\documentclass[reqno]{amsart} \usepackage{hyperref,amssymb,mathrsfs} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 04, pp. 1--16.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2008 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2008/04\hfil Problems without initial conditions] {Problems without initial conditions for degenerate implicit evolution equations} \author[M. Bokalo, Y. Dmytryshyn\hfil EJDE-2008/04\hfilneg] {Mykola Bokalo, Yuriy Dmytryshyn} % in alphabetical order \address{Mykola Bokalo \newline Department of Differential Equations\\ Ivan Franko National University of Lviv\\ Lviv, Ukraine} \email{mm\_bokalo@franko.lviv.ua} \address{Yuriy Dmytryshyn \newline Department of Differential Equations\\ Ivan Franko National University of Lviv\\ Lviv, Ukraine} \email{yuree@yandex.ru} \thanks{Submitted December 15, 2007. Published January 2, 2008.} \subjclass[2000]{34A09, 34G20, 35B15, 35K65, 47J35} \keywords{Problems without initial conditions; degenerate implicit equations; \hfill\break\indent nonlinear evolution equations; almost periodic solutions} \begin{abstract} We study some sufficient conditions for the existence and uniqueness of a solution to a problem without initial conditions for degenerate implicit evolution equations. We also establish a condition of Bohr's and Stepanov's almost periodicity of solutions for this problem. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \section{Introduction} Problems for an implicit evolution equation of the form $$\label{FirstEq1} \bigl({\mathcal{B}} u(t)\bigr)'+{\mathcal{A}}\bigl(t, u(t)\bigr)=f(t),\quad t\in S,$$ where ${\mathcal{A}}(t,\cdot)$ and ${\mathcal{B}}$ are operators from a Banach space $V$ to its dual $V'$, $S$ is an interval in $\mathbb{R}$, sometimes known as Sobolev equation (see, e.g., \cite{Bahuguna,Showalter69}), has been studied extensively by many authors. See, for example, \cite{Bahuguna}-\cite{Showalter} and references therein. Note that in the case where ${\mathcal{B}}$ is linear and ${\mathcal{A}}$ is linear or nonlinear, the monographs by Showalter \cite{Showalter77,Showalter} give many sufficient conditions to existence and uniqueness of solutions of the Cauchy problem for equation \eqref{FirstEq1}. More recently in the papers \cite{Kuttler_Shillor, Kuttler} the Cauchy problem for the inclusion of the form \eqref{FirstEq1} was considered as ${\mathcal{A}}$ may be set-valued. The existence of almost periodic solutions of abstract differential equations of the type \eqref{FirstEq1} (when ${\mathcal{B}}=I$) has been studied in several works; see for example \cite{Hu,Levitan,Pankov,Zaidman}. A problem without initial conditions for the equation of the form \eqref{FirstEq1} (when ${\mathcal{B}}=I$ and ${\mathcal{A}}$ is almost linear) was investigated in \cite{Showalter80,Showalter} in the class of integrable functions on $(-\infty, T)$, $T\in{\mathbb{R}}$. In \cite{Bokalo} the similar problem was considered (when ${\mathcal{B}}=I$ and ${\mathcal{A}}$ is nonlinear) in the class of locally integrable functions on $(-\infty, T]$. In this paper, we generalize the results of \cite{Bokalo} and \cite{Pankov} for the case of degenerate implicit equation \eqref{FirstEq1}, that is, when ${\mathcal{B}}$ may vanish on non-zero vectors. We obtain sufficient conditions to existence (Theorems~\ref{T2},~\ref{T3}) and uniqueness (Theorem~\ref{T1}) of solutions of a problem without initial conditions for \eqref{FirstEq1} independent of an additional assumption on the behavior of the solution and data-in at $-\infty$. We also establish the existence of periodic (Theorem~\ref{T_Temp4}) and almost periodic by Bohr and Stepanov (Theorem~\ref{T4}) solutions of \eqref{FirstEq1}. We shall introduce here some of the notions that we shall use hereafter. We denote by $\|\cdot\|_{X}$ the norm (seminorm) of the norm (seminorm) space $X$ and by $(\cdot, \cdot)_{Y}$ the scalar product in the Hilbert space $Y$. By $X'$ we denote the dual space of $X$. The duality pairing between $X$ and $X'$ is denoted by $\langle\cdot,\cdot\rangle_{X}$. By $L^{q}_{\rm loc}(S; X)$, where $q\in[1, +\infty)$ and $S$ is an unbounded connected subset of $\mathbb{R}$, we denote the space of (equivalence classes of) measurable functions in $S$, with values in $X$ such that its restrictions on any compact $K\subset S$ belong to $L^{q}(K; X)$. We denote by $\mathscr{D}'(S; X)$ the space of $X_{w}$ valued distributions on $\text{int}\, S$, which we regard extended on all $S$ by zero. It is known that the space $L^{q}_{\rm loc}(S; X)$ can be identified with some subspace of $\mathscr{D}'(S; X)$. For $v\in L^{q}_{\rm loc}(S; X)$, we denote by $v'$ the derivative in the sense of $\mathscr{D}'(S; X)$ \cite{Gajewski}. Throughout the paper the symbol $\hookrightarrow$ means a continuous imbedding. Our paper is organized as follows. Section~\ref{Sect2} is devoted to some preliminary facts needed in the sequel. In Section~\ref{Sect3} we state a problem and formulate main results. We prove our main results in Section~\ref{Sect4}. The last section is devoted to a simple example of applications of our results. \section{Preliminary results}\label{Sect2} Let $V$ be a separable reflexive Banach space. Assume that ${\mathcal{B}}:V\to V'$ is a linear, continuous, symmetric (i.e., $\langle {\mathcal{B}} v_{1}, v_{2}\rangle_{V}=\langle {\mathcal{B}} v_{2}, v_{1}\rangle_{V}\quad \forall\, v_{1},\, v_{2}\in V$) and monotone (i.e., $\langle {\mathcal{B}} v, v\rangle_{V}\geqslant0\quad \forall\,v\in V$) operator. Then $\langle{\mathcal{B}}\cdot,\cdot\rangle_{V}$ is a semiscalar product and $\|\cdot\|_{V_{\mathcal{B}}}:=\langle {\mathcal{B}}\cdot,\cdot\rangle_{V}^{1/2}$ is a seminorm on $V$. We denote the completion of the seminorm space $\{V\,,\,\|\cdot\|_{V_{\mathcal{B}}}\}$ by $V_{\mathcal{B}}$ and the dual Hilbert space by $V'_{\mathcal{B}}$. Note that $V\hookrightarrow V_{\mathcal{B}}$ is dense. By restriction of functionals we have $V_{\mathcal{B}}'\hookrightarrow V'$. The operator ${\mathcal{B}}$ has a unique continuous linear extension ${\mathcal{B}}: V_{\mathcal{B}}\to V_{\mathcal{B}}'$. The scalar product on $V'_{\mathcal{B}}$ satisfies $(w, {\mathcal{B}} v)_{V_{\mathcal{B}}'}= \langle w, v\rangle_{V},\quad w\in V'_{\mathcal{B}},\quad v\in V.$ Hence, taking $w={\mathcal{B}} v$, $$\label{NewEq2} \|{\mathcal{B}} v\|_{V'_{\mathcal{B}}}= \|v\|_{V_{\mathcal{B}}},\quad v\in V_{\mathcal{B}}.$$ We define the norm on the range of ${\mathcal{B}}:V\to V'$ by $\|w\|_{W}:= \inf\{\|v\|_{V}:\ v\in V,\,{\mathcal{B}} v=w\},\quad w\in \text{Rg}\,{\mathcal{B}}.$ The normed linear space $W=\{\text{Rg}\,{\mathcal{B}},\,\|\cdot\|_{W}\}$ is a reflexive Banach space. Note that $W\hookrightarrow V_{\mathcal{B}}'$. These results are due to the books by Showalter \cite{Showalter77,Showalter}. Throughout the rest of this paper $S:={\mathbb{R}}$ or $S:=(-\infty, T]$, where $T<+\infty$, unless the contrary is explicitly stated. \begin{lemma} \label{L1} Let $v\in L^{p}_{\rm loc}(S; V)$, $({\mathcal{B}} v)'\in L^{p'}_{\rm loc}(S; V')$, where $p\in[2; +\infty)$ and $p'=p/(p-1)$. Then $v\in C(S;V_{\mathcal{B}})$ and the function $\|v(\cdot)\|_{V_{\mathcal{B}}}$ is absolutely continuous on each closed subinterval of $S$. Furthermore, $$\label{Eqf1} \frac{1}{2} \frac{d}{dt}\|v(t)\|^2_{V_{\mathcal{B}}}=\bigl\langle\bigl({\mathcal{B}} v(t)\bigr)', v(t)\bigr\rangle_{V}\quad \text{ for a.e. }t\in S.$$ \end{lemma} \begin{proof} Let $t_{1}$, $t_{2}\in S$ be any numbers such that $t_{1}0$ for $\tau>0$ and $\int^{+\infty}\frac{d\tau}{\chi(\tau)}<+\infty$. Then $z(\cdot)\equiv0$. \end{lemma} \begin{lemma}[\cite{Bokalo2}, p.~60] \label{L4} Let $y\in C(S)$, $z\in L^{1}_{\rm loc}(S)$ be such that $y(t_{2})-y(t_{1})+\int_{t_{1}}^{t_{2}}z(t)\,dt\leqslant0$ for any $t_{1}$, $t_{2}\in S$. Then $y(t_{2})\theta(t_{2})-y(t_{1})\theta(t_{1}) -\int_{t_{1}}^{t_{2}}y(t)\theta'(t)\,dt +\int_{t_{1}}^{t_{2}}z(t)\theta(t)\,dt\leqslant0$ for any $\theta\in C^{1}(S)$ and $t_{1}$, $t_{2}\in S$. \end{lemma} \section{Statement of the problem and main results}\label{Sect3} Throughout this section $S,$ $V$, $V_{\mathcal{B}}$ and ${\mathcal{B}}$ are the same as in Section~\ref{Sect2} and $p\in(1,+\infty)$. Assume that a family of operators ${\mathcal{A}}(t,\cdot):V\to V'$, $t\in S$, is given such that \renewcommand{\theenumi}{\roman{enumi}} \begin{enumerate} \item \label{(i)} for each measurable function $v: S\to V$ the function $w(\cdot)={\mathcal{A}}\bigl(\cdot,v(\cdot)\bigr)$ is measurable on $S$; \item \label{(ii)} ${\mathcal{A}}\bigl(\cdot,v(\cdot)\bigr)\in L^{p'}_{\rm loc}(S; V')$ whenever $v\in L^{p}_{\rm loc}(S; V)$, where $p'=p/(p-1)$. \end{enumerate} \noindent Consider the problem: for every $f\in{L^{p'}_{\rm loc}(S; V')}$, find a function $u$ in ${L^{p}_{\rm loc}(S; V)}\cap C(S;V_{\mathcal{B}})$ such that $$\label{ProblemP} \bigl({\mathcal{B}} u(t)\bigr)'+{\mathcal{A}}\bigl(t,u(t)\bigr)=f(t) \quad\text{in }\mathscr{D}'(S; V').$$ We call this problem a \emph{Problem without initial conditions for degenerate implicit evolution equation} (\ref{ProblemP}) or Problem~(\ref{ProblemP}) for short. \begin{theorem}[Uniqueness] \label{T1} Assume that $p>2$ and \begin{enumerate} \setcounter{enumi}{2} \item \label{(iii)} for a.e. $t\in S$ and each $v$, $w\in V$, $v\neq w$, $\langle{\mathcal{A}}(t,v)-{\mathcal{A}}(t,w), v-w\rangle_{V}>\gamma(t) \varphi\bigl(\|v-w\|^2_{V_{\mathcal{B}}}\bigr),$ where $\gamma\in L^{1}_{\rm loc}(S)$, $\gamma(t)\geqslant0$ for a.e. $t\in S$, $\int_{-\infty}^{a}\gamma(\tau)\,d\tau=+\infty$ for some $a\in S$, $\varphi\in C\bigl([0, +\infty)\bigr)$, $\varphi(0)=0$, $\varphi(\tau)>0$ for $\tau>0$ and $\int_{1}^{+\infty}\frac{d\tau}{\varphi(\tau)}<+\infty$. \end{enumerate} Then there is at most one solution of Problem \eqref{ProblemP}. \end{theorem} \begin{remark}\label{R1} Clearly, conditions of Theorem \ref{T1} are satisfied by the functions $\gamma(t)\equiv\gamma_{0}$, $t\in S$, and $\varphi(\tau)=\tau^{\mu}$, $\tau\geqslant0$, where $\gamma_{0}>0$ and $\mu>1$ are some constants. \end{remark} \begin{theorem}[Existence] \label{T2} Let $p>2$ and suppose the embedding $V\hookrightarrow V_{\mathcal{B}}$ is compact. Assume that \begin{enumerate} \setcounter{enumi}{3} \item \label{(iv)} there exist $\alpha_{1}\in L^{\infty}_{\rm loc}(S)$ and $\alpha_{2}\in L^{p'}_{\rm loc}(S)$, $p'=p/(p-1)$, such that $\|{\mathcal{A}}(t,v)\|_{V'}\leqslant \alpha_{1}(t)\|v\|^{p-1}_{V}+\alpha_{2}(t),\quad v\in V,\;a.e.\;t\in S;$ \item \label{(v)} $\langle{\mathcal{A}}(t,v_{1})- {\mathcal{A}}(t,v_{2}),v_{1}-v_{2} \rangle_{V}\geqslant0$ for all $v_{1}, v_{2}\in V$, a.e. $t\in S;$ \item \label{(vi)} there exist $\beta_{1}\in L^{\infty}_{\rm loc}(S)$, $\mathop{\rm ess \inf}_ {t\in[a, b]} \beta_{1}(t)>0$ for any $[a, b]\subset S$, and $\beta_{2}\in L^{1}_{\rm loc}(S)$ such that $\langle{\mathcal{A}}(t,v),v\rangle_{V}\geqslant \beta_{1}(t)\|v\|^{p}_{V}-\beta_{2}(t),\quad v\in V,\text{a.e. }t\in S;$ \item \label{(vii)} for almost every $t\in S$ and every vectors $v_{1}$, $v_{2}\in V$ the real-valued function $s\mapsto\langle{\mathcal{A}}(t,v_{1}+sv_{2}),v_{2}\rangle_{V}$ is continuous on ${\mathbb{R}}$. \end{enumerate} Then Problem \eqref{ProblemP} has at least one solution and each its solution for any numbers $t_{1}$, $t_{2}\in S$ $(t_{1}0$, satisfies the estimate $$\label{Equation5} \begin{split} &\max_{t\in[t_{1},t_{2}]}\|u(t)\|^2_{V_{\mathcal{B}}} +\overline{\beta}(t_{1}-\delta, t_{2}) \int_{t_{1}}^{t_{2}} \|u(t)\|^p_{V}\,dt\\ &\leqslant C_{1}\bigl(\delta\cdot \overline{\beta}(t_{1}-\delta, t_{2})\bigr)^{\frac{2}{2-p}} +C_{2}\bigl(\overline{\beta}(t_{1}-\delta, t_{2})\bigr)^{\frac{1}{1-p}}\int_{t_{1}-\delta}^{t_{2}} \|f(t)\|^{p'}_{V'}\,dt\\ &\quad +2\int_{t_{1}-\delta}^{t_{2}}\beta_{2}(t)\,dt, \end{split}$$ where $\overline{\beta}(t_{1}-\delta, t_{2})= \mathop{\rm ess\,inf}_{t \in [t_{1}-\delta, t_{2}]} \beta_{1}(t)$, $C_{1}$, $C_{2}$ are positive constants depending only on ${\mathcal{B}}$ and $p$. \end{theorem} \begin{remark} \label{rmk2} \rm The family of operators ${\mathcal{A}}(t,\cdot)$ satisfies condition \eqref{(i)} in the context of conditions \eqref{(v)} and \eqref{(vii)} if we assume that the function $w(\cdot)={\mathcal{A}}\bigl(\cdot,v\bigr)$ is weakly measurable on $S$ for each $v\in V$ (see, e.g., \cite{Gajewski,Showalter}). Condition \eqref{(ii)} is an immediate consequence of conditions \eqref{(i)} and \eqref{(iv)}. \end{remark} \begin{theorem}[Existence and uniqueness] \label{T3} Assume that $p>2$ and the family of operators ${\mathcal{A}}(t,\cdot):V\to V'$, $t\in S$, satisfies conditions \eqref{(iv)}, \eqref{(vi)}, \eqref{(vii)} and \begin{enumerate} \setcounter{enumi}{7} \item \label{(viii)} there exists $K_{1}>0$ such that for each $v$, $w\in V$, $v\neq w$, $\langle{\mathcal{A}}(t,v)-{\mathcal{A}}(t,w), v-w\rangle_{V}> K_{1} \|v-w\|^{q}_{V_{\mathcal{B}}},\quad \text{a.e. } t\in S,$ where $q\in(2; p]$ is some number. \end{enumerate} Then there exists a unique solution of Problem \eqref{ProblemP}. Moreover, if $u$ is a solution of Problem \eqref{ProblemP}, then for any numbers $t_{1}$, $t_{2}\in S$ $(t_{1}0$ we have the estimate $$\label{AEqu38} \begin{split} &\max_{t\in[t_{1},t_{2}]}\|u(t)\|^2_{V_{\mathcal{B}}}+ \int_{t_{1}}^{t_{2}}\beta_{1}(t) \|u(t)\|^p_{V}\,dt\\ &\leqslant C_{3}\bigl(\delta\cdot K_{1}\bigr)^{\frac{2}{2-q}} +C_{4}\!\int_{t_{1}-\delta}^{t_{2}}\beta_{1}^{\frac{1}{1-p}}(t)\, \Bigl(\|f(t)\|^{p'}_{V'}+\|{\mathcal{A}}(t,0)\|^{p'}_{V'}\Bigr)\,dt\\ &\quad+2\!\int_{t_{1}-\delta}^{t_{2}}\beta_{2}(t)\,dt, \end{split}$$ where $C_{3}$, $C_{4}$ are some positive constants depending only on ${\mathcal{B}}$ and $p$. \end{theorem} \begin{remark} \label{rmk3} \rm Clearly condition \eqref{(viii)} is satisfied in the context of the condition \begin{enumerate} \setcounter{enumi}{8} \item \label{(ix)} there exists $K_{2}>0$ such that for every $v$, $w\in V$, $\langle{\mathcal{A}}(t,v)-{\mathcal{A}}(t,w), v-w\rangle_{V}\geqslant K_{2} \|v-w\|^{p}_{V},\quad \text{a.e. }t\in S.$ \end{enumerate} \end{remark} \begin{corollary} \label{C1} Let $S=\mathbb{R}$. Suppose that the hypotheses of Theorem \ref{T3} hold and there exists a constant $C_{5}\geqslant0$ such that $\sup_{\tau\in{\mathbb{R}}}\int_{\tau}^{\tau+1}\Bigl(\beta_{1}^{\frac{1}{1-p}} (t) \bigl(\|f(t)\|_{V'}^{p'}+\|{\mathcal{A}}(t,0)\|_{V'}^{p'}\bigr) +\beta_{2}(t)\Bigr)\,dt\leqslant C_{5}.$ Then the solution $u$ for Problem \eqref{ProblemP} satisfies $$\label{AEqu46} \sup_{\tau\in \mathbb{R}}\|u(\tau)\|_{V_{\mathcal{B}}}+\sup_{\tau\in{\mathbb{R}}}\int_{\tau}^{\tau+1}\beta_{1}(t) \|u(t)\|_{V}^{p}\,dt\leqslant C_{6},$$ where $C_{6}\geqslant0$ is a constant depending only on $p$, $q$, $K_{1}$ and $C_{5}$. \end{corollary} \begin{theorem} \label{T_Temp4} Let $S={\mathbb{R}}$ and the assumptions of Theorem \ref{T3} hold. Suppose that there exists a number $\sigma>0$ such that ${\mathcal{A}}(t+\sigma,v)={\mathcal{A}}(t,v)$ and $f(t+\sigma)=f(t)$ for any $v\in V$ and a.e. $t\in{\mathbb{R}}$. Then Problem~\eqref{ProblemP}\ has a unique solution. Moreover, this solution is $\sigma$-periodic {\rm (}that is, $u(t+\sigma)=u(t)$ for a.e. $t\in {\mathbb{R}}${\rm)} and satisfies the estimate $$\label{TempEstimate} \begin{split} &\max_{t\in[0,\sigma]}\|u(t)\|^2_{V_{\mathcal{B}}}+ \int_{0}^{\sigma} \|u(t)\|^p_{V}\,dt\\ &\leqslant C_{7}\max\Bigl\{\int_{0}^{\sigma} \bigr(\|f(t)\|^{p'}_{V'}+\beta_{2}(t)\bigl)\,dt,\ \Bigl(\int_{0}^{\sigma} \bigr(\|f(t)\|^{p'}_{V'}+\beta_{2}(t)\bigl)\,dt\Bigr)^{2/p}\Bigr\}, \end{split}$$ where $C_{7}$ is some positive constant depending only on $p$, $\sigma$, ${\mathcal{B}}$ and $\mathop{\rm ess\,inf}_{t \in [0,\sigma]} \beta_{1}(t)$. \end{theorem} Following \cite{Levitan} and \cite{Pankov} we recall some definitions. \begin{definition} \label{D1} \rm A subset $Q\subset{\mathbb{R}}$ is called \emph{relatively dense} if there exists $l>0$ such that $[a, a+l]\cap Q\neq\varnothing$ for all $a\in{\mathbb{R}}$. \end{definition} Let $X$ be a complete seminorm space with the seminorm $\|\cdot\|_{X}$ or a complete metric space with the metric $d_{X}(\cdot, \cdot)$. By $BC({\mathbb{R}}; X)$ we denote the space of all bounded continuous functions $g: {\mathbb{R}} \to X$. For any $g\in C({\mathbb{R}}; X)$ and $\varepsilon>0$ define $F_{\varepsilon}(g):=\bigl\{\sigma\in{\mathbb{R}}:\ \sup_{t\in{\mathbb{R}}}\|g(t+\sigma)-g(t)\|_{X}<\varepsilon\bigr\}$ if X is the seminorm space, and $F_{\varepsilon}(g):=\bigl\{\sigma\in{\mathbb{R}}:\ \sup_{t\in{\mathbb{R}}}d_{X}\bigl(g(t+\sigma), g(t)\bigr)<\varepsilon\bigr\}$ if X is the metric space. \begin{definition} \label{D2} \rm A function $g\in C({\mathbb{R}}; X)$ is said to be \emph{Bohr almost periodic} if for any $\varepsilon>0$ the set $F_{\varepsilon}(g)$ is relatively dense in ${\mathbb{R}}$. \end{definition} Denote by $CAP({\mathbb{R}}; X)$ the set of all Bohr almost periodic functions ${\mathbb{R}}\to X$. Note that $CAP({\mathbb{R}}; X)\subset BC({\mathbb{R}}; X)$. Let $\{Y,\,\|\cdot\|_{Y}\}$ be a Banach space and $q\in[1,+\infty)$. The Banach space of Stepanov bounded on ${\mathbb{R}}$ functions, with the exponent $q$, is the space $BS^{q}({\mathbb{R}}; Y)$ which consists of all functions $g\in L^{q}_{\rm loc}({\mathbb{R}}; Y)$ having finite norm $\|g\|^{q}_{S^{q}}:=\sup_{\tau\in{\mathbb{R}}}\int_{\tau}^{\tau+1} \|g(t)\|_{Y}^{q}\,dt.$ \begin{definition} \label{D3} \rm The \emph{Bochner transform} $g^{b}(t, s)$, $t\in{\mathbb{R}}$, $s\in [0, 1]$, of a function $g(t)$, $t\in{\mathbb{R}}$, with values in $Y$, is defined by $g^{b}(t, s):=g(t + s).$ \end{definition} \begin{definition} \label{D4} \rm A function $g\in L^{q}_{{\rm loc}}({\mathbb{R}}; Y)$ is called a \emph{Stepanov almost periodic function, with the exponent} $q$, if $g^{b}\in CAP\bigl({\mathbb{R}}; L^{q}(0,1; Y)\bigr)$. \end{definition} The space of all Stepanov almost periodic functions with values in $Y$ is denoted by $S^{q}({\mathbb{R}}; Y)$. Clearly the following inclusion holds $S^{q}({\mathbb{R}}; Y)\subset BS^{q}({\mathbb{R}}; Y)$. Denote by $Y_{p, V}$ the space of all operators $A: V\to V'$ such that $\|A(v)\|_{V'}\leqslant C_{A}(\|v\|^{p-1}_{V}+1) \quad\forall\ v\in V,$ where $C_{A}>0$ is some constant depending on $A$. The space $Y_{p, V}$ is a complete metric space with respect to the metric $d_{p, V}(A_{1}, A_{2}):= \sup_{v\in V}\frac{\|A_{1}(v)-A_{2}(v)\|_{V'}}{\|v\|^{p-1}_{V}+1}, \quad A_{1}, A_{2}\in Y_{p, V}.$ \begin{theorem} \label{T4} Let $S={\mathbb{R}}$ and $p>2$. Assume that the family of operators ${\mathcal{A}}(t,\cdot):V\to V'$, $t\in \mathbb{R}$, belongs to the space $CAP({\mathbb{R}}; Y_{p, V})$, satisfies conditions \eqref{(iv)}, \eqref{(vii)}, \eqref{(ix)} and $f\in S^{p'}({\mathbb{R}}; V')$. Then Problem \eqref{ProblemP} has a unique solution and this solution belongs to the space $CAP({\mathbb{R}}; V_{\mathcal{B}})\cap S^{p}({\mathbb{R}}; V)$. \end{theorem} \section{Proof main results}\label{Sect4} We now turn to the proof of Theorems~\ref{T1}-\ref{T4} and Corollary \ref{C1}. \begin{proof}[Proof of Theorem \ref{T1}] Suppose that $u_{1}$ and $u_{2}$ are two solutions of Problem \eqref{ProblemP}, and write $w:= u_{1}-u_{2}$. By taking the difference between \eqref{ProblemP} for $u=u_{1}$ and \eqref{ProblemP} for $u=u_{2}$ we get $$\label{Eqf3} \bigl({\mathcal{B}} w(t)\bigr)'+{\mathcal{A}}\bigl(t,u_{1}(t)\bigr)- {\mathcal{A}}\bigl(t,u_{2}(t)\bigr)=0\quad\text{in} \quad \mathscr{D}'(S; V').$$ This and condition \eqref{(ii)} give us $({\mathcal{B}} w)'\in L^{p'}_{\rm loc}(S; V')$, so using Lemma \ref{L1} we obtain $$\label{Eqf4} \frac{1}{2} \frac{d}{dt}\|w(t)\|^2_{V_{\mathcal{B}}}= \bigl\langle\bigl({\mathcal{B}} w(t)\bigr)',w(t)\bigr\rangle_{V}\quad \text{for a.e. }t\in S.$$ Multiplying (\ref{Eqf3}) by $w$ we get $$\label{Eqf5} \bigl\langle\bigl({\mathcal{B}} w(t)\bigr)',w(t)\bigr\rangle_{V}+ \bigl\langle{\mathcal{A}}\bigl(t,u_{1}(t)\bigr)- {\mathcal{A}}\bigl(t,u_{2}(t)\bigr),u_{1}(t)-u_{2}(t)\bigr\rangle_{V}=0$$ for a.e. $t\in S$. From (\ref{Eqf4}) and (\ref{Eqf5}) we obtain $$\label{Temp1} \frac{1}{2} \frac{d}{dt}\|w(t)\|^2_{V_{B}}+ \bigl\langle{\mathcal{A}}\bigl(t,u_{1}(t)\bigr)- {\mathcal{A}}\bigl(t,u_{2}(t)\bigr),u_{1}(t)-u_{2}(t)\bigr\rangle_{V}=0\quad\text{a.e. on }S.$$ From (\ref{Temp1}) and \eqref{(iii)} we have $$\label{Temp8} \frac{1}{2} \frac{dy(t)}{dt}+\gamma(t)\varphi\bigl(y(t)\bigr)\leqslant0\quad \text{ for a.e. }t\in S,$$ where $y(t)=\|u_{1}(t)-u_{2}(t)\|^2_{V_{\mathcal{B}}}$. Further, from (\ref{Temp8}) we obtain $y\equiv0$ on $S$ by Lemma \ref{L3}. This and (\ref{Temp1}) imply $$\label{TempT1} \bigl\langle{\mathcal{A}}\bigl(t,u_{1}(t)\bigr)- {\mathcal{A}}\bigl(t,u_{2}(t)\bigr),u_{1}(t)-u_{2}(t)\bigr\rangle_{V}=0\quad\text{a.e. on }S.$$ From (\ref{TempT1}) and \eqref{(iii)} we get $u_{1}(t)=u_{2}(t)$ for a.e. $t\in S$. Theorem \ref{T1} is proved. \end{proof} \begin{proof}[Proof of Theorem \ref{T2}] First we obtain a priori estimate (\ref{Equation5}) for any solution of Problem \eqref{ProblemP}. Let $u$ be a solution of Problem \eqref{ProblemP}. Hence, using Lemma \ref{L1}, we get $$\label{Eqf6} \frac{1}{2} \frac{d}{dt}\|u(t)\|^2_{V_{\mathcal{B}}}= \bigl\langle\bigl({\mathcal{B}} u(t)\bigr)',u(t)\bigr\rangle_{V}$$ for a.e. $t\in S$. Take $\theta_{1}\in C^{1}({\mathbb{R}})$ with the following properties: $\theta_{1}(t)=0$ if $t\in(-\infty, -1]$, $\theta_{1}(t)=\exp(\frac{t^{2}}{t^{2}-1})$ if $t\in(-1, 0)$, $\theta_{1}(t)=1$ if $t\in[0, +\infty)$. It is clear that \label{Eqf7} \sup_{t\in(-1, +\infty)}\frac{\theta_{1}'(t)}{\theta_{1}^{\nu}(t)}0$is a constant depending only on$\nu$. Let$t_{1}$,$t_{2}\in S(t_{1}0$be any numbers. We define the function$\theta(t):= \theta_{1}(\frac{t-t_{1}}{\delta})$for each$t\in S$. It is clear that$\theta u\in{L^{p}_{\rm loc}(S; V)}$. Multiply equation \eqref{ProblemP} by$\theta u$and integrate from$t_{1}-\delta$to$\tau\in[t_{1}, t_{2}]$with respect to$t$: $$\label{Eqf8} \begin{split} &\int_{t_{1}-\delta}^{\tau}\Bigl\{ \theta(t)\bigl\langle\bigl({\mathcal{B}} u(t)\bigr)',u(t)\bigr\rangle_{V}+ \theta(t)\bigl\langle{\mathcal{A}}\bigl(t,u(t)\bigr),u(t)\bigr\rangle_{V}\Bigr\}\,dt\\ &=\int_{t_{1}-\delta}^{\tau}\theta(t)\langle f(t),u(t)\rangle_{V}\,dt. \end{split}$$ Substituting (\ref{Eqf6}) into (\ref{Eqf8}) yields $$\label{Eqf9} \begin{split} &\int_{t_{1}-\delta}^{\tau} \theta(t)\frac{d}{dt}\|u(t)\|^2_{V_{\mathcal{B}}}\,dt+ 2\int_{t_{1}-\delta}^{\tau}\theta(t) \bigl\langle{\mathcal{A}}\bigl(t,u(t)\bigr),u(t)\bigr\rangle_{V}\,dt\\ &=2\int_{t_{1}-\delta}^{\tau}\theta(t)\langle f(t),u(t)\rangle_{V}\,dt. \end{split}$$ Integrating by parts the first term of the left hand side of equality (\ref{Eqf9}) we obtain $$\label{Eqf10} \begin{split} &\|u(\tau)\|^2_{V_{\mathcal{B}}}+2\int_{t_{1}-\delta}^{\tau}\theta(t) \bigl\langle{\mathcal{A}}\bigl(t,u(t)\bigr),u(t)\bigr\rangle_{V}\,dt\\ &=\int_{t_{1}-\delta}^{\tau} \theta'(t)\|u(t)\|^2_{V_{\mathcal{B}}}\,dt+ 2\int_{t_{1}-\delta}^{\tau}\theta(t)\langle f(t),u(t)\rangle_{V}\,dt. \end{split}$$ Let us estimate the first term of the right hand side of (\ref{Eqf10}) using (\ref{Eqf7}), the continuity of the imbedding$V$in$ V_{\mathcal{B}}$and Young's inequality: $$\label{Eqf11} \begin{split} \int_{t_{1}-\delta}^{\tau}\theta'(t)\|u(t)\|^2_{V_{\mathcal{B}}}\,dt &\leqslant C_{9}\int_{t_{1}-\delta}^{\tau} \theta'(t)\|u(t)\|^2_{V}\,dt\\ & \leqslant C_{9}\int_{t_{1}-\delta}^{\tau} \frac{\theta'(t)}{\theta^{2/p}(t)}\theta^{2/p}(t)\|u(t)\|^2_{V}\,dt\\ & \leqslant\varepsilon\!\! \int_{t_{1}-\delta}^{\tau}\theta(t)\|u(t)\|^p_{V}\,dt\\ &\quad +C_{10}\varepsilon^{-\frac{p}{p-2}} \int_{t_{1}-\delta}^{t_{2}}\bigl(\theta'(t)\theta^{-2/p}(t)\bigr)^{\frac{p}{p-2}}\,dt\\ & \leqslant\varepsilon \int_{t_{1}-\delta}^{\tau}\theta(t)\|u(t)\|^p_{V}\,dt+C_{11}(\delta\cdot\varepsilon)^{-\frac{p}{p-2}}, \end{split}$$ where$\varepsilon>0$is any number,$C_{9}$,$C_{10}$³$C_{11}$are positive constants depending only on$p$and${\mathcal{B}}$. Now we estimate the second term of the right hand side of (\ref{Eqf10}) using Young's inequality $$\label{Eqf12} \begin{split} &2\int_{t_{1}-\delta}^{\tau}\theta(t)\langle f(t),u(t)\rangle_{V}\,dt\\ &\leqslant\eta\int_{t_{1}-\delta}^{\tau}\theta(t) \|u(t)\|^p_{V}\,dt+C_{12}\eta^{\frac{1}{1-p}}\int_{t_{1}-\delta}^{\tau} \theta(t)\|f(t)\|^{p'}_{V'}\,dt, \end{split}$$ where$\eta>0$is any number and$C_{12}>0$is a constant depending only on$p. Next let us estimate the second term of the left hand side of (\ref{Eqf10}) using \eqref{(vi)} \begin{align}\label{Eqf42} 2\int_{t_{1}-\delta}^{\tau}\theta(t) \bigl\langle{\mathcal{A}}\bigl(t,u(t)\bigr),u(t)\bigr\rangle_{V}\,dt & \geqslant 2\int_{t_{1}-\delta}^{\tau} \theta(t)\beta_{1}(t)\|u(t)\|^p_{V}\,dt -2\int_{t_{1}-\delta}^{\tau}\theta(t)\beta_{2}(t)\,dt\nonumber\\ & \geqslant 2\overline{\beta}(t_{1}-\delta, \tau)\int_{t_{1}-\delta}^{\tau} \theta(t)\|u(t)\|^p_{V}\,dt\\ &\quad -2\int_{t_{1}-\delta}^{\tau}\theta(t)\beta_{2}(t)\,dt.\nonumber \end{align} From (\ref{Eqf10}), using (\ref{Eqf11})-(\ref{Eqf42}) and taking\varepsilon=\eta=\frac{1}{2}\overline{\beta}(t_{1}-\delta, \tau)$, we get $$\label{Eqf13} \begin{split} &\|u(\tau)\|^2_{V_{\mathcal{B}}}+\overline{\beta}(t_{1}-\delta,\tau) \int_{t_{1}-\delta}^{\tau} \theta(t)\|u(t)\|^p_{V}\,dt\\ &\leqslant C_{13}\bigl(\delta\cdot \overline{\beta}(t_{1}-\delta, \tau)\bigr)^{\frac{2}{2-p}} +C_{14}\bigl(\overline{\beta}(t_{1}-\delta, \tau)\bigr)^{\frac{1}{1-p}}\int_{t_{1}-\delta}^{\tau} \theta(t)\|f(t)\|^{p'}_{V'}\,dt\\ &\quad+2\int_{t_{1}-\delta}^{\tau}\theta(t)\beta_{2}(t)\,dt, \end{split}$$ where$\delta>0$is any number,$C_{13}$and$C_{14}$are some positive constants depending only on${\mathcal{B}}$and$p$. Since$\tau\in[t_{1},t_{2}]$is arbitrary, we see that (\ref{Eqf13}) implies (\ref{Equation5}). Second, we construct a sequence of functions approximating a solution for Problem \eqref{ProblemP}. We assume without loss of generality that$T>0$if$S=(-\infty, T]$. Define$S_{k}:=S\cap \{t\in\mathbb{R}:\ t\geqslant -k\}$,$k\in \mathbb{N}$. Let us for each$k\in \mathbb{N}$consider the problem of finding$\hat{u}_{k}\in L^{p}_{\rm loc}(S_{k}; V)$,${\mathcal{B}}\hat{u}_{k}\in C(S_{k}; V_{\mathcal{B}}')such that \begin{subequations}\label{Eqf14} \begin{align} \bigl({\mathcal{B}}\hat{u}_{k}(t)\bigr)'+{\mathcal{A}} \bigl(t,\hat{u}_{k}(t)\bigr)&=f(t)\quad\text{in } \mathscr{D}'(S_{k}; V') \\ \lim_{t\to -k}{\mathcal{B}}\hat{u}_{k}(t)&=0\: \qquad\text{in } V'_{\mathcal{B}}. \end{align} \end{subequations} The existence and uniqueness of a solution\hat{u}_{k}$of problem (\ref{Eqf14}) follow from results of \cite[Corollary III.6.3]{Showalter}. Let us extend$\hat{u}_{k}$to$(-\infty, -k]$by zero and denote this extension by$u_{k}$. It is clear that$u_{k}$is a solution of the problem without initial conditions $$\label{Eqf16} \bigl({\mathcal{B}} u_{k}(t)\bigr)'+{\mathcal{A}}\bigl(t,u_{k}(t)\bigr)=f_{k}(t)\quad\text{in} \quad \mathscr{D}'(S; V'),$$ where$f_{k}(t)=f(t)$on$S_{k}$and$f_{k}(t)={\mathcal{A}}(t,0)$on$(-\infty, -k]$. For each$k\in \mathbb{N}$the solution of problem (\ref{Eqf16}) satisfies estimate (\ref{Equation5}), where$f$is replaced by$f_{k}$. Thus from this estimate and the definition of$f_{k}$we get $$\label{Eqf17} \int_{t_{1}}^{t_{2}} \|u_{k}(t)\|^p_{V}\,dt\leqslant C_{15}(t_{1},t_{2})$$ for any numbers$t_{1}$,$t_{2}\in S$, where$C_{15}(t_{1},t_{2})>0$is a constant dependent on$t_{1}$and$t_{2}$but independent on$k$. From this estimate and \eqref{(iv)} we obtain $$\label{Eqf18} \int_{t_{1}}^{t_{2}} \bigl\|{\mathcal{A}}\bigl(t,u_{k}(t)\bigr)\bigr\|_{V'}^{p'}\,dt\leqslant C_{16}(t_{1},t_{2}),$$ where$C_{16}(t_{1},t_{2})>0$is a constant independent on$k$. From estimates (\ref{Eqf17}) and (\ref{Eqf18}) (see, e.g., \cite{Lions,Showalter}) the existence of the subsequence of$\bigl\{u_{k}\bigr\}_{k=1}^{+\infty}$follows, which we hereafter denote by$\bigl\{u_{k}\bigr\}_{k=1}^{+\infty}$, such that \begin{gather} u_{k}(\cdot)\stackrel{k\to +\infty}{\longrightarrow} u(\cdot)\quad \text{weakly in } L^{p}_{\rm loc}(S; V),\label{Eqf21}\\ {\mathcal{A}}\bigl(\cdot,u_{k}(\cdot)\bigr)\stackrel{k\to +\infty}{\longrightarrow}\chi(\cdot)\quad \text{weakly in } L^{p'}_{\rm loc}(S; V').\label{Eqf22} \end{gather} Since the operator${\mathcal{B}}:V\to V'$is linear and continuous, it follows that its realization${\mathcal{B}}:L^{p}_{\rm loc}(S; V)\to L^{p}_{\rm loc}(S; V')$is also linear and continuous, and hence weakly continuous. From this and (\ref{Eqf21}) we have $${\mathcal{B}} u_{k}(\cdot)\stackrel{k\to +\infty}{\longrightarrow} {\mathcal{B}} u(\cdot)\quad \text{weakly in } L^{p}_{\rm loc}(S; V').\label{Temp2}$$ Finally we show that$u$is a solution for Problem \eqref{ProblemP}. To see this, let us pass to the limit as$k\to+\infty$in (\ref{Eqf16}) and use (\ref{Eqf22}), (\ref{Temp2}): $$\label{Eqf23} \bigl({\mathcal{B}} u(t)\bigr)'+\chi(t)=f(t)\quad\text{in} \quad \mathscr{D}'(S; V').$$ From (\ref{Eqf23}) we have$({\mathcal{B}} u)'\in L^{p'}_{\rm loc}(S; V')$, so by Lemma \ref{L1} we get$u\in C(S; V_{B})$. It remains to prove only that $$\chi(t)={\mathcal{A}}\bigl(t,u(t)\bigr)\quad \text{in V' for a.e. }t\in S.\label{Eqf25}$$ We will establish (\ref{Eqf25}) using the monotonicity method of Browder and Minty. Let us define $E_{k}=\int_{S}\psi(t) \bigl\langle{\mathcal{A}}\bigl(t,u_{k}(t)\bigr)- {\mathcal{A}}\bigl(t,v(t)\bigr),u_{k}(t)-v(t)\bigr\rangle_{V}\,dt,\quad k\in\mathbb{N},$ for any$\psi\geqslant0$from$\mathscr{D}(S)$and$v$from${L^{p}_{\rm loc}(S; V)}$. From \eqref{(v)} it follows that$E_{k}\geqslant 0$. Multiplying (\ref{Eqf16}) by$\psi u_{k}$,$k\in\mathbb{N}$, and integrating over$S$with respect to$t$, we obtain $$\label{Eqf26} \begin{split} & \int_{S}\Bigl\{ \psi(t)\bigl\langle\bigl({\mathcal{B}} u_{k}(t)\bigr)',u_{k}(t)\bigr\rangle_{V}+ \psi(t)\bigl\langle{\mathcal{A}}\bigl(t,u_{k}(t)\bigr),u_{k}(t)\bigr\rangle_{V}\Bigr\}\,dt\\ &=\int_{S}\psi(t)\langle f_{k}(t),u_{k}(t)\rangle_{V}\,dt. \end{split}$$ Then from (\ref{Eqf26}), using (\ref{Eqf6}) where$u$is replaced by$u_{k}$and the definition of$f_{k}$, after integrating by parts, we have $$\label{Eqf27} \begin{split} &\int_{S} \psi(t)\bigl\langle{\mathcal{A}}\bigl(t,u_{k}(t)\bigr),u_{k}(t)\bigr\rangle_{V}\,dt\\ &=\frac{1}{2}\int_{S}\psi'(t)\|u_{k}(t)\|^2_{V_{\mathcal{B}}}\,dt+ \int_{S}\psi(t)\langle f(t),u_{k}(t)\rangle_{V}\,dt. \end{split}$$ Let$t_{1}$,$t_{2}$be any real numbers such that$\mathop{\rm supp} \psi'\subset[t_{1}, t_{2}]\subset S$. From (\ref{Eqf21}) we obtain $u_{k}(\cdot)\stackrel{k\to +\infty}{\longrightarrow} u(\cdot)\quad \text{weakly in } L^{p}(t_{1}, t_{2}; V).$ Hence, using the compactness of the imbedding$V\hookrightarrow V_{\mathcal{B}}$and Lemma \ref{L2}, by dropping to a subsequence and reindexing, we get $u_{k}(\cdot)\stackrel{k\to +\infty}{\longrightarrow} u(\cdot)\quad \text{strongly in } L^{p}(t_{1}, t_{2}; V_{\mathcal{B}}).$ This and$p>2$imply $$u_{k}(\cdot)\stackrel{k\to +\infty}{\longrightarrow} u(\cdot)\quad \text{strongly in } L^{2}(t_{1}, t_{2}; V_{\mathcal{B}}).\label{Temp5}$$ From (\ref{Temp5}) we have $$\label{Temp7} \int_{S}\psi'(t)\|u_{k}(t)\|^2_{V_{\mathcal{B}}}\,dt\stackrel{k\to +\infty}{\longrightarrow} \int_{S}\psi'(t)\|u(t)\|^2_{V_{\mathcal{B}}}\,dt.$$ Passing to the limit as$k\to+\infty$in (\ref{Eqf27}) and using (\ref{Eqf21}), (\ref{Temp7}), we obtain $$\label{NewEq5} \begin{split} &\int_{S} \psi(t)\bigl\langle{\mathcal{A}}\bigl(t,u_{k}(t)\bigr),u_{k}(t) \bigr\rangle_{V}\,dt\\ & \stackrel{k\to +\infty}{\longrightarrow} \frac{1}{2}\int_{S}\psi'(t)\|u(t)\|^2_{V_{\mathcal{B}}}\,dt +\int_{S}\psi(t)\langle f(t),u(t)\rangle_{V}\,dt. \end{split}$$ Now multiply equality (\ref{Eqf23}) by$\psi u_{k}$and integrate over$S$with respect to$t$. We get $$\label{Temp6} \int_{S} \psi(t)\langle\chi(t),u(t)\rangle_{V}= \frac{1}{2}\int_{S}\psi'(t)\|u(t)\|^2_{V_{\mathcal{B}}}\,dt+ \int_{S}\psi(t)\langle f(t),u(t)\rangle_{V}\,dt.$$ From (\ref{NewEq5}) and (\ref{Temp6}) we have $$\label{Eqf28} \int_{S} \psi(t)\bigl\langle{\mathcal{A}}\bigl(t,u_{k}(t)\bigr),u_{k}(t) \bigr\rangle_{V}\,dt \stackrel{k\to +\infty}{\longrightarrow} \int_{S} \psi(t)\langle\chi(t),u(t)\rangle_{V}\,dt.$$ Using (\ref{Eqf21}), (\ref{Eqf22}) and (\ref{Eqf28}), we deduce $$\label{Eqf29} 0\leqslant\lim_{k\to\infty} E_{k}=\int_{S}\psi(t) \bigl\langle\chi(t)- {\mathcal{A}}\bigl(t,v(t)\bigr),u(t)-v(t)\bigr\rangle_{V}\,dt.$$ Setting$v=u-sw$in (\ref{Eqf29}), where$s>0$and$w\in{L^{p}_{\rm loc}(S; V)}$is any function, we obtain $$\label{Eqf30} \int_{S}\psi(t) \bigl\langle\chi(t)- {\mathcal{A}}\bigl(t,u(t)-sw(t)\bigr),w(t)\bigr\rangle_{V}\,dt\geqslant0.$$ Passing to limit as$s\to0$in (\ref{Eqf30}) and using \eqref{(vii)}, we get $$\label{Eqf31} \int_{S}\psi(t) \bigl\langle\chi(t)- {\mathcal{A}}\bigl(t,u(t)\bigr),w(t)\bigr\rangle_{V}\,dt\geqslant0.$$ Since$\psi\geqslant0$and$w$are arbitrary functions from$\mathscr{D}(S)$and${L^{p}_{\rm loc}(S; V)}$respectively, we deduce from (\ref{Eqf31}) equality (\ref{Eqf25}), as desired. This completes the proof. \end{proof} \begin{proof}[Proof of Theorem \ref{T3}] The uniqueness of a solution for Problem \eqref{ProblemP} follows directly from condition \eqref{(viii)} and Theorem \ref{T1} by taking$\gamma(t)\equiv K_{1}$,$t\in S$,$\varphi(\tau)=\tau^{q/2}$,$\tau\in[0, +\infty)$(see Remark \ref{R1}). Estimate (\ref{AEqu38}) follows from (\ref{Eqf10}) in the same manner as we establish (\ref{Equation5}) by using (\ref{Eqf11}), where$p$and$\|\cdot\|_{V}$are replaced by$q$and$\|\cdot\|_{V_{\mathcal{B}}}respectively, (\ref{Eqf12}), (\ref{Eqf42}) and \begin{align*} & \int_{t_{1}-\delta}^{\tau}\theta(t) \bigl\langle{\mathcal{A}}\bigl(t,u(t)\bigr),u(t)\bigr\rangle_{V}\,dt\\ &\geqslant K_{1}\int_{t_{1}-\delta}^{\tau} \theta(t)\|u(t)\|^q_{V_{\mathcal{B}}}\,dt +\int_{t_{1}-\delta}^{\tau}\theta(t) \bigl\langle{\mathcal{A}}(t,0),u(t)\bigr\rangle_{V}\,dt. \end{align*} The last inequality is an immediate consequence of \eqref{(viii)}. Now we prove the existence of a solution for Problem~\eqref{ProblemP}. By the same argument used in the proof of Theorem \ref{T2} it is sufficient to show that the sequence\bigl\{u_{k}\bigr\}_{k=1}^{+\infty}$, where$u_{k}(k\in \mathbb{N})$is a solution of problem (\ref{Eqf16}), satisfies $$\label{AEqu41} u_{k}(\cdot)\stackrel{k\to +\infty}{\longrightarrow} u(\cdot)\quad \text{strongly in } L^{p}(t_{1}, t_{2}; V_{\mathcal{B}})$$ for any$t_{1}$,$t_{2}\in S$. Multiplying (\ref{Eqf16}) by$v$, where$v\in{L^{p}_{\rm loc}(S; V)}$is any function, and integrating from$t_{1}$to$t_{2}$with respect to$t$, where$t_{1}$,$t_{2}\in S$,$(t_{1}-t_{1}$, it follows from (\ref{AEqu40}), using Lemma \ref{L1} and condition \eqref{(viii)}, that $\|w_{lm}(t_{2})\|^2_{V_{\mathcal{B}}}-\|w_{lm}(t_{1})\|^2_{V_{\mathcal{B}}}+ 2K_{1}\int_{t_{1}}^{t_{2}} \|w_{lm}(t)\|^q_{V_{\mathcal{B}}}\,dt\leqslant0.$ From here and Lemma \ref{L4} in the same manner as was obtained (\ref{Equation5}) we show that for any natural numbers$l$,$m>-t_{1}+\delta$$$\label{AEqu42} \max_{t\in[t_{1}, t_{2}]}\|w_{lm}(t)\|^2_{V_{\mathcal{B}}}\equiv \max_{t\in[t_{1}, t_{2}]}\|u_{l}(t)-u_{m}(t)\|^2_{V_{\mathcal{B}}}\leqslant C_{17}\delta^{\frac{2}{2-q}},$$ where$\delta>0$is any number,$C_{17}$is some positive constant depending only on$K_{1}$,${\mathcal{B}}$and$p$. Thus from (\ref{AEqu42}) it follows that$\{u_{k}\}_{k=1}^{+\infty}$is a Cauchy sequence in$C\bigl([t_{1}, t_{2}]; V_{\mathcal{B}}\bigr)$, and therefore is a Cauchy sequence in$L^{p}(t_{1}, t_{2}; V_{\mathcal{B}})$. Consequently, we conclude from (\ref{Eqf21}) and completeness of$L^{p}(t_{1}, t_{2}; V_{\mathcal{B}})$that (\ref{AEqu41}) holds, so the proof is complete. \end{proof} We remark that the Proof of Corollary \ref{C1} follows from estimate (\ref{AEqu38}). \begin{proof}[Proof of Theorem \ref{T_Temp4}] Existence and uniqueness of a solution$u$for Problem \eqref{ProblemP} follows from Theorem \ref{T3}. Note that the function$u(t+\sigma)$,$t\in \mathbb{R}$, is also a solution of this problem. The uniqueness of a solution for Problem~\eqref{ProblemP}\ implies$u(t+\sigma)=u(t)$for a.e.$t\in \mathbb{R}$. Thus a solution of Problem~\eqref{ProblemP}\ is$\sigma$-periodic. Now we prove estimate (\ref{TempEstimate}). Let$u$be a$\sigma$-periodic solution for Problem \eqref{ProblemP}. Multiplying equation \eqref{ProblemP} by$u$, using (\ref{Eqf6}) and integrating from$t_{1}\in \mathbb{R}$to$t_{2}\in \mathbb{R}$($t_{1}0$is a constant depending on$p$. Set$t_{1}=0$and$t_{2}=\sigma$. Since$u$is a$\sigma$-periodic, from (\ref{TempEqf9}) it follows that $$\label{TempEqf10} \int_{0}^{\sigma} \|u(t)\|^p_{V}\,dt\leqslant C_{19}\int_{0}^{\sigma} \Bigr(\|f(t)\|^{p'}_{V'}+\beta_{2}(t)\Bigl)\,dt,$$ where$C_{19}>0$is a constant depending on$p$and$\mathop{\rm ess\,inf}_{t\in [0, \sigma]} \beta_{1}(t)$. Let us take$\theta\in C^{1}({\mathbb{R}})$with the following properties:$\theta(t)=0$if$t\in(-\infty, -\sigma]$,$\theta(t)=\exp(-\frac{t^{2}}{(t+\sigma)^{2}})$if$t\in(-\sigma, 0)$,$\theta(t)=1$if$t\in[0, +\infty)$. From (\ref{TempEqf9}), setting$t_{1}=-\sigma$,$t_{2}=\tau\in[0; \sigma]$and using Lemma \ref{L4}, we obtain $$\label{TempEqf11} \begin{split} &\|u(\tau)\|^2_{V_{\mathcal{B}}}+ \int_{0}^{\tau} \beta_{1}(t)\|u(t)\|^p_{V}\,dt\\ &\leqslant\int_{-\sigma}^{0} \theta'(t)\|u(t)\|^2_{V_{\mathcal{B}}}\,dt+ C_{18}\int_{-\sigma}^{\sigma} \Bigr(\beta_{1}^{-\frac{1}{p-1}}(t)\,\|f(t)\|^{p'}_{V'}+\beta_{2}(t)\Bigl)\,dt. \end{split}$$ Now we estimate the first term of the right hand side of (\ref{TempEqf11}). Since the imbedding$V\hookrightarrow V_{\mathcal{B}}$is continuous, from (\ref{TempEqf10}) we see that $$\label{TempEqf12} \begin{split} \int_{-\sigma}^{0} \theta'(t)\|u(t)\|^2_{V_{\mathcal{B}}}\,dt & \leqslant C_{20}\int_{0}^{\sigma} \|u(t)\|^2_{V}\,dt\\ & \leqslant C_{21}\Bigl(\int_{0}^{\sigma} \|u(t)\|^p_{V}\,dt\Bigr)^{2/p}\\ & \leqslant C_{22}\Bigl(\int_{0}^{\sigma} \Bigr(\|f(t)\|^{p'}_{V'}+\beta_{2}(t)\Bigl)\,dt\Bigr)^{2/p}, \end{split}$$ where$C_{20}$,$C_{21}$and$C_{22}$are constants depending on$p$,$\sigma$,${\mathcal{B}}$and$\mathop{\rm ess\, inf}_{t \in [0, \sigma]} \beta_{1}(t)$. Thus estimate (\ref{TempEstimate}) follows from (\ref{TempEqf10})-(\ref{TempEqf12}). \end{proof} \begin{proof}[Proof of Theorem \ref{T4}] Note that Theorem \ref{T3} implies the existence and uniqueness of a solution$u$for Problem~\eqref{ProblemP}. Define$u_{\sigma}(t):= u(t+\sigma)$,$w_{\sigma}(t):= u(t+\sigma)-u(t)$,$f_{\sigma}(t):= f(t+\sigma)$and${\mathcal{A}}_{\sigma}(t,\cdot):= {\mathcal{A}}(t+\sigma, \cdot)$,$t\in \mathbb{R}$, for any$\sigma\neq0$. Clearly$u_{\sigma}$is a solution for Problem~\eqref{ProblemP} with${\mathcal{A}}$replaced by${\mathcal{A}}_{\sigma}$and$f$replaced by$f_{\sigma}$. Taking the difference between \eqref{ProblemP} for$u=u_{\sigma}$and \eqref{ProblemP} for$u$we obtain $$\label{AEqu47} \bigl({\mathcal{B}} w_{\sigma}(t)\bigr)'+{\mathcal{A}}_{\sigma}\bigl(t,u_{\sigma}(t)\bigr) -{\mathcal{A}}\bigl(t,u(t)\bigr)=f_{\sigma}(t)-f(t)\quad\text{in } \mathscr{D}'(\mathbb{R}; V').$$ Let$\theta_{1}\in C^{1}({\mathbb{R}})$be the same as in proof of Theorem \ref{T2} and$\tau\in \mathbb{R}$,$\delta>0$be any numbers. Multiplying (\ref{AEqu47}) by$\theta w_{\sigma}$, where$\theta(t)=\theta_{1}(\frac{t-\tau}{\delta})$,$t\in \mathbb{R}$, and integrating from$\tau-\delta$to$\tau+1$with respect to$twe get \begin{align}\label{AEqu48} &\int_{\tau-\delta}^{\tau+1} \theta(t)\frac{d}{dt}\|w_{\sigma}(t)\|^2_{V_{\mathcal{B}}}\,dt+ 2\int_{\tau-\delta}^{\tau+1}\theta(t) \bigl\langle{\mathcal{A}}\bigl(t,u_{\sigma}(t)\bigr)- {\mathcal{A}}\bigl(t,u(t)\bigr),w_{\sigma}(t)\bigr\rangle_{V}\,dt\nonumber\\ &= 2\int_{\tau-\delta}^{\tau+1}\theta(t) \bigl\langle{\mathcal{A}}\bigl(t,u_{\sigma}(t)\bigr)- {\mathcal{A}}_{\sigma}\bigl(t,u_{\sigma}(t)\bigr),w_{\sigma}(t)\bigr\rangle_{V}\,dt\\ &\quad+2\int_{\tau-\delta}^{\tau+1}\theta(t)\langle f_{\sigma}(t)-f(t),w_{\sigma}(t)\rangle_{V}\,dt.\nonumber \end{align} From (\ref{AEqu48}), using \eqref{(ix)} and the estimates similar to (\ref{Eqf11}), (\ref{Eqf12}), in the same way as was shown (\ref{Equation5}), we obtain $$\label{AEqu49} \begin{split} &\|w_{\sigma}(\tau+1)\|^2_{V_{\mathcal{B}}}+ \int_{0}^{1} \|w_{\sigma}(s+\tau)\|^p_{V}\,ds\\ &=\|w_{\sigma}(\tau+1)\|^2_{V_{\mathcal{B}}}+ \int_{\tau}^{\tau+1} \|w_{\sigma}(t)\|^p_{V}\,dt\\ &\leqslant C_{23}\,\delta ^{\frac{2}{2-p}} +C_{24}\int_{\tau-\delta}^{\tau+1} \bigl\|{\mathcal{A}}_{\sigma}\bigl(t,u_{\sigma}(t)\bigr)- {\mathcal{A}}\bigl(t,u_{\sigma}(t)\bigr)\bigr\|^{p'}_{V'}\,dt\\ &\quad+C_{24}\int_{\tau-\delta}^{\tau+1} \|f_{\sigma}(t)-f(t)\|^{p'}_{V'}\,dt \end{split}$$ for any\tau\in{\mathbb{R}}$and$\delta>0$, where$C_{23}$,$C_{24}$are some positive constants depending only on${\mathcal{B}}$,$K_{2}$and$p$. Let$\varepsilon>0$be any number. Fix$\delta\in\mathbb{N}$large enough that $$\label{AEqu51} C_{23}\delta ^{\frac{2}{2-p}}<\frac{\varepsilon}{2}.$$ Since${\mathcal{A}}\in BC({\mathbb{R}}; Y_{p, V})$, it follows that $$\label{NewTemp57} \begin{split} \sup_{\tau\in{\mathbb{R}}}\int_{\tau}^{\tau+1} \|{\mathcal{A}}(t,0)\|^{p'}_{V'}\,dt&\leqslant \sup_{t\in{\mathbb{R}}} \|{\mathcal{A}}(t,0)\|^{p'}_{V'}\\ &\leqslant \sup_{t\in{\mathbb{R}}}\Bigl(\sup_{v\in V}\frac{\|{\mathcal{A}}(t,v)\|_{V'}}{\|v\|^{p-1}_{V}+1}\Bigr)^{p'}\\ &=\sup_{t\in{\mathbb{R}}}\Bigl(d_{p, V}\bigl({\mathcal{A}}(t,\cdot), 0\bigr)\Bigr)^{p'} \leqslant C_{25}, \end{split}$$ where$C_{25}$is some positive constant. Thus (\ref{NewTemp57}), the assumptions of the theorem and Corollary \ref{C1} imply $$\label{AEqu52} \sup_{\tau\in{\mathbb{R}}}\int_{\tau}^{\tau+1} \|u_{\sigma}(t)\|_{V}^{p}\,dt\leqslant C_{26},$$ where$C_{26}\geqslant0$is some constant independent on$\sigma$. From (\ref{AEqu52}) it follows that $$\label{AEqu53} \begin{split} &\int_{\tau-\delta}^{\tau+1}\bigl\|{\mathcal{A}}_{\sigma}\bigl(t,u_{\sigma}(t)\bigr)- {\mathcal{A}}\bigl(t,u_{\sigma}(t)\bigr)\bigr\|^{p'}_{V'}\,dt\\ &\leqslant \sup_{t\in{\mathbb{R}}}\sup_{v\in V}\frac{\|{\mathcal{A}}_{\sigma}(t,v)-{\mathcal{A}}(t,v)\|_{V'}^{p'}}{\|v\|^{p}_{V}+1} \sum_{i=0}^{\delta}\int_{\tau-i}^{\tau+1-i} \bigl(\|u_{\sigma}(t)\|_{V}^{p}+1\bigr)\,dt\\ &\leqslant C_{27}\, \Bigl(\sup_{t\in{\mathbb{R}}}\,d_{p, V}\bigl({\mathcal{A_{\sigma}}} (t,\cdot), {\mathcal{A}}(t,\cdot)\bigr)\Bigr)^{p'}, \end{split}$$ where$C_{27}$is positive constant depending only on$p$,$\delta$and$C_{26}$. Since$f\in S^{p'}({\mathbb{R}}; V'), it follows that \begin{align}\label{TempAEqu60} \int_{\tau-\delta}^{\tau+1} \|f_{\sigma}(t)-f(t)\|^{p'}_{V'}\,dt&=\sum_{i=0}^{\delta}\int_{\tau-i}^{\tau+1-i} \|f_{\sigma}(t)-f(t)\|^{p'}_{V'}\,dt\nonumber\\ &\leqslant (\delta+1)\,\sup_{s\in{\mathbb{R}}}\int_{s}^{s+1} \|f_{\sigma}(t)-f(t)\|^{p'}_{V'}\,dt\\ &=(\delta+1)\,\|f_{\sigma}-f\|^{p'}_{S^{p'}}.\nonumber \end{align} Take\varepsilon_{0}>0$such that $$\label{NewTemp61} C_{24}\bigl(C_{27}+(\delta+1)\bigr){\varepsilon_{0}}^{p'}<\frac{\varepsilon}{2}.$$ Define $U_{\varepsilon}:=\bigl\{\sigma: \sup_{\tau\in{\mathbb{R}}} \|w_{\sigma}(\tau)\|^{2}_{V_{\mathcal{B}}}+ \sup_{\tau\in{\mathbb{R}}}\int_{0}^{1} \|w_{\sigma}(t+\tau)\|_{V}^{p}\,dt<\varepsilon\bigr\}$ for any$\varepsilon>0$. Since$f^{b}\in CAP\bigl({\mathbb{R}}; L^{p'}(0,1; V')\bigr)$and${\mathcal{A}}\in CAP({\mathbb{R}}; Y_{p, V})$, we see that the set$G_{\varepsilon_{_{0}}}:=\bigl\{\sigma\in{\mathbb{R}}: \|f_{\sigma}-f\|_{S^{p'}}+\sup_{t\in{\mathbb{R}}}\,d_{p, V}\bigl({\mathcal{A_{\sigma}}}(t,\cdot), {\mathcal{A}}(t,\cdot)\bigr)<\varepsilon_{0}\bigr\}$is relatively dense in${\mathbb{R}}$(see, e.g., \cite[Property~I.VII]{Levitan}). Then from (\ref{AEqu49}), (\ref{AEqu51}) and (\ref{AEqu53})-(\ref{NewTemp61}) it follows that$\sigma\in U_{\varepsilon}$whenever$\sigma\in G_{\varepsilon_{_{0}}}$. Thus the proof is complete. \end{proof} \section{Example} Let$\Omega$,$\Omega_{1}$be bounded domains in${\mathbb{R}^{n}}$,$n\in\mathbb{N}$, such that$\Omega_{1}\subset\Omega$,$\Omega_{0}:=\Omega\setminus\Omega_{1}$,$\partial\Omega$be a$C^{1}$manifold,$S:={\mathbb{R}}$, and$2