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 t
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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