The existence of positive solutions for some nonlinear boundary value problems with linear mixed boundary conditions

In this paper we study the existence of positive solutions of the equation (φ(x′))′+a(t)f(x(t))=0, where φ:R→R is an increasing homeomorphism and homomorphism and φ(0)=0, subject to linear mixed boundary conditions by a simple application of a fixed point index theorem in cones.