Existence of uncountably many bounded positive solutions for a third order nonlinear neutral delay difference equation

Abstract

This paper deals with the solvability of the third order nonlinear neutral delay difference equation Delta(2)(a(n)Delta(x(n) + beta(n)x(n-tau))) + Delta(2)f(n, x(n-b1n), x(n-b2n), ... , x(n-bkn)) + Delta g(n, x(n-c1n), x(n-c2n), ... , x(n-ckn)) = h(n, x(n-d1n), x(n-d2n), ... , x(n-dkn)), n >= n(0) relative to the sequence {beta(n)}(n is an element of Nn0) subset of R. Using the Krasnoselskii's fixed point theorem and Schaucler's fixed point theorem, a few sufficient conditions of the existence of uncountably many bounded positive solutions for the equation are presented. Seven examples are included to demonstrate the advantages and effectiveness of the results presented in this paper. (C) 2010 Elsevier Ltd. All rights reserved.