\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 14, 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-2005/14\hfil A paratingent equation] {Existence of solutions to a paratingent equation with delayed argument} \author[L. Boudjenah\hfil EJDE-2005/14\hfilneg] {Lotfi Boudjenah} \address{D\'{e}partement de math\'{e}matiques, Universit\'{e} d'Oran, BP 1524, Oran 31000, Algeria} \email{lotfi.boudjenah@univ-oran.dz} \date{} \thanks{Submitted December 6, 2004. Published January 30, 2005.} \subjclass[2000]{34A60, 49J24, 49K24} \keywords{Convex delayed argument; differential inclusion; paratingent; \hfill\break\indent set-valued function; upper semi-continuity} \begin{abstract} In this work we prove the existence of solutions of a class of paratingent equations with delayed argument, \begin{equation*} (Pt\;x)(t)\subset F([x]_{t})\quad\hbox{for } t\geq 0 \end{equation*} with the initial condition $x(t)=\xi (t)$ for $t\leq 0$. We use a fixed point theorem to obtain a solution and then provide an estimate for the solution. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} The first works on differential inclusions were published in 1934-35 by Marchaud \cite{m1} and Zaremba \cite{z1}. They used terms of contingent or paratingent equations. Later, Wasewski and his collaborators published a series of works and developed the elementary theory of differential inclusions \cite{w1,w2}. Within few years after the first publications, the differential inclusions resulted to be the basic tool in the optimal control theory. Starting from the pioneering work of Myshkis \cite{m2}, there exists the whole series of papers devoted to paratingent and contingent differential inclusions with delay; see for example Campu \cite{c2,c3} and Kryzowa \cite{k2}. After this, many works appear on differential inclusions with delay, for example Deimling \cite{d1}, Haddad \cite{h1,h2,h3,h4} Kamenskii et al. \cite{k1} and Zygmunt \cite{z2}. Recent results for differential inclusions with a finite delay $r>0$ in spaces of Banach were obtained by Syam \cite{s2} and Castaing-Ibrahim \cite{c3}. Recently, Raczynski has successfully applied differential inclusions to simulation and modelling theory \cite{r1,r2,r3}. A more extended survey on differential inclusions can be found in the book of Aubin and Cellina \cite{a1}, the book of K. Deimling \cite{d1}, the book of M. Kamenskii \cite{k1} and the book of G. V. Smirnov \cite{s1}. In this work we study the existence of the solutions of the paratingent equation with delayed argument, \begin{gather*} (Pt\;x)(t)\subset F([x]_{t})\quad\mbox{for }t\geq 0, \\ x(t)=\xi (t)\quad \mbox{for }t\leq 0\,. \end{gather*} \section{Preliminaries} Let $(E,\rho )$ and $(E',\rho ')$ two metric spaces. By $\mathop{\rm Comp}E$, we denote the set of all the nonempty and compact subsets of $E$. When $E$ is a vector space, $\mathop{\rm Conv}E$ denotes the set of all convex elements of $\mathop{\rm Comp}E$. A set-valued map, $F:E\to \mathop{\rm Comp}E'$, is called upper semi-continuous in $E$, and denoted by u.s.c , if for any point $a\in E$ and all $\varepsilon >0$, there exists $\delta >0$ such that $x\in B(a)_{\delta }\Rightarrow F(x)\subset B(F(a))_{\varepsilon }$ where $B(a)_{\delta }=B(a,\delta )=\{x\in E :\rho (a,x)<\delta \}$ and $B(F(a))_{\varepsilon }=B(F(a),\varepsilon )=\{y\in E'$ such as $z\in F(a)$ and $\rho '(y,z)<\varepsilon \}$. (see \cite{b1}) On the upper semi-continuity of a set-valued map, we have the following lemma (see \cite{h5}). \begin{lemma} \label{lem1} Let $(E,\rho )$ and $(E',\rho ')$ be two metric spaces. A set-valued map, $F:E\to \mathop{\rm Comp}E'$, is u.s.c if and only if, for all sequences $\{x_{i}\}\in E$ and $\{y_{i}\}\in E'$ such that $\{x_{i}\}\to x_{0}$ and $\{y_{i}\}\in F(x_{i})$, there exists a subsequence $\{y_{i_{k}}\}$ of $\{y_{i}\}$ which converges to $y_{0}\in F(x_{0})$. \end{lemma} Let $C$ the space of continuous functions $x:R\to R^{n}$ with the topology defined by an almost uniform convergence (i.e. a uniform convergence on each compact interval of $\mathbb{R}$). It is well know that the almost uniform convergence in $C$ is equivalent to the convergence by the metric $\rho$ defined as follows \begin{equation*} \rho (x,y)=\sum_{i=1}^{\infty }\frac{1}{2^{i}}\min \{(1,\sup |x(t)-y(t)|),-i\leq t\leq i\}\quad \text{for }x,y\in C. \end{equation*} Then $C$ is a metric locally convex linear topological space. Let $\beta <0$ be a fixed real number and let $I=[0,\infty [ \subset \mathbb{R}$. If $x\in C$, the symbol $[x]_{t}$ denotes the restriction of $x$ on the interval $[\beta ,t]$ when $t\in I$ and $\|x\|_{t}=max\{|x(s)|,\beta \leq s\leq t\}$ with $|x|=\max\{ |x_{1}|,|x_{2}|,\dots ,|x_{n}|\}$ for $x=( x_{1},x_{2},\dots ,x_{n}) \in \mathbb{R}^n$. Let $G$ denote the metric space whose elements are functions $[x]_{t}, [y]_{u},\dots$, where $t\in I$, $u\in I$, the distance between two functions $[x]_{t}$, $[y]_{u}$ , being understood as a distance of their graphs in $R \times \mathbb{R}^n$ in the Hausdorff sense. \subsection*{Paratingent of a function} Having a function $x\in C$ and $t\in I$, the set of limit points \begin{equation*} \lim \frac{x(u_{i})-x(s_{i})}{u_{i}-s_{i}}=\alpha\,, \end{equation*} where $u_{i}\in I$, $s_{i}\in I$, $u_{i}\neq s_{i}$ ($i=1,2,\dots )$, and $\lim u_{i}=\lim s_{i}=t$, is called the paratingent of $x$ at the point $t$ and denoted by $(Ptx)(t)$. It is easy to see that $(Ptx)$ maps the interval $I$ to the family of the nonempty and closed subsets of $\mathbb{R}^n$ (see \cite{b2}). \subsection*{Paratingent equation with a delayed argument} Let a set-valued map $F:G\to \mathop{\rm Comp}\mathbb{R}^{n}$, be a relation of the form $$\label{e1} (Pt\,x)(t)\subset F([x]_{t})\quad \mbox{where }t\in I, \; x\in C.$$ is called paratingent equation with a delayed argument. Every function $x\in C$ satisfying \eqref{e1} will be called the solution of these equation. The generalized problem of Cauchy for \eqref{e1} consists in the search for a solution of \eqref{e1} which will be satisfy the initial condition $$\label{e2} x(t)=\xi (t)\quad \text{for } t\in [ \beta ,0]$$ where the function $\xi \in C$ , called the initial function, is given in advance (i.e. the solution of \eqref{e1} must contain a certain curve given in advance). \section{Existence of solutions} To show that the paratingent equation with delayed argument \eqref{e1} with the initial condition \eqref{e2} has at least one solution on interval $[0,T]$ ($T>0$ an arbitrary real positive number), we assume the following hypothesis: \begin{itemize} \item[(H1)] The set-valued mapping $F:G\to \mathop{\rm Conv}R^{n}$ is upper semi-continuous and satisfies the condition $$\label{e3} F([x]_{t})\subset \overline{B}(0,w(t,\|x\|_{t}))\quad \text{for }t\geq 0$$ where $\overline{B}(0,r)$ denotes the closed ball with center at $0$ of $\mathbb{R}^{n}$ and radius $r$, $w(t,y)$ is a continuous function from $I\times I$ to $I$, increasing in $y$ and such that the ordinary differential equation $y'=w(t,y)$, with the initial condition $y(0)=A$ (an arbitrary real positive number) has a maximal solution on all intervals $I$ and for all $A$. \end{itemize} \begin{theorem} \label{mainthm} Under the hypothesis (H1), for each $\zeta$, the paratingent equation with delayed argument \eqref{e1}--\eqref{e2} has a solution on $[0,T]$, with arbitrary $T>0$. \end{theorem} For the proof of this theorem we need some lemmas. First we will state Opial's theorem \cite{l1}. \begin{lemma} \label{lem2} Let $w(t,y)$ a continuous function from $I\times I$ to $I$, increasing with respect to $y$ and $M(t)$ a maximal solution of the ordinary differential equation $y' =w(t,y)$, with the initial condition $y(t_{0})=y_{0}$, on the interval $[t_{0},T]$, where $T>t_{0}$ ($T$ an arbitrary positive real number). Let $m(t)$ be function which is continuous and increasing on $[t_{0},T]$ and such that $m'(t)\leq w(t,m(t))$ almost everywhere on $[t_{0},T]$. If $m(t_{0})\leq y_{0}$, then $m(t)\leq M(t)$ for all $t\in [ t_{0},T]$. \end{lemma} \begin{lemma} \label{lem3} Let $x,y\in C$. If for all $t\geq 0$, $$\label{e4} (Pty)(t)\subset \overline{B}(0,w(s,\|x\|_{t})$$ Then for all $t\geq 0$ and for all $h\geq 0$ we have $$\label{e5} |y(t+h)-y(t)|\leq \int_{t}^{t+h}w(s,\|x\|_{s})ds$$ \end{lemma} \begin{proof} Let $T$ be fixed in $I$, \begin{equation*} Q(h)=\int_{t}^{t+h}w(s,\;\|x\|_{s})ds\;+2\varepsilon (h+1)\,, \end{equation*} and $R(h)=|y(t+h)-y(t)|$. It is suffices to prove that for each $\varepsilon >0$ and each $h>0$ we have \label{e6} R(h)0$such that \eqref{e6} is not satisfied, and let$h_{0}$the lower bound of the set$\{h>0:R(h)\geq Q(h)\}$. Since$R(0)=0$and$Q(0)=2\epsilon $, we have$R(0)0$. If$R(h_{0})>Q(h_{0}$), there would be exist a real number$h'\in ]0,h_{0}[$such that$R(h')=Q(h_{0})$, contrary to the definition of$h_{0}$. Therefore, we obtain $$\label{e7} R(h_{0})=Q(h_{0})=|y(t+h_{0})-y(t)|.$$ Let$\{h_{i}\}$,$i=1,2,\dots $, be an increasing sequence of positives numbers converging to$h_{0}$. We have$R(h_{i})\varepsilon +w(t+h_{0},\|x\|_{t+h_{o}})\,. \end{equation*} Passing to limit, as $i\to \infty$, we have \begin{equation*} \lim\frac{|y(t+h_{0})-y(t+h_{i})|}{h_{0}-h_{i}}\geq \varepsilon +w(t+h_{0},\|x\|_{t+h_{o}})>w(t+h_{0},\|x\|_{t+h_{o}})\,. \end{equation*} However, \begin{equation*} \lim\frac{|y(t+h_{0})-y(t+h_{i})|}{h_{0}-h_{i}}\in (Pt\,x)(t+h_{0})\,; \end{equation*} thus we obtain a contradiction with hypothesis \eqref{e4}. Therefore, \eqref{e5} must be true for all $t\in I$ and all $h>0$. \end{proof} \begin{lemma} \label{lem4} If $x\in C$ and $(Pt\,x)(t)\subset \overline{B}(0,w(t,\|x\|_{t}))$ for $t\in I$, then for all $t>0$ we have $\|x\|_{t}\leq M(t)$ where $M(t)$ is the maximal solution of the ordinary differential equation $y'=w(t,y)$, with the initial condition $y(0)=\|x\|_{0}$. \end{lemma} \begin{proof} If $t\in I$ and $u\in [ 0,t]$, we have \begin{equation*} |x(u)|=|x(u)-x(0)+x(0)|\leq |x(0)|+|x(u)-x(0)|. \end{equation*} However, $|x(0)|\leq \max\{|x(s)|,\beta \leq s\leq 0\}$, and according to Lemma \ref{lem3} we obtain \begin{equation*} |x(u)-x(0)|\leq \int_{0}^{u}w(s,\|x\|_{s})ds\,. \end{equation*} Then \begin{equation*} |x(u)|\leq \|x\|_{0}+\int_{0}^{u}w(s,\|x\|_{s})ds\,. \end{equation*} Letting $\|x\|_{0}=\mu$, we obtain \begin{equation*} \max\{|x(u)|,\,\beta \leq s\leq 0\}\leq \mu+\int_{0}^{u}w(s,\|x\|_{s})ds; \end{equation*} however, \begin{equation*} \|x\|_{t}\leq \mu +\int_{0}^{u}w(s,\|x\|_{s})ds=\mu +\int_{0}^{t}w(s,\|x\|_{s})ds\,. \end{equation*} If we assume $\lambda (t)=\|x\|_{t}$, we have \begin{equation*} \lambda (t)\leq \mu +\int_{0}^{t}w(s,\|x\|_{s})ds\,. \end{equation*} After derivation, we obtain $\lambda '(t)\leq w(t,\lambda (t))$. From this and using lemma \ref{lem2}, we obtain $\lambda (t)\leq M(t)$ for $t\geq 0$, where $M(t)$ is the maximal solution of the ordinary differential equation: $y'=w(t,y)$, with the initial condition $y(0)=\mu$. Finally we have $\|x\|_{t}\leq M(t)$, for $t\geq 0$. \end{proof} \begin{lemma} \label{lem5} Let $x,y\in C$ such that $\|x\|_{t}\leq M(t)$ for $t\in I$, where $M(t)$ is the maximal solution of the ordinary differential equation: $z' =w(t,z)$, with the initial condition $z(0)=\|y\|_{0}$. If $(Pty)(t)\subset \overline{B}(0,w(t,\|x\|_{t}))$ for all $t\in I$; then $\|y\|_{t}\leq M(t)$ for all $t\in I$. \end{lemma} \begin{proof} If $t\in I$ and $u\in [ 0,t]$, we have \begin{equation*} |y(u)|=|y(u)-y(0)+y(0)|\leq |y(0)|+|y(u)-y(0)|\,. \end{equation*} However, $|y(0)|\leq max\{|y(s)|,\beta \leq s\leq 0\}$, and in view of Lemma \ref{lem3} we have \begin{equation*} |y(u)-y(0)|\leq \int_{0}^{u}w(s,\|x\|_{s})ds\,. \end{equation*} So that \begin{equation*} |y(u)|\leq \|y\|_{0}+\int_{0}^{u}w(s,\|x\|_{s})ds\,. \end{equation*} From the preceding inequality and hypothesis $\|x\|_{t}\leq M(t)$, we obtain \begin{equation*} |y(u)|\leq \|y\|_{0}+\int_{0}^{u}w(s,\,M(s))ds\,. \end{equation*} Then \begin{equation*} \max\{|y(s)|,\beta \leq s\leq 0\}\leq \|y\|_{0}+\int_{0}^{u}w(s,\,M(s))ds\,; \end{equation*} in other words, \begin{equation*} \|y\|_{u}\leq \|y\|_{0}+\int_{0}^{u}w(s,\,M(s))ds\,. \end{equation*} If we pose $\lambda (u)=\|y\|_{u}$ and $\|y\|_{0}=\eta$, we obtain \begin{equation*} \lambda (u)\leq \eta +\int_{0}^{u}w(s,\,M(s))ds\,. \end{equation*} After derivation, we have $\lambda '(u)\leq w(u,M(u))=M'(u)$ for $u\geq 0$. Given that $\lambda (0)=M(0)=\eta$, and that the functions $\lambda$ and $M$ are positive on $I$, it follows that $\lambda (t)\leq M(t)$ for $t\geq 0$; i.e., \begin{equation*} \|y\|_{t}\leq M(t),\quad\mbox{for }t\geq 0. \end{equation*} \end{proof} \begin{lemma} \label{lem6} Under the hypotheses of Lemma \ref{lem5}, the function $y$ satisfies locally the Lipschitz condition $|y(t)-y(t')| \leq \Omega _{T} |t-t'|$ where $\Omega _{T}=\max\{w(s,M(T)):s\in [ 0,T]\}$, $t,t'\in [ 0,T]$, and $T$ is an arbitrary positive number. \end{lemma} \begin{proof} Let $T$ an arbitrary positive number and $t',t\in [ 0,T]$. According to Lemma \ref{lem3}, we have \begin{equation*} |y(t)-y(t')|\leq \int_{t'}^{t} w(s,\|x\|_{s})ds \end{equation*} However, in view of Lemma \ref{lem5}, we have $\|x\|s\leq M(s)$ for $s\in [0,T]$. Therefore, \begin{equation*} |y(t)-y(t')\leq \int_{t'}^{t} w(s,\|x\|_{s})ds \leq \int_{t'}^{t}w(s,M(s))ds\leq \int_{t'}^{t}w(s,M(T))ds \end{equation*} we obtain $|y(t)-y(t')|\leq \Omega _{T}|t-t'|$ where $\Omega_{T}=\max\{w(s,M(T)),s\in [ 0,T]\}$. \end{proof} Before proving the main theorem, we will still need some lemmas by Zygmunt \cite{z2}. \begin{lemma} \label{lem7} Let $x,y$ be functions in $C$ and $\{x_{i}\}$, $\{y_{i}\}$, $i=1,2,\dots$ be subsequences of functions in $C$. If $x_{i}\to x$, $y_{i}\to y$, $(Pt\,y_{i})(t)\subset F([x_{i}]_{t})$ for $t>0$, and $y_{i}(t)=\xi (t)$ for $t\leq 0$, $i=1,2,\dots$. Then $(Pt\,y)(t)\subset F([x]_{t})$ for $t\geq 0$, and $y(t)=\xi (t)$ for $t\leq 0$. \end{lemma} \begin{lemma} \label{lem8} Let $x,y$ be functions in $C$ and $F:G\to \mathop{\rm Conv}R^{n}$ be an upper semicontinuous set-valued map. Define $G(t)=F([x]_{i})$ for $t\geq 0$. Then the two following statements are equivalent. \begin{itemize} \item[(P1)] $(Pt\,y)(t)\subset G(t)$ \item[(P2)] For all $t\in I$ and all $\varepsilon >0$, there exists $\delta >0$ such that for all $\tau \in I$, all $\sigma \in I$, and $\tau \neq \sigma$, we have $\{|\tau -t|<\delta$ and $|\sigma -t|<\delta \}\Rightarrow \frac{y(\sigma ) -y(\tau )}{\sigma -\tau }\in \overline{G(t)_{\varepsilon }}$, where $\overline{G(t)_{\varepsilon }}$ is the closure of the $\epsilon$-neighborhood of $G(t)$. \end{itemize} \end{lemma} \begin{lemma} \label{lem9} Let $x$, $\xi$ be two functions in $C$ and $F:G\to \mathop{\rm Conv}R^{n}$ be an upper semicontinuous set-valued map. Let us define $G(t)=F[x]_{t}$ for $t\geq 0$. Then there exist a function $y\in C$ such that $(Pt\,y)(t)\subset G(t)$ for $t\geq 0$ and $y(t)=\xi (t)$ for $t\leq 0$. \end{lemma} The proof of the three lemmas above can be found in \cite{z2}. Now we shall prove the main theorem. \begin{proof}[Proof of Theorem \protect\ref{mainthm}] Let $T>0$ be an arbitrary fixed real number. Let us consider the family $\Phi$ of functions $x\in C$ satisfying the following three conditions: \begin{gather} x(t) =\xi (t),\quad \mbox{for }t\in [ \beta ,0] \label{e8} \\ \|x\|_{t} \leq M(t),\quad \mbox{for } t\in [ 0,T] \label{e9} \\ |x(t)-x(t')| \leq \Omega _{T}|t-t'|,\quad\mbox{for t} \in [ 0,T] \label{e10} \end{gather} where $\Omega _{T}=\max\{w(s,M(T)),s\in [ 0,T]\}$ and $M(t)$ is the maximal solution of the ordinary differential equation: $y'=w(t,y)$, with the initial condition $y(0)=(0)$. We shall show that $\Phi$ is a nonempty, compact and convex subset of the space $C$. \noindent(i) $\Phi$ is nonempty, it contains the function \begin{equation*} f(t)=\begin{cases} \xi (t) & \mbox{for }t\in [ \beta ,0] \\ \xi (0) & \mbox{for }t\in [ 0,T] \end{cases} \end{equation*} (ii) That $\Phi$ is compact, follows from Arzela's Theorem: its elements are uniformly bounded and equicontinuous. \noindent(iii) It is easy to establish that $\Phi$ is convex. Let us consider the map $L:\Phi \to C$ such that for $x\in \Phi$, \begin{equation*} L(x)=\{y\in C: y(t)=\xi (t) \mbox{for $t\in[ \beta ,0]$ and $(Pt\,y)(t)\subset F([x]_{t})$ for $t\in [ 0,T]$}\}. \end{equation*} For each fixed function $x$ in $\Phi$, the set $L(x)$ is nonempty according by Lemma \ref{lem9}, convex by Lemma \ref{lem8}. and closed by Lemma \ref{lem7}. Now we show that if for all $x\in \Phi$, $F([x]_{t})\subset \overline{B}(0,w(t,\|x\|_{t}))$ for $t\in [ 0,T]$, then $L(x)$ is compact. Let $y\in L(x)$, i.e., $y(t)=\xi (t)$ for $t\in [ \beta ,0]$ and $(Pt\,y)(t)\subset F([x]_{t})$ for $t\in [ 0,T]$. Let us show that $y\in \Phi$, i.e. that $y$ verified the conditions \eqref{e8}, \eqref{e9} and \eqref{e10}. (i) Obviously we have $y(t)=\xi (t)$ for $t\in [ \beta ,0]$.\newline (ii) From hypotheses $(Pt\,y)(t)\subset F([x]_{t})$ for $t\in [ 0,T]$ and $F([x]_{t})\subset \overline{B}(0,w(t,\|x\|_{t}))$ for $t\in [ 0,T]$, we obtain $(Pt\,y)(t)\subset \overline{B}(0,w(t,\|x\|_{t}))$ for $t\in [ 0,T]$. According to Lemma \ref{lem5}, we have $\|y\|_{t}\leq M(t)$for $t\in [ 0,T]$. \newline (iii) Finally, in view of Lemma \ref{lem6}, we have $|y(t)-y(t')|\leq \Omega _{T}|t-t'|$ for $t\in [ 0,T]$. Moreover, since $L(x)\subset \Phi$, all elements of $L(x)$ are uniformly bounded and equicontinuous; since $L(x)$ is closed, it is compact. Therefore, $L$ maps $\Phi$ in the family of the nonempty, compact and convex subsets of $\Phi$. Let us show that the application $L$ is upper semi-continuous. Let $x_{i}$, $x$, $y_{i}$ , $i=1,2,\dots$, an elements of $\Phi$ such that $x_{i}\to x$ and $y_{i}\in L(x_{i})$. Since $\Phi$ is compact, from sequence $\{y_{i}\}$ $i=1,2,\dots$, we can extract a subsequence $\{y_{i}\}$ which converges to a certain function $y$. According to Lemma \ref{lem7}, we have $(Pt\,y)(t)\subset F([x]_{t})$ for $t\in [ 0,T]$ and $y(t)=\xi (t)$ for $t\in [ \beta ,0]$. Therefore, $y\in L(x)$ and by applying Lemma \ref{lem1}, we show the upper semi-continuity of the map $L$. Using the Glicksberg Ky Fan theorem on the fixed point for multimaps in locally convex spaces \cite{b3}, the map $L$ has a fixed point in $\Phi$. Therefore, there exists a function $x_{0}\in \Phi$ such that $x_{0}\in L(x_{0})$, i.e., we have \begin{equation*} (Pt\,x_{0})(t)\subset F([x_{0}]_{t}) \end{equation*} for $t\in \lbrack 0,T]$, and $x_{0}(t)=\xi (t)$ for $t\in \lbrack \beta ,0]$. In other words, $x_{0}$ is a solution of the paratingent equation with delayed argument \eqref{e1} with the initial condition \eqref{e2}. Moreover, we have an estimate of the solution $x_{0}$, \begin{equation*} \Vert x_{0}\Vert _{t}\leq M(t)\quad \mbox{for }t\in \lbrack 0,T]. \end{equation*} \end{proof} \noindent\textbf{Remark.} Kryzowa \cite{k2} assumed that $F([x]_{t})\subset \overline{B}(0,M(t)+N(t)\|x\|_{t})$ and Zygmunt \cite{z2} assumed that $F([x]_{t})\subset \overline{B}(0,M(t)+N(t)\|x\|_{t}^{\alpha })$ with $M(t),N(t)\geq 0$ real-valued continuous functions and $0<\alpha \leq 1$ for $t\geq 0$. In our work we have assumed that $F$ satisfies condition \eqref{e3} which is more general than those of Kryszowa and Zygmunt. \subsection*{Acknowledgement} The author would like to thank an anonymous referee for his/her helpful suggestions for improving the original manuscript. \begin{thebibliography}{99} \bibitem{a1} J. P. Aubin, A. 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