Existence, uniqueness and stability analysis of a coupled fractional-order differential systems involving Hadamard derivatives and associated with multi-point boundary conditions

In this paper, we examine the consequences of existence, uniqueness and stability of a multi-point boundary value problem defined by a system of coupled fractional differential equations involving Hadamard derivatives. To prove the existence and uniqueness, we use the techniques of fixed point theory. Stability of Hyers-Ulam type is also discussed. Furthermore, we investigate variations of the problem in the context of different boundary conditions. The current results are verified by illustrative examples.

In this article, we extend the boundary value problem of Wang et al. [36] to nonlinear coupled system of Hadamard fractional differential equations having the Hadamard derivative value of the unknown function at T is propositional to the sum of Hadamard integral values of the unknown function on the strips (1, υ), (1, θ ) and multi-points values of the unknown functions with different strip lengths (1, υ), (1, θ ) and with different multipoint δ j , γ j , j = 1, 2, . . . , k -2. In [36], a monotone iterative method was applied to study the existence of positive solutions for Hadamard fractional differential equations complemented with non-local multi-point discrete and Hadamard integral boundary conditions.
In the present paper, inspired by [36], we introduce and investigate the existence and stability of solutions for a coupled system of nonlinear Hadamard-type fractional differential equations: for τ ∈ [1, T] := H, enhanced with boundary conditions defined by and where H D (·) denotes the Hadamard fractional derivatives (HFDs) of order (·), 2 < , ς ≤ 3 and 0 < 1 , ς 1 ≤ 1. H I (·) denotes the Hadamard fractional integrals (HFIs) of order (·), 0 < 2 , ς 2 < 1, f , g : H × R 2 → R are continuous functions, α 1 , α 2 , β 1 and β 2 are real constants and ξ j , υ j , j = 1, 2, . . . , k -2 are positive real constants. Notice that the multi-point strip boundary conditions in (2) are new and can be regarded as the HFDs value of unknown functions with the right end-point T is proportional with the sum of HFIs of unknown functions with different strip lengths (1, υ), (1, θ ) and with different multi-point values of unknown functions with δ j , γ j , j = 1, 2, . . . , k -2. By using the fixed point theory, we obtain the existence and uniqueness results. Furthermore, we investigate the Hyers-Ulam-type stability. We also discuss some variants of the given problem. Examples are given to support the theoretical outcomes. Section 2 focuses on the basic principles of fractional calculus with the accompanying fundamental definitions and lemmas. The consequences of existence and uniqueness can be explored in Sect. 3 using fixed point theorems of Leray-Schauder, Krasnoselskii, and Banach. Section 4 addresses the stability of Hyers-Ulam solutions and establishes sufficient conditions for stability. In Sect. 5, we consider two new problems analog to (1)-(2). Section 6 gives examples of verifying the results.

Preliminaries
In this section, we recall some preliminary concepts related to our work concerning Hadamard fractional calculus.

Definition 1 ([2]) The Hadamard fractional integral of order
The Hadamard derivative of fractional order for a function h ∈ AC n δ [b, c] is defined as where n -1 < < n, n = [ ] + 1, [ ] denotes the integer part of the real number and log(·) = log e (·). Recall that the Hadamard fractional derivative is the left-inverse operator to Hadamard fractional integral in the space . Then the solution of the Hadamard fractional differential equation H D y(τ ) = 0 is given by and the following formula holds: where a i ∈ R, i = 1, 2, . . . , m and n -1 < < n.

Lemma 6
Let h 1 , h 2 ∈ C [1, T]. Then the solution of the linear system of FDEs: augmented with the boundary conditions: and is given by and where and ϑ = ϑ 1 ϑ 4ϑ 2 ϑ 3 .
Next, we present the hypotheses that we need in the sequel. Let the functions h 1 , h 2 : H × R 2 → R be continuous.
(F 1 ) There exist real constants ϕ i , ϕ i ≥ 0, i = 1, 2 and ϕ 0 > 0, ϕ 0 > 0 such that for all w i ∈ R and i = 1, 2. (F 2 ) There exist positive constants ϑ i , ϑ i , with i = 1, 2 such that for all τ ∈ H, w i , w i ∈ R and i = 1, 2. Then either the set ψ(F ) is unbounded or F has at least one fixed point.
Proof We define that in the first step the operator : Y × Z → Y × Z is completely continuous, indicating the continuity of the h 1 and h 2 functions of the 1 and 2 operators. The operator is also continuous. Let ⊂ Y × Z be a bounded set to show the uniformly bounded operator . Then there exist positive constants U 1 and U 2 such that for all (y, z) ∈ . Then we have, for any (y, z) ∈ , which yields when taking the τ ∈ H norm and using (19) and (20), Likewise we obtain by using (21) and (22). We deduce that 1 and 2 are uniformly bound from inequalities (27) and (28), which means that the operator is uniformly bounded. Next we show 's equicontinuity. Let τ 1 , τ 2 ∈ H with τ 1 < τ 2 . Then we have as τ 2 → τ 1 independent of (y, z) with respect to the boundedness of h 1 and h 2 . And because of the equicontinuity of 1 and 2 , the operator is equicontinuous. Thus, by the Arzelà-Ascoli theorem the operator is compact. The set is finally shown to be bounded.
which yields, when taking the norm for τ ∈ H, We also obtain From (29) and (30), we get which yields, with y, z = y + z , This means that ( ) is bounded. Thus, by Lemma 7, the operator has at least one fixed point. This indicates that the BVP (1)-(2) has at least one solution on H.
It follows, thus, from the condition that is a contraction operator. This shows that is a contraction. Hence, by Banach's fixed point theorem, the operator has a unique fixed point which corresponds to a unique solution of problem (1)- (2). This completes the proof.

Variants of the problem
Further, we can solve some problems similar to problem (1)-(2) by using the methodology employed in the previous section. For example, we consider two new problems by replacing the condition. Note that the boundary conditions (2) include the strips of the different lengths when modifying the strips in boundary conditions like the same lengths (2), then the problem reduces to the form with 1 < υ < δ 1 < δ 2 < · · · < δ k-2 < T.