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

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^’Guérékata [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 [𝑎,𝑏], the Caputo fractional order derivative of 𝑓 is defined by ^𝑡𝑎𝐷^𝛼𝑓(𝑡)=1Γ(𝑛−𝛼)^𝑡𝑎(𝑡−𝑠)^{𝑛−𝛼−1}𝑓^(𝑛)(𝑠)𝑑𝑠,(2.1) where 𝑛=[𝛼]+1 and [𝛼] denotes the integer part of 𝛼.

Lemma 2.2. Let 𝛼>0, then ^𝑡𝑎𝐷^{𝑡𝑎−𝛼}𝐷^𝛼𝑦(𝑡)=𝑦(𝑡)+𝑐₀+𝑐₁𝑡+𝑐₂𝑡²+⋯+𝑐_𝑛−1𝑡^𝑛−1,(2.2) for some 𝑐_𝑖∈𝑅, i=0,1,…,n−1,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 ^𝑏𝑎𝐷^−𝛼𝑓=1Γ(𝛼)^𝑏𝑎(𝑏−𝑡)^𝛼−1𝑓(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,𝑦∈𝑀,(ii)𝐴 is compact and continuous;(iii)𝐵 is a contraction mapping,then there exists 𝑧∈𝑀 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 𝑦(𝐷0)^𝛼−𝑦(𝑦𝑡)_=𝑓𝑡,𝑦(𝑡),^𝑡0𝑘(𝑡,𝑠,𝑦(𝑠))𝑑𝑠,^𝑇0ℎ₁(𝑡,𝑠,𝑦(𝑠))𝑑𝑠,(2.4)(0)=^𝑇0𝑔(𝑦(𝑠))𝑑𝑠,𝑦(𝑇)−𝑦^(𝑇)=^𝑇0ℎ(𝑦(𝑠))𝑑𝑠,(2.5) where 1<𝛼≤2, 𝐷^𝛼 is the Caputo fractional derivative and the nonlinear functions 𝑓∶[0,𝑇]×𝑋×𝑋×𝑋→𝑋, 𝑘,ℎ₁∶[0,𝑇]×[0,𝑇]×𝑋→𝑋 and 𝑔,ℎ∶𝑋→𝑋 satisfy the following hypotheses: (H1)there exists constants 𝐺₁,𝐺₂ such that ‖ℎ(𝑦)‖≤𝐺₁,‖𝑔(𝑦)‖≤𝐺₂ for 𝑦∈𝑋,(H2)there exists constants 𝑏₁,𝑏₂ such that ‖ℎ(𝑥)−ℎ(𝑦)‖≤𝑏₁‖𝑥−𝑦‖ and ‖𝑔(𝑥)−𝑔(𝑦)‖≤𝑏₂‖𝑥−𝑦‖,∀𝑥,𝑦∈𝑋,(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))𝑑𝑠‖≤𝑝(𝑡)‖𝑥−𝑦‖ and ‖∫^𝑡0𝑘(𝑡,𝑠,𝑦)𝑑𝑠‖≤𝑝₁(𝑡)‖𝑦‖, for every 𝑡,𝑠∈[0,𝑇] and 𝑥,𝑦∈𝑋,(H4)there exists continuous functions 𝑞∶[0,𝑇]→𝑅⁺ and 𝑞₁∶[0,𝑇]→𝑅⁺ such that ‖∫^𝑇0(ℎ₁(𝑡,𝑠,𝑥)−ℎ₁(𝑡,𝑠,𝑦)𝑑𝑠‖≤𝑞(𝑡)‖𝑥−𝑦‖ and ‖∫^𝑇0ℎ₁(𝑡,𝑠,𝑦)𝑑𝑠‖≤𝑞₁(𝑡)‖𝑦‖𝑓𝑜𝑟𝑒𝑣𝑒𝑟𝑦𝑡,𝑠∈[0,𝑇]𝑎𝑛𝑑𝑥,𝑦∈𝑋(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(‖𝑥₁−𝑥₂‖+‖𝑦₁−𝑦₂‖+‖𝑧₁−𝑧₂‖) and 𝑁₁=sup_{𝑡∈[0,𝑇]}‖𝑓(𝑡,0,0,0)‖, for every 𝑡∈[0,𝑇] and 𝑥₁,𝑦₁,𝑧₁,𝑥₂,𝑦₂,𝑧₂∈𝑋, where 𝐾∶𝑅⁺→(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<𝛼≤2 and 𝑓∶𝐽×𝑋→𝑋, 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)𝑇^𝑇0ℎ(𝑦(𝑠))𝑑𝑠+1−(1+𝑡)𝑇^𝑇0−(1+𝑡)𝑔(𝑦(𝑠))𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑓𝑠,𝑦(𝑠),^𝑠0+(1+𝑡)𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑇^𝑇0(𝑇−𝑠)^𝛼−2Γ(𝛼−1)𝑓𝑠,𝑦(𝑠),^𝑠0+𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑓𝑠,𝑦(𝑠),^𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠.(2.7)

Proof. By Lemma 2.2, we reduce the problem (2.4)-(2.5) to an equivalent integral equation 𝑦𝑦((𝑡𝑡))==^𝑡0_𝑡0𝐼^𝛼𝑓+𝐶₁+𝐶₂𝑡,(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑓𝑠,𝑦(𝑠),^𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠+𝐶₁+𝐶₂𝑡.(2.8) In view of the relations ^𝑐𝐷^𝛼𝐼^𝛼𝑦(𝑡)=𝑦(𝑡) and 𝐼^𝛼𝐼^𝛽𝑦(𝑡)=𝐼^𝛼+𝛽𝑦(𝑡), for 𝛼,𝛽>0, we obtain 𝑦^(𝑡)=^𝑡0(𝑡−𝑠)^𝛼−2Γ(𝛼−1)𝑓𝑠,𝑦(𝑠),^𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠+𝐶₂.(2.9) Applying the boundary condition (2.5), we find that 𝑦(0)=𝐶₁,𝑦(𝑇)=^𝑇0(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑓𝑠,𝑦(𝑠),^𝑠0+𝐶₁𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠𝑦+𝐶₂𝑇,_(0)=𝐶₂,𝑦^(𝑇)=^𝑇0(𝑇−𝑠)^𝛼−2Γ(𝛼−1)𝑓𝑠,𝑦(𝑠),^𝑠0+𝐶₂𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠.,(2.10) that is, 𝐶₂=1𝑇^𝑇0ℎ(𝑦(𝑠))𝑑𝑠−1𝑇^𝑇0−1𝑔(𝑦(𝑠))𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑓𝑠,𝑦(𝑠),^𝑠0+1𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−2Γ(𝛼−1)𝑓𝑠,𝑦(𝑠),^𝑠0𝐶𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠,₁=1𝑇^𝑇0ℎ(𝑦(𝑠))𝑑𝑠+1−1𝑇^𝑇0−1𝑔(𝑦(𝑠))𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑓𝑠,𝑦(𝑠),^𝑠0+1𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−2Γ(𝛼−1)𝑓𝑠,𝑦(𝑠),^𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠.(2.11) Therefore the solution of (2.4)-(2.5) is 𝑦(𝑡)=(1+𝑡)𝑇^𝑇0ℎ(𝑦(𝑠))𝑑𝑠+1−(1+𝑡)𝑇^𝑇0−(1+𝑡)𝑔(𝑦(𝑠))𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑓𝑠,𝑦(𝑠),^𝑠0+(1+𝑡)𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−2Γ(𝛼−1)𝑓𝑠,𝑦(𝑠),^𝑠0+𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑓𝑠,𝑦(𝑠),^𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠,(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 𝐹∶𝐶→𝐶 by 𝐹𝑦(𝑡)=(1+𝑡)𝑇^𝑇0ℎ(𝑦(𝑠))𝑑𝑠+1−(1+𝑡)𝑇^𝑇0−(1+𝑡)𝑔(𝑦(𝑠))𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑓𝑠,𝑦(𝑠),^𝑠0+(1+𝑡)𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−2Γ(𝛼−1)𝑓𝑠,𝑦(𝑠),^𝑠0+𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑓𝑠,𝑦(𝑠),^𝑠0𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠.(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 𝐹𝐵𝑟⊂𝐵𝑟, where 𝐵𝑟={𝑦∈𝐶∶‖𝑦‖≤𝑟}. For 𝑦∈𝐵𝑟, we have ‖𝐹𝑦(𝑡)‖≤(1+𝑡)𝑇^𝑇0‖ℎ(𝑦(𝑠))‖𝑑𝑠+1−(1+𝑡)𝑇^𝑇0+(1+𝑡)‖𝑔(𝑦(𝑠))‖𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−2Γ(𝛼−1)‖‖‖𝑓𝑠,𝑦(𝑠),^𝑠0+(1+𝑡)𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏‖‖‖𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−1Γ(𝛼)‖‖‖𝑓𝑠,𝑦(𝑠),^𝑠0+𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏‖‖‖𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1Γ(𝛼)‖‖‖𝑓𝑠,𝑦(𝑠),^𝑠0‖𝐹𝑦(𝑡)‖≤(1+𝑡)𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏‖‖‖𝑑𝑠,𝑇^𝑇0‖ℎ(𝑦(𝑠))‖𝑑𝑠+1−(1+𝑡)𝑇^𝑇0+(1+𝑡)‖𝑔(𝑦(𝑠))‖𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−2Γ(𝛼−1)‖‖‖𝑓𝑠,𝑦(𝑠),^𝑆0+(1+𝑡)−𝑓(𝑠,0,0,0)+𝑓(𝑘𝑠,(0𝑠,,0𝜏,,0𝑦)(‖‖‖𝜏)𝑑)𝑠𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑇^𝑇0(𝑇−𝑠)^𝛼−1Γ(𝛼)‖‖‖𝑓𝑠,𝑦(𝑠),^𝑆0+−𝑓(𝑠,0,0,0)+𝑓(𝑘𝑠,(0𝑠,,0𝜏,,0𝑦)(‖‖‖𝜏)𝑑)𝑠_𝑡0𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏(𝑡−𝑠)^𝛼−1Γ(𝛼)‖‖‖𝑓𝑠,𝑦(𝑠),^𝑆0‖𝐹𝑦(𝑡)‖≤(1+𝑡)−𝑓(𝑠,0,0,0)+𝑓𝑘((𝑠𝑠,,𝜏0,,𝑦0,(0𝜏)))‖‖‖𝑑𝑑𝜏𝑠,,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑇𝐺¹𝑇+1−(1+𝑡)𝑇𝐺²+(1+𝑡)𝑇𝑇^𝑇0(𝑇−𝑠)^𝛼−2Γ(𝛼−1)‖‖‖𝑓𝑠,𝑦(𝑠),^𝑆0−𝑓(𝑠,0,0,0)‖‖‖𝑑𝑠𝑘+(𝑠(1,𝜏+,𝑦𝑡)(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑇^𝑇0(𝑇−𝑠)^𝛼−2+Γ(𝛼−1)‖𝑓(𝑠,0,0,0)‖𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1Γ(𝛼)‖‖‖𝑓𝑠,𝑦(𝑠),^𝑆0−𝑓(𝑠,0,0,0)‖‖‖𝑑𝑠𝑘+(𝑠,_𝑡0𝜏,𝑦(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏(𝑡−𝑠)^𝛼−1+1+𝑡Γ(𝛼)‖𝑓(𝑠,0,0,0)‖𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−1Γ(𝛼)‖‖‖𝑓𝑠,𝑦(𝑠),^𝑆0−𝑓(𝑠,0,0,0)‖‖‖𝑑𝑠+𝑘(1𝑠,+𝜏,𝑡𝑦(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑇^𝑇0(𝑇−𝑠)^𝛼−1≤(1+𝑡)Γ(𝛼)‖𝑓(𝑠,0,0,0)‖𝑑𝑠𝑇𝐺¹𝑇+1−(1+𝑡)𝑇𝐺²𝑇+(1+𝑡)𝑇^𝑇0(𝑇−𝑠)^𝛼−2𝐾‖𝑦(𝑠)‖+‖‖‖_𝑆0Γ(𝛼−1)𝐿(𝑠)+(1+𝑡)𝑁₁𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏‖‖‖+‖‖‖^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏)‖‖‖𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−2Γ(𝛼−1)𝑑𝑠+(1𝑇+𝑡)^𝑇0(𝑇−𝑠)^𝛼−1𝐾‖𝑦(𝑠)‖+‖‖‖_𝑠0Γ(𝛼)𝐿(𝑠)+(1+𝑡)𝑁₁𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏‖‖‖+‖‖‖^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏)‖‖‖𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑑𝑠+^𝑡0(𝑡−𝑠)^𝛼−1𝐾‖𝑦(𝑠)‖+‖‖‖_𝑠0Γ(𝛼)𝐿(𝑠)+𝑁₁𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏‖‖‖+‖‖‖^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏)‖‖‖𝑑𝑠^𝑡0(𝑡−𝑠)^𝛼−1≤(1+𝑡)𝐺₁Γ(𝛼)𝑑𝑠+1−(1+𝑡)𝑇𝐺²𝑇+(1+𝑡)𝑁¹_𝑇0𝑇(𝑇−𝑠)^𝛼−2Γ(𝛼−1)𝑑𝑠+^𝑇0(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑑𝑠+𝑁¹^𝑡0(𝑡−𝑠)^𝛼−1+(1+𝑡)Γ(𝛼)𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−2Γ(𝛼−1)𝐿(𝑠)𝐾‖𝑦‖+𝑝¹+(1+𝑡)(𝑠)‖𝑦‖+𝑞₁(𝑠)‖𝑦‖𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−1Γ(𝛼)𝐿(𝑠)𝐾‖𝑦‖+𝑝¹+(𝑠)‖𝑦‖+𝑞₁(𝑠)‖𝑦‖𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1Γ(𝛼)𝐿(𝑠)𝐾‖𝑦‖+𝑝¹‖𝐹𝑦(𝑡)‖≤(1+𝑡)𝐺₁(𝑠)‖𝑦‖+𝑞₁(𝑠)‖𝑦‖𝑑𝑠,+1−(1+𝑡)𝑇𝐺²+(1+𝑡)𝑁₁𝑇𝑇Γ𝑇(𝛼^𝛼−1)+Γ(𝛼𝑇+^𝛼1)+𝑁¹𝑇^𝛼+(1+𝑡)Γ(𝛼+1)𝑇^𝑇0(𝑇−𝑠)^𝛼−2Γ(𝛼−1)𝐿(𝑠)1+𝑝¹+(1+𝑡)(𝑠)+𝑞₁(𝑠)𝐾(‖𝑦‖)𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−1Γ(𝛼)𝐿(𝑠)1+𝑝¹+(𝑠)+𝑞₁(𝑠)𝐾(‖𝑦‖)𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1Γ(𝛼)𝐿(𝑠)1+𝑝¹(𝑠)+𝑞₁(𝑠)𝐾(‖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+1−1𝑇𝐺²𝑇+(1+𝑡)𝑁¹𝑇Γ𝑇(𝛼^𝛼−1)+Γ(𝛼𝑇+^𝛼1)+𝑁¹𝑇^𝛼+(1+𝑡)𝑀₁Γ(𝛼+1)𝑇^𝑇0(𝑇−𝑠)^𝛼−2Γ(𝛼−1)𝐾(‖𝑦‖)𝑑𝑠+(1+𝑇𝑡)𝑀¹^𝑇0(𝑇−𝑠)^𝛼−1+𝑀₁Γ(𝛼)𝐾(‖𝑦‖)𝑑𝑠^𝑡0(𝑡−𝑠)^𝛼−1≤(1+𝑡)𝐺₁Γ(𝛼)𝐾(‖𝑦‖)𝑑𝑠+1−1𝑇𝐺²𝑇+(1+𝑡)𝑁¹𝑇Γ𝑇(𝛼^𝛼−1)+Γ(𝛼𝑇+^𝛼1)+𝑁¹𝑇^𝛼+(1+𝑡)𝑀₁Γ(𝛼+1)𝐾(𝑟)𝑇^𝑇0(𝑇−𝑠)^𝛼−2Γ(𝛼−1)𝑑𝑠+(1+𝑡)𝑀¹𝐾(𝑟)𝑇^𝑇0(𝑇−𝑠)^𝛼−1+𝑀₁Γ(𝛼)𝑑𝑠𝐾(𝑟)^𝑡0(𝑡−𝑠)^𝛼−1≤(1+𝑡)𝐺₁Γ(𝛼)𝑑𝑠+1−1𝑇𝐺²𝑇+(1+𝑡)𝑁¹𝑇Γ𝑇(𝛼^𝛼−1)+Γ(𝛼𝑇+^𝛼1)+𝑁¹𝑇^𝛼+(1+𝑡)𝑀₁Γ(𝛼+1)𝐾(𝑟)𝑇Γ𝑇(𝛼^𝛼−1)+Γ(𝛼𝑇+^𝛼1)+𝑀¹𝐾(𝑟)𝑇^𝛼≤(1+𝑡)𝐺₁Γ(𝛼+1)+(𝑇−1)𝐺₂+(1+𝑡)𝑇𝑁¹+𝑀₁𝐾(𝑟)𝑇^𝛼−1+𝑁Γ(𝛼)+Γ(𝛼𝑇+^𝛼1)₁+𝑀₁𝐾(𝑟)𝑇^𝛼‖𝐹𝑦(𝑡)‖≤(1+𝑇)𝐺₁Γ(𝛼+1),+(𝑇−1)𝐺₂+(1+𝑇)𝑇𝑁¹+𝑀₁𝐾(𝑟)𝑇^𝛼−1+𝑁Γ(𝛼)+Γ(𝛼𝑇+^𝛼1)₁+𝑀₁𝐾(𝑟)𝑇^𝛼≤𝐺₁Γ(𝛼+1)(1+𝑇)+𝐺₂(𝑇−1)+𝐶⁰𝑁₁+𝑀₁𝐾(𝑟)Γ(𝛼+1)𝑇^2−𝛼,(3.3) where 𝐶₀=2𝑇²+𝑇+𝛼(𝑇+1).
Now, take 𝑥,𝑦∈𝐶 and for each 𝑡∈[0,𝑇], we obtain ‖𝐹𝑥(𝑡)−𝐹𝑦(𝑡)‖≤(1+𝑡)𝑇^𝑇0‖ℎ(𝑥)−ℎ(𝑦)‖𝑑𝑠+1−(1+𝑡)𝑇^𝑇0+(1+𝑡)‖𝑔(𝑥)−𝑔(𝑦)‖𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−1‖‖‖𝑓𝑠,𝑥(𝑠),_𝑠0Γ(𝛼)−𝑓𝑠,𝑦(𝑠),𝑘_𝑠0(𝑠,𝜏,𝑥(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑥(𝜏))𝑑𝜏+(1+𝑡)𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏‖‖‖𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−2‖‖‖𝑓𝑠,𝑥(𝑠),Γ(_𝑠0𝛼−1)−𝑓𝑠,𝑦(𝑠),𝑘_𝑠0(𝑠,𝜏,𝑥(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑥(𝜏))𝑑𝜏+𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏‖‖‖𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1‖‖‖𝑓𝑠,Γ𝑥((𝛼𝑠)),_𝑠0−𝑓𝑠,𝑦(𝑠),𝑘_𝑠0(𝑠,𝜏,𝑥(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑥(𝜏))𝑑𝜏𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏‖‖‖𝑑𝑠,(3.4) by using (H1)–(H5), we get ‖𝐹𝑥(𝑡)−𝐹𝑦(𝑡)‖≤𝑏¹(1+𝑡)𝑇^𝑇0‖𝑥−𝑦‖𝑑𝑠+𝑏₂1−(1𝑇+𝑡)^𝑇0+(1+𝑡)‖𝑥−𝑦‖𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−2𝐾‖𝑥(𝑠)−Γ𝑦(𝛼(𝑠−)‖1+)𝐿‖‖‖(𝑠_𝑠0)+‖‖‖(𝑘(𝑠,𝜏,𝑥(𝜏))−𝑘(𝑠,𝜏,𝑦(𝜏)))𝑑𝜏_𝑇0‖‖‖+(1+𝑡)ℎ₁(𝑠,𝜏,𝑥(𝜏))−ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏‖‖‖𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−1𝐾‖𝑥(𝑠)−𝑦Γ(𝑠()𝛼‖)+𝐿‖‖‖(𝑠_𝑠0)+‖‖‖(𝑘(𝑠,𝜏,𝑥(𝜏))−𝑘(𝑠,𝜏,𝑦(𝜏)))𝑑𝜏_𝑇0‖‖‖ℎ₁(𝑠,𝜏,𝑥(𝜏))−ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏‖‖‖𝑑𝑠+^𝑡0(𝑡−𝑠)^𝛼−1𝐾‖𝑥(𝑠)−𝑦(𝑠)‖+‖‖‖_𝑠0Γ(𝛼)𝐿(𝑠)+‖‖‖(𝑘(𝑠,𝜏,𝑥(𝜏))−𝑘(𝑠,𝜏,𝑦(𝜏)))𝑑𝜏_𝑇0‖‖‖≤𝑏ℎ₁(𝑠,𝜏,𝑥(𝜏))−ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏‖‖‖𝑑𝑠₁(1+𝑡)𝑇^𝑇0‖𝑥−𝑦‖𝑑𝑠+𝑏₂1−(1𝑇+𝑡)^𝑇0+(1+𝑡)‖𝑥−𝑦‖𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−2+(1+𝑡)Γ(𝛼−1)𝐿(𝑠)𝐾(‖𝑥−𝑦‖+𝑝(𝑠)‖𝑥−𝑦‖+𝑞(𝑠)‖𝑥−𝑦‖)𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−1+Γ(𝛼)𝐿(𝑠)𝐾(‖𝑥−𝑦‖+𝑝(𝑠)‖𝑥−𝑦‖+𝑞(𝑠)‖𝑥−𝑦‖)𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1≤𝑏Γ(𝛼)𝐿(𝑠)𝐾(‖𝑥−𝑦‖+𝑝(𝑠)‖𝑥−𝑦‖+𝑞(𝑠)‖𝑥−𝑦‖)𝑑𝑠₁(1+𝑡)𝑇^𝑇0‖𝑥−𝑦‖𝑑𝑠+𝑏₂1−1𝑇^𝑇0+(1+𝑡)‖𝑥−𝑦‖𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−2+(1+𝑡)Γ(𝛼−1)𝐿(𝑠)(1+𝑝(𝑠)+𝑞(𝑠))𝐾(‖𝑥−𝑦‖)𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−1+Γ(𝛼)𝐿(𝑠)(1+𝑝(𝑠)+𝑞(𝑠))𝐾(‖𝑥−𝑦‖)𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1Γ(𝛼)𝐿(𝑠)(1+𝑝(𝑠)+𝑞(𝑠))𝐾(‖𝑥−𝑦‖)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(‖𝑥−𝑦‖)≤𝑤‖𝑥−𝑦‖, (𝑤>0), then ‖𝐹𝑥(𝑡)−𝐹𝑦(𝑡)‖≤𝑏¹(1+𝑡)𝑇^𝑇0‖𝑥−𝑦‖𝑑𝑠+𝑏₂1−1𝑇^𝑇0+𝑤𝑀_∗‖𝑥−𝑦‖𝑑𝑠(1+𝑡)𝑇^𝑇0(𝑇−𝑠)^𝛼−2+𝑤𝑀_∗Γ(𝛼−1)‖𝑥−𝑦‖𝑑𝑠(1+𝑡)𝑇^𝑇0(𝑇−𝑠)^𝛼−1+𝑤𝑀_∗Γ(𝛼)‖𝑥−𝑦‖𝑑𝑠^𝑡0(𝑡−𝑠)^𝛼−1≤𝑏Γ(𝛼)‖𝑥−𝑦‖𝑑𝑠₁(1+𝑇)+𝑏₂(𝑇−1)+𝑤𝑀^∗𝐶₁(1+𝑇)Γ(𝛼+1)𝑇^2−𝛼‖𝑥−𝑦‖,(3.6) where 𝐶₁=2𝑇²+𝑇+𝛼(1+𝑇).
As 𝑏₁(1+𝑇)+𝑏₂(𝑇−1)+(𝑤𝑀^∗𝐶₁(1+𝑇))/(Γ(𝛼+1)𝑇^2−𝛼)<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 ‖‖‖𝑓𝑡,𝑦(𝑡),^𝑡0𝑘(𝑡,𝜏,𝑦(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑡,𝜏,𝑦(𝜏))𝑑𝜏‖‖‖≤𝜓(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={𝑦∈𝐶∶‖𝑦‖≤𝑟}. We define the operators 𝐴 and 𝐵 as (𝐴𝑥)(𝑡)=1Γ(𝛼)^𝑡0(𝐵𝑥)(𝑡)=(1+𝑡)(𝑡−𝑠)^𝛼−1𝑓𝑡,𝑥(𝑠),^𝑠0𝑘(𝑠,𝜏,𝑥(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑥(𝜏))𝑑𝜏𝑑𝑠,𝑇^𝑇0ℎ(𝑦(𝑠))𝑑𝑠+1−(1+𝑡)𝑇^𝑇0𝑔(𝑦(𝑠))𝑑𝑠+(1+𝑡)𝑇^𝑇0(𝑇−𝑠)^𝛼−2𝑓𝑠,𝑥(𝑠),_𝑠0Γ(𝛼−1)+(1+𝑡)𝑘(𝑠,𝜏,𝑥(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑥(𝜏))𝑑𝜏𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑓𝑠,𝑥(𝑠),^𝑠0𝑘(𝑠,𝜏,𝑥(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑥(𝜏))𝑑𝜏𝑑𝑠.(3.8) Let us observe that if 𝑥,𝑦∈𝐵𝑟, then 𝐴𝑥+𝐵𝑦∈𝐵𝑟, ‖𝐴𝑥+𝐵𝑦‖=‖‖‖^𝑡0(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑓𝑠,𝑥(𝑠),^𝑠0+(1+𝑡)𝑘(𝑠,𝜏,𝑥(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑥(𝜏))𝑑𝜏𝑑𝑠𝑇^𝑇0ℎ(𝑦(𝑠))𝑑𝑠+1−(1+𝑡)𝑇^𝑇0𝑔(𝑦(𝑠))𝑑𝑠+(1+𝑡)𝑇^𝑇0(𝑇−𝑠)^𝛼−2𝑓𝑠,𝑦(𝑠),_𝑠0Γ(𝛼−1)𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠+(1+𝑡)_𝑇0𝑇(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑓𝑠,𝑦(𝑠),^𝑠0≤𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏𝑑𝑠_𝑡0‖‖‖(𝑡−𝑠)^𝛼−1Γ(𝛼)‖‖‖𝑓𝑠,𝑥(𝑠),^𝑠0+(1+𝑡)𝑘(𝑠,𝜏,𝑥(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑥(𝜏))𝑑𝜏‖‖‖𝑑𝑠𝑇^𝑇0‖ℎ(𝑦(𝑠))‖𝑑𝑠+1−(1+𝑡)𝑇^𝑇0+(1+𝑡)‖𝑔(𝑦(𝑠))‖𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−2‖‖‖𝑓𝑠,𝑦(𝑠),Γ_𝑠0(𝛼−1)+(1+𝑡)𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏‖‖‖𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−1‖‖‖𝑓𝑠,𝑦(𝑠),Γ_𝑠0(𝛼)≤‖𝜓‖_𝐿1𝑘(𝑠,𝜏,𝑦(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑦(𝜏))𝑑𝜏‖‖‖𝑑𝑠^𝑡0(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑑𝑠+(1+𝑡)𝐺¹+1−(1+𝑡)𝑇𝐺²𝑇+(1+𝑡)‖𝜓‖^𝐿1𝑇^𝑇0(𝑇−𝑠)^𝛼−2+(1+𝑡)‖𝜓‖_𝐿1Γ(𝛼−1)𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−1≤‖𝜓‖_𝐿1Γ(𝛼)𝑑𝑠𝑇^𝛼Γ(𝛼+1)+(1+𝑡)𝐺¹+1−1𝑇𝐺²𝑇+(1+𝑡)𝑇^𝛼−1‖𝜓‖_𝐿1𝑇Γ(𝛼)+(1+𝑡)𝑇^𝛼‖𝜓‖_𝐿1≤‖𝜓‖_𝐿1𝑇Γ(𝛼+1)𝑇^𝛼Γ(𝛼+1)+(1+𝑡)𝑇^𝛼−1‖𝜓‖_𝐿1𝑇Γ(𝛼)+(1+𝑡)𝑇^𝛼‖𝜓‖_𝐿1𝑇Γ(𝛼+1)+(1+𝑇)𝐺¹+(𝑇−1)𝐺₂≤𝐺₁(1+𝑇)+𝐺₂(𝑇−1)+𝐶²𝑇^𝛼−2Γ(𝛼+1)‖𝜓‖^𝐿1,(3.9) where 𝐶₂=2𝑇²+𝑇(𝛼+1)+𝑇.
Now we prove that 𝐵𝑥 is contraction mapping, ‖‖𝐵𝑥₁−𝐵𝑥₂‖‖≤(1+𝑡)𝑇^𝑇0(𝑇−𝑠)^𝛼−1Γ(𝛼)‖‖‖𝑓𝑠,𝑥¹−𝑓𝑠,𝑥₂(𝑠),^𝑠0𝑘𝑠,𝜏,𝑥₁(𝜏)𝑑𝜏,^𝑇0ℎ₁𝑠,𝜏,𝑥₁(𝜏)𝑑𝜏+(1+𝑡)(𝑠)^𝑠0𝑘𝑠,𝜏,𝑥₂(𝜏)𝑑𝜏,^𝑇0ℎ₁𝑠,𝜏,𝑥₂(𝜏)𝑑𝜏‖‖‖𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−2Γ(𝛼−1)‖‖‖𝑓𝑠,𝑥¹−𝑓𝑠,𝑥₂(𝑠),^𝑠0𝑘𝑠,𝜏,𝑥₁(𝜏)𝑑𝜏,^𝑇0ℎ₁𝑠,𝜏,𝑥₁(𝜏)𝑑𝜏≤(1+𝑡)(𝑠),^𝑠0𝑘𝑠,𝜏,𝑥₂(𝜏)𝑑𝜏,^𝑇0ℎ₁𝑠,𝜏,𝑥₂(𝜏)𝑑𝜏‖‖‖𝑑𝑠𝑇^𝑇0(𝑇−𝑠)^𝛼−2Γ(𝛼−1)𝐿(𝑠)(1+𝑝(𝑠)+𝑞(𝑠))𝐾‖‖𝑥¹+−𝑥₂‖‖𝑑𝑠_𝑇0(𝑇−𝑠)^𝛼−1Γ(𝛼)𝐿(𝑠)(1+𝑝(𝑠)+𝑞(𝑠))𝐾‖‖𝑥¹−𝑥₂‖‖𝑑𝑠.(3.10) Let 𝐾(‖𝑥₁−𝑥₂‖)≤𝑤‖𝑥₁−𝑥₂‖, we obtain ‖‖𝐵𝑥₁−𝐵𝑥₂‖‖≤(1+𝑡)𝑤𝑀^∗𝑇‖‖𝑥¹−𝑥₂‖‖^𝑇0(𝑇−𝑠)^𝛼−2Γ(𝛼−1)𝑑𝑠+^𝑇0(𝑇−𝑠)^𝛼−1‖‖𝐵𝑥₁Γ(𝛼)𝑑𝑠,−𝐵𝑥₂‖‖≤(1+𝑇)𝑤𝑀^∗𝑇Γ𝑇(𝛼^𝛼−1)+Γ(𝛼𝑇+^𝛼1)‖‖𝑥¹≤𝑤𝑀_∗−𝑥₂‖‖(1+𝑇)(𝛼+𝑇)Γ(𝛼+1)𝑇^2−𝛼‖‖𝑥₁−𝑥₂‖‖.(3.11) It is clear that 𝐵 is contraction mapping, since 𝑥(𝑡) is continuous, then 𝐴𝑥 is continuous ‖𝐴𝑥(𝑡)‖=‖‖‖1Γ(𝛼)^𝑡0≤‖𝜓‖_𝐿1(𝑡−𝑠)^𝛼−1𝑓𝑠,𝑥(𝑠),^𝑠0𝑘(𝑠,𝜏,𝑥(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑥(𝜏))𝑑𝜏𝑑𝑠‖‖‖^𝑡0(𝑡−𝑠)^𝛼−1‖𝐴𝑥(𝑡)‖≤𝑇_𝛼Γ(𝛼)𝑑𝑠‖𝜓‖_𝐿1Γ(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 𝑥∈𝐵𝑟. Using the fact that 𝑓 is bounded on the compact set 𝐽×𝐵𝑟, thus sup_{(𝑡,𝑠)∈𝐽×𝐵𝑟}‖𝑓(𝑠,𝑥(𝑠),∫^𝑠0𝑘(𝑠,𝜏,𝑥(𝜏))𝑑𝜏,∫^𝑇0ℎ₁(𝑠,𝜏,𝑥(𝜏))𝑑𝜏)‖=𝑐₀<∞,we get ‖‖𝐴𝑥𝑡₁−𝐴𝑥𝑡₂‖‖=‖‖‖1Γ(𝛼)^𝑡1₀𝑡₁−𝑠^𝛼−1𝑓𝑠,𝑥(𝑠),^𝑠0𝑘(𝑠,𝜏,𝑥(𝜏))𝑑𝜏,_𝑇0ℎ₁(𝑠,𝜏,𝑥(𝜏))𝑑𝜏𝑑𝑠−1Γ(𝛼)^𝑡2₀𝑡₂−𝑠^𝛼−1≤𝑓1𝑠,𝑥(𝑠),^𝑠0𝑘(𝑠,𝜏,𝑥(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑥(𝜏))𝑑𝜏𝑑𝑠‖‖‖Γ(𝛼)‖‖‖^𝑡1₀𝑡¹𝑓𝑠,𝑥(𝑠−),𝑠^𝛼−1_𝑠0−𝑡₂−𝑠^𝛼−1+‖‖‖𝑘(𝑠,𝜏,𝑥(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑥(𝜏))𝑑𝜏𝑑𝑠_𝑡2𝑡1‖‖‖≤𝑐𝑡₂−𝑠^𝛼−1𝑓𝑠,𝑥(𝑠),^𝑠0𝑘(𝑠,𝜏,𝑥(𝜏))𝑑𝜏,^𝑇0ℎ₁(𝑠,𝜏,𝑥(𝜏))𝑑𝜏𝑑𝑠₀‖‖‖Γ(𝛼+1)2𝑡²−𝑡₁^𝛼+𝑡_1𝛼−𝑡_2𝛼.(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)(𝑡)=110+10+1||𝑦(𝑡)||+^𝑡0||𝑦(𝑡)||10𝑒^|𝑦(𝑡)|+𝑡𝑑𝑡+¹₀||𝑦(𝑡)||𝑒^−𝑡10+||𝑦(𝑡)||²𝑑𝑡,(3.14) with integral boundary conditions 𝑦(0)−𝑦^(0)=¹⁰110+||𝑦(𝑡)||𝑑𝑡,𝑦(1)−𝑦^(1)=¹⁰110+𝑒^{−|𝑦(𝑡)|}𝑑𝑡.(3.15) Here, ‖𝑔(𝑦(𝑡))‖=‖‖‖1‖ℎ(𝑦(𝑡))‖=‖‖‖10+1||𝑦(𝑡)||‖‖‖≤110,‖𝑔(𝑥)−𝑔(𝑦)‖≤1100‖𝑥−𝑦‖,10+𝑒^{−|𝑦(𝑡)|}‖‖‖≤1‖‖‖10,‖ℎ(𝑥)−ℎ(𝑦)‖≤1100‖𝑥−𝑦‖,_𝑡0(𝑘(𝑡,𝑠,𝑥)−𝑘(𝑡,𝑠,𝑦))𝑑𝑠‖‖‖≤110𝑒^𝑡‖𝑥−𝑦‖,‖‖‖‖‖^𝑡₀𝑘(𝑡,𝑠,𝑦)𝑑𝑠‖‖‖‖‖≤1‖‖‖10+𝑡‖𝑦(𝑡)‖,_𝑡0‖‖𝑓ℎ𝑡,¹𝑥(𝑡₁,𝑠,𝑥)−ℎ₁(𝑡,𝑠,𝑦)𝑑𝑠‖‖‖≤101𝑒^𝑡‖𝑥−𝑦‖,‖‖‖‖‖^𝑡₀ℎ₁(𝑡,𝑠,𝑦)𝑑𝑠‖‖‖‖‖≤101+𝑡‖𝑦(𝑡)‖,𝑦₁,𝑧₁−𝑓𝑡,𝑥₂,𝑦₂,𝑧₂‖‖≤110+𝑡‖‖𝑥¹𝑓(𝑡,0,0,0)=1−𝑥₂‖‖+‖‖𝑦₁−𝑦₂‖‖+‖‖𝑧₁−𝑧₂‖‖,10.(3.16) Hence, the conditions (H1)–(H5) hold with 𝐺₁=𝐺₂=0.1, 𝑏₁=𝑏₂=0.01, 𝑀^∗1=0.12, 𝑤=0.1, 𝐶_𝑜=6, 𝑁₁=0.1, 𝑀^∗=0.12, and 𝐶₁=6, thus 𝑏₁(1+𝑇)+𝑏₂(𝑇−1)+𝑤𝑀^∗𝐶₁(1+𝑇)Γ(𝛼+1)𝑇^2−𝛼<1⟺0.01(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.