A Study of Fractional Differential Equation with a Positive Constant Coefficient via Hilfer Fractional Derivative

Vu Ho,Hoa Ngo Van

doi:10.1155/2020/2749138

Abstract

The aim of the paper is to consider the existence and uniqueness of solution of the fractional differential equation with a positive constant coefficient under Hilfer fractional derivative by using the fixed-point theorem. We also prove the bounded and continuous dependence on the initial conditions of solution. Besides, Hyers–Ulam stability and Hyers–Ulam–Rassias stability are discussed. Finally, we provide an example to demonstrate our main results.

Highlights

In recent years, the study of the fractional differential and integral equation (FDE and IDE for short) has become the topic of the applied mathematics
FDE and IDE have been used as a tool mathematical to the modeling of many phenomena in various fields for example, in theory of signal processing, physics, economics, and chaotic dynamics. e reader can refer to the books or the papers
The authors considered the properties of solutions of this problem such as the boundedness of solution and the continuous dependence of solutions on the initial conditions

Summary

Introduction

The study of the fractional differential and integral equation (FDE and IDE for short) has become the topic of the applied mathematics. By using the ψ− Hilfer fractional derivative, Sousa and Oliveira [17] studied the existence and uniqueness of solution of the initial valued problem for FDEs. e continuous dependence of solution on the initial condition was considered. [14, 15], Sousa et al [16], and Kharade et al [23], in this paper, we investigate the existence and uniqueness of solutions and some properties of solutions of the following fractional differential equation with the constant coefficient λ > 0: H 0+ Hyers–Ulam–Rassias stable, with respect to φ ∈ C([0, a], R), if there exists a positive constant Cζ,φ such that, for any ε > 0 and for each ξ ∈ C1− c,g([0, a]) satisfying the inequality,

Results

Conclusion