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\fancyhead[CE]{M.  HOSSEINI FARAHI,  R.  ALLAHYARI AND M.  HASSANI} 
\fancyhead[CO]{SOLVABILITY OF INFINITE SYSTEMS OF FRACTIONAL
EQUATION 
IN THE HAHN SEQUENCE SPACE}



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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 
Solvability Of Infinite Systems Of Fractional
Equation 
In The Hahn Sequence Space\\}
%{\bf Solvability Of Infinite Systems Of Fractional
%Equation \\ If so, Write It Here} 


\let\thefootnote\relax\footnote{\scriptsize Received:   }

{\bf Masoomeh Hosseini Farahi}\vspace*{-2mm}\\
\vspace{2mm} {\small  Islamic Azad University} \vspace{2mm}

{\bf  Reza  Allahyari $^*$\let\thefootnote\relax\footnote{$^*$Corresponding Author}}\vspace*{-2mm}\\
\vspace{2mm} {\small  Islamic Azad University} \vspace{2mm}

{\bf  Mahmoud Hassani}\vspace*{-2mm}\\
\vspace{2mm} {\small  Islamic Azad University} \vspace{2mm}
\end{center}

\vspace{4mm}


{\footnotesize
\begin{quotation}
{\noindent \bf Abstract.} We define the Hausdorff MNC in the Hahn sequence space. Then, by applying the MNC we consider the solvability of BVP of fractional type by nonlocal integral boundary conditions in the Hahn sequence space. Eventually, we provide one example to inquire the performance of main results.
\end{quotation}
\begin{quotation}
\noindent{\bf AMS Subject Classification:}  47H09; 26A33; 47H10; 34A12.

\noindent{\bf Keywords and Phrases:} Fractional differential equations, Measure of noncompactness, Meir-Keeler
condensing operator, Sequence spaces.
\end{quotation}}

\section{Introduction}
\label{intro} 
Recently, the implication of MNC has been utilized in sequence spaces for different classes of differential equations (\cite{Alotaibi,ban2,Hazarika,Mursaleencan,Mursaleen,Mursal.bil,Mursal.Mohi,Mursal.riz}). Aghajani et al. \cite{Pourhadi} investigated the solvability of  infinite systems of second order differential equations in $l_1$-spaces. Afterwards, Mohiuddine et al. \cite{Moh.sir}, Bana\'{s} et al. \cite{ban.murs} focused in these systems in the $l_p$ space.
\par In this paper, we present the Hausdorff MNC in Hahn sequence space. By applying this MNC, we consider the solvability of infinite systems of BVP  fractional
type by nonlocal integral boundary conditions in the Hahn sequence space.
%\begin{equation}\label{mea1}\end{equation}$$\left\{\begin{array}{cc}
%  u(x)=f(x,u(x),v(x),\int g(y,u(y),\frac{\partial u}{\partial x_1}(y),\dots,\frac{\partial u}{\partial x_n}(y),v(y))dy) &\quad \\
% v(x)=f(x,v(x),u(x),\int g(y,v(y),\frac{\partial v}{\partial x_1}(y),\dots,\frac{\partial v}{\partial x_n}(y),u(y))dy) &\quad \\
%\end{array}\right.$$
Then, we presented one example to inquire the performance of main results.\par
Suppose that $(\mho,\| \cdot \|)$ is real Banach space by zero element $0$. $D(\nu,\sigma)$ is the ball centered at $\nu$ by radius $\sigma$. For $\emptyset\neq \mathfrak{L}\subseteq\mho$, we denote by $\overline{\mathfrak{L}}$ closure and $\text{Conv}\mathfrak{L}$ is closed convex hull of $\mathfrak{L}$, $\emptyset\neq{\mathfrak{N}}_\mho\subseteq\mho$ is the family of all relatively compact and $\emptyset\neq{\mathfrak{M}}_\mho\subseteq\mho$ is the family of  bounded.
 \begin{definition}\cite{1aga.banas}
The function $\widetilde{\mu} :\mathfrak{M}_\mho \rightarrow [0,\infty)$ is measure of noncompactness (MNC) in $\mho$ if it fulfills:
\begin{itemize}
\item[$1^{\circ}$] $\mathfrak{N}_{\mho}\supseteq\{\mathfrak{L}\in \mathfrak{M}_{\mho}: \widetilde{\mu}(\mathfrak{L})=0 \}=\ker\widetilde{\mu}\neq\emptyset$.
\item[$2^{\circ}$] $\mathfrak{L}\subset \mathfrak{R}\ \Rightarrow\ \widetilde{\mu}(\mathfrak{L})\leq \widetilde{\mu}(\mathfrak{R})$.
\item[$3^{\circ}$] $\widetilde{\mu}(\overline{\mathfrak{L}})=\widetilde{\mu}(\mathfrak{L})=\widetilde{\mu}(\text{Conv}\mathfrak{L})$.
\item[$4^{\circ}$] $\widetilde{\mu}(\zeta \mathfrak{L} + (1 -\zeta)\mathfrak{R} )\leq \zeta\widetilde{\mu}(\mathfrak{L}) + (1 -\zeta)\widetilde{\mu}(\mathfrak{R})$ for
$0\leq\zeta\leq 1$.
\item[$5^{\circ}$] If $\mathfrak{L}_n\in \mathfrak{M}_\mho$, $\mathfrak{L}_n=\overline{\mathfrak{L}_n}$, and $\mathfrak{L}_{n+1}\subset \mathfrak{L}_{n}$ for $n\in\mathbb{N}$ and $\displaystyle{\lim
_{n\rightarrow\infty}} \widetilde{\mu}(\mathfrak{L}_{n})=0$, then
$\emptyset\neq \mathfrak{L}_{\infty}=\displaystyle{\bigcap_{n=1}^{\infty}}\mathfrak{L}_n$.
\end{itemize}
\end{definition}
%In the following, we denote by $\mathfrak{M}_X$, the collection of all bounded subsets of the metric space $(X,d)$.
 \begin{definition}\cite{bban}
Let $(\mathfrak{L},d)$ is metric space and $\mathfrak{A}\in\mathfrak{M}_\mathfrak{L}$. The Kuratowski MNC of $\mathfrak{A}$, is
$$\beta(\mathfrak{A})=\inf\Big(0<\varepsilon: \displaystyle\bigcup_{\imath=1}^m K_\imath\supseteq\mathfrak{A},  K_\imath\subset \mathfrak{L}, \varepsilon>\text{diam}(K_\imath)\ (\imath=1,2,\ldots,m);\ m\in\mathbb{N}\Big),$$
where $\text{diam}(K_\imath)=\sup\{d(\upsilon,\nu): \nu,\upsilon\in K_\imath\}$.
\end{definition}
The Hausdorff MNC $\chi(\mathfrak{A})$, is
$$\chi(\mathfrak{A})=\inf\Big(\varepsilon>0:\mathfrak{A}\subset\displaystyle\bigcup_{\imath=1}^m D(\nu_\imath,\sigma_\imath), \nu_\imath\in \mathfrak{L},  \sigma_\imath<\varepsilon\  (\imath=1,2,\ldots,m);\ m\in\mathbb{N}\Big).$$
%\begin{lemma}\cite{bban}
%Let $Q$, $Q_1$, and $Q_2$ be bounded subsets of a metric space $(X,d)$. Then
%\begin{itemize}
%\item[$1^{\circ}$] $\chi(Q)=0$ if and only if $Q$ is totally bounded,
%\item[$2^{\circ}$] $Q_1\subset Q_2\quad \text{implies that}\quad  \chi(Q_1)\leq \chi(Q_2)$,
%\item[$3^{\circ}$] $\chi(\overline{Q})=\chi(Q)$,
%\item[$4^{\circ}$] $\chi(Q_1\cup Q_2)=\max\{\chi(Q_1),\chi(Q_2)\}$.
%\end{itemize}
%\end{lemma}
%In the case of a normed space $(X,\| \cdot \|)$, the function $\chi:\mathfrak{M}_X \rightarrow \mathbb{R}_{+}$ has some additional properties connected with the linear structure, for example, we have
%\\$i)$  $\chi(Q_1+Q_2)\leq \chi(Q_1)+ \chi(Q_2),$
%\\$ii)$  $\chi(Q+x)=\chi(Q)$ for all $x\in X$,
%\\$iii)$  $\chi(\lambda Q)=|\lambda|\chi(Q)$ for all $\lambda\in \mathbb{C}$,
%\\$iv)$  $\chi(Q)=\chi(\text{Conv}Q)$.
%Meir and Keeler  \cite{Meir} described the concept of Meir--Keeler contractive mapping and proved some fixed point theorems for this kind of mapping.
\begin{definition}\cite{shole}
Let $\mho$ is Banach space, $\emptyset\neq \mathfrak{Q}\subseteq \mho,$ also, $\widetilde{\mu}$ is MNC in $\mho$. The operator $\mathfrak{H}:\mathfrak{Q}\rightarrow \mathfrak{Q}$ is called a Meir--Keeler condensing operator if $\forall$ $0<\varepsilon$, $\exists$ $0<\delta$  so that $$\varepsilon\leq\widetilde{\mu}(\mathfrak{L})<\delta+\varepsilon\quad \text{implies}\quad \widetilde{\mu}(\mathfrak{H}(\mathfrak{L}))<\varepsilon,$$ 
$\forall$ bounded $\mathfrak{L}\subseteq \mathfrak{Q}$.
\end{definition}
\begin{theorem}\cite{shole}\label{meiir}
Let $\mho$ is Banach space, $\emptyset\neq \mathfrak{D}=\overline{\mathfrak{D}}\subseteq \mho$ is bounded, convex and $\widetilde{\mu}$ is an MNC in $\mho$. If $\mathfrak{H}:\mathfrak{D}\rightarrow \mathfrak{D}$ is a Meir--Keeler condensing operator and continuous, then $\mathfrak{H}$ has fixed point.
\end{theorem}
%The space $C(I=[0,T], \mho)$ have the standard norm
%$$\|\nu\|_c:=\sup\{\|\nu(\wp)\|:\wp\in I\},\qquad \nu\in C(I, \mho).$$
\begin{proposition}\cite{bban}\label{bban}
If $\Upsilon\subseteq C(I,\mho)$ is equicontinuous and bounded, then the function $\chi(\Upsilon(.))$ is continuous and
$$\displaystyle\sup_{\wp\in I}\chi(\Upsilon(\wp))=\chi(\Upsilon),\quad \chi\big(\int_{0}^{\wp}\Upsilon(\Im)d\Im\big)\leq\int_{0}^{\wp}\chi(\Upsilon(\Im))d\Im.$$
\end{proposition}
\begin{definition}(\cite{Pod})
Let $f:\mathbb{R_+}\rightarrow \mathbb{R},$ the Caputo fractional derivative of order $0<\alpha$ is
\begin{footnotesize}$$\frac{\int_{0}^{\wp}\frac{f^{(\mathfrak{m})}(\Im)}{(\wp-\Im)^{\alpha-\mathfrak{m}+1}}d\Im}{\Gamma(\mathfrak{m}-\alpha)}= ^cD^{\alpha}f(\wp),$$\end{footnotesize}
where $\mathfrak{m}-1=[\alpha]$.
\end{definition}
\begin{definition}(\cite{Pod})
Let $h:(0,\infty)\rightarrow \mathbb{R}$, the R-L (Riemann-–Liouville)  fractional integral of order $o$ is
$$\frac{1}{\Gamma(\alpha)}\int_{0}^{\wp}\frac{h(\wp)}{(\wp-\Im)^{1-o}}d\Im=I_{0_+}^{o}h(\wp).$$
%provides that the right side is pointwise defined on $(0,\infty)$.
\end{definition}
\begin{lemma}\cite{Ntouyas}
Let $\xi>0$,$\gamma\neq\frac{\Gamma(\xi+1)}{\vartheta^\xi}$ and $f_\imath\in C([0, 1]).$ Then, the solution of the FDE
\begin{equation}\label{mea1}
\begin{cases}
^CD_{0_+}^{\alpha}\nu(\wp)=f_\imath(\wp,\nu(\wp)),\ 0\leq\wp\leq1,\ \alpha\in(0,1],\ 0<\vartheta<1,\ \gamma,\xi\in\mathbb{R}&\quad \\
\nu(0)=\gamma I_{0_+}^\xi \nu(\vartheta)=\gamma \int_{0}^{\vartheta}\frac{(\vartheta-\Im)^{\xi-1}}{\Gamma(\xi)}\nu(\Im)d\Im,
\end{cases}\end{equation}
is
\begin{footnotesize}$$\nu(\wp)=\frac{1}{\Gamma(\alpha)}\int_{0}^{\wp}(\wp-\Im)^{\alpha-1}f_\imath(\Im,\nu(\Im))d\Im+
\frac{\gamma(\Gamma(\xi+1)}{\Gamma(\xi+1)-\gamma \vartheta^\xi}\int_{0}^{\vartheta}\frac{(\vartheta-\Im)^{\xi+\alpha-1}}{\Gamma(\xi+\alpha)}f_\imath(\Im,\nu(\Im))d\Im.$$\end{footnotesize}
\end{lemma}
%\begin{definition}\cite{chang 10}
%Let $X$ be a nonempty set. An element $(x,y)\in X\times X$ is called a coupled fixed point of a mapping $G:X\times X\rightarrow X$ if $G(x,y)=x$ and $G(y,x)=y$.
%\end{definition}
%Here we quote a useful theorem in \cite{Akhm 7} concerning the
%construction of a measure of noncompactness on a finite product space.
%\begin{theorem}\label{jj} Let $\widetilde{\mu}_1,\widetilde{\mu}_2,\ldots,\widetilde{\mu}_n$ be measures of noncompactness in Banach spaces $E_1, E_2,\ldots, E_n$, respectively.
%Moreover, suppose that the function $F:[0,\infty)^n\rightarrow[0,\infty)$ is convex and $F(x_1, x_2,\ldots,x_n) = 0$ if and only if $x_i = 0$ for
%$i = 1, 2,\ldots, n$. Then
%$$\widetilde{\widetilde{\mu}}(X) = F(\widetilde{\mu}_1(X_1),\widetilde{\mu}_2(X_2),\ldots,\widetilde{\mu}_n(X_n)),$$
%defines a measure of noncompactness in $E_1\times E_2\times ,\ldots \times E_n$, where $X_i$ denotes the natural projection of $X$ into $E_i$, for
%$i = 1, 2,\ldots, n$.
%\end{theorem}
%As a result from Theorem \ref{jj} above, we have the following example which is presented in \cite{agha.sabz}.
%\begin{example}\label{exa1} Let $\widetilde{\mu}$ be a measure of noncompactness on a Banach space $E$. Take $F(x,y) =x+y$ for any
%$(x,y)\in\mathbb{R}_+^2$. Then all the conditions of
%Theorem \ref{jj} are satisfied. Therefore, $\widetilde{\widetilde{\mu}}(X) = \widetilde{\mu}(X_1)+\widetilde{\mu}(X_2)$ defines a measure of noncompactness in the space $E\times E$
%where $X_i$, $i = 1, 2$ denote the natural projections of $X$.
%\end{example}
 %Denote with $\Psi$ the family of increasing functions $\psi:\mathbb{R}_+\rightarrow\mathbb{R}_+$ continuous in $t=0$ such that\\
%$\bullet$ $\psi(t)=0$ if and only if $t=0$,\\
%$\bullet$ $\psi(t+s)\leq\psi(t)+\psi(s)$.
%\begin{definition}\cite{Amiri1}\label{d1} Let $C$ be a nonempty subset of a Banach space $E$ and $\widetilde{\mu}$ be
%an arbitrary measure of noncompactness on $E$. Also, suppose $\psi:\mathbb{R}_+\rightarrow\mathbb{R}_+$ is an increasing mapping such that $\psi(t)=0$ if and only if $t=0$. We say that an operator
%$T:C\rightarrow C$ is an $(\alpha,\beta)$-generalized Meir-Keeler condensing operator
%if for any $\varepsilon>0$, $\delta>0$ exists such that
%\begin{equation}\label{gmkc}
%\varepsilon\leq\beta(\widetilde{\mu}(X))\psi(\widetilde{\mu}(X))<\varepsilon+\delta\quad\text{implies}\quad\alpha(\widetilde{\mu}(T(X)))\psi(\widetilde{\mu}(T(X)))<\varepsilon
%\end{equation}
%for any bounded subset $X$ of $C$, where
%$\alpha:\mathbb{R}_{+}\rightarrow[1,+\infty)$ and $\beta:\mathbb{R}_+\rightarrow(0,1]$ are mappings.
%\end{definition}
%\begin{theorem}\cite{Amiri1}\label{tg}
%Let $C$ be a nonempty, bounded, closed, and convex subset of a Banach space $E$ and let $\widetilde{\mu}$ be
%an arbitrary measure of noncompactness on $E$. If $T:C\rightarrow C$ is a continuous and %$(\alpha,\beta)$-generalized Meir-Keeler condensing operator,
%then $T$ has at least one fixed point in the set $C$ and the set of all fixed
%points of $T$ in $C$ is compact.
%\end{theorem}
%\begin{definition}
 %A function $\theta:\mathbb{R}_+\rightarrow \mathbb{R}_+$ is called a strictly $L$-function if $\theta(0)=0$, $\theta(s)>0$ for $s\in(0,+\infty)$, and for any  $s>0$, $\delta>0$ exists such that $\theta(t)<s$, for all $t\in[s,s+\delta]$.
%\end{definition}
% \begin{theorem}\cite{Amiri1}\label{cc1}
%Let $\alpha$, $\beta$, and $\psi$ be as Definition \ref{d1}, $C$ be a nonempty, bounded, closed and convex subset of a Banach space $E$, and let $T:C\rightarrow C$ be a continuous operator such that $$\alpha(\widetilde{\mu}(T(X)))\psi(\widetilde{\mu}(T(X)))\leq\theta\Big(\beta(\widetilde{\mu}(X))\psi(\widetilde{\mu}(X))\Big)$$
%for any $X\subseteq C$, where $\widetilde{\mu}$ is an arbitrary measure of noncompactness on $E$ and $\theta$ is a strictly L-function. Then, $T$ has at least one fixed point.
%\end{theorem}
%Now, we present a coupled fixed point theorem using strictly L-functions.
%\begin{theorem}\cite{Amiri1}\label{t1}
%Let $E$, $C$,  $\beta$, $\theta$ and $\widetilde{\mu}$ be as Theorem \ref{cc1} and let $\alpha:\mathbb{R}_{+}\rightarrow[1,+\infty)$ be an increasing map. Also, suppose that $\psi\in \Psi$ and $G:C\times C\rightarrow C$, is a continuous mapping satisfying
%\begin{footnotesize}\begin{equation}\label{f1}\alpha\Big(\widetilde{\mu}(G(X_1\times X_2))+\widetilde{\mu}(G(X_2\times X_1))\Big)\psi(\widetilde{\mu}(G(X_1\times X_2)))\qquad\qquad\qquad\qquad\qquad\end{equation}$$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\leq\frac{1}{2}\theta\Big(\beta(\frac{\widetilde{\mu}(X_1)+\widetilde{\mu}(X_2)}{2})\psi(\frac{\widetilde{\mu}(X_1)+\widetilde{\mu}(X_2)}{2})\Big)$$\end{footnotesize}
%for all subsets $X_1$,$X_2$  of $C$. Then $G$ has at least a coupled fixed point.
%\end{theorem}
\section{Hahn Sequence space}
By $\omega=\mathbb{C}^N,$ ($N=\{0, 1, 2,\ldots\}$ and $\mathbb{C}$ is complex field) we denote the space of all complex-valued or real sequences.
\\Each linear subspace of $\omega$ is called a sequence space.
% For $z=(z_k)\in\omega$, we shall employ the sequence spaces $c_0$ (null), $c$ (convergent) and $l_\infty$ (bounded) sequences $z=(z_k)$ with complex terms, by norm
%\begin{footnotesize}$$\|z\|_\infty=\displaystyle{\sup_{k\in\mathbb{N}}}|z_k|.$$\end{footnotesize}
\par In \cite{Hahn} Hahn  defined the $BK$-−space $h$ of all sequences $\nu=(\nu_k)$ so that
\begin{footnotesize}$$h=\big\{\nu:\sum_{k=1}^{\infty}k|\Delta \nu_k|<\infty\ \mbox{and}\ \displaystyle{\lim
_{k\rightarrow\infty}}\nu_k=0\big\},$$\end{footnotesize}
where $\Delta \nu_k=\nu_k-\nu_{k+1}$, $\forall$ $k\in \mathbb{N},$ by norm
\begin{footnotesize}$$\|\nu\|_h=\sum_{k=1}^{\infty}k|\Delta \nu_k|+\sup_{k}|\nu_k|.$$\end{footnotesize}
%(and also \cite{Goes}). Goes and Goes \cite{Goes} proved that the space $h$ is a $BK$--space.
Hahn showed that $h$ is Banach space and $h\subset l_1\bigcap \int c_0$.
%\begin{lemma}\cite{Hahn} \
%\\$(a)$ $h$ is a Banach space.
%\\$(b)$ $h\subset l_1\bigcap \int c_0$.
%\end{lemma}
%Now, we determine the Hausdorff MNC $\chi$ in Hahn Banach space.
\begin{lemma}\cite{Mursaleen} Let $\mathfrak{L}$ is normed space and $\mathfrak{A}\subseteq \mathfrak{L}$ is bounded, where $\mathfrak{L}$ is $l_p$ $( p\in[1,\infty))$ or $c_0$. If $R_n:\mathfrak{L}\rightarrow \mathfrak{L}$ is operator $R_n(\nu)=(\nu_0,\nu_1,\ldots,\nu_n,0,0,\ldots)$,
so\begin{footnotesize} $$\chi(\mathfrak{A})=\displaystyle\lim_{n\rightarrow\infty}\big\{\displaystyle\sup_{\nu\in \mathfrak{A}}\|(I-R_n)\nu\|\big\}.$$\end{footnotesize}
\end{lemma}
\begin{theorem}
Let $\mathfrak{A}\subseteq h$ be bounded. So the Huasdorff  MNC $\chi$ in the Banach space $h$ defined by:
\begin{footnotesize}\begin{equation}\label{koko}
 \chi(\mathfrak{A}):=\displaystyle\lim_{n\rightarrow\infty}\Big\{\displaystyle\sup_{\nu\in \mathfrak{A}}\Big\{\sum_{k\geq n}(k|\Delta \nu_k|)+\sup_{k}|\nu_k|\Big\}\Big\}.
\end{equation}\end{footnotesize}
\end{theorem}
\begin{proof}
Define the operator $R_n:h\rightarrow h$  by $R_n(\nu)=(\nu_1,\nu_2,\ldots,\nu_n, 0, 0,\ldots)$ for $\nu= (\nu_1,\nu_2,\ldots )\in h$. Then
\begin{footnotesize}\begin{equation}\label{uy}
 \mathfrak{A}\subset R_n\mathfrak{A}+(I-R_n)\mathfrak{A},
\end{equation}\end{footnotesize}
from (\ref{uy}), we get
\begin{footnotesize}\begin{eqnarray*}
\chi (\mathfrak{A})&\leq& \chi(R_n\mathfrak{A})+\chi((I-R_n)\mathfrak{A})=\chi((I- R_n)\mathfrak{A})
\\&\leq&\mbox{diam}((I- R_n)\mathfrak{A})=\displaystyle\sup_{\nu\in \mathfrak{A}}\|(I- R_n)\nu\|,
\end{eqnarray*}\end{footnotesize}
where
\begin{footnotesize}$$\|(I- R_n)\nu\|=\sum_{k=1}^{\infty}(k|\Delta \nu_k|)+\sup_{k}|\nu_k|,$$\end{footnotesize}
when $n$ is sufficiently large. So
\begin{footnotesize}\begin{equation}\label{uyt}
\chi (\mathfrak{A})\leq \displaystyle\lim_{n\rightarrow\infty}\displaystyle\sup_{\nu\in \mathfrak{A}}\|(I- R_n)\nu\|.
\end{equation}\end{footnotesize}
reciprocally, suppose that $\varepsilon>0$ and $\{z_1,z_2,\ldots, z_j\}$ be a $[\chi(\mathfrak{A}) + \varepsilon]$-net of $\mathfrak{A}.$ So
\begin{footnotesize}$$ \mathfrak{A} \subset \{z_1,z_2,\ldots, z_j\}+ [\chi(\mathfrak{A}) + \varepsilon]D(h),$$\end{footnotesize}
where $D(h)$ is unit ball of $h$.
So
\begin{footnotesize}$$\displaystyle\sup_{\nu\in \mathfrak{A}}\|(I- R_n)\nu\|\leq \displaystyle\sup_{1\leq \imath\leq j}\|(I- R_n)z_\imath\|+[\chi(\mathfrak{A})+ \varepsilon],$$\end{footnotesize}
then
\begin{footnotesize}\begin{equation}\label{uytg}
\displaystyle\lim_{n\rightarrow\infty}\displaystyle\sup_{\nu\in \mathfrak{A}}\|(I- R_n)\nu\|\leq\chi (\mathfrak{A})+\varepsilon.
\end{equation}\end{footnotesize}
Since $\varepsilon$ is arbitrary, by (\ref{uyt}) and (\ref{uytg}), relation (\ref{koko}) holds.
\end{proof}
\section{Application}
Now, we study the solvability of  Equation  (\ref{mea1}) in the Hahn sequence space. We give one example to show the performance of main results.
%\begin{equation}\label{mea1}
%\begin{cases}
%^CD_{0_+}^{\alpha}x(t)=f(t,x(t)),\ t\in[0,1],\ 0<\alpha\leq 1,\ 0<\vartheta<1,\ \gamma,\xi\in\mathbb{R}&\quad \\
%x(0)=\gamma I_{0_+}^\xi x(\vartheta)\vartheta=\gamma \int_{0}^{\vartheta}\frac{(\vartheta-s)^{\xi-1}}{\Gamma(\xi)}x(s)ds,
%\end{cases}\end{equation}
%is obtained as
%\begin{footnotesize}$$x(t)=\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}f(s,x(s))ds+
%\frac{\gamma(\Gamma(\xi+1)}{\Gamma(\xi+1)-\gamma \vartheta^\xi}\int_{0}^{\vartheta}\frac{(\vartheta-s)^{\xi+\alpha-1}}{\Gamma(\xi+\alpha)}f(s,x(s))ds$$\end{footnotesize}
\par Consider:
\\$(a)$ Let $f_\imath\in C(I\times \mathbb{R^\infty},\mathbb{R}),$ ($\imath\in\mathbb{N}$) are functions. The function $f:I\times h\rightarrow h$ is defined by
$$(z,\nu)\rightarrow (f\nu)(\Im)=(f_1(\Im,\nu(\Im)),f_2(\Im,\nu(\Im)),f_3(\Im,\nu(\Im)),\ldots),$$  so that the family of functions $((f\nu)(\Im))_{\Im\in I}$ is equicontinuous, where $I=[0,1].$
\\$(b)$ The following inequalities hold:
$$|f_\imath(\Im,\nu(\Im))|\leq|a_\imath(\Im)| |\nu_\imath(\Im)|,$$
$$|\Delta f_\imath(\Im,\nu(\Im))|\leq|a_\imath(\Im)| |\Delta \nu_\imath(\Im)|,$$
where $a_\imath:I\rightarrow \mathbb{R}$ are continuous and $(a_\imath(\Im))_{\imath\in\mathbb{N}}$ is equibounded.
\\Put
\begin{footnotesize}$$A=\sup_{\imath\in \mathbb{N}}\sup_{\Im\in I}|a_\imath(\Im)|,$$\end{footnotesize}
\begin{theorem}\label{teo}
By having the hypotheses $(a)$, $(b)$ and \begin{footnotesize}$A\Big(\frac{1}{\alpha\Gamma(\alpha)}+\frac{\gamma(\Gamma(\xi+1))\vartheta^{\xi+\alpha}}{(\Gamma(\xi+1)-\gamma \vartheta^\xi)\xi+\alpha}\Big)<1$\end{footnotesize}. The  Equation  (\ref{mea1}) admits at least one solution $\nu=(\nu_k)\in C(I,h)$ for each $\wp\in I$.
\end{theorem}
\begin{proof}
Define the operator $F:C(I,h)\rightarrow C(I,h)$ as
\begin{footnotesize}$$(F\nu)(\wp)=\frac{1}{\Gamma(\alpha)}\int_{0}^{\wp}(\wp-\Im)^{\alpha-1}f_\imath(\Im,\nu(\Im))d\Im+
\frac{\gamma(\Gamma(\xi+1))}{\Gamma(\xi+1)-\gamma \vartheta^\xi}\int_{0}^{\vartheta}\frac{(\vartheta-\Im)^{\xi+\alpha-1}}{\Gamma(\xi+\alpha)}f_\imath(\Im,\nu(\Im))d\Im.$$\end{footnotesize}
Also, $C(I,h)$ is equipped by norm
\begin{footnotesize}$$\|\nu\|_{c(I,h)}=\sup_{\wp\in I}\|\nu(\wp)\|_h.$$\end{footnotesize}
By, using our assumptions, we get\vspace{.2cm}\\ \begin{footnotesize}
$\|(F\nu)(\wp)\|_h$
\begin{eqnarray*}
&=&\displaystyle{\sum_{k=1}^{\infty}}k|\Delta(\frac{1}{\Gamma(\alpha)}\int_{0}^{\wp}(\wp-\Im)^{\alpha-1}f_k(\Im,\nu(\Im))d\Im+
\frac{\gamma(\Gamma(\xi+1))}{\Gamma(\xi+1)-\gamma \vartheta^\xi}\int_{0}^{\vartheta}\frac{(\vartheta-\Im)^{\xi+\alpha-1}}{\Gamma(\xi+\alpha)}f_k(\Im,\nu(\Im))d\Im)|
\\&&+\displaystyle{\sup_{k}}|\frac{1}{\Gamma(\alpha)}\int_{0}^{\wp}(\wp-\Im)^{\alpha-1}f_k(\Im,\nu(\Im))d\Im+
\frac{\gamma(\Gamma(\xi+1))}{\Gamma(\xi+1)-\gamma \vartheta^\xi}\int_{0}^{\vartheta}\frac{(\vartheta-\Im)^{\xi+\alpha-1}}{\Gamma(\xi+\alpha)}f_k(\Im,\nu(\Im))d\Im|
\\&\leq&\displaystyle{\sum_{k=1}^{\infty}}k|a_k(\Im)||\Delta \nu_k(\Im)|\Big(\frac{1}{\Gamma(\alpha)}\int_{0}^{\wp}(\wp-\Im)^{\alpha-1}d\Im
+\frac{\gamma(\Gamma(\xi+1))}{\Gamma(\xi+1)-\gamma \vartheta^\xi}\int_{0}^{\vartheta}\frac{(\vartheta-\Im)^{\xi+\alpha-1}}{\Gamma(\xi+\alpha)}d\Im\Big)
\\&&+\displaystyle{\sup_{k}}|a_k(\Im)||\nu_k(\Im)|\Big(\frac{1}{\Gamma(\alpha)}\int_{0}^{\wp}(\wp-\Im)^{\alpha-1}d\Im
+\frac{\gamma(\Gamma(\xi+1))}{\Gamma(\xi+1)-\gamma \vartheta^\xi}\int_{0}^{\vartheta}\frac{(\vartheta-\Im)^{\xi+\alpha-1}}{\Gamma(\xi+\alpha)}d\Im\Big)
\\&\leq& A\Big(\frac{\wp^\alpha}{\alpha\Gamma(\alpha)}+\frac{\gamma(\Gamma(\xi+1))\vartheta^{\xi+\alpha}}{(\Gamma(\xi+1)-\gamma \vartheta^\xi)\xi+\alpha}\Big)\displaystyle{\sum_{k=1}^{\infty}}k|\Delta \nu_k(\Im)|+\displaystyle{\sup_{k}}|\nu_k(\Im)|.
\end{eqnarray*}\end{footnotesize}
By supremum on $\wp$ in $[0, 1]$, we get
\begin{footnotesize}$$\|F\nu\|_{C(I,h)}\leq A\Big(\frac{1}{\alpha\Gamma(\alpha)}+\frac{\gamma(\Gamma(\xi+1))\vartheta^{\xi+\alpha}}{(\Gamma(\xi+1)-\gamma \vartheta^\xi)\xi+\alpha}\Big)\|\nu\|_{C(I,h)}.$$\end{footnotesize}
The above inequality can be written as
\begin{footnotesize}\begin{equation}\label{well}
\sigma\leq A\Big(\frac{1}{\alpha\Gamma(\alpha)}+\frac{\gamma(\Gamma(\xi+1))\vartheta^{\xi+\alpha}}{(\Gamma(\xi+1)-\gamma \vartheta^\xi)\xi+\alpha}\Big)\sigma.
\end{equation} \end{footnotesize}
Let $\sigma_0$ be optimal solution of (\ref{well}). Take
\begin{footnotesize}$$D=D(\nu^0,\sigma_0)=\{\nu=(\nu_\imath)\in C(I,h):\ \|\nu\|_{C(I,h)}\leq \sigma_0\}.$$\end{footnotesize}
Clearly, $\overline{D}=D$ is convex, bounded and $F$ is bounded on $D$.
\par Let $y\in D$ and $\varepsilon> 0$. By applying $(a)$, $\exists$ $0<\delta$ so that if $\nu\in D$ and\begin{footnotesize} $\|\nu-y\|_{C(I,h)}\leq\delta$\end{footnotesize} then\begin{footnotesize} $\|(f\nu)-
(fy)\|_{C(I,h)}\leq\frac{\varepsilon}{A\Big(\frac{1}{\alpha\Gamma(\alpha)}+\frac{\gamma(\Gamma(\xi+1))\vartheta^{\xi+\alpha}}{(\Gamma(\xi+1)-\gamma \vartheta^\xi)\xi+\alpha}\Big)}$.\end{footnotesize} Hence, for each $\wp$ in $[0, 1]$, we have\vspace{.2cm}\\ \begin{footnotesize}
$\|(F\nu)(\wp)-(Fy)(\wp)\|_h$
\begin{eqnarray*}
&=&\displaystyle{\sum_{k=1}^{\infty}}k|(\frac{1}{\Gamma(\alpha)}\int_{0}^{\wp}(\wp-\Im)^{\alpha-1}\Delta(f_k(\Im,\nu(\Im))-f_k(\Im,y(\Im)))d\Im
\\&&+\frac{\gamma(\Gamma(\xi+1))}{\Gamma(\xi+1)-\gamma \vartheta^\xi}\int_{0}^{\vartheta}\frac{(\vartheta-\Im)^{\xi+\alpha-1}}{\Gamma(\xi+\alpha)}\Delta(f_k(\Im,\nu(\Im))-f_k(\Im,y(\Im)))d\Im)|
\\&&+\displaystyle{\sup_{k}}|\frac{1}{\Gamma(\alpha)}\int_{0}^{\wp}(\wp-\Im)^{\alpha-1}(f_k(\Im,\nu(\Im))-f_k(\Im,y(\Im)))d\Im
\\&&+\frac{\gamma(\Gamma(\xi+1))}{\Gamma(\xi+1)-\gamma \vartheta^\xi}\int_{0}^{\vartheta}\frac{(\vartheta-\Im)^{\xi+\alpha-1}}{\Gamma(\xi+\alpha)}(f_k(\Im,\nu(\Im))-f_k(\Im,y(\Im)))d\Im|
\\&\leq&\Big(\frac{1}{\alpha\Gamma(\alpha)}+\frac{\gamma(\Gamma(\xi+1))\vartheta^{\xi+\alpha}}{(\Gamma(\xi+1)-\gamma \vartheta^\xi)\xi+\alpha}\Big)\displaystyle{\sup_{\wp\in I}}\bigg(\displaystyle{\sum_{k=1}^{\infty}}k|\Delta(f_k(\wp,\nu(\wp))-f_k(\wp,y(\wp)))|+\displaystyle{\sup_{k}}|(f_k(\wp,\nu(\wp))-f_k(\wp,y(\wp)))|\bigg)
\\&=&\Big(\frac{1}{\alpha\Gamma(\alpha)}+\frac{\gamma(\Gamma(\xi+1))\vartheta^{\xi+\alpha}}{(\Gamma(\xi+1)-\gamma \vartheta^\xi)\xi+\alpha}\Big)\|(f\nu)-(fy)\|_{C(I,h)}\leq\varepsilon.
\end{eqnarray*}\end{footnotesize}
Then
\begin{footnotesize}$$\|(F\nu)-(Fy)\|_{C(I,h)}\leq\varepsilon.$$\end{footnotesize}
So, $F$ is continuous.
\par Now, we prove that $(F\nu)$ is continues in $(0, 1)$. Let $\wp_1\in(0, 1)$,  $\wp>\wp_1$ and $\varepsilon>0$,
so that if $|\wp-\wp_1| <\varepsilon$, then, we can write\vspace{.2cm}\\ \begin{footnotesize}
$\|(F\nu)(\wp)-(F\nu)(\wp_1)\|_h$
\begin{eqnarray*}
&\leq&\displaystyle{\sum_{k=1}^{\infty}}k|\frac{1}{\Gamma(\alpha)}\Delta\Big(\int_{0}^{\wp}(\wp-\Im)^{\alpha-1}f_k(\Im,\nu(\Im))d\Im-\int_{0}^{\wp_1}(\wp_1-\Im)^{\alpha-1}f_k(\Im,\nu(\Im))d\Im\Big)|
\\&&+\displaystyle{\sup_{k}}|\frac{1}{\Gamma(\alpha)}\Big(\int_{0}^{\wp}(\wp-\Im)^{\alpha-1}f_k(\Im,\nu(\Im))d\Im-\int_{0}^{\wp_1}(\wp_1-\Im)^{\alpha-1}f_k(\Im,\nu(\Im))d\Im\Big)|
\\&\leq&\frac{1}{\Gamma(\alpha)}\displaystyle{\sum_{k=1}^{\infty}}k|a_k(\Im)||\Delta \nu_k(\Im)||\Big(\int_{0}^{\wp}(\wp-\Im)^{\alpha-1}d\Im-\int_{0}^{\wp_1}(\wp_1-\Im)^{\alpha-1}d\Im\Big)|
\\&&+\frac{1}{\Gamma(\alpha)}\displaystyle{\sup_{k}}|a_k(\Im)||\nu_k(\Im)||\Big(\int_{0}^{\wp}(\wp-\Im)^{\alpha-1}d\Im-\int_{0}^{\wp_1}(\wp_1-\Im)^{\alpha-1}d\Im\Big)|
\\&\leq&\frac{A}{\Gamma(\alpha)}\Big(\displaystyle{\sum_{k=1}^{\infty}}k|\Delta \nu_k(\Im)|+\displaystyle{\sup_{k}}|\nu_k(\Im)|\Big)\big(\frac{\wp_{1}^\alpha}{\alpha}-\frac{\wp^\alpha}{\alpha}\big),
\end{eqnarray*}\end{footnotesize}
since $\wp>\wp_1$ and $0\leq\alpha<1$  we have $\frac{\wp_{1}^\alpha}{\alpha}-\frac{\wp^\alpha}{\alpha}\leq0$.
This proves that $(F\nu)$ is continues on $(0, 1)$.
\par Finally, we show that $F$ satisfies in Theorem \ref{meiir}. By Proposition \ref{bban} and (\ref{koko}), Hausdorff MNC for $D\subset C(I,h)$ is defined by
\begin{footnotesize}$$\chi_{C(I,h}(D)=\displaystyle\sup_{\wp\in I}\chi_{h}(D(\wp)),$$\end{footnotesize}
where \begin{footnotesize}$D(\wp)=\{\nu(\wp):\ \nu\in D\}$.\end{footnotesize} Therefore, we get\vspace{.2cm}\\
\begin{footnotesize}$\chi_{h}(FD)(\wp)$
\begin{eqnarray*}
&=&\displaystyle\lim_{n\rightarrow\infty}\{\displaystyle\sup_{\nu\in D}\Bigg(
\displaystyle{\sum_{k\geq n}}k|\Delta(\frac{1}{\Gamma(\alpha)}\int_{0}^{\wp}(\wp-\Im)^{\alpha-1}f_k(\Im,\nu(\Im))d\Im+
\frac{\gamma(\Gamma(\xi+1))}{\Gamma(\xi+1)-\gamma \vartheta^\xi}\int_{0}^{\vartheta}\frac{(\vartheta-\Im)^{\xi+\alpha-1}}{\Gamma(\xi+\alpha)}f_k(\Im,\nu(\Im))d\Im)|
\\&&+\displaystyle{\sup_{k}}|\frac{1}{\Gamma(\alpha)}\int_{0}^{\wp}(\wp-\Im)^{\alpha-1}f_k(\Im,\nu(\Im))d\Im+
\frac{\gamma(\Gamma(\xi+1))}{\Gamma(\xi+1)-\gamma \vartheta^\xi}\int_{0}^{\vartheta}\frac{(\vartheta-\Im)^{\xi+\alpha-1}}{\Gamma(\xi+\alpha)}f_k(\Im,\nu(\Im))d\Im|\Bigg)
\\&\leq&\displaystyle\lim_{n\rightarrow\infty}\{\displaystyle\sup_{\nu\in D}\Bigg(
\displaystyle{\sum_{k\geq n}}k|a_k(\Im)||\Delta \nu_k(\Im)|\Big(\frac{1}{\Gamma(\alpha)}\int_{0}^{\wp}(\wp-\Im)^{\alpha-1}d\Im
+\frac{\gamma(\Gamma(\xi+1))}{\Gamma(\xi+1)-\gamma \vartheta^\xi}\int_{0}^{\vartheta}\frac{(\vartheta-\Im)^{\xi+\alpha-1}}{\Gamma(\xi+\alpha)}d\Im\Big)
\end{eqnarray*}\begin{eqnarray*}&&+\displaystyle{\sup_{k}}|a_k(\Im)||\nu_k(\Im)|\Big(\frac{1}{\Gamma(\alpha)}\int_{0}^{\wp}(\wp-\Im)^{\alpha-1}d\Im
+\frac{\gamma(\Gamma(\xi+1))}{\Gamma(\xi+1)-\gamma \vartheta^\xi}\int_{0}^{\vartheta}\frac{(\vartheta-\Im)^{\xi+\alpha-1}}{\Gamma(\xi+\alpha)}d\Im\Big)\Bigg)
\\&\leq& A\Big(\frac{\wp^\alpha}{\alpha\Gamma(\alpha)}+\frac{\gamma(\Gamma(\xi+1))\vartheta^{\xi+\alpha}}{(\Gamma(\xi+1)-\gamma \vartheta^\xi)\xi+\alpha}\Big)\displaystyle\lim_{n\rightarrow\infty}\{\displaystyle\sup_{\nu\in D}\Bigg(
\displaystyle{\sum_{k\geq n}}k|\Delta \nu_k(\Im)|+\displaystyle{\sup_{k}}|\nu_k(\Im)|\Bigg).
\end{eqnarray*}\end{footnotesize}
Then, we have
\begin{footnotesize}$$\displaystyle{\sup_{\wp\in I}}\chi_{h}(FD)(\wp)\leq A\Big(\frac{1}{\alpha\Gamma(\alpha)}+\frac{\gamma(\Gamma(\xi+1))\vartheta^{\xi+\alpha}}{(\Gamma(\xi+1)-\gamma \vartheta^\xi)\xi+\alpha}\Big)\chi_{C(I,h)}(D).$$\end{footnotesize}
This implies that
\begin{footnotesize}\begin{equation}\label{meir}
\chi_{C(I,h)}(FD)<A\Big(\frac{1}{\alpha\Gamma(\alpha)}+\frac{\gamma(\Gamma(\xi+1))\vartheta^{\xi+\alpha}}{(\Gamma(\xi+1)-\gamma \vartheta^\xi)\xi+\alpha}\Big)\chi_{C(I,h)}(D)<\varepsilon.
\end{equation} \end{footnotesize}
Then\begin{footnotesize}
$$\chi_{C(I,h)}(D)<\frac{1}{A\Big(\frac{1}{\alpha\Gamma(\alpha)}+\frac{\gamma(\Gamma(\xi+1))\vartheta^{\xi+\alpha}}{(\Gamma(\xi+1)-\gamma \vartheta^\xi)\xi+\alpha}\Big)}\varepsilon.$$\end{footnotesize}
Let us choose \begin{footnotesize}$\delta=\varepsilon(\frac{1}{A\Big(\frac{1}{\alpha\Gamma(\alpha)}+\frac{\gamma(\Gamma(\xi+1))\vartheta^{\xi+\alpha}}{(\Gamma(\xi+1)-\gamma \vartheta^\xi)\xi+\alpha}\Big)}-1)$.\end{footnotesize} So, $F$ is a Meir--Keeler condensing operator on $D\subset h.$
By using Theorem \ref{meiir}, $F$ has a fixed point in $D$, thus the  Equation (\ref{mea1}) has at least one solution in $C(I,h)$.
\end{proof}
\begin{example}
Consider the  Equation
\begin{footnotesize}\begin{equation}\label{example}
\begin{cases}
^CD_{0_+}^{\alpha}\nu(\wp)=\displaystyle{\sum_{j=\imath}^{\infty}}\frac{2j+1}{(j^2+1)(j^2+2j+2)}e^{-3\Im}\sin(\Im+e^\Im)\nu_j(\Im),\ \wp\in[0,1],\ \alpha=1,\ \vartheta=\frac{1}{3},\ \gamma=\frac{1}{4},\xi=\frac{3}{2}&\quad \\
\nu(0)=\gamma I_{0_+}^\xi \nu(\frac{1}{3})=\gamma \int_{0}^{\frac{1}{3}}\frac{(\frac{1}{3}-\Im)^{\frac{3}{2}-1}}{\Gamma(\frac{3}{2})}\nu(\Im)d\Im,
\end{cases}\end{equation}\end{footnotesize}
where $\alpha=1$, $\xi=\frac{3}{2}$, $\gamma=\frac{1}{4}$ and $\vartheta=\frac{1}{3}$.
\\ Take \begin{footnotesize}$f_i(\wp,\nu(\wp))=\displaystyle{\sum_{j=\imath}^{\infty}}\frac{2j+1}{(j^2+1)(j^2+2j+2)}e^{-3\Im}\sin(\Im+e^\Im)\nu_j(\Im)$.\end{footnotesize} Therefore, (\ref{example}) is a special case of (\ref{mea1}). Clearly,
\begin{footnotesize}$\displaystyle{\sum_{j=\imath}^{\infty}}\frac{2j+1}{(j^2+1)(j^2+2j+2)}e^{-3\Im}\sin(\Im+e^\Im)\nu_j(\Im)$ ($\imath\in \mathbb{N}$)\end{footnotesize}
is continuous on $I=[0,1]$. Notice that, for any $\wp\in I$, if $\nu(\wp)=(\nu_\imath(\wp))\in h,$ then $(f_\imath(\Im,\nu(\Im)))\in h$. Let $\varepsilon> 0$ and $\nu(\wp)=(\nu_\imath(\wp))\in
h$. So, by taking $y(\wp)=(y_\imath(\wp))\in  h$ with
$\|\nu(\wp)-y(\wp)\|_{h}\leq\delta(\varepsilon):=2\varepsilon$, we have
\begin{footnotesize} $$\|f(\wp,\nu(\wp))-f(\wp,y(\wp))\|_{h}\leq\frac{1}{2}\|\nu(\wp)-y(\wp)\|_{h}=\varepsilon,$$\end{footnotesize}
which implies that condition $(a)$ holds. Now, for condition $(b)$, we have
\begin{footnotesize}\begin{eqnarray*}|f_\imath(\Im,\nu(\Im))|&\leq&|\displaystyle{\sum_{j=\imath}^{\infty}}\frac{2j+1}{(j^2+1)(j^2+2j+2)}e^{-3\Im}\sin(\Im+e^\Im)\nu_j(\Im)\\
&\leq&|a_i(\Im)||\nu_i(\Im)|.
 \end{eqnarray*}\end{footnotesize}
and
\begin{footnotesize}\begin{eqnarray*}|\Delta f_\imath(\Im,\nu(\Im))|&\leq&|\Delta\displaystyle{\sum_{j=\imath}^{\infty}}\frac{2j+1}{(j^2+1)(j^2+2j+2)}e^{-3\Im}\sin(\Im+e^\Im)\nu_j(\Im)\\
  &\leq&|a_i(\Im)||\Delta \nu_\imath(\Im)|.
 \end{eqnarray*}\end{footnotesize}
$a_\imath(\wp)=\frac{e^{-3\wp}}{2}$ are continuous and $(a_\imath(\wp))_{\imath\in\mathbb{N}}$ is equibounded, by $A\leq\frac{1}{2}$
and \begin{footnotesize}$$A\Big(\frac{1}{\alpha\Gamma(\alpha)}+\frac{\gamma(\Gamma(\xi+1))\vartheta^{\xi+\alpha}}{(\Gamma(\xi+1)-\gamma \vartheta^\xi)\xi+\alpha}\Big)=0.508<1.$$\end{footnotesize}
Then Theorem \ref{teo} grantees  that   Equation (\ref{example}) has at least one solution in $C([0,1],h)$.
\end{example}

\vspace{4mm}\noindent{\bf Acknowledgements}\\
\noindent The authors appreciate anonymous reviewers for their remarks to cultivate the study.


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{\small

\noindent{\bf Masoomeh Hosseini Farahi }

\noindent Department of Mathematics

\noindent Assistant Professor of Mathematics

\noindent  Mashhad Branch, Islamic Azad
University, Mashhad, Iran


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

{\small
\noindent{\bf  Reza Allahyari }

\noindent  Department of Mathematics

\noindent Associate Professor of Mathematics

\noindent Mashhad Branch, Islamic Azad
University, Mashhad, Iran

\noindent E-mail:  rezaallahyari@mshdiau.ac.ir}\\

{\small
\noindent{\bf  Mahmoud Hassani }

\noindent  Department of Mathematics

\noindent Associate Professor of Mathematics

\noindent Mashhad Branch, Islamic Azad
University, Mashhad, Iran

\noindent E-mail:   hassani@mshdiau.ac.ir}\\
\end{document}