About this Journal Submit a Manuscript Table of Contents
International Journal of Differential Equations
Volume 2011 (2011), Article ID 304570, 15 pages
http://dx.doi.org/10.1155/2011/304570
Research Article

Existence and Uniqueness Theorem of Fractional Mixed Volterra-Fredholm Integrodifferential Equation with Integral Boundary Conditions

1Department of Mathematics, Faculty of Science, Dohuk University, Kurdistan, Iraq
2Department of Mathematics, Faculty of Science, Zakho University, Kurdistan, Iraq
3Department of Mathematics and Science, College of Education and Basic Sciences, Ajman University of Science and Technology, UAE

Received 7 May 2011; Accepted 24 May 2011

Academic Editor: Shaher Momani

Copyright © 2011 Shayma Adil Murad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the existence and uniqueness of the solutions of mixed Volterra-Fredholm type integral equations with integral boundary condition in Banach space. Our analysis is based on an application of the Krasnosel'skii fixed-point theorem.

1. Introduction

In the last century, notable contributions have been made to both the theory and applications of the fractional differential equations. For the theory part, Momani and Hadid have investigated the local and global existence theorem of both fractional differential equation and fractional integrodifferential equations; see [16]. Fractional-order differential equations have recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering.

Integrodifferential equations with integral boundary conditions are often encountered in various applications; it is worthwhile mentioning the applications of those conditions in the study of population dynamics and cellular systems. For a detailed description of the integral boundary conditions, we refer the reader to a recent paper [7]. In [8], Tidke studied the problem of existence of global solutions to nonlinear mixed Volttera-Fredholm integrodifferential equations with nonlocal condition.

Ahmad and Nieto [9] studied some existence results for boundary value problem involving a nonlinear integrodifferential equation of fractional order with integral equation.

Very recently NGuérékata [10] discussed the existence of solutions of fractional abstract differential equations with nonlocal initial condition. Anguraj et al. [11] studied the existence and uniqueness theorem for the nonlinear fractional mixed Volterra-Fredholm integrodifferential equation with nonlocal initial condition.

Motivated by these works, we study in this paper the existence of solution of boundary value problem for fractional integrodifferential equations ( in the case 1<𝛼2) in Banach spaces by using Banach and Krasnosel'skii fixed-point theorems.

2. Preliminaries

First of all, we recall some basic definitions; see [1215].

Definition 2.1. For a function 𝑓 given on the interval [𝑎,𝑏], the Caputo fractional order derivative of 𝑓 is defined by 𝑡𝑎𝐷𝛼1𝑓(𝑡)=Γ(𝑛𝛼)𝑡𝑎(𝑡𝑠)𝑛𝛼1𝑓(𝑛)(𝑠)𝑑𝑠,(2.1) where 𝑛=[𝛼]+1 and [𝛼] denotes the integer part of 𝛼.

Lemma 2.2. Let 𝛼>0, then 𝑡𝑎𝐷𝑡𝑎𝛼𝐷𝛼𝑦(𝑡)=𝑦(𝑡)+𝑐0+𝑐1𝑡+𝑐2𝑡2++𝑐𝑛1𝑡𝑛1,(2.2) for some 𝑐𝑖𝑅, i=0,1,,n1,𝑛=[𝛼]+1.

Definition 2.3. Let 𝑓 be a function which is defined almost everywhere (a.e) on [𝑎,𝑏], for 𝛼>0, we define 𝑏𝑎𝐷𝛼1𝑓=Γ(𝛼)𝑏𝑎(𝑏𝑡)𝛼1𝑓(𝑡)𝑑𝑡,(2.3) provided that the integral (Lebesgue) exists.

Theorem 2.4 (Krasnosel’skii fixed point theorem). Let 𝑀 be a closed-convex bounded nonempty subset of a Banach space 𝑋. Let 𝐴 and 𝐵 be two operators such that (i)𝐴𝑥+𝐵𝑦=𝑀, whenever 𝑥,𝑦𝑀,(ii)𝐴 is compact and continuous;(iii)𝐵 is a contraction mapping,then there exists 𝑧𝑀 such that 𝑧=𝐴𝑧+𝐵𝑧.
Let 𝑋 be a Banach space with the norm . Let 𝐶=([0,𝑇],𝑋) be Banach space of all continuous functions 𝜓[0,𝑇]𝑋, with supermum norm 𝜓=sup{𝜓(𝑠)𝑠[0,𝑇]}. Consider the fractional mixed Volttera-Fredholm integrodifferential equation with boundary conditions, which has the form 𝐷𝛼𝑦(𝑡)=𝑓𝑡,𝑦(𝑡),𝑡0𝑘(𝑡,𝑠,𝑦(𝑠))𝑑𝑠,𝑇01,(𝑡,𝑠,𝑦(𝑠))𝑑𝑠(2.4)𝑦(0)𝑦(0)=𝑇0𝑔(𝑦(𝑠))𝑑𝑠,𝑦(𝑇)𝑦(𝑇)=𝑇0(𝑦(𝑠))𝑑𝑠,(2.5) where 1<𝛼2, 𝐷𝛼 is the Caputo fractional derivative and the nonlinear functions 𝑓[0,𝑇]×𝑋×𝑋×𝑋𝑋, 𝑘,1[0,𝑇]×[0,𝑇]×𝑋𝑋 and 𝑔,𝑋𝑋 satisfy the following hypotheses: (H1)there exists constants 𝐺1,𝐺2 such that (𝑦)𝐺1,𝑔(𝑦)𝐺2 for 𝑦𝑋,(H2)there exists constants 𝑏1,𝑏2 such that (𝑥)(𝑦)𝑏1𝑥𝑦 and 𝑔(𝑥)𝑔(𝑦)𝑏2𝑥𝑦,𝑥,𝑦𝑋,(2.6)(H3)There exists continuous functions 𝑝[0,𝑇]𝑅+=[0,) and 𝑝1[0,𝑇]𝑅+ such that 𝑡0(𝑘(𝑡,𝑠,𝑥)𝑘(𝑡,𝑠,𝑦))𝑑𝑠𝑝(𝑡)𝑥𝑦 and 𝑡0𝑘(𝑡,𝑠,𝑦)𝑑𝑠𝑝1(𝑡)𝑦, for every 𝑡,𝑠[0,𝑇] and 𝑥,𝑦𝑋,(H4)there exists continuous functions 𝑞[0,𝑇]𝑅+ and 𝑞1[0,𝑇]𝑅+ such that 𝑇0(1(𝑡,𝑠,𝑥)1(𝑡,𝑠,𝑦)𝑑𝑠𝑞(𝑡)𝑥𝑦 and 𝑇01(𝑡,𝑠,𝑦)𝑑𝑠𝑞1(𝑡)𝑦𝑓𝑜𝑟𝑒𝑣𝑒𝑟𝑦𝑡,𝑠[0,𝑇]𝑎𝑛𝑑𝑥,𝑦𝑋(H5)there exists continuous function 𝐿[0,𝑇]𝑅+, and 𝑁1is positive constant such that 𝑓(𝑡,𝑥1,𝑦1,𝑧1)𝑓(𝑡,𝑥2,𝑦2,𝑧2)𝐿(𝑡)𝐾(𝑥1𝑥2+𝑦1𝑦2+𝑧1𝑧2) and 𝑁1=sup𝑡[0,𝑇]𝑓(𝑡,0,0,0), for every 𝑡[0,𝑇] and 𝑥1,𝑦1,𝑧1,𝑥2,𝑦2,𝑧2𝑋, where 𝐾𝑅+(0,) is continuous nondecreasing function satisfying 𝐾(𝛾(𝑡)𝑥)𝛾(𝑡)𝐾(𝑥), where 𝛾 is a continuous function 𝛾[0,𝑇]𝑅+.

Lemma 2.5. Let 1<𝛼2 and 𝑓𝐽×𝑋𝑋, where 𝐽=[0,𝑇], be a continuous function, then the solution of fractional differential equation (2.4) with the boundary condition (2.5) is 𝑦(𝑡)=(1+𝑡)𝑇𝑇0(𝑦(𝑠))𝑑𝑠+1(1+𝑡)𝑇𝑇0𝑔(𝑦(𝑠))𝑑𝑠(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼1𝑓Γ(𝛼)𝑠,𝑦(𝑠),𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01+(𝑠,𝜏,𝑦(𝜏))𝑑𝜏(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼2𝑓Γ(𝛼1)𝑠,𝑦(𝑠),𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01+(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠𝑡0(𝑡𝑠)𝛼1𝑓Γ(𝛼)𝑠,𝑦(𝑠),𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠.(2.7)

Proof. By Lemma 2.2, we reduce the problem (2.4)-(2.5) to an equivalent integral equation 𝑦(𝑡)=𝑡0𝐼𝛼𝑓+𝐶1+𝐶2𝑡,𝑦(𝑡)=𝑡0(𝑡𝑠)𝛼1𝑓Γ(𝛼)𝑠,𝑦(𝑠),𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠+𝐶1+𝐶2𝑡.(2.8) In view of the relations 𝑐𝐷𝛼𝐼𝛼𝑦(𝑡)=𝑦(𝑡) and 𝐼𝛼𝐼𝛽𝑦(𝑡)=𝐼𝛼+𝛽𝑦(𝑡), for 𝛼,𝛽>0, we obtain 𝑦(𝑡)=𝑡0(𝑡𝑠)𝛼2Γ𝑓(𝛼1)𝑠,𝑦(𝑠),𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠+𝐶2.(2.9) Applying the boundary condition (2.5), we find that 𝑦(0)=𝐶1,𝑦(𝑇)=𝑇0(𝑇𝑠)𝛼1Γ𝑓(𝛼)𝑠,𝑦(𝑠),𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠+𝐶1+𝐶2𝑦𝑇,(0)=𝐶2,𝑦(𝑇)=𝑇0(𝑇𝑠)𝛼2𝑓Γ(𝛼1)𝑠,𝑦(𝑠),𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠+𝐶2.,(2.10) that is, 𝐶2=1𝑇𝑇01(𝑦(𝑠))𝑑𝑠𝑇𝑇01𝑔(𝑦(𝑠))𝑑𝑠𝑇𝑇0(𝑇𝑠)𝛼1𝑓Γ(𝛼)𝑠,𝑦(𝑠),𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01+1(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠𝑇𝑇0(𝑇𝑠)𝛼2𝑓Γ(𝛼1)𝑠,𝑦(𝑠),𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01𝐶(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠,1=1𝑇𝑇01(𝑦(𝑠))𝑑𝑠+1𝑇𝑇01𝑔(𝑦(𝑠))𝑑𝑠𝑇𝑇0(𝑇𝑠)𝛼1𝑓Γ(𝛼)𝑠,𝑦(𝑠),𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01+1(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠𝑇𝑇0(𝑇𝑠)𝛼2𝑓Γ(𝛼1)𝑠,𝑦(𝑠),𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠.(2.11) Therefore the solution of (2.4)-(2.5) is 𝑦(𝑡)=(1+𝑡)𝑇𝑇0(𝑦(𝑠))𝑑𝑠+1(1+𝑡)𝑇𝑇0𝑔(𝑦(𝑠))𝑑𝑠(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼1𝑓Γ(𝛼)𝑠,𝑦(𝑠),𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01+(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼2𝑓Γ(𝛼1)𝑠,𝑦(𝑠),𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01+(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠𝑡0(𝑡𝑠)𝛼1𝑓Γ(𝛼)𝑠,𝑦(𝑠),𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠,(2.12) which completes the proof.

3. The Main Result

Theorem 3.1. If the hypotheses (H1)–(H5) are satisfied, then the fractional integrodifferential equation (2.4)-(2.5) has a unique solution on 𝐽.

Proof. Define 𝐹𝐶𝐶 by 𝐹𝑦(𝑡)=(1+𝑡)𝑇𝑇0(𝑦(𝑠))𝑑𝑠+1(1+𝑡)𝑇𝑇0𝑔(𝑦(𝑠))𝑑𝑠(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼1𝑓Γ(𝛼)𝑠,𝑦(𝑠),𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01+(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼2𝑓Γ(𝛼1)𝑠,𝑦(𝑠),𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01+(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠𝑡0(𝑡𝑠)𝛼1𝑓Γ(𝛼)𝑠,𝑦(𝑠),𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠.(3.1) We show that 𝐹 has a fixed point on Br. This fixed point is then a solution of (2.4)-(2.5). Firstly, we show that 𝐹𝐵𝑟𝐵𝑟, where 𝐵𝑟={𝑦𝐶𝑦𝑟}. For 𝑦𝐵𝑟, we have 𝐹𝑦(𝑡)(1+𝑡)𝑇𝑇0(𝑦(𝑠))𝑑𝑠+1(1+𝑡)𝑇𝑇0+𝑔(𝑦(𝑠))𝑑𝑠(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼2𝑓Γ(𝛼1)𝑠,𝑦(𝑠),𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01+(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼1𝑓Γ(𝛼)𝑠,𝑦(𝑠),𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01+(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠𝑡0(𝑡𝑠)𝛼1𝑓Γ(𝛼)𝑠,𝑦(𝑠),𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠,𝐹𝑦(𝑡)(1+𝑡)𝑇𝑇0(𝑦(𝑠))𝑑𝑠+1(1+𝑡)𝑇𝑇0+𝑔(𝑦(𝑠))𝑑𝑠(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼2𝑓Γ(𝛼1)𝑠,𝑦(𝑠),𝑆0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01+(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑓(𝑠,0,0,0)+𝑓(𝑠,0,0,0)𝑑𝑠(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼1𝑓Γ(𝛼)𝑠,𝑦(𝑠),𝑆0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01+(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑓(𝑠,0,0,0)+𝑓(𝑠,0,0,0)𝑑𝑠𝑡0(𝑡𝑠)𝛼1𝑓Γ(𝛼)𝑠,𝑦(𝑠),𝑆0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑓(𝑠,0,0,0)+𝑓(𝑠,0,0,0)𝑑𝑠,𝐹𝑦(𝑡)(1+𝑡)𝑇𝐺1𝑇+1(1+𝑡)𝑇𝐺2𝑇+(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼2𝑓Γ(𝛼1)𝑠,𝑦(𝑠),𝑆0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑓(𝑠,0,0,0)𝑑𝑠+(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼2+Γ(𝛼1)𝑓(𝑠,0,0,0)𝑑𝑠𝑡0(𝑡𝑠)𝛼1𝑓Γ(𝛼)𝑠,𝑦(𝑠),𝑆0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑓(𝑠,0,0,0)𝑑𝑠+𝑡0(𝑡𝑠)𝛼1Γ+(𝛼)𝑓(𝑠,0,0,0)𝑑𝑠1+𝑡𝑇𝑇0(𝑇𝑠)𝛼1𝑓Γ(𝛼)𝑠,𝑦(𝑠),𝑆0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑓(𝑠,0,0,0)𝑑𝑠+1+𝑡𝑇𝑇0(𝑇𝑠)𝛼1Γ(𝛼)𝑓(𝑠,0,0,0)𝑑𝑠(1+𝑡)𝑇𝐺1𝑇+1(1+𝑡)𝑇𝐺2𝑇+(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼2𝐾Γ(𝛼1)𝐿(𝑠)𝑦(𝑠)+𝑆0+𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑇01+(𝑠,𝜏,𝑦(𝜏))𝑑𝜏)𝑑𝑠(1+𝑡)𝑁1𝑇𝑇0(𝑇𝑠)𝛼2(Γ(𝛼1)𝑑𝑠+1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼1𝐾(Γ(𝛼)𝐿(𝑠)𝑦𝑠)+𝑠0+𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑇01(+𝑠,𝜏,𝑦(𝜏))𝑑𝜏)𝑑𝑠(1+𝑡)𝑁1𝑇𝑇0(𝑇𝑠)𝛼1Γ(𝛼)𝑑𝑠+𝑡0(𝑡𝑠)𝛼1𝐾Γ(𝛼)𝐿(𝑠)𝑦(𝑠)+𝑠0+𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑇01(𝑠,𝜏,𝑦(𝜏))𝑑𝜏)𝑑𝑠+𝑁1𝑡0(𝑡𝑠)𝛼1Γ(𝛼)𝑑𝑠(1+𝑡)𝐺1+1(1+𝑡)𝑇𝐺2𝑇+(1+𝑡)𝑁1𝑇𝑇0(𝑇𝑠)𝛼2Γ(𝛼1)𝑑𝑠+𝑇0(𝑇𝑠)𝛼1Γ(𝛼)𝑑𝑠+𝑁1𝑡0(𝑡𝑠)𝛼1+Γ(𝛼)𝑑𝑠(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼2Γ(𝛼1)𝐿(𝑠)𝐾𝑦+𝑝1(𝑠)𝑦+𝑞1+((𝑠)𝑦𝑑𝑠1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼1Γ(𝛼)𝐿(𝑠)𝐾𝑦+𝑝1(𝑠)𝑦+𝑞1+(𝑠)𝑦𝑑𝑠𝑡0(𝑡𝑠)𝛼1Γ(𝛼)𝐿(𝑠)𝐾𝑦+𝑝1(𝑠)𝑦+𝑞1((𝑠)𝑦𝑑𝑠,𝐹𝑦𝑡)(1+𝑡)𝐺1+1(1+𝑡)𝑇𝐺2𝑇+(1+𝑡)𝑁1𝑇𝑇𝛼1+𝑇Γ(𝛼)𝛼Γ(𝛼+1)+𝑁1𝑇𝛼+Γ(𝛼+1)(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼2Γ(𝛼1)𝐿(𝑠)1+𝑝1(𝑠)+𝑞1+(𝑠)𝐾(𝑦)𝑑𝑠(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼1Γ(𝛼)𝐿(𝑠)1+𝑝1(𝑠)+𝑞1+(𝑠)𝐾(𝑦)𝑑𝑠𝑡0(𝑡𝑠)𝛼1𝐿Γ(𝛼)(𝑠)1+𝑝1(𝑠)+𝑞1𝐾((𝑠)𝑦)𝑑𝑠.(3.2) Since we have 𝑀1=sup{𝐿(𝑡)(1+𝑝1(𝑡)+𝑞1(𝑡));𝑡[0,𝑇]}, and (1((1+𝑡)/𝑇))<(1(1/𝑇)), we get (1+𝑡)𝐺1+11𝑇𝐺2𝑇+(1+𝑡)𝑁1𝑇𝑇𝛼1+𝑇Γ(𝛼)𝛼Γ(𝛼+1)+𝑁1𝑇𝛼+Γ(𝛼+1)(1+𝑡)𝑀1𝑇𝑇0(𝑇𝑠)𝛼2Γ(𝛼1)𝐾(𝑦)𝑑𝑠+(1+𝑡)𝑀1𝑇𝑇0(𝑇𝑠)𝛼1Γ(𝛼)𝐾(𝑦)𝑑𝑠+𝑀1𝑡0(𝑡𝑠)𝛼1Γ(𝛼)𝐾(𝑦)𝑑𝑠(1+𝑡)𝐺1+11𝑇𝐺2𝑇+(1+𝑡)𝑁1𝑇𝑇𝛼1+𝑇Γ(𝛼)𝛼Γ(𝛼+1)+𝑁1𝑇𝛼+Γ(𝛼+1)(1+𝑡)𝑀1𝐾(𝑟)𝑇𝑇0(𝑇𝑠)𝛼2Γ(𝛼1)𝑑𝑠+(1+𝑡)𝑀1𝐾(𝑟)𝑇𝑇0(𝑇𝑠)𝛼1Γ(𝛼)𝑑𝑠+𝑀1𝐾(𝑟)𝑡0(𝑡𝑠)𝛼1Γ(𝛼)𝑑𝑠(1+𝑡)𝐺1+11𝑇𝐺2𝑇+(1+𝑡)𝑁1𝑇𝑇𝛼1+𝑇Γ(𝛼)𝛼Γ(𝛼+1)+𝑁1𝑇𝛼+Γ(𝛼+1)(1+𝑡)𝑀1𝐾(𝑟)𝑇𝑇𝛼1Γ+𝑇(𝛼)𝛼Γ+𝑀(𝛼+1)1𝐾(𝑟)𝑇𝛼Γ(𝛼+1)(1+𝑡)𝐺1+(𝑇1)𝐺2+(1+𝑡)𝑇𝑁1+𝑀1𝑇𝐾(𝑟)𝛼1+𝑇Γ(𝛼)𝛼+𝑁Γ(𝛼+1)1+𝑀1𝑇𝐾(𝑟)𝛼,Γ(𝛼+1)𝐹𝑦(𝑡)(1+𝑇)𝐺1+(𝑇1)𝐺2+(1+𝑇)𝑇𝑁1+𝑀1𝑇𝐾(𝑟)𝛼1+𝑇Γ(𝛼)𝛼+𝑁Γ(𝛼+1)1+𝑀1𝑇𝐾(𝑟)𝛼Γ(𝛼+1)𝐺1(1+𝑇)+𝐺2𝐶(𝑇1)+0𝑁1+𝑀1𝐾(𝑟)Γ(𝛼+1)𝑇2𝛼,(3.3) where 𝐶0=2𝑇2+𝑇+𝛼(𝑇+1).
Now, take 𝑥,𝑦𝐶 and for each 𝑡[0,𝑇], we obtain 𝐹𝑥(𝑡)𝐹𝑦(𝑡)(1+𝑡)𝑇𝑇0(𝑥)(𝑦)𝑑𝑠+1(1+𝑡)𝑇𝑇0+𝑔(𝑥)𝑔(𝑦)𝑑𝑠(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼1𝑓Γ(𝛼)𝑠,𝑥(𝑠),𝑠0𝑘(𝑠,𝜏,𝑥(𝜏))𝑑𝜏,𝑇01(𝑠,𝜏,𝑥(𝜏))𝑑𝜏𝑓𝑠,𝑦(𝑠),𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01+((𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼2𝑓Γ(𝛼1)𝑠,𝑥(𝑠),𝑠0𝑘(𝑠,𝜏,𝑥(𝜏))𝑑𝜏,𝑇01(𝑠,𝜏,𝑥(𝜏))𝑑𝜏𝑓𝑠,𝑦(𝑠),𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01+(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠𝑡0(𝑡𝑠)𝛼1𝑓Γ(𝛼)𝑠,𝑥(𝑠),𝑠0𝑘(𝑠,𝜏,𝑥(𝜏))𝑑𝜏,𝑇01(𝑠,𝜏,𝑥(𝜏))𝑑𝜏𝑓𝑠,𝑦(𝑠),𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠,(3.4) by using (H1)–(H5), we get 𝑏𝐹𝑥(𝑡)𝐹𝑦(𝑡)1(1+𝑡)𝑇𝑇0𝑥𝑦𝑑𝑠+𝑏21(1+𝑡)𝑇𝑇0+𝑥𝑦𝑑𝑠(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼2𝐾Γ(𝛼1)𝐿(𝑠)𝑥(𝑠)𝑦(𝑠)+𝑠0+(𝑘(𝑠,𝜏,𝑥(𝜏))𝑘(𝑠,𝜏,𝑦(𝜏)))𝑑𝜏𝑇01(𝑠,𝜏,𝑥(𝜏))1+((𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼1𝐾Γ(𝛼)𝐿(𝑠)𝑥(𝑠)𝑦(𝑠)+𝑠0+(𝑘(𝑠,𝜏,𝑥(𝜏))𝑘(𝑠,𝜏,𝑦(𝜏)))𝑑𝜏𝑇01(𝑠,𝜏,𝑥(𝜏))1(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠+𝑡0(𝑡𝑠)𝛼1𝐾Γ(𝛼)𝐿(𝑠)𝑥(𝑠)𝑦(𝑠)+𝑠0+(𝑘(𝑠,𝜏,𝑥(𝜏))𝑘(𝑠,𝜏,𝑦(𝜏)))𝑑𝜏𝑇01(𝑠,𝜏,𝑥(𝜏))1𝑏(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠1(1+𝑡)𝑇𝑇0𝑥𝑦𝑑𝑠+𝑏21(1+𝑡)𝑇𝑇0+𝑥𝑦𝑑𝑠(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼2+Γ(𝛼1)𝐿(𝑠)𝐾(𝑥𝑦+𝑝(𝑠)𝑥𝑦+𝑞(𝑠)𝑥𝑦)𝑑𝑠(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼1+Γ(𝛼)𝐿(𝑠)𝐾(𝑥𝑦+𝑝(𝑠)𝑥𝑦+𝑞(𝑠)𝑥𝑦)𝑑𝑠𝑡0(𝑡𝑠)𝛼1𝐿𝑏Γ(𝛼)(𝑠)𝐾(𝑥𝑦+𝑝(𝑠)𝑥𝑦+𝑞(𝑠)𝑥𝑦)𝑑𝑠1(1+𝑡)𝑇𝑇0𝑥𝑦𝑑𝑠+𝑏211𝑇𝑇0+𝑥𝑦𝑑𝑠(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼2+Γ(𝛼1)𝐿(𝑠)(1+𝑝(𝑠)+𝑞(𝑠))𝐾(𝑥𝑦)𝑑𝑠(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼1Γ+(𝛼)𝐿(𝑠)(1+𝑝(𝑠)+𝑞(𝑠))𝐾(𝑥𝑦)𝑑𝑠𝑡0(𝑡𝑠)𝛼1Γ(𝛼)𝐿(𝑠)(1+𝑝(𝑠)+𝑞(𝑠))𝐾(𝑥𝑦)𝑑𝑠.(3.5) Since we have 𝑀(𝑡)=𝐿(𝑡)(1+𝑝(𝑡)+𝑞(𝑡), 𝑀=sup{𝑀(𝑡)𝑡[0,𝑇]}, and, Let 𝐾(𝑥𝑦)𝑤𝑥𝑦, (𝑤>0), then 𝑏𝐹𝑥(𝑡)𝐹𝑦(𝑡)1(1+𝑡)𝑇𝑇0𝑥𝑦𝑑𝑠+𝑏211𝑇𝑇0+𝑥𝑦𝑑𝑠𝑤𝑀(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼2+Γ(𝛼1)𝑥𝑦𝑑𝑠𝑤𝑀(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼1Γ(𝛼)𝑥𝑦𝑑𝑠+𝑤𝑀𝑡0(𝑡𝑠)𝛼1𝑏Γ(𝛼)𝑥𝑦𝑑𝑠1(1+𝑇)+𝑏2(𝑇1)+𝑤𝑀𝐶1(1+𝑇)Γ(𝛼+1)𝑇2𝛼𝑥𝑦,(3.6) where 𝐶1=2𝑇2+𝑇+𝛼(1+𝑇).
As 𝑏1(1+𝑇)+𝑏2(𝑇1)+(𝑤𝑀𝐶1(1+𝑇))/(Γ(𝛼+1)𝑇2𝛼)<1, therefore 𝑓 is a contraction. Thus, the conclusion of the theorem is followed by the contraction mapping principle.

Theorem 3.2. Assume that (H1)–(H5) hold with 𝑓𝑡,𝑦(𝑡),𝑡0𝑘(𝑡,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01(𝑡,𝜏,𝑦(𝜏))𝑑𝜏𝜓(𝑡),where𝜓(𝑡)𝐿1(𝐽).(3.7)
Then the boundary value problem (2.4)-(2.5) has at least one element on [0,𝑇].

Proof. Consider 𝐵𝑟={𝑦𝐶𝑦𝑟}. We define the operators 𝐴 and 𝐵 as 1(𝐴𝑥)(𝑡)=Γ(𝛼)𝑡0(𝑡𝑠)𝛼1𝑓𝑡,𝑥(𝑠),𝑠0𝑘(𝑠,𝜏,𝑥(𝜏))𝑑𝜏,𝑇01(𝑠,𝜏,𝑥(𝜏))𝑑𝜏𝑑𝑠,(𝐵𝑥)(𝑡)=(1+𝑡)𝑇𝑇0(𝑦(𝑠))𝑑𝑠+1(1+𝑡)𝑇𝑇0𝑔(𝑦(𝑠))𝑑𝑠+(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼2𝑓Γ(𝛼1)𝑠,𝑥(𝑠),𝑠0𝑘(𝑠,𝜏,𝑥(𝜏))𝑑𝜏,𝑇01+(𝑠,𝜏,𝑥(𝜏))𝑑𝜏𝑑𝑠(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼1𝑓Γ(𝛼)𝑠,𝑥(𝑠),𝑠0𝑘(𝑠,𝜏,𝑥(𝜏))𝑑𝜏,𝑇01(𝑠,𝜏,𝑥(𝜏))𝑑𝜏𝑑𝑠.(3.8) Let us observe that if 𝑥,𝑦𝐵𝑟, then 𝐴𝑥+𝐵𝑦𝐵𝑟, 𝐴𝑥+𝐵𝑦=𝑡0(𝑡𝑠)𝛼1Γ𝑓(𝛼)𝑠,𝑥(𝑠),𝑠0𝑘(𝑠,𝜏,𝑥(𝜏))𝑑𝜏,𝑇01+(𝑠,𝜏,𝑥(𝜏))𝑑𝜏𝑑𝑠(1+𝑡)𝑇𝑇0(𝑦(𝑠))𝑑𝑠+1(1+𝑡)𝑇𝑇0𝑔(𝑦(𝑠))𝑑𝑠+(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼2𝑓Γ(𝛼1)𝑠,𝑦(𝑠),𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠+(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼1𝑓Γ(𝛼)𝑠,𝑦(𝑠),𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠𝑡0(𝑡𝑠)𝛼1𝑓Γ(𝛼)𝑠,𝑥(𝑠),𝑠0𝑘(𝑠,𝜏,𝑥(𝜏))𝑑𝜏,𝑇01+(𝑠,𝜏,𝑥(𝜏))𝑑𝜏𝑑𝑠(1+𝑡)𝑇𝑇0(𝑦(𝑠))𝑑𝑠+1(1+𝑡)𝑇𝑇0+(𝑔(𝑦(𝑠))𝑑𝑠1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼2𝑓Γ(𝛼1)𝑠,𝑦(𝑠),𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01(+𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼1Γ𝑓(𝛼)𝑠,𝑦(𝑠),𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,𝑇01(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠𝜓𝐿1𝑡0(𝑡𝑠)𝛼1Γ(𝛼)𝑑𝑠+(1+𝑡)𝐺1+1(1+𝑡)𝑇𝐺2𝑇+(1+𝑡)𝜓𝐿1𝑇𝑇0(𝑇𝑠)𝛼2+(Γ(𝛼1)𝑑𝑠1+𝑡)𝜓𝐿1𝑇𝑇0(𝑇𝑠)𝛼1Γ(𝛼)𝑑𝑠𝜓𝐿1𝑇𝛼Γ(𝛼+1)+(1+𝑡)𝐺1+11𝑇𝐺2𝑇+(1+𝑡)𝑇𝛼1𝜓𝐿1+𝑇Γ(𝛼)(1+𝑡)𝑇𝛼𝜓𝐿1𝑇Γ(𝛼+1)𝜓𝐿1𝑇𝛼+Γ(𝛼+1)(1+𝑡)𝑇𝛼1𝜓𝐿1+𝑇Γ(𝛼)(1+𝑡)𝑇𝛼𝜓𝐿1𝑇Γ(𝛼+1)+(1+𝑇)𝐺1+(𝑇1)𝐺2𝐺1(1+𝑇)+𝐺2𝐶(𝑇1)+2𝑇𝛼2Γ(𝛼+1)𝜓𝐿1,(3.9) where 𝐶2=2𝑇2+𝑇(𝛼+1)+𝑇.
Now we prove that 𝐵𝑥 is contraction mapping, 𝐵𝑥1𝐵𝑥2(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼1Γ𝑓(𝛼)𝑠,𝑥1(𝑠),𝑠0𝑘𝑠,𝜏,𝑥1(𝜏)𝑑𝜏,𝑇01𝑠,𝜏,𝑥1(𝜏)𝑑𝜏𝑓𝑠,𝑥2(𝑠)𝑠0𝑘𝑠,𝜏,𝑥2(𝜏)𝑑𝜏,𝑇01𝑠,𝜏,𝑥2+(𝜏)𝑑𝜏𝑑𝑠(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼2𝑓Γ(𝛼1)𝑠,𝑥1(𝑠),𝑠0𝑘𝑠,𝜏,𝑥1(𝜏)𝑑𝜏,𝑇01𝑠,𝜏,𝑥1(𝜏)𝑑𝜏𝑓𝑠,𝑥2(𝑠),𝑠0𝑘𝑠,𝜏,𝑥2(𝜏)𝑑𝜏,𝑇01𝑠,𝜏,𝑥2(𝜏)𝑑𝜏𝑑𝑠(1+𝑡)𝑇𝑇0(𝑇𝑠)𝛼2𝐿𝑥Γ(𝛼1)(𝑠)(1+𝑝(𝑠)+𝑞(𝑠))𝐾1𝑥2+𝑑𝑠𝑇0(𝑇𝑠)𝛼1𝑥Γ(𝛼)𝐿(𝑠)(1+𝑝(𝑠)+𝑞(𝑠))𝐾1𝑥2.𝑑𝑠(3.10) Let 𝐾(𝑥1𝑥2)𝑤𝑥1𝑥2, we obtain 𝐵𝑥1𝐵𝑥2(1+𝑡)𝑤𝑀𝑇𝑥1𝑥2𝑇0(𝑇𝑠)𝛼2Γ(𝛼1)𝑑𝑠+𝑇0(𝑇𝑠)𝛼1Γ,(𝛼)𝑑𝑠𝐵𝑥1𝐵𝑥2(1+𝑇)𝑤𝑀𝑇𝑇𝛼1+𝑇Γ(𝛼)𝛼𝑥Γ(𝛼+1)1𝑥2𝑤𝑀(1+𝑇)(𝛼+𝑇)Γ(𝛼+1)𝑇2𝛼𝑥1𝑥2.(3.11) It is clear that 𝐵 is contraction mapping, since 𝑥(𝑡) is continuous, then 𝐴𝑥 is continuous 1𝐴𝑥(𝑡)=Γ(𝛼)𝑡0(𝑡𝑠)𝛼1𝑓𝑠,𝑥(𝑠),𝑠0𝑘(𝑠,𝜏,𝑥(𝜏))𝑑𝜏,𝑇01(𝑠,𝜏,𝑥(𝜏))𝑑𝜏𝑑𝑠𝜓𝐿1𝑡0(𝑡𝑠)𝛼1(𝑇Γ(𝛼)𝑑𝑠𝐴𝑥𝑡)𝛼𝜓𝐿1.Γ(𝛼+1)(3.12) Hence, 𝐴 is uniformly bounded on 𝐵𝑟. Now, let us prove that 𝐴𝑥(𝑡) is equicontinuous, let 𝑡1,𝑡2[0,𝑇] and 𝑥𝐵𝑟. Using the fact that 𝑓 is bounded on the compact set 𝐽×𝐵𝑟, thus sup(𝑡,𝑠)𝐽×𝐵𝑟𝑓(𝑠,𝑥(𝑠),𝑠0𝑘(𝑠,𝜏,𝑥(𝜏))𝑑𝜏,𝑇01(𝑠,𝜏,𝑥(𝜏))𝑑𝜏)=𝑐0<,we get 𝑡𝐴𝑥1𝑡𝐴𝑥2=1Γ(𝛼)𝑡10𝑡1𝑠𝛼1𝑓𝑠,𝑥(𝑠),𝑠0𝑘(𝑠,𝜏,𝑥(𝜏))𝑑𝜏,𝑇011(𝑠,𝜏,𝑥(𝜏))𝑑𝜏𝑑𝑠Γ(𝛼)𝑡20𝑡2𝑠𝛼1𝑓𝑠,𝑥(𝑠),𝑠0𝑘(𝑠,𝜏,𝑥(𝜏))𝑑𝜏,𝑇011(𝑠,𝜏,𝑥(𝜏))𝑑𝜏𝑑𝑠Γ(𝛼)𝑡10𝑡1𝑠𝛼1𝑡2𝑠𝛼1𝑓𝑠,𝑥(𝑠),𝑠0𝑘(𝑠,𝜏,𝑥(𝜏))𝑑𝜏,𝑇01+(𝑠,𝜏,𝑥(𝜏))𝑑𝜏𝑑𝑠𝑡2𝑡1𝑡2𝑠𝛼1𝑓𝑠,𝑥(𝑠),𝑠0𝑘(𝑠,𝜏,𝑥(𝜏))𝑑𝜏,𝑇01𝑐(𝑠,𝜏,𝑥(𝜏))𝑑𝜏𝑑𝑠02𝑡Γ(𝛼+1)2𝑡1𝛼+𝑡1𝛼𝑡2𝛼.(3.13)
So 𝐴 is relatively compact. By Arzela-Ascoli theorem, 𝐴 is compact. Now we conclude the result of the theorem of Krasnosel’skii theorem.

Example 3.3. Consider the following fractional mixed Volterra-Fredholm integrodifferential equation: 𝑦(1.5)1(𝑡)=+110||𝑦||+10+(𝑡)𝑡0||||𝑦(𝑡)10𝑒|𝑦(𝑡)|+𝑡𝑑𝑡+10||||𝑒𝑦(𝑡)𝑡||||10+𝑦(𝑡)2𝑑𝑡,(3.14) with integral boundary conditions 𝑦(0)𝑦(0)=101||||10+𝑦(𝑡)𝑑𝑡,𝑦(1)𝑦(1)=10110+𝑒|𝑦(𝑡)|𝑑𝑡.(3.15) Here, 1𝑔(𝑦(𝑡))=||||110+𝑦(𝑡)110,𝑔(𝑥)𝑔(𝑦)1100𝑥𝑦,(𝑦(𝑡))=10+𝑒|𝑦(𝑡)|1110,(𝑥)(𝑦)100𝑥𝑦,𝑡01(𝑘(𝑡,𝑠,𝑥)𝑘(𝑡,𝑠,𝑦))𝑑𝑠10𝑒𝑡𝑥𝑦,𝑡01𝑘(𝑡,𝑠,𝑦)𝑑𝑠10+𝑡𝑦(𝑡),𝑡01(𝑡,𝑠,𝑥)11(𝑡,𝑠,𝑦)𝑑𝑠10𝑒𝑡𝑥𝑦,𝑡011(𝑡,𝑠,𝑦)𝑑𝑠𝑓10+𝑡𝑦(𝑡),𝑡,𝑥1𝑦1,𝑧1𝑓𝑡,𝑥2,𝑦2,𝑧21𝑥10+𝑡1𝑥2+𝑦1𝑦2+𝑧1𝑧2,1𝑓(𝑡,0,0,0)=.10(3.16) Hence, the conditions (H1)–(H5) hold with 𝐺1=𝐺2=0.1, 𝑏1=𝑏2=0.01, 𝑀1=0.12, 𝑤=0.1, 𝐶𝑜=6, 𝑁1=0.1, 𝑀=0.12, and 𝐶1=6, thus 𝑏1(1+𝑇)+𝑏2(𝑇1)+𝑤𝑀𝐶1(1+𝑇)Γ(𝛼+1)𝑇2𝛼<10.01(2)+(0.1)(0.12)6(2)Γ(2.5)<1.(3.17) We conclude from the above example that the integrodifferential equation has unique solution.

References

  1. M. Amairi, M. Aoun, S. Najar, and M. N. Abdelkrim, “A constant enclosure method for validating existence and uniqueness of the solution of an initial value problem for a fractional differential equation,” Applied Mathematics and Computation, vol. 217, no. 5, pp. 2162–2168, 2010. View at Publisher · View at Google Scholar
  2. Z. Drici, F. A. McRae, and J. V. Devi, “Fractional differential equations involving causal operators,” Communications in Applied Analysis., vol. 14, no. 1, pp. 81–88, 2010.
  3. S. B. Hadid, “Local and global existence theorems on differential equations of non-integer order,” Journal of Fractional Calculus, vol. 7, pp. 101–105, 1995. View at Zentralblatt MATH
  4. R. W. Ibrahim, “Existence results for fractional boundary value problem,” International Journal of Contemporary Mathematical Sciences, vol. 3, no. 33-36, pp. 1767–1774, 2008. View at Zentralblatt MATH
  5. S. M. Momani, “Local and global existence theorems on fractional integro-differential equations,” Journal of Fractional Calculus, vol. 18, pp. 81–86, 2000. View at Zentralblatt MATH
  6. S. M. Momani and S. B. Hadid, “On the inequalities of integro-differential fractional equations,” International Journal of Applied Mathematics, vol. 12, no. 1, pp. 29–37, 2003. View at Zentralblatt MATH
  7. B. Ahmad, A. Alsaedi, and B. S. Alghamdi, “Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions,” Nonlinear Analysis Real world Applications, vol. 9, no. 4, pp. 1727–1740, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. H L. Tidke, “Existence of global solutions to nonlinear mixed Volterra-Fredholm integrodifferential equations with nonlocal conditions,” Electronic Journal of Differential Equations, vol. 2009, pp. No. 55–7, 2009. View at Zentralblatt MATH
  9. B. Ahmad and J. J. Nieto, “Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions,” Boundary Value Problems, vol. 2009, Article ID 708576, 11 pages, 2009. View at Zentralblatt MATH
  10. G. M. N'Guérékata, “A Cauchy problem for some fractional abstract differential equation with non local conditions,” Nonlinear Analysis: Theory , Method and Applications, vol. 70, no. 5, pp. 1873–1876, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. A. Anguraj, P. Karthikeyan, and J. J. Trujillo, “Existence of solutions to fractional mixed integrodifferential equations with nonlocal initial condition,” Advances in Difference Equations, vol. 2011, Article ID 690653, 12 pages, 2011. View at Publisher · View at Google Scholar
  12. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006.
  13. M. A. Krasnosel'skiĭ, “Two remarks on the method of successive approximations,” Uspekhi Matematicheskikh Nauk, vol. 10, no. 1(63), pp. 123–127, 1955.
  14. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
  15. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science, Yverdon, Switzerland, 1993.