Solutions To The Second Order Ordinary Differential Equation Periodic Boundary Value Problems

Ordinary differential equation periodic boundary value problems are more general ones in boundary value problems. Many researchers studied these problems. In this paper,we dis-cuss the periodic boundary value problems for two classes of ordinary differential equation. One of which is the periodic boundary value problems for the ordinary differential equation system. The other is the periodic boundary value problems for a classes of ordinary differen-tial equation with concave nonlinearities. The existence and multiplicity of positive solutions for the two problems are obtained by using the fix point theorem, the variational argument and the critical point theorem.In Chapter 2, we discuss the problem: and obtain the following results.Theorem 2.2 If the following assumptions hold:(A1)(?) (?) f(t,u)/u=+∞;(A2) Exist p∈C([0,T],R+),q∈(R+,R+),satisfies(A3)(?)q(u)/u=0. then the problem (2.2) has at least one positive solution.Theorem 2.3 If the conditions (A1), (A2) hold and f,q satisfy the following conditions:(A4)(?) (?) f(t,u)/u=+∞;(A5) Exist 0<p≤1 such that then the problem (2.2) has at least two posi-tive solutions. In Chapter 3,we discuss the periodic boundary value problems for the second ordinary differential equation: and we have the result.Theorem 3.1 Suppose that f satisfies the following conditions:(F1) There existsμ0>λ1 such that (?)f(s)/s=μ0；(F2)There exist constantsμ1,μ2,μ3,μ4>0 withμ1<λ1<μ0<μ2 such that Then there existsλ*>0 such that the problem(3.1)has at leaSt two nontriVial T-periodic Solutions for入∈(0,λ*).