\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 47, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2014/47\hfil Fuzzy differential equations] {Fuzzy differential equations under dissipative and compactness type conditions} \author[T. Donchev, A. Nosheen \hfil EJDE-2014/47\hfilneg] {Tzanko Donchev, Ammara Nosheen} % in alphabetical order \address{Tzanko Donchev \newline Department of Mathematics, "Al. I. Cuza" University, Ia\c{s}i 700506, Romania} \email{tzankodd@gmail.com} \address{Ammara Nosheen \newline Abdus Salam School of Mathematical Sciences, 68-B, New Muslim Town, Lahore, Pakistan} \email{hafiza\_amara@yahoo.com} \thanks{Submitted July 27, 2012. Published February 19, 2014.} \subjclass[2000]{34A07, 34G20} \keywords{Lyapunov-like function; compact perturbations; fuzzy sets} \begin{abstract} Fuzzy differential equation with right-hand side defined as a sum of two almost continuous functions is studied. The first function satisfies dissipative-type condition with respect to Lyapunov-like function. The second maps bounded sets into relatively compact sets. The existence of solution is proved with aid of Schauder's fixed point theorem. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Starting from \cite{KB}, the theory of fuzzy differential equations is rapidly developed due to many applications in the real world problems. Notice only the basic work in this direction \cite{K,LM,PH,PS}. As it is shown in \cite{K}, the set of fuzzy numbers is not locally compact. It means that the classical Peano theorem is (probably) no longer valid and some extra conditions along with continuity of right-hand side are needed. In \cite{WS} the existence of solutions of fuzzy differential equation with uniformly continuous right-hand side is proved under compactness-type condition. The existence and uniqueness of solution under dissipative-type conditions when the right-hand side is continuous is studied in \cite{GS,Lu,SW}. In this paper we study fuzzy differential equation whose right-hand side is a sum of two almost continuous functions, one satisfies dissipative-type condition, and another maps bounded sets into relatively compact sets. To the authors knowledge there are not related results in the literature. We study the fuzzy differential equation \begin{equation}\label{4.1} \dot x(t) = f(t,x) + g(t,x);\ x(0) = x_0,\ t \in I, \end{equation} where $f: I \times \mathbb{E}\to \mathbb{E}$ satisfies dissipative-type condition and $g: I \times \mathbb{E}\to \mathbb{E}$ satisfies compactness-type assumption. Here and further in the paper $I=[0,1]$. $\mathbb{E}= \{ x: \mathbb{R}^n \to [0,1]; x\text{ satisfies (1)--(4)}\}$ is the space of fuzzy numbers: \begin{itemize} \item[(1)] $x$ is normal i.e. there exists $y_0\in \mathbb{R}^n$ such that $x(y_0)=1$, \item[(2)] $x$ is fuzzy convex i.e. $x(\lambda y + (1 - \lambda )z)\ge \min \{ x(y) , x(z)\}$ whenever $y,z \in \mathbb{R}^n$ and $\lambda \in [0,1]$, \item[(3)] $x$ is upper semicontinuous i.e. for any $y_0 \in \mathbb{R}^n$ and $\varepsilon >0$ there exists $\delta (y_0,\varepsilon)>0$ such that $x(y) < x(y_0)+\varepsilon$ whenever $|y-y_0|<\delta,\ y\in \mathbb{R}^n$, \item[(4)] The closure of the set $\{ y\in \mathbb{R}^n;\ x(y)>0\}$ is compact. \end{itemize} The set $[x]^\alpha= \{ y\in \mathbb{R}^n;\ x(y)\geq \alpha\}$ is called $\alpha$-level set of $x$. It follows from (1)--(4) that the $\alpha$-level sets $[x]^\alpha$ are convex compact subsets of $\mathbb{R}^n$ for all $\alpha\in (0,1]$. The fuzzy zero is defined by \[ \hat{{0}}(y)=\begin{cases} 0 &\text{if } y\neq 0, \\ 1 &\text{if } y=0. \end{cases} \] The metric in $\mathbb{E}$ is defined by $ D(x,y)=\sup _{\alpha\in (0,1]} D_H([x]^\alpha, [y]^\alpha)$, where \[ D_H(A,B)= \max\{ \max_{a\in A}\min_{b\in B}|a-b|, \max_{b\in B}\min_{a\in A}|a-b|\} \] is the Hausdorff distance between the convex compact subsets of $\mathbb{R}^n$. The map $F:I\times \mathbb{E} \to \mathbb{E}$ is said to be continuous at $(s,y)$ when for every $\varepsilon>0$ there exists $\delta > 0$ such that $D(F(s,y),F(t,x))< \varepsilon$ for every $t\in I$ and $x\in \mathbb{E}$ with $|t-s|+ D(x,y)< \delta$. The map $F:I\times \mathbb{E} \to \mathbb{E}$ is said to be almost continuous if there exists a sequence $\{ I_k \}_{k =1}^\infty$ of pairwise disjoint compact sets with $\operatorname{meas}(I_k)> 0$ and $\operatorname{meas}\big(\cup_{k = 1}^\infty {I_k}\big) = \operatorname{meas}(I)$ such that $F:I_k\times \mathbb{E} \to \mathbb{E}$ is continuous for every $k$. Since $I_k$ is compact for every $k$, one has that $\cup_{k=1}^nI_k$ is also compact and hence $ (0,1)\setminus \cup_{k=1}^n I_k= \cup_{i=1}^\infty (a_i,b_i)$ is open, because every open set in $\mathbb{R}$ is a union of countable sets of pairwise disjoint open intervals. Throughout this paper both $f:I\times \mathbb{E} \to \mathbb{E}$ and $g:I\times \mathbb{E} \to \mathbb{E}$ are assumed to be almost continuous. \begin{remark}\label{rem4.1} Due to Lusin's theorem (see e.g.~\cite{LT} for short proof) $\Lambda:I\to \mathbb{E}$ is strongly measurable if and only if it satisfies Lusin property, i.e. for all $\varepsilon > 0$ there exists $I_\varepsilon \subset I$ with $ \operatorname{meas}(I \backslash I_\varepsilon)\le \varepsilon$ such that $ \Lambda:I_\varepsilon \to \mathbb{E}$ is continuous. \end{remark} A mapping $\Upsilon: I\to \mathbb{E}$ is said to be differentiable at $t\in I$ if for sufficiently small $h>0$ the differences $\Upsilon(t + h) - \Upsilon(t)$, $\Upsilon(t) - \Upsilon(t-h)$ (in sense of Hukuhara) exist and there exists $\dot \Upsilon(t) \in \mathbb{E}$ such that the limits $ \lim_{h \to 0^ + } \frac{\Upsilon(t + h) - \Upsilon(t)}{h}$ and $ \lim_{h \to 0^ + } \frac{\Upsilon(t) - \Upsilon(t-h)}{h}$ exist, and are equal to $\dot \Upsilon(t)$. At the end points of $I$ we consider only the one sided derivative. The integral of fuzzy function $\Upsilon: I \to \mathbb{E}$ is defined levelwise, i.e. there exists $\Lambda: I \to \mathbb{E}$ such that $ [\Lambda(t)]^\alpha = \int_0^t [\Upsilon(s)]^\alpha ds$, where the integral is in Auman sense. Every such function $\Lambda(\cdot)$ is absolutely continuous (AC). The sequence of strongly measurable functions $ \{ y_n(\cdot)\}_{n=1}^\infty$ is said to be integrally bounded if there exists $\lambda(t) \in L_1(I,\mathbb{R}^+)$ (non negative valued integrable function) such that $D(y_n(t),\hat 0)\leq \lambda(t)$ for every $n$ and a.a. $t\in I$. The Caratheodory function $v: I\times \mathbb{R}^+\to \mathbb{R}^+$ is said to be Kamke function if it is integrally bounded on the bounded sets, $v(t,0)=0$ and the unique solution of $\dot r(t)= v(t,r(t))$ with $r(0)=0$ is $r(t)\equiv 0$. \section{Fuzzy differential equation under dissipative-type condition} In this section we consider the fuzzy differential equation \begin{equation}\label{4e} \dot x(t) = f(t,x),\quad x(0) = x_0, \end{equation} where $f: I\times \mathbb{E}\to \mathbb{E}$ satisfies dissipative-type condition. We extend the results of \cite{PS} to the case of fuzzy differential equations with almost continuous right-hand side. We need the following hypothesis: \begin{itemize} \item[(F1)] $D(f(t,x),\hat 0) \le \lambda(t)(1 + D(x,\hat 0))$ for some $\lambda(t) \in L_1(I,\mathbb{R}^+)$. \item[(F2)] There exists a Lyapunov-like function $ W: \mathbb{E}\times \mathbb{E}\to \mathbb{R}^+$ for \eqref{4e}. \end{itemize} A continuous map $ W: \mathbb{E}\times \mathbb{E}\to \mathbb{R}^+$ is said to be Lyapunov-like function for \eqref{4e} if the following conditions hold (cf.~\cite{LL}): \begin{itemize} \item[(1)] $W(x,x)= 0$, $W(x,y)>0$ for $x\neq y$ and $ \lim_{m\to \infty} W(x_m,y_m)=0$ implies $ \lim_{m\to \infty}D(x_m,y_m)=0$, \item[(2)] There exists a constant $L>0$ such that $$ |W(x_1,y_1)-W(x_2,y_2)|\leq L\left(D(x_1,x_2)+ D(y_1,y_2)\right), $$ \item[(3)] There exists a Kamke function $v:I\times \mathbb{R}^+ \to \mathbb{R}^+$ such that $$ \lim_{h\to 0^+} h^{-1} \left[ W \left( x + h f(t,x),y + h f(t,y) \right) - W(x,y)\right] \leq v(t,W(x,y)) $$ for any $x,y \in \mathbb{E}$. \end{itemize} \begin{lemma}\label{l4.3} Let {\rm (F1)} holds, then for $\varepsilon > 0$ and $\delta > 0$ there exists an AC function $x_\varepsilon(t)$ such that $D(\dot x_\varepsilon(t),f(t,x_\varepsilon(t))) \leq \varepsilon$ for all $t \in I_\varepsilon \subset I$, where $I_\varepsilon$ is a compact set with measure greater than $1- \delta$. \end{lemma} \begin{proof} Since $f:I \times \mathbb{E} \to \mathbb{E}$ is almost continuous there exists a sequence $\{ I_k \}_{k = 1}^\infty$ of pairwise disjoint compact sets such that $\operatorname{meas}\big(\cup_{k=1}^\infty I_k\big) = \operatorname{meas}(I)$ and $f:I_k \times \mathbb{E} \to \mathbb{E}$ is continuous for every $k$. For large $n$ we have $ \operatorname{meas}(I_\delta)> 1 - \delta$, where $ I_\delta= \big( \cup_{k = 1}^n I_k \big)$. Let the needed solution $x_\varepsilon(\cdot)$ be defined on $[0,\tau]$ where $\tau \leq 1$ ($\tau = 0$ is possible). If $\tau = 1$ then we have done, otherwise two cases would be possible: (i) $\tau \in (a_l,b_l) \text{ where } (0,1)\setminus{I_\delta}= \cup _{l=1}^\infty (a_l,b_l)$. In this case we extend $x_\varepsilon(\cdot)$ on $[\tau,b_l)$ by $x_\varepsilon(t) = x_\varepsilon(\tau)$ and denote $\tau_1 = b_l > \tau$, (ii) $\tau \notin \cup_{i=1}^\infty [a_i,b_i)$ then we define $$ x_\varepsilon(t) = x_\varepsilon(\tau)+(t-\tau)f(\tau,x_\varepsilon(\tau)),\quad t \in [\tau,\tau_1]\cap I_\delta. $$ Since $f(\cdot, x_\varepsilon(\cdot))$ is continuous on $I_\delta$, then $ D(\dot x_\varepsilon(t) = f(t,x_\varepsilon(t)),f(\tau,x_\varepsilon(\tau))) \leq \varepsilon,\ \forall t \in [\tau, \tau_1] \cap I_\delta$. One can continue by induction. Suppose the largest interval on which $x_\varepsilon(\cdot)$ satisfies lemma conditions is $[0,\bar\tau)$. Since $ D(f(t,x_\varepsilon), \hat 0) \leq \lambda(t)(1+D(x_\varepsilon(t),\hat 0))$, one has that $$ D(\dot x_\varepsilon(t),\hat 0) \leq \lambda(t)(1+D(x_\varepsilon(t),\hat0)) + \varepsilon \quad \text{for } t \in [0,\bar\tau). $$ Consequently, \begin{gather*} D(x_\varepsilon(t),\hat0) \leq e^{\int _0^{\bar\tau} \lambda(s)ds} D(x_0,\hat0)+\varepsilon, \\ D(\dot x_\varepsilon(t),\hat0) \leq \lambda(t)(1+N_\varepsilon)+\varepsilon, \end{gather*} where $$ N_\varepsilon = e^{\int _0^{\bar\tau} \lambda(s)ds}\big(D(x_0,\hat0) + 2\big). $$ Therefore, $D(\dot x_\varepsilon(t),\hat0) \in L_1(I,\mathbb{R}^+)$. Furthermore, since $x_\varepsilon(\cdot)$ is AC, then one can conclude that $x_\varepsilon(\cdot)$ is uniformly continuous on $[0,\bar\tau)$. Thus $\lim_{t\uparrow \bar\tau}x_\varepsilon(t)=x(\bar\tau)$ exists, which is a contradiction to the fact that $[0,\tau]$ is maximum interval of existence. If $\bar\tau=1$ then the proof is complete. If $\bar\tau< 1$ then we can continue this process by defining $$ x_\varepsilon(t)=x_\varepsilon(\bar \tau)+(t-\bar \tau)f(\bar \tau, x_\varepsilon(\bar \tau)),\ t \in [\bar{\tau},\tilde \tau] $$ for $\bar \tau \notin \cup _{l=1}^\infty [a_l,b_l)$ or $\tilde \tau = b_l$ if $\bar{\tau} \in [a_l,b_l)$ for some $l$, therefore there exists a $\tilde \tau_1 >\tilde \tau$ such that $x_\varepsilon(\cdot)$ satisfies the conclusion of the lemma on $[0, \tilde \tau_1]$. Continuing in the same way the so defined $x_\varepsilon(\cdot)$ will satisfy the conclusion of the lemma on $[0,1]$ \end{proof} \begin{theorem} \label{th1} Let {\rm (F1)} and {\rm (F2)} hold, then \eqref{4e} admits unique solution. \end{theorem} \begin{proof} Denote $ \chi_n(t) = \lambda(t)(1 + N_\varepsilon) + \frac{\varepsilon}{2^n}$, where $N_\varepsilon$ is from Lemma \ref{l4.3}. Let $ I_{\delta_n} = \cup_{n=1}^{k_{\delta_n}} I_n$ be such that $ \operatorname{meas}(I_{\delta_n}) >1-\frac{\delta}{2^n}$ and $f: I_n \times \mathbb{E} \to \mathbb{E}$ is continuous. Consider the sequence of approximate solutions $\{ x_n(\cdot)\}_{n = 0}^\infty$ where $x_n(\cdot)$ is the AC function defined in Lemma \ref{l4.3} when $\varepsilon$ is replaced by $\frac{\varepsilon}{2^n}$. Therefore $ D(\dot x_n(t),f(t,x_n(t)))\leq \eta_n(t)$, where $$ \eta_n(t) = \begin{cases} \varepsilon/2^n & \text{if }t \in I_{\delta_n},\\ \chi_n(t) & \text{if }t \notin I_{\delta_n}. \end{cases} $$ We have to prove that $ \{x_n(\cdot)\}_{n=0}^\infty$ is a Cauchy sequence. To this end we take $\{x_n(\cdot)\}$, $\{x_m(\cdot)\}$, where $n 1- \frac{\delta}{2^n}$. If $t\in J_n$, then \begin{align*} &D^+ W(x_n(t),x_m(t)) \\ &= \lim_{h \to 0^ + } \frac{W(x_n(t+h),x_m(t+h))- W(x_n(t),x_m(t))}{h}\\ &\leq\lim_{h \to 0^ + } \frac{W(x_n(t)+h\dot x_n(t), x_m(t)+h\dot x_m(t))- W(x_n(t),x_m(t))+o(h)}{h} \\ &\leq \lim_{h \to 0^ + } \frac{W(x_n(t)+h\dot x_n(t),x_m(t)+ h\dot x_m(t))- W(x_n(t),x_m(t))}{h} \\ &\leq \lim_{h \to 0^ + } \frac{W(x_n(t)+ h f(t,x_n(t)), x_m(t)+ h f(t,x_m(t)))- W(x_n(t),x_m(t))}{h} \\ &\quad + \lim_{h \to 0^ + } \frac{Lh\left[ D(\dot x_n(t), f(t,x_n(t)))+ D(\dot x_m(t),f(t,x_m(t)))\right]}{h}\\ &\leq v(t,D(x_n(t),x_m(t)))+ \frac{2L\varepsilon}{2^n}. \end{align*} For almost all $t \notin J_n$, we have \begin{align*} &D^+ W(x_n(t),x_m(t))\\ &=\lim_{h \to 0^ + } \frac{W(x_n(t+h),x_m(t+h))- W(x_n(t),x_m(t))}{h} \\ &\leq \lim_{h \to 0^ + } \frac{W(x_n(t)+h\dot x_n(t), x_m(t)+h\dot x_m(t))- W(x_n(t),x_m(t))+o(h)}{h} \\ &\leq \lim_{h \to 0^ + } \frac{W(x_n(t)+h\dot x_n(t),x_m(t)+h \dot x_m(t))- W(x_n(t),x_m(t))}{h} \\ &\leq \lim_{h \to 0^ + } \frac{W(x_n(t)+ h f(t,x_n(t)), x_m(t)+ h f(t,x_m(t)))- W(x_n(t),x_m(t))}{h} \\ &\quad + \lim_{h \to 0^ + } \frac{Lh\left[ D(\dot x_n(t), f(t,x_n(t)))+ D(\dot x_m(t),f(t,x_m(t)))\right]}{h}\\ &\leq v(t,D(x_n(t),x_m(t)))+ 2L \chi_n(t). \end{align*} Consequently, $ D^+ W(x_n(t),x_m(t)) \leq v(t,D(x_n(t),x_m(t)))+ 2L\eta_n(t)$, because $n< m$. Thus $W(x_n(t),x_m(t)) \leq r_n(t)$, where $r_n(t)$ is the maximal solution of $ \dot r(t)= v(t,r(t))+ 2L\eta_n(t)$. Clearly $\eta_n(\cdot)$ is integrally bounded (as a sequence of real valued functions), and $\lim_{n \to \infty}\eta_n(t)=0$ for almost all $t \in I$. Since $v(\cdot,\cdot)$ is Kamke function, then $\lim_{n \to \infty}r_n(t)=0$ uniformly on $I$. Therefore there exists a sequence of continuous real valued functions $S_n(t)$ with $\lim_{n \to \infty}D(x_n(t),x_m(t)) \leq S_n(t)$ for all $m \geq n$ and $\lim_{n \to \infty}S_n(t)=0$ uniformly on $I$. Thus the sequence $\{x_n(\cdot)\}_{n=1}^\infty$ is a Cauchy sequence and hence $\lim_{n \to \infty}x_n(t)=x(t)$ uniformly on $I$. Consequently $ f(t,x_n(t))\to f(t,x(t))$ for a.a. $t\in I$. Furthermore, $ D(f(t,x_n(t)),\hat 0)\leq \chi_n(t)\leq \chi_1(t)$. Due to dominated convergence theorem we get \begin{equation} \label{4e1} x(t)=x_0+\int_0^t f(s,x(s))ds. \end{equation} The proof is complete thanks to Lemma \ref{l4.4} given below. \end{proof} \begin{lemma}\label{l4.4} If $f: I \times \mathbb{E} \to \mathbb{E}$ is almost continuous and integrally bounded then every solution of \eqref{4e} is a solution of \eqref{4e1} and vice versa. \end{lemma} \begin{proof} The space $\mathbb{E}$ can be embedded as a closed convex cone in a Banach space $\mathbb{X}$. The embedding map $j: \mathbb{E} \to \mathbb{X}$ is an isometry and isomorphism. From (cf\cite{DK}) we know that $ j(\dot x(t))= \frac{d}{dt}j(x(t))$. The fact that every solution of \eqref{4e1} is at the same time a solution of \eqref{4e} is tautology because $ \int _0^t \dot x(s)ds =\int_0^t f(s, x(s))ds$. Let $x:I \to \mathbb{E}$ be a solution of \eqref{4e1}. Since $x:I \to \mathbb{E}$ is continuous, therefore $f: I \times \mathbb{E} \to \mathbb{E}$ satisfies Lusin property and hence $$ g(t) = \frac{d}{dt}\Big(\int_0^t g(s)ds \Big) $$ for a.a. $t \in I$. i.e. $\dot x(t) = g(t) = f(t, x(t))$. Evidently, $x:I \to \mathbb{E}$ is AC, i.e. $x(\cdot)$ is a solution of \eqref{4e}. \end{proof} \begin{remark} \label{r4.1} \rm Let us consider the equation \begin{equation}\label{4.2} \dot x_n = f(t,x_n(t))+\varphi_n(t),\ x_n(0)=x_0. \end{equation} If $\{\varphi_n(\cdot)\}_{n=1}^\infty$ is integrally bounded and $\lim_{n \to \infty} \varphi_n(t)=0$, then $\lim_{n \to \infty} x_n(t)=x(t)$, where $\dot x(t) = f(t,x(t)),\ x(0)=x_0.$ Therefore the solution of \eqref{4.2} depends continuously on the right-hand side. \end{remark} \section{Compact perturbations of dissipative fuzzy system} In this section we prove the existence of solution of the differential equation \eqref{4.1}. We will use the additional hypotheses: \begin{itemize} \item[(F3)] $W(x+z,y+z) = W(x,y)$ for any fuzzy number $z$. \item[(G1)] $g(t,\cdot)$ maps the bounded subsets of $\mathbb{E}$ into relatively compact subsets of $\mathbb{E}$ for a.a. $t\in I$. \item[(G2)] $D(g(t,x),\hat 0) \le \nu(t)(1 + D(x,\hat 0))$, where $\nu(\cdot) \in L_1(I,\mathbb{R}^+)$. \end{itemize} Condition (F3) is essential here. Notice that it holds automatically if $W(x,y) = \zeta (D(x,y))$, where $\zeta$ is some continuous function such that $W(x,y)$ is Lyapunov-like function for \eqref{4e}. If $x(\cdot)$ is a solution of \eqref{4.1} then $D(\dot x(t), \hat 0) \leq \big(\lambda(t) + \nu(t)\big) ( 1+D(x(t) ,\hat 0))$. Therefore, $$ D(x(t), \hat 0) \leq D(x_0, \hat 0) + e^{\int _0^t(\lambda(s) + \nu(s))ds} \Big(D(x_0, \hat 0)+\int_0^t [\lambda(s) + \nu(s)]ds\Big). $$ We can assume without loss of generality that $D(x(t), \hat 0) \le N$ and $D(\dot x(t), \hat 0) \leq \gamma(t)$, where $\gamma(t) = (\lambda(t) + \nu(t))(1+N)$ is Lebesgue integrable. Let $A = \{y \in \mathbb{E}:D(y,x_0) \leq N\}$. It follows from (G1) that $g(t,A)\subset K(t)$, where $K(t)\subset \mathbb{E}$ is a convex compact set for a.a. $t\in I$. \begin{theorem}\label{th4.3} Let {\rm (F1), (F2), (F3), (G1), (G2)} hold, then the differential equation \eqref{4.1} admits a solution. \end{theorem} We need the following lemma for proving Theorem \ref{th4.3}. \begin{lemma} \label{l4.5} Let $\{\varphi_n(\cdot)\}_{n=1}^\infty$ be an integrally bounded (by an integrable function $c(\cdot)$) sequence of strongly measurable functions from $I$ to $\mathbb{E}$ such that \[ \overline{\rm co} \big\{ \cup_{i=1}^\infty \{ \varphi_i(t)\}\big\}= K(t) \] is compact for a.a. $t\in I$ and \begin{equation}\label{4*} \dot x_n(t) = f(t,x_n(t))+\varphi_n(t),\ x_n(0)=x_0. \end{equation} Passing to subsequence, if necessarily, $x_n(\cdot)$ converges uniformly to $x(\cdot)$, such that $$ \dot x(t) \in f(t,x(t))+ K(t). $$ \end{lemma} \begin{proof} Clearly $D(\varphi_n(t), \hat 0)\leq c(t)$ implies that $ z_n(t) = \int_0^t \varphi_n(s)ds$ is equicontinuous sequence. Furthermore, \[ \int_0^t \big[ \cup_{n=1}^\infty \varphi_n(s)\big]ds \subset \int_0^t K(s)ds= R(t), \] where $ \overline{\cup_{t\in [0,1]}\{R(t)\}}$ is a compact subset of $\mathbb{E}$. Then the sequence $ z_n(t) = \int_0^t \varphi_n(s)ds$ is $C(I,\mathbb{E})$ precompact. By Arzela Ascoli theorem, passing to subsequence we have $z_n(t) \to z(t)$ uniformly on $I$. As we pointed out, $\mathbb{E}$ can be embedded as a closed convex cone in a Banach space $\mathbb{X}$ with a continuous embedding map $j: \mathbb{E} \to \mathbb{X}$. Thus $j(K)\subset \mathbb{X}$ is compact. Then due to Diestel criterion (see proposition 9.4 of \cite{D}) the set $\{j(\varphi_n(\cdot))\}_{n=1}^\infty$ is weakly precompact in $L_1(I, \mathbb{X})$. Thus passing to subsequence in $L_1(I, \mathbb{X})$ we have $j(\varphi_n(t)) \rightharpoonup s(t)$. Since $s(t) \in j(K)$, then there exists $\varphi(t)$ such that $j(\varphi(t))=s(t)$ and $ z(t)= \int_0^t \varphi(s)ds$. We denote for convenience $y(t) = j(x(t))$, $y_n(t) = j(x_n(t))$, $p(t) = j(z(t))$, $\psi(t) = j(\varphi(t))$, $y(t)-p(t)= u(t)$, $y_n(t)- p_n(t)=u_n(t)$ and $q(t,y) = j(f(t,x))$. Consider the functions $y_n(t) - p_n(t)=u_n(t)$. We have \begin{align*} &W(u(t)+h\dot u(t),u_n(t)+h\dot u_n(t)) \\ &= W(u(t)+ h q(t,y(t)), u_n(t)+ h q(t,y_n(t)))+o(h)\\ &= W(u(t)+ p_n(t)+h q(t,y(t)),y_n(t)+h q(t,y_n(t)))+o(h) \\ &= W(u(t)+p_n(t)+h q(t,y(t)-p(t)+p_n(t)),y_n(t)+h q(t,y_n(t)))\\ &\quad + h |q(t,y(t))-q(t,y(t)-p(t)+p_n(t))| + o(h). \end{align*} Consequently, \begin{align*} &\lim_{h \to 0^ + } \frac {W(u(t)+h(\dot u(t), u_n(t)+ h\dot u_n(t)) - W(u(t),u_n(t))}{h}\\ &= \lim_{h \to 0^+ } \frac{W(u(t+h), u_n(t+h))- W(u(t), u_n(t))}{h}\\ &= \lim_{h \to 0^ + } \frac{W(u(t)+h\dot u(t), u_n(t)+ h\dot u_n(t))-W(u(t),u_n(t))}{h}\\ &\leq v(t,|u(t)- u_n(t)|)+ |q(t,y(t)-p(t)+p_n(t))-q(t,y(t))|. \end{align*} Thus \begin{align*} D^+ W(y(t)-p(t),y_n(t)-p_n(t))\leq v(t,|y(t)-p(t)-(y_n(t)-p_n(t))|)\\ + |q(t,y(t))-q(t,y(t)-p(t)+p_n(t))|. \end{align*} The latter implies that $$ W(y(t)-p(t),y_n(t)-p_n(t)) \leq r_n(t), $$ where $$ \dot r_n(t) = v(t,r_n(t))+|q(t,y(t)-p(t)+p_n(t))-q(t,y(t))|,\ r_n(0)=0. $$ Since $v(\cdot,\cdot)$ is Kamke function and since $$ \lim _{n \to \infty} |q(t,y(t)-p(t)+p_n(t))-q(t,y(t))|=0 \text{ for a.a } t \in I, $$ one has that $\lim _{n \to \infty} r_n(t)=0$, which implies that $\lim _{n \to \infty} W(y(t)-p(t),y_n(t)-p_n(t))=0$. Thus $y_n(t) \to y(t)$ uniformly on $I$, where $\dot y(t)= q(t,y(t))+\psi(t)$, i.e $\dot x(t)= f(t,x(t))+\varphi(t)$. \end{proof} \begin{proof}[Proof of Theorem \ref{th4.3}] Consider the set $$ Q=\{ z(\cdot) \in C(I,K):\ D(\dot z(t), \hat 0) \leq \gamma(t),\ z(0)=x_0\}. $$ It is easy to see that $Q \subset C(I, \mathbb{E})$ is closed, bounded and convex. Consider the map $\xi: z(\cdot) \to x_z(\cdot)$, where $x_z(\cdot)$ is the unique solution of $$ \dot x_z(t)= f(t,x_z(t))+g(t,z(t));\ x_z(0)=x_0,\, t \in I. $$ Due to Remark \ref{r4.1} the map $\xi: Q\to Q$ is continuous. Furthermore, $ \overline{\xi(Q)} \subset Q$ is compact by Lemma \ref{l4.5}. It follows from Schauder's theorem that there exist a fixed point $z(\cdot) \in Q$ such that $\xi(z)=z$. This function $z(\cdot)$ is a solution of \eqref{4.1}. \end{proof} Notice that the linear growth conditions \textbf{(F1), (G2)} can be relaxed in order to prove only local existence, i.e. we can assume that $f:I \times \mathbb{E} \to \mathbb{E}$ is integrally bounded on the bounded sets. In that case, Theorem \ref{th1} is formulated as follows. \begin{theorem} \label{th4} Let $f:I \times \mathbb{E} \to \mathbb{E}$ be integrally bounded on the bounded sets. Then under (F2) there exists $a>0$ such that the system \eqref{4e} admits unique solution on $[0,a]$. \end{theorem} \begin{proof} Let $M>0$. There exists an integrable function $\zeta: I \to \mathbb{R}^+$ with $$ \sup_{|x-x_0|\leq M}|f(t,x)|\leq \zeta(t). $$ Let $a>0$ be such that $ \int_0^a (\zeta(t)+ \varepsilon)dt\leq M$. On the interval $[0,a]$ every $\delta$ solution $x_\delta(t)$ satisfies $ |x_\delta(t)|\leq M$ and $|\dot x_\delta(t)|\leq \zeta(t)+ \varepsilon$. Therefore, one can continue as in the proof of Theorem \ref{th1}. \end{proof} Theorem \ref{th4.3} can be obviously formulated as: \begin{theorem}\label{th5} Let $f:I \times \mathbb{E} \to \mathbb{E}$ and $g:I \times \mathbb{E} \to \mathbb{E}$ be integrally bounded on the bounded set. Then under {\rm (F2), (F3), (G1)} there exists $a>0$ such that the system \eqref{4.1} admits a solution on $[0,a]$. \end{theorem} \begin{proof} As in the proof of Theorem \ref{th4} we can see that there exists $a>0$ and $\varepsilon>0$ such that every $\varepsilon$-solution of \eqref{4.1} is extendable on $[0,a]$ and $|x_\varepsilon(t)-x_0|\leq M$. Let $g(t,x_0+ M \mathbb{B})\subset A(t)$, where $A(t)\subset \mathbb{E}$ is a convex compact set. It follows from Theorem \ref{th4} that for every strongly measurable $\varphi(t)\in A(t)$, the fuzzy differential equation $$ \dot x(t) = f(t,x(t))+\varphi(t),\quad x(0)=x_0 $$ admits unique solution on $[0,a]$. One can then continue as in the proof of Theorem \ref{th4.3}, proving of course the corresponding variant of Lemma \ref{l4.3}. \end{proof} \section{Conclusion} As it is pointed out in the introduction the space $\mathbb{E}$ is not locally compact. This implies that it would be very difficult (if it is possible at all) to prove analogue of the classical Peano theorem, when the right-hand side of \eqref{4e} is only jointly continuous. On the other hand up to author's knowledge there is no example of such a system without solutions. In authors opinion it is very interesting open question to give an example of fuzzy differential equation without local solution, when the right-hand side is jointly continuous. In optimal control problems the controls are measurable functions and it is one of the main motivation to study differential equations with almost continuous right-hand sides. In this paper we proved existence (and uniqueness) of the solution of \eqref{4e} under as weak as it is possible dissipative-type condition w.r.t. Lyapunov-like function. We also show the existence of solution when the right-hand side is the sum of a function satisfying such condition along with almost continuous function mapping bounded sets into relatively compact ones. For example such function is $g(t,\cdot)$ which takes values in a locally compact set $\mathbb{E}_K\subset\mathbb{E}$. It seems that it is impossible to relax compactness-type assumptions on $g$ without using stronger dissipative-type conditions on $f$. We refer the reader to the paper \cite{A}, where it is shown by example that if $v(\cdot,\cdot)$ is a Kamke function, then it is possible that the function $w(t,r)= v(t,r)+L(t)r$ is not a Kamke function. Of course in our proof we essentially used (F3), which is in general not valid for arbitrary Lyapunov-like function. It is an open question does the solution exists, when the last condition is dispensed with? Now we give a simple example of fuzzy system which satisfies our conditions. \begin{example}\label{ex1} \rm Consider the system of crisp first equation and fuzzy second: \begin{align*} \dot x= - \sqrt[3]{x} + f(t,x,y),\quad x(0)=0 \\ \dot y(t)= g(t,x,y),\quad y(0)= y_0. \end{align*} Here $x$ is crisp variable, $f:I\times \mathbb{R}\times \mathbb{E}\to \mathbb{R}$ is continuous and Lipschitzian on $x$ and on $y$. Furthermore $g:I\times \mathbb{R}\times \mathbb{E}\to \mathbb{E}$ is continuous, Lipschitzian on $x$ and takes values in a locally compact subset of $\mathbb{E}$. If some growth condition holds, than the system satisfies all the conditions of Theorem \ref{th4.3}. \end{example} \subsection*{Acknowledgments} The first author is supported by a grant from the Romanian National Authority for Scientific Research, CNCS--UEFISCDI, project number PN-II-ID-PCE-2011-3-0154. The second author is partially supported by Higher Education Commission, Pakistan. \begin{thebibliography}{l1} \bibitem{A} Augustinowicz, A.; Some remarks on comparison functions, \emph{Ann. Polon. Marg.} \textbf{96} (2009) 97--106. \bibitem{D} Deimling, K.; \emph{Multivalued Differential Equations}, De Gruyter, Berlin, (1992). \bibitem{DK} Diamond, P.; Kloeden, P.; Characterization of compact subsets of fuzzy sets, \emph{Fuzzy Sets and Systems} \textbf{29} (1989) 341--348. \bibitem{GS} Gong, Z.; Shao, Y.; Global existence and uniqueness of solutions for fuzzy differential equations under dissipative-type conditions, \emph{Comput. Math. Appl.} \textbf{56} (2008), 2716--2723. \bibitem{K} Kaleva, O.; Fuzzy differential equations, \emph{Fuzzy Sets and Systems} \textbf{24} (1987), 301--317. \bibitem{KB} Kandel, A.; Byatt, W.; Fuzzy differential equations, \emph{Proc. Int. Conf. Cybern. and Society}, Tokyo, Nov. (1978), 1213--1216. \bibitem{LL} Lakshmikantham, V.; Leela, S.; \emph{Nonlinear Differential Equations in Abstract Spaces}, Pergamon Press, New York, (1981). \bibitem{LM} Lakshmikantham, V.; Mohapatra, R; \emph{Theory of Fuzzy Differential Equations and Inclusions}, Taylor - Francis, London, (2003). \bibitem{LT} Loeb, P.; Talnila, E.; Lusin's theorem and Bochner integration, \emph{Scientiae Mathematicae Japonicae Online} \textbf{10} (2004), 55-62. \bibitem{Lu} Lupulescu, V.; Initial value problem for fuzzy differential equations under dissipative conditions, \emph{Information Sciences} \textbf{178} (2008), 4523--4533. \bibitem{PH} Park, J.; Han, H.; Existence and uniqueness theorem for a solution of fuzzy differential equation, \emph{Intern. J. Math. Math. Sci.} \textbf{22} (1999), 271--279. \bibitem{PS} Plotnikov, A.; Skripnik, N.; \emph{Differential Equations with Clear and Fuzzy Multivalued Right--hand Side, Assymptotical Methods}, Astropoint, Odessa, (2009) (in Russian). \bibitem{SW} Song, S.; Wu, C.; Xue, X.; Existence and uniqueness of Cauchy problems for fuzzy differential equations under fuzzy dissipative conditions, \emph{Comput. Math. Appl.} \textbf{51} (2006), 1483--1492. \bibitem{WS} Wu, C.; Song, S.; Existence theorem to Cauchy problem for fuzzy differential equations under compactness-type conditions, \emph{Information Sciences} \textbf{108} (1998), 123-134. \end{thebibliography} \end{document}