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

Abstract

We prove the existence of a positive solution for the three point boundary value problem on time scale T given by y ΔΔ +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 Δ∇+ f( x, y)=0, and y ∇Δ+ f( x, y)=0 satisfying similar three point boundary conditions.