Odd Periodic Solutions Of 2n-order Ordinary Differential Equations

Based on the Leray-Schauder fixed point theorem,the Fourier analysis method and the fixed point index theory of cone map,the paper discussed the existence and uniqueness of odd 27r-periodic solutions of 2nth-order ordinary differential equations.The main results are as follow:1,When the nonlinearity does not contain the derivative term,By applying the fixed point index theory in cones,the results of existence odd 2?-periodic solutions is obtained under the conditions that the nonlinearity may be superlinear or sublinear growth.2,When the nonlinearity contain the even order derivative,By applying the Leray-Schauder fixed point theorem and Fourier analysis method,the results of existence of odd 27r-periodic solutions is obtained under the nonlinearity f satisfies superlinear growth conditions.3,Under the nonlinearity f satisfies unilateral growth condition and Nagumo type growth condition,By applying the Leray-Schauder fixed point theorem and Fourier analysis method,the existence and uniqueness results of odd 2?-periodic solutions of fully 2nth-order ordinary differential equations is obtained.Here the Nagumo type condition request the growth of f(t,xo,x1,…,x2n-1)on x2n-1 which cannot be hyperquadric.4,Under the conditions that the nonlinearity may be superlinear and sublinear growth,By picking the right cone,the existence results of odd 27r-periodic solutions of fully 2nth-order ordinary differential equations is obtained via the fixed point index theorem of cone map.For the case of superlinear growth,we obtain it when the nonlinearity satisfies Nagumo type condition.