Eigenvalue problems for a three-point boundary-value problem on a time scale

E. Kaufmann, University of Arkansas at Little Rock, Little Rock, USA
Y. Raffoul, University of Dayton, Dayton, U.S.A.

E. J. Qualitative Theory of Diff. Equ., No. 2. (2004), pp. 1-10.

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.

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