(s,r)
iv. if x E
U(t,r), and
C, then {(p, U(p,s)x)
p
_>
s
is continuous.
U
G.E. PARKER
32
Such systems arise as a result of having unique solutions to the evolution
equation
y’(t)
A(t)(y(t)), y(s)
x, where x is a point of C and if t
O,
A(t) is a function from a subset of C into H.
The study of evolution systems has in several papers been made through the
study of semigroups (see for instance Kato[5]).
A strongly continuous semigroup
of continuous transformations on C is a family T
functions from C into C so that if t and s are in
i.
T(t+s)
ii.
T(0)
{ T(t):
t > 0
} of
continuous
[0, ), then
T(t)(T(s)),
I, and
(D2
iii. if x is an element of C,
is continuous.
By the relationship U(t,s)
{(p,T(p)(x))
p > 0 }
T(t-s), each semigroup is a specialized evolution
One approach to the study of evolution systems has been to study semi-
system.
groups and try to generalize to evolution systems properties the semigroups
display.
A second approach (see Ball [i], or Neveu [8]) has been to embed U into
a semigroup T on
[0, =)xC by the
action
T(t)(s,x)
(t+s,U(t+s,s)x) and study T.
Goldstein ([4]), however, has shown that even if U has linear, non-expansive maps
and analytic trajectories, T may not even have Lipschitz maps and hence may fail
to satisfy the hypothesis which is typically used in theorems about semigroups.
This paper gives an alternative approach to studying evolution systems
through semigroups.
Here the evolution systems is not studied by analogy but
rather by identifying a class of evolution systems for which semigroups are
actual structural components.
Results concern the structure of
such evolution systems as they relate to the existing theory for
evolution systems.
Examples are given that illustrate under what
conditions these evolution systems occur.
2.
DEFINITIONS and THEOREMS.
In addition to the notation established in the introduction,
the following notation will be used.
L(H,H) will denote the
normed vector space of continuous linear transformations on H with,
for f in L(H,H),
fl
glb {m: if p
H,
Ilf (p) I
__<
m
II ell
}.
SEMIGROUP STRUCTURE UNDERLYING EVOLUTIONS
R(H,H)
L(H,H) whose elements have
will denote the subset of
inverses that are in L(H,H).
will be denoted by j.
33
The identity function on the numbers
The domain (range)
of a function f will be
Df(Rf)
denoted
Suppose that G is a family of functions so that
DEFINITION:
>_ 0, G(t)
C-> C.
The statement that G is strongly continuous
if
t
at
the point x of C means that g
continuous,
{(t,G(t)(x))
x
t >
0} is
will be called the trajectory of G from x.
g
X
Suppose that U is a strongly continuous evolution
DEFINITION:
system of continuous
tranformations on C.
The statement that
A is the infinitesimal enerator for U means that A
is
so that if
-i
lim
h
t >
{(x,y)
O, A(t)
(U(t+h,t) (x)
x g
{A(t):t > O}
C and y
x) }.
h/O
Theorem i establishes a way of combining a semigroup with a
one parameter family so that the result exhibits evolution system
structure.
Conditions are given under which the frm of the infinitesimal
generator can be guaranteed.
THEOREM i.
Suppose that T is a strongly continuous semi-
group of continuous transformations on C with infinitesimal generator B and G is a strongly continuous
so that if t > 0, G(t)
is
family of functions on C
Lipschitz and has a Lipschitz inverse
and if a and b are numbers so that a < b,
on C;
k is the Lipschitz norm of
define
U(t,s)
G(p)}
G(t)T(t-s)G(s)
{k: a< p < b and
is a bounded set.
For t > s > O,
-i
Then U is a strongly continuous evolution system of continuous
transformations on C.
G(s)
is
g
Suppose, in addition, that C=H, x
R(H,H) for each s, and
in the domain of B and
t is a number so that
the trajectory of G from
g
H,
G(t)-ix
G(t)-ix
is
34
G.E. PARKER
Then, if A denotes the infinitesimal generator
differentiable at t.
G(t)BG(t)
and A(t)x
for U, x is in the domain of A(t)
-i
(x)+gy’(t)
G(t)-Ix.
where y
To see that U is an evolution system of continuous
-i
is
transformations on C, note that (i) U(t,s) --G(t)T(t-s)G(s)
PROOF:
the composition of continuous functions each mapping C into
(ii) U(s,s)= G(s)T(s-s)G(s)
C;
-i
I; and (iii) U(t,s)U(s,r)
-I
U(t,r).
G(t)T(t-r)G(r)
G(t)T(t-s)G(s)-l(G(s)T(s-r)G(r)-
For strong continuity, suppose that s
{t n}
n=l
llG(tn)T(tn-S)G(s)-l(x)
-G(t n )T(t-s)G(s)
-i
G(t)T(t-s)G(s)
I
G(t )T(t-s)G(s)
n
continuous at
I
(x)
G(t)T(t-s)G(s)-l(x) I
+
I
converges to t in [s, ).
-i
<
+
IG( t n
I
-I
(x)
_> 0,
U(t ,s)xn
<
and
U(t,s)xll
llG(tn)T(tn -s)G(s)
G(t )T(t-s)G(s)
-i
n
llr(t
x g C,
-I
(x)
(x)
-s)G(s)-l(x)-r(t-s)G(s)-l(x)
Lip
-i
(x) -G(t)T(t-s)G(s)
G(s)-l(x), {IG(tn) I}
(x)
I
T is strongly
is a bounded number
n=l
sequence, and G is strongly continuous at T(t-s)G(s)
-I
(x).
Thus
each term of the final inequality can be made arbitrarily small
and each trajectory of U is continuous.
With G satisfying the additional assumptions,
-i
an element of H so that y
G(t)
-i
consider lira
h
(U (t+h t) (x) -x)
h/0
x)
lim
suppose x is
(x) is in the domain of B and
-i
lim h
h*0
(G(t+h)T(h)G(t) -i (x)
h-l(G(t+h)T(h)G(t)-l(x) G(t+h)G(t)-l(x)) +
h-l(G(t+h)G(t)-l(x) G(t)G(t)-l(x)) lim G(t+h) (h-l(T(h)-I)(G(t) -I
h*0
(x))) + lim h
-i
h/0
(g (t+h)
Y
g (t))
G(t)BG(t)
Y
h/0
Hence x is in the domain of A(t)
and A(t) (x)
-i
(x) + g ’(t).
Y
-i
G(t)BG(t)
(x) +
g ’(t).
Y
Theorem 2 provides a partial converse to Theorem i.
thrust of the theorem is that a non-linear problem may,
The
through
35
SEMIGROUP STRUCTURE UNDERLYING EVOLUTIONS
the structure from Theorem l, be exchanged for a linear problem.
Suppose that {A(t)}
THEOREM 2:
>0
A(t)(v(t)), v(s)
unique solution to v’(t)
x in the domain of A(s).
G with domain [0,
is so that there is a
t
x for each point
Suppose also that there is a function
and range in R(H,H)
and a strongly continuous
semigroup T of continuous transformations on H with infinitesimal
generator B so that
if t > 0 and =E H,
i.
ii.
if t g
[0,
then T(t)(x)g D
then G(t) (D)
D
B
D
if x
iii.
and
x
[0, oo)
Yx
G(t) (x)
y (t)
iv.
B
then
{IG(r)
if t > s > 0,
B
A(t)
H is defined by
y x (t)
s < r
G(t)B(x)
A(t)G(t) (x)
_< t}
is
and
a bounded set.
Then if U is the evolutionsystem generated by A and t
-i
U(t,s)
G(t)T(t-s)G(s)
Suppose that t > s and that x e
PROOF:
there is a unique function v so that
v(s)
f
f
x
G(t)(D
B)
ability
(g ’(t)
Y
T(t-s)G(s)
llh
+
r(t-s)
is a subset of D
-i
(G(t+h)r(t+h-s)G(s)
G(t)BT(t-s)G(s)-l(x)II
(x))
-I
(x)
and
B
From assumption iii. on the strong differenti-
and denoting T(t-s)G(s)
llh
x
(t)) makes sense since by assumption
DA(t)
can be written
x.
(x).f (s)
I
and if t > s, R
of G,
-i
-l
I II
(x))
that A(t)(f
A(s)
Consider
h
A(t) (f x (t))
(G(t+h)r(t+h-s)G(s)
A(t)G(t)r(t-s)G(s)
(x))
(t))
G(t)T(t-s)G(s)
x is in D
G(t)T(t-s)G(s)
)+ H defined by f (t)
x
I h-l( f x (t+h)
Notice
A(t) (v(t)) and
Also, U(t,s)(x) is defined to be v(t).
x.
[0,
x
v’ (t)
By assumption,
DA(s).
(G(t+h) (T(t-s)G(s)
-i
(x))
I G(t+h)(Br(t-s)G(s)-l(x))
x by y
-i
(x)
the above norm
-i
G(t)r(t-s)G(s)
(x))-
(N-I(T(h) (T(t-s)G(s)
(x)) I +
-i
_< fIG(t+h)
G(t+h)(B(r(t-s)G(s)
-i
-i
G(t+h) (B(T(t-s)G(s)
-i
(x)))
g’
G(t)Br(t-s)G(s)-l(x))II
The first term converges to 0 by the differentiability of T at
(x))
(t)
II
G.E. PARKER
36
-i
T(t-s)G(s)
IG(r)
(x) and the boundedness of
[t, t+m]}
r
the second converges to 0 by the strong differentiability of G at
-i
T(t-s)G(s)
(x)
and the third converges to 0 by the strong contin-i
uity of G at BT(t-s)G(s)
(x)
(t)
Thus f
eA(t) (f X (t))
X
and by the uniqueness of solutions U(t,s)(x)=v(t)=f
G(t)r(t-s)G(s)-l(x)
X
(s)=x
f
X
(t)
In earlier work on nonlinear evolution systems (see for
instance Crandall
[3]) the assumption of accretiveness on the argu-
ments of A provides existence of resolvents for A from which product
formulas for the evolution system can be constructed.
Theorem 3 shows that if the linear differentiability of the hypothesis of Theorem 2 is strengthened,
then the resolvent structure
for the underlying semigroup is carried forward to the evolution
system.
THEOREM 3:
in
[0, )/ R(H,H)
L(H,H) and that T is a strongly
uous
is
Suppose that G:
continuous
is differentiable
semigroup of contin-
transformations on H with generator B so that If t>0,
a Lipschitz
transformation on H,
{(I
that the Lipschitz norms for
Define
set.
U(t,s)
to be
if 0 < m < k,
-i
and that there is a number M so
tB) -i
G(t)T(t-s)G(s)
infinitesimal generator for U.
(l-tB)
< t
0
-i
< M} is a bounded
and let A denote the
Then if t > O,
there is k> 0 so that
mA(t)) -I is a Lipschitz transformation with
(I
domain H and Lipschitz norm no greater than
(IG(t)
(I
IG(t)-ll
mB)-i
IG’ (t)
IG( t)-I
/ (I
mlG(t)
I(I
mB)-i
).
PROOF
Initially note that the differentiability assumed for
Theorem 3 is in the norm topology of L(H,H),
t
> O, G’(t) is in L(H,H).
written as
that is,
if
Hence, from Theorem i, A(t) can be
T(t)BG(t) -I + G’ (t)G(t) -I
Pick x from H.
For I
mA(t)
37
SEMIGROUP STRUCTURE UNDERLYING EVOLUTIONS
to be invertible with domain H there must be exactly one point y so
that
Fm:
(I
mA(t))(y)
Fm(p)
H-H by
For m> 0, define a function
x.
mB)-l(mG(t)-iG ’(t)G(t)-l(p)
G(t)(l
-I
c>0, let M denote an upper bound for {(I -nB)
II Fro(P)
Thus
II
Fm(q)
m <
min{(IG(t)
IG(t)-llmlG’(t) )-i
M
G(t)-l(x)).
Given
__< c}.
mB)-ll IG(t)-ll21G’(t)l
I( I
mlG(t)
__<
n
+
P
c} implies that F
m
q
is a strict
contraction and has a unique fixed point y.
-mB)-l(mG(t)-lG ’(t)G(t)-l(y)+(t)-l(x)), it
mg)G(t)-l(y) mG(t)-lG ’(t)G(t)-l(y)+(t)-l(x),
(I
G(t)-l(y) m(BG(t)-l(y) + G(t)-lG (t)G(t)-l(y)) G(t)-l(x), and
G(t)(I
Since y
y
m((G(t)BG(t)
G’(t)G(t)-l)(y))
(I -mA(t))
Thus y
For
II(I
+
(I
Yl
mA(t))
-i
(x
-i
3
x.
mA(t))
I)
(I
(x).
-i
(x
l)
and
mA(t))
-i
(x
Y2
(I
2)
mA(t))
G(t)(l
-i
mB)-lllG(t) -I
2
IG,(t)
))
x
x
i
(x2)
mB)
(mG(t)-iG ’(t)G(t)-l(yl) + G(t)-l(xl )) G(t)(l mB)
(mG(t)-lG ’(t)G(t)-l(y 2) + G(t)-l(x2))II < IG(t)] I(l
(mlG(t)-llm]G’(t)l flY I- Y2]I + IG(t)-ll x I x211 )"
Thus
-ii)/( I
I( I
mA(t))(y)
that is, (I
x
the calculations shows that any such point must be a fixed point of
Reversing
Fm.
-1
follows that
-i
mB)
-i
mIG(t)
2
OCCURRENCES.
There appear to be several possibilities for the application of the
evolution structure identified in the preceding theorems.
Example 1.
In [7], Neuberger studied semigroups of Lipschitz transformations so
that if T is one of them, then there is c > 0 so that if x and y are elements
of H, and t > 0, then
G(t)
e
-ct
T(t)(x)
T(t)(y)II
< e
ct
x
y
By taking
I, Neuberger’s special evolution system which produced product
formulas for the semigroup can be seen to be an instance of the theory.
Example 2.
Suppose that T is a semigroup of continuous transformations on H
with infinitesimal generator B and G is a group of continuous linear trans-
G.E. PARKER
38
formations on H with generator C.
with generator A so that B
Define U by
PROOF:
Then there is an evolution system U on H
+ C C A(O).
G(t)-iT(t-s)G(s) -I.
U(t,s)
to Theorem i is satisfied and
B(x) + lim h -i (G(h)(x)
A(0)(x)
G(O)BG(0)
A(0)(x)
x)
-i
If x e D
(x) +
the hypothesis
B+C
(0).
gG(O)_l(x
Hence
B(x) + C(x).
h+0
Thus, even though such sums may fail to guarantee the generation of semigroups
(see Chernoff [2]), they must always support the generation of globally defined
evolutions.
Generation theories for evolution systems typically require that
be dense in C and that (I
mA(t))
-i
be Lipschitz with domain C.
D
s < t A (s)
These
restrictions need not apply for the theory discussed in Theorem i to be viable.
Example 3.
from
[0, oo)
Let H be the Banach space of bounded uniformly continuous functions
into R with for f e
(A(s) (f))(x)
fll
H,
I
lub
f’(x) if x
0 if x
By taking T to be the semigroup generated by
and G defined by
(G(t) (f)) (x)
U, defined by U(t,s)
I
G(t)T(t-s)G(s)
{If(x)
_<
x > 0
}.
Define A(s) by
s
> s
{(f,g)
f s H, g s H, and g
f’}
f(x) if x > t
2f(x)
-i
f(t) if x < t
has generator A.
Notice that the two
conditions on the domain of A(s) are provided separately by the differentiability
of T and G respectively.
0 < a <
bDA(a)
Here U(t,s) is defined globally although
is nowhere dense in any neighborhood and I
mA(t) does not
have range H.
Even when existing theories cover a given evolution system, the structures
studied here may provide alternative computation possibilities.
Example 4.
y’(t)
Consider the ordinary differential equation
A/((Bt + C)y(t)) +-Dy(t)/(Bt + C), y(s)
x, D/B
_I_
2
A typical solution might involve solving for y 2 in the associated linear equation.
However, by letting G solve the linear term, a semigroup generator can be found
-I
by purely algebraic means from which U(t,s)
can be built to
G(t)T(t-s)G(s)
39
SEMIGROUP STRUCTURE UNDERLYING EVOLUTIONS
Here the forms in which the two methods give the
solve the original equation.
solution are computationally different.
Example 5.
Suppose that A is a linear group generator and that C is a continuous
function from H into the compact subset K of H.
Consider, for x e H, the
problem
y’(t)
(A + C)(y(t)), y(0)
x
which can be solved by the method of convergence of approximate solutions as
employed by Martin[6].
Using the approach suggested in the theorems, let G denote the group gener-
ated by A and look for a function f from [0, d] into H so that
y
{(t,G(t)(f(t)))
0 < t < d} solves the problem.
G(t)-iCG(t)(f(t)).
a solution it must be true that f’(t)
X
C([0, m),H),
For such a function to be
Letting
this in turn translates into a fixed point problem for the
function F defined by, for k e X,
(F(k))(t)
x
+
CG(j) is a compact map on [O,d] xH.
and G(j)
lim
h->O
h
-i
G(t+h)(h
G(j)
(G(t+h)(f(t+h))
-i
CG(j)k
0
-i
(f(t+h)
f(t)
G(t)G(t)-iCG(t)(f(t)).
G(t)(f(t))
lim
h/0
If f is a fixed point for F, then
h-l(G(h)G(t)(f(t))
AG(t)(f(t)) + G(t)(f’(t))
Hence y’(t)
G(t)(f(t))
+
AG(t)(f(t)) +
(A + C)(y(t)) and y(0)
x.
Notice that
the burden for solution is shifted from evaluating the convergence of approximate
solutions to finding a fixed point associated with a compact map.
4.
0UESTIONS.
In Example 4, when G is produced by solving the linear problem
-Du(t)/(Bt+c), finding the appropriate associated semigroup generator
u’(t)
depends on finding a function
In this case f(p)
f so that A/((Bt
A/p works.
+ C)x)
cD/Bf(c-D/B(Bt + C)x).
Any systematic application for the evolution
system in this paper must include an analysis of the conditions under which a
family
{B(t)}
can be algebraically resolved to the form
{G(t)fG(t)
-i
}t
< 0"
t <0
Theorem i identifies a closed form for the evolution generator in case G is
a linear family.
Can other forms of the generator for G(t)T(t-s)G(s)
identified when G is not linear?
-I
be
G.E. PARKER
40
REFERECES
I.
Ball, John M. "Measurability and Continuity Conditions for Nonlinear
Evolutionary Processes." .P.rp.ceedinFs AMS, 53(1976), 353-358.
2.
Chernoff, Paul R. "Product Formulas, Nonlinear Semigroups, and Addition
of Unbounded Operators." Memoirs AMS, no. 140(1974).
3.
Crandall, M. G., and Pazy, A. "Nonlinear Evolution Equations in Banach
Spaces." Israel J. Math., 11(1972), 57-94.
4.
Goldstein, J. A.
voor Wiskunde
"An Example
of a Nonlinear Semigroup."
Nieuw Archief
(3), XXIII(1974), 170-174.
5.
Kato, T.
6.
Martin, R. H., Jr.
Banach Spaces.
7.
Neuberger, J. W.
"Nonlinear Semigroups and Evolution Equations."
Japan, 19(1967), 508 ff.
J. Math.
So___c.
Nonlinear Operators and Differential Equations in
Wiley, New York, 1976.
"Product Integral Formulae for Nonlinear Expansive
Semigroups and Non-Expansive Evolution Systems." J. of Mat_h. and Mech.,
19(1969), 403-409.
8.
Neveu, J. "The’orie des semi-grousde Markov", Univ. Calif. Publ. Statistics
2(1958), 319-394.