The Periodic Solutions For A Class Of Third Order Nonlinear Differential Equations

In this paper, by using the monotone iterative method of upper and lower solutions, Leray-Schauder fixed point theorem and the fixed point index theory in cones, we deal with the existence and uniqueness of 2? periodic solutions and the existence of positive 2? periodic solutions to the third order nonlinear differential equation u'"(t)=f(t,u(t),u'(t)),t?R. where f:R×R~2?R is a continuous function.The main results of this paper are as follows:1. With the existence of solutions for corresponding third-order linear differ-ential equation, we build a new maximum principle, and obtain the existence 2? periodic solutions by the method of monotone iterative of upper and lower solutions for the third order nonlinear differential equation.2. Under the linear growth conditions, we obtain the existence and uniqueness of 2? periodic solutions by using the Leray-Schauder fixed point theorem of completely continuous operators.3. Using the Leray-Schauder fixed point theorem of completely continuous oper-ators, we obtain the existence and uniqueness of a third order nonlinear differential equation without growth restriction.4. Applying the fixed-point index theory in cone, we obtain the existence of positive 2? periodic solutions under the case of superlinear and sublinear conditions.