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\begin{document}
\title[Darbo's Fixed Point Theorem via \ Function Sequences]{AN APPLICATION
OF FUNCTION SEQUENCES TO THE DARBO'S THEOREM WITH INTEGRAL TYPE
TRANSFORMATIONS}
\author[V. Karakaya]{Vatan Karakaya}
\address[V. Karakaya]{Department of Mathematical Engineering, Yildiz
Technical University, Davutpasa Campus, Esenler, 34210 Istanbul,Turkey}
\address{Department of Mathematics, Ahi Evran University, Ba\u{g}ba\c{s}\i\
Campus, 40100 K\i r\c{s}ehir,Turkey}
\email{\texttt{vkkaya@yahoo.com}}
\author[D. Sekman]{Derya Sekman}
\address[D. Sekman]{ Department of Mathematics, Ahi Evran University, Ba\u{g}%
ba\c{s}\i\ Campus, 40100 K\i r\c{s}ehir,Turkey}
\email{deryasekman@gmail.com}
\keywords{Fixed point, integral type mapping, measure of noncompactness,
function sequences, shifting distance property }
\subjclass [2010] {47H08, 47H10.}
\begin{abstract}
The purpose of this work is to apply function sequences to Darbo's theorem
with the integral type transformation by changing the roles of function
sequences with function classes used in fixed point theory and to examine
the existence of fixed points. Also, an interesting example will be shown.
\end{abstract}
\maketitle
\section{Introduction}
Fixed point theory has been extensively studied both in nonlinear analysis
and applied mathematics such as game theory, economy, medicine, biology,
physics and so on. The fixed point theory presents a powerful method to
guarantee the existence and uniqueness of the solutions of the problems in
these areas. The three most important fixed point theorems used in nonlinear
analysis are Schauder \cite{schauder}, Brouwer \cite{bro} and Darbo \cite%
{darbo} theorems. The Schauder's theorem generalizes the Brouwer's theorem
for finite dimensional spaces to infinite dimensional spaces. Beside,
Darbo's theorem holds effective a facility to the set value mapping for
noncompact operators. Recently, many researchers have used the measure of
noncompactness concept and the Darbo's fixed point theorem to solve the
integral equation classes as given in \cite{aga,huda,deryam}. Contraction
type transformations, which is the starting Banach Contraction Principle
\cite{banach} is the most useful method of fixed point theory. For many
years, many researchers have made a lot of generalizations of the Banach
Contraction Principle. Berzig \cite{berzig} introduced the concept of
shifting distance functions and established fixed point theorem which
generalized Banach contraction principle. Then, Samadi and Ghaemi \cite%
{samadi} presented some generalizations of Darbo's theorem associated with
measure of noncompactness by using the notion of shifting distance functions
and given an application of the integral equation of mixed type. Later, Cai
and Liang \cite{chai} obtained new generalizations of Darbo fixed point
theorem by using integral conditions and extended the existing results on
the problem. Recently, Karakaya et al. \cite{proje} introduced new concept
which is function sequences with shifting distance property and gave new
type theorem about it. In this paper, our aim is to introduce a
generalization of integral type of Darbo's fixed point theorem with the help
of function sequences having shifting distance property defined by Karakaya
et al. \cite{proje} and also to give some interesting example.\bigskip
\section{Preliminaries}
Let $C$ be a nonempty subset of a Banach space $X$. We define $\overline{C}$
and $Conv(C)$ the closure and closed convex hull of $C,$ respectively. Also,
we denote by $M_{X}$ which is the family of all nonempty bounded subsets of $%
X$ and $N_{X}$ that is subfamily consisting of all relatively compact
subsets of $X$. Throughout this work, we will show uniform convergence
according to $n$ in function sequences with the symbol "$\rightrightarrows $
". Also we denote $%
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
,$ $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
,$ $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}$ which are natural number, reel number and positive reel number,
respectively.
\bigskip
\begin{definition}[see \protect\cite{kuro}]
\label{darbo}A mapping $\mu :M_{X}\rightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}$ is called a measure of noncompactness if it satisfy the following
conditions:
$(M_{1})$ The family Ker $\mu =\left\{ E\in M_{X}:\mu (E)=0\right\} $ is
nonempty and Ker $\mu \subseteq N_{X},$
$(M_{2})$ $E\subseteq F\Rightarrow \mu (E)\leq \mu (F),$
$(M_{3})$ $\mu (\overline{E})=\mu (E)$, where $\overline{E}$ denotes the
closure of $E,$
$(M_{4})$ $\mu (conv$ $E)=$ $\mu (E)$,
$(M_{5})$ $\mu (\lambda E+(1-\lambda )F)\leq \lambda \mu (E)+(1-\lambda )\mu
(F)$ for $\lambda \in \left[ 0,1\right] ,$
$(M_{6})$ If $\left\{ E_{n}\right\} $ is a sequence of closed sets in $M_{X}$
such that $E_{n+1}\subseteq E_{n}$ and $\lim\limits_{n\rightarrow \infty
}\mu (E_{n})=0$, then the following intersection is nonempty%
\begin{equation*}
E_{\infty }=\underset{n=1}{\overset{\infty }{\cap }}E_{n},
\end{equation*}%
for $n=1,2,\cdots .$ If $(M_{4})$ holds, then $E_{\infty }\in $ Ker$\mu .$
Let $\lim\limits_{n\rightarrow \infty }\mu (E_{n})=0.$ As $E_{\infty
}\subseteq E_{n}$ for each $n=0,1,2,...;$ by the monotonicity of $\mu ,$ we
obtain
\begin{equation*}
\mu (E_{\infty })\leq \lim\limits_{n\rightarrow \infty }\mu (E_{n})=0.
\end{equation*}%
So, by $(M_{1})$, we get that $E_{\infty }$ is nonempty and$\ E_{\infty }\in
$ Ker$\mu $.
\end{definition}
\begin{theorem}[see \protect\cite{schauder}]
\label{sch}Let $C$ be a closed and convex subset of a Banach space $X$. Then
every compact, continuous map $T:C\rightarrow C$\ has at least one fixed
point.
\end{theorem}
\begin{theorem}[see \protect\cite{darbo}]
\label{darb}Let $C$ be a nonempty, bounded, closed and convex subset of a
Banach space $X$ and let\ $T:C\rightarrow C$ be a continuous mapping.
Suppose that there exists a constant $k\in \left[ 0,1\right) $ such that%
\begin{equation*}
\mu (T(E))\leq k\mu (E)
\end{equation*}%
for any subset $E$ of $C$, then $T$ has a fixed point.
\end{theorem}
\begin{definition}[see \protect\cite{berzig}]
\label{berzig}Let $\psi ,\phi :[0,\infty )\rightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ be two functions. The pair $\left( \psi ,\phi \right) $ is said to be
shifting distance function, if the following conditions hold:
$(i)$ for $u,v\in \lbrack 0,\infty ),$ if $\psi (u)\leq \phi (v)$, then $%
u\leq v,$
$(ii)$ for $\left\{ u_{k}\right\} ,\left\{ v_{k}\right\} \subset \lbrack
0,\infty )$ with $\underset{k\rightarrow \infty }{lim}u_{k}=\underset{%
k\rightarrow \infty }{lim}v_{k}=w$, if $\psi (u_{k})\leq \phi (v_{k})$ for
all $k\in
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
$, then $w=0$.
\end{definition}
\begin{theorem}[see \protect\cite{samadi}]
Let $C$ be a nonempty, bounded, closed and convex subset of the Banach space
$X$. Suppose that $T:C\rightarrow C$ is a continuous mapping such that%
\begin{equation}
\psi (\mu (TE))\leq \phi (\mu (E))
\end{equation}%
for any nonempty $E\subset $ $C$, where $\mu $ is an arbitrary measure of
noncompactness. Then, $T$ has a fixed point in $C.$
\end{theorem}
\begin{theorem}[see \protect\cite{chai}]
Let $C$ be a nonempty, bounded, closed and convex subset of the Banach space
$X$. Suppose that $T:C\rightarrow C$ is a continuous mapping such that%
\begin{equation}
\psi \left( \int_{0}^{\mu (TE)}\varphi (t)dt\right) \leq \phi \left(
\int_{0}^{\mu (E)}\varphi (t)dt\right)
\end{equation}%
for any nonempty $E\subset $ $C$, where $\mu $ is an arbitrary measure of
noncompactness and $\psi ,\phi :[0,\infty )\rightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ are a pair of shifting distance functions. Moreover, let $\varphi :\left[
0,\infty \right) \rightarrow \left[ 0,\infty \right] $ be a
Lebesgue-integrable function, which is summable on each compact subset of $%
\left[ 0,\infty \right) $ and $\int_{0}^{\varepsilon }\varphi (t)dt>0$ for
each $\varepsilon >0.$ Then, $T$ has at least one fixed point in $C.$
\end{theorem}
\begin{definition}[see \protect\cite{proje}]
\label{shift}Let $\psi _{n},\phi _{n}:[0,\infty )\rightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ be two function sequences. Let the pair $\left( \psi ,\phi \right) $ be
shifting distance function. The pair $\left( \psi _{n},\phi _{n}\right) $ is
said to be function sequences with shifting distance property which satisfy
the following conditions:
$(i)$ for $u,v\in \lbrack 0,\infty ),$ if $\psi _{n}(u)\rightrightarrows
\psi (u),\phi _{n}(v)\rightrightarrows \phi (v)$ and\ $\psi (u)\leq \phi (v)$%
, then $u\leq v,$
$(ii)$ for $\left\{ u_{k}\right\} ,\left\{ v_{k}\right\} \subset \lbrack
0,\infty )$ with $\underset{k\rightarrow \infty }{lim}u_{k}=\underset{%
k\rightarrow \infty }{lim}v_{k}=w$, if $\psi _{n}(u_{k})\rightrightarrows
\psi (u_{k}),\phi _{n}(v_{k})\rightrightarrows \phi \left( v_{k}\right) $
and $\psi (u_{k})\leq \phi \left( v_{k}\right) $ \ for all $k\in
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
$, then $w=0$.
\end{definition}
\begin{lemma}[see \protect\cite{proje}]
\label{proje}Let $\psi _{n},\phi _{n}:[0,\infty )\rightarrow R$ be two
function sequences . Assume that the function sequences hold following
conditions:
$(i)$ if $\left\{ \psi _{n}\right\} $ upper semi-continuous function
sequences and $\psi _{n}\leq \psi _{n+1}$, then $\psi $_{n}$\rightarrow \psi
$ is uniform convergence according to $n$,
$(ii)$ if $\left\{ \phi _{n}\right\} $ lower semi-continuous function
sequences and $\phi _{n}\geq \phi _{n+1}$, then $\phi _{n}\rightarrow \phi $
is uniform convergence according to $n$.
Then, $\left( \psi _{n},\phi _{n}\right) $ is called the function sequences
having shifting distance property.
\end{lemma}
\begin{theorem}[see \protect\cite{proje}]
\label{main}Let $C$ be a nonempty, bounded, closed and convex subset of the
Banach space $X$. Suppose that $T:C\rightarrow C$ is a continuous mapping
such that%
\begin{equation}
\psi _{n}(\mu (TE))\leq \phi _{n}(\mu (E)) \label{2}
\end{equation}%
for any nonempty $E\subset $ $C$, where $\mu $ is an arbitrary measure of
noncompactness and $\psi _{n},\phi _{n}:[0,\infty )\rightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ be the function sequences with shifting distance property. Then, $T$ has a
fixed point in $C.$
\end{theorem}
\section{Main Results}
\begin{theorem}
\label{main copy(1)}Let $C$ be a nonempty, bounded, closed and convex subset
of the Banach space $X$. Suppose that $T:C\rightarrow C$ is a continuous
mapping such that%
\begin{equation}
\psi _{n}\left( \int_{0}^{\mu (TE)}\varphi (t)dt\right) \leq \phi _{n}\left(
\int_{0}^{\mu (E)}\varphi (t)dt\right) \label{10}
\end{equation}%
for any nonempty $E\subset $ $C$, where $\mu $ is an arbitrary measure of
noncompactness and $\psi _{n},\phi _{n}:[0,\infty )\rightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ be the function sequences with shifting distance property. Furthermore,
let $\varphi :\left[ 0,\infty \right) \rightarrow \left[ 0,\infty \right] $
be a Lebesgue-integrable function, which is summable on each compact subset
of $\left[ 0,\infty \right) $ and $\int_{0}^{\varepsilon }\varphi (t)dt>0$
for each $\varepsilon >0.$ Then, $T$ has a fixed point in $C.$
\end{theorem}
\begin{proof}
We define a sequence $\left\{ E_{k}\right\} $ such that $E_{0}=E$ and $%
E_{k}=Conv\left( TE_{k-1}\right) $ for all $k\geq 1.$ Then we get
\begin{eqnarray*}
TE_{0} &=&TE\subseteq E=E_{0} \\
E_{1} &=&Conv\left( TE_{0}\right) \subseteq E=E_{0}.
\end{eqnarray*}%
If this process is continued, we have
\begin{equation*}
E_{0}\supseteq E_{1}\supseteq E_{2}\supseteq \cdots \supseteq E_{k}\supseteq
\cdots .
\end{equation*}%
If there exists an integer $k\geq 0$ such that $\mu \left( E_{k}\right) =0$,
then $E_{k}$ is relatively compact and since%
\begin{equation*}
TE_{k}\subseteq Conv\left( TE_{k}\right) =E_{k+1}\subseteq E_{k},
\end{equation*}%
Theorem \ref{sch} implies that $T$ has a fixed point on the set $E_{k}$ for
all $k\geq 0.$ Now, we assume that $\mu (E_{k})>0$ for all $k\geq 0.$ By
using (\ref{10}) we have%
\begin{eqnarray}
\text{ \ }\psi _{n}\left( \int_{0}^{\mu (E_{k+1})}\varphi (t)dt\right)
&=&\psi _{n}\left( \int_{0}^{\mu (Conv\left( TE_{k}\right) )}\varphi
(t)dt\right) \label{3} \\
&=&\psi _{n}\left( \int_{0}^{\mu \left( TE_{k}\right) }\varphi (t)dt\right)
\notag \\
&\leq &\phi _{n}\left( \int_{0}^{\mu \left( E_{k}\right) }\varphi
(t)dt\right) . \notag
\end{eqnarray}
Suppose that (\ref{10}) holds. Then we get that $\left\{ \int_{0}^{\mu
\left( E_{k}\right) }\varphi (t)dt\right\} $ is a decreasing sequence of
positive real numbers by $(ii)$ of Definition \ref{shift} and there exists $%
r\geq 0$ such that $\int_{0}^{\mu \left( E_{k}\right) }\varphi
(t)dt\rightarrow r$ as $k\rightarrow \infty .$ By using together with (\ref%
{3}) and Lemma \ref{proje}, we get $\psi _{n}\left( \int_{0}^{\mu \left(
E_{k+1}\right) \ }\varphi (t)dt\right) \rightrightarrows \psi \left(
\int_{0}^{\mu \left( E_{k+1}\right) \ }\varphi (t)dt\right) $ and
\begin{equation}
\psi \left( \int_{0}^{\mu \left( E_{k+1}\right) \ }\varphi (t)dt\right)
=\psi \left( \int_{0}^{\mu \left( TE_{k}\right) \ }\varphi (t)dt\right) .
\label{5}
\end{equation}%
Also, if $\int_{0}^{\mu \left( E_{k}\right) }\varphi (t)dt\rightarrow r$ as $%
k\rightarrow \infty ,$ then $\int_{0}^{\mu \left( E_{k+1}\right) }\varphi
(t)dt\rightarrow r$ as $k\rightarrow \infty .$ Hence we have%
\begin{eqnarray*}
\lim_{k\rightarrow \infty }\psi \left( \int_{0}^{\mu \left( E_{k+1}\right) \
}\varphi (t)dt\right) &=&\lim_{k\rightarrow \infty }\psi \left(
\int_{0}^{\mu \left( TE_{k}\right) \ }\varphi (t)dt\right) \leq
\lim_{k\rightarrow \infty }\phi \left( \int_{0}^{\mu \left( E_{k}\right)
}\varphi (t)dt\right) \\
\psi (r) &\leq &\phi (r).
\end{eqnarray*}
From $(ii)$ of Definition \ref{shift}, we obtain $r=0$. Hence, we have $%
\int_{0}^{\mu \left( E_{k}\right) }\varphi (t)dt\rightarrow 0$ as $%
k\rightarrow \infty .$ On the other hand, since $E_{k+1}\subseteq E_{k},$ $%
TE_{k}\subseteq E_{k}$ and $\int_{0}^{\mu \left( E_{k}\right) }\varphi
(t)dt\rightarrow 0$ as $k\rightarrow \infty .$ Using $(M_{6})$ of Definition %
\ref{darbo}, $E_{\infty }=\underset{k=1}{\overset{\infty }{\cap }}E_{k}$ is
nonempty, closed, convex, and invariant under $T.$ Hence the mapping $T$
belong to Ker $\mu .$ Therefore, Schauder's fixed point theorem implies that
$T$ has a fixed point in $E_{\infty }\subset $ $E$.
\end{proof}
\begin{example}
We denote the following function sequences by
\begin{equation*}
\psi _{n}(u)=\frac{4n(1+u)+2u+1}{2n+1},\phi _{n}(v)=\frac{n^{2}(2+v)+1}{n^{2}%
}.
\end{equation*}
It holds the conditions of Definition \ref{shift}. We assume that
\begin{equation*}
u=\int_{0}^{\mu (TE)}\varphi (t)dt,\text{ }v=\int_{0}^{\mu (E)}\varphi (t)dt,
\end{equation*}
we get
\begin{equation*}
\int_{0}^{\mu (TE)}\varphi (t)dt\leq \frac{1}{2}\int_{0}^{\mu (E)}\varphi
(t)dt.
\end{equation*}%
If we take $\varphi (t)=1,$ according to Darbo's fixed point theorem, $T$
has a fixed point.
\end{example}
\begin{remark}
If we take $\varphi (t)=1$ for $t\in \left[ 0,\infty \right) $ in Theorem %
\ref{main copy(1)} such that
\begin{equation*}
\psi _{n}\left( \int_{0}^{\mu (TE)}\varphi (t)dt\right) =\psi _{n}(\mu
(TE))\leq \phi _{n}(\mu (E))=\phi _{n}\left( \int_{0}^{\mu (E)}\varphi
(t)dt\right) ,
\end{equation*}%
then we obtain Theorem 3.4 given in \cite{proje}.
\end{remark}
\begin{remark}
Taking $\varphi (t)=1,$ $\psi _{n}\left( t\right) =I_{n}\left( t\right) $
and $\phi _{n}(t)=kI_{n}\left( t\right) $ such that $I_{n}\rightrightarrows
I $ in Theorem \ref{main copy(1)}, then we have%
\begin{equation*}
\mu (TE)=\psi _{n}\left( \int_{0}^{\mu (TE)}\varphi (t)dt\right) \leq \phi
_{n}\left( \int_{0}^{\mu (E)}\varphi (t)dt\right) =k\mu (E),
\end{equation*}%
so we get Darbo's fixed point theorem, where $k\in \left[ 0,1\right) $.
\end{remark}
If we take $\left( \psi _{n}\right) =\left( I_{n}\right) $ such that $%
\underset{n\rightarrow \infty }{\lim }I_{n}=I$ uniformly convergence for all
$n\in
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
$ \ in Theorem \ref{main copy(1)}, we obtain the following result.
\begin{corollary}
Let $C$ be a nonempty, bounded, closed and convex subset of the Banach space
$X$. Suppose that $T:C\rightarrow C$ is a continuous function such that
\begin{equation*}
I_{n}\left( \int_{0}^{\mu (TE)}\varphi (t)dt\right) \leq \phi _{n}\left(
\int_{0}^{\mu (E)}\varphi (t)dt\right)
\end{equation*}%
for any nonempty subset of $E\subset C$, where $\mu $ is an arbitrary
measure of noncompactness and $\phi _{n}:[0,\infty )\rightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ be a function sequences such that
$(a)$ for $u,v\in \lbrack 0,\infty ),$ if $I_{n}\left( u\right) \leq \phi
_{n}(v)$, then $u\leq v,$
$(b)$ for $\left\{ u_{k}\right\} ,\left\{ v_{k}\right\} \subset \lbrack
0,\infty )$ with $\underset{k\rightarrow \infty }{lim}u_{k}=\underset{%
k\rightarrow \infty }{lim}v_{k}=w$, if $I_{n}\left( u_{k}\right) \leq \phi
_{n}(v_{k})$ for all $n,k\in
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
$, then $w=0$.
Also let $\varphi :\left[ 0,\infty \right) \rightarrow \left[ 0,\infty
\right) $ be a Lebesgue-integrable function, which is summable on each
compact subset of $\left[ 0,\infty \right) $ and $\int_{0}^{\varepsilon
}\varphi (t)dt>0$ for each $\varepsilon >0.$ Then, $T$ has a fixed point in $%
C.$
\begin{theorem}
\label{cor copy(1)}Let $C$ be a nonempty, bounded, closed and convex subset
of the Banach space $X$. Suppose that $T:C\rightarrow C$ is a continuous
mapping such that%
\begin{equation}
\psi _{n}\left( \int_{0}^{\mu (TE)}\varphi (t)dt\right) \leq \psi _{n}\left(
\int_{0}^{\mu (E)}\varphi (t)dt\right) -\phi _{n}\left( \int_{0}^{\mu
(E)}\varphi (t)dt\right) \label{4}
\end{equation}%
for any nonempty $E\subset C$, where $\mu $ is an arbitrary measure of
noncompactness and $\psi _{n},\phi _{n}:[0,\infty )\rightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}$ be a pair having shifting distance property. Also the pair $\left(
\psi ,\phi \right) $ is two nondecreasing and continuous functions
satisfying $\psi (t)=$ $\phi (t)$ if and only if $t=0$. Furthermore, let $%
\varphi :\left[ 0,\infty \right) \rightarrow \left[ 0,\infty \right) $ be a
Lebesgue-integrable function, which is summable on each compact subset of $%
\left[ 0,\infty \right) $ and $\int_{0}^{\varepsilon }\varphi (t)dt>0$ for
each $\varepsilon >0.$ Then, $T$ has a fixed point in $C.$
\end{theorem}
\end{corollary}
\begin{proof}
Suppose that (\ref{4}) holds. If by taking limit on (\ref{4}), we have
\begin{equation}
\psi \left( \int_{0}^{\mu (TE)}\varphi (t)dt\right) \leq \psi \left(
\int_{0}^{\mu (E)}\varphi (t)dt\right) -\phi \left( \int_{0}^{\mu
(E)}\varphi (t)dt\right) . \label{7}
\end{equation}
Along with that, by using hypothesis in expression, we suppose that
\begin{equation*}
\psi \left( \int_{0}^{\mu (E)}\varphi (t)dt\right) =\phi \left(
\int_{0}^{\mu (E)}\varphi (t)dt\right) .
\end{equation*}%
Then we get $\int_{0}^{\mu (E)}\varphi (t)dt.$ By using the conditions of
Theorem \ref{main copy(1)} , $E$ is relatively compact and then Theorem \ref%
{sch} implies that $T$ has a fixed point in $C$. Conversely, we suppose that
$\mu (E)=0.$ Then in (\ref{7})
\begin{equation*}
\psi \left( \int_{0}^{\mu (E)}\varphi (t)dt\right) =\phi \left(
\int_{0}^{\mu (E)}\varphi (t)dt\right) .
\end{equation*}%
Since $\int_{0}^{\mu (E)}\varphi (t)dt=0,$ it is easy to see that $E$ is
relatively compact. From the condition of Theorem \ref{sch}, we say that $T$
has a fixed point in $C.$ Also since $\left( \psi ,\phi \right) \in
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+},$ $\int_{0}^{\mu (TE)}\varphi (t)dt=0.$ So by repeating the conditions
of Theorem \ref{main copy(1)}, we obtain that $T$ belong to Ker $\mu $. \ As
a result, mapping $T$ has a fixed point in $E_{\infty }\subset E.$
\end{proof}
\begin{theorem}
\bigskip Let $C$ be a nonempty, bounded, closed and convex subset of the
Banach space $X$. Suppose that $T:C\rightarrow C$ is a continuous mapping
such that%
\begin{equation}
\psi _{n}\left( \int_{0}^{\mu (TE)}\varphi (t)dt\right) \leq \phi _{n}\left(
\int_{0}^{\mu (E)}\varphi (t)dt\right) -\theta _{n}\left( \int_{0}^{\mu
(E)}\varphi (t)dt\right) \label{3.6}
\end{equation}%
for any nonempty $E\subset C$, where $\mu $ is an arbitrary measure of
noncompactness and $\psi _{n},\phi _{n},\theta _{n}:[0,\infty )\rightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}$ be function sequences having shifting distance property, triplet
functions $\left( \psi _{n},\phi _{n},\theta _{n}\right) $ $\rightarrow
\left( \psi ,\phi ,\theta \right) $ \ be uniform convergence in $n$ and $%
\left\{ \psi _{n}\right\} $ be sequences of continuous functions. Also, the
pair $\left( \phi _{n},\theta _{n}\right) $ $\rightrightarrows \left( \phi
,\theta \right) $ is two nondecreasing and continuous functions.
Furthermore, let $\varphi :\left[ 0,\infty \right) \rightarrow \left[
0,\infty \right) $ be a Lebesgue-integrable function, which is summable on
each compact subset of $\left[ 0,\infty \right) $ and $\int_{0}^{\varepsilon
}\varphi (t)dt>0$ for each $\varepsilon >0.$ Moreover, assume that the
following conditions hold:
$i)$ $\theta _{n}(t)\rightrightarrows \theta (t)=0$ if and only if $t=0$ and
$\theta _{n}\geq 0$ for all $n$,
$ii)$ for any sequence in $\left\{ a_{k}\right\} $ in $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}$ with $a_{k}\rightarrow t>0,$%
\begin{equation*}
\psi _{n}\left( t\right) -\lim_{k\rightarrow \infty }\sup \phi _{n}\left(
a_{k}\right) +\lim_{k\rightarrow \infty }\inf \theta _{n}\left( a_{k}\right)
>0.
\end{equation*}
\end{theorem}
\begin{proof}
\bigskip By the similar idea used in Theorem \ref{main copy(1)}, we assume $%
\mu (E_{k})>0$ for all $k\geq 0.$ By using (\ref{3.6}), we have
\begin{eqnarray}
\psi _{n}\left( \int_{0}^{\mu (E_{k+1})}\varphi (t)dt\right) &=&\psi
_{n}\int_{0}^{\mu (Conv\left( TE_{k}\right) )}\varphi (t)dt \notag \\
&=&\psi _{n}\left( \int_{0}^{\mu \left( TE_{k}\right) }\varphi (t)dt\right)
\\
&\leq &\phi _{n}\left( \int_{0}^{\mu \left( E_{k}\right) }\varphi
(t)dt\right) -\theta _{n}\left( \int_{0}^{\mu \left( E_{k}\right) }\varphi
(t)dt\right) . \notag
\end{eqnarray}
Since $\theta _{n}>0$ for all $n,$ we have
\begin{equation*}
\text{ \ }\psi _{n}\left( \int_{0}^{\mu (E_{k+1})}\varphi (t)dt\right) \leq
\phi _{n}\left( \int_{0}^{\mu \left( E_{k}\right) }\varphi (t)dt\right) .
\end{equation*}
Also, from Definition \ref{shift}, we get \ the following inequality%
\begin{equation*}
\int_{0}^{\mu (E_{k+1})}\varphi (t)dt\leq \int_{0}^{\mu \left( E_{k}\right)
}\varphi (t)dt
\end{equation*}
Thus $\left\{ \int_{0}^{\mu \left( E_{k}\right) }\varphi (t)dt\right\} $ is
positive but decreasing sequence. Therefore, there exists $s\geq 0$ such
that $\lim\limits_{k\rightarrow \infty }\int_{0}^{\mu \left( E_{k}\right)
}\varphi (t)dt=s.$ Since $\left\{ \psi _{n}\right\} $ is sequence of
continuous functions and having shifting distance property, also let $\left(
\psi _{n},\phi _{n},\theta _{n}\right) $ $\rightrightarrows \left( \psi
,\phi ,\theta \right) $, we have
\begin{eqnarray*}
\lim \sup_{k\rightarrow \infty }\psi \left( \int_{0}^{\mu (E_{k+1})}\varphi
(t)dt\right) &\leq &\lim \sup_{k\rightarrow \infty }\phi \left(
\int_{0}^{\mu \left( E_{k}\right) }\varphi (t)dt\right) +\lim
\sup_{k\rightarrow \infty }-\theta \left( \int_{0}^{\mu \left( E_{k}\right)
}\varphi (t)dt\right) \\
\psi \left( \lim \sup_{k\rightarrow \infty }\int_{0}^{\mu (E_{k+1})}\varphi
(t)dt\right) &\leq &\lim \sup_{k\rightarrow \infty }\phi \left(
\int_{0}^{\mu \left( E_{k}\right) }\varphi (t)dt\right) +\lim
\sup_{k\rightarrow \infty }-\theta \left( \int_{0}^{\mu \left( E_{k}\right)
}\varphi (t)dt\right) \\
\psi \left( s\right) &\leq &\lim \sup_{k\rightarrow \infty }\phi \left(
\int_{0}^{\mu \left( E_{k}\right) }\varphi (t)dt\right) +\lim
\inf_{k\rightarrow \infty }\theta \left( \int_{0}^{\mu \left( E_{k}\right)
}\varphi (t)dt\right) .
\end{eqnarray*}
Equivalently, we have%
\begin{equation*}
\text{\ }\psi \left( s\right) -\lim \sup_{k\rightarrow \infty }\phi \left(
\int_{0}^{\mu \left( E_{k}\right) }\varphi (t)dt\right) +\lim
\inf_{k\rightarrow \infty }\theta \left( \int_{0}^{\mu \left( E_{k}\right)
}\varphi (t)dt\right) \leq 0.
\end{equation*}
Hence $\lim\limits_{k\rightarrow \infty }\int_{0}^{\mu \left( E_{k}\right)
}\varphi (t)dt=s=0$ and from the definition of $\varphi (t),$ we get $%
\lim\limits_{k\rightarrow \infty }\mu (E_{k})=0.$ As a result, we can say
that $T$ has a fixed point in $C.$ Hence this completes the proof.
\end{proof}
\bigskip
\begin{acknowledgement}
This work was supported by the Ahi Evran University Scientific Research
Projects Coordination Unit. Project Number: RKT. A3.17.001.
\end{acknowledgement}
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\end{document}