For the second-order three-point boundary value problem x″(t)+f(t,x,x′)=0,0≤t≤1,x(0)=0,x′(1)=αx′(η),f:[0,1]×[0,∞)×R→[0,∞)is continuous,0＜α＜1,η∈(0,1).The associated Green′s function for the above problem is given first,and then,by using the extension of Krasnoselskii''s fixed point theorem in a cone,growth conditions are imposed on nonlinearity f which ensure the existence of at least one positive solution.