\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
 \begin{equation}\label{FirstEq1}
 \bigl({\mathcal{B}} u(t)\bigr)'+{\mathcal{A}}\bigl(t, u(t)\bigr)=f(t),\quad t\in S,
 \end{equation}
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 $w={\mathcal{B}} v$,
 \begin{equation}\label{NewEq2}
 \|{\mathcal{B}} v\|_{V'_{\mathcal{B}}}=
 \|v\|_{V_{\mathcal{B}}},\quad v\in V_{\mathcal{B}}.
 \end{equation}
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,
 \begin{equation}\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{equation}
 \end{lemma}

 \begin{proof}
Let $t_{1}$, $t_{2}\in S$ be any numbers such that $t_{1}<t_{2}$.
In view of the assumptions we have $v\in L^{p}(t_{1}, t_{2}; V)$ and
$({\mathcal{B}} v)'\in L^{p'}(t_{1}, t_{2}; V')$. With the same proof as
that of \cite[Proposition~1.2, p.~106]{Showalter} we obtain
$v\in C\bigl([t_{1}, t_{2}]; V_{\mathcal{B}}\bigr)$,
the function $t\mapsto\|v(t)\|_{V_{\mathcal{B}}}$ is absolutely continuous
on $[t_{1}, t_{2}]$ and \eqref{Eqf1} holds for a.e. $t\in[t_{1}, t_{2}]$.
Since $t_{1}$, $t_{2}\in S$ are arbitrary, the conclusion of
Lemma \ref{L1} follows.
 \end{proof}

 \begin{lemma}  \label{L2}
Let $1<p<+\infty$. Assume that the inclusion $V\hookrightarrow
V_{\mathcal{B}}$ is compact and define
\[
 U_{p}:=\{u\in L^{p}_{{\rm loc}}(S; V):\
  ({\mathcal{B}} v)'\in L^{p'}_{{\rm loc}}(S; V')\}.
\]
Then the imbedding $U_{p}
 \hookrightarrow L^{p}_{\rm loc}(S; V_{\mathcal{B}})$ is compact.
 \end{lemma}

 \begin{proof}
Let us first prove that $W\hookrightarrow V'_{\mathcal{B}}$ is compact. To
do this, assume that $\bigl\{w_{n}\bigr\}_{n=1}^{+\infty}\subset W$ is any bounded
sequence. The definition of the space $W$
implies for each $n\in\mathbb{N}$ the existence of $v_{n}\in V$ such that
$w_{n}={\mathcal{B}} v_{n}$ and
$\|v_{n}\|_{V}<\|w_{n}\|_{W}+1$. Since $\bigl\{w_{n}\bigr\}_{n=1}^{+\infty}$
is bounded in $W$, it follows that $\bigl\{v_{n}\bigr\}_{n=1}^{+\infty}$
is bounded in $V$. Then, the compactness of the imbedding
$V\hookrightarrow V_{\mathcal{B}}$ implies the existence of a subsequence
$\bigl\{v_{n_{k}}\bigr\}_{k=1}^{+\infty}$ of
$\bigl\{v_{n}\bigr\}_{n=1}^{+\infty}$ which is strongly convergent in
the space $V_{\mathcal{B}}$.
Since the operator ${\mathcal{B}}: V_{\mathcal{B}}\to
 V_{\mathcal{B}}'$ is continuous, it follows that
 $\bigl\{{\mathcal{B}} v_{n_{k}}\bigr\}_{k=1}^{+\infty}$ is strongly
convergent in $V_{\mathcal{B}}'$. But
$w_{n_{k}}={\mathcal{B}} v_{n_{k}}$, $k\in\mathbb{N}$. Thus the sequence
$\bigl\{w_{n_{k}}\bigr\}_{k=1}^{+\infty}$ is strongly convergent in
$V_{\mathcal{B}}'$. Hence the imbedding
$W\hookrightarrow V'_{\mathcal{B}}$ is compact.

 Now we show the compactness of the imbedding $U_{p}
 \hookrightarrow L^{p}_{{\rm loc}}(S;  V_{\mathcal{B}})$.
Let $\bigl\{u_{n}\bigr\}_{n=1}^{+\infty}$ be any bounded
sequence in $U_{p}$; that is, for every $t_{1}$, $t_{2}\in S$,
 $t_{1}<t_{2}$, the sequences of restrictions to $(t_{1}, t_{2})$ of
the elements of $\bigl\{u_{n}\bigr\}_{n=1}^{+\infty}$ and
$\bigl\{({\mathcal{B}} u _{n})'\bigr\}_{n=1}^{+\infty}$ are bounded
sequences in $L^{p}(t_{1}, t_{2}; V)$ and $L^{p'}(t_{1}, t_{2}; V')$
respectively.
Let $t_{1}$, $t_{2}\in S$ with $t_{1}<t_{2}$.
Since the operator ${\mathcal{B}}:V\to  W$ is linear and continuous,
we have that  ${\mathcal{B}}:L^{p}(t_{1}, t_{2}; V)\to L^{p}(t_{1}, t_{2}; W)$
is also linear and continuous (see, e.g., \cite{Showalter}).
Thereby, the sequence
$\bigl\{{\mathcal{B}} u _{n}\bigr\}_{n=1}^{+\infty}$ is bounded in
$L^{p}(t_{1}, t_{2}; W)$. The compactness of the imbedding $W\hookrightarrow
 V_{\mathcal{B}}'$, and Lions-Aubin's
theorem (see, e.g., \cite{Lions} or \cite[p.~106]{Showalter}), imply the
existence of a subsequence
$\bigl\{{\mathcal{B}} u_{n_{k}}\bigr\}_{k=1}^{+\infty}$
 of $\bigl\{{\mathcal{B}} u_{n}\bigr\}_{n=1}^{+\infty}$,
which is strongly convergent in $L^{p}(t_{1}, t_{2}; V'_{\mathcal{B}})$.
 From (\ref{NewEq2}) it follows that
$\bigl\{u_{n_{k}}\bigr\}_{k=1}^{+\infty}$ is strongly convergent
in $L^{p}(t_{1}, t_{2}; V_{\mathcal{B}})$. Thus Lemma \ref{L2} is proved.
 \end{proof}

 \begin{lemma}[{\cite[Lemma 1.1]{Bokalo}}]  \label{L3}
Let $z$ be a nonnegative absolutely continuous function on
each closed subinterval of $S$ and
\[
 z'(t)+\beta(t)\chi\bigl(z(t)\bigr)\leqslant0\quad\text{for a.e. }t\in S,
\]
where $\beta\in L^{1}_{\rm loc}(S)$, $\beta(t)\geqslant0$
for a.e. $t\in S$, $\int_{-\infty}\beta(t)\,dt=+\infty$,
$\chi\in C\bigl([0,+\infty)\bigr)$, $\chi(0)=0$, $\chi(\tau)>0$
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
 \begin{equation}\label{ProblemP}
 \bigl({\mathcal{B}} u(t)\bigr)'+{\mathcal{A}}\bigl(t,u(t)\bigr)=f(t)
\quad\text{in }\mathscr{D}'(S; V').
 \end{equation}
 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 $t_{1}$,
$t_{2}\in S$ $(t_{1}<t_{2})$, $\delta>0$, satisfies the estimate
 \begin{equation}\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}
 \end{equation}
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 $t_{1}$, $t_{2}\in S$  $(t_{1}<t_{2})$ and $\delta>0$ we have the
estimate
 \begin{equation}\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}
 \end{equation}
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
 \begin{equation}\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},
 \end{equation}
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
 \begin{equation}\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}
 \end{equation}
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
 \begin{equation}\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').
 \end{equation}
This and condition \eqref{(ii)} give us
 $({\mathcal{B}} w)'\in L^{p'}_{\rm loc}(S; V')$,
 so using Lemma \ref{L1} we obtain
 \begin{equation}\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.
 \end{equation}
Multiplying (\ref{Eqf3}) by $w$ we get
 \begin{equation}\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
 \end{equation}
 for a.e. $t\in S$.
 From (\ref{Eqf4}) and (\ref{Eqf5}) we obtain
 \begin{equation}\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.
 \end{equation}
 From (\ref{Temp1}) and \eqref{(iii)}
 we have
 \begin{equation}\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,
 \end{equation}
 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
 \begin{equation}\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.
 \end{equation}
 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
 \begin{equation}\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}
 \end{equation}
 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
 \begin{equation}\label{Eqf7}
 \sup_{t\in(-1,
+\infty)}\frac{\theta_{1}'(t)}{\theta_{1}^{\nu}(t)}<C_{8}(\nu),
 \end{equation}
 where $0<\nu<1$, $C_{8}(\nu)>0$ is a constant depending only on
 $\nu$.

 Let $t_{1}$, $t_{2}\in S$ $(t_{1}<t_{2})$, $\delta>0$ 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 $t$:
 \begin{equation}\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}
 \end{equation}
Substituting (\ref{Eqf6}) into (\ref{Eqf8}) yields
 \begin{equation}\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}
 \end{equation}
 Integrating by parts the first term of the left hand side of equality (\ref{Eqf9})
 we obtain
 \begin{equation}\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}
 \end{equation}
 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:
 \begin{equation}\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}
 \end{equation}
 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
 \begin{equation}\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}
 \end{equation}
 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
 \begin{equation}\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}
 \end{equation}
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
 \begin{equation}\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'),
 \end{equation}
 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
 \begin{equation}\label{Eqf17}
 \int_{t_{1}}^{t_{2}} \|u_{k}(t)\|^p_{V}\,dt\leqslant
C_{15}(t_{1},t_{2})
 \end{equation}
 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
 \begin{equation}\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}),
 \end{equation}
 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
 \begin{equation}
{\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}
 \end{equation}

 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}):
 \begin{equation}\label{Eqf23}
 \bigl({\mathcal{B}} u(t)\bigr)'+\chi(t)=f(t)\quad\text{in}
 \quad \mathscr{D}'(S; V').
 \end{equation}
 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
 \begin{equation}
 \chi(t)={\mathcal{A}}\bigl(t,u(t)\bigr)\quad
\text{in $V'$ for a.e. }t\in S.\label{Eqf25}
 \end{equation}
 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
 \begin{equation}\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}
 \end{equation}
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
 \begin{equation}\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}
 \end{equation}
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
 \begin{equation}
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}
 \end{equation}
 From (\ref{Temp5}) we have
 \begin{equation}\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.
 \end{equation}
Passing to the limit as $k\to+\infty$ in (\ref{Eqf27})
 and using (\ref{Eqf21}), (\ref{Temp7}), we obtain
 \begin{equation}\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}
 \end{equation} 
 
Now multiply equality (\ref{Eqf23}) by $\psi u_{k}$ and integrate
 over $S$ with respect to $t$. We get
 \begin{equation}\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.
 \end{equation}
 From (\ref{NewEq5}) and (\ref{Temp6}) we have
 \begin{equation}\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.
 \end{equation}
Using (\ref{Eqf21}), (\ref{Eqf22}) and (\ref{Eqf28}),
 we deduce
 \begin{equation}\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.
 \end{equation}
Setting $v=u-sw$ in (\ref{Eqf29}), where $s>0$ and $w\in{L^{p}_{\rm loc}(S; V)}$ is any
 function, we obtain
 \begin{equation}\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.
 \end{equation}
Passing to limit as $s\to0$ in (\ref{Eqf30}) and using
 \eqref{(vii)}, we get
 \begin{equation}\label{Eqf31}
 \int_{S}\psi(t)
 \bigl\langle\chi(t)-
{\mathcal{A}}\bigl(t,u(t)\bigr),w(t)\bigr\rangle_{V}\,dt\geqslant0.
 \end{equation}
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
 \begin{equation}\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}})
 \end{equation}
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 $t_{1}$, $t_{2}\in S$,
$(t_{1}<t_{2})$ are any numbers, we obtain
 \begin{equation}\label{AEqu39}
 \int_{t_{1}}^{t_{2}}\bigl\langle\bigl({\mathcal{B}} u_{k}(t)\bigr)',
v(t)\bigr\rangle_{V}dt+
 \int_{t_{1}}^{t_{2}}\bigl\langle{\mathcal{A}}\bigl(t,u_{k}(t)\bigr),
v(t)\bigr\rangle_{V}dt
=\int_{t_{1}}^{t_{2}}\langle f_{k}(t),v(t)\rangle_{V}dt.
 \end{equation}
Let $l$, $m\in\mathbb{N}$ be any numbers. Taking the difference
 between (\ref{AEqu39}) for $k=l$ and (\ref{AEqu39}) for $k=m$, and then
setting $v=u_{l}-u_{m}$, we get
 \begin{equation}\label{AEqu40}
 \begin{split}
& \int_{t_{1}}^{t_{2}}\bigl\langle\bigl({\mathcal{B}}
w_{lm}(t)\bigr)',w_{lm}(t)\bigr\rangle_{V}\,dt+
 \int_{t_{1}}^{t_{2}}\bigl\langle{\mathcal{A}}\bigl(t,u_{l}(t)\bigr)-
{\mathcal{A}}\bigl(t,u_{m}(t)\bigr),w_{lm}(t)\bigr\rangle_{V}\,dt\\
&=\int_{t_{1}}^{t_{2}}\langle
f_{l}(t)-f_{m}(t),w_{lm}(t)\rangle_{V}\,dt,
 \end{split}
 \end{equation}
where $w_{lm}:= u_{l}-u_{m}$.
Since $f_{l}(t)=f_{m}(t)$ for a.e. $t\in[t_{1}, t_{2}]$ whenever
$l$, $m>-t_{1}$, 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 $l$, $m>-t_{1}+\delta$
 \begin{equation}\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}},
 \end{equation}
 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 $t_{2}\in \mathbb{R}$
($t_{1}<t_{2}$) with respect to $t$, we obtain
 \begin{equation}\label{TempEqf8}
 \frac{1}{2}\int_{t_{1}}^{t_{2}}
 \frac{d}{dt}\|u(t)\|^2_{V_{\mathcal{B}}}\,dt+
 \int_{t_{1}}^{t_{2}}\bigl\langle{\mathcal{A}}\bigl(t,u(t)\bigr),u(t)\bigr\rangle_{V}\,dt=
 \int_{t_{1}}^{t_{2}}\langle
 f(t),u(t)\rangle_{V}\,dt.
 \end{equation}
 From (\ref{TempEqf8}), using \eqref{(vi)} and Young's inequality for the
right hand side of (\ref{TempEqf8}), we get
 \begin{equation}\label{TempEqf9}
 \begin{split}
 & \|u(t_{2})\|^2_{V_{\mathcal{B}}}-\|u(t_{1})\|^2_{V_{\mathcal{B}}}+
 \int_{t_{1}}^{t_{2}}
 \beta_{1}(t)\|u(t)\|^p_{V}\,dt\\
 &\leqslant C_{18}\int_{t_{1}}^{t_{2}}
 \Bigr(\beta_{1}^{-\frac{1}{p-1}}(t)\,\|f(t)\|^{p'}_{V'}+\beta_{2}(t)\Bigl)\,dt,
 \end{split}
 \end{equation}
where $C_{18}>0$ 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
 \begin{equation}\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,
 \end{equation}
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
  \begin{equation}\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}
  \end{equation}
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
 \begin{equation}\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}
  \end{equation}
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
  \begin{equation}\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').
  \end{equation}
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 $t$
we 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
 \begin{equation}\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}
  \end{equation}
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
 \begin{equation}\label{AEqu51}
C_{23}\delta ^{\frac{2}{2-p}}<\frac{\varepsilon}{2}.
 \end{equation}
Since ${\mathcal{A}}\in BC({\mathbb{R}}; Y_{p, V})$, it follows
that
 \begin{equation}\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}
  \end{equation}
where $C_{25}$ is some positive constant. Thus (\ref{NewTemp57}),
the assumptions of the theorem and Corollary \ref{C1} imply
 \begin{equation}\label{AEqu52}
  \sup_{\tau\in{\mathbb{R}}}\int_{\tau}^{\tau+1}
  \|u_{\sigma}(t)\|_{V}^{p}\,dt\leqslant C_{26},
 \end{equation}
where $C_{26}\geqslant0$ is some constant independent on $\sigma$.
 From (\ref{AEqu52}) it follows that
 \begin{equation}\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}
  \end{equation}
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
 \begin{equation}\label{NewTemp61}
 C_{24}\bigl(C_{27}+(\delta+1)\bigr){\varepsilon_{0}}^{p'}<\frac{\varepsilon}{2}.
 \end{equation}
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 $2<p<+\infty$.
Set $V:= W^{1,p}_{0}(\Omega)$, then $V' = W^{-1,p'}(\Omega)$, where
$p'=p/(p-1)$. Define the operators
${\mathcal{A}}:W^{1,p}_{0}(\Omega)\to W^{-1,p'}(\Omega)$ by
\[
 \langle{\mathcal{A}}(u), v\rangle_{W^{1,p}_{0}(\Omega)}
 :=\int_{\Omega}\sum_{i=1}^{n}\, \Bigl|
\frac{\partial u(x)}{\partial x_{i}}\Bigr|^{p-2}
\frac{\partial u(x)}{\partial x_{i}}\,\frac{\partial v(x)}{\partial x_{i}}\,dx,
\quad u,v\in W^{1,p}_{0}(\Omega),
\]
 and ${\mathcal{B}}:W^{1,p}_{0}(\Omega)\to W^{-1,p'}(\Omega)$ by
\[
 \langle{\mathcal{B}}(u), v\rangle_{W^{1,p}_{0}(\Omega)}
:=\int_{\Omega_{1}}u(x)v(x)\,dx,\quad u,v\in W^{1,p}_{0}(\Omega).
\]
 Then $V_{\mathcal{B}}\cong\{L^{2}(\Omega),\,\|\cdot\|_{V_{\mathcal{B}}}\}$ and
$V'_{\mathcal{B}}=L^{2}(\Omega_{1})$, which we identify as the subspace
of $L^{2}(\Omega)$ whose elements are zero a.e. on
$\Omega_{0}$ (see, e.g., \cite{Showalter77,Showalter}).

 Let $f\in L^{p'}_{\rm loc}\bigl({\mathbb{R}}; L^{p'}(\Omega)\bigr)$.
Then the operators ${\mathcal{A}}$, ${\mathcal{B}}$ and $f$ satisfy the
hypothesis of Theorem \ref{T3} (see, e.g., \cite{Bokalo,Showalter}).
Thus there exists a unique generalized solution
$u\in L^{p}_{\rm loc}\bigl({\mathbb{R}}; W^{1,p}_{0}(\Omega)\bigr)
\cap C({\mathbb{R}}; V_{\mathcal{B}})$ of the problem without initial
conditions
\begin{subequations}\label{AEqu55}
\begin{align}
 \frac{\partial}{\partial t}u(x, t)
  -\sum_{i=1}^{n}\, \frac{\partial}{\partial x_{i}}\Bigl(\Bigl|
  \frac{\partial u(x,t)}{\partial x_{i}}\Bigr|^{p-2}
  \frac{\partial u(x,t)}{\partial x_{i}}\Bigr)&=f(x, t),&
  (x, t)&\in \Omega_{1}\times{\mathbb{R}},\\
 -\sum_{i=1}^{n}\, \frac{\partial}{\partial x_{i}}\Bigl(\Bigl|
  \frac{\partial u(x,t)}{\partial x_{i}}\Bigr|^{p-2}
  \frac{\partial u(x,t)}{\partial x_{i}}\Bigr)&=f(x, t),&
  (x, t)&\in \Omega_{0}\times{\mathbb{R}},\\
 u(s, t)&=0,&(s, t)&\in \partial\Omega\times{\mathbb{R}}.
\end{align}
\end{subequations}
Furthermore, if the set
\[
 \bigl\{\sigma:\ \sup_{\tau\in{\mathbb{R}}}\int_{\tau}^{\tau+1}
 \int_{\Omega}
|f(x,t+\sigma)-f(x,t)|^{p'}\,dx\,dt<\varepsilon\bigr\}
\]
is relatively dense in $\mathbb{R}$;
that is, if $f\in S^{p'}\bigl({\mathbb{R}}; L^{p'}(\Omega)\bigr)$,
then Theorem \ref{T4} implies that the solution $u$ for problem
(\ref{AEqu55}) is almost periodic by Stepanov as an element of
$BS^{p}\bigl({\mathbb{R}}; W^{1,p}_{0}(\Omega)\bigr)$ and by Bohr
as an element of $BC({\mathbb{R}}; V_{\mathcal{B}})$.

 Note that more general examples can be obtained similarly as
in \cite{Showalter77} and \cite{Showalter} by a corresponding choice
of the operators $\mathcal{A}$ and $\mathcal{B}$.

\subsection*{Acknowledgements} The authors thank to Prof. Showalter for his
critical review of the original manuscript and suggestions.

 \begin{thebibliography}{00}

\bibitem{Bahuguna} D. Bahuguna, R. Shukla;
 Approximations of solutions to nonlinear Sobolev type evolution equations,
  \emph{Electron. J. Differential Equations}, Vol. \textbf{2003}~(2003),
 No.~31, 1-16.

 \bibitem{Bokalo} M. M. Bokalo;
Problem without initial conditions for classes of
nonlinear parabolic equations, \emph{J. Sov. Math.},
  \textbf{51}~(1990), No.~3, 2291-2322.

\bibitem{Bokalo2} M. M. Bokalo; Well-posedness of problems without initial
conditions for nonlinear parabolic variational inequalities,
  \emph{Nonlinear boundary value problems}, \textbf{8}~(1998), 58-63.

\bibitem{Gajewski} H. Gajewski, K. Gr\"{o}ger and K. Zacharias;
 \emph{Nichtlineare operatorgleichungen und operatordifferentialgleichungen},
Akademie-Verlag, Berlin, 1974.

 \bibitem{Hu} Z. Hu;
Boundeness and Stepanov's almost periodicity of solutions,
  \emph{Electron. J. Differential Equations}, Vol.~\textbf{2005}~(2005),
No.~35, 1-7.

 \bibitem{Kuttler_Shillor} K. Kuttler and M. Shillor;
 Set-valued pseudomonotone mappings and
degenerate evolution inclusions, \emph{Communications in
Contemporary mathematics}, Vol. \textbf{1}~(1999), No. 1, 87-133.

\bibitem{Kuttler}
K. Kuttler; Non-degenerate implicit evolution inclusions,
\emph{Electron. J. Differential Equations}, Vol. \textbf{2000}~(2000),
 No. 34, 1-20.

\bibitem{Levitan} B. M. Levitan, V. V. Zhikov;
\emph{Almost periodic functions and differential equations},
Cambridge Univ. Press, Cambridge, 1982.

\bibitem{Lions} J. L. Lions;
\emph{Quelques m\'{e}thodes de r\'{e}solution des
probl\`{e}mes aux limites non lin\'{e}aires}, Paris, Dunod, 1969.

\bibitem{Pankov} A. A. Pankov;
\emph{Bounded and almost periodic solutions of
nonlinear operator differential equations}, Kluwer, Dordrecht,
1990.

\bibitem{Showalter69} R. E. Showalter;
 Partial differential equations of Sobolev-Galpern type,
\emph{Pacific J. Math}., \textbf{31}~(1969), 787--793.

\bibitem{Showalter77} R. E. Showalter;
\emph{Hilbert space methods for partial differential equations}, Monographs and Studies in Mathematics,
Vol. 1, Pitman, London, 1977.

\bibitem{Showalter80} R. E. Showalter;
Singular nonlinear evolution equations,
\emph{Rocky Mountain J. Math}., \textbf{10}~(1980), No.~3, 499--507.

\bibitem{Showalter} R. E. Showalter;
 \emph{Monotone operators in Banach space and
nonlinear partial differential equations}, Amer. Math. Soc.,
Vol.~49, Providence, 1997.

\bibitem{Zaidman} S. Zaidman;
 A non linear abstract differential equation with
almost-periodic solution, \emph{Riv. Mat. Univ. Parma},
\textbf{4}~(1984), 331-336.

 \end{thebibliography}

\end{document}