A study on multiterm hybrid multi-order fractional boundary value problem coupled with its stability analysis of Ulam–Hyers type

In this research work, a newly-proposed multiterm hybrid multi-order fractional boundary value problem is studied. The existence results for the supposed hybrid fractional differential equation that involves Riemann–Liouville fractional derivatives and integrals of multi-orders type are derived using Dhage’s technique, which deals with a composition of three operators. After that, its stability analysis of Ulam–Hyers type and the relevant generalizations are checked. Some illustrative numerical examples are provided at the end to illustrate and validate our obtained results.


Introduction
Most researchers in the history of mathematics place the origin of fractional calculus in a work by Leibniz where he introduces the notation of the nth derivative of an arbitrary function y; that is, d n y dx n with n ∈ N. But does it make sense to extend the values of n in that expression to other numeric fields?. The idea of fractional derivative materialized in 1695, when L'Hopital asked what d n y dx n means if n = 1 2 . After such an idea having appeared, many extended definitions of this concept have been constructed under two conceptions: global (classical) and local. In the first conception, the fractional derivative is defined as integral, Fourier or Mellin transformations, which means that its nature is not local and has a memory effect. The second conception of fractional derivative is based on a local definition through certain incremental ratios. The global formulation is associated with the appearance of the fractional calculus itself, going back to the pioneering work of Euler, Laplace, Lacroix, Fourier, Abel, Liouville, etc. until the establishment of the classic definitions of Riemann-Liouville and Caputo. Thus, the classical theory of fractional calculus constitutes a mathematical analysis tool applied to the investigation of arbitrary order integrals and derivatives, which extends the concepts of integer-order differentiation and n-fold integration.
Furthermore, the study of the practical and theoretical elements of fractional differential equations has became a base of academic advanced research [1][2][3][4]. Many fractional differential equations, particularly boundary value problems, have gathered the research interests of researchers in applied mathematics, theoretical physics, and engineering due to their nonlocality and their powerful flexibility in modeling complex scientific and physical phenomena that show the memory effect. The dynamics and behavior of certain physical systems can be explained better with respect to fractional derivatives and fractional integrals than for classical integer-order systems. In recent years, the great potential of these integrals and derivatives has been revealed in various fields of natural sciences and technology, such as biology, fluid mechanics, biomathematics, physics, image processing, chemistry, and entropy theory .
Recently, the fractional formulation of boundary value problems related to hybrid differential equations has received the interests of the most researchers. The 1th order hybrid differential equations involving first and second kind of disturbances have been discussed, using the Riemann-Liouville derivatives in [37,38]. In [39], the authors turned to the existence property of solutions to hybrid fractional differential equations by terms of both types applying the Caputo derivative. In 2015, similar results for fractional initial value problems involving hybrid integro-differential equations are established [40] by Sitho et al. The existence problems of mild solution for hybrid fractional differential equations involving the Caputo fractional derivative of arbitrary order are investigated in [41] by Mahmudov in 2017. For similar research, refer to [42][43][44].
In [45], Ben Chikh et al. proved the unique solution's existence and various stability's types for a boundary value problem involving Riemann-Liouville integrals and then in [46], they implemented same results for a newly-formulated four-point Caputoconformable fractional problem involving boundary conditions of the Riemann-Liouville conformable type (for more background information about conformable derivative, refer to [47,48]) formulated as Due to the importance and flexibility of hybrid differential equations in modeling of electromagnetic waves, deflection of a curved beam, gravity driven by flows etc., Baleanu et al. [49] designed a hybrid fractional boundary value problem of thermostat control model and discussed required existence specifications of its solutions in the form where γ ∈ (1, 2], γ -1 ∈ (0, 1], η ∈ I, c D 1 = d dt , k > 0. Inspired by the above previous work, we investigate in this work a generalized hybrid problem: indeed, we prove the existence of solution for the hybrid fractional differential equation including a finite number of Riemann-Liouville derivatives and Riemann-Liouville integrals of different orders of the following form: . . , n and t ∈ J := [0, K]. Also, D τ represents the τ th Riemann-Liouville fractional derivative, I η denotes the ηth Riemann-Liouville integral, and where k i > 0 and the maps H, andˆ : [0, K] × R n+1 → R and A : [0, K] × R n+1 → R/{0} are continuous. In the above suggested structure given by (3), we have several nonlinear functions depending on their components. This type of hybrid fractional boundary value problem can be employed in description and modeling non-homogeneous physical processes. The Dhage technique, based on some nonlinear operators, will be used here regarding the existence property of given fractional boundary value problem (3). In spite of some previous standard work regarding solutions of the fractional differential equation, we here aim to study some qualitative properties of solutions to a novel hybrid fractional boundary value problem which is a more complicated system. Naturally, if one can analyze the behavior of such a hybrid system, then we will be able to simulate other real phenomena based on these hybrid fractional differential equations.
The scheme of the paper is organized in such an order: In the next section, we present some essential fractional calculus definitions and notions that will be applied. Next, the existence results for the multiterm hybrid fractional differential equation are established

Essential preliminaries
Some essential fractional calculus definitions and notions that will be used later are presented in this section.

Definition 2.1 ([1])
The ς th Riemann-Liouville fractional integral of a given mapping ψ : (0, ∞) → R is expressed as The ς th Riemann-Liouville fractional derivative of a given function ψ : (0, ∞) → R is expressed as From the definition of the Riemann-Liouville fractional derivative, we get the following.

Results regarding to the existence property
We turn to the investigation of our required existence criteria in the current situation. The notation C = C(J, R) represents the space of all continuous mappings from J = [0, K] to R with actions (C, · , ·) is a Banach algebra. The next lemma is key.

Lemma 3.1 Assume thatχ is a continuous function on
Then the solution of the hybrid fractional boundary value problem is expressed as satisfies the following equation: where the nonzero constant i , i ∈ {1, 2, 3, 4} is defined by Proof According to the first equation in (4), we obtain Let us now take the Riemann-Liouville fractional integral of order k to (7), Thus, we have Let us apply the Riemann-Liouville fractional derivative and integral of order γ , q, respectively on both sides of (8) By substituting the values γ = β * 1 , γ = β * 2 , q = m * 1 and q = m * 2 into the above one and using the second condition in (4), we have and Let us now substitute the constants' value of A 1 and A 2 into (8) by which Eq. (5) is derived and our proof is ended.
Some essential hypotheses are presented as follows.
and by Lemma 3.1, the solution of the multiterm hybrid fractional boundary value problem (3) corresponds to the equation We build the operators A, C : C → C and B : B R → C by and where A, K and H are illustrated before. Then the integral equation (14) can be expressed in a form which is denoted by We will prove that all of A, B, and C fulfill all items of Theorem 2.6.
STEP I: We first prove that A and C are Lipschitz on C. Assume that ρ, v ∈ C. Then, from (H2), for t ∈ J, we obtain Now, for C : C − → C, u, v ∈ C, we obtain Hence, C : C → C involves the same property on C with constant n i=1 n j=0 K ϑ i +k j (ϑ i + k j + 1) > 0.
STEP II: In this step, we prove the complete continuity of B formulated on B R . First of all, assume that {ρ n } is a sequence in B R which converges to a point ρ ∈ B R . From and by the Lebesque dominated convergence theorem, we immediately get To check the uniform boundedness of B(B R ) in B R , for any ρ ∈ B R , we get for all ρ ∈ B R with Q * illustrated in (11). This yields the required result in this part for B on B R .
Let us now prove that B(B R ) is equi-continuous in C. Assume that t 1 < t 2 ∈ J. Then for any ρ ∈ B R we have Hence, we get This implies that Thus, from the Arzela-Ascoli theorem, we arrive at the complete continuity of B on B R . STEP III: The (H3) of Theorem 2.6 is fulfilled. Assume that ρ ∈ C and v ∈ B R are arbitrary elements via ρ = AρBv + Cρ. Then, by (11) and (18), we get A(t, 0, 0, . . . , 0) H(t, 0, 0, . . . , 0) and thus In consequence, We obtain STEP IV: We prove that l * 1 * + l * 2 < 1, in which the item (4) of Theorem 2.6 occurs.
by the above calculations, we obtain Therefore, all items of Theorem 2.6 are fulfilled, and so it is found a solution for ρ = AρBρ + Cρ and also for the multiterm hybrid fractional boundary value problem (3) on J. This ends our argument.

Definition 4.1 ([51]) The multiterm hybrid fractional boundary value problem (19) is
Ulam-Hyers stable whenever some c ∈ R + exists so that ∀ε > 0 and ∀v * ∈ C as a solution function satisfying the inequality there exists another solution function ρ ∈ C for the multiterm hybrid fractional boundary value problem (19) with  (19) is named generalized Ulam-Hyers stable if ϕ I σ i K ∈ C R + (R + ) exists with ϕ n i=1 I σ i K (0) = 0 so that, for any solution function v * ∈ C of inequality (20), another function ρ ∈ C exists satisfying the multiterm hybrid fractional boundary value problem (19) is valid.  (19) is Ulam-Hyers-Rassias stable which is dependent on ϕ : [0, K] → R + whenever ∃c ϕ ∈ R + so that ∀ε > 0 and ∀v * ∈ C as a solution of the inequality there exists another solution function ρ(t) ∈ C of the multiterm hybrid fractional boundary value problem (19) satisfying (19) is said to be generalized Ulam-Hyers-Rassias stable depending on ϕ : [0, K] → R + if ∃c ϕ ∈ R + so that ∀ε > 0 and ∀v * ∈ C as a solution of the inequality

Definition 4.4 ([51]) The multiterm hybrid fractional boundary value problem
another solution ρ(t) ∈ C exists for the multiterm hybrid fractional boundary value problem (19) satisfying Remark 4.5 ( [51]) v * (t) ∈ C is named as a solution for (20) iff some function g ∈ C exists which is dependent on v * and (i) |g(t)| < ε, Theorem 4.6 Letˆ : [0, K] × R n+1 → R + be continuous and ∃L * ∈ R + so that If the second condition of (H2) holds, then the multiterm hybrid fractional boundary value problem (19) is Ulam-Hyers stable on [0, K] and accordingly is generalized Ulam-Hyers stable if Proof For ε > 0, and every solution v * (t) ∈ C of the inequality there is found a function g with We get where W and V are defined in (12) and (13) .
If we put Now, we discuss the Ulam-Hyers-Rassias stability of solution to the problem (19). with For the sake of simplicity, we take Then v * (t)ρ(t) ≤ εc ϕ ϕ(t).

Numerical examples
Some illustrative numerical examples will be given in this section to apply and validate our theoretical results.
The conditions of Theorem 4.6 imply that the aforementioned problem (31) is Ulam-Hyers stable and also accordingly is generalized Ulam-Hyers stable.

Conclusion
The existence results for the proposed multiterm hybrid fractional boundary value problem that involves the Riemann-Liouville operators of finitely many orders have been successfully investigated. With the help of three operators having specific properties, we implemented the defined method in Dhage's technique for ensuring the existence of solutions. The stability criteria in different versions are checked for a special case. Some relevant numerical examples are provided to validate our obtained theoretical results. The supposed hybrid fractional boundary value problem (3) is thoroughly abstract and general but involves some special formats by assuming some specific parameters. One can extend it to the differential inclusion by terms of multi-valued version of Dhage's technique in future work. In the next work, one can use generalized fractional operators with singular or non-singular kernels to model real hybrid systems such as the thermostat equation, the pantograph equation, and the Langevin equation, and to analyze their qualitative behaviors theoretically and numerically.