Multiple Positive Solutions of Second-Order Nonlinear Difference Systems with Repulsive Singularities

Shengjun Li,Fang Zhang,Liliana Guran

doi:10.1155/2021/9954156

Shengjun Li, Fang Zhang + Show 1 more

Open Access

https://doi.org/10.1155/2021/9954156

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Abstract

We study the existence of positive solutions for second-order nonlinear repulsive singular difference systems with periodic boundary conditions. Our nonlinearity may be singular in its dependent variable. The proof of the main result relies on a fixed point theorem in cones and a nonlinear alternative principle of Leray-Schauder; the result is applicable to the case of a weak singularity as well as the case of a strong singularity. An example is given; some recent results in the literature are improved and generalized.

Highlights

Difference systems are widely used in modeling real-life phenomena [1] and references therein
We establish two different existence results of positive periodic solutions for (1) and (2) and proof of the existence of positive solutions; the first one is based on an application of a nonlinear alternative of Leray-Schauder, which has been used by many authors [19, 27, 28] and references therein; the second one is based on a fixed point theorem in cones
We study the periodic problem for nonlinear difference systems with a singularity of repulsive type in the case of γ∗ ≥ 0

Summary

Introduction

Difference systems are widely used in modeling real-life phenomena [1] and references therein. We establish the existence positive solutions for the following nonlinear difference systems:. The existence of positive periodic solutions of the singular differential equations has been established with a weak force condition (corresponds to the case 0 < λ < 1 in (5)) [13,14,15]. We establish two different existence results of positive periodic solutions for (1) and (2) and proof of the existence of positive solutions; the first one is based on an application of a nonlinear alternative of Leray-Schauder, which has been used by many authors [19, 27, 28] and references therein; the second one is based on a fixed point theorem in cones. For a given function p defined on Z1⁄20, T, we denote its maximum and minimum by p∗ and p∗, respectively

Preliminaries

Main Results

Conclusions