Existence and Ulam stability results of a coupled system for terminal value problems involving ψ-Hilfer fractional operator

The work reported in this paper deals with the study of a coupled system for fractional terminal value problems involving ψ-Hilfer fractional derivative. The existence and uniqueness theorems to the problem at hand are investigated. Besides, the stability analysis in the Ulam–Hyers sense of a given system is studied. Our discussion is based upon known fixed point theorems of Banach and Krasnoselskii. Examples are also provided to demonstrate the applicability of our results.

On the other hand, many interesting and recent results on the existence and stability of a coupled system for different categories of FDEs have been investigated, see the following studies [6][7][8]26] and the references therein.
Terminal value problems (TVPs) for FDEs for the time being play a major role in the modeling of numerous phenomena in engineering, science, and simulation. In short, the existence results for ordinary and fractional TVPs have been studied by many investigators, see [5, 13-15, 27, 28, 35]. For example, Benchohra et al. in [13] obtained the existence and uniqueness of solution to the fractional implicit TVP under the terminal condition where ρ D θ,η a + is the fractional derivative of order (θ , η) in the Hilfer-Katugampola sense (0 < θ < 1, 0 ≤ η ≤ 1), ρ > 0, and f : (a, T] × R → R is a certain function. Motivated by the aforementioned works, the target of this work is to investigate the existence, uniqueness, and Ulam-Hyers stability of solutions of a coupled system for fractional TVPs involving generalized Hilfer fractional derivative of the type ⎧ ⎨ ⎩ D θ 1 ,η 1 ;ψ a + y(t) = f 1 (t, x(t)), a < t ≤ T, a > 0, D θ 2 ,η 2 ;ψ a + x(t) = f 2 (t, y(t)), a < t ≤ T, a > 0, (3) under the terminal conditions ⎧ ⎨ ⎩ y(T) = w 1 ∈ R, where 0 < θ i < 1, 0 ≤ η i ≤ 1, D θ i ,η i ;ψ a + (i = 1, 2) is the Hilfer fractional derivative of order θ i and type η i with respect to ψ and f : (a, T] × R → R is a certain function under the conditions listed later.
As far as we know, no papers about a coupled system for fractional TVPs exist in the literature, specifically for those encompassing the generalized fractional derivative in the ψ-Hilfer sense. Moreover, the results of the problem at hand are obtained under minimal assumptions on nonlinear functions f 1 , f 2 .
The rest of the structure of this paper is as follows. In Sect. 2, we briefly state some essential definitions and the results that are applied throughout the paper. Section 3 studies the existence and uniqueness results on ψ-Hilfer FDEs with the terminal conditions via fixed point techniques of Banach and Krasnoselskii. The stability analysis in the concept Ulam-Hyers of the proposed system is investigated in Sect. 4. At the end, some examples are included to illustrate the applicability of the obtained results in Sect. 5.

Auxiliary results
Let [a, T] ⊂ R + with (0 < a < T < ∞), we also consider C[a, T] the Banach space of realvalued continuous functions defined on [a, T] with the norm Obviously, C 1-ς ;ψ [a, T] and C n 1-ς ;ψ [a, T] are Banach spaces endowed with the norms Let us introduce the following space: endowed with the norm defined by It is easy to perceive that, for σ ∈ E, (E, σ E ) is a Banach space. Then, for (σ , ρ) ∈ E × E, Then the left-sided ψ-RL fractional integral of order θ of a function σ w.r.t. ψ is described by Then the left-sided ψ-fractional derivatives in the concepts ψ-RL and ψ-Caputo of order θ of a function σ w.r.t. ψ are described by d dt ] n , and n = [θ ] + 1.

Existence and uniqueness results
To shorten the length of equations, we set K ς if and only if y fulfills the following fractional integral equation: Proof Assume that y ∈ C ς 1-ς ,ψ [a, T] is the solution of the TVP for ψ-Hilfer FDEs (7)- (8). According to the definitions of C ς 1-ς ,ψ (a, T] and D ς ;ψ a + , applying Lemma 1, we have Thanks to the definition of C n 1-ς ,ψ [a, T], we have Take advantage of Lemma 5 to get It follows from the assumption y ∈ C ς 1-ς ,ψ [a, T], Lemma 3, and equation (7) that Equating both sides of equations (12) and (13), we find that Using the terminal condition y(T) = w, we get Now, from equations (14) and (15), we conclude that Hence y(t) satisfies the TVP for ψ-Hilfer FDEs (7)- (8).
Before we present our main results, we consider that the following assumptions are satisfied: ( In the forthcoming theorem, by using Theorem 2, we prove the unique solution of a coupled system for ψ-Hilfer terminal FDEs (3)-(4). In view of Theorem 3, we get the following lemma.
1-ς 2 ,ψ (a, T] satisfy the following coupled system of fractional integral equations: According to Lemma (7), we consider the operators N 1 : E → E and N 2 : That is, Therefore, we define N : For the sake of brevity, we set where A i = max t∈[a,T] |f i (t, 0)|, i = 1, 2. Now, via Theorems 2, 1, we obtain the existence and uniqueness results of a coupled system for ψ-Hilfer FDEs (3)-(4).

Theorem 4
Assume that (H 1 ) and (H 2 ) hold. If Λ f 1 < 1 and Λ f 2 < 1, then ψ-Hilfer coupled system (3)-(4) has a unique solution in Proof Define the closed, bounded, convex, and nonempty set The analysis of proof will be presented in three steps.

Hence, inequality (23) becomes
which leads to Similarly, we can get that It follows from (24) and (25) that This proves N S R ⊂ S R .
Step (2): The operator N is a contraction. Let (y, x), (y * , x * ) ∈ S R and t ∈ (a, T]. Applying (H 2 ) we have By the same technique, we can also get In view of the conditions Λ f 1 < 1 and Λ f 2 < 1, we get Thus N is a contraction mapping. In accordance with Theorem 2, N has a unique fixed point ( y, x) ∈ E × E.
Step ( Multiplying both sides of the last system by D ς 1 ,ψ a + , D ς 2 ,ψ a + respectively, it follows from Lemmas 6 and 3 that , y(s)).
Since ς i ≥ θ i (i = 1, 2) and by (H 1 ), we get Hence, D As a sequel to the steps outlined above, we infer that the ψ-Hilfer coupled system (3)-(4) has a unique solution in C We exhibit now the next result, which relies on Theorem 1.

Theorem 5 Assume that (H 1 )-(H 2 ) hold. Then the ψ-Hilfer coupled system (3)-(4) has at least one solution.
Proof Let K ⊂ S R ⊆ E × E be a bounded set, and we define the operators F 1 , From the above-mentioned operators, we are able to write N 1 = F 1 + G 1 and N 2 = F 2 + G 2 . Thus, the operator N can be expressed as The proof will be divided into several stages as follows: Stage (1): N is continuous.

Stage(3): F(K) is equicontinuous in K.
Let (y, x) ∈ K and t 1 , t 2 ∈ (a, T] with t 1 < t 2 . Then we have which leads to Again applying the same reasoning, we have This exhibits that F(K) is equicontinuous. Stages 1-3 show that F is relatively compact on K. By E(= C 1-ς ;ψ ) type Arzelá-Ascoli theorem, F is compact on K.

Lemma 8 Let ( y, x) ∈ E × E be the solution of inequalities
Proof Thanks to Remark 1, we have .
Similarly, we have where . Inequalities (38) and (39) can be rewritten again as follows: Now, we will represent the relations in (40) as matrices as follows: After simple computations of the above inequality, we can write By compiling the above two inequalities, we get For = max{ 1 , 2 } and This proves that the ψ-Hilfer coupled system (3)-(4) is U-H stable. Moreover, we could put into writing inequality (41) as where ϕ( ) = λ with ϕ(0) = 0. This shows that the ψ-Hilfer coupled system (3)-(4) is G-U-H stable.

Examples
Here, we provide some illustrative examples to validate the obtained results.