Positive Solutions for a System of Hadamard-Type Fractional Differential Equations with Semipositone Nonlinearities

Youzheng Ding,Donal O’Regan,Jiqiang Jiang,Jiafa Xu

doi:10.1155/2020/9742418

Abstract

In this paper, we use the fixed-point index and nonnegative matrices to study the existence of positive solutions for a system of Hadamard-type fractional differential equations with semipositone nonlinearities.

Highlights

We study the existence of positive solutions for the following system of Hadamard-type fractional boundary value problems:
Fractional problems arise in many applications in aerodynamics, signal and image processing, biophysics, blood ow phenomena, etc
Where f has a semipositone nonlinearity, i.e., it is bounded below and can be sign-changing, and in [2], the authors used the Guo–Krasnoselskii’s fixed-point theorem to investigate the existence of positive solutions for fractional multipoint boundary value problems:

Summary

Introduction

We study the existence of positive solutions for the following system of Hadamard-type fractional boundary value problems: Where α ∈ (2, 3], Dα is the Hadamard fractional derivative of order α, δ t(d/dt) (i.e., if u is ξ or η, δu(t) t(d/dt)u(t); δu(1) limt⟶1+ t(d/dt)u(t) (t(d/dt) u(t)|t 1) etc.), and the nonlinearities fi(i 1, 2) satisfy the semipositone condition: (H0) fi ∈ C([1, e] × R6+, R), and there exists M > 0 such that fi t, x1, x2, x3, y1, y2, y3 ≥ − M, for t, x1, x2, x3, y1, y2, y3 ∈ [1, e] × R6+, (2) In [1], the authors used the Krasnoselskii–Zabreiko xed-point theorem to study the existence of positive solutions

Objectives

Results

Conclusion