Periodic-like Solutions For Integer Order And Fractional Order Differential Systems

The periodic-like function includes a variety of periodic function, anti-periodic function and almost periodic function. Periodic system not only widespread existence in astronomy and economics, but also in ecology, communication theories, control theories etc. Thanks to it manifests the system’s regular change, the periodic-like solutions of differential equations have attracted much attention nowadays. But the behavior of periodic-like orbits is a few important branch of the research of functional differential equation theory. Particularly, in the past many years, it obtain overall developments and abundant results. However, there are still many unknown potential research for us to do.In this dissertation, the existence of periodic-like solutions for several kind of functional differential equations are investigated. This dissertation is organized by four parts.In Chapter1, we simply introduce the development of periodic-like solutions for the function differential equations and the motivation of this paper.In Chapter2, the existence and uniqueness of periodic solutions of the following non-linear neutral differential equation are show by using the Krasnoselskii’s fixed point theorem. Moreover, we also prove the asymptotic stability of the periodic solution. Meanwhile, some examples are given to illus-trate the main results.In Chapter3, using the Leray-Schauder degree theory, the new results on the existence and uniqueness of anti-periodie solutions are established for a kind of nonlinear high order differential equations with multiple deviating arguments of the form where Ti, e:R→R are continuous functions and T-periodic,f, gi:R×R→R are continuous functions and T-periodic in their first arguments, n>2is an integer, T>0and i=1,2,…, m. Clearly, when n=2and f(t,x(t))=f(x(t)), it reduces to which has been known as the delayed Rayleigh equation with multiple deviating arguments. In a mechanical problem,f usually represents a damping or friction term, gi (i=1,2,…,m) represent a series of the restoring forces, e is an externally applied force and Ti is the time lag of the restoring force.In Chapter4, this paper studies the existence and uniqueness of almost periodic mild solutions to fractional delayed differential equations of the form where1<α<2, A:D(A)(?) X→X is a linear densely defined operator of sectional type on a complex Banach space X and f:R×X→X is jointly continuous. Let f(t, x) be almost periodic in t∈R uniformly for x. Under some additional assumptions on A and f, the existence and uniqueness of a almost periodic mild solution to above equation is obtained by using the Banach fixed-point principle. The obtaining results extent corresponding results in time delay with respect to almost periodic mild solutions for fractional differential equations.