Trichotomy of nonoscillatory solutions to second-order neutral difference equation with quasi-difference

Agata Bezubik,Magdalena Nockowska-Rosiak,Małgorzata Migda,Ewa Schmeidel

doi:10.1186/s13662-015-0531-6

Abstract

In this paper the nonlinear second-order neutral difference equation of the following form: \(\Delta ( a_{n}\Delta(x_{n}-p_{n}x_{n-1}) ) + q_{n}f(x_{n-\tau})=0\) is considered. By suitable substitution the above equation is transformed into a new one, which is a third-order non-neutral difference equation. Using results obtained for the new equation, the asymptotic properties of the neutral difference equation are studied. Some classification of nonoscillatory solutions is presented, as well as an estimation of the solutions. Finally, we present necessary and sufficient conditions for the existence of solutions to both considered equations being asymptotically equivalent to the given sequences.

Highlights

In this paper we consider the difference equation in the following form: an + qnf =, n ∈ Nmax{,τ}, ( )where is the forward difference operator defined by yn = yn+ – yn, are sequences of positive real numbers, τ is a nonnegative integer, and the function f : N → R
Where is the forward difference operator defined by yn = yn+ – yn, are sequences of positive real numbers, τ is a nonnegative integer, and the function f : N → R
The purpose of this paper is to study the asymptotic properties of the neutral difference equation ( )

Summary

Introduction

By a solution to ( ) we mean a sequence (xn) which satisfies ( ) for n sufficiently large. We obtain necessary and sufficient conditions for the existence of solutions asymptotically equivalent to the given sequences. Lemma Assume that (H ), (H ), and the following conditions: (H∗ ) zg(z) > for all z = ; (H∗ ) g : R → R is a continuous function; are satisfied.

Objectives

Results

Conclusion