On a fractional Caputo\u2013Hadamard problem with boundary value conditions via different orders of the Hadamard fractional operators

Sina Etemad,Fethiye Muge Sakar,Shahram Rezapour

doi:10.1186/s13662-020-02741-7

Abstract

We investigate the existence of solutions for a Caputo–Hadamard fractional integro-differential equation with boundary value conditions involving the Hadamard fractional operators via different orders. By using the Krasnoselskii’s fixed point theorem, the Leray–Schauder nonlinear alternative, and the Banach contraction principle, we prove our main results. Also, we provide three examples to illustrate our main results.

Highlights

1 Introduction In recent decades, it has become clear to researchers that studying different types of fractional differential equations is of particular importance
By using main ideas of the aforementioned articles, we investigate the Caputo–Hadamard fractional integro-differential equation of different orders: κCHD1+ + (1 – κ)CHD1+ w(t) = αψ t, w(t) + βH I1μ+ φ t, w(t), (1)
5 Conclusions It is known that we should increase our ability for studying of different types of fractional integro-differential equations

Summary

Introduction

It has become clear to researchers that studying different types of fractional differential equations is of particular importance. In 2014, Ahmad et al investigated the existence of solutions for the nonlinear fractional q-difference equation equipped with four-point nonlocal integral boundary conditions By using main ideas of the aforementioned articles, we investigate the Caputo–Hadamard fractional integro-differential equation of different orders: κCHD1+ + (1 – κ)CHD1+ w(t) = αψ t, w(t) + βH I1μ+ φ t, w(t) , (1) The Hadamard fractional integral of a continuous function w : (a, b) → R of order is defined by (H Ia0+ w)(t) = w(t) and

Results

Conclusion