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

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.

Some new contributions to Langevin FDEs have been investigated (see [9][10][11][12][13]) and the references referred to in that). ere 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][15][16]). e generalized Riemann-Liouville definition with respect to another function was first presented by Osler [17]. en, Kilbas et al. [2] presented important characteristics of this operator. e 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]. is 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][21][22][23][24][25][26][27][28][29]).
Ahmad and Nieto [31] considered the following Caputotype Langevin FDE with boundary conditions: C D α C D κ + λ υ(ρ) � g(ρ, υ(ρ)), In 2020, Laadjal et al. [32] studied a Caputo-type multiterm Langevin FDE with boundary conditions of the form: In this work, we study two classes of generalized Caputo Langevin equations with two various fractional orders. is 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): the symbol C D θ;φ denotes the generalized fractional derivative in the Caputo sense of order θ ∈ α, κ, κ i , 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. e studied results expand and generalize many results by selecting special cases of the φ function. e paper is organized as follows: In Section 2, 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.

Preliminaries and Lemmas
We are beginning this portion by endowment with some essential definitions and results required for forthcoming analysis.

The Linear Variant of FDPs (1.4) and (1.5)
is section deals with the linear variant of FDPs (1.4) and (1.5). For simpleness, we denote C D α;φ 0 + and I α;φ 0 + by C D α;φ and I α;φ , respectively. Lemma 5. Let 1 < κ ≤ 2, 0 < α ≤ 1, and f ∈ C(℧, R). en, υ is a solution of the following linear Langevin-type FDP: if and only if υ satisfies Mathematical Problems in Engineering Proof. Applying the operator I α;φ on both sides of the first (20) and using Lemma 2, we get where c 0 ∈ R. Next, applying the operator I κ;φ on both sides of (22) and using Lemma 2, again, we obtain where c 1 , c 2 ∈ R. In view of (23), we have where we used the fact that Using the initial conditions of (23), we find that c 1 � c 2 � 0 and Substituting the values of c 0 , c 1 , and c 2 in (23), we obtain the solution given by (21), where e converse follows by direct calculation. Hence, the proof is achieved.
. en, υ is a solution of the following linear Langevin-type multiterm FDP: if and only if υ satisfies 4 Mathematical Problems in Engineering Proof. Applying the operator I α;φ on both sides of the first (28) and using Lemma 2, we get where c 0 ∈ R. Next, applying the operator I κ;φ on both sides of (30) and using Lemma 2, again, we obtain where c 1 , c 2 ∈ R. In view of (31), we have Using the condition υ φ ′ (0) � 0 in (32) and υ(0) � 0 and υ(1) � 0 in (31), we find that c 1 � c 2 � 0 and which implies Substituting the values of c 0 , c 1 , and c 2 in (31), we obtain the solution given by (29). e converse follows by direct calculation. Hence, the proof is achieved.
Proof. Let E r 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: for ρ ∈ ℧. Let us define two operators (1)).
In order to show that F 1 + F 2 has a fixed point, we prove that F 1 and F 2 satisfy the hypotheses of Sadovskii's theorem.

Mathematical Problems in Engineering 7
which is c contractive since c 1 < 1.
Step 4. F is condensing. Due to the fact that F 1 is continuous, a c is a contraction and F 2 is compact, it follows from Lemma 1 that F: E r ⟶ E r with F � F 1 + F 2 is a condensing on E r . 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 ℧.
e second result is based on Banach's fixed point theorem.
Theorem 3 (uniqueness). Let g: ℧ × R ⟶ R is a continuous function, and there exists L g > 0 such that then the FDP (5) has a unique solution on ℧, where c 1 is defined in eorem 2.

Remark 1.
Proofs of Corollaries 1 and 2 can be obtained easily using the same arguments as in the previous theorems.
us, we omit the details.

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
Υ f > 0 such that for each ε > 0 and every solution ω ∈ C(℧, R) of the inequalities there exists a solution υ ∈ C(℧, R) of FDP (5) that satisfies Remark 4. ω ∈ C(℧, R) satisfies the inequality (45) if and only if there exists a function Π ∈ C(℧, R) with Lemma 7. We suppose that 1 < κ ≤ 2, 0 < α ≤ 1 and 1 ≤ κ − κ i < 2, and ω ∈ C(℧, R) is a solution of the inequality (45). en, ω satisfies where Proof. We suppose that ω is a solution of (45). By Lemma 6 and (ii) of Remark 4, we have , en, the solution of FDP (50) is Mathematical Problems in Engineering Again by (i) of Remark 4, it is implied that

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 of φ function.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare no conflicts of interest related to this work.