PROBLEM ON

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 ofa mild solution of a nonlocal Cauchy problem for a semilinear functional-differential evolu- tion 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 isa generalization and continuation of those re-

ported in references

[2-8].

Key

words: Abstract Cauchy

Problem,

Evolution Equation,

Func-

tional-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 ofthe 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 theU.S.A. (C)1997byNorth Atlantic SciencePublishingCompany 265

t(8)-4-(g(Utl,...,t

_P

^{))(8)- (8),}

⁸

^{I--r,0], (1.2)}

where0

<

t_I

< < tp ^<_

^a

^{(p N); A}

is the infinitesimal

generator

ofa

C

_{o semi-} group of

operators

on a

general

Bauschspace;

f,

g and are given functions satisfy- ingsome assumptions, and

ut(s): ⁼ u(t + s)

^{for t}

[0, a],

^s

[-

r,

0].

Theorems about the existence, uniqueness, and stability ofsolutions of differen- tial and functional-differential abstract evolution Cauchy problems were studied pre- viously 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 re- ported 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 resultsof Hale

[9],

^Thompson

[11],

^and

^Akca,

Shakhmurow and Arslan

[1]

^on the existence, unique- ness, and continuous dependence ofthe functional-differential evolution Cauchy pro- blem.

2. Prehminaries

We

assume that

E

isa Banachspace with norm

I1" ^{11; -A}

^isthe infinitesimalgener- ator ofa

C

_Osemigroup

{T(t)} _>

₀ ^on

E, D(A)

^{isthedomain of}

A;

and

0<t

l<...<tp<_a ^(pN)

M"

sup

II ^T(t)II

BL(E,E)"

(2.1)

E[0,a]

In

the sequel theoperator norm

![. ^I[

^{BL(E,E) willbe denoted by}

^]] "b!

For

a continuous function w:

[- aidE,

^we

^denote

^{by w afunction longing to}

C([- ^v, 0],E)

given by theformula

wt(s): w(t + s)

^fort

[0, a],

^s

[-

r,

0].

Let f: [0, a]

^x

C([-

r,

01, E)E. ^We

require the following assumptions:

Assumption

(A1): ^For

^{every wE}

C([-

r,

a], E)

^and t E

[0, a], f wt) e C([O, a], E).

Assumption

(A2):

There exists aconstant

L >

⁰ such that:

II ^{f(t, wt)- f(t, vt)II _< L} II

^w-

II

_c([-

for w, E

C([

r,

a], E),

E

[0, a].

Let

g:

[C([ v, 0], E)]PC([

^v,

0], E). ^We

^apply the assumption:

Assumption

(A3):

There exists aconstant

K >

⁰ such that:

))(s)- (g(tl., ^))(S)II --< ^K II - ^I[

^([-

[[ (g(wt

1,

wt, wt,

for

w, C([

r,

a], E), sE[-r,0].

Moreover,

^werequire the assumption"

Assumption

(A4):

^E

C([-

^r,

0], E).

A

function uE

C([-

r,

a], E)

^satisfying the conditions:

() ()- T()(0)- T()[((1"’" )1(01]

-t- / ^{T(t-s)f(S, Us)ds 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 ofthe nonlocal Cauchy problem

(1.1)-(1.2).

3. Existence and Uniqueness of

a

Mild Solution

Theorem 3.1:

Assume

that the

functions

^{f,g, and} satisfy Assumptions

(A1)-(A4).

Additionally, suppose that:

M(an + h’) <

^1.

(3.1)

Then the nonlocal Cauchy problem

(1.1)-(1.2)

^{has a} unique mild solution.

Proof:

mula:

Introduce an operator

F

on the Banach space

C([-r,a],E)

^{by the for-}

(Fw)(t)"

where w

e C([

^r,

a], E).

It

is easy tosee that

(t) (g(wtl,...,w

_p

^))(t),

^t

^{[-- r,O),}

T(t)(O)- T(t)[(g(wtl,..., Wtp))(O)]

+ f T(t- s)f(s, ws)ds

^t

e [0, a],

0

F: C([

r,

a], E)---C([

r,

a], E). (3.2)

Now,

^we will show that

F

is a contraction on

C([

r,

a], E).

consider two differences"

For

this purpose

and

(gw)(t) (gv )(t) (g(tl,...,t))(t)-(g(wtl,...

_p ^w _p

^))(t)

for

, e C([ , a], E),

^t

e , 0)

(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

(3.4)

for w, G

C([

^r,

a], E),

From (3.3)

and Assumption

(A3):

t

[0, a].

II ^{(Fw)(t)- (Fo)(t)II K} II

^w

II

c([- ,a],E) for

w,@ e C([

r,

a], E), te[-r,O).

Moreover,

by

(3.4), (2.1),

^Assumption

(A2)

and Assumption

(A3)"

II ^(Fw)(t)- (F )(t) I[ _< II ^T(t) II II ^(g(wt

_1,

^..., Wtp))(O)- ^{(g(tl, ..,} tp))(O)II

+ / ^{II Z(t- )II II f(s,}

^w

^{s)- f(s, s)II}

^ds

0

<_ MK II

^w

II

c([-

,

o1, E)^-[-

ML f

₀

^II

^w

^II

^c([-

^,

^],E)ds

(3.6)

<_ M(aL + K)II

^0-

II

C([-

,

a], E) for

w, e C([

r,

a], E), te[0,a].

Formulas

(3.5)

^and

(3.6)imply

the inequality

II ^{Fw- F} II

c([- r,a],E) ^q

II

^w-

II

_C([-

,

],E) for

w, e C([

r,

a], E),

(3.7)

where q:

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 assump-}

tions 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 proofofTheorem 3.1 is complete.

4. Continuous Dependence of

a

Mild Solution

Theorem 4.1"

Suppose

that the

functions f

^{and g satisfy} Assumptions

(A1)-(A3)

^and

M(aL + It’) <

^1.

^Then, for

^each

1, 2

^E

^C([-

^r,

^0],E),

^and

^for

^the corresponding mild solutions

Ul,

^u2

of

^{the problems}

u’(t) + Au(t) f(t, ut)

^t

e [0, a],

...,u

))(s)-i(s), se[-r,O] (i-1,2), tt(8)

^-l-

(g(ttt

1’

p

(4.1)

the inequality

II

tl U2

II

c([- r,.]E)

<-- MeaML( ^{II 1- 2 II}

^c([-,o1,

^{E)+ K II}

^Ul-

^{u2 II}

^C([

^(4.2)

is true.

1 then

Additionally,

if ^K < MeaM---,

]1

_Ul

_u2 II

C([-r,a],E)

--

^1-

^{MeaML KMe}

^aML

^{II 1 -2 II}

C([-r,0],E)"

(4.3)

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 ofproblems

(4.1).

Consequently,

and

ul(t u2(t T(t)[l(0 2(0)]

T(t) I(g((tl)tl,... ^{(tl) tp))(O)} (g((t2)tl,... ^(t2) _tp

+ J

₀

^{T(t s)[f(s, (u 1)s) f(s, (U2)s)]ds}

^{for t}

^{e [0, a],}

(4.4)

ul(t u2(t el(t) 2(t)

-- (g((u2)tl,..., ^(U2)

_P

^{))(t) (g((t} 1)tl,. ^{(t 1)t}

_P

for t

[- r,O).

(4.5)

From (4.4), (2.1),

^Assumption

(A2)and

^Assumption

(A3)"

II ^{ttl(T)- tt2(’r)II _< M} II 1 --2 II

C([- r,0],E)^@

MK II

^Ul

^U2 II

c([- r,a],E)

-t- ML / ^II

^{Ul U2}

^II

^{C([-r,s],E)ds}

0

Therefore,

_< M II 1 2 II

c([-

,

01,E)

+ ^MK II

^{Ul t2}

^][

C([-^r,a],E)

+ ^ML / ^II

^Ul

^{u2 II}

^c([-

^,

^{sl,E)ds for 0}

^_<

^r

^_<

^t

^_<

^a.

0

sup

II ttl(T)- u2(7)l[

e

[o,t]

<_ M II 1 2 II

c([-r,o],E)

+ ^MK II

Ul

u2 II

C([- r,a],E)

-t-

ML / ^II

^{ttl tt2}

^!1

^{c([-,.], E)ds for t(}

^{[0, a].}

0

Simultaneously, by

(4.5)

and Assumption

(A3):

1[ ttl(t) tt2(t)1[ -- ^{II 1 2}

^for

^11C([-

^t

^{e I-r,}

^r,0l,

^0).

^E)

^{+ K II ttl U2 [1C([-}

^{r,al, E)}

^(4.7)

Since

M _> 1,

^formulas

(4.6)

^and

(4.7)

^imply:

II Ul u2 II

_c([-r,tl,E)

<_ M II 1 2 II

_C([-r,01, E)

-t- MK II Ul U2 IIc([-

r,a], E)

+ ^ML j ^II

^Ul

^{u2 II}

C([- r,s],E)^{d8 for t}E

[0, a].

0

(4.8)

From (4.8)

^and Gronwall’s inequality:

[M ^{II 1 2 II}

^{C([-r,01, E)}

^{-t- MK II}

^Ul

^{u2 II}

^c([-r,a],

E)] ^eaML"

Therefore, (4.2)

^holds. Finally, inequality

(4.3)is

^a consequence of inequality

(4.2).

TheproofofTheorem 4.1 is complete.

Remark 4.1: If

K -0,

inequality

(4.2)

^is reduced to the classical inequality

II ^ttl

^u2

^]l

C([-r,a],E)

-- ^{MeaML II 1 2 IIc([-}

^r,01,E),

which is characteristic for the continuous dependence of the semilinear functional- differential evolutionCauchy problemwiththe classical initial condition.

5. Remarks

1.

Let

0<t

l<...<tp<_a ^(pEN).

Theorems 3.1 and4.1 canbe appliedfor g defined by the formula:

p

(g(wtl,...,

^w _P

^))(s) E ^CkW(tk

^q-

^s)

^{for wE}

^C([

^r,

^{a], E),}

^{sE r,}

^0],

k=l

where

ck(k ^{1,..., p)}

^{are given constantssuch that}

M aL+ Ickl

^<1.

k=l

2.

Let

0<t₁

<...<tp<_a ^(pEN)

and let

ek(k ^1,..., p)

be given positive constants such that"

0

<

_I

₁

and t_{k 1}

< tk

ek

(k ^2,..., p).

(5.1)

Theorems 3.1 and 4.1 canbe applied for g definedby the formula:

P k

kCkJ

((,..., ^))()- }2 ⁽ + ⁾

P k=l

tk-e

k

fo

c([- ..]. E). [- . ^0].

wherec_k

(k ^1,..., p)

^are given constants satisfying condition

(5.1). Indeed,

II (g(Wtl,’", ^Wt

_P

^))(S) (g(tl,..., t

_P

^))(S)II

k

k--1

tk

ek

_< cl II

^w-

II

c([-,o1,,) ^{for 8}

e [- , 0].

k=l

References [1]

[4]

[6]

[7]

Akca, H., Shakhmurov, V.B.

^and

Arslan, G.,

Differential-operator equations with bounded delay, Nonlinear Times and Digest 2

(1995),

^179-190.

Byszewski,

L.

and

Lakshmikantham, V.,

Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space, Appl. Anal. 40

(1990),

^11-19.

Byszewski,

L.,

^{Theorems about} the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem,

J.

Math. Anal. Appl. 162

(1991),

494-505.

Byszewski,

L.,

^Existence and uniqueness of mild and classical solutions of semi- linear functional-differential evolution nonlocal Cauchy problem, Selected Prob.

in

Math., Cracow

University ofTechnology, Anniversary

Issue

6

(1995),

^25-33.

[5]

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L., Differential

^and

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Cracow

UniversityofTechnology,

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Poland 1995.

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L., On

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(1997),

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L.,

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Byszewski,

L.,

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(1997),

^{in press.}

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Theory

of

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Differential

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New York,

Berlin, Heidelberg ^1977.

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J.H.,

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R.J.,

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