Positive solutions of a nonlinear fourth-order dynamic eigenvalue problem on time scales (Q437630)

Summary: Let \(\mathbb T\) be a time scale and \(a, b \in \mathbb T, a < \rho^2(b)\). We study the nonlinear fourth-order eigenvalue problem on \(\mathbb T, u^{\Delta^4}(t) = \lambda h(t)f(u(t), u^{\Delta^2}(t)), t \in [a, \rho^2(b)]_{\mathbb T}, u(a) = u^\Delta(\sigma(b)) = u^{\Delta^2}(a) = u^{\Delta^3}(\rho(b)) = 0\) and obtain the existence and nonexistence of positive solutions when \(0 < \lambda \leq \lambda^\ast\) and \(\lambda > \lambda^\ast\), respectively,for some \(\lambda^\ast\). The main tools to prove the existence results are the Schauder fixed point theorem and the upper and lower solution method.