一类一阶常微分系统周期边值问题正解的存在唯一性

Existence and uniqueness of positive solutions for the periodic BVP of a class of first-order ordinary differential systems

本文研究了一阶常微分系统周期边值问题{-u′(x)+a(x)u(x)=g(v(x)),0<x<1,-v′(x)+b(x)v(x)=f(u(x)),0<x<1,u(0)=u(1),v(0)=v(1)}的正解的存在唯一性,其中a,b∈C([0,1],[0,∞))且在[0,1]的任何子区间上不恒为0,f,g:R→R连续,f(0)≥0,g(0)≥0且f(t),g(t)关于t∈[0,∞)单调递增.主要结果的证明基于Schauder不动点定理和Leray-Schauder度理论.

In this paper,we study the existence and uniqueness of positive solutions for the periodic boundary value problem of the following first-order ordinary differential equations:{-u′(x)+a(x)u(x)=g(v(x)),0<x<1,-v′(x)+b(x)v(x)=f(u(x)),0<x<1,u(0)=u(1),v(0)=v(1),where a,b∈C([0,1],[0,∞))do not vanish identically on any subinterval of[0,1],f,g:R→R are continuous functions,f(0)≥0,g(0)≥0,and f(t),g(t)are nondecreasing on[0,∞).The proof of the main results are based on the Schauder fixed point theorem and the Leray-Schauder degree theory.