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Necessary Conditions for the Solutions of Second Order Non-linear Neutral Delay Difference Equations to Be Oscillatory or Tend to Zero  [PDF]
R. N. Rath,J. G. Dix,B. L. S. Barik,B. Dihudi
International Journal of Mathematics and Mathematical Sciences , 2007, DOI: 10.1155/2007/60907
Abstract: We find necessary conditions for every solution of the neutral delay difference equation Δ(rnΔ(yn−pnyn−m))
Positive Solutions of a Second-Order Nonlinear Neutral Delay Difference Equation  [PDF]
Zeqing Liu,Wei Sun,Jeong Sheok Ume,Shin Min Kang
Abstract and Applied Analysis , 2012, DOI: 10.1155/2012/172939
Abstract: The purpose of this paper is to study solvability of the second-order nonlinear neutral delay difference equation . By making use of the Rothe fixed point theorem, Leray-Schauder nonlinear alternative theorem, Krasnoselskill fixed point theorem, and some new techniques, we obtain some sufficient conditions which ensure the existence of uncountably many bounded positive solutions for the above equation. Five nontrivial examples are given to illustrate that the results presented in this paper are more effective than the existing ones in the literature. 1. Introduction It is well known that the oscillation, nonoscillation, asymptotic behavior, and existence of solutions for second-order difference equations with delays have been widely studied in many papers over the last 20 years, see, for example, [1–9] and the references cited therein. Recently, Cheng [5] considered the second-order neutral delay linear difference equation with positive and negative coefficients and investigated the existence of a nonoscillatory solution of (1.1) under the condition by using the Banach fixed point theorem. M. Migda and J. Migda [9] and Luo and Bainov [8] discussed the asymptotic behaviors of nonoscillatory solutions for the second-order neutral difference equation with maxima and the second-order neutral difference equation Cheng and Chu [2] got sufficient and necessary conditions of the oscillatory solutions for the second-order difference equation Li and Yeh [6] established some oscillation criteria of the second-order delay difference equation Using the Leray-Schauder nonlinear alternative theorem, Agarwal et al. [1] studied the existence of nonoscillatory solutions for the discrete equation under the condition . Very recently, Liu et al. [7] utilized the Banach contraction principle to establish the global existence and multiplicity of bounded nonoscillatory solutions for the second-order nonlinear neutral delay difference equation Motivated by the results in [1–9], in this paper, we discuss the solvability of the second-order nonlinear neutral delay difference equation where , ,?? , , ,?? and It is clear that (1.1)–(1.7) are special cases of (1.8). By utilizing the Rothe fixed point theorem, Leray-Schauder nonlinear alternative theorem, Krasnoselskill fixed point theorem, and a few new techniques, we prove the existence of uncountably many bounded positive solutions of (1.8). Five examples are constructed to illuminate our results which extend essentially the corresponding results in [1, 7]. 2. Preliminaries Throughout this paper, we assume that is the forward
ON THE ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF NONLINEAR NEUTRAL DELAY DIFFERENCE EQUATIONS
非线性中立型时滞差分方程解的渐近性

Sui Min YANG,Shi Zhong LIN,Yuan Hong YU,
杨绥民
,林诗仲,俞元洪

系统科学与数学 , 1999,
Abstract: The aim of this paper is to deduce oscillatory and asymptotic behavior of the solutions of the nonlinear neutral delay difference equation Results obtained generalize some theorems in the literatures.
On periodic linear neutral delay differential and difference equations  [cached]
Christos G. Philos,Ioannis K. Purnaras
Electronic Journal of Differential Equations , 2006,
Abstract: This article concerns the behavior of the solutions to periodic linear neutral delay differential equations as well as to periodic linear neutral delay difference equations. Some new results are obtained via two appropriate distinct roots of the corresponding (so called) characteristic equation.
Oscillatory Behavior of Quasilinear Neutral Delay Dynamic Equations on Time Scales  [cached]
Han Zhenlai,Sun Shurong,Li Tongxing,Zhang Chenghui
Advances in Difference Equations , 2010,
Abstract: By means of the averaging technique and the generalized Riccati transformation technique, we establish some oscillation criteria for the second-order quasilinear neutral delay dynamic equations , , where , and the time scale interval is . Our results in this paper not only extend the results given by Agarwal et al. (2005) but also unify the oscillation of the second-order neutral delay differential equations and the second-order neutral delay difference equations.
Oscillation and nonoscillation of nonlinear neutral delay difference equations  [cached]
E. Thandapani,P. Mohan Kumar
Tamkang Journal of Mathematics , 2007, DOI: 10.5556/j.tkjm.38.2007.323-333
Abstract: In this paper, the authors establish some sufficient conditions for oscillation and nonoscillation of the second order nonlinear neutral delay difference equation $$ Delta^2 (x_n-p_nx_{n-k}) + q_nf(x_{n-ell}) = 0, ~~n ge n_0 $$ where $ {p_n} $ and $ {q_n} $ are non-negative sequences with $ 0$
Frequent Oscillatory Behavior of Delay Partial Difference Equations with Positive and Negative Coefficients  [cached]
Li Hua Xu,Jun Yang
Advances in Difference Equations , 2010, DOI: 10.1155/2010/606149
Abstract: This paper is concerned with a class of nonlinear delay partial difference equations with positive and negative coefficients, which also contains forcing terms. By making use of frequency measures, some new oscillatory criteria are established.
Frequent Oscillatory Behavior of Delay Partial Difference Equations with Positive and Negative Coefficients  [cached]
Xu LiHua,Yang Jun
Advances in Difference Equations , 2010,
Abstract: This paper is concerned with a class of nonlinear delay partial difference equations with positive and negative coefficients, which also contains forcing terms. By making use of frequency measures, some new oscillatory criteria are established.
Oscillatory behavior of second order unstable type neutral difference equations
E. Thandapani,S. Pandian,R. K. Balasubramanian
Tamkang Journal of Mathematics , 2005, DOI: 10.5556/j.tkjm.36.2005.57-68
Abstract: This paper deals with the oscillatory behavior of all bounded/ unbounded solutions of second order neutral type difference equation of the form $$ Delta (a_n(Delta_c y_n+py_{n-k}))^alpha)-g_nf(y_{sigma(n)})=0, $$ where $ p $ is real, $ alpha $ is a ratio of odd positive integers, $ k $ is a positive integer and $ {sigma(n)} $ is a sequence of integers. Examples are provided to illustrate the results.
Solvability of a Second Order Nonlinear Neutral Delay Difference Equation
Zeqing Liu,Liangshi Zhao,Jeong Sheok Ume,Shin Min Kang
Abstract and Applied Analysis , 2011, DOI: 10.1155/2011/328914
Abstract: This paper studies the second-order nonlinear neutral delay difference equation Δ[Δ(
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