Positive solutions of a three-point boundary-value problem on a time scale

Abstract

Let T be a time scale such that 0, T ∈ T. We consider the second order dynamic equation on a time scale u∆∇(t) + a(t)f(u(t)) = 0, t ∈ (0,T) ∩ T, u(0) = 0, αu(ƞ) = u(T), where ƞ ∈ (0,ρ(T)) ∩ T, and 0 < α < T/ƞ. We apply a cone theoretic fixed point theorem to show the existence of positive solutions.