Vol. 51, No. 1, 1974

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Oscillatory solutions and multi-point boundary value functions for certain nth-order linear ordinary differential equations

Marvin Stanford Keener

Vol. 51 (1974), No. 1, 187–202

Consider the n-th order linear differential equation

 (n)  n∑−1     (k)
y  +    pk(t)y   = 0,

where pk(t) C[α,). This study explores some of the relationships between multi-point boundary value functions for (1) and the character of oscillatory solutions of (1). In particular, it is supposed for (1) that a certain (n1) point boundary value problem has no nontrivial solution and that two nontrivial solutions with (n 1) zeros in common are constant multiples of each other. Under these conditions it is shown that there exists an integer i,1 i n1, such that for each a > α and every integer l,1 l i 1, there is an oscillatory solution of (1) with a zero of exact multiplicity l at t = a. Furthermore, any solution of (1) with a zero at t = a of multiplicity l i is nonoscillatory.

Mathematical Subject Classification 2000
Primary: 34C10
Received: 24 November 1972
Revised: 30 April 1973
Published: 1 March 1974
Marvin Stanford Keener