Journal
of Applied Mathematics
and Stochastic Analysis, 10:3
(1997),
265-271.
ON A MILD SOLUTION OF A SEMILINEAR
FUNCTIONAL-DIFFERENTIAL EVOLUTION
NONLOCAL PROBLEM
LUDWIK BYSZEWSKI
Cracow University of Technology
Institute
of Mathematics
Warszawska 2, 31-155 Cracow, Poland
HAYDAR AKCA
Akdeniz University
Department of Mathematics
A ntalya 0720, Turkey
(Received February, 1997; Revised May, 1997)
The existence, uniqueness, and continuous dependence of a mild solution of
a nonlocal Cauchy problem for a semilinear functional-differential evolution equation in a general Banach space are studied. Methods of a C o
semigroup of operators and the Banach contraction theorem are applied.
The result obtained herein is a generalization and continuation of those reported in references [2-8].
Key words: Abstract Cauchy Problem, Evolution Equation, Functional-Differential Equation, Nonlocal Condition, Mild Solution, Existence
and Uniqueness of the Solution, Continuous Dependence of the Solution, a
C O Semigroup, the Banach Contraction Theorem.
AMS subject classifications: 34G20, 34K30, 34K99, 47D03, 47tt10.
1. Introduction
In this paper
we study the existence, uniqueness, and continuous dependence of a
mild solution of a nonlocal Cauchy problem for a semilinear functional-differential
evolution equation. Methods of functional analysis concerning a C O semigroup of
operators and the Banach theorem about the fixed point are applied. The nonlocal
Cauchy problem considered here is of the form:
u’(t) + Au(t)
f(t, ut)
t
e [0, a],
(1.1)
1The paper was supported by NATO grant: SA.11-1-05-OUTREACH (CRG.
960586) 1189/96/473.
Printed in the U.S.A.
(C)1997 by North
Atlantic Science Publishing Company
265
L. BYSZEWSKI and H. AKCA
266
t(8)-4-(g(Utl,...,t P ))(8)- (8),
8
I--r,0],
(1.2)
where 0 < t I <
< tp <_ a (p N); A is the infinitesimal generator of a Co semigroup of operators on a general Bausch space; f, g and are given functions satisfying some assumptions, and ut(s): = u(t + s) for t [0, a], s [- r, 0].
Theorems about the existence, uniqueness, and stability of solutions of differential and functional-differential abstract evolution Cauchy problems were studied previously by Byszewski and Lakshmikantham [2], by Byszewski [3-8], and by Lin and
Liu [10]. The result obtained herein is a generalization and continuation of those reported in references [2-8].
If the case of the nonlocal condition considered in this paper is reduced to the
classical initial condition, the result of the paper is reduced to previous results of Hale
[9], Thompson [11], and Akca, Shakhmurow and Arslan [1] on the existence, uniqueness, and continuous dependence of the functional-differential evolution Cauchy problem.
2. Prehminaries
assume that E is a Banach space with norm I1" 11; -A is the infinitesimal generator of a C O semigroup {T(t)} > 0 on E, D(A) is the domain of A;
We
0<t l<...<tp<_a (pN)
and
M"
sup
I T(t)II BL(E,E)"
E [0, a]
the
the
norm
In
sequel
operator
I[ BL(E,E) will be denoted by ]]
For a continuous function w: [- aidE, we denote by w a function
![.
C([- v, 0],E)
(2.1)
"b!
given by the formula
wt(s):
w(t + s) for
t
[0, a],
s
[- r, 0].
Let f: [0, a] x C([- r, 01, E)E. We require the following assumptions:
Assumption (A1): For every w E C([- r, a], E) and t E [0, a],
wt) e C([O, a], E).
f
Assumption
(A2):
There exists a constant L
_< L I w-
I c([-
C([ r, a], E),
E [0, a].
I f(t, wt)- f(t, vt)II
for w,
E
> 0 such that:
Let g: [C([ v, 0], E)]PC([ v, 0], E). We apply the assumption:
Assumption (A3): There exists a constant K > 0 such that:
[[ (g(wt 1,
wt, ))(s)- (g(tl., wt, ))(S)II --< K I
for
w,
-
C([ r, a], E), sE[-r,0].
I[ ([-
longing to
Evolution Nonlocal Problem
267
Moreover,
we require the assumption"
Assumption (A4):
E C([- r, 0], E).
A function u E C([- r, a], E) satisfying the conditions:
()
T()[((1"’")1(01]
()- T()(0)-
/ T(t-s)f(S, Us)ds
-t-
te[O,a],
0
(ii)
u(s) 2t-(g(utl,...
t
p
))(8)
(8),
8
e [-- r, 0),
is said to be a mild solution of the nonlocal Cauchy problem
(1.1)-(1.2).
3. Existence and Uniqueness of a Mild Solution
Theorem 3.1: Assume that the
Additionally, suppose that:
functions f,g, and
satisfy Assumptions
(A1)-(A4).
M(an + h’) < 1.
Cauchy problem (1.1)-(1.2) has
(3.1)
a unique mild solution.
Then the nonlocal
Proof: Introduce an operator F on the Banach space C([-r,a],E) by the formula:
))(t), t [-- r,O),
(t)
(g(wtl,...,w p
(Fw)(t)"
T(t)(O)-
T(t)[(g(wtl,..., Wtp))(O)]
+ f0 T(t- s)f(s, ws)ds
t
e [0, a],
where w e C([ r, a], E).
It is easy to see that
r, a], E)---C([
F: C([
Now, we will show that F is a contraction on
consider two differences"
(gw)(t) (gv )(t)
for
,
(3.2)
r, a], E).
C([ r, a], E). For
this purpose
w ))(t)
(g(tl,...,t))(t)-(g(wtl,...
p
p
e C([
, a], E),
t
e
, 0)
and
(Fw)(t) (Fv )(t)
+
T(t)
[(g(tl,..., tp))(O)- (g(wtl,..., Wtp))(O)]
/ T(t- s)[f(s, ws)- f(s, s)]ds
0
for w,
From (3.3) and Assumption
C([ r, a], E),
(A3):
G
t
[0, a].
(3.4)
L. BYSZEWSKI and H. AKCA
268
I (Fw)(t)- (Fo)(t)II
for w,@
KI w
I c([- ,a],E)
e C([ r, a], E), te[-r,O).
Moreover, by (3.4), (2.1), Assumption (A2) and Assumption (A3)"
I (Fw)(t)- (F )(t) I[
+
_< I T(t) I I (g(wt 1, ..., Wtp))(O)- (g(tl, .., tp))(O)II
/ I Z(t- )II I f(s,
w s)-
f(s, s)II ds
(3.6)
0
<_ MK I w
I c([- , o1, E) -[- ML
fI
w
I c([- , ], E) ds
0
I C([- , a], E)
<_ M(aL + K)II 0for
Formulas
(3.5)
w,
e C([ r, a], E), te[0,a].
and (3.6)imply the inequality
I Fw- F I c([- r,a],E)
for w,
where q:
q
I w- I C([- , ], E)
(3.7)
e C([ r, a], E),
M(aL + K).
Since, from (3.1), q E(0,1), then (3.7) shows that F is a contraction on
C([- r,a],E). Consequently, by (3.2) and (3.7), operator F satisfies all the assumptions of the nanach contraction theorem. Therefore, in space C([-r,a],E) there is
only one fixed point of F and this point is the mild solution of the nonlocal Cauchy
problem (1.1)-(1.2).
The proof of Theorem 3.1 is complete.
4. Continuous Dependence of a Mild Solution
Suppose that the functions f and g satisfy Assumptions (A1)-(A3) and
Then, for each 1, 2 E C([- r, 0],E), and for the corresponding
mild solutions Ul, u 2 of the problems
Theorem 4.1"
M(aL + It’) <
1.
u’(t) + Au(t) f(t, ut) t e [0, a],
tt(8) -l- (g(ttt 1’ ...,u p ))(s)-i(s), se[-r,O] (i-1,2),
(4.1)
the inequality
I tl U2 I c([- r,.]E)
<-- MeaML( I 1- 2 I c([- ,o1, E)+ K I Ul- u2 I C([
(4.2)
Evolution Nonlocal Problem
-
MeaML I
u2(t
T(t)[l(0
is true.
Additionally,
269
1
then
if K < MeaM--------,
]1 Ul u2 I C([- r,a], E)
I
(4.3)
-2 C([-r,0],E)"
1- KMe aML 1
Proof: Let i (i- 1,2) be arbitrary functions belonging to C([ r, 0], E), and
let u (i- 1,2) be the mild solutions of problems (4.1).
Consequently,
ul(t
I(g((tl)tl,... (tl) tp))(O)
T(t)
+
J
s)[f(s, (u 1)s)
T(t
2(0)]
(4.4)
(g((t2)tl,... (t2) tp
f(s, (U2)s)]ds for t e [0, a],
0
and
-
el(t) 2(t)
u2(t
ul(t
(4.5)
(g((u2)tl,..., (U2) P ))(t) (g((t 1)tl,. (t 1)t P
for t
From (4.4), (2.1), Assumption
I ttl(T)- tt2(’r)II
[- r,O).
(A2)and Assumption (A3)"
_< M I 1 --2 I C([- r,0],E) @ MK I Ul U2 I c([- r,a],E)
-t- ML
/
I Ul
I
U2 C([- r, s], E) ds
0
_< M I 1 2 I c([- , 01, E) + MK I Ul t2 ][ C([- r, a], E)
+ ML
/I
Ul
,
u2 I c([- sl, E) ds for 0 _< r _< t _< a.
0
Therefore,
I ttl(T)- u2(7)l[
<_ M I 1 2 I c([- r,o], E) + MK I Ul
sup
e [o,t]
-t- ML
/I
ttl
tt2 !1 c([- ,.], E) ds for t
0
Simultaneously, by
(4.5)
u2 I C([- r,a],E)
and Assumption
(A3):
(
[0, a].
270
1[ ttl(t) tt2(t)1[
Since M
-
L. BYSZEWSKI and H. AKCA
_> 1, formulas (4.6)
I 1 2 11C([- r,0l, E) + K I ttl
for t
(4.7)
and
U2
[1C([- r, al, E)
e I-r, 0).
(4.7)
imply:
I Ul
u2 I c([- r, tl, E)
<_ M I 1 2 I C([- r, 01, E) -t- MK I Ul U2 IIc([- r, a], E)
+ ML
j
I Ul
(4.8)
u2 I C([- r,s],E) d8 for t E [0, a].
0
From (4.8) and Gronwall’s inequality:
[M I 1
2 I C([- r, 01, E) -t- MK I Ul
-
u2 I c([- r, a],
E)]eaML"
Therefore, (4.2) holds. Finally, inequality (4.3)is
a consequence of inequality
The proof of Theorem 4.1 is complete.
Remark 4.1: If K -0, inequality (4.2) is reduced to the classical inequality
I ttl
u2 ]l C([- r, a], E)
(4.2).
MeaML I 1 2 IIc([- r, 01, E),
which is characteristic for the continuous dependence of the semilinear functionaldifferential evolution Cauchy problem with the classical initial condition.
5. Remarks
1. Let
0<t l<...<tp<_a (pEN).
Theorems 3.1 and 4.1 can be applied for g defined by the formula:
p
(g(wtl,...,
where
ck(k
w
P
E CkW(tk q- s) for
))(s)
wE
C([
r, a], E), s E
1,..., p) are given constants such that
M aL+
Ickl
(5.1)
<1.
k=l
2. Let
0<t 1
and let
ek(k
r, 0],
k=l
<...<tp<_a (pEN)
1,..., p) be given positive constants such that"
0
<
I
1 and t k
1
< tk ek (k
2,..., p).
Evolution Nonlocal Problem
271
Theorems 3.1 and 4.1 can be applied for g defined by the formula:
P
((,...,
fo
where c k
(k
P
}2
))()-
k=l
k
kCkJ
(+)
.
tk-e k
[- 0].
c([- ..]. E).
1,..., p) are given constants satisfying condition (5.1). Indeed,
I (g(Wtl,’", Wt P ))(S) (g(tl,..., t P ))(S)II
k
k--1
_<
cl
k=l
tk ek
I w- I c([-,o1,,)
for
8
,
e [- 0].
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[1]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
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