Boundary Value Problem For Integrodifferential Equations In Banach Space

In the first chapter ,the periodic boundary value problems (PBVP) for second order integrodifferential equations in Banach Space are considered .The conclusion is as follows :Suppose that the following assumptions are satisfied :(A1) are the lower and upper solutions of (PBVP)(1.3.1) - (1.3.2) ,and satisfy ,where Vi E is bounded , , a is the Kuratowski's measure of noncompactness of a bounded subset in E.Using the monotone iterative technique, we consider a new situation that the lower solutions are not less than the upper solutions, on the basis of comparison result and fixed point theorems of increasing operator. We have obtained the existence of the maximal and minimal solutions of PBVP (1.3.1) - (1.3.2) .Theorem 1.3.1 Assume that E is a real Banach space , P is a regular cone,and (A1), (A2), are satisfied,further:Then (PBVP) (1.3.1) - (1.3.2) exists the minimum solution u(t) and maximum solution u*(t) in [w0,v0]. Further ,let:where H(t,s) is giving by (1.3.7) - (1.3.11).Then uniformly in J.Theorem 1.3.2 Assume that E is a real Banach space ,P is a normal cone in E,andsatisty (A1), (A2), (A3),Then,the conclusion of the theorem 1.3.1 is also correct.As a application,we study the periodic boundary value problems (PBVP) for one type ofthird order integrodifferential equations in Banach Space .In the second chapter ,we consider the periodic boundary value problems for second order integro differential equations with Caratheodory functions.Where R is a Caratheodory function, k(t, s)u(s)ds,Using the method of the lower and upper solutions and the theory of topological degree,wehave obtained the existence of the solutions between the lower and upper solutions.