On a class of boundary value problems with oscillatory properties in a part of the spectrum. (Q1432196)

The author considers the two-point boundary value problem \[ x^{(n)}(t)+ p_1(t) x^{(n-1)}(t)+ p_2(t) x^{(n-2)}(t)+\cdots+ p_n(t) x(t)=\lambda x(t)\quad\text{ for }t\in [a,b], \] \[ x^{(k_i)}(a)+ \sum_{k< k_i} \gamma_{ik}x^{(k)}(a)= 0,\quad i= 1,2,\dots, m, \] \[ x^{(k_i)}(b)+ \sum_{k< k_i} \gamma_{ik} x^{(k)}(b)= 0,\quad i= m+1,m+2,\dots, n, \] with \(p_i\in L_1[a,b]\), \(0< m< n\), \(0\leq k_m<\cdots< k_1\leq n-1\), \(0\leq k_n<\cdots< k_{m+2}< k_{m+1}\leq n-1\). For this class of problems, all eigenvalues are real and simple, the \(n\)th eigenfunction \(x_n\) has exactly \(n\) zeros in \((a,b)\), the zeros of \(x_n\) and \(x_{n+1}\) alternate, and the functions \(x_1,\dots, x_n\) form a Chebyshev system. The author shows that there exists a broader class of boundary value problems with the same oscillation properties beginning with some (not necessarily the first) number. The properties of this new class are discussed.