Existence Of Periodic Solutions For Several Kinds Of Neutral Integro-differential Equations With Infinite Delay

Boundary value problem of singular diferential equations in abstract spaceis an important branch of diferential equations. Now it is very active and devel-oped rapidly in recent years since it has wide application in biology, biomedicine,economics and space technology. At the same time, many experts and scholarsinvestigated the periodic solutions for neutral functional diferential equations withfinite or infinite delay in the past twenty years.From the beginning of this century, a lot of delay problems have been pro-posed in many science branches such as circuit signal system, ecological system,fuel recycling system, genetic problems. For instance, the delay can be describedas commercial sales problems, wealth distribution, the cyclical economic crisis ofcapitalism in the economic phenomenon of social science. Recently, functional dif-ferential equation with infinite delay has been extensively studied by some scholarssuch as Zuxiu Zheng, Senlin Li, Xilin Fu et al. However, there are still some prob-lems to be further investigated. As we know, periodic phenomena widely existin nature and social society. For example, economic crisis occurred periodically;species number is also periodically changed etc. So the theory on periodic solu-tions of functional diferential equations has become a important research area inthe diferential equations and its applications. Therefore, the existence of periodicsolutions of functional diferential equations, not only has theoretical significance,but also has an important applications. This paper is divided into three chapters. By using matrix measure and Schauder fixed point theorem, the first chapter discusses the existence of periodic solutions of the following integro-differential equations with infinite delay where t G R,x∈Rn, A(t, x)concerning(t, x) is continuous, A(t, x), Q(s-t) is n×n continuous function matrix, f∈C(R×Rn×BCh, Rn)(BCh is defined in section second), A(t+ω,x)=A(t,x),xt(s)=x(t+s),s∈(-∞,0]. The equations here are more general. The results we obtained generalize that in the literature.In the second chapter, by using the Sadovskii fixed point theorem, we discusses the existence of solutions of the following integral differential equation boundary value problem where J=[0,+∞), a>0, b>0, ki>0,0<ξ1<ξ2<…<ξn <+∞,f∈Our results extended relevant conclusions.By using the Schauder fixed point theorem, the third chapter discusses the existence of anti-periodic solutions of the following neutral functional differential equations with infinite delay where x∈Rn,t G R. Define||X(t)||=t∈R/sup│x(t)│, A=(aij)n×n, K(s) is n×n periodic and continuous function matrix, function G and function Q are anti-periodic and continuous function matrix from R×Rn×Rn to Rn, also A(t+T, x)=A(t, x),G(t+T,-u,-v)=-G(t, u, v),g(t+T)=g(t), Q(t+T,-x,-y)=-Q(t,x,y),u,v,x,y G Rn.