International Journal of Differential Equations

International Journal of Differential Equations / 2011 / Article
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Fractional Differential Equations 2011

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Research Article | Open Access

Volume 2011 |Article ID 304570 | 15 pages | https://doi.org/10.1155/2011/304570

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

Academic Editor: Shaher Momani
Received07 May 2011
Accepted24 May 2011
Published23 Aug 2011

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 [1–6]. 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 N’GuΓ©rΓ©kata [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<𝛼≀2) in Banach spaces by using Banach and Krasnosel'skii fixed-point theorems.

2. Preliminaries

First of all, we recall some basic definitions; see [12–15].

Definition 2.1. For a function 𝑓 given on the interval [π‘Ž,𝑏], the Caputo fractional order derivative of 𝑓 is defined by π‘‘π‘Žπ·π›Ό1𝑓(𝑑)=ξ€œΞ“(π‘›βˆ’π›Ό)π‘‘π‘Ž(π‘‘βˆ’π‘ )π‘›βˆ’π›Όβˆ’1𝑓(𝑛)(𝑠)𝑑𝑠,(2.1) where 𝑛=[𝛼]+1 and [𝛼] denotes the integer part of 𝛼.

Lemma 2.2. Let 𝛼>0, then π‘‘π‘Žπ·π‘‘π‘Žβˆ’π›Όπ·π›Όπ‘¦(𝑑)=𝑦(𝑑)+𝑐0+𝑐1𝑑+𝑐2𝑑2+β‹―+π‘π‘›βˆ’1π‘‘π‘›βˆ’1,(2.2) for some π‘π‘–βˆˆπ‘…, i=0,1,…,nβˆ’1,𝑛=[𝛼]+1.

Definition 2.3. Let 𝑓 be a function which is defined almost everywhere (a.e) on [π‘Ž,𝑏], for 𝛼>0, we define π‘π‘Žπ·βˆ’π›Ό1𝑓=ξ€œΞ“(𝛼)π‘π‘Ž(π‘βˆ’π‘‘)π›Όβˆ’1𝑓(𝑑)𝑑𝑑,(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 𝐡 be two operators such that (i)𝐴π‘₯+𝐡𝑦=𝑀, whenever π‘₯,π‘¦βˆˆπ‘€,(ii)𝐴 is compact and continuous;(iii)𝐡 is a contraction mapping,then there exists π‘§βˆˆπ‘€ such that 𝑧=𝐴𝑧+𝐡𝑧.
Let 𝑋 be a Banach space with the norm β€–β‹…β€–. Let 𝐢=([0,𝑇],𝑋) be Banach space of all continuous functions πœ“βˆΆ[0,𝑇]→𝑋, with supermum norm β€–πœ“β€–=sup{β€–πœ“(𝑠)β€–βˆΆπ‘ βˆˆ[0,𝑇]}. Consider the fractional mixed Volttera-Fredholm integrodifferential equation with boundary conditions, which has the form π·π›Όξ‚΅ξ€œπ‘¦(𝑑)=𝑓𝑑,𝑦(𝑑),𝑑0ξ€œπ‘˜(𝑑,𝑠,𝑦(𝑠))𝑑𝑠,𝑇0β„Ž1ξ‚Ά,(𝑑,𝑠,𝑦(𝑠))𝑑𝑠(2.4)𝑦(0)βˆ’π‘¦ξ…žξ€œ(0)=𝑇0𝑔(𝑦(𝑠))𝑑𝑠,𝑦(𝑇)βˆ’π‘¦ξ…žξ€œ(𝑇)=𝑇0β„Ž(𝑦(𝑠))𝑑𝑠,(2.5) where 1<𝛼≀2, 𝐷𝛼 is the Caputo fractional derivative and the nonlinear functions π‘“βˆΆ[0,𝑇]×𝑋×𝑋×𝑋→𝑋, π‘˜,β„Ž1∢[0,𝑇]Γ—[0,𝑇]×𝑋→𝑋 and 𝑔,β„ŽβˆΆπ‘‹β†’π‘‹ satisfy the following hypotheses: (H1)there exists constants 𝐺1,𝐺2 such that β€–β„Ž(𝑦)‖≀𝐺1,‖𝑔(𝑦)‖≀𝐺2 for π‘¦βˆˆπ‘‹,(H2)there exists constants 𝑏1,𝑏2 such that β€–β„Ž(π‘₯)βˆ’β„Ž(𝑦)‖≀𝑏1β€–π‘₯βˆ’π‘¦β€– and ‖𝑔(π‘₯)βˆ’π‘”(𝑦)‖≀𝑏2β€–π‘₯βˆ’π‘¦β€–,βˆ€π‘₯,π‘¦βˆˆπ‘‹,(2.6)(H3)There exists continuous functions π‘βˆΆ[0,𝑇]→𝑅+=[0,∞) and 𝑝1∢[0,𝑇]→𝑅+ such that β€–βˆ«π‘‘0(π‘˜(𝑑,𝑠,π‘₯)βˆ’π‘˜(𝑑,𝑠,𝑦))𝑑𝑠‖≀𝑝(𝑑)β€–π‘₯βˆ’π‘¦β€– and β€–βˆ«π‘‘0π‘˜(𝑑,𝑠,𝑦)𝑑𝑠‖≀𝑝1(𝑑)‖𝑦‖, for every 𝑑,π‘ βˆˆ[0,𝑇] and π‘₯,π‘¦βˆˆπ‘‹,(H4)there exists continuous functions π‘žβˆΆ[0,𝑇]→𝑅+ and π‘ž1∢[0,𝑇]→𝑅+ such that β€–βˆ«π‘‡0(β„Ž1(𝑑,𝑠,π‘₯)βˆ’β„Ž1(𝑑,𝑠,𝑦)π‘‘π‘ β€–β‰€π‘ž(𝑑)β€–π‘₯βˆ’π‘¦β€– and β€–βˆ«π‘‡0β„Ž1(𝑑,𝑠,𝑦)π‘‘π‘ β€–β‰€π‘ž1(𝑑)β€–π‘¦β€–π‘“π‘œπ‘Ÿπ‘’π‘£π‘’π‘Ÿπ‘¦π‘‘,π‘ βˆˆ[0,𝑇]π‘Žπ‘›π‘‘π‘₯,π‘¦βˆˆπ‘‹(H5)there exists continuous function 𝐿∢[0,𝑇]→𝑅+, and 𝑁1is positive constant such that ‖𝑓(𝑑,π‘₯1,𝑦1,𝑧1)βˆ’π‘“(𝑑,π‘₯2,𝑦2,𝑧2)‖≀𝐿(𝑑)𝐾(β€–π‘₯1βˆ’π‘₯2β€–+‖𝑦1βˆ’π‘¦2β€–+‖𝑧1βˆ’π‘§2β€–) and 𝑁1=supπ‘‘βˆˆ[0,𝑇]‖𝑓(𝑑,0,0,0)β€–, for every π‘‘βˆˆ[0,𝑇] and π‘₯1,𝑦1,𝑧1,π‘₯2,𝑦2,𝑧2βˆˆπ‘‹, where πΎβˆΆπ‘…+β†’(0,∞) is continuous nondecreasing function satisfying 𝐾(𝛾(𝑑)π‘₯)≀𝛾(𝑑)𝐾(π‘₯), where 𝛾 is a continuous function π›ΎβˆΆ[0,𝑇]→𝑅+.

Lemma 2.5. Let 1<𝛼≀2 and π‘“βˆΆπ½Γ—π‘‹β†’π‘‹, 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(𝑑)=(1+𝑑)π‘‡ξ€œπ‘‡0ξ‚΅β„Ž(𝑦(𝑠))𝑑𝑠+1βˆ’(1+𝑑)π‘‡ξ‚Άξ€œπ‘‡0βˆ’π‘”(𝑦(𝑠))𝑑𝑠(1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’1π‘“ξ‚΅ξ€œΞ“(𝛼)𝑠,𝑦(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Ά+(𝑠,𝜏,𝑦(𝜏))π‘‘πœ(1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’2π‘“ξ‚΅ξ€œΞ“(π›Όβˆ’1)𝑠,𝑦(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Ά+ξ€œ(𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‘π‘ π‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’1π‘“ξ‚΅ξ€œΞ“(𝛼)𝑠,𝑦(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Ά(𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‘π‘ .(2.7)

Proof. By Lemma 2.2, we reduce the problem (2.4)-(2.5) to an equivalent integral equation 𝑦(𝑑)=𝑑0𝐼𝛼𝑓+𝐢1+𝐢2ξ€œπ‘‘,𝑦(𝑑)=𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1π‘“ξ‚΅ξ€œΞ“(𝛼)𝑠,𝑦(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Ά(𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‘π‘ +𝐢1+𝐢2𝑑.(2.8) In view of the relations 𝑐𝐷𝛼𝐼𝛼𝑦(𝑑)=𝑦(𝑑) and 𝐼𝛼𝐼𝛽𝑦(𝑑)=𝐼𝛼+𝛽𝑦(𝑑), for 𝛼,𝛽>0, we obtain π‘¦ξ…žξ€œ(𝑑)=𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’2Ξ“π‘“ξ‚΅ξ€œ(π›Όβˆ’1)𝑠,𝑦(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Ά(𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‘π‘ +𝐢2.(2.9) Applying the boundary condition (2.5), we find that 𝑦(0)=𝐢1ξ€œ,𝑦(𝑇)=𝑇0(π‘‡βˆ’π‘ )π›Όβˆ’1Ξ“π‘“ξ‚΅ξ€œ(𝛼)𝑠,𝑦(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Ά(𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‘π‘ +𝐢1+𝐢2𝑦𝑇,ξ…ž(0)=𝐢2,π‘¦ξ…žξ€œ(𝑇)=𝑇0(π‘‡βˆ’π‘ )π›Όβˆ’2π‘“ξ‚΅ξ€œΞ“(π›Όβˆ’1)𝑠,𝑦(𝑠),𝑠0π‘˜ξ€œ(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Ά(𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‘π‘ +𝐢2.,(2.10) that is, 𝐢2=1π‘‡ξ€œπ‘‡01β„Ž(𝑦(𝑠))π‘‘π‘ βˆ’π‘‡ξ€œπ‘‡0βˆ’1𝑔(𝑦(𝑠))π‘‘π‘ π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’1π‘“ξ‚΅ξ€œΞ“(𝛼)𝑠,𝑦(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Ά+1(𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‘π‘ π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’2π‘“ξ‚΅ξ€œΞ“(π›Όβˆ’1)𝑠,𝑦(𝑠),𝑠0π‘˜ξ€œ(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1𝐢(𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‘π‘ ,1=1π‘‡ξ€œπ‘‡0ξ‚€1β„Ž(𝑦(𝑠))𝑑𝑠+1βˆ’π‘‡ξ‚ξ€œπ‘‡0βˆ’1𝑔(𝑦(𝑠))π‘‘π‘ π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’1π‘“ξ‚΅ξ€œΞ“(𝛼)𝑠,𝑦(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Ά+1(𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‘π‘ π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’2π‘“ξ‚΅ξ€œΞ“(π›Όβˆ’1)𝑠,𝑦(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1(𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‘π‘ .(2.11) Therefore the solution of (2.4)-(2.5) is 𝑦(𝑑)=(1+𝑑)π‘‡ξ€œπ‘‡0ξ‚΅β„Ž(𝑦(𝑠))𝑑𝑠+1βˆ’(1+𝑑)π‘‡ξ‚Άξ€œπ‘‡0βˆ’π‘”(𝑦(𝑠))𝑑𝑠(1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’1π‘“ξ‚΅ξ€œΞ“(𝛼)𝑠,𝑦(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Ά+(𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‘π‘ (1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’2π‘“ξ‚΅ξ€œΞ“(π›Όβˆ’1)𝑠,𝑦(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Ά+ξ€œ(𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‘π‘ π‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’1π‘“ξ‚΅ξ€œΞ“(𝛼)𝑠,𝑦(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Ά(𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‘π‘ ,(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 πΉβˆΆπΆβ†’πΆ by 𝐹𝑦(𝑑)=(1+𝑑)π‘‡ξ€œπ‘‡0ξ‚΅β„Ž(𝑦(𝑠))𝑑𝑠+1βˆ’(1+𝑑)π‘‡ξ‚Άξ€œπ‘‡0βˆ’π‘”(𝑦(𝑠))𝑑𝑠(1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’1π‘“ξ‚΅ξ€œΞ“(𝛼)𝑠,𝑦(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Ά+(𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‘π‘ (1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’2π‘“ξ‚΅ξ€œΞ“(π›Όβˆ’1)𝑠,𝑦(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Ά+ξ€œ(𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‘π‘ π‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’1π‘“ξ‚΅ξ€œΞ“(𝛼)𝑠,𝑦(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Ά(𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‘π‘ .(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 πΉπ΅π‘ŸβŠ‚π΅π‘Ÿ, where π΅π‘Ÿ={π‘¦βˆˆπΆβˆΆβ€–π‘¦β€–β‰€π‘Ÿ}. For π‘¦βˆˆπ΅π‘Ÿ, we have ‖𝐹𝑦(𝑑)‖≀(1+𝑑)π‘‡ξ€œπ‘‡0ξ‚΅β€–β„Ž(𝑦(𝑠))‖𝑑𝑠+1βˆ’(1+𝑑)π‘‡ξ‚Άξ€œπ‘‡0+‖𝑔(𝑦(𝑠))‖𝑑𝑠(1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’2β€–β€–β€–π‘“ξ‚΅ξ€œΞ“(π›Όβˆ’1)𝑠,𝑦(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Άβ€–β€–β€–+(𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‘π‘ (1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’1β€–β€–β€–π‘“ξ‚΅ξ€œΞ“(𝛼)𝑠,𝑦(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Άβ€–β€–β€–+ξ€œ(𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‘π‘ π‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’1β€–β€–β€–π‘“ξ‚΅ξ€œΞ“(𝛼)𝑠,𝑦(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Άβ€–β€–β€–(𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‘π‘ ,‖𝐹𝑦(𝑑)‖≀(1+𝑑)π‘‡ξ€œπ‘‡0ξ‚΅β€–β„Ž(𝑦(𝑠))‖𝑑𝑠+1βˆ’(1+𝑑)π‘‡ξ‚Άξ€œπ‘‡0+‖𝑔(𝑦(𝑠))‖𝑑𝑠(1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’2β€–β€–β€–π‘“ξ‚΅ξ€œΞ“(π›Όβˆ’1)𝑠,𝑦(𝑠),𝑆0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Άβ€–β€–β€–+(𝑠,𝜏,𝑦(𝜏))π‘‘πœβˆ’π‘“(𝑠,0,0,0)+𝑓(𝑠,0,0,0)𝑑𝑠(1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’1β€–β€–β€–π‘“ξ‚΅ξ€œΞ“(𝛼)𝑠,𝑦(𝑠),𝑆0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Άβ€–β€–β€–+ξ€œ(𝑠,𝜏,𝑦(𝜏))π‘‘πœβˆ’π‘“(𝑠,0,0,0)+𝑓(𝑠,0,0,0)𝑑𝑠𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1β€–β€–β€–π‘“ξ‚΅ξ€œΞ“(𝛼)𝑠,𝑦(𝑠),𝑆0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Άβ€–β€–β€–(𝑠,𝜏,𝑦(𝜏))π‘‘πœβˆ’π‘“(𝑠,0,0,0)+𝑓(𝑠,0,0,0)𝑑𝑠,‖𝐹𝑦(𝑑)‖≀(1+𝑑)𝑇𝐺1𝑇+1βˆ’(1+𝑑)𝑇𝐺2𝑇+(1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’2β€–β€–β€–π‘“ξ‚΅ξ€œΞ“(π›Όβˆ’1)𝑠,𝑦(𝑠),𝑆0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Άβ€–β€–β€–(𝑠,𝜏,𝑦(𝜏))π‘‘πœβˆ’π‘“(𝑠,0,0,0)𝑑𝑠+(1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’2+ξ€œΞ“(π›Όβˆ’1)‖𝑓(𝑠,0,0,0)‖𝑑𝑠𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1β€–β€–β€–π‘“ξ‚΅ξ€œΞ“(𝛼)𝑠,𝑦(𝑠),𝑆0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Άβ€–β€–β€–ξ€œ(𝑠,𝜏,𝑦(𝜏))π‘‘πœβˆ’π‘“(𝑠,0,0,0)𝑑𝑠+𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1Ξ“+ξ‚€(𝛼)‖𝑓(𝑠,0,0,0)‖𝑑𝑠1+π‘‘π‘‡ξ‚ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’1β€–β€–β€–π‘“ξ‚΅ξ€œΞ“(𝛼)𝑠,𝑦(𝑠),𝑆0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Άβ€–β€–β€–ξ‚€(𝑠,𝜏,𝑦(𝜏))π‘‘πœβˆ’π‘“(𝑠,0,0,0)𝑑𝑠+1+π‘‘π‘‡ξ‚ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’1≀Γ(𝛼)‖𝑓(𝑠,0,0,0)‖𝑑𝑠(1+𝑑)𝑇𝐺1𝑇+1βˆ’(1+𝑑)𝑇𝐺2𝑇+(1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’2πΎξ‚΅β€–β€–β€–β€–ξ€œΞ“(π›Όβˆ’1)𝐿(𝑠)𝑦(𝑠)β€–+𝑆0β€–β€–β€–+β€–β€–β€–ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‡0β„Ž1β€–β€–β€–ξ‚Ά+(𝑠,𝜏,𝑦(𝜏))π‘‘πœ)𝑑𝑠(1+𝑑)𝑁1π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’2(Ξ“(π›Όβˆ’1)𝑑𝑠+1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’1𝐾(β€–β€–β€–ξ€œΞ“(𝛼)𝐿(𝑠)‖𝑦𝑠)β€–+𝑠0β€–β€–β€–+β€–β€–β€–ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‡0β„Ž1(β€–β€–β€–ξ‚Ά+𝑠,𝜏,𝑦(𝜏))π‘‘πœ)𝑑𝑠(1+𝑑)𝑁1π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’1ξ€œΞ“(𝛼)𝑑𝑠+𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1πΎξ‚΅β€–β€–β€–ξ€œΞ“(𝛼)𝐿(𝑠)‖𝑦(𝑠)β€–+𝑠0β€–β€–β€–+β€–β€–β€–ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‡0β„Ž1β€–β€–β€–ξ‚Ά(𝑠,𝜏,𝑦(𝜏))π‘‘πœ)𝑑𝑠+𝑁1ξ€œπ‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’1Ξ“(𝛼)𝑑𝑠≀(1+𝑑)𝐺1+ξ‚΅1βˆ’(1+𝑑)𝑇𝐺2𝑇+(1+𝑑)𝑁1π‘‡ξ‚΅ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’2ξ€œΞ“(π›Όβˆ’1)𝑑𝑠+𝑇0(π‘‡βˆ’π‘ )π›Όβˆ’1ξ‚ΆΞ“(𝛼)𝑑𝑠+𝑁1ξ€œπ‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’1+Ξ“(𝛼)𝑑𝑠(1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’2ξ€·Ξ“(π›Όβˆ’1)𝐿(𝑠)𝐾‖𝑦‖+𝑝1(𝑠)‖𝑦‖+π‘ž1ξ€Έ+((𝑠)‖𝑦‖𝑑𝑠1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’1ξ€·Ξ“(𝛼)𝐿(𝑠)𝐾‖𝑦‖+𝑝1(𝑠)‖𝑦‖+π‘ž1ξ€Έ+ξ€œ(𝑠)‖𝑦‖𝑑𝑠𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1ξ€·Ξ“(𝛼)𝐿(𝑠)𝐾‖𝑦‖+𝑝1(𝑠)‖𝑦‖+π‘ž1(ξ€Έ(𝑠)‖𝑦‖𝑑𝑠,‖𝐹𝑦𝑑)‖≀(1+𝑑)𝐺1+ξ‚΅1βˆ’(1+𝑑)𝑇𝐺2𝑇+(1+𝑑)𝑁1π‘‡ξ‚΅π‘‡π›Όβˆ’1+𝑇Γ(𝛼)𝛼Γ(𝛼+1)+𝑁1𝑇𝛼+Ξ“(𝛼+1)(1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’2ξ€·Ξ“(π›Όβˆ’1)𝐿(𝑠)1+𝑝1(𝑠)+π‘ž1ξ€Έ+(𝑠)𝐾(‖𝑦‖)𝑑𝑠(1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’1ξ€·Ξ“(𝛼)𝐿(𝑠)1+𝑝1(𝑠)+π‘ž1ξ€Έ+ξ€œ(𝑠)𝐾(‖𝑦‖)𝑑𝑠𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1𝐿Γ(𝛼)(𝑠)1+𝑝1(𝑠)+π‘ž1𝐾((𝑠)‖𝑦‖)𝑑𝑠.(3.2) Since we have 𝑀1=sup{𝐿(𝑑)(1+𝑝1(𝑑)+π‘ž1(𝑑));π‘‘βˆˆ[0,𝑇]}, and (1βˆ’((1+𝑑)/𝑇))<(1βˆ’(1/𝑇)), we get ≀(1+𝑑)𝐺1+ξ‚€11βˆ’π‘‡ξ‚πΊ2𝑇+(1+𝑑)𝑁1π‘‡ξ‚΅π‘‡π›Όβˆ’1+𝑇Γ(𝛼)𝛼Γ(𝛼+1)+𝑁1𝑇𝛼+Ξ“(𝛼+1)(1+𝑑)𝑀1π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’2Ξ“(π›Όβˆ’1)𝐾(‖𝑦‖)𝑑𝑠+(1+𝑑)𝑀1π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’1Ξ“(𝛼)𝐾(‖𝑦‖)𝑑𝑠+𝑀1ξ€œπ‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’1Ξ“(𝛼)𝐾(‖𝑦‖)𝑑𝑠≀(1+𝑑)𝐺1+ξ‚€11βˆ’π‘‡ξ‚πΊ2𝑇+(1+𝑑)𝑁1π‘‡ξ‚΅π‘‡π›Όβˆ’1+𝑇Γ(𝛼)𝛼Γ(𝛼+1)+𝑁1𝑇𝛼+Ξ“(𝛼+1)(1+𝑑)𝑀1𝐾(π‘Ÿ)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’2Ξ“(π›Όβˆ’1)𝑑𝑠+(1+𝑑)𝑀1𝐾(π‘Ÿ)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’1Ξ“(𝛼)𝑑𝑠+𝑀1ξ€œπΎ(π‘Ÿ)𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1Ξ“(𝛼)𝑑𝑠≀(1+𝑑)𝐺1+ξ‚€11βˆ’π‘‡ξ‚πΊ2𝑇+(1+𝑑)𝑁1π‘‡ξ‚΅π‘‡π›Όβˆ’1+𝑇Γ(𝛼)𝛼Γ(𝛼+1)+𝑁1𝑇𝛼+Ξ“(𝛼+1)(1+𝑑)𝑀1𝐾(π‘Ÿ)π‘‡ξ‚΅π‘‡π›Όβˆ’1Ξ“+𝑇(𝛼)𝛼Γ+𝑀(𝛼+1)1𝐾(π‘Ÿ)𝑇𝛼Γ(𝛼+1)≀(1+𝑑)𝐺1+(π‘‡βˆ’1)𝐺2+(1+𝑑)𝑇𝑁1+𝑀1𝑇𝐾(π‘Ÿ)π›Όβˆ’1+𝑇Γ(𝛼)𝛼+𝑁Γ(𝛼+1)1+𝑀1𝑇𝐾(π‘Ÿ)𝛼,β€–Ξ“(𝛼+1)𝐹𝑦(𝑑)‖≀(1+𝑇)𝐺1+(π‘‡βˆ’1)𝐺2+(1+𝑇)𝑇𝑁1+𝑀1𝑇𝐾(π‘Ÿ)π›Όβˆ’1+𝑇Γ(𝛼)𝛼+𝑁Γ(𝛼+1)1+𝑀1𝑇𝐾(π‘Ÿ)𝛼Γ(𝛼+1)≀𝐺1(1+𝑇)+𝐺2𝐢(π‘‡βˆ’1)+0𝑁1+𝑀1𝐾(π‘Ÿ)Ξ“(𝛼+1)𝑇2βˆ’π›Ό,(3.3) where 𝐢0=2𝑇2+𝑇+𝛼(𝑇+1).
Now, take π‘₯,π‘¦βˆˆπΆ and for each π‘‘βˆˆ[0,𝑇], we obtain ‖𝐹π‘₯(𝑑)βˆ’πΉπ‘¦(𝑑)‖≀(1+𝑑)π‘‡ξ€œπ‘‡0ξ‚΅β€–β„Ž(π‘₯)βˆ’β„Ž(𝑦)‖𝑑𝑠+1βˆ’(1+𝑑)π‘‡ξ‚Άξ€œπ‘‡0+‖𝑔(π‘₯)βˆ’π‘”(𝑦)‖𝑑𝑠(1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’1β€–β€–β€–π‘“ξ‚΅ξ€œΞ“(𝛼)𝑠,π‘₯(𝑠),𝑠0π‘˜ξ€œ(𝑠,𝜏,π‘₯(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Άξ‚΅ξ€œ(𝑠,𝜏,π‘₯(𝜏))π‘‘πœβˆ’π‘“π‘ ,𝑦(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Άβ€–β€–β€–+((𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‘π‘ 1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’2β€–β€–β€–π‘“ξ‚΅ξ€œΞ“(π›Όβˆ’1)𝑠,π‘₯(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,π‘₯(𝜏))π‘‘πœ,𝑇0β„Ž1(ξ‚Άξ‚΅ξ€œπ‘ ,𝜏,π‘₯(𝜏))π‘‘πœβˆ’π‘“π‘ ,𝑦(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Άβ€–β€–β€–+ξ€œ(𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‘π‘ π‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’1β€–β€–β€–π‘“ξ‚΅ξ€œΞ“(𝛼)𝑠,π‘₯(𝑠),𝑠0π‘˜ξ€œ(𝑠,𝜏,π‘₯(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Άξ‚΅ξ€œ(𝑠,𝜏,π‘₯(𝜏))π‘‘πœβˆ’π‘“π‘ ,𝑦(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Άβ€–β€–β€–(𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‘π‘ ,(3.4) by using (H1)–(H5), we get 𝑏‖𝐹π‘₯(𝑑)βˆ’πΉπ‘¦(𝑑)‖≀1(1+𝑑)π‘‡ξ€œπ‘‡0β€–π‘₯βˆ’π‘¦β€–π‘‘π‘ +𝑏2ξ‚΅1βˆ’(1+𝑑)π‘‡ξ‚Άξ€œπ‘‡0+β€–π‘₯βˆ’π‘¦β€–π‘‘π‘ (1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’2πΎξ‚΅β€–β€–β€–ξ€œΞ“(π›Όβˆ’1)𝐿(𝑠)β€–π‘₯(𝑠)βˆ’π‘¦(𝑠)β€–+𝑠0β€–β€–β€–+β€–β€–β€–ξ€œ(π‘˜(𝑠,𝜏,π‘₯(𝜏))βˆ’π‘˜(𝑠,𝜏,𝑦(𝜏)))π‘‘πœπ‘‡0ξ€·β„Ž1(𝑠,𝜏,π‘₯(𝜏))βˆ’β„Ž1ξ€Έβ€–β€–β€–ξ‚Ά+((𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‘π‘ 1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’1πΎξ‚΅β€–β€–β€–ξ€œΞ“(𝛼)𝐿(𝑠)β€–π‘₯(𝑠)βˆ’π‘¦(𝑠)β€–+𝑠0β€–β€–β€–+β€–β€–β€–ξ€œ(π‘˜(𝑠,𝜏,π‘₯(𝜏))βˆ’π‘˜(𝑠,𝜏,𝑦(𝜏)))π‘‘πœπ‘‡0ξ€·β„Ž1(𝑠,𝜏,π‘₯(𝜏))βˆ’β„Ž1ξ€Έβ€–β€–β€–ξ‚Άξ€œ(𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‘π‘ +𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1πΎξ‚΅β€–β€–β€–β€–ξ€œΞ“(𝛼)𝐿(𝑠)π‘₯(𝑠)βˆ’π‘¦(𝑠)β€–+𝑠0β€–β€–β€–+β€–β€–β€–ξ€œ(π‘˜(𝑠,𝜏,π‘₯(𝜏))βˆ’π‘˜(𝑠,𝜏,𝑦(𝜏)))π‘‘πœπ‘‡0ξ€·β„Ž1(𝑠,𝜏,π‘₯(𝜏))βˆ’β„Ž1‖‖‖≀𝑏(𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‘π‘ 1(1+𝑑)π‘‡ξ€œπ‘‡0β€–π‘₯βˆ’π‘¦β€–π‘‘π‘ +𝑏2ξ‚΅1βˆ’(1+𝑑)π‘‡ξ‚Άξ€œπ‘‡0+β€–π‘₯βˆ’π‘¦β€–π‘‘π‘ (1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’2+Ξ“(π›Όβˆ’1)𝐿(𝑠)𝐾(β€–π‘₯βˆ’π‘¦β€–+𝑝(𝑠)β€–π‘₯βˆ’π‘¦β€–+π‘ž(𝑠)β€–π‘₯βˆ’π‘¦β€–)𝑑𝑠(1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’1+ξ€œΞ“(𝛼)𝐿(𝑠)𝐾(β€–π‘₯βˆ’π‘¦β€–+𝑝(𝑠)β€–π‘₯βˆ’π‘¦β€–+π‘ž(𝑠)β€–π‘₯βˆ’π‘¦β€–)𝑑𝑠𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1𝐿≀𝑏Γ(𝛼)(𝑠)𝐾(β€–π‘₯βˆ’π‘¦β€–+𝑝(𝑠)β€–π‘₯βˆ’π‘¦β€–+π‘ž(𝑠)β€–π‘₯βˆ’π‘¦β€–)𝑑𝑠1(1+𝑑)π‘‡ξ€œπ‘‡0β€–π‘₯βˆ’π‘¦β€–π‘‘π‘ +𝑏2ξ‚€11βˆ’π‘‡ξ‚ξ€œπ‘‡0+β€–π‘₯βˆ’π‘¦β€–π‘‘π‘ (1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’2+Ξ“(π›Όβˆ’1)𝐿(𝑠)(1+𝑝(𝑠)+π‘ž(𝑠))𝐾(β€–π‘₯βˆ’π‘¦β€–)𝑑𝑠(1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’1Ξ“+ξ€œ(𝛼)𝐿(𝑠)(1+𝑝(𝑠)+π‘ž(𝑠))𝐾(β€–π‘₯βˆ’π‘¦β€–)𝑑𝑠𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1Ξ“(𝛼)𝐿(𝑠)(1+𝑝(𝑠)+π‘ž(𝑠))𝐾(β€–π‘₯βˆ’π‘¦β€–)𝑑𝑠.(3.5) Since we have 𝑀(𝑑)=𝐿(𝑑)(1+𝑝(𝑑)+π‘ž(𝑑), π‘€βˆ—=sup{𝑀(𝑑)βˆΆπ‘‘βˆˆ[0,𝑇]}, and, Let 𝐾(β€–π‘₯βˆ’π‘¦β€–)≀𝑀‖π‘₯βˆ’π‘¦β€–, (𝑀>0), then 𝑏‖𝐹π‘₯(𝑑)βˆ’πΉπ‘¦(𝑑)‖≀1(1+𝑑)π‘‡ξ€œπ‘‡0β€–π‘₯βˆ’π‘¦β€–π‘‘π‘ +𝑏2ξ‚€11βˆ’π‘‡ξ‚ξ€œπ‘‡0+β€–π‘₯βˆ’π‘¦β€–π‘‘π‘ π‘€π‘€βˆ—(1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’2+Ξ“(π›Όβˆ’1)β€–π‘₯βˆ’π‘¦β€–π‘‘π‘ π‘€π‘€βˆ—(1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’1Ξ“(𝛼)β€–π‘₯βˆ’π‘¦β€–π‘‘π‘ +π‘€π‘€βˆ—ξ€œπ‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’1≀𝑏Γ(𝛼)β€–π‘₯βˆ’π‘¦β€–π‘‘π‘ 1(1+𝑇)+𝑏2(π‘‡βˆ’1)+π‘€π‘€βˆ—πΆ1(1+𝑇)Ξ“(𝛼+1)𝑇2βˆ’π›Όξ‚Ήβ€–π‘₯βˆ’π‘¦β€–,(3.6) where 𝐢1=2𝑇2+𝑇+𝛼(1+𝑇).
As 𝑏1(1+𝑇)+𝑏2(π‘‡βˆ’1)+(π‘€π‘€βˆ—πΆ1(1+𝑇))/(Ξ“(𝛼+1)𝑇2βˆ’π›Ό)<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 β€–β€–β€–π‘“ξ‚΅ξ€œπ‘‘,𝑦(𝑑),𝑑0ξ€œπ‘˜(𝑑,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Άβ€–β€–β€–(𝑑,𝜏,𝑦(𝜏))π‘‘πœβ‰€πœ“(𝑑),whereπœ“(𝑑)∈𝐿1(𝐽).(3.7)
Then the boundary value problem (2.4)-(2.5) has at least one element on [0,𝑇].

Proof. Consider π΅π‘Ÿ={π‘¦βˆˆπΆβˆΆβ€–π‘¦β€–β‰€π‘Ÿ}. We define the operators 𝐴 and 𝐡 as 1(𝐴π‘₯)(𝑑)=Ξ“ξ€œ(𝛼)𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1π‘“ξ‚΅ξ€œπ‘‘,π‘₯(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,π‘₯(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Ά(𝑠,𝜏,π‘₯(𝜏))π‘‘πœπ‘‘π‘ ,(𝐡π‘₯)(𝑑)=(1+𝑑)π‘‡ξ€œπ‘‡0ξ‚΅β„Ž(𝑦(𝑠))𝑑𝑠+1βˆ’(1+𝑑)π‘‡ξ‚Άξ€œπ‘‡0𝑔(𝑦(𝑠))𝑑𝑠+(1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’2π‘“ξ‚΅ξ€œΞ“(π›Όβˆ’1)𝑠,π‘₯(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,π‘₯(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Ά+(𝑠,𝜏,π‘₯(𝜏))π‘‘πœπ‘‘π‘ (1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’1π‘“ξ‚΅ξ€œΞ“(𝛼)𝑠,π‘₯(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,π‘₯(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Ά(𝑠,𝜏,π‘₯(𝜏))π‘‘πœπ‘‘π‘ .(3.8) Let us observe that if π‘₯,π‘¦βˆˆπ΅π‘Ÿ, then 𝐴π‘₯+π΅π‘¦βˆˆπ΅π‘Ÿ, β€–β€–β€–ξ€œβ€–π΄π‘₯+𝐡𝑦‖=𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1Ξ“π‘“ξ‚΅ξ€œ(𝛼)𝑠,π‘₯(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,π‘₯(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Ά+(𝑠,𝜏,π‘₯(𝜏))π‘‘πœπ‘‘π‘ (1+𝑑)π‘‡ξ€œπ‘‡0ξ‚΅β„Ž(𝑦(𝑠))𝑑𝑠+1βˆ’(1+𝑑)π‘‡ξ‚Άξ€œπ‘‡0𝑔(𝑦(𝑠))𝑑𝑠+(1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’2π‘“ξ‚΅ξ€œΞ“(π›Όβˆ’1)𝑠,𝑦(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Ά(𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‘π‘ +(1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’1π‘“ξ‚΅ξ€œΞ“(𝛼)𝑠,𝑦(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Άβ€–β€–β€–β‰€ξ€œ(𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‘π‘ π‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’1β€–β€–β€–π‘“ξ‚΅ξ€œΞ“(𝛼)𝑠,π‘₯(𝑠),𝑠0π‘˜ξ€œ(𝑠,𝜏,π‘₯(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Άβ€–β€–β€–+(𝑠,𝜏,π‘₯(𝜏))π‘‘πœπ‘‘π‘ (1+𝑑)π‘‡ξ€œπ‘‡0ξ‚΅β€–β„Ž(𝑦(𝑠))‖𝑑𝑠+1βˆ’(1+𝑑)π‘‡ξ‚Άξ€œπ‘‡0+(‖𝑔(𝑦(𝑠))‖𝑑𝑠1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’2β€–β€–β€–π‘“ξ‚΅ξ€œΞ“(π›Όβˆ’1)𝑠,𝑦(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1(ξ‚Άβ€–β€–β€–+𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‘π‘ (1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’1Ξ“β€–β€–β€–π‘“ξ‚΅ξ€œ(𝛼)𝑠,𝑦(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,𝑦(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Άβ€–β€–β€–(𝑠,𝜏,𝑦(𝜏))π‘‘πœπ‘‘π‘ β‰€β€–πœ“β€–πΏ1ξ€œπ‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’1Ξ“(𝛼)𝑑𝑠+(1+𝑑)𝐺1+ξ‚΅1βˆ’(1+𝑑)𝑇𝐺2𝑇+(1+𝑑)β€–πœ“β€–πΏ1π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’2+(Ξ“(π›Όβˆ’1)𝑑𝑠1+𝑑)β€–πœ“β€–πΏ1π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’1≀Γ(𝛼)π‘‘π‘ β€–πœ“β€–πΏ1𝑇𝛼Γ(𝛼+1)+(1+𝑑)𝐺1+ξ‚€11βˆ’π‘‡ξ‚πΊ2𝑇+(1+𝑑)π‘‡π›Όβˆ’1β€–πœ“β€–πΏ1+𝑇Γ(𝛼)(1+𝑑)π‘‡π›Όβ€–πœ“β€–πΏ1≀𝑇Γ(𝛼+1)β€–πœ“β€–πΏ1𝑇𝛼+Ξ“(𝛼+1)(1+𝑑)π‘‡π›Όβˆ’1β€–πœ“β€–πΏ1+𝑇Γ(𝛼)(1+𝑑)π‘‡π›Όβ€–πœ“β€–πΏ1𝑇Γ(𝛼+1)+(1+𝑇)𝐺1+(π‘‡βˆ’1)𝐺2≀𝐺1(1+𝑇)+𝐺2𝐢(π‘‡βˆ’1)+2π‘‡π›Όβˆ’2Ξ“(𝛼+1)β€–πœ“β€–πΏ1,(3.9) where 𝐢2=2𝑇2+𝑇(𝛼+1)+𝑇.
Now we prove that 𝐡π‘₯ is contraction mapping, ‖‖𝐡π‘₯1βˆ’π΅π‘₯2‖‖≀(1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’1Γ‖‖‖𝑓(𝛼)𝑠,π‘₯1ξ€œ(𝑠),𝑠0π‘˜ξ€·π‘ ,𝜏,π‘₯1ξ€Έξ€œ(𝜏)π‘‘πœ,𝑇0β„Ž1𝑠,𝜏,π‘₯1ξ€Έξ‚Άξ‚΅(𝜏)π‘‘πœβˆ’π‘“π‘ ,π‘₯2ξ€œ(𝑠)𝑠0π‘˜ξ€·π‘ ,𝜏,π‘₯2ξ€Έξ€œ(𝜏)π‘‘πœ,𝑇0β„Ž1𝑠,𝜏,π‘₯2ξ€Έξ‚Άβ€–β€–β€–+(𝜏)π‘‘πœπ‘‘π‘ (1+𝑑)π‘‡ξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’2‖‖‖𝑓Γ(π›Όβˆ’1)𝑠,π‘₯1ξ€œ(𝑠),𝑠0π‘˜ξ€·π‘ ,𝜏,π‘₯1ξ€Έξ€œ(𝜏)π‘‘πœ,𝑇0β„Ž1𝑠,𝜏,π‘₯1ξ€Έξ‚Άξ‚΅(𝜏)π‘‘πœβˆ’π‘“π‘ ,π‘₯2ξ€œ(𝑠),𝑠0π‘˜ξ€·π‘ ,𝜏,π‘₯2ξ€Έξ€œ(𝜏)π‘‘πœ,𝑇0β„Ž1𝑠,𝜏,π‘₯2‖‖‖≀(𝜏)π‘‘πœπ‘‘π‘ (1+𝑑)π‘‡ξ‚Έξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’2𝐿‖‖π‘₯Ξ“(π›Όβˆ’1)(𝑠)(1+𝑝(𝑠)+π‘ž(𝑠))𝐾1βˆ’π‘₯2β€–β€–ξ€Έ+ξ€œπ‘‘π‘ π‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’1ξ€·β€–β€–π‘₯Ξ“(𝛼)𝐿(𝑠)(1+𝑝(𝑠)+π‘ž(𝑠))𝐾1βˆ’π‘₯2β€–β€–ξ€Έξ‚Ή.𝑑𝑠(3.10) Let 𝐾(β€–π‘₯1βˆ’π‘₯2β€–)≀𝑀‖π‘₯1βˆ’π‘₯2β€–, we obtain ‖‖𝐡π‘₯1βˆ’π΅π‘₯2‖‖≀(1+𝑑)π‘€π‘€βˆ—π‘‡β€–β€–π‘₯1βˆ’π‘₯2β€–β€–ξ‚Έξ€œπ‘‡0(π‘‡βˆ’π‘ )π›Όβˆ’2Ξ“ξ€œ(π›Όβˆ’1)𝑑𝑠+𝑇0(π‘‡βˆ’π‘ )π›Όβˆ’1Ξ“ξ‚Ή,β€–β€–(𝛼)𝑑𝑠𝐡π‘₯1βˆ’π΅π‘₯2‖‖≀(1+𝑇)π‘€π‘€βˆ—π‘‡ξ‚Έπ‘‡π›Όβˆ’1+𝑇Γ(𝛼)𝛼‖‖π‘₯Ξ“(𝛼+1)1βˆ’π‘₯2β€–β€–β‰€π‘€π‘€βˆ—(1+𝑇)(𝛼+𝑇)Ξ“(𝛼+1)𝑇2βˆ’π›Όβ€–β€–π‘₯1βˆ’π‘₯2β€–β€–.(3.11) It is clear that 𝐡 is contraction mapping, since π‘₯(𝑑) is continuous, then 𝐴π‘₯ is continuous β€–β€–β€–1‖𝐴π‘₯(𝑑)β€–=Ξ“ξ€œ(𝛼)𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1π‘“ξ‚΅ξ€œπ‘ ,π‘₯(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,π‘₯(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Άβ€–β€–β€–(𝑠,𝜏,π‘₯(𝜏))π‘‘πœπ‘‘π‘ β‰€β€–πœ“β€–πΏ1ξ€œπ‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’1(𝑇Γ(𝛼)𝑑𝑠‖𝐴π‘₯𝑑)β€–β‰€π›Όβ€–πœ“β€–πΏ1.Ξ“(𝛼+1)(3.12) Hence, 𝐴 is uniformly bounded on π΅π‘Ÿ. Now, let us prove that 𝐴π‘₯(𝑑) is equicontinuous, let 𝑑1,𝑑2∈[0,𝑇] and π‘₯βˆˆπ΅π‘Ÿ. Using the fact that 𝑓 is bounded on the compact set π½Γ—π΅π‘Ÿ, thus sup(𝑑,𝑠)βˆˆπ½Γ—π΅π‘Ÿβˆ«β€–π‘“(𝑠,π‘₯(𝑠),𝑠0βˆ«π‘˜(𝑠,𝜏,π‘₯(𝜏))π‘‘πœ,𝑇0β„Ž1(𝑠,𝜏,π‘₯(𝜏))π‘‘πœ)β€–=𝑐0<∞,we get ‖‖𝑑𝐴π‘₯1ξ€Έξ€·π‘‘βˆ’π΄π‘₯2ξ€Έβ€–β€–=β€–β€–β€–1ξ€œΞ“(𝛼)𝑑10𝑑1ξ€Έβˆ’π‘ π›Όβˆ’1π‘“ξ‚΅ξ€œπ‘ ,π‘₯(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,π‘₯(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Ά1(𝑠,𝜏,π‘₯(𝜏))π‘‘πœπ‘‘π‘ βˆ’Ξ“ξ€œ(𝛼)𝑑20𝑑2ξ€Έβˆ’π‘ π›Όβˆ’1π‘“ξ‚΅ξ€œπ‘ ,π‘₯(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,π‘₯(𝜏))π‘‘πœ,𝑇0β„Ž1‖‖‖≀1(𝑠,𝜏,π‘₯(𝜏))π‘‘πœπ‘‘π‘ β€–β€–β€–ξ€œΞ“(𝛼)𝑑10𝑑1ξ€Έβˆ’π‘ π›Όβˆ’1βˆ’ξ€·π‘‘2ξ€Έβˆ’π‘ π›Όβˆ’1ξ‚„π‘“ξ‚΅ξ€œπ‘ ,π‘₯(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,π‘₯(𝜏))π‘‘πœ,𝑇0β„Ž1ξ‚Άβ€–β€–β€–+β€–β€–β€–ξ€œ(𝑠,𝜏,π‘₯(𝜏))π‘‘πœπ‘‘π‘ π‘‘2𝑑1𝑑2ξ€Έβˆ’π‘ π›Όβˆ’1π‘“ξ‚΅ξ€œπ‘ ,π‘₯(𝑠),𝑠0ξ€œπ‘˜(𝑠,𝜏,π‘₯(𝜏))π‘‘πœ,𝑇0β„Ž1‖‖‖≀𝑐(𝑠,𝜏,π‘₯(𝜏))π‘‘πœπ‘‘π‘ 0ξ€Ί2𝑑Γ(𝛼+1)2βˆ’π‘‘1𝛼+𝑑1π›Όβˆ’π‘‘2𝛼.ξ€Έξ€»(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(𝑑)=+110||𝑦||+ξ€œ10+(𝑑)𝑑0||||𝑦(𝑑)10𝑒|𝑦(𝑑)|+𝑑𝑑𝑑+1ξ€œ0||||𝑒𝑦(𝑑)βˆ’π‘‘||||10+𝑦(𝑑)2𝑑𝑑,(3.14) with integral boundary conditions 𝑦(0)βˆ’π‘¦ξ…žξ€œ(0)=101||||10+𝑦(𝑑)𝑑𝑑,𝑦(1)βˆ’π‘¦ξ…žξ€œ(1)=10110+π‘’βˆ’|𝑦(𝑑)|𝑑𝑑.(3.15) Here, β€–β€–β€–1‖𝑔(𝑦(𝑑))β€–=||||‖‖‖≀110+𝑦(𝑑)110,‖𝑔(π‘₯)βˆ’π‘”(𝑦)‖≀‖‖‖1100β€–π‘₯βˆ’π‘¦β€–,β€–β„Ž(𝑦(𝑑))β€–=10+π‘’βˆ’|𝑦(𝑑)|‖‖‖≀1110,β€–β„Ž(π‘₯)βˆ’β„Ž(𝑦)β€–β‰€β€–β€–β€–ξ€œ100β€–π‘₯βˆ’π‘¦β€–,𝑑0‖‖‖≀1(π‘˜(𝑑,𝑠,π‘₯)βˆ’π‘˜(𝑑,𝑠,𝑦))𝑑𝑠10𝑒𝑑‖‖‖‖‖‖π‘₯βˆ’π‘¦β€–,π‘‘ξ€œ0‖‖‖‖‖≀1π‘˜(𝑑,𝑠,𝑦)π‘‘π‘ β€–β€–β€–ξ€œ10+𝑑‖𝑦(𝑑)β€–,𝑑0ξ€·β„Ž1(𝑑,𝑠,π‘₯)βˆ’β„Ž1‖‖‖≀1(𝑑,𝑠,𝑦)𝑑𝑠10𝑒𝑑‖‖‖‖‖‖π‘₯βˆ’π‘¦β€–,π‘‘ξ€œ0β„Ž1‖‖‖‖‖≀1(𝑑,𝑠,𝑦)𝑑𝑠‖‖𝑓10+𝑑‖𝑦(𝑑)β€–,𝑑,π‘₯1𝑦1,𝑧1ξ€Έξ€·βˆ’π‘“π‘‘,π‘₯2,𝑦2,𝑧2‖‖≀1ξ€·β€–β€–π‘₯10+𝑑1βˆ’π‘₯2β€–β€–+‖‖𝑦1βˆ’π‘¦2β€–β€–+‖‖𝑧1βˆ’π‘§2β€–β€–ξ€Έ,1𝑓(𝑑,0,0,0)=.10(3.16) Hence, the conditions (H1)–(H5) hold with 𝐺1=𝐺2=0.1, 𝑏1=𝑏2=0.01, π‘€βˆ—1=0.12, 𝑀=0.1, πΆπ‘œ=6, 𝑁1=0.1, π‘€βˆ—=0.12, and 𝐢1=6, thus 𝑏1(1+𝑇)+𝑏2(π‘‡βˆ’1)+π‘€π‘€βˆ—πΆ1(1+𝑇)Ξ“(𝛼+1)𝑇2βˆ’π›Ό<1⟺0.01(2)+(0.1)(0.12)6(2)Ξ“(2.5)<1.(3.17) We conclude from the above example that the integrodifferential equation has unique solution.

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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.

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