Eigenvalue problems for 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 us a cone theoretic fixed point theorem to obtain intervals for $\lambda$ for which the second order dynamic equation on a time scale, \begin{gather*} u^{\Delta\nabla}(t) + \lambda a(t)f(u(t)) = 0, \quad t \in (0,T) \cap \mathbb{T},\\ u(0) = 0, \quad \alpha u(\eta) = u(T), \end{gather*} where $\eta \in (0, \rho(T)) \cap \mathbb{T}$, and $0 < \alpha <T/\eta$, has a positive solution.