# On the Generalization of a Solution for a Class of Integro-Differential Equations with Nonseparated Integral Boundary Conditions

Authors: Yanyuan Xing; Feng Jiao; Fang Liu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, the existence and uniqueness results of the generalization nonlinear fractional integro-differential equations with nonseparated type integral boundary conditions are investigated. A natural formula of solutions is derived and some new existence and uniqueness results are obtained under some conditions for this class of problems by using standard fixed point theorems and Leray–Schauder degree theory, which extend and supplement some known results. Some examples are discussed for the illustration of the main work.


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## Body

## 1. Introduction

Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. Characteristics of the fractional derivatives make the fractional-order models more realistic and practical than the classical integral-order models. Fractional differential equations have gained considerable importance due to their application in various sciences, such as physics, polymer rheology, aerodynamics, capacitor theory, chemistry, biology, control theory, and electrodynamics of complex medium. The initial and boundary value problems for nonlinear fractional differential equations arise from the study of models of viscoelasticity, electrochemistry, porous media, and electromagnetics. In consequence, the subject of fractional differential equations is gaining much importance and attention [1–4]. The recent development in the theory and methods for fractional differential equations indicates its popularity. For more details, we refer the reader to [5–9] and the references cited therein.Moreover, the existence and uniqueness of solutions for fractional differential equations have been mathematically studied from different methods [10–15], yielding methods for solving fractional differential equations [16–19]. As we all know, boundary value problems of fractional differential equations have been investigated for many years. Now, there are many papers dealing with the problem for different kinds of boundary conditions such as periodic or antiperiodic boundary condition [20, 21], multipoint boundary condition [22, 23], and integral boundary condition [24–28] as well as stability and convergence analysis [29–32]. Integral boundary conditions have various applications in applied fields such as blood flow problems, chemical engineering, thermoelasticity, underground water flow, and population dynamics. For a detailed description of some recent work on the integral boundary conditions, we refer the reader to some recent papers [33–35] and the references therein [36–39].In [20], Ahmad and Nieto investigated the fractional differential equations with antiperiodic fractional boundary conditions as the following form:(1)DCtαut=ft,ut,t∈0,T,T>0,1<α≤2,u0=−uT,DCtβu0=−CDtβuT,0<β<1,where DCtα and DCtβ denote the Caputo fractional derivative of order α, β; f:0,T×ℝ⟶ℝ is a given continuous function; and T is a fixed positive constant. The results are based on some standard fixed point principles.Recently, in [24], the author discussed the nonlinear fractional differential equations with nonseparated type integral boundary conditions(2)DCtαut=ft,ut,t∈0,T,T>0,1<α≤2,u0+λ1uT=μ1∫0Tgs,usds,u′0+λ2u′T=μ2∫0Ths,usds,where DCtα denotes the Caputo fractional derivative of order α, f,g,h:0,T×ℝ⟶ℝ are given continuous function, and λ1, λ2, μ1, μ2 are suitably chosen real constants with λ1 ≠ −1, λ2 ≠ −1. By applying the Leray–Schauder degree theory and some standard fixed point theorems, some new existence and uniqueness results are obtained.Motivated by the abovementioned papers and many known results, in this paper, we concentrate on the existence and uniqueness of solutions for the nonlinear fractional integro-differential equations and inclusions of orderα ∈ (1, 2], with nonseparated type integral boundary conditions(3)DCtαut=ft,ut,φut,ψut,t∈0,T,1<α≤2,u0+μ1uT=σ1∫0Tgs,usds,T>0,DCtβu0+μ2CDtβuT=σ2∫0Ths,usds,0<β<1,where DCtα and DCtβ denote the Caputo fractional derivative of order α, β; f:0,T×ℝ×ℝ×ℝ⟶ℝ is a given continuous function satisfying some assumptions that will be specified later; Γ is the Euler gamma function; and μ1 ≠ −1, μ2 ≠ 0, κ, ξ: [0, T] × [0, T] ⟶ [0, ∞), φ, ψ are linear operators defined by(4)φut=∫0tκt,susds,ψut=∫0tξt,susds,g,h:0,T×ℝ⟶ℝ,σ1,σ2∈ℝ. Here, C:=C0,T,ℝ denotes the Banach space of all continuous functions from [0, T] to ℝ endowed with a topology of uniform convergence with the norm u=suput,t∈0,T.To the best of our knowledge, no paper has considered the generalization of nonlinear fractional integro-differential equations with nonseparated type integral boundary conditions (3). Our purpose here is to give some existence and uniqueness results for solution to (3).Compared with the previous research problems, (3) has more general integral boundary value conditions. This paper is organized as follows: in Section 2, we present the notations and give some preliminary results via a sequence of definitions and lemmas. In Section 3, we prove new existence and uniqueness results for problem (3). These results are based on fixed point theorems and Leray–Schauder degree theory. In Section 4, two examples are demonstrated which support the theoretical analysis.


## 2. Preliminaries and Lemmas

In this section, we present some basic notations, definitions, and preliminary results which will be used throughout this paper. Let us recall some definitions of fractional calculus. For more details, see [1, 2].Definition 1.
The fractional integral of orderα with the lower limit zero for a function f:0,∞⟶ℝ is defined as(5)Iαft=1Γα∫0tt−sα−1fsds,t>0,n−1<α<n,provided the integral exists.Definition 2.
For a functionf:0,∞⟶ℝ with the lower limit zero, the Caputo derivative of fractional order α is defined as(6)DCtαft=1Γn−α∫0tt−sn−α−1fnsds,t>0,n−1<α<n,where n = [α] + 1 and [α] denote the integer part of the real number α.Definition 3.
The Riemann–Liouville fractional derivative of orderα with the lower limit zero for a function f(t) is defined by(7)DRtαft=1Γn−αddtn∫0tt−sn−α−1fsds,n=α+1,where n = [α] + 1 and [α] denote the integer part of real number α, provided that the right side is pointwise defined on (0, ∞).Lemma 1.
Forα > 0, the general solution of the fractional differential equation DCtαut=0 is given by(8)ut=c0+c1t+c2t2+⋯+cn−1tn−1,where ci∈ℝ,i=0,1,…,n−1n=α+1.
In view of Lemma1, it follows that(9)IαDCtαut=ut+c0+c1t+c2t2+⋯+cn−1tn−1,for some ci∈ℝ,i=0,1,…,n−1n=α+1.
In the following, we derive a natural formula of solution to the integral boundary value problem for integro-differential equation (3).Lemma 2.
Assume thaty∈C0,T,ℝ,T>0,1<α≤2. A function u(t) is a solution of the boundary value problem(10)DCtαut=yt,t∈0,T,1<α≤2,u0+μ1uT=σ1∫0Tgs,usds,T>0,DCtβu0+μ2DCtβuT=σ2∫0Ths,usds,0<β<1,if and only if u is a solution of the integral equation(11)ut=∫0tt−sα−1Γαysds−μ11+μ1∫0TT−sα−1Γαysds+Γ2−βμ1T−1+μ1t1+μ1T1−β∫0TT−sα−β−1Γα−βysds+σ11+μ1∫0Tgs,usds−σ2Γ2−βTβ−1μ21+μ1μ1T−1+μ1t∫0Ths,usds.Proof.
Assume thaty satisfies (10). Using Lemma 1, for some constants c0,c1∈ℝ, we have(12)ut=Iαyt−c0−c1t=1Γα∫0tt−sα−1ysds−c0−c1t.
Using the facts thatDCtβc=0 (c is a constant), DCtβt=t1−β/Γ2−β, and DCtαItβyt=Iβ−αyt, we get(13)DCtβut=1Γα−β∫0tt−sα−β−1ysds−c1t1−βΓ2−β.
Applying the boundary conditions for problem (3), we find that(14)c0=μ11+μ1∫0TT−sα−1Γαysds−μ1TβΓ2−β1+μ1∫0TT−sα−β−1Γα−βysds−σ11+μ1∫0Tgs,usds+μ1σ2Γ2−βTβμ21+μ1∫0Ths,usds,c1=Γ2−βT1−β1Γα−β∫0TT−sα−β−1ysds−σ2μ2∫0Ths,usds.
Substituting the value ofc0 and c1 in (12), we obtain the unique solution of (10) which is given by(15)ut=∫0tt−sα−1Γαysds−μ11+μ1∫0TT−sα−1Γαysds+Γ2−βμ1T−1+μ1t1+μ1T1−β∫0TT−sα−β−1Γα−βysds+σ11+μ1∫0Tgs,usds−σ2Γ2−βTβ−1μ21+μ1μ1T−1+μ1t∫0Ths,usds.
Conversely, we assume thatu is a solution of the integral equation (11), and in view of the relations DCtαItβyt=yt, for α > 0, we get(16)DCtαut=yt,t∈0,T,1<α≤2.
Moreover, it can easily be verified that the boundary conditions(17)u0+μ1uT=σ1∫0Tgs,usds,DCtβu0+μ2DCtβuT=σ2∫0Ths,usds,are satisfied. The proof is completed.
By Lemma2, problem (3) is reduced to the fixed point problem(18)u=Φu,where Φ:C⟶C is given by(19)Φut=∫0tt−sα−1Γαfs,us,φus,ψusds−μ11+μ1∫0TT−sα−1Γαfs,us,φus,ψusds+Γ2−βμ1T−1+μ1t1+μ1T1−β∫0TT−sα−β−1Γα−βfs,us,φus,ψusds+σ11+μ1∫0Tgs,usds−σ2Γ2−βTβ−1μ21+μ1μ1T−1+μ1t∫0Ths,usds.


## 3. Main Results

In this section, we will show the existence and uniqueness of solutions for problem (3). Now we state some known fixed point theorems which are needed to prove the existence of solutions for equation (3).Theorem 1.
LetX be a Banach space. Assume that Φ: X ⟶ X is a completely continuous operator and the set V = {u ∈ X ∣ u = μΦu, 0 < μ < 1} is bounded. Then, Φ has a fixed point in X.Theorem 2.
LetX be a Banach space. Assume that Ω is an open bounded subset of X with 0 ∈ Ω and let Φ:Ω¯⟶X be a completely continuous operator such that(20)Φu≤u,∀u∈∂Ω.
Then Φ has a fixed point inΩ¯.Theorem 3.
Suppose thatf:0,T×ℝ×ℝ×ℝ⟶ℝ is a jointly continuous function and maps bounded subsets of 0,T×ℝ×ℝ×ℝ into relative compact subsets of ℝ, κ, ξ : [0, T] × [0, T] ⟶ [0, ∞) is continuous with(21)k0=maxκt,s:t,s∈0,T×0,T,(22)ξ0=maxξt,s:t,s∈0,T×0,T,K=maxκ0,ξ0,and g,h:0,T×ℝ⟶ℝ are continuous functions. Furthermore, there exist positive constants Ci(i = 1, …, 5) such that(H1) ft,u1,u2,u3−ft,v1,v2,v3≤C1u1−v1+C2u2−v2+C3u3−v3, ∀t ∈ [0, T], ui,vi∈ℝ,i=1,2,3(H2) gt,u−gt,v≤C4u−v, ht,u−ht,v≤C5u−v, ∀u,v∈ℝ
Then the boundary value problem (3) has a unique solution provided(23)r1=1+μ11+μ1TαΓα+1+Γ2−βTαΓα−β+1C1+C2+C3TK+σ11+μ1C4T+σ2μ2Γ2−βTβ+1C5<1.Proof.
Settingsupt∈0,Tft,0,0,0=M1, supt∈0,Tgt,0=M2, and supt∈0,Tht,0=M3. For a positive number r, let Br=u∈C:u≤r and r≥r2/1−r1, with r1 is given by (23), we will show that ΦBr ⊂ Br, where Φ is defined by (19), and(24)r2=1+μ11+μ1TαΓα+1+Γ2−βTαΓα−β+1M1+σ11+μ1M2T+σ2μ2Γ2−βTβ+1M3.
First, ∀u(t) ∈ Br, there exists {un} ⊂ Br, and when n ⟶ ∞, un ⟶ u, it is easy to know that(25)Φunt−Φut⟶0.
Then Φ is continuous onBr.
Furthermore, foru ∈ Br, t ∈ [0, T], we have(26)Φut≤∫0tt−sα−1Γαfs,us,φus,ψusds+μ11+μ1∫0TT−sα−1Γαfs,us,φus,ψusds+Γ2−βTβ∫0TT−sα−β−1Γα−βfs,us,φus,ψusds+σ11+μ1∫0Tgs,usds+σ2μ2Γ2−βTβ∫0Ths,usds≤∫0tt−sα−1Γαfs,us,φus,ψus−fs,0,0,0+fs,0,0,0ds+μ11+μ1∫0TT−sα−1Γαfs,us,φus,ψus−fs,0,0,0+fs,0,0,0ds+Γ2−βTβ∫0TT−sα−β−1Γα−βfs,us,φus,ψus−fs,0,0,0+fs,0,0,0ds+σ11+μ1∫0Tgs,us−gs,0+gs,0ds+σ2μ2Γ2−βTβ∫0Ths,us−hs,0+hs,0ds≤1+μ11+μ1TαΓα+1+Γ2−βTαΓα−β+1C1+C2+C3TKr+M1+σ11+μ1TC4r+M2+σ2μ2Γ2−βTβTC5r+M3≤1+μ11+μ1TαΓα+1+Γ2−βTαΓα−β+1C1+C2+C3TK+σ11+μ1C4T+σ2μ2Γ2−βTβ+1C5r+1+μ11+μ1TαΓα+1+Γ2−βTαΓα−β+1M1+σ11+μ1M2T+σ2μ2Γ2−βTβ+1M3,≤r1r+r2≤r.
Now, foru, v∈C and for each t ∈ [0, T], we obtain(27)Φut−Φvt≤∫0tt−sα−1Γαfs,us,φus,ψus−fs,vs,φvs,ψvsds+μ11+μ1∫0TT−sα−1Γαfs,us,φus,ψus−fs,vs,φvs,ψvsds+Γ2−βTβ∫0TT−sα−β−1Γα−βfs,us,φus,ψus−fs,vs,φvs,ψvsds+σ11+μ1∫0Tgs,us−gs,vsds+σ2μ2Γ2−βTβ∫0Ths,us−hs,vsds≤1+μ11+μ1TαΓα+1+Γ2−βTαΓα−β+1C1+C2+C3TK+σ11+μ1C4T+σ2μ2Γ2−βTβ+1C5u−v≤r1u−v.
Observe thatr1 depends only on the parameters involved in the problem. As r1 < 1, then Φ is a contraction map. Hence, the conclusion of the theorem follows by the contraction mapping principle, and Φ has a unique fixed point u. That is, the boundary value problem (3) has a unique solution. This completes the proof.
Our next existence results are based on Krasnoselskii’s fixed point theorem [40].Theorem 4.
LetM be a closed convex and nonempty subset of a Banach space X. Let A and B be the 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 existsz ∈ M such that z = Az + Bz.Theorem 5.
Assume (H1) and (H2) hold, f:0,T×ℝ×ℝ×ℝ⟶ℝ is a jointly continuous function. Further, we assume that(H3) ft,u,φu,ψu≤pt,gt,u≤qt,ht,u≤νt,∀t,u,φu,ψu∈0,T×ℝ×ℝ×ℝ,p,q,ν∈C0,T,ℝ+;(H4) μ1/1+μ1Tα/Γα+1+Γ2−βTα/Γα−β+1C1+C2+C3TK+σ1/1+μ1C4T+σ2/μ2Γ2−βTβ+1C5<1.
Then, problem (3) has at least one solution on [0, T].Proof.
Letsupt∈0,Tpt=p, supt∈0,Tqt=q, and supt∈0,Tνt=ν, we fix(28)R¯=1+μ11+μ11Γα+1+Γ2−βΓα−β+1Tαφ+σ11+μ1Tψ+σ2μ2Γ2−βTβ+1ν,and considering BR¯=u∈C:u≤R¯, we define the operators Φ1 and Φ2 on BR¯ as(29)Φ1ut=∫0tt−sα−1Γαfs,us,φus,ψusds,Φ2ut=−μ11+μ1∫0TT−sα−1Γαfs,us,φus,ψusds+Γ2−βμ1T−1+μ1t1+μ1T1−β∫0TT−sα−β−1Γα−βfs,us,φus,ψusds+σ11+μ1∫0Tgs,usds−σ2Γ2−βTβ−1μ21+μ1μ1T−1+μ1t∫0Ths,usds.
Foru,v∈BR¯, we find that(30)Φ1u+Φ2v≤1+μ11+μ11Γα+1+Γ2−βΓα−β+1Tαp+σ11+μ1Tq+σ2μ2Γ2−βTβ+1ν≤R¯.
Thus,Φ1u+Φ2v∈BR¯. It follows from the assumptions (H1) and (H2) that Φ2 is a contraction mapping if(31)μ11+μ1TαΓα+1+Γ2−βTαΓα−β+1C1+C2+C3TK+σ11+μ1C4T+σ2μ2Γ2−βTβ+1C5<1.
Moreover, the continuity off implies that the operator Φ1 is continuous. Also, Φ1 is uniformly bounded on BR¯ as(32)Φ1≤pTαΓα+1.
Now, we prove compactness of the operator Φ1. In view of (H3), we define(33)supt,u∈0,T×BR¯ft,u,φu,ψu=fmax,and consequently, we have(34)Φ1ut2−Φ1ut1=∫0t1t2−sα−1−t1−sα−1Γαfs,us,φus,ψusds+∫t1t2t2−sα−1Γαfs,us,φus,ψusds≤fmaxΓα+12t2−t1α+t1α−t2α,which is independent of u and tends to zero as t2 − t1 ⟶ 0. So Φ1 is relatively compact on BR¯. Hence, by the Arzelá–Ascoli theorem, Φ1 is compact on BR¯. Thus, all the assumptions of Theorem 4 are satisfied. Therefore, the conclusion of Theorem 4 applies that the fractional boundary value problem (3) has at least one solution on [0, T]. This completes the proof.
As an immediate consequence of Theorem5, we have the following.Corollary 1.
Assume thatf:0,T×ℝ×ℝ×ℝ⟶ℝ is a jointly continuous function. If there exists nonnegative functions, ai(t) ∈ L[0, T](i = 0, 1, 2, 3), bi(t), ci(t) ∈ C[0, T](i = 0, 1), 0 < ρj < 1(j = 1, 2, 3), 0 < θ1, θ2 < 1, and κ0, ξ0 are given by (21) and (22) such that(H5) ft,u,φu,ψu≤a0t+a1tuρ1+a2tφuρ2+a3tψuρ3(H6) gt,u≤b0t+b1tuθ1,ht,u≤c0t+c1tuθ2
For allt ∈ [0, T], u,φu,ψu∈ℝ, then the boundary value problem (3) has at least one solution.Proof.
Let us define a ball in the Banach spaceB=u∈Cu≤Λ, where Λ is fixed later. Setting(35)Iaα=maxt∈0,TIαait,i=0,1,2,3,IaαT=maxt∈0,TIαaiT,i=0,1,2,3,ρ=maxi=1,2,3ρi,bm=maxi=1,2bit,cm=maxi=1,2cit,Φut≤∫0tt−sα−1Γαa0s+a1suρ1+a2sφuρ2+a3sψuρ3ds+μ11+μ1∫0TT−sα−1Γαa0s+a1suρ1+a2sφuρ2+a3sψuρ3ds+Γ2−βTβ∫0TT−sα−β−1Γα−βa0s+a1suρ1+a2sφuρ2+a3sψuρ3ds+σ11+μ1Tb0s+b1suθ1+σ2μ2Γ2−βTβ+1c0s+c1suθ2ds≤Iαa0t+Iαa1tΛρ1+Iαa2tTρ2k0ρ2Λρ2+Iαa3tTρ3ξ0ρ3Λρ3+μ11+μ1Iαa0T+Iαa1TΛρ1+Iαa2TTρ2k0ρ2Λρ2+Iαa3TTρ3ξ0ρ3Λρ3+Γ2−βTβIα−βa0T+Iα−βa1TΛρ1+Iα−βa2TTρ2k0ρ2Λρ2+Iα−βa3TTρ3ξ0ρ3Λρ3+σ11+μ1Tbm1+Λθ1+σ2μ2Γ2−βTβ+1cm1+Λθ2≤Iaα1+Tρ2k0ρ2+Tρ3ξ0ρ3Λρ+Iαa0+μ11+μ1IaαT1+Tρ2k0ρ2+Tρ3ξ0ρ3Λρ+Iαa0T+Γ2−βTβIaα−βT1+Tρ2k0ρ2+Tρ3ξ0ρ3Λρ+Iα−βa0T+σ11+μ1Tbm1+Λθ1+σ2μ2Γ2−βTβ+1cm1+Λθ2=Iαa0+Iαa0T+Γ2−βTβIα−βa0T+σ11+μ1Tbm1+Λθ1+σ2μ2Γ2−βTβ+1cm1+Λθ2+1+Tρ2k0ρ2+Tρ3ξ0ρ3ΛρIaα+μ11+μ1IaαT+Γ2−βTβIaα−βT.
Choosing Λ sufficient large, then Φ :B ⟶ B. On the other hand, the continuity of f implies that the operator Φ is continuous. Also, since Φ : B ⟶ B, we have Φ(B) which is uniformly bounded on B.
Let(36)M=maxt,u,φu,ψu∈0,T×B×B×Bft,u,φu,ψu,and consequently, we obtain(37)Φut2−Φut1=∫0t1t2−sα−1−t1−sα−1Γαfs,us,φus,ψusds+∫t1t2t2−sα−1Γαfs,us,φus,ψusds+Γ2−βTβ−1t1−t2∫0TT−sα−β−1Γα−βfs,us,φus,ψusds−σ2Γ2−βt1−t2μ2Tβcm1+u≤MΓα+12t2−t1α+t1α−t2α+MT−sα−1Γ2−βΓα−β+1t1−t2+σ2Γ2−βμ2Tβcm1+ut1−t2.
It follows that Φ is equicontinuous, so Φ(B) is relatively compact on B. Hence, Φ(B) is relatively compact on B by Arzelá–Ascoli theorem. Thus, by Schauder fixed-point theorem, problem (3) has at least one solution.Theorem 6.
Assume that there exist positive constantspi, qi(i = 1, 2, 3) such that ft,u,φu,ψu≤p1/Tαu+q1, gt,u≤p2/Tu+q2, and ht,u≤p3/Tβ+1u+q3 for all t ∈ [0, T], u ∈ C[0, T]. If(38)1+μ11+μ11Γα+1+Γ2−βΓα−β+1p1+σ11+μ1p2+σ2μ2Γ2−βTβ+1p3<1,then the boundary value problem (3) has at least one solution.Proof.
In view of the fixed point problem (19), we just need to prove the existence of at least one solution u∈ℝ satisfying (19). Define a suitable ball BS∈C with radius S > 0 as(39)BS=u∈C:maxt∈0,Tut<S,where S will be fixed later. Then, it is sufficient to show that ℱ:BS¯⟶C satisfies(40)u≠λℱu,∀u∈∂BS,∀λ∈0,1.
Then, by the Arzelá–Ascoli theorem,hλu=u−Hλ,u=u−λℱu is completely continuous. If (40) is true, then the following Leray–Schauder degrees are well defined and by the homotopy invariance of topological degree, it follows that(41)deghλ,BS,0=degI−λℱ,BS,0=degh1,BS,0=degh0,BS,0=degI,BS,0=1≠0,0∈BS,where I denotes the unit operator. By the nonzero property of Leray–Schauder degree, we have h1t=u−ℱu=0 for at least one u ∈ BS. In order to prove (40), we assume that u=λℱu for some λ ∈ [0, 1] and for all t ∈ [0, T] so that(42)ut=λℱut≤∫0tt−sα−1Γαfs,us,φus,ψusds+μ11+μ1∫0TT−sα−1Γαfs,us,φus,ψusds+Γ2−βTβ∫0TT−sα−β−1Γα−βfs,us,φus,ψusds+σ11+μ1∫0Tgs,usds+σ2μ2Γ2−βTβ∫0Ths,usds≤1+μ11+μ1TαΓα+1+2−βTαΓα−β+1p1Tαu+q1+σ11+μ1Tp2Tu+q2+σ2μ2Γ2−βTβTp3Tβ+1u+q3≤1+μ11+μ11Γα+1+Γ2−βΓα−β+1p1+σ11+μ1p2+σ2μ2Γ2−βp3u+1+μ11+μ1TαΓα+1+Γ2−βTαΓα−β+1q1+σ11+μ1Tq2+σ2μ2Γ2−βTβ+1q3=h1u+h2,which by taking norm u=suput,t∈0,T and solving for u, we have(43)u≤h21−h1,where(44)h1=1+μ11+μ11Γα+1+Γ2−βΓα−β+1p1+σ11+μ1p2+σ2μ2Γ2−βp3,h2=1+μ11+μ1TαΓα+1+Γ2−βTαΓα−β+1q1+σ11+μ1Tq2+σ2μ2Γ2−βTβ+1q3.
Setting(45)S=h21−h1+1,it follows that (40) holds. This completes the proof.


## 4. Examples

Example 1.
Consider the following nonlinear fractional integro-differential equation with nonseparated type integral boundary conditions ofα=3/2, β=1/2, and T = 1:(46)DCt3/2ut=1t+102ut1+ut+cos2t+1120∫0te−t3+t2usds+1140∫0114+t2usds,u0+u1=∫01us50+usds,DCt1/2u0+DCt1/2u1=∫011t+102us1+usds.
Here,μ1 = μ2 = σ1 = σ2 = 1 and(47)φut=∫0te−t3+t2usds,ψut=∫0114+t2usds,gs,us=us50+us,hs,us=1t+102us1+us.
Foru,v∈ℝ and t ∈ [0, 1], we have(48)ft,u,φu,ψu−ft,v,φv,ψv≤1t+102u−v+1120φu−φv+1140ψu−ψv≤1100u−v+φu−φv+ψu−ψv,gs,u−gs,v≤150u−v,hs,u−hs,v≤1100u−v.
AsC1=C2=C3=1/100 and C4=1/50,C5=1/100,K=1/9, we obtain(49)r1=1+μ11+μ1TαΓα+1+Γ2−βTαΓα−β+1C1+C2+C3TK+σ11+μ1C4T+σ2μ2Γ2−βTβ+1C5=2π+π21100+1450+1100+π200≈0.043<1.
Thus, all the assumptions of Theorem3 hold. Consequently, the conclusion of Theorem 3 implies that problem (46) has a unique solution.Example 2.
Consider the following integro-differential fractional boundary value problem(50)DCt3/2ut=e−t10ut1+ut+∫0te−s−t200ut1+utds+∫01e−ts+132ut1+utds,u0+u1=∫01us50+usds,DCt1/2u0+DCt1/2u1=∫011t+102us1+usds.
Here,α=3/2, β=1/2, T = 1, μ1 = μ2 = σ1 = σ2 = 1, T = 1, and(51)K=supt∈0,1κ0,ξ0=supt∈0,1∫0te−s−t200ds,∫0te−ts+132ds=0.08.
Since(52)ft,u,φu,ψu≤3e−t10,gt,u≤1,ht,u≤1100.
Clearly,C1=1/10, C2=1/200, C3=1/32, C4=1/50, and C5=1/100. Furthermore,(53)μ11+μ1TαΓα+1+Γ2−βTαΓα−β+1C1+C2+C3TK+σ11+μ1C4T+σ2μ2Γ2−βTβ+1C5=23π+π2110+1200+132×0.08+1100+π200≈0.15<1.
Thus, by Theorem5, the integro-differential boundary value problem (50) has at least one solution on [0, 1].



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*Source: 8679465-2020-03-28.xml*