n阶常微分方程正周期解的存在性

Existence of positive periodic solutions for nth-order ordinary differential equations

利用锥上的不动点指数理论,讨论n阶变系数常微分方程u(n)(t)+a(t)u(t)=f(t,u(t),u'(t),…,u(n-1)(t))正周期解的存在性,其中n≥2,a(t):R→(0,∞)连续以ω为周期,f:R×[0,∞)×Rn-1→R连续,f(t,x0,x1,…,xn-1)关于t以ω为周期。在假设f关于x0满足超线性或次线性增长条件下,获得了正ω周期解的存在性。

By using the fixed point index theory of cones, the existence of positive periodic solutions for the nth-order ordinary differential equation u（n）（t）＋a（t）u（t）=f（t,u（t）,u＇（t）,…,u（n-1）（t））is concerned, where n≥2,a（t）：R→（0,∞）is a continuous function which is ω -periodic in t f：R×[0,∞）×Rn-1→R is a continuous function and f （t,x0,x1,…,xn-1）is ω -periodic in t. Some exist-ence results of positive ω - periodic solutions are obtained when f satisfies some super - linear or sublinear growth conditions on x0 ,x1 ,... ,xn-1.