OSCILLATORY BEHAVIOR OF THIRD ORDER ADVANCED DIFFERENCE EQUATIONS

Abstract

In this paper, we provide sufficient conditions for the third-order advanced difference equations of the form

to be oscillatory or to have property-B. In contrast to the current results, we established sufficient conditions for all solutions of the studied equation to be oscillatory. We provide examples to illustrate the results.

Keywords: Oscillation, Third-order difference equation, Property-B, Advanced argument.

Mathematics subject classification:  39A05, 39A21, 39A99.

1. Introduction

In this paper, we investigate the oscillatory behavior of third-order advanced difference equations of  the form

(1.1)

where n₀ is a non-negative  integer and we assuming the following conditions hold

[H₁]{d( )},{r( )},{h( )} and {q( )}are positive real sequences for and ;

[H₂] is an increasing sequence such that for all ;

[H₃] is a sequence of positive integers for all ;

[H₄] ;                                   

[H₅] f is continuous, non-decreasing real-valued function such that  and  for ab > 0;

 [H₆] There exists  which is positive integer satisfies  for all ;

A solution of equation (1.1), we mean a real sequence {x( )} that satisfies (1.1) for all . A non-trivial solution of (1.1) is said to be oscillatory, if it is neither eventually positive nor eventually negative, otherwise it is non-oscillatory. A difference equation is said to be oscillatory (non-oscillatory) if all of its solutions are oscillatory (non-oscillatory).

The study of oscillatory behavior of difference equations has received considerable attention over the past decades due to its theoretical significance and applications. A number of works have addressed oscillation criteria for various classes of difference equations; see, for instance, [1]–[3],[5],[10],[14]. In particular, the oscillatory and asymptotic properties of third-order difference equations have attracted increasing interest among researchers [4],[7]–[9],[12]–[15]. Several authors have developed oscillation results for third-order equations by employing comparison principles, averaging techniques, and related tools .[[3],[6],[11]-[16]] the authors used for oscillation of all solutions of third order difference equations was established with the help of comparison method and averaging technique.