The Finite Spectrum of Fourth-Order Boundary Value Problems with Transmission Conditions

Abstract

A class of fourth-order boundary value problems with transmission conditions are investigated. By constructing we prove that these class of fourth order problems consist of finite number of eigenvalues. Further, we show that the number of eigenvalues depend on the order of the equation, partition of the domain interval, and the boundary conditions (including the transmission conditions) given.

Finite spectrum of2nth order boundary value problems

“…We outline the proof here. The details are similar to those given in [14] for the second order case and in [1,6] for the fourth order case and hence omitted.…”

“…This we do by constructing a characteristic function ( ) which is a polynomial in : This construction is complicated and involves a partition of the interval J and a construction of nonnegative coe¢ cients r and w which are not positive on any common subinterval of J: Our main results reduce to known results when n = 1 [11] and n = 2 [1][2][3][4]6]. Our construction has its roots in these papers and also uses an inductive scheme:…”