Existence Solutions for Quasilinear Evolution Integrodifferential Equations with Infinite Delay

International Journal of Engineering and Technical Research (IJETR) ISSN: 2321-0869, Volume-2, Issue-4, April 2014

Existence Solutions for Quasilinear Evolution Integrodifferential Equations with Infinite Delay Francis Paul Samuel, Tumaini RukikoLisso, Kayiita Zachary Kaunda ď€

Abstract: The paper is concerned with the existence and uniqueness solution of quasi linear evolution integrodifferential equations with in finite delay in Banach spaces. The results a r e obtained by C0-semi group of linear bounded o p e r a t o r and Banach f i x e d point theorem.

Keywords: semigroup; mild a n d classical solution; Banach f i x e d pointtheorem; Infinitedelay.

2000 Subject Classification: 34G20, 47D03, 47H10,47H20,34B15.

I. INTRODUCTION Quasilinearevolution equations form savery important class of evolution equations as many time dependent phenomena inphysics,chemistry, biology and engineering can be represented by such evolution equations.

Several authorshave studied the existence of solutions of abstract quasilinear evolution equations in Banach space [1,2,3,7,8,9,10,12,14].

Oka [10]and Okaand Tanaka[11]discussed the existence of solutions of quasilinear integrodifferential equations in Banach spaces.

Kato [6] studied the non homo geneousevolution equations and Balachandran and Paul Samuel[2]proved the existence and uniqueness of mild and classical delay integro differential equation with non local condition. The problem of existence of solutions of evolution equations in Banach space has been studied by several authors [4,5,8,9].The aim of this paper is to prove the existence and uniqueness of midland classical solutions of quasilinear functional integro differential system with infinitedelay đ?‘˘â€˛ đ?‘Ą + đ??´ đ?‘Ą, đ?‘˘ đ?‘˘ đ?‘Ą đ?‘Ą

= � �, �� +

đ?‘˜ đ?‘Ą − đ?‘ đ?‘”(đ?‘ , đ?‘˘đ?‘ )đ?‘‘đ?‘ (1.1) 0

đ?‘˘0 = đ?œ‘, on [−đ?œ‡, 0](1.2)

wheređ?‘˘đ?‘Ą (đ?œƒ) = đ?‘˘(đ?‘Ą + đ?œƒ), đ?œƒ ∈ [âˆ’Âľ, 0],For đ?‘Ą ∈ [0, đ?‘‡],we denotebyđ??¸đ?‘Ą theBanach space of all continuous functionsfrom[âˆ’Âľ, đ?‘Ą]tođ?‘‹endowed with the supremum norm

ď Ł

Et

277

ď€­ď ´ ď‚Łď ą ď‚Łt

X

, x ďƒŽ Et .

đ?‘“ : [0, đ?‘‡] Ă— đ??¸0 đ?‘Ąđ?‘œđ?‘‹; đ?‘”: [0, đ?‘‡ ] Ă— Letthe functions đ??¸0 to đ?‘‹ and đ?‘˜: [0, đ?‘‡] → [0, đ?‘‡] is a real value dcontinuous function. Here we see that đ?‘Ą ∈ đ??¸0 ,we assume that forđ?‘˘ ∈ đ??¸đ?‘‡, đ?‘“(¡, đ?‘˘ (¡)) and đ?‘”(¡, đ?‘˘ (¡)) âˆś [0, đ?‘‡] → đ?‘‹are bounded L1 function. Further assume that there is a subset đ??ľ ofđ?‘‹ such that for (đ?‘Ą, đ?‘˘) ∈ [0, đ?‘‡ ] Ă— đ??¸đ?‘‡with đ?‘˘(đ?‘Ą) ∈ đ??ľforđ?‘Ą ∈ [0, đ?‘‡], đ??´(đ?‘Ą, đ?‘˘)isalinear operator in đ?‘‹.Alsođ?œ‘ ∈ đ??¸0 isLipschitz continuous with constant đ??żđ?œ‘ The results obtained in this paper are generalization so the results given by Pazy [13],Kato [6,7]and Balachandranand Paul Samuel[3].

II. PRELIMINARIES Let đ?‘‹and đ?‘Œ betwoBanach spaces such that Y is densely and continuous lyembedded inđ?‘‹. For any Banach spacesđ?‘?thenormofđ?‘?isdenoted by

. or . z .The space of all

bounded linear operators from X toY is denoted byđ??ľ(đ?‘‹, đ?‘Œ)andđ??ľ(đ?‘‹, đ?‘‹)iswritten asđ??ľ(đ?‘‹).We recall some definitions and known facts from Pazy [13].

Definition2.1.Letđ?‘†bealinear operatorinđ?‘‹and letđ?‘Œ beasubspace ofđ?‘‹.Theoperatorđ?‘†Ëœdefinedby đ??ˇ(đ?‘†Ëœ) = {đ?‘Ľ ∈ đ??ˇ(đ?‘†) ∊ đ?‘Œ âˆś đ?‘†đ?‘Ľ ∈ đ?‘Œ}and đ?‘†Ëœđ?‘Ľ = đ?‘†đ?‘Ľđ?‘“đ?‘œđ?‘&#x; đ?‘Ľ ∈ đ??ˇ(đ?‘†Ëœ)iscalledthepartofđ?‘†inđ?‘Œ. {đ?‘Ľ + đ?‘Ś âˆś đ?‘Ľ đ?œ– đ??ľ, đ?‘Ś đ?œ– đ??¸}.

Definition2.2.Let B be a subset of đ?‘‹ and for every 0 ≤ đ?‘Ą ≤ đ?‘‡and đ?‘? ∈ đ??ľ , let đ??´(đ?‘Ą, đ?‘?) be the i nfinitesimalgeneratorofaC0 -semigroupđ?‘†đ?‘Ą, đ?‘?(đ?‘ ), đ?‘ ≼ 0, on đ?‘‹ . The family of operators {đ??´(đ?‘Ą, đ?‘?)}, (đ?‘Ą, đ?‘?) ∈ [0, đ?‘‡] Ă— đ??ľ, isstable ifthere areconstants đ?‘€â‰Ľ1 and đ?‘? ,knownasstabilityconstants,suchthat đ?œŒ(đ??´(đ?‘Ą, đ?‘?)) ⊃ (đ?‘?, ∞)đ?‘“ or (đ?‘Ą, đ?‘?) ∈ [0, đ?‘‡] Ă— đ??ľ , where đ?œŒ(đ??´(đ?‘Ą, đ?‘?)) is theresolventset of đ??´(đ?‘Ą, đ?‘?) and

ďƒ• Manuscript received April 19, 2014.

Francis Paul Samuel, Department of Mathematics and Physics, University of Eastern Africa, Baraton,Eldoret 2500 - 30100 Kenya.

Tumaini RukikoLisso, ,Department of Mathematics and Physics, University of Eastern Africa, Baraton,Eldoret 2500 - 30100 Kenya.

Kayiita Zachary Kaunda, Department of Mathematics and Physics, University of Eastern Africa, Baraton,Eldoret 2500 - 30100 Kenya.

 sup ď Ł (ď ą )

k j 1

R(ď Ź : A(t j , b j )) ≤M(Îťâˆ’Ď‰)−kforÎť >ω every finite

sequences 0 ≤ đ?‘Ą1 ≤ đ?‘Ą2 ≤. . . ≤ đ?‘Ąđ?‘˜ ≤ đ?‘‡, đ?‘?đ?‘— ∈ đ??ľ.

Definition 2.3. đ??żđ?‘’đ?‘Ą đ?‘†đ?‘Ą, đ?‘?(đ?‘ ), đ?‘ ≼ 0be theC0- semigroup generatatedby đ??´(đ?‘Ą, đ?‘?), (đ?‘Ą, đ?‘?) ∈ đ??ź Ă— đ??ľ .A subspace đ?‘Œ of đ?‘‹ iscalled đ??´(đ?‘Ą, đ?‘?) -admissibleif đ?‘Œ isinvariantsubspaceofđ?‘†đ?‘Ą, đ?‘?(đ?‘ )andtherestrictionofđ?‘†đ?‘Ą, đ?‘?(đ?‘ )tođ?‘Œisa C0-semigroup inđ?‘Œ.

Let đ??ľ ⊂ đ?‘‹beasubset ofX such thatforevery (đ?‘Ą, đ?‘?) ∈ [0, đ?‘‡] Ă— đ??ľ, đ??´(đ?‘Ą, đ?‘?) istheinfinitesimal generator ofaC0-semigroup đ?‘†đ?‘Ą, đ?‘?(đ?‘ ), đ?‘ ≼ 0onđ?‘‹. Wemakethe followingassumptions: (E1) Thefamily{đ??´(đ?‘Ą, đ?‘?)}, (đ?‘Ą, đ?‘?) ∈ [0, đ?‘‡] Ă— đ??ľisstable.

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Existence Solutions for Quasilinear Evolution Integrodifferential Equations with Infinite Delay

(E2) đ?‘Œ is đ??´(đ?‘Ą, đ?‘?) − admissible for (đ?‘Ą, đ?‘?) ∈ [0, đ?‘‡] Ă— đ??ľ andthe family ofparts {đ??´Ëœ(đ?‘Ą, đ?‘?)}, (đ?‘Ą, đ?‘?) ∈ [0, đ?‘‡] Ă— đ??ľ đ??´Ëœ(đ?‘Ą, đ?‘?)ofđ??´(đ?‘Ą, đ?‘?)inđ?‘Œ ,isstable inđ?‘Œ .

(E3) For (đ?‘Ą, đ?‘?) ∈ [0, đ?‘‡] Ă— đ??ľ, đ??ˇ(đ??´(đ?‘Ą, đ?‘?)) ⊃ đ?‘Œ, đ??´(đ?‘Ą, đ?‘?) isa bounded linear operator from đ?‘Œ to đ?‘‹and đ?‘Ą → đ??´(đ?‘Ą, đ?‘?)iscontinuousin theđ??ľ(đ?‘Œ, đ?‘‹)norm

u (t )  U u (t , 0)ď Ś (0)  ďƒ˛ U u (t , s)[ f ( s, us ) (2.1) t

0

 ďƒ˛ k ( s  ď ´ ) g (ď ´ , uď ´ )]ds s

. for everyđ?‘? ∈ đ??ľ.

0

u0  ď Ś on [ ď ­ , 0].

(E4) There isaconstantđ??ż > 0suchthat

A(t , b1 )  A(t , b2 ) Y  X  L b1  b2

Next weprovethe existence of local classical solutions of the quasilinear problem (1.1)-(1.2).

For amildsolution of(1.1)-(1.2) we mean a function đ?‘˘ ∈ đ??¸đ?‘‡with values in B satisfying the integral equation

X

holdsforevery đ?‘?1, đ?‘?2 ∈ đ??ľ and 0 ≤ đ?‘Ą ≤ đ?‘‡ . Let đ??ľ be a subset of đ?‘‹ and {đ??´(đ?‘Ą, đ?‘?)}, (đ?‘Ą, đ?‘?) ∈ [0, đ?‘‡] Ă— đ??ľ be a family of operators satisfying the conditions (E1)-E4). If đ?‘˘ ∈ đ??ś([0, đ?‘‡] âˆś đ?‘‹)has value sinBthen there is a unique evolution system đ?‘ˆđ?‘˘(đ?‘Ą, đ?‘ ),0 ≤ đ?‘ ≤ đ?‘Ą ≤ đ?‘‡, in đ?‘‹ satisfying, (see [13,Theorem 5.3.1and Lemma 6.4.2, pp. 135,201-202]

(i) U u (t , s) ď‚Ł Meď ˇ (t  s ) for 0 ď‚Ł s ď‚Ł t ď‚Ł T .

Afunction u ∈ET suchthatu(t)∈Y∊Bfor 1 t∈(0,T],u∈C ((0,T]:X)and satisfies (1.1)-(1.2) in X is calleda classical solution of(1.1)-(1.2) on[0,T], where đ?‘˘ ∈ đ??ś1(0, đ?‘‡] âˆś đ?‘‹),space of all continuously differentiable functions from [0, đ?‘‡] to đ?‘‹ and đ?‘Œ isa đ??´(đ?‘Ą, đ?‘?) − admissible subspace ofX. Further there exists a constantđ??¸0 > 0such that forevery đ?‘˘, đ?‘Ł ∈ đ??ś([0, đ?‘‡] âˆś đ?‘‹)with values inB and every w∈Y,wehave

U u (t , s) w  U v (t , s) w  E0 w Y

where M and ω are stability constants.  U u (t , s) w  A( s, u ( s))U u (t , s) w for ď‚śt w ďƒŽ Y , for 0 ď‚Ł s ď‚Ł t ď‚Ł T . ď‚ś (iii) U u (t , s) w  U u (t , s) A( s, u ( s)) w for ď‚śs w ďƒŽ Y , for 0 ď‚Ł s ď‚Ł t ď‚Ł T . (ii)

ďƒ˛

t

s

u (ď ´ )  v(ď ´ ) dď ´ . (2.2)

III. EXISTENCE RESULTS Inthissectionwe provetheexistenceanduniqueness resultforaclassicalsolutionto(1.1)-(1.2). Letđ?œ‘Ëœ ∈ đ??¸đ?‘‡begivenbyđ?œ‘Ëœ(đ?‘Ą) = đ?œ‘(đ?‘Ą)forđ?‘Ą ∈ [âˆ’Âľ, 0]andđ?œ‘Ëœ(đ?‘Ą) = đ?œ‘(0)forđ?‘Ą ∈ [0, đ?‘‡].Denote

Br (ď Ś (0))  {x ďƒŽ X : x  ď Ś (0) ~ ďƒŚ~ďƒś B2 r ďƒ§ ď Ś0 ďƒˇ  {ď Ł ďƒŽ E0 : ď Ł  ď Ś0 ďƒ¨ ďƒ¸

(E5) For everyđ?‘˘ ∈ đ??ś([0, đ?‘‡]: đ?‘‹)satisfying đ?‘˘(đ?‘Ą) ∈ đ??ľ for0 ≤ đ?‘Ą ≤ đ?‘‡,wehave

X

ď‚Ł r}, ď‚Ł 2r.

E0

Theorem2.1Let đ??ľ and đ?‘‰ beopensubsetsof đ?‘‹ and đ??¸0 respectivelyand thefamilyđ??´(đ?‘Ą, đ?‘?)oflinear operatorsforđ?‘Ą ∈ [0, đ?‘‡], đ?‘? ∈ đ??ľđ?‘&#x;(đ?œ‘(0)) satisfyingassumptions (E1)–(E9)andđ??´(đ?‘Ą, đ?‘?)đ?œ‘(0) ∈ đ?‘Œwith

đ?‘ˆđ?‘˘(đ?‘Ą, đ?‘ )đ?‘Œ ⊂ đ?‘Œ, 0 ≤ đ?‘ ≤ đ?‘Ą ≤ đ?‘‡and đ?‘ˆđ?‘˘(đ?‘Ą, đ?‘ )isstrongly continuousinđ?‘Œfor0 ≤ đ?‘ ≤ đ?‘Ą ≤ đ?‘‡

A(t , b)ď Ś (0) Y ď‚Ł C A , C A  0

(E6) Every closedconvexand bounded subsetofđ?‘Œ is alsoclosedinđ?‘‹.

forall (đ?‘Ą, đ?‘?) ∈ [0, đ?‘‡] Ă— đ??ľ. There existsapositiveconstant đ?‘‡0 suchthat thequasilinearproblem(1.1)-(1.2) hasauniqueclassical solution.

Furtherweassumethat (E7) đ?‘“ âˆś [0, đ?‘‡] Ă— đ??¸0to đ?‘‹ is continuousand there exist constantsđ??šđ??ż > 0and đ??š0 > 0such that | đ?‘“ đ?‘Ą, đ?œ‘1 − đ?‘“ đ?‘Ą, đ?œ‘2 | ≤ đ??šđ??ż ( đ?‘Ą − đ?‘ + | đ?œ‘1 − đ?œ‘2 |

U u (t , s)

B (Y )

ď‚Ł K1 ,

Takeđ?‘&#x; > 0suchthatđ??ľđ?‘&#x;(đ?œ‘(0)) ⊂ đ??ľđ?‘Žđ?‘›đ?‘‘đ??ľ2đ?‘&#x;(đ?œ‘0 ) ⊂ đ?‘‰.

F0  max f (t , uo ) .

(E8) đ?‘”: [0, đ?‘‡] Ă— đ??¸0to đ?‘‹ iscontinuousand thereexist constantsđ??şđ??ż > 0and đ??ş0 > 0such that

ďƒ˛

t

0

g (t , ď Ś1  g ( s, ď Ś2 ) ds ď‚Ł GL  t  s  ď Ś1  ď Ś2 X



E0

Proof: Letđ?‘†beanonemptyclosedsubset ofđ??ś([0, đ?‘‡]: đ?‘‹) definedby

,

đ?‘† = đ?œ“ ∈ đ??śđ?‘‡0 âˆś đ?œ“0 = đ?œ‘, đ?‘“đ?‘œđ?‘&#x;đ?‘Ą ∈ âˆ’Âľ, 0 , đ?œ“ đ?‘Ą ∈ đ??ľđ?‘&#x; đ?œ‘ 0 .

G0  max ďƒ˛ g ( s, uo ) ds. t

0

(E9) Thereal-valued functionđ?‘˜iscontinuouson[0, đ?‘‡] andthere existsapositiveconstantđ??žđ?‘‡ suchthat

We easilydeduce thatđ?‘†is a closed, convex and bounded subsetofđ??śđ?‘‡0. Take đ?œ“ ∈ đ?‘†.Nowfor đ?œƒ ∈ [âˆ’Âľ, 0],wehavethefollowingtwocases.

k (t ) ≤đ??žđ?‘‡ forđ?‘Ą ∈ [0, đ?‘‡].Wenote thatthe condition (E6)isalwayssatisfied ifđ?‘‹andđ?‘Œ are reflexiveBanach spaces.

Case(i): Ifđ?‘Ą + đ?œƒ ≤ 0wehave

278

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International Journal of Engineering and Technical Research (IJETR) ISSN: 2321-0869, Volume-2, Issue-4, April 2014 ~

~

ď š t ď€¨ď ą   ď Ś 0 (ď ą )

ďƒ˛

 ď š (t  ď ą )  ď Ś (ď ą ) X

X

0

ď‚Ł K1 ďƒ˛ [ f ( s, us )  f ( s, u0 )  f ( s, u0 ) t

X

0

 Lď Ś T0 ď‚Ł r.

~

 ď š (t  ď ą )  ď Ś (ď ą ) X

s

0

k ( s  ď ´ )[ g (ď ´ , uď ´ )  g (ď ´ , u0 )  g (ď ´ , u0 )]dď ´ ds

ď‚Ł K1 ďƒ˛ [ f ( s, us )  f ( s, ď Ś ) t

X

0

 ď Ś (t  ď ą )  ď Ś (0)

 KT ďƒ˛

X

 ď Ś (0)  ď Ś (ď ą )

ďƒ˛



Case(ii): Ifđ?‘Ą + đ?œƒ ≼ 0wehave ~

s

0

 ď Ś (t  ď ą )  ď Ś (ď ą )

ď š t ď€¨ď ą   ď Ś0 (ď ą )

U u (t , s)[ f ( s, us )  ďƒ˛ k ( s  ď ´ ) g ( s, uď ´ )dď ´ ] ds

t

s

0

 g (ď ´ , uď ´ )  g (ď ´ , ď Ś ) dď ´ ]ds

r ď‚Ł .

2

ď‚Ł r  Lď Ś T0 ď‚Ł 2r.

(since−đ?œƒ ≤ đ?‘Ą ≤ đ?‘‡0).

Using the result for Thus, forđ?œ“ ∈ đ?‘†, đ?œ“đ?‘Ą ∈ đ??ľ2đ?‘&#x;(đ?œ‘).Defineđ??ť: đ?‘† → đ?‘†by

Hu (t )  ď Ś (0)

ď‚Ł

X

r r   r.

2 2

u, v ďƒŽ S , we consider

Hu (t )  Hv(t )  U u (t , 0)ď Ś (0)  U v (t , 0)ď Ś (0)  ďƒ˛ U u (t , s)[ f ( s, us )  ďƒ˛ k ( s  ď ´ ) g ( s, uď ´ )dď ´ ] ds

Hu (t )  ď Ś (0)  U u (t ,0)ď Ś (0)  ď Ś (0)

t

s

0

0

 ďƒ˛ U v (t , s)[ f ( s, vs )  ďƒ˛ k ( s  ď ´ ) g ( s, vď ´ )dď ´ ] ds

 ďƒ˛ U u (t , s)[ f (s, us )  ďƒ˛ k (s  ď ´ ) f (s, uď ´ )dď ´ ]ds 0

u ďƒŽ S , us ďƒŽ B2r (ď Ś ). Thus, for u ďƒŽ S

So, H is well defined for

Firstweshowthatđ??ť iswelldefined andđ??ťđ?‘˘(0) = đ?œ‘(0).Forđ?‘Ą ≼ 0,wehave

0

t ď‚ł 0, we get

and

ďƒŹU u (t , 0)ď Ś (0) ďƒŻ t ďƒŻďƒŻď€Ť ďƒ˛ U u (t , s)[ f ( s, us ) Hu (t )  ďƒ­ 0s ďƒŻď€Ť ďƒ˛ k ( s  ď ´ ) g ( s, uď ´ )dď ´ ]ds, t ďƒŽ [0, T0 ] ďƒŻ 0 t ďƒŽ [ ď ­ , 0]. ďƒŻďƒŽď Ś (t ),

s

 f ( s , u0 )

 K1[2r ( FL  KT GL )  F0  KT G0 ]T0

X

 r  L(ď€­ď ą ) ď‚Ł r  Lď Ś t

t

X

t

s

0

0

Let

Taking thenorm, weget

Hu (t )  ď Ś (0) ď‚Ł U u (t , 0)ď Ś (0)  ď Ś (0)

I1  U u (t , 0)ď Ś (0)  U v (t , 0)ď Ś (0)

X

 ďƒ˛ U u (t , s )[ f ( s, us )  ďƒ˛ k ( s  ď ´ ) f ( s, uď ´ )dď ´ ] ds

ď‚Ł E0 ď Ś (0)

Integrating(ii),weobtain

ď‚Ł E0 ď Ś (0)

t

s

0

0

U u (t ,0)ď Ś (0)  ď Ś (0)  ďƒ˛ U u (t , s) A( s, u ( s))ď Ś ( s)ds. t

ďƒ˛

t

0

X

X

u ( s)  v( s) ds X

u  v T0 .

Also let

0

Thus, we have U u (t ,0)ď Ś (0)  ď Ś (0)

ď‚Ł ďƒ˛ U u (t , s) A( s, u ( s)) t

X

0

X

ď Ś ( s) ds.

X

ď‚Ł C A K1T0 r ď‚Ł .

2 Also we have

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Existence Solutions for Quasilinear Evolution Integrodifferential Equations with Infinite Delay

I 2  ďƒ˛ U u (t , s )[ f ( s, us )  t

0

ďƒ˛

s

0

u ' (t )  A(t , u )u (t )  f (t , ut )  ďƒ˛ k (t  s) g (s, us )ds (3.1) t

k ( s  ď ´ )[ g ( s, uď ´ )dď ´ ]

0

u (0)  ď Ś (0).

U u (t , s )[ f ( s, vs )  ďƒ˛ k ( s  ď ´ )[ g ( s, vď ´ )dď ´ ] s

(3.2)

0

U v (t , s )[ f ( s, vs )  ďƒ˛ k ( s  ď ´ )[ g ( s, uď ´ )dď ´ ]ds

Denote đ??´Ëœ(đ?‘Ą) =đ??´(đ?‘Ą, đ?‘˘(đ?‘Ą)), đ??šËœ(đ?‘Ą)=đ?‘“ (đ?‘Ą, đ?‘˘đ?‘Ą) and đ??şËœ(đ?‘Ą)=đ?‘”(đ?‘Ą, đ?‘˘đ?‘Ą)ds,then equation (3.1)canbewrittenas

ď‚Ł K1 ďƒ˛ f ( s, us )  f ( s, vs )

u ' (t )  A(t , u)u(t )  F (t )  ďƒ˛ k (t  s) G(s)ds

s

0

~

t

~

0

0

u (0)  ď Ś (0).

 ďƒ˛ k ( s  ď ´ )[ g ( s, uď ´ )  g (ď ´ , vď ´ )]dď ´ ] ds s

0

s

0

0

f (t , ď Ł )  f ( s, ď Ł )

ď‚´ďƒ˛ u (ď ´ )  v(ď ´ ) dď ´ ]ds t

t

ďƒ˛

0

and

s

X

ď‚Ł K1[ FL ďƒ˛ us  vs ds  KT ďƒ˛ GL us  vs ds ]

(3.4)

X

g (t , ď Ł )  g ( s, ď Ł )

 FL t  s , X

ds  GL t  s

k (t ) ď‚Ł KT .

0

 E0 [2r ( FL  KT GL )

Henceforeach∈>0there existsaδ>0suchthatif |t−s|≤δimpliesthat

 F0 KT G0 ] u  v T02

f (t , ď Ł )  f ( s, ď Ł )

ď‚Ł K1[ FL ďƒ˛ sup u ( s  ď ą  v( s  ď ą ) ds t

ď ą

ďƒ˛

X

 KT GL ďƒ˛ sup u ( s  ď ą  v( s  ď ą ) ds t

0

t

0

t

0

(3.3)

whereuistheunique solution fixedpointofH inS. Nowweassumethat(E7)-(E9)wehave

 E0 ďƒ˛ [ f ( s, vs )  ďƒ˛ k ( s  ď ´ ) g (ď ´ , uď ´ )dď ´ t

~

t

ď ą

t

0

X

X

g (t , ď Ł )  g ( s, ď Ł )

ď‚Ł ď Ľ, X

ds ď‚Ł ď Ľ .

 K1 FLT0 u  v  K1KT GLT0 u  v

Thus, đ?‘“(đ?‘Ą, đ?œ’) ∈ đ??¸đ?‘‡0and đ?‘”(đ?‘Ą, đ?œ’) ∈ đ??¸đ?‘‡0forfixedđ?œ’.HencefromPazy[[13]Theorem 5.5.2],we getaunique function đ?‘Ł ∈ đ??ś1((0, đ?‘‡0]; đ?‘‹)satisfying (3.3)-(3.4) inđ?‘‹andđ?‘Łgivenby

 E0 [2r ( FL  KT GL )  F0

v(t )  U u (t , 0)ď Ś (0)  ďƒ˛ U u (t , s)[ f ( s, us

 E0 [2r ( FL  KT GL )  F0  KT G0 ] u  v T02

t

0

 KT G0 ]T02 u  v

 ďƒ˛ k ( s  ď ´ ) g ( s, uď ´ )dď ´ ]ds, t ďƒŽ [0, T0 ], s

 ( K1[ FL  KT GL ]  T0 E0 [2r ( FL  KT GL )

0

 F0  KT G0 )T0 u  v .

wheređ?‘ˆđ?‘˘ (đ?‘Ą, đ?‘ ),0 ≤ đ?‘ ≤ đ?‘Ą ≤ đ?‘‡0isthe evolution system generated by the family đ??´(đ?‘Ą, đ?‘˘(đ?‘Ą)), đ?‘Ą ∈ [0, đ?‘‡0]. The uniqueness of đ?‘Ł impliesthat đ?‘Ł ≥ đ?‘˘ on [0, đ?‘‡0]. Thus đ?‘˘ is a unique local classicalof(1.1)-(1.2).

Hence,

I1  I 2  Hu (t )  Hv(t ) ď‚Ł ( E0 ď Ś (0)

X

REFERENCES

 K1[ FL  KT GT ]

[1]

 T0 E0 [2r ( FL  KT GT )

[2]

 F0  KT G0 ])T0 u  v ď‚Ł ď —T0 u  v ď‚Ł

1 u v n

ET0

[3] ET0

[4]

.

Thus đ??ť isacontractionfromđ?‘†tođ?‘†.So,bythe Banach contract ionmapping theorem, đ??ť has aunique fixedpoint đ?‘˘ ∈ đ?‘†which satisfies the integral equation. Hence it is amild solution of(1.1)-(1.2). Now, we consider the following evolution equation

[5]

[6] [7]

[8]

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International Journal of Engineering and Technical Research (IJETR) ISSN: 2321-0869, Volume-2, Issue-4, April 2014 [9]

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