\documentclass[10pt]{studiamnew} \usepackage{graphicx} \usepackage{amsmath} \usepackage{amssymb} \sloppy \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \theoremstyle{definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{problem}[theorem]{Problem} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{solution}[theorem]{Solution} \newtheorem{notation}[theorem]{Notation} \renewcommand{\theequation}{\thesection.\arabic{equation}} \numberwithin{equation}{section} \begin{document} % \setcounter{page}{1} \setcounter{firstpage}{1} \setcounter{lastpage}{17} \renewcommand{\currentvolume}{??} \renewcommand{\currentyear}{??} \renewcommand{\currentissue}{??} % \title[On Hilfer Fractional BVP with nonlocal boundary conditions]{Existence of solution for Hilfer fractional differential problem with nonlocal boundary condition in Banach spaces} \author[H.A. Wahash$^*$]{Hanan A. Wahash} \address{``Department of Mathematics'',\\ Dr.Babasaheb Ambedkar Marathwada University,\\ Aurangabad - 431001, (M.S) India} \email{hawahash86@gmail.com\footnote{Corresponding author}} % \author[M.S. Abdo]{Mohammed S. Abdo} \address{``Department of Mathematics'',\\ $^1$Dr.Babasaheb Ambedkar Marathwada University,\\ Aurangabad - 431001, (M.S) India, \\ $^2$ Hodeidah University, Al-Hodeidah 31141, Yemen} \email{msabdo1977@gmail.com} % \author[S.K. Panchal]{Satish K. Panchal} \address{``Department of Mathematics'',\\ Dr.Babasaheb Ambedkar Marathwada University,\\ Aurangabad - 431001, (M.S) India} \email{drpanchalskk@gmail.com} % \author[S.P. Bhairat]{Sandeep P. Bhairat} \address{``Faculty of Engineering Mathematics'',\\ Institute of Chemical Technology Mumbai,\\ Marathwada Campus, Jalna - 431 203 (M.S) India.} \email{sp.bhairat@marj.ictmumbai.edu.in} % \subjclass{34A08, 26A33, 34A12, 34A40} \keywords{fractional differential equations, Hilfer fractional derivatives, Existence, Fixed point theorem} \begin{abstract} This paper is devoted to study the existence of a solution to Hilfer fractional differential equation with nonlocal boundary condition in Banach spaces. We use the equivalent integral equation to study the considered Hilfer differential problem with nonlocal boundary condition. The M\"{o}nch type fixed point theorem and the measure of the noncompactness technique are the main tools in this study. We demonstrate the existence of a solution with a suitable illustrative example. \end{abstract} \maketitle \section{Introduction} The calculus of arbitrary order has been extensively studied in the last four decades. It has been proved to be an adequate tool in almost all branches of science and engineering. Because of its widespread applications, fractional calculus is becoming an integral part of applied mathematics research. Indeed, fractional differential equations have been found useful to describe abundant phenomena in physics and engineering, and the modest amount of work in this direction has taken place, see \cite{ABLZ,AG,KD} and references therein. For basic development and theoretical applications of fractional differential equations, see \cite{HI,KL1}. In the past two decades, the fractional differential equations are extensively studied for existence, uniqueness, continuous dependence and stability of the solution. For some fundamental results in existence theory of various fractional differential problems with initial and boundary conditions, see survey papers \cite{ABLZ,AG}, the monograph \cite{KL1}, the research papers \cite{AP1,AP2,SPN,SP5,DBN,DB2,KD,FK,KM,FMG,HLT,VE,WZ} and references therein. Recently, in \cite{WZ}, Wang and Zhang obtained some existence of the solutions of IVP for the class of Hilfer FDEs: \begin{equation} D_{0^{+}}^{\mu ,\nu }z(t)=f(t,z(t)),\text{ \ }0<\mu <1,\,0\leq \nu \leq 1,% \text{ }t\in (a,b]\qquad \ \ \ \ \label{e8.1a} \end{equation}% \begin{equation} I_{a^{+}}^{1-\gamma }z(a^{+})=\sum_{k=1}^{m}\lambda _{k}z(\tau _{k}),\text{ }% \tau _{k}\in (a,b],\ \mu \leq \gamma =\mu +\nu (1-\mu ), \label{e8.1b} \end{equation}% by using fixed point theorems of Krasnoselskii and Schauder. In the year 2018, Thabet et al. \cite{SA} investigated the existence of a solution to BVP for Hilfer FDEs: \begin{equation} D_{a^{+}}^{\mu ,\nu }z(t)=f\left( t,z(t),Sz(t)\right) ,0<\mu <1,0\leq \nu \leq 1,\text{\qquad\ \ }t\in (a,b], \label{11} \end{equation}% \begin{equation} I_{a^{+}}^{1-\gamma }\left[ uz(a^{+})+vz(b^{-})\right] =w,\text{ }\ \mu \leq \gamma =\mu +\nu (1-\mu ),\text{ }u,v,w\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion , \label{12} \end{equation}% by using the M\"{o}nch fixed point theorem. Motivated by works cited above, in this paper, we consider the nonlocal boundary value problem for a class of Hilfer fractional differential equations (HNBVP): \begin{equation} D_{a^{+}}^{\mu ,\nu }z(t)=f(t,z(t)),\text{ \ }0<\mu <1,\,0\leq \nu \leq 1,t\in (a,b],\qquad \ \ \ \ \ \ \ \qquad \label{e8.1} \end{equation}% \begin{equation} I_{a^{+}}^{1-\gamma }cz(a^{+})+I_{a^{+}}^{1-\gamma }dz(b^{-})=\sum_{k=1}^{m}\lambda _{k}z(\tau _{k}),\tau _{k}\in (a,b],\ \mu \leq \gamma =\mu +\nu (1-\mu ), \label{e8.2} \end{equation}% where $D_{a^{+}}^{\mu ,\nu }$ is the generalized Hilfer fractional derivative of order $\mu $ and type $\nu $, $I_{a^{+}}^{1-\gamma }$ is the Riemann-Liouville fractional integral of order $1-\gamma $, $f:(a,b]\times E\rightarrow E$ be a function such that $f(t,z)\in C_{1-\gamma }([a,b],E)$ for any $z\in C_{1-\gamma }([a,b],E)$, $E$ is a Banach space, $c,d\in \mathbb{R}$, and $\tau _{k}$ $(k=1,2,...,m)$ are prefixed points satisfying $% a<\tau _{1}<\tau _{2}<...<\tau _{m}1,$ is the Banach space of measurable functions on $J_{1}$ with the $L^{p}$ norm where \begin{equation*} \left\Vert p\right\Vert _{L^{p}}=\left( \int_{a}^{b}\left\vert p(s)\right\vert ^{p}ds\right) ^{\frac{1}{p}}<\infty . \end{equation*}% Let $L^{\infty }(J_{1},E)$ be the Banach space of measurable functions $% z:J_{1}\longrightarrow E$ which are bounded and equipped with the norm $% \left\Vert z\right\Vert _{L^{\infty }}=\inf \{e>0:\left\Vert z\right\Vert \leq e,$ a.e $t\in J_{1}\}.$ Moreover, for a given set $\mathcal{V}$ of functions $v:J_{1}\longrightarrow E$ let us denote by% \begin{equation*} \mathcal{V(}t)=\{v(t):v\in \mathcal{V};t\in J_{1}\}, \end{equation*}% \begin{equation*} \mathcal{V(}J_{1})=\{v(t):v\in \mathcal{V};t\in J_{1}\}. \end{equation*} \begin{definition} \cite{KL1} Let $\mu >0.$ The left sided Riemann-Liouville fractional integral of order $\mu $ of $g\in L^{1}(J_{1},E)$ is defined by \begin{equation} I_{a^{+}}^{\mu }g(t)=\frac{1}{\Gamma (\mu )}\int_{a}^{t}(t-s)^{\mu -1}g(s)ds,\quad t>a, \label{d1} \end{equation}% where $\Gamma (\cdot )$ is the Euler's Gamma function and $a\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion .$ \end{definition} \begin{definition} \cite{KL1} Let $n-1<\mu a, \label{d2} \end{equation}% and \begin{equation*} ^{C}D_{a^{+}}^{\mu }g(t)=\frac{1}{\Gamma (n-\mu )}\int_{a}^{t}(t-s)^{n-\mu -1}g^{(n)}(s)ds,\text{ }t>a, \end{equation*}% respectively, where $n=[\mu ]+1,$ and $[\mu ]$ denotes the integer part of $% \mu .$ \end{definition} \begin{definition} \label{7} \cite{HI} The left sided Hilfer fractional derivative of function $% g\in L^{1}(J_{1},E)$ of order $0<\mu <1$ and type $0\leq \nu \leq 1$ is denoted as $D_{a^{+}}^{\mu ,\nu }$ and defined by \begin{equation} D_{a^{+}}^{\mu ,\nu }g(t)=I_{a^{+}}^{\nu (1-\mu )}DI_{a^{+}}^{(1-\nu )(1-\mu )}g(t),\text{ }D=\frac{d}{dt}. \label{d3} \end{equation}% where $I_{a^{+}}^{\mu }$ and $D_{a^{+}}^{\mu }$ are Riemann-Liouville fractional integral and derivative defined by \eqref{d1} and \eqref{d2}, respectively. \end{definition} \begin{remark} \label{rem8.a} From Definition \ref{7}, we observe that: \begin{itemize} \item[(i)] The operator $D_{a^{+}}^{\mu ,\nu }$ can be written as \begin{equation*} D_{a^{+}}^{\mu ,\nu }=I_{a^{+}}^{\nu (1-\mu )}DI_{a^{+}}^{(1-\gamma )}=I_{a^{+}}^{\nu (1-\mu )}D^{\gamma },~~~~~~~~\gamma =\mu +\nu (1-\mu )% \text{.} \end{equation*} \item[(ii)] The Hilfer fractional derivative can be regarded as an interpolator between the Riemann-Liouville derivative ($\nu =0$) and Caputo derivative ($\nu =1$) as \begin{equation*} D_{a^{+}}^{\mu ,\nu }=% \begin{cases} DI_{a^{+}}^{(1-\mu )}=~D_{a^{+}}^{\mu },~~~~~~~~~~if~\nu =0; \\ I_{a^{+}}^{(1-\mu )}D=~^{C}D_{a^{+}}^{\mu },~~~~~~~~if~\nu =1.% \end{cases}% \end{equation*} \item[(iii)] In particular, if $\gamma =\mu +\nu (1-\mu ),$ then \begin{equation*} (D_{a^{+}}^{\mu ,\nu }g)(t)=\Big(I_{a^{+}}^{\nu (1-\mu )}\Big(% D_{a^{+}}^{\gamma }g\Big)\Big)(t), \end{equation*}% where $\Big(D_{a^{+}}^{\gamma }g\Big)(t)=\frac{d}{dt}\Big(I_{a^{+}}^{(1-\nu )(1-\mu )}g\Big)(t).$ \end{itemize} \end{remark} \begin{definition} \cite{KL1} Let $0\leq \gamma <1.$\ The weighted spaces $C_{\gamma }(J_{1},E)$ and $C_{1-\gamma }^{n}(J_{1},E)$ are defined by \begin{equation*} C_{\gamma }(J_{1},E)=\{g:J_{2}\rightarrow E:(t-a)^{\gamma }g(t)\in C(J_{1},E)\}, \end{equation*}% and \begin{equation*} C_{\gamma }^{n}(J_{1},E)=\{g:J_{2}\rightarrow E,\text{ }g\in C^{n-1}(J_{1},E):g^{(n)}(t)\in C_{\gamma }(J_{1},E)\},\,n\in \mathbb{% %TCIMACRO{\U{2115} }% %BeginExpansion \mathbb{N} %EndExpansion } \end{equation*}% with the norms% \begin{equation*} {\Vert g\Vert }_{C_{\gamma }}={\Vert }(t-a)^{\gamma }{{g}\Vert }_{C}=\sup \{\left\vert (t-a)^{\gamma }{g(t)}\right\vert :t\in J_{1}\}, \end{equation*}% and \begin{equation} {\Vert g\Vert }_{C_{1-\gamma }^{n}}=\sum_{k=0}^{n-1}{\Vert g^{(k)}\Vert }% _{_{C}}+{\Vert g^{(n)}\Vert }_{C_{1-\gamma }}, \label{n1} \end{equation}% respectively. Furthermore we recall following weighted spaces \begin{equation} C_{1-\gamma }^{\mu ,\nu }(J_{1},E)=\big\{g\in {C_{1-\gamma }}% (J_{1},E):D_{a^{+}}^{\mu ,\nu }g\in {C_{1-\gamma }}(J_{1},E)\big\},\quad \gamma =\mu +\nu (1-\mu ) \label{w1} \end{equation}% and% \begin{equation*} C_{1-\gamma }^{\gamma }(J_{1},E)=\big\{g\in {C_{1-\gamma }}% (J_{1},E):D_{a^{+}}^{\gamma }g\in {C_{1-\gamma }}(J_{1},E)\big\},\quad \gamma =\mu +\nu (1-\mu ). \end{equation*}% where Let $0<\mu <1,0\leq \nu \leq 1$ and $\gamma =\mu +\nu (1-\mu )$. Clearly, $D_{a^{+}}^{\mu ,\nu }g=I_{a^{+}}^{\nu (1-\mu )}D_{a^{+}}^{\gamma }g $ and $C_{1-\gamma }^{\gamma }(J_{1},E)\subset C_{1-\gamma }^{\mu ,\nu }(J_{1},E).$ \end{definition} \begin{lemma} \label{def8.5} \cite{KD} If $\mu >0$ and $\nu >0,$ and $g\in L^{1}(J_{1},E)$ for $t\in \lbrack a,b]$, then the following properties hold: \begin{equation*} \Big(I_{a^{+}}^{\mu }I_{a^{+}}^{\nu }g\Big)(t)=\Big(I_{a^{+}}^{\mu +\nu }g% \Big)(t)\,\,\text{and }\Big(D_{a^{+}}^{\mu }I_{a^{+}}^{\nu }g\Big)(t)=g(t). \end{equation*}% In particular, if $g\in C_{\gamma }(J_{1},E)$ or $g\in C(J_{1},E)$, then the above properties hold for each $t\in J_{2}$ or $t\in J_{1}$ respectively. \end{lemma} \begin{lemma} \label{Le1}\cite{KL1} Let $\mu >0$ and $\delta >0$. Then for $t>a,$ we have \begin{description} \item[(i)] $I_{a^{+}}^{\mu }(t-a)^{\delta -1}=\frac{\Gamma (\delta )}{\Gamma (\delta +\mu )}(t-a)^{\delta +\mu -1},$\newline \item[(ii)] $D_{a^{+}}^{\mu }(t-a)^{\mu -1}=0,\quad \mu \in (0,1).$ \end{description} \end{lemma} \begin{lemma} \label{def8.8} \cite{HI} Let $\mu >0$, $\nu >0$ and $\gamma =\mu +\nu (1-\mu ).$ If $g\in C_{1-\gamma }^{\gamma }(J_{1},E)$, then\newline \begin{equation*} I_{a^{+}}^{\gamma }D_{a^{+}}^{\gamma }g=I_{a^{+}}^{\mu }D_{a^{+}}^{\mu ,\nu }g,~D_{a^{+}}^{\gamma }I_{a^{+}}^{\mu }g=D_{a^{+}}^{\nu (1-\mu )}g. \end{equation*} \end{lemma} \begin{lemma} \label{Le2} \cite{HI} Let $0<\mu <1,$ $0\leq \nu \leq 1$ and $g\in C_{1-\gamma }(J_{1},E).$ Then% \begin{equation*} I_{a^{+}}^{\mu }D_{a^{+}}^{\mu ,\nu }g(t)=g(t)-\frac{I_{a^{+}}^{(1-\nu )(1-\mu )}g(a)}{\Gamma (\mu +\nu (1-\mu ))}(t-a)^{\mu +\nu (1-\mu )-1}. \end{equation*}% Moreover, if $\ \gamma =\mu +\nu (1-\mu ),$ $g\in C_{1-\gamma }(J_{1},E)$ and $I_{a^{+}}^{1-\gamma }g\in C_{1-\gamma }^{n}(J_{1},E),$ then \begin{equation*} I_{a^{+}}^{\gamma }D_{a^{+}}^{\gamma }g(t)=g(t)-\frac{I_{a^{+}}^{1-\gamma }g(a)}{\Gamma (\gamma )}(t-a)^{\gamma -1}. \end{equation*} \end{lemma} \begin{lemma} \label{def8.7} \cite{HLT} If $0<\mu \leq \gamma <1$ and $g\in C_{\gamma }(J_{1},E)$, then \begin{equation*} (I_{a^{+}}^{\mu }g)(a)=\lim_{t\rightarrow a^{+}}I_{a^{+}}^{\mu }g(t)=0. \end{equation*} \end{lemma} \begin{lemma} \cite{MH} Let $E$ be a Banach space and let$\ \Upsilon _{E}$ be the bounded subsets of $E$. The Kuratowski measure of noncompactness is the map ${\Large % \alpha }:\Upsilon _{E}\longrightarrow \lbrack 0,\infty )$defined by% \begin{equation*} {\Large \alpha }(\mathcal{S})=\inf \{\varepsilon >0:\mathcal{S}\subset \cup _{i=1}^{m}\mathcal{S}_{i}\text{ and the diam }(\mathcal{S}_{i})\leq \varepsilon \};\mathcal{S}\subset \Upsilon _{E}. \end{equation*} \end{lemma} \begin{lemma} \cite{GD,GC} For all nonempty subsets $\mathcal{S}_{1},\mathcal{S}% _{2}\subset E$. The Kuratowski measure of noncompactness ${\Large \alpha }(% \mathcal{\cdot })$ satisfies the following properties: \end{lemma} \begin{enumerate} \item ${\Large \alpha }(\mathcal{S})=0\Longleftrightarrow \overline{\mathcal{% S}}$ is compact ($\mathcal{S}$ is relatively compact); \item ${\Large \alpha }(\mathcal{S})={\Large \alpha }(\overline{\mathcal{S}}% )={\Large \alpha }(conv\mathcal{S}),$ where where $\overline{\mathcal{S}}$ and $conv\mathcal{S}$ denote the closure and convex hull of the bounded set $% \mathcal{S}$ respectively; \item $\mathcal{S}_{1}\subset \mathcal{S}_{2}\Longrightarrow {\Large \alpha }% (\mathcal{S}_{1})\leq {\Large \alpha }(\mathcal{S}_{2});$ \item ${\Large \alpha }(\mathcal{S}_{1}+\mathcal{S}_{2})\leq {\Large \alpha }% (\mathcal{S}_{1})+{\Large \alpha }(\mathcal{S}_{2}),$ where $\mathcal{S}_{1}+% \mathcal{S}_{2}=\{s_{1}+s_{2}:s\in \mathcal{S}_{1},s\in \mathcal{S}_{2}\};$ \item ${\Large \alpha }(\kappa \mathcal{S})=\left\vert \kappa \right\vert {\Large \alpha }(\overline{\mathcal{S}}),$ $\kappa \in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ;$ \end{enumerate} \begin{lemma} \label{2.13} \cite{MH} Let $\mathbb{B}$ be a bounded, closed and convex subset of a Banach space $E$ such that $0\in \mathbb{B}$, and let ${\large \mathcal{T}}$ be a continuous mapping of $\mathbb{B}$ into itself. If for every subset ${\large \mathcal{V}}$ of $\mathbb{B}$% \begin{equation*} \mathcal{V=}\overline{co}\mathcal{T}(\mathcal{V})\text{ or }\mathcal{V=T}(% \mathcal{V})\cup \{0\}\Longrightarrow {\Large \alpha }(\mathcal{V}){\large \mathcal{=}}0 \end{equation*}% holds. Then ${\large \mathcal{T}}$ has a fixed point. \end{lemma} \begin{lemma} \cite{SZ} Let $\mathbb{B}$ be a bounded, closed and convex subset of a Banach space $C(J_{1},E)$, $F$ is a continuous function on $J_{1}\times J_{1} $; and a function $f:J_{1}\times E\longrightarrow E$ satisfying the Carath\'{e}odory conditions, and assume there exists $\rho \in $ $% L^{P}(J_{1},% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{+})$ such that, for each $t\in J_{1}$ and each bounded set $\mathbb{B}% ^{\ast }\subset E$; one has% \begin{equation*} \underset{r\longrightarrow 0^{+}}{\lim }{\Large \alpha }(f(J_{t,r}\times \mathbb{B}^{\ast }))\leq \rho (t){\Large \alpha }(\mathbb{B}^{\ast }),\text{% where }J_{t,r}\in \lbrack t-r,t]\cap J_{1}. \end{equation*}% If $\mathcal{V}$ is an equicontinuous subset of $\mathbb{B}$; then% \begin{equation*} {\Large \alpha }\bigg(\bigg\{\int_{J_{1}}F(t,s)f(s,z(s))ds:z\in \mathcal{V% \bigg\}\bigg)}\leq \int_{J_{1}}\left\Vert F(t,s)\right\Vert \rho (s){\Large % \alpha }(\mathcal{V(}s\mathcal{)})ds. \end{equation*} \end{lemma} \begin{lemma} \label{le}\cite{FK} Let $\gamma =\mu +\nu (1-\mu )$ where $0<\mu <1$ and $% 0\leq \nu \leq 1.$ Let $f:(a,b]\times %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion \rightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion $ be a function such that $f(t,z)\in C_{1-\gamma }[a,b]$ for any $z\in C_{1-\gamma }[a,b].$ If $z\in C_{1-\gamma }^{\gamma }[a,b],$ then $z$ satisfies IVP \begin{equation*} D_{a^{+}}^{\mu ,\nu }z(t)=f(t,z(t)),\text{ \ }0<\mu <1,\,0\leq \nu \leq 1,% \text{ }t\in \lbrack a,b], \end{equation*}% \begin{equation*} I_{a^{+}}^{1-\gamma }z(0^{+})=z_{a},\ \ \mu \leq \gamma \end{equation*}% if and only if $z$ satisfies the Volterra integral equation \begin{equation} z(t)=\frac{z_{a}}{\Gamma (\gamma )}(t-a)^{\gamma -1}+\frac{1}{\Gamma (\mu )}% \int_{a}^{t}(t-s)^{\mu -1}f(s,z(s))ds,\quad t>a. \label{s3} \end{equation} \end{lemma} \section{Main results} Now we prove the existence of solution of HNBVP \eqref{e8.1}-\eqref{e8.2} in $C_{1-\gamma }^{\gamma }(J_{1},E)\subset C_{1-\gamma }^{\mu ,\nu }(J_{1},E)$ under measure of noncompactness technique and a fixed point theorem of Monch type. \begin{definition} A function $z\in $ $C_{1-\gamma }^{\gamma }(J_{1},E)$ is said to be a solution of HNBVP \eqref{e8.1}-\eqref{e8.2} if $z$ satisfies the fractional differential equation $D_{a^{+}}^{\mu ,\nu }z(t)=f(t,z(t))$ on $J_{2}$, and the nonlocal boundary condition $\displaystyle I_{a^{+}}^{1-\gamma }\left[ cz(a^{+})+dz(b^{-})\right] =\sum_{k=1}^{m}\lambda _{k}z(\tau _{k}).$ \end{definition} In the beginning, we need the following axiom lemma: \begin{lemma} \label{lee1} Let $0<\mu <1$, $0\leq \nu \leq 1$ where $\gamma =\mu +\nu (1-\mu )$, and $f:J_{2}\times E\rightarrow E$ be a function such that $% f(t,z)\in C_{1-\gamma }(J_{1},E)$ for any $z\in C_{1-\gamma }(J_{1},E).$ If $% z\in C_{1-\gamma }^{\gamma }(J_{1},E),$ then $z$ satisfies HNBVP \eqref{e8.1}% -\eqref{e8.2} if and only if $z$ satisfies the following integral equation% \begin{eqnarray} z(t) &=&\frac{(t-a)^{\gamma -1}}{\Gamma (\gamma )}\frac{1}{\left( c+d-A\right) }\sum_{k=1}^{m}\frac{\lambda _{k}}{\Gamma (\mu )}\int_{a}^{\tau _{k}}(\tau _{k}-s)^{\mu -1}f(s,z(s))ds \notag \\ &&-\frac{(t-a)^{\gamma -1}}{\Gamma (\gamma )}\frac{d}{\left( c+d-A\right) }% \frac{1}{\Gamma (1-\gamma +\mu )}\int_{a}^{b}(b-s)^{-\gamma +\mu }f(s,z(s))ds \notag \\ &&+\frac{1}{\Gamma (\mu )}\int_{a}^{t}(t-s)^{\mu -1}f(s,z(s))ds, \label{ee3} \end{eqnarray}% where $\displaystyle{A=\sum_{k=1}^{m}\lambda _{k}\frac{(\tau _{k}-a)^{\gamma -1}}{\Gamma (\gamma )}}$, and $c+d\neq A$. \end{lemma} Proof: \ In view of Lemma \ref{le}, the solution of \eqref{e8.1} can be written as% \begin{equation} z(t)=\frac{I_{a^{+}}^{1-\gamma }z(a^{+})}{\Gamma (\gamma )}(t-a)^{\gamma -1}+% \frac{1}{\Gamma (\mu )}\int_{a}^{t}(t-s)^{\mu -1}f(s,z(s))ds,\quad t>a. \label{e8.3} \end{equation} Applying $I_{a^{+}}^{1-\gamma }$ on both sides of \eqref{e8.3} and taking the limit $t\rightarrow b^{-}$, we obtain \begin{equation} I_{a^{+}}^{1-\gamma }z(b^{-})=I_{a^{+}}^{1-\gamma }z(a^{+})+\frac{1}{\Gamma (1-\gamma +\mu )}\int_{a}^{b}(b-s)^{-\gamma +\mu }f(s,z(s))ds. \label{e8.4} \end{equation}% Now, we substitute $t=\tau _{k}$ in (\ref{e8.3}) and multiply by $\lambda _{k}$ to obtain% \begin{equation} \lambda _{k}z(\tau _{k})=\lambda _{k}\left[ \frac{I_{a^{+}}^{1-\gamma }z(a^{+})}{\Gamma (\gamma )}(\tau _{k}-a)^{\gamma -1}+\frac{1}{\Gamma (\mu )}% \int_{a}^{\tau _{k}}(\tau _{k}-s)^{\mu -1}f(s,z(s))ds\right] . \label{e8.3a} \end{equation}% Using the nonlocal boundary condition (\ref{e8.2}) with (\ref{e8.4}) and (% \ref{e8.3a}), we have% \begin{eqnarray*} I_{a^{+}}^{1-\gamma }z(a^{+}) &=&\frac{1}{c}\sum_{k=1}^{m}\lambda _{k}z(\tau _{k})-\frac{d}{c}I_{a^{+}}^{1-\gamma }z(a^{+}) \\ &&+\frac{d}{c\Gamma (1-\gamma +\mu )}\int_{a}^{b}(b-s)^{-\gamma +\mu }f(s,z(s))ds. \end{eqnarray*}% Therefore, by (\ref{e8.3a}), we have \begin{eqnarray} I_{a^{+}}^{1-\gamma }z(a^{+}) &=&\frac{1}{c}\sum_{k=1}^{m}\lambda _{k}\frac{% I_{a^{+}}^{1-\gamma }z(a^{+})}{\Gamma (\gamma )}(\tau _{k}-a)^{\gamma -1} \notag \\ &&+\frac{1}{c}\sum_{k=1}^{m}\frac{\lambda _{k}}{\Gamma (\mu )}\int_{a}^{\tau _{k}}(\tau _{k}-s)^{\mu -1}f(s,z(s))ds \notag \\ &&-\frac{d}{c}I_{a^{+}}^{1-\gamma }z(a^{+})-\frac{d}{c}\frac{1}{\Gamma (1-\gamma +\mu )}\int_{a}^{b}(b-s)^{-\gamma +\mu }f(s,z(s))ds. \notag \\ &=&\frac{1}{\left( c+d-A\right) }\sum_{k=1}^{m}\frac{\lambda _{k}}{\Gamma (\mu )}\int_{a}^{\tau _{k}}(\tau _{k}-s)^{\mu -1}f(s,z(s))ds \notag \\ &&-\frac{d}{\left( c+d-A\right) }\frac{1}{\Gamma (1-\gamma +\mu )}% \int_{a}^{b}(b-s)^{-\gamma +\mu }f(s,z(s))ds, \label{t1} \end{eqnarray}% Submitting \eqref{t1} into \eqref{e8.3}, we obtain% \begin{eqnarray} z(t) &=&\frac{(t-a)^{\gamma -1}}{\Gamma (\gamma )}\frac{1}{\left( c+d-A\right) }\sum_{k=1}^{m}\frac{\lambda _{k}}{\Gamma (\mu )}\int_{a}^{\tau _{k}}(\tau _{k}-s)^{\mu -1}f(s,z(s))ds \notag \\ &&-\frac{(t-a)^{\gamma -1}}{\Gamma (\gamma )}\frac{d}{\left( c+d-A\right) }% \frac{1}{\Gamma (1-\gamma +\mu )}\int_{a}^{b}(b-s)^{-\gamma +\mu }f(s,z(s))ds \notag \\ &&+\frac{1}{\Gamma (\mu )}\int_{a}^{t}(t-s)^{\mu -1}f(s,z(s))ds. \label{E5} \end{eqnarray} Conversely, applying $I_{a^{+}}^{1-\gamma }$ on both sides of \eqref{ee3}, using Lemma \ref{def8.5} and \ref{Le1}, some simple computations gives \begin{eqnarray*} &&I_{a^{+}}^{1-\gamma }\big(cz(a^{+})+dz(b^{-})\big) \\ &=&\frac{c}{\left( c+d-A\right) }\sum_{k=1}^{m}\frac{\lambda _{k}}{\Gamma (\mu )}\int_{a}^{\tau _{k}}(\tau _{k}-s)^{\mu -1}f(s,z(s))ds \\ &&-\frac{cd}{\left( c+d-A\right) }\frac{1}{\Gamma (1-\gamma +\mu )}% \int_{a}^{b}(b-s)^{-\gamma +\mu }f(s,z(s))ds \\ &&+\frac{d}{\left( c+d-A\right) }\sum_{k=1}^{m}\frac{\lambda _{k}}{\Gamma (\mu )}\int_{a}^{\tau _{k}}(\tau _{k}-s)^{\mu -1}f(s,z(s))ds \\ &&-\frac{d^{2}}{\left( c+d-A\right) }\frac{1}{\Gamma (1-\gamma +\mu )}% \int_{a}^{b}(b-s)^{-\gamma +\mu }f(s,z(s))ds \\ &&+\frac{d}{\Gamma (1-\gamma +\mu )}\int_{a}^{b}(b-s)^{-\gamma +\mu }f(s,z(s))ds. \end{eqnarray*}% Which implies \begin{eqnarray*} &&I_{a^{+}}^{1-\gamma }\big(cz(a^{+})+dz(b^{-})\big) \\ &=&\left( \frac{c}{\left( c+d-A\right) }+\frac{d}{\left( c+d-A\right) }% \right) \sum_{k=1}^{m}\frac{\lambda _{k}}{\Gamma (\mu )}\int_{a}^{\tau _{k}}(\tau _{k}-s)^{\mu -1}f(s,z(s))ds \\ &&-\left( d-\frac{cd}{\left( c+d-A\right) }-\frac{d^{2}}{\left( c+d-A\right) }\right) \int_{a}^{b}\frac{(b-s)^{-\gamma +\mu }}{\Gamma (1-\gamma +\mu )}% f(s,z(s))ds \\ &=&\frac{c+d}{\left( c+d-A\right) }\sum_{k=1}^{m}\frac{\lambda _{k}}{\Gamma (\mu )}\int_{a}^{\tau _{k}}(\tau _{k}-s)^{\mu -1}f(s,z(s))ds \\ &&-\frac{Ad}{\left( c+d-A\right) }\frac{1}{\Gamma (1-\gamma +\mu )}% \int_{a}^{b}(b-s)^{-\gamma +\mu }f(s,z(s))ds. \end{eqnarray*}% From (\ref{e8.3a}) and (\ref{t1}), we conclude that% \begin{equation*} I_{a^{+}}^{1-\gamma }\big(cz(a^{+})+dz(b^{-})\big)=\sum_{k=1}^{m}\lambda _{k}z(\tau _{k}), \end{equation*}% which shows that the nonlocal boundary condition (\ref{e8.2}) is satisfied. \newline Next, applying $D_{a^{+}}^{\gamma }$ on both sides of \eqref{ee3} and using Lemma \ref{Le1} and \ref{def8.8}, we have \begin{equation} D_{a^{+}}^{\gamma }z(t)=D_{a^{+}}^{\nu (1-\mu )}f\big(t,z(t)\big). \label{e8.9} \end{equation} Since $z\in C_{1-\gamma }^{\gamma }(J_{1},E)$ and by definition of $% C_{1-\gamma }^{\gamma }(J_{1},E)$, we have $D_{a^{+}}^{\gamma }z\in C_{1-\gamma }(J_{1},E)$, therefore, $D_{a^{+}}^{\nu (1-\mu )}f=DI_{a^{+}}^{1-\nu (1-\mu )}f\in C_{1-\gamma }(J_{1},E).$ For $f\in C_{1-\gamma }(J_{1},E)$, it is clear that $I_{a^{+}}^{1-\nu (1-\mu )}f\in C_{1-\gamma }(J_{1},E)$. Hence $f$ and $I_{a^{+}}^{1-\nu (1-\mu )}f$ satisfy the hypothesis of Lemma \ref{Le2}. Now, by applying $I_{a^{+}}^{\nu (1-\mu )}$ on both sides of \eqref{e8.9}, we have% \begin{equation*} {\large I_{a^{+}}^{\nu (1-\mu )}}D_{a^{+}}^{\gamma }z(t)={\large % I_{a^{+}}^{\nu (1-\mu )}}D_{a^{+}}^{\nu (1-\mu )}f\big(t,z(t)\big). \end{equation*}% Using Remark~\ref{rem8.a} (i), relation (\ref{e8.9}) and Lemma \ref{Le2}, we get% \begin{equation*} D_{a^{+}}^{\mu ,\nu }z(t)=f\big(t,z(t)\big)-\frac{I_{a^{+}}^{1-\nu (1-\mu )}f% \big(a,z(a)\big)}{\Gamma (\nu (1-\mu ))}(t-a)^{\nu (1-\mu )-1},\,\text{for all}\,\,t\in J_{2}. \end{equation*}% \ By Lemma \ref{def8.7}, we have $I_{a^{+}}^{1-\nu (1-\mu )}f\big(a,z(a)\big)% =0$. Therefore $D_{a^{+}}^{\mu ,\nu }z(t)=f\big(t,z(t)\big)$. This completes the proof. To prove the existence of solutions for the problem at hand, let us make the following hypotheses. \begin{itemize} \item[ (H1)] The function $f:J_{2}\times E\rightarrow E$ satisfies the Carath% \`{e}odory conditions. \item[ (H2)] $f:J_{2}\times E\rightarrow E$ is a function such that $f(\cdot ,z(\cdot ))\in C_{1-\gamma }^{\nu (1-\mu )}(J_{1},E)$ for any $z\in C_{1-\gamma }(J_{1},E)$ and there exists $\rho \in L^{p}(J_{1},% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{+})$ with $p>\frac{1}{\mu }$ and $p>\frac{1}{\gamma }$ such that% \begin{equation*} \left\Vert f\big(t,z\big)\right\Vert \leq \rho (t)\left\Vert z\right\Vert , \end{equation*}% for each $t\in J_{2},$ and all $z\in E.$ \item[ (H3)] The following inequalities% \begin{eqnarray*} \mathcal{G} &:&\mathcal{=}\bigg(\frac{1}{\Gamma (\gamma )}\frac{\left( \Lambda _{q,\mu ,\gamma }\right) ^{\frac{1}{q}}}{\left( c+d-A\right) }% \sum_{k=1}^{m}\frac{\lambda _{k}}{\Gamma (\mu )}(\tau _{k}-a)^{\gamma +\mu -1} \\ &&+\big(\frac{1}{\Gamma (\gamma )}\left\vert \frac{d}{\left( c+d-A\right) }% \right\vert \frac{\left( \Delta _{q,\mu ,\gamma }\right) ^{\frac{1}{q}}}{% \Gamma (1-\gamma +\mu )}+\frac{\left( \Lambda _{q,\mu ,\gamma }\right) ^{% \frac{1}{q}}}{\Gamma (\mu )}\big)(b-a)^{\mu }\bigg)\left\Vert \rho \right\Vert _{L^{p}}<1, \end{eqnarray*}% and% \begin{eqnarray*} L^{\ast } &:&=\bigg(\frac{m}{\Gamma (\gamma )}\frac{(b-a)^{\gamma -1}}{% \left( c+d-A\right) }\sum_{k=1}^{m}\frac{\lambda _{k}(\tau _{k}-a)^{\mu }}{% \Gamma (\mu +1)} \\ &&+\big(\frac{1}{\Gamma (\gamma )}\left\vert \frac{d}{\left( c+d-A\right) }% \right\vert \frac{1}{\Gamma (-\gamma +\mu )}+\frac{1}{\Gamma (\mu +1)}\big)% (b-a)^{\mu }\bigg)\left\Vert \rho \right\Vert _{L^{p}}<1 \end{eqnarray*}% hold, where $q>1,$ $\frac{1}{p}+\frac{1}{q}=1$ and% \begin{equation*} \Lambda _{q,\mu ,\gamma }:=\frac{\Gamma (q(\mu -1)+1)\Gamma (q(\gamma -1)+1)% }{\Gamma (q(\mu +\gamma -2)+2)}, \end{equation*} \begin{equation*} \Delta _{q,\mu ,\gamma }:=\frac{\Gamma (q(\mu -\gamma )+1)\Gamma (q(\gamma -1)+1)}{\Gamma (q(\mu -1)+2)}. \end{equation*} \end{itemize} Now, we are ready to prove the existence of solutions for the HNBVP % \eqref{e8.1}-\eqref{e8.2}, which is based on fixed point theorem of M\"{o}% nch's type. \begin{theorem} \label{th8.1} Assume that (H1)-(H3) are satisfied. Then HNBVP \eqref{e8.1}-% \eqref{e8.2} has at least one solution in $C_{1-\gamma }^{\gamma }(J_{1},E)\subset C_{1-\gamma }^{\mu ,\nu }(J_{1},E)$. \end{theorem} \begin{proof} Transform the problem \eqref{e8.1}-\eqref{e8.2} into a fixed point problem. Define the operator ${\large \mathcal{T}}:C_{1-\gamma }(J_{1},E)\longrightarrow C_{1-\gamma }(J_{1},E)$ as% \begin{eqnarray} {\large \mathcal{T}}z(t) &=&z(t)=\frac{(t-a)^{\gamma -1}}{\Gamma (\gamma )}% \frac{1}{\left( c+d-A\right) }\sum_{k=1}^{m}\frac{\lambda _{k}}{\Gamma (\mu )% }\int_{a}^{\tau _{k}}(\tau _{k}-s)^{\mu -1}f(s,z(s))ds \notag \\ &&-\frac{(t-a)^{\gamma -1}}{\Gamma (\gamma )}\frac{d}{\left( c+d-A\right) }% \frac{1}{\Gamma (1-\gamma +\mu )}\int_{a}^{b}(b-s)^{-\gamma +\mu }f(s,z(s))ds \notag \\ &&+\frac{1}{\Gamma (\mu )}\int_{a}^{t}(t-s)^{\mu -1}f(s,z(s))ds. \label{e8.10} \end{eqnarray}% Clearly, from Lemma \ref{lee1}, the fixed points of ${\large \mathcal{T}}$ are solutions to \eqref{e8.1}-\eqref{e8.2}. Let $\mathbb{B}_{R}=\left\{ z\in C_{1-\gamma }(J_{1},E):\left\Vert z\right\Vert _{C_{1-\gamma }}\leq R\right\} $. We shall show that $\mathcal{T}$ satisfies the conditions of M% \"{o}nch's fixed point theorem. The proof will be given in the following four steps: Step1: We show that $\mathcal{T}(\mathbb{B}_{R})\subset \mathbb{B}_{R}$. From the hypothesis $(H_{2})$ and H\"{o}lder's inequality, we have% \begin{eqnarray} &&\left\vert (\mathcal{T}z)(t)(t-a)^{1-\gamma }\right\vert \notag \\ &=&\frac{1}{\Gamma (\gamma )}\frac{1}{\left( c+d-A\right) }\sum_{k=1}^{m}% \frac{\lambda _{k}}{\Gamma (\mu )}\int_{a}^{\tau _{k}}(\tau _{k}-s)^{\mu -1}\left\vert f(s,z(s))\right\vert ds \notag \\ &&+\frac{1}{\Gamma (\gamma )}\left\vert \frac{d}{\left( c+d-A\right) }% \right\vert \frac{1}{\Gamma (1-\gamma +\mu )}\int_{a}^{b}(b-s)^{-\gamma +\mu }\left\vert f(s,z(s))\right\vert ds \notag \\ &&+\frac{(t-a)^{1-\gamma }}{\Gamma (\mu )}\int_{a}^{t}(t-s)^{\mu -1}\left\vert f(s,z(s))\right\vert ds \notag \\ &\leq &\frac{1}{\Gamma (\gamma )}\frac{1}{\left( c+d-A\right) }\sum_{k=1}^{m}% \frac{\lambda _{k}}{\Gamma (\mu )}\int_{a}^{\tau _{k}}(\tau _{k}-s)^{\mu -1}(s-a)^{\gamma -1}\rho (s)\left\Vert z\right\Vert _{C_{1-\gamma }}ds \notag \\ &&+\frac{1}{\Gamma (\gamma )}\left\vert \frac{d}{\left( c+d-A\right) }% \right\vert \int_{a}^{b}\frac{(b-s)^{-\gamma +\mu }}{\Gamma (1-\gamma +\mu )}% (s-a)^{\gamma -1}\rho (s)\left\Vert z\right\Vert _{C_{1-\gamma }}ds \notag \\ &&+\frac{(t-a)^{1-\gamma }}{\Gamma (\mu )}\int_{a}^{t}(t-s)^{\mu -1}(s-a)^{\gamma -1}\rho (s)\left\Vert z\right\Vert _{C_{1-\gamma }}ds \notag \\ &\leq &\frac{1}{\Gamma (\gamma )}\sum_{k=1}^{m}\frac{\lambda _{k}}{\Gamma (\mu )}\left( \int_{a}^{\tau _{k}}\frac{(\tau _{k}-s)^{(\mu -1)q}}{\left( c+d-A\right) }(s-a)^{(\gamma -1)q}ds\right) ^{\frac{1}{q}}\left\Vert \rho \right\Vert _{L^{p}}\left\Vert z\right\Vert _{C_{1-\gamma }} \notag \\ &&+\frac{1}{\Gamma (\gamma )}\left\vert \frac{d}{\left( c+d-A\right) }% \right\vert \left( \int_{a}^{b}\frac{(b-s)^{(-\gamma +\mu )q}}{\Gamma (1-\gamma +\mu )}(s-a)^{(\gamma -1)q}ds\right) ^{\frac{1}{q}} \notag \\ &&\times \left\Vert \rho \right\Vert _{L^{p}}\left\Vert z\right\Vert _{C_{1-\gamma }}+\frac{(t-a)^{1-\gamma }}{\Gamma (\mu )} \notag \\ &&\times \left( \int_{a}^{t}(t-s)^{(\mu -1)q}(s-a)^{(\gamma -1)q}ds\right) ^{% \frac{1}{q}}\left\Vert \rho \right\Vert _{L^{p}}\left\Vert z\right\Vert _{C_{1-\gamma }}. \label{q1} \end{eqnarray}% Since $q>1,$ $p>\frac{1}{\mu }$ and $\frac{1}{p}+\frac{1}{q}=1,$ the change of variable $s=a-u(\tau _{k}-a)$ yields \begin{equation} \left( \int_{a}^{\tau _{k}}(\tau _{k}-s)^{(\mu -1)q}(s-a)^{(\gamma -1)q}ds\right) ^{\frac{1}{q}}\leq \left( \Lambda _{q,\mu ,\gamma }\right) ^{% \frac{1}{q}}(\tau _{k}-a)^{\gamma +\mu -1}, \label{e1} \end{equation}% the change of variable $s=a-u(b-a)$ gives \begin{equation} \left( \int_{a}^{b}(b-s)^{(-\gamma +\mu )q}(s-a)^{(\gamma -1)q}ds\right) ^{% \frac{1}{q}}\leq \left( \Delta _{q,\mu ,\gamma }\right) ^{\frac{1}{q}% }(b-a)^{\mu }, \label{e2} \end{equation}% and the change of variable $s=a-u(t-a)$ gives us \begin{equation} \left( \int_{a}^{t}(t-s)^{(\mu -1)q}(s-a)^{(\gamma -1)q}ds\right) ^{\frac{1}{% q}}\leq \left( \Lambda _{q,\mu ,\gamma }\right) ^{\frac{1}{q}}(t-a)^{\gamma +\mu -1}. \label{e3} \end{equation} Substitution of (\ref{e1}),(\ref{e2}) and (\ref{e3}) into (\ref{q1}) leads% \begin{eqnarray*} &&\left\vert (\mathcal{T}z)(t)(t-a)^{1-\gamma }\right\vert \\ &\leq &\frac{1}{\Gamma (\gamma )}\frac{1}{\left( c+d-A\right) }\sum_{k=1}^{m}% \frac{\lambda _{k}}{\Gamma (\mu )}\left( \Lambda _{q,\mu ,\gamma }\right) ^{% \frac{1}{q}}(\tau _{k}-a)^{\gamma +\mu -1}\left\Vert \rho \right\Vert _{L^{p}}\left\Vert z\right\Vert _{C_{1-\gamma }} \\ &&+\frac{1}{\Gamma (\gamma )}\left\vert \frac{d}{\left( c+d-A\right) }% \right\vert \frac{1}{\Gamma (1-\gamma +\mu )}\left( \Delta _{q,\mu ,\gamma }\right) ^{\frac{1}{q}}(b-a)^{\mu }\left\Vert \rho \right\Vert _{L^{p}}\left\Vert z\right\Vert _{C_{1-\gamma }} \\ &&+\frac{(t-a)^{1-\gamma }}{\Gamma (\mu )}\left( \Lambda _{q,\mu ,\gamma }\right) ^{\frac{1}{q}}(t-a)^{\gamma +\mu -1}\left\Vert \rho \right\Vert _{L^{p}}\left\Vert z\right\Vert _{C_{1-\gamma }}. \end{eqnarray*} For any $z\in \mathbb{B}_{R},$ we obtain \begin{align*} {\Vert {\large \mathcal{T}}z\Vert }_{C_{1-\gamma }}& \leq \bigg(\frac{1}{% \Gamma (\gamma )}\frac{\left( \Lambda _{q,\mu ,\gamma }\right) ^{\frac{1}{q}}% }{\left( c+d-A\right) }\sum_{k=1}^{m}\frac{\lambda _{k}}{\Gamma (\mu )}(\tau _{k}-a)^{\gamma +\mu -1} \\ & +\big(\frac{1}{\Gamma (\gamma )}\left\vert \frac{d}{\left( c+d-A\right) }% \right\vert \frac{\left( \Delta _{q,\mu ,\gamma }\right) ^{\frac{1}{q}}}{% \Gamma (1-\gamma +\mu )}+\frac{\left( \Lambda _{q,\mu ,\gamma }\right) ^{% \frac{1}{q}}}{\Gamma (\mu )}\big)(b-a)^{\mu }\bigg)\left\Vert \rho \right\Vert _{L^{p}}R. \end{align*} By (H3), we have $\Vert {\large \mathcal{T}}z\Vert _{C_{1-\gamma }}\leq \mathcal{G}R\leq R,$ that is, ${\large \mathcal{T(}}\mathbb{B}_{R})\subset \mathbb{B}_{R}.$ Step 2. We shall prove that ${\large \mathcal{T}}$ is completely continuous.% \newline The operator ${\large \mathcal{T}}$ is continuous. Let $\{z_{n}\}_{n\in %TCIMACRO{\U{2115} }% %BeginExpansion \mathbb{N} %EndExpansion }$ is a sequence such that $z_{n}\rightarrow z$ in $\mathbb{B}_{R}$. Then for each $t\in J_{2},$ we have% \begin{eqnarray*} &&\left\vert \big((\mathcal{T}z_{n})(t)-(\mathcal{T}z)(t)\big)% (t-a)^{1-\gamma }\right\vert \\ &=&\frac{1}{\Gamma (\gamma )}\frac{1}{\left( c+d-A\right) }\sum_{k=1}^{m}% \frac{\lambda _{k}}{\Gamma (\mu )}\int_{a}^{\tau _{k}}(\tau _{k}-s)^{\mu -1}\left\vert f(s,z_{n}(s))-f(s,z(s))\right\vert ds \\ &&+\frac{1}{\Gamma (\gamma )}\left\vert \frac{d}{\left( c+d-A\right) }% \right\vert \int_{a}^{b}\frac{(b-s)^{-\gamma +\mu }}{\Gamma (1-\gamma +\mu )}% \left\vert f(s,z_{n}(s))-f(s,z(s))\right\vert dds \\ &&+\frac{(t-a)^{1-\gamma }}{\Gamma (\mu )}\int_{a}^{t}(t-s)^{\mu -1}\left\vert f(s,z_{n}(s))-f(s,z(s))\right\vert dds \\ &\leq &\frac{1}{\Gamma (\gamma )}\frac{1}{\left( c+d-A\right) }\sum_{k=1}^{m}% \frac{\lambda _{k}}{\Gamma (\mu )}\int_{a}^{\tau _{k}}(\tau _{k}-s)^{\mu -1}(s-a)^{\gamma -1}ds \\ &&\times \left\Vert f\big(\cdot ,z_{n}(\cdot )\big)-f\big(\cdot ,z(\cdot )% \big)\right\Vert _{C_{1-\gamma }} \\ &&+\frac{1}{\Gamma (\gamma )}\left\vert \frac{d}{\left( c+d-A\right) }% \right\vert \frac{1}{\Gamma (1-\gamma +\mu )}\int_{a}^{b}(b-s)^{-\gamma +\mu }(s-a)^{\gamma -1}ds \\ &&\times \left\Vert f\big(\cdot ,z_{n}(\cdot )\big)-f\big(\cdot ,z(\cdot )% \big)\right\Vert _{C_{1-\gamma }} \\ &&+\frac{(t-a)^{1-\gamma }}{\Gamma (\mu )}\int_{a}^{t}(t-s)^{\mu -1}(s-a)^{\gamma -1}ds\left\Vert f\big(\cdot ,z_{n}(\cdot )\big)-f\big(\cdot ,z(\cdot )\big)\right\Vert _{C_{1-\gamma }}. \end{eqnarray*} Thus,% \begin{eqnarray*} &&\left\vert \big((\mathcal{T}z_{n})(t)-(\mathcal{T}z)(t)\big)% (t-a)^{1-\gamma }\right\vert \\ &\leq &\frac{1}{\left( c+d-A\right) }\frac{\mathcal{B}(\gamma ,\mu )}{\Gamma (\mu )\Gamma (\gamma )}\sum_{k=1}^{m}\frac{\lambda _{k}(\tau _{k}-a)^{\gamma -1+\mu }}{\Gamma (\mu )}\left\Vert f\big(\cdot ,z_{n}(\cdot )\big)-f\big(% \cdot ,z(\cdot )\big)\right\Vert _{C_{1-\gamma }} \\ &&+\left\vert \frac{d}{\left( c+d-A\right) }\right\vert \frac{(b-a)^{\mu }}{% \Gamma (\mu +1)}\left\Vert f\big(\cdot ,z_{n}(\cdot )\big)-f\big(\cdot ,z(\cdot )\big)\right\Vert _{C_{1-\gamma }} \\ &&+\frac{(b-a)^{\mu }}{\Gamma (\mu )}\frac{\mathcal{B}(\gamma ,\mu )}{\Gamma (\mu )}\left\Vert f\big(\cdot ,z_{n}(\cdot )\big)-f\big(\cdot ,z(\cdot )\big)% \right\Vert _{C_{1-\gamma }}. \end{eqnarray*}% By (H1) and the Lebesgue dominated convergence theorem, we have \begin{equation*} \Vert (\mathcal{T}z_{n}-\mathcal{T}z)\Vert _{C_{1-\gamma }}\longrightarrow 0~~as~~n\longrightarrow \infty , \end{equation*}% which means that operator $\mathcal{T}$ is continuous on $\mathbb{B}_{R}$.% \newline Step 3. $\mathcal{T}(\mathbb{B}_{R})$ is relatively compact.\newline From Step 1, we have ${\large \mathcal{T(}}\mathbb{B}_{R})\subset \mathbb{B}% _{R}.$ It follows that ${\large \mathcal{T(}}\mathbb{B}_{R})$ is uniformly bounded i.e.\ $\mathcal{T}$ maps $\mathbb{B}_{R}$ into itself. Moreover, we show that operator $\mathcal{T}$ is equicontinuous on $\mathbb{B}_{R}$. Indeed, for any $a