Triple positive solutions for some second-order boundary value problem on a measure chain

In this paper, we study the existence of three positive solutions for the second-order two-point boundary value problem on a measure chain, x(Delta Delta) (t) + p(t) f (t, x, (sigma(t)), x(Delta)(t)) = 0, t epsilon [t(1), t(2)], a(1)x(t(1)) - a(2)x(Delta)(t(1)) = 0, a(3)x(sigma(t(2))) + a(4)x(Delta)(sigma(t(2))) = 0, where f : [t(1), sigma (t(2))] x [0, infinity) x R -> [0, infinity) is continuous and p : [t(l), sigma (t(2))] - [0, infinity) a nonnegative function that is allowed to vanish on some subintervals of [t(1), sigma (t(2))] of the measure chain. The method involves applications of anew fixed-point theorem due to Bai and Ge [Z.B. Bai, W.G. Ge, Existence of three positive solutions for some second order boundary-value problems, Comput. Math. Appl. 48 (2004) 699-707]. The emphasis is put on the nonlinear term f involved with the first order delta derivative x(Delta)(t). (C) 2007 Elsevier Ltd. All rights reserved.