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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 153, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/153\hfil Existence of mild solutions]
{Existence of mild solutions for a neutral fractional equation with
fractional nonlocal conditions}

\author[N.-e. Tatar\hfil EJDE-2012/153\hfilneg]
{Nasser-eddine Tatar}  % in alphabetical order

\address{Nasser-eddine Tatar \newline
King Fahd University of Petroleum and Minerals,
Department of Mathematics and Statistics,
Dhahran 31261, Saudi Arabia}
\email{tatarn@kfupm.edu.sa}

\thanks{Submitted May 5, 2012. Published September 7, 2012.}
\subjclass[2000]{26A33, 34K40, 35L90, 35L70, 35L15, 35L07}
\keywords{Cauchy problem; cosine family; fractional derivative; \hfill\break\indent
 mild solutions; neutral second-order abstract problem}

\begin{abstract}
 The existence of mild solutions in an appropriate space
 is established for a second-order abstract problem of neutral type
 with derivatives of non-integer order in the nonlinearity as well
 as in the initial conditions.
 We introduce some new spaces taking into account the minimum
 requirements of regularity.
\end{abstract}

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



\section{Introduction}

In this article we study the  neutral second-order abstract
differential problem
\begin{equation}
\begin{gathered}
\frac{d}{dt}[ u'(t)+g(t,u(t))]
=Au(t)+f(t,u(t),D^{\alpha }u(t)),\quad t\in I=[0,T] \\
u(0)=u^0+p(u,D^{\beta }u(t)), \\
u'(0)=u^1+q(u,D^{\gamma }u(t))
\end{gathered} \label{e1}
\end{equation}
with $0\leq \alpha ,\beta ,\gamma \leq 1$. Here the prime denotes time
differentiation and $D^{\kappa }$, $\kappa =\alpha ,\beta ,\gamma $ denotes
fractional time differentiation (in the sense of Riemann-Liouville). The
operator $A$ is the infinitesimal generator of a strongly continuous cosine
family $C(t)$, $t\geq 0$ of bounded linear operators in the
Banach space $X$ and $f$, $g$ are nonlinear functions from
$\mathbb{R}^{+}\times X\times X$ to $X$ and $\mathbb{R}^{+}\times X$ to $X$,
respectively, $u^0$ and $u^1$ are given initial data in $X$. The
functions $p:[ C(I;X)] ^{2}\to X$,
$q:[C(I;X)] ^{2}\to X$ are given continuous functions.

This problem has been studied in case $\alpha ,\beta ,\gamma $ are $0$ or $1$
(see \cite{1,2,3,4,10,11,22}).
Well-posedness has been established using different
fixed point theorems and the theory of strongly continuous cosine families
in Banach spaces. We refer the reader to \cite{6,20,21} for a good account on the
theory of cosine families. A similar problem to this one with Caputo
derivative has been studied by the present author in \cite{19}. Here the
situation with the Riemann-Liouville fractional derivative is completely
different. The singularity at zero inherent to the Riemann-Liouville
derivative brings new challenges and difficulties. Moreover the underlying
spaces are different. Indeed, Caputo derivative needs more regularity as it
uses the first derivative of the function in question in its definition
whereas the Riemann-Liouville does not require as much smoothness.

Problems with fractional derivatives are very convenient to model hereditary
phenomena in many fields of sciences and engineering
\cite{5,7,8,9,12,14,15,16,17}.
They can be used, for instance, as damping to reduce the effect of
vibrations in mechanical structures or to reduce noise in signals.

Here we consider the neutral case ($g\not{\equiv}0$) and prove existence and
uniqueness of mild solutions under different conditions on the different
data. Even in the case $g\equiv 0$ (and $p=q=0$) these conditions are
different from the ones assumed in \cite{18}. In particular, this work may be
viewed as an extension of the works in \cite{10,11} to the fractional order case
and of (18) to the neutral case.

The next section of this paper contains some notation and preliminary
results needed in our proofs. Section 3 treats the existence of a mild
solution in an appropriate ``fractional'' space.

\section{Preliminaries}

In this section we present some notation, assumptions and results needed in
our proofs later.

\begin{definition} \label{def1} \rm
The integral
\[
(I_{a+}^{\kappa }h)(x)=\frac{1}{\Gamma (\alpha )}
\int_{a}^{x}\frac{h(t)dt}{(x-t)^{1-\kappa }},\quad x>a
\]
is called the Riemann-Liouville fractional integral of $h$ of order
$\kappa >0$ when the right side exists.
\end{definition}

Here $\Gamma $ is the usual Gamma function
\[
\Gamma (z):=\int_0^{\infty }e^{-s}s^{z-1}ds,\quad z>0.
\]

\begin{definition} \label{def2} \rm
The (left hand) Riemann-Liouville fractional derivative of $h$ of order
$\kappa >0$ is defined by
\[
(D_{a}^{\kappa }h)(x)=\frac{1}{\Gamma (n-\kappa )}(\frac{d}{dx})
^{n}\int_a^x \frac{h(t)dt}{(x-t)^{\kappa -n+1}},\quad x>a,\;
n=[\kappa ]+1
\]
whenever the right side is pointwise defined.
In particular
\[
(D_{a}^{\kappa }h)(x)=\frac{1}{\Gamma (1-\kappa )}\frac{d}{dx}\int_{a}^{x}
\frac{h(t)dt}{(x-t)^{\kappa }},\quad x>a,\;0<\kappa <1
\]
and
\[
(D_{a}^{\kappa }h)(x)=\frac{1}{\Gamma (2-\kappa )}(\frac{d}{dx})
^{2}\int_{a}^{x}\frac{h(t)dt}{(x-t)^{\kappa -1}},\quad x>a,\;1<\kappa <2.
\]
\end{definition}

\begin{lemma} \label{lem1}
Let $0<\alpha$, $\beta <0$ and $\varphi \in L^1(a,b)$ be such that
$$
I^{n-\alpha }\varphi \in AC^{n}([a,b])
:=\big\{ \phi :[a,b]\to \mathbb{R}\text{ and }(D^{n-1}\phi )(x)\in AC[a,b]\big\}.
$$
 Then
\[
I_{a+}^{\alpha }I_{a+}^{\beta }\varphi =I_{a+}^{\alpha +\beta }\varphi
-\sum_{k=0}^{n-1}\frac{\varphi _{n+\beta }^{(n-k-1)}(a)}{\Gamma
(\alpha -k)}(x-a)^{\alpha -k-1}
\]
where $\varphi _{n+\beta }(x)=I_{a+}^{n+\beta }\varphi (x)$ and
$n=[-\beta ]+1$.
\end{lemma}

If $0<\kappa <1$, $n=1$ and $\gamma \leq \kappa $ then
\[
I_{a+}^{\kappa -\gamma }D_{a+}^{\kappa }\varphi =D_{a+}^{\gamma }\varphi -
\frac{\varphi _{1-\kappa }(a)}{\Gamma (\kappa -\gamma )}(x-a)^{\kappa
-\gamma -1}.
\]
See \cite{12,13,14,16,17}
for more on fractional derivatives and fractional
integrals. We will also need the following lemmas. The first one can be
found in \cite{17}.

\begin{lemma} \label{lem2}
If $h(x)\in AC^{n}[a,b]$, $\alpha >0$ and $n=[\alpha ]+1$, then
\begin{align*}
(D_{a}^{\alpha }h)(x)
&=\sum_{k=0}^{n-1}\frac{h^{(k)}(a)}{\Gamma
(1+k-\alpha )}(x-a)^{k-\alpha }+\frac{1}{\Gamma (n-\alpha )}
\int_{a}^{x}\frac{h^{(n)}(t)dt}{(x-t)^{\alpha -n+1}} \\
&=:\sum_{k=0}^{n-1}\frac{h^{(k)}(a)}{\Gamma (1+k-\alpha )}
(x-a)^{k-\alpha }+(^{C}D_{a}^{\alpha }h)(x),\quad x>a.
\end{align*}
\end{lemma}

The expression $(^{C}D_{a}^{\alpha }h)(x)$ is known as the
fractional derivative of $h$ of order $\alpha $ in the sense of Caputo.

We will assume the following condition.
\begin{itemize}
\item[(H1)] $A$ is the infinitesimal generator of a strongly continuous
cosine family $C(t)$, $t\in \mathbb{R}$, of bounded linear operators in the
Banach space $X$.
\end{itemize}

The associated sine family $S(t)$, $t\in \mathbb{R}$ is defined by
\[
S(t)x:=\int_0^{t}C(s)x\,ds,\quad t\in \mathbb{R},\;x\in X.
\]
It is known (see \cite{20,21,22}) that there exist constants
$M\geq 1$ and $\omega \geq 0$ such that
\[
| C(t)| \leq Me^{\omega | t| },\;t\in \mathbb{R}
\quad\text{and}\quad | S(t)-S(t_0)| \leq M|
\int_{t_0}^{t}e^{\omega | s| }ds| ,\;t,t_0\in
\mathbb{R}.
\]
For simplicity we will write $| C(t)| \leq \tilde{M}$ and
$| S(t)| \leq \tilde{N}$ on $I=[0,T]$ (of course $\tilde{M}\geq 1$ and
$\tilde{N}\geq 1$ depend on $T)$.

Let us define
\[
E:=\{x\in X:C(t)x\text{ is once continuously differentiable on }\mathbb{R}\}.
\]


\begin{lemma}[\cite{20,21,22}] \label{lem3}
Assume {\rm (H1)} is satisfied. Then
\begin{itemize}
\item[(i)] $S(t)X\subset E$, $t\in \mathbb{R}$,

\item[(ii)] $S(t)E\subset D(A)$, $t\in \mathbb{R}$,

\item[(iii)] $\frac{d}{dt}C(t)x=AS(t)x$, $x\in E$, $t\in \mathbb{R}$,

\item[(iv)] $\frac{d^{2}}{dt^{2}}C(t)x=AC(t)x=C(t)Ax$,
$x\in D(A)$, $t\in \mathbb{R}$.
\end{itemize}
\end{lemma}

\begin{lemma}[\cite{20,21,22}] \label{lem4}
Suppose that {\rm (H1)} holds, $v:\mathbb{R}\to X$ is a continuously
differentiable function, and $q(t)=\int_0^{t}S(t-s)v(s)\,ds$. Then,
$q(t)\in D(A)$, $q'(t)=\int_0^{t}C(t-s)v(s)\,ds$ and
$q''(t)=\int_0^{t}C(t-s)v'(s)\,ds+C(t)v(0)=Aq(t)+v(t)$.
\end{lemma}

Now we make clear what we mean by a mild solution of \eqref{e1}.

\begin{definition} \label{def3} \rm
A continuous function $u$, such that $D^{\eta }u$
($\eta =\max \{\alpha,\beta ,\gamma \})$ exists and is continuous on $I$,
satisfying the integro-differential equation
\begin{align*}
u(t)
&=C(t)[ u^0+p(u,D^{\beta }u(t))] \\
&\quad +S(t)[ u^1+q(u,D^{\gamma }u(t))-g(0,u^0+p(u,D^{\beta }u(t)))] \\
&\quad -\int_0^{t}C(t-s)g(s,u(s)) ds
+\int_0^{t}S(t-s)f(s,u(s),D^{\alpha }u(s))\,ds,\quad t\in I
\end{align*}
is called a mild solution of problem \eqref{e1}.
\end{definition}

\section{Existence of mild solutions}

In this section we prove the existence and uniqueness of a mild solution
in the space
\begin{equation}
C_{\eta }^{RL}([0,T]):=\{ v\in C([0,T]):D^{\eta }v\in C([0,T])\}
\label{e2}
\end{equation}
equipped with the norm $\| v\| _{\eta }:=\| v\|
_{C}+\| D^{\eta }v\| _{C}$ where $\| .\| _{C}$ is the
sup norm in $C([0,T])$ and $\eta =\max \{\alpha ,\beta ,\gamma \}$. For the
initial data we define
\begin{equation}
E_{\eta }:=\{x\in X:D^{\eta }C(t)x\text{ is continuous on }\mathbb{R}^{+}\}.
\label{e3}
\end{equation}

\begin{lemma} \label{lem5}
If $R(t)$ is a linear operator such that $I^{1-\nu }R(t)x\in C^1([0,T])$,
$T>0$, then, for $0<\nu <1$, we have
\[
D^{\nu }\int_0^{t}R(t-s)x\,ds=\int_0^{t}D^{\nu
}R(t-s)x\,ds+\lim_{t\to 0^{+}}I^{1-\nu }R(t)x,\quad
x\in X,\; t\in [0,T].
\]
\end{lemma}


\begin{proof}
By Definition \ref{def2} and Fubini's theorem we have
\begin{align*}
&D^{\nu }\int_0^{t}R(t-s)x\,ds\\
&=\frac{1}{\Gamma (1-\nu )}\frac{d}{dt}
\int_0^{t}\frac{d\tau }{(t-\tau )^{\nu }}\int_0^{\tau
}R(\tau -s)x\,ds \\
&=\frac{1}{\Gamma (1-\nu )}\frac{d}{dt}\int_0^{t}ds
\int_{s}^{t}\frac{R(\tau -s)x}{(t-\tau )^{\nu }}d\tau \\
&=\frac{1}{\Gamma (1-\nu )}\int_0^{t}ds\frac{\partial }{\partial t}
\int_{s}^{t}\frac{R(\tau -s)x}{(t-\tau )^{\nu }}d\tau
+\frac{1}{\Gamma (1-\nu )}\lim_{s\to t^{-}}\int_{s}^{t}\frac{R(\tau
-s)x}{(t-\tau )^{\nu }}d\tau .
\end{align*}
These steps are justified by the assumption
$I^{1-\nu }R(t)x\in C^1([0,T])$.
 Moreover, a change of variable $\sigma =\tau -s$ leads to
\begin{align*}
&D^{\nu }\int_0^{t}R(t-s)x\,ds\\
&=\frac{1}{\Gamma (1-\nu )}
\int_0^{t}ds\frac{\partial }{\partial t}\int_0^{t-s}
\frac{R(\sigma )x}{(t-s-\sigma )^{\nu }}d\sigma 
 +\frac{1}{\Gamma (1-\nu )}\lim_{t\to 0^{+}}\int_0^{t}
\frac{R(\sigma )x}{(t-\sigma )^{\nu }}d\sigma .
\end{align*}
This is exactly the formula stated in the lemma.
\end{proof}


\begin{lemma} \label{lem6}
If $g$ is a continuous function such that $I^{1-\nu }g(t)\in C^1([0,T])$,
$T>0$ and $R(t)$ is continuous, then, for
$0<\nu <1$, we have
\[
D^{\nu }\int_0^{t}R(t-s)g(s)\,ds=\int_0^{t}R(t-s)D^{\nu}g(s)\,ds,\quad t\in [0,T].
\]
\end{lemma}



\begin{proof}
By Definition \ref{def2}, we have
\begin{align*}
D^{\nu }\int_0^{t}R(t-s)g(s)\,ds
&=\frac{d}{dt}I^{1-\nu}\int_0^{t}R(t-s)g(s)\,ds \\
&=\frac{d}{dt}I^{1-\nu }\int_0^{t}R(s)g(t-s)\,ds\\
&=\frac{1}{\Gamma(1-\nu )}\frac{d}{dt}\int_0^{t}\frac{d\tau }{(t-\tau )^{\nu }}
\int_0^{\tau }R(s)g(\tau -s)\,ds.
\end{align*}
Then, Fubini's theorem and the continuity of $g$ allow us to write
\begin{align*}
D^{\nu }\int_0^{t}R(t-s)g(s)\,ds
&=\frac{1}{\Gamma (1-\nu )}\frac{d}{
dt}\int_0^{t}R(s)\int_{s}^{t}\frac{g(\tau -s)}{(t-\tau)^{\nu }}d\tau ds \\
&=\frac{1}{\Gamma (1-\nu )}\frac{d}{dt}\int_0^{t}R(s)
\int_0^{t-s}\frac{g(\sigma )}{(t-s-\sigma )^{\nu }}d\sigma ds\\
&=\int_0^{t}R(s)\frac{d}{dt}I^{1-\nu }g(t-s)\,ds \\
&=\int_0^{t}R(s)D^{\nu }g(t-s)\,ds
\end{align*}
which is the desired relation. Note that we have used the continuity of $g$
to deduce that the value (or the limit) of the inner integral in the second
line of the relation is zero at $s=t$.\end{proof}


\begin{corollary} \label{coro7}
For the sine family $S(t)$ associated with the cosine family $C(t)$,
$x\in X$, $t\in [0,T]$ and $0<\nu <1$, we have
\[
D^{\nu }\int_0^{t}S(t-s)x\,ds=\int_0^{t}D^{\nu
}S(t-s)x\,ds=\int_0^{t}I^{1-\nu }C(t-s)x\,ds.
\]
\end{corollary}

\begin{proof}
First, from Lemma \ref{lem2} as $S(t)x$ is absolutely continuous on $[0,T]$, we have
\begin{align*}
\frac{d}{dt}I^{1-\nu }S(t)x
&=D^{\nu }S(t)x=\frac{1}{\Gamma (1-\nu )}\Big[
\frac{S(0)x}{t^{\nu }}+\int_0^{t}(t-s)^{-\nu }\frac{dS(s)}{ds}x\,ds\Big] \\
&=\frac{1}{\Gamma (1-\nu )}\int_0^{t}(t-s)^{-\nu }C(s)x\,ds\\
&=I^{1-\nu }C(t)x.
\end{align*}

Now from the continuity of $C(t)$ it is clear that $I^{1-\nu }C(t)x$ is
continuous on $[0,T]$ and therefore $I^{1-\nu }S(t)x\in C^1([0,T])$. We
can therefore apply Lemma \ref{lem4} to obtain
\[
D^{\nu }\int_0^{t}S(t-s)x\,ds=\int_0^{t}D^{\nu}S(t-s)x\,ds
+\lim_{t\to 0^{+}}I^{1-\nu }S(t)x,\quad x\in X,\;t\in [0,T].
\]
Next, we claim that $\lim_{t\to 0^{+}}I^{1-\nu }S(t)x=0$. This
follows easily from
\[
| I^{1-\nu }S(t)x| \leq \frac{1}{\Gamma (1-\nu )}
\int_0^{t}(t-s)^{-\nu }| S(s)x| ds
\leq \frac{t^{1-\nu }}{\Gamma (2-\nu )}\sup_{0\leq t\leq T}| S(t)x| .
\]
\end{proof}


On the functins  $f,g$, $p$ and $q$ we assume the following conditions.
\begin{itemize}

\item[(H2)] 
\begin{itemize}
\item[(i)] $f(t,.,.):X\times X\to X$ is continuous for a.e.
$t\in I$.

\item[(ii)] For every $(x,y)\in X\times X$, the function $f(.,x,y):I\to X$
is strongly measurable.

\item[(iii)] There exists a nonnegative continuous function $K_{f}(t)$ and a
continuous nondecreasing positive function $\Omega _{f}$ such that
\[
\| f(t,x,y)\| \leq K_{f}(t)\Omega _{f}(\| x\|
+\| y\| )
\]
for $(t,x,y)\in I\times X\times X$.

\item[(iv)] For each $r>0$, the set $f(I\times B_{r}(0,X^{2}))$ is relatively
compact in $X$.
\end{itemize}
\item[(H3]
\begin{itemize}
\item[(i)] $g\in C_{\eta }^{RL}([0,T])$.

\item[(ii)] There exist a nonnegative continuous function $K_{g}(t)$ and a
continuous nondecreasing positive function $\Omega _{g}$ such that
\[
\| g(t,x)\| _{\eta }\leq K_{g}(t)\Omega _{g}(\|
x\| )
\]
for $(t,x)\in I\times X$.
\item[(iii)] The family of functions $\{t\to g(t,u)\, u\in B_{r}(0,C(I;X))\}$
 is equicontinuous on $I$.

\item[(iv)] For each $r>0$, the set $g(I\times B_{r}(0,X))$ is relatively compact
in $X$.
\end{itemize}

\item[(H4)] $u^0+p:[ C(I;X)] ^{2}\to E_{\eta }$
(takes its values in $E_{\eta }$, see \eqref{e3}) and
$q:[ C(I;X)]^{2}\to X$ are completely continuous.

\end{itemize}
The positive constants $N_{p}$, $N_{q}$, $\tilde{N}_{p}$ and $N_{g}$ will
denote bounds for $\| u^0+p(u,v)\| $, $\| q(u,v)\|$,
$\| D^{\eta }C(t)[ u^0+p(u,I^{1-\beta }v(t))] \| $
and the term $\| g(0,u^0+p(u,I^{\eta -\beta }v(t)))
\| $, respectively. Note that
$\| D^{\eta }C(t)[u^0+p(u,I^{1-\beta }v(t))] \| $ is finite by the
assumption in the next theorem. By $B_{r}(x,X)$ we will denote the closed
ball in $X$ centered at $x$ and of radius $r$. Let also
\begin{gather}
C_{1}:=\tilde{M}N_{p}
+\Big(\tilde{N}+\frac{\tilde{M}T^{1-\eta }}{\Gamma (2-\eta )}\Big)N_{q}
+\Big(\tilde{N}+\frac{\tilde{M}T^{1-\eta }}{\Gamma (2-\eta )}\Big)
(\| u^1\| +N_{g})+\tilde{N}_{p}\label{e4}
\\
C_3:=\max \Big\{ 1,\frac{T^{\eta -\alpha }}{\Gamma (\eta -\alpha +1)}\Big\} ,\quad
C_2:=C_{1}C_3  \label{e5}
\\
H(t):=2\tilde{M}K_{g}(t)+\tilde{N}K_{f}(t)+\tilde{M}(I^{1-\eta }K_{f})(t).
\label{e6}
\end{gather}

We are now ready to state and prove our main result.

\begin{theorem} \label{thm1}
Assume that {\rm (H1)--(H4)} hold. If $0<\eta <1$ and
\[
\int_0^{t}H(s)\,ds<\frac{1}{C_3}\int_{C_2}^{\infty }
\frac{ds}{\Omega _{f}(s)+\Omega _{g}(s)}
\]
then problem \eqref{e1} admits a mild solution $u\in C_{\eta }^{RL}([0,T])$.
\end{theorem}

\begin{proof}
Note that by our assumptions and for $u\in C_{\eta }^{RL}([0,T])$
(see \eqref{e2}), the maps
\begin{equation}
\begin{aligned}
\Phi (u,v)(t)
&:=C(t)[ u^0+p(u,I^{\eta -\beta }v(t))] \\
&\quad +S(t)[ u^1+q(u,I^{\eta -\gamma }v(t))-g(0,u^0+p(u,I^{\eta
-\beta }v(t)))] \\
&\quad -\int_0^{t}C(t-s)g(s,u(s))
ds+\int_0^{t}S(t-s)f(s,u(s),I^{\eta -\alpha
}v(s))\,ds,
\end{aligned}
\label{e7}
\end{equation}
for $t\in I$, and
\begin{equation}
\begin{aligned}
\Psi (u,v)(t)
&:=D^{\eta }C(t)[ u^0+p(u,I^{\eta -\beta }v(t))] \\
&\quad +D^{\eta }S(t)[ u^1+q(u,I^{\eta -\gamma }v(t)
)-g(0,u^0+p(u,I^{\eta -\beta }v(t)))] \\
&\quad -\int_0^{t}C(t-s)D^{\eta }g(s,u(s))\,ds
\\
&\quad +\int_0^{t}I^{1-\eta }C(t-s)f(s,u(s),I^{\eta
-\alpha }v(s))\,ds,\quad t\in I
\end{aligned}
\label{e8}
\end{equation}
are well-defined, and map $[ C([0,T])] ^{2}$ into $C([0,T])$. We
obtained these mappings from the definition of a mild solution taking into
account Lemma \ref{lem6}
(note that $I^{1-\eta }g\in C^1([0,T])$ by our assumption (H3)(i))
and Corollary \ref{coro7}.
In addition to that we passed from $D^{\kappa }u$,
$\kappa =\alpha ,\beta ,\gamma $ to $D^{\eta }u=:v$ through the formula
$$
D_{a+}^{\kappa }u(t)=I_{a+}^{\eta -\kappa }D_{a+}^{\eta }u(t)(t)
+\frac{u_{1-\eta }(0)}{\Gamma (\eta -\kappa )}t^{\eta -\kappa -1}
$$
and noticing that $u_{1-\eta }(0)=0$.

We would like to apply the Leray-Schauder Alternative (which states that
either the set of solutions (below) is unbounded or we have a fixed point
in $D$ (containing zero) a convex subset of $X$ provided that the mappings
$\Phi $ and $\Psi $ are completely continuous). To this end we first prove
that the set of solutions $(u_{\lambda },v_{\lambda })$ of
\begin{equation}
(u_{\lambda },v_{\lambda })
=\lambda (\Phi (u_{\lambda },v_{\lambda }),\Psi
(u_{\lambda },v_{\lambda })),\;0<\lambda <1  \label{e9}
\end{equation}
is bounded. Then, we prove that this map is completely continuous. Therefore
there remains the alternative which is the existence of a fixed point. We
observe first from \eqref{e7} that
\begin{align*}
\| u_{\lambda }(t)\|
&\leq \tilde{M}N_{p}+\tilde{N}(\|
u^1\| +N_{q}+N_{g})+\tilde{M}\int_0^{t}K_{g}(s)
\Omega _{g}(\| u_{\lambda }(s)\| )\,ds \\
&\quad +\tilde{N}\int_0^{t}K_{f}(s)\Omega _{f}(\| u_{\lambda
}(s)\| +\frac{s^{\eta -\alpha }}{\Gamma (\eta -\alpha +1)}\sup_{0\leq
\tau \leq s}\| v_{\lambda }(\tau )\| )\,ds
\end{align*}
and from \eqref{e8},
\begin{align*}
&\| v_{\lambda }(t)\|\\
&\leq \tilde{N}_{p}+\frac{\tilde{M}t^{1-\eta }}{\Gamma (2-\eta )}
[ \| u^1\| +N_{q}+N_{g}]
 +\tilde{M}\int_0^{t}K_{g}(s)\Omega _{g}(\| u_{\lambda }(s)\| )\,ds \\
&\quad +\tilde{M}\int_0^{t}(I^{1-\eta }K_{f})(s)\Omega _{f}
\Big(\sup_{0\leq \tau \leq s}\| u_{\lambda }(\tau )\|
 +\frac{s^{\eta -\alpha }}{\Gamma (\eta -\alpha +1)}\sup_{0\leq \tau \leq s}\|
v_{\lambda }(\tau )\| \Big)\,ds
\end{align*}
where $\tilde{N}_{p}$ and $N_{g}$ are bounds for the expressions
$D^{\eta }C(t)[ u^0+p(u,I^{\eta -\beta }v(t))] $ and
$g( 0,u^0+p(u,I^{\eta -\beta }v(t)))$, respectively. Clearly
\begin{equation}
\begin{aligned}
&\| u_{\lambda }(t)\| +\| v_{\lambda }(t)\| \\
&\leq C_{1}+2\tilde{M}\int_0^{t}K_{g}(s)\Omega _{g}(\| u_{\lambda}(s)\| )\,ds 
 +\int_0^{t}(\tilde{N}K_{f}(s)+\tilde{M}(I^{1-\eta}K_{f})(s))\\
&\quad \times \Omega _{f}
\Big(\sup_{0\leq \tau \leq s}\| u_{\lambda }(\tau
)\| +\frac{s^{\eta -\alpha }}{\Gamma (\eta -\alpha +1)}\sup_{0\leq
\tau \leq s}\| v_{\lambda }(\tau )\| \Big)\,ds
\end{aligned} \label{e10}
\end{equation}
where $C_{1}$ is given in \eqref{e4}.
If we put
$$
\Theta _{\lambda }(t)=\max \{1,T^{\eta -\alpha }/\Gamma (\eta -\alpha +1)\}
\sup_{0\leq \tau \leq t}(\| u_{\lambda }(\tau )\| +\| v_{\lambda }(\tau
)\| ),
$$ then \eqref{e10} yields
\begin{equation}
\begin{aligned}
\Theta _{\lambda }(t)
&\leq \max \{1,T^{\eta -\alpha }/\Gamma (\eta -\alpha +1)\}
\Big\{ C_{1}+2\tilde{M}\int_0^{t}K_{g}(s)\Omega _{g}(
\Theta _{\lambda }(s))\,ds \\
&\quad +\int_0^{t}\Big(\tilde{N}K_{f}(s)+\tilde{M}(I^{1-\eta
}K_{f})(s)\Big)\Omega _{f}(\Theta _{\lambda }(s))\,ds\Big\}
\\
&\leq \max \big\{ 1,\frac{T^{\eta -\alpha }}{\Gamma (\eta -\alpha +1)}\big\}
\Big\{ C_{1}+\int_0^{t}H(s)[ \Omega _{g}(
\Theta _{\lambda }(s))+\Omega _{f}(\Theta _{\lambda }(s))
] ds\Big\} \\
&\leq C_2+C_3\int_0^{t}H(s)[ \Omega _{g}(\Theta
_{\lambda }(s))+\Omega _{f}(\Theta _{\lambda }(s))] ds
\end{aligned}\label{e11}
\end{equation}
where $H(s)$, $C_2$ and $C_3$ are as in the paragraph preceding the
statement of the theorem (see \eqref{e5} and \eqref{e6}).
Let us denote by $\varphi _{\lambda }(t)$ the right hand side of
\eqref{e11}. Then
$\varphi _{\lambda}(0)=C_2$,
 $\Theta _{\lambda }(t)\leq \varphi _{\lambda }(t)$ and
\[
\varphi _{\lambda }'(t)\leq C_3H(t)[ \Omega _{g}(
\varphi _{\lambda }(t))+\Omega _{f}(\varphi _{\lambda}(t))] ,\quad t\in I.
\]
We infer that
\begin{equation}
\int_{C_2}^{\varphi _{\lambda }(t)}\frac{ds}{\Omega
_{f}(s)+\Omega _{g}(s)}\leq C_3\int_0^{t}H(s)\,ds.  \label{e12}
\end{equation}
This relation, together with our hypotheses, shows that
$\Theta_{\lambda }(t)$ and thereafter the set of solutions of \eqref{e9}
is bounded in $[ C(I;X)] ^{2}$.

It remains to show that the maps $\Phi $ and $\Psi $ are completely
continuous. From our hypotheses it is immediate that
\begin{align*}
\Phi _{1}(u,v)(t)
&:=C(t)[ u^0+p(u,I^{\eta -\beta }v(t))] \\
&\quad +S(t)[ u^1+q(u,I^{\eta -\gamma }v(t))
-g(0,u^0+p(u,I^{\eta -\beta }v(t)))]
\end{align*}
is completely continuous. To apply Ascoli-Arzela Theorem we need to check
that
\[
(\Phi -\Phi _{1})(B_{r}^{2}):=\{(\Phi -\Phi _{1})(u,v):(u,v)\in B_{r}^{2}\}
\]
is equicontinuous on $I$. Let us observe that
\begin{align*}
&\| (\Phi -\Phi _{1})(u,v)(t+h)-(\Phi -\Phi _{1})(u,v)(t)\| \\
&\leq \int_0^{t}\| (C(t+h-s)-C(t-s))g(s,u(s))\| ds \\
&\quad +\int_{t}^{t+h}\| C(t+h-s)g(s,u(s)) \| ds \\
&\quad +\int_0^{t}\| (S(t+h-s)-S(t-s))f(s,u(s),I^{\eta -\alpha }v(s))\| ds \\
&\quad +\int_{t}^{t+h}\| S(t+h-s)f(s,u(s),I^{\eta-\alpha }v(s))\| ds.
\end{align*}
By (H1) and (H3), for $t\in I$ and $\varepsilon >0$ given, there
exists $\delta >0$ such that
\[
\| (C(s+h)-C(s))g(t-s,u(t-s))\| <\varepsilon
\]
for $s\in [0,t]$ and $u\in B_{r}$, when $| h| <\delta $. This
together with (H2), (H3) and the fact that $S(t)$ is Lipschitzian imply that
\begin{align*}
&\| (\Phi -\Phi _{1})(u,v)(t+h)-(\Phi -\Phi _{1})(u,v)(t)\|\\
& \leq \varepsilon t+\tilde{M}\Omega _{g}(r)
\int_{t}^{t+h}K_{g}(s)\,ds \\
&\quad +N_1h\Omega _{f}
\Big(r+\frac{T^{\eta -\alpha }r}{\Gamma (\eta -\alpha +1)}\Big)
\int_0^{t}K_{f}(s)\,ds\\
&\quad +\tilde{N}\Omega _{f}\Big(r+\frac{T^{\eta -\alpha }r}{\Gamma (\eta -\alpha +1)}\Big)
\int_{t}^{t+h}K_{f}(s)\,ds
\end{align*}
for some positive constant $N_1$. The equicontinuity is therefore
established.

On the other hand, for $t\in I$, as $(s,\xi )\to C(t-s)\xi $ is
continuous from $[0,t]\times \overline{g(I\times X)}$ to $X$ and
$[0,t]\times \overline{g(I\times X)}$ is relatively compact in $X$.
The set
\[
\big\{ \Phi _2u(t):=\int_0^{t}C(t-s)g(s,u(s))\,ds,\;u\in
B_{r}(0,X)\big\}
\]
is relatively compact in $X$ as well. We infer that $\Phi _2$ is
completely continuous. As for $\Phi _3:=\Phi -\Phi _{1}+\Phi _2$ we
decompose it as follows
\begin{align*}
\Phi_3(u,v)(t)
&=\sum_{i=1}^{k-1}\int_{s_i}^{s_{i+1}}(S(s)-S(s_i))f(t-s,u(t-s)
,I^{\eta -\alpha }v(t-s))\,ds \\
&\quad +\sum_{i=1}^{k-1}\int_{s_i}^{s_{i+1}}S(s_i)f(
t-s,u(t-s),I^{\eta -\alpha }v(t-s))\,ds
\end{align*}
and select the partition $\{s_i\}_{i=1}^{k}$ of $[0,t]$ in such a manner
that, for a given $\varepsilon >0$
\[
\| (S(s)-S(s'))f(t-s,u(t-s),I^{\eta -\alpha
}v(t-s))\| <\varepsilon ,
\]
for $(u,v)\in B_{r}^{2}(0,X)$, when $s,s'\in [s_i,s_{i+1}]$ for
some $i=1,\dots,k-1$. This is possible in as much as
\[
\{f(t-s,u(t-s),I^{\eta -\alpha }v(t-s)
),\;s\in [0,t],\;(u,v)\in B_{r}^{2}(0,X)\}
\]
is bounded (by (H2) (iii)) and the operator $S$ is uniformly Lipschitz on $I$.
This leads to
\[
\Phi _3(u,v)(t)\in \varepsilon
B_{T}(0,X)+\sum_{i=1}^{k-1}(s_{i+1}-s_i)\overline{\operatorname{co}(U(t,s_i,r))}
\]
where
\begin{align*}
&U(t,s_i,r) \\
&:=\big\{S(s_i)f(t-s,u(t-s),I^{\eta -\alpha }v(
t-s)),\;s\in [0,t],\;(u,v)\in B_{r}^{2}(0,X)\big\}
\end{align*}
and $\operatorname{co}U(t,s_i,r))$ designates its convex hull.
Therefore $\Phi _3(B_{r}^{2})(t)$ is relatively compact in $X$.
By Ascoli-Arzela Theorem, $\Phi _3(B_{r}^{2})$ is relatively compact in
$C(I;X)$ and consequently $\Phi _3$ is completely continuous.
Similarly we may prove that $\Psi $ is completely continuous.

We conclude that $(\Phi ,\Psi )$ admits a fixed point in
$[C([0,T])] ^{2}$.
\end{proof}


\begin{remark} \label{rmk1} \rm
In the same way we may treat the more general case
\begin{gather*}
\frac{d}{dt}[ u'(t)+g(t,u(t))]
=Au(t)+f(t,u(t),D^{\alpha _{1}}u(t),\dots,D^{\alpha _{n}}u(t)), \\
u(0)=u^0+p(u,D^{\beta _{1}}u(t),\dots,D^{\beta _{m}}u(t)), \\
u'(0)=u^1+q(u,D^{\gamma _{1}}u(t),\dots,D^{\gamma _{r}}u(t))
\end{gather*}
where $0\leq \alpha _i,\beta _j,\gamma _k\leq 1$, $i=1,\dots,n$,
$j=1,\dots,m$, $k=1,\dots,r$.
\end{remark}

\subsection*{Example}
Consider the  problem
\begin{equation}
\begin{gathered}
\begin{aligned}
&\frac{\partial }{\partial t}[ u_{t}(t,x) +G(t,x,u(t,x))] \\
&=u_{xx}(t,x) +F(t,x,u(t,x),D^{\alpha }u(t,x)),\quad t\in I=[0,T],\;x\in [a,b]
\end{aligned}
\\
u(t,a)=u(t,b)=0,\quad t\in I \\
u(0,x)=u^0(x)+\int_0^TP(u(s),D^{\beta}u(s))(x)\,ds,\quad x\in [a,b] \\
u'(0)=u^1(x)+\int_0^TQ(u(s),D^{\gamma }u(s))(x)\,ds,\quad x\in [a,b]
\end{gathered}  \label{e13}
\end{equation}
in the space $X=L^{2}([0,\pi ])$. This problem can be reformulated in the
abstract setting \eqref{e1}. To this end we define the operator
 $Ay=y''$ with domain
\[
D(A):=\{y\in H^{2}([0,\pi ]):y(0)=y(\pi )=0\}.
\]
The operator $A$ has a discrete spectrum with $-n^{2}$, $n=1,2,\dots$. as
eigenvalues and $z_{n}(s)=\sqrt{2/\pi }\sin (ns)$, $n=1,2,\dots $, as their
corresponding normalized eigenvectors. So we may write
\[
Ay=-\sum_{n=1}^{\infty }n^{2}(y,z_{n})z_{n},\quad y\in D(A).
\]
Since $-A$ is positive and self-adjoint in $L^{2}([0,\pi ])$, the operator
$A $ is the infinitesimal generator of of a strongly continuous cosine family
$C(t)$, $t\in \mathbb{R}$ which has the form
\[
C(t)y=\sum_{n=1}^{\infty }\cos (nt)(y,z_{n})z_{n},\quad y\in X.
\]
The associated sine family is
\[
C(t)y=\sum_{n=1}^{\infty }\frac{\sin (nt)}{n}(y,z_{n})z_{n},\quad y\in X.
\]
One can also consider more general non-local conditions by allowing the
Lebesgue measure $ds$ to be of the form $d\mu (s)$ and $d\eta (s)$ for
non-decreasing functions $\mu $ and $\eta $ (or even more general: $\mu $
and $\eta $ of bounded variation); that is,
\begin{gather*}
u(0,x)=u^0(x)+\int_0^TP(u(s),D^{\beta}u(s))(x)d\mu (s), \\
u_{t}(0,x)=u^1(x)+\int_0^TQ(u(s),D^{\gamma}u(s))(x)d\eta (s).
\end{gather*}
These (continuous) non-local conditions cover, of course, the discrete cases
\begin{gather*}
u(0,x)=u^0(x)+\sum_{i=1}^{n}\alpha _iu(t_i,x)
 +\sum_{i=1}^{m}\beta _i{}D^{\beta }u(t_i,x), \\
u_{t}(0,x)=u^1(x)+\sum_{i=1}^{r}\gamma _iu(t_i,x)
+\sum_{i=1}^{k}\lambda _iD^{\gamma }u(t_i,x)
\end{gather*}
which have been extensively studied by several authors in the integer order
case.

For $u,v\in C([0,T];X)$ and $x\in [a,b]$, defining the operators
\begin{gather*}
p(u,v)(x):=\int_0^TP(u(s),v(s))(x)\,ds, \\
q(u,v)(x):=\int_0^TQ(u(s),v(s))(x)\,ds, \\
g(t,u)(x):=G(t,x,u(t,x)), \\
f(t,u,v)(x):=F(t,x,u(t,x),v(t,x)),
\end{gather*}
allows us to write \eqref{e13} abstractly as
\begin{gather*}
\frac{d}{dt}[ u'(t)+g(t,u(t),u'(t))] =Au(t)+f(t,u(t),D^{\alpha }u(t)), \\
u(0)=u^0+p(u,D^{\beta }u(t)), \\
u'(0)=u^1+q(u,D^{\gamma }u(t)).
\end{gather*}
Under appropriate conditions on $F$, $G$, $P$ and $Q$ which make
(H2)--(H4) hold for the corresponding functions $f$, $g$, $p$ and $q$, 
Theorem \ref{thm1}
ensures the existence of a mild solution to problem \eqref{e13}.

Some special cases of this problem may be found in models of some phenomena
with hereditary properties (see \cite{5,7,8,9,15}).

\subsection*{Acknowledgments}
 The author is very grateful for the financial
support provided by King Fahd University of Petroleum and Minerals through
the project No. IN 100007.

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\end{document}