# Analysis of a Coupled System of Nonlinear Fractional Langevin Equations with Certain Nonlocal and Nonseparated Boundary Conditions

Authors: Zaid Laadjal; Qasem M. Al-Mdallal; Fahd Jarad
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this article, we use some fixed point theorems to discuss the existence and uniqueness of solutions to a coupled system of a nonlinear Langevin differential equation which involves Caputo fractional derivatives of different orders and is governed by new type of nonlocal and nonseparated boundary conditions consisting of fractional integrals and derivatives. The considered boundary conditions are totally dissimilar than the ones already handled in the literature. Additionally, we modify the Adams-type predictor-corrector method by implicitly implementing the Gauss–Seidel method in order to solve some specific particular cases of the system.


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## Body

## 1. Introduction

The fractional calculus is the ramification of mathematics concerning the integrals and derivatives of functions with arbitrary orders. It has a long history that goes back to more than three hundred years. Nonetheless, researchers discovered the importance and effectiveness of this calculus just a mere in the last few decades. It turned out that the fractional integrals and derivatives are very good tools in modeling some phenomena. This was concluded simply because of the amazing results obtained when some of the researchers used the implements in the fractional calculus for the sake of understanding real world problems happening in the environment surrounding. Recently, differential equations of fractional order have been applied in various fields like physical, biology, chemistry, control theory, electrical circuits, blood flow phenomena, and signal and image processing; for more details, see [1–3] and references cited therein.In 1908, Langevin [4] formulated his famous equation containing derivative of integer order. This equation describes the evolution of certain physical phenomena in fluctuating environments [5]. The Langevin equation was used in large part to describe some phenomena such as anomalous transport [6]. The Langevin equation has been recently extended to the fractional order by Lim et al. [7]. They acquainted a new form of Langevin equations involving two different fractional order for the sake of describing the viscoelastic anomalous diffusion in the complex liquids. We refer the reader to Subsection 2.1 in [3] and the references cited therein for further details. Uranagase and Munakata [8] discussed the generalized Langevin equation with emphasis on a mechanical random force whose time evolution is not natural due to the presence of a projection operator in a propagator. Lozinski et al. [9] discussed the applications of Langevin and Fokker–Planck equations in polymer rheology and stochastic simulation techniques for solving this equation. Laadjal et al. [10] presented the existence and uniqueness of solutions for the multiterm fractional Langevin equation with boundary conditions.Using the tools in mathematical analysis and the theory of fixed points, discussing the qualitative specification encapsuling the behaviors of solutions of differential equations in fractional derivatives settings has attracted the attention of many scientists. To get an update about the works in the literature, we ask the readers to investigate [11] and the references cited there. On the top of this, classes of systems of fractional differential equations with separated (or nonseparated) boundary conditions have been studied intensively in literatures [12–14].Motivated by what are mentioned above and the recent development on Langevin equations, in this paper, we discuss the existence and the uniqueness of solutions to a coupled system of fractional Langevin equations in the form as follows:(1)Dcα1Dcβ1+λψ1t=ft,ψ1t,ψ2t,t∈J,0<α1≤1<β1≤2,Dcα2Dcβ2+kψ2t=gt,ψ1t,ψ2t,t∈J,0<α2≤1<β2≤2,subject to a new type of nonlocal nonseparated boundary conditions as follows:(2)ψ10=a0,ψ20=b0,ψ1′0=ψ2′0=0,ψ1ξ=acDpψ2μ1,ξ∈0,1,μ1∈J,0<p<β2,ψ2η=bIqψ1μ2,η∈0,1,μ2∈J,q≥0,where J=0,1,λ,k,a0,b0,a,b are real constants, Dci where i=α1,β1,α2,β2,p are the Caputo fractional derivatives of order α1,β1,α2,β2,andp, respectively, f,g:J×ℝ×ℝ⟶ℝ are given functions, and Iq is the Riemann–Liouville fractional integral of order q. By using the Banach contraction principle and Leray–Schauder alternative fixed point theorem, we investigate the existence of solutions for problems (1) and (2). We remark that the boundary value problem discussed here is distinctive of the ones discussed in literatures [12–14].This article is organized as follows. In Section2, we present some definitions, theorems, and related lemmas used in next sections. Section 3 discusses the existence and uniqueness of the system under consideration. In Section 4, we furnish some numerical examples. Section 5 is devoted to our concluding remarks.


## 2. Preliminaries

Definition 1.
(see [1, 2]). Let a,b∈ℝ−∞<a<b<∞. The Riemann–Liouville fractional integral of order α∈ℝ+ for a function f∈L1a,b is defined by(3)Iaαft=1Γα∫att−sα−1fsds,forα>0,andIa0ft=ft,where Γ is the Euler Gamma function.Definition 2 (see [1, 2]).
The Caputo fractional derivative of orderα∈ℝ+ for a function f∈Cna,b is defined by(4)Dcaαft=Ian−αDnft,forα>0,andDca0ft=ft,where n−1<α≤n, n∈ℕ, and Dn=dn/dtn.Proposition 3 (see [2]).
Forβ>0 and α>0 and f∈L1a,b, we have the following properties:(5)IaαIaβft=IaβIaαft=Iaα+βft.Iaαt−aμ=Γμ+1Γα+μ+1t−aα+μ,μ>−1.DcaαIaβft=Iaβ−αft,hereβ≥α>0.Proposition 4 (see [2]).
Letα>0 with n−1<α≤n and f∈Cna,b. Then,(6)IaαDcaαft=ft−∑k=0n−1ckt−ak,where ck=fka/k!. In particular, when 0<α≤1, we have(7)I0αDc0αft=ft−f0.Proposition 5 (see [2]).
Letμ>0 and α>0 with n−1<α≤n. Then,(8)Dcaαt−aμ−1=ΓμΓμ−αt−aμ−α−1,μ>n,Dcaαt−ak=0,k=0,1,…,n−1.Theorem 6 (Leray–Schauder alternative [15]).
LetT:X⟶X be a completely continuous operator (i.e., a map that is restricted to any bounded set in X is compact). Let MT=u∈X:u=mTuforsome0<m<1. Then, either the set MT is unbounded or T has at least one fixed point.For the sake of simplicity, henceforwards we will writeIα and Dcα instead of I0α and Dc0α, respectively.


## 3. Main Results

In this section, we will discuss the existence and uniqueness of the solution to systems (1)-(2).Lemma 7.
Letv,w∈C0,1,ℝ, 0<α1,α2≤1, 1<β1,β2≤2, q≥0, 0<p<β2, α¯1:=α1+β1+q,α¯2:=α2+β2−p, and(9)Δ=ξβ1ηβ2Γβ1+1Γβ2+1−abμ1β2−pμ2β1+qΓβ1+q+1Γβ2−p+1≠0.
Then, the solution of the following coupled system of fractional Langevin equations is as follows:(10)Dcα1Dcβ1+λψ1t=vt,(11)Dcα2Dcβ2+kψ2t=wt,equipped with the boundary condition (2) which is equivalent to the coupled system of the following integral equations:(12)ψ1t=Iα1+β1vt−λIβ1ψ1t+tβ1A1aIα¯2wμ1−akIβ2−pψ2μ1−Iα1+β1vξ+λIβ1ψ1ξ+A2bIα¯1vμ2−bλIβ1+qψ1μ2−Iα2+β2wη+kIβ2ψ2η+A3+a0,(13)ψ2t=Iα2+β2wt−kIβ2ψ2t+tβ2B1bIα¯1vμ2−bλIβ1+qψ1μ2−Iα2+β2wη+kIβ2ψ2η+B2aIα¯2wμ1−akIβ2−pψ2μ1−Iα1+β1vξ+λIβ1ψ1ξ+B3+b0,where(14)A1=1ΔΓβ1+1ηβ2Γβ2+1,A2=1ΔΓβ1+1aμ1β2−pΓβ2−p+1,A3=1ΔΓβ1+1−a0ηβ2Γβ2+1+aμ1β2−pΓβ2−p+1ba0μ2qΓq+1−b0,B1=1ΔΓβ2+1ξβ1Γβ1+1,B2=1ΔΓβ2+1bμ2q+β1Γq+β1+1,B3=1ΔΓβ2+1−ba0μ2q+β1Γq+β1+1+ξβ1Γβ1+1ba0μ2qΓq+1−b0.Proof.
Applying the operatorIα1+β1 and Iα2+β2 on (10) and (11), respectively, we get(15)ψ1t=Iα1+β1vt−λIβ1ψ1t+c2tβ1Γβ1+1+c1t+c0,(16)ψ2t=Iα2+β2wt−kIβ2ψ2t+c¯2tβ2Γβ2+1+c¯1t+c¯0,where ci,c¯i∈ℝi=0,1,2. From the boundary conditions ψ10=a0,ψ20=b0, and ψ1′0=ψ2′0=0, we get c0=a0, c¯0=b0, and c1=c¯1=0, respectively. Next, using the nonlocal integral conditions ψ1ξ=aDcpψ2μ1 and ψ2η=bIqψ1μ2, we obtain that(17)Iα1+β1vξ−λIβ1ψ1ξ+c2ξβ1Γβ1+1+a0=aIα¯2wμ1−akIβ2−pψ2μ1+ac¯2μ1β2−pΓβ2−p+1,(18)Iα2+β2wη−kIβ2ψ2η+c¯2ηβ2Γβ2+1+b0=bIα¯1vμ2−bλIβ1+qψ1μ2+c2bμ2β1+qΓβ1+q+1+ba0μ2qΓq+1.
Solving the above system, we find that(19)c2=1Δηβ2Γβ2+1aIα¯2wμ1−akIβ2−pψ2μ1−Iα1+β1vξ+λIβ1ψ1ξ+aμ1β2−pΓβ2+1−pbIα¯1vμ2−bλIβ1+qψ1μ2−Iα2+β2wη+kIβ2ψ2η−a0ηβ2Γβ2+1+aμ1β2−pΓβ2−p+1ba0μ2qΓq+1−b0c¯2=1Δξβ1Γβ1+1bIα¯1vμ2−bλIβ1+qψ1μ2−Iα2+β2wη+kIβ2ψ2η+bμ2q+β1Γq+β1+1aIα¯2wμ1−akIβ2−pψ2μ1−Iα1+β1vξ+λIβ1ψ1ξ−a0bμ2q+β1Γq+β1+1+ξβ1Γβ1+1ba0μ2qΓq+1−b0,where Δ is the determinant of the matrix associated with systems (17)-(18) in the two variables c2 and c¯2, and it is given by (9).
Substituting the values ofc0,c¯0,c1,c¯1,c2, and c¯2 in (15) and (16), we obtain the system of the integral equations (12) and (13). The proof is completed.The Banach spaceE=C0,1,ℝ is defined with the norm ψ1E=supt∈Jψ1t.So, the spaceE×E=ψ1,ψ2,s.tψ1,ψ2∈E×E with the norm ψ1,ψ2E×E=ψ1E+ψ2E is Banach space.Now, let us define the operatorN:E×E⟶E×E, by Nψ1,ψ2t=N1ψ1,ψ2t,N2ψ1,ψ2t, where(20)N1ψ1,ψ2t=1Γα1+β1∫0tt−sα1+β1−1fs,ψ1s,ψ2sds−λΓβ1∫0tt−sβ1−1ψ1sds+tβ1A1aΓα¯2∫0μ1μ1−sα¯2−1gs,ψ1s,ψ2sds−akΓβ2−p×∫0μ1μ1−sβ2−p−1ψ2sds−1Γα1+β1∫0ξξ−sα1+β1−1fs,ψ1s,ψ2sds+λΓβ1∫0ξξ−sβ1−1ψ1sds+A2bΓα¯1∫0μ2μ2−sα¯1−1fs,ψ1s,ψ2sds−bλΓβ1+q∫0μ2μ2−sβ1+q−1ψ1sds−1Γα2+β2∫0ηη−sα2+β2−1gs,ψ1s,ψ2sds+kΓβ2∫0ηη−sβ2−1ψ2sds+A3+a0,N2ψ1,ψ2t=1Γα2+β2∫0tt−sα2+β2−1gs,ψ1s,ψ2sds−kΓβ2∫0tt−sβ2−1ψ2sds+tβ2B1bΓα¯1∫0μ2μ2−sα¯1−1fs,ψ1s,ψ2sds−bλΓβ1+q×∫0μ2μ2−sβ1+q−1ψ1sds−1Γα2+β2∫0ηη−sα2+β2−1gs,ψ1s,ψ2sds+kΓβ2∫0ηη−sβ2−1ψ2sds+B2aΓα¯2∫0μ1μ1−sα¯2−1gs,ψ1s,ψ2sds−akΓβ2−p∫0μ1μ1−sβ2−p−1ψ2sds−1Γα1+β1∫0ξξ−sα1+β1−1fs,ψ1s,ψ2sds+λΓβ1∫0ξξ−sβ1−1ψ1sds+B3+b0.Note that the coupleψ1,ψ2 is a fixed point of the operator N if only if ψ1,ψ2 is a solution of systems (1)-(2). Consider the following hypotheses:H1. f,g:J×ℝ×ℝ⟶ℝ are continuous functions, and there exist real positive constants σi,τii=0,1,2 such that(21)ft,ζ,κ≤σ0+σ1ζ+σ2κ,gt,ζ,κ≤τ0+τ1ζ+τ2κ,for allt,ζ,κ∈J×ℝ×ℝ.H2. There exist constants K,L>0 such that(22)ft,ζ,κ−ft,ζ¯,κ¯≤Kζ−ζ¯+κ−κ¯,gt,ζ,κ−gt,ζ¯,κ¯≤Lζ−ζ¯+κ−κ¯,for allt,ζ,κ,t,ζ¯,κ¯∈J×ℝ×ℝ.H3. There exist F0,G0>0 such that F0=supt∈Jft,0,0 and G0=supt∈Jgt,0,0.Remark 8.
From conditionsH1 and H2 for all ψ1,ψ2∈E×E , we get(23)ft,ψ1t,ψ2t≤Kψ1,ψ2E×E+F0,gt,ψ1t,ψ2t≤Lψ1,ψ2E×E+G0.
For computation convenience, we set the following constants:(24)Q1=1+A1ξα1+β1Γα1+β1+1+A2bμ2α¯1Γα¯1+1,Q2=aA1μ1α¯2Γα¯2+1+A2ηα2+β2Γα2+β2+1,(25)Q3=bB1μ2α¯1Γα¯1+1+B2ξα1+β1Γα1+β1+1,Q4=1+B1ηα2+β2Γα2+β2+1+B2aμ1α¯2Γα¯2+1,(26)Q5=1+A1ξβ1λΓβ1+1+A2bλμ2β1+qΓβ1+q+1,Q6=A1akμ1β2−pΓβ2−p+1+A2kηβ2Γβ2+1,(27)Q7=B2λξβ1Γβ1+1+bλB1μ2β1+qΓβ1+q+1,Q8=B2akμ1β2−pΓβ2−p+1+1+B1ηβ2kΓβ2+1,(28)ρ1=maxQ5,Q6,ρ2=maxQ7,Q8.
In the following step, we present the following result about the uniqueness of solutions for problems (1)-(2) by applying the Banach contraction principle.Theorem 9.
Assume thatH2 and H3 hold. If(29)Q1+Q3K+Q2+Q4L+ρ1+ρ2<1,where Q1,Q2,Q3,andQ4 and ρ1andρ2 are, respectively, defined by (24), (25), and (28), then the boundary value problems (1)-(2) have a unique solution.Proof. Choose a positive real constantR where(30)R≥Q1+Q3F0+Q2+Q4G0+A3+B3+a0+b01−Q1+Q3K+Q2+Q4L+ρ1+ρ2.LetBR=ψ1,ψ2∈E×Es.t.ψ1,ψ2E×E≤R. First, we prove that NBR⊆BR.For allψ1,ψ2∈BR, we have(31)N1ψ1,ψ2t≤KR+F0Γα1+β1∫0tt−sα1+β1−1ds+λψ1EΓβ1∫0tt−sβ1−1ds+tβ1A1aLR+G0Γα¯2∫0μ1μ1−sα¯2−1ds+akψ2EΓβ2−p∫0μ1μ1−sβ2−p−1ds+KR+F0Γα1+β1∫0ξξ−sα1+β1−1ds+λψ1EΓβ1∫0ξξ−sβ1−1ds+A2bKR+F0Γα¯1∫0μ2μ2−sα¯1−1ds+bλψ1EΓβ1+q∫0μ2μ2−sβ1+q−1ds+LR+G0Γα2+β2∫0ηη−sα2+β2−1ds+kψ2EΓβ2∫0ηη−sβ2−1ds+A3+a0≤KR+F0Γα1+β1+1+aA1LR+G0μ1α¯2Γα¯2+1+A1KR+F0ξα1+β1Γα1+β1+1+bA2KR+F0μ2α¯1Γα¯1+1+A2LR+G0ηα2−β2Γα2−β2+1+|A3|+|a0|+λ1+A1ξβ1Γβ1+1+bλA2μ2β1+qΓβ1+q+1ψ1E+kA2ηβ2Γβ2+1+akA1μ1β2−pΓβ2−p+1ψ2E≤K1+A1ξα1+β1Γα1+β1+1+aA1μ1α¯2LΓα¯2+1+bA2μ2α¯1KΓα¯1+1+A2ηα2+β2LΓα2+β2+1R+F01+A1ξα1+β1Γα1+β1+1+aA1μ1α¯2G0Γα¯2+1+bA2μ2α¯1F0Γα¯1+1+A2ηα2+β2G0Γα2+β2+1+A3+a0+ρ1ψ1,ψ2E×E≤Q1F0+Q2G0+A3+a0+Q1K+Q2L+ρ1R.On the other hand, we have(32)N2ψ1,ψ2t≤LR+G0Γα2+β2+1+bB1μ2α¯1KR+F0Γα¯1+1+B1LR+G0ηα2+β2Γα2+β2+1+aB2LR+G0μ1α¯2Γα¯2+1+B2KR+F0ξα1+β1Γα1+β1+1+B3+b0+bλB1μ2β1+qΓβ1+q+1+λB2ξβ1Γβ1+1ψ1E+kΓβ2+1+kB1ηβ2Γβ2+1+akB2μ1β2−pΓβ2−p+1ψ2E≤Q4G0+Q3F0+B3+b0+Q4L+Q3KR+ρ2ψ1E+ψ2E≤Q4G0+Q3F0+B3+b0+Q4L+Q3K+ρ2R.Consequently,(33)Nψ1,ψ2≤Q1+Q3F0+Q2+Q4G0+A3+B3+a0+b0+Q1+Q3K+Q2+Q4L+ρ1+ρ2R,≤R.Therefore,NBR⊆BR. Next, we prove that N is a contraction. Let ψ1,ψ2,ψ¯1,ψ¯2∈E×E. For all t∈J, we have(34)N1ψ1,ψ2−N1ψ¯1,ψ¯2t≤Kψ1−ψ¯1E+ψ2−ψ¯2EΓα1+β1+1tα1+β1+λψ1−ψ¯1EΓβ1+1tβ1+tβ1A1aLψ1−ψ¯1E+ψ2−ψ¯2EΓα¯2+1μ1α¯2+akψ2−ψ¯2EΓβ2−p+1μ1β2−p+Kψ1−ψ¯1E+ψ2−ψ¯2EΓα1+β1+1ξα1+β1+λx−x¯EΓβ1+1ξβ1+A2bKψ1−ψ¯1E+ψ2−ψ¯2EΓα¯1+1μ2α¯1+bλψ1−ψ¯1EΓβ1+q+1μ2β1+q+Lψ1−ψ¯1E+ψ2−ψ¯2EΓα2+β2+1ηα2+β2+kψ2−ψ¯2EΓβ2+1ηβ2+A3≤Kψ1−ψ¯1,ψ2−ψ¯2E×EΓα1+β1+1+λψ1−ψ¯1EΓβ1+1+aA1Lψ1−ψ¯1,ψ2−ψ¯2E×EΓα¯2+1μ1α¯2+akA1ψ2−ψ¯2EΓβ2−p+1μ1β2−p+A1Kψ1−ψ¯1,ψ2−ψ¯2E×EΓα1+β1+1ξα1+β1+A1λψ1−ψ¯1EΓβ1+1ξβ1+A2bKψ1−ψ¯1,ψ2−ψ¯2E×EΓα¯1+1μ2α¯1+A2bλψ1−ψ¯1EΓβ1+q+1μ2β1+q+A2Lψ1−ψ¯1,ψ2−ψ¯2E×EΓα2+β2+1ηα2+β2+A2kψ2−ψ¯2EΓβ2+1ηβ2≤Q1K+Q2L+ρ1ψ1,ψ2−ψ¯1,ψ¯2E×E.In the other respect, we have(35)N2ψ1,ψ2t−N2ψ¯1,ψ¯2t≤Lψ1−ψ¯1E+ψ2−ψ¯2EΓα2+β2+1+kψ2−ψ¯2EΓβ2+1+B1bKψ1−ψ¯1E+ψ2−ψ¯2EΓα¯1+1μ2α¯1−1+B1bλψ1−ψ¯1EΓβ1+q+1μ2β1+q+B1Lψ1−ψ¯1E+ψ2−ψ¯2EΓα2+β2+1ηα2+β2+B1kKψ2−ψ¯2EΓβ2+1ηβ2+B2aLψ1−ψ¯1E+ψ2−ψ¯2EΓα¯2+1μ1α¯2+B2akψ2−ψ¯2EΓβ2−p+1μ1β2−p+B2Kψ1−ψ¯1E+ψ2−ψ¯2EΓα1+β1+1ξα1+β1+B2λψ1−ψ¯1EΓβ1+1ξβ1,≤LΓα2+β2+1+B1bKΓα¯1+1μ2α¯1−1+B1LΓα2+β2+1ηα2+β2+B2aLμ1α¯2Γα¯2+1+B2Kξα1+β1Γα1+β1+1ψ1−ψ¯1,ψ2−ψ¯2E×EB1bλμ2β1+qΓβ1+q+1+B2λξβ1Γβ1+1ψ1−ψ¯1Ek1+B1Kηβ2Γβ2+1+B2akμ1β2−pΓβ2−p+1ψ2−ψ¯2E≤Q4L+Q3K+ρ2ψ1,ψ2−ψ¯1,ψ¯2E×E.Consequently,(36)Nψ1,ψ2−Nx¯,y¯≤Q1+Q3K+Q2+Q4L+ρ1+ρ2ψ1,ψ2−ψ¯1,ψ¯2E×E.Therefore, the operatorN has a unique fixed point. Thus, we conclude that problems (1)-(2) have a unique solution on 0,1. The proof is complete.Now, we apply the Leray-Schauder alternative theoerm to obtain the following result about the existence of solutions for problems (1)-(2).Theorem 10.
Assume thatH1 holds. If(37)S<1,andS¯<1,where(38)S=Q1σ1+Q2τ1+Q3σ1+Q4τ1+Q5+Q8,(39)S¯=Q1σ2+Q2τ2+Q3σ2+Q4τ2+Q6+Q7,then the boundary value problems (1)-(2) have at least one solution on 0,1.Proof. First, we show that the operatorN is completely continuous.Becausef and g are continuous functions, N is continuous operator as well. Let Λ be any nonempty bounded subset of E×E. Then, there exists r>0 such that for any ψ1,ψ2∈Λ, ψ1,ψ2E×E≤r. Notice that from the condition (H1) for all ψ1,ψ2∈Λ, we have(40)ft,ψ1t,ψ2t≤σ0+σ1ψ1t+σ2ψ2t,≤σ0+maxσ1,σ2ψ1,ψ2E×E≤σ0+rmaxσ1,σ2gt,ψ1t,ψ2t≤τ0+rmaxτ1,τ2.Next, we prove thatNΛ is uniformly bounded. Let ψ1,ψ2∈Λ. Indeed, for any t∈0,1, we have(41)N1ψ1,ψ2t≤σ0+rmaxσ1,σ2Γα1+β1∫0tt−sα1+β1−1ds+λψ1EΓβ1∫0tt−sβ1−1ds+tβ1A1aτ0+rmaxτ1,τ2Γα¯2∫0μ1μ1−sα¯2−1ds+akψ2EΓβ2−p∫0μ1μ1−sβ2−p−1ds+σ0+rmaxσ1,σ2Γα1+β1∫0ξξ−sα1+β1−1ds+λψ1EΓβ1∫0ξξ−sβ1−1ds+A2bσ0+rmaxσ1,σ2Γα¯1∫0μ2μ2−sα¯1−1ds+bλψ1EΓβ1+q∫0μ2μ2−sβ1+q−1ds+τ0+rmaxτ1,τ2Γα2+β2∫0ηη−sα2+β2−1ds+kψ2EΓβ2∫0ηη−sβ2−1ds+A3+a0,≤1+A1ξα1+β1σ0+rmaxσ1,σ2Γα1+β1+1+aA1τ0+rmaxτ1,τ2μ1α¯2Γα¯2+1+bA2σ0+rmaxσ1,σ2μ2α¯1Γα¯1+1+A2τ0+rmaxτ1,τ2ηα2+β2Γα2+β2+1+A3+a0+λ1+A1ξβ1Γβ1+1+bλA2μ2β1+qΓβ1+q+1ψ1E+kA2ηβ2Γβ2+1+akA1μ1β2−pΓβ2−p+1ψ2E≤1+A1ξα1+β1Γα1+β1+1+bA2μ2α¯1Γα¯1+1σ0+rmaxσ1,σ2+aA1μ1α¯2Γα¯2+1+A2ηα2+β2Γα2+β2+1τ0+rmaxτ1,τ2+A3+a0+ρ1r≤+∞.Similarly,(42)N2ψ1,ψ2t=1+B1ηα2+β2τ0+rmaxτ1,τ2Γα2+β2+1+bB1μ2α¯1σ0+rmaxσ1,σ2Γα¯1+1+aB2τ0+rmaxτ1,τ2μ1α¯2Γα¯2+1+B2σ0+rmaxσ1,σ2ξα1+β1Γα1+β1+1+B3+b0+bλB1μ2β1+qΓβ1+q+1+λB2ξβ1Γβ1+1ψ1E+k+kB1ηβ2Γβ2+1+akB2μ1β2−pΓβ2−p+1ψ2E≤bB1μ2α¯1Γα¯1+1+B2ξα1+β1Γα1+β1+1σ0+rmaxσ1,σ2+aB2μ1α¯2Γα¯2+1+1+B1ηα2+β2Γα2+β2+1τ0+rmaxτ1,τ2+B3+b0+ρ2r<+∞.Consequently,ψ1,ψ2E×E<+∞for anyψ1,ψ2∈Λ. Therefore, NΛ is uniformly bounded.Now, we show thatN is equicontinuous on Λ. Let ψ1,ψ2∈Λ. For any t2,t1∈J, where t2>t1, we have(43)N1ψ1,ψ2t2−N1ψ1,ψ2t1≤σ0+rmaxσ1,σ2Γα1+β1∫0t1t2−sα1+β1−1−t1−sα1+β1−1ds+∫t1t2t2−sα1+β1−1ds+λψ1EΓβ1∫0t1t2−sβ1−1−t1−sβ1−1ds+∫t1t2t2−sβ1−1ds+t2β1−t1β1A1aτ0+rmaxτ1,τ2Γα¯2∫0μ1μ1−sα¯2−1ds+akψ2EΓβ2−p∫0μ1μ1−sβ2−p−1ds+σ0+rmaxσ1,σ2Γα1+β1∫0ξξ−sα1+β1−1ds+λψ1EΓβ1∫0ξξ−sβ1−1ds+A2bσ0+rmaxσ1,σ2Γα¯1∫0μ2μ2−sα¯1−1ds+bλψ1EΓβ1+q∫0μ2μ2−sβ1+q−1ds+τ0+rmaxτ1,τ2Γα2+β2∫0ηη−sα2+β2−1ds+kψ2EΓβ2∫0ηη−sβ2−1ds+A3≤σ0+rmaxσ1,σ2Γα1+β1+1−t2−t1α1+β1+t2α1+β1−t1α1+β1+t2−t1α1+β1+λψ1EΓβ1+1−t2−t1β1+t2β1−t1β1+t2−t1β1+t2β1−t1β1A1aμ1α¯2Γα¯2+1+A2ηα2+β2Γα2+β2+1τ0+rmaxτ1,τ2+A1akμ1β2−pΓβ2−p+1+A2kηβ2Γβ2+1ψ2E+A1λξβ1Γβ1+1++A2bλμ2β1+qΓβ1+q+1ψ1E+A2bμ2α¯1Γα¯1+1+A1ξα1+β1Γα1+β1+1σ0+rmaxσ1,σ2+A3≤σ0+rmaxσ1,σ2Γα1+β1+1t2α1+β1−t1α1+β1+λrΓβ1+1t2β1−t1β1+t2β1−t1β1A1aμ1α¯2Γα¯2+1+A2ηα2+β2Γα2+β2+1τ0+rmaxτ1,τ2+rmaxA1akμ1β2−pΓβ2−p+1+A2kηβ2Γβ2+1,A1λξβ1Γβ1+1+A2bλμ2β1+qΓβ1+q+1+A2bμ2α¯1Γα¯1+1+A1ξα1+β1Γα1+β1+1σ0+rmaxσ1,σ2+A3⟶0,ast2⟶t1.Analogously,(44)N2ψ1,ψ2t2−N2ψ1,ψ2t1≤τ0+rmaxτ1,τ2Γα2+β2+1t2α2+β2−t1α2+β2+krΓβ2+1t2β2−t1β2+t2β2−t1β2bB1μ2α¯1Γα¯1+1+B2ξα1+β1Γα1+β1+1σ0+rmaxσ1,σ2+rmaxB2λξβ1Γβ1+1+bλB1μ2β1+qΓβ1+q+1,B2akμ1β2−pΓβ2−p+1+B1kηβ2Γβ2+1+B1ηα2+β2Γα2+β2+1+B2aμ1α¯2Γα¯2+1τ0+rmaxτ1,τ2+B3⟶0,ast2⟶t1,which imply that Nψ1,ψ2t2−Nψ1,ψ2t1⟶0 as t2⟶t1. Thus, the operator N is equicontinuous. Hence, by Arzela–Ascoli theorem, we deduce that the operator N is completely continuous.Finally, we will verify that the setMN=ψ1,ψ2∈E×E:ψ1,ψ2=mNψ1,ψ2for some0<m<1 is bounded. For all ψ1,ψ2∈MN and for any t∈J, we have mNψ1,ψ2=mN1ψ1,ψ2,mN2ψ1,ψ2. Then,(45)ψ1t=mN1ψ1,ψ2t≤σ0+σ1ψ1E+σ2ψ2EΓα1+β1+1+A1aμ1α¯2Γα¯2+1+A2ηα2+β2Γα2+β2+1×τ0+τ1ψ1E+τ2ψ2E+A1akμ1β2−pΓβ2−p+1+A2kηβ2Γβ2+1ψ2E+A1λξβ1Γβ1+1+A2bλμ2β1+qΓβ1+q+1+λΓβ1+1ψ1E+A2bμ2α¯1Γα¯1+1+A1ξα1+β1Γα1+β1+1σ0+σ1ψ1E+σ2ψ2E+A3+a0.This yields that(46)ψ1E=mN1ψ1,ψ2≤Q1σ0+σ1ψ1E+σ2ψ2E+Q2τ0+τ1ψ1E+τ2ψ2E+Q5ψ1E+Q6ψ2E+A3+a0.In the same way, we deduce that(47)ψ2E=mN2ψ1,ψ2≤Q4τ0+τ1ψ1E+τ2ψ2E+Q3σ0+σ1ψ1E+σ2ψ2E+Q7ψ2E+Q8ψ1E+B3+b0.Hence, we have(48)ψ1,ψ2E×E=ψ1E+ψ2E≤Q1σ0+σ1ψ1E+σ2ψ2E+Q2τ0+τ1ψ1E+τ2ψ2E+Q5ψ1E+Q6ψ2E+Q4τ0+τ1ψ1E+τ2ψ2E+Q3σ0+σ1ψ1E+σ2ψ2E+Q7ψ2E+Q8ψ1E+A3+|B3|+a0+b0,which yields(49)ψ1,ψ2E×E≤Q3σ0+Q4τ0+Q1σ0+Q2τ0+A3+B3+a0+b01−maxS,S¯,where S and S¯ are given by (36) and (38), respectively, which proves that MN is bounded. Thus, as a consequence of Leray–Schauder alternative theorem, N has more than one fixed point. Hence, the boundary value problems (1)-(2) have one solution at the very least on 0,1.


## 4. Applications and Numerical Examples

In this section, we solve the integral equations (12) and (13) using the Adams-type predictor-corrector method with step size h=0.01 (for details, see [16–18] and the references therein). Briefly, we aim to approximate the solution of the following fractional initial value problem:(50)Dαyt=ft,yt,yt0=y0,t∈t0,T,α∈0,1,at the grid points tm=t0+mh,m≥0, with h is a uniform step size. It is found that(51)ytm≈Ytm=yt0+1Γα∑j=0mRm,jfj,where(52)Rm,j=W0m,0,ifj=0,W1m,j−1+W0m,j,if1≤j≤m−1,W1m,m−1,ifj=m.W1m,j=−α+j−mh−j+m−1α+m−jhm−jααα+1W0m,j=α+j−m+1hm−jα+h−j+m−1α+1/hαα+1.To measure the accuracy of the present algorithm, we calculated the residual function as follows:(53)Rt≔DαYt−ft,Yt,at the grid points tm for m>0, i.e.,(54)Rtm=1Γ1−α∫t0tmtm−τ−αY′τdτ−ftm,Ytm,=1Γ1−α∑j=0m−1∫tjtj+1tm−τ−αY′τdτ−ftm,Ytm,=h−α2Γ1−α∑j=0m−1m−j−1−αY′tj+1−m−j−αY′tj−ftm,Ytm,after using the trapezoidal rule.Consider the following coupled system of fractional Langevin equations:(55)Dcα1Dcβ1+18ψ1t=ψ2t71+t21+ψ2t−ψ1t5+ln1+t,t∈0,1,Dcα2Dcβ2+17ψ2t=ψ1t141−t+ψ12t−3sinψ2t5+t−1−t2.Equipped with the nonlocal nonseparated fractional integral and fractional derivative boundary conditions,(56)ψ10=1,ψ20=3,ψ1′0=ψ2′0=0,ψ1910=14Dc1/4ψ215,ψ21=2I5/4ψ118.Here,λ=1/8,k=1/7,a=1/4,b=2,a0=1,b0=3,p=1/4,q=5/4,ξ=9/10,η=1,μ1=1/5,μ2=1/8, and(57)ft,ψ1,ψ2=ψ271+t21+ψ2−ψ15+ln1+t,gt,ψ1,ψ2=ψ1141−t+ψ12−3sinψ25+t−1−t2.On the other part, fort,ζ,κ,t,ζ¯,κ¯∈0,1×ℝ×ℝ, we get(58)ft,ζ,κ−ft,ζ¯,κ¯≤15ζ−ζ¯+κ−κ¯,gt,ζ,κ−gt,ζ¯,κ¯≤35ζ−ζ¯+κ−κ¯,ft,ζ,κ≤σ0+15ζ+17κ,gt,ζ,κ≤τ0+114ζ+35κ,so σ1=1/5,σ2=1/7,τ1=1/14,τ2=3/5,K=1/5, and L=3/5.Case i.
In order to illustrate Theorem9, we take α1=1/2,β1=3/2,α2=3/4, and β2=7/4. Thus, α¯1=13/4, and α¯2=9/4. By using the Matlab program, we found that(59)Δ≃0.399301327423794,A1≃1.171331813010056,A2≃0.031689241829108,A3≃−1.262241885268691,B1≃1.000100634730081,B2≃0.002312772254050,B3≃−2.871400693622515,Q1≃00.974398268143493,Q2≃0.043889448160305,Q3≃0.001217044521056,Q4≃0.601838570591574,Q5≃0.188078540898721,Q6≃0.005629381698062,Q7≃3.713645478256922×10−4,Q8≃0.177657802566698,ρ1≃0.188078540898721,ρ2≃0.0.177657802566698,Q1+Q3K+Q2+Q4L+ρ1+ρ2≃0.948296217249456<1.
Thus, the hypothesis of Theorem9 holds. Then, problems (55)-(56) have a unique solution on 0,1. The behavior of the solutions ψ1t and ψ2t for Case i is presented in Figure 1. Table 1 displays the residuals Rψ1tm and Rψ2tm for the couple of equations given in (55) which clearly indicates the accuracy of the present algorithm.Figure 1
Graphs of the approximated solutionsψ1t and ψ2t for Case i.Table 1
ResidualsRψ1tm and Rψ2tm for example 1.
tmRψ1tmRψ2tm0.20.64265×10−42.33263×10−40.23.61491×10−31.51331×10−40.62.96055×10−37.53023×10−30.35.12564×10−32.02533×10−31.03.69052×10−35.32055×10−3Case ii.
In order to illustrate Theorem10, we take α1=1/4,β1=5/4,α2=1/2, and β2=2.
Thus,α¯1=11/4 and α¯2=5/4.
By using the Matlab program, we found that(60)Δ≃0.386787685580080,A1≃1.140949096832007,A2≃0.021215616584967,A3≃−1.201812441154092,B1≃1.000159825315441,B2≃0.004297601923856,B3≃−2.873555329197462,Q1≃1.485098417429794,Q2≃0.056422691832900,Q3≃0.004245829991826,Q4≃0.601861587859298,Q5≃0.220678979659357,Q6≃0.003030802369281,Q7≃8.31262586410917×10−4,Q8≃0.142874266998083,ρ1≃0.220678979659357,ρ2≃0.142874266998083,S≃0.708442401834063<1,S¯≃0.611596096688384<1.
Thus, all the conditions of Theorem10 are satisfied. Then, problems (55)-(56) have one solution at the least 0,1. In addition, both solutions ψ1t and ψ2t for Case ii are sketched in Figure 2.Figure 2
Graphs of the approximated solutionsψ1t and ψ2t for Case ii.


## 5. Conclusion

The most important features of differential equations subject to either initial or boundary conditions are the existence and uniqueness of their solutions. In this paper, we discussed the existence and uniqueness of solutions of specific type of the couple system of the Langevin differential equation in the framework of Caputo fractional derivatives and under the suzerainty of nonlocal and nonseparated boundary conditions. The boundary value problem we studied contained 6 different parameters. Because of the complexity, we were forced to use computer programs in order to find examples that would support our results. We discussed these examples from the theoretical point and solved numerically using the Adams-type predictorcorrector method by implicitly implementing the Gauss–Seidel method.It is recommended to consider the same problem in the frame of other fractional derivatives especially the ones with no singular kernels and compare their results to the ones discussed in this paper.



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