On a class of Langevin equations in the frame of Caputo function-dependent-kernel fractional derivatives with antiperiodic boundary conditions

1 Laboratory of Mathematics and Applied Sciences University of Ghardaia, Algeria 2 Department of Mathematics, Hodeidah University, Al-Hudaydah, Yemen 3 Department of Mathematical Sciences, Faculty of Sciences, Princess Nourah bint Abdulrahman University, Riyadh 11671, Saudi Arabia 4 Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, Saudi Arabia 5 Department of Medical Research, China Medical University, Taichung 40402, Taiwan 6 Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan


Introduction
The theme of fractional calculus (FC) has appeared as a broad and interesting research point due to its broad applications in science and engineering. FC is now greatly evolved and embraces a wide scope of interesting findings. To obtain detailed information on applications and recent results about this topic, we refer to [1][2][3][4][5] and the references therein.
Some researchers in the field of FC have realized that innovation for new FDs with many non-singular or singular kernels is very necessary to address the need for more realistic modeling problems in different fields of engineering and science. For instance, we refer to works of Caputo and Fabrizio [6], Losada and Nieto [7] and Atangana-Baleanu [8]. The class of FDs and fractional integrals (FIs) concerning functions is a considerable branch of FC. This class of operators with analytic kernels is a new evolution proposed in [1,9,10]. Every one of these operators is appropriate broader to cover various kinds of FC and catch diversified behaviors in fractional models. Joining the previous ideas yields another, significantly wide, which is a class of function-dependent-kernel fractional derivatives. This covers both of the two preceding aforesaid classes see [11,12].
The considered problem in this work is more general, in other words, when we take certain values of function ϑ, the problem (1.3) is reduced to many problems in the frame of classical fractional operators. Also, the gained results here are novel contributes and an extension of the evolution of FDEs that involving a generalized Caputo operator, especially, the study of stability analysis of Ulam-Hyers type of fractional Langevin equations is a qualitative addition to this work. Besides, analysis of the results was restricted to a minimum of assumptions.
Here is a brief outline of the paper. Section 2 provides the definitions and preliminary results required to prove our main findings. In Section 3, we establish the existence, uniqueness, and stability in the sense of Ulam for the system (1.3). In Section 5, we give some related examples to light the gained results.

Preliminaries and lemmas
We start this part by giving some basic definitions and results required for fractional analysis. Consider the space of real and continuous functions U = C(J, R) space with the norm z ∞ = sup{|z(ς)| : ς ∈ J}.
From the above definition, we can express ϑ-Caputo FD by formula Also, the ϑ-Caputo FD of order α 1 of ω is defined as For more details see [9,Theorem 3].
Theorem 2.8. (Kransnoselskii's FPT [46]). Let E be a Banach space. Let S is a nonempty convex, closed and bounded subset of E and let A 1 , A 2 be mapping from S to E such that: Then there exists z ∈ S such that z = A 1 z + A 2 z.

Main results
This portion interests in the existence, uniqueness, and Ulam stability of solutions to the suggested problem (1.3).
The next auxiliary lemma, which attentions the linear term of a problem (1.3), plays a central role in the afterward findings.
has a unique solution defined by and Proof. Applying the RL operator I α 2 ;ϑ a + to (3.1) it follows from Lemma 2.5 that Again, we apply the RL operator I α 1 ;ϑ a + and use the results of Lemma 2.5 to get where c 0 , c 1 , c 2 , c 3 , c 4 ∈ R. By utilizing the boundary conditions in (3.1) and (3.7), we obtain Now, by using the conditions D α 1 ;ϑ a + z(a) = I δ;ϑ a + z(γ) and D α 1 ;ϑ Now, we shall need to the following lemma: The functions µ and ν are continuous functions on J and satisfy the following properties: where µ and ν are defined by Lemma 3.1.
Here, we give the following hypotheses: For simplicity, we denote As a result of Lemma 3.1, we have the subsequent lemma:  Proof. Thanks to Lemma 3.1, we consider the operator G : U → U defined by (3.13). Thus, G is well defined as F is a continuous. Then the fixed point of G coincides with the solution of FLDE (1.3). Next, the Theorem 2.7 will be used to prove that G has a fixed point. For this end, we show that G is a contraction.
(3.14) which implies that Gz ≤ R, i.e., Now, let z, κ ∈ U. Then, for every ς ∈ J, using (H2), we can get Using the above arguments, we get As ∆ < 1, we derive that G is a contraction. Hence, by Theorem 2.7, G has a unique fixed point which is a unique solution of FLDE (1.3). This ends the proof. Now, we apply the Theorem 2.8 to obtain the existence result.

Theorem 3.5. Let us assume (H1)-(H3) hold. Then FLDE (1.3) has at least one solution on
where it is supposed that Proof. By the assumption (H3), we can fix where B ρ = {z ∈ U : z ≤ ρ}. Let us split the operator G : U → U defined by (3.13) as G = G 1 + G 2 , where G 1 and G 2 are given by and (3.17) The proof will be split into numerous steps: Step 1: Hence Step 2: G 2 is a contraction map on B ρ . Due to the contractility of G as in Theorem 3.4, then G 2 is a contraction map too.
Step 3: G 1 is completely continuous on B ρ . From the continuity of F (·, z(·)), it follows that G 1 is continuous. Since we get G 1 z ≤ N which emphasize that G 1 uniformly bounded on B ρ . Finally, we prove the compactness of G 1 . For z ∈ B ρ and ς ∈ J, we can estimate the operator derivative as follows: where we used the fact Hence, for each ς 1 , ς 2 ∈ J with a < ς 1 < ς 2 < b and for z ∈ B ρ , we get which as (ς 2 − ς 1 ) tends to zero independent of z. So, G 1 is equicontinuous. In light of the foregoing arguments along with Arzela-Ascoli theorem, we derive that G 1 is compact on B ρ . Thus, the hypotheses of Theorem 2.8 holds, so there exists at least one solution of (1.3) on J.

Ulam-Hyers stability analysis for the ϑ-Caputo FLDE (1.3)
In the current section, we are interested in studying Ulam-Hyers (U-H) and the generalized Ulam-Hyers stability types of the problem (1.3).
Let ε > 0. We consider the next inequality:   Proof. For ε > 0 andz ∈ C(J, R) be a function which fulfills the inequality (4.1). Let z ∈ U the unique solution of By Lemma 3.1, we have Since we have assumed that z is a solution of (4.1), hence by Remark 4.3 Again by Lemma 3.1, we have Using part (i) of Remark 4.3 and (H2), we get where ∆ is defined by (3.11). In consequence, it follows that If we let c F = (ϑ(b)−ϑ(a)) α 1 +α 2 , then, the U-H stability condition is satisfied.

Examples
This section is intended to illustrate the reported results with relevant examples.

Conclusions
In this reported article, we have considered a class of nonlinear Langevin equations involving two different fractional orders in the frame of Caputo function-dependent-kernel fractional derivatives with antiperiodic boundary conditions. The existence and uniqueness results are established for the suggested problem. Our perspective is based on properties of ϑ-Caputo's derivatives and applying of Krasnoselskii's and Banach's fixed point theorems. Moreover, we discuss the Ulam-Hyers stability criteria for the at-hand problem. Some related examples illustrating the effectiveness of the theoretical results are presented. The results obtained are recent and provide extensions to some known results in the literature. Furthermore, they cover many fractional Langevin equations that contain classical fractional operators.