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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 129, 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}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2009/129\hfil Existence and uniqueness of solutions]
{Existence and uniqueness of solutions for the fractional
integro-differential equations in Banach spaces}

\author[J. Wu, Y. Liu \hfil EJDE-2009/129\hfilneg]
{Jun Wu, Yicheng Liu}  % in alphabetical order


\address{Jun Wu \newline
College of Mathematics and Computer Science, Changsha University
of Science Technology,  Changsha, 410114,  China}
\email{junwmath@hotmail.com}

\address{Yicheng Liu \newline
Department of Mathematics and System Sciences, College of Science,
 National University of Defense Technology, Changsha, 410073, China}
\email{liuyc2001@hotmail.com}


\thanks{Submitted August 21, 2009. Published October 7, 2009.}
\thanks{J. Wu was supported by grants 08C117 from  the Scientific Research
Fund of Hunan \hfill\break\indent
Provincial Education Department, and 1004132 from
the Scientific Research Fund for the \hfill\break\indent
Doctoral Program of CSUST.}
 
\subjclass[2000]{34K45} 
\keywords{Fractional integro-differential equations;
  nonlocal condition; \hfill\break\indent equivalent norms}

\begin{abstract}
 In this article, we established the existence and
 uniqueness of solutions for fractional integro-differential
 equations with nonlocal conditions in Banach spaces.
 Krasnoselskii-Krein-type conditions are used for obtaining
 the main result.
\end{abstract}

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

\section{Introduction}

In this article, we are interesting in the existence and uniqueness
of solutions for the Cauchy problem with a Caputo fractional
 derivative and nonlocal conditions:
\begin{gather}\label{ivp1}
D^qx(t)=f(t,x(t),[\theta x](t)),\quad q\in(0,1)\; t\in I:=[0,1],\\
\label{ivp2} x(0)+g(x)=x_0,
\end{gather}
where $q\in (0,1)$, $f:I\times X\times X\to X$,
$g:C(I,X)\to X$, $\theta: X\to X$ defined as
$$
[\theta x](t)=\int_0^tk(t,s,x(s))ds,
$$
and $k:\Delta\times X\to X$,  $\Delta=\{(t,s):0\leq s\leq t\leq
1\}$. Here, $(X,\|\cdot\|)$ is a Banach space and $C=C(I,X)$
denotes the Banach space of all bounded continuous functions from
$I$ into $X$ equipped with the norm $\|\cdot\|_C$.

The study of fractional differential equations and inclusions is
linked to the wide applications of  fractional calculus in
physics, continuum mechanics, signal processing, and
electromagnetics. The theory of fractional differential equations
has seen considerable development, see for example the monographs of Kilbas et
al. \cite{Kilbas} and Lakshmikantham et al. \cite{Laks-book}.

 Recently, existence and uniqueness criteria for the various
fractional (integro-)differential equations were
considered by Ahmad and Nieto \cite{ahmad-Nieto},
Bhaskar\cite{B-Laks-Lee2}, Lakshmikantham and Leela et al
\cite{Laks-Lee1,Laks-Lee2}. For more information in
this fields, see \cite{ahmad-Siva, anguraj} and the references
therein.

As indicated in many previous articles,  the nonlocal condition
$x(0)+g(x)=x_0$ generalizes the Cauchy condition $x(0)=x_0$, and
can be applied in physics with better cases than the Cauchy
condition. The term $g(x)$ denotes the nonlocal effects, which
describe the diffusion phenomenon of the a small amount in a
transparent tube, with the general form
$g(x)=\sum_{i=1}^{p}c_ix(t_i)$. Also, the problem
\eqref{ivp1}-\eqref{ivp2} includes many classical formulations.
For example, $g(x)=x_0-x(T)$ becomes  a periodic
boundary problem, $g(x)=x_0+x(T)$  becomes an antiperiodic
boundary problem, while  $g(x)=0$ becomes  a Cauchy problem.

In \cite{ahmad-Siva}, the authors presented
some existence and uniqueness results for the problem
\eqref{ivp1}-\eqref{ivp2}, when $f(t,x(t),[\theta
x](t))=p(t,x(t))+\int_0^tk(t,s,x(s))ds$.
In \cite{anguraj}, the authors
presented some existence and uniqueness results for the problem
\eqref{ivp1}-\eqref{ivp2}, when
$f(t,x(t),[\theta x](t))=\int_0^tk(t,s,x(s))ds$.
The aim of this paper is to present
some  existence results for the problem
\eqref{ivp1}-\eqref{ivp2} for
 some Krasnoselskii-Krein-type conditions.  Our methods are
based on the equivalence of norms and a fixed point theorem.

\section{Main results}

For the next theorem, we sue the following assumptions:

\begin{itemize}

\item[(F1)] $f$ is continuous and there exist constants
$\alpha,\beta\in(0,1]$, $L_1, L_2>0$ such that for
$t\in I$ and $x_i,y_i\in X$,
$$
\|f(t,x_1,y_1)-f(t,x_2,y_2)\|\leq L_1\|x_1-y_1\|^\alpha
+L_2\|x_2-y_2\|^\beta\,;
$$
\item[(F2)] $k$ is continuous and there exist
$\beta_1\in (0,1]$, $h\in L^1(I)$  such that
$$
\|k(t,s,x)-k(t,s,y)\|\leq h(s)\|x-y\|^{\beta_1},\quad (t,s)\in \Delta,\;
  x,y\in X\,;
$$
\item[(G)] $g$  is bounded, continuous, and there exists a
constant $b\in (0,1)$  such that
$\|g(u)-g(v)\|\leq b\|u-v\|$.
\end{itemize}

\begin{theorem} \label{thm1}
 Under Assumptions {\rm (F1), (F2), (G)}, Problem
\eqref{ivp1}-\eqref{ivp2} has a unique solution.
\end{theorem}

For special cases of $f$, we obtain the following  corollaries.

\begin{corollary} \label{coro1}
Let $f(t,x(t),[\theta x](t))=p(t,x(t))+\int_0^tk(t,s,x(s))ds$.
Assume {\rm (F2), (G)} and that
$p$ is continuous and there exist constants
$\beta\in(0,1]$, $L>0$ such that
$$
\|p(t,x)-p(t,y)\|\leq L\|x-y\|^\beta\quad t\in I,\;  x,y\in X.
$$
Then \eqref{ivp1}-\eqref{ivp2} has a unique solution.
\end{corollary}

\begin{corollary} \label{coro2}
 Assume {\rm (F1), (G)} and that
$k(t,s,x(s))=\gamma(t,s)x(s)$ and $\gamma\in C(\Delta)$.
 Then  \eqref{ivp1}-\eqref{ivp2} has a unique solution.
\end{corollary}

For the next theorem, we use the assumptions:
\begin{itemize}

\item[(F1')] $f$ is continuous and there exist constants
$p_1,p_2\in[0,q)$, $L_1, L_2, C>0$  such that
$$
\|f(t,x,y)\|\leq \frac{L_1}{t^{p_1}}\|x\|+\frac{L_2}{t^{p_2}}\|y\|+C,
\quad  t\in I, \; x,y\in X\,;
$$
\item[(F2')] $k$ is continuous and there exist
$h\in L^1(I)$, $K>0$ such that
$$
\|k(t,s,x)\|\leq h(s)\|x\|+K,\quad (t,s)\in \Delta,\;  x,y\in X.
$$
\end{itemize}

\begin{theorem} \label{thm2}
 Assume {\rm (F1'), F(2'), (G)}.
 Then  \eqref{ivp1}-\eqref{ivp2} has at least one solution.
\end{theorem}

We remark that Theorem \ref{thm1} extends  \cite[Theorem 2.1]{ahmad-Siva}
 and \cite[Theorem 2.1]{anguraj}.


 \section{Proof of Theorem \ref{thm1}}

The following lemma, due to Krasnoselskii,   plays an important
role in the proof of the existence part of Theorem \ref{thm1}.

\begin{lemma}[\cite{Krasnoselskii}] \label{lem3.1}
 Let $M$ be a closed convex and nonempty subset of a Banach space $X$.
Let $A, B$ be two operators such that
(1) $Ax+By\in M$ whenever $x,y\in M$;
(2) $A$ is compact and continuous;
(3) $B$ is a contraction mapping.
Then there exists $z\in M$ such that $z=Az+Bz$.
\end{lemma}

\begin{proof}[Proof of Theorem \ref{thm1}]
First, we transform the Cauchy problem \eqref{ivp1}-\eqref{ivp2}
into  fixed point problem with  $F:C(I,X)\to C(I,X)$ defined by
\begin{equation} \label{e33}
Fx(t)= x_0-g(x)+\frac{1}{\Gamma(q)}\int_{0}^{t}(t-s)^{q-1}f(s,x(s),
[\theta x](s))ds.
\end{equation}
Let $F=A+B$, with
\begin{gather}
 Ax(t) = \frac{1}{\Gamma(q)}\int_{0}^{t}(t-s)^{q-1}f(s,x(s),
[\theta x](s))ds; \label{e34}\\
  Bx(t)= x_0-g(x).  \label{e35}
\end{gather}
Define the norm $\|\cdot\|_k$ in $C(I,X)$,  for $u\in C(I,X)$
 and for some $k\in \mathbb{N}$,  by
$$
\|u\|_k=\max\{e^{-kt}\|u(t)\|: t\in I\}.
$$
Note that the norms $\|\cdot\|_C$ and
$\|\cdot\|_k$ are equivalent.

We prove Theorem \ref{thm1} in the following two steps.

 \noindent\textbf{Step 1: Existence.}
Let $P=\sup_{x\in X}\|g(x)\|$, $M_0=\sup_{t\in I}\|\int_0^tk(t,s,0)ds\|$,
 $M_1=\sup_{t\in I}\|f(t,0,0)\|$ and
 $Q=\|x_0\|+P+\frac{M_1}{\Gamma(q+1)}+3$. Choose a $k_1\in N$ such that
  $$
\frac{1}{k_1^q}(L_1Q^\alpha+L_2(\|h\|_{L^1}Q^{\beta_1}+M_0)^\beta)<3.
$$
Setting $B_Q=\{u\in C(I,X):  \|u\|_{k_1}\leq
 Q\}$. For $u\in B_Q$,  noting the assumption (F2), we have
\begin{align*}
\|[\theta u](t)\|
&\leq \int_0^t\|k(t,r,u(r))-k(t,r,0)+k(t,r,0)\|dr\\
&\leq \|h\|_{L^1}\sup_{r\in[0,t]}\|x(r)\|^{\beta_1}+M_0\\
&\leq \|h\|_{L^1}e^{k_1t}Q^{\beta_1}+M_0.
\end{align*}
Thus
$$
\|\theta u\|_{k_1}\leq \|h\|_{L^1}Q^{\beta_1}+M_0.
$$
By assumption (F1), for $u\in B_Q$, we obtain
\begin{align*}
\|Fu(t)\|
&\leq  \|x_0\|+P +\frac{1}{\Gamma(q)}\int_{_{0}}^{^{t}}(t-s)^{q-1}
\|f(s,u(s),[\theta u](s))-f(s,u(s),0)\|ds\\
&\quad +\frac{1}{\Gamma(q)}\int_{_{0}}^{^{t}}(t-s)^{q-1}
\|f(s,u(s),0)-f(s,0,0)\|ds\\
&\quad + \frac{1}{\Gamma(q)}\int_{_{0}}^{^{t}}(t-s)^{q-1}
\|f(s,0,0)\|ds\\
&\leq  \|x_0\|+P
+\frac{L_2}{\Gamma(q)}\int_{_{0}}^{^{t}}(t-s)^{q-1}
\|[\theta u](s)\|^\beta ds\\
&\quad +\frac{L_1}{\Gamma(q)}\int_{_{0}}^{^{t}}(t-s)^{q-1}
\|u(s)\|^\alpha ds+ \frac{M_1}{\Gamma(q+1)}\\
&\leq  \|x_0\|+P+ \frac{M_1}{\Gamma(q+1)}
+\frac{L_2}{\Gamma(q)}\int_{_{0}}^{^{t}}(t-s)^{q-1}
e^{\beta k_1s}ds\|\theta u\|_{k_1}^\beta\\
&\quad +\frac{L_1}{\Gamma(q)}\int_{_{0}}^{^{t}}(t-s)^{q-1}
e^{\alpha k_1s}ds\|u\|_{k_1}^\alpha \\
&\leq  \|x_0\|+P+ \frac{M_1}{\Gamma(q+1)}
+\frac{L_1}{\Gamma(q)}\int_{_{0}}^{^{t}}(t-s)^{q-1}
e^{k_1s}ds\|u\|_{k_1}^\alpha
\\
&\quad +\frac{L_2}{\Gamma(q)}\int_{_{0}}^{^{t}}(t-s)^{q-1}
e^{k_1s}ds(\|h\|_{L^1}Q^{\beta_1}+M_0)^\beta\\
&\leq  \|x_0\|+P+ \frac{M_1}{\Gamma(q+1)}
+e^{k_1t}[\frac{L_1}{k_1^q}Q^\alpha+\frac{L_2}{k_1^q}(\|h\|_{L^1}Q^{\beta_1}+M_0)^\beta].
\end{align*}
Thus
$$
\|Fu\|_{k_1}\leq \|x_0\|+P+ \frac{M_1}{\Gamma(q+1)}
+\frac{L_1}{k_1^q}Q^\alpha+\frac{L_2}{k_1^q}(\|h\|_{L^1}Q^{\beta_1}
+M_0)^\beta<Q.
$$
This implies $F(B_Q)\subset B_Q$.

On the other hand, for $u\in B_Q$ and $t_1,t_2\in J (t_1<t_2)$, we
deduce that
\begin{align*}
&\|Au(t_2)-Au(t_1)\|\\
&= \frac{1}{\Gamma(q)} \|\int_{0}^{^{t_2}}(t_2-s)^{q-1}
f(s,u(s),[\theta u](s))ds-\int_{0}^{^{t_1}}(t_1-s)^{q-1}f(s,u(s),[\theta u](s))ds\|\\
&\leq \frac{M}{\Gamma(q+1)}[2(t_2-t_1)^q+(t_1)^{q}-(t_2)^{q}]\\
&\leq \frac{2M}{\Gamma(q+1)}(t_2-t_1)^q,
\end{align*}
where $M=\sup\{\|f(t,x,y)\|: (t, x, y)\in I\times B_Q\times
\theta(B_Q)\}$. This means $A(B_Q)$ is equicontinuous set.  By
Ascoli-Arzela theorem, we easily deduce that $A(B_Q)$ is
relatively compact set. It follows from the continuousness of $f$
that $A$ is complete continuous.

By Assumption (G), it is easy to see that $B$ is contraction
mapping. Following the Lemma \ref{lem3.1} (Krasnoselskii's fixed point
theorem), we conclude that $F$ has a fixed point in $B_Q$. Thus
there exists a solution of
Cauchy problem \eqref{ivp1}-\eqref{ivp2}.

 \noindent\textbf{Step 2: Uniqueness.}
Let $\varphi(t)$ and $\psi(t)$ be two solutions
of  Cauchy problem \eqref{ivp1}-\eqref{ivp2}, and set
$m(t)=\|\varphi(t)-\psi(t)\|$.

First, we prove that $m(0)=0$. Indeed, by the definition of
operator $B$ and assumption $(G)$, we see that $B$ is contraction
on $C(I,X)$. Thus there exists a unique $y(t)$ such that
$By(t)=x_0+g(y)$. On the other hand, noting that
$\varphi(0)=x_0+g(\varphi)$ and   $\psi(0)=x_0+g(\psi)$, we
obtain $\varphi(0)=\psi(0)$.

 Next, we  prove $m(t)\equiv 0$ for $t\in I$ by contraction. If
$m(t)\neq0$ for some $t\in I$. Setting $t_*=\min\{t\in I: m(t)\neq
0\}$, then $m(t)\equiv0$ for $t\in[0,t_*]$. Thus $m(t)\equiv0$ for
$t\in I$ if and only if $t_*=1$. If $t_*<1$, then we can choose
positive numbers $\varepsilon_0$ and $k_2\in N$ such that
$$
\frac{e^{k_2\varepsilon_0}}{k_2^q}(L_1m_{\varepsilon_0}^{\alpha-1}+
L_2\|h\|_{L^1}^\beta m_{\varepsilon_0}^{\beta\beta_1-1})<1,
$$
where
$m_{\varepsilon_0}=\max\{\|\varphi(t)-\psi(t)\|
:t\in[t_*,t_*+\varepsilon_0]\}$.

Redefine the norm $\|\cdot\|_{k_2}$ on the interval
$[t_*,t_*+\varepsilon_0]$ by
$$
\|u\|_{k_2}=\sup\{e^{-k_2(t-t_*)}\|u(t)\|: t\in
[t_*,t_*+\varepsilon_0]\},
$$
 then the norms $\|\cdot\|_{k_2}$ and
$\|\cdot\|_{C}$ are equivalent on $[t_*,t_*+\varepsilon_0]$. Since
$\varphi(0)=\psi(0)$, we claim that $g(\varphi)=g(\psi)$. Thus
there exists $t_1\in[t_*,t_*+\varepsilon_0]$ such that
\begin{align*}
0&< m_{\varepsilon_0}
=\|\varphi(t_1)-\psi(t_1)\|\\
&=\|F\varphi(t_1)-F\psi(t_1)\|\\
&\leq  \frac{1}{\Gamma(q)}
\int_{t_*}^{^{t_1}}(t_1-s)^{q-1}\|f(s,\varphi(s),[\theta\varphi](s))
 -f(s,\psi(s),[\theta\psi](s))\|ds\\
&\leq \frac{L_1}{\Gamma(q)}
\int_{t_*}^{^{t_1}}(t_1-s)^{q-1}\|\varphi(s)
 -\psi(s)\|^\alpha ds\\
&\quad +\frac{L_2}{\Gamma(q)}
\int_{t_*}^{^{t_1}}(t_1-s)^{q-1}\|[\theta\varphi](s)
-[\theta\psi](s)\|^\beta ds\\
&\leq \frac{1}{\Gamma(q)}
\int_{t_*}^{^{t_1}}(t_1-s)^{q-1}[L_1m^\alpha(s)
+L_2\|h\|_{L^1}^\beta\sup_{r\in[0,s]}m^{\beta\beta_1}(r)]
ds \\
&\leq \frac{L_1}{\Gamma(q)}
\int_{t_*}^{^{t_1}}(t_1-s)^{q-1}e^{\alpha k_2(s-t_*)}ds
\|\varphi-\psi\|_{k_2}^\alpha\\
&\quad +\frac{L_2\|h\|_{L^1}^\beta}{\Gamma(q)}
\int_{t_*}^{^{t_1}}(t_1-s)^{q-1}e^{\beta\beta_1
k_2(s-t_*)}ds\|\varphi-\psi\|_{k_2}^{\beta\beta_1}\\
&\leq
\frac{e^{k_2\varepsilon_0}}{k_2^q}(L_1m_{\varepsilon_0}^{\alpha}+
L_2\|h\|_{L^1}^\beta
m_{\varepsilon_0}^{\beta\beta_1})
< m_{\varepsilon_0}.
\end{align*}
This is impossible. Thus  $t_*=1$ and we conclude that
$\varphi(t)\equiv\psi(t)$ for $t\in[0,1]$.
The proof  is complete.
\end{proof}

 \section{Proof of Theorem \ref{thm2}}

 Define an operator $H:C(I,R^+)\to C(I,R^+)$ by
 $$
Hx(t)=\frac{1}{\Gamma(q)}\int_{0}^{^{t}}(t-s)^{q-1}
(as^{-p_1}+bs^{-p_2})\sup_{r\in[0,s]}x(r)ds,
$$
 where $p_1,p_2\in[0,q)$ are constants and $a=L_1$, $b=L_2\|h\|_{L^1}$.

\begin{lemma} \label{lem4.1}
 There exist an increasing function $b\in C(I,R^+)$
and a $\delta \in(0,1)$ such that $Hb(t)\leq \delta b(t)$.
\end{lemma}

\begin{proof}
  We choose a positive number $\eta\in I$ such that
$$
\frac{a\eta^{q-p_1}B(q,1-p_1)}{\Gamma(q)}
 +\frac{b\eta^{q-p_2}B(q,1-p_2)}{\Gamma(q)}+a\eta^{q-p_1}
 +b\eta^{q-p_2}<1,
$$
where $B(\cdot,\cdot)$ is the Beta function
$B(x,y)=\int_0^1(1-s)^{x-1}s^{y-1}ds$.
Let
$$
\delta=\frac{a\eta^{q-p_1}B(q,1-p_1)}{\Gamma(q)}
+\frac{b\eta^{q-p_2}B(q,1-p_2)}{\Gamma(q)}+a\eta^{q-p_1}+b\eta^{q-p_2}
$$
and define an increasing function $b:I \to \mathbb{R}$ by
$$
b(t)=\begin{cases}
1, & \text{if  } t\in[0,\eta], \\
e^{(t-\eta)/\eta}, & \text{if  } t\in(\eta,1].
\end{cases}
$$
We claim that $Hb(t)\leq \delta  b(t)$ for $t\in [0,1]$.
For $t\in[0, \eta]$, recalling that
$B(x,y)=\int_0^1(1-s)^{x-1}s^{y-1}ds$, we have
\begin{align*}
Hb(t)
&=\frac{1}{\Gamma(q)}\int_{0}^{^{t}}(t-s)^{q-1}(as^{-p_1}+bs^{-p_2})ds\\
&= \frac{a}{\Gamma(q)}t^{q-p_1}\int_0^1(1-z)^{q-1}z^{1-p_1-1}dz
 +\frac{b}{\Gamma(q)}t^{q-p_2}\int_0^1(1-z)^{q-1}z^{1-p_2-1}dz\\
&= \frac{aB(q,1-p_1)}{\Gamma(q)}t^{q-p_1}
 +\frac{bB(q,1-p_2)}{\Gamma(q)}t^{q-p_2}\\
&\leq \frac{aB(q,1-p_1)}{\Gamma(q)}\eta^{q-p_1}
 +\frac{bB(q,1-p_2)}{\Gamma(q)}\eta^{q-p_2}<\delta b(t).
\end{align*}
For  $t\in(\eta, 1]$, we have
\begin{align*}
Hb(t)&= \frac{1}{\Gamma(q)}
\int_{0}^{^{t}}(t-s)^{q-1}(as^{-p_1}+bs^{-p_2})b(s)ds\\
&= \frac{1}{\Gamma(q)}
\int_{0}^{^{\eta}}(t-s)^{q-1}(as^{-p_1}+bs^{-p_2})ds\\
&\quad +\frac{1}{\Gamma(q)}
\int_{_{\eta}}^{^{t}}(t-s)^{q-1}(as^{-p_1}+bs^{-p_2})e^{\frac{s-\eta}{\eta}}ds\\
&\leq \frac{1}{\Gamma(q)}
\int_{0}^{^{\eta}}(\eta-s)^{q-1}(as^{-p_1}+bs^{-p_2})ds\\
&\quad +\frac{1}{\Gamma(q)}
\int_{_{\eta}}^{^{t}}(t-s)^{q-1}(as^{-p_1}+bs^{-p_2})
 e^{\frac{s-\eta}{\eta}}ds\\
&\leq  \frac{a\eta^{q-p_1}B(q,1-p_1)}{\Gamma(q)}
 +\frac{b\eta^{q-p_2}B(q,1-p_2)}{\Gamma(q)}\\
&\quad +\frac{1}{\Gamma(q)} \int_{_{\eta}}^{^{t}}(t-s)^{q-1}(as^{-p_1}
 +bs^{-p_2}) e^{-\frac{t-s}{\eta}}dse^{\frac{t-\eta}{\eta}}\\
&\leq [\frac{a\eta^{q-p_1}B(q,1-p_1)}{\Gamma(q)}
 +\frac{b\eta^{q-p_2}B(q,1-p_2)}{\Gamma(q)}
 +a\eta^{q-p_1}+b\eta^{q-p_2}]e^{\frac{t-\eta}{\eta}}\\
&= \delta  b(t).
\end{align*}
The proof  is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2}]
 As in the proof of Theorem \ref{thm1}, we  prove the operator $F$ admits
a fixed point.
Define the norm $\|\cdot\|_b$ in $C(I,X)$,  for $u\in C(I,X)$,
by
$$
\|u\|_b=\max\{\frac{1}{b(t)}\|u(t)\|: t\in I\}.
$$
Then the norms $\|\cdot\|_C$ and $\|\cdot\|_b$ are equivalent.
Let $P=\sup_{x\in X}\|g(x)\|$ ,
 $$
Q=\frac{1}{1-\delta}(\|x_0\|+P
+\frac{C}{\Gamma(q+1)}+\frac{L_2KB(q,1-p_2)}{\Gamma(q)}),
$$
and $B_Q=\{u\in C(I,X): \|u\|_b\leq Q\}$. For $u\in B_Q$,
noting the assumption (F2'), we have
\[
\|[\theta u](t)\|\leq  \int_0^t\|k(t,r,u(r))\|dr \leq
\|h\|_{L^1}\sup_{r\in[0,t]}\|x(r)\|+K.
\]
By the assumption (F1') and Lemma \ref{lem4.1}, for $u\in B_Q$, we
obtain
\begin{align*}
\|Fu(t)\|&\leq  \|x_0\|+P +\frac{1}{\Gamma(q)}\int_{_{0}}^{^{t}}(t-s)^{q-1}
\|f(s,u(s),[\theta u](s))\|ds\\
&\leq  \|x_0\|+P +\frac{1}{\Gamma(q)}\int_{_{0}}^{^{t}}(t-s)^{q-1}
(L_1s^{-p_1}\|u(s)\|+L_2s^{-p_2}\|\theta u(s)\|+C)ds\\
&\leq  \|x_0\|+P +\frac{1}{\Gamma(q)}\int_{_{0}}^{^{t}}(t-s)^{q-1}
(L_1s^{-p_1}+L_2\|h\|_{L^1}s^{-p_2})\sup_{r\in[0,s]}\|u(s)\|ds\\
&\quad +\frac{1}{\Gamma(q)}\int_{_{0}}^{^{t}}(t-s)^{q-1}(L_2Ks^{-p_2}+C)ds\\
&\leq  \frac{1}{\Gamma(q)}\int_{_{0}}^{^{t}}(t-s)^{q-1}
(L_1s^{-p_1}+L_2\|h\|_{L^1}s^{-p_2})b(s)ds\|u\|_b\\
&\quad +\|x_0\|+P +\frac{C}{\Gamma(q+1)}+\frac{L_2KB(q,1-p_2)}{\Gamma(q)}\\
&\leq \delta  b(t)\|u\|_b +\|x_0\|+P
+\frac{C}{\Gamma(q+1)}+\frac{L_2KB(q,1-p_2)}{\Gamma(q)}.
\end{align*}
Thus
$$
\|Fu\|_b\leq \delta  Q +\|x_0\|+P
+\frac{C}{\Gamma(q+1)}+\frac{L_2KB(q,1-p_2)}{\Gamma(q)}=Q.
$$
This implies $F(B_Q)\subset B_Q$.

Similar arguments as in the proof of Theorem \ref{thm1} show that $A$ is
completely continuous and $B$ is contraction mapping. Thus, by Lemma
\ref{lem3.1}, we conclude that $F$ has a fixed point in $B_Q$. Thus there
exists a solution of
Cauchy problem \eqref{ivp1}-\eqref{ivp2}.
The proof is complete.
\end{proof}


\begin{thebibliography}{0}

\bibitem{ahmad-Nieto} B. Ahmad, J. J. Nieto;
\emph{Existence results for
nonlinear boundary value problems of fractional
integrodifferential equations with integral boundary conditions},
Boundary Value Problems, 2009, dio:10.1155/2009/708576.

\bibitem{ahmad-Siva} B. Ahmad, S. Sivasundaram;
\emph{Some existence results for
fractional integrodifferential equations with nonlinear
conditions}, Communications in Applied Analysis, 12 (2008), 107-112.

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

\bibitem{B-Laks-Lee2} T. G. Bhaskar, V. Lakshmikantham, S. Leela;
\emph{Fractional  differential equations with
 Krasnoselskii - Krein - type condition}. Nonlinear Analysis:
Hybrid Systems (2009),  doi: 10.1016/ j.nahs.2009.06.010.

\bibitem{Kilbas}A. A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo;
\emph{Theory and Applications of Fractional
Differential Equations}. North-Holland Mathematics Studies, 204.
Elsevier Science, Amsterdam, 2006.

\bibitem{Krasnoselskii} M. A. Krasnoselskii;
\emph{Some problems of nonlinear analysis}. Amer. Math. Soc. Trans.,
1958(10),345-409.

\bibitem{Laks-Lee1}V. Lakshmikantham, S. Leela;
\emph{Nagumo-type uniqueness result for fractional
 differential equations}. Nonlinear Analysis 71 (2009), 2886-2889.

\bibitem{Laks-Lee2} V. Lakshmikantham, S. Leela;
\emph{A Krasnoselskii-Krein-type uniqueness result for fractional
 differential equations}. Nonlinear Analysis 71 (2009), 3421-3424.

\bibitem{Laks-book}V. Lakshmikantham, S. Leela, J. Vasundhara Devi;
\emph{Theory of  fractional Dynamical systems}, Cambridge Academic
Publishers, Cambridge, 2009.

\end{thebibliography}

\end{document}