Stability And Oscillation Of Solution For Difference Equations

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 coefficientsis 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) becomesThe 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 equationThe sufficent conditions which guarantee the stability of zero solutions of neutral difference equation without impulsesare given in [13]. When c = 0, the equation (2.1.1) is the discrete form of the implusive differential equationThe 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 delayare 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 delayhas 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.