Existence results for a general class of sequential hybrid fractional differential equations

In this paper, we study a class of nonlinear boundary value problems (BVPs) consisting of a more general class of sequential hybrid fractional differential equations (SHFDEs) together with a class of nonlinear boundary conditions at both end points of the domain. The nonlinear functions involved depend explicitly on the fractional derivatives. We study the necessary conditions required for the unique solution to the suggested BVP under the Caratheodory conditions using the technique of measure of noncompactness and degree theory. We also develop conditions for uniqueness results and also on stability analysis.


Introduction
The existence theory for solutions of BVPs of hybrid fractional differential equations and SHFDEs has attracted the attention of many researchers, we refer to [1][2][3][4][5][6][7][8][9][10][11] and the references therein for the recent development in this particular area of interest. In most of these studies, BVPs with lower order fractional derivatives together with either constant or linear boundary conditions are considered. However, in many situations, there are possibilities to have nonlinear conditions at the boundary, and the differential equations may be of higher order involving functions that depend explicitly on the fractional order derivatives. For example, in case of head flow problems, there are possibilities to have some source or sink on both sides of the boundary (at x = 0 and x = 1) which may be nonlinear functions and a controller at x = ζ 0 (0 < ζ 0 < 1). Such situation may have importance in application point of view and also in theoretical development. The purpose of this paper is to investigate existence results for BVPs involving nonlinear boundary conditions at both end points, that is, we study the following class of three point © The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Proposition 2.4
If T : 0 → E satisfies Lipschitz with constant k, then T is * * -Lipschitz with the same constant k.
The following theorem [14] will be used in the sequel.
If is a bounded set in E, that is, there exists r > 0 such that ⊂ B r (0), then the topological degree that is, T has a fixed point in B r (0).
Organization of the paper. This article consists of five sections. The first section explains the importance of the article and the related literature. In the second section, we study sufficient conditions for the existence and uniqueness of solutions to the hybrid fractional differential equations (1). Third section is reserved for the Hyers-Ulam stability of problem (1). Section 4 explains the application of the results, and finally the conclusion of the article is given in Sect. 5.

Existence criteria
This section of the article is reserved for the existence and uniqueness of solution of hybrid problem (1) with the help of the fixed point approach. For these, we first transform the suggested problem into an integral form of the problem.

Lemma 3.3 Under conditions (H 1 ) and (H 3 ), the operator A is * * -Lipschitz with zero
constant. Further A satisfies the following growth condition: Proof By (H 1 ), the continuity of h i , with respect to u for each fixed t ∈ I implies the continuity of the operator A for each fixed t ∈ I. Moreover, for each u ∈ E, using (H 3 ), we obtain f t, u(t), D ω-1 u(t) ≤ * * (t) u δ + ξ , g t, u, I γ u ≤ ρ(t) + ν u q + I γ u q .

Hyers-Ulam stability
In this section, we present the Hyers-Ulam stability analysis for the hybrid fractional differential equation (1). For more related problems to the Hyers-Ulam stability, the readers may take help from the references in [15][16][17][18][19][20] and the literature. (13) is said to be Hyers-Ulam stable if there exists a constant ζ > 0 such that, for given ϕ > 0 and for each solution u of the inequality