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

Abstract

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.

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

“…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.…”

Some Existence and Dependence Criteria of Solutions to a Fractional Integro-Differential Boundary Value Problem via the Generalized Gronwall Inequality

“…The parameters K 1 = 0.9, K 2 = 0.5, σ = 2.44, q 1 = 3.44, q 2 = 2.55, and θ = 2.33 are supposed according to the above suggested CFBVP (17). Besides, functionsh 1 ,h 2 : [0, 1] × R → R displayed as Finally, by taking into account the above constants, we find that ξ ≈ 1.2981 and ∆ (1) +L * ∆(2) + R∆ (3) ≈ 0.1848 < 1.…”

“…(see [5][6][7][8][9][10][11][12] and references therein). Since theoretical findings and outcomes can support the arrival at a profound understanding for the arbitrary-order models, a large number of mathematicians have tended to discuss the existence and dependence aspects of solutions of various structures of fractional equations (see [13][14][15][16][17][18][19][20][21][22][23][24][25]). In [26] a discrete Gronwall inequality was introduced to provide a stability bound.…”

Two fractional hybrid and non‐hybrid boundary value problems

“…What is more important to researchers today than anything else is to understand some qualitative properties of solutions for these fractional dynamical systems modeled based on different complicated fractional boundary value problems (BVPs) of hybrid or non‐hybrid type. In this direction, a lot of works have been published about different types of fractional integro‐differential and differential equations, 1‐6 integro‐differential equations involving the Caputo–Fabrizio derivative, 7‐11 hybrid differential equations, 12‐19 approximate solutions of different fractional differential equations, 20‐23 fractional mathematical modelings, 24‐31 and numerical techniques in different fields of fractional calculus 32‐45 …”