Spectrum of a linear fourth-order differential operator and its applications

In this paper, we apply the disconjugacy theory and Elias's spectrum theory to study the positivity and the spectrum structure of the linear operator u⁽⁴⁾+Mu coupled with the clamped beam boundary conditions (1.2). We also study the positivity and the spectrum structure of the more general operator u⁽⁴⁾+p(t)u coupled with (1.2). As the applications of our results on positivity and spectrum of fourth-order linear differential operators, we show the existence of nodal solutions for the corresponding nonlinear problems via Rabinowitz's global bifurcation theorem.