Neutral Functional Differential Equations With Infinite Delay Periodic Solution And Stability

As we all know, the systematic research of FDE began in 1850s. FDE and DDE have been extensively developed since 1959,and each branch has been set up a complete theory system.Now,more and more scholars study FDE and explore further developments.Also,FDE with infinite delay is one of the fields of great interest to people.In fact,FDE with infinite delay has undergone a rapid development since 1870s.Hale and Kato gave a normal and set up the B phase space theory in 1978.Under the basic theory,people studied the stability,boundedness and periodic solution of RFDE.For example:in [4],Huang Qichang introduced the concept of uniformly forgetful functional , discussed the boundedness and stability of solution; [5]-[8] discussed the existence of periodic solutions,generalized the results of RFDE with finite delay.However,for NFDE with infinite delay , few people discuss it,and many problems have not been solved.So there are some very interesting developments.lt is worth while generalizing the results of FDE with finite delay or RFDE with infinite delay to NFDE with infinite delay.Because of the difficulty of infinite delay,we may discuss neutral Volterra integro-differential equations,and obtain simple results.A brief description of the organization of this paper follows.There are four chapters in this paper .In the first chapter ,by using the character of operator D and Liapunov functional, we deal with the stability of solutions of linear NFDE of D-operator type with infinite delay,generalize the results of RFDE. In chapter 2 ,by using matrix measure and Schauder fixed theorem ,we discuss the existence and uniform stability of unique periodic solutions on neutral Volterra integro-differential equations with infinite delay.involving the results that occured. In chapter 3,we use the tool of chapter 2.and initiate systematic study the stability of solutions of simple Volterra equations,and give succinct conditions.In chapter 4,by means of constructing a kind of Liapunov functional ,we obtain h uniform stability of the solutions of a scalar Volterra equation with infinite delay.