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\fancyhead[CE]{A. Salim, S. Krim, S. Abbas and  M. Benchohra} 
\fancyhead[CO]{ON DEFORMABLE IMPLICIT FRACTIONAL DIFFERENTIAL EQUATIONS IN $b$-METRIC SPACES}



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{\noindent Journal of Mathematical Extension \\
Journal Pre-proof}\\
ISSN: 1735-8299\\
URL: http://www.ijmex.com\\
Original Research Paper\\
‎\vspace*{9mm}
‎
\begin{center}

{\Large \bf 
On Deformable Implicit Fractional Differential Equations in b-Metric Spaces\\}
%{\bf Do You Have a Subtitle? \\ If so, Write It Here} 


\let\thefootnote\relax\footnote{\scriptsize Received: January 2008; Accepted: February 2009 }

{\bf A. Salim$^*$\let\thefootnote\relax\footnote{$^*$Corresponding Author}}\vspace*{-2mm}\\
\vspace{2mm} {\small  Hassiba Benbouali University of Chlef} \vspace{2mm}

{\bf  S. Krim}\vspace*{-2mm}\\
\vspace{2mm} {\small  Djillali Liabes University of Sidi
	Bel-Abbes} \vspace{2mm}

{\bf  S. Abbas}\vspace*{-2mm}\\
\vspace{2mm} {\small  University of Sa\"{\i}da--Dr. Moulay Tahar} \vspace{2mm}

{\bf  M. Benchohra}\vspace*{-2mm}\\
\vspace{2mm} {\small  Djillali Liabes University of Sidi
	Bel-Abbes} \vspace{2mm}

\end{center}

\vspace{4mm}


{\footnotesize
\begin{quotation}
{\noindent \bf Abstract.} In this paper, we prove some existence and uniqueness results for some classes of deformable implicit fractional
differential equations in $b$-Metric spaces with initial conditions. We base our arguments on some some fixed point theorems. Finally, we provide an example to illustrate our results.
\end{quotation}
\begin{quotation}
\noindent{\bf AMS Subject Classification:} 26A33; 34A08; 34K37.

\noindent{\bf Keywords and Phrases:} Fixed point, implicit differential equations, existence, uniqueness, deformable fractional derivative, nonlocal conditions.
\end{quotation}}

 \section{Introduction}
Recently, fractional differential equations have been used in engineering, mathematics, physics, and other applied disciplines. The existence of solutions to the ordinary and fractional differential equations with various conditions has received much attention;
see the monographs \cite{ABGH,ABN,ABN1,SaKiMa,Tar,Zh1} and the papers \cite{SaLaAhBeNi,SaAbBeKa2,HeSaBeKa,DeHaSaBe,BeAiSaBeTu,BeBouKaSa}. Several results of implicit fractional differential equations have been recently provided, see \cite{ABGH,SaBeGrLa2,LaSaLaAbAhBe},
and the references therein.

\vskip0.3cm In \cite{Cz1,Cz2}, Czerwik introduced the concept of $b$-metric. Following these early studies, numerous problems with differential equations in $b$-metric spaces has been intensively researched; see \cite{CC,AAK2,AAK3,KrAbBeKa} as well as the included references.

\vskip0.3cm In \cite{ZuUA}, Zulfeqarr \textit{et al.} proposed the novel notion of deformable fractional derivative, employing the limit technique as in the usual derivative. It was termed "deformable" due to its inherent ability of continuously deforming function to derivative. Deformable derivatives can be thought of as fractional order derivatives.

\vskip0.3cm The author of \cite{MeNgu1} investigated further properties of the new concept of deformable derivative and used the results to study the following Cauchy problem with non-local condition:
$$
\begin{gathered}
	{\mathfrak{D}}_{0}^{\alpha} x(t)=f(t, x(t)), \quad t \in(0, T], \\
	x(0)+g(x)=x_{0},
\end{gathered}
$$
where ${\mathfrak{D}}_{0}^{\alpha}$ is the deformable derivative of order $\alpha \in(0,1)$, and $g: \mathcal{C} \rightarrow \mathbb{R}$ is a continuous function. Their arguments are based on Krasnoselskii’s  fixed point theorem.

\vskip0.3cm In \cite{MePan}, Meraj and Pandey studied the existence and uniqueness of mild solution for the following initial value problem:
\begin{align*}
	{\mathfrak{D}}_{0}^{\alpha} x(t) &=A x(t)+f(t, x(t)), \quad t \in J, \\
	x(0) &=x_{0},
\end{align*}
where $A: D(A) \subset X \rightarrow X$ is an infinitesimal generator of a $C_{0}$-semigroup $T(t)(t \geqslant 0)$ on a suitable space $X, x_{0} \in X$, and $J=[0, b], b>0$ is a constant.  The results are
obtained with the help of semigroup theory, Banach fixed point theorem, and Schauder fixed point theorem.

\vskip0.3cm In \cite{MeNgu2}, by using Weissinger’s
and Krasnoselskii’s fixed point theorems, Mebrat and N'Gu\'er\'ekata studied the existence of solutions for the following problem:
$$
\begin{gathered}
	{\mathfrak{D}}_{0}^{\alpha} \chi(\theta)=\Phi(\chi(\theta))+{\Psi}(\theta, \chi(\theta))+\int_{0}^{\theta} \Upsilon(\theta, s, \chi(s)) d s,\ \theta \in \Theta, \\
	\chi(0)=\chi_{0},
\end{gathered}
$$
where ${\Psi}: \Theta \times E \rightarrow E, \Upsilon: \Theta \times \Theta \times E \rightarrow E$ are continuous functions.

\vskip0.3cm Motivated by the above-mentioned papers, first we study the existence and uniqueness
of solutions for the implicit problem with
nonlinear fractional differential equation involving the deformable fractional derivative:
\begin{equation}\label{e1}
	\left({\mathfrak{D}}_{0}^{\alpha} \chi\right)(\theta)= {\Psi}\left({\theta},{\chi}({\theta}), {\mathfrak{D}}_{0}^{\alpha} \chi({\theta}) \right); \ \theta\in \Theta:=[0,{\kappa}],
\end{equation}
with the initial condition
\begin{equation}\label{e2}
	\chi(0)=\chi_0,
\end{equation}
where $  0 < {\alpha}< 1$,  ${\mathfrak{D}}_{0}^{\alpha}$ is the deformable fractional derivative defined in \cite{ZuUA}, $ {\Psi} : \Theta\times \R^2 \to \R $ is a given function to be specified later and $\chi_0\in\R$.
\vskip0.3cm  Next,  we discuss the existence of solutions for the following problem of deformable implicit nonlocal fractional differential equations:
\begin{equation}\label{e3}
	\begin{cases}
		\left({\mathfrak{D}}_{0}^{\alpha} \chi\right)(\theta)= {\Psi}\left({\theta},{\chi}({\theta}), {\mathfrak{D}}_{0}^{\alpha} \chi({\theta}) \right); \ \theta\in \Theta,
		&\\
		\chi(0)+\overline{\psi}(\chi)=\chi_0,
	\end{cases}
\end{equation}
where $\overline{\psi}: C({\Theta},\R)\to{\Bbb R}$ is a continuous function.
%%%%%%%%%%%%%%%%%%%%
\section{Preliminaries}
First, we give the definitions and the notations   that we will use throughout this paper. We denote by
$C(\Theta,\R)$  the Banach space of all continuous functions
from $\Theta$ into $\R$  with the following norm
$$ \|\chi\|_{\infty}=\Sup_{ \theta\in \Theta}\{|\chi(\theta)|\}.$$

Consider the space $X^p_b(0,{\kappa})$, $(b\in\R, \ 1\leq p \leq \infty )$ of those complex-valued Lebesgue measurable functions ${\Psi}$ on $[0,{\kappa}]$ for which $ \| {\Psi} \|_{X^p_b}<\infty$, where the norm is given by:
$$
\|{\Psi}\|_{X_{b}^{p}}=\left(\Int_{0}^{{\kappa}}|\theta^{b}{\Psi}(\theta)|^{p}\frac{d\theta}{\theta}\right)^{\frac{1}{p}},   \  \ (1\leq p< \infty,  b\in \R).
$$
\begin{definition}[The deformable fractional derivative \cite{MeNgu1,ZuUA}]
	Let ${\Psi}:[0,+\infty) \longrightarrow \R$ be a given function, then the non-conformable fractional derivative of ${\Psi}$ of order $\alpha$ is defined by
	$$
	\left({\mathfrak{D}}_{0}^{\alpha}{\Psi}\right)(\theta)=\lim _{\varepsilon \rightarrow 0} \frac{(1+\varepsilon\beta)\Psi\left(\theta+\varepsilon \alpha\right)-{\Psi}(\theta)}{\varepsilon},
	$$
	where $\alpha+\beta=1$ and $\alpha \in(0,1].$  If the deformable fractional derivative of ${\Psi}$ of order $\alpha$ exists, then we simply say that ${\Psi}$ is $\alpha$-differentiable.
\end{definition}
\begin{definition}[The $\alpha$-fractional integral \cite{MeNgu1,MeNgu2}]
	For $\alpha \in(0,1]$ and a continuous function ${\Psi}$, let
	$$
	\left({}{\mathcal{J}}_{0^+}^{\alpha}{\Psi}\right)(\theta)= \dfrac{1}{\alpha}e^{\frac{-\beta}{\alpha}\theta}\int_{0}^{\theta} e^{\frac{\beta}{\alpha}\tau}{\Psi}(\tau) d\tau.
	$$	
\end{definition}

\begin{lemma}[\cite{MeNgu1,MeNgu2}]\label{lemcons}
	If $\alpha,\alpha_1 \in(0,1]$ such that $\alpha+\beta=1$, $\Psi$ and $\Phi$ are two $\alpha$-differentiable functions at a point $\theta$ and $m,n$ are two given numbers,  then the improved conformable fractional derivative satisfies the following properties:
	\begin{itemize}
		\item ${\mathfrak{D}}_{0}^{\alpha}(\lambda)=\beta\lambda$, for any constant $\lambda$;
		\item ${\mathfrak{D}}_{0}^{\alpha}(m \Psi+n \Phi)=m {\mathfrak{D}}_{0}^{\alpha}(\Psi)+n {\mathfrak{D}}_{0}^{\alpha}(\Phi)$;
		\item ${\mathfrak{D}}_{0}^{\alpha}(\Psi \Phi)= \Phi{\mathfrak{D}}_{0}^{\alpha}(\Psi)+\alpha \Psi \Phi^\prime$;
		\item  ${}{\mathcal{J}}_{0^+}^{\alpha}\ {}{\mathcal{J}}_{0^+}^{\alpha_1}\Psi={}{\mathcal{J}}_{0^+}^{\alpha+\alpha_1}\Psi$.
	\end{itemize}
\end{lemma}


\begin{lemma}[\cite{MeNgu1,MeNgu2}]\label{lem2.6}
	If $\alpha \in(0,1]$, $f$ is continuous function, then we have:
	\begin{itemize}
		\item $\left({}{\mathcal{J}}_{0^+}^{\alpha}\ {\mathfrak{D}}_{0}^{\alpha}(\Psi)\right)(\theta)=\Psi(\theta)-e^{\frac{-\beta}{\alpha}\theta}\Psi(0);$
		\item ${\mathfrak{D}}_{0}^{\alpha}\left({}{\mathcal{J}}_{0^+}^{\alpha}\Psi\right)(\theta)=\Psi(\theta).$
	\end{itemize}
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}
	Let $\Phi\in L^1(\Theta)$ and $0<\alpha\leq 1$. Then the initial value problem
	\begin{equation}\label{ee2}
		\begin{cases}
			\left({\mathfrak{D}}_{0}^{\alpha} \chi\right)(\theta)= \Phi(\theta); \ \theta\in \Theta:=[0,{\kappa}],\\
			\chi(0)=\chi_0,
		\end{cases}
	\end{equation}
	has a unique solution defined by
	\begin{equation}\label{13}
		\chi(\theta)=\chi_0e^{\frac{-\beta}{\alpha}\theta}+\dfrac{1}{\alpha}e^{\frac{-\beta}{\alpha}\theta}\int_{0}^{\theta} e^{\frac{\beta}{\alpha}\tau}\Phi(\tau) d\tau.
	\end{equation}
\end{lemma}
\begin{proof} Applying the $\alpha$-fractional integral of order $\alpha$ to both sides the equation
	$$\left({\mathfrak{D}}_{0}^{\alpha} \chi\right)(\theta)= \Phi(\theta),$$
	and by using Lemma \ref{lem2.6} and if $\theta\in \Theta$, we get
	\begin{equation}\label{14}
		\chi(\theta)-\chi(0)e^{\frac{-\beta}{\alpha}\theta}=\dfrac{1}{\alpha}e^{\frac{-\beta}{\alpha}\theta}\int_{0}^{\theta} e^{\frac{\beta}{\alpha}\tau}\Phi(\tau) d\tau.
	\end{equation}
	From the initial conditions, we get
	\begin{equation}\label{13}
		\chi(\theta)=\chi_0e^{\frac{-\beta}{\alpha}\theta}+\dfrac{1}{\alpha}e^{\frac{-\beta}{\alpha}\theta}\int_{0}^{\theta} e^{\frac{\beta}{\alpha}\tau}\Phi(\tau) d\tau
	\end{equation}
	Conversely, we can easily show by Lemma \ref{lemcons} that if $\chi$ verifies equation (\ref{13}) then it satisfied the problem (\ref{ee2}).
\end{proof}


\begin{definition}[\cite{AAK2}]
	Let ${\cal H}$ be a  set and ${\varepsilon}\leq1$ be a given real number. A distance function ${\delta}: {\cal H}\times {\cal H} \rightarrow \R_{+}^* $ is called a b-metric if the following conditions hold for all $\xi_1,\xi_2,\xi_3\in {\cal H}$:
	\begin{itemize}
		\item[(1)] ${\delta}(\xi_1, \xi_2) = 0$  if and only if $\xi_1 = \xi_2,$
		\item[(2)] $ {\delta}(\xi_1, \xi_2) = {\delta}(\xi_2, \xi_1),$
		\item[(3)] $ {\delta}(\xi_1, \xi_2) \leq {\varepsilon}[{\delta}(\xi_1, \xi_3) + {\delta}(\xi_3, \xi_2)].$
	\end{itemize}
	Then, the pair $({\cal H}, {\delta},{\varepsilon})$ is called a b-metric space with parameter ${\varepsilon}.$
\end{definition}

Consider $\tilde{\Xi}$ the set of continuous and increasing function
${\psi}:\R_{+}^*\rightarrow \R_{+}^*$ satisfying the
property: ${\psi}({\varepsilon}\chi)\leq {\varepsilon}{\psi}(\chi)\leq {\varepsilon}\chi$, for ${\varepsilon}>1$ and ${\psi}(0)=0$.
We denote by ${\Xi}$ the family of all nondecreasing functions ${\eta}:\R_{+}^*\rightarrow [0,\frac{1}{{\varepsilon}^2})$ for some ${\varepsilon}\geq1$.\\

\begin{definition}[\cite{AAK2}]
	Let $({\cal H}, {\delta}, {\varepsilon})$ be a b-metric space, ${\mathfrak S}:{\cal H}\rightarrow {\cal H}$ is said to be
	a generalized ${\omega}$-${\psi}$-Geraghty mapping whenever there exists ${\omega}:{\cal H}\times {\cal H}\rightarrow\R_{+}^*$ such
	that  $${\omega}(\xi_1,\xi_2) {\psi}({\varepsilon}^3d({\mathfrak S}(\xi_1),{\mathfrak S}(\xi_2))\leq{\eta}({\psi}({\delta}(\xi_1,\xi_2)){\psi}({\delta}(\xi_1,\xi_2),$$
	for $\xi_1,\xi_2\in {\cal H}$, where $ {\eta}\in{\Xi}$.
	
	
\end{definition}

\begin{definition}[\cite{AAK2}]
	Let ${\cal H}$ be a non empty set, ${\mathfrak S}:{\cal H}\rightarrow {\cal H}$ and ${\omega}:{\cal H}\times {\cal H}\rightarrow\R_{+}^*$ be given mappings. The operator ${\mathfrak S}$ is orbital ${\omega}$-admissible if for $\chi\in {\cal H}$, we have
	$${\omega}(\chi,{\mathfrak S}(\chi))\geq1\Rightarrow {\omega}({\mathfrak S}(\chi),{\mathfrak S}^2(\chi))\geq1.$$
\end{definition}
%The following fixed point theorem plays a key role in the proof of our main results.

\begin{theorem}[\cite{AAK2}]\label{T0}
	Let $({\cal H},{\delta})$ be a complete b-metric space and   $\aleph:{\cal H}\rightarrow {\cal H}$ be a
	generalized ${\omega}$-${\psi}$-Geraghty mapping where
	\begin{description}
		\item[$(a)$]  $\aleph$ is ${\omega}$-admissible;
		\item[$(b)$] there exists $\chi_0\in {\cal H}$ where ${\omega}(\chi_0,\aleph(\chi_0))\geq1$;
		\item[$(c)$] If $\left({\chi_n}\right)_{n\in N}\subset {\cal H}$ with $\chi_n\rightarrow\chi$ and
		${\omega}(\chi_n,\chi_{n+1})\geq1$, then ${\omega}(\chi_n,\chi)\geq1,$
	\end{description}
	
	Then $\aleph$ admit a fixed point. Further, if
	
	\begin{description}
		\item[$(d)$] for all fixed points $\chi,\chi'$ of $\aleph$, either  $$ {\omega}(\chi,\chi')\geq1\ \ or \ \ {\omega}(\chi',\chi)\geq1,$$
	\end{description}
	
	Then $\aleph$ admit a unique fixed point.
\end{theorem}


\section{Existence of Solutions for the First Problem}
Let $( C(\Theta),{\delta},2)$ be the complete b-metric space with ${\varepsilon}=2$, where ${\delta}: C(\Theta)\times  C(\Theta)\rightarrow \R_{+}^*$, is given by:
$${\delta}(\chi,\Im)=\|(\chi-\Im)^2\|_\infty:=\sup_{\theta\in \Theta}|\chi(\theta)-\Im(\theta)|^2.$$


In this section, we establish some existence results for problem (\ref{e1})-(\ref{e2}).
\begin{definition}
	By a solution of problem (\ref{e1})-(\ref{e2}), we mean a continuous function $\chi\in C(\Theta)$ given by
	\begin{equation}
		\chi(\theta)= \chi_0e^{\frac{-\beta}{\alpha}\theta}+\dfrac{1}{\alpha}e^{\frac{-\beta}{\alpha}\theta}\int_{0}^{\theta} e^{\frac{\beta}{\alpha}\tau}\Phi(\tau) d\tau,
	\end{equation}
	where ${\Phi}\in C(\Theta)$ such that ${\Phi}(\theta)={\Psi}(\theta,\chi(\theta),{\Phi}(\theta)).$
\end{definition}

The hypotheses:
\begin{description}
	\item[$(H_{1})$] There exist ${\gamma_1}:C(\Theta)\times C(\Theta)\rightarrow \R_{+}^*$ and ${\gamma_2}:\Theta\rightarrow (0,1)$  where for  $\chi,\Im,\chi_1,\Im_1\in C(\Theta)$ and $\theta\in \Theta$, we have
	$$|{\Psi}(\theta,\chi,\Im)-{\Psi}(\theta,\chi_1,\Im_1)|\leq {\gamma_1}(\chi,\Im)\|\chi-\Im\|_{\infty}+{\gamma_2}(\theta)\|\chi_1-\Im_1\|_{\infty}$$	
	
	with
	$$\left\|\dfrac{1}{\alpha}e^{\frac{-\beta}{\alpha}\theta}\int_{0}^{\theta} e^{\frac{\beta}{\alpha}\tau}\frac{{\gamma_1}(\chi,\Im)}{1-{\gamma_2}^*}  d\tau\right\|_{\infty}^2\leq{\psi}(\|(\chi-\Im)^2\|_{\infty}).
	$$
	\item[$(H_{2})$] There exist ${\psi}\in\tilde{\Xi}$ and $\overline{{\lambda}}_0\in C(\Theta)$ and a function ${\gamma_3}:C(\Theta)\times C(\Theta)\rightarrow \mathbb{R}$, where
	$$ {\gamma_3}\left(\overline{{\lambda}}_0(\theta),\dfrac{1}{\alpha}e^{\frac{-\beta}{\alpha}\theta}\int_{0}^{\theta} e^{\frac{\beta}{\alpha}\tau}\Phi(\tau) d\tau\right)\geq0$$
	where ${\Phi}\in C(\Theta)$ such that ${\Phi}(\theta)={\Psi}(\theta,\overline{{\lambda}}_{0}(\theta),{\Phi}(\theta)).$
	\item[$(H_{3})$] For each $\theta\in \Theta,$ and $\chi,\Im\in C(\Theta)$, we have:
	$${\gamma_3}(\chi(\theta),\Im(\theta))\geq0$$
	implies
	
	$${\gamma_3}\left(\dfrac{1}{\alpha}e^{\frac{-\beta}{\alpha}\theta}\int_{0}^{\theta} e^{\frac{\beta}{\alpha}\tau}\Phi(\tau) d\tau,\dfrac{1}{\alpha}e^{\frac{-\beta}{\alpha}\theta}\int_{0}^{\theta} e^{\frac{\beta}{\alpha}\tau}\Phi'(\tau) d\tau\right)\geq0,$$
	where ${\Phi},{\Phi}'\in C(\Theta)$ such that $${\Phi}(\theta)={\Psi}(\theta,\chi({\theta}),{\Phi}(\theta))$$ and $${\Phi}'(\theta)={\Psi}(\theta,\Im({\theta}),{\Phi}'(\theta)).$$
	
	
	\item[$(H_{4})$] If $\left({\chi_n}\right)_{n\in N}\subset C(\Theta)$ with $\chi_n\rightarrow \chi$ and ${\gamma_3}(\chi_n,\chi_{n+1})\geq1$, then
	$${\gamma_3}(\chi_n,\chi)\geq1.$$
	
	\item[$(H_{5})$] For all fixed solutions $\chi,\chi'$ of (\ref{e1})-(\ref{e2}), either
	$$ {\gamma_3}(\chi(\theta),\chi'(\theta))\geq0,$$
	or
	$$ {\gamma_3}(\chi'(\theta),\chi(\theta))\geq0.$$
\end{description}

\begin{theorem} \label{T2}
	Assume that the hypotheses $(H_{1})$-$(H_{4})$ hold.
	Then the problem (\ref{e1})-(\ref{e2}) has a least one solution. And, if $(H_5)$ holds, then the solution is unique.
\end{theorem}
\begin{proof} Let ${\mathcal{K}}:\ C(\Theta) \to C(\Theta)$ be the operator given by:
	\begin{equation}\label{N2}
		({\mathcal{K}}\chi)(\theta)=\chi_0e^{\frac{-\beta}{\alpha}\theta}+\dfrac{1}{\alpha}e^{\frac{-\beta}{\alpha}\theta}\int_{0}^{\theta} e^{\frac{\beta}{\alpha}\tau}\Phi(\tau) d\tau,
	\end{equation}
	where ${\Phi}\in C(\Theta)$ such that ${\Phi}(\theta)={\Psi}(\theta,\chi_\theta,{\Phi}(\theta)).$\\
	The function ${\omega}:C(\Theta)\times C(\Theta)\rightarrow \R_{+}^*$ is given by:
	$$\left\{
	\begin{array}{ll}
		{\omega}(\chi,\chi')=1; & if\ {\gamma_3}(\chi(\theta),\chi'(\theta))\geq0,\ \theta\in \Theta, \\
		{\omega}(\chi,\chi')=0; & eles.
	\end{array}
	\right.
	$$
	
	First, we demonstrate that ${\mathcal{K}}$ is a generalized ${\omega}$-${\psi}$-Geraghty operator:\\
	For any $\chi,\chi'\in C(\Theta)$.Then, for each $\theta\in \Theta$, we obtain
	$$|({\mathcal{K}}\chi)(\theta)-({\mathcal{K}}\chi')(\theta)|\leq \dfrac{1}{\alpha}e^{\frac{-\beta}{\alpha}\theta}\int_{0}^{\theta} e^{\frac{\beta}{\alpha}\tau}|{\Phi}({\vartheta})-{\Phi}'({\vartheta})|d{\vartheta},$$
	where ${\Phi},\ {\Phi}'\in C(\Theta)$ such that
	$${\Phi}({\theta})={\Psi}({\theta},\chi({\theta}),{\Phi}({\theta}))\ and\ {\Phi'}({\theta})={\Psi}({\theta},\chi({\theta}),{\Phi}'({\theta})).$$
	From $(H_{1})$ we have
	$$\|{\Phi}-{\Phi'}\|_\infty\leq\frac{{\gamma_1}(\chi,\chi')}{1-{\gamma_2}^*}\|(\varkappa_2-\varkappa_2')^2\|_\infty^{\frac{1}{2}},$$
	where ${\gamma_2}^*=\sup_{\theta\in \Theta}|{\gamma_2}(\theta)|$.\\
	Next, we have
	$$|({\mathcal{K}}\chi)(\theta)-({\mathcal{K}}\chi')(\theta)|\leq \frac{1}{\alpha}e^{\frac{-\beta}{\alpha}\theta}\int_{0}^{\theta} e^{\frac{\beta}{\alpha}\tau}\frac{{\gamma_1}(\chi,\chi')}{1-{\gamma_2}^*}\|(\chi-\chi')^2\|_\infty^{\frac{1}{2}} d{\vartheta}.$$
	Thus
	\begin{align*}
		&{\omega}(\varkappa_2,\varkappa_2') |({\mathcal{K}}\varkappa_2)(\theta)-({\mathcal{K}}\varkappa_2')(\theta)|^2\\
		&\quad \leq \|(\chi-\chi')^2\|_\infty  {\omega}(\chi,\chi')\left\|\frac{1}{\alpha}e^{\frac{-\beta}{\alpha}\theta}\int_{0}^{\theta} e^{\frac{\beta}{\alpha}\tau}\frac{{\gamma_1}(\chi,\chi')}{1-{\gamma_2}^*}d{\vartheta}\right\|_\infty^2  \\
		&\quad  \leq\|(\chi-\chi')^2\|_\infty{\psi}(\|(\chi-\chi')^2\|_\infty).
	\end{align*}
	Hence
	$${\omega}(\chi,\chi') {\psi}(2^3d({\mathcal{K}}(\chi),{\mathcal{K}}(\chi'))\leq{\eta}({\psi}({\delta}(\chi,\chi')){\psi}({\delta}(\chi,\chi')),$$
	where ${\eta}\in{\Xi}$, ${\psi}\in\tilde{\Xi}$, with ${\eta}(\theta)=\frac{1}{8}\theta$, and ${\psi}(\theta)=\theta.$ So, ${\mathcal{K}}$ is generalized ${\omega}$-${\psi}$-Geraghty operator.\\
	
	Let $\chi,\chi'\in C(\Theta)$ such that $${\omega}(\chi,z\chi')\geq1.$$
	Thus,  for each $\theta \in \Theta$, we have $${\gamma_3}(\chi(\theta),\chi'(\theta))\geq0.$$
	This implies from $(H_{3}$) that $${\gamma_3}({\mathcal{K}}\chi(\theta),{\mathcal{K}}\chi'(\theta))\geq0,$$
	which gives $${\omega}({\mathcal{K}}(\chi),{\mathcal{K}}(\chi'))\geq1.$$
	Hence, ${\mathcal{K}}$ is a ${\omega}$-admissible.
	
	Now, by $(H_{2})$,  there exist  $\overline{{\lambda}}_0\in C(\Theta)$ such that $${\omega}(\overline{{\lambda}}_0,\aleph(\overline{{\lambda}}_0))\geq1.$$
	Thus, by $(H_{4})$, if $\left({\overline{{\lambda}}_n}\right)_{n\in N}\subset {\cal H}$ with $\overline{{\lambda}}_n\rightarrow\overline{{\lambda}}$ and
	${\omega}(\overline{{\lambda}}_n,\overline{{\lambda}}_{n+1})\geq1$, then $${\omega}(\overline{{\lambda}}_n,\overline{{\lambda}})\geq1.$$
	By Theorem \ref{T0}, we conclude that ${\mathcal{K}}$ has a fixed
	point $\chi$ which is a solution of (\ref{e1})-(\ref{e2}).\\
	
	Further, $(H_5)$ implies that if $\chi$ and $\chi'$ are fixed points of ${\mathcal{K}}$, then either $$ {\gamma_3}(\chi,\chi')\geq 0\ \ or \ \  {\gamma_3}(\chi',\chi)\geq0.$$
	Thus
	$$ {\omega}(\chi,\chi')\geq1\ \ or \  \ {\omega}(\chi',\chi)\geq1,$$
	Then, (\ref{e1})-(\ref{e2}) has a unique solution.
\end{proof}
\section{Existence of Solutions for the Second Problem}
In this section, we establish some existence results for problem (\ref{e3}).

\vskip0.3cm Let us introduce the following hypotheses:
\begin{itemize}
	
	\item[$(H_{6})$] There exist constant $\varsigma>0$ such that
	$$|\overline{\psi}(\chi)-\overline{\psi}(\Im)|\leq \varsigma\|\chi-\Im\|_{\infty},$$
	for each $\chi,\Im\in C({\Theta}).$
	\item[$(H_{7})$] There exist $\overline{M}:C(\Theta)\times C(\Theta)\rightarrow \R_{+}^*$ and $\overline{N}:\Theta\rightarrow (0,1)$  
	such that for each $\chi,\Im,\chi_1,\Im_1\in C(\Theta)$ and $\theta\in \Theta$
	$$|{\Psi}(\theta,\chi,\Im)-{\Psi}(\theta,\chi_1,\Im_1)|\leq \overline{M}(\chi,\Im)\|\chi-\Im\|_{\infty}+\overline{N}(\theta)\|\chi_1-\Im_1\|_{\infty}$$	
	with
	$$\left\|\varsigma+\dfrac{1}{\alpha}e^{\frac{-\beta}{\alpha}\theta}\int_{0}^{\theta} e^{\frac{\beta}{\alpha}\tau}\frac{\overline{M}(\chi,\Im)}{1-\overline{N}^*}  d\tau\right\|_{\infty}^2\leq{\psi}(\|(\chi-\Im)^2\|_{\infty}).
	$$
	\item[$(H_{8})$] There exist ${\psi}\in\tilde{\Xi}$ and $\overline{{\lambda}}_0\in C(\Theta)$ and a function ${\gamma_3}:C(\Theta)\times C(\Theta)\rightarrow \mathbb{R}$, such that
	$$ {\gamma_3}\left(\overline{{\lambda}}_0(\theta),-\overline{\psi}(\overline{{\lambda}}_0)+e^{\frac{-\beta}{\alpha}\theta}+\dfrac{1}{\alpha}e^{\frac{-\beta}{\alpha}\theta}\int_{0}^{\theta} e^{\frac{\beta}{\alpha}\tau}\Phi(\tau) d\tau\right)\geq0$$
	where ${\Phi}\in C(\Theta)$ such that ${\Phi}(\theta)={\Psi}(\theta,\overline{{\lambda}}_{0}(\theta),{\Phi}(\theta)).$
	\item[$(H_{9})$] For each $\theta\in \Theta,$ and $\chi,\Im\in C(\Theta)$, we have:
	$${\gamma_3}(\chi(\theta),\Im(\theta))\geq0$$
	implies	
	\begin{align*}		
	&{\gamma_3}\left(-\overline{\psi}(\chi)+\dfrac{1}{\alpha}e^{\frac{-\beta}{\alpha}\theta}\int_{0}^{\theta} e^{\frac{\beta}{\alpha}\tau}\Phi(\tau) d\tau,-\overline{\psi}(\Im)\right.\\
	&\left.+\dfrac{1}{\alpha}e^{\frac{-\beta}{\alpha}\theta}\int_{0}^{\theta} e^{\frac{\beta}{\alpha}\tau}\Phi'(\tau) d\tau\right)\geq0,
	\end{align*}
	where ${\Phi},{\Phi}'\in C(\Theta)$ such that $${\Phi}(\theta)={\Psi}(\theta,\chi({\theta}),{\Phi}(\theta))$$ and $${\Phi}'(\theta)={\Psi}(\theta,\Im({\theta}),{\Phi}'(\theta)).$$
\end{itemize}
\begin{theorem} \label{T22}
	Assume that the hypotheses $(H_{4})$ and  $(H_{6})$-$(H_{9})$ hold.
	Then the problem (\ref{e3}) has a least one solution. And, if $(H_5)$ holds, then the solution is unique.
\end{theorem}
\begin{proof} Let ${\mathcal{K}'}:\ C(\Theta) \to C(\Theta)$ be the operator given by:
	\begin{equation}\label{N2}
		({\mathcal{K}'}\chi)(\theta)=(\chi_0-\overline{\psi}(\chi))e^{\frac{-\beta}{\alpha}\theta}+\dfrac{1}{\alpha}e^{\frac{-\beta}{\alpha}\theta}\int_{0}^{\theta} e^{\frac{\beta}{\alpha}\tau}\Phi(\tau) d\tau,
	\end{equation}
	where ${\Phi}\in C(\Theta)$ such that ${\Phi}(\theta)={\Psi}(\theta,\chi_\theta,{\Phi}(\theta)).$\\
	Clearly, the fixed points of the operator ${\mathcal{K}'}$ are solution of the problem (\ref{e3}). By repeating the same process of Theorem \ref{T2}, we can easily show all the conditions
	of Theorem \ref{T22} are satisfied by ${\mathcal{K}'}$. 
\end{proof}
%%%%%%%
\section{An example}
\begin{example}
	Consider the following problem which is an example of our problem (\ref{e1})-(\ref{e2}):
	\begin{equation}\label{ex2}
		\begin{cases}
			\left({\mathfrak{D}}_{0}^{\frac{1}{2}}{\xi}\right)(\theta)=\frac{1+\sin(|{\xi}(\theta)|)}{4(1+|{\xi}(\theta)|)}+\frac{1}{4\left(1+\left| \left({\mathfrak{D}}_{0}^{\frac{1}{2}} {\xi}\right)(\theta)\right|\right)},\\
			{\xi}(0)=0.
		\end{cases}
	\end{equation}
	Set
	$${\Psi}({\theta},{{\xi}},\Im)=\frac{1+\sin(|{\xi}|)}{4(1+|{\xi}|)}+\frac{1}{4(1+|\Im|)},$$
	where ${\theta}\in\Theta:=[0,1],\ {{\xi}},\Im\in {\Bbb R}.$
	
	Let $( C(\Theta),{\delta},2)$ be the complete b-metric space with ${\varepsilon}=2$, such that ${\delta}: C(\Theta)\times  C(\Theta)\rightarrow \R_{+}^*$, is defined by:
	$${\delta}({\xi},\Im)=\|({\xi}-\Im)^2\|_\infty:=\sup_{\theta\in \Theta}|{\xi}(\theta)-\Im(\theta)|^2.$$
	For any ${\xi},\bar{{\xi}}\in C(\Theta),\ \Im,\bar{\Im}\in {\Bbb R}$ and $\theta\in\Theta$. If $|{\xi}(\theta)|\leq|\Im(\theta)|,$ then
	\begin{align*}
		&|\Psi(\theta,{\xi},\bar{{\xi}})-\Psi(\theta,\Im,\bar{\Im})|\\
		&\leq\left|\frac{1+\sin(|{\xi}(\theta)|)}{4(1+|{\xi}(\theta)|+|\bar {\xi}(\theta)|)}-\frac{1+\sin(|\Im(\theta)|)}{4(1+|\Im(\theta)|+|\bar \Im(\theta)|)} \right| \\
		&\quad+\frac{ |\bar {\xi}(\theta)-\bar \Im(\theta)|}{4}\\
		&\leq\frac{1}{4}||{\xi}(\theta)|-|\Im(\theta)||+\frac{1}{4}|\sin(|{\xi}(\theta)|)-\sin(|\Im(\theta)|)| \\
		&\quad+\left||{\xi}(\theta)|\sin(|\Im(\theta)|)-|\Im(\theta)|\sin(|{\xi}(\theta)|)\right|\\
		&\quad+\frac{ |\bar {\xi}(\theta)-\bar \Im(\theta)|}{4}\\
		&\leq|{\xi}(\theta)-\Im(\theta)|+\frac{1}{4}|\sin(|{\xi}(\theta)|)-\sin(|\Im(\theta)|)| \\
		&\quad+\frac{ |\bar {\xi}(\theta)-\bar \Im(\theta)|}{4}\\
		&\quad+\left||\Im(\theta)|\sin(|\Im(\theta)|)-|\Im(\theta)| \sin(|{\xi}(\theta)|)\right|\\
		&=|{\xi}(\theta))-\Im(\theta)|+(1+|\Im(\theta)|)\left|\sin(|{\xi}(\theta)|)-\sin(|\Im(\theta)|)\right|\\
		&\quad+\frac{ |\bar {\xi}(\theta)-\bar \Im(\theta)|}{4}\\
		&\leq|{\xi}(\theta)-\Im(\theta)|+\frac{1}{2}(1+|\Im(\theta)|)  \\
		&\quad\times  \left|\sin\left(\frac{||{\xi}(\theta)|-|\Im(\theta)|}{2}\right)\right|\left|\cos\left(\frac{|{\xi}(\theta)|+|\Im(\theta)||}{2}\right)\right|\\
		&\leq(2+\|\Im\|_{\infty}) \|{\xi}-\Im\|_{\infty}+\frac{ \|\bar {\xi}-\bar \Im\|_\infty}{4}.
	\end{align*}
	The case when $|\Im(\theta)|\leq|{\xi}(\theta)|,$ we get
	$$
	|\Psi(\theta,{\xi},\bar{{\xi}})-\Psi(\theta,\Im,\bar{\Im})|\leq (2+\|{\xi}\|_{\infty})\|{\xi}-\Im\|_{\infty} +\frac{ \|\bar {\xi}-\bar \Im\|_\infty}{4}.
	$$
	Hence
	$$
	|\Psi(\theta,{\xi},\bar{{\xi}})-\Psi(\theta,\Im,\bar{\Im})|\leq\Min\{2+\|{\xi}\|_{\infty},2+\|\Im\|_{\infty}\} \|{\xi}-\Im\|_{\infty}
	+\frac{\|\bar {\xi}-\bar \Im\|_\infty}{4},
	$$
	Thus, hypothesis $(H_1)$ is satisfied with
	$${\gamma_1}({\xi},\Im)=\Min\{2+\|{\xi}\|_{\infty},2+\|\Im\|_{\infty}\}$$
	and  $${\gamma_2}(\theta)=\frac{1}{4}.$$
	
	\vskip0.3cm Let $\lambda(\theta)=\frac{1}{8}\theta,$ $\phi(\theta)=\theta,$
	${\varpi}:C(\Theta)\times C(\Theta)\to{\Bbb R}^{\ast}_+$ with
	$$
	\begin{cases}
		{\varpi}({\xi},\Im)=1; \ if \ \delta({\xi}(\theta),\Im(\theta))\geq0,\ \theta\in \Theta, \\
		{\varpi}({\xi},\Im)=0;\ else,
	\end{cases}
	$$
	and $\delta:C(\Theta)\times C(\Theta)\to {\Bbb R}$ with $\delta({\xi},\Im)=\|{\xi}-\Im\|_{\infty}.$\\
	Hypothesis $(H_2)$ is satisfied with $\overline{\mu}_0(\theta)={\xi}_0.$ Also, $(H_3)$ holds from the definition of the function $\delta.$
	\vskip0.3cm Since all requirements of Theorem \ref{T2} are verified, then we conclude the existence and the uniqueness of solutions for problem $(\ref{ex2})$.
\end{example}


\begin{center}
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\end{center}



{\small

\noindent{\bf Abdelkrim Salim }

\noindent Faculty of Technology, Hassiba Benbouali University of Chlef

\noindent Assistant Professor of Mathematics

\noindent P.O. Box 151 Chlef 02000,  Algeria

\noindent E-mail: salim.abdelkrim@yahoo.com, a.salim@univ-chlef.dz}\\

{\small
\noindent{\bf  Salim Krim  }

\noindent  Laboratory of Mathematics, Djillali Liabes University of Sidi
Bel-Abbes

\noindent P.O. Box 89, Sidi Bel-Abbes 22000, Algeria


\noindent E-mail: salimsalimkrim@gmail.com}\\

{\small
	
	\noindent{\bf Sa\"{\i}d Abbas }
	
	\noindent Department of Electronics, University of Sa\"{\i}da--Dr. Moulay Tahar
	
	\noindent Professor of Mathematics
	
	\noindent P.O. Box 138,  EN-Nasr, 20000 Sa\"{\i}da, Algeria
	
	\noindent E-mail: abbasmsaid@yahoo.fr, said.abbas@univ-saida.dz}\\

{\small
	
	\noindent{\bf Mouffak Benchohra}
	
	\noindent  Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbes 
		
	\noindent Assistant Professor of Mathematics
	
	\noindent P.O. Box 89, Sidi Bel-Abbes 22000, Algeria
	
	\noindent E-mail: benchohra@yahoo.com}\\

\end{document}