Existence of positive solutions for boundary value problem of second-order functional differential equation

Abstract

We use a fixed point index theorem in cones to study the existence of positive solutions for boundary value problems of second-order functional differential equations of the form $$\left\{ \begin{array}{ll} y''(x)+r(x)f(y(w(x)))=0,&0<x<1,\\ \alpha y(x)-\beta y'(x)=\xi (x),&a\leq x\leq 0,\\ \gamma y(x)+\delta y'(x)=\eta (x),&1\leq x\leq b; \end{array}\right.$$ where $w(x)$ is a continuous function defined on $[0,1]$ and $r(x)$ is allowed to have singularities on $[0,1]$. The result here is the generalization of a corresponding result for ordinary differential equations.