Positive solutions for three-point boundary value problem on the half-line

In this paper we consider the existence of positive solutions for the following boundary value problem on the half-line {(ρ(t)x′(t))′+f(t,x(t),x′(t))=0,t∈[0,+∞),x(0)=αx(ξ),limt→∞x(t)=0. By applying fixed-point theorems, we obtain a variety of existence results. In particular, the nonlinear term is involved with the first-order derivative.