\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 19, pp. 1--8.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2008 Texas State University - San Marcos.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2008/19\hfil Nonlocal integrodifferential equations] {Quasilinear nonlocal integrodifferential equations in Banach spaces} \author[Q. Dong, G. Li, J. Zhang\hfil EJDE-2008/19\hfilneg] {Qixiang Dong, Gang Li, Jin Zhang } % in alphabetical order \address{Qixiang Dong \newline School of Mathematical Science \\ Yangzhou University \\ Yangzhou 225002, China } \email{qxdongyz@yahoo.com.cn} \address{Gang Li \newline School of Mathematical Science \\ Yangzhou University \\ Yangzhou 225002, China} \email{gangli@yzvod.com} \address{Jin Zhang (Corresponding Author) \newline College of Mathematical Science\\ Yangzhou University, Yangzhou 225002, China} \email{jzhangmath@163.com} \thanks{Submitted October 23, 2007. Published February 5, 2008.} \thanks{Supported by grant 10571150 from National Natural Science Foundation of China. Q. Dong \hfill\break\indent is also supported by the Ph. D. scientific research innovation project of Jiangsu Province, \hfill\break\indent China.} \subjclass[2000]{34K05, 34K30} \keywords{Nonlocal conditions; mild solution; integrodifferential equation; \hfill\break\indent Hausdorff measure of noncompactness} \begin{abstract} In this paper, we study the existence of mild solutions for quasilinear integrodifferential equations with nonlocal conditions in Banach spaces. The results are established by using Hausdorff's measure of noncompactness. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} In this paper, we discuss the existence of mild solution of the following nonlinear integrodifferential equation with nonlocal condition \begin{gather} \frac{du(t)}{dt}=A(t,u)u+\int_{0}^{t}f(t,s,u(s))ds,\quad t\in[0,b], \label{eq1} \\ u(0)=g(u)+u_0, \label{eq2} \end{gather} where $f: [0,b]\times [0,b]\times\mathbb{X}\to\mathbb{X}$ and $A:[0,b]\times\mathbb{X}\to\mathbb{X}$ are continuous functions, $g: \mathcal {C}([0,b];\mathbb{X})\to \mathbb{X}, u_0\in\mathbb{X}$ and $\mathbb{X}$ is a real Banach space with norm $\|\cdot\|$. The notion of ``nonlocal condition'' has been introduced to extend the study of the classical initial value problems; see, e.g. \cite{ben,bys3,chan,don,x2}. It is more precise for describing nature phenomena than the classical condition since more information is taken into account, thereby decreasing the negative effects incurred by a possibly erroneous single measurement taken at the initial time. The study of abstract nonlocal initial value problems was initiated by Byszewski,we refer to some of the papers below. Byszewski \cite{bys1,bys2} , Byszewski and Lasmikauthem \cite{bys4} give the existence and uniqueness of mild solutions and classical solutions when $f$ and $g$ satisfy Lipschitz-type conditions. Subsequently, many authors are devoted to studying of nonlocal problems. See \cite{anic,bahu,fan,jack,l,x3} for the references and remarks about the advantage of the nonlocal problems over the classical initial value problems. This article is motivated by the recent paper of Chandrasekaran \cite{chan}. We use some hypotheses in \cite{chan}, and using the method of Hausdorff's measure of noncompactness, we give the existence of mild solutions of quasilinear integrodifferential equations with nonlocal conditions \eqref{eq1}--\eqref{eq2}. Our results improve and extend some corresponding results in \cite{bahu,bys2,bys3,chan,l}. \section{Preliminaries} Throughout this paper $\mathbb{X}$ will represent a Banach space with norm $\|\cdot\|$. Denoted $\mathcal {C}([0,b];\mathbb{X})$ by the space of $\mathbb{X}$-valued continuous functions on $[0,b]$ with the norm $\|u\|=\sup\{\|u(t)\|, t\in [0,b]\}$ for u $\in \mathcal {C}([0,b];\mathbb{X})$, and denoted $\mathcal {L}(0,b;\mathbb{X})$ by the space of $\mathbb{X}$-valued Bochner integrable functions on $[0,b]$ with the norm $\|u\|_{\mathcal {L}}=\int_{0} ^{b} \|u(t)\|dt$. The Hausdroff's measure of noncompactness $\beta_{\mathbb{Y}}$ is defined by $\beta_{\mathbb{Y}}(B)=\inf\{r>0, B\text{ can be covered by finite number of balls with radii }r\}$ for bounded set $B$ in a Banach space $\mathbb{Y}$. \begin{lemma}[\cite{ban}] \label{lem2.1} Let $\mathbb{Y}$ be a real Banach space and $B, C \subseteq \mathbb{Y}$ be bounded, with the following properties: \begin{enumerate} \item $B$ is pre-compact if and only if $\beta_{\mathbb{X}}(B)=0$; \item $\beta_{\mathbb{Y}}(B)=\beta_{\mathbb{Y}}({\overline{B}}) =\beta_{\mathbb{Y}} (conv B)$, where $\overline{B}$ and $\mathop{\rm conv }B$ mean the closure and convex hull of $B$ respectively; \item $\beta_{\mathbb{Y}}(B)\leq \beta_{\mathbb{Y}}(C)$, where $B\subseteq C$; \item $\beta_{\mathbb{Y}}(B+C)\leq \beta_{\mathbb{Y}}(B) + \beta_{\mathbb{Y}}(C)$, where $B + C=\{x+y: x\in B, y\in C\}$; \item $\beta_{\mathbb{Y}}(B\cup C)\leq \max\{\beta_{\mathbb{Y}}(B), \beta_{\mathbb{Y}}(C)\}$; \item $\beta_{\mathbb{Y}}(\lambda B)\leq |\lambda|\beta_{\mathbb{Y}}(B)$ for any $\lambda\in \mathbb{R}$; \item If the map $Q: D(Q)\subseteq \mathbb{Y}\to \mathbb{Z}$ is Lipschitz continuous with constant $k$, then $\beta_{\mathbb{Z}}(QB)\leq k\beta_{\mathbb{Y}}(B)$ for any bounded subset $B\subseteq D(Q)$, where $\mathbb{Z}$ be a Banach space; \item $\beta_{\mathbb{Y}}(B) = \inf \{ d_{\mathbb{Y}}(B, C); C\subseteq\mathbb{Y} \text{ is precompact }\} =\inf \{ d_{\mathbb{Y}}(B, C); C\subseteq\mathbb{Y}\\ \text{ is finite valued}\}$, where $d_{\mathbb{Y}}(B, C)$ means the nonsymmetric (or symmetric) Hausdorff distance between $B$ and $C$ in $\mathbb{Y}$; \item If $\{W_n\}^{+\infty}_{n=1}$ is decreasing sequence of bounded closed nonempty subsets of $\mathbb{Y}$ and $\lim_{n\to\infty}\beta_{\mathbb{Y}}(W_n)=0$, then $\bigcap_{n=1}^{+\infty}W_n$ is nonempty and compact in $\mathbb{Y}$. \end{enumerate} \end{lemma} The map $Q: W\subseteq \mathbb{Y}\to\mathbb{Y}$ is said to be a $\beta_{\mathbb{Y}}$-contraction if there exists a positive constant $k<1$ such that $\beta_{\mathbb{Y}}(Q(B))\leq k\beta_{\mathbb{Y}}(B)$ for any bounded closed subset $B\subseteq W$, where $\mathbb{Y}$ is a Bananch space. \begin{lemma}[Darbo-Sadovskii \cite{ban}] \label{lem2.2} If $W\subseteq\mathbb{Y}$ is bounded closed and convex, the continuous map $Q: W\to W$ is a $\beta_{\mathbb{Y}}$-contraction, then the map $Q$ has at least one fixed point in $W$. \end{lemma} In this paper we denote by $\beta$ the Hausdorff's measure of noncompactness of $\mathbb{X}$ and denote $\beta_\mathcal {C}$ by the Hausdorff's measure of noncompactness of $\mathcal{C}([a,b];\mathbb{X})$. To discuss the existence, we need the following Lemmas in this paper. \begin{lemma}[\cite{ban}] \label{lem2.3} If $W\subseteq\mathcal{C}([0,b];\mathbb{X})$ is bounded, then $\beta(W(t))\leq \beta_\mathcal {C}(W)$ for all $t\in [0,b]$, where $W(t)=\{u(t);u\in W\}\subseteq\mathbb{X}$. Furthermore if $W$ is equicontinuous on $[a,b]$, then $\beta(W(t))$ is continuous on $[a,b]$ and $\beta_\mathcal{C}(W)=\sup\{\beta(W(t)),\ t\in [a,b]\}$. \end{lemma} \begin{lemma}[\cite{k}] \label{lem2.4} If $\{u_n\}_{n=1}^{\infty}\subset \mathcal {L}^{1}(a,b;\mathbb{X})$ is uniformly integrable, then the function $\beta(\{u_n(t)\}_{n=1}^{\infty})$ is measurable and \begin{equation} \beta\Big(\Big\{\int_{0}^{t} u_{n}(s)ds\Big\}_{n=1}^{\infty}\Big) \leq 2\int_{0}^{t}\beta\big(\big\{u_n(s)\big\} _{n=1}^{\infty}\big)ds. \label{eq3} \end{equation} \end{lemma} \begin{lemma}[\cite{ban}] \label{lem2.5} If $W\subseteq\mathcal{C}([0,b];\mathbb{X})$ is bounded and equicontinuous, then $\beta(W(s))$ is continuous and \begin{equation} \beta(\int_{0}^{t}W(s)ds)\leq\int_{0}^{t}\beta(W(s))ds. \label{equ4} \end{equation} \end{lemma} From \cite{chan}, we know that for any fixed $u\in\mathcal {C}([0,b];\mathbb{X})$ there exist a unique continuous function $U_u:\ [0,b]\times [0,b]\to B(\mathbb{X})$ defined on $[0,b]\times [0,b]$ such that \begin{equation} U_u(t,s)=I+\int_{s}^{t}A_u(\omega)U_u(\omega,s)d\omega, \label{eq5} \end{equation} where $B(\mathbb{X})$ denote the Banach space of bounded linear operators from $\mathbb{X}$ to $\mathbb{X}$ with the norm $\|Q\|=\sup\{\|Qu\|:\ \|u\|=1\}$, and $I$ stands for the identity operator on $\mathbb{X}$, $A_{u}(t)=A(t,u(t))$. From (\ref{eq5}), we have \begin{gather*} U_u(t,t)=I,\quad U_u(t,s)U_u(s,r)=U_u(t,r),\quad (t,s,r)\in [0,b]\times [0,b]\times [0,b],\\ \frac{\partial U_u(t,s)}{\partial t}=A_u(t)U_u(t,s)\quad \text{for almost all } t\in [0,b],\; \forall s\in [0,b]. \end{gather*} \begin{definition} \label{def2.6} \rm A continuous function $u(t)\in \mathcal {C}([0,b];\mathbb{X})$ such that \begin{equation} u(t)=U_{u}(t,0)u_0+U_u(t,0)g(u)+\int_{0}^{t} U_u(t,s)\int_{0}^{s} f(s,\tau,u(\tau))dsd\tau \label{eq6} \end{equation} and $u(0)=g(u)+u_0$ is called a mild solution of \eqref{eq1}--\eqref{eq2}. \end{definition} The evolution family $\{U_u(t,s)\}_{0\leq s\leq t\leq b}$ is said to be equicontinuous if $(t,s)\to \{U_u(t,s)x:\ x\in B\}$ is equicontinuous for $t>0$ and for all bounded subset $B$ in $\mathbb{X}$. The following Lemma is obvious. \begin{lemma} \label{lem2.7} If the evolution family $\{U_u(t,s)\}_{0\leq s\leq t\leq b}$ is equicontinuous and $\eta\in\mathcal{L}(0,b;\mathbb{R}^{+})$, then the set $\{\int_{0}^{t}U_{u}(t-s,s)u(s)ds, \|u(s)\|\leq \eta(s)\text{ for a.e. }s\in [0,b]\}$ is equicontinuous for $t\in [0,b]$. \end{lemma} In section 3, we give some existence results when $g$ is compact and $f$ satisfies the conditions with respect to Hauadorff's measure of noncompactness. In section 4, we use the different method to discuss the case when $g$ is Lipschitz continuous and $f$ satisfies the conditions with the Hauadorff's measure of noncompactness. In this paper, we denote $M=\sup\{\|U_u(t,s)\|: (t,s)\in [0,b]\times [0,b]\}$ for all $u\in\mathbb{X}$. Without loss of generality, we let $u_0=0$. \section{The existence results for compact $g$} In this section by using the usual techniques of the Hausdorff's measure of noncompactness and its applications in differential equations in Banach spaces (see, e.g. \cite{ban,bot,k}), we give some existence results of the nonlocal problem \eqref{eq1}--\eqref{eq2}. Here we list the following hypotheses: \begin{itemize} \item[(HA)]: The evolution family $\{U_u(t,s)\}_{0\leq s\leq t\leq b}$ generated by $A(t,u)$ is equicontinuous, and $\|U_u(t,s)\|\leq M$ for almost all $t, s\in [0,b]$. \item[(Hg)] \begin{enumerate} \item $g: \mathcal {C}([0,b];\mathbb{X})\to\mathbb{X}$ is continuous and compact; \item There exist $N>0$ such that $\|g(u)\|\leq N$ for all $u\in\mathcal {C}([0,b];\mathbb{X})$. \end{enumerate} \item[(Hf)] \begin{enumerate} \item $f: [0,b]\times[0,b]\times\mathbb{X}\to\mathbb{X}$ satisfies the {\it Carath\'{e}odory-type } condition; i.e., $f(\cdot,\cdot,u)$ is measurable for all $u\in\mathbb{X}$ and $f(t,s,\cdot)$ is continuous for a.e. $t,s\in [a,b]$; \item There exist two functions $h:[0,b]\times\mathbb{R}^{+}\to\mathbb{R}^{+}$ and $ q: [0,b]\times \mathbb{R}^{+}\to\mathbb{R}^{+}$such that $h(\cdot,r)\in\mathcal {L}(0,b;\mathbb{R}^{+})$ for every $r\geq 0$, $h(t,\cdot)$ is continuous and increasing,$q(s)\in\mathcal {L}(0,b;\mathbb{R}^{+})$ , and $\|f(t,s,u)\|\leq q(t)h(s,\|u\|)$ for a.e. $t\in [0,b]$, and all $u\in \mathcal{C}([0,b];\mathbb{X})$, and for all positive constants $K_1, K_2$, the scalar equation \begin{equation} m(t)=K_1+K_2\int_{0}^{t}h(s,m(s))ds,\,\, t\in [0,b] \label{eq6b} \end{equation} has at least one solution; \item There exist $\eta\in\mathcal {L}(0,b;\mathbb{R}^{+})$, $\zeta\in\mathcal {L}(0,b; \mathbb{R}^{+})$ such that $\beta(f(t,s,D))\leq\eta(t)\zeta(s)\beta(D)$ for a.e. $t, s\in [0,b]$, and for any bounded subset $D\subset \mathcal {C}([0,b],\mathbb{X})$. Here we let $\int_{0}^{t}\eta(s)ds\leq K$ \end{enumerate} \end{itemize} Now, we give an existence result under the above hypotheses. \begin{theorem} \label{thm3.1} Assume the hypotheses {\rm (HA), (Hf), (Hg)} are satisfied, then the nonlocal initial value problem \eqref{eq1}--\eqref{eq2} has at least one mild solution. \end{theorem} \begin{proof} Let $m(t)$ be a solution of the scalar equation \begin{equation} m(t)=MN+RM\int_{0}^{t} h(s,m(s))ds, \label{eq7} \end{equation} where $R=\int_{0}^{t}q(s)ds$. Defined a map $Q: \mathcal {C}([0,b];\mathbb{X})\to\mathcal {C}([0,b];\mathbb{X})$ by \begin{equation} (Qu)(t)=U_u(t,0)g(u)+\int_{0}^{t}U_u(t,s)\int_{0}^{s}f(s,\tau,u(\tau))d\tau ds,\quad t\in [0,b] \label{eq8} \end{equation} for all $u\in \mathcal {C}([0,b];\mathbb{X})$. We can show that $Q$ is continuous by the usual techniques (see, e.g. \cite{n1,n2}). We denote by $W_0=\{u\in \mathcal {C}([0,b];\mathbb{X}),\,\|u(t)\|\leq m(t)\text{ for all }t\in [0,b]\}$. Then $W_0\subseteq \mathcal {C}([0,b];\mathbb{X})$ is bounded and convex. Define $W_1=\overline{{\rm conv}}K(W_0)$, where $\overline{{\rm conv}}$ means the closure of the convex hull in $\mathcal {C}([0,b];\mathbb{X})$. As $U_u(t,s)$ is equicontinuous, $g$ is compact and $W_0\subseteq\mathcal {C}([0,b];\mathbb{X})$ is bounded, due to Lemma \ref{lem2.7} and hypothesis (Hf)(2), $W_1\subseteq\mathcal {C}([0,b];\mathbb{X})$ is bounded closed convex nonempty and equicontinuous on $[0,b]$. For any $u\in Q(W_0)$, we know \begin{align*} \|u(t)\|&\leq MN+M\int_{0}^{t}\int_{0}^{s}q(s)h(\tau,m(\tau))d\tau ds \\ &\leq MN+M\int_{0}^{t}h(\tau,m(\tau))d\tau\int_{0}^{t}q(s)ds\\ &\leq MN+MR\int_{0}^{t}h(s,m(s))ds\\ &= m(t) \end{align*} for $t\in [0,b]$. It follows that $W_1\subset W_0$. We define $W_{n+1}=\overline{{\rm conv}} Q(W_n)$, for $n=1,2,\dots$. Form above we know that $\{W_n\}_{n=1}^{\infty}$ is a decreasing sequence of bounded, closed, convex, equicontinuous on [0,b] and nonempty subsets in $\mathcal {C}([0,b],\mathbb{X})$. Now for $n\geq 1$ and $t\in [0,b]$, $W_n(t)$ and $Q(W_n(t))$ are bounded subsets of $\mathbb{X}$, hence, for any $\varepsilon>0$, there is a sequence $\{u_k\}_{k=1}^{\infty}\subset W_n$ such that (see, e.g. \cite{bot}, pp.125) \begin{align*} \beta(W_{n+1}(t) &= \beta(QW_n(t))\\ &\leq 2\beta(\int_{0}^{t} U_u(t,s)\int_{0}^{s} f(s,\tau,\{u_k(\tau)\}_{k=1}^{\infty})d\tau ds)+\varepsilon \\ &\leq 2M\int_{0}^{t}\beta(\int_{0}^{s}f(s,\tau,\{u_k(\tau)\}_{k=1}^{\infty})d\tau)ds+\varepsilon \\ &\leq 4M\int_{0}^{t}\int_{0}^{s}\beta(f(s,\tau,\{u_k(\tau)\}_{k=1}^{\infty}))d\tau ds+\varepsilon\\ &\leq 4M\int_{0}^{t}\int_{0}^{s}\eta(s)\zeta(\tau)\beta(\{u_k(\tau)\}_{k=1}^{\infty})d\tau ds+\varepsilon\\ &\leq 4M\int_{0}^{t}\zeta(\tau)\beta(W_n(\tau))d\tau\int_{0}^{t}\eta(s)ds+\varepsilon\\ &\leq 4MK\int_{0}^{t}\zeta(s)\beta(W_n(s)) ds+\varepsilon. \end{align*} Since $\varepsilon>0$ is arbitrary, it follows from the above inequality that \begin{equation} \beta(QW_{n+1}(t))\leq 4MK\int_{0}^{t}\zeta(s)\beta(W_n(s))ds \label{eq10} \end{equation} for all $t\in [0,b]$. Because $W_n$ is decreasing for $n$, we define $$ \alpha(t)=\lim_{n\to\infty} \beta(W_n(t)) $$ for all $t\in [0,b]$. From (\ref{eq10}), we have $$ \alpha(t)\leq 4MK\int_{0}^{t}\zeta(s)\alpha(s)ds $$ for $t\in [0,b]$, which implies that $\alpha(t)=0$ for all $t\in [0,b]$. By Lemma \ref{lem2.3}, we know that $\lim_{n\to\infty}\beta_\mathcal {C}(W_n)=0$. Using Lemma \ref{lem2.1}, we know that $W=\bigcap_{n=1}^{\infty}W_n$ is convex compact and nonempty in $\mathcal {C}([0,b];\mathbb{X})$ and $Q(W)\subset W$. By the famous Schauder's fixed point theorem, there exists at least one mild solution $u$ of the initial value problem \eqref{eq1}--\eqref{eq2}, where $u\in W$ is a fixed point of the continuous map $Q$. \end{proof} \begin{remark} \label{rmk3.2} \rm If the function $f$ is compact or Lipschitz continuous (see, e.g. \cite{bys1,n1,x1}), then (Hf)(3) is automatically satisfied. \end{remark} In some of the early related results in references and above result, it is supposed that the map $g$ is uniformly bounded. We indicate here that this condition can be released. In fact, if $g$ is compact, then it must be bounded on bounded set. Here we give an existence result under another growth condition of $f$ (see, \cite{don,x3}), when $g$ is not uniformly bounded. Precisely, we replace the hypothesis (Hf)(2) by \begin{itemize} \item[(Hf)(2')] There exists a function $p\in\mathcal {L}(0,b;\mathbb{R}^{+})$ and a increasing function $\psi: \mathbb{R}^{+}\to\mathbb{R}^{+}$ such that $\|f(t,s,u)\|\leq p(t)\psi(\|u\|)$, for a.e. $t\in [0,b]$, and all $u\in\mathcal {C}([0,b];\mathbb{X})$. \end{itemize} \begin{theorem} \label{thm3.3} Suppose that {\rm (HA), (Hf)(1), (Hf)(2'), (Hf)(3), (Hg)(1)} are satisfied. Then the equation \eqref{eq1}--\eqref{eq2} has at least one mild solution if \begin{equation} \lim_{k\to\infty}\sup\frac{M}{k}(\varphi(k)+b\psi(k)\int_{0}^{b}p(s)ds)<1, \label{eq11} \end{equation} where $\varphi(k)=\sup\{\|g(u)\|,\ \|u\|\leq k\}$. \end{theorem} \begin{proof} The inequality (\ref{eq11}) implies that there exists a constant $k>0$ such that $$ M(\varphi(k)+b\psi(k)\int_{0}^{b}p(s)ds)0$, there is a sequence $\{u_k\}_{k=1}^{\infty}\subset B$, such that $$ \beta(Q_2B(t))\leq 2\beta(\{Q_2u_{k}(t)\}_{n=1}^{\infty})+\varepsilon. $$ Note that $B$ and $Q_2B$ are equicontinuous, we can get from Lemma \ref{lem2.1}, Lemma \ref{lem2.4}, Lemma \ref{lem2.5} and (Hf)(3) that \begin{align*} \beta(Q_2B(t)) &\leq 2M\int_{0}^{t}\beta(\int_{0}^{s}f(s,\tau,\{u_k(\tau)\} _{k=1}^{\infty})d\tau)ds+\varepsilon\\ &\leq 4M\int_{0}^{t}\int_{0}^{s}\beta(f(s,\tau,\{u_k(\tau)\}_{k=1}^{\infty}))d\tau ds+\varepsilon\\ &\leq 4M\int_{0}^{t}\int_{0}^{s}\eta(s)\zeta(\tau)\beta(\{u_k(\tau)\}_{k=1}^{\infty}) d\tau ds+\varepsilon\\ &\leq 4M\int_{0}^{t}\zeta(\tau)\beta(B(\tau))d\tau\int_{0}^{t}\eta(s)ds+\varepsilon.\\ &\leq 4MK\int_{0}^{t}\zeta(\tau)\beta(B(\tau))d\tau+\varepsilon\\ &\leq 4MK\beta_\mathcal {C}(B)\int_{0}^{b}\zeta(s)ds+\varepsilon \end{align*} Taking supremum in $t\in [0,b]$, we have $$ \beta_\mathcal {C} (Q_2B)\leq 4MK\beta_\mathcal {C}(B)\int_{0}^{b}\zeta(s)ds+\varepsilon. $$ Since $\varepsilon>0$ is arbitrary, we have \begin{equation} \beta_\mathcal {C}(Q_2B)\leq 4MK\beta_\mathcal {C}(B)\int_{0}^{b}\zeta(s)ds \label{eq14} \end{equation} for any bounded $B\subset W$. Now, for any subset $B\subset W$, due to Lemma \ref{lem2.1}, (\ref{eq13}) and (\ref{eq14}) we have \begin{align*} \beta_\mathcal {C}(QB) &\leq \beta_\mathcal {C}(Q_1B)+\beta_\mathcal {C}(Q_2B)\\ &\leq (ML+4MK\int_{0}^{b}\zeta(s)ds)\beta_\mathcal {C}(B). \end{align*} By (\ref{eq12}) we know that $Q$ is a $\beta_\mathcal{C}$-contraction on $W$. By Lemma \ref{lem2.2}, there is a fixed point $u$ of $Q$ in $W$, which is a solution of \eqref{eq1}--\eqref{eq2}. This completes the proof. \end{proof} Now we give an existence result without the uniform boundedness of $g$. \begin{theorem} \label{thm4.2} Suppose that {\rm (HA), (Hf)(1), (Hf)(2'), (Hf)(3), (Hg)(1')} are satisfied. Then the equation \eqref{eq1}--\eqref{eq2} has at least one mild solution if \eqref{eq12} and the following condition are satisfied \begin{equation} ML+bM\int_{0}^{b}p(s)ds\lim_{k\to\infty}\sup\frac{\psi(k)}{k}<1. \label{eq15} \end{equation} \end{theorem} \begin{proof} From (\ref{eq15}) and the fact that $L<1$, there exists a constant $k>0$ such that $$ M(kL+bM\int_{0}^{b}p(s)ds\psi(k)+\|g(0)\|)