Existence and approximation of solutions to fractional order hybrid differential equations

Abstract

We consider fractional hybrid differential equations involving the Caputo fractional derivative of order 0 < α < 1. Using fixed point theorems developed by Dhage et al. in Applied Mathematics Letters 34, 76-80 (2014), we prove the existence and approximation of mild solutions. In addition, we provide a numerical example to illustrate the results obtained. Primary 34A08; secondary 34A38; 34A45 MSC:

Approximate solutions for a fractional hybrid initial value problem via the Caputo conformable derivative

“…In this manuscript, we intend to study the existence and approximation of solutions for a fractional hybrid multi-term integro-differential equation in the Caputo conformable setting. It is notable that the main motivation for this work is taken from a valuable paper published by Somjaiwang and Ngiamsunthorn in 2016 [32]. They formulated a fractional hybrid IVP involving the Caputo derivative as follows:…”

“…where d ds is the Caputo derivative with respect to s ∈ J = [s 0 , s 0 + k] of fractional order ∈ (0, 1] with s 0 ≥ 0 and k > 0. The authors dealt with the existence of mild solutions for this hybrid IVP and also they obtained an approximation of solutions by the help of an iterative sequence [32].…”

“…By extending the hybrid initial problem mentioned in [32] to the newly defined fractional operator introduced by [30], we intend to investigate the existence and approximation of solutions for the following Caputo conformable fractional hybrid initial value problem:…”

“…Notice that our proposed hybrid initial problem is an extended version of the aforementioned hybrid problem (1). Also, it is necessary to mention that we apply the same techniques used in the paper published by Somjaiwang and Ngiamsunthorn [32] but for a new structure of the multiterm integro-differential equation equipped with newly-defined Caputo and Riemann-Liouville conformable derivative operator given by (2). In other words, in the current research work, we discuss two results on the existence and approximation of solutions simultaneously.…”

On Some Qualitative Properties of Integrable Solutions for Cauchy-type Problem of Fractional Order

“…t ∈ [t 0 , t 0 + a], u(0) = u 0 , has been discussed in [19] with the Riemann-Liouville fractional derivative and in [16] studied the existence of mild solutions for the above hybrid differential equation using the Caputo fractional derivative. This was extended by [24] to give approximation of solutions to Caputo fractional order hybrid differential equations. The authors (see [15]) studied the Cauchy-type problem:…”