\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 38, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/38\hfil Semilinear functional differential equations] {Semilinear functional differential equations of fractional order with state-dependent delay} \author[M. A. Darwish, S. K. Ntouyas\hfil EJDE-2009/38\hfilneg] {Mohamed Abdalla Darwish, Sotiris K. Ntouyas} % in alphabetical order \address{Mohamed Abdalla Darwish\newline Department of Mathematics, Faculty of Science\\ Alexandria University at Damanhour, 22511 Damanhour, Egypt} \email{mdarwish@ictp.it, darwishma@yahoo.com} \address{Sotiris K. Ntouyas\newline Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece} \email{sntouyas@cc.uoi.gr} \thanks{Submitted January 8, 2009. Published March 10, 2009.} \subjclass[2000]{26A33, 26A42, 34K30} \keywords{Functional differential equations; fractional derivative; \hfill\break\indent fractional integral; existence; state-dependent delay; infinite delay; fixed point} \begin{abstract} In this paper we study the existence of solutions for the initial value problem for semilinear functional differential equations of fractional order with state-dependent delay. The nonlinear alternative of Leray-Schauder type is the main tool in our analysis. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \section{Introduction} Recently in \cite{DaNt2}, existence results were proved for an initial value problem for functional differential equations of fractional order with state-dependent delay \begin{gather}\label{ee1} D^{\beta}y(t)=f(t,y_{\rho(t,y_{t})}), \quad t\in J=[0,b],\; 0<\beta<1, \\ \label{ee2} y(t)=\varphi(t), \quad t\in (-\infty,0] \end{gather} as well as for neutral functional differential equations of fractional order with state-dependent delay \begin{gather}\label{ee3} D^{\beta}[y(t)-g(t,y_{\rho(t,y_{t})})]=f(t,y_{\rho(t,y_{t})}), \quad \hbox{for } t\in J, \\ \label{ee4} y(t)=\varphi(t),\quad t\in(-\infty,0], \end{gather} where $D^\beta$ is the standard Riemman-Liouville fractional derivative, $f:J \times \mathcal{B}\to \mathbb{R}$, $g: J\times \mathcal{B}\to \mathbb{R}$ and $\rho:J\times \mathcal{B}\to (-\infty, b]$ are appropriate given functions, $\varphi\in \mathcal{B}$, $\varphi(0)=0$, $g(0,\varphi)=0$ and $\mathcal{B}$ is called a {\em phase space}. The purpose of this paper is to extend the results of \cite{DaNt2} by studying the existence of solutions for initial value problems for a functional semilinear differential equations of fractional order with state-dependent delay, as well as, for a neutral functional semilinear differential equations of fractional order with state-dependent delay. In particular, in Section 3, we consider the following initial value problem for a functional semilinear differential equations of fractional order with state-dependent delay \begin{gather}\label{e1} D^{\beta}y(t)=Ay(t)+f(t,y_{\rho(t,y_{t})}), \quad t\in J=[0,b],\quad 0<\beta<1, \\ \label{e2} y(t)=\varphi(t), \quad t\in (-\infty,0], \end{gather} while in Section 4, we consider the following initial value problem for a neutral functional semilinear differential equations of fractional order with state-dependent delay, \begin{gather}\label{e3} D^{\beta}[y(t)-g(t,y_{\rho(t,y_{t})})]=A[y(t)-g(t,y_{\rho(t,y_{t})})] +f(t,y_{\rho(t,y_{t})}), \quad t\in J, \\ \label{e4} y(t)=\varphi(t),\quad t\in(-\infty,0], \end{gather} where $D^\beta$ is the standard Riemman-Liouville fractional derivative. Here, $f:J \times \mathcal{B}\to E$, $g: J\times \mathcal{B}\to E$ and $\rho:J\times \mathcal{B}\to (-\infty, b]$ are appropriate given functions, $\varphi\in \mathcal{B}$, $\varphi(0)=0$, $g(0,\varphi)=0$, $A:D(A)\subseteq E\to E$ is the infinitesimal generator of a strongly continuous semigroup $\{T(t)\}_{t\ge 0}$, and $\mathcal{B}$ is called a {\em phase space} that will be defined later (see Section 2). The notion of the phase space $\mathcal{B}$ plays an important role in the study of both qualitative and quantitative theory for functional differential equations. A usual choice is a semi-normed space satisfying suitable axioms, which was introduced by Hale and Kato \cite{HaKa} (see also Kappel and Schappacher \cite{KaSc} and Schumacher \cite{Sch}). For a detailed discussion on this topic we refer the reader to the book by Hino {\em et al} \cite{HiMuNa}. While functional differential equations have been used in modelling a panorama of natural phenomena as discussed in the books by Kolmanovskii and Myshkis \cite{KoMy} and Hale and Lunel \cite{HaLu}, it has been only recently that fractional differential equations have begun to see extensive utilization in modelling problems that arise in engineering and other sciences, including viscoelasticity, electrochemistry, control, porous media flow, physics, mechanics and others \cite{GaRaKa,Hi,KiSrTr,MiRo,Pod,SaKiOl,YuGa}. On the other hand, functional differential equations with state-dependent delay appear frequently in applications as model of equations and for this reason the study of this type of equation has received a significant amount of attention in the last years, we refer to \cite{AiFrWu,Ba,CaFaGa,DoDrLi,He1,He2,He3} and the references therein. In part, differential equations of fractional order play a very important role in describing some real world problems. For example some problems in physics, mechanics and other fields can be described with the help of fractional differential equations, see \cite{GaRaKa,GlNo,Hi,Mai,SaKiOl,SaKa,SaMaHa} and references therein. The theory of differential equations of fractional order has recently received a lot of attention and now constitutes a significant branch of nonlinear analysis. Numerous research papers and monographs have appeared devoted to fractional differential equations, for example see \cite{Ag,BeHeNtOu,DaNt1,DaNt3,DeRo,KiSrTr,La,LaDe,LaVa1,LaVa2,LaVa3,Pod,YuGa}. Our approach is based on the nonlinear alternative of Leray-Schauder type \cite{DuGr}. These results can be considered as a contribution to this emerging field. \section{Preliminaries} In this section, we introduce notation, definitions, and preliminary facts which are used throughout this paper. By $C(J,E)$ we denote the Banach space of continuous functions from $J$ into $E$ with the norm $$\|y\|_{\infty}:=\sup\{|y(t)|: t\in J\}.$$ Now, we recall some definitions and facts about fractional derivatives and fractional integrals of arbitrary orders, see \cite{KiSrTr,MiRo,Pod,SaKiOl}. \begin{definition} \rm The fractional primitive of order $\beta>0$ of a function $h: (0,b]\to E$ is defined by $$I^{\beta}_0h(t)=\int_0^t\frac{(t-s)^{\beta-1}}{\Gamma(\beta)}h(s)ds,$$ provided the right hand side exists pointwise on $(0,b]$, where $\Gamma$ is the gamma function. \end{definition} For instance, $I^{\beta}h$ exists for all $\beta>0$, when $h\in C((0,b],E)\cap L^1((0,b],E);$ note also that when $h\in C([0,b],E)$ then $I^{\beta}h\in C([0,b],E)$ and moreover $I^{\beta}h(0)=0$. \begin{definition} \rm The fractional derivative of order $\beta>0$ of a continuous function $h: (o,b]\to E$ is given by $\frac{d^{\beta}h(t)}{dt^{\beta}} = \frac{1}{\Gamma(1-\beta)}\frac {d}{dt}\int_a^t(t-s)^{-\beta}h(s)ds = \frac{d}{dt}I_a^{1-\beta}h(t).$ \end{definition} In this paper, we will employ an axiomatic definition for the phase space $\mathcal{B}$ which is similar to those introduced in \cite{HiMuNa}. More precisely, $\mathcal{B}$ will be a linear space of all functions from $(-\infty,0]$ to $E$ endowed with a seminorm $\|\cdot\|_{\mathcal{B}}$ satisfying the following axioms: \begin{itemize} \item[(A)] If $y:(-\infty,b]\to E$, $b>0$, is continuous on $J$ and $y_{0}\in\mathcal{B}$, then for every $t\in J$ the following conditions hold: \begin{itemize} \item[(i)] $y_{t}\in\mathcal{B}$, \item[(ii)] $\|y_{t}\|_{\mathcal{B}}\le K(t)\sup \{|y(s)| : 0 \le s\le t\}+ M(t)\| y_{0}\|_{\mathcal{B}}$, \item[(iii)] $|y(t)|\leq H\|y_t\|_{\mathcal{B}}$, \end{itemize} where $H>0$ is a constant, $K:[0,\infty)\to [1, \infty)$ is continuous, $M:[0,\infty)\to [1, \infty)$ is locally bounded and $H,\;K,\;M$ are independent of $y(\cdot)$. \item[(A1)] For the function $y(\cdot)$ in $(A)$, $y_{t}$ is a $\mathcal{B}$-valued continuous function on $[0, b]$. \item[(A2)] The space $\mathcal{B}$ is complete. \end{itemize} The next lemma is a consequence of the phase space axioms and is proved in \cite{He1}. \begin{lemma}\label{lh1} Let $\varphi\in \mathcal{B}$ and $I=(\gamma,0]$ be such that $\varphi_t\in \mathcal{B}$ for every $t\in I$. Assume that there exists a locally bounded function $J^{\varphi}:I\to [0,\infty)$ such that $\|\varphi_t\|_{\mathcal{B}}\le J^{\varphi}(t)\|\varphi\|_{\mathcal{B}}$ for every $t\in I$. If $y:(\infty, b]\to \mathbb{R}$ is continuous on $J$ and $y_0=\varphi$, then $$\|y_t\|_{\mathcal{B}}\le (M_b+J^{\varphi}(\max\{\gamma,-|s|\})\|\varphi\|_{\mathcal{B}}+K_b \sup\{|y(\theta)|:\theta\in [0,\max\{0,s\}]\},$$ for $s\in (\gamma,b]$, where we denoted $K_b=\sup_{t\in J}K(t)$ and $M_b=\sup_{t\in J}M(t)$. \end{lemma} \section{Main Result} In this section, the nonlinear alternative of Leray-Schauder type is used to investigate the existence of solutions of problem \eqref{e1}-\eqref{e2}. Let us start by defining what we mean by a solution of problem \eqref{e1}-\eqref{e2}. \begin{definition}\label{d1} \rm A function $y:(-\infty,b]\to E$ is said to be a solution of \eqref{e1}-\eqref{e2} if $y_0=\varphi, y_{\rho(s,y_s)}\in \mathcal{B}$ for every $s\in J$ and $$y(t)=\frac{1}{\Gamma(\beta)}\int_0^t (t-s)^{\beta-1}T(t-s) f(s,y_{\rho(s,y_{s})})\,ds, \quad t\in J.$$ \end{definition} In what follows we assume that $\rho:J\times \mathcal{B}\to (-\infty,b]$ is continuous and $\varphi\in \mathcal{B}$ and the following hypotheses are satisfied \begin{itemize} \item[(H1)] $A$ is the infinitesimal generator of a strongly continuous semigroup of bound\-ed linear operators $T(t), t\ge 0$ in $E$, which is compact for $t>0$, and there exist constant $M\ge 1$ such that $\|T(t)\|_{B(E)}\le M, t\ge 0$; \item[(H2)] $f:J\times\mathcal{B}\to E$ is a continuous function; \item[(H3)] there exists $p\in C([0,b],\mathbb{R}^+)$ and $\Omega:[0,\infty)\to (0,\infty)$ continuous and nondecreasing such that $$|f(t,u)|\leq p(t)\Omega(\|u\|_{\mathcal{B}})$$ for $t\in [0,b]$ and each $u\in \mathcal{B}$; \item[(H4)] there exists a number $K_0>0$ such that $$\frac{K_0}{(M_b+\tilde J^{\varphi})\|\varphi\|_{\mathcal{B}} +MK_b\Omega(K_0)\|I^{\beta}p\|_{\infty}}> 1;$$ \item[(H5)] the function $t\to\varphi_t$ is well defined and continuous from the set $\mathcal{R}(\rho^-)=\{\rho(s,\psi):(s,\psi)\in J\times B, \rho(s,\psi)\leq 0\}$ into $\mathcal{B}$. Moreover, there exists a continuous and bounded function $J^{\varphi}:\mathcal{R}(\rho^-)\to (0,\infty)$ such that $\|\varphi_t\|_{\mathcal{B}}\le J^{\varphi}(t)\|\varphi\|_{\mathcal{B}}$ for every $t\in \mathcal{R}(\rho^-)$. \end{itemize} \begin{remark} \rm The hypothesis (H5) is adapted from \cite{He1}, where we refer for remarks concerning this hypothesis. \end{remark} \begin{theorem}\label{t1} Assume that the hypotheses {\rm (H1)--(H5)} hold. If $\rho(t,\psi)\le t$ for every $(t,\psi)\in J\times\mathcal{B}$, then the \eqref{e1}-\eqref{e2} has at least one solution on $(-\infty,b]$. \end{theorem} \begin{proof} Let $Y=\{u\in C(J, E):u(0)=\varphi(0)=0\}$ endowed with the uniform convergence topology and $N:Y\to Y$ be the operator defined by $$Ny(t)=\frac{1}{\Gamma(\beta)}\int_0^t (t-s)^{\beta-1}T(t-s)f(s, \bar y_{\rho(s,\bar{y}_{s})})\,ds, \quad t\in J,$$ where $\bar y:(-\infty,b]\to E$ is such that $\bar y_0=\varphi$ and $\bar y=y$ on $J$. From axiom (A) and our assumption on $\varphi$, we infer that $Ny(\cdot)$ is well defined and continuous. Let $\bar \varphi:(-\infty,b]\to E$ be the extension of $\varphi$ to $(-\infty, b]$ such that $\bar{\varphi}(\theta)=\varphi(0)=0$ on $J$ and $\tilde J^{\varphi}=\sup\{J^{\varphi}:s\in \mathcal{R}(\rho^{-})\}$. We will prove that $N(\cdot)$ is completely continuous from $B_r(\bar\varphi|_J,Y)$ to $B_r(\bar\varphi|_J,Y)$. \noindent{\bf Step 1:} $N$ is continuous on $B_r(\bar\varphi|_J,Y)$. This was proved in \cite[p. 515, Step 3]{He1}. \noindent{\bf Step 2:} $N$ maps bounded sets into bounded sets. If $y\in B_r(\bar\varphi|_J,Y)$, from Lemma \ref{lh1} follows that $$\|\bar y_{\rho(t,\bar y_{t})}\|_{\mathcal{B}} \le r^*:=(M_b+\tilde J^{\varphi})\|\varphi\|_{\mathcal{B}}+K_br$$ and so \begin{align*} |(Ny)(t)|&= \frac{M}{\Gamma(\beta)}\int_0^t (t-s)^{\beta-1}T(t-s)f(s, \bar y_{\rho(s,\bar{y}_{s})})\,ds\\ &\leq \frac{M}{\Gamma(\beta)}\int_0^t (t-s)^{\beta-1} p(s)\Omega(\|\bar{y}_{\rho(s,\bar{y}_{s})}\|_{\mathcal{B}}) \,ds\\ &\leq \frac{M}{\Gamma(\beta)}\|p\|_{\infty}\Omega(r^*)\int_0^t(t-s)^{\beta-1}\, ds\\ &\leq \frac{Mb^{\beta}}{\Gamma(\beta+1)}\|p\|_{\infty}\Omega(r^*). \end{align*} Thus $$\|Ny\|_{\infty}\le \frac{Mb^{\beta}}{\Gamma(\beta+1)}\|p\|_{\infty}\Omega(r^*):=\ell.$$ \noindent{\bf Step $3$:} $N$ maps bounded sets into equicontinuous sets of $B$. Let $t_1,\;t_2\in(0,b]$ with $t_10$ be given. Now let $\tau_1,\tau_2\in J$ with $\tau_2>\tau_1$. We consider two cases $\tau_1>\epsilon$ and $\tau_1\le \epsilon$. \noindent{\bf Case 1.} If $\tau_1>\epsilon$ then \begin{align*} &|({N}y)(t_2)-({N}y)(t_1)|\\ &\leq \frac{1}{\Gamma(\beta)}\int_0^{t_1-\epsilon} [(t_2-s)^{\beta-1}T(t_2-s)-(t_1-s)^{\beta-1}T(t_1-s)]|f(s,\bar{y}_{\rho(s,\bar{y}_{s})})|\,ds\\ &\quad +\frac{1}{\Gamma(\beta)}\int_{t_1-\epsilon}^{t_1}\left[(t_2-s)^{\beta-1}T(t_2-s)-(t_1-s)^{\beta-1}T(t_1-s)\right]|f(s,\bar{y}_{\rho(s,\bar{y}_{s})})|\,ds\\ &\quad +\frac{1}{\Gamma(\beta)}\int_{t_1}^{t_2}(t_2-s)^{\beta-1}T(t_2-s)|f(s,\bar{y}_{\rho(s,\bar{y}_{s})})|\,ds\\ &\leq \frac{\|p\|_{\infty}\Omega(r^*)}{\Gamma(\beta)} \Big(\Big|\int_0^{t_1-\epsilon}[(t_2-s)^{\beta-1} -(t_1-s)^{\beta-1}]T(t_2-s)\,ds\Big|\\ &\quad +\Big|\int_0^{t_1-\epsilon}(t_1-s)^{\beta-1}T(t_1-\epsilon-s) [T(t_2-t_1-\epsilon)-T(\epsilon)]ds\Big|\\ &\quad +\Big|\int_{t_1-\epsilon}^{t_1} [(t_2-s)^{\beta-1}-(t_1-s)^{\beta-1}]T(t_2-s)\,ds\Big|\\ &\quad +\Big|\int_{t_1-\epsilon}^{t_1} (t_1-s)^{\beta-1}T(t_1-\epsilon-s)[T(t_2-t_1-\epsilon)-T(\epsilon)]\,ds\Big| +\int_{t_1}^{t_2} \! (t_2-s)^{\beta-1}\, ds\Big)\\ &\leq \frac{\|p\|_{\infty}\Omega(r^*)}{\Gamma(\beta)} \Big(M\int_0^{t_1-\epsilon}[(t_2-s)^{\beta-1} -(t_1-s)^{\beta-1}]\, ds \\ &\quad +M\|T(t_2-t_1-\epsilon)-T(\epsilon)\|_{B(E)} \int_0^{t_1-\epsilon}(t_2-s)^{\beta-1}\, ds\\ &\quad +M\int_{t_1-\epsilon}^{t_1} [(t_2-s)^{\beta-1}-(t_1-s)^{\beta-1}]ds\\ &\quad +M\|T(t_2-t_1-\epsilon)-T(\epsilon)\|_{B(E)} \int_{t_1-\epsilon}^{t_1}(t_2-s)^{\beta-1}\, ds+M\int_{t_1}^{t_2}(t_2-s)^{\beta-1}\, ds\Big), \end{align*} where we have used the semigroup identities $$T(\tau_2-s)=T(\tau_2-\tau_1+\epsilon)T(\tau_1-s-\epsilon), \quad T(\tau_1-s)=T(\tau_1-s-\epsilon)T(\epsilon).$$ \noindent{\bf Case 2.} Let $\tau_1\le \epsilon$. For $\tau_2-\tau_1<\epsilon$ we get \begin{align*} |({N}y)(t_2)-({N}y)(t_1)| &\leq \frac{1}{\Gamma(\beta)}\Big|\int_0^{t_2} (t_2-s)^{\beta-1}T(t_2-s) f(s,\bar{y}_{\rho(s,\bar{y}_{s})}) \,ds\\ &\quad -\int_0^{t_1}(t_2-s)^{\beta-1}T(t_2-s) f(s,\bar{y}_{\rho(s,\bar{y}_{s})}) \,ds\Big|\\ &\leq M\frac{\|p\|_{\infty}\Omega(r^*)}{\Gamma(\beta)} \Big(\int_0^{2\epsilon}(t_2-s)^{\beta-1}\,ds +\int_0^{\epsilon}(t_1-s)^{\beta-1}\, ds\Big). \end{align*} Note equicontinuity follows since (i). $T(t), t \geq 0$ is a strongly continuous semigroup and (ii). $T(t)$ is compact for $t>0$ (so $T(t)$ is continuous in the uniform operator topology for $t>0$) \cite{Pa}. From the steps $1$ to $3$, together with Arzel\'a-Ascoli theorem, it suffices to show that $N$ maps $B_{\alpha}$ into a precompact set in $E$. Let $00$, the set $Y_{\epsilon}(t)=\{{N}_{\epsilon}(y)(t):y\in B_{\alpha}\}$ is precompact in $E$ for every $\epsilon$, $0<\epsilon0$ such that $$\frac{\|\xi\|_{\infty}} {(M_b+\tilde J^{\varphi})\|\varphi\|_{\mathcal{B}}+\frac{K_b}{1-K_b d_1} \big\{d_1(M_b+\tilde J^{\varphi})\|\varphi\|_{\mathcal{ B}}+d_2+M\Omega(\|\xi\|_{\infty})\|I^{\beta}p\|_{\infty}\big\}}>1.$$ \end{itemize} If $\rho(t,\psi)\le t$ for every $(t,\psi)\in J\times\mathcal{B}$, then the \eqref{e3}-\eqref{e4} has at least one solution on $(-\infty,b]$. \end{theorem} \begin{proof} Consider the operator $N_0: C((-\infty,b],E)\to C((-\infty,b],E)$ defined by, $$N_0(y)(t)=\begin{cases} \varphi(t),& \text{if } t\in (-\infty,0], \\ g(t,y_{\rho(t,y_{t})})\\ +\frac{1}{\Gamma(\beta)}\int_{0}^{t}(t-s)^ {\beta-1}T(t-s)f(s, y_{\rho(s,y_{s})})ds, &\text{if } t\in[0,b]. \end{cases}$$ In analogy to Theorem \ref{t1}, we consider the operator $N_1:Y\to Y$ defined by $$(N_1y)(t)=\begin{cases} 0, & t\le 0\\ g(t, \bar{y}_{\rho(s,\bar{y}_{s})})+\frac{1}{\Gamma(\beta)} \int_{0}^{t}(t-s)^{\beta-1}T(t-s)f(s, \bar{y}_{\rho(s,\bar{y}_{s})})ds, & t\in [0,b]. \end{cases}$$ We shall show that the operator $N_1$ is continuous and completely continuous. Using (H6) it suffices to show that the operator $N_2: Y\to Y$, defined by $$N_2(y)(t)=\frac{1}{\Gamma(\beta)}\int_{0}^{t}(t-s)^{\beta-1}T(t-s)f(s, \bar{y}_{\rho(s,\bar{y}_{s})})\, ds,\quad t\in [0,b],$$ is continuous and completely continuous. This was proved in Theorem \ref{t1}. {\em We now show there exists an open set $U\subseteq Y$ with $y\ne \lambda N_1(y)$ for $\lambda\in (0,1)$ and $y\in \partial U.$} Let $y\in Y$ and $y=\lambda N_1(y)$ for some $0<\lambda<1$. Then $y(t)=\lambda\Big[g(s, \bar y_{\rho(s,\bar{y}_{s})})\,+\frac{1}{\Gamma(\beta)}\int_{0}^{t}(t-s)^{\beta-1}T(t-s)f(s, \bar y_{\rho(s,\bar{y}_{s})})\,ds\Big],\quad t\in[0,b],$ and \begin{align*} |y(t)| &\leq d_1\Big[(M_b+\tilde J^{\varphi})\|\varphi\|_{\mathcal{B}}+K_b\sup\{|y(s)|:s\in [0,t]\}\Big]+d_2 \\ &\quad+\frac{M}{\Gamma(\beta)}\int_0^t(t-s)^{\beta-1}p(s) \Omega((M_b+\tilde J^{\varphi})\|\varphi\|_{\mathcal{B}} +K_b\sup\{|\bar y(s)|:s\in [0,t]\})ds, \end{align*} for $t\in(0,b]$. If $\mu(t)=\sup\{|y(s)|:s\in [0,t]\}$ then \begin{align*} \mu(t)&\leq d_1(M_b+\tilde J^{\varphi})\|\varphi\|_{\mathcal{B}}+d_1K_b\mu(t)+d_2 \\ &\quad +\frac{M}{\Gamma(\beta)}\int_0^t(t-s)^{\beta-1}p(s)\Omega((M_b+\tilde J^{\varphi})\|\varphi\|_{\mathcal{B}}+K_b\mu(s))\, ds, \end{align*} or \begin{align*} \mu(t) &\leq \frac{1}{1-K_b d_1}\big[d_1(M_b+\tilde J^{\varphi})\|\varphi\|_{\mathcal{ B}}+d_2\big]\\ &\quad +\frac{1}{1-K_b d_1}\frac{M}{\Gamma(\beta)}\int_0^t(t-s)^{\beta-1}p(s)\Omega((M_b+\tilde J^{\varphi})\|\varphi\|_{\mathcal{B}}+K_b\mu(s))\, ds, \end{align*} for $t\in (0,b]$. If $\xi(t)=(M_b+\tilde J^{\varphi})\|\varphi\|_{\mathcal{B}}+K_b\mu(s)$ then we have \begin{align*} \xi(t)&\leq (M_b+\tilde J^{\varphi})\|\varphi\|_{\mathcal{B}}\\ &\quad +\frac{K_b}{1-K_b d_1}\Big\{d_1(M_b+\tilde J^{\varphi})\|\varphi\|_{\mathcal{ B}}+d_2+\frac{M}{\Gamma(\beta)}\int_0^t(t-s)^{\beta-1}p(s)\Omega(\xi(s))\, ds\Big\}\\ &\leq (M_b+\tilde J^{\varphi})\|\varphi\|_{\mathcal{B}}\\ &\quad +\frac{K_b}{1-K_b d_1}\Big\{d_1(M_b+\tilde J^{\varphi})\|\varphi\|_{\mathcal{ B}}+d_2+M\Omega(\|\xi\|_{\infty})\|I^{\beta}p\|_{\infty}\Big\}. \end{align*} Consequently, $$\frac{\|\xi\|_{\infty}}{(M_b+\tilde J^{\varphi})\|\varphi\|_{\mathcal{B}}+\frac{K_b}{1-K_b d_1} \big\{d_1(M_b+\tilde J^{\varphi})\|\varphi\|_{\mathcal{ B}}+d_2+M\Omega(\|\xi\|_{\infty})\|I^{\beta}p\|_{\infty}\big\}}\le 1.$$ By (H7), there exists $L^*$ such that $\|y\|_{\infty}\ne L^*$. Set  U_1=\{y\in Y: \|y\|_{\infty}