# Qualitative Analysis for Multiterm Langevin Systems with Generalized Caputo Fractional Operators of Different Orders

Authors: Saeed M. Ali; Mohammed S. Abdo
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this research work, we study two types of fractional boundary value problems for multi-term Langevin systems with generalized Caputo fractional operators of different orders. The existence and uniqueness results are acquired by applying Sadovskii’s and Banach’s fixed point theorems, whereas the guarantee of the existence of solutions is shown by Ulam–Hyer’s stability. Our reported results cover many outcomes as special cases. An example is provided to illustrate and validate our results.


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## Body

## 1. Introduction

Fractional order models are more convenient than integer-order ones as fractional derivatives give superb tools for the portrayal of memory and inherited processes. Fractional differential equations (FDEs) have been applied in numerous fields, such as engineering, physical science, chemistry, financial matters, electrodynamics, aerodynamics, and dynamical systems (see [1–8]).Some new contributions to Langevin FDEs have been investigated (see [9–13]) and the references referred to in that). There are many definitions of fractional integrals and derivatives, e.g., Riemann–Liouville type, Caputo type, Hadamard type, Hilfer type, and Erdelyi–Kober type, etc. With regard to the exciting development of local fractional calculus, it has been applied to deal with numerous different nondifferentiable problems in many applied fields (see [14–16]).The generalized Riemann–Liouville definition with respect to another function was first presented by Osler [17]. Then, Kilbas et al. [2] presented important characteristics of this operator. The Caputo version called φ-Caputo fractional derivative has been introduced by Almeida [18]. Some amazing properties and generalized Laplace transform for the same operator were introduced by Jarad and Abdeljawad [19]. This recently defined fractional operator could model more precisely the process utilizing differential kernel. In order to evolve these definitions, special kernels and some kinds of operators are selected to apply on FDEs; for more details, we refer to some recent results associated with this development (see [20–29]).In 2018, Almeida et al. [30] considered the following φ-Caputo type FDE with initial conditions:(1)CDa+α;φυρ=gρ,υρ,ρ∈a,b,υa=υa,υφka=υak,k=1,…,n−1.Ahmad and Nieto [31] considered the following Caputo-type Langevin FDE with boundary conditions:(2)CDαCDκ+λυρ=gρ,υρ,υ0=γ1,υ1=γ2.In 2020, Laadjal et al. [32] studied a Caputo-type multiterm Langevin FDE with boundary conditions of the form:(3)CDαCDκ+λυρ−∑i=1mξiCDαCDκi+λiυρ=gρ,υρ,υ0=υφ′0=υ1=0.In this work, we study two classes of generalized Caputo Langevin equations with two various fractional orders. This work is inspired by the recent works of Laadjal et al. [32] and Almeida et al. [30]. Precisely, we consider the following Langevin-type fractional differential problems (FDPs):(4)CDα;φCDκ;φ+μυρ=gρ,υρ,υ0=υφ′0=υ1=0,(5)CDα;φCDκ;φ+μυρ−∑i=1mξiCDα;φCDκi;φ+μiυρ=gρ,υρ,υ0=υφ′0=υ1=0,where ρ∈℧=0,1, 0<κ1<…<κm≤1, 1<κ≤2,0<α≤1, 1≤κ−κi<2, i=1,2,…,m, m∈ℕ, ξi,μi∈ℝ, the symbol CDθ;φ denotes the generalized fractional derivative in the Caputo sense of order θ∈α,κ,κi, and g:℧×ℝ⟶ℝ is a given continuous nonlinear function.Here, we investigate the existence and uniqueness of solutions for FDPs (1.4) and (1.5) involving generalized Caputo fractional derivatives of various orders. Moreover, the guarantee of the existence of solutions is proved by UH stability. The studied results expand and generalize many results by selecting special cases of theφ function.The paper is organized as follows: In Section2, we give some definitions and lemmas that are used in the research paper. Section 3 derives equivalent fractional integral equations to the linear variants of Langevin FDPs (1.4) and (1.5). Section 4 deals with the qualitative analysis of proposed problems. In Section 5, we give an example to substantiate the main outcomes.


## 2. Preliminaries and Lemmas

We are beginning this portion by endowment with some essential definitions and results required for forthcoming analysis.We consider the Banach spaceC℧,ℝ with the norm υ∞=maxυρ,ρ∈℧.Letυ:℧⟶ℝ be an integrable function and φ∈Cn℧,ℝ an increasing function such that φ′ρ≠0, for any ρ∈℧.Definition 1 (see [2]).
Theφ-Riemann–Liouville fractional integral of a function υ of order θ is described by(6)ℑa+θ;φυρ=1Γθ∫aρφ′ςφρ−φςθ−1υςdς.Definition 2 (see [2]).
Theφ-Riemann–Liouville fractional derivative of a function υ of order θ is described by(7)Da+θ;φυρ=1φ′ρddρnℑa+n−θ;φυρ,where n=α+1, n∈ℕ.Definition 3 (see [18]).
Theφ-Caputo fractional derivative of a function υ∈Cn℧,ℝ of order θ is described by(8)CDa+θ;φυρ=ℑa+n−θ;φυφnρ,where υφnρ  =  1/φ′ρd/dρnυρ and n=α+1, n∈ℕ.Lemma 1 (see [2, 18]).
Letr1,r2>0. Then,(1)
ℑa+r1;φφρ−φar2−1ρ=Γr2/Γr1+r2φρ−φar1+r2−1(2)
CDa+r1;φφρ−φar2−1ρ=Γr2/Γr2−r1φρ−φar1+r2−1(3)
CDa+r1;φφρ−φakρ=0, for r1>k∈ℕLemma 2 (see [18]).
Ifυ∈Cn℧,ℝ and r1∈n−1,n, then(1)
ℑa+r1;φCDa+r1;φυρ=υρ−∑k=0n−1υφna+/k!φρ−φak.
In particular,
ifr1∈0,1, we have(9)ℑa+r1;φCDa+r1;φυρ=υρ−υa.
Ifr1∈1,2, we have(10)ℑa+r1;φCDa+r1;φυρ=υρ−υa−υφ′aφρ−φa.
Moreover, ifυ∈C℧,ℝ, then(11)CDa+r1;φℑa+r1;φυρ=υρ.Lemma 3 (see [18]).
Let0<θ<1. Then, the unique solution of the following linear FDP is as follows:(12)CD0+θ;φυρ=gρ,ρ∈0,ρ,υ0=υ0,
is obtained as(13)υρ=υ0+1Γθ∫0ρφ′ςφρ−φςθ−1gςdς.Lemma 4.
Let1<θ<2. Then, the unique solution of the following linear FDP is as follows:(14)CD0+θ;φυρ=gρ,ρ∈0,ρ,υ0=υ0,υφ′0=υ1,
is obtained as(15)υρ=υ0+φρ−φ0υ1−ℑ0+θ−1;φgρ+ℑ0+θ;φgρ.Proof.
In view of Lemma2, we have(16)υρ=c0+c1φρ−φ0+1Γθ∫0ρφ′ςφρ−φςθ−1gςdς,where c0,c1∈ℝ. By conditions of (14), we get c0=υ0 and(17)υφ′ρ=υ′ρφ′ρ=c1+1Γθ−1∫0ρφ′ςφρ−φςθ−2gςdς,
which implies(18)c1=υ1−1Γθ−1∫0ρφ′ςφρ−φςθ−2gςdς.
Substituting the value ofc0 and c1 in (16), we get(19)υρ=υ0+φρ−φ0υ1−1Γθ−1∫0ρφ′ςφρ−φςθ−2gςdς+1Γθ∫0ρφ′ςφρ−φςθ−1gςdς,
which is identical to (15).Theorem 1 (see [33]).
LetX is a Banach space. The map P+Q is a λ-set contraction with 0≤λ<1, and thus also condensing, if (i) P,Q:D⊆X⟶X are operators on X; (ii) P is λ contractive; (iii) Q is compact.
To apply Sadovskii’s and Banach’s fixed point theorems, we will suffice herewith reference to [34, 35].


## 3. The Linear Variant of FDPs (1.4) and (1.5)

This section deals with the linear variant of FDPs (1.4) and (1.5). For simpleness, we denoteCD0+α;φ and ℑ0+α;φ by CDα;φ and ℑα;φ, respectively.Lemma 5.
Let1<κ≤2,0<α≤1, and f∈C℧,ℝ. Then, υ is a solution of the following linear Langevin-type FDP:(20)CDα;φCDκ;φ+μυρ=fρ,υ0=υφ′0=υ1=0,
if and only ifυ satisfies(21)υρ=1Γα+κ∫0ρφ′ςφρ−φςα+κ−1fςdς−μΓκ∫0ρφ′ςφρ−φςκ−1υςdς+XφκρXφκ1μΓκ∫01φ′ςφ1−φςκ−1υςdς−1Γα+κ∫01φ′ςφ1−φςα+κ−1fςdς,where Xφκρ=φρ−φ0κ.Proof.
Applying the operatorℑα;φ on both sides of the first (20) and using Lemma 2, we get(22)CDκ;φ+μυρ=ℑα;φfρ+c0,where c0∈ℝ.
Next, applying the operatorℑκ;φ on both sides of (22) and using Lemma 2, again, we obtain(23)υρ=c1+c2Xφ1ρ+ℑα+κ;φfρ−μℑκ;φυρ+c0Γκ+1Xφκρ,where c1,c2∈ℝ. In view of (23), we have(24)υφ′ρ=υ′ρφ′ρ=c2Xφ1ρ+ℑα+κ−1;φfρ−μℑκ−1;φυρ+c0ΓκXφκ−1ρ,where we used the fact that(25)ℑθ;φfρφ′=1φ′ρddρℑθ;φfρ=1φ′ρddρ1Γθ∫0ρXφθ−1ρ,ςfςdς=1Γθ−1∫0ρXφθ−2ρ,ςfςdς=ℑθ−1;φfρ,where Xφθρ,ς=φ′ςφρ−φςθ. Using the initial conditions of (23), we find that c1=c2=0 and(26)c0=Γκ+1Xφκ1μℑκ;φυ1−ℑα+κ;φf1.
Substituting the values ofc0,c1, and c2 in (23), we obtain the solution given by (21), where(27)ℑr;φhη=1Γr∫0ρφ′ςφη−φςr−1hςdς,r∈κ,α+κ,h∈f,υ,η∈ρ,1.
The converse follows by direct calculation. Hence, the proof is achieved.Lemma 6.
Let1<κ≤2,0<α≤1, 1≤κ−κi<2, and f∈C℧,ℝ. Then, υ is a solution of the following linear Langevin-type multiterm FDP:(28)CDα;φCDκ;φ+μυρ−∑i=1mξiCDα;φCDκi;φ+μiυρ=fρ,υ0=υφ′0=υ1=0,
if and only ifυ satisfies(29)υρ=∑i=1mξiΓκ−κi∫0ρφ′ςφρ−φςκ−κi−1υςdς+1Γα+κ∫0ρφ′ςφρ−φςα+κ−1fςdς−σΓκ∫0ρφ′ςφρ−φςκ−1υςdς+XφκρXφκ1σΓκ∫01φ′ςφ1−φςκ−1υςdς−∑i=1mξiΓκ−κi∫01φ′ςφ1−φςκ−κi−1υςdς−1Γα+κ∫01φ′ςφ1−φςα+κ−1fςdς,where σ=μ−∑i=1mξiμi.Proof.
Applying the operatorℑα;φ on both sides of the first (28) and using Lemma 2, we get(30)ℑα;φCDα;φCDκ;φ+μυρ−∑i=1mξiℑα;φCDα;φCDκi;φ+μiυρ=ℑα;φfρ,CDκ;φ+μυρ−∑i=1mξiCDκi;φ+μiυρ=ℑα;φfρ−συρ+c0,where c0∈ℝ.
Next, applying the operatorℑκ;φ on both sides of (30) and using Lemma 2, again, we obtain(31)υρ=c1+c2φρ−φ0+∑i=1mξiℑκ−κi;φυρ−υ0+ℑα+κ;φfρ−σℑκ;φυρ+c0Γκ+1Xφκρ,where c1,c2∈ℝ. In view of (31), we have(32)υφ′ρ=υ′ρφ′ρ=c2Xφ1ρ+∑i=1mξiℑκ−κi−1;φυρ−υ0+ℑα+κ−1;φfρ−σℑκ−1;φυρ+c0ΓκXφκ−1ρ.
Using the conditionυφ′0=0 in (32) and υ0=0 and υ1=0 in (31), we find that c1=c2=0 and(33)0=∑i=1mξiℑκ−κi;φυ1+ℑα+κ;φf1−σℑκ;φυ1+c0Γκ+1Xφκ1,
which implies(34)c0=Γκ+1Xφκ1σℑκ;φυ1−∑i=1mξiℑκ−κi;φυ1−ℑα+κ;φf1.
Substituting the values ofc0,c1, and c2 in (31), we obtain the solution given by (29). The converse follows by direct calculation. Hence, the proof is achieved.


## 4. Qualitative Analysis

### 4.1. Existence and Uniqueness Results

This subsection proves the existence and uniqueness results for FDPs (4) and (5) by applying Sadovskii’s theorem [34] and Banach’s theorem [35].Theorem 2 (existence).
Letg:℧×ℝ⟶ℝ is a continuous function. We assume that
(H1): there exists a functionp∈C℧,ℝ+ such that gρ,x≤pρ, for ρ,x∈℧×ℝ.
(H2):γ1:=2∑i=1mξiXφκ−κi1/Γκ−κi+1+σXφκ1/Γκ+1<1.
Then, the FDP (1.5) has at least one solution on℧.Proof.
LetEr be a closed bounded and convex subset of C℧,ℝ, where r is a fixed constant. By virtue of Lemma 6, we define an operator f:C℧,ℝ⟶C℧,ℝ as follows:(35)fυρ=∑i=1mξiℑκ−κi;φυρ+ℑα+κ;φgρ,υρ−σℑκ;φυρ+XφκρXφκ1σℑκ;φυ1−∑i=1mξiℑκ−κi;φυ1−ℑα+κ;φg1,υ1,
forρ∈℧. Let us define two operators F1,F2:C℧,ℝ⟶C℧,ℝ by(36)f1υρ=∑i=1mξiℑκ−κi;φυρ−σℑκ;φυρ+XφκρXφκ1σℑκ;φυ1−∑i=1mξiℑκ−κi;φυ1,F2υρ=ℑα+κ;φgρ,υρ−XφκρXφκ1ℑα+κ;φg1,υ1.
We observe thatFυρ=F1υρ+F2υρ, ρ∈℧.
In order to show thatF1+F2 has a fixed point, we prove that F1 and F2 satisfy the hypotheses of Sadovskii’s theorem. This will be provided in several steps:Step 1.
FEr⊂Er.
Let us selectr≥γ2/1−γ1, where γ2:=2pXφα+κ1/Γα+κ+1. For any υ∈Er, we have(37)Fυ≤supρ∈℧∑i=1mξiℑκ−κi;φυρ+σℑκ;φυρ+XφκρXφκ1σℑκ;φυ1+∑i=1mξiℑκ−κi;φυ1+ℑα+κ;φgρ,υρ+XφκρXφκ1ℑα+κ;φg1,υ1≤2υ∑i=1mξiφ1−φ0κ−κiΓκ−κi+1+σφ1−φ0κΓκ+1+2pφ1−φ0α+κΓα+κ+1≤2r∑i=1mξiXφκ−κi1Γκ−κi+1+σXφκ1Γκ+1+2pXφα+κ1Γα+κ+1=rγ1+γ2≤rγ1+r1−γ1=r,
which implies thatF1+F2Er⊂Er.Step 2.
F2 is compact. We note that F2 is uniformly bounded from Step 1. Let ρ1,ρ2∈℧ with ρ1≤ρ2 and υ∈Er. Then, we obtain(38)F2υρ2−F2υρ1=1Γα+κ∫0ρ2Xφα+κ−1ρ2,sgs,υsds−Xφκρ2Xφκ11Γα+κ∫01Xφα+κ−11,sgs,υsds−1Γα+κ∫0ρ1Xφα+κ−1ρ1,sgs,υsds+Xφκρ1Xφκ11Γα+κ∫01Xφα+κ−11,sgs,υsds≤Xφκρ1−Xφκρ2Xφκ11Γα+κ∫01Xφα+κ−11,sgs,υsds+1Γα+κ∫0ρ1Xφα+κ−1ρ1,s−Xφα+κ−1ρ2,sgs,υsds+1Γα+κ∫ρ1ρ2Xφα+κ−1ρ2,sgs,υsds≤φρ1−φ0κ−φρ2−φ0κΓα+κ+1φ1−φ0α+κp+2pΓα+κ+1φρ2−φρ1α+κ≤2pΓα+κ+1φρ2−φρ1α+κ.
From the continuity ofφ, F2υρ2−F2υρ1⟶0 as ρ2⟶ρ1. Thus, F2 is equicontinuous. So, by the Arzela–Ascoli theorem, F2Er is a relatively compact set.Step 3.
F1 is γ contractive. Let υ1,υ2∈Er. Then, we have(39)F1υ1−F1υ2=supρ∈℧|∑i=1mξiℑκ−κi;φυ1ρ−υ1ρ−σℑκ;φυ1ρ−υ1ρ+XφκρXφκ1σℑκ;φυ11−υ11−∑i=1mξiℑκ−κi;φυ11−υ11|≤∑i=1mξiℑκ−κi;φs1υ1−υ2+σℑκ;φs1υ1−υ2+σℑκ;φs1υ1−υ2+∑i=1mξiℑκ−κi;φs1υ1−υ2≤2∑i=1mξiXφκ−κi1Γκ−κi+1+σXφκ1Γκ+1υ1−υ2=γ1υ1−υ2,
which isγ contractive since γ1<1.Step 4.
F is condensing.
Due to the fact thatF1 is continuous, a γ is a contraction and F2 is compact, it follows from Lemma 1 that F:Er⟶Er with F=F1+F2 is a condensing on Er. From the previous arguments, we conclude through Sadovskii’s theorem that F has a fixed point. As a result, FDP (5) has a solution on ℧.
The second result is based on Banach’s fixed point theorem.Theorem 3 (uniqueness).
Letg:℧×ℝ⟶ℝ is a continuous function, and there exists Lg>0 such that(40)gρ,x−gρ,x∗≤Lgx−x∗,ρ∈℧,x,x∗∈ℝ.
If(41)σ≔γ1+2Xφα+κ1Γα+κ+1Lg<1,
then the FDP (5) has a unique solution on ℧, where γ1 is defined in Theorem 2.Proof.
We apply Banach’s theorem to prove thatF defined by (35) has a fixed point. For this end, we show that F is a contraction. Let υ,υ∗∈C℧,ℝ and ρ∈℧. Then,(42)Fυρ−Fυ∗ρ≤∑i=1mξiℑκ−κi;φυρ−υ∗ρ+ℑα+κ;φgρ,υρ−gρ,υ∗ρ+σℑκ;φυρ−υ∗ρ+XφκρXφκ1σℑκ;φυ1−υ∗1+∑i=1mξiℑκ−κi;φυ1−υ∗1+XφκρXφκ1ℑα+κ;φg1,υ1−g1,υ∗1≤2∑i=1mξiXφκ−κi1Γκ−κi+1+σXφκ1Γκ+1υ−υ∗+2Xφα+κ1Γα+κ+1Lgυ−υ∗=γ1+2Xφα+κ1Γα+κ+1Lgυ−υ∗,
which implies(43)Fυ−Fυ∗≤συ−υ∗.
Asσ<1, it follows that F is a contraction. As a result of Banach’s theorem, there is a unique fixed point υ∈C℧,ℝ such that Fυ=υ. Therefore, the FDP (5) has a unique solution on ℧.Corollary 1 (existence).
Letg:℧×ℝ⟶ℝ is a continuous function, and there exists a function p∈C℧,ℝ+ such that gρ,x≤pρ, for ρ,x∈℧×ℝ. If γ3<1, where γ3=2μXφκ1/Γκ+1,
then the FDP (4) has at least one solution on ℧.Corollary 2 (uniqueness).
Letg:℧×ℝ⟶ℝ is a continuous function, and there exists Mg>0 such that(44)gρ,x−gρ,x∗≤Mgx−x∗,ρ∈℧,x,x∗∈ℝ.
IfΛ2<1, where Λ2=γ3+2Xφα+κ1/Γα+κ+1Lg, then the FDP (4) has a unique solution on ℧, where γ3 is defined in Corollary 1.Remark 1.
Proofs ofCorollaries 1 and 2 can be obtained easily using the same arguments as in the previous theorems. Thus, we omit the details.Remark 2.
Ifφρ=ρ, then the obtained results on FDP (1.5) include the results of Laadjal et al. [32].Remark 3.
The Langevin FDPs (4) and (5) are new to the literature on Langevin-type multiterm FDPs and include many problems, as a special case, for φρ=logρ and φρ=ρρ; our problems are reduced to Hadamard-type problems and Katugampola problems, respectively.


### 4.2. UH Stability Analysis

In this subsection, we discuss the UH stability of the considered problem.Definition 4.
FDP (5) is UH stable if there exists a constant Υf>0 such that for each ε>0 and every solution ω∈C℧,ℝ of the inequalities(45)CDα;φCDκ;φ+μωρ−∑i=1mξiCDα;φCDκi;φ+μiωρ−gρ,ωρ≤ε,ρ∈℧,
there exists a solutionυ∈C℧,ℝ of FDP (5) that satisfies(46)ωρ−υρ≤Υfε.Remark 4.
ω∈C℧,ℝ satisfies the inequality (45) if and only if there exists a function Π∈C℧,ℝ with(i)
Πρ≤ε, ρ∈℧(ii)
For allρ∈℧(47)CDα;φCDκ;φ+μωρ−∑i=1mξiCDα;φCDκi;φ+μiωρ=gρ,ωρ+Πρ.Lemma 7.
We suppose that1<κ≤2,0<α≤1 and 1≤κ−κi<2, and ω∈C℧,ℝ is a solution of the inequality (45). Then, ω satisfies(48)ωρ−Wρ−ℑα+κ;φgρ,ωρ≤2εXφα+κ1Γα+κ+1,where(49)Wρ=∑i=1mξiℑκ−κi;φωρ−σℑκ;φωρ+XφκρXφκ1σℑκ;φω1+∑i=1mξiℑκ−κi;φω1+ℑα+κ;φg1,ω1.Proof.
We suppose thatω is a solution of (45). By Lemma 6 and (ii) of Remark 4, we have(50)CDα;φCDκ;φ+μωρ−∑i=1mξiCDα;φCDκi;φ+μiωρ=gρ,ωρ+Πρ,ω0=ωφ′0=ω1=0.
Then, the solution of FDP (50) is(51)ωρ=∑i=1mξiℑκ−κi;φωρ+ℑα+κ;φgρ,ωρ−σℑκ;φωρ+XφκρXφκ1σℑκ;φω1−∑i=1mξiℑκ−κi;φω1−ℑα+κ;φg1,ω1+ℑα+κ;φΠρ−XφκρXφκ1ℑα+κ;φΠ1.
Again by (i) of Remark4, it is implied that(52)ωρ−Wρ−ℑα+κ;φgρ,ωρ≤ℑα+κ;φΠρ+XφκρXφκ1ℑα+κ;φΠ1≤εXφα+κρΓα+κ+11+XφκρXφκ1≤2εXφα+κ1Γα+κ+1.Theorem 4.
Under the hypotheses of Theorem3, the solution of the FDP (5) is UH stable.Proof.
We suppose thatω∈C℧,ℝ is a solution of the inequality (45) and υ∈C℧,ℝ is a unique solution of FDP (5). From Lemma 6, we obtain υρ=Vρ+ℑα+κ;φgρ,υρ, where(53)Vρ=∑i=1mξiℑκ−κi;φυρ+ℑα+κ;φgρ,υρ−σℑκ;φυρ+XφκρXφκ1σℑκ;φυ1−∑i=1mξiℑκ−κi;φυ1−ℑα+κ;φg1,υ1.
Clearly, ifυ0=ω0, υφ′0=ωφ′0, υ1=ω1, then(54)∑i=1mξiℑκ−κi;φυρ=∑i=1mξiℑκ−κi;φωρ,ℑα+κ;φgρ,υρ=ℑα+κ;φgρ,ωρ,ℑκ;φυρ=ℑκ;φωρ,∑i=1mξiℑκ−κi;φυ1=∑i=1mξiℑκ−κi;φω1.
Hence, we getVρ=Wρ.
Using Lemma7 and due to the fact that x+y≤x+y, for any ρ∈℧, we have(55)ωρ−υρ=ωρ−Vρ−ℑα+κ;φgρ,υρ≤ωρ−Wρ−ℑα+κ;φgρ,ωρ+ℑα+κ;φgρ,ωρ−gρ,υρ≤2εXφα+κ1Γα+κ+1+ℑα+κ;φLgωρ−υρ≤2εXφα+κ1Γα+κ+1+Xφα+κ1LgΓα+κ+1ωρ−υρ,
which implies(56)1−Xφα+κ1LgΓα+κ+1ωρ−υρ≤2εXφα+κ1Γα+κ+1.
From (41), we get Xφα+κ1Lg/Γα+κ+1<1. It follows that(57)ωρ−υρ≤Υfε,where(58)Υf=2Xφα+κ1Γα+κ+1−Xφα+κ1Lg.
Hence, the FDP (5) is UH stable in C℧,ℝ.Example 1.
We consider the generalized Caputo-type Langevin FDP:(59)CD14;φCD74;φ+56υρ−13CDα;φCD15;φ+12υρ=1100cosρ+tan−1υρ,ρ∈0,1,υ0=υφ′0=υ1=0,where α=1/4,κ=7/4,μ=5/6,m=1,ξ1=1/3,κ1=1/5,μ1=1/2, 1≤κ−κ1=31/20<2,φρ=ρ/3,℧=0,1, and(60)gρ,υ=cosρ+tan−1υ.
We observe that(61)gρ,υ1−gρ,υ2≤1100tan−1υ1−tan−1υ2≤1100υ1−υ2=Lgυ1−υ2,Xφκ1=φ1−φ0κ=1374,σ=μ−ξ1μ1=23,
(62)σ=29×311/20Γ51/20+49×33/4Γ11/7+1900<1.
Clearly, all the hypotheses of Theorem3 hold, and hence, FDP (59) has a unique solution on 0,1. Furthermore, we have(63)gρ,υ≤cosρ+1≔pρ∈C0,1,ℝ+,
which satisfies the assumption (H1) of Theorem2. Moreover, we can find that(64)γ1=29×311/20Γ51/20+49×33/4Γ11/7<1.
Consequently, by Theorem1, the FDP (59) has at least one solution on 0,1. Moreover, we have(65)Υf=2001799>0.
From Theorem3, the FDP (59) is UH stable on 0,1.


## 4.1. Existence and Uniqueness Results

This subsection proves the existence and uniqueness results for FDPs (4) and (5) by applying Sadovskii’s theorem [34] and Banach’s theorem [35].Theorem 2 (existence).
Letg:℧×ℝ⟶ℝ is a continuous function. We assume that
(H1): there exists a functionp∈C℧,ℝ+ such that gρ,x≤pρ, for ρ,x∈℧×ℝ.
(H2):γ1:=2∑i=1mξiXφκ−κi1/Γκ−κi+1+σXφκ1/Γκ+1<1.
Then, the FDP (1.5) has at least one solution on℧.Proof.
LetEr be a closed bounded and convex subset of C℧,ℝ, where r is a fixed constant. By virtue of Lemma 6, we define an operator f:C℧,ℝ⟶C℧,ℝ as follows:(35)fυρ=∑i=1mξiℑκ−κi;φυρ+ℑα+κ;φgρ,υρ−σℑκ;φυρ+XφκρXφκ1σℑκ;φυ1−∑i=1mξiℑκ−κi;φυ1−ℑα+κ;φg1,υ1,
forρ∈℧. Let us define two operators F1,F2:C℧,ℝ⟶C℧,ℝ by(36)f1υρ=∑i=1mξiℑκ−κi;φυρ−σℑκ;φυρ+XφκρXφκ1σℑκ;φυ1−∑i=1mξiℑκ−κi;φυ1,F2υρ=ℑα+κ;φgρ,υρ−XφκρXφκ1ℑα+κ;φg1,υ1.
We observe thatFυρ=F1υρ+F2υρ, ρ∈℧.
In order to show thatF1+F2 has a fixed point, we prove that F1 and F2 satisfy the hypotheses of Sadovskii’s theorem. This will be provided in several steps:Step 1.
FEr⊂Er.
Let us selectr≥γ2/1−γ1, where γ2:=2pXφα+κ1/Γα+κ+1. For any υ∈Er, we have(37)Fυ≤supρ∈℧∑i=1mξiℑκ−κi;φυρ+σℑκ;φυρ+XφκρXφκ1σℑκ;φυ1+∑i=1mξiℑκ−κi;φυ1+ℑα+κ;φgρ,υρ+XφκρXφκ1ℑα+κ;φg1,υ1≤2υ∑i=1mξiφ1−φ0κ−κiΓκ−κi+1+σφ1−φ0κΓκ+1+2pφ1−φ0α+κΓα+κ+1≤2r∑i=1mξiXφκ−κi1Γκ−κi+1+σXφκ1Γκ+1+2pXφα+κ1Γα+κ+1=rγ1+γ2≤rγ1+r1−γ1=r,
which implies thatF1+F2Er⊂Er.Step 2.
F2 is compact. We note that F2 is uniformly bounded from Step 1. Let ρ1,ρ2∈℧ with ρ1≤ρ2 and υ∈Er. Then, we obtain(38)F2υρ2−F2υρ1=1Γα+κ∫0ρ2Xφα+κ−1ρ2,sgs,υsds−Xφκρ2Xφκ11Γα+κ∫01Xφα+κ−11,sgs,υsds−1Γα+κ∫0ρ1Xφα+κ−1ρ1,sgs,υsds+Xφκρ1Xφκ11Γα+κ∫01Xφα+κ−11,sgs,υsds≤Xφκρ1−Xφκρ2Xφκ11Γα+κ∫01Xφα+κ−11,sgs,υsds+1Γα+κ∫0ρ1Xφα+κ−1ρ1,s−Xφα+κ−1ρ2,sgs,υsds+1Γα+κ∫ρ1ρ2Xφα+κ−1ρ2,sgs,υsds≤φρ1−φ0κ−φρ2−φ0κΓα+κ+1φ1−φ0α+κp+2pΓα+κ+1φρ2−φρ1α+κ≤2pΓα+κ+1φρ2−φρ1α+κ.
From the continuity ofφ, F2υρ2−F2υρ1⟶0 as ρ2⟶ρ1. Thus, F2 is equicontinuous. So, by the Arzela–Ascoli theorem, F2Er is a relatively compact set.Step 3.
F1 is γ contractive. Let υ1,υ2∈Er. Then, we have(39)F1υ1−F1υ2=supρ∈℧|∑i=1mξiℑκ−κi;φυ1ρ−υ1ρ−σℑκ;φυ1ρ−υ1ρ+XφκρXφκ1σℑκ;φυ11−υ11−∑i=1mξiℑκ−κi;φυ11−υ11|≤∑i=1mξiℑκ−κi;φs1υ1−υ2+σℑκ;φs1υ1−υ2+σℑκ;φs1υ1−υ2+∑i=1mξiℑκ−κi;φs1υ1−υ2≤2∑i=1mξiXφκ−κi1Γκ−κi+1+σXφκ1Γκ+1υ1−υ2=γ1υ1−υ2,
which isγ contractive since γ1<1.Step 4.
F is condensing.
Due to the fact thatF1 is continuous, a γ is a contraction and F2 is compact, it follows from Lemma 1 that F:Er⟶Er with F=F1+F2 is a condensing on Er. From the previous arguments, we conclude through Sadovskii’s theorem that F has a fixed point. As a result, FDP (5) has a solution on ℧.
The second result is based on Banach’s fixed point theorem.Theorem 3 (uniqueness).
Letg:℧×ℝ⟶ℝ is a continuous function, and there exists Lg>0 such that(40)gρ,x−gρ,x∗≤Lgx−x∗,ρ∈℧,x,x∗∈ℝ.
If(41)σ≔γ1+2Xφα+κ1Γα+κ+1Lg<1,
then the FDP (5) has a unique solution on ℧, where γ1 is defined in Theorem 2.Proof.
We apply Banach’s theorem to prove thatF defined by (35) has a fixed point. For this end, we show that F is a contraction. Let υ,υ∗∈C℧,ℝ and ρ∈℧. Then,(42)Fυρ−Fυ∗ρ≤∑i=1mξiℑκ−κi;φυρ−υ∗ρ+ℑα+κ;φgρ,υρ−gρ,υ∗ρ+σℑκ;φυρ−υ∗ρ+XφκρXφκ1σℑκ;φυ1−υ∗1+∑i=1mξiℑκ−κi;φυ1−υ∗1+XφκρXφκ1ℑα+κ;φg1,υ1−g1,υ∗1≤2∑i=1mξiXφκ−κi1Γκ−κi+1+σXφκ1Γκ+1υ−υ∗+2Xφα+κ1Γα+κ+1Lgυ−υ∗=γ1+2Xφα+κ1Γα+κ+1Lgυ−υ∗,
which implies(43)Fυ−Fυ∗≤συ−υ∗.
Asσ<1, it follows that F is a contraction. As a result of Banach’s theorem, there is a unique fixed point υ∈C℧,ℝ such that Fυ=υ. Therefore, the FDP (5) has a unique solution on ℧.Corollary 1 (existence).
Letg:℧×ℝ⟶ℝ is a continuous function, and there exists a function p∈C℧,ℝ+ such that gρ,x≤pρ, for ρ,x∈℧×ℝ. If γ3<1, where γ3=2μXφκ1/Γκ+1,
then the FDP (4) has at least one solution on ℧.Corollary 2 (uniqueness).
Letg:℧×ℝ⟶ℝ is a continuous function, and there exists Mg>0 such that(44)gρ,x−gρ,x∗≤Mgx−x∗,ρ∈℧,x,x∗∈ℝ.
IfΛ2<1, where Λ2=γ3+2Xφα+κ1/Γα+κ+1Lg, then the FDP (4) has a unique solution on ℧, where γ3 is defined in Corollary 1.Remark 1.
Proofs ofCorollaries 1 and 2 can be obtained easily using the same arguments as in the previous theorems. Thus, we omit the details.Remark 2.
Ifφρ=ρ, then the obtained results on FDP (1.5) include the results of Laadjal et al. [32].Remark 3.
The Langevin FDPs (4) and (5) are new to the literature on Langevin-type multiterm FDPs and include many problems, as a special case, for φρ=logρ and φρ=ρρ; our problems are reduced to Hadamard-type problems and Katugampola problems, respectively.


## 4.2. UH Stability Analysis

In this subsection, we discuss the UH stability of the considered problem.Definition 4.
FDP (5) is UH stable if there exists a constant Υf>0 such that for each ε>0 and every solution ω∈C℧,ℝ of the inequalities(45)CDα;φCDκ;φ+μωρ−∑i=1mξiCDα;φCDκi;φ+μiωρ−gρ,ωρ≤ε,ρ∈℧,
there exists a solutionυ∈C℧,ℝ of FDP (5) that satisfies(46)ωρ−υρ≤Υfε.Remark 4.
ω∈C℧,ℝ satisfies the inequality (45) if and only if there exists a function Π∈C℧,ℝ with(i)
Πρ≤ε, ρ∈℧(ii)
For allρ∈℧(47)CDα;φCDκ;φ+μωρ−∑i=1mξiCDα;φCDκi;φ+μiωρ=gρ,ωρ+Πρ.Lemma 7.
We suppose that1<κ≤2,0<α≤1 and 1≤κ−κi<2, and ω∈C℧,ℝ is a solution of the inequality (45). Then, ω satisfies(48)ωρ−Wρ−ℑα+κ;φgρ,ωρ≤2εXφα+κ1Γα+κ+1,where(49)Wρ=∑i=1mξiℑκ−κi;φωρ−σℑκ;φωρ+XφκρXφκ1σℑκ;φω1+∑i=1mξiℑκ−κi;φω1+ℑα+κ;φg1,ω1.Proof.
We suppose thatω is a solution of (45). By Lemma 6 and (ii) of Remark 4, we have(50)CDα;φCDκ;φ+μωρ−∑i=1mξiCDα;φCDκi;φ+μiωρ=gρ,ωρ+Πρ,ω0=ωφ′0=ω1=0.
Then, the solution of FDP (50) is(51)ωρ=∑i=1mξiℑκ−κi;φωρ+ℑα+κ;φgρ,ωρ−σℑκ;φωρ+XφκρXφκ1σℑκ;φω1−∑i=1mξiℑκ−κi;φω1−ℑα+κ;φg1,ω1+ℑα+κ;φΠρ−XφκρXφκ1ℑα+κ;φΠ1.
Again by (i) of Remark4, it is implied that(52)ωρ−Wρ−ℑα+κ;φgρ,ωρ≤ℑα+κ;φΠρ+XφκρXφκ1ℑα+κ;φΠ1≤εXφα+κρΓα+κ+11+XφκρXφκ1≤2εXφα+κ1Γα+κ+1.Theorem 4.
Under the hypotheses of Theorem3, the solution of the FDP (5) is UH stable.Proof.
We suppose thatω∈C℧,ℝ is a solution of the inequality (45) and υ∈C℧,ℝ is a unique solution of FDP (5). From Lemma 6, we obtain υρ=Vρ+ℑα+κ;φgρ,υρ, where(53)Vρ=∑i=1mξiℑκ−κi;φυρ+ℑα+κ;φgρ,υρ−σℑκ;φυρ+XφκρXφκ1σℑκ;φυ1−∑i=1mξiℑκ−κi;φυ1−ℑα+κ;φg1,υ1.
Clearly, ifυ0=ω0, υφ′0=ωφ′0, υ1=ω1, then(54)∑i=1mξiℑκ−κi;φυρ=∑i=1mξiℑκ−κi;φωρ,ℑα+κ;φgρ,υρ=ℑα+κ;φgρ,ωρ,ℑκ;φυρ=ℑκ;φωρ,∑i=1mξiℑκ−κi;φυ1=∑i=1mξiℑκ−κi;φω1.
Hence, we getVρ=Wρ.
Using Lemma7 and due to the fact that x+y≤x+y, for any ρ∈℧, we have(55)ωρ−υρ=ωρ−Vρ−ℑα+κ;φgρ,υρ≤ωρ−Wρ−ℑα+κ;φgρ,ωρ+ℑα+κ;φgρ,ωρ−gρ,υρ≤2εXφα+κ1Γα+κ+1+ℑα+κ;φLgωρ−υρ≤2εXφα+κ1Γα+κ+1+Xφα+κ1LgΓα+κ+1ωρ−υρ,
which implies(56)1−Xφα+κ1LgΓα+κ+1ωρ−υρ≤2εXφα+κ1Γα+κ+1.
From (41), we get Xφα+κ1Lg/Γα+κ+1<1. It follows that(57)ωρ−υρ≤Υfε,where(58)Υf=2Xφα+κ1Γα+κ+1−Xφα+κ1Lg.
Hence, the FDP (5) is UH stable in C℧,ℝ.Example 1.
We consider the generalized Caputo-type Langevin FDP:(59)CD14;φCD74;φ+56υρ−13CDα;φCD15;φ+12υρ=1100cosρ+tan−1υρ,ρ∈0,1,υ0=υφ′0=υ1=0,where α=1/4,κ=7/4,μ=5/6,m=1,ξ1=1/3,κ1=1/5,μ1=1/2, 1≤κ−κ1=31/20<2,φρ=ρ/3,℧=0,1, and(60)gρ,υ=cosρ+tan−1υ.
We observe that(61)gρ,υ1−gρ,υ2≤1100tan−1υ1−tan−1υ2≤1100υ1−υ2=Lgυ1−υ2,Xφκ1=φ1−φ0κ=1374,σ=μ−ξ1μ1=23,
(62)σ=29×311/20Γ51/20+49×33/4Γ11/7+1900<1.
Clearly, all the hypotheses of Theorem3 hold, and hence, FDP (59) has a unique solution on 0,1. Furthermore, we have(63)gρ,υ≤cosρ+1≔pρ∈C0,1,ℝ+,
which satisfies the assumption (H1) of Theorem2. Moreover, we can find that(64)γ1=29×311/20Γ51/20+49×33/4Γ11/7<1.
Consequently, by Theorem1, the FDP (59) has at least one solution on 0,1. Moreover, we have(65)Υf=2001799>0.
From Theorem3, the FDP (59) is UH stable on 0,1.


## 5. Conclusions

In this paper, we have given some results dealing with the existence and stability of solutions for two types of fractional boundary value problems for multiterm Langevin equations with generalized Caputo fractional operators of different orders. As an initial step, we obtained the equivalent solutions associated with linear problems by applying the instruments of advanced fractional calculus and characterizing a fixed point problem. Once the fixed point problem is available, the existence and uniqueness theorems are established via Banach’s and Sadovskii’s fixed point techniques, whereas the guarantee of the existence of solutions has been shown by the vigorous techniques, such as UH stability.We do not apply any significant bearing to the complex transformations, and our outcomes are characteristic of the integral operators’ theory of such kind. Indeed, our methodology is straightforward and can without much of a stretch be applied to an assortment of real-world problems. For the justification of the main results, we have given an example. As a special case, the reported results are new, and we have generalized many results with various values ofφ function.



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