Odd periodic solutions for 2n2nth-order ordinary differential equations

This paper deals with the existence of periodic solutions for the 2n2nth-order ordinary differential equation u(2n)(t)=f(t,u(t),u″(t),…,u(2n−2)(t)),u(2n)(t)=f(t,u(t),u″(t),…,u(2n−2)(t)), where the nonlinear term f:R×Rn→Rf:R×Rn→R is a continuous odd function and f(t,x0,x1,…,xn−1)f(t,x0,x1,…,xn−1) is 2π2π-periodic in tt. Some existence results for odd 2π2π-periodic solutions are obtained under the condition that ff satisfies some linear, superlinear or sublinear growth conditions on x0,x1,…,xn−1x0,x1,…,xn−1.