Existence and approximation of solutions to fractional order hybrid differential equations

Dussadee Somjaiwang,Parinya Sa Ngiamsunthorn

doi:10.1186/s13662-016-0999-8

Abstract

We consider fractional hybrid differential equations involving the Caputo fractional derivative of order $0<\alpha<1$ . 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.

Highlights

1 Introduction Fractional differential equations are of interest in many areas of applications, such as economics, signal identification and image processing, optical systems, aerodynamics, biophysics, thermal system materials and mechanical systems, control theory
There are several results that investigate the existence of solutions of various classes of fractional differential equations
Much attention has been focused on the study of the existence and multiplicity of solutions as well as positive solutions for boundary value problem of fractional differential equations [ – ]

Summary

Introduction

Fractional differential equations are of interest in many areas of applications, such as economics, signal identification and image processing, optical systems, aerodynamics, biophysics, thermal system materials and mechanical systems, control theory (see [ – ]). The existence result for solutions has been generalized to the fractional order hybrid differential equation dα x(t) – f t, x(t) = g t, x(t) dtα a.e. t ∈ [t , t + a],. Herzallah and Beleanu [ ] discussed the existence of mild solutions for the above fractional order hybrid differential equation Section is devoted to a proof of the existence and approximation of mild solutions of fractional order hybrid differential equations The function x ∈ C(J, R) is called a mild solution of the fractional nonlinear hybrid ordinary differential equation The following hybrid fixed point result of [ ] is often applied to establish the existence and approximation of solutions of various differential and integral equations.

The graphs of these two functions are shown in Figure

Mg π