WEIGHTED PSEUDO PERIODIC SOLUTIONS OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS

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1 Elecronic Journal of Differenial Equaions, Vol , No. 9, pp. 7. ISSN: URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu fp ejde.mah.xsae.edu WEIGHTED PSEUDO PERIODIC SOLUTIONS OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS ZHINAN XIA Absrac. In his aricle, we inroduced and explore he properies of wo ses of funcions: weighed pseudo periodic funcions of class r, and weighed Sepanov-like pseudo periodic funcions of class r. We show he exisence and uniqueness of weighed pseudo periodic soluion of class r ha are soluions o neural funcional differenial equaions. Oher applicaions o parial differenial equaions and scalar reacion-diffusion equaions wih delay are also given.. Inroducion The exisence periodic soluions is one of he mos ineresing and imporan opics in he qualiaive heory of differenial equaions. Many auhors have made imporan conribuions o his heory. Recenly, in [, 9], he concep of weighed pseudo periodiciy, weighed Sepanov-like pseudo periodiciy, is inroduced and sudied, respecively. On he oher hand, o sudy issues relaed o delay differenial equaions, Diagana [5] inroduce he funcions called pseudo almos periodic of class r, for more on his opic and relaed applicaions in differenial equaions, we refer he reader o [2, 3, 4,, ]. Moivaed by he above menioned papers, in his paper, we inroduce new class of funcions called weighed pseudo periodic of class r, weighed Sepanov-like pseudo periodic of class r, respecively. We sysemaically explore he properies of hese funcions in general Banach space including composiion resuls and is applicaions in differenial equaions. In recen years, neural funcional differenial equaions have araced a grea deal of aenion of many mahemaicians due o heir significance and applicaions in physics, mahemaical biology, conrol heory, and so on. The general asympoic behavior of soluions have been one of he mos aracing opics in he conex of neural funcional differenial equaions [6, 9, 2, 3, 4, 5, 8]. However, o he bes of our knowledge, he sudies on he weighed pseudo periodic soluions of neural funcional differenial equaions is quie new and an unreaed opic. This is one of he key moivaions of his sudy. 2 Mahemaics Subjec Classificaion. 35R, 35B4. Key words and phrases. Weighed pseudo periodiciy; weighed Sepanov-like pseudo periodic; neural funcional differenial equaion; Banach conracion mapping principle. c 24 Texas Sae Universiy - San Marcos. Submied March 6, 24. Published Sepember 6, 24.

2 2 Z. XIA EJDE-24/9 The paper is organized as follows. In Secion 2, some noaions and preliminary resuls are presened. Nex, we propose new class of funcions called weighed pseudo periodic funcions of class r, weighed Sepanov-like pseudo periodic funcions of class r, explore he properies of hese funcions and esablish he composiion heorems. Secions 3 is devoed o he exisence and uniqueness of weighed pseudo periodic soluions of class of r o neural funcional differenial equaions. In secion 4, we presen applicaions o parial differenial equaions and scalar reacion-diffusion equaions wih delay. 2. Preliminaries and basic resuls Le X,, Y, be wo Banach spaces and N, Z, R be he sand ses of naural numbers, inegers, real numbers, respecively. To faciliae he discussion below, we furher inroduce he following noaion: CR, X resp. CR Y, X: he se of coninuous funcions from R o X resp. from R Y o X. BCR, X resp. BCR Y, X: he Banach space of bounded coninuous funcions from R o X resp. from R Y o X wih he remum norm; L p R, X: he space of all classes of equivalence wih respec o he equaliy almos everywhere on R of measurable funcions f : R X such ha f L p R, R; L p loc R, X: sand for he space of all classes of equivalence of measurable funcions f : R X such ha he resricion of f o every bounded subinerval of R is in L p R, X. C: he space C[ r, ], X endowed wih he norm ψ C on [ r, ]. [DA]: he domain of A when i is endowed wih graph norm, x [DA] = x + Ax for each x DA. 2.. Weighed pseudo periodic of class r. In his subsecion, we inroduce he new class of funcions called weighed pseudo ani-periodic of class r, weighed pseudo periodic funcions of class r, and invesigae he properies of hese funcions. Definiion 2.. A funcion f CR, X is said o be ani-periodic if here exiss a ω R\{} wih he propery ha f + ω = f for all R. The leas posiive ω wih his propery is called he ani-periodic of f. The collecion of hose funcions is denoed by P ωap R, X. Definiion 2.2. A funcion f CR, X is said o be periodic if here exiss a ω R\{} wih he propery ha f + ω = f for all R. The leas posiive ω wih his propery is called he periodic of f. The collecion of hose ω-periodic funcions is denoed by P ω R, X. Noe ha if f P ωap R, X, hen f P 2ω R, X. Le U be he se of all funcions ρ : R, which are posiive and locally inegrable over R. For a given T > and each ρ U, se Define µt, ρ := ρd. U := {ρ U : lim µt, ρ = }, T

3 EJDE-24/9 WEIGHTED PSEUDO PERIODIC SOLUTIONS 3 I is clear ha U B U U. U B := {ρ U : ρ is bounded and inf ρx > }. x R Definiion 2.3. Le ρ, ρ 2 U. The funcion ρ is said o be equivalen o ρ 2 i.e., ρ ρ 2 if ρ ρ 2 U B. I is rivial o show ha is a binary equivalence relaion on U. The equivalence class of a given weigh ρ U is denoed by clρ = {ϱ U : ρ ϱ}. I is clear ha U = ρ U clρ. Le ρ U, τ R be given, and defined ρ τ by ρ τ = ρ + τ for R. Define [2] U T = {ρ U : ρ ρ τ for each τ R}. I is easy o see ha U T conains many of weighs, such as, + 2 /2 + 2, e, and + n wih n N ec. For ρ, ρ 2 U, define W P P R, X, ρ, ρ 2 := W P P R, X, r, ρ, ρ 2 { := f BCR, X : lim T W P P R Y, X, r, ρ, ρ 2 { := f BCR Y, X : lim T } uniformly for u Y. { f BCR, X : lim T µt, ρ µt, ρ µt, ρ } f ρ 2 d =, } fθ ρ 2 d =, fθ, u ρ 2 d = Definiion 2.4. Le ρ, ρ 2 U. A funcion f CR, X is called weighed pseudo ani-periodic for ω R\{} if i can be decomposed as f = g + ϕ, where g P ωap R, X and ϕ W P P R, X, ρ, ρ 2. Denoe by W P P ωap R, X, ρ, ρ 2 he se of such funcions. Definiion 2.5. Le ρ, ρ 2 U. A funcion f CR, X is called weighed pseudo periodic for ω R\{} if i can be decomposed as f = g + ϕ, where g P ω R, X and ϕ W P P R, X, ρ, ρ 2. Denoe by W P P ω R, X, ρ, ρ 2 he se of such funcions. If ρ ρ 2, W P P ωap R, X, ρ, ρ 2, and W P P ω R, X, ρ, ρ 2 coincide wih he weighed pseudo ani-periodic, and he weighed pseudo periodic funcion respecively, inroduce by []. Definiion 2.6. Le ρ, ρ 2 U. A funcion f CR, X is called weighed pseudo ani-periodic of class r for ω R\{} if i can be decomposed as f = g + ϕ, where g P ωap R, X and ϕ W P P R, X, r, ρ, ρ 2. The se of hese funcions is denoe by W P P ωap R, X, r, ρ, ρ 2. Definiion 2.7. Le ρ, ρ 2 U. A funcion f CR, X is called weighed pseudo periodic of class r for ω R\{} if i can be decomposed as f = g+ϕ, where g P ω R, X and ϕ W P P R, X, r, ρ, ρ 2. Denoe by W P P ω R, X, r, ρ, ρ 2 he se of such funcions.

4 4 Z. XIA EJDE-24/9 Remark 2.8. If r =, hen he weighed pseudo ani-periodic funcion of class r reduces o he weighed pseudo ani-periodic funcion, he weighed pseudo periodic funcion of class r reduces o he weighed pseudo periodic funcion. Tha is, W P P ωap R, X,, ρ, ρ 2 = W P P ωap R, X, ρ, ρ 2, and W P P ω R, X,, ρ, ρ 2 = W P P ω R, X, ρ, ρ 2. Nex, we show some properies of he space W P P ω R, X, r, ρ, ρ 2. resuls hold for W P P ωap R, X, r, ρ, ρ 2. Similarly Lemma 2.9. Le f BCR, X, hen f W P P R, X, r, ρ, ρ 2, ρ, ρ 2 U, µt,ρ 2 T > µt,ρ < if and only if for every ε >, lim ρ 2 d =, T µt, ρ MT,ε,f where MT, ε, f := { [, T ] : fθ ε}. Proof. The proof is similar as []. Sufficiency: From he saemen of he lemma i is clear ha for any ε >, here exiss T > such ha for T > T, Then so T µt, ρ µt, ρ = µt, ρ + µt, ρ f µt, ρ MT,ε,f MT,ε,f ρ 2 d < fθ ρ 2 d [,T ]\MT,ε,f MT,ε,f µt, ρ 2 ε + T > µt, ρ ε, T lim T µt, ρ Tha is, f W P P R, X, r, ρ, ρ 2. ρ 2 d + ε f. fθ ρ 2 d ε µt, ρ fθ ρ 2 d ρ 2 d fθ ρ 2 d =. Necessiy: Suppose on he conrary ha here exiss ε > such ha ρ 2 d µt, ρ MT,ε,f does no converge o as T. Then here exiss δ > such ha for each n, ρ 2 d δ for some T n n. µt n, ρ Then MT n,ε,f Tn ρ 2 µt n, ρ n fθ d

5 EJDE-24/9 WEIGHTED PSEUDO PERIODIC SOLUTIONS 5 µt n, ρ ε µt n, ρ MT n,ε,f MT n,ε,f ρ 2 ρ 2 d ε δ, fθ d which conradics he fac ha f W P P R, X, r, ρ, ρ 2, and he proof is complee. Le W P P R, R +, r, ρ, ρ 2 = {f W P P R, R, r, ρ, ρ 2 : f, R}. Lemma 2.. Le α >, hen f W P P R, R +, r, ρ, ρ 2 if and only if f α W P P R, R +, r, ρ, ρ 2, where f α = [f] α, ρ, ρ 2 U, T > µt,ρ 2 µt,ρ <. Proof. By Lemma 2.9, f W P P R, R +, r, ρ, ρ 2 if and only if for every ε >, lim ρ 2 d =, T µt, ρ MT,ε,f where MT, ε, f := { [, T ] : fθ ε}. I is equivalen o for every ε >, lim ρ 2 d =, T µt, ρ MT,ε,f α where MT, ε, f α := { [, T ] : f α θ ε}. So f α W P P R, R +, r, ρ, ρ 2. Lemma 2.. Le ϕ n ϕ uniformly on R where each ϕ n W P P R, X, r, ρ, ρ 2, ρ, ρ 2 U, if T > µt,ρ 2 µt,ρ <, hen ϕ W P P R, X, r, ρ, ρ 2. Proof. For T >, µt, ρ µt, ρ + µt, ρ ϕθ ρ 2 d µt, ρ 2 µt, ρ ϕ n ϕ + µt, ρ µt, ρ 2 T > µt, ρ ϕ n ϕ + ϕ n θ ϕθ ρ 2 d ϕ n θ ρ 2 d µt, ρ ϕ n θ ρ 2 d ϕ n θ ρ 2 d, Le T and hen n in he above inequaliy, i follows ha ϕ W P P R, X, r, ρ, ρ 2. By carrying ou similar argumens as hose in he proof of [2, Lemma 4.], we conclude he following.

6 6 Z. XIA EJDE-24/9 Lemma 2.2. Le ρ, ρ 2 U T, ϕ W P P R, X, r, ρ, ρ 2, hen ϕ τ belongs o W P P R, X, r, ρ, ρ 2 for τ R. Using similar ideas as in [7, 8], one can easily show he following resul. Lemma 2.3. If ρ, ρ 2 U T and inf T > µt,ρ 2 µt,ρ = δ >, hen he decomposiion of weighed pseudo periodic funcion of class r is unique. By Lemma 2.3, i is obvious ha W P P ω R, X, r, ρ, ρ 2,, ρ, ρ 2 U T and inf T > µt,ρ 2 µt,ρ = δ > is a Banach space when endowed wih he norm. Lemma 2.4. Le ρ, ρ 2 U T, u W P P ω R, X, r, ρ, ρ 2, hen u belongs o W P P ω R, C, r, ρ, ρ 2. Proof. Suppose ha u = α+β, where α P ω R, X and β W P P R, X, r, ρ, ρ 2, hen u = α + β and α P ω R, C, r, ρ, ρ 2. On he oher hand, for T >, we see ha µt, ρ µt, ρ µt, ρ µt, ρ + µt, ρ [ r r θ [ 2r,] τ [ r,] θ [ 2r, r] ] βθ + τ ρ 2 d βθ ρ 2 d βθ + βθ ρ 2 + rd βθ ρ 2 d +r µt + r, ρ µt, ρ µt + r, ρ r T + βθ ρ 2 d. µt, ρ βθ ρ 2 d βθ ρ 2 ρ 2 + r ρ 2 Since ρ, ρ 2 U T implies ha here exiss η > such ha ρ + r/ρ η, ρ r/ρ η, ρ 2 + r/ρ 2 η. For T > r, r +r r +r µt + r, ρ = ρ d + ρ d ρ d + ρ d r T r r +r hen = ρ rd + [ µt, ρ 2η 2 µt + r, ρ + µt, ρ +r r ρ + rd 2ηµT, ρ, τ [ r,] ] βθ + τ ρ 2 d βθ ρ 2 d βθ ρ 2 d. d

7 EJDE-24/9 WEIGHTED PSEUDO PERIODIC SOLUTIONS 7 Noe ha β W P P R, X, r, ρ, ρ 2, ρ, ρ 2 U T, hen β W P P R, C, r, ρ, ρ 2. Therefore, u W P P ω R, C, r, ρ, ρ 2. Similarly as [3, Theorem 3.9], we have he following composiion heorem for weighed pseudo periodic funcion of class r. Theorem 2.5. Assume ha ρ, ρ 2 U, r, f W P P ω R Y, X, r, ρ, ρ 2 and here exiss a funcion L f : R [, + saisfying A f, u f, v L f u v, for all R, u, v Y ; A2 lim T A3 lim T µt,ρ µt,ρ L f θ ρ 2 d < ; ξρ 2 d = for each funcion L f θ ξ W P P R, R, ρ, ρ 2. Then f, h W P P ω R, X, r, ρ, ρ 2 if h W P P ω R, Y, r, ρ, ρ 2. Remark 2.6. Noe ha A2 and A3 are verified by many funcions. Concree examples include consan funcions, and funcions in W P P ω R, R, r, ρ, ρ Weighed Sepanov-like pseudo periodic of class r. In his subsecion, we inroduce he new class of funcions called weighed S p -pseudo ani-periodic of class r, weighed S p -pseudo periodic funcions of class r, and invesigae he properies of hese funcions. Le p [,. The space BS p R, X of all Sepanov bounded funcions, wih he exponen p, consiss of all measurable funcions f : R X such ha f b L R, L p [, ]; X, where f b is he Bochner ransform of f defined by f b, s := f + s, R, s [, ]. BS p R, X is a Banach space wih he norm [7] f S p = f b L R,L p = R + /p. fτ dτ p I is clear ha L p R, X BS p R, X L p loc R, X and BSp R, X BS q R, X for p q. For ρ, ρ 2 U, define he weighed ergodic space in BS p R, X S p W P P R, X, r, ρ, ρ 2 := { f BS p R, X : lim T ρ 2 µt, ρ θ+ /p } fs p ds d =. Definiion 2.7. Le ρ, ρ 2 U. A funcion f BS p R, X is said o be weighed Sepanov-like pseudo ani-periodic of class r or weighed S p -pseudo aniperiodic of class r if here exiss ϕ S p W P P R, X, r, ρ, ρ 2 such ha he funcion g = f ϕ saisfies g + ω + g = a.e. R. The collecion of such funcions is denoed by S p W P P ωap R, X, r, ρ, ρ 2 Definiion 2.8. Le ρ, ρ 2 U. A funcion f BS p R, X is said o be weighed Sepanov-like pseudo periodic of class r or weighed S p -pseudo periodic of class r if here exiss ϕ S p W P P R, X, r, ρ, ρ 2 such ha he funcion g = f ϕ saisfies g + ω g = a. e. R. Denoe by S p W P P ω R, X, r, ρ, ρ 2 he collecion of such funcions. Nex, we show some properies of he space S p W P P ω R, X, r, ρ, ρ 2. Similarly resuls hold for S p W P P ωap R, X, r, ρ, ρ 2. θ

8 8 Z. XIA EJDE-24/9 Lemma 2.9. Le ρ, ρ 2 U T, hen W P P ω R, X, r, ρ, ρ 2 S p W P P ω R, X, r, ρ, ρ 2. Proof. If f W P P ω R, X, r, ρ, ρ 2, le f = f + f 2, where f P ω R, X, f 2 W P P R, X, r, ρ, ρ 2. Then f 2 W P P R, R +, r, ρ, ρ 2. By Lemma 2., f 2 p W P P R, R +, r, ρ, ρ 2. Noe ha f 2 +σ p W P P R, R +, r, ρ, ρ 2 for each σ [, ], hen lim T µt, ρ f 2 θ + σ p ρ 2 d =. by Lebesgue s dominaed convergence heorem, one has T f 2 θ + σ p ρ 2 d dσ, T, µt, ρ i.e., T ρ 2 µt, ρ which means ha i.e., µt, ρ ρ 2 T µt, ρ ρ 2 so h 2 W P P R, R +, r, ρ, ρ 2, where h 2 = f 2 θ + σ p dσ d, T, f 2 θ + σ p dσ d, T ; h 2 θ d, T, f 2 + σ p dσ, R. By Lemma 2., h /p 2 W P P R, R +, r, ρ, ρ 2 ; i.e., T /pd ρ 2 f 2 θ + σ dσ p, T, µt, ρ which means ha f 2 S p W P P R, X, r, ρ, ρ 2, hen f S p W P P ω R, X, r, ρ, ρ 2. The proof is complee. Theorem 2.2. Assume ρ, ρ 2 U, f = f + f 2 S p W P P ω R Y, X, r, ρ, ρ 2 wih f 2 S p W P P R Y, X, r, ρ, ρ 2, f + ω, u f, u = a. e. R, u X, and here exiss a funcion L f : R [, + saisfying: + /p A fs, u fs, v ds p Lf u v, for all R, u, v Y ; A2 lim T A3 lim T µt,ρ µt,ρ L f θ ρ 2 d < ; ξρ 2 d = for each funcion L f θ ξ b W P P R, L p [, ], R, ρ, ρ 2 ; A4 f is uniform coninuous on bounded se K Y for any R. Then f, h S p W P P ω R, X, r, ρ, ρ 2 if h S p W P P ω R, Y, r, ρ, ρ 2.

9 EJDE-24/9 WEIGHTED PSEUDO PERIODIC SOLUTIONS 9 The proof of he above heorem is similar o he proof of [2, Theorem 3.2] and i is omied here. I is no difficul o see ha Theorem 2.5, Theorem 2.2 hold for W P P ωap R, X, r, ρ, ρ 2, S p W P P ωap R, X, r, ρ, ρ 2 respecively. 3. Neural funcional differenial equaions As an applicaion, he main goal of his secion is o esablish sufficien crieria for he exisence, uniqueness of he weighed pseudo periodic soluion o a class of neural funcional differenial equaions of he form d d [u + f, u ] = Au + g, u, R, 3. where A is he infiniesimal generaor of a semigroup of linear operaors on X, u C is defined by u θ = u + θ for θ [ r, ], where r is a nonnegaive consan. Firs, we recall he definiion of he so called exponenial dichoomy of a semigroup. Definiion 3.. [6] A semigroup T is said o be exponenial dichoomy if here exis projecion P and consans M, δ > such ha each T commues wih P, KerP is invarian wih respec o T, T : ImQ ImQ is inverible and T P x Me δ x for, 3.2 T Qx Me δ x for, 3.3 where Q := I P and T := T for. To sudy 3., we make he following assumpions: H The operaor A : DA X X is he infiniesimal generaor of semigroup T which has an exponenial dichoomy. H2 f W P P ω R C, X, r, ρ, ρ 2, f is DA-valued, here exiss a posiive consan L f such ha f, ψ f, ψ 2 [DA] L f ψ ψ 2 C, for all R, ψ i C, i =, 2. H3 g W P P ω R C, X, r, ρ, ρ 2 and here exiss a coninuous funcion L g : R R + such ha g, ψ g, ψ 2 L g ψ ψ 2 C, for all R, ψ i C, i =, 2. H3 g = g + g 2 S p W P P ω R C, X, r, ρ, ρ 2 wih g 2 S p W P P R C, X, r, ρ, ρ 2, g + ω, ψ g, ψ = a.e. R, ψ C, saisfying i g is uniform coninuous on bounded se K C for any R. ii here exiss a posiive consan L g such ha + /p g, ψ g, ψ 2 ds p Lg ψ ψ 2 C, for all R, ψ i C, i =, 2. H4 ρ, ρ 2 U T, inf T > µt,ρ 2 µt,ρ = δ > and T > µt,ρ 2 µt,ρ <.

10 Z. XIA EJDE-24/9 Definiion 3.2 [6]. A coninuous funcion u is said o be a mild soluion of 3. provided ha he funcion s AT sp fs, u s is inegrable on,, s AT sqfs, u s is inegrable on, for R and u = f, u AT sp fs, u s ds + + T sp gs, u s ds AT sqfs, u s ds T sqgs, u s ds, R. Lemma 3.3. Assume ha H, H2, H4 hold, if u W P P ω R, X, r, ρ, ρ 2, hen Λ f = Λ 2 f = AT sp fs, u s ds W P P ω R, X, r, ρ, ρ 2. AT sqfs, u s ds W P P ω R, X, r, ρ, ρ 2. Proof. By Lemma 2.4 and Theorem 2.5, fs, u s := hs W P P ω R, X, r, ρ, ρ 2 and h is DA-valued. Le hs = h s + h 2 s where h P ω R, X and h 2 W P P R, X, r, ρ, ρ 2. Then where Λ h = Λ f = for R. From h P ω R, X, AT sp h sds + := Λ h + Λ 2 h 2, Λ h + ω = AT sp h sds, Λ 2 h 2 = +ω AT sp h 2 sds AT sp h 2 sds, AT + ω sp h sds = Λ h ; hen Λ h P ω R, X. Nex, we show ha Λ 2 h 2 W P P R, X, r, ρ, ρ 2 ; ha is, T lim Λ 2 h 2 θ ρ 2 d =. T µt, ρ In fac, for T >, one has µt, ρ = = µt, ρ µt, ρ µt, ρ µt, ρ Λ 2 h 2 θ ρ 2 d θ AT θ sp h 2 sds ρ 2 d AT sp h 2 θ sds ρ 2 d Me δs Ah 2 θ s ds ρ 2 d Me δs h 2 θ s [DA] ds ρ 2 d

11 EJDE-24/9 WEIGHTED PSEUDO PERIODIC SOLUTIONS where Me δs Φ T sds, Φ T s = µt, ρ h 2 θ s [DA] ρ 2 d. Since ρ, ρ 2 U T, by Lemma 2.2, we have h 2 s W P P R, [DA], r, ρ, ρ 2 for each s R; hence lim T Φ T s = for all s R. Then Λ 2 h 2 belongs o W P P R, X, r, ρ, ρ 2 by using he Lebesgue dominaed convergence heorem, so Λ f W P P ω R, X, r, ρ, ρ 2. The proof of Λ 2 f is similar o ha of Λ f, one makes use of 3.3 raher han 3.2. This complees he proof. Lemma 3.4. Assume ha H, H4 hold, if φ S p W P P ω R, X, r, ρ, ρ 2, hen Γ φ = Γ 2 φ = T sp φsds W P P ω R, X, r, ρ, ρ 2, T sqφsds W P P ω R, X, r, ρ, ρ 2. Proof. By φ S p W P P ω R, X, r, ρ, ρ 2, we le φs = φ s + φ 2 s, where φ 2 S p W P P R, X, r, ρ, ρ 2 and φ + ω φ = a.e. R, hen Γ φ = T sp φ sds+ T sp φ 2 sds := Γ φ +Γ 2 φ 2. Firs, we show ha Γ 2 φ 2 W P P R, X, r, ρ, ρ 2. Consider he inegrals Y n = Fix n N and R, we have Y n + h Y n n+ n M M In view of φ 2 L p loc R, X, we ge lim h n+ n n n n+ T sp φ 2 sds. T sp φ 2 + h s φ 2 s ds n n+ n φ 2 s + h φ 2 s ds φ 2 s + h φ 2 s p ds /p. φ 2 s + h φ 2 s p ds =, which yields lim h Y n + h Y n =. This means ha Y n is coninuous. By Hölder s inequaliy, one has Y n n n n n T sp φ 2 s ds Me δs φ 2 s ds Me δn n n φ 2 s ds

12 2 Z. XIA EJDE-24/9 Since n= n+ Me δn φ 2 s ds n Me δn n+ n Me δn φ 2 S p. /p φ 2 s p ds Me δn φ 2 S p M e δ φ 2 S p < +, i follows ha n= Y n converges uniformly on R. Le Y = n= Y n for R. Then Y = Γ 2 φ 2 = T sp φ 2 sds, R. I is obvious ha Y BCR, X. So, we only need o show ha T lim ρ 2 Y θ d =. 3.4 T µt, ρ In fac, by Hölder inequaliy, Y n for some consan M > ; hen T ρ 2 µt, ρ M µt, ρ n M n n+ Me δs φ 2 s ds n φ 2 s ds M n+ /p, φ 2 s ds p ρ 2 n Y n θ d θ n+ θ n /p φ 2 s p ds d, and hence Y n W P P R, X, r, ρ, ρ 2 since φ 2 S p W P P R, X, r, ρ, ρ 2. By Lemma 2., 3.4 holds, whence Γ 2 φ 2 W P P R, X, r, ρ, ρ 2. From φ + ω φ = a.e. R, one has Γ φ + ω = +ω T + ω sp φ sds = Γ φ, a.e. R. Hence Γ φ W P P ω R, X, r, ρ, ρ 2. The proof of Γ 2 φ is similar o ha of Γ φ, one uses 3.3 raher han 3.2. This complees he proof. Theorem 3.5. Assume ha H H4 hold and g saisfy he condiions A2 A3, if ϑ := L f + 2ML f + M e δ s L g sds + M e δ s L g sds <, δ R R

13 EJDE-24/9 WEIGHTED PSEUDO PERIODIC SOLUTIONS 3 hen 3. has a unique mild soluion of W P P ω ype. Proof. Define F : W P P ω R, X, r, ρ, ρ 2 W P P ω R, X, r, ρ, ρ 2 as Fu = f, u AT sp fs, u s ds + AT sqfs, u s ds + T sp gs, u s ds T sqgs, u s ds, R. 3.5 If u W P P ω R, X, r, ρ, ρ 2, hen u W P P ω R, C, r, ρ, ρ 2 by Lemma 2.4; herefore by Theorem 2.5, gs, u s W P P ω R, X, r, ρ, ρ 2 S p W P P ω R, X, r, ρ, ρ 2. By Lemmas 3.3 and 3.4, i is no difficul o see ha F is well defined. For any u, v W P P ω R, X, r, ρ, ρ 2, we have Fu Fv f, u f, v AT sqfs, u s fs, v s ds T sp gs, u s gs, v s ds T sqgs, u s gs, v s ds L f u u C + ML f AT sp fs, u s fs, v s ds e δ s u s u s C ds + ML f e δ s u s u s C ds + M + M L f + 2ML f δ + M R ϑ u v, e δ s L g s u s u s C ds + M R e δ s L g sds e δ s L g sds u v e δ s L g s u s u s C ds hen F is a conracion since ϑ <. By he Banach conracion mapping principle, F has a unique fixed poin in W P P ω R, X, r, ρ, ρ 2, which is he unique W P P ω soluion o 3.. In H3, if L g L g, I is no difficul o see ha g saisfy he condiions A2 A3 and ϑ = L f + 2ML f + L g /δ. Theorem 3.6. Assume ha H, H2, H3, H4 hold. If Θ := L f + 2ML f + 2ML g δ e δ <, hen 3. has a unique mild soluion of W P P ω ype.

14 4 Z. XIA EJDE-24/9 Proof. Define he operaor F as in 3.5. Le u W P P ω R, X, r, ρ, ρ 2, hen i is no difficul o see ha gs, u s S p W P P ω R, X, r, ρ, ρ 2 by Theorem 2.2, so Γ is well defined by Lemma 3.3 and Lemma 3.4. Le u, v W P P ω R, X, r, ρ, ρ 2, one has Fu Fv L f u v C + ML f e δ s u s v s C ds + ML f e δ s u s v s C ds + M + M e δ s gs, u s gs, v s ds L f u v C + ML f e δ s u s v s C ds + ML f e δ s u s v s C ds + M + M L f + 2ML f δ + M k= e δs g s, u s g s, v s ds k L f + 2ML f δ + M k= u v + M k L f + 2ML f δ L f + 2ML f δ Θ u v. k= k+ k e δs g s, u s g s, v s ds u v + M e δk k k= e δ s gs, u s gs, v s ds e δs g s, u s g s, v s ds e δs g s, u s g s, v s ds k e δk k+ gs, u s gs, v s p ds k + ML g e δk + M k= + 2ML g e δ u v k= /p gs, u s gs, v s p ds /p e δk u v By he Banach conracion mapping principle, F has a unique fixed poin in W P P ω R, X, r, ρ, ρ 2, which is he unique W P P ω soluion o 3.. Remark 3.7. I is easy o see ha similar resuls of Theorem 3.5 and Theorem 3.6 hold for W P P ωap R, X, r, ρ, ρ 2 mild soluion, ha is 3. has a unique W P P ωap mild soluion, in his case, f W P P ωap R C, X, r, ρ, ρ 2 in H2, g W P P ωap R C, X, r, ρ, ρ 2 in H3, g S p W P P ωap R C, X, r, ρ, ρ 2 in H3. 4. Examples In his secion, we provide some examples o illusrae our main resuls.

15 EJDE-24/9 WEIGHTED PSEUDO PERIODIC SOLUTIONS 5 Example 4.. Consider he parial differenial equaion [ π ] u, ξ + bs, η, ξu + s, ηdηds r = 2 ξ 2 u, ξ + a u, ξ + r a su + s, ξds, u, = u, π =,, ξ R [, π], 4. where a W P P ω R, R, r, ρ, ρ 2, ρ = e, ρ 2 = + 2 and he following condiions hold: The funcions b, i ζ bτ, η, ζ, i =, 2 are Lebesgue measurable, bτ, η, = i bτ, η, π = for every τ, η and { π π i 2dηdτdζ, } N := max ζ i bτ, η, ζ : i =,, 2 <. r Under hese condiions, le X = L 2 [, π], R, L 2 and define he operaor A on X by Au = u wih DA = {u X : u X, u = uπ = }. I is well known ha A is he infiniesimal generaor of a semigroup T on X such ha T e for every. Define he funcions f, g : R C X by f, ψξ := π r g, ψξ := a ψ, ξ + bs, η, ξψs, η dη ds, r a sψs, ξ ds, hen 4. can be rewrien as an absrac sysem of he form 3., where u = u,. By a sraighforward esimaion ha uses i, one can show ha f is DA- valued, and he following hold: Af, N r /2, R, g, a + r /2 /2, asds 2 R. See [3, 5] for more deails. The nex resul is a consequence of Theorem 3.5. Theorem 4.2. Under he previous assumpions, 4. as a unique weighed pseudo periodic soluion whenever 3 N r + 2 a + 2r /2 /2 asds 2 <. Example 4.3. Consider he following scalar reacion-diffusion equaion wih delay r r 2 u, x = u, x + g, u r, x, x2 u, = u, π =, uτ, x = ϕτ, x, τ [ r, ], x [, π], 4.2 where g = g + g 2 S p W P P ω R C, R, r, ρ, ρ 2 wih g P ω R C, R, g 2 S p W P P R C, R, r, ρ, ρ 2, ρ = e, ρ 2 = + 2.

16 6 Z. XIA EJDE-24/9 By Theorem 3.6, one has he following resul. Theorem 4.4. Assume ha here exiss a posiive consan L g such ha for i =, 2, + /p g, ψ g, ψ 2 ds p Lg ψ ψ 2 C, for all R, ψ i C. Then here exiss a unique W P P ω soluion of 4.2 if 2L g < e. Acknowledgemens. This research was pored by Zhejiang Provincial Naural Science Foundaion of China under Gran No. LQ3A5. References [] N. S. Al-lslam, S. M. Alsulami, T. Diagana; Exisence of weighed pseudo ani-periodic soluions o some non-auonomous differenial equaions, Appl. Mah. Compu , [2] R. P. Agarwal, C. Cuevas, H. Soo, M. El-Gebeily; Asympoic periodiciy for some evoluion equaions in Banach spaces, Nonlinear Anal. 74 2, [3] R. P. Agarwal, T. Diagana, E. M. Hernández; Weighed pseudo almos periodic soluions o some parial neural funcional differenial equaions, J. Nonlinear Convex Anal. 8 27, [4] C. Cuevasa, E. M. Hernández; Pseudo-almos periodic soluions for absrac parial funcional differenial equaions, Appl. Mah. Le , [5] T. Diagana, E. M. Hernández; Exisence and uniqueness of pseudo almos periodic soluions o some absrac parial neural funcional-differenial equaions and applicaions, J. Mah. Anal. Appl , [6] T. Diagana, R. P. Agarwal; Exisence of pseudo almos auomorphic soluions for he hea equaion wih S p -pseudo almos auomorphic coefficiens, Bound. Value Probl , -9. [7] T. Diagana; Double-weighed pseudo almos periodic funcions, Afr. Diaspora J. Mah. 2 2, [8] T. Diagana; Exisence of doubly-weighed pseudo almos periodic soluions o non-auonomous differenial equaions, Elecron. J. Differenial Equaions 28 2, -5. [9] W. Dimbour, G. M. N Guérékaa; S-asympoically ω-periodic soluions o some classes of parial evoluion equaions, Appl. Mah. Compu [] H. S. Ding, W. Long, G. M. N Guérékaa; Exisence of pseudo almos periodic soluions for a class of parial funcional differenial equaions, Elecron. J. Differenial Equaions 4 23, -4. [] K. Ezzinbi, H. Toure, I. Zabsonre; Pseudo almos periodic soluions of class r for neural parial funcional differenial equaions, Afrika Ma , [2] E. M. Hernández, H. Henríquez; Exisence of periodic soluions of parial neural funcionaldifferenial equaions wih unbounded delay, J. Mah. Anal. Appl [3] E. Hernández, H. Henríquez; Pseudo-almos periodic soluions for non-auonomous neural differenial equaions wih unbounded delay, Nonlinear Anal. Real World Appl [4] E. M. Hernández, M. L. Pelicer; Asympoically almos periodic and almos periodic soluions for parial neural differenial equaions, Appl. Mah. Le [5] H. Henríquez, M. Pierri, P. Táboas; Exisence of S-asympoically ω-periodic soluions for absrac neural equaions, Bull. Ausral. Mah. Soc , [6] A. Lunardi; Analyic Semigroups and Opimal Regulariy in Parabolic Problems, vol.6 of Progress in Nonlinear Differenial Equaions and Their Applicaions, Birkhäuser, Basel, Seizerland, 995. [7] A. Pankov; Bounded and Almos Periodic Soluions of Nonlinear Operaor Differenial Equaions, Kluwer, Dordrech, 99. [8] M. Pierri, V. Rolnik; On pseudo S-asympoically priodic funcions, Bull. Aus. Mah. Soc ,

17 EJDE-24/9 WEIGHTED PSEUDO PERIODIC SOLUTIONS 7 [9] Z. N. Xia; Weighed Sepanov-like pseudoperiodiciy and applicaions, Absr. Appl. Anal. Volume 24, Aricle ID 98869, 4 pages. [2] R. Zhang, Y. K. Chang, G. M. N Guérékaa; Weighed pseudo almos auomorphic soluions for non-auonomous neural funcional differenial equaions wih infinie delay in Chinese, Sci. Sin. Mah , [2] L. Zhang, H. Li; Weighed pseudo-almos periodic soluions for some absrac differenial equaions wih uniform coninuiy, Bull. Aus. Mah. Soc. 82 2, Zhinan Xia Deparmen of Applied Mahemaics, Zhejiang Universiy of Technology, Hangzhou, Zhejiang, 323, China address: xiazn299@zju.edu.cn