Positive solutions of singular Hadamard-type fractional differential equations with infinite-point boundary conditions or integral boundary conditions

A M A El-Sayed,F M Gaafar

doi:10.1186/s13662-019-2315-x

Abstract

We establish the existence of positive solutions to a class of a singular nonlinear Hadamard-type fractional differential equations with infinite-point boundary conditions (BCs) or integral BCs. Our analysis is based on Leray–Schauder type continuation. Several examples are given to illustrate our results.

Highlights

1 Introduction Our aim in this article is to study the problem of existence of continuous solutions of the following singular Hadamard fractional differential equation: HDγ v(t) + f t, v(t), HDδv(t), v (t) = 0, a.e. t ∈ (1, e), (1)
Together with either the functional integral boundary condition given by e φ (ξ )
We first find the existence of positive solutions of the problem (1) subject to the multi-point boundary conditions m v(1) = 0, v(e) = v0 + λ ajv φ(ηj)

Summary

Introduction

1 Introduction Our aim in this article is to study the problem of existence of continuous solutions of the following singular Hadamard fractional differential equation: HDγ v(t) + f t, v(t), HDδv(t), v (t) = 0, a.e. t ∈ (1, e), (1) We first find the existence of positive solutions of the problem (1) subject to the multi-point boundary conditions m v(1) = 0, v(e) = v0 + λ ajv φ(ηj) . Theorem 1 Let γ > 0, n – 1 < γ < n, : (d1) The Hadamard fractional differential equation HDγ v(t) = 0 is valid if and only if n v(t) = ci(log t)γ –i, i=1 where ci ∈ R

Results

Conclusion