Frequent Oscillatory Behavior of Delay Partial Difference Equations with Positive and Negative Coefficients
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Research Article Frequent Oscillatory Behavior of Delay Partial Difference Equations with Positive and Negative Coefficients Li Hua Xu1 and Jun Yang1, 2 1 2
College of Science, Yanshan University, Qinhuangdao, Hebei 066004, China Mathematics Research Center in Hebei Province, Shijiazhuang 050000, China
Correspondence should be addressed to Li Hua Xu, [email protected] Received 12 November 2009; Accepted 11 February 2010 Academic Editor: Binggen Zhang Copyright q 2010 L. H. Xu and J. Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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.
1. Introduction Partial difference equations are difference equations that involve functions with two or more independent integer variables. Such equations arise from considerations of random walk problems, molecular structure problems, and numerical difference approximation problems. Recently, there have been a large number of papers devoted to partial difference equations, and the problem of oscillatory of solutions and frequent oscillatory solutions for partial difference equations is receiving much attention. In 1, authors considered oscillatory behavior of the partial difference equations with positive and negative coefficients of the form
Am1,n Am,n1 − Am,n pm, nAm−k,n−l − qm, nAm−k ,n−l 0,
but they have not discussed frequent oscillations of this equation.
1.1
2
Advances in Difference Equations
In 2, authors considered oscillatory behavior for nonlinear partial difference equations with positive and negative coefficients of the form ωfm, n, Am−σ,n−τ , Am−u,n−v Am−1,n Am,n−1 − Am,n pAmk,nl − qAmk ,nl 0, ωm,n fm, n, Am−σ,n−τ , Am−u,n−v Am−1,n Am,n−1 − Am,n pmn Amk,nl − qmn Amk ,nl 0 1.2 In 3, authors considered frequent oscillation in the nonlinear partial difference equation um,n um1,n um,n1 pm,n |um−k1 ,n−l1 |α sgn um−k1 ,n−l1 qm,n |um−k2 ,n−l2 |β sgn um−k2 ,n−l2 0.
1.3
In 4, authors considered oscillations of the partial difference equations with several nonlinear terms of the form, um1,n um,n1 − um,n
h
pi m, n|um−ki ,n−li |αi sgn um−ki ,n−li 0,
1.4
i 1
and in 5 authors considered frequent oscillations of these equations. In 6, authors considered unsaturated solutions for partial difference equations with forcing terms Δ1 u i − 1, j Δ2 u i, j − 1 P1 i, j u i − 1, j P2 i, j u i, j − 1 P3 i, j u i, j f i, j . 1.5 Let Z be the set of integers, Zk, l {i ∈ Z | i k, k 1, . . . , l}, and Zk, ∞ {i ∈ Z | i k, k 1, . . .}. In this paper, we will consider the equation of the following form: um1,n um,n1 − um,n
h i 1
pi m, num−ki ,n−li −
g
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