The generalized U–H and U–H stability and existence analysis of a coupled hybrid system of integro-differential IVPs involving φ-Caputo fractional operators

We investigate the existence and uniqueness of solutions to a coupled system of the hybrid fractional integro-differential equations involving φ-Caputo fractional operators. To achieve this goal, we make use of a hybrid fixed point theorem for a sum of three operators due to Dhage and also the uniqueness result is obtained by making use of the Banach contraction principle. Moreover, we explore the Ulam–Hyers stability and its generalized version for the given coupled hybrid system. An example is presented to guarantee the validity of our existence results.

The notion of the Ulam-Hyers (U-H) stability has been taken into consideration in several publications. The announced stability analysis is a simple manner in this regard. Such a species of stability was developed by Ulam [15]. Later, it was developed by Hyers [16,17]. Recently, Ben Chikh et al. [18] considered a multi-order BVP via integral conditions and studied the U-H stability for this system. Samina et al. [19] reviewed the qualitative properties of a coupled system of fractional hybrid differential equations by terms of the U-H stability. At the same time, Ahmad et al. [20] derived similar results on the U-H stability for a coupled system of fractional hybrid BVPs with finite delays.
On the other side, ϕ-fractional operators were introduced by Kilbas [2] as a generalization of Riemann-Liouville (Riem-Lio) operators. These fractional operators are not quite the same as the other classical fractional operators; this is so because their kernel appears with respect to another increasing function ϕ. Several generalized FODs and their applications were introduced by Agarwal [21].
In 2017, Almeida [22] proposed a kind of Caputo FOD with some applied specifications and after that, he studied the existence results for two distinct ϕ-fractional models by these new derivatives [23,24]. Also, In 2020, Derbazi et al. [25] investigated a ϕ-fractional initial value problem by using a monotone iterative technique and then Wahash et al. studied a singular structure of fractional differential equations based on the newly-defined ϕ-derivatives and presented a modified Picard iterative method [26]. Abdo et al. [27] obtained some results in two directions of the existence and the U-H stability for a mixed structure of ϕ-Hilfer fractional intgro-differential equations.
The novelty of our suggested problem in comparison to problems (1) and (2) is that, in this paper, we consider a kind of general case of initial value problem in a configuration of a hybrid coupled system illustrated by (3). Indeed, fractional operators used in our problem are considered as generalized ones with respect to an increasing function ϕ, which implies that we can cover a wide range of fractional operators in our IVP (3) subject to the generalized kernels. This feature of ϕ-operators shows the importance and usefulness of these kinds of fractional operators compared to other ones. Also, unlike the two papers mentioned above, we here extend our problem to a hybrid coupled system of fractional integro-differential initial value problems with different sequential orders based on the generalized ϕ-RL-integral operators and in addition to the establishment of the existence and uniqueness results, we investigate the stability of the suggested coupled system in terms of the Ulam-Hyers stability and the generalized Ulam-Hyers one. In future work, we can implement these techniques on different boundary value problems equipped with complicated integral multi-point boundary conditions. The structure of this research work is as follows: Sect. 2 provides the auxiliary definitions along with desired lemmas. Section 3 is devoted to generalized U-H and U-H stability and the existence analysis for the given system of integro-differential IVPs (3)-(4). Moreover, we present a concrete example to emphasize the validity of the obtained outcomes.

Basic preliminaries
To achieve the desired fundamental purposes, we first review several basic auxiliary notions that are required throughout the manuscript.
The collection C = C(J, R) is designated as a collection consisting of continuous realvalued functions υ : J → R. Apparently, C is a Banach space along with the supremum norm υ = sup υ(z) : z ∈ J and is a Banach algebra under the action "·" defined by (υ · )(z) = υ(z) · (z) for υ, ∈ C and z ∈ J = [a, b]. Given the Banach algebra C, consider the product space E = C × C which is a vector space equipped with the coordinate-wise addition and scalar multiplication. Define a norm · in the product linear space E by (υ, ) = υ + .
Then the normed linear space (E, (·, ·) ) is a Banach space which further becomes a Banach algebra. The multiplication action between the members of E is illustrated by We start by characterizing ϕ-Riem-Lio fractional integrals and derivatives.
where denotes the standard Euler-Gamma function.
Equation (6) turns to the Riem-Lio and Hadamard fractional integrals by taking ϕ(z) = z and ϕ(z) = ln z, respectively. Moreover, the Cauchy formula for m-fold integrals can be obtained by considering ϕ(z) = z and α = 1: The left-sided ϕ-Caputo fractional derivative of an existing function υ ∈ C m (J, R) w.r.t. a non-decreasing function ϕ such that ϕ (z) = 0, for all z ∈ J in the fractional framework is represented as follows: For simplicity, we have From the definition, it is clear that Notice that, if υ ∈ C m (J, R), then the αth ϕ-Caputo fractional derivative of υ is determined by

Lemma 2.5 ([23])
Assuming α > 0, following assertions hold: If υ ∈ C m (J, R) and m -1 < α < m, then The following definition will be used in the sequel. We shall make use of the hybrid fixed point result due to Dhage [30,32] and the contraction principle due to Banach as a fundamental apparatus for demonstrating the existenceuniqueness result of the coupled solutions of the proposed system given in this paper. Theorem 2.8 ([30, 32]) Let X be a convex, bounded and closed set contained in the Banach algebra C and the operators P, S : C → C and Q : X → C be such that: (s1) P and S are Lipschitz maps with Lipschitz constants L P and L S , respectively; (s2) Q is continuous and compact; Then the operator equation υ = PυQυ + Sυ possesses a solution in X.

Main result
To start for verifying the main results, the following assumptions are required for us in the sequel: (HYP0) The real functions H 1 and H 2 are bounded on J × R × R subject to bounds M H 1 and M H 2 , respectively. And there exist M K i > 0 for i = 1, 2 such that (HYP1) There exist L H 1 > 0 and L H 2 > 0 such that for all z ∈ J and υ,ῡ, ,¯ ∈ R.
(HYP3) There exist bounded functions L F k , L G k : J → R + with bounds L F k and L G j such that for z ∈ J and υ,ῡ, ,¯ ∈ R.
with the initial condition then it satisfies the following hybrid fractional integral equation: Proof Applying the νth ψ-Riem-Lio fractional integral on both sides of (9) and using Lemma 2.5, we obtain which implies Using the initial condition υ(a) = 0, we have c 1 = υ(a) K 1 (a,υ(a), (a)) = 0. Now, substituting the value of c 1 in (13), we get The proof is finished.

Lemma 3.2 If a function
∈ C m (J, R) is taken as a solution for the hybrid fractional integro-differential equation (15) with the initial condition then it satisfies the following hybrid fractional integral equation: Proof The proof is similar to above.

Notation 3.3 For simplicity, take
and Choose ρ > 0 so that and specify a subset X of the Banach space C × C by Evidently, X is a convex, bounded and closed set contained in the Banach space C × C = E. Characterize the operators P = (P 1 , and ⎧ ⎨ ⎩ Q 1 (υ, ) = I ν;ϕ a + H 1 (z, υ(z), (z)), z ∈ J, Q 2 (υ, ) = I μ;ϕ a + H 2 (z, υ(z), (z)), z ∈ J, In this case, the coupled system of the given hybrid integral equations (20)- (21) can be represented in the framework of a system of operator equations as which further taking into account the multiplication given in (5) reduces to for z ∈ J. This further implies that ⎧ ⎨ ⎩ P 1 (υ, )(z)Q 1 (υ, )(z) + S 1 (υ, )(z) = υ(z), z ∈ J, Presently, we demonstrate in the following steps that all three operators P, Q and S follow the assertions of Theorem 2.8.
Step I: We first show that P = (P 1 , P 2 ) and S = (S 1 , S 2 ) are Lipschitzian on E with Lipschitz constants L P = (L K 1 + L K 2 ) and L S = (B 1 L F k + B 2 L G j ), respectively. Let (υ, ), (ῡ,¯ ) ∈ E be arbitrary. Then, using (HYP2), we have -¯ for all z ∈ J. Operating the supremum norm over z, we get -¯ for all (υ, ), (ῡ,¯ ) ∈ E. Along the same lines, we get -¯ for all (υ, ), (ῡ,¯ ) ∈ E. Accordingly, employing the definition of operator P, we get P(υ, ) -P(ῡ,¯ ) = P 1 (υ, ), P 2 (υ, ) -P 1 (ῡ,¯ ), P 2 (ῡ,¯ ) = P 1 (υ, ) -P 1 (ῡ,¯ ), P 2 (υ, ) -P 2 (ῡ,¯ ) ≤ P 1 (υ, ) -P 1 (ῡ,¯ ) + P 2 (υ, ) -P 2 (ῡ,¯ ) for all (υ, ), (ῡ,¯ ) ∈ E, where L P = (L K 1 + L K 2 ). Similarly, due to the definition of S and using (HYP3), we get -¯ for all z ∈ J. Operating the supremum over z, we get for all (υ, ), (ῡ,¯ ) ∈ E. Similarly using (HYP3), we can confirm that S 2 is also Lipschitzian with Lipschitz constant B 2 L G j ; that is, for all (υ, ), (ῡ,¯ ) ∈ E. Hence, from (23)- (24) it follows that for all (υ, ), (ῡ,¯ ) ∈ E. In consequence, S = (S 1 , S 2 ) is a Lipschitz map subject to the constant Step II: Now we show that Q = (Q 1 , Q 2 ) is a continuous and compact operator from X into E. To deduce the continuity of Q, we regard {(υ n , n )} n∈N as a sequence of points contained in X going to (υ, ) ∈ X. Then the dominated convergence result propounded by Lebesgue yields for all z ∈ J. Hence Q(υ n , n ) = (Q 1 (υ n , n ), Q 2 (υ n , n )) converges to Q(υ, ) pointwise on J. In the next, the compactness of Q is explored on X. Firstly, to ensure the uniform boundedness, by assuming (υ, ) ∈ X and applying (HYP0), we get Operating the supremum in terms of z in the above, we arrive at for all (υ, ) ∈ X. Hence Q 1 is a uniformly bounded operator on X. In a similar phase, we can guarantee that Q 2 involves the uniform boundedness specification on X subject to bound A 2 M H 2 . Accordingly, Q will be a uniformly bounded operator on X, because we have Next, to confirm the equicontinuity of Q, let (υ, ) ∈ X be an arbitrary point and let r, q ∈ J subject to r < q. Then we have This implies Q 1 (υ, )(q) -Q 1 (υ, )(r) → 0 as r → q uniformly for all (υ, ) ∈ X. Similarly, Q 2 (υ, )(q) -Q 2 (υ, )(r) → 0 as r → q uniformly for all (υ, ) ∈ X. Hence, it follows that uniformly for all (υ, ) ∈ X. Now, it is understood that Q has the equicontinuity feature on the Banach space E. In consequence, Q will be relatively compact and thus the conclusion of a result due to Arzelá-Ascoli shows that Q is completely continuous and in the final step, Q is compact on X.
Step IV: At last, we have From the above estimate and by (18), we obtain and so the hypothesis (s4) of Theorem 2.8 is obeyed. Accordingly, the operators P, Q and S obey all four assertions of Theorem 2.8 and thus the equation P(υ, )Q(υ, ) + S(υ, ) = (υ, ) possesses a solution in X. Consequently, the generalized coupled hybrid system of integro-differential IVPs (3)-(4) involves a mild coupled solution formulated on J. This finishes the argument.

Uniqueness via the Banach contraction principle
This section is devoted to demonstrating the uniqueness subject for the proposed coupled system of ϕ-Caputo integro-differential IVPs (3)-(4) by making use of Theorem 2.9.

Lemma 3.5
If the functions F k , G j : J × R 2 → R, K 1 , K 2 : J × R 2 → R\{0} and H 1 , H 2 : J × R 2 → R are continuous, then the coupled system of ϕ-Caputo integro-differential IVPs (3)-(4) is equivalent to the nonlinear fractional integral equations which take the form and for all z ∈ J.
Let (υ, ), (υ, ) ∈ E. Applying (HYP0)-(HYP3), we have subject to 1 given in (32). By the same technique, we can also get subject to 2 given in (32). In view of the condition 2 i=1 i < 1 and we see that G is a contraction. In the light of Theorem 2.9, G possesses a fixed point uniquely which guarantees that the system of the coupled integro-differential IVPs (3)-(4) involves a solution uniquely.

U-H stability and its generalized U-H version
In the current subsection, we are interested in studying U-H and the generalized U-H stability types of the proposed system of the coupled integro-differential IVPs (3)-(4).

Example
In this section, we present an illustrative coupled system of the given coupled hybrid integro-differential IVPs (3)-(4) to ensure the correctness of results obtained above.

Conclusion
In this research article, we investigate the existence and uniqueness of solutions to a coupled hybrid system of fractional integro-differential equations involving ϕ-Caputo fractional operators. To achieve the goals, we make use of a hybrid fixed point theorem for a sum of three operators due to Dhage and at the same time the uniqueness result is obtained by making use of the contraction principle. Moreover, we explore the Ulam-Hyers stability and its generalized version for the given coupled hybrid system. An example is presented to confirm the viability of our obtained results.