Periodic Boundary Value Problems Of Nonlinear Ordinary Differential Equations

In last few years, all sorts of nolinear problems have resulted from mathematics, physics, chemistry, biology, medicine, economics, engineering, sybernetics and so on. With solving these problems, many important methods and theory such as partial ordering method, topological degree method, the theory of cone and the variational method have been developed gradually. They become a very effective theoretical tool to solve many nonlinear problems in the fields of the science and technology.This paper mainly investigates the existence of solutions for periodic boundary value problems of nolinear differential equations by using the theory of cone and topological de-gree method. The existence and uniqueness of positive solutions for differential equations have been considered extensively since twenty years ago([20]-[32]). This paper discusses the existence of solutions for periodic boundary value problems more generally on the basis of above references.Chapter 1 investigates the existence of multiple positive solutions for impulsive periodic boundary value problemwhere p∈((?),(?)),f : J×R→R⁺,R⁺ = [0, +∞),R^- = (-∞,0],0 = t₀ < t₁ < t₂<…kk+1=2Ï€,I_i∈C[R, R⁺],I_i^*∈C[R, R^-].In the papers [1]-[8], the writer investigated this problem by using lower and upper solutions and monotone iterative technique. In the paper [9], the writer considered this problem by using topological degree method. However, the nonlinear f was continuous function in these papers. In this paper, f is a C arath(?)odory function, then we gain 2n-1 positive solutions of the equation by constructing a special cone and using the fixed point theorem of cone expansion and compression, improve the ruselt in paper [10]. Chapter 2 investigates the existence of positive solution for singular integro-differential periodic boundary value problem in Banach spacewhere 0 < k < (?),f is singular at t = 0, t = 2Ï€and u =θ.f∈C[(0, 2Ï€)×P \ {θ}×E×P×P,P].In last few years, many writers considered this problem by using lower and upper solutions and monotone iterative technique. As far as we know, Considering this problem by using fixed point theorem is rare. This paper gains the existence of positive solution of the equation by constructing a special closed convex subset and using Monch fixed point theorem in abstract space.Chapter 3 investigates the existence of multiple nontrivial solutions for periodic boundary value problem of a coupled systemwhere J=[0,T],0 < k < ((?))², f,g∈C[J×R×R,R],T > 0. In the papers [21]-[26], writers investigated the two-point or three-point boundary value problem of differential system. As far as we know, Considering the periodic boundary value system is rare. This paper gains the existence of multiple nontrivial solutions by using the lack of direction property and homotopy invariance of topological degree, improves the result in paper [27].Chapter 4 investigates the existence of multiple nontrivial periodic solutions for a class of second order neutral functional differential equation with a statement-dependent deviation variableIn the papers [29][30], all results of the researches on the second order NFDE are derived under the condition that delay r is a constant. In the paper [31], writer changedÏ„into a function about t and gained the existence of one periodic solution by using the continuation theorem of coincidence degree. This paper gains two nontrivial periodic solutions by using the lack of direction property and additivity of coincidence degree.