Methods of extending lower order problems to higher order problems in the context of smallest eigenvalue comparisons

Abstract

The theory of $u_{0}$-positive operators with respect to a cone in a Banach space is applied to the linear differential equations $u^{(4)}+\lambda_{1} p(x)u=0$ and $u^{(4)}+\lambda_{2} q(x)u=0$, $0\leq x\leq 1$, with each satisfying the boundary conditions $u(0)=u^{\prime}(r)=u^{\prime \prime}(r)=u^{\prime \prime \prime}(1)=0$, $0<r<1$. The existence of smallest positive eigenvalues is established, and a comparison theorem for smallest positive eigenvalues is obtained. These results are then extended to the $n$th order problem using two different methods. One method involves finding sign conditions for the Green's function for $-u^{(n)}=0$ satisfying the higher order boundary conditions, and the other involves making a substitution that allows us to work with a variation of the fourth order problem.

Keywords

• eigenvalue
• cone
• boundary value problem
• comparison theorem