An Intemational Joumal Available online at www.sciencedirect.com computers & .o,..c= #-~ o,..o~, mathematics with applications ELSE~ZR Computers and Mathematics with Applications 52 (2006) 439-448 www.elsevier.com/locate/camwa

Oscillation of Solutions of Neutral Difference Equations with a Nonlinear Neutral Term

XIAOYAN LIN Department of Mathematics, University Huaihua, 418008, P.R. China and School of Mathematical Sciences and Computing Technology , Hunan 410083, P.R. China xiaoyanlin98©hotmail, com (Received October 2005; revised and accepted February 2006)

Abstract--The oscillatory behavior of solutions of the neutral difference equations with nonlinear neutral term, _ -Fqn n-a=O, n>_no, is studied in the case when c~ :fi 1. A necessary and sufficient oscillatory conditions for the case 0 < a < 1 and an almost "sharp" oscillatory and nonoseillatory criteria for the case a > i are obtained. (~) 2006 Elsevier Ltd. All rights reserved.

Keywords--Neutral difference equation, Sublineax neutral term, Superlinear neutral term, Os- cillation, Nonoscillation.

1. INTRODUCTION Consider the neutral difference equation with nonlinear neutral term,

A(x~ -- p• • "~_~) + q~_~ = 0, n > no, (1.1) where a and fl are quotient of odd positive integers, T is positive integer, a is nonnegative integer and {p~} and {q~} are two sequences of nonnegative real numbers. When a = fl = 1, equation (1.1) reduces to a linear neutral difference equation,

A(x~ - p~x~_~) + q,~xn-a = O, n >_ no, (1.2) and many oscillatory criteria for equation (1.2) have been obtained in literature, see [1-5] and the references cited therein. When a ¢ 1 or ~ ¢ 1, we only find one paper [6] which dealt with

This work is partially supported by the NNSF of China.

0898-1221/06/$ - see front matter (~) 2006 Elsevier Ltd. All rights reserved. Typeset by Jt~IS-TEX doi:10.1016/j.camwa.2006.02.009 440 X. LIN the oscillatory behavior of solutions of equation (1.1) with Pn - P > 0. The result obtained there is the following.

THEOREM A. (See [6].) Assume that a • (0, 1) and p,~ -- p > 0. Then, every solution of equation (1.1) oscillates if and only if

qs ---- oo. (1.3)

$~nO

Besides the linear case when a = j3 = 1 and the case mentioned in Theorem A, we find no results in literature on the oscillation of solutions of equation (1.1) in the following two cases: (i) a 6 (0, 1),/3 • (0, oo); (ii) a • (1, oo), fl • (0, oo). In this paper, our purpose is to obtain necessary and sufficient oscillatory condition for equa- tion (1.1) in Case (i) and an almost "sharp" oscillatory conditions for equation (1.1) in Case (ii). As is customary, a solution {x~} of equation (1.1) is said to be eventually positive if x,~ > 0 for all large n, and eventually negative if xn < 0 for all large n. It is said to be oscillatory if it is neither eventually positive nor eventually negative.

2. EQUATION (1.1) WITH SUBLINEAR NEUTRAL TERM (a < 1)

THEOREM 2.1. Assume that a 6 (0, 1) and the following Condition (H) holds. (H) There exist p.,p* 6 (0, oo) such that for large n,

p. <_p. <_p*. (2.1)

Then, every solution of equation (1.1) oscillates if and only if (1.3) holds.

PROOF. SUFFICIENCY. Suppose to the contrary that {xn} is an eventually positive solution of equation (1.1). Then, there exists an integer nl > no such that p. < pn _< p* and Xn-r--a > 0 for n > nl. Set

Zn ~ Xn -- Pn x~n-r" (2.2)

Then, by (1.1), (1.3), and (2.2), we have

Az~ = -q ....x ~ G 0 (5 0), n _ nl. (2.3)

This shows that { Z n},~=,~,O0 is nonincreasing, and so there exists an integer n2 > nl such that

(a) z. < 0, n > us, or (b) zn>O,n>n2. If Case (a) holds, then by (2.2) and (2.3),

z~_¢ > - z~+._¢)1/~ > _ z..) , n>_n2q-•--T. (2.4) Download English Version: https://daneshyari.com/en/article/470195

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