Positive Solutions of a Nonlinear Fourth-Order Dynamic Eigenvalue Problem on Time Scales

Abstract

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