吳冬梅，陶元紅

（延邊大學(xué) 理學(xué)院數(shù)學(xué)系，吉林 延吉 133002）

帶有正負(fù)系數(shù)的非線性偏差分方程的不飽和解

吳冬梅，陶元紅?

（延邊大學(xué) 理學(xué)院數(shù)學(xué)系，吉林 延吉 133002）

利用數(shù)列的頻密測(cè)度的定義及其相關(guān)性質(zhì)，對(duì)一類(lèi)帶有正負(fù)系數(shù)的時(shí)滯偏差分方程的不飽和性問(wèn)題做了深入的討論,得出此類(lèi)方程的解具備不飽和性的充分條件.由于不飽和解一定是振動(dòng)解，本文的結(jié)論不僅準(zhǔn)確刻畫(huà)了解的振動(dòng)頻率，而且完善了解的振動(dòng)性描述.

偏差分方程；頻密測(cè)度；不飽和性

經(jīng)典的振動(dòng)概念已經(jīng)不能準(zhǔn)確描述數(shù)列的振動(dòng)頻率，為了更細(xì)致地刻畫(huà)數(shù)列的振動(dòng)性，田傳俊等[1]首次引進(jìn)了數(shù)列的頻密測(cè)度的概念，并由此定義了數(shù)列的頻密振動(dòng)性.朱志強(qiáng)等[2]又定義了數(shù)列的頻密正振動(dòng)和負(fù)振動(dòng)概念，并且利用這些概念刻畫(huà)了無(wú)窮雙序列的不飽和性概念，更加完善了對(duì)數(shù)列的頻密振動(dòng)的描述.目前已有一些關(guān)于差分方程的解的頻密振動(dòng)性結(jié)果，見(jiàn)文獻(xiàn)[3-9].

本文討論如下一類(lèi)非線性偏差分方程的不飽和解：

1 預(yù)備知識(shí)

2 主要結(jié)論

[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,陶元紅.一類(lèi)非線性時(shí)滯偏差分方程的不飽和解[J].延邊大學(xué)學(xué)報(bào):自然科學(xué)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]陶元紅,吳冬梅.一類(lèi)中立型偏差分方程的差頻密振動(dòng)性[J].延邊大學(xué)學(xué)報(bào):自然科學(xué)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

國(guó)家自然科學(xué)基金資助項(xiàng)目（11161049）*通訊作者

畢和平