Oscillations of Difference Equations with Several Oscillating Coefficients

L Berezansky,A Domoshnitsky,G E Chatzarakis,I P Stavroulakis

doi:10.1155/2014/392097

Abstract

We study the oscillatory behavior of the solutions of the difference equationΔx(n)+∑i=1mpi(n)x(τi(n))=0,n∈N0[∇xn-∑i=1mpinxσin=0, n∈N]where(pi(n)),1≤i≤mare real sequences with oscillating terms,τi(n)[σi(n)],1≤i≤mare general retarded (advanced) arguments, andΔ[∇]denotes the forward (backward) difference operatorΔx(n)=x(n+1)-x(n)[∇x(n)=x(n)-x(n-1)]. Examples illustrating the results are also given.

Highlights

We study the oscillatory behavior ∑mi=1 pi (n) x (σi (n)) = 0, n ∈ N]
We study the oscillatory behavior of the solutions of the difference equation m
Strong interest in (ER) is motivated by the fact that it represents a discrete analogue of the differential equation

Summary

Introduction

We study the oscillatory behavior of the solutions of the difference equation m. By a solution of (ER), we mean a sequence of real numbers {x(n)}n≥−w which satisfies (ER) for all n ∈ N0. By a solution of the advanced difference equation (EA), we mean a sequence of real numbers {x(n)}n∈N0 which satisfies (EA) for all n ∈ N. In the last few decades, the oscillatory behavior of all solutions of difference equations has been extensively studied when the coefficients pi(n) are nonnegative. The authors study further (ER) [(EA)] and derive new sufficient oscillation conditions when neither (14) [(17)] nor (20) [(23)] is satisfied (cf [6–8] and the references cited therein in the case of the equations (ER) [(EA)] with nonnegative coefficients pi, 1 ≤ i ≤ m).

Retarded Equations

Advanced Equations

Examples