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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 178, 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 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/178\hfil Integro-differential inclusions]
{Existence of solutions for some Hammerstein type
integro-differential inclusions}

\author[N. T. Hoai, N. V. Loi \hfil EJDE-2007/178\hfilneg]
{Nguyen Thi Hoai, Nguyen Van Loi}  % in alphabetical order

\address{Nguyen Thi Hoai \newline
Faculty of mathematics, Voronezh State Pedagogical University, Russia}
\email{nthoai0682@yahoo.com}

\address{Nguyen Van Loi \newline
Faculty of mathematics, Voronezh State Pedagogical University, Russia}
\email{loitroc@yahoo.com}


\thanks{Submitted December 27, 2006. Published December 17, 2007.}
\subjclass[2000]{47H04, 34A60, 47H10}
\keywords{Multivalued map; differential inclusion; fixed point}

\begin{abstract}
 In the present work we obtain existence results for Hammerstein
 type integro-differential inclusions in a finite dimensional
 space for the cases when the integral multifunction satisfies
 upper Caratheodory conditions and when it is almost lower
 semicontinuous.
\end{abstract}

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

\section{Introduction}

 Integral inclusions of the Hammerstein type have been studied in the
articles
\cite{ApZa1,ApZa2,BuLya,CaPa,Coff,GlaSpe,HuPa2,Loi,LoOb,Lyap,Oreg,Papa}
and others. In a finite-dimentional space the inclusion was been
studied in \cite{ApZa1,BuLya,GlaSpe,Loi,Lyap,Oreg}.  O'Regan
\cite{Oreg} investigated solvability of the inclusions in
$\mathbb{R}$;  Glashoff and  Sprekels \cite{GlaSpe} considered the
integral inclusions, arising in the theory of thermostats;
Bulgakov and  Lyapin \cite{BuLya} studied the properties of the
set of solutions of the inclusions of Vollterra-Hammerstein type.
The existence of solutions of the Hammerstein's integral
inclusions in $\mathbb{R}^n$ was established in \cite{ApZa1,Loi}.
For the inclusions in Banach space the problem of existence of
solutions was considered in \cite{CaPa,HuPa2,LoOb}.

In the present work, applying the fixed point principle of
Bohnenblust -  Karlin we give existence results for the
Hammerstein type integro-differential inclusion
\begin{equation}\label{eq1}
  u(t)\in\int_{a}^{b}K(t,s)F(s,u(s),u'(s))ds.
\end{equation}
in $\mathbb{R}^n$.

 Let $X,Y$  be normed spaces; $P(Y)\ [C(Y),
K(Y), Cv(Y), Kv(Y)]$ denote the collections of all nonempty
[respectively, nonempty: closed, compact, closed convex and
compact convex] subsets of $Y$. Recall (see, e.g.
\cite{BGMO,HuPa1,KOZ}) that, a multimap $F:\ X\to{P(Y)}$ is
said to be upper semicontinuous (u.s.c.) [lower semicontinuous
(l.s.c.)] if the set
$F_{+}^{-1}(V)=\{x\in{X}\mid{F(x)\subset{V}}\}$ is open [closed]
for every open [respectively, closed] subset $V\subset{Y}$. A
multimap $F$ is said to be compact if the set $F(X)$ is relatively
compact in $Y$.

Let $C([a,b],\mathbb{R}^n)\ [C^{1}([a,b],\mathbb{R}^n),
L^1([a,b],\mathbb{R}^n)]$ denote the collections of all continuous
[respectively, continuously differentiable, integrable] functions
on $[a,b]$ with values in $\mathbb{R}^n$.

Let $F: [a,b]\times\mathbb{R}^n\times\mathbb{R}^n\to{Kv{(\mathbb{R}^n)}}$
 be a multimap, satisfying the following assumptions:
\begin{itemize}
\item[(F1)] For every $x\in\mathbb{R}^n\times\mathbb{R}^n$ the
multifunction $F(\cdot,x): [a,b]\to{Kv({\mathbb{R}^n})}$ has
a measurable selection; i.e., there exists a measurable function
$f(\cdot)\in L^{1}([a,b],\mathbb{R}^{n})$ such that
$f(t)\in{F(t,x)}$ for a.e. $t\in{[a,b]}$;

\item[(F2)] For a.e. $t\in{[a,b]}$ the multimap
$F(t,\cdot):\mathbb{R}^n\times\mathbb{R}^n\to{Kv(\mathbb{R}^n)}$
is u.s.c.;

\item [(F3)]  For every bounded subset
$\Omega\subset\mathbb{R}^n\times\mathbb{R}^n$ there exists a positive
function $\vartheta_{\Omega}(\cdot)\in{L^1([a,b],\mathbb{R})}$ such that
$$
\|F(t,x)\|_{\mathbb{R}^n}\leq\vartheta_{\Omega}(t),
$$
for all $x\in\Omega$ and a.e. $t\in{[a,b]}$, where
$\|F(t,x)\|_{\mathbb{R}^n}=
\max\{\|y\|_{\mathbb{R}^n}:y\in{F(t,x)}\}$.

\end{itemize}
It is known (see, e.g. \cite{BGMO}) that under these conditions the
superposition multioperator
\begin{gather*}
\wp_{F}: C([a,b],\mathbb{R}^n\times\mathbb{R}^n)\to{Cv(L^{1}([a,b],
\mathbb{R}^n))},\\
\wp_{F}(u)=\{f\in{L^{1}([a,b],\mathbb{R}^n): f(s)\in{F(s,u(s))},
\text{ for a.e. }s\in{[a,b]}}\},
\end{gather*}
is well defined and closed; i.e., it has a closed graph.

For every function $u\in{C^{1}([a,b],\mathbb{R}^n)}$ the function
\begin{gather*}
v: [a,b]\to{\mathbb{R}^n\times\mathbb{R}^n},\\
v(s)=(u(s),u'(s)),
\end{gather*}
is continuous. And hence the multioperator
\begin{gather*}
\wp_{F}^{1}: C^{1}([a,b],\mathbb{R}^n)\to{Cv(L^{1}([a,b],\mathbb{R}^n))},\\
\wp_{F}^{1}(u)=\wp_{F}(v),
\end{gather*}
is defined and closed.
Consider the linear operator
\begin{gather*}
A:L^{1}([a,b],\mathbb{R}^n)\to{C^{1}([a,b],\mathbb{R}^n)},\\
A(f)(t)=\int_{a}^{b}K(t,s)f(s)ds,
\end{gather*}
where $K:\ [a,b]\times{[a,b]}\to{L(\mathbb{R}^n)}$ and
$L(\mathbb{R}^n)$ denotes the collection of all linear operators
in $\mathbb{R}^n$. The following statement can be easily verified.

\begin{theorem}\label{th1}
Let the kernel $K: [a,b]\times{[a,b]}\to{L(\mathbb{R}^n)}$
satisfy the following assumptions:
\begin{itemize}
\item[(K1)]  the function
$K(\cdot,s)x:\ [a,b]\to\mathbb{R}^n$ is differentiable on
$[a,b]$ for all $x\in\mathbb{R}^n$ and a.e. $s\in{[a,b]}$; i.e.,
there exists $K_{t}'(t,s)\in{L(\mathbb{R}^n)}$ such that:
$$
\lim_{\Delta{t}\to{0}}\frac{K(t+\Delta{t},s)x-
K(t,s)x}{\Delta{t}}=K_{t}'(t,s)x,
$$
for all $t\in{[a,b]}, x\in\mathbb{R}^n$ and a.e. $s\in{[a,b]}$;

\item[(K2)] there exists $K>0$ such that
$$
\|K(t,s)\|_{L}\leq{K},\quad
\|K_{t}'(t,s)\|_{L}\leq{K},\quad
\big\|\frac{K(t+\Delta t,s)-K(t,s)}{\Delta t}\big\|\leq K,
$$
for all $t,t+\Delta t \in{[a,b]}$ and a.e. $s\in{[a,b]}$;

\item[(K3)] for every $t\in{[a,b]}$ the functions
$s\mapsto{K(t,s)x}$ and $s\mapsto{K_{t}'(t,s)x}$ are integrable
for all $x\in\mathbb{R}^n$;

\item[(K4)] there exist a positive function
$\omega(\cdot)\in{L^{1}([a,b],\mathbb{R})}$ and a function
$\eta(\cdot)\in C([a,b],\mathbb{R})$ such that
$$
\|K_{t}'(t_2,s)-K_{t}'(t_1,s)\|_{L}\leq\omega(s)|\eta(t_2) - \eta(t_1)|,
$$
for all $t_{1},t_{2}\in{[a,b]}$ and a.e. $s\in{[a,b]}$.

\end{itemize}
Then the operator $A$ is completely continuous.
\end{theorem}

Following \cite[Theorem 1.5.30]{BGMO} we obtain the following result.

\begin{theorem}\label{th2}
Let multimap $F: [a,b]\times\mathbb{R}^n\times\mathbb{R}^n
\to{Kv{(\mathbb{R}^n)}}$
satisfy the assumptions {\rm (F1)--(F3)} and the operator $A$ satisfy
conditions {\rm (K1)--(K4)}. Then the multioperator
$A\circ\wp_{F}^{1}$ is closed.
\end{theorem}

Consider the integral multioperator
\begin{gather*}
\Gamma=A\circ\wp_{F}^{1}:\ C^{1}([a,b],\mathbb{R}^n)
\to{Kv(C^{1}([a,b],\mathbb{R}^n))},\\
\Gamma(u)(t)=\int_{a}^{b}K(t,s)F(s,u(s),u'(s))ds.
\end{gather*}
Applying \cite[Theorem 1.2.48]{BGMO}, Theorem \ref{th1} and
Theorem \ref{th2} we obtain the following theorem.

\begin{theorem}\label{th3}
Let the conditions {\rm (K1)--(K4)} and {\rm (F1)--(F3)}
hold. Then multioperator $\Gamma$ is u.s.c. and the restriction of
$\Gamma$ to any bounded subset
$\Omega\subset{C^{1}([a,b],\mathbb{R}^n)}$ is compact; i.e., the
set $\Gamma(\Omega)$ is relatively compact.
\end{theorem}

Consider now the multioperator $\Gamma$ when the multimap
$F:{[a,b]}\times\mathbb{R}^n\times\mathbb{R}^n\to{K(\mathbb{R}^n)}$
is almost lower semicontinuous (a.l.s.c.). Recall
(see, e.g. \cite{BGMO,KOZ}) that $F$ is said to be an a.l.s.c. multimap
if there exists a sequence of disjoint
compact sets $\{I_m\}, {I_m}\subset{{[a,b]}}$ such that:
\begin{itemize}
\item[(i)]  meas$([a,b]\setminus\bigcup_{m}I_{m})=0$;
\item[(ii)] the restriction of $F$ on each set
${J_m}={I_m}\times{\mathbb{R}^n\times\mathbb{R}^n}$ is l.s.c.
\end{itemize}
We also assume that $F$ satisfies the condition of boundedness
(F3). Then the superposition multioperator
$$
\wp_{F}^{1}:
C^{1}([a,b];\mathbb{R}^n)\to{C(L^{1}([a,b];\mathbb{R}^n))}
$$
is l.s.c. (see \cite{BGMO,Deim,KOZ}).
Consider again the multioperator
$$
A\circ\wp_{F}^{1}:
C^{1}([a,b];\mathbb{R}^n)\to{P(C^{1}([a,b];\mathbb{R}^n))},
$$
where the operator $A$ is given by the above conditions
{\rm (K1)--(K4)}. From \cite[Theorem 1.3.11]{BGMO} it follows
easily that the multioperator $\Gamma=A\circ\wp_{F}^{1}$ is l.s.c.
The following statement can be easily verified.

\begin{theorem}\label{th4}
Let {\rm (F1), (F3)} and {\rm (K1)--(K4)} hold. Then for any bounded subset
$\Omega\subset{C^{1}([a,b];\mathbb{R}^n)}$ the set $\Gamma(\Omega)$
is relatively compact.
\end{theorem}

 Let $E$ be a Banach space. A nonempty subset
$M\subset{L^{1}([a,b];E)}$ is said to be decomposable provided for
every $f,g\in{M}$ and each Lebesgue measurable subset
$m\subset{[a,b]}$,
$$
{f\cdot{k_m}+h\cdot{k_{([a,b]\setminus{m})}}}\in{M},
$$
where $k_{m}$ is the characteristic function of the set $m$ (see,
e.g. \cite{BGMO,Frys,KOZ} for further details).

\begin{theorem}[\cite{Frys}] \label{th5}
Let $X$ be a separable metric space and $E$ a Banach space. Then every
 l.s.c. multimap
$G: X\to{P(L^{1}([a,b];E))}$ with closed decomposable values
 has a continuous selection; i.e., there exists a continuous map
$g: X\to{L^{1}([a,b];E)}$ such that
$g(x)\in{G(x)}$ for all $x\in{X}$.
\end{theorem}

It is clear that for every $u\in{C^{1}([a,b];\mathbb{R}^n)}$, the
set $\wp_{F}^{1}(u)$ is closed and decomposable. Then the
multioperator $\wp_{F}^{1}$ has a continuous selection
$$
\ell: C^{1}([a,b];\mathbb{R}^n)\to{L^{1}([a,b];\mathbb{R}^n)},\quad
\ell(u)\in{\wp_{F}^{1}(u)}.
$$
Therefore, the continuous operator
$\gamma:C^{1}([a,b];\mathbb{R}^n)\to{C^{1}([a,b];\mathbb{R}^n)}$,
$$
\gamma(u)(t)=\int_{a}^{b}K(t,s)\ell(u)(s)ds
$$
is a continuous selection for the multioperator $\Gamma$. By
virtue of Theorem \ref{th4}, the operator $\gamma$ is completely continuous
and its fixed points are also fixed points of the multioperator $\Gamma$.

\section{Main results}

In this section, we give some existence results of solutions of the
inclusion \eqref{eq1}.

\begin{theorem} \label{th6}
Let the conditions {\rm (K1)--(K4)} and {\rm (F1)--(F2)} hold. Assume that:
\begin{itemize}
\item[(F3')]  there exists a positive function
$\omega\in{L^{1}([a,b],\mathbb{R})}$ such that
$$
\|F(t,x,y)\|_{\mathbb{R}^n}\leq\omega(t)(1+\|x\|_{\mathbb{R}^n}
+\|y\|_{\mathbb{R}^n}),
$$
for all $x,y\in\mathbb{R}^n$ and a.e. $t\in{[a,b]}$;

\item[(F4)] $2K\int_{a}^{b}\omega(t)dt<1$, where $K$ is the
 constant from the condition {\rm (K2)}.

\end{itemize}
Then the inclusion \eqref{eq1} has at least one solution.
\end{theorem}

\begin{proof}
 It is easy to see that from (F3') we obtain (F3).
Consider the multioperator $\Gamma$ on the ball
$T=T({\|u\|}_{C^{1}}\leq{\rho})$. We have
$$
\|\Gamma(u)\|_{C^{1}}=\max\big\{\big\|\int_{a}^{b}K(t,s)f(s)ds
\big\|_{C^{1}}: f\in\wp_{F}^{1}(u)\big\},
$$
where
\begin{align*}
\big\|\int_{a}^{b}K(t,s)f(s)ds\big\|_{C^{1}}
&=\max\big\{\big\|\int_{a}^{b}K(t,s)f(s)ds\big\|_{\mathbb{R}^n}:
t\in{[a,b]}\big\}\\
&\quad +\max\big\{\big\|\int_{a}^{b}K_{t}'(t,s)f(s)ds\big\|_{\mathbb{R}^n}:
t\in{[a,b]}\big\}.
\end{align*}
It is clear that
\[
\big\|\int_{a}^{b}K(t,s)f(s)ds\big\|_{\mathbb{R}^n}\leq
\int_{a}^{b}\|K(t,s)\|_{L}\|f(s)\|_{\mathbb{R}^n}ds
 \leq K\int_{a}^{b}\|f(s)\|_{\mathbb{R}^n}ds,
\]
and
\[
\big\|\int_{a}^{b}K_{t}'(t,s)f(s)ds\big\|_{\mathbb{R}^n}\leq
\int_{a}^{b}\|K_{t}'(t,s)\|_{L}\|f(s)\|_{\mathbb{R}^n}ds
 \leq K\int_{a}^{b}\|f(s)\|_{\mathbb{R}^n}ds.
\]
Since
$f(s)\in{F(s,u(s),u'(s))}$ for a.e. $s\in{[a,b]}$ we have
\begin{align*}
\|f(s)\|_{\mathbb{R}^n}&\leq\|F(s,u(s),u'(s))\|_{\mathbb{R}^n}\\
& \leq\omega(s)(1+\|u(s)\|_{\mathbb{R}^n}+\|u'(s)\|_{\mathbb{R}^n})\\
& \leq\omega(s)(1+\|u\|_{C^{1}})\\
& \leq\omega(s)(1+\rho),
\end{align*}
for a.e. $s\in{[a,b]}$.
Consequently,
$$
K\int_{a}^{b}\|f(s)\|_{\mathbb{R}^n}\:ds\leq{K(1+\rho)
\int_{a}^{b}\omega(s)ds}.
$$
And hence
$$
\big\|\int_{a}^{b}K(t,s)f(s)ds\big\|_{C^{1}}\leq
2K(1+\rho)\int_{a}^{b}\omega(s)ds.
$$
The last inequality is true for all $f\in\wp_{F}^{1}(u)$, and so we obtain
$$
\|\Gamma(u)\|_{C^{1}}\leq{2K(1+\rho)\int_{a}^{b}\omega(s)ds}.
$$
Choose $\rho$ so that
\[
\rho\geq\frac{2K{\int_{a}^{b}\omega(s)ds}}{1-2K\int_{a}^{b}\omega(s)ds}.
\]
Then $\|\Gamma(u)\|_{C^{1}}\leq{\rho}$. Consider the upper
semicontinuous multioperator $\Gamma: T\to{Kv(T)}$.
 By  Theorem \ref{th3}, the multioperator $\Gamma$ is
compact. From the Bohnenblust-Karlin Theorem (see, e.g.
\cite{BGMO}) it follows that the multioperator $\Gamma$ has at
least one fixed point $u^{*}\in{T}$: $u^{*}\in\Gamma(u^{*})$. The
function $u^{*}$ is a solution of the inclusion \eqref{eq1}.
\end{proof}

\begin{theorem}\label{th7}
Let the conditions {\rm (K1)--(K4)} and $(F_L)$ hold. Assume that
there exist two numbers
$\lambda,\beta\in\mathbb{R};\beta>0$ and a positive function
$\omega\in{L^{1}([a,b],\mathbb{R})}$ such that:
\begin{itemize}
\item[(F3'')] $\|F(t,x,y)-\lambda{(x+y)}\|_{\mathbb{R}^n}\leq
\beta(\|x\|_{\mathbb{R}^n}+\|y\|_{\mathbb{R}^n})+\omega(t)$,
for all $x,y\in\mathbb{R}^n$ and a.e. $t\in{[a,b]}$;

\item[(F5)] $2K(b-a)(\beta+|\lambda|)<1$, where $K$ is the constant from
(K2).

\end{itemize}
Then inclusion \eqref{eq1} has at least one solution.
\end{theorem}

For the proof we need the following result (see, e.g. \cite{Kras}).

\begin{lemma}\label{lm1}
Let $A$ be nonlinear and  $B$ be linear completely continuous operators
in a Banach space $E$. If on the sphere $S=S(\|x\|=\rho)$ the following
inequality holds
$$
\|Ax-Bx\|<\|x-Bx\|.
$$
Then the equation $x=Ax$ in the ball $T(\|x\|\leq{\rho})$ has at least
one solution.
\end{lemma}

\begin{proof}[Proof of Theorem \ref{th7}]
 It is easy to see that from (F3'') we have
$$
\|F(t,x,y)\|_{\mathbb{R}^n}\leq(\beta+|\lambda|)(\|x\|_{\mathbb{R}^n}+
\|y\|_{\mathbb{R}^n})+\omega(t),
$$
for all $x,y\in\mathbb{R}^n$
and a.e. $t\in{[a,b]}$. And hence we
obtain (F3).
Consider a linear operator
$B:{C^{1}([a,b];\mathbb{R}^n)}\to{C^{1}([a,b];\mathbb{R}^n)}$,
$$
Bu(t)=\lambda\int_{a}^{b}K(t,s)(u(s)+u'(s))ds.
$$
It is clear that $B$ is completely continuous.
Consider the multioperator $\Gamma$ and the operator $B$ on
$S=S({\|u\|}_{C^{1}}=\rho)$. For each function $u\in{S}$ we have
$$
\|\Gamma{u}-B{u}\|_{C^{1}}=\sup\big\{\big\|\int_{a}^{b}K(t,s)[f(s)-\lambda
(u(s)+u'(s))]ds\big\|_{C^{1}}:f\in\wp_{F}(u)\big\}.
$$
On the other hand
\begin{align*}
&\big\|\int_{a}^{b}K(t,s)[f(s)-\lambda{(u(s)+u'(s)}]ds\big\|_{C^{1}}\\
&=\max_{t\in{[a,b]}}\big\|\int_{a}^{b}K(t,s)[f(s)-\lambda(u(s)+u'(s))]ds
 \big\|_{\mathbb{R}^n}\\
&\quad +\max_{t\in{[a,b]}}\big\|\int_{a}^{b}K_{t}'(t,s)[f(s)
 -\lambda{(u(s)+u'(s))}]ds\big\|_{\mathbb{R}^n}\\
&\leq\max_{t\in{[a,b]}}\int_{a}^{b}\|K(t,s)\|_{L}
 \|f(s)-\lambda(u(s)+u'(s))\|_{\mathbb{R}^n}ds\\
&\quad +\max_{t\in{[a,b]}}\int_{a}^{b}\|K_{t}'(t,s)\|_{L}
 \|f(s)-\lambda(u(s)+u'(s))\|_{\mathbb{R}^n}ds\\
&\leq {2K\int_{a}^{b}\|f(s)-\lambda(u(s)+u'(s))\|_{\mathbb{R}^n}ds}.
\end{align*}
Since $f(s)\in{F(s,u(s),u'(s))}$ for a.e. $s\in{[a,b]}$ we have
\begin{align*}
\|f(s)-\lambda(u(s)+u'(s))\|_{\mathbb{R}^n}
&\leq \|F(s,u(s),u'(s))-\lambda(u(s)+u'(s))\|_{\mathbb{R}^n}\\
&\leq\beta(\|u(s)\|_{\mathbb{R}^n}+\|u'(s)\|_{\mathbb{R}^n})+\omega(s)\\
&\leq\beta\|u\|_{C^{1}}+\omega(s)=\beta\rho+\omega(s),\:
\end{align*}
for a.e. $s\in{[a,b]}$. Therefore,
\begin{align*}
\big\|\int_{a}^{b}K(t,s)[f(s)-\lambda(u(s)+u'(s))]ds\big\|_{C^{1}}
&\leq 2K\int_{a}^{b}(\beta\rho+\omega(s))ds \\
&\leq 2K\beta\rho(b-a)+2K\int_{a}^{b}\omega(s)ds.
\end{align*}
The above inequality holds for all $f\in\wp_{F}^{1}(u)$, and so we
obtain
$$
\|\Gamma{u}-Bu\|_{C^{1}}\leq{2K\beta\rho(b-a)+2K\int_{a}^{b}\omega(s)ds},
\quad\forall{u\in{S}}.
$$
On the other hand for each $t\in{[a,b]}$ we have
\begin{align*}
\|u(t)-B(u)(t)\|_{\mathbb{R}^n}
&=\big\|u(t)-\lambda\int_{a}^{b}K(t,s)(u(s)+u'(s))ds \big\|_{\mathbb{R}^n}\\
&\geq\|u(t)\|_{\mathbb{R}^n}-\|\lambda\int_{a}^{b}
 K(t,s)(u(s)+u'(s))ds\|_{\mathbb{R}^n}\\
&\geq\|u(t)\|_{\mathbb{R}^n}-|\lambda|\int_{a}^{b}\|K(t,s)\|_{L}
 \|u(s)+u'(s)\|_{\mathbb{R}^n}ds\\
&\geq\|u(t)\|_{\mathbb{R}^n}-|\lambda|\int_{a}^{b}K\|u\|_{C^{1}}ds\\
&\geq\|u(t)\|_{\mathbb{R}^n}-K\rho|\lambda|(b-a).
\end{align*}
Analogously,
$$
\|u'(t)-(B(u))'(t)\|_{\mathbb{R}^n}\geq\|u'(t)\|_{\mathbb{R}^n}
-K\rho|\lambda|(b-a).
$$
And hence we obtain
\begin{align*}
\|u-Bu\|_{C^{1}}
&=\max_{t\in{[a,b]}}\|u(t)-Bu(t)\|_{\mathbb{R}^n}+
\max_{t\in{[a,b]}}\|u'(t)-(Bu)'(t)\|_{\mathbb{R}^n}\\
&\geq\|u\|_{C^{1}}-2K\rho|\lambda|(b-a) \\
&=\rho(1-2K|\lambda|(b-a)).
\end{align*}
Choose $\rho$ so that
$\rho>{\frac{2K{\int_{a}^{b}\omega(s)ds}}{1-2K(b-a)(\beta+|\rho|)}}$.
Then $\|\Gamma{u}-Bu\|_{C^{1}}<\|u-Bu\|_{C^{1}}$, for all ${u\in{S}}$.
Let $\gamma$ be an arbitrary continuous selection of the multioperator
$\Gamma$. Then on the sphere $S$ we have
$$
\|\gamma{u}-Bu\|_{C^{1}}\leq\|\Gamma{u}-Bu\|_{C^{1}}<\|u-Bu\|_{C^{1}}\,.
$$
By  Lemma \ref{lm1}, the operator $\gamma$ has at least one fixed
point in the ball $T(\|u\|_{C^{1}}<\rho)$: $u_{*}=\gamma(u_{*})$.
The function $u_{*}$ is a solution of the inclusion \eqref{eq1}.
\end{proof}

\begin{theorem} \label{thm2.4}
Let the conditions {\rm (K1)--(K4), (F1), (F3'),  (F4)} hold.
Then  inclusion \eqref{eq1} has at least one solution.
\end{theorem}

\begin{proof}
 From the proof of Theorem \ref{th6} it follows that with the conditions
(F3') and (F4) we can choose a number $\rho >0$ such
that the multioperator $\Gamma$ maps the ball
$ T (\|u\|_{C^{1}}\leq\rho)$ into itself.
Let $\gamma$ be an arbitrary continuous
selection of the multioperator $\Gamma$. Then the operator $\gamma$
maps the ball $ T (\|u\|_{C^{1}}\leq\rho)$ into itself.
Consider the completely continuous operator
$\gamma:{T}\to{T}$.
From the Schauder fixed point theorem, the operator $\gamma$ has at
least one fixed point on $T$, i.e. there exists a function
$u_{*}\in{T}$ such that: $u_{*}=\gamma(u_{*})$.
The function $u_{*}$ is a solution of
the inclusion \eqref{eq1}.
\end{proof}

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\end{thebibliography}


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