Positive Solutions of a Second-Order Nonlinear Neutral Delay Difference Equation

Abstract

The purpose of this paper is to study solvability of the second-order nonlinear neutral delay difference equation $\mathrm{\Delta }(a(n,y_{a_{1n}},\dots ,y_{a_{rn}})\mathrm{\Delta }(y_n+b_n{y}_{n-\tau }))+f(n,y_{f_{1n}},\dots ,y_{f_{kn}})=c_n,\forall n\ge n_0$. By making use of the Rothe fixed point theorem, Leray-Schauder nonlinear alternative theorem, Krasnoselskill fixed point theorem, and some new techniques, we obtain some sufficient conditions which ensure the existence of uncountably many bounded positive solutions for the above equation. Five nontrivial examples are given to illustrate that the results presented in this paper are more effective than the existing ones in the literature.