Periodic Solutions Of Functional Differential Equations With Infinite Delay

In this paper, the periodic solutions of higher dimensional neutral functional differential equations with infinite delay of the followingand one-dimension functional differential equation with infinite delay of the followingare considered.In Section 1 of Chapter 3, let |x| = . min(B) ( max(B) ) denotesminimal ( maximal) characteristic value of symmetry matrix B.Then some conditions on the existence and uniqueness of periodic solutions of equation (1) are obtained by using continuation theorem in coincidence degree. Especially, when Q(t) = 0,A(t,x) =A(t),the conditions which guarantee the existence of unique and stable periodic solu-tion are derived. If |x| = , then (A) = max ( ) . Accordingly, when , Theorem 3.1 and Theorem 3.2 of [21] are not applica-ble.Therefore, the results of this paper are different from those of [21].In Section 2 of Chapter 3, some conditions on the existence and uniqueness of periodic solutions of equation (2) are obtained by using matrix measure and Leray-Schauder fixed point theorem. Especially, when g(s, x) = 0, A(t, x) = A(t), the conditions which guarantee the existence of unique stable or attractive periodic solution are derived. Then, these results generalize those of [21].Furthermore,[21,Theorem 3.3]required o > max{3, }, while Theorem 3.2.3 in this section needs a > . Therefore, Theorem 3.2.3 in this section has improved [21,Theorem 3.3].In Chapter 4, some conditions on the existence and uniqueness of periodic solutions of equation (3) are obtained by using continuation theorem in coincidence degree again.