CUBO A Mathematical Journal Vol.16, No¯ 01, (21–35). March 2014 Weighted pseudo almost automorphic solutions of fractional functional differential equations Syed Abbas School of Basic Sciences, Indian Institute of Technology Mandi, Mandi, H.P. - 175001, India. sabbas.iitk@gmail.com, abbas@iitmandi.ac.in ABSTRACT In this paper we discuss the existence of weighted pseudo almost automorphic solution of fractional order functional differential equations. Using the fixed point theorem we establish existence and uniqueness of solution to the problem under consideration. The results obtained extend the theory of almost automorphic solutions to a more general class of weighted pseudo almost automorphic solutions. These extensions allow to treat infinite dimensional dynamics such as fractional wave and heat equation which are presented in the paper. At the end we give several example to illustrate the analytical findings. RESUMEN En este art́ıculo discutimos la existencia de una solución seudo casi automórfica con peso de ecuaciones diferenciales funcionales de orden fraccional. Usando el teorema del punto fijo, establecemos la existencia y unicidad de la solución del problema en estudio. Los resultados obtenidos extienden la teoŕıa de soluciones casi automórficas a clases más generales de soluciones seudo casi automórficas con peso. Estas extensiones permiten estudiar dinámicas infinito-dimensional como la onda fraccionaria y la ecuación del calor, las cuales se presentan en este art́ıculo. Al final, mostramos varios ejemplos para ilustrar los resultados anaĺıticos obtenidos. Keywords and Phrases: Fractional differential equation, Fixed point theorem, Almost auto- morphic functions, Abstract differential equations. 2010 AMS Mathematics Subject Classification: 34K40, 34K14. 22 Syed Abbas CUBO 16, 1 (2014) 1 Introduction In this work we consider the following functional differential equations of fractional order α ∈ (1,2), Dαt u(t) = Au(t) + D α−1 t f(t,u(t),ut), t ∈ R, u(t) = φ(t), t ∈ (−∞,0], ut(θ) = u(t + θ), θ ∈ (−∞,0], (1) where f : R × X × X → X, φ ∈ C0((−∞,0],R) and A : D(A) ⊂ X → X is a linear densely defined operator of sectorial type on a complex Banach space X. With motivation coming from a wide range of engineering and physical applications, fractional differential equations have recently attracted great attention of mathematicians and scientists. This kind of equations is a generalization of ordinary differential equations to arbitrary non integer orders. The origin of fractional calculus goes back to Newton and Leibniz in the seventieth century. It is widely and efficiently used to describe many phenomena arising in engineering, physics, economy and science. Recent investigations have shown that many physical systems can be represented more accurately through fractional derivative formulation [36]. Fractional differential equations find numerous applications in the field of visco- elasticity, feedback amplifiers, electrical circuits, electro analytical chemistry, fractional multipoles, neuron modelling encompassing different branches of physics, chemistry and biological sciences [39]. Many physical processes appear to exhibit fractional order behavior that may vary with time or space. The fractional calculus has allowed the operations of integration and differentiation to any fractional order. The order may take on any real or imaginary value. The existence and uniqueness of solutions to fractional differential equations have been shown by many authors [1, 2, 4, 5, 7, 11, 12, 16, 9, 10, 17, 18, 23, 37, 28, 31, 32, 37, 39]. Agarwal et. al. [6] have shown the existence of weighted pseudo almost periodic solutions of semilinear fractional differential equations. Since Bohr [15] introduced the concept of almost periodic functions, there have been many important generalizations of this functions in the past few decades. The generalization includes pseudo almost periodic functions [41], where the function can be decomposable in two part. These functions are further generalized to weighted pseudo almost periodic function by Diagana, where the weighted mean of the second component is zero [20]. Another direction of generalization is almost automorphic functions introduced by Bochner [14]. The pseudo almost automorphic functions are natural generalization of almost automorphic functions [14] and introduced by Liang et. al. [33]. These functions are further generalized by Blot et.al. [13] and named weighted pseudo almost automorphic. The authors in [13] have proved very important properties of these functions including composition theorem and completeness property. The study of weighted pseudo almost automorphic solutions of various kind of differential equations are very new and an attractive area of research. For more details on theory and applications of these functions we refer to [13] and references therein. The existence and uniqueness of almost automorphic and pseudo almost automorphic solutions have been established by many authors, for example [3, 13, 26] and references therein. CUBO 16, 1 (2014) Weighted pseudo almost automorphic solutions of fractional . . . 23 The problem considered in this work is motivated by the work of Claudio Cuevas and Carlos Lizama [17] work in which they have considered the following fractional differential equations Dαt u(t) = Au(t) + D α−1 t f(t,u(t)), t ∈ R, (2) and proved the existence of almost automorphic solutions under certain assumptions. In this paper we discuss existence and uniqueness of weighted pseudo almost automorphic solutions of problem (1). The concept of Stepanov like pseudo almost periodicity is introduced by Diagana [21, 22], which is a generalization of pseudo almost periodicity. Further Stepanov like almost automorphy has been introduced by N’Guerekata and Pankov [29]. 2 Preliminaries Denote B(X) be the Banach space of all linear and bounded operators on X endowed with the norm ‖ · ‖B(X) and C = C(R,X) the set of all continuous functions from R to X. Let U the collection of all positive integrable functions ρ : R → R. For each ρ ∈ U define m(r,ρ) = ∫r −r ρ(s)ds. Denote U∞ : The set of all ρ ∈ U such that limr→∞ m(r,ρ) = ∞ Ub : The set of all bounded ρ ∈ U∞ such that inft∈R ρ(t) > 0. Now we state the definitions of weighted almost automorphic functions. Definition 2.1. A continuous function f : R → X is called almost automorphic if for every real sequence (sn), there exists a subsequence (snk) such that g(t) = lim n→∞ f(t + snk) is well defined for each t ∈ R and lim n→∞ g(t − snk) = f(t) for each t ∈ R. The set of all almost automorphic functions from R to X are denoted by AA(X). The set of all almost automorphic functions from R to X are denoted by AA(X) and it is a Banach space equipped with the sup norm ‖f‖∞ = sup t∈R ‖f(t)‖. Definition 2.2. A continuous function f : R ×X → R is called almost automorphic in t uniformly for x in compact subsets of X if for every compact subset K of X and every real sequence (sn), there exists a subsequence (snk) such that g(t,x) = lim n→∞ f(t + snk,x) 24 Syed Abbas CUBO 16, 1 (2014) is well defined for each t ∈ R, x ∈ K and lim n→∞ g(t − snk,x) = f(t,x) for each t ∈ R, x ∈ K. Denote by AA(R × X) the set of all such functions. We denote by AA0(X) = { f ∈ BC(R,X) : lim r→∞ 1 m(r,ρ) ∫r −r ρ(ξ)‖f(ξ)‖dξ = 0 } , and by AA0(R × X × X,X) the set of all continuous functions f : R × X × X → X such that f(.,u,φ) ∈ AA0(X) and lim r→∞ 1 m(r,ρ) ∫r −r ρ(ξ)‖f(ξ,u,φ)‖dξ = 0, uniformly in (u,φ) ∈ X × X. Definition 2.3. A mapping f ∈ BC(R,X) is called weighted pseudo almost automorphic if it can be written as f = f1 + f2, where f1 ∈ AA(X) and f2 ∈ AA0(X). The functions f1 and f2 are called the almost automorphic and the weighted ergodic pertur- bation components of f respectively. The set of all such functions will be denoted by PAA(X). Remark 2.4. A classical example of pseudo almost automorphic function is f(t) = sin 1 2 + cost + cos √ 2t + 1 1 + t2 . t ∈ R. One can easily see that this function is not almost periodic. Example: Consider the function f(t) = sin 1 2 + cost + cos √ 2t + eαt It is well known that the function sin 1 2+cos t+cos √ 2t is almost automorphic. Now consider the weight function ρ defined by ρ(t) = 1 t < 0 and ρ(t) = e−βt t ≥ 0 for some β > 0. It is easy to verify that m(r,ρ) = ∫r −r ρ(t)dt = ∫0 −r ρ(t)dt + ∫r 0 ρ(t)dt = r + 1 − e−βr β . Thus limr→∞ m(r,ρ) = ∞ which implies that ρ ∈ U∞. Further ∫r −r eαtρ(t)dt = ∫0 −r eαtdt + ∫r 0 eαte−βtdt = 1 − e−αr α + ∫r 0 e(α−β)tdt = 1 − e−αr α + e(α−β)r − 1 α − β . (3) CUBO 16, 1 (2014) Weighted pseudo almost automorphic solutions of fractional . . . 25 Thus for α ≤ β, we have lim r→∞ 1 m(r,ρ) ∫r −r eαtρ(t)dt = 0. Hence eαt ∈ PAA0(R,ρ) and so f(t) ∈ WPAA(R). It is also interesting to note that lim r→∞ 1 2r ∫r −r eαtdt = lim r→∞ eαr − e−αr 2αr = ∞. This implies that f(t) does not belongs to PAA(R), the space of all pseudo almost automorphic functions. Definition 2.5. A continuous mapping f : R × X × X → X is called weighted pseudo almost automorphic in t ∈ R uniformly in (x,φ) ∈ X × X if it can be written as f = f1 + f2, where f1 ∈ AA(R × X × X,X) and f2 ∈ AA0(R × X × X,X). We denote the set of all weighted pseudo almost automorphic functions f : R × X × X → X by WPAA(R × X × X). The following theorems are from [13]. Theorem 2.6. The decomposition of a weighted pseudo almost automorphic function is unique for any ρ ∈ Ub. Theorem 2.7. Let WPAA(R,ρ) 3 f = g + φ where ρ ∈ U∞ and assume that f(t,u) is uniformly continuous in any bounded subset K of X uniformly in t ∈ R and g(t,u) is uniformly continuous in any bounded subset K of X uniformly in t ∈ R. Then if u ∈ WPAA(R,ρ), implies f(·,u(·)) ∈ WPAA(R,ρ). The above theorem holds if both functions f,g are Lipschitz continuous in u uniformly in t ∈ R. The weight one functions that is ρ = 1, are called pseudo almost automorphic. 3 Weighted pseudo almost automorphic solutions Assumptions: Let us consider the the following assumptions: (A1) The function f : R × X × X → X is weighted pseudo almost automorphic with respect to t uniformly in (u,φ) ∈ X × X, and there exists 0 < L < 1, such that ‖f(t,u,φ) − f(t,v,ψ)‖ ≤ L(‖u − v‖ + ‖φ − ψ‖. (A2) The function f is bounded. Lemma 3.1. Let {S(t)}t>0 ⊂ B(X) be a strongly continuous family of bounded and linear operators such that ‖S(t)‖ ≤ φ(t) for almost all t ∈ R+ with φ ∈ L1(R+). If f : R → X is a weighted pseudo almost automorphic function then ∫t −∞ S(t − s)f(s)ds ∈ WPAA(X). 26 Syed Abbas CUBO 16, 1 (2014) A closed and linear operator A is said to be sectorial of type ω and angle θ if there exists 0 < θ < π 2 ,M > 0 and ω ∈ R such that its resolvent exists outside the sector ω + Sθ := {ω + λ : λ ∈ C, |arg(−λ)| < θ}, and ‖(λ − A)−1‖ ≤ M |λ − ω| , λ 6∈ ω + Sθ. Sectorial operators are well studied in the literature. For a recent reference including several examples and properties we refer the reader to [30]. Note that an operator A is sectorial of type ω if and only if ωI − A is sectorial of type 0. The equation 1 can be thought as a limiting case of the following equation v′(t) = ∫t 0 (t − s)α−2 Γ(α − 1) Av(s)ds + f(t,u(t),ut), t ≥ 0, vt(θ) = φ(t),t ∈ (−∞,0), (4) in the sense that the solutions are asymptotic to each other as t → ∞. If we consider that the operator A is sectorial of type ω with θ ∈ [0,π(1 − α 2 )), then problem 4 is well posed [19]. Thus we can use variation of parameter formulae to get v(t) = Sα(t)u0 + ∫t 0 Sα(t − s)f(s,u(s),us)ds, t ≥ 0, where Sα(t) = 1 2πi ∫ γ eλtλα−1(λαI − A)−1dλ, t ≥ 0, where the path γ lies outside the sector ω+Sθ. If Sα(t) is integrable then the solution is given by u(t) = ∫t −∞ Sα(t − s)f(s,u(s),us)ds. Now one can easily see that v(t) − u(t) = Sα(t)u0 − ∫ ∞ t Sα(s)f(t − s,u(t − s),ut−s). Hence for f ∈ Lp(R+ × X × X,X), p ∈ [1,∞) we have v(t) − u(t) → 0 as t → ∞. Definition 3.2. A function u : R → X is said to be a mild solution to 1 if the function Sα(t − s)f(s,u(s),us) is integrable on (−∞,t) for each t ∈ R and u(t) = ∫t −∞ Sα(t − s)f(s,u(s),us)ds, for each t ∈ R. CUBO 16, 1 (2014) Weighted pseudo almost automorphic solutions of fractional . . . 27 Recently, Cuesta in [19], Theorem-1, has proved that if A is a sectorial operator of type ω < 0 for some M > 0 and θ ∈ [0,π(1 − α 2 )), then there exists C > 0 such that ‖Sα(t)‖ ≤ CM 1 + |ω|tα for t ≥ 0. Also the following relation [17], Theorem-3.4 holds, ∫ ∞ 0 dt 1 + |ω|tα = |ω| −1 α π αsin π α for α ∈ (1,2). Define the operator Fu(t) = ∫t −∞ Sα(t − s)f(s,u(s),us)ds, t ∈ R. First thing we observe about the operator F is boundedness and continuity. Indeed, ‖Fu‖ ≤ ∫t −∞ ‖Sα(t − s)‖ × ‖f(s,u(s),us)‖ds ≤ ∫ ∞ 0 ‖Sα(s)‖‖f(t − s,u(t − s),ut−s)‖ds ≤ CM ∫ ∞ 0 1 1 + |ω|sα ‖f(t − s,u(t − s),ut−s)‖ds ≤ CM‖f‖ ∫ ∞ 0 1 1 + |ω|sα ds = CM‖f‖ω− 1α π αsin π α (5) Thus F is bounded. Further, we have ‖Fu(t + h) − Fu(t)‖ = ∥ ∥ ∥ ∫t+h −∞ Sα(t + h − s)f(s,u(s),us)ds − ∫t −∞ Sα(t − s)f(s,u(s),us)ds ∥ ∥ ∥ ≤ ∫t −∞ ‖Sα(t − s)‖ × ‖f(s + h,u(s + h),us+h) − f(s,u(s),us)‖ds ≤ ∫ ∞ 0 ‖Sα(s)‖ × ‖f(t − s + h,u(t − s + h),ut−s+h) − f(t − s,u(t − s),ut−s)‖ds ≤ CMsup t∈R ‖f(t − s + h,u(t − s + h),ut−s+h) − f(t − s,u(t − s),ut−s)‖ × ∫ ∞ 0 1 1 + |ω|sα ds = CMω− 1 α π αsin π α sup t∈R ‖f(t − s + h,u(t − s + h),ut−s+h) − f(t − s,u(t − s),ut−s)‖, (6) 28 Syed Abbas CUBO 16, 1 (2014) which goes to zero as h → 0 and hence F is continuous. It is easy to see that the operator F maps WPAA(X) to WPAA(X), which we represent in the form of a lemma as follows. Lemma 3.3. The operator F maps WPAA(X) to WPAA(X) if f ∈ WPAA(X). Proof: As f ∈ WPAA(X), we can decompose it into two part f1 ∈ AA(X) and f2 ∈ AA0(X). Now define the operators F1u(t) = ∫t −∞ Sα(t − s)f1(s,u(s),us)ds, t ∈ R and F2u(t) = ∫t −∞ Sα(t − s)f2(s,u(s),us)ds, t ∈ R. Also for every sequence tn there exists a subsequence tnk such that f1(t + tnk,u,ψ) → g1(t,u,ψ), g1(t − tnk,u,ψ) → f1(t,u,ψ), u,ψ ∈ D, where D is a compact subset of X × X. F1u(t + tnk) = ∫t+tn k −∞ Sα(t + tnk − s)f1(s,u(s),us)ds = ∫t −∞ Sα(t − s)f1(s + tnk,u(s + tnk),u(s+tn k ))ds → ∫t −∞ Sα(t − s)g1(s,u(s),us)ds = (F∗u)(t). (7) Thus (F1u)(t + tnk) → (F ∗u)(t). Similarly one can get (F∗u)(t − tnk) → (F1u)(t). Now we need to show lim r→∞ 1 m(r,ρ) ∫r −r ∫t −∞ ρ(s)|Sα(t − s)|‖f2(s,u(s),us)‖dsdt = 0. Consider 1 m(r,ρ) ∫r −r ∫t −∞ ρ(s)‖Sα(t − s)‖‖f2(s,u(s),us)‖dsdt ≤ I1(r) + I2(r), where I1(r) = 1 m(r,ρ) ∫r −r dt ∫t −r ρ(s)‖Sα(t − s)‖‖f2(s,u(s),us)‖ds CUBO 16, 1 (2014) Weighted pseudo almost automorphic solutions of fractional . . . 29 and I2(r) = 1 m(r,ρ) ∫r −r dt ∫−r −∞ ρ(s)‖Sα(t − s)‖‖f2(s,u(s),us)‖ds. Thus we have I1(r) ≤ 1 m(r,ρ) ∫r −r ρ(ξ)‖f2(ξ,u(ξ),uξ)‖dξ ∫r s ‖Sα(t − ξ)‖dt ≤ 1 m(r,ρ) ∫r −r ρ(ξ)‖f2(ξ,u(ξ),uξ)‖dξ ∫r−s 0 ‖Sα(t)‖dt ≤ 1 m(r,ρ) ∫r −r ρ(ξ)‖f2(ξ,u(ξ),uξ)‖dξ ∫ ∞ 0 ‖Sα(t)‖dt ≤ CM m(r,ρ) ∫r −r ρξ‖f2(ξ,u(ξ),uξ)‖dξ |ω| −1 α π αsin π α ≤ M1 m(r,ρ) ∫r −r ρ(ξ)‖f2(ξ,u(ξ),uξ)‖dξ, (8) for some positive constant M1. The above calculations imply that lim r→∞ I1(r) = 0 as f2 ∈ AA0(R × X × X). Now consider I2(r) ≤ 1 m(r,ρ) ∫r −r dt ∫ ∞ t+r ρ(t − s)‖Sα(s)‖‖f2(t − s,u(t − s),ut−s)‖ds ≤ 1 m(r,ρ) ∫r −r dt ∫ ∞ 2r ρ(t − s)‖Sα(s)‖‖f2(t − s,u(t − s),ut−s)‖ds ≤ ‖f2‖∞ ∫ ∞ 2r ‖Sα(s)‖ds. (9) From the above analysis we get lim r→∞ I2(r) = 0. Thus we have lim r→∞ 1 2r ∫r −r ‖F2(u)(t)‖dt = 0. Hence the result is proved. Theorem 3.4. Problem (1) has a unique solution in WPAA(X) under the assumption (A1) pro- vided that 2L|ω| −1 α π αsin π α < 1. 30 Syed Abbas CUBO 16, 1 (2014) Proof: In order to prove that the operator F has a fixed point, consider ‖Fu1(t) − Fu2(t)‖ ≤ ∫t −∞ ‖Sα(t − s)‖‖f(s,u1(s),u1s) − f(s,u2(s),u2s)‖ds ≤ L ∫t −∞ ‖Sα(t − s)‖ ( ‖u1(s) − u2(s)‖ + ‖u1s − u1s‖B(X) ) ds ≤ 2L‖u1 − u2‖∞ ∫ ∞ 0 ‖Sα(t)‖dt. (10) Thus for 2L ∫ ∞ 0 |Sα(t)|dt < 1, the problem (1) has an unique solution. We have mentioned that ∫ ∞ 0 1 1 + |ω|tα = |ω| −1 α π αsin π α for α ∈ (1,2). Thus the above condition reduces to 2CML|ω| −1 α π α sin π α < 1. Remark 3.5. One can easily show that for f Stepanov almost automorphic, the problem (1) has a unique stepanov almost automorphic solutions under the same condition as in both Theorems. Remark 3.6. It is to note that for differential equation du(t) dt = Au(t) + f(t,u(t)), t ∈ R u(0) = u0, (11) where A generates an exponentially stable C0 semigroup {T(t)}t≥0, we can conclude that, if f if Lipschitz continuous, bounded and Weyl almost automorphic or Weyl pseudo almost automorphic, then there exists a unique Weyl almost automorphic or Weyl pseudo almost automorphic solution accordingly of the problem provided that LfM1 δ < 1, where ‖T(t)‖ ≤ M1e−δt for some M ≥ 1 and δ > 0. 4 Examples Example-1: Consider the following fractional order partial differential equation for α ∈ (1,2), ∂αu(t,x) ∂tα = ∂2u(t,x) ∂x2 + ∂α−1 ∂tα−1 (g(t,u(t,x),u(t − τ,x))), τ > 0, t ∈ R, x ∈ (0,π) u(t,0) = u(t,π) = 0, t ∈ R, u(t,x) = φ(t,x) t ∈ [−τ,0], (12) where g is a weighted pseudo almost automorphic function in t. Also assume that g satisfies Lipschitz condition in both variable with Lipschitz constant Lg. Using the transformation u(t)x = CUBO 16, 1 (2014) Weighted pseudo almost automorphic solutions of fractional . . . 31 u(t,x) and A = ∂ 2 ∂x2 with D(A) = { u ∈ L2((0,π),R), u ′ ∈ L2((0,π),R), u ′′ ∈ L2((0,π),R),u(0) = u(π) = 0 } , the above equation can be transform into dαu(t) dtα = Au(t) + dα−1 dtα−1 g(t,u(t),ut(−τ)), (13) t ∈ R and u(t) = φ(t) t ∈ [−τ,0]. It is to note that A generates an analytic semigroup {T(t) : t ≥ 0 on X, where X = L2((0,π),R). Further A has discrete spectrum with eigenvalues of the form −k2;k ∈ N, and corresponding normalized eigenfunctions given by zk(x) = ( 2π) 1 2 sin(kx). As A is analytic, let us assume that it is sectorial of type ω1 and let the following relation holds 2Lg|ω1| −1 α π αsin π α < 1. Thus under all the required assumption on g, the existence of weighted almost automorphic solu- tions is ensured accordingly. Example-2: Consider the following fractional order delay relaxation oscillation equation for α ∈ (1,2), ∂αu(t,x) ∂tα = ∂2u(t,x) ∂x2 − pu(t,x) + ∂α−1 ∂tα−1 (f(t,u(t,x),u(t − τ,x))), τ > 0, t ∈ R, x ∈ (0,π) u(t,0) = u(t,π) = 0, t ∈ R, u(t,x) = φ(t,x) t ∈ [−τ,0], (14) where p > 0 and f is a weighted pseudo almost automorphic function in t. Also assume that f satisfies Lipschitz condition in both variable with Lipschitz constant Lf. Using the transformation u(t)x = u(t,x) and define Au = ∂ 2 u ∂x2 − pu, u ∈ D(A), where D(A) = { u ∈ L2((0,π),C), u ′ ∈ L2((0,π),C), u ′′ ∈ L2((0,π),C),u(0) = u(π) = 0 } , the above equation can be transform into dαu(t) dtα = Au(t) + dα−1 dtα−1 g(t,u(t),ut(−τ)), (15) t ∈ R and u(t) = φ(t) t ∈ [−τ,0]. It is to note that A generates an analytic semigroup {T(t) : t ≥ 0 on X, where X = L2((0,π),R). Hence pI−A is sectorial of type ω = −p < 0. Further A has discrete spectrum with eigenvalues of the form −k2;k ∈ N, and corresponding normalized eigenfunctions given by zk(x) = ( 2 π ) 1 2 sin(kx). As A is analytic. Let us assume that 2Lf|ω| −1 α π αsin π α < 1. 32 Syed Abbas CUBO 16, 1 (2014) Thus under all the required assumption on f, the existence of weighted almost automorphic solu- tions is ensured accordingly. Example-3: Consider the following abstract differential equations of fractional order over a complex Banach space (X,‖ · ‖), dαu(t) dtα = Au(t) + dα−1 dtα−1 (g(t,u(t)) + Ku(t)), (16) t ∈ R, where Ku(t) = ∫t −∞ k(t − s)u(s)ds and A : D(A) ⊂ X → X is a linear densely defined operator of sectorial type on a complex Banach space X. We assume that g is weighted pseudo almost automorphic in t uniformly in u and k satisfy |k(t)| ≤ Ce−bt for some C,b > 0. For u weighted pseudo almost automorphic, it is not difficult to see that Ku(t) is also weighted pseudo almost automorphic. Let us assume that g satisfy Lipschitz condition with Lipschitz constant Lg. Now for u1,u2 ∈ X, consider ‖g(t,u1(t)) − g(t,u2(t))‖ + ‖Ku1(t) − Ku2(t)‖ ≤ Lg‖u1 − u2‖ + ∫t −∞ |k(t − s)||u1(s) − u2(s)|ds ≤ Lg‖u1 − u2‖ + ‖u1 − u2‖ ∫ ∞ 0 |k(s)|ds ≤ Lg‖u1 − u2‖ + C b ‖u1 − u2‖ ≤ ( Lg + C b ) ‖u1 − u2‖. (17) Thus we have ‖g(t,u1) − g(t,u2)‖ + ‖Ku1 − Ku2‖ ≤ ( Lg + C b ) ‖u1 − u2‖. Considering t − s = s1 we have Ku(t) = ∫ ∞ 0 k(s1)u(t + s1) = ∫ ∞ 0 k(s)ut(s). Thus if we take g1(t,u(t),ut) = g(t,u(t)) + Ku(t), the above equation is similar to (1). From the above analysis, one can easily see that g1 satisfies Lipschitz condition with Lipschitz constant Lg + C b . Further assume that A is sectorial of type ω2 and the following condition hold 2(Lg + C b )|ω2| −1 α π αsin π α < 1. One can easily see that for u ∈ WPAA(X),Ku(t) ∈ WPAA(X). Thus we can apply our result to ensure the existence of weighted almost automorphic solution for g weighted almost automorphic. Received: November 2012. 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