带有正负系数的非线性偏差分方程的不饱和解

吴冬梅，陶元红

（延边大学 理学院数学系，吉林 延吉 133002）

带有正负系数的非线性偏差分方程的不饱和解

吴冬梅，陶元红∗

（延边大学 理学院数学系，吉林 延吉 133002）

利用数列的频密测度的定义及其相关性质，对一类带有正负系数的时滞偏差分方程的不饱和性问题做了深入的讨论,得出此类方程的解具备不饱和性的充分条件.由于不饱和解一定是振动解，本文的结论不仅准确刻画了解的振动频率，而且完善了解的振动性描述.

偏差分方程；频密测度；不饱和性

经典的振动概念已经不能准确描述数列的振动频率，为了更细致地刻画数列的振动性，田传俊等[1]首次引进了数列的频密测度的概念，并由此定义了数列的频密振动性.朱志强等[2]又定义了数列的频密正振动和负振动概念，并且利用这些概念刻画了无穷双序列的不饱和性概念，更加完善了对数列的频密振动的描述.目前已有一些关于差分方程的解的频密振动性结果，见文献[3-9].

本文讨论如下一类非线性偏差分方程的不饱和解：

1 预备知识

2 主要结论

[1]Tian C J,Xie S L,Cheng S S.Measures for oscillatory se⁃quence[J].Computers Mathematics with Applications,1998,36:10-12,149-161.

[2]Zhu Z Q,Cheng S S.Frequently oscillatory solutions of neutral difference equations[J].Southeast Asian Bulletin of Mathematics,2005,29(3):627-634.

[3]Cheng S S.Partial Difference Equations[J].Advances in Discrete Mathematics and Applications Taylor&Francis,London,UK,2003.

[4]Tian C J,Cheng S S,Xie S L.Frequently Oscillatory Crite⁃ria for a Delay Difference Equation[J].Funkcialaj Ekva⁃cioj,2003,46:421-439.

[5]Zhu Z Q,Cheng S S.Frequently oscillatory solutions for multi-level partial difference equations[J].Internat Math Forum,2006,31:1497-1509.

[6]Yang J,Zhang Y J.“Frequent oscillatory solutions of a nonlinear partial difference equation”[J].Journal of Math⁃ematical Analysis and Applications,2009,224:492-499.

[7]李冬梅,陶元红.一类非线性时滞偏差分方程的不饱和解[J].延边大学学报:自然科学版,2010,36(2):95-100.

[8]Tao Y H,Li X D.Frequently Oscillatory Solutions for Nonlinear Delay Partial Difference Equations[J].Journal of Natural Science of Hei Longjiang Univercity,2010,27(5):591-595.

[9]陶元红,吴冬梅.一类中立型偏差分方程的差频密振动性[J].延边大学学报:自然科学版,2011,37(1):42-45.

Unsaturated Solutions of Nonlinear Partial Difference Equation with Positive and Negative Coefficients

WU Dongmei，TAO Yuanhong＊
（Department of Mathematics，College of Sciences，Yanbian University，Yanji133002，China）

By employing the concept and the properties of frequency measures of sequences,the unsaturated solutions of the nonlinear partial difference equation with positive and negative coefficients are discussed.Some new criteria of un⁃saturated solutions are established and the sufficient conditions of solutions to be unsaturated are presented.Since the unsaturated solutions are oscillatory,the results not only well describe the oscillation frequency of the solution but also generalize the description of oscillation.

partial difference equation；frequency measure；unsaturated solution

O 177.3

A

1674-4942（2011）04-0355-04

2011-09-28

国家自然科学基金资助项目（11161049）*通讯作者

毕和平