\documentclass[11pt,twoside]{article} \usepackage{amsmath, amsthm, amscd, amsfonts, amssymb, graphicx, color} \usepackage[bookmarksnumbered, colorlinks]{hyperref} \usepackage{float} \usepackage{lipsum} \usepackage{afterpage} \usepackage[labelfont=bf]{caption} \usepackage[nottoc,notlof,notlot]{tocbibind} %\renewcommand\bibname{References} \def\bibname{\Large \bf References} \usepackage{lipsum} \usepackage{fancyhdr} \pagestyle{fancy} \fancyhf{} \renewcommand{\headrulewidth}{0pt} \fancyhead[LE,RO]{\thepage} \thispagestyle{empty} %\afterpage{\lhead{new value}} \fancyhead[CE]{M. HOSSEINI FARAHI, R. ALLAHYARI AND M. HASSANI} \fancyhead[CO]{SOLVABILITY OF INFINITE SYSTEMS OF FRACTIONAL EQUATION IN THE HAHN SEQUENCE SPACE} %\topmargin=-1.6cm \textheight 17.5cm% \textwidth 12cm % \topmargin 8mm % \oddsidemargin 20mm % \evensidemargin 20mm % \footskip=24pt % \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{xca}[theorem]{Exercise} %\theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \renewenvironment{proof}{{\bfseries \noindent Proof.}}{~~~~$\square$} \makeatletter \def\th@newremark{\th@remark\thm@headfont{\bfseries}} \makeatletter %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % If you want to insert other packages. Insert them here %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\long\def\symbolfootnote[#1]#2{\begingroup% %\def\thefootnote{\fnsymbol{footnote}}\footnote[#1]{#2}\endgroup} \def \thesection{\arabic{section}} \begin{document} %\baselineskip 9mm %\setcounter{page}{} \thispagestyle{plain} {\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 $\theta(t)0$ 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) 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. % BibTeX users please use one of %\bibliographystyle{spbasic} % basic style, author-year citations %\bibliographystyle{spmpsci} % mathematics and physical sciences %\bibliographystyle{spphys} % APS-like style for physics %\bibliography{} % name your BibTeX data base % Non-BibTeX users please use \begin{center} \begin{thebibliography}{99} % Enter references in alphabetical order and according to the following format. \bibitem{1aga.banas} Aghajani, A., Bana\'{s}, J. and Jalilian, Y., Existence of solution for a class of nonlinear Volterra singular integral equation, Comput. Math. Appl. \textbf{62} (2011) 1215--1227. \bibitem{Alotaibi} Alotaibi, A., Mursaleen, M. and Alamri, B.A.S., Solvability of second order linear differential equations in the sequence space $n(\phi)$, Adv. Differ. Equ. \textbf{2018:377} (2018) 8 pages. \bibitem{shole} Aghajani, A., Mursaleen, M. and Shole Haghighi, A., Fixed point theorems for Meir-Keeler condensing operators via measure of noncompactness, Acta Math. Sci. \textbf{35B} (2015) 552--566. \bibitem{Pourhadi} Aghajani, A. and Pourhadi, E., Application of measure of noncompactness to $l_1$-solvability of infinite systems of second order differential equations. Bull. Belg. Math. Soc. Simon Stevin \textbf{22}(1) (2015) 105--118. %\bibitem{agha.sabz} A. Aghajani, and N. Sabzali, Existence of coupled fixed points via measure of noncompactness and applications, \textit{J. Nonlinear Convex Anal.} 15(5) (2014) 941--952. % \bibitem{Akhm 7} R. R. Akhmerov, M. I. Kamenski, A. S. Potapov, A. E. Rodkina and B. N. Sadovskii, \textit{Measures Of Noncompactness And Condensing Operators,} Birkhauser Verlag. Basel 1992. %\bibitem{Amiri1} Amiri Kayvanloo, H., Khanehgir, M., and Allahyari, R., A generalization of Darbo's theorem with application to the solvability of systems of integral-differential equations in Sobolev spaces. International Journal of Nonlinear Analysis and Applications, (2021) 12(1), 287-300. \bibitem{bban} Bana\'{s}, J. and Goebel, K., \newblock Measures of noncompactness in Banach spaces, Lecture notes in pure and applied mathematics, \textbf{60} Marcel Dekker, New York, 1980. %\bibitem{ban1} Bana\'{s}, J. and Lecko, M., \newblock Solvability of infinite systems of differential equations in Banach sequence spaces, J. Comput. %Appl. Math. \textbf{137} (2001) 363--375. \bibitem{ban2} Bana\'{s}, J. and Mursaleen, M., \newblock Sequence spaces and measure of noncompactness with applications to differential and integral equation, Springer, India 2014. \bibitem{ban.murs} Bana\'{s}, J., Mursaleen, M. and Rizvi, S.M.H., Existence of solutions to a boundary-value problem for an infinite system of differential equations. Electron. J. Differ. Equ. 2017, 262 (2017) 12 pages. %\bibitem {Rao} W. Chandrasekhara Rao, The Hahn sequence spaces I, Bull. Calcutta Math. Soc. 82(1990), 72–78. %\bibitem{chang 10} S. S. Chang, Y. J. Cho and N. J. Huang, Coupled fixed point theorems with application, \textit{J. Korean %Math. Soc}. 33 (1996) 575--585. %\bibitem{Goes} G. Goes and S., Goes, Sequences of bounded variation and sequences of Fourier coefficients I, Math. Z., 118(1970), 93–102. \bibitem{Hahn} H. Hahn, Über Folgen linearer Operationen, Monatsh. Math., 32(1922), 3–88. \bibitem{Hazarika} Hazarika, B., Das, A., Arab, R. and Mursaleen, M., Solvability of the infinite system of integral equations in two variables in the sequence spaces $c_0$ and $l_1$. J. Comput. Appl. Math. \textbf{326} (2012) 183--192. %\bibitem{Meir} Meir, A. and Keeler E.A., Theorem on contraction mappings. J. Math. Anal. Appl. \textbf{28} (1969) 326--329. \bibitem{Moh.sir} Mohiuddine, S.A., Srivastava, H.M. and Alotaibi, A., Application of measures of noncompactness to the infinite system of second-order differential equations in $l_p$ spaces. Adv. Differ. Equ. \textbf{2016:317} (2016) 13 pages. \bibitem{Mursaleencan} Mursaleen, M., Application of measure of noncompactness to infinite system of differential equations. Can. Math. Bull. \textbf{56} (2013) 388--394. \bibitem{Mursaleen} Mursaleen, M., Some geometric properties of a sequence space related to $l_p,$ Bull. Aust. Math. Soc. \textbf{67}, 2 (2003) 343--347. \bibitem{Mursal.bil} Mursaleen, M., Bilalov, B. and Rizvi, S.M.H., Applications of measures of noncompactness to infinite system of fractional differential equations. Filomat \textbf{31} (11) (2017) 3421--3432. \bibitem{Mursal.Mohi} Mursaleen, M. and Mohiuddine, S.A., Applications of measures of noncompactness to the infinite system of differential equations in $l_p$ spaces, Nonlinear Anal. \textbf{75} (2012) 2111--2115. \bibitem{Mursal.riz} Mursaleen, M. and Rizvi, S.M.H., Solvability of infinite system of second order differential equations in $c_0$ and $l_1$ by Meir–Keeler condensing operator. Proc. Am. Math. Soc. \textbf{144} (10) (2016) 4279--4289. \bibitem{Ntouyas} Ntouyas, S.K., Obaid, M.: A coupled system of fractional differential equations with nonlocal integral boundary conditions. Adv. Diff. Equ. 2012(1), 1–8 (2012 \bibitem{Pod} I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of Their Applications, Elsevier, vol. 198, 1998. %\bibitem{Sargent} Sargent, W.L.C., Some sequence spaces related to the $l^p$ spaces, J. London Math. %Soc. \textbf{35} (1960) 161--171. %\bibitem{vakeel} Vakeel A. Khan, A new type of difference sequence spaces, Applied Sci. \textbf{12} (2010) 102--108. \end{thebibliography} \end{center} {\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}