\def\papertitle{Solvability for a class of nonlinear fractional relaxation differential equations}
\def\paperauthorA{Adel Lachouri$^{1}$, Abdelouaheb Ardjouni $^{2}$}
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\title{\papertitle}
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{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!$^1$ Laboratory of Applied Mathematics, Annaba University, Algeria\\
\!\!\!\!\!\!\!\!\!$^2$Department of Mathematics and Informatics, Souk
Ahras University, Algeria}

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\begin{abstract}
This paper is concerned with a class of nonlinear fractional relaxation differential equations. The existence and uniqueness results are obtained by applying some fixed point theorems. Finally, we provide an example to illustrate the obtained results.
\end{abstract}



\section{Introduction}
The concept of fractional calculus is a generalization of the ordinary
differentiation and integration to arbitrary non integer order. Fractional
differential equations with and without delay arise from a variety of
applications including in various fields of science and engineering such as
applied sciences, practical problems concerning mechanics, the engineering
technique fields, economy, control systems, physics, chemistry, biology,
medicine, atomic energy, information theory, harmonic oscillator, nonlinear
oscillations, conservative systems, stability and instability of geodesic on
Riemannian manifolds, dynamics in Hamiltonian systems, etc. In particular,
problems concerning qualitative analysis of linear and nonlinear fractional
differential equations with and without delay have received the attention of
many authors, see \cite{ref1,bai,ref2,sal} and the
references therein.\\
Inspired and motivated by the studied works in \cite{ref1,bai,ref2,sal}, in this
paper, we study the existence and uniqueness of solution for the following nonlinear fractional relaxation differential equation
\begin{equation}
	\left\{ 
	\begin{array}{c}
		^{C}D^{\alpha }\left( u\left( t\right) -g\left( t,u\left( t\right) \right)
		\right) +\omega u\left( t\right) =f\left( t,u\left( t\right) \right) ,\text{ 
		}\omega >0,t\in \left( 0,T\right]  \\ 
		u\left( 0\right) =u_{0}\in 
		%TCIMACRO{\U{211d} }%
		%BeginExpansion
		\mathbb{R}
		%EndExpansion
		,%
	\end{array}%
	\right.   \label{1}
\end{equation}%
where $^{C}D^{\alpha }$ is the Caputo fractional derivative of order $\alpha
\in \left( 0,1\right] $, $f:\left[ 0,T\right] \times 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\rightarrow 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ and $\ g$ $:\left[ 0,T\right] \times 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\rightarrow 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ are nonlinear continuous functions. To show the existence and uniqueness
of solutions, we transform (\ref{1}) into an integral equation and then use
the Banach and Krasnoselskii fixed point theorems. Finally, we provide an
example to illustrate our obtained results.

The rest of this paper is organized as follows. Some definitions from
fractional calculus theory are recalled in Section 2. In Section 3, we prove
the existence and uniqueness of solutions for (\ref{1}). Finally, in Section
4, we give an example to illustrate the usefulness of our main results.


\section{Preliminaries}

In this section we present some basic definitions, notations and results of
fractional calculus which are used throughout this paper.

Let $T>0$, $J=\left[ 0,T\right] $. By $C\left( J,%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\right) $ we denote the Banach space of all continuous functions from $J$
into $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ with the norm%
\[
\left\Vert u\right\Vert =\sup \left\{ u\left( t\right) :t\in J\right\} .
\]

\begin{definition}[\protect\cite{ki}]
	The fractional integral of order $\alpha >0$ of a function $u:J\rightarrow 
	%TCIMACRO{\U{211d} }%
	%BeginExpansion
	\mathbb{R}
	%EndExpansion
	$ is given by%
	\[
	I^{\alpha }u\left( t\right) =\frac{1}{\Gamma \left( \alpha \right) }%
	\int_{0}^{t}\left( t-s\right) ^{\alpha -1}u\left( s\right) ds,
	\]%
	provided the right side is pointwise defined on $J$.
\end{definition}

\begin{definition}[\protect\cite{ki}]
	\textit{The Caputo fractional derivative of order }$\alpha >0$ of function $%
	u:J\rightarrow 
	%TCIMACRO{\U{211d} }%
	%BeginExpansion
	\mathbb{R}
	%EndExpansion
	$ is given by%
	\[
	^{C}D^{\alpha }u(t)=I^{n-\alpha }D^{\left( n\right) }u\left( t\right) =\frac{%
		1}{\Gamma \left( n-\alpha \right) }\int_{0}^{t}\left( t-s\right) ^{n-\alpha
		-1}u^{\left( n\right) }\left( s\right) ds,
	\]%
	where $n=\left[ \alpha \right] +1$, provided the right side is pointwise
	defined on $J$.
\end{definition}

\begin{definition}[\protect\cite{bai}]
	The two-parameter function of the Mittag-Leffler type is defined by the
	series expansion%
	\[
	E_{\alpha ,\beta }\left( z\right) =\sum_{n=0}^{\infty }\frac{z^{n}}{\Gamma
		(\alpha n+\beta )},\ \alpha >0,\ \beta \in 
	%TCIMACRO{\U{2102} }%
	%BeginExpansion
	\mathbb{C}
	%EndExpansion
	,\ z\in 
	%TCIMACRO{\U{2102} }%
	%BeginExpansion
	\mathbb{C}
	%EndExpansion
	.
	\]%
	For $\beta =1$, we obtain the Mittag-Leffler function in one parameter%
	\[
	E_{\alpha }\left( z\right) =\sum_{n=0}^{\infty }\frac{z^{n}}{\Gamma (\alpha
		n+1)},\ \alpha >0,\ z\in 
	%TCIMACRO{\U{2102} }%
	%BeginExpansion
	\mathbb{C}
	%EndExpansion
	.
	\]
\end{definition}

\begin{lemma}[\protect\cite{bai}]
	For $0<\alpha \leq 1$, the Mittag-Leffler type function $E_{\alpha ,\alpha
	}(-\omega t^{\alpha })$ satisfies%
	\[
	0\leq E_{\alpha ,\alpha }(-\omega t^{\alpha })\leq \frac{1}{\Gamma \left(
		\alpha \right) },\text{ }t\in \left[ 0,\infty \right) ,\text{ }\omega \geq 0,
	\]%
	and%
	\[
	\lim_{t\rightarrow 0^{+}}E_{\alpha ,\alpha }(-\omega t^{\alpha })=E_{\alpha
		,\alpha }(0)=\frac{1}{\Gamma \left( \alpha \right) }.
	\]
\end{lemma}

\begin{lemma}[\protect\cite{conc}]
	For $t\in \left[ 0,\infty \right) $ and $0<\alpha \leq 1$, the one-parameter
	Mittag-Leffler function $E_{\alpha ,1}(-t^{\alpha })$ is decreasing function
	of $t$ and it is bounder from above by $1$, that is 
	\[
	E_{\alpha ,1}(-\omega t^{\alpha })\leq 1.
	\]%
	Furthermore, it is to be noted that%
	\[
	\lim_{t\rightarrow \infty }E_{\alpha ,1}(-\omega t^{\alpha })=0.
	\]
\end{lemma}

Lastly in this section, we state the fixed point theorems which enable us to
prove the existence and uniqueness of a solution of (\ref{1}).

\begin{theorem}[Banach's fixed point theorem \protect\cite{s}]
	\label{th2}Let $\Omega $ be a non-empty closed subset of a Banach space $%
	(S,\left\Vert .\right\Vert )$, then any contraction mapping $\Phi $ of $%
	\Omega $ into itself has a unique fixed point.
\end{theorem}

\begin{theorem}[Krasnoselskii's fixed point theorem \protect\cite{s}]
	Let $\Omega $ be a non-empty bounded closed convex subset of a Banach space $%
	(S,\left\Vert .\right\Vert )$. Suppose that $F_{1}$ and $F_{2}$ map $\Omega $
	into $S$ such that
	
	(i) $F_{1}u+F_{2}v\in \Omega $ for all $u,v\in \Omega $,
	
	(ii) $F_{1}$ is continuous and compact,
	
	(iii) $F_{2}$ is a contraction.
	
	\noindent Then there is a $u\in \Omega $ with $F_{1}u+F_{2}u=u$.
\end{theorem}

\section{Existence and uniqueness}

Let us start by defining what we mean by a solution of the problem (\ref{1}).

\begin{definition}
	A function $u\in C^{1}\left( J,%
	%TCIMACRO{\U{211d} }%
	%BeginExpansion
	\mathbb{R}
	%EndExpansion
	\right) $ is said to be a solution of problem (\ref{1}) if $u$ satisfies $%
	^{C}D^{\alpha }\left( u\left( t\right) -g\left( t,u\left( t\right) \right)
	\right) +\omega u\left( t\right) =f\left( t,u\left( t\right) \right) $ for
	any $t\in J$ and $u\left( 0\right) =u_{0}$.
\end{definition}

For the existence of solutions for the problem (\ref{1}), we need the
following auxiliary lemma.

\begin{lemma}
	\label{lm01}Let $u\in C\left( J,%
	%TCIMACRO{\U{211d} }%
	%BeginExpansion
	\mathbb{R}
	%EndExpansion
	\right) $ and $u^{\prime }$ exists, then $u$ is a solution of the initial
	value problem (\ref{1}) if and only if it is a solution of the integral
	equation%
	\begin{align}
		u(t)& =(u_{0}-g(t,u_{0}))E_{\alpha ,1}(-\omega t^{\alpha })+g\left(
		t,u\left( t\right) \right)  \nonumber \\
		& +\int_{0}^{t}\left( t-s\right) ^{\alpha -1}E_{\alpha ,\alpha }(-\omega
		(t-s)^{\alpha })\left[ f\left( s,u\left( s\right) \right) -\omega g\left(
		s,u\left( s\right) \right) \right] ds.  \label{ope}
	\end{align}
\end{lemma}

\begin{proof}
	It is easy to prove by the Laplace transform.
\end{proof}

In the following subsections we prove existence, as well as existence and
uniqueness results for the problem (\ref{1}) by using a variety of fixed
point theorems.

The following assumptions will be used in our main results

(H1) There exists a constant $L_{f}\in 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}$ such that%
\begin{equation}
	\left\vert f(t,u)-f(t,v)\right\vert \leq L_{f}\left\vert u-v\right\vert , 
	\nonumber
\end{equation}%
for $t\in J$, $u,v\in 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$.

(H2) There exists a constant $L_{g}\in \left( 0,1\right) $ such that%
\[
\left\vert g(t,u)-g(t,v)\right\vert \leq L_{g}\left\vert u-v\right\vert ,
\]%
for $t\in J$, $u,v\in 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$.

\subsection{Existence and uniqueness results via Banach's fixed point theorem%
}

\begin{theorem}
	\label{th1}Assume that the assumptions (H1)--(H2) are satisfied. If%
	\begin{equation}
		L_{g}+\frac{T^{\alpha }}{\Gamma \left( \alpha +1\right) }\left(
		L_{f}+L_{g}\omega \right) <1.  \label{cdt}
	\end{equation}%
	Then there exists a unique solution for the problem (\ref{1}) on $J$.
\end{theorem}

\begin{proof}
	We define the operator $\Phi :C\left( J,%
	%TCIMACRO{\U{211d} }%
	%BeginExpansion
	\mathbb{R}
	%EndExpansion
	\right) \rightarrow C\left( J,%
	%TCIMACRO{\U{211d} }%
	%BeginExpansion
	\mathbb{R}
	%EndExpansion
	\right) $ by%
	\begin{align*}
		\left( \Phi u\right) \left( t\right) & =(u_{0}-g(t,u_{0}))E_{\alpha
			,1}(-\omega t^{\alpha })+g\left( t,u\left( t\right) \right) \\
		& +\int_{0}^{t}\left( t-s\right) ^{\alpha -1}E_{\alpha ,\alpha }(-\omega
		(t-s)^{\alpha })\left[ f\left( s,u\left( s\right) \right) -\omega g\left(
		s,u\left( s\right) \right) \right] ds.
	\end{align*}%
	Clearly, the fixed points of operator $\Phi $ are solutions of problem (\ref%
	{1}). For any $u,v\in C\left( \left[ 0,T\right] ,%
	%TCIMACRO{\U{211d} }%
	%BeginExpansion
	\mathbb{R}
	%EndExpansion
	\right) $ and $t\in J$, we have%
	\begin{align*}
		& \left\vert \left( \Phi u\right) \left( t\right) -\left( \Phi v\right)
		\left( t\right) \right\vert \\
		& \leq \left\vert g\left( t,u\left( t\right) \right) -g\left( t,v\left(
		t\right) \right) \right\vert \\
		& +\int_{0}^{t}\left( t-s\right) ^{\alpha -1}E_{\alpha ,\alpha }(-\omega
		(t-s)^{\alpha })\left\vert f\left( s,u\left( s\right) \right) -f\left(
		s,v\left( s\right) \right) \right\vert ds \\
		& +\omega \int_{0}^{t}\left( t-s\right) ^{\alpha -1}E_{\alpha ,\alpha
		}(-\omega (t-s)^{\alpha })\left\vert g\left( s,u\left( s\right) \right)
		-g\left( s,v\left( s\right) \right) \right\vert ds.
	\end{align*}%
	By (H1) and (H2), we have%
	\begin{align*}
		& \left\vert \left( \Phi u\right) \left( t\right) -\left( \Phi v\right)
		\left( t\right) \right\vert \\
		& \leq L_{g}\left\Vert u-v\right\Vert +\frac{T^{\alpha }L_{f}}{\Gamma \left(
			\alpha +1\right) }\left\Vert u-v\right\Vert +\frac{T^{\alpha }L_{g}\omega }{%
			\Gamma \left( \alpha +1\right) }\left\Vert u-v\right\Vert ,
	\end{align*}%
	thus%
	\[
	\left\Vert \Phi u-\Phi v\right\Vert \leq \left( L_{g}+\frac{T^{\alpha }}{%
		\Gamma \left( \alpha +1\right) }\left( L_{f}+L_{g}\omega \right) \right)
	\left\Vert u-v\right\Vert .
	\]%
	From (\ref{cdt}), $\Phi $ is a contraction. As a consequence of Banach's
	fixed point theorem, we get that $\Phi $ has a unique fixed point which is a
	unique solution of the problem (\ref{1}) on $J$.
\end{proof}

\subsection{Existence results via Krasnoselskii's fixed point theorem}

\begin{theorem}
	Assume (H2) and the following hypotheses
	
	(H3) There exist $p_{1}\in C\left( J,%
	%TCIMACRO{\U{211d} }%
	%BeginExpansion
	\mathbb{R}
	%EndExpansion
	^{+}\right) $ such that%
	\[
	\left\vert f\left( t,u\right) \right\vert \leq p_{1}\left( t\right) ,
	\]%
	for $t\in J$ and each $u\in 
	%TCIMACRO{\U{211d} }%
	%BeginExpansion
	\mathbb{R}
	%EndExpansion
	$.
	
	(H4) There exist $p_{2}\in C\left( J,%
	%TCIMACRO{\U{211d} }%
	%BeginExpansion
	\mathbb{R}
	%EndExpansion
	^{+}\right) $ such that%
	\[
	\left\vert g\left( t,u\right) \right\vert \leq p_{2}\left( t\right) ,
	\]%
	for $t\in J$ and each $u\in 
	%TCIMACRO{\U{211d} }%
	%BeginExpansion
	\mathbb{R}
	%EndExpansion
	$.
	
	Then the problem (\ref{1}) has at least one solution in $\Omega $.
\end{theorem}

\begin{proof}
	Let us fix 
	\[
	\rho \geq \left\vert u_{0}\right\vert +q+p_{2}^{\ast }+\frac{T^{\alpha }}{%
		\Gamma \left( \alpha +1\right) }\left( p_{1}^{\ast }+\omega p_{2}^{\ast
	}\right) ,
	\]%
	where $p_{1}^{\ast }=\sup_{t\in J}p_{1}\left( t\right) $, $p_{2}^{\ast
	}=\sup_{t\in J}p_{2}\left( t\right) $ and $q=\sup_{t\in J}\left\vert
	g(t,u_{0})\right\vert $. Consider the non-empty closed convex subset%
	\[
	\Omega =\left\{ u\in C\left( J,%
	%TCIMACRO{\U{211d} }%
	%BeginExpansion
	\mathbb{R}
	%EndExpansion
	\right) ,\ \left\Vert u\right\Vert \leq \rho \right\} ,
	\]%
	and define two operators $F_{_{1}}$ and $F_{_{2}}$ on $\Omega $, as follows%
	\[
	\left( F_{1}u\right) (t)=\int_{0}^{t}\left( t-s\right) ^{\alpha -1}E_{\alpha
		,\alpha }(-\omega (t-s)^{\alpha })\left[ f\left( s,u\left( s\right) \right)
	-\omega g\left( s,u\left( s\right) \right) \right] ds,
	\]%
	and%
	\[
	\left( F_{2}u\right) (t)=(u_{0}-g(t,u_{0}))E_{\alpha ,1}(-\omega t^{\alpha
	})+g\left( t,u\left( t\right) \right) .
	\]%
	We shall use the Krasnoselskii fixed point theorem to prove there exists at
	least one fixed point of the operator $F_{_{1}}+F_{2}$ in $\Omega $. The
	proof will be given in the following steps.
	
	\textbf{Step 1. }We prove that $F_{_{1}}u+F_{2}v\in \Omega $ for all $u,v\in
	\Omega $.
	
	\textbf{Step 2.} We prove that $F_{1}$ is compact and continuous.
	
	\textbf{Step 3.} We prove that $F_{2}:\Omega \rightarrow C\left( J,%
	%TCIMACRO{\U{211d} }%
	%BeginExpansion
	\mathbb{R}
	%EndExpansion
	\right) $ is a contraction mapping.
	
	Clearly, all the hypotheses of the Krasnoselskii fixed point theorem are
	satisfied. Hence, there exists a fixed point $u\in \Omega $ such that $%
	u=F_{_{1}}u+F_{_{2}}u$ which is a solution of the problem (\ref{1}).
\end{proof}

\begin{example}
	We consider the fractional initial value problem 
	\begin{equation}
		\left\{ 
		\begin{array}{l}
			^{C}D^{\frac{1}{2}}\left( u\left( t\right) -\frac{1}{4}t\sin (u\left(
			t\right) )\right) +\frac{1}{2}u\left( t\right) =\frac{1}{\left( \exp
				(t)+4\right) \left( \left\vert u\left( t\right) \right\vert +1\right) },%
			\text{ }t\in J=\left[ 0,1\right] , \\ 
			u\left( 0\right) =1,%
		\end{array}%
		\right.   \label{P1}
	\end{equation}%
	where $T=1$, $u_{0}=1$, $\alpha =\omega =\frac{1}{2}$, $g(t,u)=\frac{1}{4}%
	t\sin (u)$ and $f(t,u)=\frac{1}{\left( \exp (t)+4\right) \left( \left\vert
		u\right\vert +1\right) }$. For each $u,v\in 
	%TCIMACRO{\U{211d} }%
	%BeginExpansion
	\mathbb{R}
	%EndExpansion
	$ and $t\in J$, we have%
	\begin{align*}
		\left\vert f\left( t,u\right) -f\left( t,v\right) \right\vert & =\left\vert 
		\frac{1}{\left( \exp (t)+4\right) \left( \left\vert u\right\vert +1\right) }-%
		\frac{1}{\left( \exp (t)+4\right) \left( \left\vert v\right\vert +1\right) }%
		\right\vert  \\
		& \leq \frac{\left\vert u-v\right\vert }{\left( \exp (t)+4\right) \left(
			1+\left\vert u\right\vert \right) \left( 1+\left\vert v\right\vert \right) }
		\\
		& \leq \frac{1}{5}\left\vert u-v\right\vert ,
	\end{align*}%
	and%
	\begin{equation*}
		\left\vert g(t,u)-g(t,v)\right\vert \leq \frac{1}{4}\left\vert
		u-v\right\vert .
	\end{equation*}%
	Hence, assumptions (H1) and (H2) are satisfied with $L_{f}=\frac{1}{5}$ and $%
	L_{g}=\frac{1}{4}$. The condition%
	\begin{equation*}
		L_{g}+\frac{T^{\alpha }}{\Gamma \left( \alpha +1\right) }\left( L_{g}\omega
		+L_{f}\right) \simeq 0.62<1,
	\end{equation*}%
	is satisfied. It follows from Theorem \ref{th1} that the problem (\ref{P1})
	has a unique solution on $J$.
\end{example}
\section{conclusions}
We can conclude that the main results of this work have been successfully achieved, that is, through the Banach contraction principle and the Krasnoselskii fixed point theorem , we scrutinized the existence and uniqueness of solutions for a class of nonlinear fractional relaxation differential equations. 
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{\sc A. Ardjouni and A. Djoudi}, {\em Positive solutions for first-order nonlinear Caputo-Hadamard fractional relaxation differential equations}, Kragujevac Journal of Mathematics, 45(2021), 897--908.
\bibitem{bai} 
{\sc Z. Bai, S. Zhang, S. Sun and C. Yin}, {\em  Monotone iterative method
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\bibitem{ref2}
{\sc M. Belaid, A. Ardjouni and A. Djoudi}, {\em Positive solutions for nonlinear fractional relaxation differential equations}, J. Fract. Calc. Appl , 11(2020), 1--10.

\bibitem{conc} 
{\sc M. Concezzi and R. Spigler}, {\em Some analytical and numerical
	properties of the Mittag-Leffler functions}, Fractional Calculus and Applied
Analysis, 18(1) (2015), 64--94.
\bibitem{ki}
{\sc A. A. Kilbas, H. M. Srivastava and J. J. Trujillo}, {\em Theory and Applications of Fractional Differential Equations}, Elsevier Science B. V.,Amsterdam, (2006).
\bibitem{sal} 
{\sc A. Seemab and M. U. Rehman}, {\em Existence and stability analysis by
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	differential equations}, Dynamic Systems and Applications 27(3) (2018),
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\bibitem{s}
{\sc D. R. Smart}, {\em Fixed Point Theorems}, Cambridge Tracts in Mathematics, no. 66, Cambridge University Press, London-New York, (1974).
	
\end{thebibliography}


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