The Study Of Asymptotic Behavior Of Non-oscillatory Solutions To Neutral-Type Lower-Order Difference Equations

In recent years, with the development of science and technology, the theoryof difference equations has been widely applied to many fields, such as moderneconomics, biology, physics, dynamical systems theory and control engineering,and it has become an indispensable mathematical tool. In practice, people haveproposed many mathematical models described by difference equations. Theoryof stability, asymptotic theory, vibration theory and the existence of positivesolutions of difference equations theory is one of the important contents ofqualitative theory, the study of which is of great theoretical significance andpossesses practical values. This paper studies the oscillation and asymptotic behavior of the solutions to low order difference equations and the existence of non-oscillatory solutions to higher order nonlinear neutral difference equations, respectively.First of all, this paper studies the asymptotic behavior of solutions to third-order neutral difference equationThis paper presents some conditions under which all non-oscillatory solutions to the above equation possess the property xn=cn+o(n) for some c∈R as well as sufficient conditions under which all non-oscillatory solutions are asymptotically linear, the results obtained generalize the previous conclusions in literatures.Secondly, this article studies the fifth-order nonlinear differential equation By constructing auxiliary functions and using the Schauder’s fixed point theorem, the oscillation and asymptotic behavior of solutions are discussed and a sufficient condition to ensure the existence of a bounded non-oscillatory solution is given, which is the generalization to the previous results.Finally, this paper studies the high-order nonlinear neutral difference equationThe existence of non-oscillatory solutions is discussed. According to the different values of cn, we divided it into five parts to finish the discussion. We firstly construct a bounded closed convex subset l∞of one Banach space, and with the help of Krasnoselskii fixed point theorem, we obtain a sufficient condition to ensure the existence of non-oscillatory solutions. In March2003, Zhou Yong and Huang Y.Q. studied the high-order nonlinear neutral difference equationsThe results obtained in that literature is a special case of our paper.