摘要
随着经济的发展和科技的进步，差分方程在生态学、经济学、数论以及物理等多个领域都有着广泛的应用，因此对差分方程的研究日益引起人们的普遍重视。关于差分方程解的稳定性和振动性问题，长期以来倍受国内外学者的关注，并做了大量的工作。不仅研究了一般的线性差分方程，而且对中立型的、非线性的、时滞的差分方程都做了深入研究。
本文可分为三部分，第一部分，主要讨论了一类具有正负系数的非线性中立型时滞差分方程
Δ(x_n-c_nx_(n-k))+P_nf(X_n-1)-q_ng(x_(n-r)=0，n∈N(0)， (1.1.1)解的一致稳定性，其中k，l，r∈N(1)，f，g，∈C(R，R)，且f(0)=g(0)=0；{C_n}为实数序列。{p_n}，{q_n}为非负实数序列，进而讨论了此类方程解的一致渐近稳定性，并给出了例子。当f(x)≡x，g(x)≡x时，方程(1.1.1)变为
Δ(x_n-c_nx(n-k))+p_nx_(n-1)-q_nx_(n-r)=0，n∈N(0)． (1.1.2)文[6]对方程(1.1.2)零解的全局吸引性做了具体讨论。本章结论的一些特殊情形与文[6]的结论完全吻合。
目前对脉冲微分方程解的振动性和稳定性，很多学者都进行了研究。然而对脉冲差分方程的研究成果却很少。在第二部分，我们主要讨论了一类非线性中立型脉冲差分方程零解稳定性，其中c∈(-1，1)，k,r∈N(1)，且r≥k；f：{N(0)\{n_j}}×R→R，{n_j}是一个严格单调递增的非负整数序列，I_j：R→R。
对应于(2.1.1)的非脉冲中立型差分方程
Δ(x_n-cx_(n-r))=f(n，x_(n-k))，(2.1.2)零解的稳定性条件，在文[13]中已给出。当c≡0时，方程(2.1.1)是脉冲微分方程
关于差分方程解的称定性和振动性
的离散形式，文【n，12】对(2 .1，3)的稳定性已进行了深入研究，还有许多文献对脉冲
徽分方程的稳定性进行了讨论，如文【7一101，而对脉冲差分方程的研究却十分少，仅见
文献【l‘l刀，本章结果当。二0是脉冲徽分方程11‘，‘21离散形式的相关结果.当几二0
时，本章结果即为文献【13」中的结果.
第三部分，研究了一阶变时滞非线性差分方程
二(n+1)一二(。)+p。f(:(:(。)))=0，n EN(0).
(3 .1 .1)
解的振动准则，其中{p。}是非负实数序列，::N(0)”z，o簇。一《司续k，
哎为。
im:(n)==
方程(3.1.1)的特殊情形
二(n+1)一:(n)+pof(二(。一无))=0，
(3 .1.2)
二(。+1)一二(。)+P。:(。一七)=0，(3 .1.3)
在文【1冬24】中已被讨论，文【l例中举反例说明了非线性方程(3.1.2)和其对应的线性
方程(3.1.3)，即使有条件
lim
钻~今0
=1，
(3 .1.4)
也可能有不同的振动性.为此，我们不可能由(3 .1。3)的振动性直接导出(3.L2)的振
动性.因此，即使变时滞线性方程
:(。+‘)一x(n)+艺二(n)二(。一k‘(n))=o
的娘动性在文【25】中已被讨堆，但我们建立方程(3. LI)的振动准则还是很有必要的·
字大推广和改进了一些已有的结论·
Nonlinear difference equations are of paramount importance in applications where the (n +1)st generation of the system depends on the previous k generations. Such equations also appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations which model various diverse phenomena in biology, ecology, physiology, physics, engineering and economics.
The paper consists of three parts. In the first part, the uniform stablity of neutral delay difference equation with positive and nonnegative coefficients
is considered , where k,r,l N(1) and f,g C(R,R), f(0) = g(0) = 0; {cn}is a sequence of real numbers and {pn}, {qn}are sequences of nonnegative real numbers. We also discuss the uniformly asymptotic stablity of this equation and give an example. When f(x) = x, g(x) = z, the equation (1.1.1) becomes
The global attractivity of zero solutions of the equation (1.1.2) is considered in [6]. The results of the chapter and those of [6] are totally concident in the special conditions.
Recently stability and oscillation of implusive differential equations have been studied by many scholars, but there has been little research on difference equations. In the second part, we discuss stability of zero solutions of a class of nonlinear neutral implusive difference equation
The sufficent conditions which guarantee the stability of zero solutions of neutral difference equation without impulses
are given in [13]. When c = 0, the equation (2.1.1) is the discrete form of the implusive differential equation
The stability of (2.1.3) has been studied in [11,12], and there are many literatures in which the stability of the implusive differential equations has been discussed, such as [7-10]. However the study for the implusive difference equations is quite little, and we only refers to [14-17]. When c = 0, the results of this paper are the correlative results of the discrete form of implusive differential equations [11,12]. When Ij = 0, the results of this paper are those in [13].
In the third part ,some oscillation ciiteria for all solutions of nonlinear difference equation with variable delay
are obtained, where {pn} is a sequence of nonnegative real number, : N(0) - Z,0 n - (n) k, lim (n) = .
The special cases of Eq.(3.1.1) have been discussed in [18-24]. The counterexample in [19] shows that the nonlinear equations and the corresponding linear equations may have different oscillation behavior. Therefore, even though the oscillation of the linear difference with variable delay
has been discussed in [25], it is valuable to establish the oscillation criteria of Eq.(3.1.1). In this paper,we extend some well-known results.
引文
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16．张勤勤，周展．不稳定型差分方程在脉冲扰动下的稳定性．湖南大学学报．2002，29：4-9．
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