Positive solutions to a new kind Sturm-Liouville-like four-point boundary value problem

In this paper, we considered the following four-point boundary value problem {x ''(t) + h(t) f(t, x(t), x'(t)) = 0, 0<T<1, x'(0) - alpha(1)chi(xi) = 0, x'(1) + alpha(2 chi)(eta) - 0, where 0 < alpha(1) < 1/xi, 0 < alpha(2) < 1/1-n , 0 < xi < eta < 1; alpha(1)alpha(2)eta - alpha(1)alpha(2)xi + alpha(1) + alpha(2) > 0: By applying fixed-point theorems, we obtain a variety of existence results. In particular, our four-point boundary condition is a new kind Sturm-Liouville-like boundary condition, which has rarely been considered up to now. (C) 2010 Published by Elsevier Inc.