# Multipoint BVP for the Langevin Equation underφ-Hilfer Fractional Operator

Authors: Mohammed A. Almalahi; Satish K. Panchal; Fahd Jarad
Journal: Journal of Function Spaces
 (2022)

---

## Abstract

In this research paper, we consider a class of boundary value problems for a nonlinear Langevin equation involving two generalized Hilfer fractional derivatives supplemented with nonlocal integral and infinite-point boundary conditions. At first, we derive the equivalent solution to the proposed problem at hand by relying on the results and properties of the generalized fractional calculus. Next, we investigate and develop sufficient conditions for the existence and uniqueness of solutions by means of semigroups of operator approach and the Krasnoselskii fixed point theorems as well as Banach contraction principle. Moreover, by means of Gronwall’s inequality lemma and mathematical techniques, we analyze Ulam-Hyers and Ulam-Hyers-Rassias stability results. Eventually, we construct an illustrative example in order to show the applicability of key results.


---

## Body

## 1. Introduction

In recent decades, the subject of fractional calculus (FC) becomes a very significant tool to characterize memory phenomena in many branches of engineering and sciences. Some properties of solutions like the existence and uniqueness of solutions for fractional boundary value problems (FBVPs) have been widely investigated [1–5], and a broad rundown of references is given in that regard. The importance of FBVPs comes from their applicability in several fields like science and engineering. Langevin [6] devised an equation to describe the progression of physical processes in fluctuating settings in 1908, which is known as the Langevin equation. Mainardi and Pironi in 1990 [7] presented the fractional Langevin equation (FLE). Yukunthorn et al. [8] via Banach’s, Krasnoselskii’s, and Leray-Schauder’s nonlinear alternative and Leray-Schauder studied the existence and uniqueness of solution for a sequential nonlinear fractional Caputo-Langevin equation. Baghani [9] discussed the solvability of initial value problems for nonlinear Langevin equation involving two fractional orders. Fazli and Nieto [10] by means of coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces investigated the existence and uniqueness of solutions for the nonlinear Langevin equation involving two fractional orders with antiperiodic boundary conditions. Salem et al. [11] studied the existence and uniqueness of solution for fractional integrodifferential Langevin equation involving two fractional orders with three-point multiterm fractional integral boundary conditions. For further works on characteristics of solutions, such as existence, uniqueness, and stability results for fractional Langevin equations (FLE), see [12–15]. We also include some recent works on qualitative analysis of similar situations with the generalized fractional operators (see [16–20]). Samet et al. [21, 22] introduced a new concept of fixed point theorems for mappings in complete metric spaces. System stability is one of the most essential qualitative characteristics of solutions to FDEs. However, there are few results of Ulam-Hyers (UH) and generalized Ulam-Hyers (GUH) stability of solutions of FDEs in the literature. On the other hand, Guo et al. [23] studied the existence and Hyers-Ulam stability of the virtually periodic solution to the fractional differential equation with impulse and fractional Brownian motion under nonlocal conditions via the semigroups of operator approach and the Mönch fixed point method. Li et al. [24] by using Krasnoselskii’s fixed point method investigated the existence and Hyers-Ulam stability of random impulsive stochastic functional differential equations with finite delay. Shu and Shi [25] studied the mild solution of impulsive fractional evolution equations under Caputo fractional derivative.Recently, Almalahi and Panchal [26] by means of some fixed point theorems studied the qualitative properties of solution for the following problem:
(1)HDa+q,δ;φ−βuϰ=fϰ,uϰ,σ∈J≔a,b,ua=0,ub=∑i=1mκiIa+ζ,ϕuτi,τi∈a,b,where DHa+q,δ;φ is the φ-Hilfer FD of order q∈1,2 with type δ∈0,1, γ=q+2δ−qδ, β<0,m∈ℕ and Ia+ζ,φ is the φ-RL fractional integral of order ζ>0,β,κi∈ℝ,a<τ1<τ2<⋯⋯<b, and φ∈C1J is an increasing function with φ′ϰ≠0, for all ϰ∈J.The recent works regarding of the FLE can be found in [27–29]. Li et al. [27] via some fixed point techniques and degree theory obtained some existence results of Caputo-type Langevin FBVPs with infinite-point boundary conditions
(2)cD0+q1cD0+q2−βuϰ=fϰ,uϰ,ϰ∈0,1,u0=0,cD0+q2u0=0,cD0+q2u1=∑i=1mκicD0+q2uτi,where Dc0+q1,Dc0+q2 are the Caputo FDs of order q1∈0,1,q2∈1,2.Seemab et al. in [29] discussed the existence and uniqueness of solution for the following problem:
(3)cD0+q1;φcD0+q2;φ−βuϰ=fϰ,uϰ,cD0+θ,φuϰ,ϰ∈0,b,u0=0,uη=0,ub=μI0+σ,φuϰ,μ>0,where Dca+m,φ denotes the φ-Caputo FD of order to m∈q1,q2,θ,q1∈1,2,q2,θ∈0,1.Nuchpong et al. [28] by using Banach, Leray–Schauder, and Krasnoselskii fixed point theorems discussed the existence and uniqueness results of φ-Hilfer-type Langevin FBVP nonlocal integral boundary conditions
(4)HD0+q1,δ1;φHD0+q2,δ2;φ−βuϰ=fϰ,uϰ,σ∈J≔a,b,ua=0,ub=∑i=1mκiIa+ζ,ϕuτi,τi∈a,b.Motivated by the aforementioned discussions, in this research paper, we study the existence and uniqueness as well as Ulam-Hyers stability results for Langevin-type generalized FDEs with infinite-point boundary conditions of the form(5)HD0+q1,δ1;φHD0+q2,δ2;φ−βuϰ=fϰ,uϰ,I0+θ,φuϰ,HD0+q1,δ1;φuϰ,ϰ∈J≔0,b,u0=0,HD0+q2,δ2;φu0=0,ub=∑i=1∞ηiI0+σi,φuλi, where DH0+q1,δ1;φ,DH0+q2,δ2;φ are the φ-Hilfer FDs of order q1∈0,1,q2∈1,2,2<q1+q2≤3 with type δ1,δ2∈0,1 and I0+k,φ is a φ-RL fractional integral of order k∈θ,σi such that θ,σi>0,β∈ℝ,ηi∈0,1,0<λ1<λ2<⋯<1, and φ∈C1J is an increasing function with φ′ϰ≠0, for all ϰ∈J, and f:0,b×ℝ3⟶ℝ is a given function.Observe that the results that will be obtained according to the problem (5) cover results of [27–29] as follows:
(i)
Li et al. in [27] (for a=0,δ1=δ2=1, and φϰ=ϰ)(ii)
Nuchpong et al. [28] results (for θ=0,i=1,2,..,n, and 0<q1,q2≤1)(iii)
Seemab et al., in [29] (for a=0,δ1=δ2=1, and θ=0,i=1,η=μ)We anticipate that the results presented in this paper will be groundbreaking and contribute to the existence of knowledge on the Langevin equation. The results acquired in this study are applicable to a wide range by choosing suitable values of functionφ and may be used to a variety of other challenges.The arrangement of this paper is as per the following. In Section2, we will give some definitions and lemmas to demonstrate our primary outcomes. In Section 3, Krasnoselskii and Banach fixed point techniques are used to acquire the existence and uniqueness of solutions of the suggested problem (5). In Section 4, we analyze the stability results in Ulam-Hyers sense. In the last section, we introduce a numerical model to represent the fundamental results.


## 2. Auxiliary Results

In this section, to analyze our main results, we present here some important definitions and auxiliary lemmas. Assume thatJ≔0,b and CJ≔CJ,ℝ denote the Banach space of all continuous functions defined from J into ℝ with the norm u=supuϰ:ϰ∈J. Let φ∈C1J be an increasing function with φ′ϰ≠0, for all ϰ∈J.Definition 1 (see [4]).
Letq>0 and g:0,∞⟶ℝ. Then, the following representation
(6)I0+q,φgϰ=1Γq∫0ϰφ′sφϰ−φsq−1gsdsis called φ-RL fractional integral of g of order q.Definition 2 (see [31]).
Letq∈n−1,n,n∈ℕ, and the functions g,φ∈CnJ. Then, the φ-Hilfer FD of g with order q and type 0≤δ≤1 is defined by
(7)HD0+q,δ,φgϰ=I0+δn−q;φgφnI0+1−δn−q,φgϰ=I0+δn−q;φgφnI0+n−γ,φgϰ=I0+δn−q;φDa+γ;φgϰ,γ=q+δn−q,where
(8)D0+γ;φgϰ=gφnI0+1−δn−q;φgϰ,gφn=1φ′ϰddϰn.Remark 3.
In Definition2, type δ and function φ allow DH0+q,δ,φ to interpolate continuously between the Riemann-Liouville FD and the Caputo FD. More precisely, we have the following:
(i)
φ-Hilfer FD corresponds to the Riemann-Liouville FD for (δ=0,φϰ=ϰ), i.e.,(9)HD0+q,0,ϰgϰ=gϰnI0+n−q,ϰgϰ=DnI0+n−qgϰ,ϰ>0(ii)
φ-Hilfer FD corresponds to the Caputo FD for (δ=1,φϰ=ϰ), i.e.,(10)HD0+q,1,ϰgϰ=I0+n−q,ϰgϰngϰ=I0+n−qDngϰ,ϰ>0(iii)
φ-Hilfer FD corresponds to the φ-Riemann-Liouville FD for δ=0, i.e.,(11)HD0+q,0,φgϰ=gφnI0+n−q,φgϰ=1φ′ϰddϰnI0+n−q,φgϰ=DφnI0+n−q,φgϰ,ϰ>0(iv)
φ-Hilfer FD corresponds to the φ-Caputo FD for δ=1, i.e.,
HD0+q,1,φgϰ=I0+n−q,φgφngϰ=I0+n−q,φ1/φ′ϰd/dϰngϰ=I0+n−q,φDφngϰ,ϰ>0Theorem 4 (see [4]).
Letg∈CJ be a function. Then, I0+q;φg0=limϰ⟶0+I0+q;φgϰ=0.Lemma 5 (see [4, 30]).
Letq,δ>0 and η>0. Then,
(12)I0+q,φI0+δ,φgϰ=I0+q+δ,φgϰ,I0+q,φφϰ−φ0η−1=ΓηΓq+ηφϰ−φ0q+η−1,(13)HD0+q,δ,φφϰ−φ0γ−1=0,γ=q+δn−q.Lemma 6 (see [30]).
Ifg∈CnJ,q∈n−1,n, and 0≤δ≤1, then
(14)I0+q;φHD0+q,δ,φgϰ=gϰ−∑k=1nφϰ−φ0γ−kΓγ−k+1gφn−kIa+1−δn−q;φg0,HD0+q,δ,φI0+q;φgϰ=gϰ.Theorem 7 (see [31]) (Banach theorem).
LetO be a closed subset of Banach space H, and the operator Q:O⟶O be a strict contraction that means Qu−Qv≤Lu−v for some 0<L<1, u,v∈O. Then, Q has a fixed point in O.Theorem 8 (see [32]) (Krasnoselskii’s theorem).
LetO be a nonempty, closed, convex, and bounded subset from Banach space H. If there exist two operators Q,Q⋆ such that (i) for all u,v∈H, imply Qu+Q⋆v∈H, (ii) Q is compact and continuous, and (iii) Q⋆ is a contraction mapping, then there exists a function z∈O such that z=Qz+Q⋆z.


## 3. Equivalent Integral Equations for Problem (5)

In order to convert the problem (5) into a fixed point problem, we will present the following lemma with linear function hϰ.Lemma 9.
Forj=1,2, let γj=qj+jδj−qjδj, q1∈0,1,q2∈1,2,δj∈0,1,h∈CJ. Then, the function u∈CJ is a solution of the linear problem
(15)HD0+q1,δ1;φHD0+q2,δ2;φuϰ−βuϰ=hϰ,ϰ∈0,b,u0=0,HD0+q2,δ2;φu0=0,ub=∑i=1∞ηiI0+σi,φuλi,if and only if
(16)uϰ=φϰ−φ0γ2−1ΘΓγ2∑i=1∞ηiI0+q2+σi,φβuλi+I0+q1+q2+σi,φhλi−I0+q2,φβub−I0+q2+q1,φhb+I0+q2,φβuϰ+I0+q2+q1,φhϰ,where
(17)Θ=φb−φ0γ2−1Γγ2−∑i=1∞ηiφλi−φ0γ2+σi−1Γγ2+σi≠0.Proof.
Letu∈CJ be a solution of the problem (15). Take I0+q1,φ taking into consideration that q1∈0,1 on both sides of the following equation:
(18)HD0+q1,δ1;φHD0+q2,δ2;φuϰ−βuϰ=hϰ.
Using Lemma6, we have
(19)HD0+q2,δ2;φuϰ=βuϰ+c0Γγ1φϰ−φ0γ1−1+I0+q1,φhϰ,where c0 is an arbitrary constant. By conditions u0=0 and DH0+q2,δ2;φu0=0, we obtain c0=0 and hence, (19) reduces to
(20)HD0+q2,δ2;φuϰ=βuϰ+I0+q1,φhϰ.
Taking againI0+q2,φ taking into consideration that q2∈1,2 on both sides of (20) and using Lemmas 6 and 5, we have
(21)uϰ=c1Γγ2φϰ−φ0γ2−1+c2Γγ2−1φϰ−φ0γ2−2+I0+q2,φβuϰ+I0+q2+q1,φhϰ.
According to the conditionu0=0, we obtain c2=0 and hence, (21) reduces to
(22)uϰ=c1Γγ2φϰ−φ0γ2−1+I0+q2,φβuϰ+I0+q2+q1,φhϰ.
By replacingϰ with b in equation (22), we get
(23)ub=c1Γγ2φb−φ0γ2−1+I0+q2,φβub+I0+q2+q1,φhb.
Replacing againϰ with λi in equation (22) with multiplication by ∑i=1∞ηiI0+σi,φ with the use of semigroup property defined in Lemma 5, we get
(24)∑i=1∞ηiI0+σi,φuλi=∑i=1∞ηiI0+σi,φc1Γγ2φλi−φ0γ2−1+∑i=1∞ηiI0+σi,φI0+q2,φβuλi+∑i=1∞ηiI0+σi,φI0+q2+q1,φhλi.
Now, by equations (23) and (24) and second condition ub=∑i=1∞ηiI0+σi,φuλi, we get
(25)c1Γγ2φb−φ0γ2−1+I0+q2,φβub+I0+q2+q1,φhb=∑i=1∞ηiI0+σi,φc1Γγ2φλi−φ0γ2−1+∑i=1∞ηiI0+q2+σi,φβuλi+∑i=1∞ηiI0+q2+q1+σi,φhλi.
Hence,(26)c1=1Θ∑i=1∞ηiI0+q2+σi,φβuλi+I0+q2+q1+σi,φhλi−I0+q2,φβub−I0+q2+q1,φhb.
Puttingc1 into (22), we get (16). Conversely, we assume that the solution u satisfies (16). Then, one can get u0=0 and DH0+q2,δ2;φu0=0. Furthermore, applying I0+σi,φ on both sides of (16) replacing ϰ by λi and multiplying by ∑i=1∞ηi, we get
(27)∑i=1∞ηiI0+σi,φuλi=∑i=1∞ηiφλi−φ0γ2+σi−1ΘΓγ2+σi∑i=1∞ηiI0+q2+σi,φβuλi+∑i=1∞ηiI0+q2+q1+σi,φhλi−I0+q2,φβub−I0+q2+q1,φhb+∑i=1∞ηiI0+q2+σi,φβuλi+∑i=1∞ηiI0+q2+q1+σi,φhλi=1Θφb−φ0γ2−1Γγ2−Θ∑i=1∞ηiI0+q2+σi,φβuλi+∑i=1∞ηiI0+q2+q1+σi,φhλi−I0+q2,φβub−I0+q2+q1,φhb+∑i=1∞ηiI0+q2+σi,φβuλi+∑i=1∞ηiI0+q2+q1+σi,φhλi=ub.
Thus, all conditions are satisfied. Next, applyingDH0+q2,δ2;φ on both sides of (16), we have
(28)HD0+q2,δ2;φuϰ=HD0+q2,δ2;φφϰ−φ0γ2−1ΘΓγ2∑i=1∞ηiI0+q2+σi,φβuλi+I0+q2+q1+σi,φhλi−I0+q2,φβub−I0+q2+q1,φhb+HD0+q2,δ2;φI0+q2,φβuϰ+HD0+q2,δ2;φI0+q2+q1,φhϰ.
Using Lemmas5 and 6, we get
(29)HD0+q2,δ2;φuϰ−βuϰ=I0+q1,φhϰ.
ApplyingDH0+q1,δ1;φ on both sides of equation (29), we have
(30)HD0+q1,δ1;φHD0+q2,δ2;φuϰ−βuϰ=HD0+q1,δ1;φI0+q1,φhϰ.
Using Lemma5, we get
(31)HD0+q1,δ1;φHD0+q2,δ2;φuϰ−βuϰ=hϰ.
The proof is completed.Lemma 10.
Forj=1,2, let γj=qj+jδj−qjδj, q1∈0,1,q2∈1,2,δj∈0,1, and f:J×ℝ3⟶ℝ be a continuous function. Then, the solution of the problem (5) is given by
(32)uϰ=φϰ−φ0γ2−1ΘΓγ2∑i=1∞ηiI0+q2+σi,φβuλi+∑i=1∞ηiI0+q2+q1+σi,φfλi,uλi,I0+θ,φuλi,HD0+q1,δ1;φuλi−I0+q2,φβub−I0+q2+q1,φfb,ub,I0+θ,φub,HD0+q1,δ1;φub+I0+q2,φβuϰ+I0+q2+q1,φfϰ,uϰ,I0+θ,φuϰ,HD0+q1,δ1;φuϰ.
In order to simplify our analysis, we will use the following notation:(33)ΠAB=φA−φ0BΓB+1,where A∈λi,b and B∈q2+σi,q2,q2+q1+σi,q2+q1.


## 4. Existence and Uniqueness Solution

In this section, we will discuss the existence and uniqueness of solutions forφ-Hilfer FDE (5) by applying Theorems 8 and 7. To demonstrate our main results, the following assumptions must be fulfilled.(H1) f:J×ℝ3⟶ℝ is a continuous and there exists a constant number μf>0 such that
(34)fϰ,u,v,z−fσ,u^,v^,z^≤μfu−u^+v−v^+z−z^,ϰ∈J,u,u^,v,v^,z,z^∈(H2) f:J×ℝ3⟶ℝ is continuous and there exists ωf∈CJ such that
(35)fϰ,uϰ,vϰ,zϰ≤ωfϰ,ϰ,u,v,z∈J×ℝ3,with supϰ∈Jωfϰ=ωf∗.Theorem 11.
Assume that (H1) and (H2) hold. Then, the problem (5) has at least one solution, provided that M+V<1, where
(36)M=φb−φ0γ2−1ΘΓγ2∑i=1∞ηiβΠλiq2+σi+βΠbq2,V=φb−φ0γ2−1ΘΓγ2μf1−μf1+1Γθ+1∑i=1∞ηiΠλiq2+q1+σi+Πbq2+q1.Proof.
Consider the continuous operatorQ:CJ⟶CJ, which is defined by
(37)Quϰ=φϰ−φ0γ2−1ΘΓγ2∑i=1∞ηiI0+q2+σi,φβuλi+∑i=1∞ηiI0+q2+q1+σi,φfλi,uλi,I0+θ,φuλi,HD0+q1,δ1;φuλi−I0+q2,φβub−I0+q2+q1,φfb,ub,I0+θ,φub,HD0+q1,δ1;φub+I0+q2,φβuϰ+I0+q2+q1,φfϰ,uϰ,I0+θ,φuϰ,HD0+q1,δ1;φuϰ.
Clearly, the fixed points of the operatorQ defined by (37) is a solution of the problem (5). Now, we will prove that the operator Q has a fixed point by using Theorem 8. Let Br be a closed ball defined by
(38)Br=u∈CJ:u≤r,with
(39)r≥φb−φ0γ2−1/ΘΓγ2∑i=1∞ηiΠλiq2+q1+σi+φb−φ0γ2−1/ΘΓγ2Πbq2+q1+Πbq2+q1ωf∗1−φb−φ0γ2−1/ΘΓγ2∑i=1∞ηiΠλiq2+σi+φb−φ0γ2−1/ΘΓγ2Πbq2+Πbq2β.
LetQ1,Q2∈Br be two operators such that Q1+Q2=Q, where
(40)Q1uϰ=I0+q2,φβuϰ+I0+q2+q1,φfϰ,uϰ,I0+θ,φuϰ,HD0+q1,δ1;φuϰ,Q2uϰ=φϰ−φ0γ2−1ΘΓγ2∑i=1∞ηiI0+q2+σi,φβuλi+∑i=1∞ηiI0+q2+q1+σi,φfλi,uλi,I0+θ,φuλi,HD0+q1,δ1;φuλi−I0+q2,φβub−I0+q2+q1,φfb,ub,I0+θ,φub,HD0+q1,δ1;φub.
In order to achieve conditions of Theorem8, the proof is divided into the following steps.
Step 1.Q∈Br for all u∈Br. For any u∈Br,ϰ∈J, we have
(41)Q1u=supϰ∈JI0+q2,φβuϰ+I0+q2+q1,φfϰ,uϰ,I0+θ,φuϰ,HD0+q1,δ1;φuϰ≤supϰ∈JI0+q2,φβuϰ+I0+q2+q1,φfϰ,uϰ,I0+θ,φuϰ,HD0+q1,δ1;φuϰ≤βΠbq2u+Πbq2+q1ωf∗,Q2u≤φϰ−φ0γ2−1ΘΓγ2∑i=1∞ηiβuΠλiq2+σi+ωf∗Πλiq2+q1+σi+βuΠbq2+ωf∗Πbq2+q1.
Due to the fact thatφϰ is an increasing function, we have φϰ−φ0γ2−1≤φb−φ0γ2−1 and hence,
(42)Q2u≤φb−φ0γ2−1ΘΓγ2∑i=1∞ηiβuΠλiq2+σi+ωf∗Πλiq2+q1+σi+βuΠbq2+ωf∗Πbq2+q1.
Thus,(43)Qu≤Q1u+Q2u≤φb−φ0γ2−1ΘΓγ2∑i=1∞ηiβuΠλiq2+σi+φb−φ0γ2−1ΘΓγ2∑i=1∞ηiωf∗Πλiq2+q1+σi+φb−φ0γ2−1ΘΓγ2βuΠbq2+φb−φ0γ2−1ΘΓγ2ωf∗Πbq2+q1+βΠbq2u+Πbq2+q1ωf∗≤φb−φ0γ2−1ΘΓγ2∑i=1∞ηiΠλiq2+σi+φb−φ0γ2−1ΘΓγ2Πbq2+Πbq2βu+φb−φ0γ2−1ΘΓγ2∑i=1∞ηiΠλiq2+q1+σi+φb−φ0γ2−1ΘΓγ2Πbq2+q1+Πbq2+q1ωf∗≤r.
Thus,QBr⊂Br.
Step 2.Q1 is compact and continuous.
Letun be a sequence such that un⟶u in Br. Then,
(44)Q1un−Q1u≤I0+q2,φβunϰ−uϰ+I0+q2+q1,φfϰ,unϰ,I0+θ,φunϰ,HD0+q1,δ1;φunϰ−fϰ,uϰ,I0+θ,φuϰ,HD0+q1,δ1;φuϰ≤Πbq2un−u+μf1−μf1+1Γθ+1Πbq2+q1un−u⟶0asun⟶u.
That meansQ1 is continuous. Moreover, the operator Q1 is bounded on Br due to Step 1. Thus, Q1 is uniformly bounded on Br. Next, we will prove that Q1 is equicontinuous. Let ϰ1,ϰ2∈J such that ϰ1<ϰ2. Then,
(45)Q1uϰ2−Q1uϰ1=I0+q2,φβuϰ2−I0+q2,φβuϰ1+I0+q2+q1,φfϰ2,uϰ2,I0+θ,φuϰ2,HD0+q1,δ1;φuϰ2−I0+q2+q1,φfϰ1,uϰ1,I0+θ,φuϰ1,HD0+q1,δ1;φuϰ1≤βΓq2∫0ϰ1φ′sφϰ2−φsq2−1−φϰ1−φsq2−1usds+βΓq2∫ϰ1ϰ2φ′sφϰ2−φsq2−1usds+1Γq2+q1∫0ϰ1φ′sφϰ2−φsq2+q1−1−φϰ1−φsq2+q1−1fs,us,I0+θ,φus,HD0+q1,δ1;φusds+1Γq2+q1∫ϰ1ϰ2φ′sφϰ2−φsq2+q1−1fs,us,I0+θ,φus,HD0+q1,δ1;φusds≤uβΓq2+1φϰ2−φ0q2−φϰ1−φ0q2+ωf∗Γq2+q1+1φϰ2−φ0q2+q1−φϰ1−φ0q2+q1.
Forϰ2⟶ϰ1 and continuity of φ, we obtain
(46)Q1uϰ2−Q1uϰ1⟶0asϰ2⟶ϰ1.
Hence,Q1 is equicontinuous. As a result of the Arzelá-Ascoli theorem, we deduce that the operator Q1 is compact in Br. Thus, Q1 is completely continuous.
Step 3.Q2 is contraction mapping. For u,u^∈Br and ϰ∈J, we obtain
(47)Q2u−Q2u^≤φb−φ0γ2−1ΘΓγ2∑i=1∞ηiI0+q2+σi,φβuλi−u^λi+∑i=1∞ηiI0+q2+q1+σi,φfλi,uλi,I0+θ,φuλi,HD0+q1,δ1;φuλi−fλi,u^λi,I0+θ,φu^λi,HD0+q1,δ1;φu^λi+I0+q2,φβub−u^b+I0+q2+q1,φfb,ub,I0+θ,φub,HD0+q1,δ1;φub−fb,u^b,I0+θ,φu^b,HD0+q1,δ1;φu^b≤φb−φ0γ2−1ΘΓγ2∑i=1∞ηiβΠλiq2+σiu−u^+∑i=1∞ηiμf1−μf1+1Γθ+1Πλiq2+q1+σiu−u^+βΠbq2u−u^+μf1−μf1+1Γθ+1Πbq2+q1u−u^≤M+Vu−u^.
Thus, we conclude thatQ2 is a contraction.
According to the above steps and Theorem8, we infer that problem (5) has at least one solution on J.Theorem 12.
Assume that (H1) holds. If
(48)Q1≔φb−φ0γ2−1ΘΓγ2∑i=1∞ηiβΠλiq2+σi+μf1−μf1+1Γθ+1Πλiq2+q1+σi+1+φb−φ0γ2−1ΘΓγ2βΠbq2+μf1−μf1+1Γθ+1Πbq2+q1<1,then the problem (5) has a unique solution on J.Proof.
We noted that the fixed points of the operatorQ defined in (37) are a solution of problem (5). Define a closed ball set Bϖ as
(49)Bϖ=u∈CJ:u≤ϖ,with ϖ≥Q2/1−Q11−Q1, where
(50)Q2=φb−φ0γ2−1ΘΓγ2∑i=1∞ηiΠλiq2+q1+σi+1+φb−φ0γ2−1ΘΓγ2Πbq2+q1P,and P=sups∈Jfs,0,0,0.First, we show thatQBϖ⊂Bϖ. By (33) and condition (H1), we have
(51)I0+q2+σi,φβuλi=βΓq2+σi∫0λiφ′sφλi−φsq2+σi−1usds≤βφλi−φ0q2+σiΓq2+σi+1u=βΠλiq2+σiu,I0+q2+q1+σi,φfλi,uλi,I0+θ,φuλi,HD0+q1,δ1;φuλi≤1Γq2+q1+σi∫0λiφ′sφλi−φsq2+q1+σi−1×fs,us,I0+θ,φus,HD0+q1,δ1;φus−fs,0,0,0+fs,0,0,0ds≤1Γq2+q1+σi∫0λiφ′sφλi−φsq2+q1+σi−1ds×μf1−μfu+I0+θ,φu+P≤Πλiq2+q1+σiμf1−μf1+1Γθ+1u+P.Thus, foru∈Bϖ, we obtain
(52)Quϰ≤φb−φ0γ2−1ΘΓγ2∑i=1∞ηiI0+q2+σi,φβuλi+I0+q2+q1+σi,φfλi,uλi,I0+θ,φuλi,HD0+q1,δ1;φuλi+I0+q2,φβub+I0+q2+q1,φfb,ub,I0+θ,φub,HD0+q1,δ1;φub+I0+q2,φβuϰ+I0+q2+q1,φfϰ,uϰ,I0+θ,φuϰ,HD0+q1,δ1;φuϰ≤φb−φ0γ2−1ΘΓγ2∑i=1∞ηiβuΠλiq2+σi+μf1−μf1+1Γθ+1u+PΠλiq2+q1+σi+βuΠbq2+μf1−μf1+1Γθ+1u+PΠbq2+q1+βuΠbq2+μf1−μf1+1Γθ+1u+PΠbq2+q1≤φb−φ0γ2−1ΘΓγ2∑i=1∞ηiβϖΠλiq2+σi+μf1−μf1+1Γθ+1ϖ+PΠλiq2+q1+σi+1+φb−φ0γ2−1ΘΓγ2βϖΠbq2+1+φb−φ0γ2−1ΘΓγ2μf1−μf1+1Γθ+1ϖ+PΠbq2+q1≤ϖΘΓγ2∑i=1∞ηiβΠλiq2+σi+μf1−μf1+1Γθ+1Πλiq2+q1+σi+PΘΓγ2∑i=1∞ηiΠλiq2+q1+σi+1+φb−φ0γ2−1ΘΓγ2ϖβΠbq2+μf1−μf1+1Γθ+1Πbq2+q1+1+φb−φ0γ2−1ΘΓγ2PΠbq2+q1≤Q1ϖ+Q2≤ϖ.Thus,QBϖ⊂Bϖ.Next, we prove thatQ is contraction map. Indeed, for u,u^∈Bϖ and ϰ∈J, we obtain
(53)Qu−Qu^≤φb−φ0γ2−1ΘΓγ2∑i=1∞ηiI0+q2+σi,φβuλi−u^λi+∑i=1∞ηiI0+q2+q1+σi,φfλi,uλi,I0+θ,φuλi,HD0+q1,δ1;φuλi−fλi,u^λi,I0+θ,φu^λi,HD0+q1,δ1;φu^λi+I0+q2,φβub−u^b+I0+q2+q1,φfb,ub,I0+θ,φub,HD0+q1,δ1;φub−fb,u^b,I0+θ,φu^b,HD0+q1,δ1;φu^b+I0+q2,φβuϰ−u^ϰ+I0+q2+q1,φfϰ,uϰ,I0+θ,φuϰ,HD0+q1,δ1;φuϰ−fϰ,u^ϰ,I0+θ,φu^ϰ,,HD0+q1,δ1;φu^ϰ≤φb−φ0γ2−1ΘΓγ2∑i=1∞ηiβΠλiq2+σiu−u^+∑i=1∞ηiμf1−μf1+1Γθ+1Πλiq2+q1+σiu−u^+βΠbq2u−u^+μf1−μf1+1Γθ+1Πbq2+q1u−u^+Πbq2u−u^+μf1−μf1+1Γθ+1Πbq2+q1u−u^≤Q1u−u^.From (48), Q is a contraction map. Hence, in view of Theorem 7, we conclude that the problem (5) has a unique solution on J.


## 5. Stability Analysis

In 1940 [33], the problem of stability of functional equations was created by Ulam’s question regarding the stability of group homomorphisms. In the following year, Hyers [34] gave a positive interpretation of the Ulam question within the Banach spaces, and this was the first major advance and a step towards more solutions in this field. Since then, many papers have been published regarding various generalizations of the Ulam problem and Hyers theory. In 1978, Rassias [35] succeeded in extending Hyers’ theory of mappings between Banach spaces. Rassias’s result attracted many mathematicians around the world who began their investigations of the problems of stability of functional equations.In this regard, we discuss the stability results in the frame of Ulam-Hyers- (UH-) Rassias (UHR). Letϵ>0 and a continuous function αϕ:J⟶ℝ+ such that it satisfies the following inequalities:
(54)HD0+q1,δ1;φHD0+q2,δ2;φu^ϰ−βu^ϰ−fϰ,u^ϰ,I0+θ,φu^ϰ,HD0+q1,δ1;φu^ϰ≤ϵ(55)HD0+q1,δ1;φHD0+q2,δ2;φu^ϰ−βu^ϰ−fϰ,u^ϰ,I0+θ,φu^ϰ,HD0+q1,δ1;φu^ϰ≤ϵαϕϰ.Lemma 13 (see [36]).
Letq>0 and u,v be two nonnegative functions locally integrable on J. Assume that g is nonnegative and nondecreasing, and let φ∈C1J be an increasing function such that φ′t≠0 for all t∈J. If
(56)uϰ≤vϰ+gt∫0ϰφ′sφϰ−φsq−1usds,ϰ∈J,then
(57)uϰ≤vϰ+∫0t∑n=1∞gϰΓpnΓnqφ′sφϰ−φsnq−1vsds,ϰ∈J.
Ifv is a nondecreasing function on J. then, we have
(58)uϰ≤vϰEqgϰΓpφϰ−φ0q,ϰ∈J.Definition 14.
We say that the problem (5) is UH stable if for everyu^∈CJthat satisfies an inequality (54) andu∈CJ is a solution of the problem (5), there exists a constant number 0<T∈ℝ such that
(59)u^ϰ−uϰ≤Tϵ,ϰ∈J,ϵ>0.Definition 15.
We say that the problem (5) is UHR stable with respect to nondecreasing function αϕϰ if for every u^∈CJ that satisfies an inequality (55) and u∈CJ is a solution of the problem (5), there exists 0<N∈ℝ such that
(60)u^ϰ−uϰ≤Nϵαϕϰ,ϰ∈J,ϵ>0.Remark 16.
A functionu^∈CJ satisfies an inequality (54) if and only if there exist a functions z∈CJ such that
(61)zϰ≤ϵ,ϰ∈J,HD0+q1,δ1;φHD0+q2,δ2;φu^ϰ−βu^ϰ=fϰ,u^ϰ,I0+θ,φu^ϰ,HD0+q1,δ1;φu^ϰ+zϰ.Lemma 17.
Letγj=qj+jδj−qjδjj=1,2, q1∈0,1,q2∈1,2, and 0≤δj≤1. If a function u^∈CJ satisfies (54), then u^ satisfies
(62)u^ϰ−Au^−I0+q1+q2,φfϰ,u^ϰ,I0+θ,φu^ϰ,HD0+q1,δ1;φu^ϰ≤ϵP,where
(63)Au^≔φϰ−φ0γ2−1ΘΓγ2∑i=1∞ηiI0+q2+σi,φβu^λi+∑i=1∞ηiI0+q2+q1+σi,φfλi,u^λi,I0+θ,φu^λi,DH0+q1,δ1;φu^λi−I0+q2,φβu^b−I0+q2+q1,φfb,u^b,I0+θ,φu^b,DH0+q1,δ1;φu^b+I0+q2,φβu^ϰ,P≔φb−φ0γ2−1ΘΓγ2∑i=1∞ηiΠλiq2+q1+σi+φb−φ0γ2−1ΘΓγ2+1Πbq2+q1.Proof.
By Remark16, we have
(64)u^ϰ=φϰ−φ0γ2−1ΘΓγ2∑i=1∞ηiI0+q2+σi,φβu^λi+∑i=1∞ηiI0+q2+q1+σi,φfλi,u^λi,I0+θ,φu^λi,DH0+q1,δ1;φu^λi+zλi−I0+q2,φβu^b−I0+q2+q1,φfb,u^b,I0+θ,φu^b,DH0+q1,δ1;φu^b+zb+I0+q2,φβu^ϰ+I0+q2+q1,φfϰ,u^ϰ,I0+θ,φu^ϰ,DH0+q1,δ1;φu^ϰ+zϰ.
Then, we get
(65)u^ϰ−Au^−I0+q2+q1,φfϰ,u^ϰ,I0+θ,φu^ϰ,DH0+q1,δ1;φu^ϰ≤φϰ−φ0γ2−1ΘΓγ2∑i=1∞ηiI0+q2+q1+σi,φzλi+I0+q2+q1,φzb+I0+q2+q1,φzϰ.
It follows that(66)u^ϰ−Au^−I0+q2+q1,φfϰ,u^ϰ,I0+θ,φu^ϰ,DH0+q1,δ1;φu^ϰ≤εP.Theorem 18.
Let us assume thatH1 holds. Then,
(67)DH0+q1,δ1;φDH0+q2,δ2;φuϰ−βuϰ=fϰ,uϰ,I0+θ,φuϰDH0+q1,δ1;φuϰis UH stable.Proof.
Letε>0 and u^∈CJ be a function that satisfies an inequality (54) and let u∈CJ be a solution of
(68)DH0+q1,δ1;φDH0+q2,δ2;φuϰ−βuϰ=fϰ,uϰ,I0+θ,φuϰ,HD0+q1,δ1;φuϰ,ϰ∈0,b,u0=u0=0,D0+q2,δ2;φHu0=DH0+q2,δ2;φu^0=0,ub=u^b=∑i=1∞ηiI0+σi,φu^λi.
Then, by using Theorem12, we have
(69)uϰ=Au+I0+q2+q1,φfϰ,uϰ,I0+θ,φuϰ,DH0+q1,δ1;φuϰ,where
(70)Au≔φϰ−φ0γ2−1ΘΓγ2∑i=1∞ηiI0+q2+σi,φβuλi+∑i=1∞ηiI0+q2+q1+σi,φfλi,uλi,I0+θ,φuλi,DH0+q1,δ1;φuλi−I0+q2,φβub−I0+q2+q1,φfb,ub,I0+θ,φub,DH0+q1,δ1;φub+I0+q2,φβuϰ.Since(71)u0=u^0=0,DH0+q2,δ2;φu0=DH0+q2,δ2;φu^0=0,ub=u^b=∑i=1∞ηiI0+σi,φu^λi,we can easily prove that Au=Au^. Hence, according to (H1) and Lemma 17, then for each ϰ∈J,(72)u^ϰ−uϰ=u^ϰ−Au^−I0+q2+q1,φfϰ,uϰ,I0+θ,φuϰ,DH0+q1,δ1;φuϰ−I0+q2+q1,φfϰ,u^ϰ,I0+θ,φu^ϰ,DH0+q1,δ1;φu^ϰ+I0+q2+q1,φfϰ,u^ϰ,I0+θ,φu^ϰ,DH0+q1,δ1;φu^ϰ≤u^ϰ−Au^−I0+q2+q1,φfϰ,u^ϰ,I0+θ,φu^ϰ,DH0+q1,δ1;φu^ϰ+I0+q2+q1,φfϰ,u^ϰ,I0+θ,φu^ϰ,DH0+q1,δ1;φu^ϰ−fϰ,uϰ,I0+θ,φuϰ,DH0+q1,δ1;φuϰ≤Pε+I0+q2+q1,φfϰ,u^ϰ,I0+θ,φu^ϰ,DH0+q1,δ1;φu^ϰ−fϰ,uϰ,I0+θ,φuϰ,DH0+q1,δ1;φuϰ≤Pε+μf1−μf1+1Γθ+1I0+q2+q1,φu^ϰ−uϰ,Thus, by Lemma13, we get
(73)u^ϰ−uϰ≤Pε+∫0ϰ∑n=1∞μf/1−μf1+1/Γθ+1nΓnq2+q1φ′sφϰ−φsnq2+q1−1Pεds<PεEq2+q1μf1−μf1+1Γθ+1φb−φ0q2+q1.It follows that(74)u^ϰ−uϰ≤Tε.where T=PEq2+q1μf/1−μf1+1/Γθ+1φb−φ0q2+q1. Hence, the problem (5) is UH stable.Corollary 19.
Under the assumptions of Theorem18, if there exists a function ϕf∈CJ, then the problem (5) is generalized UH stable.
In the forthcoming theorem, we discuss Ulam-Hyers-Rassias stability. For that, the following hypothesis must be satisfied.
(H3) There exists an increasing function αϕ∈CJ, and there exists R>0 such that for any ϰ∈J,
(75)I0+ξ,φαϕϰ≤Rαϕϰ,where ξ∈q1+q2,q2+q1+σi.Remark 20.
A functionu^∈CJ satisfies an inequality (55) if and only if there exist a functions z∈CJ such that
(76)zϰ≤εαϕϰ,ϰ∈J,DH0+q1,δ1;φDH0+q2,δ2;φu^ϰ−βu^ϰ=fϰ,u^ϰ,I0+θ,φu^ϰ,DH0+q1,δ1;φu^ϰ+zϰ.Lemma 21.
Letγj=qj+jδj−qjδjj=1,2, q1∈0,1,q2∈1,2, and 0≤δj≤1. If a function u^∈CJ satisfies (55), then u^ satisfies
(77)u^ϰ−Au^−I0+q1+q2,φfϰ,u^ϰ,I0+θ,φu^ϰ,DH0+q1,δ1;φu^ϰ≤εP1Rαϕϰ,where
(78)Au^≔φϰ−φ0γ2−1ΘΓγ2∑i=1∞ηiI0+q2+σi,φβu^λi+∑i=1∞ηiI0+q2+q1+σi,φfλi,u^λi,I0+θ,φu^λi,DH0+q1,δ1;φu^λi−I0+q2,φβu^b−I0+q2+q1,φfb,u^b,I0+θ,φu^b,DH0+q1,δ1;φu^b+I0+q2,φβu^ϰ,P1≔φb−φ0γ2−1ΘΓγ2∑i=1∞ηi+1+1.Proof.
Indeed, by Remark20 and Theorem 12, one can easily prove that
(79)u^ϰ−Au^−I0+q1+q2,φfϰ,u^ϰ,I0+θ,φu^ϰ,DH0+q1,δ1;φu^ϰ≤εP1Rαϕϰ.Theorem 22.
Assume that (H1) and (H3) hold. Then,
(80)DH0+q1,δ1;φDH0+q2,δ2;φuϰ−βuϰ=fϰ,uϰ,I0+θ,φuϰ,DH0+q1,δ1;φuϰ,is UHR and generalized UHR stable.Proof.
By the same technique in Theorem18, one can prove that
(81)u^ϰ−uϰ≤u^ϰ−Au^−I0+q2+q1,φfϰ,u^ϰ,I0+θ,φu^ϰ,DH0+q1,δ1;φu^ϰ+I0+q2+q1,φfϰ,u^ϰ,I0+θ,φu^ϰ,DH0+q1,δ1;φu^ϰ−fϰ,uϰ,I0+θ,φuϰ,DH0+q1,δ1;φuϰ≤εP1Rαϕϰ+I0+q2+q1,φfϰ,u^ϰ,I0+θ,φu^ϰ,DH0+q1,δ1;φu^ϰ−fϰ,uϰ,I0+θ,φuϰ,DH0+q1,δ1;φuϰ≤εP1Rαϕϰ+μf1−μf1+1Γθ+1I0+q2+q1,φu^ϰ−uϰ.
Thus, by (H3) and Lemma 13, we have
(82)u^ϰ−uϰ≤εP1Rαϕϰ+εP1R∫0ϰ∑n=1∞μf/1−μf1−μf1+1/Γθ+1nΓnq2+q1φ′sφϰ−φsnq2+q1−1αϕsds≤εP1Rαϕϰ1+∑n=1∞μf1−μf1+1Γθ+1Rn.
It follows that(83)u^ϰ−uϰ≤Nεαϕϰ.where N=P1R1+∑n=1∞μf/1−μf1−μf1+1/Γθ+1Rn. Hence, the problem (5) is UHR stable as well as generalized UHR stable.


## 6. An Example

Consider the Langevin-type fractional integrodifferential equation(84)DH0+q1,δ1;φDH0+q2,δ2;φuϰ−βuϰ=fϰ,uϰ,I0+θ,φuϰ,DH0+q1,δ1;φuϰ,ϰ∈0,1,u0=0,D0+q2,δ2;φHu0=0,ub=∑i=1∞ηiI0+σi,φuλi,such that q1=1/2,δ1=1/3,q2=3/2,δ2=1/3,γ2=5/3,θ=1/2,β=1/10,b=1,λi=1/2i,ηi=2/8i, σi=2/2i+1, φϰ=ϰ, and
(85)fϰ,uϰ,I0+θ,φuϰ,DH0+q1,δ1;φuϰ=ϰ50+ϰuϰ+I0+1/5,ϰuϰ+DH0+1/2,1/3;ϰuϰ.Clearly, for eachu,u^∈CJ, we have
(86)fϰ,uϰ,I0+1/5,ϰuϰ,DH0+1/2,1/3;ϰuϰ−fϰ,u^ϰ,I0+1/5,ϰu^ϰ,DH0+1/2,1/3;ϰu^ϰ≤150uϰ−u^ϰ+I0+1/5,ϰuϰ−I0+1/5,ϰu^ϰ+DH0+1/2,1/3;ϰuϰ−DH0+1/2,1/3;ϰu^ϰ,with μf=1/50. By the given data, we can get Θ≈0.89≠0 and hence, Q1≈0.2148<1. Thus, all assumptions in Theorem 12 hold. Then, problem (84) has a unique solution. Moreover, we have
(87)fϰ,uϰ,vϰ,zϰ≤ϰ50+ϰ=ωfϰ,forallϰ,u,v∈J×ℝ2.Thus, all the assumptions in Theorem11 hold. Then, problem (84) has at least one solution. Furthermore, for ε>0, we find that
(88)DH0+q1,δ1;φDH0+q2,δ2;φu^ϰ−βu^ϰ−fϰ,u^ϰ,I0+θ,φu^ϰ,DH0+q1,δ1;φu^ϰ≤εis satisfied. Then, equation (67) is Ulam-Hyers stable with
(89)u^ϰ−uϰ≤Tε,where
(90)T=PEq2+q1μf1−μf1+1Γθ+1φb−φ0q2+q1>0,and Λf≈0.042<1. Finally, we consider αϕϰ=φϰ−φ0, for ϰ∈0,1]. Then, αϕ:0,1⟶ℝ is nondecreasing continuous function. Hence, by Lemma 5, we get
(91)I0+ξ,φαϕϰ=I0+ξ,φφϰ−φ0=I0+ξ,φφϰ−φ02−1=Γ2Γξ+2φϰ−φ0ξ+1=φϰ−φ0ξΓξ+2αϕϰ≤φ1−φ0ξΓξ+2αϕϰ=Rαϕϰ,forallϰ∈J,where R=φ1−φ0ξ/Γξ+2>0 for ξ∈q1+q2,q2+q1+σi and φϰ=ϰ. Therefore, Theorem 22 applicable. Furthermore, for ε>0 and a continuous function αϕ:J⟶ℝ+, we find that
(92)DH0+q1,δ1;φDH0+q2,δ2;φu^ϰ−βu^ϰ−fϰ,u^ϰ,I0+θ,φu^ϰ,DH0+q1,δ1;φu^ϰ≤εαϕϰis satisfied. Then, equation (80) is UHR stable with
(93)u^ϰ−uϰ≤Nεαϕϰ,where
(94)N=P1R1+∑n=1∞μf1−μf1+1Γθ+1Rn>0.


## 7. Conclusion

Because the idea of fractional operators in the context ofφ-Hilfer is innovative and important, several academics have explored and established various qualitative features of FDE solutions incorporating such operators. To extend these qualitative features, we developed and investigated sufficient conditions for the existence and uniqueness of solutions, as well as Ulam-Hyers stability results for a nonlinear fractional integrodifferential Langevin equation involving φ-Hilfer FD with respect to an increasing function, for a nonlinear fractional integrodifferential Langevin equation involving φ-Hilfer FD.Our technique was based on reducing the problem (5) to a fractional integral equation and using certain conventional Banach-type and Krasnoselskii-type fixed point theorems. Furthermore, we investigated the stability data in the Ulam-Hyers sense using mathematical analytic tools. To support the main results, an example was given.In fact, our outcomes generalize those in [26–29] and cover many results not study yet. Due to the wide recent investigations and applications of the Langevin equation, we believe that acquired results here will be important for future investigations on the theory of fractional calculus.



---

*Source: 2798514-2022-05-14.xml*