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

Abstract

Let $mathbb{T}$ be a time scale such that $0, T in mathbb{T}$. We consider the second order dynamic equation on a time scale $$displaylines{ u^{Delta abla}(t) + a(t)f(u(t)) = 0, quad t in (0,T) cap mathbb{T},cr u(0) = 0, quad alpha u(eta) = u(T), }$$ where $eta in (0, ho(T)) cap mathbb{T}$, and $0 < alpha <T/eta$. We apply a cone theoretic fixed point theorem to show the existence of positive solutions.

Keywords

• Time scale
• cone
• boundary-value problem
• positive solutions.