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\fancyhead[CE]{W. RAHOU, A. SALIM, J. E. LAZREG AND  M. BENCHOHRA} 
\fancyhead[CO]{IMPLICIT CAPUTO TEMPERED FRACTIONAL BOUNDARY PROBLEMS}



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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 
Implicit Caputo Tempered Fractional Differential Boundary Value Problems with Infinite Delay in Banach Spaces\\}



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

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

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

{\bf  J. E. Lazreg}\vspace*{-2mm}\\
\vspace{2mm} {\small  Djillali Liabes University of Sidi
	Bel-Abbes} \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.} The purpose of this article is to study the existence and Ulam-Hyers stability results for a class of boundary value problems with Caputo tempered fractional derivative and infinite delay. The results are based on M\"{o}nch's fixed point theorem. An illustrative example is given to demonstrate the applicability of our  results.
\end{quotation}
\begin{quotation}
\noindent{\bf AMS Subject Classification:} 26A33; 34A08.

\noindent{\bf Keywords and Phrases:} Caputo tempered fractional derivative; implicit boundary problem; existence; fixed point; measure of noncompactness; infinite delay; Ulam stability.
\end{quotation}}

\section{Introduction}
Fractional calculus, an approach that involves extending differentiation and integration to non-integer orders, has gained significant interest in both theory and applications across various research fields. This has made it an important tool in tackling complex issues. To fully comprehend this approach, it is recommended to consult monographs such as \cite{ABGH,ABLNZ,SaKiMa,Zh1,ABN3,ABN2,GrDu} and papers like \cite{ChBoSaBe,DeHaSaBe2}. In recent years, there has been a noticeable increase in research on fractional calculus, where authors have explored different outcomes for varying conditions and forms of fractional differential equations and inclusions. Additional information can be found in papers like \cite{ABGH,KrSaAbBe1,KrSaAbBe2,BeLaRaSa1}, as well as their respective references.\\

In recent years, tempered fractional calculus has emerged as a noteworthy class of fractional calculus operators. It is capable of generalizing various forms of fractional calculus and possesses analytic kernels. This class is viewed as an extension of fractional calculus since it can describe the transition between normal and anomalous diffusion. The definitions of fractional integration with weak singular and exponential kernels were established by Buschman's seminal work \cite{Busch}, and further elaboration on this topic can be found in \cite{Li1,Sab1,Shiri1,Obei1,Almeida1,Medved1}. Even though the Caputo tempered fractional derivative has not received much attention in the literature, it has the potential to make a substantial contribution to this discipline. By studying the Caputo tempered fractional derivative, we aim to gain a better understanding of its properties and potential applications in this unique mathematical notion. Through this study, we hope to contribute to the advancement of fractional calculus.\\

While solving differential equations precisely is difficult or impossible in several situations, along with nonlinear analysis and optimization, we investigate approximate solutions. It is important to stress that only stable estimates are acceptable. As a result, numerous methodologies for stability analysis are employed such as Lyapunov and exponential stability. Ulam, a mathematician, first raised the stability issue in functional equations in a $1940$ lecture at Wisconsin University. S.M. Ulam posed the question, "Under what conditions does an additive mapping exist near an approximately additive mapping$?$" \cite{Ula}. The succeeding year, Hyers addressed Ulam's issue for additive functions defined on Banach spaces in \cite{Hye}. Rassias \cite{Ras} showed the existence of unique linear mappings close to approximation additive mappings in $1978$, generalizing Hyers' results. In comparison to Lyapunov and exponential stability analysis, Ulam-Hyers stability analysis focuses on the behavior of a function under perturbations, rather than the stability of a dynamical system or equilibrium point.  The authors of \cite{BeLaRaSa1,AhBaYu6,AhBaYu7} investigated the Ulam stabilities of fractional differential problems with different conditions. Considerable focus has been given to investigating the stability of various types of functional equations, specifically Ulam-Hyers and Ulam-Hyers-Rassias stability. This can be observed through the book by Abbas {\it et al.} \cite{ABGH}, as well as the research conducted by Luo {\it et al.} \cite{LuLuQi} and Rus \cite{Rus}, which delved into the stability of operatorial equations using the Ulam-Hyers approach. \\

In \cite{AnMa}, the authors considered the following fractional impulsive neutral integro-differential systems with infinite delay:
$$
\begin{cases}
	D_{{\theta}}^{q}\left({\xi}({\theta})-{\chi}\left({\theta}, {\xi}_{{\theta}}\right)\right)=A({\theta}, {\xi})\left({\xi}({\theta})-{\chi}\left({\theta}, {\xi}_{{\theta}}\right)\right)\\
	\qquad\qquad\qquad\qquad\qquad+{\Psi}\left({\theta}, {\xi}_{{\theta}}, \Int_{0}^{{\theta}} \widetilde{\Psi}\left({\theta}, s, {\xi}_{s}\right) d s\right), {\theta} \in [0, b],  {\theta} \neq {\theta}_{{\jmath}}, \\
	\left.\Delta {\xi}\right|_{{\theta}={\theta}_{{\jmath}}}=I_{{\jmath}}\left({\xi}\left({\theta}_{{\jmath}}^{-}\right)\right), \quad {\theta}={\theta}_{{\jmath}}; {\jmath}=1,\ldots,m, \\
	{\xi}(0)+g({\xi})=\Lambda, \quad \Lambda \in B_{\vartheta},
\end{cases}
$$
where $0<q<1$, $D_{{\theta}}^{q}$ is the Caputo fractional derivative and ${\xi}_{{\theta}}(.)$ denote ${\xi}_{{\theta}}(s)={\xi}({\theta}+s), s \in(-\infty, 0]$. The results are obtained by a fixed point theorem.\\

In \cite {KrSaBe1}, the authors considered a class of problems for nonlinear Caputo tempered implicit fractional differential equations with boundary conditions and delay:
\begin{equation*}
	_{0}^{C} D^{\beta, \gamma }_{{\theta}}{\xi}({\theta})=\Psi\left({\theta},{\xi}_{\theta}, \ _{0}^{C} D^{\beta, \gamma}_{{\theta}}{\xi}({\theta})\right),  \quad  {\theta} \in {\Xi}:=[0, {\varkappa}],
\end{equation*}
\begin{equation*}
	{\xi}({\theta})=\Lambda({\theta}), \quad  {\theta} \in [-\kappa,0],
\end{equation*}
\begin{equation*}
	\delta_{1}{\xi}(0)+\delta_{2}{\xi}({\varkappa})=\delta_3,
\end{equation*}
where $0 < \beta < 1$, $\gamma \geq 0$, $_{0}^{C} D^{\beta, \gamma }_{{\theta}}$ is the Caputo tempered fractional derivative, $\Psi : \Xi \times C([-\kappa, 0], \mathbb{R}) \times \mathbb{R} $ is a
continuous function,  $ \xi  \in C([-\kappa, 0], R)$, $0 < {\varkappa} < +\infty$, $\delta_1, \delta_2, \delta_3$ are real constants, and $\kappa > 0$ is the time delay. Their arguments are based on Banach, Schauder and Schaefer fixed point theorems.\\

Motivated by the above-mentioned works, we investigate the existence and  results for the following boundary value problem with Caputo tempered fractional derivative and infinite delay:
\begin{equation}\label{t1'}
	_{0}^{C} D^{\sigma, \varrho }_{{\theta}}{\xi}({\theta})={\Psi}\left({\theta},{\xi}_{\theta}, \ _{0}^{C} D^{\sigma, \varrho }_{{\theta}}{\xi}({\theta})\right),  \quad  {\theta} \in {\Xi}:=[0, {\varkappa}],
\end{equation}
\begin{equation}\label{t2'}
	{\xi}({\theta})=\Lambda({\theta}), \quad  {\theta} \in (-\infty,0],
\end{equation}
\begin{equation}\label{t3'}
	\delta_{1}{\xi}(0)+\delta_{2}{\xi}({\varkappa})= \delta_{3},
\end{equation}
where $_{0}^{C} D^{\sigma, \varrho }_{{\theta}}$ represents the Caputo tempered  fractional derivative of order $\sigma \in (0, 1)$, $\varrho \geq 0$, ${\Psi}:\Xi \times  {\Bbbk} \times E \rightarrow E$ is a given function, $\Lambda \in {{\Bbbk}}$, $ \delta_{1}, \delta_{2},  \delta_{3}$ are real constants and ${\Bbbk}$ is called a phase space. We denote by ${\xi}_{{\theta}}$ the element of  ${\Bbbk}$ defined by
$${\xi}_{{\theta}}(s)={\xi}({\theta}+s): s \in (-\infty,0]. $$

The following is how this paper is organized. Section 2 contains definitions and lemmas that will be utilized throughout the work. Section 3 derives the existence results for the problem $(\ref{t1'})$-$(\ref{t3'})$. The fourth section discusses the Ulam-Hyers stability results for our problem. In the final part, we present an example to demonstrate our main results.

\section{Preliminaries}
This section aims to present the notations, definitions, and earlier findings that are essential for understanding the content of this paper. Specifically, we use the notation $C(\Xi, E)$ to denote the Banach space of continuous functions from $\Xi := [0, {\varkappa}]$ into $E$, with
$$\|{\xi}\|_{\infty}=\sup \{\|{\xi}({\theta})\| : {\theta} \in \Xi\}.$$
Let the space $({\Bbbk}, \|\cdot\|_{\Bbbk})$  a seminormed
linear space of functions mapping $(-\infty, 0]$ into $E$ and verifying the following axioms which were derived from Hale and Kato's originals \cite{HaKa}:
\begin{description}
	\item[$(Ax_1)$] If ${\xi}:(-\infty,0]\rightarrow E$ and ${\xi}_0 \in {\Bbbk},$ then
	there exist constants ${\wp_1},{\wp_2},{\wp_3}>0$, such that for each ${{\theta}}\in {\Xi},$ we have:
	\begin{description}
		\item[$(i)$] ${\xi}_{{\theta}}$ is in ${\Bbbk};$
		\item[$(ii)$] $\|{\xi}_{{\theta}}\|_{\Bbbk}\leq {\wp_1} \|{\xi}_0\|_{\Bbbk}+ {\wp_2} \Sup_{{\theta}\in[0, {\varkappa}]}\|{\xi}({\theta})\|;$
		\item[$(iii)$] $\|{\xi}({\theta})\|\leq {\wp_3}\|{{\xi}}_{{\theta}}\|_{\Bbbk}.$
	\end{description}
	\item[$(Ax_2)$] For the function ${\xi}(\cdot)\  in\  (Ax_1)$, ${\xi}_{{\theta}}$ is a ${\Bbbk}-$valued continuous function on ${\Xi}.$
	\item[$(Ax_3)$] The space ${\Bbbk}$ is complete.
\end{description}
Consider the space
\begin{align*}
	\Upsilon_E = \big\{{\xi}:(-\infty, {\varkappa}] \rightarrow E: {\xi}|_{(-\infty, 0]} \in  {\Bbbk},  {\xi}|_{[0, {\varkappa}]} \in C(\Xi, E)\big\}.
\end{align*}
$\Upsilon_E$ is a Banach space with the norm
$$\|{\xi}\|_{\Upsilon_E}=\sup_{{\theta}\in(-\infty, {\varkappa}]}{\|{\xi}({\theta})\|}.$$

\begin{definition}[\cite{Li1,Sab1,Shiri1}]
	Let the function  ${\Psi}\in C(\Xi, E)$, $\varrho\geq0$. The Riemann-Liouville tempered
	fractional integral of order $\sigma$ is defined by
	\begin{equation}\label{w1}
		{ }_{0}{I}_{{\theta}}^{\sigma, \varrho} {\Psi}({\theta})=e^{-\varrho {\theta}}\ {}_{0}{I}_{{\theta}}^{\sigma} \left( e^{\varrho {\theta}} {\Psi}({\theta})\right)=\frac{1}{\Gamma(\sigma)} \int_{0}^{{\theta}} \frac{e^{-\varrho({\theta}-s)} {\Psi}(s)}{({\theta}-s)^{1-\sigma}} ds,
	\end{equation}
	where ${}_{0}{I}_{{\theta}}^{\sigma}$ denotes the Riemann-Liouville fractional integral \cite{KiSrTr}, defined by
	
	\begin{equation}\label{w2}
		{}_{0}{I}_{{\theta}}^{\sigma} {\Psi}({\theta})=\frac{1}{\Gamma(\sigma)} \int_{0}^{{\theta}} \frac{{\Psi}(s)}{({\theta}-s)^{1-\sigma}} ds.
	\end{equation}
	Obviously, the tempered fractional integral (\ref{w1}) reduces to the Riemann-Liouville fractional integral (\ref{w2}) if $\varrho=0$.
\end{definition}

\begin{definition}[\cite{Li1,Sab1}]
	For ${\jmath}-1<\sigma<{\jmath}$; ${\jmath} \in \mathbb{N}, \varrho \geq 0$, the Riemann-Liouville tempered fractional derivative is defined by
	$$
	{ }_{0} {D}_{{\theta}}^{\sigma, \varrho} {\Psi}({\theta})=e^{-\varrho {\theta}}{ }_{0} {D}_{{\theta}}^{\sigma}\left(e^{\varrho {\theta}} {\Psi}({\theta})\right)=\frac{e^{-\varrho {\theta}}}{\Gamma({\jmath}-\sigma)} \frac{d^{{\jmath}}}{d {\theta}^{{\jmath}}} \int_{0}^{{\theta}} \frac{e^{\varrho s} {\Psi}(s)}{({\theta}-s)^{\sigma-{\jmath}+1}} d s,
	$$
	where ${ }_{0} {D}_{{\theta}}^{\sigma}\left(e^{\varrho {\theta}} {\Psi}({\theta})\right)$ denotes the Riemann-Liouville fractional derivative \cite{KiSrTr}, given by
	\begin{align*}
	{ }_{0} {D}_{{\theta}}^{\sigma}\left(e^{\varrho {\theta}} {\Psi}({\theta})\right)&=\frac{d^{{\jmath}}}{d {\theta}^{{\jmath}}}\left({ }_{0} {I}_{{\theta}}^{{\jmath}-\sigma}\left(e^{\varrho {\theta}} {\Psi}({\theta})\right)\right)\\
	&=\frac{1}{\Gamma({\jmath}-\sigma)} \frac{d^{{\jmath}}}{d {\theta}^{{\jmath}}} \int_{0}^{{\theta}} \frac{\left(e^{\varrho s} {\Psi}(s)\right)}{({\theta}-s)^{\sigma-{\jmath}+1}} ds.
	\end{align*}
\end{definition}


\begin{definition}[\cite{Li1,Shiri1}] \label{def2.3}
	For ${\jmath}-1<\sigma<{\jmath}$; ${\jmath} \in \mathbb{N}, \varrho \geq 0$, the Caputo tempered fractional derivative is defined as
\begin{align*}
	{ }_{0}^{C} {D}_{{\theta}}^{\sigma, \varrho} {\Psi}({\theta})&=e^{-\varrho {\theta}}\ { }_{0}^{C} {D}_{{\theta}}^{\sigma}\left(e^{\varrho {\theta}} {\Psi}({\theta})\right)\\&=\frac{e^{-\varrho {\theta}}}{\Gamma({\jmath}-\sigma)} \int_{0}^{{\theta}} \frac{1}{({\theta}-s)^{\sigma-{\jmath}+1}} \frac{d^{{\jmath}}\left(e^{\varrho s} {\Psi}(s)\right)}{d s^{{\jmath}}} d s,
\end{align*}
	where ${ }_{0}^{C} {D}_{{\theta}}^{\sigma, \varrho}\left(e^{\varrho {\theta}} {\Psi}({\theta})\right)$ denotes the Caputo fractional derivative \cite{KiSrTr}, given by
	$$
	{ }_{0}^{C} {D}_{{\theta}}^{\sigma}\left(e^{\varrho {\theta}} {\Psi}({\theta})\right)=\frac{1}{\Gamma({\jmath}-\sigma)} \int_{0}^{{\theta}} \frac{1}{({\theta}-s)^{\sigma-{\jmath}+1}} \frac{d^{{\jmath}}\left(e^{\varrho s} {\Psi}(s)\right)}{d s^{{\jmath}}} ds.
	$$
\end{definition}

\begin{lemma}[\cite{Li1}]\label{lemcons}
	For a constant $C$,
	$$
	{ }_{0} {D}_{{\theta}}^{\sigma, \varrho} C=C e^{-\varrho {\theta}}{ }_{0} {D}_{{\theta}}^{\sigma} e^{\varrho {\theta}}, \quad{ }_{0}^{C} {D}_{{\theta}}^{\sigma, \varrho} C=C e^{-\varrho {\theta} }\ { }_{0}^{C} {D}_{{\theta}}^{\sigma} e^{\varrho {\theta}} .
	$$
	Obviously, ${ }_{0} {D}_{{\theta}}^{\sigma, \varrho}(C) \neq{ }_{0}^{C} {D}_{{\theta}}^{\sigma, \varrho}(C)$. And, ${ }_{0}^{C} {D}_{{\theta}}^{\sigma, \varrho}(C)$ is no longer equal to zero, being different from ${ }_{0}^{C} {D}_{{\theta}}^{\sigma}(C) \equiv 0$.
\end{lemma}
\begin{lemma}[\cite{Li1,Shiri1}]\label{lem2.6}
	Let ${\Psi} \in C^{{\jmath}}(\Xi, E)$ and ${\jmath}-1<\sigma<{\jmath}$; ${\jmath} \in \mathbb{N}$. Then, the Caputo tempered fractional derivative and the Riemann-Liouville tempered fractional integral have the composite properties
	$$
	{ }_{0} {I}_{{\theta}}^{\sigma, \varrho}\left[{ }_{0}^{C} {D}_{{\theta}}^{\sigma, \varrho} {\Psi}({\theta})\right]={\Psi}({\theta})-\sum_{k=0}^{{\jmath}-1} e^{-\varrho {\theta}} \frac{{\theta}^{k}}{k !}\left[\left.\frac{d^{k}\left(e^{\varrho {\theta}} {\Psi}({\theta})\right)}{d {\theta}^{k}}\right|_{{\theta}=0}\right],
	$$
	and
	$$
	{ }_{0}^{C} {D}_{{\theta}}^{\sigma, \varrho}\left[{ }_{0} {I}_{{\theta}}^{\sigma, \varrho} {\Psi}({\theta})\right]={\Psi}({\theta}), \text { for } \sigma \in(0,1).
	$$
\end{lemma}

\subsection{Measure of Noncompactness}
\begin{definition}[\cite{2}]
	Let ${\mathfrak W}$ be a Banach space and let ${\Theta}_{\mathfrak W}$ be the family of bounded subsets of ${\mathfrak W}$. The Kuratowski measure of noncompactness is the map $ \zeta: {\Theta}_{\mathfrak W} \longrightarrow [ 0,\infty)$ defined by
	$$\zeta({\Omega})=inf\left\{\varepsilon>0: {\Omega}\subset\bigcup_{j=1}^{m} {\Omega}_{j} , diam({\Omega}_j)\leq\varepsilon\right\},$$  where ${\Omega}\in{\Theta}_{\mathfrak W}.$
\end{definition}
The map $\zeta$ satisfies the following properties:
\begin{itemize}
	\item $\zeta({\Omega})=0 \Leftrightarrow \overline{{\Omega}} $ is compact (${\Omega}$ is relatively compact);
	\item $\zeta({\Omega})=\zeta(\overline{{\Omega}});$
	\item ${\Omega}_1\subset {\Omega}_2 \Rightarrow \zeta({\Omega}_1)\leq\zeta({\Omega}_2);$
	\item $\zeta({\Omega}_1+{\Omega}_2)\leq\zeta({\Omega}_1)+\zeta({\Omega}_2);$
	\item $\zeta(c {\Omega})=|c|\zeta({{\Omega}}),$  $c \in \R;$
	\item $\zeta(conv {\Omega})=\zeta({\Omega}).$
\end{itemize}
\begin{lemma}[\cite{3}]\label{lem2.7}
	Let $B\subset \Upsilon_{E}$ be a bounded and equicontinuous set. Then
	\begin{enumerate}
		\item[$a)$] The function ${\theta}\rightarrow\zeta(B({\theta}))$ is continuous on $\Xi$, and $$\zeta_{\Upsilon_{E}}(B)=\Sup_{{\theta}\in (-\infty, {\varkappa}]}\zeta(B({\theta})).$$
		\item[$b)$] $\zeta\left(\displaystyle\int^{{\varkappa}}_{0}{{\xi}(s) dy: {\xi} \in B}\right)\leq \displaystyle\int^{{\varkappa}}_{0}\zeta(B(s))ds$, where
		$$
		B({\theta})=\{{\xi}({\theta}): {\xi} \in B\}, {\theta} \in {{\Xi}}.
		$$
	\end{enumerate}
\end{lemma}

%\subsection{Some Fixed Point Theorems}
\begin{theorem}[M\"{o}nch's fixed point Theorem \cite{kGo}]
	Let $D$ be a non-empty, closed, bounded and convex subset of a Banach space ${{\mathfrak W}}$ such that $0 \in D$ and let $\mathcal{H}: D \longrightarrow D$ be a continuous  mapping. If the implication
	$$ B= \overline{conv} \mathcal{H}(B) \ or \ B=\mathcal{H}(B) \cup \{0\} \Longrightarrow
	\zeta(B)=0,$$
	holds for every subset $B$ of $D$, then $\mathcal{H}$ has at least one fixed point.
\end{theorem}	

\section{Existence Results}
Consider the following fractional differential problem:
\begin{equation}\label{t10}
	_{0}^{C} D^{\sigma, \varrho}_{{\theta}}{\xi}({\theta})=\mu({\theta}),  \quad if \ {\theta} \in {\Xi}, \ 0<\sigma< 1, \varrho \geq 0,
\end{equation}
\begin{equation}
	{\xi}({\theta})=\Lambda({\theta}), \quad if \ {\theta} \in (-\infty, 0],
\end{equation}
\begin{equation}\label{t20}
	{\delta_1} {\xi}(0)+{\delta_2} {\xi}({\varkappa})= {\delta_3},
\end{equation}
where $\mu: \Xi \rightarrow E$ is a continuous function and $\Lambda \in {{\Bbbk}}$.
\begin{lemma}
	Let $\sigma \in (0, 1)$ and $\mu: \Xi \rightarrow E$ be continuous. Then, the problem
	$(\ref{t10})$-$(\ref{t20})$ has a unique solution given by:
	\begin{align}\label{sol}
		{{\xi}}({\theta})=\left\{\begin{array}{l}
			\Frac{\delta_{3}e^{-\varrho {\theta}}}{\delta_1+ \delta_2 e^{-\varrho {\varkappa}}}-
			\Frac{\delta_{2}e^{-\varrho {\theta}}\Int_{0}^{{\varkappa}}{e^{-\varrho({\varkappa}-s)}({\varkappa}-s)^{\sigma-1}\mu(s)ds}}{\Gamma(\sigma)(\delta_1+ \delta_2 e^{-\varrho {\varkappa}})}\\
			+ \Frac{1}{\Gamma(\sigma)}\Int_{0}^{{\theta}}{e^{-\varrho({\theta}-s)}({\theta}-s)^{\sigma-1}\mu(s)ds}
			, \ {\theta}\in \Xi,\\\\
			\Lambda({\theta}), \ \ \ \qquad {\theta} \in (-\infty, 0].
		\end{array}\right.
	\end{align}
\end{lemma}
\begin{proof}
	Suppose that ${\xi}$ satisfies $(\ref{t10})$-$(\ref{t20})$. Then, by applying the Riemann-Liouville tempered fractional integral of order $\sigma$ and by Lemma $\ref{lem2.6},$ we get
	$$_{0}I^{\sigma, \varrho}_{{\theta}} \ _{0}^{C} D^{\sigma, \varrho}_{{\theta}}{\xi}({\theta})=\  _{0}I^{\sigma, \varrho}_{{\theta}} \mu({\theta}).$$
	This implies that
	\begin{align*}
		{\xi}({\theta})-{\xi}(0) e^{-\varrho {\theta}}=
		\Frac{1}{\Gamma(\sigma)}\Int_{0}^{{\theta}}{e^{-\varrho ({\theta}-s)}({\theta}-s)^{\sigma-1}\mu(s)ds}.
	\end{align*}
	Then,
	\begin{align*}
		{\xi}({\theta})={\xi}(0) e^{-\varrho {\theta}}
		+ \Frac{1}{\Gamma(\sigma)}\Int_{0}^{{\theta}}{e^{-\varrho ({\theta}-s)}({\theta}-s)^{\sigma-1}\mu(s)ds}.
	\end{align*}
	For ${\theta}={\varkappa},$ we have
	\begin{align*}
		{\xi}({\varkappa})={\xi}(0) e^{-\varrho {\varkappa}}
		+ \Frac{1}{\Gamma(\sigma)}\Int_{0}^{{\varkappa}}{e^{-\varrho ({\varkappa}-s)}({\varkappa}-s)^{\sigma-1}\mu(s)ds}.
	\end{align*}
	From condition $(\ref{t20}),$ we get
	\begin{align*}
		\delta_1 {\xi}(0)+ \delta_2 \left({\xi}(0) e^{-\varrho {\varkappa}}
		+ \Frac{1}{\Gamma(\sigma)}\Int_{0}^{{\varkappa}}{e^{-\varrho ({\varkappa}-s)}({\varkappa}-s)^{\sigma-1}\mu(s)ds}\right)=\delta_3.
	\end{align*}
	Thus,
	\begin{align*}
		{\xi}(0)=\Frac{\delta_3-{\Frac{\delta_2}{\Gamma(\sigma)}}{\Int_{0}^{{\varkappa}}{e^{-\varrho ({\varkappa}-s)}({\varkappa}-s)^{\sigma-1}\mu(s)ds}}}{\delta_1+\delta_2 e^{-\varrho {\varkappa}}}.
	\end{align*}
	Finally, we obtain
	\begin{align*}
		{\xi}({\theta})&=\Frac{\delta_3 e^{-\varrho {\theta}}}{\delta_1+\delta_2 e^{-\varrho {\varkappa}}}-\Frac{\delta_2 e^{-\varrho {\theta}}}{\Gamma(\sigma)(\delta_1+\delta_2 e^{-\varrho {\varkappa}})}\Int_{0}^{{\varkappa}}{e^{-\varrho ({\varkappa}-s)}({\varkappa}-s)^{\sigma-1}\mu(s)ds}\\
		&+\Frac{1}{\Gamma(\sigma)}\Int_{0}^{{\theta}}{e^{-\varrho ({\theta}-s)}({\theta}-s)^{\sigma-1}\mu(s)ds}.
	\end{align*}
	Conversely, we can easily show by Definition $\ref{def2.3}$, Lemma $\ref{lemcons}$ and Lemma $\ref{lem2.6}$ that if ${\xi}$ verifies $(\ref{sol})$, then it satisfied the problem $(\ref{t10})$-$(\ref{t20}).$
\end{proof}
\begin{definition}
	By a solution of the problem $(\ref{t1'})$-$(\ref{t3'}),$ we mean a function ${\xi} \in \Upsilon_E$ that satisfies the equation $(\ref{t1'})$ and the conditions $(\ref{t2'})$-$(\ref{t3'})$.
\end{definition}

\begin{lemma}
	Let ${\Psi}:\Xi \times {\Bbbk}\times E \rightarrow E$ be a continuous function. Then, the problem $(\ref{t1'})$-$(\ref{t3'})$ is equivalent to the following integral equation:
	\begin{align}
		{{\xi}}({\theta})=\left\{\begin{array}{l}
			\Frac{\delta_{3}e^{-\varrho {\theta}}}{\delta_1+ \delta_2 e^{-\varrho {\varkappa}}}-
			\Frac{\delta_{2}e^{-\varrho {\theta}}\Int_{0}^{{\varkappa}}{e^{-\varrho({\varkappa}-s)}({\varkappa}-s)^{\sigma-1}{\Psi}(s, {\xi}_s, \widehat{\Psi}(s))ds}}{\Gamma(\sigma)(\delta_1+ \delta_2 e^{-\varrho {\varkappa}})}\\
			+ \Frac{1}{\Gamma(\sigma)}\Int_{0}^{{\theta}}{e^{-\varrho({\theta}-s)}({\theta}-s)^{\sigma-1}{\Psi}(s, {\xi}_s, \widehat{\Psi}(s))ds}
			, \ {\theta}\in \Xi,\\\\
			\Lambda({\theta}), \ \ \ \qquad {\theta} \in (-\infty, 0].
		\end{array}\right.
	\end{align}
	where  $\widehat{\Psi} \in C(\Xi,E)$ satisfies the following functional equation
	$$ \widehat{\Psi}({\theta})= {\Psi}({\theta},{\xi}_{\theta},\widehat{\Psi}({\theta})).$$
\end{lemma}

Let us set the following assumptions:
\begin{itemize}
	\item[(\emph{A1})] The function ${\Psi}: \Xi \times {\Bbbk}\times E \rightarrow E$ is  continuous.
	\item[(\emph{A2})] There exist constants $\lambda>0$ and $0<L<1$ such that
	$$\| {\Psi}({\theta},{\alpha},{\beta})-{\Psi}({\theta},\bar{{\alpha}},\bar{{\beta}})\| \leq \lambda\| {\alpha}-\bar{{\alpha}}\|_{\Bbbk}+ L \| {\beta}-\bar{{\beta}}\|,$$
	for any ${\alpha}, \bar{{\alpha}} \in {\Bbbk},$ ${\beta}, \bar{{\beta}} \in E$ and ${\theta}\in \Xi$.
	\item[(\emph{A3})] For each ${\theta}\in \Xi$ and bounded sets $B_{1} \subseteq {\Bbbk},$ $B_{2}\subseteq E$, we have
	$$\zeta({\Psi}({\theta},B_{1},B_{2}))\leq \lambda \Sup_{s\in (-\infty, 0]}\zeta(B_{1}(s))+L \zeta(B_{2}).$$
\end{itemize}
\begin{remark}[\cite{Appell}]\label{remark1}
	It is worth noting that the hypotheses $({A2})$ and $({A3})$  are equivalent.
\end{remark}

We are now in a position to prove the existence result of the problem $(\ref{t1'})$-$(\ref{t3'})$ based on M\"{o}nch's fixed point theorem.

\begin{theorem}\label{theo1'}
	Assume that the hypotheses $(A1)$-$(A2)$ hold. If
	\begin{equation*}
		\Frac{\lambda {\varkappa}^{\sigma}  ({\wp_2}+1)\left(\delta_2 +\delta_1+\delta_{2} e^{-\varrho {\varkappa}}\right)}{\Gamma(\sigma+1)(1-L)(\delta_{1}+\delta_{2}e^{-\varrho {\varkappa}})}< 1,
	\end{equation*}
	then the implicit fractional problem $(\ref{t1'})$-$(\ref{t3'})$ has at least one solution.
\end{theorem}

\begin{proof}
	Transform problem $(\ref{t1'})$-$(\ref{t3'})$ into a fixed point problem by considering the operator $A:\Upsilon_E \longrightarrow \Upsilon_E$  by
	\begin{align*}
		{A{\xi}}({\theta})=\left\{\begin{array}{l}
			\Frac{\delta_{3}e^{-\varrho {\theta}}}{\delta_1+ \delta_2 e^{-\varrho {\varkappa}}}-
			\Frac{\delta_{2}e^{-\varrho {\theta}}\Int_{0}^{{\varkappa}}{e^{-\varrho({\varkappa}-s)}({\varkappa}-s)^{\sigma-1} \widehat{\Psi}(s)ds}}{\Gamma(\sigma)(\delta_1+ \delta_2 e^{-\varrho {\varkappa}})}	\\
			+ \Frac{1}{\Gamma(\sigma)}\Int_{0}^{{\theta}}{e^{-\varrho({\theta}-s)}({\theta}-s)^{\sigma-1} \widehat{\Psi}(s)ds}
			, \ {\theta}\in \Xi,\\\\
			\Lambda({\theta}), \ \ \ \qquad {\theta} \in (-\infty, 0].
		\end{array}\right.
	\end{align*}
	Let ${w}: (-\infty, {\varkappa}] \rightarrow E$ be a function given by
	\begin{align*}
		{{w}}({\theta})=\left\{\begin{array}{l}
			\Frac{\delta_{3}e^{-\varrho {\theta}}}{\delta_1+ \delta_2 e^{-\varrho {\varkappa}}}
			, \ {\theta}\in \Xi,\\\\
			\Lambda({\theta}), \ \ \ \qquad {\theta} \in (-\infty, 0].
		\end{array}\right.
	\end{align*}
	For each ${z} \in C(\Xi, E)$, with ${z}(0)=0$, we denote by $\bar{z}$ the function defined by
	\begin{align*}
		{\bar{z}}({\theta})=\left\{\begin{array}{l}
			{z}({\theta}), \ {\theta}\in \Xi,\\\\
			0, \ \ \ \qquad {\theta} \in (-\infty, 0],
		\end{array}\right.
	\end{align*}
	where
	\begin{align*}
		{z}({\theta})&=-
		\Frac{\delta_{2}e^{-\varrho {\theta}}}{\Gamma(\sigma)(\delta_1+ \delta_2 e^{-\varrho {\varkappa}})}
		\Int_{0}^{{\varkappa}}{e^{-\varrho({\varkappa}-s)}({\varkappa}-s)^{\sigma-1} \widehat{\Psi}(s)ds}\\
		&+ \Frac{1}{\Gamma(\sigma)}\Int_{0}^{{\theta}}{e^{-\varrho({\theta}-s)}({\theta}-s)^{\sigma-1} \widehat{\Psi}(s)ds}.
	\end{align*}
	If ${\xi}(\cdot)$ satisfies the integral equation
	\begin{align*}
		{\xi}({\theta})&=\Frac{\delta_{3}e^{-\varrho {\theta}}}{\delta_1+ \delta_2 e^{-\varrho {\varkappa}}}-
		\Frac{\delta_{2}e^{-\varrho {\theta}}}{\Gamma(\sigma)(\delta_1+ \delta_2 e^{-\varrho {\varkappa}})}
		\Int_{0}^{{\varkappa}}{e^{-\varrho({\varkappa}-s)}({\varkappa}-s)^{\sigma-1} \widehat{\Psi}(s)ds}\\
		&+ \Frac{1}{\Gamma(\sigma)}\Int_{0}^{{\theta}}{e^{-\varrho({\theta}-s)}({\theta}-s)^{\sigma-1} \widehat{\Psi}(s)ds},
	\end{align*}
	we can decompose ${\xi}(\cdot)$ as ${\xi}({\theta})={w}({\theta})+{\bar{z}}({\theta})$, for ${\theta} \in \Xi$, which implies that ${\xi}_{\theta}={w}_{\theta}+{\bar{z}_{{\theta}}}$ for every ${\theta} \in \Xi$.\\
	
	Set $$C_0=\{{z} \in C(\Xi, E): {z}_0=0\},$$
	and let $\|\cdot\|_T$ be the norm in $C_0$ defined by
	$$\|{z}\|_T=\|{z}_0\|_{{\Bbbk}}+\Sup_{{\theta} \in \Xi}\|{z}({\theta})\|=\Sup_{{\theta} \in \Xi}\|{z}({\theta})\|, \qquad {z} \in C_0,$$
	where $(C_0, \|\cdot\|_T)$ is a Banach space. Let define the operator $S: C_0 \rightarrow C_0$ by
	\begin{align*}
		{S{z}}({\theta})&=-
		\Frac{\delta_{2}e^{-\varrho {\theta}}}{\Gamma(\sigma)(\delta_1+ \delta_2 e^{-\varrho {\varkappa}})}
		\Int_{0}^{{\varkappa}}{e^{-\varrho({\varkappa}-s)}({\varkappa}-s)^{\sigma-1} \widehat{\Psi}(s)ds}\\
		&+ \Frac{1}{\Gamma(\sigma)}\Int_{0}^{{\theta}}{e^{-\varrho({\theta}-s)}({\theta}-s)^{\sigma-1} \widehat{\Psi}(s)ds}.
	\end{align*}
	To show that the operator $A$ has a fixed point is equivalent to show that $S$ has a fixed point. The proof will be given in several steps.\\
	
	\textbf{Step 1:} The operator $S:C_0 \longrightarrow C_0$ is continuous.\\
	Let $\{{z}_{{\jmath}}\}_{{\jmath}\in \mathbb{N}}$ be a sequence such that ${z}_{{\jmath}} \longrightarrow {z}$ in $C_0$. Then, for ${\theta} \in \Xi$, we have
	\begin{align*}
		&\|S{z}_{{\jmath}}({\theta})-S{z}({\theta})\|\\
		 &\quad\leq \Frac{\delta_{2}e^{-\varrho {\theta}}}{\Gamma(\sigma)(\delta_1+ \delta_2 e^{-\varrho {\varkappa}})}
		\Int_{0}^{{\varkappa}}{e^{-\varrho({\varkappa}-s)}({\varkappa}-s)^{\sigma-1} \|\widehat{\Psi}_{{\jmath}}(s)-\widehat{\Psi}(s)\|ds}\\
		&\qquad+\Frac{1}{\Gamma(\sigma)}\Int_{0}^{{\theta}}{e^{-\varrho({\theta}-s)}({\theta}-s)^{\sigma-1} \|\widehat{\Psi}_{{\jmath}}(s)-\widehat{\Psi}(s)\|ds},
	\end{align*}
	where $\widehat{\Psi}_{\jmath}$ and $\widehat{\Psi}$ are two functions satisfying the following functional equations:
	$$\widehat{\Psi}_{\jmath}({\theta})={\Psi}({\theta},{w}_{\theta}+{\bar{z}_{{\jmath}{\theta}}},\widehat{\Psi}_{\jmath}({\theta})),$$
	and
	$$\widehat{\Psi}({\theta})={\Psi}({\theta},{w}_{\theta}+{\bar{z}_{{\theta}}},\widehat{\Psi}({\theta})).$$
	By $(A2)$, we have
	\begin{align*}
		\|\widehat{\Psi}_{{\jmath}}({\theta})-\widehat{\Psi}({\theta})\|&=\|{\Psi}({\theta},{w}_{\theta}+\bar{z}_{{\jmath}{\theta}},\widehat{\Psi}_{\jmath}({\theta}))-{\Psi}({\theta},{w}_{\theta}+\bar{z}_{{\theta}},\widehat{\Psi}({\theta}))\|\\
		&\leq \lambda \|\bar{z}_{{\jmath}{\theta}}-\bar{z}_{{\theta}}\|_{{\Bbbk}}+ L \|\widehat{\Psi}_{{\jmath}}({\theta})-\widehat{\Psi}({\theta})\|.
	\end{align*}
	Thus,
	\begin{align*}
		\|\widehat{\Psi}_{{\jmath}}({\theta})-\widehat{\Psi}({\theta})\| \leq \Frac{ \lambda}{1-L} \|\bar{z}_{{\jmath}{\theta}}-\bar{z}_{{\theta}}\|_{{\Bbbk}}.
	\end{align*}
	\begin{align*}
		&\|S{z}_{{\jmath}}({\theta})-S{z}({\theta})\| \\
		&\quad\leq \Frac{\lambda \wp_2 \delta_{2}e^{-\varrho {\theta}}}{\Gamma(\sigma)(\delta_1+ \delta_2 e^{-\varrho {\varkappa}})(1-L)}
		\Int_{0}^{{\varkappa}}{e^{-\varrho({\varkappa}-s)}({\varkappa}-s)^{\sigma-1} \|\bar{z}_{{\jmath}}-\bar{z}\|_T \ ds}\\
		&\qquad+ \Frac{\lambda \wp_2}{(1-L)\Gamma(\sigma)}\Int_{0}^{{\theta}}{e^{-\varrho({\theta}-s)}({\theta}-s)^{\sigma-1} \|\bar{z}_{{\jmath}}-\bar{z}\|_T \ ds}.
	\end{align*}
	By applying the Lebesgue dominated convergence theorem, we get
	\begin{align*}
		\|S{z}_{{\jmath}}({\theta})-S{z}({\theta})\| \longrightarrow 0  \quad \text{as} \quad {\jmath} \longrightarrow \infty,
	\end{align*}
	which implies that
	\begin{align*}
		\|S{z}_{{\jmath}}-S{z}\|_{{\varkappa}} \longrightarrow 0  \quad \text{as} \quad {\jmath} \longrightarrow \infty.
	\end{align*}
	Hence, the operator $S$ is continuous.\\
	
	Let ${\kappa}>0$ such that
	$${\kappa} \geq \Frac{{\varkappa}^{\sigma}\left(q_{1}^{*}+\lambda {\wp_1}\|\Lambda\|_{\Bbbk}+\Frac{\lambda{\wp_2} \delta_3}{\delta_1+\delta_2 e^{-\varrho {\varkappa}}}\right)\left(\delta_2+\delta_1+\delta_2 e^{-\varrho {\varkappa}}\right)}{{ \Gamma(\sigma+1)(\delta_1+\delta_2 e^{-\varrho {\varkappa}})(1-L)-\lambda  {\varkappa}^{\sigma} {\wp_2}\left(\delta_2 +\delta_1+\delta_2 e^{-\varrho {\varkappa}}\right) }}
	$$
	where
	$q_{1}({\theta})=\|{\Psi}({\theta},0,0)\|,$  with $q_{1} \in C(\Xi,E), $ such that
	$$q_{1}^{*}=\Sup_{ {\theta}\in \Xi}q_{1}({\theta}).$$
	Define the ball $${\digamma}_{\kappa}=\{{z} \in C_0 : \|{z}\|_{{\varkappa}}\leq {\kappa} \}.$$
	It is clear that ${\digamma}_{\kappa}$ is a bounded, closed and convex subset of $C_0$.\\
	
	\textbf{Step 2:} $S({\digamma}_{\kappa}) \subset {\digamma}_{\kappa}$. \\
	Let ${\xi} \in {\digamma}_{\kappa}$ and ${\theta} \in \Xi.$ Then,
	\begin{align*}
		\|S{z}({\theta})\|&\leq \Frac{\delta_{2}e^{-\varrho {\theta}}}{\Gamma(\sigma)(\delta_1+ \delta_2 e^{-\varrho {\varkappa}})}
		\Int_{0}^{{\varkappa}}{e^{-\varrho({\varkappa}-s)}({\varkappa}-s)^{\sigma-1} \|\widehat{\Psi}(s)\|ds}\\
		&+ \Frac{1}{\Gamma(\sigma)}\Int_{0}^{{\theta}}{e^{-\varrho({\theta}-s)}({\theta}-s)^{\sigma-1} \|\widehat{\Psi}(s)\|ds}.
	\end{align*}
	From hypothesis $(A2)$, we have
	\begin{align*}
		\|\widehat{\Psi}({\theta})\|&=\|{\Psi}({\theta},{w}_{\theta}+{\bar{z}_{{\theta}}},\widehat{\Psi}({\theta}))\|\\
		&\leq q_{1}({\theta})+\lambda \|{w}_{\theta}+{\bar{z}_{{\theta}}}\|_{\Bbbk}+L\|\widehat{\Psi}({\theta})\|\\
		&\leq q_{1}^{*}+\lambda [\|{w}_{\theta}\|_{\Bbbk}+\|{\bar{z}_{{\theta}}}\|_{\Bbbk}]+L\|\widehat{\Psi}({\theta})\|\\
		&\leq q_{1}^{*}+\lambda {\wp_2} {\kappa}+\lambda {\wp_1}\|\Lambda\|_{\Bbbk}+\Frac{{\wp_2} \delta_3}{\delta_1+\delta_2 e^{-\varrho {\varkappa}}} +L\|\widehat{\Psi}({\theta})\|.
	\end{align*}
	Then,
	\begin{align*}
		\|\widehat{\Psi}({\theta})\|\leq \frac{q_{1}^{*}+\lambda {\wp_2} {\kappa}+\lambda {\wp_1}\|\Lambda\|_{\Bbbk}+\Frac{\lambda{\wp_2} \delta_3}{\delta_1+\delta_2 e^{-\varrho {\varkappa}}}}{1-L}.
	\end{align*}
	Finally, we have
	\begin{align*}
		\|S{z}({\theta})\|&\leq \Frac{\delta_2 e^{-\varrho {\theta}} {\varkappa}^{\sigma}\left(q_{1}^{*}+\lambda {\wp_2} {\kappa}+\lambda {\wp_1}\|\Lambda\|_{\Bbbk}+\Frac{\lambda{\wp_2} \delta_3}{\delta_1+\delta_2 e^{-\varrho {\varkappa}}}\right)}{\Gamma(\sigma+1)(\delta_1+\delta_2 e^{-\varrho {\varkappa}})(1-L)}\\
		&+\Frac{ {\varkappa}^{\sigma}\left(q_{1}^{*}+\lambda {\wp_2} {\kappa}+\lambda {\wp_1}\|\Lambda\|_{\Bbbk}+\Frac{\lambda{\wp_2} \delta_3}{\delta_1+\delta_2 e^{-\varrho {\varkappa}}}\right)}{\Gamma(\sigma+1)(1-L)}\\
		&\leq \Frac{{\varkappa}^{\sigma}\left(q_{1}^{*}+\lambda {\wp_2} {\kappa}+\lambda {\wp_1}\|\Lambda\|_{\Bbbk}+\Frac{\lambda{\wp_2} \delta_3}{\delta_1+\delta_2 e^{-\varrho {\varkappa}}}\right)\left(\delta_2+\delta_1+\delta_2 e^{-\varrho {\varkappa}}\right)}{{\Gamma(\sigma+1)(\delta_1+\delta_2 e^{-\varrho {\varkappa}})(1-L)}}\\
		&\leq {\kappa}.
	\end{align*}
	Hence, $S({\digamma}_{\kappa}) \subset {\digamma}_{\kappa}$.\\
	
	\textbf{Step 3:} $A({\digamma}_{\kappa})$ is equicontinuous.\\
	Let ${\theta}_{1}, {\theta}_{2} \in \Xi$, where ${\theta}_{1} < {\theta}_{2}$ and ${\xi} \in {\digamma}_{\kappa}.$ Then,
	\begin{align*}
		&\|S{z}({\theta}_{2})-S{z}({\theta}_{1})\|\\
		&\quad=\Bigg\|-
		\Frac{\delta_{2}e^{-\varrho t_2}}{\Gamma(\sigma)(\delta_1+ \delta_2 e^{-\varrho {\varkappa}})}
		\Int_{0}^{{\varkappa}}{e^{-\varrho({\varkappa}-s)}({\varkappa}-s)^{\sigma-1} \widehat{\Psi}(s)ds}\\
		&\qquad+ \Frac{1}{\Gamma(\sigma)}\Int_{0}^{t_2}{e^{-\varrho(t_2-s)}(t_2-s)^{\sigma-1} \widehat{\Psi}(s)ds}\\
		&\qquad+\Frac{\delta_{2}e^{-\varrho t_1}}{\Gamma(\sigma)(\delta_1+ \delta_2 e^{-\varrho {\varkappa}})}
		\Int_{0}^{{\varkappa}}{e^{-\varrho({\varkappa}-s)}({\varkappa}-s)^{\sigma-1} \widehat{\Psi}(s)ds}\\
		&\qquad+ \Frac{1}{\Gamma(\sigma)}\Int_{0}^{t_1}{e^{-\varrho(t_1-s)}(t_1-s)^{\sigma-1} \widehat{\Psi}(s)ds}\Bigg\|\\
		&\quad\leq \Frac{\delta_{2}\left\|e^{-\varrho {\theta}_{2}}-e^{-\varrho {\theta}_{1}}\right\|}{\Gamma(\sigma)(\delta_1+ \delta_2 e^{-\varrho {\varkappa}})}
		\Int_{0}^{{\varkappa}}{e^{-\varrho({\varkappa}-s)}({\varkappa}-s)^{\sigma-1} \|\widehat{\Psi}(s)\|ds}\\
		&\qquad+\Frac{1}{\Gamma(\sigma)}\Int_{0}^{{\theta}_{1}}{\left[e^{-\varrho({\theta}_{2}-s)}({\theta}_{2}-s)^{\sigma-1}-e^{-\varrho({\theta}_{1}-s)}({\theta}_{1}-s)^{\sigma-1}\right]\|\widehat{\Psi}(s)\|ds}\\
		&\qquad+\Frac{1}{\Gamma(\sigma)}\Int_{{\theta}_{1}}^{{\theta}_{2}}{e^{-\varrho({\theta}_{2}-s)}({\theta}_{2}-s)^{\sigma-1} \|\widehat{\Psi}(s)\|ds}\\
		&\quad\leq \Frac{\delta_{2} {\varkappa}^{\sigma}\left\|e^{-\varrho {\theta}_{2}}-e^{-\varrho {\theta}_{1}}\right\|\left(q_{1}^{*}+\lambda {\wp_2} {\kappa}+\lambda {\wp_1}\|\Lambda\|_{\Bbbk}+\Frac{\lambda{\wp_2} \delta_3}{\delta_1+\delta_2 e^{-\varrho {\varkappa}}}\right)}{\Gamma(\sigma+1)(1-L)(\delta_1+ \delta_2 e^{-\varrho {\varkappa}})}\\
		&\qquad+\Frac{\left(q_{1}^{*}+\lambda {\wp_2} {\kappa}+\lambda {\wp_1}\|\Lambda\|_{\Bbbk}+\Frac{\lambda{\wp_2} \delta_3}{\delta_1+\delta_2 e^{-\varrho {\varkappa}}}\right)({\theta}_{2}^{\sigma}-{\theta}_{1}^{\sigma})}{(1-L)\Gamma(\sigma+1)}\\
		&\qquad+\Frac{\left(q_{1}^{*}+\lambda {\wp_2} {\kappa}+\lambda {\wp_1}\|\Lambda\|_{\Bbbk}+\Frac{\lambda{\wp_2} \delta_3}{\delta_1+\delta_2 e^{-\varrho {\varkappa}}}\right)({\theta}_{2}-{\theta}_{1})^{\sigma}}{(1-L)\Gamma(\sigma+1)}.
	\end{align*}
	As ${\theta}_{1} \longrightarrow {\theta}_{2}$, the right-hand side of the inequality above tend to zero. Therefore, the operator $S$ is equicontinuous.\\
	
	
	\textbf{Step 4:} The implication of M\"{o}nch's theorem is satisfied.\\
	Let $B$ be a subset of ${\digamma}_{\kappa}$ such that $B = A(B) \cup \{0\}.$ Therefore, the function  ${\theta}\longrightarrow b({\theta})=\zeta(B({\theta})) $ is continuous  on $\Xi$. Then, for ${\theta} \in \Xi $, we have
	\begin{align*}
		b({\theta})&=\zeta (B({\theta}))\\
		&=\zeta\big\{S{z}({\theta}), \ \ {\xi}\in B\big\}\\
		&=\zeta \Bigg\{-
		\Frac{\delta_{2}e^{-\varrho {\theta}}}{\Gamma(\sigma)(\delta_1+ \delta_2 e^{-\varrho {\varkappa}})}
		\Int_{0}^{{\varkappa}}{e^{-\varrho({\varkappa}-s)}({\varkappa}-s)^{\sigma-1} \widehat{\Psi}(s)ds}\\
		&+ \Frac{1}{\Gamma(\sigma)}\Int_{0}^{{\theta}}{e^{-\varrho({\theta}-s)}({\theta}-s)^{\sigma-1} \widehat{\Psi}(s)ds}, \ \ {\xi}\in B         \Bigg\}\\
		&\leq\Frac{\delta_{2}e^{-\varrho {\theta}}}{\Gamma(\sigma)(\delta_1+ \delta_2 e^{-\varrho {\varkappa}})}
		\Int_{0}^{{\varkappa}}{e^{-\varrho({\varkappa}-s)}({\varkappa}-s)^{\sigma-1} \{\zeta(\widehat{\Psi}(s))ds, \ \ {\xi}\in B\}}\\
		&+\Frac{1}{\Gamma(\sigma)}\Int_{0}^{{\theta}}{({\theta}-s)^{\sigma-1} \{\zeta(\widehat{\Psi}(s))ds, \ \ {\xi}\in B\}}.
	\end{align*}
	By condition $(A3),$ we have
	\begin{align*}
		\zeta(\widehat{\Psi}({\theta}))&=\zeta({\Psi}({\theta}, {w}_{\theta}+{\bar{z}_{{\theta}}}, \widehat{\Psi}({\theta}))\\
		&\leq \lambda \Sup_{{\theta} \in (-\infty, 0]} \zeta({w}_{\theta}+{\bar{z}_{{\theta}}})+L \zeta(\widehat{\Psi}({\theta}))\\
		&\leq \lambda \Sup_{{\theta} \in (-\infty, {\varkappa}]} \zeta({w}_{\theta}+{\bar{z}_{{\theta}}})+L \zeta(\widehat{\Psi}({\theta})).
	\end{align*}
	Thus,
	\begin{align*}
		\zeta(\widehat{\Psi}({\theta}))\leq \Frac{\lambda}{1-L}\Sup_{{\theta} \in (-\infty, {\varkappa}]} \zeta({w}_{\theta}+{\bar{z}_{{\theta}}}).
	\end{align*}
	Then,
	\begin{align*}
		\zeta(B({\theta}))&\leq \Frac{ \lambda \delta_{2}\Int_{0}^{{\varkappa}}{({\varkappa}-s)^{\sigma-1}  \{\Sup_{{\theta} \in (-\infty, {\varkappa}]} \zeta({w}_{s}+{\bar{z}_{s}})ds, \ \ {\xi}\in B\}}}{(1-L)\Gamma(\sigma)(\delta_1+ \delta_2 e^{-\varrho {\varkappa}})}\\
		&\quad+\Frac{\lambda}{(1-L)\Gamma(\sigma)}\Int_{0}^{{\theta}}{({\theta}-s)^{\sigma-1} \{\Sup_{{\theta} \in (-\infty, {\varkappa}]} \zeta({w}_{s}+{\bar{z}_{s}})ds, \ \ {\xi}\in B\}}\\
		&\leq \left[\Frac{ \lambda {\varkappa}^{\sigma} \delta_{2}}{(1-L)\Gamma(\sigma+1)(\delta_1+ \delta_2 e^{-\varrho {\varkappa}})}+
		\Frac{\lambda {\varkappa}^{\sigma}}{(1-L)\Gamma(\sigma+1)}\right]\zeta_{\Upsilon_{E}}(B)\\
		&\leq \left[\Frac{\lambda {\varkappa}^{\sigma}\left(\delta_2 +\delta_1+\delta_{2} e^{-\varrho {\varkappa}}\right)}{(1-L)\Gamma(\sigma+1)(\delta_{1}+\delta_{2}e^{-\varrho {\varkappa}})}\right]\zeta_{\Upsilon_{E}}(B).
	\end{align*}
	Therefore,
	\begin{align*}
		\zeta_{\Upsilon_{E}}(B)\leq \left[\Frac{\lambda {\varkappa}^{\sigma}\left(\delta_2 +\delta_1+\delta_{2} e^{-\varrho {\varkappa}}\right)}{(1-L)\Gamma(\sigma+1)(\delta_{1}+\delta_{2}e^{-\varrho {\varkappa}})}\right]\zeta_{\Upsilon_{E}}(B),
	\end{align*}
	which implies that $ \zeta_{\Upsilon_{E}}(B)=0$. We conclude by M\"{o}nch fixed point theorem that the operator $S$ has at least one fixed point which is the fixed point of the operator $A$ and the solution of the problem $(\ref{t1'})$-$(\ref{t3'}).$
\end{proof}

\section{Ulam-Hyers Stability }
In this section, we will establish the Ulam stability for the problem $(\ref{t1'})$-$(\ref{t3'}).$

\begin{definition}[\cite{ABGH}]
	Problem $(\ref{t1'})$-$(\ref{t3'})$  is Ulam-Hyers stable if there exists a real number $C_{{\Psi}}>0$ such that for each $\varepsilon>0$ and for each solution ${\xi} \in \Upsilon_E$
	of the inequality
	\begin{eqnarray}\label{ineqd}
		\left\|_{0}^{C} D^{\sigma, \varrho}_{{\theta}}{\xi}({\theta})- {\Psi}\left({\theta},{\xi}_{{\theta}}, \ _{0}^{C} D^{\sigma, \varrho}_{{\theta}}{\xi}({\theta})\right)\right\|<\varepsilon, \quad  {\theta} \in \Xi,
	\end{eqnarray}
	there exists a solution $\bar{{\xi}} \in \Upsilon_E$ of the problem $(\ref{t1'})$-$(\ref{t3'})$ with
	\begin{eqnarray*}
		\|{\xi}({\theta})- \bar{{\xi}}({\theta})\|<C_{{\Psi}} \varepsilon, \quad {\theta} \in \Xi.
	\end{eqnarray*}
\end{definition}
\begin{definition}[\cite{ABGH}]
	Problem $(\ref{t1'})$-$(\ref{t3'})$ is generalized Ulam-Hyers stable if there exists $\Lambda_{{\Psi}} \in C({\mathbb{R_+}},{\mathbb{R_+}})$, $\Lambda_{{\Psi}}(0)=0$ such that for each solution
	${\xi} \in \Upsilon_E$ of the inequality $(\ref{ineqd})$ there exists a solution $\bar{{\xi}} \in \Upsilon_E$ of the problem $(\ref{t1'})$-$(\ref{t3'})$ with
	\begin{eqnarray*}
		\|{\xi}({\theta})- \bar{{\xi}}({\theta})\|<\Lambda_{{\Psi}} \varepsilon, \quad {\theta} \in \Xi.
	\end{eqnarray*}
\end{definition}
\begin{remark}
	A function ${\xi} \in \Upsilon_E$ is a solution of the inequality $(\ref{ineqd})$ if and only if there exists a function $\ell \in  C(\Xi, E)$ (which depend on ${\xi}$)
	such that
	\begin{enumerate}
		\item  $\|\ell({\theta})\|\leq \varepsilon$, \quad $ for \ each \ {\theta} \in \Xi$.
		\item $_{0}^{C} D^{\sigma, \varrho}_{{\theta}}{\xi}({\theta})= {\Psi}\left({\theta},{\xi}_{{\theta}}, \ _{0}^{C} D^{\sigma, \varrho}_{{\theta}}{\xi}({\theta})\right)+\ell({\theta})$, \quad $ for \ each \ {\theta} \in \Xi.$
	\end{enumerate}
\end{remark}

\begin{lemma}
	The solution of the following perturbed problem
	\begin{equation*}
		_{0}^{C} D^{\sigma, \varrho}_{{\theta}}{\xi}({\theta})={\Psi}\left({\theta},{\xi}_{\theta}, \ _{0}^{C} D^{\sigma, \varrho}_{{\theta}}{\xi}({\theta})\right)+\ell({\theta}),  \quad  \ {\theta} \in {\Xi}:=[0, {\varkappa}],
	\end{equation*}
	\begin{equation*}
		{\xi}({\theta})=\Lambda({\theta}), \quad \ {\theta} \in (-\infty, 0],
	\end{equation*}
	\begin{equation*}
		\delta_1 {\xi}(0)+ \delta_2 {\xi}({\varkappa})= \delta_3,
	\end{equation*}
	is given by
	\begin{align*}
		{{\xi}}({\theta})=\left\{\begin{array}{l}
			\Frac{\delta_{3}e^{-\varrho {\theta}}}{\delta_1+ \delta_2 e^{-\varrho {\varkappa}}}-
			\Frac{\delta_{2}e^{-\varrho {\theta}}\Int_{0}^{{\varkappa}}{e^{-\varrho({\varkappa}-s)}({\varkappa}-s)^{\sigma-1}\widehat{\Psi}(s)ds}}{\Gamma(\sigma)(\delta_1+ \delta_2 e^{-\varrho {\varkappa}})}	\\
			+ \Frac{1}{\Gamma(\sigma)}\Int_{0}^{{\theta}}{e^{-\varrho({\theta}-s)}({\theta}-s)^{\sigma-1} \widehat{\Psi}(s)ds}\\
			-
			\Frac{\delta_{2}e^{-\varrho {\theta}}}{\Gamma(\sigma)(\delta_1+ \delta_2 e^{-\varrho {\varkappa}})}
			\Int_{0}^{{\varkappa}}{e^{-\varrho({\varkappa}-s)}({\varkappa}-s)^{\sigma-1}\ell(s)ds}\\
			+ \Frac{1}{\Gamma(\sigma)}\Int_{0}^{{\theta}}{e^{-\varrho({\theta}-s)}({\theta}-s)^{\sigma-1} \ell(s)ds}\, \ {\theta}\in \Xi,\\\\
			\Lambda({\theta}), \ \ \ \ \ \ \ \qquad {\theta} \in (-\infty, 0].
		\end{array}\right.
	\end{align*}
	Moreover, the solution satisfies the following inequality
	
	\begin{align*}
		\Bigg\|{\xi}({\theta})-\Bigg[\Frac{\delta_{3}e^{-\varrho {\theta}}}{\delta_1+ \delta_2 e^{-\varrho {\varkappa}}}-
		\Frac{\delta_{2}e^{-\varrho {\theta}}}{\Gamma(\sigma)(\delta_1+ \delta_2 e^{-\varrho {\varkappa}})}
		\Int_{0}^{{\varkappa}}{e^{-\varrho({\varkappa}-s)}({\varkappa}-s)^{\sigma-1}\widehat{\Psi}(s)ds}\\
		+ \Frac{1}{\Gamma(\sigma)}\Int_{0}^{{\theta}}{e^{-\varrho({\theta}-s)}({\theta}-s)^{\sigma-1} \widehat{\Psi}(s)ds}\Bigg]\Bigg\|
		\quad\leq \Frac{ {\varkappa}^{\sigma}\varepsilon}{\Gamma(\sigma+1)}\left[\frac{\delta_2 }{\delta_1+\delta_2 e^{-\varrho {\varkappa}}}+1\right].
	\end{align*}
\end{lemma}
\begin{theorem}\label{theo3'}
	Assume that the conditions $(A1)$-$(A2)$ hold. If
	\begin{equation*}
		\Frac{\lambda {\varkappa}^{\sigma} \left(\delta_1+\delta_{2} e^{-\varrho {\varkappa}}+\delta_2\right)}{\Gamma(\sigma+1)(1-L)(\delta_{1}+\delta_{2} e^{-\varrho {\varkappa}})}< 1,
	\end{equation*}
	then the problem $(\ref{t1'})$-$(\ref{t3'})$ is Ulam-Hyers stable.
\end{theorem}
\begin{proof}
	Let ${\xi} \in \Upsilon_E$ be a solution of the inequality $(\ref{ineqd})$ and $\bar{{\xi}} \in \Upsilon_E$  the  solution of the problem  $(\ref{t1'})$-$(\ref{t3'}).$ Then,
	\begin{align*}
		\|{\xi}({\theta})-\bar{{\xi}}({\theta})\|&\leq \Frac{ {\varkappa}^{\sigma}\varepsilon}{\Gamma(\sigma+1)}\left[\frac{\delta_2}{\delta_1+\delta_2 e^{-\varrho {\varkappa}}}+1\right]\\
		&+ \Frac{\delta_{2}e^{-\varrho {\theta}}\Int_{0}^{{\varkappa}}{e^{-\varrho({\varkappa}-s)}({\varkappa}-s)^{\sigma-1}\|\widehat{\Psi}(s)-\widetilde{\Psi}(s)\|ds}}{\Gamma(\sigma)(\delta_1+ \delta_2 e^{-\varrho {\varkappa}})}
		\\
		&+
		\Frac{1}{\Gamma(\sigma)}\Int_{0}^{{\theta}}{e^{-\varrho({\theta}-s)}({\theta}-s)^{\sigma-1}\|\widehat{\Psi}(s)-\widetilde{\Psi}(s)\|ds},
	\end{align*}
	where $\widehat{\Psi}$ and $\widetilde{\Psi}$ are two functions satisfying the following functional equations:
	$$\widehat{\Psi}({\theta})={\Psi}({\theta}, {\xi}_{\theta}, \widehat{\Psi}({\theta})),$$
	and
	$$\widetilde{\Psi}({\theta})={\Psi}({\theta}, \bar{{\xi}}_{\theta}, \widetilde{\Psi}({\theta})).$$
	From hypothesis $(A2)$, we have
	\begin{align*}
		\|\widehat{\Psi}({\theta})-\widetilde{\Psi}({\theta})\|&=\|{\Psi}({\theta}, {\xi}_{\theta}, \widehat{\Psi}({\theta}))-{\Psi}({\theta}, \bar{{\xi}}_{\theta}, \widetilde{\Psi}({\theta}))\|\\
		& \leq \lambda \|{\xi}_{\theta}-\bar{{\xi}}_{\theta}\|_{{\Bbbk}}+L\|\widehat{\Psi}({\theta})-\widetilde{\Psi}({\theta})\|,
	\end{align*}
	which implies that
	\begin{align*}
		\|\widehat{\Psi}({\theta})-\widetilde{\Psi}({\theta})\|\leq \Frac{\lambda}{1-L}\|{\xi}_{\theta}-\bar{{\xi}}_{\theta}\|_{{\Bbbk}}.
	\end{align*}
	Then,
	\begin{align*}
		\|{\xi}({\theta})-\bar{{\xi}}({\theta})\|&\leq  \Frac{ {\varkappa}^{\sigma}\varepsilon}{\Gamma(\sigma+1)}\left[\frac{\delta_2  }{\delta_1+\delta_2 e^{-\varrho {\varkappa}}}+1\right]\\
		&+  \Frac{\lambda {\varkappa}^{\sigma}\delta_{2}}{\Gamma(\sigma+1)\left(\delta_1+ \delta_2 e^{-\varrho {\varkappa}}\right)(1-L)}\|{\xi}-\bar{{\xi}}\|_{\Upsilon_E}\\
		&+\Frac{\lambda {{\varkappa}}^{\sigma}}{\Gamma(\sigma+1)(1-L)} \|{\xi}-\bar{{\xi}}\|_{\Upsilon_E}.
	\end{align*}
	Thus,
	\begin{align*}
		\|{\xi}-\bar{{\xi}}\|_{\Upsilon_E} \leq
		\Frac{\Frac{ {\varkappa}^{\sigma}\varepsilon}{\Gamma(\sigma+1)}\left[\frac{\delta_2 }{\delta_1+\delta_2 e^{-\varrho {\varkappa}}}+1\right]}{1-\Frac{\lambda {{\varkappa}}^{\sigma}(\delta_1+\delta_2 e^{-\varrho {\varkappa}}+\delta_2  )}{\Gamma(\sigma+1)(1-L)(\delta_1+\delta_2 e^{-\varrho {\varkappa}})}}:= C_f \varepsilon.
	\end{align*}
	Consequently, the problem $(\ref{t1'})$-$(\ref{t3'})$ is Ulam-Hyers stable.
	If we take $\Lambda_{{\Psi}}(\varepsilon)= C_{{\Psi}} \varepsilon$ and $\Lambda_{{\Psi}}(0)=0,$ then we get the generalized  Ulam-Hyers stability of the problem $(\ref{t1'})$-$(\ref{t3'}).$
\end{proof}
\section{An Example}
Set
$$E=l^{1}=\left\{{\xi}=({\xi}_1, {\xi}_2,\ldots, {\xi}_{\jmath},\ldots),  \sum_{{\jmath}=1}^{\infty}|{\xi}_{\jmath}|<\infty\right\},$$
where $E$ is a Banach space with the norm $\|{\xi}\|=\displaystyle\sum_{{\jmath}=1}^{\infty}|{\xi}_{\jmath}|.$\\
Consider the following implicit problem:	

\begin{equation}\label{te1'}
	_{0}^{C} D^{\frac{1}{2}, 4}_{{\theta}}{\xi}({\theta})
	={\Psi}\left({\theta},{\xi}_{\theta}, \ _{0}^{C} D^{\frac{1}{2}, 4}_{{\theta}}{\xi}({\theta})\right),\ {\theta} \in [0, 1],
\end{equation}
\begin{equation}
	{\xi}({\theta})= \Lambda({\theta}), \quad \ {\theta} \in (-\infty, 0],
\end{equation}
\begin{equation}\label{te2'}
	{\xi}(0)+{\xi}(1)=0,
\end{equation}
where $\Lambda \in {{\Bbbk}}_\gamma$, $\gamma$ is a positive real constant and
\begin{equation}
	{\Bbbk}_\gamma=\{{\xi}\in C((-\infty,0],E):\ \Lim_{{{\theta}}\rightarrow-\infty}e^{\gamma{{\theta}}}{{\xi}}({{\theta}})\ exists\  in\ E \}.
\end{equation}
The norm of ${\Bbbk}_\gamma$ is given by
$$\|{{\xi}}\|_\gamma=\Sup_{{{\theta}}\in(-\infty,0]}e^{\gamma{{\theta}}}\|{{\xi}}({{\theta}})\|.$$
Let ${{\xi}}:(-\infty,1]\rightarrow E$ be such that ${{\xi}}_0\in {\Bbbk}_\gamma.$
Then
\begin{align*}
	\Lim_{{s}\rightarrow -\infty}e^{\gamma{s}}{{\xi}}_{{{\theta}}}({s})&=
	\Lim_{{s}\rightarrow -\infty}e^{\gamma{s}}{{\xi}}({\theta}+s)\\
	&= \Lim_{{s}\rightarrow -\infty}e^{\gamma({s}-{\theta})}{{\xi}}({s})\\
	&=e^{-\gamma {\theta}} \Lim_{{s}\rightarrow -\infty}e^{\gamma{s}}{{\xi}}_0({s})\\
	&<\infty.
\end{align*}
Hence ${{\xi}}_{{\theta}}\in {\Bbbk}_\gamma.$ Finally we prove that
$$\|{\xi}_{{\theta}}\|_{\gamma}\leq {{\wp_1}} \|{\xi}_0\|_{\gamma}+ {\wp_2} \sup_{{{\theta}}\in [0,{\varkappa}]}\|{{\xi}}({{\theta}})\|,$$
where ${\wp_1}={\wp_2}=1.$ We have
$$\|{\xi}_{{\theta}}(s)\|=\|{\xi}({{\theta}}+s)\|.$$
If ${\theta}+s\leq 0,$ we get$$\|{\xi}_{{\theta}}(s)\|\leq \Sup_{{{\theta}}\in (-\infty,0]}\|{{\xi}}(s)\|.$$
For ${{\theta}+s}\geq 0,$ then we have
$$\|{\xi}_{{\theta}}(s)\|\leq \Sup_{{s}\in [0,{\varkappa}]}\|{\xi}(s)\|.$$
Thus, for all ${{\theta}+s}\in (-\infty, 1],$ we get
$$\|{\xi}_{{\theta}}(s)\|\leq \Sup_{{s}\in (-\infty,0]}\|{{\xi}}(s)\|+\Sup_{s\in [0,{\varkappa}]}\|{\xi}(s)\|.$$
Then
$$\|{\xi}_{{\theta}}\|_{\gamma}\leq  \|{\xi}_0\|_{\gamma}+\Sup_{s\in [0,{\varkappa}]}\|{\xi}(s)\|.$$
It is clear that $({\Bbbk}_\gamma,\|\cdot\|)$ is a Banach space.
We can conclude that ${\Bbbk}_\gamma$ a phase space. \\

Set
\begin{align*}
	{\Psi}_{\jmath}\left({\theta},{\xi}_{{\jmath}{\theta}}, \ _{0}^{C} D^{\frac{1}{2}, 4}_{{\theta}}{\xi}_{\jmath}({\theta})\right)=\Frac{e^{-\pi {\theta}}\left(\|{\xi}_{{\theta}}\|_{{\Bbbk}}+\left|_{0}^{C} D^{\frac{1}{2}, 4}_{{\theta}}{\xi}_{{\jmath}}({\theta})\right|\right)}
	{110 \left(1+\|{\xi}_{{\theta}}\|_{{\Bbbk}}+\left\|_{0}^{C} D^{\frac{1}{2}, 4}_{{\theta}}{\xi}({\theta})\right\|\right)},
\end{align*}
for ${\theta} \in [0, 1], \ {\xi}\in C((-\infty, 1], E),$ where
$$
{\xi}=({\xi}_1,{\xi}_2,\ldots,{\xi}_{\jmath},\ldots),
$$
$$
{\Psi}=({\Psi}_{1},{\Psi}_{2},\ldots,{\Psi}_{{\jmath}},\ldots),
$$
and
$$
_{0}^{C} D^{\frac{1}{2}, 4}_{{\theta}}{\xi}=\left({}_{0}^{C} D^{\frac{1}{2}, 4}_{{\theta}}{\xi}_1\ ,\ _{0}^{C} D^{\frac{1}{2}, 4}_{{\theta}}{\xi}_2\ ,\ldots,\ _{0}^{C} D^{\frac{1}{2}, 4}_{{\theta}}{\xi}_{\jmath},\ldots\right).
$$
Clearly, ${\Psi}$ is a  continuous function, then the hypothesis $(A1)$ is satisfied.\\
For any ${\alpha}, \bar{{\alpha}} \in {\Bbbk}$, ${\beta}, \bar{{\beta}} \in E $ and ${\theta} \in [0, 1],$ we have
\begin{align*}
	\|{\Psi}({\theta},{\alpha},{\beta})-{\Psi}({\theta},\bar{{\alpha}}, \bar{{\beta}})\|\leq \frac{1}{110}\left[\|{\alpha}- \bar{{\alpha}}\|_{{\Bbbk}}+\|{\beta}- \bar{{\beta}}\|\right].
\end{align*}
Then, the hypothesis $(A2)$ is satisfied with $\lambda=L=\frac{1}{110}$. Also we have
\begin{align*}
	\Frac{\lambda {\varkappa}^{\sigma} ({\wp_2}+1)\left(\delta_2 +\delta_1+\delta_{2} e^{-\varrho {\varkappa}}\right)}{\Gamma(\sigma+1)(1-L)(\delta_{1}+\delta_{2} e^{-\varrho {\varkappa}})} <
	\Frac{6}{109 \sqrt{\pi}}<1,
\end{align*}
for ${\varkappa}=1$, $\sigma=\frac{1}{2}$, ${\wp_1}={\wp_2}=1$ and $\delta_1= \delta_2=1$. By Theorem $\ref{theo1'}$,  the problem $(\ref{te1'})$-$(\ref{te2'})$ has at least one solution. Moreover, we have
\begin{align*}
	\Frac{\lambda {\varkappa}^{\sigma} \left(\delta_1+\delta_{2} e^{-\varrho {\varkappa}}+\delta_2\right)}{\Gamma(\sigma+1)(1-L)(\delta_{1}+\delta_{2} e^{-\varrho {\varkappa}})}<\Frac{3}{109 \sqrt{\pi}}< 1.
\end{align*}
Thus, by Theorem $\ref{theo3'},$ the problem $(\ref{te1'})$-$(\ref{te2'})$ is Ulam-Hyers stable.

\section*{Declarations}

\noindent\textbf{Ethical approval:} This article does not contain any studies with human participants or animals performed by any of the authors.\\

\noindent\textbf{Competing interests:} It is declared that authors has no competing interests.\\

\noindent\textbf{Author's contributions:} The study was carried out in collaboration of all authors. All authors read and approved the final manuscript.\\


\noindent\textbf{Funding:} Not available.\\

\noindent\textbf{Availability of data and materials:} Data sharing not applicable to this paper as no data sets were generated or analyzed during the current study.\\

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\end{center}

{\small
	
	\noindent{\bf Wafaa Rahou}
	
	\noindent  Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbes
	
	
	\noindent P.O. Box 89, Sidi Bel-Abbes 22000, Algeria
	
	\noindent E-mail: wafaa.rahou@yahoo.com}\\

{\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 Jamal Eddine Lazreg}
	
	\noindent  Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbes
	
	\noindent Full Professor of Mathematics
	
	\noindent P.O. Box 89, Sidi Bel-Abbes 22000, Algeria
	
	\noindent E-mail: lazregjamal@yahoo.fr}\\

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



\end{document}