\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 22, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/22\hfil Existence solutions] {Existence of solutions for differential inclusions on closed moving constraints in Banach spaces} \author[A. M. Gomaa\hfil EJDE-2009/22\hfilneg] {Adel Mahmoud Gomaa} \address{Adel Mahmoud Gomaa\hfill\break Mathematics Department, Faculty of Science, Helwan University, Cairo, Egypt} \email{gomaa5@hotmail.com} \thanks{Submitted December 1, 2008. Published January 27, 2009.} \subjclass[2000]{32F27, 32C35, 35N15} \keywords{Differential inclusions; moving constraints; existence solutions} \begin{abstract} In this paper, we prove the existence of solutions to a multivalued differential equation with moving constraints. We use a weak compactness type condition expressed in terms of a strong measure of noncompactness. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \section{Introduction} In this paper we study the existence of solutions to the multivalued differential equation with moving constraints $$\label{eP} \begin{gathered} \dot x(t) \in F\big(t,x(t)\big) \quad \text{a.e. on }I, \\ x(t) \in \Gamma(t) \quad \forall t \in [0,T], \\ x(0)=x_0 \in \Gamma(0). \end{gathered}$$ Where $F:[0,T]\times E\to P_{ck}(E)(P_{ck}$ is the family of nonempty convex compact subsets of $E)$ and $\Gamma: [0,T] \to P_f(E)$, $(P_f(E)$ is the family of closed subsets of $E)$. Problem \eqref{eP} has been studied by many authors; see for example \cite{C.V, AC, 407.418, 43.50, 507.514} when $F$ is lower semicontinuous, and \cite{23.30, C.V} when $F$ is upper semicontinuous with $\Gamma$ is independent of $t$. For $\Gamma$ depending on $t$, we refer to \cite{AC, A.M.G, 1.48}. In \cite{ont} we consider the differential inclusions $\dot{x}(t)\in A(t)x(t)+F(t,x(t))$, $x(0)=x_{0}$ where $\{A(t):0\le t\le T\}$ is a family of densely defined closed linear operators generating a continuous evolution operator ${{S}}(t,s)$ and $F$ is a multivalued function with closed convex values in Banach spaces. there, we show how that this results can be used in abstract control problems. Also in \cite{wea} we consider the Cauchy problem \begin{gather*} \dot{x}(t)=f\big(t,x(t)\big), \quad t \in [0,T] \\ x(0)=x_{0}, \end{gather*} where $f:[0,T]\times E\to E$ and $E$ is a Banach space. In \cite{China, Polon}, we study nonlinear differential equations. In \cite{1172} we study some differential inclusions with delay and their topological properties. Much work has been done in the study of topological properties of solution for differential inclusions; see \cite{59.65, 91.110, 255.263, 197.223, 363.379, onth, 155.168}. In this paper we to prove the existence of solutions to \eqref{eP} by using a measure of strong noncompactness, $\gamma$, (see the next section). Since the Kuratowksi measure of noncompactness and the ball measure of noncomactness are measures of strong noncomactness and we can construct many measures such $\gamma$ as in \cite{93.102}, in this paper Theorem \ref{thm3} is a generalization of results for example Szufla \cite{507.514} and Ibrahim-Gomaa \cite{A.M.G}. In Theorem \ref{thm4}, the assumption on noncompactness is weaker than that of Benabdellah-Castaing and Ibrahim \cite{1.48}. \section{Preliminaries} Let $E$ be a Banach space, $E^*$ its topological dual space, $E_w$ the Banach space $E$ endowed with the weak topology, $B(0,1)$ unit ball of $E$, $I=[0,T]$, $(T>0)$, and $\lambda$ be the Lebesgue measure on $I$ . Consider ${B}$ is the family of all bounded subsets of $E$ and $C(I, E)$ is the space of all weakly continuous functions from $I$ to $E$ endowed with the topology of weak uniform convergence. \begin{definition} \label{def1.1} \rm By a measure of strong noncompactness, $\gamma$, we will understand a function $\gamma :{B}\to \mathbb{R}^{+}$ such that, for all $U, V\in {{B}}$, \begin{itemize} \item[(M1)] $U\subset V\Longrightarrow \gamma (U)\le \gamma (V)$, \item[(M2)] $\gamma (U\cup V)\le \max (\gamma (U),\gamma (V))$, \item[(M3)] $\gamma (\overline{\rm conv}U)=\gamma (U)$, \item[(M4)] $\gamma (U+V)\le \gamma (U)+\gamma (V)$, \item[(M5)] $\gamma (cU)=|c|\gamma (U),\quad c\in \mathbb{R}$, \item[(M6)] $\gamma (U)=0\Longleftrightarrow U$ is relatively compact in $E$, \item[(M7)] $\gamma (U\cup \{x\})=\gamma (U)$, $x\in E$. \end{itemize} \end{definition} \begin{definition} \label{def1.2} \rm For any nonempty bounded subset $U$ of $E$ the weak measure of noncompactness, $\beta$, and the Kuratowski's measure of noncompactness, $\alpha$, is defined as: $\alpha (U)=\inf \{\varepsilon >0: U \text{ admits a finite number of sets with diameter less than \varepsilon}.\}$ \end{definition} For the properties of $\beta$ and $\alpha$ we refer the reader to \cite{BG, KD}. \begin{definition} \label{def1.3} \rm By a Kamke function we mean a function $w:I\times {\mathbb{R}^{+}} \to \mathbb{R}^{+}$ such that: \begin{itemize} \item[(i)] $w$ satisfies the Caratheodry conditions, \item[(ii)] for all $t\in I$; $w(t,0)=0$, \item[(iii)] for any $c\in (0,b]$, $u\equiv 0$ is the only absolutely continuous function on $[0,c]$ which satisfies $\dot{u}(t)\le w\big(t,u(t)\big)$ a.e. on $[0,c]$ and such that $u(0)=0$. \end{itemize} \end{definition} \begin{lemma}[\cite{JR, BG}] \label{lem1} If $\gamma :{{B}\to \mathbb{R}^{+}}$ satisfies conditions {\rm (M2), (M4), (M6)}, then, for any nonempty $U\in {B}$, $\gamma (U)\le \gamma (B(0,1))\alpha (U)$ \end{lemma} \begin{lemma}[\cite{387.404, 607.614}] \label{lem2} If $\gamma$ is a measure of weak (strong) noncompactness and $A\subset C(I,E)$ be a family of strongly equicontinuous functions, then $\gamma (A(I))=\sup \{\gamma (A(t)): t\in I\}$. \end{lemma} \section{Main Results} \begin{theorem}\label{thm3} Let $\Gamma :I\to P_{f}(E)$ be a set-valued function such that its graph, $G$, is left closed and $F:I\times E\to P_{ck}(E)$ be a scalarly measurable set-valued function such that for any $t\in I$, $F(t,.)$ is upper semicontinuous on $E$. Suppose that $F$ satisfies the following conditions: \begin{itemize} \item[(A1)] For each $\varepsilon >0$, there exists a closed subset $I_{\varepsilon }$ of $I$ with $\lambda (I-I_{\varepsilon })<\varepsilon$ such that for any nonempty bounded subset $Z$ of $E$, one has $\gamma (F(J\times Z))\le \sup_{t\in J}w(t,\gamma (Z))$ for any compact subset $J$ of $I_{\varepsilon}$; \item[(A2)] there is $\mu \in L^{1}(I,\mathbb{R}^{+})$, such that $\Vert F(t,x)\Vert <\mu (t)(1+\Vert x\Vert)$, for all $(t,x)\in G$; \item[(A3)] for each $(t,x)\in ([0,T[\times E)\cap G$ and $\varepsilon >0$ there is $(t_{\varepsilon },x_{\varepsilon })\in G$ such that $01$, a sequence $(\theta _{n})_{n\in \mathbb{N}}$ of right continuous functions $\theta _{n}:I\to I$ such that $\theta _{n}(0)=0$, $\theta _{n}(T)=T$ and $\theta _{n}(t)\in [t-\varepsilon _{n},t]$, and a sequence $(x_{n})_{n\in \mathbb{N}}$ from $I$ to $E$ with \begin{itemize} \item[(i)] for all $t\in I$, $x_{n}(t)=x_{0}+\int_{0}^{t}\dot{x}_{n}(s)\,ds$, where $\dot{x}_{n}\in L^{1}(I,E)$; \item[(ii)] for all $t\in I,x_{n}(\theta _{n}(t))\in \Gamma (\theta _{n}(t))$; \item[(iii)] $\dot{x}_{n}(t)\in F(t,x_{n}(\theta _{n}(t))+\varepsilon _{n}B(0,1)$ a.e on $I$; \item[(iv)] $\Vert \dot{x}_{n}(t)\Vert \le m\mu (t)+1$, a.e on $I$. \end{itemize} From (iv) the sequence $(x_{n})$ is equicontinuous in $C(I,E)$. For each $t\in I$, set $A(t)=\{x_{n}(t):n\in \mathbb{N}\}\quad \text{and}\quad \rho (t)=\gamma (A(t)).$ We claim that $(x_{n})_{n\in \mathbb{N}}$ is relatively compact in the space $C(I,E)$. So we will show that $\rho \equiv 0$. Since for each $(t,\tau )\in I\times I$, we have $\gamma \{(x_{n})(\tau ):n\in \mathbb{N}\}\le \gamma \{(x_{n})(t):n\in \mathbb{N}\}+\gamma \{(x_{n})(\tau )-(x_{n})(t):n\in \mathbb{N}\}$ and $\gamma \{(x_{n})(t):n\in \mathbb{N}\}\le \gamma \{(x_{n})(\tau ):n\in \mathbb{N}\}+\gamma \{(x_{n})(t)-(x_{n})(\tau ):n\in \mathbb{N}\},$ then, from Lemma \ref{lem1}, $|\rho (\tau )-\rho (t)|\le \gamma (B(0,1))\alpha (\{x_{n}(t)-x_{n}(\tau ): n\in \mathbb{N}\}),$ which implies $|\rho (\tau )-\rho (t)|\le 2\gamma (B(0,1))| \int_{t}^{\tau }(m\mu(s)+1)\,ds|.$ It follows that $\rho$ is an absolutely continuous and hence differentiable a.e. on $I$. Let $\varepsilon >0$. Since $\varepsilon _{n}\to 0$ as $n\to \infty$, then we can find $n_{0}\in \mathbb{N}$ such that $T\varepsilon _{n}<\frac{\varepsilon }{\gamma (B(0,1))}$, for all $n\ge n_{0}$. Now let $(t,\tau )\in I\times I$ with $t\le \tau$. In view of Condition (iii) and properties of $\gamma$ ((M4), (M7)), we have \begin{align*} \rho (\tau )-\rho (t) &\leq \gamma (\int_{t}^{\tau }\dot{x}_{n}(s)\,ds:n\in N) \\ &\le \gamma (\cup _{n\in \mathbb{N}}\int_{t}^{\tau }F(s,\theta _{n}(s))ds)+\gamma (\{\varepsilon _{n}B(0,1)(\tau -t):n\in \mathbb{N}\}) \\ &=\gamma (\cup _{n\in \mathbb{N}}\int_{t}^{\tau }F(s,\theta _{n}(s))ds)+\gamma (\{\varepsilon _{n}B(0,1)(\tau -t):n\ge n_{0}\}) \\ &\le \gamma (\cup _{n\in \mathbb{N}}\int_{t}^{\tau }F(s,\theta _{n}(s))ds)+\frac{\varepsilon }{\gamma (B(0,1))}\gamma (B(0,1)) \\ &\le \gamma (\cup _{n\in \mathbb{N}}\int_{t}^{\tau }F(s,\theta _{n}(s))ds)+\varepsilon . \end{align*} Thus, $$\rho (\tau )-\rho (t)\le \gamma (\cup _{n\in \mathbb{N}}\int_{t}^{\tau }F(s,\theta _{n}(s))ds). \label{a}$$ Since $\rho$ is continuous and $w$ is Caratheodory we can find a closed subset $I_{\varepsilon }$ of $I$, $\delta >0$, $\eta >0$ ($\eta <\delta$) and for $s_{1},s_{2}\in I_{\varepsilon }$; $r_{1},r_{2}\in [0,2T]$ such that if $|s_{1}-s_{2}|<\delta$, $|r_{1}-r_{2}|<\delta$, then $|w(s_{1},r_{1})-w(s_{2},r_{2})|<\varepsilon$ and if $|s_{1}-s_{2}|<\eta$, then $|\rho (s_{1})-\rho (s_{2})|<\frac{\delta }{2}$. Consider the following partition, of $[t,\tau ]$, $t=t_{0}0$, there exists a closed subset $I_{\varepsilon }$ of $I$ with $\lambda (I-I_{\varepsilon })<\varepsilon$ such that for any nonempty bounded subset $Z$ of $E$, one has $\gamma (F(G\cap (I\times Z)))\le \sup_{t\in I}w(t,\gamma (Z)),$ for any compact subset $J$ of $I_{\varepsilon}$; \item[(A2')] there is a positive number $c$ such that \Vert F(t,x)\Vert 0 there is (t_{\varepsilon },x_{\varepsilon })\in G such that 00,\tau \in I) be the set of all points (x,\theta ) where \theta :[0,\tau ]\to [0,\tau]  is an increasing right continuous function with \theta (0)=0,\theta (\tau )=\tau  and for all t\in ]0,\tau [, \theta (t)\in [t-\varepsilon ,t] and x:[0,\tau ]\to E is such that: \begin{itemize} \item[(i)] for all t\in [0,\tau ], x(t)=x_{0}+\int_{0}^{t}\dot{x}(s)\,ds, where \dot{x}\in L^{1}(I, E); \item[(ii)] for all t\in [0,\tau ], x\big(\theta(t)\big)\in \Gamma (\theta(t)); \item[(iii)] for all t\in [0,\tau ], \dot{x}(t)\in F\big(t,x(\theta(t)\big) +\varepsilon \overline{B(0,1)}, a.e. \end{itemize} Let \varepsilon \in ]0,1] and (\theta ,x)\in A_{\varepsilon }([0,\tau ]). Then by (A2') and the fact that, for all t\in [0,\tau ], \theta (t)\in [t-\varepsilon ,t], we have \begin{align*} \Vert x(\theta (t))\Vert &\leq \Vert x_{0}\Vert +\int_{0}^{\theta (t)}\Vert \dot{x}(s)\Vert \,ds\le \Vert x_{0}\Vert +\int_{0}^{t}\Vert \dot{x}(s)\Vert \,ds \\ &\leq \Vert x_{0}\Vert +\varepsilon T+\int_{0}^{t}c(1+\Vert x(\theta ((s))\Vert )\,ds. \end{align*} By Gronwall's lemma, we obtain \Vert x(\theta (t))\Vert \le (\Vert x_{0}\Vert +T)e^{cT} which gives us  \Vert x(\theta (t))\Vert +1\le (1+\Vert x_{0}\Vert +T)e^{cT}.  Consequently we get for all t\in [0,\tau ], $$F(t,x(\theta (t)))\subseteq pc\overline{B(0,1)} \label{**}$$ where p=(1+\Vert x_{0}\Vert +T)e^{cT}. Let A_{\varepsilon }=\bigcup_{\tau \in I}A_{\varepsilon }([0,\tau ]). Obviously A_{\varepsilon }\ne \emptyset . Partially order A_{\varepsilon } such that for any (\theta _{i},x_{i})\in A_{\varepsilon }([0,\tau _{i}]) \subseteq A_{\varepsilon } (i=1,2) (\theta _{1},x_{1})\le (\theta _{2},x_{2})\Longleftrightarrow \tau _{1}\le \tau _{2},\theta _{1}=\theta _{2}|_{[0,\tau _{1}]} and x_{1}=x_{2}|_{[0,\tau _{2}]}. Let C be a subset of A_{\varepsilon } such that each two elements of it are comparable that is there exists a subset \mathbb{N}'\subseteq \mathbb{N} such that C=\{(\theta _{j},x_{j}):j\in \mathbb{N}'\}\subseteq A_{\varepsilon } and each (\theta _{n},x_{n}), (\theta _{m},x_{n})\in C we have (\theta _{n},x_{n})\le (\theta _{m},x_{m}) or (\theta _{m},x_{m})\le (\theta _{n},x_{n}). Now we prove that C has an upper bound. Let \tau =\sup_{j\in \mathbb{N}'}\tau _{j}. Also let \theta :[0,\tau ]\to [0,\tau ] is such that, for each j\in \mathbb{N}',\theta |_{[0,\tau _{j}]}=\theta _{j} and x:[0,\tau [\to E with x|_{[0,\tau _{j}]}=x_{j}, for each j\in \mathbb{N}'. Let \{\tau _{k_{n}}\} be increasing sequence in \mathbb{N}' such that \tau =\sup_{n\in \mathbb{N}}\tau _{k_{n}} and for any n,m\in \mathbb{N},\;m0 such that \delta _{\varepsilon }<\inf(\varepsilon ,T-\tau _{\varepsilon }). Then by (A3') there exists (\hat{t},\hat{x})\in G such that 0<\hat{t}-\tau _{\varepsilon }\le \delta _{\varepsilon } and \[ \frac{\hat{x}-x_{\varepsilon }}{\hat{t}-\tau _{\varepsilon }}\in F(\tau _{\varepsilon },x_{\varepsilon }(\tau _{\varepsilon }))+\varepsilon \overline{B(0,1)}. Let $\hat y\in F(\tau_{\varepsilon},x_{\varepsilon}(\tau_{\varepsilon}))+\varepsilon \overline {B(0,1)}$ such that $\hat x-x_{\varepsilon}(\tau_{ \varepsilon})=(\hat t-\tau_{\varepsilon})\hat y$. If $\hat{\theta}:[0,\hat{t}]\to [0,\hat{t}]$ and $\tilde{x} :[0,\hat{t}]\to E$ are defined as: $\hat{\theta}(t)=\begin{cases} \theta _{\varepsilon } & \text{if }t\in [0,\tau_{\varepsilon}] \\ \tau _{\varepsilon } & \text{if }t\in ]\tau_{\varepsilon},\hat t] \\ \hat{t} & \text{if } t=\hat t, \end{cases} \qquad \tilde{x}(t)=\begin{cases} x_{\varepsilon } & \text{if }t\in [0,\tau_{\varepsilon}] \\ \hat{x} & \text{if }t\in [\tau_{\varepsilon}, \hat t] \end{cases}$ Then it is easy to check that \cite[p. 10.25]{1.48} $(\hat{\theta},\tilde{x})\in A_{\varepsilon }([0,\hat{t}])$ and $(\theta _{\varepsilon },x_{\varepsilon })<(\hat{\theta},\tilde{x})$. This contradicts the fact that $(\theta _{\varepsilon },x_{\varepsilon })$ is maximal. Now there exist $p>1$, (from (\ref{**})) a sequence $(\theta _{n})_{n\in \mathbb{N}}$ of right continuous functions $(\theta)_{n}:I\to I$ such that $\theta _{n}(0)=0$, $\theta _{n}(T)=T$ and $\theta _{n}(t)\in [t-\varepsilon _{n},t]$, if we have decreasing sequence $(\varepsilon _{n})$ such that $0<\varepsilon _{n}\le 1\;\varepsilon _{n}\to 0$ as $n\to \infty$ and $T\varepsilon _{n}<\frac{\varepsilon }{\gamma (B(0,1))}$, for all $n\ge n_{0}$ we can define a sequence $(x_{n})$ of approximated solutions as the follows: $\forall t\in I,\quad x_{n}(t)=x_{0}+\int_{0}^{t}\dot{x}_{n}(s)\,ds$, where $\dot{x}_{n}\in L^{1}(I, E)$. $(\theta _{n}(t),x_{n}(\theta _{n}(t)))\in G$. $\dot{x}_{n}(t)\in F\big(t,x_{n}(\theta _{n}(t)\big) +\varepsilon _{n}B(0,1)$, a.e on $I$. $\Vert \dot{x}_{n}(t)\Vert \le pc+1$, a.e on $I$. By the same arguments used in the proof of Theorem \ref{thm3} we can prove that the sequence $(x_{n})$ converges to an absolutely continuous function $x$ which is a solution for problem \eqref{eP}. \end{proof} \section{Conclusion} Let us remark that, if we replace $\gamma$ in (A1') by $\alpha$, the condition \begin{itemize} \item[(A4)] For each $\varepsilon >0$, there exists a closed subset $I_{\varepsilon }$ of $I$ with $\lambda (I-I_{\varepsilon })<\varepsilon$ such that for almost all $t\in I_{\varepsilon }$ and for any nonempty bounded subset $Z$ of $E$, one has $\inf_{\delta >0}\alpha (F(G\cap (([t-\delta ,t]\cap I)\times Z)))\le w(t,\alpha (Z))$ \end{itemize} implies Condition (A1') and the converse is not true. Indeed Let $\varepsilon >0$. Since $w$ is Caratheodory function, we can find a closed subset $I_{\varepsilon }$ of $I$ with $\lambda (I-I_{\varepsilon })<\varepsilon$ such that $w$ is continuous on $I_{\varepsilon }$ and Condition (A4) holds on $I_{\varepsilon }$. Let $Z$ be a nonempty bounded subset of $E$. It follows from (A4) that, for any $\tau >0$ and any $t\in I_{\varepsilon }$, there exists a $\delta _{\tau ,t}$ such that $\alpha (F(G\cap (([t-\delta _{\tau ,t},t]\cap I)\times Z)))\le w(t,\alpha (Z))+\tau$. Let $\tau$ be arbitrary but fixed, $J$ be a compact subset of $I_{\varepsilon }$. The collection $\{(t-\frac{\delta _{t}}{2},t+\frac{\delta_{t}}{2}):t\in J\}$ is an open cover for $J$. By compactness of $J$, there exist $t_{1}',t_{2}'\dots ,t_{n}'$ such that $J\subseteq \cup _{i=1}^{n}(t_{i}'-\frac{\delta _{t_{i}'}}{2 },t_{i}'+\frac{\delta _{t_{i}'}}{2})\subseteq \cup _{i=1}^{n}[t_{i}'-\frac{\delta _{t_{i}'}}{2},t_{i}' +\frac{\delta _{t_{i}'}}{2}]$. Now if $J_{i}=J\cap [t_{i}'-\frac{\delta _{t_{i}'}}{2},t_{i}'+\frac{\delta _{t_{i}'}}{2}]$ and $t_{i}=\text{max}J_{i}, 1\le i\le n$, then there exist $t_{1},t_{2}\dots t_{n}\in J$ such that $J_{i}\subseteq [t_{i}-\delta _{t_{i}},t_{i}]$ and $J\subseteq \cup _{i=1}^{n}[t_{i}-\delta _{t_{i}},t_{i}]$. This implies that, \begin{align*} \alpha (F(G\cap (J\times Z))) &\leq \alpha (\cup _{i=1}^{n}F(G\cap (([t_{i}-\delta _{t_{i}},t_{i}]\cap I)\times Z))) \\ &\leq \max_{1\le i\le n}\alpha (F(G\cap (([t_{i}-\delta _{t_{i}},t_{i}]\cap I)\times Z))) \\ &\leq \max_{1\le i\le n}w(t_{i},\alpha (Z))+\tau \le \max_{t\in J}w(t,\alpha (Z))+\tau \end{align*} Since $\tau$ is arbitrary, Condition (A1') holds. To show that the converse is not true we give an example. Let $f:[0,1]\times B(0,1)\to E$ be the single valued function defined by $f(t,x)=k(t)x$, where $k:[0,1]\to \mathbb{R}$, $k(t)=\begin{cases} 1 & \text{if t is irrational} \\ 1/t^2 & \text{if t is rational} \end{cases}$ Let also $w(t,s)=k(t)s$, for all $(t,s)\in I\times \mathbb{R}^{+}$. Clearly, $w$ is a Kamke function. Let $\varepsilon >0$ and choose a closed subset $I_{\varepsilon }$ of $I$ such that $\lambda (I-I_{\varepsilon })<\varepsilon$ and $k$ is continuous on $I_{\varepsilon }$. Then for any compact subset $J$ of $I_{\varepsilon }$ and any bounded subset $Z$ of $E$, \begin{align*} \alpha (f(G\cap (J\times Z)))\le \alpha (f(J\times Z)) &= \alpha \big(\cup _{t\in J,x\in Z}f\{(t,x)\}\big) \\ &= \alpha \big( \cup _{t\in J}k(t)Z\big) =\sup_{t\in J}k(t)\alpha (Z) \\ &= \sup_{t\in J}w\big(t,\alpha (Z)\big). \end{align*} Then Condition (A1') holds as the measure $\gamma$ replaced by the measure $\alpha$. But for each $t\in (0,1)$ and each nonempty subset $Z$ of $E$ we have $\alpha \big(f([t-\delta ,t]\times Z)\big) =\alpha \big(\cup _{s\in [t-\delta ,t]}k(s)Z\big) =\alpha (Z)\cdot \big (\sup_{s\in [t-\delta ,t]}k(s)\big) =\frac{\alpha (Z)}{(t-\delta )^{2}}$. %\end{align*} Thus, $\inf_{\delta >0}\alpha (F(([t-\delta ,t]\cap I)\times Z))=\frac{\alpha (Z)}{ t^{2}}$. So if t is irrational then $\inf_{\delta >0}\alpha (F(([t-\delta ,t]\cap I)\times Z))=\frac{\alpha (Z)}{ t^{2}}>\alpha (Z)=k(t)\alpha (Z)=w(t,\alpha (Z))$. Then (A4) does not hold and consequently Theorem \ref{thm4} is a generalization of the following theorem. \begin{theorem}[Benabdellah-Castaing and Ibrahim \cite{1.48}] Let $F$ and $\Gamma$ be as in Theorem \ref{thm4} except $F$ satisfies Condition {\rm (A4)} instead of {\rm (A1')}. 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