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\begin{document}
\title[Hyers-Ulam stability and controllability of abstract equation and
application]{Coupled system of sequential partial $\sigma \left( .,.\right)
- $Hilfer fractional differential equations with weighted double phase
operator : Existence, Hyers-Ulam stability and controllability}
\address{University M'Hamed Bougara of Boumerdes, Algeria}
\author{Nadir Benkaci-Ali}
\email{radians\_2005@yahoo.fr}
\subjclass{34B15-34B16-34B18.}
\keywords{Control, sequential PDE, Hyers-Ulam stability, fixed point}

\begin{abstract}
\ In this paper, we are concerned by a sequential partial Hilfer fractional
differential system with weighted double phase operator. First, we introduce
the concept of Hyers-Ulam stability with respect to an operator $L$ for an
abstract equation of the form $u=LFu$ in Banach lattice by using the fixed
point arguments and spectral theory. Then, we prove the controllability and
apply the previous results obtained for abstract equation to prove existence
and Hyers-Ulam stability of a coupled system of sequential fractional
partial differential equations involving a weighted double phase operator.
Finally, example illustrating the main results is constructed. This work
contains several new ideas, and gives a unified approach applicable to many
types of differential equations.
\end{abstract}

\maketitle

\section{$\protect\bigskip $Introduction}

Fractional order and Hilfer fractional order differential equations
involving a $p$-Laplacian operator are of great importance and are
interesting class of problems. Such kinds of problems have been studied by
many authors, see, \cite{bed, ben2, ben3, man, va}. At the same time, the
studies of Hyers-Ulam stability have attracted a great deal of attention in
the last ten years, (see \cite{am, ar, jin, in, has1, has, lo, se, ulam}),
and the references therein. 

In \cite{zh}, the authors discussed the existence of positive solutions for
the double phase differential equation 
\begin{equation*}
-D_{p,q}\left( u\right) \left( x\right) =f\left( x,u\right) ,\text{ }x\in
\Omega \subset 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{n},
\end{equation*}%
with double phase differential operator $D_{p,q}\left( u\right) =\Delta
_{p}u+a.\Delta _{q}u$.

In \cite{sh}, existence and uniquness of solutions to sequential fractional
differential equation 
\begin{equation*}
\lambda D^{\alpha }u\left( t\right) +D^{\beta }u\left( t\right) =f\left(
t,u\left( t\right) \right)
\end{equation*}%
was investigated.

In \cite{ho}, the authors worked on the existence and Hyers--Ulam stability
for the following sequential fractional differential system :%
\begin{equation*}
\begin{array}{c}
\left[ ^{c}D_{q}^{\nu }+r.^{c}D_{q}^{\sigma }\right] u\left( t\right)
=f\left( t,u\left( t\right) ,u\left( \alpha t\right) ,^{c}D_{q}^{\sigma
}\left( \alpha t\right) \right) ,\text{ }t\in \left( 0,T\right)%
\end{array}%
\end{equation*}%
where $D^{\nu },$ $D^{\sigma }$ are the Caputo fractional derivatives of
orders $\nu \in (1,2]$ and $\sigma \in (0,1]$ respectively.

Motivated by the works\ mentioned above, in this paper, we give the
existence, Hyers-Ulam stability and controllability results for the abstract
equation $LFu=u$ and there application to a coupled sequential partial
Hilfer fractional differential system with weighted double phase partial
differential operator :%
\begin{equation}
\left\{ 
\begin{array}{l}
\left( \zeta _{1}\left( t\right) .D_{0^{+},t}^{\alpha +1,\omega ,\sigma
}+D_{0^{+},t}^{\alpha ,\omega ,\sigma }\right) \left( \dfrac{\partial }{%
\partial x}\left( \phi \left( \theta _{1}\left( x\right) \dfrac{\partial
u_{1}}{\partial x}\right) \right) \right) \left( t,x\right) +f_{1}\left(
t,x,u_{1},u_{2}\right) =0,\ t,x>0, \\ 
\\ 
\left( \zeta _{2}\left( t\right) .D_{0^{+},t}^{\alpha +1,\omega ,\sigma
}+D_{0^{+},t}^{\alpha ,\omega ,\sigma }\right) \left( \dfrac{\partial }{%
\partial x}\left( \phi \left( \theta _{2}\left( x\right) \dfrac{\partial
u_{2}}{\partial x}\right) \right) \right) \left( t,x\right) +f_{2}\left(
t,x,u_{1},u_{2}\right) =0,\ t,x>0, \\ 
\\ 
u_{j}\left( 0,x\right) =u_{j}\left( t,0\right) =\underset{x\rightarrow
+\infty }{\lim }\dfrac{\partial u_{j}}{\partial x}\left( t,x\right) =0,\text{
}j\in \left\{ 1,2\right\} ,%
\end{array}%
\right.  \label{e1}
\end{equation}%
where $D_{0^{+},t}^{\alpha ,\omega ,\sigma }$is the partial $\sigma \left(
.,.\right) -$Hilfer fractional derivative with respect to the variable $t$
of order $\alpha $ and type $0\leq \omega \leq 1$ with $0<\alpha <1$,%
\begin{equation*}
\phi =\phi _{p^{-}}+\phi _{p^{+}}\text{ , }1<p^{-}<p^{+}
\end{equation*}%
with 
\begin{equation*}
\phi _{p^{\nu }}\left( x\right) =\left\vert x\right\vert ^{p^{\nu }-2}.x,%
\text{ for }\nu \in \left\{ -,+\right\} ,
\end{equation*}%
and for $j\in \left\{ 1,2\right\} ,$ 
\begin{equation*}
\zeta _{j}\left( t\right) =a_{j}+t,\text{ }a_{j}>0,
\end{equation*}%
The function $\sigma \left( t,x\right) $ is bounded and positive on $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\times 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}$ having a continuous and positive derivative $\dfrac{\partial \sigma }{%
\partial t}\left( t,x\right) >0$ with respect to the variable $t$ on $%
(0,+\infty )$ with $\sigma \left( 0,x\right) =0$ for all $x\geq 0$ and such
that%
\begin{equation*}
\left( \sigma ^{+}\right) ^{\alpha }\in L^{1}\left( 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\right) ,\text{ with }\sigma ^{+}\left( x\right) =\underset{t\rightarrow
+\infty }{\lim }\sigma \left( t,x\right) .
\end{equation*}

\section{$\protect\bigskip $Abstract background}

\bigskip

Let $\left( E,\Vert .\Vert \right) $ be a real Banach space$.$ A nonempty
subset $P$ of $E$ is said to be a cone if $P$ is closed and convex, $P\cap
(-P)={0}$ and for all $t\geq 0,$ $tP\subset P$ . In this situation, $P$
induces a partial order in the Banach space $E$ defined by $x\leq y$ if $%
y-x\in P.$

The mapping $L:E\rightarrow E$ is said to be bounded if it maps bounded
subsets in $E$ into bounded subsets in $E.$ $L$ is said to be compact if it
is continuous and maps bounded subsets in $E$ into relatively compact
subsets in $E.$

\begin{definition}
A normed lattice $E$ is a vector space with a norm $\left\Vert .\right\Vert $
and a partial ordering ($\leq $) under which it is a Riesz space and the
following condition holds: \newline
if $\left\vert x\right\vert \leq \left\vert y\right\vert ,$ then $\left\Vert
x\right\Vert \leq \left\Vert y\right\Vert ,$ where 
\begin{equation*}
\left\vert u\right\vert =\sup \left\{ u,-u\right\} .
\end{equation*}%
If $\left( E,\left\Vert {}\right\Vert \right) $ is complete, it is called a
Banach lattice.
\end{definition}

\bigskip

Let us recall the definition and some properties of the resolvent :\ 

\begin{definition}
\cite{ha, kr}Let $L:E\rightarrow E$ be a bounded and linear operator. The
resolvent set of $L$ is the set%
\begin{equation*}
\rho \left( L\right) =\left\{ \lambda \in 
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
:\lambda I-L\text{ is invertible in }Q\left( E\right) \right\} ,
\end{equation*}%
where $Q\left( E\right) $ is the unital Banach algebra defined by%
\begin{equation*}
Q\left( E\right) =\left\{ f:E\rightarrow E:f\text{ is linear and bounded}%
\right\}
\end{equation*}%
and $I:E\rightarrow E$ is the identity.\newline
\newline
The resolvent of $L$ is $r_{L}:\rho \left( L\right) \rightarrow Q\left(
E\right) $ defined by%
\begin{equation*}
r_{L}\left( \lambda \right) =\left( \lambda I-L\right) ^{-1}\in Q\left(
E\right) .
\end{equation*}%
\newline
\newline
The spectrum of $L,$ $\sigma \left( L\right) =%
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
\backslash \rho \left( L\right) $ is non-empty, compact and 
\begin{equation*}
r\left( L\right) =\underset{\lambda \in \sigma \left( L\right) }{\max }%
\left\vert \lambda \right\vert =\lim_{n\rightarrow \infty }\left\Vert
L^{n}\right\Vert ^{\frac{1}{n}},
\end{equation*}%
called the spectral radius of $L$.
\end{definition}

The serie's representation of the resolvent: If $\left\vert \lambda
\right\vert >r\left( L\right) ,$ then $\lambda \in \rho \left( L\right) $
and $r_{L}\left( \lambda \right) $ is given by 
\begin{equation*}
r_{L}\left( \lambda \right) =\overset{+\infty }{\underset{k=0}{\sum }}%
\lambda ^{-k-1}L^{k}.
\end{equation*}

\bigskip

Let $E^{+}=\left\{ u\in E,u\geq 0\right\} $ be the positive cone of a real
Banach lattice $\left( E,\left\Vert .\right\Vert ,\leq \right) .$

We consider an operator $T:E\rightarrow E$ defined by

\QTP{Body Math}
\begin{equation*}
Tu=LFu,\text{ }u\in E
\end{equation*}

where$\ L:E\rightarrow E$ is a completely continuous operator and $%
F:E\rightarrow E$ is a continuous and bounded map.

\begin{remark}
$T$ is completely continuous, because it is the composition of the
completely continuous operator $L$ and the bounded continuous map $F$.
\end{remark}

We consider the equation 
\begin{equation}
u=Tu.  \label{1}
\end{equation}

\begin{definition}
Equation (\ref{1}) is said to be Hyers-Ulam stable in $E$ with respect to $L$
( or $L$-Hyers-Ulam stable)$,$ if $T=LF$ and there exists $N>0,$ such that
the following (p$_{N}$) property is satisfied: 
\begin{equation}
\left\{ 
\begin{array}{c}
\text{For all }\epsilon >0\text{ and all }\left( v,w\right) \in E\times \bar{%
B}\left( 0,\epsilon \right) \backslash \left\{ 0\right\} , \\ 
\text{ if }v=L\left( F\left( v\right) +w\right) \text{ then }T\text{ admits
a fixed point }u\in G\text{ such that} \\ 
\text{ }\left\Vert u-v\right\Vert \leq N.\epsilon .%
\end{array}%
\right.  \tag{p$_{N}$}
\end{equation}
\end{definition}

The main tools of this work are the following Theorems:

\begin{theorem}
\cite{gra} Let $E$ be a Banach space, $C$ be a nonempty bounded convex and
closed subset of $E$, and $T:C\rightarrow C$ be a compact and continuous
map. \ Then $T$ has at least one fixed point in $C$.
\end{theorem}

\section{Main Results}

\ \ \ \ \ \ 

\subsection{\protect\bigskip Existence and Hyers-Ulam stability of abstract
equation}

\ \ 

Throughout this paper, we assume that the following hypothesis hold:%
\begin{equation}
\left\{ 
\begin{array}{c}
\text{There exists an operator }L^{\left( k\right) }:E^{+}\rightarrow E^{+}%
\text{ such that},\text{ for all }u\in E \\ 
\left\vert L\left( u\right) \right\vert \leq L^{\left( k\right) }\left(
\left\vert u\right\vert \right) ,%
\end{array}%
\right.  \label{inf}
\end{equation}%
where $L^{\left( k\right) }$ is bounded, increasing, $k-$positively
homogeneous and sub-additive on $E$, $k\in (0,1],$ with $L^{\left( k\right)
}\left( E^{+}\backslash \left\{ 0\right\} \right) \subset E^{+}\backslash
\left\{ 0\right\} .$

$F:E\rightarrow E$ is a continous mapping such that%
\begin{equation}
\left\{ 
\begin{array}{c}
\text{There exist }\left( g,h\right) \in E^{+}\backslash \left\{ 0\right\}
\times E^{+}\text{ such that }\left\Vert L^{\left( k\right) }\left( g\right)
\right\Vert <1\text{ and} \\ 
\left\vert F\left( u\right) \right\vert \leq g\left\Vert u\right\Vert ^{%
\frac{1}{k}}+h\text{, for all }u\in E\text{.}%
\end{array}%
\right.  \label{inf'}
\end{equation}

\begin{lemma}
\label{lemex}Assume that If the hypothesis (\ref{inf}) and (\ref{inf'}) hold
true, and let Then $T$ admits a fixed point $u$ in $\bar{B}\left( 0,r\right) 
$, $r>r_{0},$ where 
\begin{equation*}
r_{0}=\dfrac{\left\Vert L^{\left( k\right) }\left( h\right) \right\Vert }{%
1-\left\Vert L^{\left( k\right) }\left( g\right) \right\Vert }\geq 0.
\end{equation*}
\end{lemma}

\begin{proof}
Let $u\in \bar{B}\left( 0,r\right) ,$ $r>r_{0}.$ So,%
\begin{eqnarray*}
\left\vert Tu\right\vert &=&\left\vert LFu\right\vert \leq L^{\left(
k\right) }\left( \left\vert Fu\right\vert \right) \leq L^{\left( k\right)
}\left( \left\Vert u\right\Vert ^{\frac{1}{k}}.g+h\right) \\
&\leq &\left\Vert u\right\Vert .L^{\left( k\right) }\left( g\right)
+L^{\left( k\right) }\left( h\right)
\end{eqnarray*}%
this implies that%
\begin{equation*}
\left\Vert Tu\right\Vert \leq r.\left\Vert L^{\left( k\right) }\left(
g\right) \right\Vert +\left\Vert L^{\left( k\right) }\left( h\right)
\right\Vert =\left( r-r_{0}\right) .\left\Vert L^{\left( k\right) }\left(
g\right) \right\Vert +r_{0}\leq r,
\end{equation*}%
then $T\left( \bar{B}\left( 0,r\right) \right) \subset \bar{B}\left(
0,r\right) .$ From Schauder fixed point theorem, we deduce that $T$ has at
least one fixed point $u\in \bar{B}\left( 0,r\right) .$\newline
\newline
\end{proof}

\begin{lemma}
\label{lemv}Assume that hypothesis (\ref{inf}) and (\ref{inf'}) hold true.%
\newline
If $\left( v,w\right) \in E\times \bar{B}\left( 0,\epsilon \right)
\backslash \left\{ 0\right\} ,$ $\epsilon >0$ such that 
\begin{equation*}
v=L\left( F\left( v\right) +w\right) ,
\end{equation*}%
then $v\in \bar{B}\left( 0,r_{\epsilon }\right) $, with 
\begin{equation*}
r_{\epsilon }=\dfrac{\left\Vert L^{\left( k\right) }\left( h\right)
\right\Vert +\epsilon ^{k}M}{1-\left\Vert L^{\left( k\right) }\left(
g\right) \right\Vert }\text{ and }M=\sup \left\{ \left\Vert L^{\left(
k\right) }\left( x\right) \right\Vert ,x\in \bar{B}\left( 0,1\right)
\right\} .
\end{equation*}
\end{lemma}

\begin{proof}
Indeed, if $v=L\left( Fv+w\right) ,$ then%
\begin{eqnarray*}
\left\vert v\right\vert &=&\left\vert L\left( Fv+w\right) \right\vert \leq
L^{\left( k\right) }\left( \left\vert Fv\right\vert +\left\vert w\right\vert
\right) \leq L^{\left( k\right) }\left( \left\Vert v\right\Vert ^{\frac{1}{k}%
}.g+h+\left\vert w\right\vert \right) \\
&\leq &\left\Vert v\right\Vert .L^{\left( k\right) }\left( g\right)
+L^{\left( k\right) }\left( h\right) +L^{\left( k\right) }\left( \left\vert
w\right\vert \right) .
\end{eqnarray*}%
This leads%
\begin{equation*}
\left\Vert v\right\Vert \leq \left\Vert v\right\Vert .\left\Vert L^{\left(
k\right) }\left( g\right) \right\Vert +\left\Vert L^{\left( k\right) }\left(
h\right) \right\Vert +\left\Vert L^{\left( k\right) }\left( \left\vert
w\right\vert \right) \right\Vert .
\end{equation*}%
Thus%
\begin{equation*}
\left\Vert v\right\Vert \leq \frac{\left\Vert L^{\left( k\right) }\left(
h\right) \right\Vert +\left\Vert L^{\left( k\right) }\left( \left\vert
w\right\vert \right) \right\Vert }{1-\left\Vert L^{\left( k\right) }\left(
g\right) \right\Vert }\leq \frac{\left\Vert L^{\left( k\right) }\left(
h\right) \right\Vert +\epsilon ^{k}M}{1-\left\Vert L^{\left( k\right)
}\left( g\right) \right\Vert }.
\end{equation*}
\end{proof}

Let $r_{\ast }$ $=\max \left\{ r_{0},\left( r_{0}\right) ^{\frac{1}{k}%
}\left\Vert g\right\Vert +\left\Vert h\right\Vert \right\} \geq 0$, where $%
r_{0}$ is the constant given in Lemma (\ref{lemex}). We consider the
following hypothesis : \newline
\newline
There exist $\rho \in E^{+}\backslash \left\{ 0\right\} ,$ $\lambda >0$ and $%
r>r_{\ast }$ such that, for all $u,v\in \bar{B}\left( 0,r\right) ,$

\begin{equation}
\text{ }\left\vert Fu-Fv\right\vert \leq \rho \left\Vert u-v\right\Vert ,
\label{inf1}
\end{equation}%
and 
\begin{equation}
\left\vert L\left( u\right) -L\left( v\right) \right\vert \leq \lambda
L_{+}\left\vert u-v\right\vert \text{.}  \label{inf2}
\end{equation}%
where $L_{+}$ is a linear, bounded and strictly positive operator on $E$.

\begin{theorem}
\label{th1}Assume that hypothesis (\ref{inf}), (\ref{inf'}), (\ref{inf1})
and (\ref{inf2}) hold true, and 
\begin{equation}
\lambda \in \left( 0,\left\Vert L_{+}\left( \rho \right) \right\Vert
^{-1}\right) .  \label{a}
\end{equation}%
Then, equation (\ref{1}) is $L$-Hyers-Ulam stable in $E.$
\end{theorem}

\begin{proof}
Suppose that 
\begin{equation*}
v=L\left( F\left( v\right) +w\right) ,
\end{equation*}%
where $\left( v,w\right) \in E\times \bar{B}\left( 0,\epsilon \right)
\backslash \left\{ 0\right\} ,$ $\epsilon >0$.\newline
Let $r>$ $r_{\ast }=\max \left\{ r_{0},\left( r_{0}\right) ^{\frac{1}{k}%
}\left\Vert g\right\Vert +\left\Vert h\right\Vert \right\} $ be the constant
given in the hypothesis (\ref{inf1}).

We deduce from lemmas (\ref{lemex}) and (\ref{lemv}) that $T$ admits a fixed
point $u\in \bar{B}\left( 0,r\right) $ and $v\in \bar{B}\left( 0,r_{\epsilon
}\right) ,$ with%
\begin{equation*}
r_{\epsilon }=\dfrac{\left\Vert L^{\left( k\right) }\left( h\right)
\right\Vert +\epsilon ^{k}M}{1-\left\Vert L^{\left( k\right) }\left(
g\right) \right\Vert }\text{ and }M=\sup \left\{ \left\Vert L^{\left(
k\right) }\left( x\right) \right\Vert ,x\in \bar{B}\left( 0,1\right)
\right\} .
\end{equation*}%
\newline
Now, let $x_{0}>0$ be the unique positive solution of the algebraic equation%
\begin{equation*}
\left( r_{0}+\dfrac{M}{1-\left\Vert L^{\left( k\right) }\left( g\right)
\right\Vert }.x^{k}\right) ^{\dfrac{1}{k}}\left\Vert g\right\Vert
+\left\Vert h\right\Vert +x-r=0
\end{equation*}%
where \newline
\newline
We distinguish the following three cases :

\textbf{Case 1.} If $r<\left( r_{\epsilon }\right) ^{\frac{1}{k}}\left\Vert
g\right\Vert +\left\Vert h\right\Vert +\epsilon ,$ then $\epsilon >x_{0}$.
This leads%
\begin{equation*}
\left\Vert u-v\right\Vert \leq r+r_{\epsilon }\leq x_{0}^{-1}\left( 2r+%
\dfrac{M.x_{0}^{k}}{1-\left\Vert L^{\left( k\right) }\left( g\right)
\right\Vert }\right) .\epsilon .
\end{equation*}

\textbf{Case 2.} If $r<r_{\epsilon },$ then $\epsilon >\mu ,$ with%
\begin{equation*}
\mu =\left[ \dfrac{\left( r-r_{0}\right) \left( 1-\left\Vert L^{\left(
k\right) }\left( g\right) \right\Vert \right) }{M}\right] ^{\dfrac{1}{k}},
\end{equation*}%
and so, 
\begin{equation*}
\left\Vert u-v\right\Vert \leq 2r+\dfrac{\epsilon ^{k}M}{1-\left\Vert
L^{\left( k\right) }\left( g\right) \right\Vert }\leq \mu ^{-1}\left( 2r+%
\dfrac{M.\mu ^{k}}{1-\left\Vert L^{\left( k\right) }\left( g\right)
\right\Vert }\right) .\epsilon .
\end{equation*}

\textbf{Case 3.} If $\max \left\{ r_{\epsilon },\left( r_{\epsilon }\right)
^{\frac{1}{k}}\left\Vert g\right\Vert +\left\Vert h\right\Vert +\epsilon
\right\} \leq r,$ then $\left( Fu,\left( Fv\right) +w\right) \in \bar{B}%
\left( 0,r\right) \times \bar{B}\left( 0,r\right) ,$ and from hypothesis (%
\ref{inf2}),\ it follows that 
\begin{equation*}
\left\vert L\left( Fu\right) -L\left( Fv+w\right) \right\vert \leq \lambda
L_{+}\left\vert Fu-Fv-w\right\vert .
\end{equation*}%
And by using (\ref{inf1}), we obtain 
\begin{eqnarray*}
\left\vert u-v\right\vert &\leq &\lambda L_{+}\left\vert Fu-Fv-w\right\vert
\\
&\leq &\lambda L_{+}\left\vert Fu-Fv\right\vert +\lambda L_{+}\left(
\left\vert w\right\vert \right) \\
&\leq &\lambda .\left\Vert u-v\right\Vert L_{+}\left( \rho \right) +\lambda
\epsilon L_{+}\left( \frac{\left\vert w\right\vert }{\left\Vert w\right\Vert 
}\right)
\end{eqnarray*}%
thus%
\begin{equation*}
\left\Vert u-v\right\Vert \leq \left( \dfrac{\lambda \left\Vert
L_{+}\right\Vert }{1-\lambda .\left\Vert L_{+}\left( \rho \right)
\right\Vert }\right) .\epsilon .
\end{equation*}%
Consequently, 
\begin{equation*}
\left\Vert u-v\right\Vert \leq N.\epsilon
\end{equation*}%
where%
\begin{equation*}
N=\max \left\{ \gamma _{1}^{\prime }\left( 2r+\dfrac{M.\gamma _{2}^{\prime }%
}{1-\left\Vert L^{\left( k\right) }\left( g\right) \right\Vert }\right)
,\left( \dfrac{\lambda \left\Vert L_{+}\right\Vert }{1-\lambda .\left\Vert
L_{+}\left( \rho \right) \right\Vert }\right) \right\} ,
\end{equation*}%
with%
\begin{equation*}
\gamma _{1}^{\prime }=\max \left\{ x_{0}^{-1},\mu ^{-1}\right\} \text{ and }%
\gamma _{2}^{\prime }=\max \left( x_{0}^{k},\mu ^{k}\right) .
\end{equation*}%
Proving our claim.
\end{proof}

Now, we replace the hypothesis (\ref{inf1}) and (\ref{inf2}) by the
following conditions :

There exists $\lambda _{0}>0$ and $r>r_{\ast }$ such that, for all $u,v\in 
\bar{B}\left( 0,r\right) ,$

\begin{equation}
\left\vert F\left( u\right) -F\left( v\right) \right\vert \leq \lambda
_{0}\left\vert u-v\right\vert \text{,}  \label{inf3}
\end{equation}%
and

\begin{equation}
\left\vert L\left( u\right) -L\left( v\right) \right\vert \leq
L_{0}\left\vert u-v\right\vert ,  \label{inf4}
\end{equation}%
where $L_{0}:E\rightarrow E$ is a linear, compact and strictly positive
operator.

\begin{theorem}
\label{th2}Assume that hypothesis (\ref{inf}), (\ref{inf'}), (\ref{inf3})
and (\ref{inf4}) hold, and 
\begin{equation}
r(L_{0})<\lambda _{0}^{-1}.  \label{b}
\end{equation}%
Then equation (\ref{1}) is $L$-Hyers-Ulam stable in $E.$
\end{theorem}

\begin{proof}
Suppose that $v=L\left( F\left( v\right) +w\right) ,$ $\left( v,w\right) \in
E\times \bar{B}\left( 0,\epsilon \right) \backslash \left\{ 0\right\} ,$ $%
\epsilon >0$.\newline
Let $r>$ $r_{\ast }=\max \left\{ r_{0},\left( r_{0}\right) ^{\frac{1}{k}%
}\left\Vert g\right\Vert +\left\Vert h\right\Vert \right\} $ is the constant
given in the hypothesis (\ref{inf3}). It follows from lemmas (\ref{lemex})
and (\ref{lemv}), that $v\in \bar{B}\left( 0,r_{\epsilon }\right) $ and $T$
admits a fixed point $u\in \bar{B}\left( 0,r\right) ,$ with%
\begin{equation*}
r_{\epsilon }=\dfrac{\left\Vert L^{\left( k\right) }\left( h\right)
\right\Vert +\epsilon ^{k}M}{1-\left\Vert L^{\left( k\right) }\left(
g\right) \right\Vert }\text{ and }M=\sup \left\{ \left\Vert L^{\left(
k\right) }\left( x\right) \right\Vert ,x\in \bar{B}\left( 0,1\right)
\right\} .
\end{equation*}%
\newline
\newline
We have seen in the proof of theorem (\ref{th1}) that, if 
\begin{equation*}
r\leq \max \left\{ r_{\epsilon },\left( r_{\epsilon }\right) ^{\frac{1}{k}%
}\left\Vert g\right\Vert +\left\Vert h\right\Vert +\epsilon \right\} ,
\end{equation*}%
then $\epsilon \geq \max \left\{ \mu ,x_{0}\right\} $, where $x_{0}>0$ is
the positive solution of the algebraic equation%
\begin{equation*}
\left( r_{0}+\dfrac{M}{1-\left\Vert L^{\left( k\right) }\left( g\right)
\right\Vert }.x^{k}\right) ^{\dfrac{1}{k}}\left\Vert g\right\Vert
+\left\Vert h\right\Vert +x-r=0
\end{equation*}%
In this case, we have 
\begin{equation*}
\left\Vert u-v\right\Vert \leq \gamma _{1}^{\prime }\left( 2r+\dfrac{%
M.\gamma _{2}^{\prime }}{1-\left\Vert L^{\left( k\right) }\left( g\right)
\right\Vert }\right) .\epsilon ,
\end{equation*}%
where%
\begin{equation*}
\gamma _{1}^{\prime }=\max \left\{ x_{0}^{-1},\mu ^{-1}\right\} \text{ and }%
\gamma _{2}^{\prime }=\max \left( x_{0}^{k},\mu ^{k}\right) .
\end{equation*}%
\newline
\newline
Now, we assume that $\max \left\{ r_{\epsilon },\left( r_{\epsilon }\right)
^{\frac{1}{k}}\left\Vert g\right\Vert +\left\Vert h\right\Vert +\epsilon
\right\} \leq r.$ Then $\left( Fu,\left( Fv\right) +w\right) \in \bar{B}%
\left( 0,r\right) \times \bar{B}\left( 0,r\right) ,$ and by using hypothesis
(\ref{inf2}),\ it follows that 
\begin{equation}
\left\vert L\left( Fu\right) -L\left( Fv+w\right) \right\vert \leq
L_{0}\left\vert Fu-Fv-w\right\vert .  \label{proof0}
\end{equation}%
By using (\ref{inf3}), inequality (\ref{proof0}) leads 
\begin{eqnarray*}
\left\vert u-v\right\vert &\leq &L_{0}\left\vert Fu-Fv-w\right\vert \\
&\leq &L_{0}\left\vert Fu-Fv\right\vert +L_{0}\left( \left\vert w\right\vert
\right) \\
&\leq &\lambda _{0}.L_{0}\left( \left\vert u-v\right\vert \right) +\epsilon
.\pi _{w},
\end{eqnarray*}%
where%
\begin{equation*}
\pi _{w}=L_{0}\left( \frac{\left\vert w\right\vert }{\left\Vert w\right\Vert 
}\right) \in E^{+}\backslash \left\{ 0\right\} .
\end{equation*}%
Then 
\begin{eqnarray*}
z &=&\left\vert u-v\right\vert \leq \lambda _{0}.L_{0}\left( z\right)
+\epsilon L_{0}\left( \frac{\left\vert w\right\vert }{\left\Vert
w\right\Vert }\right) \\
&\leq &\lambda _{0}.L_{0}\left( z\right) +\epsilon \pi _{w} \\
&\leq &\lambda _{0}.L_{0}\left( \lambda _{0}.L_{0}\left( z\right) +\epsilon
\pi _{w}\right) +\epsilon \pi _{w} \\
&\leq &\lambda _{0}^{3}.L_{0}^{3}\left( z\right) +\epsilon .\left( \lambda
_{0}^{2}.L_{0}^{2}\left( \pi _{w}\right) +\lambda _{0}.L_{0}\left( \pi
_{w}\right) +\pi _{w}\right) \\
&\leq &\lambda _{0}^{n}.L_{0}^{n}\left( z\right) +\epsilon
.\tsum_{k=0}^{n-1}\lambda _{0}^{k}L_{0}^{k}\left( \pi _{w}\right) \in
E^{+}\backslash \left\{ 0\right\} ,\text{ for all }n\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
^{\ast }.
\end{eqnarray*}%
As $\lambda _{0}.r\left( L_{0}\right) =\lambda _{0}.\lim_{n\rightarrow
\infty }\sqrt[n]{\left\Vert L_{0}^{n}\right\Vert }<1$ then $%
\lim_{n\rightarrow \infty }\lambda _{0}^{n}.L_{0}^{n}\left( z\right) =0,$ $%
\lambda _{0}^{-1}\in \rho \left( L_{0}\right) $ and $\left( I-\lambda
_{0}.L_{0}\right) $ is invertible. The serie's representation of the
resolvent $r_{L_{0}}$ at $\lambda _{0}^{-1}$ is given by 
\begin{equation*}
r_{L_{0}}\left( \lambda _{0}^{-1}\right) =\left( \lambda
_{0}^{-1}I-L_{0}\right) ^{-1}=\tsum_{k=0}^{+\infty }\left( \lambda
_{0}\right) ^{k+1}L_{0}^{k}.
\end{equation*}%
Then 
\begin{equation*}
\tsum_{k=0}^{+\infty }\lambda _{0}^{k}L_{0}^{k}\left( \pi _{w}\right)
=\left( I-\lambda _{0}.L_{0}\right) ^{-1}\left( \pi _{w}\right) \in
E^{+}\backslash \left\{ 0\right\} .
\end{equation*}%
Thus, 
\begin{equation*}
\left\Vert u-v\right\Vert \leq \left\Vert \left( I-\lambda _{0}.L_{0}\right)
^{-1}\left( \pi _{w}\right) \right\Vert .\epsilon \leq \left\Vert \left(
I-\lambda _{0}.L_{0}\right) ^{-1}\right\Vert \left\Vert L_{0}\right\Vert
.\epsilon .
\end{equation*}%
Consequently, 
\begin{equation*}
\left\Vert u-v\right\Vert \leq N.\epsilon
\end{equation*}%
where%
\begin{equation*}
N=\max \left\{ \gamma _{1}^{\prime }\left( 2r_{0}+\dfrac{M.\gamma
_{2}^{\prime }}{1-\left\Vert L^{\left( k\right) }\left( g\right) \right\Vert 
}\right) ,\left\Vert \left( I-\lambda _{0}.L_{0}\right) ^{-1}\right\Vert
\left\Vert L_{0}\right\Vert \right\} .
\end{equation*}%
Proving our claim.
\end{proof}

\subsection{Existence and Hyers-Ulam stability of coupled system IVS (%
\protect\ref{e1})}

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 

\bigskip

In this section, we use the results obtained in the previous section to
prove existence and Hyers-Ulam stability of the coupled system of sequential
time $\sigma -$Hilfer fractional differential equations (\ref{e1}), where $%
D_{0^{+},t}^{\alpha ,\omega ,\sigma }$is the $\sigma -$Hilfer fractional
derivative with respect to the variable $t$ of order $\alpha $ and type $%
0\leq \omega \leq 1$ with $0<\alpha <1$,%
\begin{equation*}
\phi =\phi _{p^{-}}+\phi _{p^{+}}\text{ , }1<p^{-}<p^{+}
\end{equation*}%
with 
\begin{equation*}
\phi _{p^{\nu }}\left( x\right) =\left\vert x\right\vert ^{p^{\nu }-2}.x,%
\text{ for }\nu \in \left\{ -,+\right\} ,
\end{equation*}%
and for $j\in \left\{ 1,2\right\} ,$ 
\begin{equation*}
\zeta _{j}\left( t\right) =a_{j}+t,\text{ }a_{j}>0.
\end{equation*}%
We suppose that the following conditions hold, 
\begin{equation}
\left\{ 
\begin{array}{c}
f_{j}\in C\left( 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\times 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\times 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{2},%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\right) ,\text{ }\frac{1}{\theta _{j}}\in L^{1}\left( 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+},%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\right) \\ 
\text{ and} \\ 
\sigma ^{+}\in L^{\alpha }\text{ }\left( 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\right) ,%
\end{array}%
\right. \text{ }  \label{h0}
\end{equation}%
with 
\begin{equation*}
0<\sigma ^{+}\left( x\right) =\sup \left\{ \sigma \left( t,x\right) ,\text{ }%
t\geq 0\right\} <\infty ,\text{ }\forall x\geq 0.
\end{equation*}%
Next, we recall the definitions of $\sigma -$Hilfer fractional orders
integrals and derivatives of order $\alpha $ and type $0\leq \omega \leq 1$,
where $J\subset 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{n}$ and $\sigma :I\times J\rightarrow 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}$ is the positive function on $I\times J\subset 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\times 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}$ having a continuous and positive derivative $\dfrac{\partial \sigma }{%
\partial t}\left( t,x\right) >0$ with respect to the variable $t$ on $%
(0,+\infty )$ with $\sigma \left( 0,x\right) =0$ for all $x\geq 0$.\ \ 

\bigskip\ \ 

\begin{definition}
\cite{va} Let $a\in 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+},$ $\alpha >0$ and $J\subset 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{n}.$ Then the $\sigma -$left-sided fractional integral of a function $u$
with respect to $t$ on $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}$\ is defined by%
\begin{equation*}
I_{a^{+},t}^{\alpha ,\sigma }u(t,x)=\frac{1}{\Gamma \left( \alpha \right) }%
\int_{a}^{t}\dfrac{\partial \sigma }{\partial t}\left( t,x\right) \left(
\sigma \left( t,x\right) -\sigma \left( \tau ,x\right) \right) ^{\alpha
-1}u(\tau ,x)d\tau .
\end{equation*}%
In the case $\alpha =0$, this integral is interpreted as the identity
operator $I_{a^{+}}^{0,\sigma }u=u.$
\end{definition}

\begin{definition}
\cite{va} Let $\alpha \in \left( n-1,n\right) $ with $n\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
,$ u and $\sigma \ $two functions such that $t\mapsto u\left( t,.\right) \in
C^{n}\left( 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+},%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\right) $ and $t\mapsto \sigma \left( t,.\right) \in C^{n}\left( 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+},%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\right) .$ The $\sigma $-Hilfer fractional derivative $D_{a^{+},t}^{\alpha
,\omega ,\sigma }$ of $u$ with respect to $t$ of order $n-1<\alpha <n$ and
type $0\leq \omega \leq 1$ is defined by%
\begin{equation*}
D_{a^{+},t}^{\alpha ,\omega ,\sigma }u\left( t,x\right) =I_{a^{+},t}^{\omega
\left( n-\alpha \right) ,\sigma }\left( \frac{1}{\sigma _{t}^{\prime }(t,x)}%
\frac{\partial }{\partial t}\right) ^{n}I_{a^{+},t}^{\left( 1-\omega \right)
\left( n-\alpha \right) ,\sigma }u\left( t,x\right) ,
\end{equation*}%
where $\sigma _{t}^{\prime }(t,x)=\dfrac{\partial \sigma }{\partial t}\left(
t,x\right) .$
\end{definition}

Let's also recall the following important result (\cite{va}):

\begin{theorem}
\label{Hi} If $t\mapsto u\left( t,x\right) \in C^{n}\left( 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\right) ,$ $n-1<\beta <\alpha <n,$ $0\leq \omega \leq 1$ and $\xi
=\alpha +\omega \left( n-\alpha \right) ,$ then%
\begin{equation*}
I_{a^{+},t}^{\alpha ,\sigma }.D_{a^{+},t}^{\alpha ,\omega ,\sigma }u\left(
t,x\right) =u(t,x)-\sum_{k=1}^{n}\frac{\left( \sigma (t,x)-\sigma
(a,x)\right) ^{\xi -k}}{\Gamma \left( \xi -k+1\right) }\left( \frac{1}{%
\sigma _{t}^{\prime }(t,x)}\frac{\partial }{\partial t}\right)
^{n-k}I_{a^{+},t}^{\left( 1-\omega \right) \left( n-\alpha \right) ,\sigma
}u\left( a,x\right) .
\end{equation*}%
Moreover, $I_{a^{+},t}^{\alpha ,\sigma }I_{a^{+},t}^{\beta ,\sigma }\left(
u\right) =I_{a^{+},t}^{\alpha +\beta ,\sigma },$ $D_{a^{+},t}^{\alpha
,\omega ,\sigma }\left( D_{a^{+},t}^{\beta ,\omega ,\sigma }u\right)
=D_{a^{+},t}^{\alpha +\beta ,\omega ,\sigma }u,$ $D_{a^{+},t}^{1,\omega
,\sigma }u=D_{t}^{1}u=\dfrac{\partial u}{\partial t}$ and $%
D_{a^{+},t}^{\alpha ,\omega ,\sigma }I_{a^{+},t}^{\alpha ,\sigma }\left(
u\right) =u.$
\end{theorem}

\begin{remark}
In this paper, we assume that $\sigma :%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\times 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\rightarrow 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}$ is continuous having a positive and continuous derivative $\dfrac{%
\partial \sigma }{\partial t}\left( t,x\right) $ on $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\times 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}$ such that $\sigma \left( 0,x\right) =0,$ for all $x\in 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}.$ If $\alpha \in \left( 0,1\right) ,$ then $n=1$ and for $t,x>0$%
\begin{equation*}
I_{0^{+},t}^{\alpha ,\sigma }.D_{0^{+},t}^{\alpha ,\omega ,\sigma }u\left(
t,x\right) =u(t,x)-\frac{\left( \sigma (t,x)\right) ^{\xi -1}}{\Gamma \left(
\xi \right) }\left( I_{0^{+},t}^{\left( 1-\omega \right) \left( 1-\alpha
\right) ,\sigma }u\right) \left( 0^{+},x\right) .
\end{equation*}%
Moreover, if $u$ is continuous, then 
\begin{equation*}
\lim_{t\rightarrow 0^{+}}\left( I_{0^{+},t}^{\left( 1-\omega \right) \left(
1-\alpha \right) ,\sigma }u\right) \left( t,x\right) =0,\text{ }\forall
x\geq 0
\end{equation*}%
and so $I_{0^{+},t}^{\alpha ,\sigma }.D_{0^{+},t}^{\alpha ,\omega ,\sigma
}u\left( t,x\right) =u(t,x).$
\end{remark}

\begin{definition}
We say that IVS (\ref{e1}) has the Hyers-Ulam stability in a Banach space $%
E=G\times G$ if there exits a constant $N>0$ such that for every $\epsilon
>0,$ $v=\left( v_{1},v_{2}\right) \in E,$ if%
\begin{equation}
\left\{ 
\begin{array}{l}
\left\vert \left( \zeta _{1}\left( t\right) .D_{0^{+},t}^{\alpha +1,\omega
,\sigma }+D_{0^{+},t}^{\alpha ,\omega ,\sigma }\right) \left( \dfrac{%
\partial }{\partial x}\left( \phi \left( \theta _{1}\left( x\right) \dfrac{%
\partial v_{1}}{\partial x}\right) \right) \right) \left( t,x\right)
+f_{1}\left( t,x,v_{1},v_{2}\right) \right\vert \leq \epsilon ,\ t,x>0, \\ 
\\ 
\left\vert \left( \zeta _{2}\left( t\right) .D_{0^{+},t}^{\alpha +1,\omega
,\sigma }+D_{0^{+},t}^{\alpha ,\omega ,\sigma }\right) \left( \dfrac{%
\partial }{\partial x}\left( \phi \left( \theta _{2}\left( x\right) \dfrac{%
\partial v_{2}}{\partial x}\right) \right) \right) \left( t,x\right)
+f_{2}\left( t,x,v_{1},v_{2}\right) \right\vert \leq \epsilon ,\ t,x>0, \\ 
\\ 
v_{j}\left( 0,x\right) =v_{j}\left( t,0\right) =\underset{x\rightarrow
+\infty }{\lim }\dfrac{\partial v_{j}}{\partial x}\left( t,x\right) =0,\text{
}j\in \left\{ 1,2\right\} ,%
\end{array}%
\right. \   \label{eqstu}
\end{equation}%
then there exists a solution $u\in E$ of IVS (\ref{e1}), such that%
\begin{equation}
\left\Vert u-v\right\Vert \leq N.\epsilon .  \label{stu}
\end{equation}%
We call such $N$ a Hyers-Ulam stability constant.
\end{definition}

\bigskip

Let $E=G\times G$ be a real Banach space with

\begin{equation*}
G=\left\{ u\in C(%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\times 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+},%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
):\underset{t,x\geq 0}{\sup }\left\vert u(t,x)\right\vert <\infty \right\}
\end{equation*}%
equipped with the norm $\left\Vert \left( u,v\right) \right\Vert =\max
\left( \left\Vert u\right\Vert _{0},\left\Vert v\right\Vert _{0}\right) $
where%
\begin{equation*}
\left\Vert u\right\Vert _{0}=\sup_{t,x\in 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}}\left( \left\vert u(t,x)\right\vert \right) .
\end{equation*}

\begin{remark}
$E$ is a Banach lattice under the partial ordering ($\leq $) defined by 
\begin{equation*}
\left( u_{1},u_{2}\right) \leq \left( v_{1},v_{2}\right) \Leftrightarrow
u_{1}\left( x\right) \leq v_{1}\left( x\right) \text{ and }u_{2}\left(
x\right) \leq v_{2}\left( x\right) \text{ for all }x\geq 0.
\end{equation*}%
under which it is a Riesz space and $\left\vert \left( u,v\right)
\right\vert =\left( \left\vert u\right\vert ,\left\vert v\right\vert \right) 
$. \newline
Moreover, $E^{+}=\left\{ \left( u,v\right) \in E,\left( u,v\right) \geq
0\right\} $ is the positive cone of $\left( E,\left\Vert .\right\Vert ,\leq
\right) .$
\end{remark}

We consider the operator $T:E\rightarrow E$ defined by

\QTP{Body Math}
\begin{equation*}
T\left( u_{1},u_{2}\right) =LF\left( u_{1},u_{2}\right) ,\text{ }\left(
u_{1},u_{2}\right) \in E
\end{equation*}

where$\ $%
\begin{equation*}
L\left( u_{1},u_{2}\right) =\left( L_{1}\left( u_{1},u_{2}\right)
,L_{2}\left( u_{1},u_{2}\right) \right) \text{ and }F\left(
u_{1},u_{2}\right) =\left( F_{1}\left( u_{1},u_{2}\right) ,F_{2}\left(
u_{1},u_{2}\right) \right) ,
\end{equation*}%
such that for $j\in \left\{ 1,2\right\} $%
\begin{eqnarray*}
L_{j}\left( u_{1},u_{2}\right) \left( t,x\right) &=&\int_{0}^{x}\frac{1}{%
\theta _{j}\left( z\right) }\psi \left( \int_{z}^{+\infty
}I_{0^{+},t}^{\alpha ,\sigma }\left( \frac{1}{\zeta _{j}\left( t\right) }%
\int_{0}^{t}\left( u_{j}\right) \left( \tau ,s\right) d\tau \right) \left(
t,s\right) ds\right) dz, \\
F_{j}\left( u_{1},u_{2}\right) \left( t,x\right) &=&f_{j}\left(
t,x,u_{1}\left( t,x\right) ,u_{2}\left( t,x\right) \right) ,
\end{eqnarray*}%
where $\psi =\phi ^{-1}:%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\rightarrow 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ is the inverse function of sum of $p_{i}$-Laplacian operators $\phi
=\sum_{i=1}^{i=N}\phi _{p_{i}},$ with $\phi _{p_{i}}\left( x\right)
=\left\vert x\right\vert ^{p_{i}-2}.x$ and $\psi _{p_{i}}$ is the inverse
function of $\phi _{p_{i}}.$

We denote 
\begin{equation*}
T=\left( T_{1},T_{2}\right)
\end{equation*}%
with%
\begin{equation*}
T_{j}=L_{j}F,\text{ }j\in \left\{ 1,2\right\} .
\end{equation*}

\begin{remark}
\label{rem1}Let $p^{-}=\min \left\{ p_{1},p_{2}...p_{N}\right\} $ and $%
p^{+}=\max \left\{ p_{1},p_{2}...p_{N}\right\} $. For all $x\geq 0,$ $i\in
\left\{ 1,2...N\right\} $ $\ $%
\begin{equation*}
\phi _{p_{i}}\left( x\right) \leq \phi \left( x\right) \leq N.\phi
^{+}\left( x\right)
\end{equation*}%
where%
\begin{equation*}
\phi ^{+}\left( x\right) =\left\{ 
\begin{array}{cc}
\phi _{p^{+}}\left( x\right) & if\text{ }x\geq 1 \\ 
\phi _{p^{-}}\left( x\right) & if\text{ }x\leq 1%
\end{array}%
\right.
\end{equation*}%
and so, we conclude that 
\begin{equation}
\psi ^{+}\left( \frac{x}{N}\right) \leq \psi \left( x\right) \leq \psi
_{p_{i}}\left( x\right)  \label{psi}
\end{equation}%
\newline
where 
\begin{equation*}
\psi ^{+}\left( \frac{x}{N}\right) =\left\{ 
\begin{array}{cc}
\psi _{p^{+}}\left( \frac{x}{N}\right) & if\text{ }x\geq 1 \\ 
\psi _{p^{-}}\left( \frac{x}{N}\right) & if\text{ }x\leq 1.%
\end{array}%
\right.
\end{equation*}%
Moreover, for $x\geq y\geq 0,$ 
\begin{equation}
\left\{ 
\begin{array}{cc}
\psi _{p}\left( x+y\right) \leq \psi _{p}\left( x\right) +\psi _{p}\left(
y\right) , & \text{if }p\geq 2, \\ 
\psi _{p}\left( x+y\right) \leq \left( 2\right) ^{\dfrac{2-p}{p-1}}.\left[
\psi _{p}\left( x\right) +\psi _{p}\left( y\right) \right] , & \text{if }p<2.%
\end{array}%
\right.  \label{2p}
\end{equation}
\end{remark}

\begin{remark}
The condition (\ref{h0})\ makes that the operator $L_{j}$ is completely
continuous and $F_{j}$ is bounded for each $j\in \left\{ 1,2\right\} ,$ and
so, $T$ is completely continuous.
\end{remark}

\begin{lemma}
\label{pf}Let $h_{1},h_{2}\in C\left( 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\times 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+},%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\right) $ be continuous and bounded functions$.\ \left(
u_{1},u_{2}\right) \in C^{1}\left( 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\times 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\right) \times C^{1}\left( 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\times 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\right) $ is solution of IVS (\ref{e11})%
\begin{equation}
\left\{ 
\begin{array}{l}
\left( \zeta _{1}\left( t\right) .D_{0^{+},t}^{\alpha +1,\omega ,\sigma
}+D_{0^{+},t}^{\alpha ,\omega ,\sigma }\right) \left( \dfrac{\partial }{%
\partial x}\left( \phi \left( \theta _{1}\left( x\right) \dfrac{\partial
u_{1}}{\partial x}\right) \right) \right) \left( t,x\right) +h_{1}\left(
t,x\right) =0,\ t,x>0, \\ 
\\ 
\left( \zeta _{2}\left( t\right) .D_{0^{+},t}^{\alpha +1,\omega ,\sigma
}+D_{0^{+},t}^{\alpha ,\omega ,\sigma }\right) \left( \dfrac{\partial }{%
\partial x}\left( \phi \left( \theta _{2}\left( x\right) \dfrac{\partial
u_{2}}{\partial x}\right) \right) \right) \left( t,x\right) +h_{2}\left(
t,x\right) =0,\ t,x>0, \\ 
\\ 
u_{j}\left( 0,x\right) =u_{j}\left( t,0\right) =\underset{x\rightarrow
+\infty }{\lim }\dfrac{\partial u_{j}}{\partial x}\left( t,x\right) =0,\text{
}j\in \left\{ 1,2\right\} ,%
\end{array}%
\right.  \label{e11}
\end{equation}%
if and only if 
\begin{equation*}
u_{j}\left( t,x\right) =\int_{0}^{x}\frac{1}{\theta _{j}\left( z\right) }%
\psi \left( \int_{z}^{+\infty }I_{0^{+},t}^{\alpha ,\sigma }\left( \frac{1}{%
\zeta _{j}\left( t\right) }\int_{0}^{t}h_{j}\left( \tau ,s\right) d\tau
\right) \left( t,s\right) ds\right) dz,\text{ for }j\in \left\{ 1,2\right\} .
\end{equation*}%
$\left( u_{1},u_{2}\right) $ is fixed point of $T$ (i.e $T\left(
u_{1},u_{2}\right) =\left( u_{1},u_{2}\right) $).
\end{lemma}

\begin{proof}
First, assume that $\left( u_{1},u_{2}\right) \in E$ is a solution of IVS (%
\ref{e11}), then for each $j\in \left\{ 1,2\right\} $, The function $u_{j}$
satisfies equation%
\begin{equation*}
D_{t}^{1}\left( \left( a_{j}+t\right) .D_{0^{+},t}^{\alpha ,\omega ,\sigma } 
\left[ \dfrac{\partial }{\partial x}\left( \phi \left( \theta _{j}\left(
x\right) \dfrac{\partial u_{j}}{\partial x}\right) \right) \right] \right)
\left( t,x\right) =-h_{j}\left( t,x\right) ,
\end{equation*}%
where $\phi =\phi _{p^{-}}+\phi _{p^{+}}.$\ Integrating, we have 
\begin{equation}
D_{0^{+},t}^{\alpha ,\omega ,\sigma }\left[ \dfrac{\partial }{\partial x}%
\left( \phi \left( \theta _{j}\left( x\right) \dfrac{\partial u_{j}}{%
\partial x}\right) \right) \right] \left( t,x\right) =\frac{-1}{a_{j}+t}%
\int_{0}^{t}h_{j}\left( \tau ,x\right) d\tau ,\text{ }t>0\text{.}
\label{eq1}
\end{equation}%
Applying $I_{0^{+},t}^{\alpha ,\sigma }$ on both sides of equation (\ref{eq1}%
) and using Lemma (\ref{Hi}) and initial condition $\dfrac{\partial u_{j}}{%
\partial x}\left( 0,x\right) =0,$ we obtain 
\begin{equation*}
\dfrac{\partial }{\partial x}\left( \phi \left( \theta _{j}\left( x\right) 
\dfrac{\partial u_{j}}{\partial x}\right) \right) \left( t,x\right)
=-I_{0^{+},t}^{\alpha ,\sigma }\left( \frac{1}{\zeta _{j}\left( t\right) }%
\int_{0}^{t}h_{j}\left( \tau ,x\right) d\tau \right) \left( t,x\right)
\end{equation*}%
\newline
By integrating on $[x,+\infty \lbrack $\ and using the boundary conditions $%
u_{j}\left( t,0\right) =\underset{x\rightarrow +\infty }{\lim }\dfrac{%
\partial u_{j}}{\partial x}\left( t,x\right) =0$, we have%
\begin{equation*}
\phi \left( \theta _{j}\left( x\right) \dfrac{\partial u_{j}}{\partial x}%
\right) =\int_{x}^{+\infty }I_{0^{+},t}^{\alpha ,\sigma }\left( \frac{1}{%
\zeta _{j}\left( t\right) }\int_{0}^{t}h_{j}\left( \tau ,s\right) d\tau
\right) \left( t,s\right) ds
\end{equation*}%
and so%
\begin{equation*}
u_{j}\left( t,x\right) =\int_{0}^{x}\frac{1}{\theta _{j}\left( z\right) }%
\psi \left( \int_{z}^{+\infty }I_{0^{+},t}^{\alpha ,\sigma }\left( \frac{1}{%
\zeta _{j}\left( t\right) }\int_{0}^{t}h_{j}\left( \tau ,s\right) d\tau
\right) \left( t,s\right) ds\right) dz.
\end{equation*}%
Conversely, assume that $\left( u_{1},u_{2}\right) \in E$ such that for $%
j\in \left\{ 1,2\right\} ,$ 
\begin{equation*}
u_{j}\left( t,x\right) =\int_{0}^{x}\frac{1}{\theta _{j}\left( z\right) }%
\psi \left( \int_{z}^{+\infty }I_{0^{+},t}^{\alpha ,\sigma }\left( \frac{1}{%
\zeta _{j}\left( t\right) }\int_{0}^{t}h_{j}\left( \tau ,s\right) d\tau
\right) \left( t,s\right) ds\right) dz.
\end{equation*}%
Then $u_{j}\in C^{1}\left( 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\times 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\right) $ and verifies 
\begin{equation*}
u_{j}(x,0)=u_{j}\left( 0,x\right) =0.
\end{equation*}%
Moreover, by derivating with respect to the variable $x$, we obtain 
\begin{equation}
\dfrac{\partial u_{j}}{\partial x}\left( t,x\right) =\frac{1}{\theta
_{j}\left( x\right) }\psi \left( \int_{x}^{+\infty }I_{0^{+},t}^{\alpha
,\sigma }\left( \frac{1}{\zeta _{j}\left( t\right) }\int_{0}^{t}h_{j}\left(
\tau ,s\right) d\tau \right) \left( t,s\right) ds\right) ,  \label{eq3}
\end{equation}%
and so%
\begin{equation}
\dfrac{\partial }{\partial x}\phi \left( \theta _{j}\left( x\right) \dfrac{%
\partial u_{j}}{\partial x}\right) =-I_{0^{+},t}^{\alpha ,\sigma }\left( 
\frac{1}{\zeta _{i}\left( t\right) }\int_{0}^{t}h_{j}\left( \tau ,x\right)
d\tau \right) \left( t,x\right) .  \label{eq2}
\end{equation}%
Applying $D_{0^{+},t}^{\alpha ,\omega ,\sigma }$ on both sides of equation (%
\ref{eq2}) and using Lemma (\ref{Hi}) we have 
\begin{equation*}
\zeta _{j}\left( t\right) .D_{0^{+},t}^{\alpha ,\omega ,\sigma }\left[ 
\dfrac{\partial }{\partial x}\left( \phi \left( \theta _{j}\left( x\right) 
\dfrac{\partial u_{j}}{\partial x}\right) \right) \right] \left( t,x\right)
=-\int_{0}^{t}h_{j}\left( \tau ,x\right) d\tau ,
\end{equation*}%
so, $u_{j}$ is solution of the equation 
\begin{equation*}
D_{t}^{1}\left( \zeta _{j}\left( t\right) .D_{0^{+},t}^{\alpha ,\omega
,\sigma }\left[ \dfrac{\partial }{\partial x}\left( \phi \left( \theta
_{j}\left( x\right) \dfrac{\partial u_{j}}{\partial x}\right) \right) \right]
\right) \left( t,x\right) =-h_{j}\left( t,x\right) .
\end{equation*}%
Now, we show that $\underset{x\rightarrow +\infty }{\lim }\dfrac{\partial
u_{j}}{\partial x}\left( t,x\right) =0.$ Let $H_{j}=\sup \left\{ h_{j}\left(
t,x\right) ,t,x\geq 0\right\} $. We have 
\begin{equation*}
I_{0^{+},t}^{\alpha ,\sigma }\left( \frac{1}{\zeta _{j}\left( t\right) }%
\int_{0}^{t}h_{j}\left( \tau ,s\right) d\tau \right) \left( t,s\right) \leq
H_{j}.I_{0^{+},t}^{\alpha ,\sigma }\left( 1\right) \left( t,s\right) =\frac{%
H_{j}}{\Gamma \left( \alpha +1\right) }\sigma ^{\alpha }\left( t,s\right) ,
\end{equation*}%
then%
\begin{equation*}
\int_{x}^{+\infty }I_{0^{+},t}^{\alpha ,\sigma }\left( \frac{1}{\zeta
_{j}\left( t\right) }\int_{0}^{t}h_{j}\left( \tau ,s\right) d\tau \right)
\left( t,s\right) ds\leq \frac{H_{j}}{\Gamma \left( \alpha +1\right) }%
\int_{x}^{+\infty }\sigma ^{\alpha }\left( t,s\right) ds
\end{equation*}%
so, it follows from equation (\ref{eq3}) that%
\begin{eqnarray*}
\dfrac{\partial u_{j}}{\partial x}\left( t,x\right) &\leq &\frac{1}{\theta
_{j}\left( x\right) }\psi \left( \frac{H_{j}}{\Gamma \left( \alpha +1\right) 
}\int_{x}^{+\infty }\sigma ^{\alpha }\left( t,s\right) ds\right) \\
&\leq &\frac{1}{\theta _{j}\left( x\right) }\psi \left( \frac{H_{j}}{\Gamma
\left( \alpha +1\right) }\int_{0}^{+\infty }\left( \sigma ^{+}\right)
^{\alpha }\left( s\right) ds\right)
\end{eqnarray*}%
Since $\dfrac{1}{\theta _{i}\left( x\right) .}\in L^{1}\left( 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+},%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\right) $ then%
\begin{equation*}
\underset{x\rightarrow +\infty }{\lim }\dfrac{\partial u_{j}}{\partial x}%
\left( t,x\right) =0.
\end{equation*}%
Thus, $\left( u_{1},u_{2}\right) $ is solution of IVS (\ref{e11}). This
completes the proof.
\end{proof}

\begin{remark}
We deduce from Lemma (\ref{pf}) that, $\left( u_{1},u_{2}\right) \in
C^{1}\left( 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\times 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+},%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\right) $ is solution of IVS (\ref{e1}) if and only if $\left(
u_{1},u_{2}\right) $ is a fixed point of $T$.
\end{remark}

\bigskip

\begin{lemma}
\label{hue1}If equation (\ref{1}) is $L$-Hyers-Ulam stable in $E$ then IVS (%
\ref{e1}) has the Hyers-Ulam stability in $E.$
\end{lemma}

\begin{proof}
Aassume that equation (\ref{1}) is $L$-Hyers-Ulam stable in $E.$ \ Let $%
\epsilon >0$ and $v=\left( v_{1},v_{2}\right) \in E$ verifying inequalities (%
\ref{eqstu}). Let $w=\left( w_{1},w_{2}\right) \in \bar{B}_{E}\left(
0,\epsilon \right) $ such that%
\begin{equation*}
w_{j}\left( t\right) =-\left( \zeta _{j}\left( t\right) .D_{0^{+},t}^{\alpha
+1,\omega ,\sigma }+D_{0^{+},t}^{\alpha ,\omega ,\sigma }\right) \left( 
\dfrac{\partial }{\partial x}\left( \phi \left( \theta _{j}\left( x\right) 
\dfrac{\partial v_{j}}{\partial x}\right) \right) \right) \left( t,x\right)
-f_{j}\left( t,v_{1}\left( t\right) ,v_{2}\left( t\right) \right) ,\text{ }%
j\in \left\{ 1,2\right\} .
\end{equation*}%
We have from Lemma (\ref{pf}) that%
\begin{equation*}
v_{j}\left( x\right) =T_{j}\left( v_{1},v_{2}\right) \left( x\right)
==L_{j}\left( F\left( v_{1},v_{2}\right) +w\right) ,
\end{equation*}%
then%
\begin{equation*}
v=L\left( F\left( v\right) +w\right) .
\end{equation*}%
If $w=\left( 0,0\right) $ then $v$ is a fixed point of $T$, and so, $u=v$ is
solution of IVS (\ref{e1}) and we have%
\begin{equation*}
\left\Vert u-v\right\Vert =0\leq N.\epsilon .
\end{equation*}%
Now, if $w\in \bar{B}_{E}\left( 0,\epsilon \right) \backslash \left\{
0\right\} ,$ as (\ref{1}) is $L$-Hyers-Ulam stable then there exists a fixed
point $u$ of $T$ which is solution of IVS (\ref{e1}) such that 
\begin{equation*}
\left\Vert u-v\right\Vert \leq N.\epsilon .
\end{equation*}%
Thus, IVS (\ref{e1}) has the Hyers-Ulam stability in $E.$
\end{proof}

$\bigskip $

\begin{lemma}
Assume that 
\begin{equation}
p^{+}\geq 2.  \label{p+}
\end{equation}%
Then $L$ verifies the condition (\ref{inf}), with $L^{\left( k\right)
}=\left( L_{1}^{\left( k\right) },L_{2}^{\left( k\right) }\right) $ such
that 
\begin{equation*}
k=\frac{1}{p^{+}-1}\leq 1,
\end{equation*}%
where for $j\in \left\{ 1,2\right\} $ 
\begin{equation*}
L_{j}^{\left( k\right) }\left( u_{1},u_{2}\right) \left( t,x\right)
=\int_{0}^{x}\frac{1}{\theta _{j}\left( z\right) }\psi _{p^{+}}\left(
\int_{z}^{+\infty }I_{0^{+},t}^{\alpha ,\sigma }\left( \frac{1}{\zeta
_{j}\left( t\right) }\int_{0}^{t}u_{j}\left( \tau ,s\right) d\tau \right)
\left( t,s\right) ds\right) dz.
\end{equation*}
\end{lemma}

\begin{proof}
Let $u=\left( u_{1},u_{2}\right) \in E.$\ For $j\in \left\{ 1,2\right\} $%
\begin{eqnarray*}
\left\vert L_{j}\left( u_{1},u_{2}\right) \left( t,x\right) \right\vert
&=&\left\vert \int_{0}^{x}\frac{1}{\theta _{j}\left( z\right) }\psi \left(
\int_{z}^{+\infty }I_{0^{+},t}^{\alpha ,\sigma }\left( \frac{1}{\zeta
_{j}\left( t\right) }\int_{0}^{t}u_{j}\left( \tau ,s\right) d\tau \right)
\left( t,s\right) ds\right) dz\right\vert \\
&\leq &\int_{0}^{x}\frac{1}{\theta _{j}\left( z\right) }\psi \left(
\int_{z}^{+\infty }I_{0^{+},t}^{\alpha ,\sigma }\left( \frac{1}{\zeta
_{j}\left( t\right) }\int_{0}^{t}\left\vert u_{j}\left( \tau ,s\right)
\right\vert d\tau \right) \left( t,s\right) ds\right) dz.
\end{eqnarray*}%
By using the inequality (\ref{psi}) we find that for all $t,x\geq 0$%
\begin{equation*}
\left\vert L_{j}\left( u_{1},u_{2}\right) \left( tx\right) \right\vert \leq
\int_{0}^{x}\frac{1}{\theta _{j}\left( z\right) }\psi _{p^{+}}\left(
\int_{z}^{+\infty }I_{0^{+},t}^{\alpha ,\sigma }\left( \frac{1}{\zeta
_{j}\left( t\right) }\int_{0}^{t}\left\vert u_{j}\left( \tau ,s\right)
\right\vert d\tau \right) \left( t,s\right) ds\right) dz=L_{j}^{\left(
k\right) }\left( \left\vert u_{1}\right\vert ,\left\vert u_{2}\right\vert
\right) \left( x\right)
\end{equation*}%
and then $\left\vert L\left( u\right) \right\vert \leq L^{\left( k\right)
}\left( \left\vert u\right\vert \right) .$ Moreover, $L^{\left( k\right) }$
is bounded, increasing, $k-$positively homogeneous and verifies 
\begin{equation*}
L^{\left( k\right) }\left( E^{+}\backslash \left\{ 0\right\} \right) \subset
E^{+}\backslash \left\{ 0\right\} .
\end{equation*}%
And the condition (\ref{2p}) leads that $L^{\left( k\right) }$ is
sub-additive.
\end{proof}

\bigskip

\begin{lemma}
\label{leminf2}Assume that 
\begin{equation}
1<p^{-}\leq 2.\newline
\label{p-}
\end{equation}%
Then For all $r>0$ and for all $u,v\in \bar{B}\left( 0,r\right) ,$ 
\begin{equation*}
\left\vert L\left( u\right) -L\left( v\right) \right\vert \leq \lambda
L_{+}\left\vert u-v\right\vert \text{.}
\end{equation*}%
where%
\begin{equation*}
L_{+}=\left( L_{+,1},L_{+,2}\right)
\end{equation*}%
with 
\begin{equation*}
L_{+,j}\left( u_{1},u_{2}\right) =\int_{0}^{x}\frac{1}{\theta _{j}\left(
z\right) }\int_{z}^{+\infty }I_{0^{+},t}^{\alpha ,\sigma }\left( \frac{1}{%
\zeta _{j}\left( t\right) }\int_{0}^{t}u_{j}\left( \tau ,s\right) d\tau
\right) \left( t,s\right) dsdz,\text{ }j\in \left\{ 1,2\right\} ,
\end{equation*}%
\begin{equation*}
\lambda =\lambda \left( r\right) =\frac{1}{p^{-}-1}\left( \frac{r.\left\Vert
\left( \sigma ^{+}\right) ^{\alpha }\right\Vert _{L^{1}}}{\Gamma \left(
\alpha +1\right) }\right) ^{\dfrac{2-p^{-}}{p^{-}-1}}>0,
\end{equation*}%
and%
\begin{equation*}
\sigma ^{+}\left( x\right) =\underset{t\rightarrow \infty }{\lim }\sigma
\left( t,x\right) .
\end{equation*}
\end{lemma}

\begin{proof}
Let $r>0$ and $u,v\in \bar{B}\left( 0,r\right) ,$ for each $j\in \left\{
1,2\right\} ,$we have 
\begin{eqnarray*}
\left\vert L_{j}\left( u\right) -L_{j}\left( v\right) \right\vert
&=&\left\vert \int_{0}^{x}\frac{1}{\theta _{j}\left( z\right) }\left[ \psi
\left( \int_{z}^{+\infty }I_{0^{+},t}^{\alpha ,\sigma }\left(
B_{j}u_{j}\left( t,s\right) \right) ds\right) -\psi \left( \int_{z}^{+\infty
}I_{0^{+},t}^{\alpha ,\sigma }\left( B_{j}v_{j}\left( t,s\right) \right)
ds\right) \right] dz\right\vert \\
&\leq &\int_{0}^{x}\frac{1}{\theta _{j}\left( z\right) }\left\vert \psi
\left( \int_{z}^{+\infty }I_{0^{+},t}^{\alpha ,\sigma }\left(
B_{j}u_{j}\left( t,s\right) \right) ds\right) -\psi \left( \int_{z}^{+\infty
}I_{0^{+},t}^{\alpha ,\sigma }\left( B_{j}v_{j}\left( t,s\right) \right)
ds\right) \right\vert dz,
\end{eqnarray*}%
where%
\begin{equation*}
B_{j}u_{j}\left( t,s\right) =\frac{1}{\zeta _{j}\left( t\right) }%
\int_{0}^{t}u_{j}\left( \tau ,s\right) d\tau \leq \left\Vert u\right\Vert ,%
\text{ for all }u\in E.
\end{equation*}%
Let $t,x>0$ such that $u_{j}\neq v_{j}$ on $\left[ 0,t\right] \times \lbrack
x,+\infty \lbrack ,$ and let $\chi _{t,x}\in \left[ b_{t,x},c_{t,x}\right]
\backslash \left\{ 0\right\} $ where%
\begin{eqnarray*}
b_{t,x} &=&\min \left( \int_{z}^{+\infty }I_{0^{+},t}^{\alpha ,\sigma
}\left( B_{j}u_{j}\left( t,s\right) \right) ds,\int_{z}^{+\infty
}I_{0^{+},t}^{\alpha ,\sigma }\left( B_{j}v_{j}\left( t,s\right) \right)
ds\right) \text{ and} \\
c_{t,x} &=&\max \left( \int_{z}^{+\infty }I_{0^{+},t}^{\alpha ,\sigma
}\left( B_{j}u_{j}\left( t,s\right) \right) ds,\int_{z}^{+\infty
}I_{0^{+},t}^{\alpha ,\sigma }\left( B_{j}v_{j}\left( t,s\right) \right)
ds\right) ,
\end{eqnarray*}%
such that%
\begin{equation*}
\psi \left( \int_{z}^{+\infty }I_{0^{+},t}^{\alpha ,\sigma }\left(
B_{j}u_{j}\left( t,s\right) \right) ds\right) -\psi \left( \int_{z}^{+\infty
}I_{0^{+},t}^{\alpha ,\sigma }\left( B_{j}v_{j}\left( t,s\right) \right)
ds\right) =A\left( \chi _{t,x}\right) \int_{z}^{+\infty }I_{0^{+},t}^{\alpha
,\sigma }\left( B_{j}\left( u_{j}-v_{j}\right) \right) \left( t,s\right) ds
\end{equation*}%
where%
\begin{equation*}
A\left( \chi _{t}\right) =\frac{1}{\left( p^{+}-1\right) \left\vert \psi
\left( \chi _{t,x}\right) \right\vert ^{p^{+}-2}+\left( p^{-}-1\right)
\left\vert \psi \left( \chi _{t,x}\right) \right\vert ^{p^{-}-2}}.
\end{equation*}%
We have%
\begin{eqnarray*}
A\left( \chi _{t}\right) &=&\frac{1}{\left( p^{+}-1\right) \left( \psi
\left( \left\vert \chi _{t,x}\right\vert \right) \right) ^{p^{+}-2}+\left(
p^{-}-1\right) \left( \psi \left( \left\vert \chi _{t,x}\right\vert \right)
\right) ^{p^{-}-2}} \\
&\leq &\frac{\left( \psi \left( \left\vert \chi _{t,x}\right\vert \right)
\right) ^{2-p^{-}}}{p^{-}-1} \\
&\leq &\frac{\left( \psi _{p^{-}}\left( \left\vert \chi _{t,x}\right\vert
\right) \right) ^{2-p^{-}}}{p^{-}-1}.
\end{eqnarray*}%
Moreover,%
\begin{eqnarray*}
\left\vert \chi _{t,x}\right\vert &\leq &\left\vert c_{t,x}\right\vert \\
&\leq &\max \left( \int_{z}^{+\infty }I_{0^{+},t}^{\alpha ,\sigma }\left(
B_{j}\left( \left\vert u_{j}\right\vert \right) \left( t,s\right) \right)
ds,\int_{z}^{+\infty }I_{0^{+},t}^{\alpha ,\sigma }\left( B_{j}\left(
\left\vert v_{j}\right\vert \right) \left( t,s\right) \right) ds\right) \\
&\leq &r.\int_{z}^{+\infty }I_{0^{+},t}^{\alpha ,\sigma }\left( B_{j}\left(
1\right) \right) ds \\
&\leq &r.\int_{0}^{+\infty }I_{0^{+},t}^{\alpha ,\sigma }\left( 1\right)
ds=r.\int_{0}^{+\infty }\frac{\sigma ^{\alpha }\left( s,t\right) }{\Gamma
\left( \alpha +1\right) }ds \\
&\leq &r.\frac{\left\Vert \left( \sigma ^{+}\right) ^{\alpha }\right\Vert
_{L^{1}}}{\Gamma \left( \alpha +1\right) },
\end{eqnarray*}%
this leads%
\begin{equation*}
\left\vert \psi \left( \int_{z}^{+\infty }I_{0^{+},t}^{\alpha ,\sigma
}\left( B_{j}u_{j}\left( t,s\right) \right) ds\right) -\psi \left(
\int_{z}^{+\infty }I_{0^{+},t}^{\alpha ,\sigma }\left( B_{j}v_{j}\left(
t,s\right) \right) ds\right) \right\vert \leq \lambda \int_{z}^{+\infty
}I_{0^{+},t}^{\alpha ,\sigma }\left( B_{j}\left( \left\vert
u_{j}-v_{j}\right\vert \right) \right) \left( t,s\right) ds
\end{equation*}%
and so,%
\begin{equation*}
\left\vert L_{j}\left( u\right) -L_{j}\left( v\right) \right\vert \leq
\lambda \int_{0}^{x}\frac{1}{\theta _{j}\left( z\right) }\int_{z}^{+\infty
}I_{0^{+},t}^{\alpha ,\sigma }\left( B_{j}\left( \left\vert
u_{j}-v_{j}\right\vert \right) \right) \left( t,s\right) ds.
\end{equation*}%
Thus 
\begin{equation*}
\left\vert L\left( u\right) -L\left( v\right) \right\vert \leq \lambda
L_{+}\left\vert u-v\right\vert \text{.}
\end{equation*}
\end{proof}

\begin{remark}
\label{rem}Since $L_{+}$ is linear, bounded and strictly positive on $E$,
then Lemma (\ref{leminf2}) implies that the condition (\ref{inf2}) holds for
all $r_{\ast }>0.$ Moreover, the operator 
\begin{equation*}
L_{0}=\lambda L_{+}=\left( \lambda L_{+,1},\lambda L_{+,2}\right)
\end{equation*}%
is linear, compact and strictly positive operator, so, the condition (\ref%
{inf4}) is also satisfied.
\end{remark}

\begin{lemma}
\bigskip \label{lemrad} Let $\theta _{0}=\min \left\{ \theta _{1},\theta
_{2}\right\} .$ Then 
\begin{equation}
r\left( L_{0}\right) \leq \beta =\frac{\lambda \left\Vert \left( \sigma
^{+}\right) ^{\alpha }\right\Vert _{L^{1}}}{\Gamma \left( \alpha +1\right) }%
\int_{0}^{\infty }\frac{dt}{\theta _{0}(t)},\text{ }  \label{d}
\end{equation}%
where $r\left( L_{0}\right) $ is the spectral raidus of $L_{0}.$
\end{lemma}

\begin{proof}
Assume that (\ref{d}) holds. Let $u=\left( u_{1},u_{2}\right) \in \partial
B_{E}\left( 0,1\right) .$ For $j\in \left\{ 1,2\right\} $ 
\begin{eqnarray*}
L_{0,j}\left( u\right) &=&\lambda \int_{0}^{x}\frac{1}{\theta _{j}\left(
z\right) }\int_{z}^{+\infty }I_{0^{+},t}^{\alpha ,\sigma }\left( \frac{1}{%
\zeta _{j}\left( t\right) }\int_{0}^{t}u_{j}\left( \tau ,s\right) d\tau
\right) \left( t,s\right) dsdz \\
&\leq &\lambda \int_{0}^{x}\frac{1}{\theta _{0}\left( z\right) }%
\int_{z}^{+\infty }I_{0^{+},t}^{\alpha ,\sigma }\left( 1\right) \left(
t,s\right) dsdz \\
&\leq &\frac{\lambda \left\Vert \left( \sigma ^{+}\right) ^{\alpha
}\right\Vert _{L^{1}}}{\Gamma \left( \alpha +1\right) }\int_{0}^{\infty }%
\frac{dz}{\theta _{0}\left( z\right) },
\end{eqnarray*}%
then for all $n\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
^{\ast },$%
\begin{equation*}
L_{0}^{n}\left( \mu \right) \leq \left( \beta ^{n},\beta ^{n}\right) .
\end{equation*}%
Thus,%
\begin{equation*}
r\left( L_{0}\right) =\underset{n\rightarrow +\infty }{\lim }\sqrt[n]{%
\left\Vert L_{0}^{n}\right\Vert }\leq \beta .
\end{equation*}
\end{proof}

We consider the following hypothesis : 
\begin{equation}
\left\{ 
\begin{array}{c}
\text{There exist }\left( g_{1},g_{2}\right) \in E^{+}\backslash \left\{
0\right\} \text{ and }\left( h_{1},h_{2}\right) \in E^{+}\text{ such that }
\\ 
\left\Vert L^{\left( k\right) }\left( g_{1},g_{2}\right) \right\Vert <1\text{%
, and for all }\left( t,x,y_{1},y_{2}\right) \in 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\times 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\times 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{2} \\ 
\left\vert f_{j}\left( t,x,y_{1},y_{2}\right) \right\vert \leq g_{j}\left(
t,x\right) .\left( \max \left( \left\vert y_{1}\right\vert ,\left\vert
y_{2}\right\vert \right) \right) ^{\frac{1}{k}}+h_{j}\left( t,x\right) ,%
\text{ }\forall j\in \left\{ 1,2\right\} .%
\end{array}%
\right.  \label{inf'ex}
\end{equation}%
Let $r_{\ast }$ $=\max \left\{ r_{0},\left( r_{0}\right) ^{\frac{1}{k}%
}\left\Vert \left( g_{1},g_{2}\right) \right\Vert +\left\Vert \left(
h_{1},h_{2}\right) \right\Vert \right\} $ and 
\begin{equation*}
r_{0}=\dfrac{\left\Vert L^{\left( k\right) }\left( h_{1},h_{2}\right)
\right\Vert }{1-\left\Vert L^{\left( k\right) }\left( g_{1},g_{2}\right)
\right\Vert }.
\end{equation*}

\begin{theorem}
\label{thap}Assume that the condition (\ref{inf'ex}) holds and%
\begin{equation*}
1<p^{-}\leq 2\leq p^{+}.
\end{equation*}%
If there exist $r>r_{\ast },$ $\rho ^{\ast }>0$ and $\rho _{0}\in
G\backslash \left\{ 0\right\} $ such that for all $j\in \left\{ 1,2\right\}
, $ $f_{j}$ verifies one of the following conditions for all $t,x\in 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}$ and all $\left( x_{1},x_{2}\right) ,\left( y_{1},y_{2}\right) \in %
\left[ -r,r\right] ^{2}$;%
\begin{equation}
\left\{ 
\begin{array}{c}
\left\vert f_{j}\left( t,x,x_{1},x_{2}\right) -f_{j}\left(
t,x,y_{1},y_{2}\right) \right\vert \leq \rho _{0}\left( t\right) .\max
\left( \left\vert x_{1}-y_{1}\right\vert ,\left\vert x_{2}-y_{2}\right\vert
\right) \text{ } \\ 
\text{and} \\ 
\lambda <\left\Vert L_{+}\left( \rho _{0},\rho _{0}\right) \right\Vert ^{-1}%
\end{array}%
\right.  \label{ex1}
\end{equation}%
or%
\begin{equation}
\begin{array}{c}
\left\{ 
\begin{array}{c}
\left\vert f_{j}\left( t,x,x_{1},x_{2}\right) -f_{j}\left(
t,x,y_{1},y_{2}\right) \right\vert \leq \rho ^{\ast }.\left\vert
x_{j}-y_{j}\right\vert , \\ 
\text{and} \\ 
\dfrac{\lambda .\left\Vert \left( \sigma ^{+}\right) ^{\alpha }\right\Vert
_{L^{1}}}{\Gamma \left( \alpha +1\right) }\int_{0}^{\infty }\dfrac{dt}{%
\theta _{j}(t)}<\left( \rho ^{\ast }\right) ^{-1},%
\end{array}%
\right. \text{ }%
\end{array}
\label{ex2}
\end{equation}%
then IVS (\ref{e1}) is Hyers-Ulam stable in $E.$
\end{theorem}

\begin{proof}
We have from hypothesis (\ref{inf'ex}) and remark \ref{rem} that the
conditions (\ref{inf}), (\ref{inf'}), (\ref{inf2}) and (\ref{inf4}) hold. 
\newline
1. Assume that the condition (\ref{ex1}), this means that the hypothesis (%
\ref{inf1}) and (\ref{a}) hold with 
\begin{equation*}
\rho =\left( \rho _{1},\rho _{2}\right) =\left( \rho _{0},\rho _{0}\right) ,
\end{equation*}%
so, it follows from theorem \ref{th1}) that equation (\ref{1}) is $L$%
-Hyers-Ulam stable, and from Lemma (\ref{hue1}) that IVS (\ref{e1}) is
Hyers-Ulam stable in $E.$ \newline
2. Now, assume that $f$ verifies (\ref{ex2}). It follows from Lemma (\ref%
{lemrad}) and (\ref{ex2}) that%
\begin{equation*}
r(L_{0})\leq \beta =\dfrac{\lambda .\left\Vert \left( \sigma ^{+}\right)
^{\alpha }\right\Vert _{L^{1}}}{\Gamma \left( \alpha +1\right) }%
\int_{0}^{\infty }\dfrac{dt}{\theta _{0}(t)}<\left( \rho ^{\ast }\right)
^{-1}
\end{equation*}%
and so, the conditions (\ref{inf3}) and (\ref{b}) of theorem (\ref{th2})
hold with 
\begin{equation*}
\lambda _{0}=\rho ^{\ast }.
\end{equation*}%
Consequently, IVS (\ref{e1}) is Hyers-Ulam stable in $E.$
\end{proof}

\subsection{Existence and controllability}

\bigskip

In this section, we assume that for all $\left( t,x,u_{1},u_{2}\right) \in
\left( 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\right) ^{2}\times 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{2}:$ 
\begin{equation*}
f\left( t,x,u_{1},u_{2}\right) =G\left( t,x,u_{1},u_{2}\right) +h\left(
t,x\right) ,
\end{equation*}%
where $h\in E$ is the control function of IVS (\ref{e1}) and $G\in E^{+}$
such that, for each $j\in \left\{ 1,2\right\} ,$%
\begin{equation}
G_{j}\left( u_{1},u_{2}\right) \leq \bar{\lambda}\max \left( \left\vert
u_{1}\right\vert ^{p^{+}-1},\left\vert u_{2}\right\vert ^{p^{+}-1}\right) ,
\label{3'}
\end{equation}%
with%
\begin{equation}
\bar{\lambda}\left\Vert \frac{1}{\theta _{j}}\right\Vert
_{L^{1}}^{p^{+}-1}\left( \frac{\left\Vert \left( \sigma ^{+}\right) ^{\alpha
}\right\Vert _{L^{1}}}{\Gamma \left( \alpha +1\right) }\right) \newline
<1.  \label{4}
\end{equation}

We denote by $C_{0,\phi }^{1}\left( 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\right) $ the set%
\begin{equation*}
C_{0,\phi }^{1}\left( 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\right) =\left\{ u\in C^{1}\left( 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\right) :\phi \left( u\right) \in AC\left( 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\right) ,\text{ }u\left( 0\right) =\underset{x\rightarrow +\infty }{\lim 
}u^{\prime }\left( x\right) =0\right\} .
\end{equation*}

\begin{definition}
IVS (\ref{e1}) is said to be controllable in $E$ at $\infty ,$ if given any $%
x^{\infty }\in C_{0,\phi }^{1}\left( 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\right) \times C_{0,\phi }^{1}\left( 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\right) ,$ there exists a control function $h\in E$, such that the
solution $u$ of IVS (\ref{e1}) satisfies $\underset{x\rightarrow +\infty }{%
\lim }u\left( t,x\right) =x^{\infty }.$
\end{definition}

\bigskip

\begin{lemma}
We have $\underset{t\rightarrow \infty }{\lim }I_{0^{+},t}^{\alpha ,\sigma
}\left( \frac{t}{\zeta _{j}\left( t\right) }\right) \left( t,x\right) >0,$ $%
\forall x\geq 0.$
\end{lemma}

\begin{proof}
Let $x\geq 0.$ Since $\dfrac{\partial \sigma }{\partial t}\left( t,x\right)
>0;$ 
\begin{eqnarray*}
\underset{t\rightarrow \infty }{\lim }I_{0^{+},t}^{\alpha ,\sigma }\left( 
\frac{t}{\zeta _{j}\left( t\right) }\right) &=&\frac{1}{\Gamma \left( \alpha
\right) }\underset{t\rightarrow \infty }{\lim }\int_{0}^{\sigma \left(
t,x\right) }\frac{T\sigma _{t}^{\prime }\left( T,x\right) }{\zeta _{j}\left(
T\right) }\left( \sigma \left( t,x\right) -\sigma \left( T,x\right) \right)
^{\alpha -1}dT \\
&\geq &\frac{1}{\Gamma \left( \alpha \right) }\underset{t\rightarrow \infty }%
{\lim }\int_{\sigma \left( 1,x\right) }^{\sigma \left( t,x\right) }\frac{%
T\sigma _{t}^{\prime }\left( T,x\right) }{a_{j}+T}\left( \sigma \left(
t,x\right) -\sigma \left( T,x\right) \right) ^{\alpha -1}dT \\
&\geq &\underset{t\rightarrow \infty }{\lim }\frac{\sigma \left( 1,x\right) 
}{\Gamma \left( \alpha \right) \left( a_{j}+\sigma \left( t,x\right) \right) 
}\int_{\sigma \left( 1,x\right) }^{\sigma \left( t,x\right) }\sigma
_{t}^{\prime }\left( T,x\right) \left( \sigma \left( t,x\right) -\sigma
\left( T,x\right) \right) ^{\alpha -1}dT \\
&\geq &\underset{t\rightarrow \infty }{\lim }\frac{\sigma \left( 1,x\right) 
}{\Gamma \left( \alpha \right) \left( a_{j}+\sigma \left( t,x\right) \right) 
}\int_{\sigma \left( 1,x\right) }^{\sigma \left( t,x\right) }\left( \sigma
\left( t,x\right) -\sigma \right) ^{\alpha -1}d\sigma \\
&\geq &\frac{\sigma \left( 1,x\right) }{\Gamma \left( \alpha +1\right)
\left( a_{j}+\sigma ^{+}\left( x\right) \right) }\left( \sigma ^{+}\left(
x\right) -\sigma \left( 1,x\right) \right) ^{\alpha }>0.
\end{eqnarray*}
\end{proof}

\begin{theorem}
\label{th1c}Assume that (\ref{3'}) and (\ref{4}) hold true. Then for all $%
h\in E,$ IVS (\ref{e1}) admits a solution.
\end{theorem}

\begin{proof}
Let $h\in E$. We show that there exists $R>0$ such that $T\left( \bar{B}%
\left( 0,R\right) \right) \subset \bar{B}\left( 0,R\right) $ and then we
deduce from Schauder's theorem that the compactness of $T$ guarantees the
existence of at least one fixed point of $T$ which is, from Lemma (\ref{pf}%
), a solution of IVS (\ref{e1}). \newline
Assume on the contrary that for all $n\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
^{\ast },$ there is $u^{\left( n\right) }=\left( u_{1}^{\left( n\right)
},u_{2}^{\left( n\right) }\right) \in \bar{B}\left( 0,n\right) ,$ $\left(
t,x\right) \in 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\times 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}$ and $j\in \left\{ 1,2\right\} ,$ such that 
\begin{equation*}
n\leq \left\vert T_{j}\left( u^{\left( n\right) }\right) \left( t,x\right)
\right\vert =\left\vert \int_{0}^{x}\frac{1}{\theta _{j}\left( z\right) }%
\psi \left( \int_{z}^{+\infty }I_{0^{+},t}^{\alpha ,\sigma }\left( \frac{1}{%
\zeta _{j}\left( t\right) }\int_{0}^{t}\left( G_{j}\left( u_{1}^{\left(
n\right) },u_{2}^{\left( n\right) }\right) +h_{j}\right) \left( \tau
,s\right) d\tau \right) ds\right) dz\right\vert .
\end{equation*}%
By using the inequlity (\ref{psi}) of Remark (\ref{rem1}), it follows $;$%
\begin{eqnarray*}
1 &\leq &\frac{1}{n}\int_{0}^{x}\frac{1}{\theta _{j}\left( z\right) }\psi
_{p^{+}}\left( \int_{z}^{+\infty }I_{0^{+},t}^{\alpha ,\sigma }\left( \frac{1%
}{\zeta _{j}\left( t\right) }\int_{0}^{t}\left( G_{j}\left( u_{1}^{\left(
n\right) },u_{2}^{\left( n\right) }\right) +\left\vert h_{j}\right\vert
\right) \left( \tau ,s\right) d\tau \right) ds\right) dz \\
&\leq &\int_{0}^{x}\frac{1}{\theta _{j}\left( z\right) }\psi _{p^{+}}\left(
\int_{z}^{+\infty }I_{0^{+},t}^{\alpha ,\sigma }\left( \frac{1}{\zeta
_{j}\left( t\right) }\int_{0}^{t}\left( \frac{G_{j}\left( u_{1}^{\left(
n\right) },u_{2}^{\left( n\right) }\right) +\left\vert h_{j}\right\vert }{%
n^{p^{+}-1}}\right) \left( \tau ,s\right) d\tau \right) ds\right) dz \\
&\leq &\psi _{p^{+}}\left( \bar{\lambda}+\frac{\left\Vert h_{j}\right\Vert
_{0}}{n^{p^{+}-1}}\right) \int_{0}^{x}\frac{1}{\theta _{j}\left( z\right) }%
\psi _{p^{+}}\left( \int_{z}^{+\infty }I_{0^{+},t}^{\alpha ,\sigma }\left( 
\frac{t}{\zeta _{j}\left( t\right) }\right) ds\right) dz \\
&\leq &\psi _{p^{+}}\left( \bar{\lambda}+\frac{\left\Vert h_{j}\right\Vert
_{0}}{n^{p^{+}-1}}\right) \int_{0}^{x}\frac{1}{\theta _{j}\left( z\right) }%
\psi _{p^{+}}\left( \int_{0}^{+\infty }I_{0^{+},t}^{\alpha ,\sigma }\left(
1\right) ds\right) dz \\
&\leq &\left( \bar{\lambda}+\frac{\left\Vert h_{j}\right\Vert _{0}}{%
n^{p^{+}-1}}\right) ^{\dfrac{1}{^{p^{+}-1}}}\left\Vert \frac{1}{\theta _{j}}%
\right\Vert _{L^{1}}\left( \frac{\left\Vert \left( \sigma ^{+}\right)
^{\alpha }\right\Vert _{L^{1}}}{\Gamma \left( \alpha +1\right) }\right) ^{%
\dfrac{1}{^{p^{+}-1}}}.
\end{eqnarray*}%
Letting $n\rightarrow \infty ,$ we have 
\begin{equation*}
\bar{\lambda}\left\Vert \frac{1}{\theta _{j}}\right\Vert
_{L^{1}}^{p^{+}-1}\left( \frac{\left\Vert \left( \sigma ^{+}\right) ^{\alpha
}\right\Vert _{L^{1}}}{\Gamma \left( \alpha +1\right) }\right) \newline
\geq 1.
\end{equation*}%
This contradicts hypothesis (\ref{4}) and the proof is finished.
\end{proof}

\begin{theorem}
\label{thc}Assume that (\ref{3'}) and (\ref{4}) hold true. ThenIVS (\ref{e1}%
) is controllable.
\end{theorem}

\begin{proof}
For each $u^{\infty }=\left( u_{1}^{\infty },u_{2}^{\infty }\right) \in
C_{0}^{2}\left( 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\right) \times C_{0}^{2}\left( 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\times 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\right) ,$ let%
\begin{equation}
h\left( t,x\right) =-\frac{1}{\underset{t\rightarrow \infty }{\lim }%
I_{0^{+},t}^{\alpha ,\sigma }\left( \frac{t}{\zeta _{j}\left( t\right) }%
\right) }\left( \frac{\partial }{\partial x}\phi \left( \theta _{j}.\frac{%
\partial u_{j}^{\infty }}{\partial x}\right) \left( x\right) +\underset{%
t\rightarrow \infty }{\lim }I_{0^{+},t}^{\alpha ,\sigma }\left( \frac{1}{%
\zeta _{j}\left( t\right) }\int_{0}^{t}G_{j}\left( u_{1},u_{2}\right) \left(
\tau ,x\right) d\tau \right) \right) .  \label{2}
\end{equation}%
Let $u=\left( u_{1},u_{2}\right) \in C^{1}\left( 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\times 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\right) \times C^{2}\left( 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\times 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\right) $ be solution of IVS (\ref{e1}). We have from Lemma (\ref{pf})
that for each $j\in \left\{ 1,2\right\} ;$ 
\begin{equation*}
u_{j}\left( t,x\right) =\int_{0}^{x}\frac{1}{\theta _{j}\left( z\right) }%
\psi \left( \int_{z}^{+\infty }I_{0^{+},t}^{\alpha ,\sigma }\left( \frac{1}{%
\zeta _{j}\left( t\right) }\int_{0}^{t}\left( G_{j}\left( u_{1},u_{2}\right)
+h_{j}\right) \left( \tau ,s\right) d\tau \right) ds\right) dz.
\end{equation*}%
This means that for every $x\geq 0,$%
\begin{eqnarray*}
y_{j}\left( x\right) &=&\underset{t\rightarrow \infty }{\lim }u_{j}\left(
t,x\right) =\int_{0}^{x}\frac{1}{\theta _{j}\left( z\right) }\psi \left(
\int_{z}^{+\infty }\underset{t\rightarrow \infty }{\lim }I_{0^{+},t}^{\alpha
,\sigma }\left( \frac{1}{\zeta _{j}\left( t\right) }\int_{0}^{t}\left(
G_{j}\left( u_{1},u_{2}\right) +h_{j}\right) \left( \tau ,s\right) d\tau
\right) ds\right) dz \\
&\Rightarrow &-\frac{\partial }{\partial x}\phi \left( \theta _{j}.\frac{%
\partial y_{j}}{\partial x}\right) \left( x\right) =\underset{t\rightarrow
\infty }{\lim }I_{0^{+},t}^{\alpha ,\sigma }\left( \frac{1}{\zeta _{j}\left(
t\right) }\int_{0}^{t}\left( G_{j}\left( u_{1},u_{2}\right) +h_{j}\right)
\left( \tau ,x\right) d\tau \right) \\
&\Rightarrow &-\frac{\partial }{\partial x}\phi \left( \theta _{j}.\frac{%
\partial y_{j}}{\partial x}\right) \left( x\right) -\underset{t\rightarrow
\infty }{\lim }I_{0^{+},t}^{\alpha ,\sigma }\left( \frac{1}{\zeta _{j}\left(
t\right) }\int_{0}^{t}G_{j}\left( u_{1},u_{2}\right) \left( \tau ,x\right)
d\tau \right) =\underset{t\rightarrow \infty }{\lim }I_{0^{+},t}^{\alpha
,\sigma }\left( \frac{1}{\zeta _{j}\left( t\right) }\int_{0}^{t}h_{j}\left(
\tau ,x\right) d\tau \right) .
\end{eqnarray*}%
\newline
then%
\begin{equation}
-\frac{\partial }{\partial x}\phi \left( \theta _{j}.\frac{\partial y_{j}}{%
\partial x}\right) \left( x\right) -\underset{t\rightarrow \infty }{\lim }%
I_{0^{+},t}^{\alpha ,\sigma }\left( \frac{1}{\zeta _{j}\left( t\right) }%
\int_{0}^{t}G_{j}\left( u_{1},u_{2}\right) \left( \tau ,x\right) d\tau
\right) =\underset{t\rightarrow \infty }{\lim }I_{0^{+},t}^{\alpha ,\sigma
}\left( \frac{1}{\zeta _{j}\left( t\right) }\int_{0}^{t}h_{j}\left( \tau
,x\right) d\tau \right) .  \label{3}
\end{equation}%
Substituting (\ref{2}) into (\ref{3}), we find that 
\begin{equation*}
\frac{\partial }{\partial x}\phi \left( \theta _{j}.\frac{\partial
u_{j}^{\infty }}{\partial x}\right) \left( x\right) =\frac{\partial }{%
\partial x}\phi \left( \theta _{j}.\frac{\partial y_{j}}{\partial x}\right)
\left( x\right) ,
\end{equation*}%
and using $\underset{x\rightarrow \infty }{\lim }\dfrac{\partial
u_{j}^{\infty }}{\partial x}\left( x\right) =\underset{x\rightarrow \infty }{%
\lim }\dfrac{\partial y_{j}}{\partial x}\left( x\right) =0$ and the fact
that $\phi $ is invertible$,$ we can get%
\begin{equation*}
\frac{\partial u_{j}^{\infty }}{\partial x}\left( x\right) =\frac{\partial
y_{j}}{\partial x}\left( x\right) ,
\end{equation*}%
and also, from $u_{j}^{\infty }\left( 0\right) =y_{j}\left( 0\right) ,$ it
follows that 
\begin{equation*}
\underset{t\rightarrow \infty }{\lim }u_{j}\left( t,x\right) =y_{j}\left(
x\right) =u_{j}^{\infty }\left( x\right) .
\end{equation*}%
Thus, at the stat $\infty ,$ $u\left( \infty ,.\right) =u_{j}^{\infty },$
so, IVS (\ref{e1}) is controllable.
\end{proof}

\begin{example}
Let $\alpha =\frac{1}{2},$ $\sigma (t,x)=\frac{\pi }{4}\left(
1-e^{-t}\right) ^{2}e^{-2x},$ $\phi \left( x\right) =\left\vert x\right\vert
^{-\frac{1}{2}}.x+\left\vert x\right\vert .x.$ For $j\in \left\{ 0,1\right\}
,$\newline
\begin{eqnarray*}
f_{j}\left( t,x,x_{1},x_{2}\right) &=&G_{j}\left( t,x,x_{1},x_{2}\right)
+h_{j}\left( t,x\right) ,\text{ } \\
\theta _{j}(x) &=&1+x^{2},
\end{eqnarray*}%
and $h_{j}\left( t,x\right) \in E$ is a control function. \newline
1. If $G_{j}\left( t,x,x_{1},x_{2}\right) =g_{j}\left( t,x\right) .x_{j},$
with%
\begin{equation*}
g_{j}\left( t,x\right) =\frac{1}{\pi ^{2}}=\bar{\lambda}.
\end{equation*}%
Then $p^{-}=\frac{3}{2}<2<p^{+}=3,$ $\left\Vert \left( \sigma ^{+}\right)
^{\alpha }\right\Vert _{L^{1}}=\left\Vert \sqrt{\sigma ^{+}}\right\Vert
_{L^{1}}=\dfrac{\sqrt{\pi }}{2}$ 
\begin{equation*}
\sigma ^{+}\left( x\right) =\frac{\pi }{4}e^{-2x}.
\end{equation*}%
We have $\bar{\lambda}=\dfrac{1}{\pi ^{2}}$ and%
\begin{equation*}
\bar{\lambda}\left\Vert \frac{1}{\theta _{j}}\right\Vert
_{L^{1}}^{p^{+}-1}\left( \frac{\left\Vert \left( \sigma ^{+}\right) ^{\alpha
}\right\Vert _{L^{1}}}{\Gamma \left( \alpha +1\right) }\right) =\bar{\lambda}%
\left( \frac{\pi }{2}\right) ^{2}\left( \frac{1}{\Gamma \left( \frac{3}{2}%
\right) }\sqrt{\frac{\pi }{4}}\right) =\frac{1}{4}<1.
\end{equation*}%
So, the conditions (\ref{3'}) and (\ref{4}) of theorems (\ref{th1c}) and (%
\ref{thc}) hold true. Then IVS (\ref{e1}) is controllable. \newline
2. Assume that $G_{j}\left( t,x,x_{1},x_{2}\right) =g_{j}\left( t,x\right)
.x_{j}^{2}$ and $h_{j}\left( t,x\right) =\eta \in 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}$\newline
with%
\begin{equation*}
g_{j}\left( t,x\right) =\frac{1}{\pi ^{2}}=g^{+}
\end{equation*}%
and $\eta $ verifies$\newline
$%
\begin{equation}
\eta <\min \left\{ \dfrac{\sqrt{\pi }}{4\pi },\dfrac{\sqrt{\pi \sqrt{\pi }}}{%
2\left( 2\pi +1\right) }\right\} .  \label{eta}
\end{equation}%
We have%
\begin{eqnarray*}
L_{j}^{\left( k\right) }\left( g_{1},g_{2}\right) \left( t,x\right)
&=&\int_{0}^{x}\frac{1}{\theta _{j}\left( z\right) }\psi _{p^{+}}\left(
\int_{z}^{+\infty }I_{0^{+},t}^{\alpha ,\sigma }\left( \frac{1}{\zeta
_{j}\left( t\right) }\int_{0}^{t}g_{j}\left( \tau ,s\right) d\tau \right)
\left( t,s\right) ds\right) dz \\
&=&\int_{0}^{x}\frac{1}{1+z^{2}}\sqrt{g^{+}\left( \int_{z}^{+\infty
}I_{0^{+},t}^{\alpha ,\sigma }\left( 1\right) \left( t,s\right) ds\right) }dz
\\
&\leq &\int_{0}^{x}\frac{dz}{1+z^{2}}\sqrt{\frac{g^{+}\left\Vert \left(
\sigma ^{+}\right) ^{\alpha }\right\Vert _{L^{1}}}{\Gamma \left( \frac{3}{2}%
\right) }} \\
&\leq &\int_{0}^{x}\frac{dz}{1+z^{2}}\sqrt{\frac{g^{+}\dfrac{\sqrt{\pi }}{2}%
}{\Gamma \left( \frac{3}{2}\right) }}=\sqrt{g^{+}}.\arctan \left( x\right) ,
\end{eqnarray*}%
then%
\begin{equation*}
\left\Vert L^{\left( k\right) }\left( g_{1},g_{2}\right) \right\Vert \leq 
\frac{1}{2}<1.
\end{equation*}%
This means that \ref{inf'ex} holds. \newline
Moreover, 
\begin{eqnarray*}
L_{j}^{\left( k\right) }\left( h_{1},h_{2}\right) &\leq &\int_{0}^{x}\frac{1%
}{\theta _{j}\left( z\right) }\sqrt{\left( \int_{z}^{+\infty
}I_{0^{+},t}^{\alpha ,\sigma }\left( \frac{1}{\zeta _{j}\left( t\right) }%
\int_{0}^{t}\eta .d\tau \right) \left( t,s\right) ds\right) }dz \\
&<&\frac{\pi }{2}\sqrt{\eta }
\end{eqnarray*}%
then%
\begin{equation*}
r_{0}=\dfrac{\left\Vert L^{\left( k\right) }\left( h_{1},h_{2}\right)
\right\Vert }{1-\left\Vert L^{\left( k\right) }\left( g_{1},g_{2}\right)
\right\Vert }\leq 2\left\Vert L^{\left( k\right) }\left( h_{1},h_{2}\right)
\right\Vert <\pi \sqrt{\eta }
\end{equation*}%
and from (\ref{eta}), we have%
\begin{eqnarray*}
r_{\ast } &=&\max \left\{ r_{0},\frac{2}{\pi }\left( r_{0}\right)
^{2}+\left\Vert \left( h_{1},h_{2}\right) \right\Vert \right\} \leq \max
\left\{ \pi \sqrt{\eta },\left( 2\pi +1\right) .\eta \right\} \\
&<&\dfrac{\sqrt{\pi \sqrt{\pi }}}{2}.
\end{eqnarray*}%
Now, let $r>0$ such that 
\begin{equation*}
r_{\ast }<r<\dfrac{\sqrt{\pi \sqrt{\pi }}}{2}.
\end{equation*}%
For all $t,x\geq 0$, $\left( x_{1},x_{2}\right) \left[ -r,r\right]
^{2},\left( y_{1},y_{2}\right) \in \left[ -r,r\right] ^{2}$ we have 
\begin{eqnarray*}
\left\vert f_{j}\left( t,x,x_{1},x_{2}\right) -f_{j}\left(
t,x,y_{1},y_{2}\right) \right\vert &=&g_{j}\left( t,x\right) .\left\vert
x_{j}^{2}-y_{j}^{2}\right\vert \\
&\leq &2.r.g^{+}.\left\vert x_{j}-y_{j}\right\vert =\rho ^{\ast }.\left\vert
x_{j}-y_{j}\right\vert ,
\end{eqnarray*}%
where%
\begin{equation*}
\rho ^{\ast }=\frac{2.r}{\pi ^{2}},
\end{equation*}%
and 
\begin{eqnarray*}
\lambda &=&\frac{1}{p^{-}-1}\left( \frac{r.\left\Vert \left( \sigma
^{+}\right) ^{\alpha }\right\Vert _{L^{1}}}{\Gamma \left( \alpha +1\right) }%
\right) ^{\dfrac{2-p^{-}}{p^{-}-1}} \\
&=&\frac{4}{\sqrt{\pi }}r.
\end{eqnarray*}%
As $r<\dfrac{\sqrt{\pi \sqrt{\pi }}}{2},$ we have%
\begin{eqnarray*}
\dfrac{\rho ^{\ast }}{\Gamma \left( \alpha +1\right) }\int_{0}^{\infty }%
\dfrac{\left\Vert \left( \sigma ^{+}\right) ^{\alpha }\right\Vert _{L^{1}}}{%
\theta _{j}(t)}dt &\leq &\frac{2.r}{\pi ^{2}}\int_{0}^{\infty }\dfrac{1}{%
1+t^{2}}dt \\
&\leq &\frac{r}{\pi }<\frac{\sqrt{\pi }}{4r}=\lambda ^{-1}.
\end{eqnarray*}%
Then, hypothesis (\ref{ex2}) is also satisfied. Thus, we dduce from theorem (%
\ref{thap}) that IVS (\ref{e1}) is Hyers-Ulam stable in $E.$\newline
\end{example}

\bigskip

\bigskip 

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\end{document}