# Stability of a Nonlinear Fractional Langevin System with Nonsingular Exponential Kernel and Delay Control

Authors: Kaihong Zhao
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

Fractional Langevin system has great advantages in describing the random motion of Brownian particles in complex viscous fluid. This manuscript deals with a delayed nonlinear fractional Langevin system with nonsingular exponential kernel. Based on the fixed point theory, some sufficient criteria for the existence and uniqueness of solution are established. We also prove that this system is UH- and UHR-stable attributed to the nonlinear analysis and inequality techniques. As applications, we provide some examples and simulations to illustrate the availability of main findings.


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## Body

## 1. Introduction

To expound the random motion of particles in fluid after colliding with each other, Langevin raised the famous Langevin equation in 1908. Afterward, many random phenomena and processes were found to be described by Langevin system [1, 2]. However, the integer-order Langevin equation cannot meet the accuracy requirements in describing complex viscoelasticity. Thereby, the classical Langevin equation has been extended and modified. For example, Kubo [3, 4] put forward a general Langevin equation to simulate the complex viscoelastic anomalous diffusion process. Eab and Lim [5] applied a fractional Langevin equation to describe the single-file diffusion. Sandev and Tomovski [6] established a fractional Langevin equation model to study the motion of free particles driven by power-law noise. Furthermore, the stability of the system with practical application background is the most important dynamic characteristics. Ulam and Hyers [7, 8] proposed a concept of system stability called UH-stability in 1940s. In recent ten years, here have many works (some of them [9–30]) on UH-stability of fractional system.As far as we know, the papers on fractional Langevin system published at present are all about Caputo and Riemann–Liouville fractional derivatives. However, as the authors [31–33] pointed out, the definitions of Caputo and Riemann–Liouville fractional derivatives have singular kernels. In 2015, Caputo and Fabrizio [34] defined a nonsingular fractional derivative with exponential kernel under a more general framework, which is also called the Caputo–Fabrizio (CF) fractional derivative. The properties and applications of this novel fractional derivative have attracted the attention of many scholars. Losada and Nieto [31] systematically studied the Laplace transform of CF-fractional derivative and its antiderivative, and applied it to study the falling body problem. For more research and application of CF-fractional differential equations, readers refer to references [35–42]. To the best of my knowledge, there are no papers dealing with Ulam–Hyers type stability of CF-fractional Langevin system. Motivated by aforementioned system, this manuscript focuses on the following nonlinear fractional Langevin system with nonsingular exponential kernel and delay control(1)FD0+νCD0+μCF−λut=ft,ut,Cut,t∈0,T,Cut=gt,ut−ξt,t∈0,T,ut=ϖ1t,D0+μCFut=ϖ2t,t∈−η,0,where T>0, 0<μ,ν≤1, and λ>0 are some constants, D0+∗CF represents the ∗-order fractional derivative with nonsingular exponential kernel, the nonlinear response f∈C0,T×ℝ2,ℝ, the control function g∈C0,T×ℝ,ℝ, the delay function ξ∈C0,T,ℝ+ with η=maxξt:t∈0,T, and the initial functions ϖ1,ϖ2∈C−η,0,ℝ.Compared with previous papers on fractional Langevin equation, the influence of delay control is considered for the first time in our system (1). In fact, it is sometimes necessary and beneficial to manually control and intervene in the random motion of free particles. However, manual control is not instantaneous, but often lagging. Therefore, it is of great practical value to consider the control delay Cut in system (1). Meanwhile, for 0<α<1, the kernel functions of Caputo fractional derivative and CF-fractional derivative with α-order are written by t−s−α and e−α/1−αt−s, respectively. Obviously, t−s−α⟶∞ (singular) and e−α/1−αt−s⟶1 (nonsingular), as s⟶t. Therefore, it is of great significance to explore the dynamic properties of system (1). The highlights of this paper mainly include two aspects: (a) In the fractional Langevin equation, we consider delay control and nonsingular exponential kernel function, which have not appeared in previous studies. (b) We obtain some new and easily verifiable sufficient criteria for the solvability and stability of system (1).The structure of the remaining sections of the paper is as follows. Section2 introduces some fundamental definitions and lemmas of CF-fractional calculus. In Section 3, we obtain some criteria for the existence of solutions to system (1) by utilizing some fixed point theorems. In Section 4, we shall prove that system (1) is UH- and UHR-stable. Section 5 provides some applications to illustrate the correctness of our major outcomes. A brief summary is made in Section 6.


## 2. Preliminaries

This section gives the concepts of Caputo–Fabrizio fractional derivative and integral as well as some useful results.Definition 1.
(see [31]). For 0≤μ≤1, T>0 and u∈H10,T, the left-sided μ-order Caputo–Fabrizio fractional integral of function u is defined by(2)J0μCFut=1−μNμut+μNμ∫0tusds,where Nμ represents the normalisation constant with N0=N1=1.Definition 2.
(see [34]). For 0≤μ≤1, T>0 and u∈H10,T, the left-sided μ-order Caputo–Fabrizio fractional derivative of function u is defined by(3)D0+μCFut=Nμ1−μ∫0te−μ/1−μt−su′sds.Lemma 1 (see [31]).
Let0≤γ≤1 and h∈C0,∞. Consider the below initial value problem(4)FD0+γCwt=ht,t≥0,w0=w0.
Then, the unique solution of this IVP is read as(5)wt=w0+1−γNγht−h0+γNγ∫0thsds.Lemma 2.
LetT,λ>0, 0<μ,ν≤1, f∈C0,T×ℝ2,ℝ, g∈C0,T×ℝ,ℝ, ξ∈C0,T,ℝ+ with η=maxξt:t∈0,T, ϖ1,ϖ2∈C−η,0,ℝ. If Λ≜1−λ1−μ/Nμ≠0, then the CF-fractional Langevin equation (1) is equivalent to the following integral equation:(6)ut=ϖ10+1Λμϖ20−λϖ10Nμt+1−μ1−νNμNνFut−Fu0+λμNμ∫0tusds+μ+ν−2μνNμNν∫0tFusds+μνNμNν∫0tt−sFusds,t∈0,Tϖ1t,t∈−η,0,where Fut=ft,ut,Cut, Cut=gt,ut−ξt.Proof.
Assume thatut∈C−η,T satisfies system (1). Then, when t∈0,T, we derive from Lemma 1 that(7)D0+μCF−λut=D0+μCFu0−λu0+1−νNνft,ut,Cut−f0,u0,Cu0+νNν∫0tfτ,uτ,Cuτdτ.
Eqaution (7) gives(8)D0+μCFut=D0+μCFu0−λu0+λut+1−νNνft,ut,Cut−f0,u0,Cu0+νNν∫0tfτ,uτ,Cuτdτ.
From Lemma1, u0=ϖ10 and (8), we have(9)ut=ϖ10+1−μNμλut−ϖ10+1−νNνft,ut,Cut−f0,ϖ10,Cu0+νNν∫0tfτ,uτ,Cuτdτ+μNμ∫0tϖ20−λϖ10+λus+1−νNνfs,us,Cus+νNν∫0sfτ,uτ,Cuτdτds=1−λ1−μNμϖ10+λ1−μNμut+1−μ1−νNμNνft,ut,Cut−f0,ϖ10,Cu0+1−μνNμNν∫0tfτ,uτ,Cuτdτ+μϖ20−λϖ10Nμt+λμNμ∫0tusds+μ1−νNμNν∫0tfs,us,Cusds+μνNμNν∫0t∫0sfτ,uτ,Cuτdτds.
Exchanging the order of double integrals, the last integral term of (9) is reduced to(10)∫0t∫0sfτ,uτ,Cuτdτds=∫0tfτ,uτ,Cuτ∫τtdsdτ=∫0tt−τfτ,uτ,Cuτdτ.
It follows from (9) and (10) that(11)ut=ϖ10+1Λμϖ20−λϖ10Nμt+1−μ1−νNμNνFut−Fu0+λμNμ∫0tusds+μ+ν−2μνNμNν∫0tFusds+μνNμNν∫0tt−sFusds.
Whent∈−η,0, it is clear that ut=ϖ1t holds. Thus, system (6) holds, namely, ut∈C−η,T also satisfies system (6). And vice versa, if ut∈C−η,T satisfies integral system (6), then, when t∈0,T, we know that (8) and (7) hold by finding the fractional derivative D0+μCF at both sides of (6). Next, by finding the fractional derivative D0+νCF at both sides of (7), we easily derive the first fractional equation of (1). When t∈−η,0, let us make a supplementary definition D0+μCFut=ϖ2t, then ut=ϖ1t and D0+μCFut=ϖ2t satisfy the equation (1). Thus, we verify that ut∈C−η,T also satisfies system (1). The proof is completed.


## 3. Existence of Solutions

This section mainly studies the solvability of system (1) by using the below some important fixed point theorems.Lemma 3 (see [43]).
LetX be a Banach space and ϕ≠Y⊂X be closed convex. Assume that P and Q satisfy(i)
Pu+Qv∈Y, ∀u,v∈Y.(ii)
P is contract, and Q is compact and continuous.
Then, there exists at least anu∗∈Y satisfying u∗=Pu∗+Qu∗.Lemma 4 (see [44]).
LetX be a Banach space and ϕ≠E⊂X be closed. If ℱ:E⟶E is contract, then ℱ admits a unique fixed point u∗∈E.According to Lemma2, we take X=C−η,T,ℝ. For all u∈X, we define the norm u=suput:−η≤t≤T, then X,⋅ is a Banach space. We always argue the existence and stability of solution for system (1) on X,⋅. Throughout the paper, the following fundamental assumptions are needed:(H1)
T, μ, ν, and λ are some constants satisfying T,λ>0, 0<μ,ν≤1, and Λ≜1−λ1−μ/Nμ≠0.(H2)
f∈C0,T×ℝ2,ℝ, g∈C0,T×ℝ,ℝ, ξ∈C0,T,ℝ+ with η=maxξt:t∈0,T, ϖ1,ϖ2∈C−η,0,ℝ.Theorem 1.
Assume thatH1 and H2 are true, as well as the following conditions H3 and H4 also hold:(H3)
For allt∈0,T, u,v∈ℝ, there have some continuous functions ℳ1t, ℳ2t, ℳ3t≥0 such that (12)ft,u,v≤ℳ1t+ℳ2tv,gt,v≤ℳ3t.(H4)
0<ϑ≜λT/Nμ−λ1−μ<1.
Then, system (1) admits at least a solution u∗t∈X.Proof.
In the light of Lemma2, for all u∈X, the operators P, Q:X⟶X are defined by(13)Put=ϖ10+1Λμϖ20−λϖ10Nμt+λμNμ∫0tusds,t∈0,T,ϖ1t,t∈−η,0,and(14)Qut=1Λ1−μ1−νNμNνFut−Fu0+μ+ν−2μνNμNν∫0tFusds+μνNμNν∫0tt−sFusds,t∈0,T,0,t∈−η,0,where Fut is defined as (6). It is easy to see from (13) and (14) that Pu+Qv∈X,∀u,v∈X, that is, the condition i in Lemma 3 holds. In addition, ∀t∈−η,T, u,v∈X, when t∈0,T, one has(15)Put−Pvt=λμΛNμ∫0tus−vsds≤NμNμ−λ1−μ×λμNμ∫0tus−vsds≤λμNμ−λ1−μ∫0tds⋅u−v=λtNμ−λ1−μu−v≤λTNμ−λ1−μu−v=ϑu−v.
Whent∈−η,0, one derives from (13) that(16)Put−Pvt=ϖ1t−ϖ1t≡0.
Equations (15) and (16) mean that(17)Pu−Pv≤λTNμ−λ1−μu−v=ϑu−v.
In view ofH4 and (17), one concludes that P:X⟶X is contract.
Now, we apply Arzelá–Ascoli theorem to prove thatQ:X⟶X is completely continuous. Indeed, for all t∈−η,T, u∈X, when t∈0,T, it follows from (14) and H3 that(18)Qut≤1Λ1−μ1−νNμNνFut+Fu0+μ+ν−2μνNμNν∫0tFusds+μνNμNν∫0tt−sFusds≤MΛ21−μ1−νNμNν+μ+ν−2μνNμNν∫0tds+μνNμNν∫0tt−sds≤MΛNμNν21−μ1−ν+μ+ν−2μνT+μνT22,where M=ℳ1T+ℳ2T⋅ℳ3T and ℳiT=supℳit:0≤t≤Ti=1,2,3. When t∈−η,0, (14) gives(19)Qut≡0,
By (18) and (19), we know that Q:X⟶X is uniformly bounded.
In the meantime, for allu∈X, t1,t2∈−η,T with t1<t2, we verify that the operator Q is equicontinuous in three cases.Case 1.
When0≤t1<t2≤T, according to (14) and H3, we get(20)Qut2−Qut1≤1Λ1−μ1−νNμNνFut2−Fut1+μ+ν−2μνNμNν∫0t2Fusds−∫0t1Fusds+μνNμNν∫0t2t2−sFusds−∫0t1t1−sFusds≤1Λ1−μ1−νNμNνFut2−Fut1+μ+ν−2μνNμNν∫t1t2Fusds+μνNμNν∫t1t2t2−sFusds+∫0t1t2−t1Fusds≤1Λ1−μ1−νNμNνFut2−Fut1+μ+ν−2μνMNμNνt2−t1+2μνTMNμNνt2−t1⟶0,ast2⟶t1.Case 2.
.When−η≤t1<0<t2≤T, then t2⟶t1 means that t1⟶0− and t2⟶0+. From (14), we obtain(21)Qut2−Qut1⟶Qu0+−Qu0−=0,ast2⟶t1.Case 3.
.When−η≤t1<t2≤0, then(22)Qut2−Qut1≡0−0=0⟶0,ast2⟶t1.
From (20)–(22), one concludes that, ∀ϵ>0, t1,t2∈−η,T and u∈X, ∃δ=δϵ>0 such that Qut2−Qut1<ϵ provided that t2−t1<δ, i.e., Q:X⟶X is equicontinuous. Thus, the condition ii is also true. So, it follows from Lemmas 3 and 2 that there exists at least a fixed point u∗t∈X with u∗t=Pu∗t+Qu∗t, which satisfies system (1). The proof is completed.Theorem 2.
Assume thatH1 and H2 are true, as well as the following conditions H5 and H6 also hold.(H5)
For allt∈0,T, u,u¯,v,v¯∈ℝ, there have some continuous functions ℋ1t, ℋ2t, ℋ3t≥0 such that(23)ft,u,v−ft,u¯,v¯≤ℋ1tu−u¯+ℋ2tv−v¯,gt,v−gt,v¯≤ℋ3tv−v¯.(H6)
0<Γ<1, where Γ=1/Λ1−μ1−νℋ/NμNν+λμT/Nμ+μ+ν−2μνℋT/NμNν+μνℋT2/2NμNν, ℋ=ℋ1T+ℋ2T⋅ℋ3T and ℋiT=supℋit:0≤t≤T, i=1,2,3.
Then, system (1) admits a unique solution u∗t∈X.Proof.
According to Lemma2, an operator ℱ:X⟶X is defined by(24)ℱut=ϖ10+1Λμϖ20−λϖ10Nμt+1−μ1−νNμNνFut−Fu0+λμNμ∫0tusds+μ+ν−2μνNμNν∫0tFusds+μνNμNν∫0tt−sFusds,t∈0,Tϖ1t,t∈−η,0,where Fut is the same as (6). Then, for all u,v∈X, when t∈0,T, it follows from H5 that(25)Tut−Tvt≤1Λ1−μ1−νNμNνFut−Fvt+λμNμ∫0tus−vsds+μ+ν−2μνNμNν∫0tFus−Fvsds+μνNμNν∫0tt−sFus−Fvsds≤1Λ1−μ1−νNμNνℋ1tut−vt+ℋ2tCut−Cvt+λμNμ∫0tus−vsds+μ+ν−2μνNμNν∫0tℋ1sus−vs+ℋ2sCus−Cvsds+μνNμNν∫0tt−sℋ1sus−vs+ℋ2sCus−Cvsds≤1Λ1−μ1−νNμNνℋ1tut−vt+ℋ2tℋ3tus−ξs−vs−ξs+λμNμ∫0tus−vsds+μ+ν−2μνNμNν∫0tℋ1sus−vs+ℋ2sℋ3sut−ξt−vt−ξtds+μνNμNν∫0tt−sℋ1sus−vs+ℋ2sℋ3sus−ξs−vs−ξsds≤1Λ1−μ1−νNμNνℋ1T⋅u−v+ℋ2T⋅ℋ3T⋅u−v+λμNμ∫0tu−vds+μ+ν−2μνNμNν∫0tℋ1T⋅u−v+ℋ2T⋅ℋ3T⋅u−vds+μνNμNν∫0tt−sℋ1T⋅u−v+ℋ2T⋅ℋ3T⋅u−vds≤1Λ1−μ1−νℋNμNν+λμTNμ+μ+ν−2μνℋTNμNν+μνℋT22NμNνu−v=Γu−v,where Γ is the same as the condition H6. When t∈−η,0, (24) leads to(26)ℱut−ℱvt=ϖ1t−ϖ1t≡0.
From (25) and (26), one has(27)ℱu−ℱv≤Γu−v.
By (27) and H6, one knows that ℱ:X⟶X is a contraction mapping. Thus, it follows from Lemmas 3 and 2 that a mapping ℱ exists a unique fixed point u∗t∈X, which satisfies system (1). The proof is completed.


## 4. UH- and UHR-Stability

This section mainly establishes the UH- and UHR-stability of system (1). Let z∈X, ε>0, 0<μ,ν≤1, and φ∈C0,T,ℝ+ be nondecreasing. Consider the following two inequalities:(28)|D0+νCFD0+uCF−λWt−ft,Wt,AEt≤ε,t∈0,T,AEt=gt,Wt−ξt,t∈0,T,Wt=ϖ1t,D0+uCFWt=ϖ2t,t∈−η,0,and(29)|D0+νCFD0+uCF−λWt−ft,Wt,AEt≤φtε,t∈0,T,AEt=gt,Wt−ξt,t∈0,T,Wt=ϖ1t,D0+uCFWt=ϖ2t,t∈−η,0.Definition 3.
Problem (1) is Ulam–Hyers (UH) stable, if for all ϵ>0 and any solution W∈X of (28), there have a unique solution u∗∈X of (1) and a constant C1>0 satisfying(30)Wt−u∗t≤C1ϵ.Definition 4.
Problem (1) is generalised Ulam–Hyers (GUH) stable, if for all ϵ>0 and any solution W∈X of (28), there have a unique solution u∗∈X of (1) and function ϑ⋅∈Cℝ,ℝ+ with ϑ0=0 satisfying(31)Wt−u∗t≤ϑϵ.Definition 5.
Problem (1) is Ulam–-Hyers–Rassias (UHR) stable, if for all ϵ>0 and any solution W∈X of (29), there have a unique solution u∗∈X of (1) and constant C2>0 satisfying(32)Wt−u∗t≤C2φtϵ,t∈−η,T.Definition 6.
Problem (1) is generalised Ulam-Hyers-Rassias (GUHR) stable, if for any solution W∈X of (29), there have a unique solution u∗∈X of (1) and constant C3>0 satisfying(33)Wt−u∗t≤C3φt,t∈−η,T.Obviously, UH-stable⇒ GUH-stable, and UHR-stable ⇒ GUHR-stable.Remark 1.
W∈X satisfies the inequality (28) iff there has ϕ∈X such that (1)
ϕt≤ϵ, 0<t≤T.(2)
D0+νCFD0+μML−λWt=ft,Wt,AEt+ϕt,0<t≤T.(3)
Wt=ϖ1t,D0+μCFWt=ϖ2t,t∈−η,0.Remark 2.
W∈X satisfies the inequality (29) iff there has ψ∈X such that(1)
ψt≤φtϵ, 0<t≤T.(2)
D0+νCFD0+μML−λWt=ft,Wt,CWt+ψt,0<t≤T.(3)
Wt=ϖ1t,D0+μCFWt=ϖ2t,t∈−η,0.Theorem 3.
If all the conditions of Theorem2 are true, then system (1) is UH-stable and also GUH-stable.Proof.
By Lemma2 and Remark 1, the solution Wt of inequality (28) is formulated as(34)Wt=ϖ10+1Λμϖ20−λϖ10Nμt+1−μ1−νNμNνFWt−FW0+ϕt−ϕ0+λμNμ∫0tWsds+μ+ν−2μνNμNν∫0tFWs+ϕsds+μνNμNν∫0tt−sFWs+ϕsds,t∈0,Tϖ1t,t∈−η,0,
By Theorem2 and Lemma 2, the unique solution u∗t of (1) satisfies(35)u∗t=ϖ10+1Λμϖ20−λϖ10Nμt+1−μ1−νNμNνFu∗t−Fu∗0+λμNμ∫0tu∗sds+μ+ν−2μνNμNν∫0tFu∗sds+μνNμNν∫0tt−sFu∗sds,t∈0,Tϖ1t,t∈−η,0,
Similar to (25) and (26), we derive from (34) and (35) that(36)Wt−u∗t≤1Λ1−μ1−νNμNνFWt−Fu∗t+ϕt+ϕ0+λμNμ∫0tWs−u∗sds+μ+ν−2μνNμNν∫0tFWs−Fu∗s+ϕsds+μνNμNν∫0tt−sFWs−Fu∗s+ϕsds≤1Λ1−μ1−νNμNνℋWt−u∗t+ε+ε+λμNμ∫0tWs−u∗sds+μ+ν−2μνNμNν∫0tℋWs−u∗s+εds+μνNμNν∫0tt−sℋWs−u∗s+εds≤ΓWt−u∗t+Υε,t∈0,T,where Υ=1/Λ21−μ1−νℋ/NμNν+λμT/Nμ+μ+ν−2μνℋT/NμNν+μνℋT2/2NμNν, and(37)Wt−u∗t=ϖ1t−ϖ1t≡0,t∈−η,0.
From (36) and (37), we have(38)Wt−u∗t≤Υ1−Γε.
Thus, it follows from (36) and Definitions 3 and 4 that system (1) is UH-stable and also GUH-stable. The proof is completed.Theorem 4.
If all the conditions of Theorem2 are true, then system (1) is UHR-stable and also GUHR-stable.Proof.
According to Lemma2 and Remark 2, the solution Wt of inequality (29) is formulated as(39)Wt=ϖ10+1Λμϖ20−λϖ10Nμt+1−μ1−νNμNνFWt−FW0+ψt−ψ0+λμNμ∫0tWsds+μ+ν−2μνNμNν∫0tFWs+ψsds+μνNμNν∫0tt−sFWs+ψsds,t∈0,Tϖ1t,t∈−η,0,
Sinceφ≥0 is nondecreasing, one derives from (35) and (39) that(40)Wt−u∗t≤1Λ1−μ1−νNμNνFWt−Fu∗t+ψt+ψ0,+λμNμ∫0tWs−u∗sds+μ+ν−2μνNμNν∫0tFWs−Fu∗s+ψsds,+μνNμNν∫0tt−sFWs−Fu∗s+ψsds,≤1Λ1−μ1−νNμNνℋWt−u∗t+φtε+φtε,+λμNμ∫0tWs−u∗sds+μ+ν−2μνNμNν∫0tℋWs−u∗s+φtεds,+μνNμNν∫0tt−sℋWs−u∗s+φtεds,≤ΓWt−u∗t+Υε,t∈0,T,and(41)Wt−u∗t=ϖ1t−ϖ1t≡0,t∈−η,0,
By (40) and (41), one has(42)Wt−u∗t≤Υ1−Γφtε,t∈0,T.
Thereby, from (42) and Definitions 5 and 6, one concludes that system (1) is UHR-stable and also GUHR-stable. The proof is completed.


## 5. Illustrative Examples

In this section, we shall apply our findings to solve the existence and stability of solutions for two specific systems.


### 5.1. Theoretical Analysis

Example 1.
In (1), take T=1, μ=0.7, ν=0.4, λ=1/2, ft,u,w=t2+sin2u+sintw/1+w2, w=gt,z=costarc tanz, z=ut−ξt, ξt=2+sin2t/6, ϖ1t=2cost, ϖ2t=t, Nx=1−x+x/Γx,0<x≤1. A simple calculation gives η=0.5, ℳ1t=1+t2, ℳ2t=sint, ℳ3t=π/2cost, N0=N1=1 and(43)Λ=1−λ1−μNμ≈0.8213>0,ϑ=λTNμ−λ1−μ≈0.7254<1.
Thus, the conditionsH1–H4 all fulfil. It follows from Theorem 1 that the system of Example 1 exists at least a solution u∗t∈C−0.5,1,ℝ.Example 2.
In (1), take T=1, μ=0.6, ν=0.8, λ=1/5, ft,u,w=cost+t2/20log1+u2+w2, w=gt,z=sint+2+sin3t/30arc tanz, z=ut−ξt, ξt=2+sin2t/6, ϖ1t=2cost, ϖ2t=t, Nx=1−x+x/Γx,0<x≤1. By a direct calculation, one has η=0.5, N0=N1=1, ℋ1t=ℋ2t=t2/20, ℋ3t=2+sin3t/30, ℋ1T=ℋ2T=1/20, ℋ3T=1/10 and(44)Λ=1−λ1−μNμ≈0.9004>0,ℋ=ℋ1T+ℋ2T⋅ℋ3T=11200,Γ=1Λ1−μ1−νℋNμNν+λμTNμ+μ+ν−2μνℋTNμNν+μνℋT22NμNν≈0.2312<1.
Thus, we verify that the conditionsH1, H2, H5, and H6 hold. From Theorem 2 and Theorems 3 and 4, we conclude that the system of Example 2 has a unique solution u∗t∈C−0.5,1,ℝ, which is UH-, GUH-, UHR-, and GUHR-stable.


### 5.2. Numerical Simulation

Letvt=D0+μCF−λut, then system (1) is transformed into a system of equations as follows:(45)D0+μCFut=λut+vt,t∈0,T,D0+νCFvt=ft,ut,Cut,t∈0,T,Cut=gt,ut−ξt,t∈0,T,ut=ϖ1t,vt=ϖ2t−λϖ1t,t∈−η,0.The simulations of solution of Examples1 and 2 are shown as Figures 1 and 2, respectively. ut is the solution of Langevin system in the figures. The simulation of UH-stability of Example 2 is shown as Figure 3. It follows from the images of ε=0.1 and ε=0.01 that, when 0<ε≪1, the solution curve of the inequality (28) almost coincides with that of system (45), which shows that system (45) is UH-stable.Figure 1
The solution of Example1.Figure 2
The solution of Example2.Figure 3
The UH-stability of Example2 with ε=0.1,0.01.


## 5.1. Theoretical Analysis

Example 1.
In (1), take T=1, μ=0.7, ν=0.4, λ=1/2, ft,u,w=t2+sin2u+sintw/1+w2, w=gt,z=costarc tanz, z=ut−ξt, ξt=2+sin2t/6, ϖ1t=2cost, ϖ2t=t, Nx=1−x+x/Γx,0<x≤1. A simple calculation gives η=0.5, ℳ1t=1+t2, ℳ2t=sint, ℳ3t=π/2cost, N0=N1=1 and(43)Λ=1−λ1−μNμ≈0.8213>0,ϑ=λTNμ−λ1−μ≈0.7254<1.
Thus, the conditionsH1–H4 all fulfil. It follows from Theorem 1 that the system of Example 1 exists at least a solution u∗t∈C−0.5,1,ℝ.Example 2.
In (1), take T=1, μ=0.6, ν=0.8, λ=1/5, ft,u,w=cost+t2/20log1+u2+w2, w=gt,z=sint+2+sin3t/30arc tanz, z=ut−ξt, ξt=2+sin2t/6, ϖ1t=2cost, ϖ2t=t, Nx=1−x+x/Γx,0<x≤1. By a direct calculation, one has η=0.5, N0=N1=1, ℋ1t=ℋ2t=t2/20, ℋ3t=2+sin3t/30, ℋ1T=ℋ2T=1/20, ℋ3T=1/10 and(44)Λ=1−λ1−μNμ≈0.9004>0,ℋ=ℋ1T+ℋ2T⋅ℋ3T=11200,Γ=1Λ1−μ1−νℋNμNν+λμTNμ+μ+ν−2μνℋTNμNν+μνℋT22NμNν≈0.2312<1.
Thus, we verify that the conditionsH1, H2, H5, and H6 hold. From Theorem 2 and Theorems 3 and 4, we conclude that the system of Example 2 has a unique solution u∗t∈C−0.5,1,ℝ, which is UH-, GUH-, UHR-, and GUHR-stable.


## 5.2. Numerical Simulation

Letvt=D0+μCF−λut, then system (1) is transformed into a system of equations as follows:(45)D0+μCFut=λut+vt,t∈0,T,D0+νCFvt=ft,ut,Cut,t∈0,T,Cut=gt,ut−ξt,t∈0,T,ut=ϖ1t,vt=ϖ2t−λϖ1t,t∈−η,0.The simulations of solution of Examples1 and 2 are shown as Figures 1 and 2, respectively. ut is the solution of Langevin system in the figures. The simulation of UH-stability of Example 2 is shown as Figure 3. It follows from the images of ε=0.1 and ε=0.01 that, when 0<ε≪1, the solution curve of the inequality (28) almost coincides with that of system (45), which shows that system (45) is UH-stable.Figure 1
The solution of Example1.Figure 2
The solution of Example2.Figure 3
The UH-stability of Example2 with ε=0.1,0.01.


## 6. Summaries

It is well known that the Langevin equation is a powerful tool in describing the random motion of particles in fluid. In a particularly complex viscous liquid, the integer-order Langevin equation describing the motion of particles is no longer accurate. The manuscript focuses on the existence, uniqueness, and UH-like stability of a nonlinear fractional order Langevin equation with nonsingular exponential kernel and delay control. Theoretical analysis and numerical simulation of two examples verify the correctness and effectiveness of our main conclusions. Our findings provide mathematical theoretical support for some physical problems. Furthermore, the mathematical theories and methods used in this paper can be made use of a reference for the study of other fractional differential system. In addition, we can further study the existence, multiplicity, and stability of solutions of some impulsive nonsingular fractional Langevin equations with nonlocal or integral boundary value conditions in the future.



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