\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2018 (2018), No. 27, pp. 1--15.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2018 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2018/27\hfil $\mu$-Stepanov pseudo almost periodic functions] {Composition and convolution theorems for $\mu$-Stepanov pseudo almost periodic functions and applications to fractional integro-differential equations} \author[E. Alvarez \hfil EJDE-2018/27\hfilneg] {Edgardo Alvarez} \address{Edgardo Alvarez \newline Universidad del Norte, Departamento de Matem\'aticas y Estad\'istica, Barranquilla, Colombia} \email{edgalp@yahoo.com, ealvareze@uninorte.edu.co} \thanks{Submitted September 12, 2016. Published January 18, 2018.} \subjclass[2010]{45D05, 34A12, 45N05} \keywords{$\mu$-Stepanov pseudo almost periodic; mild solutions; \hfill\break\indent fractional integro-differential equations; composition; convolution} \begin{abstract} In this article we establish new convolution and composition theorems for $\mu$-Stepanov pseudo almost periodic functions. We prove that the space of vector-valued $\mu$-Stepanov pseudo almost periodic functions is a Banach space. As an application, we prove the existence and uniqueness of $\mu$-pseudo almost periodic mild solutions for the fractional integro-differential equation \[ D^\alpha u(t)=Au(t)+\int_{-\infty}^t a(t-s)Au(s)\,ds+f(t,u(t)), \] where $A$ generates an $\alpha$-resolvent family $\{S_\alpha(t)\}_{t\geq 0}$ on a Banach space $X$, $a\in L^1_{\rm loc}(\mathbb{R}_+)$, $\alpha>0$, the fractional derivative is understood in the sense of Weyl and the nonlinearity $f$ is a $\mu$-Stepanov pseudo almost periodic function. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} Ezzinbi et al.\ \cite{Ak-Ez-Kh15} defined the space of $\mu$-$S^p$-pseudo almost periodic functions. This space contains the space of Stepanov-like weighted pseudo almost periodic functions (see \cite{Di,Di-Mo-Gue}) and the space of $\mu$-pseudo almost periodic functions (see \cite{Blot-Ci-Ez13}). Several composition theorems and their applications in the context of Stepanov-like almost periodic, Stepanov-like pseudo almost periodic and Stepanov-like weighted pseudo almost periodic functions appear for example in \cite{Al-Li15,Ding-Long, Di-Lo-Gue11,Li-Zhang,Zhao11}. Here we generalize the composition theorem given by Zhao et al.\ for the space of Stepanov-like weighted pseudo almost periodic functions (see \cite[Th. 2.15]{Zhao11}). Also, we recover the composition result given by Ezzinbi et al.\ for $\mu$-$S^p$-pseudo almost periodic functions (see \cite[Th. 2.29]{Ak-Ez-Kh15}). Moreover, we establish another composition theorem that does not require Lipschitzian nonlinearities (Theorem \ref{compthe1} and Theorem \ref{comp.the3}). In Theorem \ref{convolutionth} we prove that the convolution of a strongly continuous family $\{S(t)\}_{t\geq0}$ with a $\mu$-$S^p$-pseudo almost periodic function $F$, $(S\ast f)(t)=\int_{-\infty}^tS(t-s)F(s)ds$, is a $\mu$-pseudo almost periodic function. We prove that the collection of $\mu$-$S^p$-pseudo almost periodic functions is a Banach space with a natural norm (Theorem \ref{mainres1}), and combine our results to prove the existence and uniqueness of $\mu$-pseudo almost periodic solutions to a class of abstract fractional differential equations \begin{equation}\label{0.1} D^\alpha u(t)=Au(t)+\int_{-\infty}^t a(t-s)Au(s)\,ds+f(t,u(t)), \end{equation} where $A$ generates an $\alpha$-resolvent family $\{S_\alpha(t)\}_{t\geq 0}$ on a Banach space $X$, $a\in L^1_{\rm loc}(\mathbb{R}_+)$, $\alpha>0$, the fractional derivative is understood in the sense of Weyl and provided that the nonlinear term $f$ is $\mu$-Stepanov pseudo almost periodic. \section{Preliminaries} Throughout this article $(X,\|\cdot\|_X)$ and $(Y,\|\cdot\|_Y)$ denote complex Banach spaces and $B(X,Y)$ the Banach space of bounded linear operators from $X$ to $Y$; when $X=Y$ we write $B(X)$. We denote by $ BC(\mathbb{R},X)$ the Banach space of $X$-valued bounded and continuous defined functions on $\mathbb R$, with norm \begin{equation} \|f\| = \sup\{\|f(t)\|_X: t \in \mathbb{R}\}. \end{equation} \begin{definition}[\cite{Bochner1}] \rm A function $f\in C(\mathbb{R},X)$ is called (Bohr) almost periodic if for each $\epsilon>0$ there exists $l=l(\epsilon)>0$ such that every interval of length $l$ contains a number $\tau$ with the property that \[ \|f(t+\tau)-f(t)\|<\epsilon\quad (t\in\mathbb{R}). \] The collection of all such functions will be denoted by $AP(\mathbb{R},X)$. \end{definition} This definition is equivalent to the so-called Bochner's criterion, namely, $f\in AP(\mathbb{R},X)$ if and only if for every sequence of reals $(s_n')$ there exists a subsequence $(s_n)$ such that $(f(\cdot+ s_n))$ is uniformly convergent on $\mathbb{R}$. \begin{definition}[\cite{Bochner1}] \rm A function $f\in C(\mathbb{R}\times Y,X)$ is called (Bohr) almost periodic in $t\in\mathbb{R}$ uniformly in $y\in K$ where $K\subset Y$ is any compact subset if for each $\epsilon>0$ there exists $l=l(\epsilon)>0$ such that every interval of length $l$ contains a number $\tau$ with the property that \[ \|f(t+\tau,y)-f(t,y)\|<\epsilon\quad (t\in\mathbb{R},\; y\in K). \] The collection of such functions will be denoted by $AP(\mathbb{R}\times Y,X)$. \end{definition} Let $\mathcal{B}$ denote the Lebesgue $\sigma$-field of $\mathbb{R}$, see \cite{Blot-Ci-Ez12}. Let $\mathcal{M}$ stand for the set of all positive measures $\nu$ on $\mathcal{B}$ satisfying $\mu(\mathbb{R})=\infty$ and $\mu([a,b])<\infty$ for all $a,b\in\mathbb{R}$. Throughout this paper will consider the following hypotheses: \begin{itemize} \item[(H1)] For all $a,b$ and $c\in \mathbb{R}$, such that $0\leq a0$ such that \[ |\tau|\leq \tau_0\Rightarrow \mu((a+\tau,b+\tau))\geq\alpha_0\mu([\tau,c+\tau]). \] \item[(H2)] For all $\tau\in\mathbb{R}$, there exist $\beta>0$ and a bounded interval $I$ such that $\mu(\{a+\tau,a\in A\})\leq\beta\mu(A)$ if $A\in \mathcal{B}$ satisfies $A\cap I=\emptyset$. \end{itemize} Note that Hypothesis (H2) implies (H1), see \cite[Lemma 2.1]{Blot-Ci-Ez13}. \begin{definition}[\cite{Blot-Ci-Ez12}] \rm Let $\mu\in \mathcal{M}$. A function $f\in BC(\mathbb{R},X)$ is said to be $\mu$-ergodic if $$ \lim_{T\to +\infty}\frac{1}{\mu([-T,T])}\int_{[-T,T]} \|f(t)\|d\mu(t)=0. $$ We denote by $\mathcal{E}(\mathbb{R},X,\mu)$ the set of such functions. A function $f\in BC(\mathbb{R}\times X,X)$ is said to be $\mu$-ergodic if $$ \lim_{T\to +\infty}\frac{1}{\mu([-T,T])}\int_{[-T,T]}\|f(t,x)\|d\mu(t)=0, $$ uniformly in $x\in X$. Denote by $\mathcal{E}(\mathbb{R}\times X,X,\mu)$ the set of such functions. \end{definition} \begin{definition}[\cite{Blot-Ci-Ez13}] \rm Let $\mu\in \mathcal{M}$. A function $f\in C(\mathbb{R},X)$ is said to be $\mu$-pseudo almost periodic if it can be decomposed as $f=g+\varphi$, where $g\in AP(\mathbb{R},X)$ and $\varphi\in \mathcal{E}(\mathbb{R},X,\mu)$. Denote by $PAP(\mathbb{R},X,\mu)$ the collection of such functions. \end{definition} \begin{definition}[\cite{Di-Mo-Gue}] \rm The Bochner transform $f^b(t,s)$ with $t\in\mathbb{R}, s\in[0,1]$ of a function $f:\mathbb{R}\to X$ is defined by \[ f^b(t,s):=f(t+s). \] \end{definition} \begin{definition}[\cite{Di-Mo-Gue}] \rm The Bochner transform $f^b(t,s,u)$ with $t\in\mathbb{R}$, $s\in[0,1]$, $u\in X$ of a function $f:\mathbb{R}\times X\to X$ is defined by \[ f^b(t,s,u):=f(t+s,u)\quad \text{for all } u\in X. \] \end{definition} \begin{definition}[\cite{Di-Mo-Gue}] \rm Let $p\in[1,\infty)$. The space $BS^p(\mathbb{R},X)$ of all Stepanov bounded functions, with exponent $p$, consist of all measurable functions $f:\mathbb{R}\to X$ such that $f^b\in L^{\infty}(\mathbb{R},L^p(0,1;X))$. This is a Banach space with the norm \[ \|f\|_{BS^p(\mathbb{R},X)}:=\|f^b\|_{L^{\infty}(\mathbb{R},L^p)} =\sup_{t\in\mathbb{R}}\Big(\int_{t}^{t+1}\|f(\tau)\|^p\,d\tau\Big)^{1/p}. \] \end{definition} \begin{definition}[\cite{Di}] \rm A function $f\in BS^p(\mathbb{R},X)$ is called Stepanov almost periodic if $f^b\in AP(\mathbb{R},L^p(0,1;X))$. We denote the set of all functions by $APS^p(\mathbb{R},X)$. \end{definition} \begin{definition}[\cite{Di}] \rm A function $f:\mathbb{R}\times X\to Y$ with $f(\cdot,u)\in BS^p(\mathbb{R},Y)$, for each $u\in X$, is called Stepanov almost periodic function in $t\in\mathbb{R}$ uniformly for $u\in X$ if, for each $\epsilon>0$ and each compact set $K\subset X$ there exists a relatively dense set $P=P(\epsilon,f,K)\subset\mathbb{R}$ such that \[ \sup_{t\in\mathbb{R}}\Big(\int_{0}^1\|f(t+s+\tau,u)-f(t+s,u)\|\,ds\Big)^{1/p} <\epsilon, \] for each $\tau\in P$ and each $u\in K$. We denote by $APS^p(\mathbb{R}\times X,Y)$ the set of such functions. \end{definition} \begin{definition}[\cite{Ak-Ez-Kh15}] \rm Let $\mu\in \mathcal{M}$. A function $f\in BS^p(\mathbb{R},X)$ is said $\mu$-Stepanov-like pseudo almost periodic (or $\mu$-$S^p$-pseudo almost periodic) if it can be expressed as $f=g+\phi$, where $g\in APS^p(\mathbb{R},X)$ and $\phi^b\in\mathcal{E}(\mathbb{R},L^p(0,1;X),\mu)$. In other words, a function $f\in L^p_{\rm loc}(\mathbb{R},X)$ is said $\mu$-$S^p$-pseudo almost periodic relatively to measure $\mu$, if its Bochner transform $f^b:\mathbb{R}\to L^p(0,1;X)$ is $\mu$-pseudo almost periodic in the sense that there exist two functions $g,\phi:\mathbb{R}\to X$ such that $f=g+\phi$, where $g\in APS^p(\mathbb{R},X)$ and $\phi^b\in\mathcal{E}(\mathbb{R},L^p(0,1;X),\mu)$, that is $\phi^b\in BC(\mathbb{R},L^p(0,1;X))$ and \[ \lim_{T\to +\infty}\frac{1}{\mu([-T,T])} \int_{[-T,T]}(\int_t^{t+1}\|\phi(s)\|^pds)^{1/p}d\mu(t)=0. \] We denote by $PAPS^p(\mathbb{R},X,\mu)$ the set of all such functions. \end{definition} \begin{definition}[\cite{Ak-Ez-Kh15}] \rm Let $\mu\in \mathcal{M}$. A function $f:\mathbb{R}\times Y\to X$ with $f(\cdot,u)\in L^p_{\rm loc}(\mathbb{R},X)$ for each $u\in Y$, is said to be $\mu$-Stepanov-like pseudo almost periodic (or $\mu$-$S^p$-pseudo almost periodic) if it can be expressed as $f=g+\phi$, where $g\in APS^p(\mathbb{R}\times Y,X)$ and $\phi^b\in\mathcal{E}(\mathbb{R}\times Y,L^p(0,1;X),\mu)$. We denote by $PAPS^p(\mathbb{R}\times Y,X,\mu)$ the set of all such functions. \end{definition} \section{Main results} For $1\leq p<\infty$, we define $\mathcal{B}:BS^p(\mathbb{R},X)\to L^{\infty}(\mathbb{R},L^p(0,1;X))$ by \[ % \label{defoperator} f\mapsto(\mathcal{B}f)(t)(s)=f^b(t,s)=f(t+s)\quad(t\in\mathbb{R},\; s\in[0,1]), \] see \cite{Al-Li15}. \begin{remark}\label{isometry} \rm It follows from its definition that the operator $\mathcal{B}$ is a linear isometry between $BS^p(\mathbb{R},X)$ and $L^{\infty}(\mathbb{R},L^p(0,1;X))$. More precisely, \[ \|\mathcal{B}f\|_{L^{\infty}(\mathbb{R},L^p)} =\|f\|_{BS^p(\mathbb{R},X)}. \] \end{remark} \begin{remark} \rm The definition of $\mu$-Stepanov-like pseudo almost periodic functions can be written using the preceding notation. Thus, for $\mu\in\mathcal{M}$, we say that a function $f$ is said to be $\mu$-Stepanov-like pseudo almost periodic (or $\mu$-$S^p$-pseudo almost periodic) if and only if $f\in \mathcal{B}^{-1}(AP(\mathbb{R},L^p(0,1;X)))+ \mathcal{B}^{-1}(\mathcal{E} (\mathbb{R},L^p(0,1;X),\mu))$. Thus, \begin{equation}\label{WPAPS^p} PAPS^p(\mathbb{R},X,\mu) =\mathcal{B}^{-1}(AP(\mathbb{R},L^p(0,1;X)))+ \mathcal{B}^{-1}(\mathcal{E}(\mathbb{R},L^p(0,1;X),\mu)). \end{equation} Also, assume that $\mu$ satisfies (H1). Since $\mathcal{B}$ is an isometry and $AP(\mathbb{R},L^p(0,1;X))\cap \mathcal{E}(\mathbb{R},L^p(0,1;X),\mu)=\{0\}$ by \cite[Cor. 2.29]{Blot-Ci-Ez13} we have that the sum is direct, that is, $$ PAPS^p(\mathbb{R},X,\mu)=\mathcal{B}^{-1}(AP(\mathbb{R},L^p(0,1;X)))\oplus \mathcal{B}^{-1}(\mathcal{E}(\mathbb{R},L^p(0,1;X),\mu)). $$ \end{remark} Based on the definition of the operator $\mathcal{B}$, next we prove that $PAPS^p(\mathbb{R},X,\mu)$ is a Banach space. \begin{theorem}\label{mainres1} If $\mu\in \mathcal{M}$ satisfies {\rm (H1)}, then $PAPS^p(\mathbb{R},X,\mu)$ is a Banach space with the norm \[ \|f\|_{PAPS^p(\mathbb{R},X,\mu)} =\|g\|_{BS^p(\mathbb{R},X)}+\|h\|_{BS^p(\mathbb{R},X)} \] where $f=g+h$ with $g\in\mathcal{B}^{-1}(AP(\mathbb{R},L^p(0,1;X)))$, $h\in\mathcal{B}^{-1}(\mathcal{E}(\mathbb{R},L^p(0,1;X),\mu))$. \end{theorem} \begin{proof} Let $(f_n)$ be a Cauchy sequence in $PAPS^p(\mathbb{R},X,\mu)$. Then \[ \|f_n-f_m\|_{PAPS^p(\mathbb{R},X,\mu)}\to0\quad\text{as }n,m\to\infty. \] Let $f_n=g_n+h_n$ and $f_m=g_m+h_m$ with $g_n,g_m\in\mathcal{B}^{-1}(AP(\mathbb{R},L^p(0,1;X)))$ and $h_n,h_m\in\mathcal{B}^{-1}(\mathcal{E}(\mathbb{R},L^p(0,1;X),\mu))$. If $n,m\to\infty$, then \begin{gather*} \|\mathcal{B}g_n-\mathcal{B}g_m\|_{L^{\infty}(\mathbb{R},L^p)} =\|g_n-g_m\|_{BS^p(\mathbb{R},X)}\leq\|f_n-f_m\|_{PAPS^p(\mathbb{R},X,\mu)}\to0,\\ \|\mathcal{B}h_n-\mathcal{B}h_m\|_{L^{\infty}(\mathbb{R},L^p)} =\|h_n-h_m\|_{BS^p(\mathbb{R},X)}\leq\|f_n-f_m\|_{PAPS^p(\mathbb{R},X,\mu)}\to0. \end{gather*} This implies that $(\mathcal{B}g_n)$ and $(\mathcal{B}h_n)$ are Cauchy sequences in $AP(\mathbb{R},L^p(0,1;X))$ and $\mathcal{E}(\mathbb{R},L^p(0,1;X),\mu)$ respectively. Since $AP(\mathbb{R},L^p(0,1;X))$ is a closed subspace of \newline $BC(\mathbb{R},L^p(0,1;X))$ then it is a Banach space. Also, it follows from \cite[Cor. 2.31]{Blot-Ci-Ez13} that $\mathcal{E}(\mathbb{R},L^p(0,1;X),\mu)$ is a Banach space. Then there exist $g\in AP(\mathbb{R},L^p(0,1;X))$ and $h\in \mathcal{E}(\mathbb{R},L^p(0,1;X),\mu)$ such that \[ \|\mathcal{B}g_n-g\|_{L^{\infty}(\mathbb{R},L^p)}\to0,\quad \|\mathcal{B}h_n-h\|_{L^{\infty}(\mathbb{R},L^p)}\to0\quad (n\to\infty). \] Let \begin{gather*} f_1:=\mathcal{B}^{-1}(\{g\})\in \mathcal{B}^{-1}(AP(\mathbb{R},L^p(0,1;X)))\\ f_2:=\mathcal{B}^{-1}(\{h\})\in \mathcal{B}^{-1}(\mathcal{E}(\mathbb{R}, L^p(0,1;X),\mu)). \end{gather*} Note that $f_1$ and $f_2$ are well defined because $\mathcal{B}$ is injective. Let $f:=f_1+f_2\in PAPS^p(\mathbb{R},X,\mu)$. Thus \begin{align*} \|f_n-f\|_{PAPS^p(\mathbb{R},X,\mu)} &=\|(g_n+h_n)-(f_1+f_2)\|_{PAPS^p(\mathbb{R},X,\mu)}\\ &=\|g_n-f_1\|_{BS^p(\mathbb{R},X)}+\|h_n-f_2\|_{BS^p(\mathbb{R},X)}\\ &=\|\mathcal{B}g_n-\mathcal{B}f_1\|_{L^{\infty}(\mathbb{R},L^p)} +\|\mathcal{B}h_n-\mathcal{B}f_2\|_{L^{\infty}(\mathbb{R},L^p)}\\ &=\|\mathcal{B}g_n-g\|_{L^{\infty}(\mathbb{R},L^p)} +\|\mathcal{B}h_n-h\|_{L^{\infty}(\mathbb{R},L^p)}\to0\,\,\,\,\,(n\to\infty). \end{align*} Hence $PAPS^p(\mathbb{R},X,\mu)$ is a Banach space. \end{proof} The following theorem is taken from \cite[Theorem 2.1]{Ch-Zh-Gue15}. \begin{theorem}\label{the2.1} Let $\mu\in\mathcal{M}$ and $I$ be a bounded interval (eventually $\emptyset$). Assume that $f(\cdot)\in BS^{p}(\mathbb{R},X)$. Then the following assertions are equivalent. \begin{itemize} \item[(a)] $f^{b}(\cdot)\in \mathcal{E}(\mathbb{R},L^p(0,1;X)),\mu)$. \item[(b)] \[ \lim_{T\to\infty}\frac{1}{\mu([-T,T]\setminus I)} \int_{\mu([-T,T]\setminus I)}\Big(\int_t^{t+1}\|f(s)\|^pds\Big)^{1/p}d\mu(t)=0. \] \item[(c)] For any $\epsilon>0$, \[ \lim_{T\to\infty}\frac{\mu\Big(t\in[-T,T]\setminus I: \big(\int_t^{t+1}\|f(s)\|^pds\big)^{1/p}>\epsilon\Big)}{\mu([-T,T]\setminus I)}=0. \] \end{itemize} \end{theorem} The following theorem about composition of Stepanov-like type pseudo almost periodic functions generalizes \cite[Theorem 2.15]{Zhao11}. \begin{theorem}\label{compthe1} Let $\mu\in\mathcal{M}$ and let $f=g+\phi\in PAPS^p(\mathbb{R}\times X,X,\mu)$ with $g\in \mathcal{B}^{-1}(AP(\mathbb{R}\times X,L^p(0,1;X)))$ and $\phi\in \mathcal{B}^{-1}(\mathcal{E}(\mathbb{R}\times X,L^p(0,1;X)),\mu)$. Assume the following conditions. \begin{itemize} \item[(a)] $f(t,x)$ is uniformly continuous in any bounded set $K'\subset X$ uniformly for $t\in \mathbb{R}$, \item[(b)] $g(t,x)$ is uniformly continuous in any bounded set $K'\subset X$ uniformly for $t\in \mathbb{R}$, \item[(c)] for every bounded subset $K'\subset X$, the set $\{f(\cdot,x):x\in K'\}$ is bounded in $ PAPS^p(\mathbb{R}\times X,X,\mu)$. \end{itemize} If $x=\alpha+\beta\in PAPS^p(\mathbb{R},X,\mu)\cap B(\mathbb{R},X)$, with $\alpha\in \mathcal{B}^{-1}(AP(\mathbb{R},L^p(0,1;X)))$, $\beta\in \mathcal{B}^{-1}(\mathcal{E}(\mathbb{R},L^p(0,1;X),\mu))$ and $Q=\overline{\{x(t):t\in \mathbb{R}\}}$, $Q_1=\overline{\{\alpha(t):t\in \mathbb{R}\}}$ are compact, then $f(\cdot,x(\cdot))\in PAPS^p(\mathbb{R},X,\mu)$. \end{theorem} \begin{proof} Let \[ f(t,x(t))=G(t)+H(t)+W(t), \] where \[ G(t)=g(t,\alpha(t)),\quad H(t)=f(t,x(t))-f(t,\alpha(t)),\quad W(t)=\phi(t,\alpha(t)). \] Since $g$ satisfies condition (b) and $Q_1=\overline{\{\alpha(t):t\in \mathbb{R}\}}$ is compact, by \cite[Prop. 1]{Am-Ma} we have $G\in \mathcal{B}^{-1}(AP(\mathbb{R},L^p(0,1;X)))$. To show that $f(\cdot,x(\cdot))\in PAPS^p(\mathbb{R},X,\mu)$ it is sufficient to show that $H,W\in\mathcal{B}^{-1}(\mathcal{E}(\mathbb{R},L^p(0,1;X)))$. First, we see that $H\in \mathcal{B}^{-1}(\mathcal{E}(\mathbb{R},L^p(0,1;X)))$. Since $x(\cdot)$ and $\alpha(\cdot)$ are bounded, we can choose a bounded subset $K'\subset X$ such that $x(\mathbb{R}),\alpha(\mathbb{R})\subset K'$. By assumption $(c)$ we have that $H(\cdot)\in BS^p(\mathbb{R},X)$ and by assumption (a) we obtain that $f$ is uniformly continuous on the bounded set $K'\subset X$ uniformly $t\in \mathbb{R}$. Then, given $\epsilon>0$, there exists $\delta>0$, such that $u,v\in K'$ and $\|u-v\|<\delta$ imply that $\|f(t,u)-f(t,v)\|\leq \epsilon$ for all $t\in \mathbb{R}$. Then, we have \[ \Big(\int_t^{t+1}\|f(s,u)-f(s,v)\|^pds\Big)^{1/p}\leq \epsilon. \] Hence, for each $t\in\mathbb{R}$, $\|\beta(s)\|_{BS^p(\mathbb{R},X)}<\delta$, $s\in [t,t+1]$ implies that for all $t\in \mathbb{R}$, \[ \Big(\int_t^{t+1}\|H(s)\|^pds\Big)^{1/p} =\Big(\int_t^{t+1}\|f(s,x(s))-f(s,\alpha(s))\|^pds\Big)^{1/p} \leq \epsilon. \] Therefore, \begin{align*} &\frac{\mu\Big(t\in[-T,T]: \big(\int_t^{t+1}\|f(s,x(s)) -f(s,\alpha(s))\|^pds\big)^{1/p}>\epsilon\Big)}{\mu([-T,T])}\\ &\leq \frac{\mu\Big(t\in[-T,T]: \big(\int_t^{t+1}\|\beta(s)\|^pds\big)^{1/p} >\delta\Big)}{\mu([-T,T])}. \end{align*} Since $\beta\in \mathcal{B}^{-1}(\mathcal{E}(\mathbb{R},L^p(0,1;X),\mu))$, then Theorem \ref{the2.1} implies that for the above mentioned $\delta$ we have $$ \lim_{T\to\infty}\frac{\mu\Big(t\in[-T,T]: \big(\int_t^{t+1}\|f(s,x(s))-f(s,\alpha(s)) \|^pds\big)^{1/p}>\epsilon\Big)}{\mu([-T,T])}=0. $$ By Theorem \ref{the2.1} we have that $H\in \mathcal{B}^{-1}(\mathcal{E}(\mathbb{R},L^p(0,1;X)))$. Now, we prove that $W\in \mathcal{B}^{-1}(\mathcal{E}(\mathbb{R},L^p(0,1;X)))$. Since $f$ and $g$ satisfy (a) and (b) respectively, then, given $\epsilon>0$, exists $\delta>0$, such that $u,v\in Q_1$, $\|u-v\|<\delta$ imply that \begin{gather*} \Big(\int_t^{t+1}\|f(s,u)-f(s,v)\|^pds\Big)^{1/p} \leq \frac{\epsilon}{16},\quad t\in\mathbb{R}, \\ \Big(\int_t^{t+1}\|g(s,u)-g(s,v)\|^pds\Big)^{1/p} \leq \frac{\epsilon}{16},\quad t\in\mathbb{R}. \end{gather*} Let $\delta_0:=\min\{\epsilon,\delta\}$. Then \begin{align*} &\Big(\int_t^{t+1}\|\phi(s,u)-\phi(s,v)\|^pds\Big)^{1/p}\\ &\leq \Big(\int_t^{t+1}\|f(s,u)-f(s,v)\|^pds\Big)^{1/p} +\Big(\int_t^{t+1}\|g(s,u)-g(s,v)\|^pds\Big)^{1/p}\\ &\leq\frac{\epsilon}{8}, \end{align*} for all $t\in \mathbb{R}$, and $u,v\in Q_1$, $\|u-v\|<\delta_0$. Since $Q_1=\overline{\{\alpha(t):t\in \mathbb{R}\}}$ is compact, there exist open balls $O_k$ ($k=1,2,\dots ,m$) with center in $u_k\in Q_1$ and radius $\delta_0$ given above, such that $\{\alpha(t):t\in \mathbb{R}\}\subset \cup_{k=1}^mO_k$. Define and choose $B_k$ such that $B_k:=\{t\in\mathbb{R}:\|\alpha(t)-u_k\|<\delta_0\}$, $k=1,2,\dots ,m$, $\mathbb{R}=\cup_{k=1}^mB_k$ and set $C_1=B_1$, $C_k=B_k\setminus (\cup_{j=1}^{k-1}B_j)$ ($k=2,3,\dots ,m$). Then $\mathbb{R}=\cup_{k=1}^mC_k$ where $C_i\cap C_j=\emptyset$, $i\neq j$, $1\leq i,j\leq m$. Let us define the function $\overline{u}:\mathbb{R}\to X$ by $\overline{u}(t)=u_k$ for $t\in C_k$, $k=1,\dots ,m$. Then $\|\alpha(t)-\overline{u}\|<\delta_0$ for all $t\in\mathbb{R}$ and \begin{align*} &\Big(\sum_{k=1}^m\int_{C_k\cap[t,t+1]}\|\phi(s,\alpha(s))-\phi(s,u_k)\|^pds \Big)^{1/p} \\ &=\Big(\int_t^{t+1}\|\phi(s,\alpha(s))-\phi(s,\overline{u}(s))\|^pds\Big)^{1/p} <\frac{\epsilon}{8}. \end{align*} Since $\phi\in \mathcal{B}^{-1}(\mathcal{E}(\mathbb{R}\times X,L^p(0,1;X)),\mu)$, there exists $T_0>0$ such that \[ \frac{1}{\mu([-T,T])}\int_{[-T,T]}\Big(\int_t^{t+1} \|\phi(s,u_k)\|^p\,d\sigma\Big)^{1/p}\,d\mu(t) <\frac{\epsilon}{8m^2}, \] for all $T>T_0$ and $1\leq k\leq m$. Therefore, \begin{align*} &\frac{1}{\mu([-T,T])}\int_{[-T,T]}\Big(\int_t^{t+1} \|W(s)\|^p\,ds\Big)^{1/p}\,d\mu(t)\\ &=\frac{1}{\mu([-T,T])}\int_{[-T,T]} \Big(\sum_{k=1}^m\int_{C_k\cap[t,t+1]}\|\phi(s,\alpha(s)) -\phi(s,u_k) \\ &\quad +\phi(s,u_k)\|^pds\Big)^{1/p}\,d\mu(t)\\ &\leq \frac{2^{1+\frac{1}{p}}}{\mu([-T,T])}\int_{[-T,T]} \Big(\int_{C_k\cap[t,t+1]}\|\phi(s,\alpha(s)) -\phi(s,\overline{u}(s))\|^pds\Big)^{1/p}\,d\mu(t)\\ &\quad +\frac{2^{1+\frac{1}{p}}}{\mu([-T,T])}\int_{[-T,T]} \Big(\sum_{k=1}^m\int_{C_k\cap[t,t+1]}\|\phi(s,u_k)\|^pds\Big)^{1/p}\,d\mu(t)\\ &<\frac{\epsilon}{2}+m^{1/p}\frac{\epsilon}{2m}<\epsilon. \end{align*} Hence $W\in \mathcal{B}^{-1}(\mathcal{E}(\mathbb{R},L^p(0,1;X)))$. The conclusion follows. \end{proof} From Theorem \ref{compthe1} we obtain the following result of \cite{Ak-Ez-Kh15}. \begin{corollary}\label{comp.the4} Let $\mu\in\mathcal{M}$ and let $f=g+\phi\in PAPS^p(\mathbb{R}\times X,X,\mu)$ that satisfies a Lipschitz condition in $x\in X$ uniformly in $t\in \mathbb{R}$, that is, there is a constant $L\geq 0$ such that $\|f(t,x)-f(t,y)\|\leq L\|x-y\|$, for all $x,y\in X$ and $t\in \mathbb{R}$. If $x\in PAP(\mathbb{R},X,\mu)$, then $f(\cdot,x(\cdot))\in PAPS^p(\mathbb{R},X,\mu)$. \end{corollary} To prove the next composition theorem, we need the following lemma. \begin{lemma}[\cite{Ding-Long}] \label{comp.the2} Suppose that \begin{itemize} \item[(a)] $f\in APS^p(\mathbb{R}\times X, X)$ with $p>1$ and there exists a function $L_f\in BS^r(\mathbb{R},\mathbb{R})$ ($r\geq\max\{p,p/p-1\}$) such that \[ \|f(t,u)-f(t,v)\|\leq L_f(t)\|u-v\|\quad t\in\mathbb{R},\; u,v\in X. \] \item[(b)] $x\in APS^p(\mathbb{R},X)$, and there exist a set $E\subset \mathbb{R}$ with $\operatorname{meas}(E)=0$ such that \[ K=\overline{\{x(t):t\in\mathbb{R}\setminus E\}} \] is compact in $X$. \end{itemize} Then there exist $q\in[1,p)$ such that $f(\cdot,x(\cdot))\in APS^q(\mathbb{R},X)$. \end{lemma} The next result of composition is new. \begin{theorem}\label{comp.the3} Let $\mu\in\mathcal{M}$, $p>1$, $f=g+\phi\in PAPS^p(\mathbb{R}\times X,X,\mu)$ with $g\in\mathcal{B}^{-1}(AP(\mathbb{R}\times X,L^p(0,1;X)))$ and $\phi\in\mathcal{B}^{-1}(\mathcal{E}(\mathbb{R}\times X,L^p(0,1;X),\mu))$. Assume that \begin{itemize} \item[(i)] there exist nonnegative functions $L_f,L_g$ in the space $ APS^r(\mathbb{R},\mathbb{R})$, with $r\geq\max\{p,p/p-1\}$, such that \[ \|f(t,u)-f(t,v)\|\leq L_f(t)\|u-v\|,\quad \|g(t,u)-g(t,v)\|\leq L_g(t)\|u-v\| \] for $t\in\mathbb{R}$ and $u,v\in X$. \item[(ii)] $h=\alpha+\beta\in PAPS^p(\mathbb{R},X,\mu)$ with $$\alpha\in\mathcal{B}^{-1}(AP(\mathbb{R},L^p(0,1;X))),\,\,\,\,\,\, \beta\in\mathcal{B}^{-1}(\mathcal{E}(\mathbb{R},L^p(0,1;X),\mu))$$ and there exist a set $E\subset \mathbb{R}$ with $\operatorname{meas}(E)=0$ such that the set $ K=\overline{\{\alpha(t):\,t\in\mathbb{R}\setminus E\}}$ is compact in $X$. \end{itemize} Then there exist $q\in[1,p)$ such that $f(\cdot,h(\cdot))\in PAPS^q(\mathbb{R},X,\mu)$. \end{theorem} \begin{proof} We can decompose \[ f(t,h(t))=g(t,\alpha(t))+f(t,h(t))-f(t,\alpha(t))+\phi(t,\alpha(t)). \] Set \[ F(t):=g(t,\alpha(t)),\quad G(t):=f(t,h(t))-f(t,\alpha(t)),\quad H(t):=\phi(t,\alpha(t)). \] Since $r\geq\frac{p}{p-1}$ then there exists $q\in[1,p)$ such that $r=\frac{pq}{p-q}$. Let $p'=p/p-q$ and $q'=p/q$. Therefore $\frac{1}{p'}+\frac{1}{q'}=1$. Since $\alpha\in APS^p(\mathbb{R},X)$ and $g\in APS^p(\mathbb{R}\times X,X)$ then by assumptions and Lemma \ref{comp.the2} we obtain that $F\in \mathcal{B}^{-1}(AP(\mathbb{R},L^q(0,1;X)))$. Next we show that $G\in \mathcal{B}^{-1}(\mathcal{E}(\mathbb{R},L^q(0,1;X),\mu))$. By H\"{o}lder inequality we have \begin{align*} \int_t^{t+1}\|G(\sigma)\|^q\,d\sigma &=\int_t^{t+1} \|f(\sigma,h(\sigma))-f(\sigma,\alpha(\sigma))\|^q\,d\sigma\\ &\leq\int_t^{t+1} L_f^q(\sigma)\|h(\sigma)-\alpha(\sigma)\|^q\,d\sigma\\ &=\int_t^{t+1} L_f^q(\sigma)\|\beta(\sigma)\|^q\,d\sigma\\ &\leq\Big(\int_t^{t+1} L_f^{qp'}(\sigma)\,d\sigma\Big)^{1/p'} \Big(\int_t^{t+1} \|\beta(\sigma)\|^{qq'}\,d\sigma\Big)^{1/q'}\\ &=\Big[\Big(\int_t^{t+1} L_f^{r}(\sigma)\,d\sigma\Big)^{1/r}\Big]^{r/p'} \Big[\Big(\int_t^{t+1} \|\beta(\sigma)\|^p\,d\sigma\Big)^{1/p}\Big]^{p/q'}\\ &\leq\|L_f\|_{BS^r}^q\Big[\Big(\int_t^{t+1} \|\beta(\sigma)\|^p\,d\sigma\Big)^{1/p}\Big]^{q}. \end{align*} Then \begin{align*} &\frac{1}{\mu([-T,T])}\int_{[-T,T]}\Big(\int_t^{t+1} \|G(\sigma)\|^q\,d\sigma\Big)^{1/q}\,d\mu(t)\\ &\leq \frac{\|L_f\|_{BS^r}}{\mu([-T,T])}\int_{[-T,T]} \Big(\int_t^{t+1} \|\beta(\sigma)\|^p\,d\sigma\Big)^{1/p}\,d\mu(t). \end{align*} Since $\beta\in\mathcal{B}^{-1}(\mathcal{E}(\mathbb{R},L^p(0,1;X),\mu))$ we obtain that $G\in \mathcal{B}^{-1}(\mathcal{E}(\mathbb{R},L^q(0,1;X),\mu))$. Next, we prove that $H\in\mathcal{B}^{-1}(\mathcal{E}(\mathbb{R},L^q(0,1;X),\mu))$. Since $\phi\in \mathcal{B}^{-1}(\mathcal{E}(\mathbb{R},L^p(0,1;X),\mu))$, for each $\epsilon>0$ there exist $T_0>0$ such that $T>T_0$ implies that \[ \frac{1}{\mu([-T,T])}\int_{[-T,T]}\Big(\int_t^{t+1} \|\phi(\sigma,u)\|^p\,d\sigma\Big)^{1/p}\,d\mu(t)<\epsilon\quad (u\in X). \] Since $K$ is compact, we can find finite open balls $O_k$ ($k=1,2,3,\dots ,m$) with center $x_k$ such that $K\subset\cup_{k=1}^mO_k$. Thus, for all $u\in K$ there exist $x_k$ such that \begin{align*} &\|\phi(t+\sigma,u)\| \\ &\leq\|\phi(t+\sigma,u)-\phi(t+\sigma,x_k)\|+\|\phi(t+\sigma,x_k)\|\\ &\leq\|f(t+\sigma,u)-f(t+\sigma,x_k)\| +\|g(t+\sigma,u) -g(t+\sigma,x_k)\| +\|\phi(t+\sigma,x_k)\|\\ &\leq L_f(t+\sigma)\epsilon+L_g(t+\sigma)\epsilon +\|\phi(t+\sigma,x_k)\|\quad (t\in\mathbb{R},\;\sigma\in[0,1]). \end{align*} Hence \[ \sup_{u\in K}\|\phi(t+\sigma,u)\| \leq L_f(t+\sigma)\epsilon+L_g(t+\sigma)\epsilon +\sum_{k=1}^m\|\phi(t+\sigma,x_k)\|. \] Since $r\geq p$ then $L_f,L_g\in APS^r(\mathbb{R},\mathbb{R}) \subset APS^p(\mathbb{R},\mathbb{R})\subset BS^p(\mathbb{R},\mathbb{R})$. By Minkowskii's inequality, we obtain \begin{align*} &\Big[\int_0^1(\sup_{u\in K}\|\phi(t+\sigma,u)\|)^p\,d\sigma\Big]^{1/p}\\ & \leq (\|L_f\|_{BS^p}+\|L_g\|_{BS^p})\epsilon + \sum_{k=1}^m\Big(\int_0^1\Big(\sup_{u\in K}\|\phi(t+\sigma,u)\|\Big)^p\,d\sigma\Big)^{1/p}. \end{align*} For $T>T_0$ we have \begin{align*} &\frac{1}{\mu([-T,T])}\int_{[-T,T]}\Big(\int_0^1 \Big(\sup_{u\in K}\|\phi(t+\sigma,u)\|\Big)^p\,d\sigma\Big)^{1/p}\,d\mu(t) \\ &\leq (\|L_f\|_{BS^p}+\|L_g\|_{BS^p}+m)\epsilon. \end{align*} Hence \[ \lim_{T\to\infty}\frac{1}{\mu([-T,T])}\int_{[-T,T]}\Big(\int_0^1 \Big(\sup_{u\in K}\|\phi(t+\sigma,u)\|\Big)^p\,d\sigma\Big)^{1/p}\,d\mu(t)=0. \] On the other hand \begin{align*} &\frac{1}{\mu([-T,T])}\int_{[-T,T]}\|H^b(t)\|_q\,d\mu(t) \\ &\leq \frac{1}{\mu([-T,T])}\int_{[-T,T]}\|H^b(t)\|_p\,d\mu(t) \\ &=\frac{1}{\mu([-T,T])}\int_{[-T,T]}\Big(\int_0^{1} \|\phi(t+\sigma,\alpha(t+\sigma))\|^p\,d\sigma\Big)^{1/p}\,d\mu(t)\\ &\leq \frac{1}{\mu([-T,T])}\int_{[-T,T]}\Big(\int_0^1 (\sup_{u\in K}\|\phi(t+\sigma,u)\|)^p\,d\sigma\Big)^{1/p}\,d\mu(t) \to0 \end{align*} as $T\to\infty$. Hence $H\in \mathcal{B}^{-1}(\mathcal{E}(\mathbb{R},L^q(0,1;X),\mu))$. It proves that $f(\cdot,h(\cdot))=F(\cdot)+[G(\cdot)+H(\cdot)]\in PAPS^q(\mathbb{R},X,\mu)$. \end{proof} We recall the following convolution theorem. \begin{theorem}[{\cite[Theorem 3.1]{Al-Li15}}] \label{maintheorem1} Let $S:\mathbb{R}\to B(X)$ be strongly continuous. Suppose that there exists a function $\phi\in L^1(\mathbb{R})$ such that \begin{itemize} \item[(a)] $\|S(t)\|\leq\phi(t), \quad t \in \mathbb{R}$; \item[(b)] $\phi(t)$ is nonincreasing; \item[(c)] $\sum_{n=1}^{\infty}\phi(n)<\infty$. \end{itemize} If $g\in APS^p(\mathbb{R},X)$, then $$ (S\ast g)(t):=\int_{-\infty}^tS(t-s)g(s)\,ds\in AP(\mathbb{R},X). $$ \end{theorem} The next result is one of the original contributions of this work. \begin{theorem}\label{convolutionth} Let $\mu\in \mathcal{M}$ be given and let $S:\mathbb{R}\to B(X)$ be strongly continuous. Suppose that there exists a function $\phi\in L^1(\mathbb{R})$ such that \begin{itemize} \item[(a)] $\|S(t)\|\leq\phi(t) \quad t \in \mathbb{R}$; \item[(b)] $\phi(t)$ is nonincreasing; \item[(c)] $\sum_{n=1}^{\infty}\phi(n)<\infty$. \end{itemize} If $f=g+h\in PAPS^p(\mathbb{R},X,\mu)$ with $g\in\mathcal{B}^{-1}(AP(\mathbb{R},L^p(0,1;X)))$ and $h\in\mathcal{B}^{-1}(\mathcal{E}(\mathbb{R},L^p(0,1;X)))$, then $$ (S\ast f)(t):=\int_{-\infty}^tS(t-s)f(s)\,ds\in PAP(\mathbb{R},X,\mu). $$ \end{theorem} \begin{proof} Since $$ (S\ast f)(t):=\int_{-\infty}^tS(t-s)f(s)\,ds=\int_{-\infty}^tS(t-s)g(s)\,ds +\int_{-\infty}^tS(t-s)h(s)\,ds, $$ and, from Theorem \ref{maintheorem1}, $(S\ast g)\in AP(\mathbb{R},X)$ it remains to show that $(S\ast h)\in \mathcal{E}(\mathbb{R},X,\mu)$. Set \begin{align*} H(t)&:=\int_{-\infty}^tS(t-s)h(s)\,ds =\int_{-\infty}^tS(s)h(t-s)\,ds, \end{align*} and \[ H_n(t):=\int_{t-n}^{t-n+1}S(t-\sigma)h(\sigma)\,d\sigma,\quad n=1,2,\dots \,. \] Note that $H_n(t)$ is continuous and \begin{align*} \|H_n(t)\|&\leq\int_{t-n}^{t-n+1}\|S(t-\sigma)\|\|h(\sigma)\|\,d\sigma\\ &=\int_{n-1}^{n}\|S(\sigma)\|\|h(t-\sigma)\|\,d\sigma\\ &\leq\int_{n-1}^{n}\phi(s)\|h(t-\sigma)\|\,d\sigma\\ &\leq\phi(n-1)\Big(\int_{n-1}^{n}\|h(t-\sigma)\|^p\,d\sigma\Big)^{1/p}. \end{align*} Hence, for $T>0$, \begin{align*} & \frac{1}{\mu([-T,T])} \int_{[-T,T]}\|H_n(t)\|\,d\mu(t) \\ & \quad \leq\phi(n-1) \frac{1}{\mu([-T,T])} \int_{[-T,T]} \Big(\int_{n-1}^{n}\|h(t-\sigma)\|^p\,d\sigma\Big)^{1/p}\,d\mu(t). \end{align*} Using the fact that the space $\mathcal{E}(\mathbb{R},X,\mu)$ is translation invariant, it follows that $t\to h(t-\sigma)$ belongs to $\mathcal{E}(\mathbb{R},X,\mu)$. The above inequality leads to $H_n\in \mathcal{E}(\mathbb{R},X,\mu)$ for each $n=1,2,\dots $. The above estimate implies \[ \|H_n(t)\|\leq\phi(n-1)\|h\|_{BS^p(\mathbb{R},X)}. \] By hypothesis we have \[ \sum_{n=1}^{\infty}\|H_n(t)\| \leq\sum_{n=1}^{\infty}\phi(n-1)\|h\|_{BS^p(\mathbb{R},X)} 0$ is defined by $$ D^{-\alpha}g(t):=\frac{1}{\Gamma(\alpha)}\int_{-\infty}^t (t-s)^{\alpha-1}g(s)ds, \quad t\in\mathbb{R}, $$ when this integral is convergent. The \textit{Weyl fractional derivative} $D^\alpha g$ of order $\alpha>0$ is defined by $$ D^\alpha g(t):=\frac{d^n}{dt^n}D^{-(n-\alpha)}g(t), \quad t\in\mathbb{R}, $$ where $n=[\alpha]+1$. It is known that $D^{\alpha} D^{-\alpha}g=g$ for any $\alpha>0$, and $D^n=\frac{d^n}{dt^n}$ holds with $n\in\mathbb{N}$. \begin{definition}[\cite{Pon-13}] \rm Let $A$ be a closed and linear operator with domain $D(A)$ defined on a Banach space $X$, and $\alpha>0$. Given $a\in L^1_{\rm loc}(\mathbb{R}_+)$, we say that $A$ is the generator of an $\alpha$-resolvent family if there exist $\omega\geq 0$ and a strongly continuous family $S_\alpha: [0,\infty)\to \mathcal{B}(X)$ such that $\{\frac{\lambda^\alpha}{1+\hat{a}(\lambda)}: \operatorname{Re} \lambda>\omega\}\subset \rho(A)$ and for all $x\in X$, \[ \big(\lambda^\alpha-(1+\hat{a}(\lambda))A\big)^{-1}x =\frac{1}{1+\hat{a}(\lambda)}\Big(\frac{\lambda^\alpha}{1+\hat{a}(\lambda)} -A\Big)^{-1}x=\int_0^\infty e^{-\lambda t}S_\alpha(t)x\,dt, \] for $\operatorname{Re} \lambda>\omega$. In this case, $\{S_\alpha(t)\}_{t\geq 0}$ is called the \emph{$\alpha$-resolvent family} generated by $A$. \end{definition} Next, we consider the existence and uniqueness of $\mu$-pseudo almost periodic mild solutions for the fractional integro-differential equations \begin{equation}\label{maineq1} D^\alpha u(t)=Au(t)+\int_{-\infty}^t a(t-s)Au(s)\,ds+f(t,u(t)), \end{equation} where $A$ generates an $\alpha$-resolvent family $\{S_\alpha(t)\}_{t\geq 0}$ on a Banach space $X$, $a\in L^1_{\rm loc}(\mathbb{R}_+)$ and $f\in PAPS^p(\mathbb{R}\times X,X,\mu)$ satisfies the Lipschitz condition. \begin{definition} \rm A function $u:\mathbb{R}\to X$ is said to be a mild solution of \eqref{maineq1} if \[ u(t)=\int_{-\infty}^{t}S_{\alpha}(t-s)f(s,u(s))\,ds\quad (t\in\mathbb{R}) \] where $\{S_\alpha(t)\}_{t\geq0}$ is the $\alpha$-resolvent family generated by $A$. \end{definition} \begin{theorem}\label{mainthe1} Let $\mu\in \mathcal{M}$, and assume {\rm (H2)} holds. Let $p>1$ and $f\in PAPS^p(\mathbb{R}\times X,X,\mu)$ be given. Suppose that \begin{itemize} \item[(H3)] There exists $L_f\geq0$ such that \[ \|f(t,u)-f(t,v)\|\leq L_f\|u-v\|, \quad t\in\mathbb{R},\;u,v\in X. \] \item [(H4)] Operator $A$ generates an $\alpha$-resolvent family $\{S_{\alpha}(t)\}_{t\geq0}$ such that $\|S_{\alpha}(t)\|\leq \varphi_{\alpha}(t)$, for all $t\geq0$, where $\varphi_{\alpha}(\cdot)\in L^1(\mathbb{R_+})$ is nonincreasing such that $\varphi_{0} := \sum_{n=0}^{\infty}\varphi_{\alpha}(n) < \infty$. \end{itemize} If $ L_f<\|\varphi_\alpha\|_1^{-1}$, then \eqref{maineq1} has a unique mild solution in $PAP(\mathbb{R},X,\mu)$. \end{theorem} \begin{proof} Consider the operator $Q:PAP(\mathbb{R},X,\mu)\to PAP(\mathbb{R},X,\mu)$ defined by \[ (Qu)(t):=\int_{-\infty}^tS(t-s)f(s,u(s))\,ds,\quad t\in\mathbb{R}. \] First, we show that $Q(PAP(\mathbb{R},X,\mu))\subset PAP(\mathbb{R},X,\mu)$. Let $u\in PAP(\mathbb{R},X,\mu)$. Since $f\in PAPS^p(\mathbb{R}\times X,X,\mu)$ and satisfy (H3) we have from Corollary \ref{comp.the4} that $f(\cdot,u(\cdot))\in PAPS^p(\mathbb{R},X,\mu)$. Then, by assumption (h4) we obtain from Theorem \ref{convolutionth} that $Qu\in PAP(\mathbb{R},X,\mu)$. Let $u,v\in PAP(\mathbb{R},X,\mu)$. By conditions (H3) and (H4) we have \begin{align*} \|Qu-Qv\|_{\infty} &=\sup_{t\in\mathbb{R}}\|(Qu)(t)-(Qv)(t)\|\\ &=\sup_{t\in\mathbb{R}}\|\int_{-\infty}^tS(t-s) [f(s,u(s))-f(s,v(s))]\,ds\|\\ &\leq L_f\sup_{t\in\mathbb{R}}\int_0^{\infty}\|S(s)\| \|u(t-s)-v(t-s)\|\,ds\\ &\leq L_f\|u-v\|_{\infty}\int_{0}^{\infty}\varphi_{\alpha}(s)\,ds\\ &=L_f\|\varphi_{\alpha}\|_1\|u-v\|_{\infty}. \end{align*} This proves that $Q$ is a contraction, so by the Banach Fixed Point Theorem we conclude that $Q$ has unique fixed point. It follows that $Qu=u\in PAP(\mathbb{R},X,\mu)$ and it is unique. Hence $u$ is the unique mild solution of \eqref{maineq1} which belongs to $PAP(\mathbb{R},X,\mu)$. \end{proof} \begin{theorem}\label{mainthe2} Let $\mu\in \mathcal{M}$. Assume that {\rm (H2)} holds. Let $p>1$ and $f=g+h\in PAPS^p(\mathbb{R}\times X,X,\mu)$ be given. Suppose that: \begin{itemize} \item [(H5)] There exist nonnegative functions $L_f(\cdot),L_g(\cdot)\in APS^{r}(\mathbb{R},\mathbb{R})$ with $r\geq\max\{p,\frac{p}{p-1}\}$ such that \[ \|f(t,u)-f(t,v)\|\leq L_f(t)\|u-v\|,\quad \|g(t,u)-g(t,v)\|\leq L_g(t)\|u-v\|, \] for $t\in\mathbb{R}$ and $u,v\in X$. \item [(H6)] Operator $A$ generates an $\alpha$-resolvent family $\{S_{\alpha}(t)\}_{t\geq0}$ such that $\|S_{\alpha}(t)\|\leq Me^{-\omega t}$, for all $t\geq0$ and \[ \|L_f\|_{BS^r}<\frac{ 1-e^{-\omega}}{M} (\frac{\omega r_0}{1-e^{-\omega r_0}})^{1/r_0} \] where $\frac{1}{r}+\frac{1}{r_0}=1$. \end{itemize} Then \eqref{maineq1} has a unique mild solution in $PAP(\mathbb{R},X,\mu)$. \end{theorem} \begin{proof} Let $u=u_1+u_2\in PAP(\mathbb{R},X,\mu)$ where $u_1\in AP(\mathbb{R},X)$ and $u_2\in \mathcal{E}(\mathbb{R},X,\mu)$. Then $u\in PAPS^p(\mathbb{R},X,\mu)$. Since the range of almost periodic functions is relatively compact set, then $K=\overline{\{u_1(t):t\in\mathbb{R}\}}$ is compact in $X$. Thus, by conditions (H5) and (H6) we have that all the hypotheses of Theorem \ref{comp.the3} fulfilled, then there exists $q\in[1,p)$ such that $f(\cdot,u(\cdot))\in PAPS^q(\mathbb{R},X,\mu)$. Consider the operator $Q:PAP(\mathbb{R},X,\mu)\to PAP(\mathbb{R},X,\mu)$ such that \[ (Qu)(t):=\int_{-\infty}^tS(t-s)f(s,u(s))\,ds,\quad (t\in\mathbb{R}). \] Since $f(\cdot,u(\cdot))\in PAPS^q(\mathbb{R},X,\mu)$ it follows from Theorem \ref{convolutionth} that $Q$ maps $PAP(\mathbb{R},X,\mu)$ into $PAP(\mathbb{R},X,\mu)$. For any $u,v\in PAP(\mathbb{R},X,\mu)$ we have \begin{align*} \|(Qu)(t)-(Qv)(t)\|&\leq\int_{-\infty}^t\|S(t-s)\|\|f(s,u(s)-f(s,v(s)))\|\,ds\\ &\leq\int_{-\infty}^tMe^{-\omega(t-s)}L_f(s)\|u(s)-v(s)\|\,ds\\ &\leq\|u-v\|\sum_{k=1}^{\infty} \int_{t-k}^{t-k+1}Me^{-\omega(t-s)}L_f(s)\,ds\\ &\leq\|u-v\|\sum_{k=1}^{\infty} \Big(\int_{t-k}^{t-k+1}M^{r_0}e^{-\omega r_0(t-s)}\Big)^{1/r_0}\,ds \|L_f(s)\|_{BS^r}\\ &=\frac{M}{1-e^{-\omega}}\Big(\frac{1-e^{-\omega r_0}}{\omega r_0}\Big)^{1/r_0} \|u-v\|\|L_f(s)\|_{BS^r}. \end{align*} From Banach contraction mapping principle we have that $Q$ has a unique fixed point in $PAP(\mathbb{R},X,\mu)$ which is the unique mild solution of Equation \eqref{maineq1}. \end{proof} \begin{example} \rm We put $A = -\varrho $ in $X=\mathbb{R}$, $a(t)= \frac{\varrho }{4}\frac{{{t^{\alpha - 1}}}}{{\Gamma (\alpha )}}$, $\varrho>0$, $0<\alpha<1$, and $f(t,u)=g(t,u)+h(t,u)$ where \[ g(t,u(t,x))=[\sin t+\sin(\sqrt{2}\,t)]\sin(u(t,x)),\quad h(t,u(t,x))=\phi(t)\sin(u(t,x)), \] and $\phi(t)$ is such that $|\phi(t)e^{t}|\leq K$ with $K>0$. Consider the measure $\mu$ whose Radon-Nikodym derivative is $\rho(t)=e^t$. Then $\mu\in\mathcal{M}$ and satisfies the (H2) (see \cite[Ex. 3.6]{Blot-Ci-Ez13}). Note that $g \in \mathcal{B}^{-1}(AP(\mathbb{R},L^p(0,1;X)))$ and $h\in \mathcal{B}^{-1}(\mathcal{E}(\mathbb{R},L^p(0,1;X),\mu))$. Hence $f \in PAPS^p(\mathbb{R}\times X,X,\mu)$. Furthermore, \[ |f(t,u)-f(t,v)|\leq L|u-v|, \] where $L:=\max\{2,K\}$. Therefore $f$ satisfies $(C1)$. Now, note that Equation \eqref{maineq1} takes the form \begin{equation}\label{4.1} {D^\alpha }u(t) = -\varrho u(t) - \frac{{{\varrho ^2}}}{4} \int_{ - \infty }^t {\frac{{{{(t - s)}^{\alpha - 1}}}}{{\Gamma (\alpha )}}u(s)ds + f(t,u(t))}, \quad t\in \mathbb{R}. \end{equation} It follows from \cite[Example 4.17]{Pon-13} that $A$ generates an $\alpha$-resolvent family $\{S_{\alpha}(t)\}_{t\geq0}$ such that \[ {\widehat S_\alpha }(\lambda ) = \frac{\lambda ^\alpha }{(\lambda ^\alpha + 2 / \varrho)^2} \frac{\lambda ^{\alpha-\alpha/2} }{(\lambda ^\alpha + 2 / \varrho)^2}\cdot \frac{\lambda ^{\alpha-\alpha/2} }{(\lambda ^\alpha + 2 / \varrho)^2}\,. \] Thus, we obtain explicitly $$ S_{\alpha}(t)=(r*r)(t) \quad t>0, $$ with $r(t)=t^{\frac{\alpha}{2}-1}E_{\alpha,\frac{\alpha}{2}} (-\frac{\varrho}{2}t^{\alpha})$, and where $E_{\alpha,\frac{\alpha}{2}}(\cdot)$ is the Mittag-Leffler function. By properties of the Mittag-Leffler function we obtain that (H4) holds. Then, by Theorem \ref{mainthe1}, \eqref{4.1} has a unique mild solution $u\in PAP(\mathbb{R},X,\mu)$ provided $\|S_{\alpha}\|<\frac{1}{2}$. Finally we note that, for $0<\alpha<1$, $\varrho>0$ may be chosen so that $\|S_{\alpha}\|<\frac{1}{2}$ as in the proof of \cite[Lemma 3.9]{Pon-13}. \end{example} \begin{thebibliography}{00} \bibitem{Ak-Ez-Kh15} A. N. Akdad, K. Ezzinbi, L. Souden; \emph{Pseudo almost periodic and automorphic mild solutions to nonautonomous neutral partial evolution equations.} Nonauton. Dyn. Syst., \textbf{2} (2015), 12--30. \bibitem{Al-Li15} E. Alvarez, C. Lizama; \emph{Weighted pseudo almost periodic solutions to a class of semilinear integro-differential equations in Banach spaces.} Adv. Difference Equ., DOI 10.1186/s13662-015-0370-5 \textbf{2015} (2015), 1–-18. \bibitem{Am-Ma} B. Amir, L. Maniar; \emph{Composition of pseudo-almost periodic functions and Cauchy problems with operator of nondense domain, Ann. Math. 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