We investigate the following fractional hybrid differential equation:

           (1.0)

where  is the Riemann-Liouville differential operator order of   for some and satisfies certain conditions. We investigate such equations in two cases:  and In the first case, we prove the existence and uniqueness of a solution of (1.0), which extends the main result of [1]. Moreover, we show that the Picard iteration associated to an operator converges to the unique solution of (1.0) for any initial guess  In particular, the rate of convergence is  In the second case, we investigate this equation in the space of k times differentiable functions. Naturally, the initial condition  is replaced by   and the existence and uniqueness of a solution of (1.0) is established. Moreover, the convergence of the Picard iterations to the unique solution of (1.0) is shown. In particular, the rate of convergence is  Finally, we provide some examples to show the applicability of the abstract results. These examples cannot be solved by the methods demonstrated in [1].

fixed point theorem, Riemann-Liouville fractional derivative, hybrid initial value problem.

Received: August 8, 2022; Revised: September 24, 2022; Accepted: October 14, 2022; Published: November 9, 2022

How to cite this article: Sahar Mohammad A. Abusalim, Solutions of fractional hybrid differential equations via fixed point theorems and Picard approximations, Advances in Differential Equations and Control Processes 29 (2022), 65-100. http://dx.doi.org/10.17654/0974324322034

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

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