\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small 
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 103, pp. 1--8.\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 2005 Texas State University - San Marcos.} 
\vspace{9mm}}

\begin{document} 

\title[\hfilneg EJDE-2005/103\hfil Periodic trajectories]
{Periodic trajectories for evolution equations in Banach spaces} 

\author[M. D. Voisei\hfil EJDE-2005/103\hfilneg]
{Mircea D. Voisei}  

\address{Mircea D. Voisei \hfill\break
Department of Mathematics, MAGC 3.734, 
The  University of Texas - Pan American, Edinburg, TX 78539, USA}
\email{mvoisei@utpa.edu}


\date{}
\thanks{Submitted August 3, 2005. Published September 28, 2005.}
\subjclass[2000]{47J35, 34C25, 35K55}
\keywords{Periodic solution, evolution equation of parabolic type}


\begin{abstract}
 The existence of periodic solutions for the evolution 
 equation 
 $$
 y'(t)+Ay(t)\ni F(t,y(t))
 $$
 is investigated under considerably simple assumptions on 
 $A$ and $F$. Here $X$ is a Banach space, the operator  
 $A$ is $m$-accretive, $-A$ generates a compact semigroup, 
 and $F$ is a Carath\'{e}odory mapping.  Two examples 
 concerning nonlinear parabolic equations are presented.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section] 
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

Consider the nonlinear evolution equation
\begin{equation} \label{e1.1}
y'(t)+Ay(t)\ni F(t,y(t)), \quad 0\leq t\leq T,
\end{equation}
where $(X,\|\cdot \|)$ is a Banach space,
$A:D(A)\subset X\to 2^{X}$ is $m$-accretive such that $-A$ generates a
compact semigroup, $F:[0,T]\times \overline{D(A)}\to X$ is
a Carath\'{e}odory mapping, i.e., $F(\cdot ,x):[0,T]\to X$
is strongly measurable for every $x\in \overline{D(A)}$ and
$F(t,\cdot ):\overline{D(A)}\to X$ is continuous for almost
every $t\in [0,T]$.

The aim of this note is to investigate the existence of a mild
periodic solution $y\in C([0,T];X)$, $y(0)=y(T)$ for \eqref{e1.1} in
general Banach spaces. Our main result is the following.

\begin{theorem} \label{thm1.1}
Assume that $A$ is $m$-accretive, $-A$ generates a compact
semigroup, $F$ is Carath\'{e}odory with
\begin{equation} \label{e1.2}
\int_{0}^{T} \sup_{x\in \overline{D(A)},\|x\|\leq r}
\|F(t,x)\|dt<\infty,
\end{equation}
for every $r>0$,
and there exist $R>0$, $b,c\in L^{1}(0,T)$, $c(t)\geq 0$, $t\in (0,T)$,
$c\neq 0$, such that
\begin{equation} \label{e1.3}
[x,y-F(t,x)]_{+}\geq c(t)\|x\|+b(t), \quad t\in (0,T),\; x\in D(A),
\; \|x\|\geq R,\; y\in Ax,
\end{equation}
 where $[x,v]_{+}=\lim\limits_{h\downarrow 0}\frac 1h (\|x+hv\|-\|x\|)$, 
$x,v\in X$.
Then \eqref{e1.1} admits at least one mild $T$-periodic solution.
\end{theorem}

This periodic problem has been intensively studied in the literature
\cite{c1,h1,s1,s2,v1,v2}. The most common argument used is to apply 
a fixed point
theorem for a suitable Poincar\'{e} operator. Generally, Schauder's 
fixed
point theorem produced remarkable results for the general Banach space
setting (Vrabie \cite{v2}, Shioji \cite{s1}).
For example the main result in Shioji \cite{s1}
has a statement similar to Theorem \ref{thm1.1}. In addition $X$ is 
separable, 
$\overline{D(A)}$ is convex, and condition \eqref{e1.3} is strengthened 
to 
$c(t)=c>0, $ $t\in [0,T]$. The earlier result of Vrabie \cite{v2} 
studies
\eqref{e1.1} under the assumptions that $A-aI$ is $m$-accretive for 
some
$a>0$ and the growth condition \eqref{e1.2} is replaced by
$$
\limsup_{r\to \infty } \frac{1}{r}\sup \{\|F(t,x)\|:
t\geq 0,x\in \overline{D(A)},\|x\|\leq r\}=m<a.
$$
The limiting case $a=0$ is studied in Hilbert spaces by Cascaval \& 
Vrabie
\cite{c1} without the growth condition on $F$ but under a supplementary
flow-invariant type condition of the form: there exists $r>0$ such that
$\{x\in X;$ $\|x\|=r\}\cap D(A)$ is nonempty and
$$
\langle x,y-F(t,x)\rangle \geq 0,
\quad \mbox{for every }x\in D(A),\; \|x\|=r,\; y\in Ax,\;0\leq t\leq T,
$$
where $\langle \cdot ,\cdot \rangle $ stands for the inner product
of $X$. Also, in \cite{v1} the case $m=a\geq 0$ is studied in general 
Banach
spaces under the condition: there exists $r>0$ such that
$$
[x,y-F(t,x)]_{+}\geq 0, \quad \mbox{for } 0\leq t\leq T,\;
x\in D(A),\; y\in Ax,\; \|x\|\geq r.
$$
Most of these new assumptions are needed for the invariance of a closed
convex ball in the Schauder fixed point theorem argument.

The method we use here relies on the Leray-Schauder topological degree
applied for the Green operator and it is neither concerned with the 
initial
value problem for \eqref{e1.1} nor involves the Poincar\'{e} operator. 
That is why,
the convexity of $\overline{D(A)}$ or the strong accretivity of $A$ are 
no
longer needed. Condition \eqref{e1.3} ensures some ``a priori'' 
estimates for the
periodic solutions of \eqref{e1.1} and that the ``periodic solution'' 
operator $%
P_{A}:L^{1}(0,T;X)\to C([0,T];X)$, which associates to
$g\in L^{1}(0,T;X)$ all solutions $y\in P_{A}g$ of the periodic
problem $y'(t)+Ay(t)\ni g(t)$, $0\leq t\leq T$, $y(0)=y(T)$, is
compact.

Next section is devoted to preliminaries and main notations. Section 3
contains the proof of our main result. This paper concludes with 
section 4
where two examples concerning nonlinear evolution equation 
of
parabolic type are presented.

\section{Preliminaries}

For notions such as $m$-accretive operator, mild solution, or
compact semigroup the reader is referred to Vrabie \cite{v3} and the
references therein. We suggest Lloyd \cite{l1} for the theory and
notations of the topological degree.

Let $(X,\Vert \cdot \Vert )$ be a real Banach space and
$A:D(A)\subset X\to 2^{X}$. If $A-\omega I$ is $m$-accretive for
some $\omega \in \mathbb{R} $ then for each $(\xi ,f)\in
\overline{D(A)}\times L^{1}(0,T;X)$ the initial value problem
\begin{equation} \label{e2.1}
y'(t)+Ay(t)\ni f(t), \quad 0\leq t\leq T, \; y(0)=\xi ,
\end{equation}
has a unique mild solution denoted by $M(\xi ,f)$.

 For $y_{i}=M(\xi _{i},f_{i})$, where $(\xi
_{i},f_{i})\in \overline{D(A)}\times L^{1}(0,T;X)$, $i=1,2$ we have
the mild solution inequality
\begin{equation} \label{e2.2}
e^{\omega t}\Vert y_{1}(t)-y_{2}(t)\Vert \le
e^{\omega s}\Vert y_{1}(s)-y_{2}(s)\Vert +\int_{s}^{t}e^{\omega
\tau }\Vert f_{1}(\tau )-f_{2}(\tau )\Vert d\tau ,
\end{equation}
 for $0\le s\le t\le T$.

The norm of a function $f$ in $L^{p}(0,T;X)$, $1\leq p<\infty $, is 
denoted by
$$
\|f\|_{L^{p}}=(\int_{0}^{T}\|f(t)\|^{p}dt)^{1/p}.
$$
We use the norm $\|y\|_{\infty }=\underset{t\in [0,T]}{\sup }\|y(t)\|$,
for $y\in C([0,T];X)$.

A family $\mathcal{G}\subset L^{1}(0,T;X)$ is called uniformly 
integrable if
for each $\varepsilon >0$ there exists $\delta (\varepsilon )>0$ such 
that
for every measurable set $E$ in $[0,T]$ whose Lebesgue measure is less 
than
$\delta (\varepsilon )$ we have $\int_{E}\Vert f(t)\Vert dt<\varepsilon 
$,
uniformly for $f\in \mathcal{G}$.

Every bounded subset of $L^{p}(0,T;X)$, $1<p\leq \infty $, is
uniformly
integrable and every uniformly integrable subset is bounded in 
$L^{1}(0,T;X)$.
We recall the following result from  Vrabie \cite[Theorem 2]{v2}).

\begin{theorem} \label{thmVrabie}
If $A:D(A)\subset X\to 2^{X}$ is an operator with $A-\omega I$
$m$-accretive for some $\omega \ge 0$ such that $-A$ generates a
compact semigroup, then for each bounded subset $B$ in 
$\overline{D(A)}$ and
each uniformly integrable $\mathcal{G}$ in $L^{1}(0,T;X)$, the set
$M(B\times \mathcal{G)}$ of all mild solution of (2.1) corresponding to
$(\xi,f)\in B\times \mathcal{G}$ is relatively compact in $C([d,T];X)$
for each $d\in (0,T)$. If, in addition, B is relatively compact in $X$,
then $M(B\times \mathcal{G})$ is relatively compact in C([0,T];X).
\end{theorem}


Consider the ``periodic solution'' operator
$P_{A}:L^{1}(0,T;X)\to 2^{L^{1}(0,T;X)}$ defined by
$(f,y)\in \mathop{\rm Graph}P_{A}$ if $y\in C([0,T];X)$, $y(0)=y(T)$,
is a mild solution of
$$
y'(t)+Ay(t)\ni f(t),\quad 0\leq t\leq T.
$$


\begin{theorem} \label{thm2.1}
If $\mathcal{G}$ is uniformly integrable in $L^{1}(0,T;X)$, $A-\omega 
I$
is $m$-accretive for some $\omega >0$, and $-A$ generates a compact
semigroup then $P_{A}(\mathcal{G})$ is relatively compact in
$C([0,T];X)$.
\end{theorem}

\begin{proof} The mild solution inequality and the periodicity offer us
the estimate
\begin{equation} \label{e2.3}
\|y\|_{\infty }\leq \frac{e^{2\omega T}}{e^{\omega T}-1}
\|f\|_{L^{1}},\quad\mbox{for every }y\in P_{A}f\,,
\end{equation}
where, without loss of generality, we assume that $0\in A0$.
Since $\mathcal{G}$ is bounded, this shows that the set of 
initial-final
data $B=\{y(0)$; $y\in P_{A}(\mathcal{G})\}$ is bounded in $X$. 
According to
Theorem \ref{thmVrabie}, this implies that $B$ is relatively compact in 
$X$
and $P_{A}(\mathcal{G})$ is relatively compact in $C([0,T];X)$.
\end{proof}

\section{Proof of Theorem \ref{thm1.1}}

In the sequel we assume that all the assumptions in Theorem \ref{thm1.1} 
hold. Without
loss of generality we may assume that, in \eqref{e1.3},
$R>\|b\|_{L^{1}}/\|c\|_{L^{1}}$. Let $K>C:=\|b\|_{L^{1}}+R+2$ and
$\rho \in C^{\infty }(\mathbb{R})$ such that $0\leq \rho \leq 1$,
$\rho (u)=1$ for $|u|\leq K$, $\rho (u)=0$ for $|u|\geq K+1$.

\begin{lemma} \label{lem3.1}
Let $y\in C([0,T];X)$ be a mild $T$-periodic solution of
\begin{equation} \label{e3.1}
y'(t)+Ay(t)\ni \rho (\|y(t)\|)F(t,y(t)), \quad t\in [0,T].
\end{equation}
Then $\|y\|_{\infty }\leq C$ or $\|y\|_{\infty }\geq K$.
\end{lemma}


\begin{proof}  Assume by contradiction that $C<\|y\|_{\infty}<K$.
Therefore, $\rho (\|y(t)\|)=1$, $0\leq t\leq T$. The mild
solution definition states that for every $0<c<T$ and
$\varepsilon>0$ there exist
\begin{itemize}
\item[(i)] $0=t_{0}<t_{1}<\dots <c\le t_{n}<T$,
$t_{k}-t_{k-1}\le \varepsilon $ for $k=1,\dots ,n$;

\item[(ii)] $f_{1},\dots,f_{n}\in X$ with
$\sum_{k=1}^{n}\int_{t_{k-1}}^{t_{k}}\Vert \rho
(\|y(t)\|)F(t,y(t))-f_{k}\Vert dt\le \varepsilon $;

\item[(iii)] $y_{0},\dots,y_{n}\in X$ satisfying
$\frac{y_{k}-y_{k-1}}{t_{k}-t_{k-1}}+Ay_{k}\ni f_{k}$ for
 $k=1,\dots ,n$,
\end{itemize}
 such that $\Vert y(t)-y_{k}\Vert \le \varepsilon $ for
$t\in [t_{k-1},t_{k}),~k=1,\dots ,n$.

 Suppose that $\underset{[0,T]}{\inf }\|y\|>R$.
For $\varepsilon <\min \{K-\|y\|_{\infty },\inf\limits_{[0,T]}\|y\|-R\}$, we
have $R\leq \|y_{k}\|\leq K$, $k=1,\dots ,n$. From \eqref{e1.3} and 
(iii) we
find
\begin{equation} \label{e3.2}
[y_{k},f_{k}-\frac{y_{k}-y_{k-1}}{t_{k}-t_{k-1}
}-F(t,y_{k})]_{+}\geq c(t)\|y_{k}\|+b(t), \quad\mbox{a.e. }t\in [0,T].
\end{equation}
Integrate on $[t_{k-1},t_{k}]$ and add from $k=1$ to $n$, to
obtain
\begin{equation} \label{e3.3}
\begin{aligned}
&\|y_{n}\|+\sum_{k=1}^n \|y_{k}\|\int_{t_{k-1}}^{t_{k}}c(t)dt\\
&\leq \|y_{0}\|+\sum_{k=1}^n \int_{t_{k-1}}^{t_{k}}\|f_{k}-F(t,y_{k})\|
+\int_{0}^{t_{n}}|b(t)|dt \\
&\leq \|y_{0}\|+\|b\|_{L^{1}}+\sum_{k=1}^{n}\int_{t_{k-1}}^{t_{k}}\Vert
\rho (\|y(t)\|)F(t,y(t))-f_{k}\Vert dt\\
&\quad +\int_{0}^{T}\|F(t,y(t))-F_{\varepsilon }(t)\|dt,
\end{aligned}
\end{equation}
 where $F_{\varepsilon }(t)=F(t,y_{k})$ if $t\in [t_{k-1},t_{k})$,
$k=1,\dots ,n$, $F_{\varepsilon }(t)=F(t,y(t))$, $t\in [t_{n},T]$.
According to (ii), this yields
\begin{equation} \label{e3.4}
\|y_{n}\|+R\int_{0}^{t_{n}}c(t)dt\leq
\|y_{0}\|+\|b\|_{L^{1}}+\varepsilon 
+\int_{0}^{T}\|F(t,y(t))-F_{\varepsilon
}(t)\|dt.
\end{equation}
Since $F$ is Carath\'{e}odory, from \eqref{e1.2} and the Lebesgue 
dominated
convergence theorem, we have $\lim\limits_{\varepsilon \to 0}
\int_{0}^{T}\|F(t,y(t))-F_{\varepsilon }(t)\|dt=0$. Let
$\varepsilon \to 0$, $c\to T$ in \eqref{e3.4}. We find
$R\leq \|b\|_{L^{1}}/\|c\|_{L^{1}}$ which is a contradiction.
Therefore, $\inf\limits_{[0,T]}\|y\|\leq R$. Eventually
shifting the time, we may assume without loss of generality, that
there exists $t_{+}\in [0,T]$, such that $\|y(0)\|=R+1$,
$\|y(t_{+})\|=\|y\|_{\infty }$, and $\|y(t)\|\geq R+1$, for every
$t\in [0,t_{+}]$. By the same argument used above we obtain
$\|y\|_{\infty }\leq R+1+\|b\|_{L^{1}}\leq C$ which is a contradiction.
 The proof is complete.
\end{proof}

For $0\leq \lambda \leq 1$, define the operators
$L_{\lambda}:C([0,T];X)\to C([0,T];X)$,
$$
L_{\lambda }v=P_{A+\omega I}(\lambda \rho (\|v\|)F(\cdot
,v)+\lambda\omega v),\quad v\in C([0,T];X),
$$
i.e., $y=L_{\lambda }v$ is the unique $T$-periodic solution of
\begin{equation} \label{e3.5}
y'(t)+Ay(t)+\omega y(t)\ni \lambda
\rho (\|v(t)\|)F(t,v(t))+\lambda \omega v(t),\quad 0\leq t\leq T.
\end{equation}
 where $\omega >0$ is specified below. Note that $L_{0}=0$ and
that Lemma \ref{lem3.1} contains an ``a priori'' estimate for the fixed 
points of
 $L_{1}$. Similarly, we provide an ``a priori'' estimate for the fixed 
points
of $L_{\lambda }$, $0<\lambda <1$.

\begin{lemma} \label{lem3.2}
For $\omega >0$ big enough, every $T$-periodic solution
$y\in C([0,T];X)$, of
\begin{equation} \label{e3.6}
y'(t)+Ay(t)+\omega (1-\lambda )y(t)\ni \lambda
\rho (\|y(t)\|)F(t,y(t)),\quad 0\leq t\leq T,\; 0<\lambda <1,
\end{equation}
satisfies $\|y\|_{\infty }\leq C$ or $\|y\|_{\infty }\geq K$.
\end{lemma}

\begin{proof} Consider
$a_{K}(t)=\sup\{ \|F(t,x)\|;\; x\in \overline{D(A)},\|x\|\leq K+1\}$, $t\in 
[0,T]$.

\noindent According to \eqref{e1.2} $a_{K}\in
L^{1}(0,T) $ and $\rho (\|x\|)\|F(t,x)\|\leq a_{K}(t)$, a.e. $t\in 
[0,T]$,
 $x\in \overline{D(A)}$.

Assume by contradiction that $C<\|y\|_{\infty }<K$. Suppose in addition 
that
$\inf\limits_{[0,T]}\|y\|>R$. Reasoning as above we find
\begin{equation} \label{e3.7}
 \omega (1-\lambda )RT+R\|c\|_{L^{1}}\leq
(1-\lambda )\|a_{K}\|_{L^{1}}+\|b\|_{L^{1}},
\end{equation}
which is absurd if $\omega >\|a_{K}\|_{L^{1}}/RT$, since
$R\|c\|_{L^{1}}>\|b\|_{L^{1}}$. Therefore, $\inf\limits_{[0,T]}\|y\|\leq R$, 
and
we may assume that there exist $0\leq s_{0}<t_{0}\leq T$ such that
$\|y(s_{0})\|=R+1$, $\|y(t_{0})\|=\|y\|_{\infty }$, and
$\|y(t)\|\geq R+1$, for every $t\in [s_{0},t_{0}]$. Similarly, we
obtain
\begin{equation} \label{e3.8}
\omega (1-\lambda)R(t_{0}-s_{0})+R\int_{s_{0}}^{t_{0}}c(t)dt
+\|y\|_{\infty }\leq R+1+(1-\lambda )\int_{s_{0}}^{t_{0}}a_{K}(t)dt
+\|b\|_{L^{1}}.
\end{equation}
This leads to
\begin{equation} \label{e3.9}
\int_{s_{0}}^{t_{0}}a_{K}(t)dt-\omega R(t_{0}-s_{0})\geq
C-R-\|b\|_{L^{1}}-1=1.
\end{equation}
Since $a_{K}\in L^{1}(0,T)$ we have
$\lim\limits_{|t-s|\to 0}\int_{s}^{t}a_{K}(t)dt=0$. Relation \eqref{e3.9}
 tells us that
$t_{0}-s_{0}\geq \delta _{0}>0$, where $\delta _{0}$ depends only
on $a_{K}$. If $\omega >\|a_{K}\|_{L^{1}}/R\delta _{0}$, then 
\eqref{e3.9}
 provides us with a
contradiction. The proof is complete.
\end{proof}

\begin{lemma} \label{lem3.3}
$H(\lambda )=L_{\lambda }$, $0\leq \lambda \leq 1$, defines a
homotopy of compact transformations in $C([0,T];X)$.
\end{lemma}

\begin{proof}  Condition \eqref{e1.2} ensures the fact that for every
$0\leq \lambda \leq 1$, the operator $C([0,T];X)\ni v\mapsto
\lambda \rho (\|v\|)F(t,v)+\lambda \omega v$ transforms bounded
subsets of $C([0,T];X)$ into locally integrable subsets of
$L^{1}(0,T;X)$. According to Theorem \ref{thm2.1}, this implies that
$L_{\lambda }:C([0,T];X)\to C([0,T];X)$ is compact, for
every $0\leq \lambda \leq 1$. For $0\leq \lambda$, $\mu \leq 1$ and
$v\in C([0,T];X)$, we have, from the mild solution inequality
combined with the periodicity, that
\begin{equation} \label{e3.10}
\|H(\lambda )v-H(\mu )v\|_{\infty }\leq |\lambda
-\mu |e^{2\omega T}/(e^{\omega T}-1)\|a_{K}\|_{L^{1}}
\end{equation}
 which shows that $H$ is a homotopy, thereby completing the proof.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
Pick $r_{0}\in (C,K)$ and let $B=\{v\in C([0,T];X)$;
 $\|v\|_{\infty }<r_{0}\}$, $S=\{v\in C([0,T];X)$;
 $\|v\|_{\infty }=r_{0}\}$. Since $C<r_{0}<K$ we know from Lemma 
\ref{lem3.2} that
$0\notin (I-L_{\lambda })(S)$, $0\leq \lambda \leq 1$. The invariance 
of the
degree with respect to $H$ gives
\begin{equation} \label{e3.11}
\deg (I-L_{1},B,0)=\deg (I-L_{0},B,0)=\deg (I,B,0)=1\neq 0,
\end{equation}
i.e., $L_{1}$ has at least one fixed point in $B$. Since
$r_{0}<K$, this fixed point of $L_{1}$ is a $T$-periodic solution
of \eqref{e1.1}.
\end{proof}

\section{Examples}

Let $\Omega $ be a bounded domain in $\mathbb{R}^{n}$, $n\geq 2$, with
smooth boundary $\partial \Omega $. First, we study the periodic
problem for the nonlinear diffusion equation
\begin{equation} \label{e4.1}
\begin{gathered}
\frac{\partial y}{\partial t}-\Delta \rho (y)=f(t,x,y(t,x))\quad
\mbox{a.e. }(t,x)\in \mathbb{R}_{+}\times \Omega,  \\
y=0 \quad\mbox{a.e. }(t,x)\in \mathbb{R}_{+}\times \partial\Omega,  \\
y(t,x)=y(t+T,x)\quad \mbox{for every }t\geq 0, \mbox{ and a.e. }
x\in \Omega .
\end{gathered}
\end{equation}


\begin{theorem} \label{thm4.1}
If $\rho \in C(\mathbb{R})\cap C^{1}(\mathbb{R}\setminus \{0\})$,
$\rho(0)=0$, and there exist $c>0$, $p>(n-2)/2$ such that
 $\rho'(r)\geq c|r|^{p-1}$ for every $r\neq 0$, and
$f:\mathbb{R}\times \Omega \times \mathbb{R}\to \mathbb{R}$,
$f=f(t,x,u)$ is $T$-periodic in $t$, $f(t,x,\cdot )$ is continuous for 
almost
every $(t,x)\in \mathbb{R}\times \Omega $,
$f(\cdot ,\cdot,u)$ is measurable for every $u\in \mathbb{R}$,
and there exist $M>0$, $a,c\in L^{1}(0,T)$, $c(t)\geq 0$, $t\in (0,T)$,
$c\neq 0$, $b,d\in L^{1}((0,T)\times \Omega )$ such that
\begin{equation} \label{e4.2}
 |f(t,x,u)|\leq a(t)|u|+b(t,x), \quad
(t,x,u)\in [0,T]\times \Omega \times \mathbb{R},
\end{equation}
 and
\begin{equation} \label{e4.3}
\begin{gathered}
f(t,x,u)u\leq 0, \quad |f(t,x,u)|\geq c(t)|u|+d(t,x), \\
(t,x)\in [0,T]\times \Omega ,\quad |u|\geq M.
\end{gathered}
\end{equation}
Then \eqref{e4.1} has at least one $T$-periodic solution.
\end{theorem}


\begin{remark} \label{rmk4.2} \rm
For the problem above, Shioji \cite{s2} considers \eqref{e4.2} and the
condition
$$
\limsup_{|u|\to \infty } \mathop{\rm ess\,sup}
_{(t,x)\in \mathbb{R}\times \Omega } \frac{f(t,x,u)}{u}<0,
$$
which is equivalent to there exist $\delta ,M>0$, $f(t,x,u)/u\leq 
-\delta $
for $(t,x,u)\in \mathbb{R}\times \Omega \times \mathbb{R}$ with 
$|u|\geq
M$. This clearly has the form of \eqref{e4.3} with $c(t)=\delta $,
$t\in [0,T]$, $d=0$.
\end{remark}


\begin{proof}[Proof of Theorem \ref{thm4.1}]
The operator $A:D(A)=\{u\in L^{1}(\Omega )$;
$\rho (u)\in W_{0}^{1,1}(\Omega )$,
$\Delta \rho (u)\in L^{1}(\Omega )\}\subset L^{1}(\Omega )\to 
L^{1}(\Omega )$
given by $Au:=-\Delta \rho (u)$, $u\in D(A)$, is $m$-accretive in
$X=L^{1}(\Omega )$, $\overline{D(A)}=L^{1}(\Omega )$, and it generates
a compact semigroup (Vrabie \cite{v3}). Define
$F:\mathbb{R}\times L^{1}(\Omega )\to L^{1}(\Omega )$ by
$F(t,u)(x):=f(t,x,u(x))$, $t\in \mathbb{R}$, $u\in L^{1}(\Omega )$.
>From \eqref{e4.2} $F$ is well defined, Carath\'{e}odory, and
satisfies \eqref{e1.2}.

Since $A0=0$, for every $u\in D(A)$ we have
\begin{equation} \label{e4.4}
[u,\Delta \rho (u)-F(t,u)]_{+}\geq [u,\Delta \rho
(u)]_{+}-[u,F(t,u)]_{+}\geq -[u,F(t,u)]_{+}.
\end{equation}
Denote by $\{u<0\}=\{x\in \Omega ;\; u(x)<0\}$,
$\{u=0\}=\{x\in \Omega ;\; u(x)=0\}$,
$\{u>0\}=\{x\in \Omega ;\; u(x)>0\}$, and
$|\Omega |$ the measure of $\Omega $.
>From \eqref{e4.2} and \eqref{e4.3}, for every $u\in L^{1}(\Omega )$
 we obtain
\begin{equation} \label{e4.5}
\begin{aligned}
&[u,F(t,u)]_{+}\\
&=\int_{\{u>0\}} f(t,x,u(x))dx-
\int_{\{u<0\}} f(t,x,u(x))dx+\int_{\{u=0\}} f(t,x,0)dx\\
&\leq 2Ma(t)+3\int_{\Omega } b(t,x)dx
+\int_{\{u>M\}} f(t,x,u(x))dx-\int_{\{u<-M\}} f(t,x,u(x))dx\\
&\leq -c(t)\|u\|_{L^{1}}+Mc(t)|\Omega |+2Ma(t)
+3\int_{\Omega } b(t,x)dx+\int_{\Omega } |d(t,x)|dx.
\end{aligned}
\end{equation}
Relations \eqref{e4.3} and \eqref{e4.4} combined prove that 
\eqref{e1.3}
 is fulfilled with $R=0$.
According to Theorem \ref{thm1.1}, \eqref{e4.1} has at least one 
$T$-periodic solution.
\end{proof}

Next, we consider the periodic problem for the nonlinear heat equation
\begin{equation} \label{e4.6}
\begin{gathered}
\frac{\partial y}{\partial t}-\Delta _{p}(y)=f(t,x,y(t,x))
\quad\mbox{a.e. }(t,x)\in \mathbb{R}_{+}\times \Omega,  \\
y=0\quad\mbox{a.e. }(t,x)\in \mathbb{R}_{+}\times \partial\Omega,  \\
y(t,x)=y(t+T,x)\quad \mbox{for every }t\geq 0,\mbox{ and a.e. }
x\in \Omega ,
\end{gathered}
\end{equation}
where $\Omega $ is a bounded domain of $\mathbb{R}^{n}$, with smooth
boundary $\partial \Omega $, $p\geq 2$, and
$$
\Delta _{p}y=\sum_{i=1}^n \frac{\partial }{\partial x_{i}}
(|\frac{\partial y}{\partial x_{i}}|^{p-2}
\frac{\partial y}{\partial x_{i}}),
$$
is the pseudo-Laplace operator.


\begin{theorem} \label{thm4.2}
Suppose that $f:\mathbb{R}\times \Omega \times \mathbb{R}\to 
\mathbb{R}$,
 $f=f(t,x,u)$ is $T$-periodic in $t$, $f(t,x,\cdot )$ is continuous
for almost every $(t,x)\in \mathbb{R}\times \Omega $,
$f(\cdot ,\cdot ,u)$ is measurable for every $u\in
\mathbb{R}$, and there exist $M>0$, $a,k\in L^{1}(0,T)$,
$a,k\geq 0$, $k\neq 0$, $b,d\in L^{1}(0,T;L^2(\Omega ))$ such
that
\begin{equation} \label{e4.7}
|f(t,x,u)|\leq a(t)|u|+b(t,x), \quad
(t,x,u)\in [0,T]\times \Omega \times \mathbb{R},
\end{equation}
 and
\begin{equation} \label{e4.8}
f(t,x,u)u\leq [c-k(t)]|u|^{p}+d(t,x)|u|, \quad
(t,x)\in [0,T]\times \Omega ,\; |u|\geq M,
\end{equation}
where $c>0$ is such that
\begin{equation} \label{e4.9}
\int_{\Omega }|\nabla u(x)|^{p}dx\geq
c\int_{\Omega }|u(x)|^{p}dx, \quad u\in W_{0}^{1,p}(\Omega ).
\end{equation}
Then \eqref{e4.6} has at least one $T$-periodic solution.
\end{theorem}

\begin{remark} \label{rmk4.4} \rm
In \cite{v2} the existence of periodic solutions for the nonlinear heat 
equation
governed by the pseudo-Laplace operator is showed under the conditions
$$
|f(t,x,u)|\leq a|u|+b,\quad f(t,x,u)u\leq \alpha |u|^{p}+\beta ,\;\;
(t,x,u)\in \mathbb{R}\times \Omega \times \mathbb{R},
$$
where $a,b,\alpha ,\beta >0$ and $\alpha <c$.
These conditions are particular cases of \eqref{e4.7}, \eqref{e4.8}
 with $a(t)=a$, $b(t,x)=b$, $k(t)=c-\alpha $, $d(t,x)=d>0$,
$(t,x)\in \mathbb{R}\times \Omega $, since for $M=\beta /d$ and
$|u|\geq M$, $d|u|\geq \beta $.
\end{remark}


\begin{proof}[Proof of Theorem \ref{thm4.2}]
 The operator $A:D(A)=\{u\in W_{0}^{1,2}(\Omega )$,
$\Delta _{p}u\in L^2(\Omega )\}\subset L^2(\Omega )\to L^2(\Omega )$
given by $Au:=-\Delta_{p}u$, $u\in D(A)$, is maximal monotone in
$X=L^2(\Omega )$, and it generates a compact semigroup (Vrabie 
\cite{v3}).
Define
$F:\mathbb{R}\times L^2(\Omega )\to L^2(\Omega )$
by $F(t,u)(x):=f(t,x,u(x))$, $t\in \mathbb{R}$, $u\in L^2(\Omega )$.
>From \eqref{e4.7} $F$ is well defined, Carath\'{e}odory, and
satisfies \eqref{e1.2}. For $u\in D(A)$, according to \eqref{e4.7},
\eqref{e4.8}, we have
\begin{equation} \label{e4.10}
\begin{aligned}
&\|u\|_{L^2}[u,Au-F(t,u)]_{+}\\
&=\langle u,-\Delta _{p}u-F(t,u)\rangle _{L^2} \\
&=\int_{\Omega }|\nabla u(x)|^{p}dx-\int_{\Omega }f(t,x,u(x))u(x)dx\\
&\geq c\int_{\Omega }|u|^{p}dx-\int_{\{|u|\geq M\}} f(t,x,u(x))u(x)dx
-\int_{\{|u|<M\}} f(t,x,u(x))u(x)dx\\
&\geq c\int_{\{|u|<M\}} |u|^{p}dx
+k(t)\int_{\{|u|\geq M\}} |u|^{p}dx-a(t)\|u\|_{L^2}^2
-\alpha (t)\|u\|_{L^2}\\
&\geq \min \{c,k(t)\}\int_{\Omega }|u|^{p}dx-\alpha (t)\|u\|_{L^2},
\end{aligned}
\end{equation}
 where
$\alpha (t)=(\int_{\Omega}|b(t,x)|^2dx)^{1/2}
+(\int_{\Omega }|d(t,x)|^2dx)^{1/2}\in L^{1}(0,T)$.
Using the inequality
$$
\int_{\Omega }|u|^{p}dx\geq \|u\|_{L^2}|\Omega |^{(2-p)/2}, \quad
u\in L^{p}(\Omega ),
$$
one gets that, for any $R>0$ and $\|u\|_{L^2}\geq R$,
\begin{equation} \label{e4.11}
[u,Au-F(t,u)]_{+}\geq c(t)\|u\|_{L^2}-\alpha (t),
\end{equation}
where $c(t)=\min \{c,k(t)\}R^{p-2}|\Omega |^{(2-p)/2}$,
$t\in [0,T] $. Clearly, $c\geq 0$, $c\neq 0$ and all the
conditions of Theorem \ref{thm1.1} are fulfilled. This provides us with
a $T$-periodic solution of \eqref{e4.6}.
\end{proof}


\begin{thebibliography}{2}
\bibitem{c1}  R. Ca\c{s}caval and I. I. Vrabie;
 \emph{Existence of periodic solutions for a class of nonlinear
evolution equations}, Rev. Mat. Univ. Complutense Madrid
 \textbf{7} (1994), 325-338.

\bibitem{h1}  N. Hirano and N. Shioji; \emph{Existence of periodic
solutions under saddle point type conditions}, J. Nonlinear Convex 
Anal.
\textbf{1} (2000), 115-128.

\bibitem{l1}  N. G. Lloyd; \emph{Degree theory}, Cambridge University 
Press
(1978).

\bibitem{s1}  N. Shioji; \emph{Periodic solutions for nonlinear 
evolution
equations in Banach spaces}, Funkc. Ekvacioj, Ser. Int. \textbf{42} 
(1999),
157-164.

\bibitem{s2}  N. Shioji; \emph{Existence of periodic solutions for
nonlinear evolution equations with pseudo monotone operators}, Proc. 
Am.
Math. Soc. \textbf{125} (1997), 2921-2929.

\bibitem{v1}  M. D. Voisei; \emph{Existence of periodic solutions for
nonlinear equations of evolution in Banach spaces}, Math. Sci. Res. 
Hot-Line
\textbf{3} (1999), 1-7.

\bibitem{v2}  I. I. Vrabie; \emph{Periodic solutions for nonlinear
evolution equations in a Banach space}, Proc. Am. Math. Soc. 
\textbf{109},
(1990), 653-661.

\bibitem{v3}  I. I. Vrabie; \emph{Compactness methods for nonlinear
evolutions. 2nd ed}, Pitman Monographs and Surveys in Pure and Applied
Mathematics, Longman (1995).

\end{thebibliography}

\end{document}