New results on the positive solutions of nonlinear second-order differential systems - Yingxin Guo (Electronic Journal of Qualitative Theory of Differential Equations, 2009)
In this paper, we study the three-point boundary value problems for systems of nonlinear second order ordinary differential equations of the form $$ \left\{\begin{aligned} &u''=-f(t,v), \ \ 0< t< 1,\\&v''=-g(t,u), \ \ 0< t< 1\\&u(0)=v(0)=0,\varsigma u(\zeta)=u(1),\varsigma v(\zeta)=v(1),\end{aligned}\right. $$ where $f:(0,1)\times [0,+\infty)\to [0,+\infty),g:[0,1]\times [0,+\infty)\to [0,+\infty),0<\zeta<1, \varsigma>0,$ and $\varsigma\zeta< 1,f$ may be singular at $t = 0$ and/or $t = 1.$ Under some rather simple conditions, by means of monotone iterative technique, a necessary and sufficient condition for the existence of positive solutions is established, a result on the existence and uniqueness of the positive solution and the iterative sequence of solution is given.