Ⅰ.The Existence Of The Solutions Of Second Order Periodic Boundary Value Problems In Banach Space Ⅱ.Periodic Boundary Value Problems For The Generalized Nonlinear Systems

I The Existence of the Solutions of Second Order Periodic Boundary Value Problems in Banach SpaceIn this dissertation, we study the following periodic boundary value problems (PBVP) for integro-differential equations in a order Banach Space E.(1.1)where We consider the normal situation, that is the lower solution is less than upper solution, on the basis of comparison result , and fixed point theorems of increasing operator, using the monotone iterative technique, we have got the extremal solutions of PBVP (1.1).Definition: Suppose is continuous on 7 }, real number M > 0 and satisfiesthen v0(t) is called a lower solution of PBVP (1.1) .Definition: Suppose C2[I, E] ,real number M > 0 and satisfiesthen w0(t) is called a upper solution of PBVP (1.1) .For convenience ,let us list the following assumptions for PBVP (1.1): (A1) v0,w0 are the lower and upper solutions for PBVP (1.1) respectively and satisfy(A2) There exist constants M > 0, N > 0,such thatwhere (A3) For any t ∈ I and equicontinuous,bounded monotone sequences B = ,such thatwhere C3 , C4 > 0.Main results:Theorem 1 Let E be a real Banach space, P be a normal cone in E. Conditions (A1)-(A3) be satisfied,let k0 < C2,and L = max{1,maxJ G(t,s)}, inequality 4L#[C3 + M+47#k0(C4 + 2N)] < 1 holds. Then there exist monotone sequences {vn(t)}, (wn(t)}, such thatuniformly on I and p(t] , r(t) are minimal and maximal solutions between VQ and W0 forPBVP(1.1).Certainly ,this paper also generalized the methods f1' to the case of second order integro-differential equation.