\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 60, 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 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/60\hfil Nonlocal fractional boundary-value problems] {Integro-differential equations of fractional order with nonlocal fractional boundary conditions associated with financial asset model} \author[B. Ahmad, S. K. Ntouyas \hfil EJDE-2013/60\hfilneg] {Bashir Ahmad, Sotiris K. Ntouyas} % in alphabetical order \address{Bashir Ahmad \newline Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia} \email{bashirahmad\_qau@yahoo.com} \address{Sotiris K. Ntouyas \newline Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece} \email{sntouyas@uoi.gr} \thanks{Submitted December 6, 2012. Published February 26, 2013.} \subjclass[2000]{34A08, 34B10, 34B15} \keywords{Fractional differential equations; integral boundary conditions;\hfill\break\indent existence; fixed point theorems; financial asset} \begin{abstract} In this article, we discuss the existence of solutions for a boundary-value problem of integro-differential equations of fractional order with nonlocal fractional boundary conditions by means of some standard tools of fixed point theory. Our problem describes a more general form of fractional stochastic dynamic model for financial asset. An illustrative example is also presented. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Formulation and basic result} Fractional calculus, regarded as a branch of mathematical analysis dealing with derivatives and integrals of arbitrary order, has been extensively developed and applied to a variety of problems appearing in sciences and engineering. It is worthwhile to mention that this branch of mathematics has played a crucial role in exploring various characteristics of engineering materials such as viscoelastic polymers, foams, gels, and animal tissues, and their engineering and scientific applications. For a recent detailed survey of the activities involving fractional calculus, we refer a recent paper by Machado, Kiryakova and Mainardi \cite{mach}. Some recent work on the topic can be found in \cite{Ravi, Agha, bvp1, BNi, BA5, ahmad, Kli, CuevS, Ford, Kirane1, Kirane2, S-T} and references therein. The underlying dynamics of equity prices following a jump process or a Levy process provide a basis for modeling of financial assets. The CGMY, KoBoL and FMLS are examples of some interesting financial models involving the dynamics of stock prices. In \cite{car}, it is shown that the prices of financial derivatives are expressible in terms of fractional derivative. In \cite{las}, the author described the dynamics of a financial asset by the fractional stochastic differential equation of order $\mu$ (representing the dynamical memory effects in the market stochastic evolution) with fractional boundary conditions. In the present paper, we study a more general model associated with financial asset. Precisely, we consider the following problem: $$\label{p} \begin{gathered} -D^{\alpha}x(t)= Af(t, x(t))+BI^{\beta}g(t, x(t)), \quad (n-1)<\alpha\leq n, \; t\in [0,1],\\ D^\delta x(0)=0, \quad D^{\delta+1}x(0)=0, \dots,\; D^{\delta+(n-2)}x(0)=0, \quad D^\delta x(1)=\int_0^\eta D^\delta x(s)ds, \end{gathered}$$ where $0<\delta\le 1$, $\alpha-\delta >n$, $0<\beta<1$, $0<\eta<1$, $D^{(\cdot)}$ denotes the Riemann-Lioville fractional derivative of order $(\cdot)$, $f, g$ are given continuous function, and $A,B$ are real constants. We remark that the problem \eqref{p} also arises in real estate asset securitization modeling \cite{tao}. By the substitution $x(t)=I^{\delta}y(t)=D^{-\alpha}y(t)$, the problem \eqref{p} takes the form $$\label{21} \begin{gathered} -D^{\alpha-\delta}y(t)= Af(t,I^{\delta}y(t))+BI^{\beta}g(t, I^{\delta}y(t)), \quad t\in [0,1], \\ y(0)=0, \quad y'(0)=0,\dots, y^{(n-2)}(0)=0,\quad y(1)=\int_0^\eta y(s)ds. \end{gathered}$$ \begin{lemma}\label{22} For any $h \in C(0,1)\cap L(0, 1)$, the unique solution of the linear fractional boundary-value problem $$\label{11} \begin{gathered} -D^{\alpha-\delta}y(t)= h(t), \quad t\in [0,1], \\ y(0)=0, \quad y'(0)=0, \dots , y^{(n-2)}(0)=0, \quad y(1)=\int_0^\eta y(s)ds, \end{gathered}$$ is $$y(t)=-I^{\alpha-\delta} h(t)+\frac{(\alpha-\delta) t^{\alpha-\delta-1}}{\alpha-\delta-\eta^{\alpha-\delta}} \Big(I^{\alpha-\delta}h(1)-I^{\alpha-\delta+1}h(\eta)\Big),$$ where $I^{(\cdot)}(\cdot)$ denotes Riemann-Liouville integral. \end{lemma} \begin{proof} It is well known that the solutions of fractional differential equation in \eqref{22} can be written as $$\label{sol} y(t)= -I^{\alpha-\delta} h(t)+c_1t^{\alpha-\delta-1}+c_2t^{\alpha-\delta-2}+c_3t^{\alpha-\delta-3} +\dots+c_nt^{\alpha-\delta-n},$$ where $c_1, c_2, \dots ,c_n \in \mathbb{R}$ are arbitrary constants \cite{Kil}. Using the given boundary conditions, we find that $c_2=0$, $c_3=0,\dots,c_n=0$ and $$c_1=\frac{\alpha-\delta}{\alpha-\delta-\eta^{\alpha-\delta}} \Big(I^{\alpha-\delta}h(1)-I^{\alpha-\delta+1}h(\eta)\Big).$$ Substituting these values in \eqref{22} yields $$y(t)=-I^{\alpha-\delta} h(t)+\frac{(\alpha-\delta) t^{\alpha-\delta-1}}{\alpha-\delta-\eta^{\alpha-\delta}} \Big(I^{\alpha-\delta}h(1)-I^{\alpha-\delta+1}h(\eta)\Big).$$ This completes the proof. \end{proof} Thus, the solution of the linear variant of the problem \eqref{p} can be written as \begin{align*} x(t)& = I^{\delta}y(t)\\&= I^{\delta}\Big[-I^{\alpha-\delta} h(t)+\frac{(\alpha-\delta) t^{\alpha-\delta-1}}{\alpha-\delta-\eta^{\alpha-\delta}} \Big(I^{\alpha-\delta}h(1)-I^{\alpha-\delta+1}h(\eta)\Big)\Big]\\ &= -I^{\alpha}h(t)+ \frac{(\alpha-\delta) }{\alpha-\delta-\eta^{\alpha-\delta}} \Big(I^{\alpha-\delta}h(1)-I^{\alpha-\delta+1}h(\eta)\Big) \int_0^t\frac{(t-s)^{\delta-1}}{\Gamma(\delta)}s^{\alpha-\delta-1}ds \\ &= -I^{\alpha}h(t)+ \frac{(\alpha-\delta) }{\alpha-\delta-\eta^{\alpha-\delta}} \Big(I^{\alpha-\delta}h(1)-I^{\alpha-\delta+1}h(\eta)\Big)\times\\ & \quad\times\Big\{\frac{t^{\alpha-1}}{\Gamma(\delta)} \int_0^1(1-\nu)^{\delta-1}\nu^{\alpha-\delta-1}d\nu\Big\}, \end{align*} where we have used the substitution $s= \nu t$ in the integral of the last term. Using the relation for Beta function $B(\cdot,\cdot)$: $$B(\beta+1, \alpha)=\int_0^1 (1-u)^{\alpha-1}u^{\beta}du =\frac{\Gamma(\alpha)\Gamma(\beta+1)}{\Gamma(\alpha+\beta+1)},$$ we obtain $$\label{vlin} x(t)= -I^{\alpha}h(t)+ \frac{\Gamma(\alpha-\delta+1)t^{\alpha-1} }{(\alpha-\delta-\eta^{\alpha-\delta})\Gamma(\alpha)} \Big(I^{\alpha-\delta}h(1)-I^{\alpha-\delta+1}h(\eta)\Big).$$ The solution of the original nonlinear problem \eqref{p} can be obtained by replacing $h$ with the right hand side of the fractional equation of \eqref{p} in \eqref{vlin}. Let ${\mathcal{C}}=C([0,1], {\mathbb R})$ denote the Banach space of all continuous functions from $[0,1] \to {\mathbb R}$ endowed with the norm defined by $\|x\|= \sup \{|x(t)|, t \in [0,1]\}$. In relation to problem \eqref{p}, we define an operator $\mathcal{U}: {\mathcal{C}} \to {\mathcal{C}}$ as \begin{align*} &(\mathcal{U} x)(t)\\ &= -A\int_0^t \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s, x(s))ds-B\int_0^t \frac{(t-s)^{\alpha+\beta-1}}{\Gamma(\alpha+\beta)}g(s, x(s))ds\\ &\quad + Q t^{\alpha-1}\Big[A\int_0^1 \frac{(1-s)^{\alpha-\delta-1}}{\Gamma(\alpha-\delta)}f(s, x(s))ds+B\int_0^1 \frac{(1-s)^{\alpha-\delta+\beta-1}}{\Gamma(\alpha-\delta+\beta)}g(s, x(s))ds\\ &\quad - A \int_0^\eta \frac{(\eta-s)^{\alpha-\delta}}{\Gamma(\alpha-\delta+1)}f(s, x(s))ds-B\int_0^\eta \frac{(\eta-s)^{\alpha-\delta+\beta}}{\Gamma(\alpha-\delta+\beta+1)}g(s, x(s))ds\Big], \end{align*} where $$Q=\frac{\Gamma(\alpha-\delta+1)}{(\alpha-\delta-\eta^{\alpha-\delta}) \Gamma(\alpha)}, \quad \alpha \ne \delta+\eta^{\alpha-\delta}.$$ For the sake of convenience, we set \label{omega} \begin{aligned} \Omega &=\sup_{t \in [0,1]}\Big\{|A|\Big[\frac{t^\alpha}{\Gamma(\alpha+1)} +|Q|t^{\alpha-1}\Big(\frac{1}{\Gamma(\alpha-\delta+1)} +\frac{\eta^{\alpha-\delta+1}}{\Gamma(\alpha-\delta+2)}\Big)\Big] \\ &\quad +|B|\Big[\frac{t^{\alpha+\beta}}{\Gamma(\alpha+\beta+1)} +|Q|t^{\alpha-1}\Big(\frac{1}{\Gamma(\alpha-\delta+\beta+1)} +\frac{\eta^{\alpha-\delta+\beta+1}}{\Gamma(\alpha-\delta+\beta+2)} \Big)\Big]\Big\}. \end{aligned} \subsection{Existence results via Banach's fixed point theorem} \begin{theorem}\label{t1} Assume that $f, g : [0,1]\times \mathbb{R} \to \mathbb{R}$ are continuous functions satisfying the condition: \begin{itemize} \item[(A1)] $|f(t,x)-f(t,y)| \le L_1 |x-y|$, $|g(t,x)-g(t,y)| \le L_2 |x-y|$, for all $t \in [0,1]$, $L_1,L_2>0$, $x, y\in \mathbb{R}$. \end{itemize} Then the boundary-value problem \eqref{p} has a unique solution if $L < 1/\Omega$, where $L=\max\{L_1, L_2\}$ and $\Omega$ is given by \eqref{omega}. \end{theorem} \begin{proof} Let us define $M=\max\{M_1, M_2\}$, where $M_1, M_2$ are finite numbers given by $\sup_{t \in [0,1]}|f(t,0)|=M_1,\,\sup_{t \in [0,1]}|g(t,0)|=M_2$. Selecting $r \ge \frac{\Omega M}{1-L\Omega}$, we show that $\mathcal{U} B_r \subset B_r$, where $B_r=\{x \in {\mathcal{C}}: \|x\|\le r \}$. Using that $|f(s,x(s))\le |f(s,x(s))-f(s,0)|+|f(s,0)|\le L_1r+M_1$, $|g(s,x(s))| \le |g(s,x(s))-g(s,0)|+|g(s,0)|\le L_2r+M_2$ for $x \in B_r$ and \eqref{omega}, it can easily be shown that \begin{align*} &\|(\mathcal{U}x)\|\\ & \le (Lr+M )\sup_{t \in [0,1]}\Big\{|A|\Big[\frac{t^\alpha}{\Gamma(\alpha+1)}+|Q|t^{\alpha-1}\Big(\frac{1}{\Gamma(\alpha-\delta+1)} +\frac{\eta^{\alpha-\delta+1}}{\Gamma(\alpha-\delta+2)}\Big)\Big] \\ &\quad +|B|\Big[\frac{t^{\alpha+\beta}}{\Gamma(\alpha+\beta+1)} +|Q|t^{\alpha-1}\Big(\frac{1}{\Gamma(\alpha-\delta+\beta+1)} +\frac{\eta^{\alpha-\delta+\beta+1}}{\Gamma(\alpha-\delta+\beta+2)} \Big)\Big]\Big\} \\ &= (Lr+M)\Omega \le r, \end{align*} which implies that $\mathcal{U} B_r \subset B_r$. Now, for $x, y \in {\mathcal{C}}$ we obtain \begin{align*} &\|\mathcal{U}x-\mathcal{U}y\| \\ & \le \sup_{t\in [0,1]}\Big\{|A|\int_0^t \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}|f(s,x(s))-f(s,y(s))|ds\\ &\quad + |B|\int_0^t \frac{(t-s)^{\alpha+\beta-1}}{\Gamma(\alpha+\beta)}|g(s,x(s))-g(s,y(s))|ds\\ &\quad +|Q| t^{\alpha-1}\Big[|A|\int_0^1 \frac{(1-s)^{\alpha-\delta-1}}{\Gamma(\alpha-\delta)}|f(s,x(s))-f(s,y(s))|ds\\ &\quad +|B|\int_0^1 \frac{(1-s)^{\alpha-\delta+\beta-1}}{\Gamma(\alpha-\delta+\beta)}|g(s,x(s))-g(s,y(s))|ds\\ &\quad + |A| \int_0^\eta \frac{(\eta-s)^{\alpha-\delta}}{\Gamma(\alpha-\delta+1)}|f(s,x(s))-f(s,y(s))|ds\\ &\quad +|B|\int_0^\eta \frac{(\eta-s)^{\alpha-\delta+\beta}}{\Gamma(\alpha-\delta+\beta+1)}|g(s,x(s))-g(s,y(s))|ds\Big]\Big\} \\ &\le L \sup_{t \in [0,1]}\Big\{|A|\Big[\frac{t^\alpha}{\Gamma(\alpha+1)}+|Q|t^{\alpha-1}\Big(\frac{1}{\Gamma(\alpha-\delta+1)} +\frac{\eta^{\alpha-\delta+1}}{\Gamma(\alpha-\delta+2)}\Big)\Big] \\ &\quad +|B|\Big[\frac{t^{\alpha+\beta}}{\Gamma(\alpha+\beta+1)}+|Q|t^{\alpha-1}\Big(\frac{1}{\Gamma(\alpha-\delta+\beta+1)} +\frac{\eta^{\alpha-\delta+\beta+1}}{\Gamma(\alpha-\delta+\beta+2)}\Big)\Big] \Big\} \\ &\quad\times\|x-y\|\\ &= L \Omega \|x-y\|. \end{align*} By the given assumption, $L<1/\Omega$. Therefore $\mathcal{U}$ is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle (Banach fixed point theorem). \end{proof} Now we present another variant of existence-uniqueness result. This result is based on the H\"older's inequality. \begin{theorem}\label{t11} Suppose that the continuous functions $f$ and $g$ satisfy the following assumptions: \begin{itemize} \item [(H1)] $|f(t,x)-f(t,y)|\le m(t)|x-y|$, $|g(t,x)-g(t,y)|\le n(t)|x-y|$, for $t\in [0,1]$, $x,y\in {\mathbb R}$, and $m,n\in L^{\frac{1}{\gamma}}([0,1], {\mathbb R}^+)$, $\gamma\in (0,\alpha-\delta-n)$. \item[(H2)] $|A|\|m\|Z_1+|B|\|n\|Z_2<1$, where \begin{align*} Z_1&= \frac{1}{\Gamma(\alpha)} \Big(\frac{1-\gamma}{\alpha-\gamma}\Big)^{1-\gamma} +\frac{|Q|}{\Gamma(\alpha-\delta)} \Big(\frac{1-\gamma}{\alpha-\delta-\gamma}\Big)^{1-\gamma} \\ &\quad+\frac{|Q|}{\Gamma(\alpha-\delta+1)} \Big(\frac{1-\gamma}{\alpha-\delta+1-\gamma}\Big)^{1-\gamma} \eta^{\alpha-\delta+1-\gamma}, \end{align*} \begin{align*} Z_2&= \frac{1}{\Gamma(\alpha+\beta)} \Big(\frac{1-\gamma}{\alpha+\beta-\gamma}\Big)^{1-\gamma} +\frac{|Q|}{\Gamma(\alpha-\delta+\beta)} \Big(\frac{1-\gamma}{\alpha-\delta+\beta-\gamma}\Big)^{1-\gamma}\\ &\quad +\frac{|Q|}{\Gamma(\alpha-\delta+\beta+1)} \Big(\frac{1-\gamma}{\alpha-\delta+\beta+1-\gamma} \Big)^{1-\gamma}\eta^{\alpha-\delta+\beta+1-\gamma}, \end{align*} \end{itemize} and $\|\mu\|=\big(\int_0^1|\mu(s)|^{\frac{1}{\gamma}}ds\big)^{\gamma}$, $\mu=m, n$. Then the boundary value problem \eqref{p} has a unique solution. \end{theorem} \begin{proof} For $x,y\in {\mathbb R}$ and for each $t\in [0,1]$, by H\"older inequality, we have \begin{align*} &\|\mathcal{U}x-\mathcal{U}y\| \\ &\le \sup_{t\in [0,1]}\Big\{|A|\int_0^t\frac{(t-s)^{\alpha-1}} {\Gamma(\alpha)}m(s)|x(s)-y(s)|ds\\ &\quad +|B|\int_0^t\frac{(t-s)^{\alpha+\beta-1}} {\Gamma(\alpha+\beta)}n(s)|x(s)-y(s)|ds\\ &\quad +|Q|\Big[|A|\int_0^1\frac{(1-s)^{\alpha-\delta-1}} {\Gamma(\alpha-\delta)}m(s)|x(s)-y(s)|ds\\ &\quad +|B|\int_0^1\frac{(1-s)^{\alpha-\delta+\beta-1}} {\Gamma(\alpha-\delta+\beta)}n(s)|x(s)-y(s)|ds\\ &\quad +|A|\int_0^{\eta}\frac{(\eta-s)^{\alpha-\delta}} {\Gamma(\alpha-\delta+1)}m(s)|x(s)-y(s)|ds\\ &\quad +|B|\int_0^{\eta}\frac{(\eta-s)^{\alpha-\delta+\beta}} {\Gamma(\alpha-\delta+\beta+1)}n(s)|x(s)-y(s)|ds\Big]\Big\}\\ &\leq \sup_{t\in [0,1]}\Big\{\frac{|A|\|m\|}{\Gamma(\alpha)} \Big(\frac{1-\gamma}{\alpha-\gamma}\Big)^{1-\gamma}t^{\alpha-\gamma}+ \frac{|B|\|n\|}{\Gamma(\alpha+\beta)} \Big(\frac{1-\gamma}{\alpha+\beta-\gamma}\Big)^{1-\gamma} t^{\alpha+\beta-\gamma}\\ &\quad +|Q|\Big[\frac{|A|\|m\|}{\Gamma(\alpha-\delta)} \Big(\frac{1-\gamma}{\alpha-\delta-\gamma}\Big)^{1-\gamma} +\frac{|B|\|n\|}{\Gamma(\alpha-\delta+\beta)} \Big(\frac{1-\gamma}{\alpha-\delta+\beta-\gamma}\Big)^{1-\gamma} \\ &\quad +\frac{|A|\|m\|}{\Gamma(\alpha-\delta+1)} \Big(\frac{1-\gamma}{\alpha-\delta+1-\gamma}\Big)^{1-\gamma} \eta^{\alpha-\delta+1-\gamma}\\ &\quad +\frac{|B|\|n\|}{\Gamma(\alpha-\delta+\beta+1)} \Big(\frac{1-\gamma}{\alpha-\delta+\beta+1-\gamma}\Big)^{1-\gamma} \eta^{\alpha-\delta+\beta+1-\gamma}\Big]\Big\}\|x-y\|\\ &\le |A|\|m\|\Big[\frac{1}{\Gamma(\alpha)} \Big(\frac{1-\gamma}{\alpha-\gamma}\Big)^{1-\gamma} +\frac{|Q|}{\Gamma(\alpha-\delta)} \Big(\frac{1-\gamma}{\alpha-\delta-\gamma}\Big)^{1-\gamma}\\ &\quad +\frac{|Q|}{\Gamma(\alpha-\delta+1)} \Big(\frac{1-\gamma}{\alpha-\delta+1-\gamma}\Big)^{1-\gamma}\Big]\|x-y\|\\ &\quad +|B|\|n\|\Big[\frac{1}{\Gamma(\alpha+\beta)} \Big(\frac{1-\gamma}{\alpha+\beta-\gamma}\Big)^{1-\gamma} +\frac{|Q|}{\Gamma(\alpha-\delta+\beta)} \Big(\frac{1-\gamma}{\alpha-\delta+\beta-\gamma}\Big)^{1-\gamma} \\ &\quad +\frac{|Q|}{\Gamma(\alpha-\delta+\beta+1)} \Big(\frac{1-\gamma}{\alpha-\delta+\beta+1-\gamma}\Big)^{1-\gamma} \eta^{\alpha-\delta+\beta+1-\gamma}\Big]\|x-y\|\\ &= [|A|\|m\|Z_1+|B|\|n\|Z_2]\|x-y\|. \end{align*} In view of condition (H2), it follows that $\mathcal{U}$ is a contraction mapping. Hence, Banach's fixed point theorem applies and $\mathcal{U}$ has a unique fixed point which is the unique solution of problem \eqref{p}. This completes the proof. \end{proof} \subsection{Existence result via Leray-Schauder Alternative} \begin{lemma}[Nonlinear alternative for single valued maps \cite{GrDu}] \label{lls} Let $E$ be a Banach space, $C$ a closed, convex subset of $E$, $U$ an open subset of $C$ and $0\in U$. Suppose that $F:\overline{U}\to C$ is a continuous, compact (that is, $F(\overline{U})$ is a relatively compact subset of $C$) map. Then either \begin{itemize} \item[(i)] $F$ has a fixed point in $\overline{U}$, or \item[(ii)] there is a $u\in \partial U$ (the boundary of $U$ in $C$) and $\lambda\in(0,1)$ with $u=\lambda F(u)$. \end{itemize} \end{lemma} \begin{theorem}\label{tls} Assume that $f, g : [0,1]\times \mathbb{R} \to \mathbb{R}$ are continuous functions. Assume that: \begin{itemize} \item[(A3)] There exist functions $p_1, p_2 \in L^1([0,1], {\mathbb R}^+)$, and nondecreasing functions $\psi_1, \psi_2: {\mathbb{R}}^+\to { \mathbb{R}}^+$ such that $|f(t,x)|\le p_1(t)\psi_1(\|x\|),\quad |g(t,x)|\le p_2(t)\psi_2(\|x\|),$ for all $(t,x) \in [0,1] \times {\mathbb R}$. \item[(A4)] There exists a constant $M>0$ such that $$\frac{M}{ |A|\Lambda_1\psi_1(M)\|p_1\|_{L^1}+|B|\Lambda_1\psi_2(M)\|p_2\|_{L^1}}> 1,$$ where \begin{gather*} \Lambda_1= \frac{1}{\Gamma(\alpha+1)}+\frac{|Q|}{\Gamma(\alpha-\delta+1)} +\frac{|Q|}{\Gamma(\alpha-\delta+2)}, \\ \Lambda_2= \frac{1}{\Gamma(\alpha+\beta+1)}+\frac{|Q|}{\Gamma(\alpha-\delta+\beta+1)} +\frac{|Q|}{\Gamma(\alpha-\delta+\beta+2)}. \end{gather*} \end{itemize} Then the boundary-value problem \eqref{p} has at least one solution on $[0,1]$. \end{theorem} \begin{proof} Consider the operator $\mathcal{U}: \mathcal{C} \to \mathcal{C}$ with $x=\mathcal{U} x$, where \begin{align*} &(\mathcal{U} x)(t)\\ &= -A\int_0^t \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s, x(s))ds-B\int_0^t \frac{(t-s)^{\alpha+\beta-1}}{\Gamma(\alpha+\beta)}g(s, x(s))ds\\ &\quad + Q t^{\alpha-1}\Big[A\int_0^1 \frac{(1-s)^{\alpha-\delta-1}}{\Gamma(\alpha-\delta)}f(s, x(s))ds+B\int_0^1 \frac{(1-s)^{\alpha-\delta+\beta-1}}{\Gamma(\alpha-\delta+\beta)}g(s, x(s))ds\\ &\quad - A \int_0^\eta \frac{(\eta-s)^{\alpha-\delta}}{\Gamma(\alpha-\delta+1)}f(s, x(s))ds-B\int_0^\eta \frac{(\eta-s)^{\alpha-\delta+\beta}}{\Gamma(\alpha-\delta+\beta+1)}g(s, x(s))ds\Big]. \end{align*} We show that $F$ maps bounded sets into bounded sets in $C([0,1], \mathbb{R})$. For a positive number $r$, let $B_r = \{x \in C([0,1], \mathbb{R}): \|x\| \le r \}$ be a bounded set in $C([0,1], \mathbb{R})$. Then \begin{align*} &|(\mathcal{U} x)(t)|\\ &\leq |A|\int_0^t \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}p_1(s)\psi_1(\|x\|)ds+|B|\int_0^t \frac{(t-s)^{\alpha+\beta-1}}{\Gamma(\alpha+\beta)}p_2(s)\psi_2(\|x\|)ds\\ &\quad + |Q| t^{\alpha-1}\Big[|A|\int_0^1 \frac{(1-s)^{\alpha-\delta-1}}{\Gamma(\alpha-\delta)}p_1(s)\psi_1(\|x\|)ds\\ &\quad +|B|\int_0^1 \frac{(1-s)^{\alpha-\delta+\beta-1}}{\Gamma(\alpha-\delta+\beta)}p_2(s)\psi_2(\|x\|)ds\\&\quad + |A| \int_0^\eta \frac{(\eta-s)^{\alpha-\delta}}{\Gamma(\alpha-\delta+1)}p_1(s)\psi_1(\|x\|)ds\\ &\quad +|B|\int_0^\eta \frac{(\eta-s)^{\alpha-\delta+\beta}}{\Gamma(\alpha-\delta+\beta+1)}p_2(s)\psi_2(\|x\|)ds\Big]\\ &\leq |A|\psi_1(r)\|p_1\|_{L^1}\Big\{\frac{1}{\Gamma(\alpha+1)}+\frac{|Q|}{\Gamma(\alpha-\delta+1)} +\frac{|Q|}{\Gamma(\alpha-\delta+2)}\Big\}\\ &\quad + |B|\psi_2(r)\|p_2\|_{L^1}\Big\{\frac{1}{\Gamma(\alpha+\beta+1)}+\frac{|Q|}{\Gamma(\alpha-\delta+\beta+1)} +\frac{|Q|}{\Gamma(\alpha-\delta+\beta+2)}\Big\}. \end{align*} Consequently \begin{align*} &\|\mathcal{U} x\|\\&\leq |A|\psi_1(r)\|p_1\|_{L^1}\Big\{\frac{1}{\Gamma(\alpha+1)}+\frac{|Q|}{\Gamma(\alpha-\delta+1)} +\frac{|Q|}{\Gamma(\alpha-\delta+2)}\Big\}\\ &\quad + |B|\psi_2(r)\|p_2\|_{L^1}\Big\{\frac{1}{\Gamma(\alpha+\beta+1)}+\frac{|Q|}{\Gamma(\alpha-\delta+\beta+1)} +\frac{|Q|}{\Gamma(\alpha-\delta+\beta+2)}\Big\}. \end{align*} Next we show that $F$ maps bounded sets into equicontinuous sets of $C([0,1], \mathbb{R})$. Let $t_1, t_2 \in [0,1]$ with \$t_1