\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 188, pp. 1--20.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/188\hfil Stepanov-like pseudo almost automorphy] {Stepanov-like pseudo-almost automorphic functions in Lebesgue spaces with variable exponents $L^{p(x)}$} \author[T. Diagana, M. Zitane \hfil EJDE-2013/188\hfilneg] {Toka Diagana, Mohamed Zitane} % in alphabetical order \address{Toka Diagana \newline Department of Mathematics, Howard University, 2441 6th Street NW, Washington, DC 20059, USA} \email{tokadiag@gmail.com} \address{Mohamed Zitane \newline Universit\'{e} Ibn Tofa\"{\i}l, Facult\'e des Sciences, D\'{e}partement de Math\'{e}matiques, Laboratoire d'An. Maths et GNC, B.P. 133, K\'{e}nitra 1400, Maroc} \email{zitanem@gmail.com} \thanks{Submitted May 23, 2013. Published August 28, 2013.} \subjclass[2000]{34C27, 35B15, 46E30} \keywords{Pseudo-almost automorphy; $S^{p, q(x)}$-pseudo-almost automorphic; \hfill\break\indent Lebesgue space with variable exponents; variable exponents} \begin{abstract} In this article we introduce and study a new class of functions called Stepanov-like pseudo-almost automorphic functions with variable exponents, which generalizes in a natural way classical Stepanov-like pseudo-almost automorphic spaces. Basic properties of these new spaces are investigated. The existence of pseudo-almost automorphic solutions to some first-order differential equations with $S^{p,q(x)}$-pseudo-almost automorphic coefficients will also be studied. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} The impetus of this article comes from three main sources. The first one is a series of papers by Liang et al \cite{J.Liang-J.Zhang-T.J.Xiao, T.J.Xiao-J.Liang-J.Zhang, XZL} in which the concept of pseudo-almost automorphy was introduced and intensively studied. Pseudo-almost automorphic functions are natural generalizations to various classes of functions including almost periodic functions, almost automorphic functions, and pseudo-almost periodic functions. The second source is a paper by Diagana \cite{Diagana3} in which the concept of $S^p$-pseudo-almost automorphy ($p \geq 1$ being a constant) was introduced and studied. Note that $S^p$-pseudo-almost automorphic functions (or Stepanov-like pseudo-almost automorphic functions) are natural generalizations of pseudo-almost automorphic functions. The spaces of Stepanov-like pseudo-almost automorphic functions are now fairly well-understood as most of their fundamental properties have recently been established through the combined efforts of several mathematicians. Some of the recent developments on these functions can be found in \cite{TDbook, TKD, fan2, fan, Z.Hu-Z.jin}. The third and last source is a paper by Diagana and Zitane \cite{Toka1} in which the class of $S^{p, q(x)}$-pseudo-almost periodic functions was introduced and studied, where $q: \mathbb{R} \mapsto \mathbb{R}$ is a measurable function satisfying some additional conditions. The construction of these new spaces makes extensive use of basic properties of the Lebesgue spaces with variable exponents $L^{q(x)}$ (see \cite{Diening, Fan1, Peter}). In this article we extend $S^p$-pseudo-almost automorphic spaces by introducing $S^{p, q(x)}$-pseudo-almost automorphic spaces (or Stepanov-like pseudo-almost automorphic spaces with variable exponents). Basic properties as well as some composition results for these new spaces are established (see Theorems \ref{T512} and \ref{T514}). To illustrate our above-mentioned findings, we will make extensive use of the newly-introduced functions to investigate the existence of pseudo-almost automorphic solutions to the first-order differential equations \begin{equation} u'(t)=A(t)u(t)+f(t), \quad t\in \mathbb{R},\label{AA} \end{equation} and \begin{equation} u'(t)=A(t)u(t)+F(t,Bu(t)), \quad t\in \mathbb{R},\label{A} \end{equation} where $A(t): D(A(t)) \subset \mathbb{X} \mapsto \mathbb{X}$ is a family of closed linear operators on a Banach space $\mathbb{X}$, satisfying the well-known Acquistapace--Terreni conditions, the forcing terms $f : \mathbb{R}\to \mathbb{X}$ is an $S^{p, q(x)}$-pseudo-almost automorphic function and $F : \mathbb{R}\times \mathbb{X} \to \mathbb{X}$ is an $S^{p, q}$-pseudo-almost automorphic function, satisfying some additional conditions, and $B: \mathbb{X} \mapsto \mathbb{X}$ is a bounded linear operator. Such result (Theorems \ref{TT} and\ref{FTO}) generalize most of the known results encountered in the literature on the existence and uniqueness of pseudo-almost automorphic solutions to Equations \eqref{AA}-\eqref{A}. \section{Preliminaries} Let $(\mathbb{X}, \|\cdot\|), (\mathbb{Y}, \|\cdot\|_\mathbb{Y})$ be two Banach spaces. Let $BC(\mathbb{R} , \mathbb{X})$ (respectively, $BC(\mathbb{R} \times \mathbb{Y}, \mathbb{X})$) denote the collection of all bounded continuous functions from $\mathbb{R}$ into $\mathbb{X}$ (respectively, the class of jointly bounded continuous functions $F: \mathbb{R} \times \mathbb{Y} \to \mathbb{X}$). The space $BC(\mathbb{R}, \mathbb{X})$ equipped with the sup norm $\|\cdot\|_\infty$ is a Banach space. Furthermore, $C(\mathbb{R}, \mathbb{Y})$ (respectively, $C(\mathbb{R} \times \mathbb{Y}, \mathbb{X})$) denotes the class of continuous functions from $\mathbb{R}$ into $\mathbb{Y}$ (respectively, the class of jointly continuous functions $F: \mathbb{R} \times \mathbb{Y} \to \mathbb{X}$). Let $B(\mathbb{X},\mathbb{Y})$ stand for the Banach space of bounded linear operators from $\mathbb{X}$ into $\mathbb{Y}$ equipped with its natural operator topology $\|\cdot\|_{B(\mathbb{X},\mathbb{Y})}$ with $B(\mathbb{X}, \mathbb{X}) :=B(\mathbb{X})$. \subsection{Pseudo-almost automorphic functions}\label{EF} \begin{definition}[\cite{bochner1, TDbook, nguerigata}]\label{d3.1}\rm A function $f\in C(\mathbb{R},\mathbb{X})$ is said to be almost automorphic if for every sequence of real numbers $(s_n')_{n \in \mathbb{N}}$ there exists a subsequence $(s_n)_{n \in \mathbb{N}}$ such that $$ g(t):=\underset{n\to\infty}{\lim}f(t+s_n) $$ is well defined for each $t\in\mathbb{R} $ and $$ f(t)=\underset{n\to\infty}{\lim}g(t-s_n) $$ for each $t\in\mathbb{R}$. \end{definition} The collection of all such functions will be denoted by $AA(\mathbb{X})$, which turns out to be a Banach space when it is equipped with the sup-norm. \begin{definition}[\cite{TDbook, J.Liang-J.Zhang-T.J.Xiao}] \label{D3.7} \rm A function $F\in C(\mathbb{R}\times \mathbb{Y},\mathbb{X})$ is said to be almost automorphic if $F(t,u)$ is almost automorphic in $t\in\mathbb{R}$ uniformly for all $u\in\ K$, where $K \subset \mathbb{Y}$ is an arbitrary bounded subset. The collection of all such functions will be denoted by $AA(\mathbb{R} \times \mathbb{X})$. \end{definition} \begin{definition}[\cite{Z.Hu-Z.jin}] \label{D3.3} \rm A function $L\in C(\mathbb{R}\times \mathbb{R},\mathbb{X})$ is called bi-almost automorphic if for every sequence of real numbers $(s_n')_n$ we can extract a subsequence $(s_n)_n$ such that $$ H(t,s):=\underset{n\to\infty}{\lim}L(t+s_n,s+s_n) $$ is well defined for each $t, s\in\mathbb{R}$, and $$ L(t,s)=\underset{n\to\infty}{\lim}H(t-s_n,s-s_n) $$ for each $t, s\in\mathbb{R}$. The collection of all such functions will be denoted by $bAA(\mathbb{R}\times \mathbb{R},\mathbb{X})$. \end{definition} \begin{proposition}[\cite{nguerigata}] \label{P3.2} Assume $f,g: \mathbb{R} \to \mathbb{X}$ are almost automorphic and $\lambda$ is any scalar. Then the following hold \begin{itemize} \item[(a)] $f+g, \lambda f, f_{\tau}(t):=f(t+\tau)$ and $\widehat{f}(t):=f(-t)$ are almost automorphic; \item[(b)] The range $R_{f}$ of $f$ is precompact, so $f$ is bounded; \item[(c)] If $\{f_n\}$ is a sequence of almost automorphic functions and $f_n\to f$ uniformly on $\mathbb{R}$, then $f$ is almost automorphic. \end{itemize} \end{proposition} Define $$ PAA_0(\mathbb{X}) := \Big\{ f \in BC(\mathbb{R} , \mathbb{X}): \lim_{T \to \infty} {\frac{1}{2T}} \int_{-T}^{T} \| f(\sigma)\|d\sigma = 0\Big\}. $$ Similarly, define $PAA_0(\mathbb{R} \times \mathbb{X})$ as the collection of jointly continuous functions $F: \mathbb{R} \times \mathbb{Y} \to \mathbb{X}$ such that $F(\cdot, y)$ is bounded for each $y \in \mathbb{Y}$ and $$ \lim_{T \to \infty} {\frac{1}{2T}} \int_{-T}^{T} \| F(s, y)\| ds = 0 $$ uniformly in $y \in \mathbb{Y}$. \begin{definition}[\cite{bochner1}] \label{D3.4} \rm A function $f\in BC(\mathbb{R},\mathbb{X})$ is said to be pseudo-almost automorphic if it can be decomposed as $f=g+\varphi$ where $g\in AA(\mathbb{X})$ and $\varphi\in PAA_0(\mathbb{X})$. The set of all such functions will be denoted by $PAA(\mathbb{X})$. \end{definition} \begin{definition}[\cite{J.Liang-J.Zhang-T.J.Xiao}] \label{D3.8} \rm A function $F\in C(\mathbb{R}\times \mathbb{Y},\mathbb{X})$ is said to be pseudo-almost automorphic if it can be decomposed as $f=G+\Phi$ where $G\in AA(\mathbb{R} \times \mathbb{X})$ and $\Phi\in AA_0(\mathbb{R} \times \mathbb{X})$. The collection of such functions will be denoted by $PAA(\mathbb{R} \times \mathbb{X})$. \end{definition} \begin{theorem}[\cite{T.J.Xiao-J.Liang-J.Zhang}] \label{T3.5} \rm The space $PAA(\mathbb{X})$ equipped with the sup-norm is a Banach space. \end{theorem} \begin{theorem}[\cite{Z.Hu-Z.jin}] \label{T3.6} \rm If $u\in PAA(\mathbb{X})$ and if $C \in B(\mathbb{X})$, then the function $t \mapsto Cu(t)$ belongs to $PAA(\mathbb{X})$. \end{theorem} \begin{theorem}[\cite{Diagana3,Z.Hu-Z.jin}] \label{T3.8} Assume $F\in PAA(\mathbb{R} \times \mathbb{X})$. Suppose that $u \mapsto F(t,u)$ is Lipschitz uniformly in $t\in\mathbb{R}$, in the sense that there exists $L>0$ such that \begin{equation} \|F(t,u)-F(t,v)\|\leq L\|u-v\|\quad \text{for all } t\in\mathbb{R}, u,v\in\mathbb{X} \end{equation} If $\Phi \in PAA(\mathbb{\mathbb{X}})$, then $F(.,\Phi(.))\in PAA(\mathbb{X})$. \end{theorem} \subsection{Evolution family and exponential dichotomy} \begin{definition}[\cite{TDbook, K.J.Negel}] \label{def2.10}\rm A family of bounded linear operators $(U(t,s))_{t\geq s}$ on a Banach space $\mathbb{X}$ is called a strongly continuous evolution family if \begin{itemize} \item [(i)] $U(t,t) = I$ for all $t\in \mathbb{R}$; \item[(ii)] $U(t,s)=U(t,r)U(r,s)$ for all $t\geq r\geq s$ and $t, r, s\in \mathbb{R}$; and \item[(iii)] the map $(t,s)\mapsto U(t,s)x$ is continuous for all $x\in\mathbb{X}, t\geq s$ and $t, s \in \mathbb{R}$. \end{itemize} \end{definition} \begin{definition}[\cite{TDbook, K.J.Negel}] \label{evolution} An evolution family $(U(t,s))_{t\geq s}$ on a Banach space $\mathbb{X}$ is called hyperbolic (or has exponential dichotomy) if there exist projections $P(t), t\in \mathbb{R}$, uniformly bounded and strongly continuous in $t$, and constants $M>0, \delta>0$ such that \begin{itemize} \item[(i)] $U(t,s)P(s)=P(t)U(t,s) $ for $t\geq s$ and $t, s\in \mathbb{R}$; \item[(ii)] The restriction $U_{Q}(t,s) : Q(s)\mathbb{X}\mapsto Q(t)\mathbb{X}$ of $U(t,s)$ is invertible for $t\geq s$ (and we set $U_{Q}(s,t):=U(t,s)^{-1}$); \item[(iii)] $\|U(t,s)P(s)\|\leq Me^{-\delta(t-s)}$, $\|U_{Q}(s,t)Q(t)\|\leq Me^{-\delta(t-s)} $ for $t\geq s$ and $t, s \in \mathbb{R}$, \end{itemize} where $Q(t):=I-P(t)$ for all $t\in \mathbb{R}$. \end{definition} \begin{definition}[\cite{K.J.Negel}] Given a hyperbolic evolution family $U(t,s)$, we define its so-called Green's function by \begin{equation} \Gamma(t,s):= \begin{cases} U(t,s)P(s),&\text{for } t\geq s,\quad t, s\in \mathbb{R},\\ U_{Q}(t,s)Q(s),&\text{for } t< s,\quad t, s\in \mathbb{R}. \end{cases} \end{equation} \end{definition} \section{Lebesgue spaces with variable exponents $L^{p(x)}$} The setting of this section follows that of Diagana and Zitane \cite{Toka1}. This section is mainly devoted to the so-called Lebesgue spaces with variable exponents $L^{p(x)}(\mathbb{R}, \mathbb{X})$. Various basic properties of these functions are reviewed. For more on these spaces and related issues we refer to Diening et al \cite{Diening}. Let $(\mathbb{X}, \|\cdot\|)$ be a Banach space and let $\Omega \subseteq \mathbb{R}$ be a subset. Let $M(\Omega, \mathbb{X})$ denote the collection of all measurable functions $f: \Omega \mapsto \mathbb{X}$. Let us recall that two functions $f$ and $g$ of $M(\Omega, \mathbb{X})$ are equal whether they are equal almost everywhere. Set $m(\Omega) := M(\Omega, \mathbb{R})$ and fix $p \in m(\Omega)$. Let $\varphi(x,t)=t^{p(x)}$ for all $x\in\Omega$ and $t\geq0$, and define \begin{gather*} \rho(u)=\rho_{p(x)}(u)=\int_{\Omega} \varphi(x,\|u(x)\|)dx =\int_{\Omega}\|u(x)\|^{p(x)}dx, \\ L^{p(x)}(\Omega, \mathbb{X})=\Big\{u \in M(\Omega, \mathbb{X}): \lim_{\lambda \to 0^{+}}\rho(\lambda u)=0\Big\}, \\ L^{p(x)}_{OC}(\Omega, \mathbb{X})=\Big\{u \in L^{p(x)}(\Omega, \mathbb{X}): \rho(u)<\infty\Big\}, \text{ and} \\ E^{p(x)}(\Omega, \mathbb{X})=\Big\{u \in L^{p(x)}(\Omega, \mathbb{X}): \text{ for all } \lambda>0, \, \rho(\lambda u)<\infty \Big\}. \end{gather*} Note that the space $L^{p(x)}(\Omega, \mathbb{X})$ defined above is a Musielak-Orlicz type space while $L^{p(x)}_{OC}(\Omega, \mathbb{X})$ is a generalized Orlicz type space. Further, the sets $E^{p(x)}(\Omega, \mathbb{X})$ and $L^{p(x)}(\Omega, \mathbb{X})$ are vector subspaces of $M(\Omega, \mathbb{X})$. In addition, $L^{p(x)}_{OC}(\Omega, \mathbb{X})$ is a convex subset of $L^{p(x)}(\Omega, \mathbb{X})$, and the following inclusions hold $$ E^{p(x)}(\Omega, \mathbb{X})\subset L^{p(x)}_{OC}(\Omega, \mathbb{X}) \subset L^{p(x)}(\Omega, \mathbb{X}). $$ \begin{definition}[\cite{Diening}] \label{def3.1} \rm A convex and left-continuous function $\psi: [0,\infty)\to [0,\infty]$ is called a $\Phi-$function if it satisfies the following conditions: \begin{itemize} \item[(a)] $\psi(0)=0$; \item[(b)] $\lim_{t \to 0^{+}}\psi(t)=0$; and \item[(c)] $\lim_{t \to \infty}\psi(t)=\infty$. \end{itemize} Moreover, $\psi$ is said to be positive whether $\psi(t) > 0$ for all $t > 0$. \end{definition} Let us mention that if $\psi$ is a $\Phi$-function, then on the set $\{t > 0: \psi(t) < \infty\}$, the function $\psi$ is of the form $$ \psi(t) =\int_0^t k(t)dt, $$ where $k(\cdot)$ is the right-derivative of $\psi(t)$. Moreover, $k$ is a non-increasing and right-continuous function. For more on these functions and related issues we refer to \cite{Diening}. \begin{example}\label{examp3.2} \rm (a) Consider the function $\varphi_p(t) = p^{-1} t^p$ for $1 \leq p < \infty$. It can be shown that $\varphi_p$ is a $\Phi$-function. Furthermore, the function $\varphi_p$ is continuous and positive. (b) It can be shown that the function $\varphi$ defined above; that is, $\varphi(x,t)=t^{p(x)}$ for all $x\in\mathbb{R}$ and $t\geq0$ is a $\Phi-$function. \end{example} For any $p \in m(\Omega)$, we define $$ p^{-}:= \operatorname{ess\,inf}_{x \in \Omega} p(x), \quad p^{+}:=\operatorname{ess\,sup}_{x\in \Omega} p(x). $$ Define $$ C_{+}(\Omega):=\Big\{p\in m(\Omega) : 1< p^{-}\leq p(x) \leq p^{+}<\infty, \text{ for each } x\in \Omega\Big\}. $$ Let $p\in C_{+}(\Omega)$. Using similar argument as in \cite[Theorem 3.4.1]{Diening}, it can be shown that $$ E^{p(x)}(\Omega, \mathbb{X})= L^{p(x)}_{OC}(\Omega, \mathbb{X})= L^{p(x)}(\Omega, \mathbb{X}). $$ In view of the above, we define the Lebesgue space $L^{p(x)}(\Omega, \mathbb{X})$ with variable exponents $p\in C_{+}(\Omega)$, by $$ L^{p(x)}(\Omega, \mathbb{X}):=\Big\{u \in M(\Omega, \mathbb{X}): \int_{\Omega} \|u(x)\|^{p(x)}dx<\infty \Big\}. $$ Define, for each $u \in L^{p(x)}(\Omega, \mathbb{X})$, $$ \|u\|_{p(x)}:=\inf \Big\{\lambda>0: \int_{\Omega} \big\|\frac{u(x)}{\lambda}\big\|^{p(x)}dx\leq1 \Big\}. $$ It can be shown that $\|\cdot\|_{p(x)}$ is a norm upon $L^{p(x)}(\Omega, \mathbb{X})$, which is referred to as the \textit{Luxemburg norm}. \begin{remark} \label{rmk3.3} \rm Let $p\in C_{+}(\Omega)$. If $p$ is constant, then the space $L^{p(\cdot)}(\Omega, \mathbb{X})$, as defined above, coincides with the usual space $L^{p}(\Omega, \mathbb{X})$. \end{remark} We now establish some basic properties for these spaces. For more on these functions and related issues we refer to \cite{Diening}. \begin{proposition}[\cite{Toka1}]\label{prop4.3} Let $p\in C_{+}(\Omega)$ and let $u, u_{k}, v \in M(\Omega, \mathbb{X})$ for $k = 1, 2, \ldots$. Then the following statements hold, \begin{itemize} \item[(a)] If $u_{k} \to u$ \emph{a.e.}, then $ \rho_{p}(u) \leq \lim_{k \to \infty}\inf(\rho_{p}(u_{k}))$; \item[(b)] If $\|u_{k}\| \to \|u\|$ \emph{a.e.}, then $\rho_{p}(u)= \lim_{k \to \infty}\rho_{p}(u_{k})$; \item[(c)] If $u_{k} \to u $ \emph{a.e.}, $\|u_{k}\|\leq \|v\|$ and $v\in E^{p(x)}(\Omega, \mathbb{X})$, then $u_{k} \to u$ in the space $L^{p(x)}(\Omega, \mathbb{X})$. \end{itemize} \end{proposition} \begin{proposition}[\cite{Diening, Peter}] \label{prop4.4} Let $p\in C_{+}(\Omega)$. If $u,v \in L^{p(x)}(\Omega, \mathbb{X})$, then the following properties hold, \begin{itemize} \item[(a)] $\|u\|_{p(x)}\geq0$, with equality if and only if $u=0$; \item[(b)] $\rho_{p}(u)\leq\rho_{p}(v)$ and $\|u\|_{p(x)}\leq \|v\|_{p(x)}$ if $\|u\|\leq \|v\|$; \item[(c)] $\rho_{p}(u \|u\|_{p(x)}^{-1})=1$ if $u\neq0$; \item[(d)] $\rho_{p}(u)\leq1$ if and only if $\|u\|_{p(x)}\leq1$; \item[(e)] If $\|u\|_{p(x)}\leq1$, then $$ [\rho_{p}(u)\Big]^{1/p^{-}} \leq \|u\|_{p(x)}\leq\Big[\rho_{p}(u)]^{1/p^{+}}. $$ \item[(f)] If $\|u\|_{p(x)}\geq1$, then $$ \Big[\rho_{p}(u)\Big]^{1/p^{+}}\leq \|u\|_{p(x)} \leq \Big[\rho_{p}(u)\Big]^{1/p^{-}}. $$ \end{itemize} \end{proposition} \begin{proposition}[\cite{Diening}] \label{prop4.5} Let $p\in C_{+}(\Omega)$ and let $u, u_{k}, v \in M(\Omega, \mathbb{X})$ for $k = 1, 2, \ldots$. Then the following statements hold: \begin{itemize} \item[(a)] If $u \in L^{p(x)}(\Omega, \mathbb{X})$ and $0\leq \|v\|\leq \|u\|$, then $v \in L^{p(x)}(\Omega, \mathbb{X})$ and $\|v\|_{p(x)}\leq \|u\|_{p(x)}$. \item[(b)] If $u_{k} \to u$ \emph{a.e.}, then $ \|u\|_{p(x)}\leq \lim_{k \to \infty}\inf(\|u_{k}\|_{p(x)})$. \item[(c)] If $\|u_{k}\| \to \|u\|$ \emph{a.e.} with $u_{k}\in L^{p(x)}(\Omega, \mathbb{X})$ and $\sup_{k }\|u_{k}\|_{p(x)}<\infty$, then $u \in L^{p(x)}(\mathbb{R}, \mathbb{X})$ and $\|u_{k}\|_{p(x)} \to \|u\|_{p(x)}$. \end{itemize} \end{proposition} Using similar arguments as in Fan et al \cite{Fan1}, we obtain the following result. \begin{proposition} \label{prop4.6} If $u, u_n \in L^{p(x)}(\Omega, \mathbb{X})$ for $k = 1, 2, \ldots$, then the following statements are equivalent: \begin{itemize} \item[(a)] $ \lim_{k\to \infty}\|u_{k}-u\|_{p(x)} = 0$; \item[(b)] $\lim_{k\to \infty}\rho_{p}(u_{k}-u) = 0$; \item[(c)] $u_{k} \to u$ and $\lim_{k\to \infty}\rho_{p}(u_{k}) = \rho_{p}(u)$. \end{itemize} \end{proposition} \begin{theorem}[\cite{Diening, Fan1}] \label{Holder1} Let $p\in C_{+}(\Omega)$. The space $(L^{p(x)}(\Omega, \mathbb{X}), \|\cdot\|_{p(x)})$ is a Banach space that is separable and uniform convex. Its topological dual is $L^{q(x)}(\Omega, \mathbb{X})$, where $p^{-1}(x)+ q^{-1}(x)=1$. Moreover, for any $u\in L^{p(x)}(\Omega, \mathbb{X})$ and $v\in L^{q(x)}(\Omega, \mathbb{R})$, we have \begin{equation}\label{Holder41} \big\|\int_{\Omega} uv dx \big\| \leq \Big(\frac{1}{p^{-}}+\frac{1}{q^{-}}\Big)\,\|u\|_{p(x)}|v|_{q(x)}. \end{equation} \end{theorem} Define $$ D_{+}(\Omega):=\Big\{p\in m(\Omega) : 1\leq p^{-}\leq p(x) \leq p^{+}<\infty, \text{ for each } x\in \Omega\Big\}. $$ \begin{corollary}[\cite{Peter}] \label{Holder2} Let $p,r\in D_{+}(\Omega)$. If the function $q$ defined by the equation $$ \frac{1}{q(x)}=\frac{1}{p(x)}+\frac{1}{r(x)} $$ is in $D_{+}(\Omega)$, then there exists a constant $C=C(p,r)\in [1,5]$ such that \[ \|uv\|_{q(x)}\leq C\|u\|_{p(x)}|v|_{r(x)}, \] for every $u\in L^{p(x)}(\Omega, \mathbb{X})$ and $v\in L^{r(x)}(\Omega, \mathbb{R})$. \end{corollary} \begin{corollary}[\cite{Diening}] \label{embedding} Let $\operatorname{meas}(\Omega)<\infty$ where $\operatorname{meas}(\cdot)$ stands for the Lebesgue measure and $p, q\in D_{+}(\Omega)$. If $q(\cdot)\leq p(\cdot)$ almost everywhere in $\Omega$, then the embedding $L^{p(x)}(\Omega, \mathbb{X})\hookrightarrow L^{q(x)}(\Omega, \mathbb{X})$ is continuous whose norm does not exceed $2(\operatorname{meas}(\Omega)+1)$. \end{corollary} \section{Stepanov-like pseudo-almost automorphic functions with variable exponents} \begin{definition}\label{def4.1} \rm The Bochner transform $f^b(t,s)$, $t\in \mathbb{R}$, $s\in[0,1]$ of a function $f: \mathbb{R} \to \mathbb{X}$ is defined by $f^b(t,s):=f(t+s)$. \end{definition} \begin{remark} \label{rmk4.2} \rm A function $\varphi(t,s)$, $t\in \mathbb{R}$, $s \in [0,1]$, is the Bochner transform of a certain function $f$, $\varphi(t,s)=f^b(t,s)$, if and only if $\varphi(t+\tau, s-\tau)=\varphi(s,t)$ for all $t\in\mathbb{R}$, $s\in [0,1]$ and $\tau\in [s-1, s]$. Moreover, if $f = h + \varphi$, then $f^b = h^b + \varphi^b$. Moreover, $(\lambda f)^b = \lambda f^b$ for each scalar $\lambda$. \end{remark} \begin{definition} \rm The Bochner transform $F^b(t,s, u)$, $t\in \mathbb{R}$, $s\in[0,1]$, $u \in \mathbb{X}$ of a function $F: \mathbb{R} \times \mathbb{X} \mapsto \mathbb{X}$, is defined by $F^b(t,s, u):=F(t+s, u)$ for each $u \in \mathbb{X}$. \end{definition} \begin{definition} \rm Let $p\in [1,\infty)$. The space $BS^p(\mathbb{X})$ of all Stepanov bounded functions, with the exponent $p$, consists of all measurable functions $f$ on $\mathbb{R}$ with values in $\mathbb{X}$ such that $f^b\in L^\infty\big(\mathbb{R}, L^p((0,1), \mathbb{X})\big)$. This is a Banach space with the norm $$ \|f\|_{S^p}=\|f^b\|_{L^\infty(\mathbb{R},L^p)} =\sup_{t\in\mathbb{R}}\Big(\int_t^{t+1}\|f(\tau)\|^p\,d\tau\Big)^{1/p}. $$ \end{definition} Note that for each $p \geq 1$, we have the following continuous inclusion: $$ (BC(\mathbb{X}), \|\cdot\|_\infty) \hookrightarrow (BS^p(\mathbb{X}), \|\cdot\|_{S^p}). $$ \begin{definition}[Diagana and Zitane \cite{Toka1}] \rm Let $p \in C_{+}(\mathbb{R})$. The space $BS^{p(x)}(\mathbb{X})$ consists of all functions $f \in M(\mathbb{R}, \mathbb{X})$ such that $\|f\|_{S^{p(x)}} < \infty$, where \begin{align*} \|f\|_{S^{p(x)}} &= \sup_{t\in\mathbb{R}}\Big[\inf \Big\{\lambda>0: \int_0^{1} \big\|\frac{f(x +t)}{\lambda}\big\|^{p(x+t)}dx\leq1 \Big\} \Big]\\ &= \sup_{t\in\mathbb{R}}\Big[\inf \Big\{\lambda>0: \int_t^{t+1} \big\|\frac{f(x)}{\lambda}\big\|^{p(x)}dx\leq1 \Big\} \Big]. \end{align*} \end{definition} Note that the space $\big(BS^{p(x)}(\mathbb{X}),\|\cdot\|_{S^{p(x)}}\big)$ is a Banach space, which, depending on $p(\cdot)$, may or may not be translation-invariant. \begin{definition}[Diagana and Zitane \cite{Toka1}] \rm If $p, q\in C_{+}(\mathbb{R})$, we then define the space $BS^{p(x), q(x)}(\mathbb{X})$ as follows \begin{align*} BS^{p(x), q(x)}(\mathbb{X}) &:= BS^{p(x)}(\mathbb{X}) + BS^{q(x)}(\mathbb{X}) \\ &= \Big\{f = h + \varphi \in M(\mathbb{R}, \mathbb{X}): h \in BS^{p(x)}(\mathbb{X}) \text{ and } \varphi \in BS^{q(x)}(\mathbb{X})\Big\}. \end{align*} \end{definition} We equip $BS^{p(x), q(x)}(\mathbb{X})$ with the norm $\|\cdot\|_{S^{p(x), q(x)}}$ defined by $$ \|f\|_{S^{p(x), q(x)}} := \inf\Big\{\|h\|_{S^{p(x)}} + \|\varphi\|_{S^{q(x)}}: f = h + \varphi\Big\}. $$ Clearly, $\big(BS^{p(x), q(x)}(\mathbb{X}),\|\cdot\|_{S^{p(x), q(x)}}\big)$ is a Banach space, which, depending on both $p(\cdot)$ and $q(\cdot)$, may or may not be translation-invariant. \begin{lemma}[Diagana and Zitane \cite{Toka1}] \rm Let $p, q\in C_{+}(\mathbb{R})$. Then the following continuous inclusion holds, $$ \Big(BC(\mathbb{R}, \mathbb{X}),\|\cdot\|_{\infty}\Big)\hookrightarrow \Big(BS^{p(x)}(\mathbb{X}),\|\cdot\|_{S^{p(x)}}\Big) \hookrightarrow \Big(BS^{p(x), q(x)}(\mathbb{X}),\|\cdot\|_{S^{p(x), q(x)}}\Big). $$ \end{lemma} \begin{proof} The fact that $\big(BS^{p(x)}(\mathbb{X}),\|\cdot\|_{S^{p(x)}}\big) \hookrightarrow \big(BS^{p(x), q(x)}(\mathbb{X}),\|\cdot\|_{S^{p(x), q(x)}}\big)$ is obvious. Thus we will only show that $\big(BC(\mathbb{R}, \mathbb{X}),\|\cdot\|_{\infty}\big) \hookrightarrow \big(BS^{p(x)}(\mathbb{X}),\|\cdot\|_{S^{p(x)}}\big)$. Indeed, let $f \in BC(\mathbb{R}, \mathbb{X}) \subset M(\mathbb{R}, \mathbb{X})$. If $\|f \|_\infty = 0$, which yields $f=0$, then there is nothing to prove. Now suppose that $\|f \|_\infty \not = 0$. Using the facts that $0 < \|\frac{f(x)}{\|f\|_{\infty}}\| \leq 1$ and that $p\in C_{+}(\mathbb{R})$ it follows that for every $t\in\mathbb{R}$, $$ \int_t^{t+1} \Big\|\frac{f(x)}{\|f\|_{\infty}}\Big\|^{p(x)}dx \leq \int_t^{t+1} 1^{p(x)}dx=1, $$ and hence $\|f\|_{\infty}\in \Big\{\lambda>0: \int_t^{t+1} \|\frac{f(x)}{\lambda}\|^{p(x)}dx\leq1 \Big\}$, which yields $$ \inf \Big\{\lambda>0: \int_t^{t+1} \big\|\frac{f(x)}{\lambda}\big\|^{p(x)}dx\leq1 \Big\}\leq\|f\|_{\infty}. $$ Therefore, $\|f\|_{S^{p(x)}}\leq \|f\|_{\infty}<\infty$. This shows that not only $f\in (BS^{p(x)}(\mathbb{X})),\|\cdot\|_{S^{p(x)}})$ but also the injection $(BC(\mathbb{R}, \mathbb{X}),\|\cdot\|_{\infty})\hookrightarrow (BS^{p(x)}(\mathbb{X}),\|\cdot\|_{S^{p(x)}})$ is continuous. \end{proof} \begin{definition} \label{Dap} \rm Let $p\geq 1$ be a constant. A function $f\in BS^{p}(\mathbb{X})$ is said to be $S^{p}$-almost automorphic (or Stepanov-like almost automorphic function) if $f^b \in AA\big(L^p((0,1),\mathbb{X})\big)$. That is, a function $f\in L^{p}_{\rm loc}(\mathbb{R},\mathbb{X})$ is said to be Stepanov-like almost automorphic if its Bochner transform $f^{b} : \mathbb{R} \to L^{p}(0,1;\mathbb{X})$ is almost automorphic in the sense that for every sequence of real numbers $(s_n')_n$, there exists a subsequence $(s_n)_n$ and a function $g\in L^{p}_{\rm loc}(\mathbb{R},\mathbb{X})$ such that $$ \Big(\int^{1}_0\|f(t+s+s_n)-g(t+s) \|^{p} ds \Big)^{1/p} \to 0 ,\quad \Big(\int^{1}_0\|g(t+s-s_n)-f(t+s) \|^{p} ds \Big)^{1/p} \to 0 $$ as $ n \to \infty $ pointwise on $\mathbb{R}$. The collection of such functions will be denoted by $S_{aa}^{p}(\mathbb{X})$. \end{definition} \begin{remark}\label{OP}\rm There are some difficulties in defining $S_{aa}^{p(x)}(\mathbb{X})$ for a function $p \in C_{+}(\mathbb{R})$ that is not necessarily constant. This is mainly due to the fact that the space $BS^{p(x)}(\mathbb{X})$ is not always translation-invariant. In other words, the quantities $f^b (t+\tau,s)$ and $f^b(t,s)$ (for $t \in \mathbb{R},\; s \in [0, 1]$) that are used in the definition of $S^{p(x)}$-almost automorphy, do not belong to the same space, unless $p$ is constant. \end{remark} \begin{remark} \label{remark4.10} \rm It is clear that if $1 \leq p 0: \int_t^{t+1} \big\|\frac{\psi(x)}{\lambda}\big\|^{q(x)}dx\leq1 \Big\} dt= 0. $$ The collection of $S^{p, q(x)}$-pseudo-almost automorphic functions will be denoted by $S_{paa}^{p, q(x)}(\mathbb{X})$. \end{definition} \begin{lemma} \label{remark5.7} Let $r,s\geq 1,p,q\in D_{+}(\mathbb{R})$. If $s0: \int_t^{t+1} \big\|\frac{\varphi(x)}{\lambda}\big\|^{q(x)}dx\leq1 \Big\}\Big]\\ &\leq 4\Big[\inf \Big\{\lambda>0: \int_t^{t+1} \big\|\frac{\varphi(x)}{\lambda}\big\|^{p(x)}dx\leq1 \Big\}\Big]. \end{align*} Using the fact that $\varphi \in S_{paa_0}^{p(x)}(\mathbb{X})$ and the previous inequality it follows that $$ \lim_{T \to \infty} {\frac{1}{2T}} \int_{-T}^{T} \inf \Big\{\lambda>0: \int_t^{t+1} \big\|\frac{\varphi(x)}{\lambda}\big\|^{q(x)}dx\leq1 \Big\} dt= 0; $$ that is, $\varphi \in S_{paa_0}^{q(x)}(\mathbb{X})$. Therefore, $f\in S_{paa}^{s,q(x)}(\mathbb{X})$. \end{proof} \begin{proposition} Let $p \geq 1$ be a constant and let $q\in C_{+}(\mathbb{R})$. If $f\in PAA(\mathbb{X})$, then $f$ is $S^{p,q(x)}$-pseudo-almost automorphic. \end{proposition} \begin{proof} Let $f \in PAA(\mathbb{X})$, that is, there exist two functions $h, \varphi: \mathbb{R} \to \mathbb{X}$ such that $f = h + \varphi$ where $h \in AA(\mathbb{X})$ and $\varphi \in PAA_0(\mathbb{X})$. Now from remark \ref{remark4.10}, $h \in AA(\mathbb{X})\subset S_{aa}^{p}(\mathbb{X})$. The proof of $\varphi\in S_{paa_0}^{q(x)}(\mathbb{X})$ was given in \cite{Toka1}. However for the sake of clarity, we reproduce it here. Using (e)-(f) of Proposition \ref{prop4.4} and the usual H\"{o}lder inequality, it follows that \begin{align*} &\int_{-T}^{T} \inf \Big\{\lambda>0: \int_0^{1} \big\|\frac{\varphi(x+t)}{\lambda}\big\|^{q(x+t)}dx\leq1 \Big\} dt \\ &\leq \int_{-T}^{T} \Big(\int_0^{1}\|\varphi(t+x)\|^{q(t+x)}\>dx\Big)^{\gamma} \> dt\\ & \leq (2T)^{1-\gamma}\Big[\int_{-T}^{T}\Big(\int_0^{1} \|\varphi(t+x)\|^{q(t+x)}\>dx\Big)\>dt\Big]^{\gamma}\\ & \leq (2T)^{1-\gamma}\Big[\int_{-T}^{T}\Big(\int_0^{1} \|\varphi(t+x)\| \|\varphi\|_{\infty}^{q(t+x)-1}\>dx\Big)\>dt\Big]^{\gamma}\\ &\leq (2T)^{1-\gamma} \Big(\|\varphi\|_{\infty}+1\Big)^{\frac{q^{+}-1}{\gamma}}\Big[\int_{-T}^{T}\Big(\int_0^{1} \|\varphi(t+x)\|\>dx\Big)\>dt\Big]^{\gamma}\\ & = (2T)^{1-\gamma} \Big(\|\varphi\|_{\infty}+1\Big)^{\frac{q^{+}-1}{\gamma}}\Big[\int_0^{1}\Big(\int_{-T}^{T} \|\varphi(t+x)\|\>dt\Big)\>dx\Big]^{\gamma}\\ & = (2T) \Big(\|\varphi\|_{\infty}+1\Big)^{\frac{q^{+}-1}{\gamma}} \Big[\int_0^{1}\Big(\frac{1}{2T}\int_{-T}^{T} \|\varphi(t+x)\|\>dt\Big)\>dx\Big]^{\gamma}, \end{align*} where \[ \gamma=\begin{cases} \frac{1}{q^{+}}& \text{if $\|\varphi\|<1$},\\ \frac{1}{q^{-}}& \text{if $\|\varphi\|\geq1$}. \end{cases} \] Using the fact that $PAA_0(\mathbb{X})$ is translation invariant and the (usual) Dominated Convergence Theorem, it follows that \begin{align*} &\lim_{T \to \infty}{\frac{1}{2T}} \int_{-T}^{T}\inf \Big\{\lambda>0: \int_0^{1} \big\|\frac{\varphi(x+t)}{\lambda}\big\|^{q(x+t)}dx\leq1 \Big\} dt\\ &\leq \Big(\|\varphi\|_{\infty}+1\Big) ^{\frac{q^{+}-1}{\gamma}}\Big[\int_0^{1}\Big(\lim_{T \to\infty} \frac{1}{2T}\int_{-T}^{T} \|\varphi(t+x)\|\>dt\Big)\>dx\Big]^{\gamma}=0. \end{align*} \end{proof} Using similar argument as in \cite{T.J.Xiao-J.Liang-J.Zhang}, the following Lemma can be established. \begin{lemma}\label{lemma5.10} Let $p,q\geq1$ be a constants. If $f = h + \varphi \in S_{paa}^{p,q}(\mathbb{X})$ such that $h^{b} \in AA\big(L^{p}((0,1), \mathbb{X})\big)$ and $\varphi^{b} \in PAA_0\big(L^{q}((0,1), \mathbb{X})\big)$, then $$ \big\{h(t+.) : t\in \mathbb{R}\big\}\subset \overline{\big\{f(t+.) : t\in \mathbb{R}\big\}}, \quad \text{in } S^{p,q}(\mathbb{X}). $$ \end{lemma} \begin{proof} We prove it by contradiction. Indeed, if this is not true, then there exist a $t_0\in \mathbb{R}$ and an $\varepsilon>0$ such that $$ \|h(t_0+\cdot)-f(t+\cdot)\|_{S^{p, q}}\geq2\varepsilon,\quad t\in \mathbb{R}. $$ Since $h^{b} \in AA\big(L^{p}((0,1), \mathbb{X})\big)$ and $\big(BS^{p}(\mathbb{X}),\|\cdot\|_{S^{p}}\big) \hookrightarrow \big(BS^{p, q}(\mathbb{X}),\|\cdot\|_{S^{p, q}}\big)$, fix $t_0\in \mathbb{R}, \varepsilon>0$ and write, $B_{\varepsilon}:=\{\tau\in\mathbb{R}; \|h(t_0+\tau+\cdot)-g(t_0+\cdot)\|_{S^{p, q}} <\varepsilon \}$. By \cite[Lemma 2.1]{T.J.Xiao-J.Liang-J.Zhang}, there exist $s_1,\dots ,s_m\in \mathbb{R}$ such that $$ \cup_{i=1}^{m}(s_i+B_{\varepsilon})=\mathbb{R}. $$ Write $$ \hat{s}_i=s_i-t_0\quad (1\leq i\leq m),\quad \eta=\max_{1\leq i\leq m}|\hat{s}_i|. $$ For $T\in \mathbb{R}$ with $|T|>\eta$; we put $$ B_{\varepsilon,T}^{(i)} =[-T+\eta-\hat{s}_i,T-\eta-\hat{s}_i]\cap(t_0+B_{\varepsilon}),\quad 1\leq i\leq m, $$ one has $\cup_{i=1}^{m}(\hat{s}_i+B_{\varepsilon,T}^{(i)}) =[-T+\eta,T-\eta]$. Using the fact that $B_{\varepsilon,T}^{(i)}\subset[-T,T]\cap(t_0+B_{\varepsilon})$, $i=1,\dots ,m$, we obtain \begin{align*} 2(T-\eta) &=\operatorname{meas}([-T+\eta,T-\eta])\\ &\leq \sum^{m}_{i=1}\operatorname{meas}(\hat{s}_i+B_{\varepsilon,T}^{(i)})\\ &=\sum^{m}_{i=1}\operatorname{meas}(B_{\varepsilon,T}^{(i)}) \\ &\leq m\, \max_{1\leq i\leq m}\big\{\operatorname{meas}(B_{\varepsilon,T}^{(i)})\big\}\\ &\leq m\, \operatorname{meas}([-T,T]\cap (t_0+B_{\varepsilon})), \end{align*} On the other hand, by using the Minkowski inequality, for any $t\in t_0+B_{\varepsilon}$, one has \begin{align*} \|\varphi(t+\cdot)\|_{S^{q}}&=\|\varphi(t+\cdot)\|_{S^{p,q}}\\ &= \|f(t+\cdot)-h(t+\cdot)\|_{S^{p,q}}\\ &\geq \|h(t_0+\cdot)-f(t+\cdot)\|_{S^{p,q}} -\|h(t+\cdot)-h(t_0+\cdot)\|_{S^{p,q}} >\varepsilon. \end{align*} Then \begin{align*} {\frac{1}{2T}} \int_{-T}^{T} \|\varphi(t+\cdot)\|_{S^{q}}\,dt &\geq {\frac{1}{2T}} \int_{[-T,T]\cap(t_0+B_{\varepsilon})} \|\varphi(t+\cdot)\|_{S^{q}}\,dt\\ &\geq\varepsilon(T-\eta)(mT)^{-1}\to \varepsilon m^{-1},\quad \text{as } T\to\infty. \end{align*} This is a contradiction, since $\varphi^{b} \in PAA_0\big(L^{q}((0,1), \mathbb{X})\big)$. \end{proof} \begin{theorem}\label{theorem5.10} Let $p,q\geq 1$ be constants. The space $S_{paa}^{p,q}(\mathbb{X})$ equipped with the norm $\|\cdot\|_{S^{p, q}}$ is a Banach space. \end{theorem} \begin{proof} It is sufficient to prove that $S_{paa}^{p,q}(\mathbb{X})$ is a closed subspace of $BS^{p,q}(\mathbb{X})$. Let $f_n = h_n +\varphi_n$ be a Cauchy sequence in $S_{paa}^{p,q}(\mathbb{X})$ with $(h_n^b)_{n\in \mathbb{N}} \subset AA\big(L^{p}((0,1), \mathbb{X})\big)$ and $(\varphi_n ^b)_{n \in \mathbb{N}} \subset PAA_0\big(L^{q}((0,1), \mathbb{X})\big)$ such that $\|f_n - f\|_{S^{p,q}}\to 0$ as $n \to\infty$. By Lemma \ref{lemma5.10}, one has $$ \{h_n(t+.) : t\in \mathbb{R}\}\subset \overline{\{f_n(t+.) : t\in \mathbb{R}\}}, $$ and hence $$ \|h_n\|_{S^{p}} = \|h_n\|_{S^{p,q}} \leq \|f_n\|_{S^{p,q}} \quad \text{for all } n \in \mathbb{N}. $$ Consequently, there exists a function $h \in S_{aa}^{p}(\mathbb{X})$ such that $\|h_n - h\|_{S^{p}}\to 0$ as $n \to \infty$. Using the previous fact, it easily follows that the function $\varphi:=f-h \in BS^{q}(\mathbb{X})$ and that $\|\varphi_n - \varphi\|_{S^{q}}=\|(f_n -h_n)-(f-h) \|_{S^{q}} \to 0$ as $n \to \infty$. Using the fact that $\varphi=(\varphi-\varphi_n)+\varphi_n$ it follows that \begin{align*} &{\frac{1}{2T}} \int_{-T}^{T}\Big(\int_0^{1} \|\varphi(\tau+t)\|^{q}d\tau\Big)^{1/q} dt\\ &\leq {\frac{1}{2T}} \int_{-T}^{T}\Big(\int_0^{1} \|\varphi(\tau+t)-\varphi_n(\tau+t)\|^{q}d\tau\Big)^{1/q} dt \\ &\quad + {\frac{1}{2T}} \int_{-T}^{T}\Big(\int_0^{1} \|\varphi_n(\tau+t)\|^{q}d\tau\Big)^{1/q} dt\\ &\leq \|\varphi_n - \varphi\|_{S^{q}} + {\frac{1}{2T}} \int_{-T}^{T}\Big(\int_0^{1} \|\varphi_n(\tau+t)\|^{q}d\tau\Big)^{1/q} dt. \end{align*} Letting $T \to \infty$ and then $n \to \infty$ in the previous inequality, we obtain that $\varphi^b \in PAA_0\big(L^{q}((0,1), \mathbb{X})\big)$; that is, $f = h + \varphi \in S_{paa}^{p,q}(\mathbb{X})$. \end{proof} Using similar arguments as in the proof of \cite[Theorem 3.4]{Z.Hu-Z.jin}, we obtain the next theorem. \begin{theorem} \label{T4} If $u\in S_{paa}^{p,q}(\mathbb{Y})$ and if $C\in B(\mathbb{Y},\mathbb{X})$, then the function $t \mapsto Cu(t)$ belongs to $S_{paa}^{p,q}(\mathbb{X})$. \end{theorem} \begin{definition} \label{OP2} Let $p \geq 1$ and $q\in C_{+}(\mathbb{R})$. A function $F: \mathbb{R}\times \mathbb{Y} \to \mathbb{X}$ with $F(.,u)\in BS^{p, q(x)}(\mathbb{X})$ for each $u\in\mathbb{Y}$, is said to be $S^{p,q(x)}$-pseudo-almost automorphic in $t\in \mathbb{R}$ uniformly in $u\in\mathbb{Y}$ if $t\mapsto F(t,u)$ is $S^{p,q(x)}$-pseudo-almost automorphic for each $u\in B$ where $B \subset \mathbb{Y}$ is an arbitrary bounded set. This means, there exist two functions $G,H : \mathbb{R}\times \mathbb{Y} \to \mathbb{X}$ such that $F=G+H$, where $G^{b}\in AA(\mathbb{Y},L^{p}((0,1), \mathbb{X}))$ and $H^{b}\in PAA_0(\mathbb{Y}, L^{q^b(x)}((0,1), \mathbb{X}))$; that is, $$ \lim_{T \to \infty} {\frac{1}{2T}} \int_{-T}^{T} \inf \Big\{\lambda>0: \int_0^{1} \big\|\frac{H(x+t,u)}{\lambda}\big\|^{q(x+t)}dx \leq1 \Big\} dt= 0, $$ uniformly in $u\in B$ where $B \subset \mathbb{Y}$ is an arbitrary bounded set. The collection of such functions will be denoted by $S_{paa}^{p,q(x)}(\mathbb{Y},\mathbb{X})$. \end{definition} Let $Lip^{r}(\mathbb{Y},\mathbb{X})$ denote the collection of functions $f: \mathbb{R}\times\mathbb{Y} \to \mathbb{X}$ satisfying: there exists a nonnegative function $L_{f}\in L^{r}(\mathbb{R})$ such that \[ \|f(t,u)-f(t,v)\|\leq L_{f}(t)\|u-v\|_\mathbb{Y} \quad \text{for all } u,v\in\mathbb{Y}, \; t \in \mathbb{R}. \] Now, we recall the following composition theorem for $S_{aa}^{p}$ functions. \begin{theorem}[\cite{Ding}] \label{T512} Let $p>1$ be a constant. We suppose that the following conditions hold: \begin{itemize} \item[(a)] $f\in S_{aa}^{p}(\mathbb{Y}, \mathbb{X}) \cap Lip^{r}(\mathbb{Y}, \mathbb{X})$ with $r\geq \max\{p, \frac{p}{p-1}\}$. \item[(b)] $\phi\in S_{aa}^{p}(\mathbb{X})$ and there exists a set $E\subset \mathbb{R}$ such that $K := \overline{\{\phi(t): t \in \mathbb{R}\setminus E\}}$ is compact in $\mathbb{X}$. \end{itemize} Then there exists $m\in [1, p)$ such that $f(\cdot,\phi(\cdot))\in S_{aa}^{m}(\mathbb{X})$. \end{theorem} To obtain a composition theorem for $S_{paa}^{p,q}$ functions, we need the following lemma. \begin{lemma}\label{L513} Let $p,q>1$ be a constants. Assume that $f=g+h\in S_{paa}^{p,q}(\mathbb{R} \times \mathbb{X})$ with $g^b\in AA(\mathbb{R} \times L^{p}((0,1), \mathbb{X})\big)$ and $h^b\in PAA_0(\mathbb{R} \times L^{q}((0,1), \mathbb{X})\big)$. If $f\in Lip^{p}(\mathbb{R},\mathbb{X})$, then $g$ satisfies \[ \Big(\int_0^1 \|g(t+s,u(s))-g(t+s,v(s))\|^p\,ds\Big)^{1/p} \leq c \|L_f\|_{S^p}\|u-v\|_\mathbb{Y}. \] for all $u,v\in \mathbb{Y}$ and $t\in\mathbb{R}$, where $c$ is a nonnegative constant. \end{lemma} \begin{proof} Let $f=g+h\in S_{paa}^{p,q(x)}(\mathbb{R} \times \mathbb{X})$ with $g^b(\cdot,u)\in AA(L^{p}((0,1), \mathbb{X})\big)$ and $h^b(\cdot,u)\in PAA_0(L^{q}((0,1), \mathbb{X})\big)$ for each $u\in \mathbb{Y}$. Using Lemma \ref{lemma5.10} it follows that \[ \{g(t+\cdot,u) : t\in \mathbb{R}\}\subset \overline{\{f(t+\cdot,u) : t\in \mathbb{R}\}} \quad \text{in }\ S^{p,q}(\mathbb{X}) \] for each $u\in \mathbb{Y}$. Since $f\in Lip^{p}(\mathbb{R},\mathbb{X})$ and $\big(BS^{p}(\mathbb{X}),\|\cdot\|_{S^{p}}\big) \hookrightarrow \big(BS^{p, q}(\mathbb{X}), \|\cdot\|_{S^{p, q}}\big)$, it follows that \begin{align*} \Big(\int_0^1\|g(t+s,u(s))-g(t+s,v(s))\|^p\,ds\Big)^{1/p} &\leq \|g(\cdot,u)-g(\cdot,v)\|_{S^p}\\ &=\|g(\cdot,u)-g(\cdot,v)\|_{S^{p,q}}\\ &\leq\|f(\cdot,u)-f(\cdot,v)\|_{S^{p,q}}\\ &\leq c\|f(\cdot,u)-f(\cdot,v)\|_{S^p}\\ &\leq c\|L_{f}\|_{S^p} \|u-v\|_\mathbb{Y}. \end{align*} for all $u,v\in \mathbb{Y}$ and $t\in\mathbb{R}$. \end{proof} \begin{theorem}\label{T514} Let $p,q>1$ be a constants such that $p\leq q$. Assume that the following conditions hold: \begin{itemize} \item[(a)] $f=g+h\in S_{paa}^{p,q}(\mathbb{R} \times \mathbb{X})$ with $g\in S_{aa}^{p}(\mathbb{R} \times \mathbb{X})\big)$ and $h\in S_{paa_0}^{q}(\mathbb{R} \times \mathbb{X})$. Moreover, $f,g\in Lip^{r}(\mathbb{R},\mathbb{X})$ with $r\geq \max\{p, \frac{p}{p-1}\}$; \item[(b)] $\phi=\alpha+\beta\in S_{paa}^{p,q}(\mathbb{Y})$ with $\alpha \in S_{aa}^{p}(\mathbb{Y})$ and $\beta \in S_{paa_0}^{q}(\mathbb{Y})$, and $K := \overline{\{\alpha(t): t \in \mathbb{R}\}}$ is compact in $\mathbb{Y}$. \end{itemize} Then there exists $m\in [1, p)$ such that $f(\cdot,\phi(\cdot))\in S_{paa}^{m,m}(\mathbb{R} \times \mathbb{X})$. \end{theorem} \begin{proof} First of all, write $$ f^{b}(\cdot,\phi^{b}(\cdot))=g^{b}(\cdot,\alpha^{b}(\cdot))+ f^{b}(\cdot,\phi^{b}(\cdot))-f^{b}(\cdot,\alpha^{b}(\cdot))+ h^{b}(\cdot,\alpha^{b}(\cdot)). $$ From Lemma \ref{L513}, one has $g\in S_{aa}^{p}(\mathbb{R} \times \mathbb{X})$. Now using the theorem of composition of $S^{p}$-almost automorphic functions (Theorem \ref{T512}), it is easy to see that there exists $m\in [1, p)$ with $\frac{1}{m}=\frac{1}{p}+\frac{1}{r}$ such that $g^{b}(\cdot,\alpha^{b}(\cdot))\in AA(\mathbb{R} \times L^{m}((0,1), \mathbb{X}))$. Set $\Phi^{b}(\cdot)=f^{b}(\cdot,\phi^{b}(\cdot))-f^{b}(\cdot,\alpha^{b}(\cdot))$. Clearly, $\Phi^{b} \in PAA_0(\mathbb{R} \times L^{m}((0,1),\mathbb{X}))$. Now, for $T>0$, \begin{align*} &{\frac{1}{2T}} \int_{-T}^{T}\Big(\int_t^{t+1} \|\Phi^{b}(s)\|^{m}ds\Big)^{1/m} dt\\ &= {\frac{1}{2T}}\int_{-T}^{T}\Big(\int_t^{t+1} \|f^{b}(s,\phi^{b}(s))-f^{b}(s,\alpha^{b}(s))\|^{m}ds\Big)^{1/m} dt\\ &\leq{\frac{1}{2T}} \int_{-T}^{T} \Big(\int_t^{t+1} \Big(L_{f}^{b}(s)\|\beta^{b}(s)\|_\mathbb{Y}\Big)^{m}ds \Big)^{1/m} dt\\ &\leq \|L_{f}^{b}\|_{S^{r}}\Big[{\frac{1}{2T}} \int_{-T}^{T} \Big(\int_t^{t+1} \|\beta^{b}(s)\|_\mathbb{Y}^{p}ds \Big)^{1/p} dt\Big]\\ &\leq \|L_{f}^{b}\|_{S^{r}}\Big[{\frac{1}{2T}} \int_{-T}^{T} \Big(\int_t^{t+1} \|\beta^{b}(s)\|_\mathbb{Y}^{q}ds \Big)^{1/q} dt\Big]. \end{align*} Using the fact that $\beta^b\in PAA_0(L^{q}((0,1), \mathbb{\mathbb{Y}})\big)$, it follows that $\Phi^{b} \in PAA_0(\mathbb{R} \times L^{m}((0,1), \mathbb{X}))$. On the other hand, since $f,g\in Lip^{r}(\mathbb{R},\mathbb{X}) \subset Lip^{p}(\mathbb{R},\mathbb{X})$, one has \begin{align*} &\Big(\int_0^{1} \|h(t+s,u(s))-h(t+s,v(s))\|^{m}ds\Big)^{1/m}\\ &\leq\Big(\int_0^{1} \|f(t+s,u(s))-f(t+\cdot,v(s))\|^{m}ds\Big)^{1/m}\\ &\quad +\Big(\int_0^{1}\|g(t+s,u(s))-g(t+s,v(s))\|^{m}ds\Big)^{1/m}\\ &\leq\Big(\int_0^{1} \Big(L_{f}(t+s) \|u(s)-v(s)\|_\mathbb{Y}\Big)^{m}ds\Big)^{1/m}\\ &\quad +\Big(\int_0^{1}\Big(L_{g}(t+s) \|u(s)-v(s)\|_\mathbb{Y}\Big)^{m}ds\Big)^{1/m}\\ &\leq \Big(\|L_{f}\|_{S^{r}} +\|L_{g}\|_{S^{r}}\Big) \|u(s)-v(s)\|_{p}. \end{align*} Since $K := \overline{\{\alpha(t): t \in \mathbb{R}\}}$ is compact in $\mathbb{Y}$, then for each $\varepsilon>0$, there exists a finite number of open balls $B_k=B(x_k,\varepsilon)$, centered at $x_k\in K$ with radius $\varepsilon$ such that $$ \{\alpha(t): t \in \mathbb{R}\}\subset\cup_{k=1}^{m}B_k. $$ Therefore, for $1\leq k\leq m$, the set $U_k=\{t \in \mathbb{R} : \alpha\in B_k \}$ is open and $\mathbb{R}=\cup_{k=1}^{m}U_k$. Now, for $2\leq k\leq m$, set $V_k=U_k-\cup_{i=1}^{k-1}U_i$ and $V_1=U_1$. Clearly, $V_i\cap V_j=\emptyset$ for all $i\neq j$. Define the step function $\overline{x}: \mathbb{R}\to \mathbb{\mathbb{Y}}$ by $\overline{x}(t)=x_k, t\in V_k, k=1,2,\dots ,m$. It easy to see that $$ \|\alpha(s)-\overline{x}(s)\|_\mathbb{Y}\leq \varepsilon,\quad\text{for all } s \in \mathbb{R}. $$ which yields \begin{align*} &{\frac{1}{2T}} \int_{-T}^{T}\Big(\int_t^{t+1} \|h(s,\alpha(s))\|^{m}ds\Big)^{1/m} dt\\ &\leq {\frac{1}{2T}} \int_{-T}^{T}\Big(\int_t^{t+1} \|h(s,\alpha(s))-h(s,\overline{x}(s))\|^{m}ds\Big)^{1/m} dt\\ &\quad +{\frac{1}{2T}} \int_{-T}^{T}\Big(\int_t^{t+1} \|h(s,\overline{x}(s))\|^{m}ds\Big)^{1/m} dt\\ &\leq\Big(\|L_{f}\|_{S^{r}} +\|L_{g}\|_{S^{r}}\Big) \varepsilon+{\frac{1}{2T}} \int_{-T}^{T}\Big(\sum_{k=1}^{m}\int_{V_k\cap [t,t+1]} \|h(s,\overline{x}(s))\|^{m}ds\Big)^{1/m} dt\\ &\leq \Big(\|L_{f}\|_{S^{r}} +\|L_{g}\|_{S^{r}}\Big) \varepsilon+{\frac{1}{2T}} \int_{-T}^{T}\Big(\sum_{k=1}^{m}\int_{V_k\cap [t,t+1]} \|h(s,\overline{x}(s))\|^{q}ds\Big)^{1/q} dt. \end{align*} Since $\varepsilon$ is arbitrary and $h^b\in PAA_0(\mathbb{R} \times L^{q}((0,1), \mathbb{X})\big)$, it follows that the function $h^{b}(\cdot,\alpha^{b}(\cdot))$ belongs to $PAA_0(\mathbb{R} \times L^{m}((0,1), \mathbb{X}))$. \end{proof} \begin{remark} \rm A general composition theorem in $S_{paa}^{p,q(x)}(\mathbb{R} \times \mathbb{X})$ is unlikely as compositions of elements of $S_{paa}^{p,q(x)}(\mathbb{R} \times \mathbb{X})$ may not be well-defined unless $q(\cdot)$ is the constant function. \end{remark} \section{Existence of pseudo-almost automorphic solutions} Let $p,q>1$ be constants such that $p\leq q$. In this section, we discuss the existence and uniqueness of pseudo-almost automorphic solutions to the first-order linear differential equation \eqref{AA} and to the semilinear equation \eqref{A}. For that, we make the following assumptions: \begin{itemize} \item[(H1)] The family of closed linear operators $A(t)$ satisfy Acquistapace--Terreni conditions. \item[(H2)] The evolution family $(U(t,s))_{t\geq s}$ generated by $A(t)$ has an exponential dichotomy with constants $M>0, \delta>0$, dichotomy projections $P(t), t\in \mathbb{R}$, and Green's function $\Gamma(t,s)$. \item[(H3)] $\Gamma(t,s)\in bAA(\mathbb{R}\times\mathbb{R},B(\mathbb{X}))$. \item[(H4)] $B: \mathbb{X} \mapsto \mathbb{X}$ is a bounded linear operator and let $\|B\|_{B(\mathbb{X})}=c$. \item[(H5)] $F =G+H\in S_{paa}^{p,q}(\mathbb{R} \times \mathbb{X}) \cap C(\mathbb{R} \times \mathbb{X}, \mathbb{X})$ with $G^b\in AA(\mathbb{R} \times L^{p}((0,1), \mathbb{X})\big)$ and $H^b\in PAA_0(\mathbb{R} \times L^{q}((0,1), \mathbb{X})\big)$. Moreover, $F, G\in Lip^{r}(\mathbb{R},\mathbb{X})$ with $$ r\geq \max\Big\{p, \frac{p}{p-1}\Big\}. $$ \end{itemize} Let us also mention that (H1) was introduced in the literature by Acquistapace and Terreni in \cite{P.Acquistapace2, P.Acquistapace1}. Among other things, from \cite[Theorem 2.3]{P.Acquistapace3} (see also \cite{P.Acquistapace1, A.Yagi2, A.Yagi1}), assumption (H1) does ensure that the family of operators $A(t)$ generates a unique strongly continuous evolution family on $\mathbb{X}$, which we will denote by $(U(t,s))_{t\geq s}$. \begin{definition} \rm Under (H1), if $f: \mathbb{R} \to \mathbb{X}$ is a bounded continuous function, then a mild solution to \eqref{AA} is a continuous function $u : \mathbb{R}\to \mathbb{X}$ satisfying \begin{equation} u(t)=U(t,s)u(s)+\int^{t}_{s}U(t,\sigma)f(\sigma) d\sigma\label{DD} \end{equation} for all $(t,s)\in \mathbb{T}:=\big\{(t,s)\in \mathbb{R}\times\mathbb{R}:\quad t\geq s\big\}$. \end{definition} \begin{definition} \rm Suppose (H1) and (H4) hold. If $F: \mathbb{R} \times \mathbb{X}\to \mathbb{X}$ is a bounded continuous function, then a mild solution to \eqref{A} is a continuous function $u : \mathbb{R}\to \mathbb{X}$ satisfying \begin{equation} u(t)=U(t,s)u(s)+\int^{t}_{s}U(t,\sigma)F(\sigma,Bu(\sigma)) d\sigma\label{D} \end{equation} for all $(t,s)\in \mathbb{T}$. \end{definition} \begin{theorem}\label{TT} Let $p>1$ be a constant and let $q\in C_{+}(\mathbb{R})$. Suppose that {\rm (H1)--(H3)} hold. If $f\in S_{paa}^{p,q(x)}(\mathbb{X})\cap C(\mathbb{R}, \mathbb{X})$, then the \eqref{AA} has a unique pseudo-almost automorphic solution given by \begin{equation} u(t)=\int^{+\infty}_{-\infty}\Gamma(t,\sigma)f(\sigma) d\sigma, \quad t\in\mathbb{R}.\label{EEE} \end{equation} \end{theorem} \begin{proof} Define the function $u: \mathbb{R} \mapsto \mathbb{X}$ by \begin{equation} u(t):=\int^{t}_{-\infty}U(t,\sigma)P(\sigma)f(\sigma) d\sigma - \int^{+\infty}_{t}U_{Q}(t,\sigma)Q(\sigma)f(\sigma) d\sigma , \quad t\in\mathbb{R}.\notag \end{equation} Let us show that $u$ satisfies \eqref{DD} for all $t\geq s$, all $t,s\in\mathbb{R}$. Indeed, applying $U(t,s)$ for all $t\geq s$, to both sides of the expression of $u$, we obtain, \begin{equation} \begin{split} U(t,s)u(s) & =\int^{s}_{-\infty}U(t,\sigma)P(\sigma)f(\sigma)d\sigma - \int^{+\infty}_{s}U_{Q}(t,\sigma)Q(\sigma)f(\sigma)d\sigma\\ & = \int^{t}_{-\infty}U(t,\sigma)P(\sigma)f(\sigma)d\sigma - \int^{t}_{s}U(t,\sigma)P(\sigma)f(\sigma)d\sigma\\ &\quad -\int^{+\infty}_{t}U_{Q}(t,\sigma)Q(\sigma)f(\sigma)d\sigma -\int^{t}_{s}U_{Q}(t,\sigma)Q(\sigma)f(\sigma)d\sigma\\ & = u(t)-\int^{t}_{s}U(t,\sigma)f(\sigma)d\sigma\notag \end{split} \end{equation} and hence $u$ is a mild solution to \eqref{AA}. Let us show that $u \in PAA(\mathbb{X})$. Indeed, since $f\in S_{paa}^{p,q(x)}(\mathbb{X})\cap C(\mathbb{R}, \mathbb{X})$, then $f=g+\varphi$, where $g^b\in AA(L^{p}((0,1), \mathbb{X})\big)$ and $\varphi^b\in PAA_0(L^{q^b(x)}((0,1), \mathbb{X})\big)$. Then $u$ can be decomposed as $u(t) =X(t)+Y(t)$, where \begin{gather*} X(t)=\int^{t}_{-\infty}U(t,s)P(s)g(s)ds +\int^{t}_{+\infty}U_{Q}(t,s)Q(s)g(s)ds,\\ Y(t)=\int^{t}_{-\infty}U(t,s)P(s)\varphi(s)ds +\int^{t}_{+\infty}U_{Q}(t,s)Q(s)\varphi(s)ds. \end{gather*} The proof that $X\in AA(\mathbb{X})$ is obvious and hence is omitted. To prove that $Y\in PAA_0(\mathbb{X})$, we define for all $n=1,2,\dots$, the sequence of integral operators \begin{align*} Y_n(t):&=\int^{t-n+1}_{t-n}U(t,s)P(s)\varphi(s)ds+ \int^{t+n}_{t+n-1}U_{Q}(t,s)Q(s)\varphi(s)ds\\ &=\int^{n}_{n-1}U(t,t-s)P(t-s)\varphi(t-s)ds +\int^{n}_{n-1}U_{Q}(t,t+s)Q(t+s)\varphi(t+s)ds \end{align*} for each $t\in\mathbb{R}$. Let $d\in m(\mathbb{R})$ such that $q^{-1}(x)+ d^{-1}(x)=1$. From exponential dichotomy of $(U(t,s))_{t\geq s}$ and H\"{o}lder's inequality (Theorem \ref{Holder1}), it follows that \begin{align*} \|Y_n(t)\| &\leq M\int^{t-n+1}_{t-n}e^{-\delta(t-s)}\|\varphi(s)\| ds + M\int^{t+n}_{t+n-1}e^{\delta(t-s)}\|\varphi(s)\| ds\\ &\leq M\big(\frac{1}{d^{-}}+\frac{1}{q^{-}}\big)\Big[\inf \Big\{\lambda>0: \int^{t-n+1}_{t-n} \Big(\frac{e^{-\delta(t-s)}}{\lambda}\Big) ^{d(s)}ds\leq1 \Big\}\Big]\\ &\quad \times\Big[\inf \Big\{\lambda>0: \int^{t-n+1}_{t-n} \big\|\frac{\varphi(s)}{\lambda}\big\| ^{q(s)}ds \leq1 \Big\}\Big]\\ &\quad +M\big(\frac{1}{d^{-}}+\frac{1}{q^{-}}\big)\Big[\inf \Big\{\lambda>0: \int^{t+n}_{t+n-1} \Big(\frac{e^{\delta(t-s)}}{\lambda}\Big) ^{d(s)}ds\leq1 \Big\}\Big]\\ &\quad \times\Big[\inf \Big\{\lambda>0: \int^{t+n}_{t+n-1} \Big\|\frac{\varphi(s)}{\lambda}\Big\| ^{q(s)}ds\leq1 \Big\}\Big]. \end{align*} Now since \begin{align*} \int^{t-n+1}_{t-n}\Big[\frac{e^{-\delta(t-s)}} {e^{-\delta(n-1)}}\Big]^{d(s)}ds& =\int^{t-n+1}_{t-n}\Big[e^{\delta(s-t+n-1)}\Big]^{d(s)}ds\\ &\leq\int^{t-n+1}_{t-n}\Big[1\Big]^{d(s)}ds \leq1 \end{align*} it follows that $$ e^{-\delta(n-1)}\in \Big\{\lambda>0: \int^{t-n+1}_{t-n} \Big(\frac{e^{-\delta(t-s)}}{\lambda}\Big) ^{d(s)}ds\leq1 \Big\}, $$ which shows that $$ \Big[\inf \Big\{\lambda>0: \int^{t-n+1}_{t-n} \Big(\frac{e^{-\delta(t-s)}}{\lambda}\Big) ^{d(s)}ds\leq1 \Big\}\Big]\leq e^{-\delta(n-1)}.$$ Consequently, \begin{align*} \|Y_n(t)\| &\leq M\big(\frac{1}{d^{-}}+\frac{1}{q^{-}}\big) e^{-\delta(n-1)}\|\varphi\|_{S^{q(x)}}+M\big(\frac{1}{d^{-}} +\frac{1}{q^{-}}\big) e^{\delta(1-n)}\|\varphi\|_{S^{q(x)}}\\ &\leq 2M\big(\frac{1}{d^{-}}+\frac{1}{q^{-}}\big) e^{-\delta(n-1)}\|\varphi\|_{S^{q(x)}}. \end{align*} Since the series $\sum_{n=1}^{\infty}e^{-\delta(n-1)}$ converges, we deduce from the well-known Weierstrass test that the series $\sum_{n=1}^{\infty}Y_n(t)$ is uniformly convergent on $\mathbb{R}$. Furthermore, \[ Y(t)=\int^{t}_{-\infty}U(t,s)P(s)\varphi(s)ds + \int^{t}_{+\infty}U_{Q}(t,s)Q(s)\varphi(s)ds =\sum_{n=1}^{\infty}Y_n(t), \] $Y\in C(\mathbb{R},\mathbb{X})$, and $$ \|Y(t)\| \leq\sum_{n=1}^{\infty}\|Y_n(t)\| \leq 2M\big(\frac{1}{d^{-}}+\frac{1}{q^{-}}\big) \sum_{n=1}^{\infty}e^{-\delta(n-1)}\|\varphi\|_{S^{q(x)}}. $$ Next, we will show that \[ \lim_{T \to \infty} {\frac{1}{2T}} \int_{-T}^{T} \|Y(s)\|\>ds= 0.\notag \] Indeed, \begin{align*} &{\frac{1}{2T}} \int_{-T}^{T}\|Y_n(t)\| \>dt\\ &\leq2M\big(\frac{1}{d^{-}}+\frac{1}{q^{-}}\big) e^{-\delta(n-1)} \Big[{\frac{1}{2T}} \int^{T}_{-T}\inf \Big\{\lambda>0: \int^{t+n}_{t+n-1} \Big\|\frac{\varphi(s)}{\lambda}\Big\| ^{q(s)}ds\leq1 \Big\}\Big]. \end{align*} Since $\varphi^{b}\in PAA_0(L^{q^b(x)}((0,1), \mathbb{X}))$, the above inequality leads to $Y_n\in PAA_0(\mathbb{X})$. Using the following inequality \begin{equation} {\frac{1}{2T}} \int_{-T}^{T}\|Y(s)\|\>ds\leq{\frac{1}{2T}} \int_{-T}^{T} \Big\|Y(s)-\sum_{n=1}^{\infty}Y_n(s)\Big\|\>dt +\sum_{n=1}^{\infty}{\frac{1}{2T}} \int_{-T}^{T} \|Y_n(s)\|\>ds,\notag \end{equation} we deduce that the uniform limit $Y(\cdot)=\sum_{n=1}^{\infty}Y_n(\cdot)\in PAA_0(\mathbb{X})$. Therefore $u\in PAA(\mathbb{X})$. It remains to prove the uniqueness of $u$ as a mild solution. This has already been done by Diagana \cite{TDbook, TKD2}. However, for the sake of clarity let us reproduce it here. Let $u,v$ be two bounded mild solutions to \eqref{AA}. Setting $w=u-v$, one can easily see that $w$ is bounded and that $w(t)=U(t,s)w(s)$ for all $(t,s)\in \mathbb{T}$. Now using property (i) from exponential dichotomy (Definition \ref{evolution}) it follows that $ P(t)w(t)=P(t)U(t,s)w(s)=U(t,s)P(s)w(s)$, and hence $$ \|P(t)w(t)\|=\|U(t,s)P(s)w(s)\|\leq Me^{-\delta(t-s)}\|w(s)\| \leq Me^{-\delta(t-s)}\|w\|_{\infty}. $$ for all $(t,s)\in \mathbb{T}$. Now, given $t\in \mathbb{R}$ with $ t\geq s$, if we let $s\to-\infty$, we then obtain that $P(t)w(t)=0$, that is, $P(t)u(t)=P(t)v(t)$. Since $t$ is arbitrary it follows that $P(t)w(t)=0$ for all $t\geq s$. Similarly, from $w(t)=U(t,s)w(s)$ for all $t\geq s$ and property (i) from exponential dichotomy (Definition \ref{evolution}) it follows that $Q(t)w(t)=Q(t)U(t,s)w(s)=U(t,s)Q(s)w(s)$, and hence $U_{Q}(s,t)Q(t)w(t)=Q(s)w(s)$ for all $t\geq s$. Moreover, $$ \|Q(s)w(s)\|=\|U_{Q}(s,t)Q(t)w(t)\|\leq Me^{-\delta(t-s)}\|w\|_{\infty}. $$ for all $t\geq s$. Now, given $s\in \mathbb{R}$ with $ t\geq s$, if we let $t\to+\infty$, we then obtain that $Q(t)w(t)=0$, that is, $Q(s)u(s)=Q(s)v(s)$. Since $s$ is arbitrary it follows that $Q(s)w(s)=0$ for all $t\geq s$. \end{proof} Using Theorem \ref{TT} one easily proves the following theorem. \begin{theorem}\label{FTO} Let $p,q>1$ be constants such that $p\leq q$. Under assumptions {\rm(H1)--(H5)}, then \eqref{A} has a unique solution whenever $\|L_{F}\|_{S^{r}}$ is small enough. And the solution satisfies the integral equation \begin{equation*} u(t)=\int^{t}_{-\infty}U(t,\sigma)P(\sigma)F(\sigma,Bu(\sigma)) d\sigma - \int^{+\infty}_{t}U_{Q}(t,\sigma)Q(\sigma)F(\sigma,Bu(\sigma)) d\sigma , \; t\in\mathbb{R}\label{E}. \end{equation*} \end{theorem} \begin{proof} Define $\Xi: PAA(\mathbb{X}) \to PAA(\mathbb{X})$ as \begin{equation} (\Xi u)(t)=\int^{t}_{-\infty}U(t,\sigma)P(\sigma) F(\sigma,Bu(\sigma)) d\sigma - \int^{+\infty}_{t}U_{Q}(t,\sigma)Q(\sigma) F(\sigma,Bu(\sigma)) d\sigma \notag \end{equation} Let $u\in PAA(\mathbb{X})\subset S_{paa}^{p,q}(\mathbb{X})$. From (H4) and Theorem \ref{T4} it is clear that $Bu(.)\in S_{paa}^{p,q}(\mathbb{X})$. Using the composition theorem for $S_{paa}^{p,q}$ functions, we deduce that there exists $m\in [1, p)$ such that $F(.,Bu(.))\in S_{paa}^{m,m}(\mathbb{X})$. applying the proof of Theorem \ref{TT}, to $f(.)=F(., Bu(.))$, one can easily see that the operator $\Xi$ maps $PAA(\mathbb{X})$ into its self. Moreover, for all $u,v \in PAA(\mathbb{X})$, it is easy to see that \begin{align*} &\|(\Xi u)(t)-(\Xi v)(t)\| \\ &\leq \int_{\mathbb{R}}\|\Gamma(t-s)\| \|F(s,Bu(s))-F(s,Bv(s))\|\>ds\\ &\leq \int^{t}_{-\infty} cMe^{-\delta (t-s)}L_{F}(s)\>ds \|u-v\|_{\infty} + \int^{+\infty}_{t}cMe^{\delta (t-s)}L_{F}(s)\>ds \|u-v\|_{\infty}\\ &\leq \sum_{n=1}^{\infty}\int^{t-n+1}_{t-n}c Me^{-\delta (t-s)}L_{F}(s)\>ds \|u-v\|_{\infty}\\ &\quad +\sum_{n=1}^{\infty}\int^{t+n}_{t+n-1} c Me^{\delta (t-s)}L_{F}(s)\>ds \|u-v\|_{\infty}\\ &\leq c M\sum_{n=1}^{\infty}\Big(\int^{t-n+1}_{t-n}e^{-r_0 \delta (t-s)}\>ds\Big)^{\frac{1}{r_0}} \|L_{F}\|_{S^{r}} \|u-v\|_{\infty}\\ &\quad +c M\sum_{n=1}^{\infty}\Big(\int^{t+n}_{t+n-1}e^{r_0 \delta (t-s)}\>ds\Big)^{\frac{1}{r_0}} \|L_{F}\|_{S^{r}} \|u-v\|_{\infty}\\ &\leq 2cM\sum_{n=1}^{\infty} \Big(\frac{e^{-r_0(n-1)\delta}-e^{-r_0n\delta}}{r_0\delta}\Big) ^{\frac{1}{r_0}} \|L_{F}\|_{S^{r}} \|u-v\|_{\infty}\\ &\leq 2cM \sqrt[r_0]{\frac{1+e^{r_0\delta}}{r_0\delta}} \sum_{n=1}^{\infty} e^{-n\delta} \|L_{F}\|_{S^{r}} \|u-v\|_{\infty}, \end{align*} for each $t\in\mathbb{R}$, where $ \frac{1}{r}+\frac{1}{r_0}=1. $ Hence whenever $\|L_{F}\|_{S^{r}}$ is small enough, that is, $$ 2cM \sqrt[r_0]{\frac{1+e^{r_0\delta}}{r_0\delta}} \sum_{n=1}^{\infty} e^{-n\delta} \|L_{F}\|_{S^{r}}<1, $$ then $\Xi$ has a unique fixed point, which obviously is the unique pseudo-almost automorphic solution to \eqref{A}. \end{proof} \begin{thebibliography}{00} \bibitem{P.Acquistapace3} P. 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