Remarks on some fourth-order boundary value problems with non-monotone increasing eigenvalue sequences - Chengyue Li; Yuhan Wu (Electronic Journal of Qualitative Theory of Differential Equations, 2016)
By $Z_{2}$-index theory, the existence and multiplicity of solutions for some fourth-order boundary value problems $$ \begin{cases} u^{(4)}+au''=\mu u+F_{u}(t,u),\ 0<t<L,\\ u(0)=u(L)=u''(0)=u''(L)=0 \end{cases}$$ at resonance are studied, where $a>0$ and $\mu\in \mathbb{R}$ is an eigenvalue of the corresponding eigenvalue problem. The difficulty caused by the non-monotone eigenvalue sequence is handled concretely.