Maximal and minimal iterative positive solutions for singular infinite-point p-Laplacian fractional differential equations

Limin Guo,Lishan Liu

doi:10.15388/na.2018.6.3

Abstract

The existence of maximal and minimal positive solutions for singular infinite-point p-Laplacian fractional differential equation is investigated in this paper. Green's function is derived, and some properties of Green's function are obtained. Based upon these properties of Green's function, the existence of maximal and minimal positive solutions is obtained, and iterative schemes are established for approximating the maximal and minimal positive solutions.

Highlights

In this paper, we consider the following singular infinite-point p-Laplacian fractional differential equations: φp D0α+u(t) + f t, u(t), D0μ+ u(t) = 0, 0 < t < 1, u(i)(0) = 0, i = 0, 1, 2, . . . , n − 2, ∞ (1) D0p+1 u(1) =ηj D0p+2 u(ξj ), j=1c Vilnius University, 2018L
The existence of maximal and minimal positive solutions is obtained by iterative sequence for the boundary value problem (1) under certain conditions
Boundary value problems of nonlinear fractional differential equations constitutes a new and important branch of differential equation theory and has attracted great research efforts worldwide, and it is a valuable tool for simulating many phenomena in various fields such as fluid flows, electrical networks, rheology, biology, chemical physics, and so on

Summary

Introduction

The existence of maximal and minimal positive solutions is obtained by iterative sequence for the boundary value problem (1) under certain conditions. In [18], the authors considered the following fractional differential equation: φp D0α+u(t) + f t, u(t) = 0, 0 < t < 1, u(0) = u (0) = u (1) = 0, where α ∈ R+, 2 < α 3, φp(s) = |s|p−2s, p > 1, (φp)−1 = φq, 1/p + 1/q = 1, f : [0, 1] × [0, +∞) → [0, +∞) is continuous, and D0α+ is the standard Riemann–Liouville derivative. Motivated by the results above, in this paper, we investigate the existence of positive solutions for a class of infinite-point singular p-Laplacian fractional differential equations. Compared with [24, 29], the fractional-order derivatives are involved in the nonlinear term and boundary condition, and at the same time, iterative solutions are obtained by iterative sequences. Compared with [7], we do obtain the existence of positive solutions, but we establish iterative sequences to approximate the maximal and minimal positive solutions

Preliminaries and lemmas

Main results

An example