Fourth order boundary value problems with finite spectrum

Finite spectrum of2nth order boundary value problems

“…Thus from the Fundamental Theorem of Algebra, ( ) has at most (2n 1)m + 1 roots or ( ) = 0 for all 2 C: Remark 1. For n = 1 Theorem 1 reduces to a result obtained in [11]; and for n = 2 it reduces to a result in [1]. P roof of T heorem 2: From Corollary 2, Corollary 3 and (3.13) we know that the maximum of degree of the characteristic function ( ) is n(m + 1).…”

“…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:…”

“…We study boundary value problems consisting of the equation Here and below M k;m (F ) denotes the k m matrices over F = C; the complex numbers, or F = R the reals, M k;m (F ) = M k (F ) when m = k; Y = [y [0] ; y [1] ; ; y [2n 1] ] T ; where T denotes transpose, is the spectral parameter, n 2 N =f1; 2; 3; g; y [j] denotes the quasi-derivatives: y [j] = y (j) ; j = 0; : : : ; n 1; y [n+j] = (py (n) ) (j) ; j = 0; : : : ; n 1; where y (j) denotes the ordinary derivative.…”