On Hyers–Ulam stability of a multi-order boundary value problems via Riemann–Liouville derivatives and integrals

In this research paper, we introduce a general structure of a fractional boundary value problem in which a 2-term fractional differential equation has a fractional bi-order setting of Riemann–Liouville type. Moreover, we consider the boundary conditions of the proposed problem as mixed Riemann–Liouville integro-derivative conditions with four different orders which cover many special cases studied before. In the first step, we investigate the existence and uniqueness of solutions for the given multi-order boundary value problem, and then the Hyers–Ulam stability is another notion in this regard which we study. Finally, we provide two illustrative examples to support our theoretical findings.

In 2016, Niyom et al. studied the boundary value problem via four-order fractional Riemann-Liouville derivatives ⎧ ⎨ ⎩ λD k (u(t)) + (1λ)D θ (u(t)) =Υ(t, u(t)) (t ∈ [0, T], k ∈ (1, 2]), under some conditions [55]. In 2017, Ntouyas et al. reviewed a boundary value problem via multiple orders of fractional derivatives and integrals ⎧ ⎨ ⎩ λD k (u(t)) + (1λ)D θ (u(t)) =Υ(t, u(t)) (t ∈ [0, T], k ∈ (1, 2]), under some conditions [15]. In 2018, Xu et al. investigated the existence of solutions and Hyers-Ulam stability for the fractional differential equations ⎧ ⎨ ⎩ λD k (u(t)) + D θ (u(t)) =Υ(t, u(t)) (t ∈ [0, T], k ∈ (1, 2]), under some conditions [39]. They considered two-term class of three-point boundary value problems with Riemann-Liouville fractional derivatives and integrals [39]. Now, by using and mixing the idea of the above-mentioned works, we consider a new category of boundary value problem including multi-order Riemann-Liouville fractional equation supplemented with different linear integro-derivative boundary conditions as follows: where 2 < θ < k, 0 < λ, μ 1 , μ 2 ≤ 1, 0 ≤ γ 1 , γ 2 < kθ , q 1 , q 2 ∈ R + , D β is the Riemann-Liouville fractional derivative of order β ∈ {k, θ , γ 1 , γ 2 }, I η denotes the Riemann-Liouville fractional integral of order η ∈ {q 1 , q 2 }, and the mapΥ : [0, T] × R → R is continuous. As many researchers would like to investigate the stability notion of different boundary value problems, this can be a motivation for us to study the stability of complicated systems supplemented with general boundary conditions. Hence more precisely, our main goal in the present manuscript is to obtain some existence criteria of the solutions for a new general boundary value problem including 2-term fractional differential equation (4) which contains multi-order Riemann-Liouville fractional derivatives and integrals. To fulfil this aim, we use the well-known standard fixed point theorems. Also, in the sequel, we investigate the Hyers-Ulam stability of the proposed problem (4) in the special case μ 1 = 1 and μ 2 = 1. Finally, we present two illustrative examples to show the validity of our theoretical findings. We believe that such proposed boundary value problem is general, and it involves many fractional dynamical systems as special cases in physics and other applied sciences.

Preliminaries
Now, let us provide some basic notions. It is known that the Riemann-Liouville fractional integral of order η of a real-valued function g : (0, ∞) → R is defined by I η g(t) = t 0 (t-s) η-1 (η) g(s) ds, provided the right-hand side is point-wise defined on (0, ∞), where is the gamma function [1]. The Riemann-Liouville fractional derivative of order k of a func- (t-s) β-n+1 ds, where n = [β] + 1, [β] denotes the integer part of real number β provided the right-hand side is point-wise defined on (0, ∞) [1]. We need the next results.
Theorem 3 (Krasnoselskii's fixed point theorem, [56]) Let M be a closed, bounded, convex, and nonempty subset of a Banach space X . Assume that A 1 and A 2 are two operators on M such that Theorem 4 (Leray-Schauder's nonlinear alternative, [57]) Let X be a Banach space, B be a closed, convex subset of X , U be an open subset of B, and 0 ∈ U . Assume that P :Ū → B is a continuous and compact map. Then either (a) P has a fixed point inŪ , or (b) there is u ∈ ∂U (the boundary of U ) and τ ∈ (0, 1) with u = τ P(u).
Theorem 6 (Banach contraction principle, [57]) Let X be a Banach space and P : X → X be a contraction. Then P has a unique fixed point.

Some existence results
Let T > 0, J = [0, T], and C = C(J, R) be the Banach space of continuous mappings with the sup norm u = sup t∈J |u(t)|. We first provide our key result.

Lemma 7 A map u 0 is a solution for boundary value problem (4) if and only if u 0 is a solution for the integral equation
Proof First, assume that u 0 is a solution for problem (4). Then we have By taking the Riemann-Liouville fractional integral of order k from both sides of equation (7), we obtain for some real constants C 1 , C 2 , and C 3 . For 2 < k < 3, the first boundary condition of (4) implies that C 3 = 0. Hence, By using the Riemann-Liouville fractional derivative and integral of order α and β respec- By replacing the values α = γ 1 , α = γ 2 , β = q 1 , and β = q 2 and using the second condition of (4), we get and which leads to and By inserting the values of constants C 1 and C 2 in (8), we find that u 0 satisfies (5). Some calculations show that the converse part holds. This completes the proof.
Based on Lemma 7, define the operator F : C → C by Note that boundary value problem (4) has solution u 0 if and only if u 0 is a fixed point of the operator Fu. To simplify calculations, we use the notations and Theorem 8 Suppose thatΥ : J × R → R is a continuous map and there exists a constant L > 0 such that |Υ(t, u) -Υ(t, u )| ≤ L|uu | for all t ∈ J and u, u ∈ R. If LW 2 + W 1 < 1, then problem (4) has a unique solution, where W 1 and W 2 are defined by (10) and (11).
Proof Put sup t∈J |Υ(t, 0)| = N < ∞ and choose where i i ∈ {1, 2, 3, 4} are defined by (7). Set B R = {u ∈ C : u ≤ R}. We show that FB R ⊂ B R . For each u ∈ B R , we have Thus, Fu ≤ R and so FB R ⊂ B R . Let u, u ∈ C. For each t ∈ J, we have Hence, Fu -Fu ≤ (LW 2 + W 1 ) uu and so F is a contraction. By using the principle of contraction, F has a unique fixed point which is the unique solution for problem (4).
Here, by using Krasnoselskii's fixed point theorem, we provide our next existence result.

Theorem 9
Suppose thatΥ : J × R → R is a continuous map and there exists a constant L > 0 such that |Υ(t, u) -Υ(t, u )| ≤ L|uu | for each t ∈ J and u, u ∈ R. If there is V(t) ∈ C(J, R + ) such thatΥ(t, u) ≤ V(t) for all (t, u) ∈ J × R and W 1 < 1, then problem (4) has at least one solution. Here, W 1 is given by (10).
and 1 , 2 , 3 , 4 , and W 1 are given by (7) and (10), respectively. For each t ∈ J, define the operators F 1 and F 2 on B r by and We show that F 2 u + F 2 u ∈ B r . Let u, u ∈ B r . Then we have and so F 1 u + F 2 u ∈ B r . Now, we prove F 1 is a contraction. For every u, u ∈ B r , we have Since W 1 < 1, F 1 is a contraction. Utilizing the continuity of the functionΥ , we find that the operator F 2 is continuous. If u ∈ B r , then This means that the operator F 2 is uniformly bounded on B r . Now, we show that F 2 is equicontinuous. Set sup t∈J,u∈B r |Υ(t, u)| = M. For each t 1 , t 2 with t 2 > t 1 and u ∈ B r , we have The right-hand side of the above inequality tends to zero independently of u as t 2 tends to t 1 . Hence, F 2 is equicontinuous, and so F 2 is relatively compact on B r . Now, by using the Arzela-Ascoli theorem, F 2 is compact on B r . Now, by using Theorem 3, boundary value problem (4) has at least one solution.
Here, by applying the Leray-Schauder theorem, we give another existence result.
where W 1 , W 2 are defined by (10) and (11), respectively. Then boundary value problem (4) has at least one solution.
Proof Consider the operator F defined by (9). We show that F maps bounded sets into bounded sets of C. Let ρ > 0 and B ρ = {u ∈ C : u ≤ ρ} be a bounded ball in C and t ∈ J. Then we have and consequently Now, we prove that the operator F maps bounded sets into equicontinuous sets of C.
Assume that t 1 , t 2 ∈ J with t 1 < t 2 and u ∈ B ρ . Then we have ω ∈ (0, 1). Let u be a solution of the last equation. For each t ∈ J, we have One can check that the operator F :Ū → C is continuous and completely continuous. In view of the choice of U , there is no u ∈ ∂U so that u = ωF u for some ω ∈ (0, 1). Now, by using the Leray-Schauder theorem, the operator F has a fixed point u ∈Ū which is a solution of boundary value problem (4).

Stability analysis
In this section, we study the Hyers-Ulam stability of the boundary value problem which is a special case of problem (4) when we take μ 1 = μ 2 = 1. (12) is called Hyers-Ulam stable whenever there exists a real constant > 0 such that, for each ε > 0 and u(t) ∈ C R ([0, T]) satisfying
Theorem 12 Suppose thatΥ : J × R → R is a continuous map and there exists a constant L > 0 such that |Υ(t, u) -Υ(t, u )| ≤ L|uu | for all t ∈ J and u, u ∈ R. Then boundary value problem (12) is Hyers-Ulam stable.

Examples
Now, we provide two examples to illustrate our main results.