一類非線性差分方程的全局漸進穩定性*

趙玉萍

（青海民族大學數學與統計學院，青海　西寧　810007）

一類非線性差分方程的全局漸進穩定性*

趙玉萍

（青海民族大學數學與統計學院，青海西寧810007）

非線性差分方程在工程技術中有廣泛應用.筆者利用特殊不等式，研究了一類非線性差分方程解的穩定性，得到了這類方程有全局漸進穩定平衡點的充分條件，給出了方程唯一的正的全局漸進穩定的平衡點.改進了N.kruse和T.Nesemann已有的研究結果.通過一個例子說明了主要結論，使差分方程的研究領域更廣、更完善.

穩定性；差分方程；非線性

0　引言

差分方程在計算機科學、經濟學、生物數學等領域有著廣泛的應用，差分方程的研究越來越受到人們的重視.近幾年，關于差分方程的穩定性理論的研究，引起了國內外學者的廣泛關注，成果較多[1－16].在文獻[2]中N.kruse和T.Nesemann研究了差分方程

漸進穩定性.筆者研究一類普遍的非線性差分方程

正解的穩定性.

1　引理

2　主要結果

[1]Agarwal.R.P.Difference Equations and Inequalities[M].Theory，Methods，and Applications，2nd ed，Monographs and Textbooks in Pure and Applied Mathematics Vol.228，Marcel Dekker，New York，2000.

[2]N.kruse and T.Nesemann.Global asymptotic stability in some discrete dynamical systems[J].Math.Anal.Appl.，1999，235：151－158.

[3]Shi.B，Wang.Z.C and Yu.J.S.Asymptotic constancy of solutions of linear parabolic Volterra difference equations[J].Comput.Math.Appl. 2006，32：65－77.

[4]Morchalo.J.On convergence of solutions of a second order nonlinear difference equations[J].Appl.Math.Comput.2003，12：59－66.

[5]X.Li.Global behavior for a fourth－order rational difference equation[J].Math.Anal.Appl.2005，312：555－563.

[6]X.Li.Qualitative properties for a fourth－order rational difference equation[J].Math.Anal.Appl.2005，311：103－111.

[7]R.P.Agarwal，C.Cuevas，H.Soto，M.El－Gebeily.Asymptotic periodicity for some evolution equations in Banach spaces[J].Nonlinear Anal. 2011，74：1769－1798.

[8]A.Caicedo，C.Cuevas，G.M.Mophou，G.M.N'Guérékata.Asymptotic behavior of solutions of some semilinear functional differential and differential equations with infinite delay in Banach spaces[J].Franklin Inst.2012，349：1－24.

[9]X.H.Nang，Y.Liu.Bounded oscillation for second－order delay difference equations with unstable type in a critical case[J].Appl.Math.Lett. 2003，16：263－268.

[10]X.H.Nang，J.S.Yu.Oscillations of delay difference equation in a critical state[J].Appl.Math.Lett.2000，13（20）：9－15.

[11]Qiong Meng，JurangYan.Bounded oscillation for second－order nonlinear neutral differential equations in critical and non－critical states[J]. Journal of Computational and Applied Mathematics，2008，211：156－172.

[12]C.Cuevas，C.Lizama.Almost asymptotic solutions to a class of semilinear fractional differential equations[J].Appl.Math.Lett.2008，21：1315－1319.

[13]G.Papaschinopoulos，C.J.Schinas.Oscillation and asymptotic stability of two systems of difference equations of rational form[J].Differ.Equ. Appl.2001（7）：601－617.

[14]L.Del Campo，M.Pinto，C.Vidal.Almost and asymptotically almost periodic solutions of abstract retarded functional difference equations in phase space[J].Differ.Equ.Appl.2001，17（6）：915－934.

[15]S.Stevic'.Global stability of a max－type equation[J].Appl.Math.Comput.2010，216：354－356.

[16]G.Papaschinopoulos，C.Schinas，V.Hatzifilippidis.Global behavior of the solutions of a max－equation and of a system of two max－equations[J].Comput.Anal.Appl.2003，5（2）：237－254.

[責任編輯 蘇 琴] [責任校對 方麗菁]

Global Asymptotic Stability for a Family of Nonlinear Difference Equations

ZHAO Yu－ping
（Department of Mathematics and Statistics，Qinghai University for Nationalities，Xining810007，China）

Nonlinear difference equations are widely used in engineering technology.By means of special inequality technique，this paper is concerned with stability of solution for a family of nonlinear difference equations，sufficient conditions that the equations have globally asymptotically stable equilibrium were given，an only positive globally asymptotically stable equilibrium was given，which improves some known results that N.kruse and T.Nesemannt had studied.An example is given to illustrate the main results.The research field of difference is wider，more perfect.

Stability；Difference equation；Nonlinear

O175.7

A

1673－8462（2015）02－0055－04

2014－11－25.

國家自然科學基金資助項目（11361047）；青海民族大學校級科研項目.

趙玉萍（1975－），女，青海湟中人，碩士，青海民族大學數學與統計學院副教授，研究方向：差分方程，微分方程的理論研究.