Positive Solutions for a System of Neumann Boundary Value Problems of Second-Order Difference Equations Involving Sign-Changing Nonlinearities

Johnny Henderson,Jiafa Xu,Zhengqing Fu,Jiqiang Jiang

doi:10.1155/2019/3203401

Abstract

In this paper, we study the existence of positive solutions for the system of second-order difference equations involving Neumann boundary conditions: -Δ2u1(t-1)=f1(t,u1(t),u2(t)), t∈[1,T]Z, -Δ2u2(t-1)=f2(t,u1(t),u2(t)), t∈[1,T]Z, Δui(0)=Δui(T)=0, i=1,2, where T>1 is a given positive integer, Δu(t)=u(t+1)-u(t), and Δ2u(t)=Δ(Δu(t)). Under some appropriate conditions for our sign-changing nonlinearities, we use the fixed point index to establish our main results.

Highlights

B ∈ Z with a < b, let [a, b]Z = {a, a+1, a+2, . . . , b−1, b}
We study the existence of positive solutions for the system of second-order difference equations involving Neumann boundary conditions: −Δ2u1(t − 1) = f1(t, u1(t), u2(t)), t ∈ [1, T]Z, −Δ2u2(t − 1) = f2(t, u1(t), u2(t)), t ∈ [1, T]Z, Δui(0) = Δui(T) = 0, i = 1, 2, where T > 1 is a given positive integer, Δu(t) = u(t+1)−u(t), and Δ2u(t) = Δ(Δu(t))
Semipositone problems arise in bulking of mechanical systems, design of suspension bridges, chemical reactions, astrophysics, combustion, and management of natural resources; for example, see [1,2,3,4]

Summary

Introduction

We study the existence of positive solutions for the system of second-order difference equations involving Neumann boundary conditions: −Δ2u1(t − 1) = f1(t, u1(t), u2(t)), t ∈ [1, T]Z, −Δ2u2(t − 1) = f2(t, u1(t), u2(t)), t ∈ [1, T]Z, Δui(0) = Δui(T) = 0, i = 1, 2, where T > 1 is a given positive integer, Δu(t) = u(t+1)−u(t), and Δ2u(t) = Δ(Δu(t)). In [5], the author used the Guo-Krasnosel’skii fixed point theorem to study the existence of at least one positive solution for the discrete fractional equation:

Results

Conclusion