Stepanov-like Pseudo Almost Automorphic. Solutions to Nonautonomous Neutral. Partial Evolution Equations

Transcription

1 Journal of Applied Mahemaic & Bioinformaic, vol.2, no.3, 2012, ISSN: (prin), (online) Scienpre Ld, 2012 Sepanov-like Peudo Almo Auomorphic Soluion o Nonauonomou Neural Parial Evoluion Equaion Mohamed Ziane 1 and Charaf Benouda 2 Abrac In hi paper, upon making ome uiable aumpion uch a he Acquiapace-Terreni condiion and exponenial dichoomy, we obain new exience and uniquene heorem of peudo almo auomorphic oluion o ome nonauonomou neural parial evoluion equaion. Some example are preened o illurae he main finding. Mahemaic Subjec Claificaion: 43A60, 34K27, 34K40, 37B55 Keyword: Evoluion family, Exponenial dichoomy, Fixed poin heorem, Sepanov-like peudo almo auomorphiy 1 Univerié Ibn Tofaïl, Faculé de Science, Déparemen de Mahémaique, Laboraoire d An. Mah e GNC, B.P. 133, Kénira 1400, Maroc, zianem@gmail.com 2 Univerié Ibn Tofaïl, Faculé de Science, Déparemen de Mahémaique, Laboraoire d An. Mah e GNC, B.P. 133, Kénira 1400, Maroc, benoudach@yahoo.fr Aricle Info: Received : Ocober 18, Revied : December 2, 2012 Publihed online : December 30, 2012

2 194 Sepanov-like Peudo Almo Auomorphic Soluion... 1 Inroducion In hi paper, we coninue he inveigaion of [6, 7, 12, 13, 14]. Tha i, we aim o eablih he exience and uniquene heorem of peudo almo auomorphic mild oluion o nonauonomou neural parial evoluion equaion wih Sepanov-like peudo almo auomorphic erm d [u()+f(,u())] = A()[u()+f(,u())]+h(), R, (1) d d [u()+f(,u())] = A()[u()+f(,u())]+g(,u()), R, (2) d where A(.) i a given family of cloed linear operaor on D(A()) aifying he well-known Acquiapace-Terreni condiion, f : R X X i peudo almo auomorphic, h : R X and g : R X X are Sepanov-like peudo almo auomorphic for p > 1 and coninuou, where X i a Banach pace. Anoher aim in hi paper i o inveigae peudo almo auomorphic mild oluion o perurbed nonauonomou neural parial evoluion equaion d [u()+f(,bu())] = A()[u()+f(,Bu())]+g(,Cu()), R, (3) d where B, C are bounded linear operaor. In recen year, he heory of almo auomorphy and i variou exenion have araced a grea deal of aenion due o heir ignificance and applicaion in area uch a phyic, mahemaical biology, conrol heory, and oher. The concep of peudo almo auomorphy, which i he cenral iue in hi work, wa inroduced in he lieraure by Xiao, Liang and Zhang [21] a a naural generalizaion of boh he claical concep of almo auomorphy and ha of peudo almo periodiciy. In [17], N Guérékaa and Pankov inroduced he noion of Sepanov-like almo auomorphic funcion. Uing hi new concep, he auor in [9, 10] ydied he compoiion of Sepanov-like almo auomorphic funcion and obained he exience and uniquene of almo auomorphic oluion o ome abrac differenial equaion. Recenly, Diagana [8] udied he baic properie of a new cla of funcion, called Sepanov-like peudo almo auomorphic funcion, which generalize boh he Sepanov-like almo auomorphy and peudo almo auomorphy.

3 M. Ziane and C. Benouda 195 We now urn o a ummary of hi work. In Secion 2, we review he concep of almo auomorphy and i generaliaion uch a peudo almo auomorphy and Sepanov-like almo auomorphy. Secion 3 and 4 are devoed o he exience and uniquene of peudo almo auomorphic oluion o nonauonomou neural parial evoluion equaion and perurbed nonauonomou neural parial evoluion equaion, repecively. In Secion 5, we provide ome example o illurae our main reul. 2 Preliminarie In hi ecion, we fix noaion and collec ome preliminary fac ha will be ued in he equel. hroughou hi paper, N, R and C and for he e of poiiveineger, realandcomplexnumber, (X,. ),(Y,. Y )andforbanach pace, B(X, Y) denoe he Banach pace of all bounded linear operaor from X ino Y equipped wih naural opology. If Y = X i i imply denoe by B(X). 2.1 peudo almo auomorphy Le C(R,X) denoe he collecion of coninuou funcion from R ino X. Le BC(R, X) denoe he Banach pace of all X-valued bounded coninuou funcion equipped wih he up norm u := up R u() for each u BC(R,X). Similarly, C(R Y, X)) denoe he collecion of all joinly coninuou funcion from R Y ino X, BC(R Y,X) denoe he collecion of all joinly bounded coninuou funcion f : R Y X. Definiion 2.1. [5, 16, 25] A funcion f C(R,X) i aid o be almo auomorphic if for every equence of real number ( n) n here exi a ubequence ( n ) n uch ha lim limf(+ n m ) = f() for each R. m n hi limi mean ha g() := lim n f(+ n ) i well defined for each R

4 196 Sepanov-like Peudo Almo Auomorphic Soluion... and f() = lim n g( n ) for each R. The collecion of all uch funcion will be denoed by AA(X). Theorem 2.2. [16] Aume f,g : R X are almo auomorphic and λ i any calar. Then he following hold rue: (1) f +g,λf,f τ () := f(+τ) and f() := f( ) are almo auomorphic. (2) The range R f of f i precompac, o f i bounded. (3) If {f n } i a equence of almo auomorphic funcion and f n f uniformly on R, hen f i almo auomorphic. Definiion 2.3. [12] A funcion f C(R R,X) i called bi-almo auomorphic if for every equence of real number ( n) n we can exrac a ubequence ( n ) n uch ha g(,) := limf(+ n,+ n ) i well defined for each n, R and f(,) = limg( n, n ) for each, R. The collecion of n all uch funcion will be denoed by baa(c(r R,X)). We e AA 0 (X) := {f BC(R,X) : lim T 1 2T T T f(σ) dσ = 0}. Definiion 2.4. [5] A funcion f BC(R,X) i aid o be peudo almo auomorphic if i can be decompoed a f = g + ϕ where g AA(X) and ϕ AA 0 (X). The e of all uch funcion will be denoed by PAA(X). Theorem 2.5. [21] Ifone equippaa(x) wihheup norm, henpaa(x). urn ou o be a Banach pace. Theorem 2.6. [12] Le u PAA(Y),B B(Y,X). If for each R,Bu() = v(), hen v PAA(X). Definiion 2.7. [15] A funcion f C(R Y,X) i aid o be almo auomorphic if f(,u) i almo auomorphic in R uniformly for all u K, where K i any bounded ube of Y. The collecion of all uch funcion will be denoed by AA(R Y,X).

5 M. Ziane and C. Benouda 197 We e AA 0 (R Y,X) := {f BC(R Y,X) : lim T 1 2T T T f(σ,u) dσ = 0 uniformly for u in any bounded ube of Y}. Definiion 2.8. [15] A funcion f C(R Y,X) i aid o be peudo almo auomorphic if i can be decompoed a f = g +ϕ where g AA(R Y,X) and ϕ AA 0 (R Y,X). The e of all uch funcion will be denoed by PAA(R Y,X). Theorem 2.9. [8, 12] Aume F PAA(R X). Suppoe ha F(,u) i Lipchiz in u X uniformly in R, in he ene ha here exi L > 0 uch ha F(,u) F(,v) L u v for all R,u,v X If Φ(.) PAA(X), hen F(.,Φ(.)) PAA(X). 2.2 Sepanov-like almo auomorphy Le L p (R,X) denoe he pace of all clae of equivalence ( wih repec o he equaliy almo everywhere on R ) of meaurable funcion f : R X uch ha f L p (R). Le L p loc (R,X) denoe he pace of all clae of equivalence of meaurable funcion f : R X uch ha he rericion of every bounded ubinerval of R i in L p (R,X). Definiion [8, 17, 19] The Bochner ranform f b (,), R, [0,1], of a funcion f : R X i defined by f b (,) := f(+). Remark [19] A funcion ϕ(,), R, [0,1], i he Bochner ranform of a cerain funcion f,ϕ(,) = f b (,), if and only if ϕ(+τ, τ) = ϕ(,) for all R, [0,1] and τ [ 1,]. Definiion [19] The Bochner ranform F b (,,u), R, [0,1], u X, of a funcion F : R X X i defined by F b (,,u) := F(+,u) for each u X.

6 198 Sepanov-like Peudo Almo Auomorphic Soluion... Definiion [19] Le p [1, ). The pace BS p (X) of all Sepanov bounded funcion, wih he exponen p, coni of all meaurable funcion f : R X uch ha f b L (R,L p (0,1;X)). Thi i a Banach pace wih he norm f S p = f b L (R,L p ) = up R ( +1 f(τ) p dτ )1 p Definiion [17] The pace S p AA(X) of Sepanov-like almo auomorphic funcion (or S p almo auomorphic funcion), coni of all f BS p (X) uch ha f b AA(L p (0,1;X)). Tha i, a funcion f L p loc (R,X) i aid o be Sepanov-like almo auomorphic if i Bochner ranform f b : R L p (0,1;X) i almo auomorphic in he ene ha for every equence of real number ( n) n here exi a ubequence ( n ) n and a funcion g L p loc (R,X) uch ha ( 1 f(+ n ) g() p d )1 p 0 and ( 1 g( n ) f() p d )1 p 0 0 a n poinwie on R Sepanov-like peudo almo auomorphy Definiion [8] A funcion f BS p (X) i aid o be Sepanovlike peudo almo auomorphic ( or S p peudo almo auomorphic ) if i can be decompoed a f = g + ϕ where g b AA(L p (0,1;X)) and ϕ b AA 0 (L p (0,1;X)). The e of all uch funcion will be denoed by S p PAA(X). Lemma [8] If f PAA(X), hen f S p PAA(X) for each 1 p <. In oher word, PAA(X) S p PAA(X). Lemma [8] The pace S p PAA(X) equipped wih he norm. S p i a Banach pace.

7 M. Ziane and C. Benouda 199 Definiion [8] A funcion F : R Y X wih F(.,u) L p (R,X) for each u Y, i aid o be Sepanov-like peudo almo auomorphic ( or S p peudo almo auomorphic ) if i can be decompoed a F = G+Φ where G b AA(R L p (0,1;Y)) and Φ b AA 0 (R L p (0,1;Y)). The collecion of uch funcion will be denoed by S p PAA(R Y). Theorem [8] Aume F S p PAA(R X). Suppoe ha F(,u) i Lipchiz in u X uniformly in R, in he ene ha here exi L > 0 uch ha F(,u) F(,v) L u v for all R,u,v X IfΦ(.) S p PAA(X) andk := {Φ() : R} icompacinx, henf(.,φ(.)) S p PAA(X). 2.4 Evoluion family and exponenial dichoomy Definiion [18] A family of bounded linear operaor (U(,)) on a Banach pace X i called a rongly coninuou evoluion family if (1) U(,) = U(,r)U(r,) and U(,) = I for all r and,r, R (2) he map (,) U(,)x i coninuou for all x X, and, R, Definiion [18, 11] An evoluion family (U(,)) on a Banach pace X i called hyperbolic ( or ha exponenial dichoomy ) if here exi projecion P(), R, uniformly bounded and rongly coninuou in, and conan M > 0,δ > 0 uch ha (1) U(,)P() = p()u(,) for and, R (2) he rericion U Q (,) : Q()X Q()X of U(,) i inverible for ( and we e U Q (,) := U(,) 1 ) (3) U(,)P() Me δ( ), U Q (,)Q() Me δ( ) for and, R. Here and below we e Q := I P.

8 200 Sepanov-like Peudo Almo Auomorphic Soluion... Definiion [18, 3] Given an hyperbolic evoluion family, we define i o-called Green funcion by U(,)P(), for,, R Γ(,) := U Q (,)Q(), for <,, R (4) 3 Neural parial evoluion equaion Throughou hi paper, we make he following baic aumpion (H 1 ) he family of cloed linear operaor A(), for R, on X wih domain D(A())(poibly no denely defined) aify he o-called Acquiapace- Terreni condiion; namely, here exi conan λ 0 0, θ ( π 2,π), M 1,M 2 0, and α,β (0,1] wih α+β > 1 uch ha and Σ θ {0} ρ(a() λ 0 ) λ, R(λ,A() λ 0 ) M 1 1+ λ (A() λ 0 )R(λ,A() λ 0 )[R(λ 0,A()) R(λ 0,A())] M 2 α λ β (5) for, R, λ Σ θ := {λ C {0} : argλ θ} (H 2 ) The evoluion family (U(,)) generaed by A() ha an exponenial dichoomy wih conan M > 0,δ > 0, dichoomy projecion P(), R, and Green funcion Γ(, ). (H 3 ) Γ(,) baa(r R,B(X)) (H 4 ) f : R X X i peudo almo auomorphic and Lipchiz wih repec o he econd argumen uniformly in he fir argumen in he ence ha here exi K f uch ha for all R and u,v X f(,u) f(,v) K f u v (H 5 ) g : R X X i Sepanov-like peudo almo auomorphic for p > 1,

9 M. Ziane and C. Benouda 201 joinly coninuou and Lipchiz wih repec o he econd argumen uniformly in he fir argumen in he ence ha here exi K g uch ha g(,u) g(,v) K g u v for all R and u,v X. LeumenionhainhecaewhenA()haaconandomainD(A()) = D, i i well-known [4, 20] ha equaion (5) can be replaced wih he following (H 1) There exi conan M 2 and 0 < α 1 uch ha (A() A())R(λ 0,A(r)) M 2 α, for,,r R (6) Le u alo menion ha (H 1 ) wa inroduced in he lieraure by Acquiapace and Terreni in [1, 2]. Among oher hing, from [3, Theorem 2.3]; ee alo [1, 23, 24]; i enure ha he family (A()) generae a unique rongly coninuou evoluion family (U(,)) on X. Definiion 3.1. A mild oluion o Equaion (1) i a coninuou funcion u : R X aifying u() = f(,u())+u(,)[u()+f(,u())]+ for all and all R. Now, we ae our fir reul. U(, σ)h(σ)dσ (7) Theorem 3.2. Aume ha (H 1 ) (H 4 ) hold and h S p AA(X) C(R,X) for p > 1. if K f < 1, hen here exi a unique oluion u PAA(X) of equaion (1) uch ha u() = f(,u())+ U(, σ)p(σ)h(σ)dσ U Q (,σ)q(σ)h(σ)dσ, R (8) Lemma 3.3. Suppoe ha (H 1 ) (H 4 ) hold. If h S p AA(X) C(R,X) for p > 1, hen he operaor Λ defined by (Λu)() = map PAA(X) ino i elf. U(,σ)P(σ)h(σ)dσ U Q (,σ)q(σ)h(σ)dσ, R

10 202 Sepanov-like Peudo Almo Auomorphic Soluion... Proof of Lemma: The proof i imilar o ha one of [12, Theorem 3.3], o, we omi i. Proof of Theorem: To prove ha u aifie equaion (7) for all, all R, we le u() = f(,u())+ U(, σ)p(σ)h(σ)dσ Muliply boh ide of (9) by U(,) for all, hen U(,)u() = U(,)f(,u())+ U Q (,σ)q(σ)h(σ)dσ = U(,)f(,u())+ U(,σ)P(σ)h(σ)dσ U Q (,σ)q(σ)h(σ)dσ U(, σ)p(σ)h(σ)dσ U(, σ)p(σ)h(σ)dσ = U(,)f(,u())+u()+f(,u()) Hence u i a mild oluion o equaion (7). U Q (,σ)q(σ)h(σ)dσ (9) U Q (,σ)q(σ)h(σ)dσ U(, σ)h(σ)dσ In PAA(X) define he operaor Γ : PAA(X) C(R,X) by eing (Γu)() = f(,u())+ U(,σ)P(σ)h(σ)dσ U Q (,σ)q(σ)h(σ)dσ. From previou aumpion one can eaily ee ha Γu i well defined and coninuou. Moreover, from Theorem (2.9) and Lemma (3.3) we deduce ha Γu PAA(X), ha i Γ : PAA(X) PAA(X). I remain o prove ha Γ i a ric conracion on PAA(X). For u,v PAA(X) we ge (Γu)() (Γv)() = f(,u()) f(,v()) K f u() v() K f u v Since K f < 1, i follow ha Γ i a ric conracion, and by he Banach fixedpoin principle, here exi a unique mild oluion o (1) which obviouly i peudo almo auomorphic.

11 M. Ziane and C. Benouda 203 Definiion 3.4. A mild oluion o Equaion (2) i a coninuou funcion u : R X aifying u() = f(,u())+u(,)[u()+f(,u())]+ U(, σ)g(σ, u(σ))dσ (10) for all and all R. Now, we ae our econd reul. Theorem 3.5. Aume ha (H 1 ) (H 5 ) hold. If (K f + 2MKg δ ) < 1, hen here exi a unique mild oluion u PAA(X) of equaion (2) uch ha u() = f(,u())+ U(, σ)p(σ)g(σ, u(σ))dσ U Q (,σ)q(σ)g(σ,u(σ))dσ, Proof In PAA(X) define he nonlinear operaor Γ : PAA(X) C(R,X) by eing (Γu)() = f(,u())+ U(, σ)p(σ)g(σ, u(σ))dσ U Q (,σ)q(σ)g(σ,u(σ))dσ, R (i) From previou aumpion one can eaily ee ha Γ i well defined and coninuou. Moreover, le u(.) PAA(X) S p PAA(X), Uing (H 5 ) and compoiion heorem on Sepanov-like peudo almo auomorphic funcion (Theorem (2.19)), we deduce ha g(.,u(.)) S p PAA(X). I i eay o check ha g(.,u(.)) C(R,Y). Applying Lemma (3.3) for h(.) = g(.,u(.)), i follow ha he operaor Γ map PAA(X) ino i elf. (ii) we will how ha Γ : PAA(X) PAA(X) ha a unique fixed poin

12 204 Sepanov-like Peudo Almo Auomorphic Soluion... For u,v PAA(X) we ge Γ(u)() Γ(v)() K f u() v() +M +M e δ( σ) g(σ,u(σ)) g(σ,v(σ)) dσ e δ( σ) g(σ,u(σ)) g(σ,v(σ)) dσ K f u v +MK g +MK g e δ( σ) u(σ) v(σ) dσ (K f + 2MK g ) u v δ e δ( σ) u(σ) v(σ) dσ Since K f + 2MKg < 1, i follow ha Γ i a ric conracion, and by he δ Banach fixed-poin principle, Γ ha a unique fixed poin in PAA(X). (iii) I remain o prove ha u aifie equaion (10) for all, all R, we le u() = f(,u())+ U(,σ)P(σ)h(σ)dσ Muliply boh ide of (11) by U(,) for all, hen U(,)u() = U(,)f(,u())+ U Q (,σ)q(σ)h(σ)dσ = U(,)f(,u())+ U(,σ)P(σ)h(σ)dσ U Q (,σ)q(σ)h(σ)dσ U(, σ)p(σ)h(σ)dσ U(, σ)p(σ)h(σ)dσ = U(,)f(,u())+u()+f(,u()) U Q (,σ)q(σ)h(σ)dσ, U Q (,σ)q(σ)h(σ)dσ U(, σ)h(σ)dσ Hence u i a mild oluion o equaion (10). The proof i complee. (11)

13 M. Ziane and C. Benouda Perurbed nonauonomou neural parial evoluion equaion In hi ecion, we dicu briefly he exience and uniquene of a peudo almo auomorphic mild oluion o perurbed nonauonomou neural parial evoluion equaion (3). For hi, we need he invariance heorem of Sepanovlike peudo almo auomorphic funncion under bounded linear operaor. Theorem 4.1. [12] Le u S p PAA(Y), B B(Y, X). If for each X,Bu() = v(), hen v S p PAA(X). To dicu he exience of peudo almo auomorphic mild oluion o (3) we need o e he following aumpion (H 6 ) Y X i coninuouly embedded, where Y := D(A()) for R ( equipped wih graph norm ). (H 7 ) f : R X Y i peudo almo auomorphic and Lipchiz wih repec o he econd argumen uniformly in he fir argumen in he ence ha here exi K f uch ha f(,u) f(,v) K f u v Y for all R and u,v X. (H 8 ) g : R X Y i Sepanov-like peudo almo auomorphic for p > 1, joinly coninuou and Lipchiz wih repec o he econd argumen uniformly in he fir argumen in he ence ha here exi K g uch ha g(,u) g(,v) K g u v Y for all R and u,v X. (H 9 ) B,C B(Y,X) wih max( B B(Y,X), C B(Y,X) ) = K. Definiion 4.2. A mild oluion o Equaion (3) i a coninuou funcion u : R Y aifying u() = f(,bu())+u(,)[u()+f(,bu())]+ for all and all R. U(, σ)g(σ, Cu(σ))dσ

14 206 Sepanov-like Peudo Almo Auomorphic Soluion... Lemma 4.3. Suppoe ha (H 1 ) (H 3 ) and (H 6 ) (H 9 ) hold. Then he operaor Ξ defined by (Ξu)() = map PAA(Y) ino i elf. U(,σ)P(σ)g(σ,Cu(σ))dσ U Q (,σ)q(σ)g(σ,cu(σ))dσ Proof Le u(.) PAA(Y) S p PAA(Y), from (H 9 ) and Theorem (4.1) i i clear ha Cu(.) S p PAA(Y). Uing (H 8 ) and compoiion heorem on Sepanov-like peudo almo auomorphic funcion (Theorem (2.19)), we deduce ha g(.,cu(.)) S p PAA(Y). I i eay o check ha g(.,cu(.)) C(R,Y). Applying Lemma (3.3) for h(.) = g(.,cu(.)), i follow ha he operaor Ξ map PAA(Y) ino i elf. Now, we ae our hird reul. Theorem 4.4. Under Aumpion (H 1 ) (H 3 ),(H 6 ) (H 9 ), if K (K f + 2MKg ) < 1, hen here exi a unique oluion u PAA(Y) of δ equaion (3) uch ha u() = f(,bu())+ U(, σ)p(σ)g(σ, Cu(σ))dσ U Q (,σ)q(σ)g(σ,cu(σ))dσ Proof The proof i imilar o ha of Theorem (3.5). So we omi i. 5 Applicaion In hi ecion we provide wih wo example o illurae our abrac reul. For ha, we fir inroduce he required background needed in he equel. Throughou he re of hi ecion, we ake X := L 2 ([0,π]) equipped wih i naural opology and le A be he operaor given by Aψ(ξ) := ψ (ξ) 2ψ(ξ), ξ [0,π],ψ D(A), where D(A) = { } ψ L 2 ([0,π]) : ψ L 2 ([0,π]),ψ(0) = ψ(π) = 0.

15 M. Ziane and C. Benouda 207 I i well known [12, 8, 18] ha A i he infinieimal generaor of an analyic emigroup (T()) 0 on X. Furhermore, A ha a dicree pecrum wih eigenvalue of he form n 2 2,n N, and correponding normalized eigenfuncion given by 2 ψ n (ξ) := π in(nξ). In addiion o he above, he following properie hold: (a) {ψ n : n N} i an orhonormal bai for X. (b) for ψ X, T()ψ = n=1 e (n2 +2) ψ,ψ n ψ n and Aψ = n=1 (n2 +2) ψ,ψ n ψ n, for every ψ D(A). (c) T() e π2, for 0. Now, define a family of linear operaor A() by D(A()) = D(A), R A()ψ(ξ) := (A+co( 1 2+in()+in(π) ))ψ(ξ), Then, A() generae an evoluion family (U(,)) uch ha U(,)ψ(ξ) = T( )e co( 1 2+in(σ)+in(πσ) )dσ ψ(ξ) ξ [0,π],ψ D(A), and U(,) e (π2 +1)( ), I i eay o verify ha A() aify (H 1 ) (H 3 ) wih M = 1,δ = π Example 5.1. Le u inveigae peudo almo auomorphic mild oluion o nonauonomou evoluion equaion 2 [u(,x)+f(,u(,x))] = x [u(,x)+f(,u(,x))] 2 +( 2+co( 1 2+in()+in(π) ))[u(,x)+f(,u(,x))]+g(,u(,x)), for R and x [0,π], u(,x) = 0, for x = 0, π and R, (12) f(,u(,x)) = 0, for x = 0, π and R. where f i peudo almo auomorphic and g i Sepanov-like peudo almo auomorphic for p > 1, joinly coninuou.

16 208 Sepanov-like Peudo Almo Auomorphic Soluion... The following propoiion i an immediae conequence of Theorem 3.5. Propoiion 5.2. Under aumpion (H 4 ),(H 5 ), Eq.(12)admi a unique peudo almo auomorphic mild oluion if L f + 2Lg π 2 +1 < 1. Example 5.3. Le u conider he following perurbed nonauonomou evoluion equaion [ u(,x)+f(, ] [ 2 x 2u(,x)) = 2 u(,x)+f(, x 2 +( 2+co( 1 2+in()+in(π) )) [u(,x)+f(, +g(, 2 x2u(,x)), for R and x [0,π], u(,x) = 0, for x = 0, π and R, f(, 2 x2u(,x)) = 0, for x = 0, π and R. ] 2 x 2u(,x)) ] 2 x 2u(,x)) (13) Le Y := D(A()) denoe he pace D(A()) endowed wih he graph norm, hen Y X i coninuouly embedded. Define he operaor B,C by Bψ = Cψ = ψ wih D(B) = D(C) = D(A), hen B,C : Y X are bounded and B B(Y,X) = C B(Y,X) = 1. The following propoiion i an immediae conequence of Theorem 4.4. Propoiion 5.4. Under aumpion (H 7 ),(H 8 ), Eq.(13)admi a unique peudo almo auomorphic mild oluion if L f + 2Lg π 2 +1 < 1. Reference [1] P. Acquiapace and B. Terreni, A unified approach o abrac linear parabolic equaion, Rend. Sem. Ma. Univ. Padova, 78, (1987), [2] P. Acquiapace, F. Flandoli and B. Terreni, Iniial boundary value problem and opimal conrol for nonauonomou parabolic yem, SIAM J. Conrol Opim., 29, (1991),

17 M. Ziane and C. Benouda 209 [3] P. Acquiapace, Evoluion operaor and rong oluion of abrac linear parabolic equaion, Differenial Inegral Equaion, 1, (1988), [4] H. Amann, Linear and quailinear parabolic problem, Birkhäuer, Berlin, [5] S. Bochner, A new approach o almo auomorphiciy, Proceeding of he Naional Academy of Science, USA, 48, (1962), [6] T. Diagana, Sepanov-like peudo-almo periodiciy and i applicaion o ome nonauonomou differenial equaion, Nonlinear Anal. TMA, 69, (2008), [7] T. Diagana, Sepanov-like peudo-almo periodic funcion and heir applicaion o differenial equaion, Commun. Mah. Anal, 3(1), (2007), [8] T. Diagana, Exience of peudo-almo auomorphic oluion o ome abrac differenial equaion wih S p -peudo-almo auomorphic coefficien, Nonlinear Anal. TMA, 70, (2009), [9] H.S. Ding, J. Liang and T.J. Xiao, Some prorerie of Sepanov-like almo auomorphic funcion and applicaion o abrac evoluion equaion, Appl. Anal., 88(7), (2009), [10] H.S. Ding, J. Liang and T.J. Xiao, Almo auomorphic mild oluion o nonauonomou emilinear evoluion equaion in Banach pace, Nonlinear Anal. TMA, 73, (2010), [11] D. Henry, Geomery Theory of Semilinear Parabolic Equaion, Springer- Verlag, [12] Z. Hu and Z. Jin, Sepanov-like peudo almo auomorphic mild oluion o nonauonomou evoluion equaion, Nonlinear Anal. TMA, 71,(2009), [13] Z. Hu, Z. Jin, Sepanov-like peudo almo periodic mild oluion o perurbed nonauonomou evoluion equaion wih infinie delay, Nonlinear Anal. TMA, 71, (2009),

18 210 Sepanov-like Peudo Almo Auomorphic Soluion... [14] Z. Hu and Z. Jin, Sepanov-like peudo almo periodic mild oluion o nonauonomou neural parial evoluion equaion, Nonlinear Anal. TMA, 75, (2012), [15] J. Liang, J. Zhang and T.J. Xiao, Compoiion of peudo-almo auomorphic and aympoically almo auomorphic funcion, J. Mah. Anal. Appl., 34076(1), (2008), [16] G.M. N Guérékaa, Topic in Almo Auomorphy, Springer-Verlag, New York, [17] G.M. N Guérékaa and A. Pankov, Sepanov-like almo auomorphic funcion and monoone evoluion equaion, Nonlinear Anal. TMA, 68, (2008), [18] K.J. Negel, One-Parameer Semigroup for Linear Evoluion Equaon, Springer-Verlag, 194, [19] A. Pankov, Bounded and Almo Periodic Soluion of Nonlinear Operaor Differenial Equaion, Kluwer, Dordrech, [20] A. Pazy, Semigroup of Linear Operaor and Applicaion o Parial Differenial Equaion, in Applied mahemaical cience, Springer-Verlag, New York, 44, [21] T.J. Xiao, J. Liang and J. Zhang, Peudo-almo auomorphic oluion o emilinear differenial equaion in Banach pace, Semigroup Forum, 76, (2008), [22] T.J. Xiao, X.X. Zhu and J. Liang, Peudo-almo auomorphic mild oluion o nonauonomou differenial equaion and applicaion, Nonlinear Anal. TMA, 70, (2009), [23] A. Yagi, Parabolic equaion in which he coefficien are generaor of infiniely differeniable emigroup II, Funkcial. Ekvac., 33, (1990), [24] A. Yagi, Abrac quailinear evoluion equaion of parabolic ype in Banach pace, Unione Ma. Ial. Sez B, 7(5), (1991),

19 M. Ziane and C. Benouda 211 [25] M. Ziane and C. Benouda, Weighed peudo-almo auomorphic oluion o a neural delay inegral equaion of advanced ype, Applied Mahemahical Science, 6(122), (2012),