Higher Order Lag Differential Equation Of Asymptotic Analysis

Difference equations appear as one of natural descriptions of observed evolution phenomena in the real world. In many cases, especially, under the impetus of the rapid development of computer science and technology, time evolving variables are often sampled and measured in the discrete manner. Therefore, difference equations became one of important methods for stablishing mathematical modeds in some scientific fields, and are widely applied to various scientific fields, for example, ecobiology,economics, control computer science and so on. In the recent years, there has appeared many new directions in the research of difference equation theory, one of which is stability theory of difference equations. In particular, it becomes one of focuses of studying the difference equation theory to find stablity criteria for the zero solution of difference equations. In this paper, by using the root-analysis for characteristic equations and the generating function method, we establish necessary and sufficient conditions for asymptotic stability of the zero solution of 6-th order linear delay difference equations of the formwhere coefficients a, b are nonzero real constants,/:; is a positive integer. On the basis of the above results,we study asymptotic behaviors of the following nonlinear delay difference equationswhere a₅,a_k+5 are real constants, g(x₅,x_n-5,x_n-k-5 is a known function. All these results are represented by parameters of the above equtions, which are easy to apply and verify.