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\fancyhead[CE]{M. S. Abdo and S. K. Panchal}
\fancyhead[CO]{Fractional integro-differential equation}
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%\baselineskip 9mm
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\thispagestyle{plain} {\noindent Journal of Mathematical Extension \newline
Vol. XX, No. XX, (2014), pp-pp (Will be inserted by layout editor)}\newline
ISSN: 1735-8299\newline
URL: http://www.ijmex.com\newline
\vspace*{9mm}

\begin{center}
{\Large \textbf{An existence result for fractional integro-differential
equations in Banach spaces}}

\let\thefootnote\relax\footnote{{\scriptsize Received: XXXX; Accepted: XXXX
(Will be inserted by editor)}}

\textbf{Mohammed S. Abdo$^*$}\vspace*{-2mm}\\[0pt]
\vspace{2mm} {\small Research Scholar at Dr. Babasaheb Ambedkar Marathwada University,\\ Aurangabad, 431004 (M.S.), India} \vspace{%
2mm}

\textbf{Satish K. Panchal\let\thefootnote\relax\footnote{$^*$Corresponding
Author}}\vspace*{-2mm}\\[0pt]
\vspace{2mm} {\small Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University,\\ Aurangabad, 431004 (M.S.), India} \vspace{%
2mm}
\end{center}

\vspace{4mm}

\begin{quotation}
{\footnotesize {\noindent \textbf{Abstract.}} In this paper, we consider a
class of nonlinear fractional integro-differential equations with fractional
derivative of Caputo sense. We shall rely on the Krasnoselskii fixed point
theorem to obtain the existence result in Banach spaces. Moreover, one of
Krasnoselskii-Krein conditions is applied to establish the result. Finally,
an illustrative example is also presented.}

{\footnotesize \noindent \textbf{AMS Subject Classification:} 26A33; 26D10;
47H10. }

{\footnotesize \noindent \textbf{Keywords and Phrases:} Integro-differential
equations; Fractional derivative and integral; Krasnoselskii fixed point
theorem.}
\end{quotation}

\section{Introduction}

This paper is concerned with the existence result for a fractional
integro-differential equations of the type

\begin{equation}
^{c}D_{0^{+}}^{\alpha }y(t)=h(y(t))+f(t,y(t))+\int_{0}^{t}K(t,s,y(s))ds,\ \
t\in \lbrack 0,1],  \label{1.1}
\end{equation}%
with the initial condition
\begin{equation}
y(0)=y_{0}.  \label{1.2}
\end{equation}%
where $0<\alpha \leq 1$, $^{c}D_{0^{+}}^{\alpha }$ denotes the Caputo
fractional derivative operator, $f:[0,1]\times X\rightarrow X$, $%
K:[0,1]\times \lbrack 0,1]\times X\rightarrow X$ and $h: C([0,1]\rightarrow
X $ appropriate functions satisfying some conditions which will be stated
later.

Fractional differential equations are linked with extensive applications
such as continuum phenomena mechanics, electrochemistry, biophysics,
biotechnology engineering and so forth. For more details see studies of Guo
et al. \cite{GL}, Kilbas et al. \cite{KL1}, Miller and Ross \cite{MR} Oldham
and Spanier \cite{OS} and many other references.

Integro-differential equations emerge in many scientific and engineering
specialties, oftentimes be an approximation to partial differential
equations, that represent a lot of the incessant phenomena. Recently, the
existence and uniqueness of solutions to fractional differential equations
have studied in \cite{AP1,AP2,AP3,AP4,AP5,BV,DR,DIE,LV}, and the various
fractional integro-differential equations have been taken into consideration
by some authors, for extra information, see \cite{AN,AS,AK,BP,BT,MO,MJ,WL,ZX}%
. For example in \cite{MJ} Momani et al. studied the local and global
uniqueness results by applying Bihari's inequality and Gronwall's inequality
for the following problem

\begin{equation*}
^{c}D^{\alpha }y(t)=f(t,y(t))+\int_{t_{0}}^{t}K(t,s,y(s))ds,  \label{y1}
\end{equation*}%
\begin{equation*}
y(0)=y_{0}  \label{y2}
\end{equation*}%
where $0<\alpha \leq 1,$ $f\in C([0,1]\times
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{n},%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{n})$, $K\in C([0,1]\times \lbrack 0,1]\times
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{n},%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{n})$ and $^{c}D^{\alpha }$ is the Caputo fractional operator. In \cite{AS}
Ahmad and Sivasundaram considered the integro-differential equations with
fractional order and nonlocal conditions

\begin{equation*}
^{c}D^{\alpha }y(t)=f(t,y(t))+\int_{0}^{t}K(t,s,y(s))ds,\text{ \ }t\in
\lbrack 0,T],
\end{equation*}%
\begin{equation*}
y(0)=y_{0}-g(y),
\end{equation*}%
where $0<\alpha <1,$ $^{c}D^{\alpha }$ is the Caputo fractional operator, $%
f:[0,T]\times X\rightarrow X$, $K:[0,T]\times \lbrack 0,T]\times
X\rightarrow X$ are jointly continuous and $g\in C([0,T],X)$ is continuous.
The authors employed the Banach contraction principle and Krasnoselskii's
fixed point theorem to establish the existence and uniqueness results. Wu
and Liu in \cite{WL} extended the results that have been obtained in \cite%
{AS,AK} by employed Krasnoselskii-Krein-type conditions. In \cite{ZX}, Zhao
discussed the collocation methods for fractional integro-differential
equations with weakly singular kernels%
\begin{equation}
^{c}D_{0}^{\alpha }y(t)=p(t)y(t)+g(t)+\int_{0}^{t}q(t,s)y(s)ds,\text{ \ }%
t\in \lbrack 0,T],  \label{p}
\end{equation}%
\begin{equation*}
y^{(i)}(0)=y_{0}^{(i)},\text{ }i=0,1,...,n-1,
\end{equation*}%
where $0<\alpha <1,$ $g(t)$ and $p(t)$ are bounded and continuous on $[0,T]$%
, and $q(t,s)$ might possess a weak singularity.

On the other hand, Heydari et al. proved the existence of a unique solution
for a class of system of nonlinear singular fractional integro-differential
equations and they also used many numerical methods including Chebyshev,
wavelet method to solving such these equations see \cite{H1,H2,H3}.

In this paper, we will prove the existence solution of the fractional
integro-differential equation (\ref{1.1}) together with the initial
condition (\ref{1.2}) via taking advantage of Krasnoselskii's fixed point
theorem on the interval $[0,1]$. The existence result obtained in Banach spaces.
Moreover, we also use one of the Krasnoselskii-Krein conditions.

The organization of this paper is as follows. In Section \ref{jj}, we
mention some known notations and definitions and also we listing the
hypotheses which have advantage on this paper. The main Section \ref{kk}
proves the existence of solution for the problem (\ref{1.1})-(\ref{1.2}) in
Banach space by Krasnoselskii fixed point theorem. Finally, an illustrative
example is presented in Section \ref{mm}.

\section{Preliminaries\label{jj}}

In this section, we mainly demonstrate some essential notations,
definitions, and Lemmas which regard to fractional calculus and fixed point
theorem. Let $J=[0,1]$, $(X,\left\Vert .\right\Vert )$ is a Banach space and$%
\ C\mathcal{(}J,X\mathcal{)}$ denotes the Banach space of all continuous
bounded functions $g:J\rightarrow X$ equipped with the norm $\left\Vert
g\right\Vert _{C\mathcal{(}J,X\mathcal{)}}=\max \{\left\vert g(t)\right\vert
:t\in J\},$ for any $g(t)\in X$. We consider the space $C^{n}\mathcal{(}J,X%
\mathcal{)}$ consisting of all real valued continuous functions which are
continuously differentiable up to order $(n-1)$ on $J,$ and $L^{1}(J)$
denotes the space of all real functions defined on $J$ which are Lebesgue
integrable. In the following, the Mittag-Leffler function is given by
\begin{equation*}
E_{\alpha ,\beta }(w)=\sum_{k=0}^{\infty }\frac{w^{k}}{\Gamma (\alpha
k+\beta )},\text{ \ }Re(\alpha ),Re(\beta )>0.
\end{equation*}

Furthermore, if $0<\alpha <2$ and $\beta >1,$ we have \cite{GK}
\begin{equation*}
E_{\alpha ,\beta }(w)\leq \frac{1}{\alpha }w^{\frac{(1-\beta )}{\alpha }%
}e^{w^{\frac{1}{\alpha }}}.
\end{equation*}

\begin{definition}
\label{e2} (\cite{KL1}). Let $\alpha >0$ and $g:J\rightarrow X$. The left
sided Riemann--Liouville fractional integral of order $\alpha $ of a
function $g$\ is defined as%
\begin{equation*}
I_{0^{+}}^{\alpha }g(t)=\frac{1}{\Gamma (\alpha )}\int_{0}^{t}(t-s)^{\alpha
-1}g(s)ds,\ \ t\in J,
\end{equation*}
provided the right-hand side is pointwisely defined, where $\Gamma (.)$ is
the Euler gamma function.
\end{definition}

\begin{definition}
(\cite{KL1}). Let $n-1<\alpha <n$ and $g\in C^{n}\mathcal{(}J,X\mathcal{)}$%
.\ The left sided Caputo fractional derivative of order $\alpha $ of a
function $g$ is defined as
\begin{equation*}
^{c}D_{0^{+}}^{\alpha }g(t)=I_{0^{+}}^{n-\alpha -1}\frac{d^{n}}{dt^{n}}g(t),%
\text{ \ }t\in J,
\end{equation*}%
where $n=[\alpha ]+1$ and $[\alpha ]$ denotes the integer part of the real
number $\alpha $.
\end{definition}

\begin{lemma}
\label{rr}(\cite{KL1,SK}). For $\alpha ,\beta >0$ and $g,p$ are appropriate
functions then, for $t\in J,$ we have

\begin{enumerate}
\item $I_{0^{+}}^{\alpha }I_{0^{+}}^{\beta }g(t)=I_{0^{+}}^{\alpha +\beta
}g(t)=I_{0^{+}}^{\beta }I_{0^{+}}^{\alpha }g(t).$

\item $I_{0^{+}}^{\alpha }(g(t)+p(t))=I_{0^{+}}^{\alpha
}g(t)+I_{0^{+}}^{\alpha }p(t).$

\item $I_{0^{+}}^{\alpha }{}^{c}D_{0^{+}}^{\alpha }g(t)=g(t)-g(0),$ $%
0<\alpha <1.$

\item $^{c}D_{0^{+}}^{\alpha }I_{0^{+}}^{\alpha }{}g(t)=g(t)$.

\item $^{c}D_{0^{+}}^{\alpha }g(t)=I_{0^{+}}^{1-\alpha }\frac{d}{dt}g(t),$ $%
0<\alpha <1.$

\item $^{c}D_{0^{+}}^{\alpha }C=0,$ where $C$ is a constant.
\end{enumerate}
\end{lemma}

\begin{lemma}
\label{tt} (\cite{ZY}) (Krasnoselskii fixed point theorem). Let $E$ be
bounded, closed and convex subset of a Banach space $X$, and let $%
T_{1},T_{2}:E\rightarrow E$ satisfying the following:

\begin{description}
\item[(1)] $T_{1}x+T_{2}y\in E,$ for every $x,y\in E.$

\item[(2)] $T_{1}$ is contraction.

\item[(3)] $T_{2}$ is compact and continuous.

\item Then, there exists $z\in E$ such that the equation $z=T_{1}z+T_{2}z$
has a solution on $E.$
\end{description}
\end{lemma}

\section{Main results\label{kk}}

In this section, we shall demonstrate the existence result of $(\ref{1.1})-(%
\ref{1.2})$. Foremost, we state the subsequent lemma without proof.

\begin{lemma}
\label{aa} The fractional integro-differential equation $(\ref{1.1})$ with
the initial condition $(\ref{1.2})$ is equivalent to the following nonlinear
integral equation%
\begin{eqnarray}
y(t) &=&y_{0}+\frac{1}{\Gamma (\alpha )}\int_{0}^{t}(t-s)^{\alpha
-1}h(y(s))ds+\frac{1}{\Gamma (\alpha )}\int_{0}^{t}(t-s)^{\alpha -1}  \notag
\\
&&\times f(s,y(s))ds+\frac{1}{\Gamma (\alpha )}\int_{0}^{t}(t-s)^{\alpha
-1}\int_{s}^{t}K(\tau ,s,y(s))d\tau ds.  \label{e1}
\end{eqnarray}%
On the other hand, each solution of the integral equation (\ref{e1}) is
likewise a solution of the problem $(\ref{1.1})-(\ref{1.2})$ and vice versa.
\end{lemma}

For reader's comfort, we list of hypotheses is supplied as follows:

\begin{description}
\item[(A1)] $h:C(J,X)\rightarrow X$ is continuous, bounded and there exists $%
0<M<1$ such that $\left\Vert h(u)-h(v)\right\Vert \leq M\left\Vert
u-v\right\Vert ,$ for $u,v\in X.$

\item[(A2)] $f:J\times X\rightarrow X$ is continuous and there exist $\beta
\in (0,1],$ $L>0$ such that
\begin{equation*}
\left\Vert f(t,u)-f(t,v)\right\Vert \leq L\left\Vert u-v\right\Vert ^{\beta
},\text{ }t\in J,\text{ }u,v\in X.
\end{equation*}

\item[(A3)] $K:D\times X\rightarrow X,$ is continuous on $D$ and there exist
$\gamma \in (0,1],$ $\rho \in L^{1}(J)$ such that
\begin{equation*}
\left\Vert K(\tau ,s,u(s))-K(\tau ,s,v(s))\right\Vert \leq \rho (\tau
)\left\Vert u-v\right\Vert ^{\gamma },(\tau ,s)\in D,\ u,v\in X,
\end{equation*}%
where $D=\{(t,s):0\leq s\leq t\leq 1\}.$
\end{description}

Now, we give an existence result based on the Krasnoselskii's fixed point
theorem.

\begin{theorem}
\label{BB} Assume that the hypotheses (A1),(A2) and (A3) hold. Then the
fractional integro-differential problem $(\ref{1.1})-(\ref{1.2})$ has a
solution in $C(J,X)$ on $J.$
\end{theorem}

\begin{proof}
Transform the problem $(\ref{1.1})-(\ref{1.2})$ into a fixed point problem.
Consider the operator $\digamma :C(J,X)\rightarrow C(J,X)$ defined by%
\begin{eqnarray*}
\digamma y(t) &=&y_{0}+\frac{1}{\Gamma (\alpha )}\int_{0}^{t}(t-s)^{\alpha
-1}h(y(s))ds+\frac{1}{\Gamma (\alpha )}\int_{0}^{t}(t-s)^{\alpha -1} \\
&&\times f(s,y(s))ds+\frac{1}{\Gamma (\alpha )}\int_{0}^{t}(t-s)^{\alpha
-1}\int_{s}^{t}K(\tau ,s,y(s))d\tau ds.
\end{eqnarray*}

Before move ahead, we need to analyze the operator $\digamma $ into sum two
operators $P+Q$ as follows
\begin{equation}
Py(t)=y_{0}+\frac{1}{\Gamma (\alpha )}\int_{0}^{t}(t-s)^{\alpha -1}h(y(s))ds
\label{a1}
\end{equation}%
and
\begin{eqnarray}
Qy(t) &=&\frac{1}{\Gamma (\alpha )}\int_{0}^{t}(t-s)^{\alpha -1}f(s,y(s))ds+%
\frac{1}{\Gamma (\alpha )}\int_{0}^{t}(t-s)^{\alpha -1}  \notag \\
&&\times \int_{s}^{t}K(\tau ,s,y(s))d\tau ds.  \label{a2}
\end{eqnarray}

For any function $z\in C(J,X)$ and for som $j\in
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
,$\ we define the norm $\left\Vert z\right\Vert _{j}=\max
\{e^{-jt}\left\Vert z(t)\right\Vert :t\in J\}.$ Notice that the norm $%
\left\Vert z\right\Vert _{j}$ is equivalent to the norm $\left\Vert
z\right\Vert _{C}$ for $z\in C(J,X)$. Now, we present the proof in several
steps:

\textbf{Step 1}: We prove that $Pz+Qz^{\ast }\in S_{r}\subset C(J,X),$ for
every $z,z^{\ast }\in S_{r}.$

Let us set
\begin{equation*}
\mu =\underset{(s,z^{\ast })\in J\times S_{r}}{\sup }\left\Vert f(s,z^{\ast
}(s))\right\Vert
\end{equation*}%
\begin{equation*}
\mu ^{\ast }=\underset{(\tau ,s,z^{\ast })\in D\times S_{r}}{\sup }%
\int_{s}^{t}\left\Vert K(\tau ,s,z^{\ast }(s))\right\Vert d\tau \text{, }%
\eta =\underset{z\in S_{r}}{\sup }\left\Vert h(z)\right\Vert
\end{equation*}
and there exists $r=\left\Vert z_{0}\right\Vert +\frac{\eta +\mu +\mu ^{\ast
}}{\Gamma (\alpha +1)}+1$ such that $S_{r}=\{z\in C(J,X):\left\Vert
z\right\Vert _{j}\leq r\}.$
From the previous assumptions, then for $z,z^{\ast }\in S_{r}$ and $t\in J$,
we have
\begin{eqnarray*}
&&\left\Vert Pz(t)+Qz^{\ast }(t)\right\Vert  \\
&\leq &\left\Vert z_{0}\right\Vert +\frac{1}{\Gamma (\alpha )}%
\int_{0}^{t}(t-s)^{\alpha -1}\left\Vert h(z(s))\right\Vert ds \\
&&+\frac{1}{\Gamma (\alpha )}\int_{0}^{t}(t-s)^{\alpha -1}\left\Vert
f(s,z^{\ast }(s))\right\Vert ds+\frac{1}{\Gamma (\alpha )}%
\int_{0}^{t}(t-s)^{\alpha -1} \\
&&\times \int_{s}^{t}\left\Vert K(\tau ,s,z^{\ast }(s))\right\Vert d\tau ds
\\
&\leq &\left\Vert z_{0}\right\Vert +\frac{1}{\Gamma (\alpha )}%
\int_{0}^{t}(t-s)^{\alpha -1}\underset{z\in S_{r}}{\sup }\left\Vert
h(z(s))\right\Vert ds \\
&&+\frac{1}{\Gamma (\alpha )}\int_{0}^{t}(t-s)^{\alpha -1}\underset{%
(s,z^{\ast })\in J\times S_{r}}{\sup }\left\Vert f(s,z^{\ast
}(s))\right\Vert ds \\
&&+\frac{1}{\Gamma (\alpha )}\int_{0}^{t}(t-s)^{\alpha -1}\underset{(\tau
,s,z^{\ast })\in D\times S_{r}}{\sup }\int_{s}^{t}\left\Vert K(\tau
,s,z^{\ast }(s))\right\Vert d\tau ds \\
&\leq &\left\Vert z_{0}\right\Vert +\frac{\eta t^{\alpha }}{\Gamma (\alpha
+1)}+\frac{\mu t^{\alpha }}{\Gamma (\alpha +1)}+\frac{\mu ^{\ast }t^{\alpha }%
}{\Gamma (\alpha +1)} \\
&\leq &\left\Vert z_{0}\right\Vert +\frac{\eta +\mu +\mu ^{\ast }}{\Gamma
(\alpha +1)}.
\end{eqnarray*}

Consequently,
\begin{equation*}
\left\Vert Pz+Qz^{\ast }\right\Vert _{j}\leq e^{-j}\left( \left\Vert
z_{0}\right\Vert +\frac{\eta +\mu +\mu ^{\ast }}{\Gamma (\alpha +1)}\right)
<r.
\end{equation*}

This means that, $Pz+Qz^{\ast }\in S_{r}.$

\textbf{Step 2}: We prove that operator $P$ is a contraction map on $S_{r}.$

Let us make $S_{r}$ as in step 1, by the preceding assumptions, then for $%
z,z^{\ast }\in S_{r}$ and for $t$ $\in J$, we have%
\begin{eqnarray*}
\left\Vert Pz(t)-Pz^{\ast }(t)\right\Vert &\leq &\frac{1}{\Gamma (\alpha )}%
\int_{0}^{t}(t-s)^{\alpha -1}\left\Vert h(z(s))-h(z^{\ast }(s))\right\Vert ds
\\
&\leq &\frac{1}{\Gamma (\alpha )}\int_{0}^{t}(t-s)^{\alpha -1}M\left\Vert
z(s)-z^{\ast }(s)\right\Vert ds \\
&\leq &M\frac{1}{\Gamma (\alpha )}\int_{0}^{t}(t-s)^{\alpha -1}e^{js}%
\underset{s\in J}{\max }e^{-js}\left\Vert z(s)-z^{\ast }(s)\right\Vert ds \\
&=&M\left[ I_{0}^{\alpha }e^{jt}\right] \left\Vert z-z^{\ast }\right\Vert
_{j} \\
&=&Mt^{\alpha }E_{1,\alpha +1}(jt) \\
&\leq &M\frac{e^{jt}}{j^{\alpha }}\left\Vert z-z^{\ast }\right\Vert _{j} \\
&\leq &Me^{jt}\left\Vert z-z^{\ast }\right\Vert _{j}..
\end{eqnarray*}

Thus,
\begin{equation*}
\left\Vert Pz-Pz^{\ast }\right\Vert _{j}\leq M\left\Vert z-z^{\ast
}\right\Vert _{j}.
\end{equation*}

Since $M<1,$ we conclude that $P$ is a contraction map on $S_{r}.$

\textbf{Step 3}: We show that operator $Q$ is completely continuous on $%
S_{r}. $

For this end, we consider $S_{r}$ defined as in step 1, and we prove that $%
(QS_{r})\ $is uniformly bounded, $(\overline{QS_{r}})$ is equicontinuous and
$Q:S_{r}\rightarrow S_{r}$ is continuous.

Firstly\textbf{, }we show that $(QS_{r})\ $is uniformly bounded. By our
hypotheses, then for $z\in S_{r}$ and $t\in J,$ we have%
\begin{eqnarray*}
&&\left\Vert Qz(t)\right\Vert \\
&\leq &\frac{1}{\Gamma (\alpha )}\int_{0}^{t}(t-s)^{\alpha -1}\left\Vert
f(s,z(s))-f(s,0)\right\Vert ds \\
&&+\frac{1}{\Gamma (\alpha )}\int_{0}^{t}(t-s)^{\alpha -1}\left\Vert
f(s,0)\right\Vert ds \\
&&+\frac{1}{\Gamma (\alpha )}\int_{0}^{t}(t-s)^{\alpha
-1}\int_{s}^{t}\left\Vert K(\tau ,s,z(s))-K(\tau ,s,0)\right\Vert d\tau ds \\
&&+\frac{1}{\Gamma (\alpha )}\int_{0}^{t}(t-s)^{\alpha
-1}\int_{s}^{t}\left\Vert K(\tau ,s,0)\right\Vert d\tau ds \\
&\leq &\frac{1}{\Gamma (\alpha )}\int_{0}^{t}(t-s)^{\alpha -1}Le^{\beta
js}\left\Vert z\right\Vert _{j}^{\beta }+\frac{1}{\Gamma (\alpha )}%
\int_{0}^{t}(t-s)^{\alpha -1}Rds \\
&&+\frac{1}{\Gamma (\alpha )}\int_{0}^{t}(t-s)^{\alpha -1}\int_{s}^{t}\rho
(\tau )d\tau e^{\gamma js}\left\Vert z\right\Vert _{j}^{\gamma }ds+\frac{1}{%
\Gamma (\alpha )}\int_{0}^{t}(t-s)^{\alpha -1}R^{\ast }ds \\
&\leq &\left( L\left\Vert z\right\Vert _{j}^{\beta }+\left\Vert \rho
\right\Vert _{L^{1}}\left\Vert z\right\Vert _{j}^{\gamma }\right) \frac{1}{%
\Gamma (\alpha )}\int_{0}^{t}(t-s)^{\alpha -1}e^{js}ds+\frac{R+R^{\ast }}{%
\Gamma (\alpha )}\int_{0}^{t}(t-s)^{\alpha -1}ds \\
&\leq &\left( Lr^{\beta }+\left\Vert \rho \right\Vert _{L^{1}}r^{\gamma
}\right) \frac{e^{jt}}{j^{\alpha }}+\frac{R+R^{\ast }}{\Gamma (\alpha +1)}%
t^{\alpha }.
\end{eqnarray*}

Thus,%
\begin{equation*}
\left\Vert Qz\right\Vert _{j}\leq \frac{Lr^{\beta }+\left\Vert \rho
\right\Vert _{L^{1}}r^{\gamma }}{j^{\alpha }}+\frac{R+R^{\ast }}{\Gamma
(\alpha +1)e^{j}}:=\ell ,
\end{equation*}%
where $R=\underset{s\in J}{\sup }\left\Vert f(s,0)\right\Vert $ and $R^{\ast
}=\underset{(\tau ,s)\in D}{\sup }\int_{s}^{t}\left\Vert K(\tau
,s,0)\right\Vert d\tau .$ This means that $QS_{r}\subset S_{\ell }$, for any
$z\in S_{r},$ i.e. the set $\{Qz:z\in S_{r}\}$ is uniformly bounded.

Next, we will prove that $(\overline{QS_{r}})$ is equicontinuous. For $z\in
S_{r}$ and for $t_{1},t_{2}\in J$ with $t_{1}\leq t_{2},$ and also let $%
\delta =\left( \frac{\Gamma (\alpha +1)\epsilon }{2(\mu +\mu ^{\ast })}%
\right) ^{\frac{1}{\alpha }}$ then, when $\left\vert t_{2}-t_{1}\right\vert
<\delta ,$ we conclude that%
\begin{eqnarray*}
&&\left\Vert Qz(t_{2})-Qz(t_{1})\right\Vert \\
&\leq &\frac{1}{\Gamma (\alpha )}\int_{0}^{t_{2}}\left\vert
(t_{2}-s)^{\alpha -1}-(t_{1}-s)^{\alpha -1}\right\vert \left\Vert
f(s,z(s))\right\Vert ds \\
&&+\frac{1}{\Gamma (\alpha )}\int_{t_{1}}^{t_{2}}(t_{2}-s)^{\alpha
-1}\left\Vert f(s,z(s))\right\Vert ds \\
&&+\frac{1}{\Gamma (\alpha )}\int_{0}^{t_{1}}\left\vert (t_{2}-s)^{\alpha
-1}-(t_{1}-s)^{\alpha -1}\right\vert \int_{s}^{t_{2}}\left\Vert K(\tau
,s,z(s))\right\Vert d\tau ds \\
&&+\frac{1}{\Gamma (\alpha )}\int_{t_{1}}^{t_{2}}(t_{2}-s)^{\alpha
-1}\int_{s}^{t_{2}}\left\Vert K(\tau ,s,z(s))\right\Vert d\tau ds \\
&&+\frac{1}{\Gamma (\alpha )}\int_{0}^{t_{1}}((t_{1}-s)^{\alpha
-1}\int_{t_{1}}^{t_{2}}\left\Vert K(\tau ,s,z(s))\right\Vert d\tau ds \\
&\leq &\frac{1}{\Gamma (\alpha )}\int_{0}^{t_{1}}((t_{1}-s)^{\alpha
-1}-(t_{2}-s)^{\alpha -1})\left\Vert f(s,z(s))\right\Vert ds \\
&&+\frac{1}{\Gamma (\alpha )}\int_{t_{1}}^{t_{2}}(t_{2}-s)^{\alpha
-1}\left\Vert f(s,z(s))\right\Vert ds \\
&&+\frac{1}{\Gamma (\alpha )}\int_{0}^{t_{1}}((t_{1}-s)^{\alpha
-1}-(t_{2}-s)^{\alpha -1})\int_{s}^{t_{2}}\left\Vert K(\tau
,s,z(s))\right\Vert d\tau ds \\
&&+\frac{1}{\Gamma (\alpha )}\int_{t_{1}}^{t_{2}}(t_{2}-s)^{\alpha
-1}\int_{s}^{t_{2}}\left\Vert K(\tau ,s,z(s))\right\Vert d\tau ds \\
&&+\frac{1}{\Gamma (\alpha )}\int_{0}^{t_{1}}(t_{1}-s)^{\alpha -1}\left[
\int_{s}^{t_{2}}\left\Vert K(\tau ,s,z(s))\right\Vert d\tau
-\int_{s}^{t_{1}}\left\Vert K(\tau ,s,z(s))\right\Vert d\tau \right] ds \\
&\leq &\left[ \frac{(t_{1}^{\alpha }-t_{2}^{\alpha })+2(t_{2}-t_{1})^{\alpha
}}{\Gamma (\alpha +1)}\right] \mu +\left[ \frac{(t_{1}^{\alpha
}-t_{2}^{\alpha })+2(t_{2}-t_{1})^{\alpha }}{\Gamma (\alpha +1)}\right] \mu
^{\ast } \\
&\leq &\frac{2(\mu +\mu ^{\ast })(t_{2}-t_{1})^{\alpha }}{\Gamma (\alpha +1)}
\\
&<&\frac{2(\mu +\mu ^{\ast })\delta ^{\alpha }}{\Gamma (\alpha +1)}=\epsilon
.
\end{eqnarray*}

where $\mu $ and $\mu ^{\ast }$ are defined as in step 1. Therefore, $(%
\overline{QS_{r}})$ is equicontinuous.

Finally, from the continuity of $f$ and $K,$ we can directly reach that
operator $Q:S_{r}\rightarrow S_{r}.$ As consequence of step 3 with
Arzela-Ascoli theorem, we easily infer that $(QS_{r})$ is relatively compact
set. Hence, the operator $Q$ is completely continuous. Thus all the
assumptions of Lemma \ref{tt} are satisfied. Consequently, the conclusion of
Krasnoselskii's fixed point theorem shows that operator $\digamma =P+Q$ has
a fixed point on $S_{r}$. So the fractional integro-differential problem $(%
\ref{1.1})-(\ref{1.2})$ has a solution $y(t)\in C(J,X)$. This proves the
required.
\end{proof}

\section{An Example\label{mm}}

Consider the following nonlinear fractional integro-differential equation%
\begin{equation}
^{c}D_{0^{+}}^{\frac{1}{2}}y(t)=\frac{1}{2}\sin y(t)+(\cos t+\sin t)\left[
y(t)\right] ^{\frac{1}{2}}+\int_{0}^{t}t\sin \left[ y(s)\right] ^{\frac{1}{3}%
}ds,  \label{3}
\end{equation}%
with the initial condition%
\begin{equation}
u(0)=0.  \label{4}
\end{equation}

Here, $\alpha =\frac{1}{2},$ $h(y(t))=\frac{1}{2}\sin (y(t)),$ $%
f(t,y(t))=(\cos t+\sin t)\left[ y(t)\right] ^{\frac{1}{2}},$ and $%
K(t,s,y(s))=t\sin \left[ y(s)\right] ^{\frac{1}{3}}.$ For $u,v\in X=%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}$ and $t\in \lbrack 0,1].$ We can see that
\begin{equation*}
\left\Vert h(u)-h(v)\right\Vert \leq \frac{1}{2}\left\Vert u-v\right\Vert ,
\end{equation*}%
\begin{equation*}
\left\Vert f(t,u)-f(t,v)\right\Vert \leq 2\left\Vert u^{\frac{1}{2}}-v^{%
\frac{1}{2}}\right\Vert \leq 2\left\Vert u-v\right\Vert ^{\frac{1}{2}},\text{
}(0<\frac{1}{2}=\beta ).
\end{equation*}%
and
\begin{equation*}
\left\Vert K(t,s,u)-K(t,s,v)\right\Vert \leq t\left\Vert u^{\frac{1}{3}}-v^{%
\frac{1}{3}}\right\Vert \leq t\left\Vert u-v\right\Vert ^{\frac{1}{3}},\text{
}(\gamma =\frac{1}{3}<1).
\end{equation*}

So, the conditions (A1), (A2) and (A3) are satisfied with $M=\frac{1}{2}%
,L=2, $ and $\rho (t)=t\in L^{1}[0,1]$. By applying Theorem \ref{BB} the
problem (\ref{3}) (\ref{4}) has a solution on $[0,1]$.

\section{Conclusions}

This paper presents a class of nonlinear integro-differential equations with
Caputo fractional derivative. By using famous Krasnoselskii's fixed point
theorem, we have developed some adequate conditions for the existence of at
least one solution to a class of nonlinear fractional integro-differential
equations. The respective result has been verified by providing a suitable
example.

\section{Acknowledgments}

The authors would like to thank the anonymous reviewers and the editor of
this journal for their helpful comments and valuable suggestions which led
to an improved presentation of this paper.

\begin{thebibliography}{99}
\bibitem{AP1} M. S. Abdo and S. K. Panchal, Effect of perturbation in the
solution of fractional neutral functional differential equations, J. Korean
Soci. Industrial Appl. Math., 22(1) (2018), 63-74.

\bibitem{AP2} M. S. Abdo and S. K. Panchal, Existence and continuous
dependence for fractional neutral functional differential equations, \textit{%
J. Mathematical Model.}, 5 (2017), 153-170.

\bibitem{AP3} M. S. Abdo and S. K. Panchal, Fractional Integro-differential Equations Involving
$\psi$-Hilfer Fractional Derivative, Adv. Appl. Math. Mech.,11(1) (2019), 1-22.

\bibitem{AP4} M. S. Abdo and S. K. Panchal, Some New Uniqueness Results of
Solutions to Nonlinear Fractional Integro-Differential Equations, Annals of
Pure and Appl. Math., 16(2) (2018), 345-352.

\bibitem{AP5} M. S. Abdo and S. K. Panchal, Weighted Fractional Neutral
Functional Differential Equations, J. Sib. Fed. Univ. Math. Phys., 11(5)
(2018), 535-549.

\bibitem{AN} B. Ahmad and J. J. Nieto, Existence results for nonlinear
boundary value problems of fractional integro-differential equations with
integral boundary conditions, \textit{Boundary Value Problems}, 2009 (2009),
708576.

\bibitem{AS} B. Ahmad and S. Sivasundaram, Some existence results for
fractional integro-differential equations with nonlinear conditions, \textit{%
Communications Appl. Anal.}, 12 (2008), 107-112.

\bibitem{AK} A. Anguraj, P. Karthikeyan and G. M. N'gu\'er \'ekata, Nonlocal
Cauchy problem for some fractional abstract integro-differential equations
in Banach spaces, \textit{Communications Math. Anal.}, 6 (2009), 31-35.

\bibitem{BP} K. Balachandran and F. P. Samuel, Existence of solutions for
quasilinear delay integro-differential equations with nonlocal condition,
\textit{Electronic J. Diff. Equ.}, 2009 (2009), 1-7.

\bibitem{BT} K. Balachandran and J. J. Trujillo, The nonlocal Cauchy problem
for nonlinear fractional integro-differential equations in Banach spaces,
\textit{Nonlinear Anal. Theory Meth. Applic.}, 72 (2010), 4587-4593.

\bibitem{BV} T. G. Bhaskar, V. Lakshmikantham and S. Leela, Fractional
differential equations with Krasnoselskii-Krein-type condition, \textit{%
Nonlinear Anal. Hybrid Sys.},3 (2009), 734-737.

\bibitem{DR} D. Delboso and L. Rodino, Existence and uniqueness for a
nonlinear fractional differential equation, \textit{J. Math. Anal. Appl.},
204 (1996), 609-625.

\bibitem{DIE} K. Diethelm, \textit{The Analysis of Fractional Differential
Equations}, Lecture Notes in Mathematics, 2004, Springer, Berlin, (2010).

\bibitem{GK} R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin,
\textit{Mittag-Leffler functions, related topics and applications}, Berlin:
Springer, (2014).

\bibitem{GL} D. Guo, V. Lakshmikantham, \textit{Nonlinear problems in
abstract cones}, 5, Academic press. (1988).

\bibitem{H1} M. H. Heydari, M. R. Hooshmandasl, F. M. Maalek Ghaini, and
Ming Li, Chebyshev wavelets method for solution of nonlinear fractional
integrodifferential equations in a large interval, Advances in Mathematical
Physics, (2013) Volume 20 1 3, A rticleI D 482083, 12 pages.

\bibitem{H2} M. H. Heydari, M. R. Hooshmandasl, C. Cattani and M. Li,
Legendre wavelest method for solving fractional population growth model in a
closed system, Mathematical Problems in Engineering, (2013) Volume 2013, A
rticleI D161030, 8 pages.

\bibitem{H3} M. H. Heydari, M. R. Hooshmandasl, F. Mohammadi and C. Cattani,
Wavelets method for solving systems of nonlinear singular fractional
Volterra integro-differential equations. Commun. Nonlinear Sci. Numer. Simu.
19(1), (2014) 37-48.

\bibitem{KL1} A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, \textit{%
Theory and Applications of Fractional Differential Equations}, North-Holland
Math. Stud., 204, Elsevier, Amsterdam, (2006).

\bibitem{LV} V. Laksmikanthahm, S. Leela, A Krasnoselskii-Krein-type
uniqueness result for fractional differential equations, \textit{Nonlinear
Anal. Th. Meth. Applic.}, 71 (2009), 3421-3424.

\bibitem{MR} K. S. Miller and B. Ross, \textit{An Introduction to the
Fractional Calculus and Differential Equations}, John Wiley, New York,
(1993).

\bibitem{MO} S. Momani, Local and global existence theorems on fractional
integro-differential equations, \textit{J. Fract. Calc.}, 18 (2000), 81--86.

\bibitem{MJ} S. Momani, A. Jameel and S. Al-Azawi, Local and global
uniqueness theorems on fractional integro-differential equations via
Bihari's and Gronwall's inequalities, \textit{Soochow Journal of Mathematics}%
, 33 (2007) 619.

\bibitem{OS} K. Oldham and J. Spanier, \textit{The fractional calculus
theory and applications of differentiation and integration to arbitrary order%
}, 111, Elsevier, (1974).

\bibitem{SK} S. G. Samko, A. A. Kilbas and O. I. Marichev, \textit{%
Fractional Integrals and Derivatives. Theory and Applications}, Gordon and
Breach, Yverdon, (1993).

\bibitem{WL} J. Wu, and Y. Liu, Existence and uniqueness of solutions for
the fractional integro-differential equations in Banach spaces, \textit{%
Electronic J Diff. Equ.}, 2009 (2009), 1-8.

\bibitem{ZY} Y. Zhou, \textit{Basic theory of fractional differential
equations}, 6, Singapore: World Scientific, (2014).

\bibitem{ZX} J. Zhao, J. Xiao and N. Ford, Collocation methods for
fractional integro-differential equations with weakly singular kernels.
Numerical Algorithms, 65 (2014) 723-743.
\end{thebibliography}


{\small \noindent\textbf{Mohammed S. Abdo} }

{\small \noindent Research Scholar At. Department of Mathematics }

{\small \noindent PhD candidate }

{\small \noindent Dr. Babasaheb Ambedkar Marathwada University }

{\small \noindent Aurangabad 431004 (M.S.), India }

{\small \noindent E-mail: msabdo1977@gmail.com}\newline

{\small \noindent\textbf{Satish K. Panchal } }

{\small \noindent Department of Mathematics }

{\small \noindent Professor of Mathematics }

{\small \noindent Dr. Babasaheb Ambedkar Marathwada University }

{\small \noindent Aurangabad 431004 (M.S.), India }

{\small \noindent E-mail: drpanchalsk@gmail.com}\newline

\end{document}