一般三阶非线性常微分方程的正周期解

Positive Periodic Solutions of General Nonlinear Third-Order Ordinary Differential Equations

用锥上的不动点指数理论,考虑一般三阶常微分方程Lu(t)=f(t,u(t),u′(t),u″(t))(t∈ℝ)正2π-周期解的存在性,其中:Lu(t)=u‴(t)+a2u″(t)+a1u′(t)+a0u(t)是三阶常微分算子;f:ℝ×[0,∞)×ℝ2→[0,∞)连续,f(t,x,y,z)关于t以2π为周期.在非线性项f满足一些易验证的不等式条件下,允许f(t,x,y,z)关于x,y,z满足超线性或次线性增长,得到了该方程正2π-周期解的存在性结果.

Using the fixed point index theory of cones,we consider the existence of positive 2π-periodic solutions for general third-order ordinary differential equation Lu(t)=f(t,u(t),u′(t),u″(t))(t∈ℝ).Where Lu(t)=u(t)+a 2u″(t)+a 1u′(t)+a 0u(t)is a third-order ordinary differential operator,f:ℝ×[0,∞)×ℝ2→[0,∞)is continuous function and f(t,x,y,z)is 2π-periodic with respect to t.Under the conditions that the nonlinear term f satisfi es some easily verifiable inequalities,some existence results of the positive 2π-periodic solutions of the equation are obtained that allow f(t,x,y,z)satisfies the superlinear or sublinear growth with respect to x,y,z.