在半无限区间上二阶方程组边值问题的正解

Positive Solutions to Boundary Value Problems of Second Order Systems on Semi-infinite Interval

半无限区间上的边值问题经常出现在应用数学的各种分支,Agarwal等人也对该类问题进行了讨论。然而,半无限区间上的非线性边值问题的一般理论还很不完善。本文讨论半无限区间上的二阶微分方程组x″(t)-k21x(t)+f(t,x(t),y(t))=0,y″(t)-k22y(t)+g(t,x(t),y(t))=0,x(0)=y(0)=0,limt→∞y(t)=0,其中f,g是非负t→∞x(t)=lim连续的函数,在具有Bielecki模的某一函数空间的一个锥K1×K2上定义积分算子A,利用锥上的Krasnoselskii不动点定理,赋予f,g一定的增长条件建立上述问题的正解存在性定理。同时,最近文献一个定理中的错误也被改正。

The boundary value problems on the semiinfinite interval occur frequently in various branches of applied mathematics.The problems have also been discussed by Agarwal et al.However,the genral theory of nonlinear boundary value problems on the semiinfinite interval is far from complete.This paper is devoted to the study of the second order system on the semiinfinite interval of the form x″(t)-k21x(t)+f(t,x(t),y(t))=0,y″(t)-k22y(t)+g(t,x(t),y(t))=0,x(0)=y(0)=0,limt→∞x(t)=limt→∞y(t)=0,where f,g are continuous and nonnegative functions.Define the integral operatora on K1×K2,where K1×K2 is a cone in the appropriate function spaces equipped with Bielecki's norm.Using the Krasnoselskii fixed point theorem on cone,we impose growth conditions on f,g,which guarantee the existence of positive solutions to the above problem.At the same time,an incorrect result in a recent paper is pointed out.