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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 134, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/134\hfil Asymptotic behavior of solutions]
{Asymptotic power type behavior of solutions to a nonlinear
fractional integro-differential equation}

\author[A. M. Ahmad, K. M. Furati, N.-E. Tatar \hfil EJDE-2017/134\hfilneg]
{Ahmad M. Ahmad, Khaled M. Furati, Nasser-Eddine Tatar}

\address{Ahmad M. Ahmad \newline
King Fahd University of Petroleum and Minerals,
Department of Mathematics and Statistics,
Dhahran 31261, Saudi Arabia}
\email{mugbil@kfupm.edu.sa}

\address{Khaled M. Furati \newline
King Fahd University of Petroleum and Minerals,
Department of Mathematics and Statistics,
Dhahran 31261, Saudi Arabia}
\email{kmfurati@kfupm.edu.sa}

\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}

\dedicatory{Communicated by Mokhtar Kirane}

\thanks{Submitted April 27, 2017. Published May 17, 2017}
\subjclass[2010]{35B40, 34A08, 26D10}
\keywords{Asymptotic behavior; fractional integro-differential equation;
\hfill\break\indent Riemann-Liouville fractional derivative; nonlocal source; 
integral inequalities}

\begin{abstract}
 This article concerns a general fractional differential equation of
 order between 1 and 2. We consider the cases where the nonlinear term
 contains or does not contain other (lower order) fractional derivatives
 (of Riemann-Liouville type). Moreover, the nonlinearity involves also
 a nonlinear non-local in time term. The case where this non-local term
 has a singular kernel is treated as well. It is proved, in all these
 situations, that solutions approach power type functions at infinity.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

We consider the  initial value problem
\begin{equation}
\begin{gathered}
( D_{0^{+}}^{\alpha +1}y) (t)=f\Big( t,(D_{0^{+}}^{\beta
}y)(t),\int_0^{t}k( t,s,( \!D_{0^{+}}^{\gamma }y)
(s)) ds\Big) ,\quad t>0, \\
( I_{0^{+}}^{1-\alpha }y) (0^{+})=a_1,\quad
(D_{0^{+}}^{\alpha }y) (0^{+})=a_2,\quad a_1,a_2\in\mathbb{R},
\end{gathered} \label{myprblm}
\end{equation}
where $D_{0^{+}}^{\alpha +1},D_{0^{+}}^{\beta }$ and $D_{0^{+}}^{\gamma }$
are the Riemann-Liouville fractional derivatives of orders $\alpha +1$,
$\beta $ and $\gamma $, respectively, $0\leq \beta \leq \alpha <1$ and
$0\leq \gamma \leq \alpha <1$. The definition of the Riemann-Liouville fractional
derivative is given in the next section. Notice that
$D_{0^{+}}^{\alpha+1}=DD_{0^{+}}^{\alpha }=( D_{0^{+}}^{\alpha }) '$,
$0<\alpha <1$.

We study the asymptotic behavior of solutions of this nonlinear fractional
integro-differential problem. Different types of the nonlinear function $f$
and the kernel $k$ are discussed. In this regard, we consider the case of
fractional and non-fractional source terms and also the case of singular
kernels.

It is of great importance to have an idea about the behavior of solutions
for large values of the time variable. Unfortunately, relatively few
problems only can be solved explicitly. Therefore there is a need to find
analytical techniques which allow us to explore the behavior of solutions
without solving the differential equations. The study of asymptotically
linear solutions to linear and nonlinear ordinary differential equations is
important in many fields like fluid mechanics, differential geometry,
bidimensional gravity, Jacobi fields, etc. see e.g. \cite{Lang1999}.

In many cases, the main idea to study the asymptotic behavior of solutions
is to establish sufficient reasonable conditions ensuring comparison or
similarity with the long-time behavior of solutions of simpler differential
equations. This important issue has attracted many researchers, see
\cite{Halanay1965,Kusano1985,Mustafa2006a,Philos2005,Rogovchenko2000}.

Recently, some papers discussed the issue of asymptotic behavior for some
types of fractional differential equations, see
\cite{Baleanu2011a,Brestovanska2014,Furati2007,Grace2015,Medved2015,Mustafa2009}.
In 2004, Momani, et al.\ \cite{Momani2004} discussed the Lyapunov
stability and asymptotic stability for solutions of the fractional
integro-differential equation
\begin{equation}
( D_{a^{+}}^{\alpha }y) (t)=f(t,y(t))+\int_{a}^{t}k(t,s,y(s))ds,\quad
0<\alpha \leq 1,\; t\geq a,\label{E2}
\end{equation}
with the initial condition $( I_{a^{+}}^{1-\alpha }y)(a^{+})=c_0\in
\mathbb{R}$. The assumptions
\begin{gather*}
| f(t,y(t))| \leq \gamma (t)| y| , \\
\int_{s}^{t}k(\sigma ,s,y(s))d\sigma \leq \delta (t)| y| , \quad
 s\in [ a,t],
\end{gather*}
where $\gamma (t)$ and $\delta (t)$ are continuous nonnegative functions and
\begin{equation*}
\sup_{t\geq a}\int_{a}^{t}(t-s)^{\alpha -1}[\gamma (s)+\delta (s)]ds<\infty ,
\end{equation*}
were imposed. The authors proved that every solution $y(t)$ of \eqref{E2}
satisfies
\begin{equation*}
| y(t)| \leq \frac{| c_0| }{\Gamma
(\alpha )}(t-a)^{\alpha -1}\exp \Big\{ \frac{1}{\Gamma (\alpha )}
\int_{a}^{t}(t-s)^{\alpha -1}[\gamma (s)+\delta (s)]ds\Big\} <\infty ,
\end{equation*}
and if
\begin{equation*}
\int_{a}^{t}(t-s)^{\alpha -1}[\gamma (s)+\delta (s)]ds=O( (t-a)^{\alpha
-1}) ,
\end{equation*}
then $| y(t)| \leq C_0(t-a)^{\alpha -1}$ where $C_0$
is a positive constant, and hence the solution of \eqref{E2} is
asymptotically stable.

Furati and Tatar \cite{Furati2005} considered  \eqref{E2} subject
to the initial condition
\begin{equation*}
\lim_{t\to a^{+}}( t^{1-\alpha }y(t)) =b,\quad b\in\mathbb{R},\;
0<\alpha <1,\; a=0,
\end{equation*}
and showed that solutions decay polynomially for some nonlinear functions
 $f$ and $k$. When $k\equiv 0$, they proved in \cite{Furati2005a}\ that solutions
of the problem exist globally and decay as a power function in the space
$C_{1-\alpha }^{\alpha }[ 0,\infty ) $ defined in
 \eqref{cca}, see Section \ref{sec2}. In 2007, the same authors considered in
\cite{Furati2007} the equation \eqref{E2} and found uniform bounds for
solutions and also provided sufficient conditions assuring decay of power
type for the solutions.

In 2015, Medve\v{d} and Posp\'{\i}\v{s}il considered in the paper
\cite{Medved2015} a more general case when the right-hand side depends
on Caputo fractional derivatives of the solution. They proved that there exists a
constant $b\in \mathbb{R}$ such that any global solution of the initial
value problem
\begin{gather*}
( ^{C}\!D_{a^{+}}^{\alpha }x) ( t)
=f\Big( t,x(t) ,x'( t) ,\dots ,x^{( n-1) }(t) ,( ^{C}\!D_{a^{+}}^{\alpha _1}x) ( t)
,\dots ,( ^{C}\!D_{a^{+}}^{\alpha _{m}}x) ( t) \Big) , \\
x^{( i) }( a) =c_i,\quad i=0,1,\dots ,n-1,\; n\in\mathbb{N},
\end{gather*}
where $t\geq a$ and $n-1<\alpha _{j}<\alpha <n$,
$j=1,2,\dots ,m$, $m\in\mathbb{N}$, is asymptotic to $bt^{r}$ with
$r=\max \{ n-1,\alpha _{m}\} $.

To the best of our knowledge, there are no similar investigations on the
asymptotic behavior of solutions for fractional integro-differential
equations of type \eqref{myprblm}.

There is a great volume of literature on the well-posedness for various
classes of fractional differential and integro-differential equations;
see \cite{Agarwal2010,Agarwal2013,Aghajani2012,Ahmad2011,Baleanu2013,
Hernandez2010,Sitho2015,Tatar2010,Wu2009}.
In fact most of the analytical investigations are on existence and
uniqueness. Several nonlinearities of the form
\begin{equation*}
f(t,y),\quad
f(t,y,D_{0^{+}}^{\beta }y),\quad
f\Big(t,y,D_{0^{+}}^{\beta }y,\int_0^{t}
k(s,t,D_{0^{+}}^{\gamma}y(s))ds\Big) ,
\end{equation*}
(with different kinds of fractional derivatives) or even more general ones
have been treated. The local existence has been proved under much weaker
conditions than those for the asymptotic behavior. For our purpose here, the
local existence holds under the simple continuity of the nonlinearities. In
this paper we will be concerned mainly with the asymptotic properties of
solutions. Therefore, the local existence (which we will assume throughout
this document) justifies our investigations. There is no need for uniqueness
as our results will apply for all possible solutions.

The rest of this paper is organized as follows. In Section 2 we present the
used notations, underlying function spaces, background material and some
preliminary results. It contains, in particular, the definitions and basic
properties of the fractional integrals and derivatives used in this paper.
Some useful lemmas and inequalities that will be used later in our proofs
are listed there. The asymptotic behavior of solutions for fractional
integro-differential equations of type \eqref{myprblm} is
studied in detail in Section 3. Finally, we illustrate our findings by an
example in the last section, Section 4.

\section{Preliminaries} \label{sec2}

In this section we briefly introduce some basic definitions, notions and
properties from the theory of fractional calculus.

\begin{definition}[\cite{Kilbas2006}] \rm
Let $-\infty \leq a<b\leq \infty $.
 The space $L^{p}( a,b) $ $(1\leq p\leq \infty)$
 consists of all (Lebesgue) real-valued measurable
functions $f$ on $( a,b) $ for which $\| f\| _{p}<\infty$,  where
\begin{gather*}
\| f\| _{p}=\Big( \int_{a}^{b}| f(s) | ^{p}ds\Big) ^{1/p},\quad 1\leq p<\infty , \\
\| f\| _{\infty }=\operatorname{ess\,sup}_{a\leq t\leq b}|f(t)|,
\end{gather*}
and $\operatorname{ess\,sup}| f( t) | $ is the essential
supremum of the function $| f( t) | $.
\end{definition}

\begin{definition}[\cite{Kilbas2006}] \label{nrm1} \rm
 We denote by $C[ a,b] $ and $C^{n}[ a,b] $,
 $n\in\mathbb{N}_0=\mathbb{N}\cup \{ 0\} $, the spaces of continuous
and $n$-times continuously differentiable functions on $[a,b]$,
with the norms
\begin{gather*}
\| f\| _{C}=\max_{t\in [ a,b] }|f( t) | , \\
\| f\| _{C^{n}}=\sum_{i=0}^{n}\| f^{(i) }\| _{C}=\sum_{i=0}^{n}\max_{t\in [ a,b]
}| f^{( i) }( t) | ,\quad n\in \mathbb{N}_0,
\end{gather*}
respectively, where $C[ a,b] =C^{0}[ a,b] $.
\end{definition}

\begin{definition}[\cite{Kilbas2006}] \label{nrm2} \rm
We denote by $C_{\gamma }[ a,b] $, $0\leq \gamma <1$,
the  weighted space of continuous functions
\begin{equation}
C_{\gamma }[ a,b] =\{ f:( a,b] \to\mathbb{R}:( t-a) ^{\gamma }f(t)\in C[ a,b] \} ,
\label{calpha}
\end{equation}
with the norm
\begin{equation*}
\| f\| _{C_{\gamma }}=\| ( t-a) ^{\gamma}f( t) \| _{C},
\end{equation*}
In particular, $C[ a,b] =C_0[ a,b] $.
\end{definition}

\begin{definition}[\cite{Kilbas2006}] \rm
For $n\in\mathbb{N}$ and $0\leq \gamma <1$, we denote by
$C_{\gamma }^{n}[ a,b] $,  the following weighted space of continuously
differentiable functions up to order $n-1$ with $n$-th
derivative in $C_{\gamma }[ a,b] $,
\begin{equation*}
C_{\gamma }^{n}[ a,b] =\{ f:( a,b] \to \mathbb{R}:
 f\in C^{n-1}[ a,b] ,\; f^{( n) }\in C_{\gamma }[ a,b] \} ,
\end{equation*}
with the norm
\begin{equation*}
\| f\| _{C_{\gamma }^{n}}
=\sum_{k=0}^{n-1}\|f^{(k)}\| _{C}+\| f^{(n)}\| _{C_{\gamma }}\,.
\end{equation*}
In particular, $C_{\gamma }[ a,b] =C_{\gamma }^{0}[a,b] $.
\end{definition}

Next we introduce some definitions, notation and properties of the
Riemann-Liouville fractional derivative.

\begin{definition} \rm
The Riemann-Liouville left-sided fractional integral of order
$\alpha >0$ is defined by
\begin{equation*}
( I_{a+}^{\alpha }u) (t)=\frac{1}{\Gamma (\alpha )}
\int_{a}^{t}(t-s)^{\alpha -1}u(s)ds,\quad a<t<b,
\end{equation*}
provided the right-hand side exists. We define $I_{a+}^{0}u=u$. The function
$\Gamma $ is the Euler gamma function defined by
$\Gamma (\alpha ) =\int_0^{\infty }t^{\alpha -1}e^{-t}dt,\;\alpha >0$.
\end{definition}

\begin{definition} \rm
The Riemann-Liouville left-sided fractional derivative of order
 $\alpha \geq 0$, is defined by
\[
( D_{a^{+}}^{\alpha }u) ( t) =D^{n}(I_{a^{+}}^{n-\alpha }u) (t),\text{ }t>a,
\]
where $D^{n}=\frac{d^{n}}{dt^{n}}$, $n=[ \alpha ] +1$,
$[\alpha ] $ is the integral part of $\alpha $. In particular, when
$\alpha =m\in\mathbb{N}_0$, it follows from the definition that
$D_{a^{+}}^{m}u=D^{m}u$.
\end{definition}

The next lemma shows that the Riemann-Liouville fractional integral and
derivative of the power functions yield power functions multiplied by
certain coefficients and with the order of the fractional derivative added
or subtracted from the power.

\begin{lemma}[\cite{Kilbas2006}]\label{ipwr}
 If $\alpha \geq 0$, $\beta >0$, then
\begin{gather*}
\big( I_{a^{+}}^{\alpha }( s-a) ^{\beta -1}\big) (
t) =\frac{\Gamma ( \beta ) }{\Gamma (\beta +\alpha )}(
t-a) ^{\beta +\alpha -1},\quad t>a, \\
\big( D_{a^{+}}^{\alpha }( s-a) ^{\beta -1}\big) (
t) =\frac{\Gamma ( \beta ) }{\Gamma (\beta -\alpha )}(
t-a) ^{\beta -\alpha -1}, \quad t>a.
\end{gather*}
\end{lemma}

The Riemann fractional integration operator $I_{a+}^{\alpha }$ has the
semigroup property expressed in the following lemma.

\begin{lemma}[\cite{Kilbas2006}]\label{semirl}
Let $\alpha >0$, $\beta>0 $  and $0\leq \gamma <1$.
 Then
\begin{equation*}
I_{a+}^{\alpha }I_{a+}^{\beta }u=I_{a+}^{\alpha +\beta }u,
\end{equation*}
almost everywhere in $[ a,b] $ for $u\in L^{p}( a,b) $
and  holds at any point in $(a,b]$  if
$u\in C_{\gamma }[ a,b] $.
 When $u\in C[ a,b]$, this relation is valid at every point in $[a,b]$.
\end{lemma}

\begin{lemma}[\cite{Kilbas2006}] \label{rldbialpha}
Let $0<\beta \leq \alpha $ and $ 0\leq\gamma <1$.
 If $u\in C_{\gamma }[ a,b] $, then
\begin{equation*}
D_{a^{+}}^{\beta }I_{a^{+}}^{\alpha }u=I_{a^{+}}^{\alpha -\beta }u
\end{equation*}
at every point in $( a,b] $.
\end{lemma}

The following result is about the composition
$I_{a+}^{\alpha}D_{a^{+}}^{\alpha }$ of the Riemann-Liouville fractional
integration and differentiation operators.

\begin{lemma}[\cite{Kilbas2006}] \label{idrl}
Let $\alpha >0$, $0\leq \gamma <1$, $n=[ \alpha ] +1$.
If $u\in C_{\gamma }$ $[ a,b] $ and
$I_{a^{+}}^{n-\alpha }u\in C_{\gamma }^{n}[ a,b] $, then
\begin{equation*}
( I_{a^{+}}^{\alpha }D_{a^{+}}^{\alpha }u) ( t)
=u( t) -\sum_{i=1}^{n}\frac{(D^{n-i}I_{a^{+}}^{n-\alpha }u)
 ( a) }{\Gamma (\alpha -i+1)}( t-a) ^{\alpha -i}
\end{equation*}
for all $t\in ( a,b] $. In particular, if $0<\alpha <1$,
$u\in C_{\gamma }$ $[ a,b] $ and
$I_{a^{+}}^{1-\alpha }u\in C_{\gamma}^{1}[ a,b] $, then
\begin{equation}
( I_{a^{+}}^{\alpha }D_{a^{+}}^{\alpha }u) ( t)
=u( t) -\frac{( I_{a^{+}}^{1-\alpha }u) (
a) }{\Gamma (\alpha )}( t-a) ^{\alpha -1},  \label{idrl1}
\end{equation}
for all $t\in ( a,b] $.
\end{lemma}

For more details about fractional integrals and fractional derivatives, the
reader is referred to the books \cite{Podlubny1998,Samko,Kilbas2006}.

Let $S\subset\mathbb{R}$. For two functions $f,g:S\to\mathbb{R}\backslash \{ 0\} $,
we write $f\propto g$ if $g/f$ is nondecreasing on $S$.

Next, we mention two lemmas, due to Pinto \cite{Pinto1990}, about some
useful nonlinear integral inequalities.

\begin{lemma}[{\cite[Theorem 1]{Pinto1990}}] \label{pinto2}
 Let $u,\lambda _i$, $i=1,\dots ,n$ be continuous and nonnegative functions
 on $I=[a,b]$ and the functions $\omega _i$, $i=1,\dots ,n$ be continuous
 nonnegative and nondecreasing on $[ 0,\infty ) $ such that
$\omega _1\propto \omega_2\propto \dots \propto \omega _n$.
Assume further that $c$ is a positive constant. If
\begin{equation*}
u(t)\leq c+\sum_{i=1}^{n}\int_{a}^{t}\lambda _i(s)\omega
_i(u(s))ds,\text{ }t\in [ a,b] ,
\end{equation*}
then, for $t\in [ a,b_1] $,
\begin{equation*}
u( t) \leq W_n^{-1}( W_n(c_{n-1})+\int_{a}^{t}\lambda_n(s)ds) ,
\end{equation*}
where
\begin{enumerate}
\item $W_i(v)=\int_{v_i}^{v}\frac{d\tau }{\omega _i(\tau )}
,v>0,v_i>0,i=1,\dots ,n$ and $W_i^{-1}$ is the inverse function of $W_i$.

\item The constants $c_i$ are given by $c_0=c$ and $c_i=W_i^{-1}
( W_i(c_{i-1})+\int_{a}^{b_1}\lambda _i(s)ds) $, $i=1,\dots ,n-1$.

\item The number $b_1\in [ a,b] $ is the largest number such that
\begin{equation*}
\int_{a}^{b_1}\lambda _i(s)ds\leq \int_{c_{i-1}}^{\infty }\frac{d\tau }{
\omega _i(\tau )},\quad i=1,\dots ,n.
\end{equation*}
\end{enumerate}
\end{lemma}

\begin{lemma}[{\cite[Theorem 4]{Pinto1990}}] \label{pinto}
Let $u$, $\lambda _i,\omega_i$, $i=1,2,3$ and $c$ be as in Lemma \ref{pinto2}.
If
\begin{equation*}
u(t)\leq c+\int_{a}^{t}\lambda _1(s)\omega_1(u(s))ds
+\int_{a}^{t}\lambda _2(s)\omega _2\Big(
\int_{a}^{s}\lambda _3(\tau )\omega _3(u(\tau ))d\tau \Big) ds,
\end{equation*}
then, for $t\in [ a,b_1] $,
\begin{equation*}
u(t)\leq W_3^{-1}( W_3(c_2)+\int_{a}^{t}\lambda _3(s)ds) ,
\end{equation*}
where $W_i,W_i^{-1}$, $i=1,2,3$ and $c_0,c_1,c_2$ are the same as
in Lemma \ref{pinto2}.
\end{lemma}

\section{Main Results}

 According to the types of the
functions $f$ and $k$, we consider the case of fractional and non-fractional
source terms and also the case of singular kernels. We discuss the
asymptotic behavior of solutions for the problem \eqref{myprblm}
 in the sense of the following definition.

\begin{definition} \rm
By a solution $y$ of \eqref{myprblm}, we mean a function
$y:( 0,b] \to\mathbb{R}$, that is continuable (continuous on $( 0,+\infty ) $),
satisfying the equation and the initial conditions in \eqref{myprblm}
and is in the space $C_{1-\alpha }^{\alpha +1}[ 0,b] $,
$0<b\leq \infty $, defined by
\begin{equation}
C_{1-\alpha }^{\alpha +1}[ 0,b] =\{ y:( 0,b]\to\mathbb{R}
: y\in C_{1-\alpha }[ 0,b] ,\; D_{0^{+}}^{\alpha +1}y\in C_{1-\alpha }[ 0,b] \} ,
\label{cca}
\end{equation}
where the space $C_{1-\alpha }[ 0,b] $ is defined in \eqref{calpha}.
\end{definition}

We assume that the functions $f$ and $k$ satisfy the hypotheses
\begin{itemize}
\item[(A1)] $f(t,u,v)$ is a $C_{1-\alpha }$ function in $
D=\{(t,u,v):t\geq 0$, $u$, $v\in \mathbb{R}\}$.

\item[(A2)] $k(t,s,u)$ is continuous in $E=\{(t,s,u):0\leq s<t<\infty $,
$u\in \mathbb{R}\}$.
\end{itemize}

Before presenting our main results we need to define the following classes
of functions:

\begin{definition} \rm
We say that a function $h:[ 0,\infty ) \to [0,\infty ) $ is of type
 $\mathcal{H}_{\sigma }$ if $h\in C[0,\infty ) $ and
$t^{\sigma }h(t)\in L^{1}( 1,\infty ) $, $\sigma \geq 0$.
\end{definition}

\begin{definition} \rm
We say that a function $g$ is of type $\mathcal{G}$ if it is continuous
nondecreasing on $[ 0,\infty ) $ and positive on
$( 0,\infty) $ with $g(v)\leq ug(\frac{v}{u})$, $u\geq 1,v>0$ and
$\int_{t_0}^{t}\frac{d\tau }{g(\tau )}\to \infty $ as
$t\to\infty $ for any $t_0>0$.
\end{definition}

The above classes are not empty. Examples showing this fact are given in the
next subsections. We will need to deal with the limit of the ratio of the
Riemann-Liouville fractional integral $I_{a^{+}}^{\alpha +1}$ of a function
and the power function $t^{\alpha }$ as $\ t\to \infty $. This is
treated in the next lemma.

\begin{lemma}\label{limit1}
Let $f\in L^{1}( a,\infty ) $, $a\geq 0$. Suppose that $u$ and $v$ are
real-valued functions defined on $[ a,\infty) $, then
\begin{equation*}
\lim_{t\to \infty }\frac{1}{t^{\alpha }}\int_{a}^{t}(t-s)^{\alpha}
f( s,u(s),v( s) ) ds
=\int_{a}^{\infty }f( s,u(s),v( s) ) ds\,.
\end{equation*}
\end{lemma}

\begin{proof}
It is sufficient to prove that
\begin{equation*}
\lim_{t\to \infty }\big| \frac{1}{t^{\alpha }}
\int_{a}^{t}(t-s)^{\alpha }f( s,u(s),v( s) )d\tau )
-\int_{a}^{\infty }f( s,u(s),v( s) ) ds\big| =0\,.
\end{equation*}
Note that
\begin{align*}
&\big| \frac{1}{t^{\alpha }}\int_{a}^{t}(t-s)^{\alpha }f(
s,u(s),v( s) ) -\int_{a}^{\infty }f( s,u(s),v(s) ) ds\big| \\
& =\big| \int_{a}^{t}(1-\frac{s}{t})^{\alpha }f(
s,u(s),v( s) ) ds-\int_{a}^{\infty }f( s,u(s),v(s) ) ds\big| \\
& =\big| \int_{a}^{\infty }( \chi _{[ a,t]
}( s) ( 1-\frac{s}{t}) ^{\alpha }-1) f(s,u(s),v( s) ) ds\big|  \\
&\leq \int_{a}^{\infty }\big| \chi _{[ a,t] }(
s) ( 1-\frac{s}{t}) ^{\alpha }-1| \big|f( s,u(s),v( s) ) | ds,
\end{align*}
where
\begin{equation*}
\chi _{[ a,t] }( s) =\begin{cases}
1, &s \in [ a,t] \\
0, & s\notin [ a,t].
\end{cases}
\end{equation*}
Since
\begin{equation*}
\lim_{t\to \infty }\chi _{[ a,t] }( s) (
1-\frac{s}{t}) ^{\alpha }=1,
\end{equation*}
 by the Dominated Convergence Theorem \cite{DeBarra2003} we obtain
\begin{align*}
&\lim_{t\to \infty }\big| \frac{1}{t^{\alpha }}
\int_{a}^{t}( t-s) ^{\alpha }f( s,u(s),v( s)
) -\int_{a}^{\infty }f( s,u(s),v( s) )ds\big| \\
& \leq \lim_{t\to \infty }\int_{a}^{\infty }
\big| \chi _{ [ a,t] }( s) ( 1-\frac{s}{t}) ^{\alpha}-1\big| | f( s,u(s),v( s) )
| ds\ \\
&=\int_{a}^{\infty }\lim_{t\to \infty }\big| \chi _{
[ a,t] }( s) ( 1-\frac{s}{t}) ^{\alpha}-1\big| | f( s,u(s),v( s) )| ds=0,
\end{align*}
which is the desired result.
\end{proof}

The following lemmas will be needed in the next subsections.

\begin{lemma}\label{limit2rl}
Let $y$ be a solution of problem \eqref{myprblm}
 with $f\in L^{1}( 0,\infty ) $. Then
\begin{align*}
&\lim_{t\to \infty }\frac{y(t)}{t^{\alpha }} \\
&=\lim_{t\to\infty }\frac{( D_{0^{+}}^{\alpha }y) (t)}{\Gamma (\alpha +1)}
\\
&=\frac{1}{\Gamma (\alpha +1)}\Big( a_2+\int_0^{\infty }f\Big(
s,( \!D_{0^{+}}^{\beta }y) (s),\int_0^{s}k(s,\tau ,(
D_{0^{+}}^{\gamma }y) (\tau ))d\tau \Big) ds\Big) .
\end{align*}
\end{lemma}

\begin{proof}
Applying $I_{0^{+}}^{1}$ to both sides of the equation in
\eqref{myprblm} yields
\begin{equation}
( D_{0^{+}}^{\alpha }y) (t)=
a_2+\int_0^{t}f\Big(
s,(D_{0^{+}}^{\beta }y)(s),\int_0^{s}k( s,\tau ,(
D_{0^{+}}^{\gamma }y) (\tau )) d\tau \Big) ds,
\label{rlgaseq01}
\end{equation}
Applying $I_{a^{+}}^{\alpha }$ to the \ref{rlgaseq01} with taking into
account Lemmas \ref{ipwr}, \ref{semirl} and \ref{idrl}, we obtain
\begin{equation}
\begin{aligned}
y(t)&=\frac{a_1t^{\alpha -1}}{\Gamma (\alpha )}
 +\frac{a_2t^{\alpha }}{\Gamma (\alpha +1)}\\
&\quad +\Big( I_{0^{+}}^{\alpha +1}f\Big( s,(D_{0^{+}}^{\beta
}y)(s),\int_0^{s}k(s,\tau ,( D_{0^{+}}^{\gamma }y) (\tau
))d\tau \Big) \Big) (t),
\end{aligned}  \label{rlgaseq02}
\end{equation}
for all $t>0$. Taking the limit of the ratio $\frac{y(t)}{t^{\alpha }}$ as
 $t $ $\to \infty $ gives the desired result with the help of Lemma
\ref{limit1}.
\end{proof}

The next two lemmas provide estimates for some integrals which will appear
later in our arguments.

\begin{lemma}\label{lzrl1}
Let $b_2,b_3$ and $b_4$ be positive constants
and $z( t) $  be a continuous and nonnegative
function on $[0,\infty )$. Assume that
\begin{equation}
z(t)\leq b_2+b_3t+b_4t\int_0^{t}( h_1(s)g_1(
z(s)) +h_2(s)g_2( z(s)) ) ds,\quad t\geq 0,\label{zz}
\end{equation}
where $h_1,h_2\in \mathcal{H}_1$ and $g_1,g_2\in \mathcal{G}$ with
$g_1\propto g_2$. Then
\begin{equation*}
z(t)\leq \begin{cases}
G_2^{-1}( d_2) ,& 0\leq t<1 \\
tG_2^{-1}( d_3) ,& t\geq 1,
\end{cases}
\end{equation*}
where
\begin{gather*}
d_2 = G_2( d_1) +\int_0^{1}h_2(s)ds,\quad
d_1=G_1^{-1}( G_1(d_0)+\frac{\int_0^{1}h_1(s)ds}{\Gamma
(\alpha +1)}) ,\\
d_0=b_2+b_3, \quad
d_3 =G_2(e_1)+b_4\int_1^{\infty }sh_2(s)ds,\\
e_1=G_1^{-1}\Big( G_1(d_4)+b_4\int_1^{\infty
}sh_1(s)ds\Big) , \\
d_4 =d_0+b_4g_1(G_2^{-1}( d_2))
\int_0^{1}h_1(s)ds+b_4g_2( G_2^{-1}( d_2))
\int_0^{1}h_2(s)ds,
\end{gather*}
and $G_i^{-1}$ is the inverse function of
$G_i( t)=\int_{t_0}^{t}\frac{d\tau }{g_i(\tau )}$, $i=1,2$, $t\geq t_0>0$.
\end{lemma}

\begin{proof}
For $0\leq t<1$, from \eqref{zz} we obtain
\begin{equation*}
z(t)\leq
b_2+b_3+b_4\int_0^{t}h_1(s)g_1(z(s))ds+b_4
\int_0^{t}h_2(s)g_2( z(s)) ds.
\end{equation*}
From Lemma \ref{pinto2} it follows
$z(t)\leq G_2^{-1}( d_2) $.
For $t\geq 1$,  from \eqref{zz} we have
\begin{equation}
\begin{aligned}
\frac{z(t)}{t} 
&\leq b_2+b_3+b_4\int_0^{t}h_1(s)g_1(z(s))ds+b_4
\int_0^{t}h_2(s)g_2( z(s)) ds  \\
&\leq d_4+b_4\int_1^{t}sh_1(s)g_1( \frac{z(s)}{s})
ds+b_4\int_1^{t}sh_2(s)g_2( \frac{z(s)}{s}) ds\,.
\end{aligned} \label{z1rlib}
\end{equation}
Notice that the hypotheses of Lemma \ref{pinto2} are satisfied with 
$b_1=\infty $ (because $\int_{t_0}^{\infty }\frac{d\tau }{g_i(\tau )}
=\infty $, $i=1,2$). Then, for $t\in [1,\infty ) $ \eqref{z1rlib} leads to
\begin{equation*}
\frac{z(t)}{t}\leq G_2^{-1}( d_3) .
\end{equation*}
This completes the proof.
\end{proof}

\begin{lemma} \label{lesszrl} 
Let $b_2,b_3$ and $b_4$ be positive constants and let $z(t)$ 
be a continuous and nonnegative function on $[0,\infty )$.
Assume further that
\begin{equation}
z(t)\leq b_2+b_3t+b_4t\int_0^{t}\Big(
h_1(s)g_1(z(s))+h_2(s)g_2\Big( \int_0^{s}h_3(\tau )g_3(z(\tau
))d\tau \Big) \Big) ds,  \label{zeqrl}
\end{equation}
for $t\geq 0$, where $h_1,h_3$ are of type $\mathcal{H}_1,h_2$ is of type 
$\mathcal{H}_0$ and $g_i$ is of type $\ \mathcal{G}$, $i=1,2,3$ with 
$g_1\propto g_2\propto g_3$. Then
\begin{equation*}
z(t)\leq\begin{cases}
G_3^{-1}( M) , &0\leq t<1 \\
tG_3^{-1}( M_1) , & t\geq 1,
\end{cases}
\end{equation*}
where
\begin{gather*}
M = G_3( d_2) +\int_0^{1}h_3(s)ds,\quad
d_2=G_2^{-1}( G_2(d_1)+b_4\int_0^{1}h_2(s)ds) , \\
d_1 = G_1^{-1}\Big( G_1(d_0)+b_4\int_0^{1}h_1(s)ds\Big) , \quad
d_0=b_2+b_3, \\
M_1 = G_3(e_2)+\int_1^{\infty }sh_3(s)ds,\quad
e_2=G_2^{-1}\Big( G_2(e_1)+b_4\int_1^{\infty }h_2(s)ds\Big), \\
e_1 =G_1^{-1}( G_1(M_2)+b_4\int_1^{\infty}sh_1(s)ds) , \\
\begin{aligned}
M_2 &=d_0+b_4g_1(G_3^{-1}( M))\int_0^{1}h_1(s)ds\\
&\quad +b_4g_2\Big( g_3( G_3^{-1}(M) ) \int_0^{1}h_3(\tau )d\tau \Big)
 \int_0^{1}h_2(s)ds\,.
\end{aligned}
\end{gather*}
\end{lemma}

\begin{proof}
For $0\leq t<1$,  from \eqref{zeqrl} we obtain
\begin{equation*}
z(t)\leq b_2+b_3+b_4\int_0^{t}\Big( h_1(s)g_1( z(s))
+h_2(s)g_2( \int_0^{s}h_3(\tau )g_3(z(\tau ))d\tau )
\Big) ds.
\end{equation*}
From Lemma \ref{pinto} it follows that
\begin{equation*}
z(t)\leq G_3^{-1}( M) \quad \text{for all }0\leq t<1\,.
\end{equation*}
For $t\geq 1$,  from $( \ref{zeqrl}) $ we have
\begin{align*}
\frac{z(t)}{t} 
&\leq d_0+b_4\int_0^{1}\Big(h_1(s)g_1(G_3^{-1}( M) )
 +h_2(s)g_2\Big(\int_0^{s}h_3(\tau )g_3(G_3^{-1}( M) )d\tau \Big)\Big) ds \\
&\quad +b_4\int_1^{t}\Big( h_1(s)g_1(z(s))+h_2(s)g_2\Big(
\int_0^{s}h_3(\tau )g_3(z(\tau ))d\tau \Big) \Big) ds \\
&\leq M_2+b_4\int_1^{t}\Big( h_1(s)g_1(z(s))+h_2(s)g_2\Big(
\int_0^{s}h_3(\tau )g_3(z(\tau ))d\tau \Big) \Big) ds\,.
\end{align*}
Let $u=u_1+u_2+u_3$, where
\begin{gather*}
u_1( t) = M_2+b_4\int_0^{t}h_1(s)g_1(z(s))ds, \\
u_2( t) = b_4\int_0^{t}h_2(s)g_2( u_3(s) ) ds,\quad
u_3( t)=\int_0^{t}h_3(s)g_3(z(s))ds,\quad t>0.
\end{gather*}
Differentiating $u$,  by the monotonicity of $g_i,i=1,2,3$, we obtain
\begin{equation}
u'( t) \leq b_4th_1(t)g_1( u( t)) +b_4h_2(t)g_2( u( t) )
+th_3(t)g_3( u( t) ) ,  \label{eqn9u}
\end{equation}
for all $t\geq 1$. Integrating both sides of \eqref{eqn9u}
over $[ 1,t] $ gives
\begin{equation}
\begin{aligned}
u( t) &\leq u( 1) +b_4\int_1^{t}sh_1(s)g_1(u( s))ds
 +b_4\int_1^{t}h_2(s)g_2(u( s) )ds   \\
&\quad +\int_1^{t}sh_3(s)g_3(u( s) )ds\,.
\end{aligned} 
\label{eqn10u}
\end{equation}
Now, since $\int_{t_0}^{\infty }\frac{d\tau }{g_i(\tau )}=\infty $, 
$i=1,2,3$ for any $t_0>0$, the hypotheses of Lemma \ref{pinto2} are
satisfied with $b_1=\infty $. Therefore, by Lemma \ref{pinto2}, the
inequality \eqref{eqn10u} leads to
\begin{equation*}
u(t)\leq G_3^{-1}( M_1) ,\quad \text{for all }t\geq 1.
\end{equation*}
The proof is now complete.
\end{proof}

Although the estimates in Lemmas \ref{lzrl1} and \ref{lesszrl} are not the
best, they ensure that all the involved integrals are bounded, which is the
most useful fact we need in the next subsections.

\subsection{Case of a non-fractional source}

In this subsection, we consider problem \eqref{myprblm}
with $\beta =\gamma =0$ and $0<\alpha <1$; that is,
\begin{equation}
\begin{gathered}
( D_{0^{+}}^{\alpha +1}y) (t)=f\Big( t,y(t),\int_0^{t}k(
t,s,y(s)) ds\Big) ,\text{ }t>0, \\
( I_{0^{+}}^{1-\alpha }y) (0^{+})=a_1,\quad (
D_{0^{+}}^{\alpha }y) (0^{+})=a_2,\quad a_1,a_2\in\mathbb{R}.
\end{gathered} \label{rlaseq1}
\end{equation}

First, we need the following condition:
\begin{itemize}
\item[(A3)] There are functions $h_1,h_3\in \mathcal{H}_1,h_2\in \mathcal{H}_0$ 
and $g_i\in \mathcal{G}$, $i=1,2,3$ with $g_1\propto g_2\propto g_3$ such that
\begin{gather}
| f(t,u,v)| \leq h_1(t)g_1\big( \frac{|
u| }{t^{\alpha -1}}\big) +h_2(t)g_2(| v| ),\quad
( t,u,v) \in D,  \label{condfrl} \\
| k(t,s,y)| \leq h_3(s)g_3( \frac{|y| }{s^{\alpha -1}}) ,\quad 
( t,s,y) \in E\,.  \label{condkrl}
\end{gather}
\end{itemize}

Now, we prove the main result in this subsection.

\begin{theorem} \label{th1asrl} 
Suppose that $f$ and $k$ satisfy {\rm (A1)--(A3)}.
Then, any solution of problem \eqref{rlaseq1} is asymptotic to $ct^{\alpha }$
as $ t\to \infty $, for some $c\in\mathbb{R}$.
\end{theorem}

\begin{proof}
Applying $I_{0^{+}}^{\alpha +1}$ to both sides of the equation in 
\eqref{rlaseq1} gives
\[
y(t)=\frac{a_1t^{\alpha -1}}{\Gamma (\alpha )}
 +\frac{a_2t^{\alpha }}{\Gamma (\alpha +1)}
+\Big( I_{0^{+}}^{\alpha +1}f\Big(
s,y(s),\int_0^{s}k(s,\tau ,y(\tau ))d\tau \Big) \Big) (t).
\]
Then, for all $t>0$,
\begin{equation}
\begin{aligned}
\frac{| y(t)| }{t^{\alpha -1}} 
&\leq \frac{|a_1| }{\Gamma (\alpha )}
 +\frac{| a_2| t}{\Gamma (\alpha +1)}
 +\frac{t}{\Gamma (\alpha +1)}\int_0^{t}\Big[
h_1(s)g_1\big( \frac{| y(s)| }{s^{\alpha -1}}\big)
 \\
&\quad +h_2(s)g_2\Big( \int_0^{s}h_3(\tau )g_3( \frac{
| y(\tau )| }{\tau ^{\alpha -1}}) d\tau \Big)\Big] ds. 
\end{aligned} \label{eqn2rl}
\end{equation}
Let us denote the right hand side of \eqref{eqn2rl}
by $z(t)$ for all $t>0$, then
\begin{equation}
\frac{| y(t)| }{t^{\alpha -1}}\leq z(t),\quad \text{for all }
t>0,  \label{eqn3rl}
\end{equation}
and consequently,
\begin{equation}
\begin{aligned}
z(t)  &\leq \frac{| a_1| }{\Gamma (\alpha )}
 +\frac{| a_2| }{\Gamma (\alpha +1)}t
 +\frac{t}{\Gamma (\alpha +1)}\int_0^{t}[ h_1(s)g_1(z(s))  \\
&\quad +h_2(s)g_2\Big( \int_0^{s}h_3(\tau )g_3(z(\tau ))d\tau
\Big) ] ds\quad \text{for all }t>0.
\end{aligned}  \label{eqn4rl}
\end{equation}
It follows from Lemma \ref{lesszrl} that
\begin{equation*}
z(t)\leq tG_3^{-1}( M_1) ,\quad \text{for all }t\geq 1,
\end{equation*}
and  from \eqref{eqn3rl} we have
\begin{equation}
\frac{| y(t)| }{t^{\alpha }}
\leq M_3:=G_3^{-1}(M_1) ,\quad \text{for all }t\geq 1.  \label{eqn5rl}
\end{equation}
Let
\begin{equation*}
J:=\int_0^{t}\Big| f\Big( s,y(s),\int_0^{s}k(s,\tau ,y(\tau
))d\tau \Big) \Big| ds,\quad t>0.
\end{equation*}
Using  assumption (A3) and \eqref{eqn3rl} we
see that
\begin{equation}
\begin{aligned}
J &\leq \int_0^{1}\Big[ h_1(s)g_1(z(s))+h_2(s)g_2(
\int_0^{s}h_3(\tau )g_3(z(\tau ))d\tau ) \Big] ds    \\
&\quad +\int_1^{t}\Big[ h_1(s)g_1(z(s))+h_2(s)g_2(
\int_0^{s}h_3(\tau )g_3(z(\tau ))d\tau ) \Big] ds,\quad 
t\geq 1.  \label{jrl}
\end{aligned}
\end{equation}
The second integral on the right-hand side of \eqref{jrl} can
be estimated using \eqref{eqn3rl} as follows
\begin{align*}
J_2 
&\leq \int_1^{t}sh_1(s)g_1(M_3)ds
+\int_1^{t}h_2(s)g_2\Big(\int_0^{1}h_3(\tau )g_3( z(\tau )) d\tau \\
&\quad +\int_1^{s}h_3(\tau )g_3(z(\tau ))d\tau \Big) ds \\
&\leq  g_1(M_3)\int_1^{t}sh_1(s)ds
 +g_2\Big(g_3(M_4)\int_0^{1}h_3(\tau )d\tau +g_3(M_3)\int_1^{t}\tau
h_3(\tau )d\tau \Big) \\
&\quad\times \int_1^{t}h_2(s)ds.
\end{align*}
for all $t\geq 1$. As $h_1,h_3\in \mathcal{H}_1,h_2\in \mathcal{H}
_0$, we deduce that $J_2$ is uniformly bounded and so is $J$.

It means that the integral
 $\int_0^{t}f( s,y(s),\int_0^{s}k(s,\tau,y(\tau ))d\tau ) ds$ 
is absolutely convergent and so
\begin{equation}
\lim_{t\to \infty }\int_0^{t}f\Big( s,y(s),\int_0^{s}k(s,\tau
,y(\tau ))d\tau \Big) ds<\infty .  \label{eqn7rl}
\end{equation}
Integrating both sides of the equation in\eqref{rlaseq1}
over the interval $[ 0,t] $ yields
\begin{equation*}
( D_{0^{+}}^{\alpha }y) (t)=a_1+\int_0^{t}f\Big(
s,y(s),\int_0^{s}k(s,\tau ,y(\tau ))d\tau \Big) ds.
\end{equation*}
Now, \eqref{eqn7rl} ensures that there is a real number
 $\hat{c}$ such that
\begin{equation*}
\lim_{t\to \infty }D_{0^{+}}^{\alpha }y(t)=\hat{c}.
\end{equation*}
By Lemma \ref{limit2rl},
\begin{equation*}
\lim_{t\to \infty }\frac{y(t)}{t^{\alpha }}=\lim_{t\to
\infty }\frac{( D_{0^{+}}^{\alpha }y) (t)}{\Gamma (\alpha +1)}=c,
\end{equation*}
$c:=\frac{\hat{c}}{\Gamma (\alpha +1)}$. This completes the proof.
\end{proof}

\subsection{Case of a singular kernel}

Consider the  problem
\begin{equation}
\begin{gathered}
D_{0^{+}}^{\alpha +1}y(t)=f\Big( t,y(t),( I_{0^{+}}^{\beta }y)
(t)\Big) ,\quad t>0,\; 0<\alpha <1,\; 0<\alpha +\beta <1, \\
( I_{0^{+}}^{1-\alpha }y) (0^{+})=a_1,\quad 
\big(D_{0^{+}}^{\alpha }y\big) (0^{+})=a_2,\quad 
a_1,a_2\in\mathbb{R}. 
\end{gathered}  \label{rlibeq1}
\end{equation}

To study the asymptotic behavior of solutions for the problem 
\eqref{rlibeq1}, we assume that the function $f$ satisfies the condition
\begin{itemize}
\item[(A4)] There are functions $h_1,h_2\in \mathcal{H}_1$ and 
$g_1,g_2\in \mathcal{G}$ with $g_1\propto g_2$ such that
\begin{equation*}
| f(t,u,v)| 
\leq h_1(t)g_1\big( \frac{| u| }{t^{\alpha -1}}\big) 
+h_2(t)g_2\big( \frac{|v| }{t^{\alpha +\beta -1}}\big) ,\quad ( t,u,v) \in
D.
\end{equation*}
\end{itemize}

\begin{theorem} \label{thm2}
Suppose that $f$ satisfies conditions {\rm (A1), )A4)}.
Then, every solution of  problem \eqref{rlibeq1} is
asymptotic to $ct^{\alpha }$ when $t\to \infty $, 
for some $c\in\mathbb{R}$.
\end{theorem}

\begin{proof}
From condition (A4), after applying $
I_{0^{+}}^{\alpha +1}$ to both sides of the equation in \eqref{rlibeq1},
we have
\begin{equation}
\begin{aligned}
t^{1-\alpha }| y(t)| 
&\leq \frac{|a_1| }{\Gamma (\alpha )}+\frac{| a_2| t}{
\Gamma (\alpha +1)}+\frac{t}{\Gamma (\alpha +1)}\int_0^{t}\Big[
h_1(s)g_1( \frac{| y(s)| }{s^{\alpha -1}})
 \\
&\quad  +h_2(s)g_2\Big( \frac{| ( I_{0^{+}}^{\beta
}y) (s)| }{s^{\alpha +\beta -1}}\Big) \Big] ds, \quad
t>0.  
\end{aligned}\label{est1}
\end{equation}
Since
\begin{equation*}
( I_{0^{+}}^{\beta }y) (t)=\frac{a_1t^{\alpha +\beta -1}}{
\Gamma (\alpha +\beta )}+\frac{a_2t^{\alpha +\beta }}{\Gamma (\alpha
+\beta +1)}+I_{0^{+}}^{\alpha +\beta +1}f\big( \tau ,y(\tau ),(
I_{0^{+}}^{\beta }y) (\tau )\big) ( s) ( t) ,\
\end{equation*}
for all $t>0$, we arrive at
\begin{align*}
&| ( I_{0^{+}}^{\beta }y) (t)| \\
&\leq \frac{
| a_1| t^{\alpha +\beta -1}}{\Gamma (\alpha +\beta )}
+\frac{| a_2| t^{\alpha +\beta }}{\Gamma (\alpha +\beta+1)}
+I_{0^{+}}^{\alpha +\beta +1}\big| f( \tau ,y(\tau ),(
I_{0^{+}}^{\beta }y) (\tau )) \big| ( s) (t) \\
&\leq \frac{| a_1| t^{\alpha +\beta -1}}{\Gamma
(\alpha +\beta )}+\frac{t^{\alpha +\beta }}{\Gamma (\alpha +\beta +1)}
\Big(| a_2| +\int_0^{t}| f( s,y(s),(I_{0^{+}}^{\beta }y) (s)) | ds\Big) ,
\end{align*}
or equivalently with the help of (A4),
\begin{equation}
\begin{aligned}
t^{1-\alpha -\beta }| ( I_{0^{+}}^{\beta }y)(t)| 
&\leq \frac{| a_1| }{\Gamma (\alpha
+\beta )}+\frac{t}{\Gamma (\alpha +\beta +1)}
\Big(|a_2| +\int_0^{t}\Big[ h_1(s)g_1\big( \frac{|y(s)| }{s^{\alpha -1}}\big)   \\
&\quad +h_2(s)g_2\Big( \frac{| (
I_{0^{+}}^{\beta }y) (s)| }{s^{\alpha +\beta -1}}\Big)
\Big] ds\Big)\quad  \forall t>0.
\end{aligned}  \label{est2}
\end{equation}
Now, let
\begin{equation}
z(t)=A_1+A_2t+A_3t\int_0^{t}\Big( h_1(s)g_1\big( \frac{
| y(s)| }{s^{\alpha -1}}\big) +h_2(s)g_2\big(
\frac{| ( I_{0^{+}}^{\beta }y) (s)| }{
s^{\alpha +\beta -1}}\big) \Big) ds,  \label{est3}
\end{equation}
for all $t>0$, where
\begin{gather*}
A_1 =\max \big\{ \frac{| a_1| }{\Gamma (\alpha )},
\frac{| a_1| }{\Gamma (\alpha +\beta )}\big\} ,\quad
A_2=\max \big\{ \frac{| a_2| }{\Gamma (\alpha +1)},
\frac{| a_2| }{\Gamma (\alpha +\beta +1)}\big\} , \\
A_3 =\max \big\{ \frac{1}{\Gamma (\alpha +1)},\frac{1}{\Gamma
(\alpha +\beta +1)}\big\} .
\end{gather*}
It is not difficult to see from the relations \eqref{est1}--\eqref{est3}, that
\begin{equation*}
t^{1-\alpha }| y(t)| ,\; 
t^{1-\alpha -\beta}| ( I_{0^{+}}^{\beta }y) (t)| \leq z(t),\quad
t>0,
\end{equation*}
and consequently, for $t>0$,
\begin{equation*}
z(t)\leq A_1+A_2t+A_3t\int_0^{t}h_1(s)g_1(z(s))ds+A_3t
\int_0^{t}h_2(s)g_2( z(s)) ds,\quad t>0.
\end{equation*}
It follows from Lemma \ref{lzrl1} that
\begin{equation*}
z(t)\leq tG_2^{-1}( d_3) ,\quad \text{for all }t\geq 1,
\end{equation*}
where $G_2^{-1}$ and $d_3$ are given in Lemma \ref{lzrl1}. Now, the
proof can be completed as the proof of Theorem \ref{th1asrl}.
\end{proof}

\subsection{Case of fractional source terms}

In this subsection we study the asymptotic behavior of solutions for
problem \eqref{myprblm} under the following condition:
\begin{itemize}
\item[(A5)] There are functions $h_1,h_3\in \mathcal{H}_1$, 
$h_2\in \mathcal{H}_0$ and $g_i\in \mathcal{G}$, $i=1,2,3$, with 
$g_1\propto g_2\propto g_3$ such that
\begin{gather*}
| f(t,u,v)| \leq h_1(t)g_1\big( \frac{|u| }{t^{\alpha -\beta -1}}\big) 
+h_2(t)g_2(|v| ),\quad ( t,u,v) \in D, \\
| k(t,s,y)| \leq h_3(s)g_3\big( \frac{|
y| }{t^{\alpha -\gamma -1}}\big) ,\quad ( t,s,y)\in E.
\end{gather*}
\end{itemize}
The main result of this subsection is as follows.

\begin{theorem}
Suppose that $f$ and $k$ satisfy conditions {\rm (A1), (A2), (A5)}.
 Then, every solution of the problem \eqref{myprblm} is asymptotic to 
$ct^{\alpha }$ when $t\to \infty $, for some $c\in\mathbb{R}$.
\end{theorem}

\begin{proof}
Here we have
\begin{gather}
\begin{aligned}
y(t)&=\frac{a_1t^{\alpha -1}}{\Gamma (\alpha )}
 +\frac{a_2t^{\alpha }}{\Gamma (\alpha +1)} \\
&\quad +\Big( I_{0^{+}}^{\alpha +1}f\Big( s,(D_{0^{+}}^{\beta
}y)(s),\int_0^{s}k(s,\tau ,( D_{0^{+}}^{\gamma }y) (\tau
))d\tau \Big) \Big) (t), 
\end{aligned} \label{eq0grl} 
\\
\begin{aligned}
\frac{| y(t)| }{t^{\alpha -1}} 
&\leq \frac{|a_1| }{\Gamma (\alpha )}
 +\frac{| a_2| t}{\Gamma (\alpha +1)}
 +\frac{t}{\Gamma (\alpha +1)}\int_0^{t}
 \Big[h_1(s)g_1\Big(\frac{| (D_{0^{+}}^{\beta }y)(s)| }{
s^{\alpha -\beta -1}}\Big)    \\
&\quad +h_2(s)g_2\Big( \int_0^{s}h_3(\tau )g_3( \frac{
| ( D_{0^{+}}^{\gamma }y) (\tau )| }{\tau
^{\alpha -\gamma -1}}) d\tau \Big) \Big] ds,\quad t>0.
\end{aligned}\label{eq2g}
\end{gather}
Applying $D_{0^{+}}^{\beta }$ and $D_{0^{+}}^{\gamma }$ to both sides of 
\eqref{eq0grl}, and taking Lemma \ref{ipwr} and Lemma \ref{rldbialpha} 
into account, we have
\begin{gather*}
\begin{aligned}
(D_{0^{+}}^{\beta }y)(t) 
&=\frac{a_1t^{\alpha -\beta -1}}{\Gamma (\alpha
-\beta )}+\frac{a_2t^{\alpha -\beta }}{\Gamma (1+\alpha -\beta )} \\
&\quad+\Big( I_{0^{+}}^{\alpha +1-\beta }f\Big( s,(D_{0^{+}}^{\beta
}y)(s),\int_0^{s}k(s,\tau ,( D_{0^{+}}^{\gamma }y) (\tau
))d\tau \Big) \Big) ( t) ,\quad t>0 ,
\end{aligned} \\
\begin{aligned}
( D_{0^{+}}^{\gamma }y) (t) 
&= \frac{a_1t^{\alpha -\gamma -1}}{\Gamma (\alpha -\gamma )}
 +\frac{a_2t^{\alpha -\gamma }}{\Gamma (1+\alpha-\gamma )} \\
&\quad +\Big( I_{0^{+}}^{\alpha +1-\gamma }f\Big( s,(D_{0^{+}}^{\beta
}y)(s),\int_0^{s}k(s,\tau ,( D_{0^{+}}^{\gamma }y) (\tau
))d\tau \Big) \Big) ( t) ,\quad  t>0 ,
\end{aligned}
\end{gather*}
respectively. Therefore for all $t>0$,
\begin{equation}
\begin{aligned}
&t^{1-( \alpha -\beta ) }| (D_{0^{+}}^{\beta}y)(t)|   \\
&\leq \frac{| a_1| }{\Gamma (\alpha -\beta )}+\frac{
| a_2| t}{\Gamma (1+\alpha -\beta )}+\frac{t}{\Gamma
(1+\alpha -\beta )}\int_0^{t}h_1(s)g_1\Big( \frac{|
(D_{0^{+}}^{\beta }y)(s)| }{s^{\alpha -\beta -1}}\Big) ds \\
&\quad +\frac{t}{\Gamma (1+\alpha -\beta )}\int_0^{t}h_2(s)g_2
\Big(\int_0^{s}h_3(\tau )g_3\Big( \frac{| (
D_{0^{+}}^{\gamma }y) (\tau )| }{\tau ^{\alpha -\gamma -1}}
\Big) d\tau \Big) ds,\quad t>0, 
\end{aligned}  \label{eq02g}
\end{equation}
and
\begin{equation}
\begin{aligned}
&t^{1-( \alpha -\gamma ) }| ( D_{0^{+}}^{\gamma}y) (t)|   \\
&\leq \frac{| a_1| }{\Gamma (\alpha -\gamma )}+\frac{
| a_2| t}{\Gamma (1+\alpha -\gamma )}+\frac{t}{\Gamma
(1+\alpha -\gamma )}\int_0^{t}h_1(s)
g_1\Big( \frac{|(D_{0^{+}}^{\beta }y)(s)| }{s^{\alpha -\beta -1}}\Big) ds
\\
&\quad +\frac{t}{\Gamma (1+\alpha -\gamma )}\int_0^{t}h_2(s)
g_2\Big(\int_0^{s}h_3(\tau )g_3\Big( \frac{| (
D_{0^{+}}^{\gamma }y) (\tau )| }{\tau ^{\alpha -\gamma -1}}
\Big) d\tau \Big) ds,\quad t>0 .
\end{aligned}  \label{eq002g}
\end{equation}
Now, let
\begin{gather*}
b_2 =| a_1| \max \big\{ \frac{1}{\Gamma (\alpha )},
\frac{1}{\Gamma (\alpha -\beta )},\frac{1}{\Gamma (\alpha -\gamma )}\big\},\quad 
b_3=| a_2| b_4, \\
b_4 =\max \big\{ \frac{1}{\Gamma (\alpha +1)},\frac{1}{\Gamma (1+\alpha
-\beta )},\frac{1}{\Gamma (1+\alpha -\gamma )}\big\} , \\
\begin{aligned}
z(t) &=b_2+b_3t+b_4t\int_0^{t}\Big[ h_1(s)g_1\Big( \frac{
| (D_{0^{+}}^{\beta }y)(s)| }{s^{\alpha -\beta -1}}\Big)  \\
&\quad +h_2(s)g_2\Big( \int_0^{s}h_3(\tau )g_3( \frac{
| ( D_{0^{+}}^{\gamma }y) (\tau )| }{\tau
^{\alpha -\gamma -1}}) d\tau \Big) \Big] ds,\quad t>0.
\end{aligned}
\end{gather*}
Then,  for all $t>0$ we obtain
\begin{equation}
\frac{| y(t)| }{t^{\alpha -1}},\;
\frac{| (D_{0^{+}}^{\beta }y)(t)| }{t^{\alpha -\beta -1}},\; 
\frac{| ( D_{0^{+}}^{\gamma }y) (t)| }{t^{\alpha -\gamma -1}}\leq z(t). 
 \label{eq3g}
\end{equation}
The remaining steps of the proof are similar to those of the proof of
Theorem \ref{th1asrl}.
\end{proof}

\section{Example}

The next example provides some functions to which Theorem \ref{th1asrl}
applies.

\begin{example} \rm
Consider the equation
\begin{equation}
( D_{0^{+}}^{\alpha +1}y) (t)=t^{\mu _1}e^{-t}\text{ }
y(t)+t^{\mu _2}e^{-t}\int_0^{t}s^{\mu _3}e^{-( s+t) }y(s)ds,\quad
 t>0,  \label{exrl}
\end{equation}
where $0<\alpha <1$, $\mu _1>-\alpha -1$, $\mu _2>-1$ and 
$\mu_3>-\alpha -1$. \ Notice that the right-hand side of the equation
\eqref{exrl} can be rewritten as
\begin{equation*}
t^{\mu _1+\alpha -1}e^{-t} \frac{y(t)}{t^{\alpha -1}}
+t^{\mu_2}e^{-t}\int_0^{t}s^{\mu _3+\alpha -1}e^{-( s+t) }\frac{
y(s)}{s^{\alpha -1}}ds,\quad t>0.
\end{equation*}
Let $h_1( t) =t^{\mu _1+\alpha -1}e^{-\rho _1t}$,
$h_2( t) =t^{\mu _2}e^{-\rho _2t}$, 
$h_3(t) =t^{\mu _3+\alpha -1}e^{-\rho _3t}$ for $t>0$,
\begin{equation*}
g_i( t) =t,\quad 0<\rho _i\leq 1,\quad i=1,2,3,\;  t>0.
\end{equation*}
Then conditions (A1)--(A3) are satisfied,
\begin{gather*}
 \int_1^{\infty }th_1( t) dt
<\int_0^{\infty}th_1( t) dt 
=\int_0^{\infty }t^{\mu _1+\alpha }e^{-\rho_1t}dt
=\frac{\Gamma (\mu _1+\alpha +1)}{\rho _1^{\mu _1+\alpha +1}}<\infty , \\
 \int_1^{\infty }h_2( t) dt
<\int_0^{\infty}h_2( t) dt
=\int_0^{\infty }t^{\mu _2}e^{-\rho _2t}dt
=\frac{\Gamma (\mu _2+1)}{\rho _2^{\mu _2+1}}<\infty , \\
\int_1^{\infty }th_3( t) dt
<\int_0^{\infty}th_3( t) dt
=\int_0^{\infty }t^{\mu _3+\alpha }e^{-\rho_3t}dt
=\frac{\Gamma (\mu _3+\alpha +1)}{\rho _3^{\mu _3+\alpha +1}}<\infty , \\
 \int_{t_0}^{\infty }\frac{1}{g_i( t) }dt
=\int_{t_0}^{\infty }\frac{1}{t}dt=\infty\quad \text{for any }t_0>0.
\end{gather*}
From Theorem \ref{th1asrl}  every solution of \eqref{exrl}, 
subject to the initial conditions given in \eqref{exrl}, is asymptotic to 
$d_1t^{\alpha }$ as $t\to \infty $, for some $d_1\in\mathbb{R}$.
\end{example}



\subsection*{Acknowledgements}
The authors would like to acknowledge the support provided by King Fahd
University of Petroleum and Minerals (KFUPM) through project number IN161010.

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\end{document}