Existence results for singular three point boundary value problems on time scales

We prove the existence of a positive solution for the three point boundary value problem on time scale T given by y + f (x, y) = 0, x∈ (0, 1] ∩ T, y(0) = 0, y(p)= y σ 2(1) , where p ∈ (0, 1) ∩ T is fixed and f (x, y) is singular at y = 0 and possibly at x = 0, y =∞. We do so by applying a fixed point theorem due to Gatica, Oliker, and Waltman [J. Differential Equations 79 (1989) 62] for mappings that are decreasing with respect to a cone. We also prove the analogous existence results for the related dynamic equations y∇∇ + f (x, y) = 0, y ∇ + f (x, y) = 0, and y∇ + f (x, y) = 0 satisfying similar three point boundary conditions.  2004 Elsevier Inc. All rights reserved