CUBO A Mathematical Journal o Vol.16, N¯02, (01–31). June 2014 Pseudo-Almost Periodic and Pseudo-Almost Automorphic Solutions to Some Evolution Equations Involving Theoretical Measure Theory Toka Diagana Khalil Ezzinbi Mohsen Miraoui Howard University, Université Cadi Ayyad, Institut supérieur des Etudes 2441 6th Street N.W., Faculté des Sciences Semlalia, technologiques de Kairouan, Washington, D.C. 20059, Département de Rakkada-3191 Kairouan, USA. Mathématiques, Tunisie. tdiagana@howard.edu BP 2390, Marrakesh, Maroc ABSTRACT Motivatedbytherecentworksbythefirstandthesecondnamedauthors,inthispaper we introduce the notion of doubly-weighted pseudo-almost periodicity (respectively, doubly-weighted pseudo-almost automorphy) using theoretical measure theory. Basic properties of these new spaces are studied. To illustrate our work, we study, under Acquistapace–Terreni conditions and exponential dichotomy, the existence of (µ,ν)- pseudo almost periodic (respectively, (µ,ν)-pseudo almost automorphic) solutions to some nonautonomous partial evolution equations in Banach spaces. A few illustrative examples will be discussed at the end of the paper. RESUMEN Motivado por los trabajos recientes del primer y segundo autor, en este art´ıculo intro- ducimoslanocióndeseudo-casiperiodicidadcondoblepeso(seudo-casiautomorf´ıacon doble peso respectivamente) usando Teor´ıa de la Medida. Se estudian las propiedades básicasde estos espaciosnuevos. Parailustrar nuestrotrabajo,bajo las condicionesde Acquistapace-Terreni y dicotom´ıa exponencial estudiamos la existencia de soluciones (respectivamente, (µ,ν) seudo-casi periódicas (µ,ν) seudo-casi automórficas) para al- gunas ecuaciones parciales de evolución autónomas en espacios de Banach. Algunos ejemplos ilustrativos se discutirán al final del art´ıculo. Keywords and Phrases: Evolution family; exponential dichotomy; Acquistapace–Terreni conditions; pseudo-almost periodic; pseudo-almost automorphic; evolution equation; nonautonomous equation; doubly-weighted pseudo-almost periodic; doubly-weighted pseudo-almost automorphy; (µ,ν)-pseudo-almostperiodicity;(µ,ν)-pseudo-almostautomorphy;neutralsystems;positivemea- sure. 2010 AMS Mathematics Subject Classification: 34C27; 34K14; 34K30; 35B15; 43A60; 47D06; 28Axx; 58D25; 65J08. CUBO 2 Toka Diagana, Khalil Ezzinbi & Mohsen Miraoui 16,2(2014) 1 Introduction Motivated by the recent works by Ezzinbi et al. [12, 13] and Diagana [30], in this paper we make extensiveuseoftheoreticalmeasuretheoryto introduceandstudythe conceptofdoubly-weighted pseudo almost periodicity (respectively, doubly-weighted pseudo almost automorphy). Obviously, thesenewnotionsgeneralizeallthedifferentnotionsofweightedpseudo-almostperiodicity(respectively,weightedpseudo-almostautomorphy)recentlyintroducedintheliterature. Incontrastwith [12, 13],here the idea consistsofusing twopositivemeasuresinsteadofone. Doingso willprovide us alargerandricherclassofweightedergodicspaces. Basicpropertiesofthesenew functionswill be studied including their translation invariance and compositions etc. To illustrate our study, we study the existence of (µ,ν)-pseudo-almost periodic (respectively, (µ,ν)-pseudo-almostautomorphic)solutionstothefollowingnonautonomousdifferentialequations, d u(t)=A(t)u(t)+F(t,u(t)), t R, (1.1) dt ∈ and d u(t)−G(t,u(t)) =A(t) u(t)−G(t,u(t)) +F(t,u(t)), t R, (1.2) dt! " ! " ∈ where A(t) : D(A(t)) X X for t R is a family of closed linear operators on a Banach ⊂ # ∈ space X, satisfying the well-known Acquistapace–Terreni conditions, and F,G : R X X are × # jointly continuous functions →satisfying some additional conditions. One should indicate that the autonomous case, i.e., A(t) = A for all t R, and the periodic case, that is, A(t+θ) =→A(t) for ∈ some θ > 0, have been extensively studied, see [8, 10, 40, 41, 53, 56] for the almost periodic case and [18, 22, 39, 42, 50, 51] for the almost automorphic case. Recently, Diagana [24, 25, 26, 32] studied the existence and uniqueness of weighted pseudo-almost periodic and weighted pseudo- almost automorphic solutions to some classes of nonautonomous partial evolution equations of type Eq. (1.1). Similarly,inDiagana[33],the existence ofpseudo-almostperiodic solutionsto Eq. (1.2)hasbeenstudiedintheparticularcasewhenG=0. Inthispaperitgoesbacktostudyingthe existenceofdoubly-weightedpseudo-almostperiodic(respectively,doubly-weightedpseudo-almost automorphic) solutions in the general case as outlined above using theoretical measure theory. The existence and uniqueness of almost periodic, almost automorphic, pseudo-almost periodic and pseudo-almost automorphic solutions is one of the most attractive topics in the quali- tative theory of ordinary or functional differential equations due to applications in the physical sciences, mathematical biology, and control theory. The concept of almost automorphy, which was introduced by Bochner [15], is an important generalization of the classical almost periodicity in the sense of Bohr. For basic results on almost periodic and almost automorphic functions we refer the reader to [7, 59, 61], where the authors give an important overview about their applications to differential equations. In recent years, the existence of almost periodic, pseudo-almost periodic, almost automorphic, and pseudo-almost automorphic solutions to dif- ferent kinds of differential equations have been extensively investigated by many people, see, e.g., CUBO Pseudo-Almost Periodic and Pseudo-Almost Automorphic ... 3 16,2(2014) [3, 4, 5, 16, 17, 19, 20, 30, 23, 24, 32, 33, 34, 35, 36, 37, 39, 43, 44, 45, 46, 48, 57, 58, 60] and the references therein. The concept of weighted pseudo-almost periodicity, which was introduced by Diagana [25, 26, 27, 29] is a natural generalization of the classical pseudo-almost periodicity due to Zhang [59, 60, 61]. A few years later, Blot et al. [11], introduced the concept of weighted pseudo-almost automorphy as a generalization of weighted pseudo-almost periodicity. More recently, Ezzinbi et al. [12, 13] presented a new approach to study weighted pseudo-almost periodic and weighted pseudo-almostautomorphicfunctionsusingtheoreticalmeasuretheory,whichturnsouttobemore general than Diagana’s approach. Let us explain the meaning of this notion as introduced by Ezzinbi et al.et al. [12, 13]. Let µ be a positive measure on R. We say that a continuous function f : R X is µ-pseudo-almost # periodic (respectively, µ-pseudo almost automorphic) if f = g+ϕ, where g is almost periodic (respectively, almost automorphic) and ϕ is ergodic with respect to the→measure µ in the sense that 1 lim ϕ(s) dµ(s)=0, r→∞µ(Qr)"Qr∥ ∥ where Q :=[−r,r] and µ(Q ):= dµ(t). r r "Qr One can observe that a ρ-weighted pseudo almost automorphic function is µ-pseudo almost automorphic, where the measure µ is absolutely continuous with respect to the Lebesgue measure and its RadonNikodym derivative is ρ, dµ(t) =ρ(t). dt Here we generalize the above-mentioned notion of µ-pseudo-almost periodicity. Fix two positive measures µ,ν in R. We say that a function f : R X is (µ,ν)-pseudo-almost periodic # (respectively, (µ,ν)-pseudo-almost automorphic) if → f=g+ϕ, where g is almost periodic (respectively, almost automorphic) and ϕ is (µ,ν)-ergodic in the sense that 1 lim ϕ(s) dµ(s)=0, r→∞ν(Qr)"Qr∥ ∥ Clearly,the(µ,µ)-pseudo-almostperiodicitycoincideswiththeµ-pseudo-almostperiodicity. More generally,the(µ,ν)-pseudo-almostperiodicitycoincideswiththeµ-pseudo-almostperiodicitywhen the measures µ and ν are equivalent. In this paper, we introduce and study properties of (µ,ν)-pseudo-almost periodic functions andmakeuseofthesenewfunctionstostudytheexistenceanduniquenessof(µ,ν)-pseudo-almost periodic (respectively,(µ,ν)- pseudo-almostautomorphic)solutionsofthe nonautonomouspartial evolution equations Eq. (1.1) and Eq. (1.2) in a Banach space. CUBO 4 Toka Diagana, Khalil Ezzinbi & Mohsen Miraoui 16,2(2014) The organization of this paper is as follows. In Section 2, we recall some definitions and lemmas of (µ,ν)-pseudo almost periodic functions, (µ,ν)-pseudo-almost automorphic functions, and the basic notations of evolution family and exponential dichotomy. In Section 3, we study the existence and uniqueness of (µ,ν)-pseudo almost periodic (respectively, (µ,ν)-pseudo almost automorphic) solutions to both Eq. (1.1) and Eq. (1.2). In Section 4, we give some examples to illustrate our abstract results. 2 Preliminaries 2.1 (µ,ν)-Pseudo-Almost Periodic and (µ,ν)-Pseudo-Almost Automor- phic Functions Let (X, ), (Y, ) be two Banach spaces and let BC(R,X) (respectively, BC(R Y,X)) be ∥·∥ ∥·∥ × the space of bounded continuous functions f : R X (respectively, jointly bounded continuous − functions f:R Y X). Obviously, the space BC(R,X) equipped with the super norm × − → → f ∞ :=sup f(t) ∥ ∥ t∈R∥ ∥ is a Banach space. Let B(X,Y) denote the Banach spaces of all bounded linear operator from X into Y equipped with natural topology with B(X,X)=B(X). Definition 2.1. [21] A continuous function f : R X is said to be almost periodic if for every # ε>0 there exists a positive number l(ε) such that every interval of length l(ε) contains a number τ such that → f(t+τ)−f(t) <ε for t R. ∥ ∥ ∈ Let AP(R,X) denote the collection of almost periodic functions from R to X. It can be easily shown that (AP(R,X), ∞) is a Banach space. ∥·∥ Definition 2.2. [38] A jointly continuous function f:R Y X is said to be almost periodic in × # t uniformly for y Y, if for every ε > 0, and any compact subset K of Y, there exists a positive ∈ number l(ε) such that every interval of length l(ε) contains a→number τ such that f(t+τ,y)−f(t,y) <ε for (t,y) R K. ∥ ∥ ∈ × We denote the set of such functions as APU(R Y,X). × Let µ,ν . If f : R X is a bounded continuous function, we define its doubly-weighted ∈ M # mean, if the limit exists, by → 1 (f,µ,ν):= lim f(t)dµ(t). M r→∞ν(Qr)"Qr CUBO Pseudo-Almost Periodic and Pseudo-Almost Automorphic ... 5 16,2(2014) It is well-known that if f AP(R,X), then its mean defined by ∈ 1 (f):= lim f(t)dt M r→∞2r "Qr exists [15]. Consequently, for every λ R, the following limit ∈ 1 a(f,λ):= lim f(t)e−iλtdt r→∞2r "Qr exists and is called the Bohr transform of f. Itiswell-knownthata(f,λ)isnonzeroatmostatcountablymanypoints[15]. Thesetdefined by σ (f):= λ R:a(f,λ)=0 b ∈ ̸ # $ is called the Bohr spectrum of f [47]. Theorem2.3. [47]Letf AP(R,X). Thenforeveryε>0thereexistsatrigonometricpolynomial ∈ n Pε(t)= akeiλkt k=1 % where a X and λ σ (f) such that f(t)−P (t) <ε for all t R. k k b ε ∈ ∈ ∥ ∥ ∈ µ(Q ) Theorem 2.4. Let µ,ν and suppose that lim r = θ . If f : R X is an almost ∈ M r→∞ν(Qr) µν # periodic function such that → 1 lim eiλtdµ(t) =0 (2.1) r→∞###ν(Qr)"Qr ### # # for all 0=λ σ (f), then the doubl#y-weighted mean of f,# b ̸ ∈ 1 (f,µ,ν)= lim f(t)dµ(t) M T→∞ν(QT)"QT exists. Furthermore, (f,µ,ν)=θ (f). µν M M Proof. The proof of this theorem was given in [30] in the case of measures of the form ρ(t)dt. For the sake of completeness we reproduce it here for positive measures. If f is a trigonometric polynomial, say, f(t) = nk=0akeiλkt where ak ∈ X−{0} and λk ∈ R for k = 1,2,...,n, then σ (f)={λ : k=1,2,...,n}. Moreover, b k & 1 µ(Q ) 1 n f(t)dµ(t) = a r + a eiλkt dµ(t) ν(Q ) 0ν(Q ) ν(Q ) k r "Qr r r "Qr$k=1 % % µ(Q ) n 1 = a r + a eiλktdµ(t) 0ν(Q ) k ν(Q ) r k=1 $ r "Qr % % CUBO 6 Toka Diagana, Khalil Ezzinbi & Mohsen Miraoui 16,2(2014) and hence n 1 µ(Q ) 1 &&&ν(Qr)"Qrf(t)dµ(t)−a0ν(Qrr)&&& ≤ k%=1∥ak∥###ν(Qr)"Qreiλktdµ(t)### & & which by Eq. (2.1) yields 1 f(t)dµ(t)−a θ 0 as r &ν(Q ) 0 µν& & r "Qr & & & and therefore (f,µ,ν)&=a0θµν =θµνM(f). &→ →∞ M If in the finite sequence of λ there exist λ = 0 for k = 1,2,...l with a X−{0} for all k nk m ∈ m=n (k=1,2,...,l), it can be easily shown that k ̸ l (f,µ,ν)=θ a =θ (f). M µν nk µνM k=1 % Now if f:R X is an arbitrary almost periodic function, then for every ε>0 there exists a # trigonometric polynomial (Theorem 2.3) P defined by ε → n Pε(t)= akeiλkt k=1 % where a X and λ σ (f) such that k k b ∈ ∈ f(t)−P (t) <ε (2.2) ε ∥ ∥ for all t R. ∈ Proceeding as in Bohr [15] it follows that there exists r such that for all r ,r >r , 0 1 2 0 1 1 P (t)dµ(t)− P (t)dµ(t) =θ (P )− (P ) =0<ε. &&ν(Qr1)"Qr1 ε ν(Qr2)"Qr2 ε && µν&&M ε M ε && & & & & In view of the above it follows that for all r ,r >r , 1 2 0 1 1 1 f(t)dµ(t) − P (t)dµ(t) f(t)−P (t) dµ(t) &&ν(Qr1)"Qr1 ν(Qr2)"Qr2 ε &&≤ ν(Qr1)"Qr1 ∥ ε ∥ & & 1 1 + P (t)dµ(t)− P (t)dµ(t) <ε. &&ν(Qr1)"Qr1 ε ν(Qr2)"Qr2 ε && & & Now for all r>r , 0 1 1 f(t)dµ(t)− P (t)dµ(t) <ε &&ν(Qr)"Qr ν(Qr)"Qr ε && & & and hence (f,µ,ν)= (P ,µ,ν)=θ (P )=θ (f). The proof is complete. ε µν ε µν M M M M CUBO Pseudo-Almost Periodic and Pseudo-Almost Automorphic ... 7 16,2(2014) Definition 2.5. [51] A continuous function f : R X is called almost automorphic if for every sequence (σn)n∈N there exists a subsequence (sn)n∈N (σn)n∈N such that ⊂ → lim f(t+s −s )=f(t) for each t R. n,m→∞ n m ∈ Equivalently, g(t):= lim f(t+s ) and f(t)= lim g(t−s ) n→∞ n n→∞ n are well defined for each t R. ∈ Let AA(R,X) denote the collection of all almost automorphic functions from R to X. It can be easily shown that (AA(R,X), . ∞) is a Banach space. ∥∥ Definition 2.6. [13] A function f : R X Y is said to be almost automorphic in t uniformly × with respect to x in X if the following two conditions hold: (i) for all x X, f(.,x) AA(R,Y), → ∈ ∈ (ii) f is uniformly continuous on each compact set K in X with respect to the second variable x, namely, for each compact set K in X, for all ε>0, there exists δ>0 such that for all x , x K, 1 2 ∈ one has x −x δ sup f(t,x )−f(t,x ) ε. 1 2 1 2 ∥ ∥≤ t∈R∥ ∥≤ Denote by AAU(R X,Y) the set of all such functions. ⇒ × Remark2.7. [13]Notethatintheabovelimitthefunctiongisjustmeasurable. Iftheconvergence in both limits is uniform in t R, then f is almost periodic. The concept of almost automorphy is ∈ then larger than almost periodicity. If fis almost automorphic, thenits range is relatively compact, thus bounded in norm. Example 2.8. [49] Let k: R R be such that 1 k(t)→=sin , t R. !2+cos(t)+cos(√2t)" ∈ Then k is almost automorphic, but it is not uniformly continuous on R. Then, it is not almost periodic. In what follows, we introduce a new concept of ergodicity, which will generalize those given in [12] and [29, 31]. Let denote the Lebesque σ-field of R and let be the set of all positive measures µ on B M B satisfying µ(R)=+ and µ([a,b])< , for all a,b R (a b). ∈ ≤ Definition 2.9. [12] Let µ,ν . The measures µ and ν are said to be equivalent there exist ∞ ∈ M ∞ constants c ,c >0 and a bounded interval Ω R (eventually ) such that 0 1 ⊂ ∅ c ν(A) µ(A) c ν(A) 0 1 ≤ ≤ for all A satisfying A Ω= . ∈B ∩ ∅ CUBO 8 Toka Diagana, Khalil Ezzinbi & Mohsen Miraoui 16,2(2014) We introduce the following new space. Definition 2.10. Let µ,ν . A bounded continuous function f : R X is said to be (µ,ν)- ∈ M ergodic if 1 lim f(s) dµ(s)=0. → r→∞ν(Qr)"Qr∥ ∥ We then denote the collection of all such functions by (R,X,µ,ν). E We are now ready to introduce the notion of (µ,ν)-pseudo-almost periodicity (respectively, (µ,ν)-pseudo-almost automorphy) for two positive measures µ,ν . ∈M Definition 2.11. Let µ,ν . A continuous function f : R X is said to be (µ,ν)-pseudo ∈ M almost periodic if it can be written in the form → f=g+h, where g AP(R,X) and h (R,X,µ,ν). The collection of such functions is denoted by ∈ ∈ E PAP(R,X,µ,ν). Definition 2.12. Let µ,ν . A continuous function f : R X is said to be (µ,ν)-pseudo ∈ M almost automorphic if it can be written in the form → f=g+h, where g AA(R,X) and h (R,X,µ,ν). The collection of such functions will be denoted by ∈ ∈ E PAA(R,X,µ,ν). We formulate the following hypotheses. (M.1) Let µ,ν such that ∈M µ(Q ) limsup r < . (2.3) r→∞ ν(Qr) ∞ (M.2) For all τ R, there exist β>0 and a bounded interval I such that ∈ µ({a+τ: a A}) βµ(A) when A satisfies A I= . ∈ ≤ ∈B ∩ ∅ Theorem2.13. Letµ,ν satisfy(M.2). ThenthespacesPAP(R,X,µ,ν)andPAA(R,X,µ,ν) ∈M are translation invariants. Proof. We show that (R,X,µ,ν) is translationinvariant. Let f (R,X,µ,ν), we will show that E ∈E t f(t+s) belongs to (R,X,µ,ν) for each s R. # E ∈ → CUBO Pseudo-Almost Periodic and Pseudo-Almost Automorphic ... 9 16,2(2014) Indeed, letting µ = µ({t+s : t A}) for A it follows from (M.2) that µ and µ are s s ∈ ∈ B equivalent (see [12]). Now 1 ν(Q ) 1 f(t+s) dµ(t) = r+|s| . f(t+s) dµ(t) ν(Q ) ∥ ∥ ν(Q ) ν(Q ) ∥ ∥ r "Qr r r+|s| "Qr ν(Q ) 1 = r+|s| . f(t) dµ (t) ν(Q ) ν(Q ) ∥ ∥ −s r r+|s| "Qr+|s| ν(Q ) cst. r+|s| . f(t) dµ(t). ≤ ν(Q ) ν(Q ) ∥ ∥ r r+|s| "Qr+|s| Since ν satisfies (M.2) and f (R,X,µ,ν), we have ∈E 1 lim f(t+s) dµ(t)=0. r→∞ν(Qr)"Qr∥ ∥ Therefore, (R,X,µ,ν) is translation invariant. Since AP(R,X) and AA(R,X) are translation E invariants, then PAP(R,X,µ,ν) and PAA(R,X,µ,ν) are translation invariants. Theorem 2.14. Let µ,ν satisfy (M.1), then ( (R,X,µ,ν), . ∞) is a Banach space. ∈M E ∥∥ Proof. It is clear that ( (R,X,µ,ν) is a vector subspace of BC(R,X). To complete the proof, it is E enough to prove that ( (R,X,µ,ν) is closed in BC(R,X). If (f ) be a sequence in ( (R,X,µ,ν) n n E E such that lim f =f n→∞ n uniformly in R. From ν(R)= , it follows ν(Q )>0 for r sufficiently large. Using the inequality r f(t) dµ(t) f(t)−f (t) dµ(t)+ f (t) dµ(t) ∞ ∥ ∥ ≤ ∥ n ∥ ∥ n ∥ "Qr "Qr "Qr we deduce that 1 µ(Q ) 1 ν(Q ) ∥f(t)∥dµ(t)≤ ν(Qr)∥f−fn∥∞+ ν(Q ) ∥fn(t)∥dµ(t), r "Qr r r "Qr then from (M.1) we have 1 lirm→s∞upν(Qr)"Qr∥f(t)∥dµ(t)≤cst.∥f−fn∥∞ for all n∈N. Since nl→im∞∥f−fn∥∞ =0, we deduce that 1 lim f(t) dµ(t)=0. r→∞ν(Qr)"Qr∥ ∥ CUBO 10 Toka Diagana, Khalil Ezzinbi & Mohsen Miraoui 16,2(2014) Lemma 2.15. [13] Let g AA(R,X) and ε > 0 be given. Then there exist s ,...,s R such 1 m ∈ ∈ that i=1 R= (s +C ), whereC :={t R: g(t)−g(0) <ε}. i ε ε ∈ ∥ ∥ ’m Theorem 2.16. Let µ,ν and f PAA(R,X,µ,ν) be such that ∈M ∈ f=g+φ, where g AA(R,X) and φ (R,X,µ,ν). If PAA(R,X,µ,ν) is translation invariant, then ∈ ∈E {g(t); t R} {f(t); t R},(theclosureoftherangeoff). (2.4) ∈ ⊂ ∈ Proof. The proofis similar to the one givenin [13]. Indeed, if we assume that (2.4) does not hold, then there exists t R such that g(t ) is not in {f(t); t R}. Since the spaces AA(R,X) and 0 0 ∈ ∈ (R,X,µ,ν) are translation invariants, we can assume that t = 0, then there exists ε > 0 such 0 E that f(t)−g(0) >2ε for all t R. Then we have ∥ ∥ ∈ φ(t) = f(t)−g(t) f(t)−g(0) − g(t)−g(0) ε ∥ ∥ ∥ ∥≥∥ ∥ ∥ ∥≥ for all t C . Therefore, ε ∈ φ(t−s ) ε, foralli {1,...,m}, and t s +C . i i ε ∥ ∥≥ ∈ ∈ Let φ be the function defined by i=m φ(t):= φ(t−s ) . i ∥ ∥ i=1 % From Lemma 2.15, we deduce that φ(t) ε forallt R. (2.5) ∥ ∥≥ ∈ Since (R,X,µ,ν) is translationinvariant,then [t φ(t−s )] (R,X,µ,ν) for all i {1,...,m}, i E ∈E ∈ then φ (R,X,µ,ν) which is a contradiction. Consequently (2.4) holds. ∈E → Theorem 2.17. Let µ,ν satisfy (M.2), then the decomposition of a (µ,ν)-pseudo almost ∈ M automorphic function in the form f=g+h, where g AA(R,X) and h (R,X,µ,ν), is unique. ∈ ∈E Proof. Suppose that f=g +φ =g +φ , where g ,g AA(R,X) and φ ,φ (R,X,µ,ν). 1 1 2 2 1 2 1 2 ∈ ∈E Then 0 = (g −g )+(φ −φ ) PAA(R,X,µ,ν) where g −g AA(R,X) and φ −φ 1 2 1 2 1 2 1 2 ∈ ∈ ∈ (R,X,µ,ν). From Theorem 2.16, we obtain (g −g )(R) {0}, therefore we have g = g and 1 2 1 2 E ⊂ φ =φ . 1 2 From Theorem 2.17, we deduce

Pseudo-Almost Periodic and Pseudo-Almost Automorphic Solutions to Some Evolution Equations ...

Similar Pseudo-Almost Periodic and Pseudo-Almost Automorphic Solutions to Some Evolution Equations ...