On stability analysis and existence of positive solutions for a general non-linear fractional differential equations

Amita Devi,Dumitru Baleanu,Aziz Khan,Anoop Kumar

doi:10.1186/s13662-020-02729-3

Abstract

In this article, we deals with the existence and uniqueness of positive solutions of general non-linear fractional differential equations (FDEs) having fractional derivative of different orders involving p-Laplacian operator. Also we investigate the Hyers–Ulam (HU) stability of solutions. For the existence result, we establish the integral form of the FDE by using the Green function and then the existence of a solution is obtained by applying Guo–Krasnoselskii’s fixed point theorem. For our purpose, we also check the properties of the Green function. The uniqueness of the result is established by applying the Banach contraction mapping principle. An example is offered to ensure the validity of our results.

Highlights

1 Introduction Fractional calculus concerns the applications of derivatives and integrals of arbitrary order
Khan et al [44] discussed the analytical study of existence and stability results of a singular non-linear fractional differential equations (FDEs) with φp-operator involving fractional integral and differential boundary conditions
Existence and stability of solutions for singular delay FDEs with fractional integral initial conditions by using the Green function and the fixed point theorem were established by the Khan et al in [48]

Summary

Introduction

Fractional calculus concerns the applications of derivatives and integrals of arbitrary order. Khan et al [44] discussed the analytical study of existence and stability results of a singular non-linear FDEs with φp-operator involving fractional integral and differential boundary conditions. The EU and HU stability of solutions for a coupled system of FDEs involving the derivative in Caputo’s sense are proved by Khan et al [45] using a Leray–Schauder-type fixed point theorem and topological degree theory. Existence and stability of solutions for singular delay FDEs with fractional integral initial conditions by using the Green function and the fixed point theorem were established by the Khan et al in [48]. Motivated by the above work, we introduce the EU and HU stability results, for nonliner FDEs involving Caputo fractional derivatives of distinct orders with φP∗ Laplacian operator:. By Eqs. (2.14) and (2.15), condition B3 is proved

Existence result

Conclusion