一类非线性二阶常微分方程Dirichlet问题正解的存在性

Existence of Positive Solutions for a Class of Nonlinear Second-Order Dirichlet Problem

摘要:

本文研究了一类非线性二阶常微分方程~Dirichlet~边值问题 $$ \left\{\begin{array}{ll} u''-\ a(t)u+f(t,u)= 0,~~\ \ \ 0< t< 1\\[2ex] \ u(0)=\ u(1)=0 \end{array} \right.\eqno $$ 正解的存在性,~其中~$f: [0,1]\times R^{+}\rightarrow R^{+}$连续,~$a:[0,1]\rightarrow R^{+}$~连续.~主要结果的证明基于锥拉伸与压缩不动点定理.~

Abstract:

~In this paper,~ we study the existence of positive solutions for a class of nonlinear second-order Dirichlet problem ~ $$ \left\{\begin{array}{ll} u''-\ a(t)u+f(t,u)= 0,~~\ \ \ 0< t< 1\\[2ex] \ u(0)=\ u(1)=0 \end{array} \right.\eqno $$\~where~$f:[0,1]\times R^{+}\rightarrow R^{+}$~is continuous,~$a:[0,1]\rightarrow R^{+}$~is continuous.~The proof of the main results is based on the fixed-point theorem of cone expansion-compression.