Existence and H-U Stability of Solution for Coupled System of Fractional-Order with Integral Conditions Involving Caputo-Hadamard Derivatives, Hadamard Integrals

In this article, the primary focus of our study is to investigate the existence, uniqueness, and Ulam-Hyers stability results for coupled fractional di ﬀ erential equations of the Caputo-Hadamard type that are supplemented with Hadamard integral boundary conditions. We employ adequate conditions to achieve existence and uniqueness results for the presented problems by utilizing the Banach contraction principle and the Leray-Schauder ﬁ xed point theorem. We also show Ulam-Hyers stability using the standard functional analysis technique. Finally, examples are used to validate the results.


Introduction
In the past two decades, scientists and researchers have published results on fractional calculus analysis and concluded that integer-order derivatives are not always reliable. The study of turbulent fluid flows, control theory, blood flow through biological tissues, porous media, and signal and image processing, among other fields, have all benefited greatly from the use of fractional calculus. The recent study on fractional calculus, including theory and applications, can be found in [1][2][3][4][5][6][7][8][9][10][11][12][13]. Their research is especially pertinent since coupled systems with fractional differential equations are used to address a wide range of real-world problems.
Over the past few decades, FDEs have also been the topic of substantial research in the field of stability analysis. Different types of stability, such as Mittag-Leffler and Lyapunov, have been researched in the literature. Very few researches have investigated the Ulam-Hyers stability of a linked system of FDEs. Ulam and Hyers [14,15] identified the novel type of stability known as Ulam-Hyers stability. Under-standing biological processes, fluid motion, semiconductors, population dynamics, heat conduction, and elasticity can all be helped by this kind of research. Meanwhile, the researchers have focused on the differences and results of mathematical models created by these operators and have used a variety of fractional derivation operators in their studies as a result of the diversity of fractional operators described by mathematicians. Different forms of fractional mathematical models, in which the effects of the order of fractional derivatives on the dynamic behavior of the solutions of the assumed systems are rigorously simulated, are some of the well-known works on this topic. The following is just one illustration: Caputo derivatives are used in [16,17], Caputo-conformable derivatives in [18,19], generalized derivatives in [20,21], quantum Caputo derivatives in [22], nonsingular Caputo-Fabrizio derivatives in [23], and nonsingular Mittag-Leffler kernel-type derivatives in [24,25]. The features of the Caputo and Hadamard operations are combined to define the Caputo-Hadamard fractional derivative, one of the fractional derivatives. By using this operator, very few fractional models and problems were produced. Examples can be seen in [26][27][28][29]. However, the Hadamard fractional derivative (HFD) is the most frequently used [30]. Butzer et al. [31] investigated a variety of properties of HFD, which are more general than HFDs. In [32], the authors investigated a hybrid fractional Caputo-Hadamard boundary value problem with hybrid Hadamard integral boundary value conditions. The authors in [33] studied topological degree theory and Caputo-Hadamard fractional boundary value problems. In [34], Etemad et al. investigated a fractional Caputo-Hadamard inclusion problem with sum boundary value conditions by using approximate endpoint property. In 2020, Etemad et al. [35] discussed a fractional Caputo-Hadamard problem with boundary value conditions via different orders of the Hadamard fractional operators: where t ∈ ½1, e, ρ, ϖ ∈ ð3, 4, δ ∈ ð1, 2, κ ∈ ð0, 1, and μ, ϑ > 0 with δ + ϑ ≠ 0 and also α, β ∈ ℝ. In 2021, Rezapour et al. [36] investigated Caputo-Hadamard fractional boundary value problem via mixed multi-order integro-derivative conditions: Recently, in [37], the authors derived existence and uniqueness results for a nonlinear coupled system of Caputo-type FDEs equipped with new coupled boundary conditions given by where C D ð·Þ 0+ denote the CFDs of order ð·Þ, α, β ∈ ð0, 1, f , g : ½0, T × R 2 e ⟶ R e are continuous functions and A is real constant. In 2022, Belbali et.al [38] existence theory and generalized Mittag-Leffler stability for a nonlinear Caputo-Hadamard fractional initial value problem using the Lyapunov method. By using main ideas of the aforementioned articles, we investigate the Caputo-Hadamard coupled system of FDEs with the Hadamard fractional integral conditions and present its existence, uniqueness, and Ulam-Hyers stability results. We study the following system: where Y 1 , Y 2 : S × R 2 ⟶ R are continuous functions; H I ρ is the HFI of order ρ defined by and H D ς denotes HFD of order ς and is defined by see [39,40], where C D ð·Þ denote the Caputo-Hadamard fractional derivatives (CHFDs) of order ð·Þ, 0 < ς, ρ ≤ 1, and ϕ is real constant. In this article, authors have extend the afore-mentioned articles [35][36][37] to nonlinear coupled system of Caputo-Hadamard fractional differential equations having the value of the sum of unknown functions P and U at the interval endpoints ½1, T being zero, whereas the value of the sum of the unknown functions on an arbitrary domain ð1, φÞ of the given interval ½1, T remains constant. The remainder of the paper is as follows: Section 2 introduces some fundamental definitions, lemmas, and theorems that support our results. In Section 3, we prove the existence and uniqueness of solutions to the given system (4) using various conditions and some regular fixed-point theorems.
Finally, examples are given to explain the main results.

Preliminaries
In this section, we discuss some relevant definitions and lemmas that will be needed later in our proof [11,39,40].
Journal of Function Spaces The left and right CHFDs of order ς are, respectively, defined by Definition 3. Let R e ðςÞ > 0, n = ½R e ðςÞ + 1 and p ∈ C½b, c. If R e ðςÞ ≠ 0 or ς ∈ ℕ, then Definition 4. Let p ∈ AC n δ ½b, c or C n δ ½b, c and ς ∈ ℂ, then Definition 5. Let K, L ∈ C½1, e and P , U ∈ ACðSÞ. Then, the solution of the following linear coupled system: Proof. Using Lemma 2.3 and the operators H I ς and H I ρ on both sides of FDEs in (11), we obtain where a 0 , b 0 ∈ R e , are arbitrary constants. Using the bound-ary Condition (11) in (15) and (16), we obtain Defined by Ω = CðS, R e Þ × CðS, R e Þ the Banach space endowed with the norm kP , Uk = sup ϑ∈S jP ðϑÞj + sup ϑ∈S j UðϑÞj, for ðP , UÞ ∈ Ω. In spite of Lemma 2.4, the following operator Ξ : Ω ⟶ Ω is associated with the problem (4): Following that, we introduce the assumptions necessary to construct the paper's primary results. Let We use the notation: For computational ease.
Journal of Function Spaces Our first existence result for the problem (4) is based on the following fixed point theorem ( [41,42]). Lemma 6. Let G : H ⟶ H be a completely continuous operator in the Banach space H , and the set Π = fP ∈ H j P = λGP , 0 < λ < 1g is bounded. Then, G has a fixed point in H .
Proof. We begin by demonstrating that the operator Ξ : Ω ⟶ Ω defined by (20) is completely continuous, i.e., that Ξ is continuous and maps any bounded subset of Ω to a relatively compact subset of Ω. Since the functions Y 1 and Y 2 are continuous, the operator Ξ : Ω ⟶ Ω is also continuous. Now, let Ψ̂r ⊂ Ω be bounded. Then, ∃ positive constants T Y 1 and T Y 2 such that So, for any ðP , UÞ ∈ Ψ̂r, ϑ ∈ S, we get Thus, Thus, the operator Ξ is uniformly bounded as a result of the preceding inequality. Let Ξ prove that it determines bounded sets into equicontinuous sets of Ω, let ϑ 1 , ϑ 2 ∈ S, ϑ 1 < ϑ 2 , and ðP , UÞ ∈ Ψr. Then, Take note that in the limit ϑ 1 ⟶ ϑ 2 , the RHS of the preceding inequalities tends to zero independently of ðP , UÞ ∈ Ψr. Then, The Arzela-Ascoli theorem implies that the operator Ξ : Ω ⟶ Ω is completely continuous. Following that, we consider the set Λ = fðP , UÞ ∈ ΩjðP , UÞ = κΞðP , UÞ, 0 < κ < 1g and demonstrate that it is bounded. Let ðP , UÞ ∈ Λ, then ðP , UÞ = κΞðP , UÞ, 0 < κ < 1. For any ϑ ∈ S, we have Using Y i ði = 1, 2Þ given by (25)-(26), we find that In consequence, we get Journal of Function Spaces Thus, in Conditions (29)-(30), we obtain T demonstrates that jP , Uj is constrained for ϑ ∈ S. As a result, the set Λ is bounded. As a result, the inference of Lemma 6 applies, and the operator Ξ has at least one fixed point, corresponding to a solution of the problem (4).
The existence of a unique solution to the problem (4) is demonstrated using Banach's contraction mapping theorem in the following result.

Ulam-Hyers Stability Results (4)
The U-H stability of the solutions to the BVPs (4) will be discussed in this section using the integral representation of their solutions defined by where ϑ 1 and ϑ 2 are given by (21) and (22). Consider the following definitions of nonlinear operators It considered the following inequalities for some b λ 1 , b λ 2 : Definition 9. The coupled system (4) is said to be U-H stable if V 1 , V 2 > 0, and there exists a unique solution ðP , UÞ ∈ CðE, R e Þ of a problem (4) with ∀ðP , UÞ ∈ CðS, R e Þ of inequality (49).
Proof. Let CðS, R e Þ × CðS, R e Þ be the solution to (4) that satisfies (21) and (22). Let ðP , UÞ be any solution that meets Condition (49): It follows that where Y 1 and Y 2 are defined in (25)-(26), respectively. As an outcome, we deduce from operator Ξ's fixed-point property, which is defined by (21) and (22) From the above Equations (54) and (55), it follows that with Hence, the problem (4) is U-H stable.