Existence Of Periodic Solutions Of Fourth-Order Ordinary Differential Equations

In this thesis,by using Leray-Schauder fixed point theorem and Schauder fixed point theorem of completely continuous operators,Banach contraction mapping principle,upper and lower solution method,and fixed point index theory in cones,we deal with the existence and uniqueness of the periodic solutions of fourth-order ordinary differential equation u(4)(t)=f(t,u(t),u'(t),u"(t),u'"(t)),t?R,where f:R×R4?R is a continuous function.The main result are as follows:1.Under the condition that the nonlinear term f satisfies linear growth,the existence and uniqueness of the solution of periodic fourth-order nonlinear differ-ential equation are obtained by using Leray-Schauder fixed point theorem of the completely continuous operator.2.Under the condition of two parameter nonresonance,by using Schauder fixed point theorem and Banach contraction mapping principle,the existence and uniqueness of periodic solutions for the fourth order nonlinear differential equations are obtained,our conclusion extend and improve known result.3.With the help of Nagumo condition,the existence of periodic solutions for the fourth-order nonlinear differential equations is obtained by using truncated function technique and upper and lower solution method.4.Under the conditions that the nonlinear term f satisfies some easily verifiable inequalities a nd satisfies the superlinear or sublinear growth,by selecting a suitable cone and using the fixed point index theory of cones,the existence of the positive periodic solutions for the fourth order nonlinear differential equations are obtained,our conclusion extend and improve corresponding result of known literature.