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, Huaihua University Huaihua, Hunan 418008, P.R. China and School of Mathematical Sciences and Computing Technology Central South University Changsha, 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 Download Persian Version: https://daneshyari.com/article/470195 Daneshyari.com.