Zeros of the Green's function for the de la Vall\'ee-Poussin problem

Yu. V. Pokornyi

Sbornik. Mathematics, Tome 199 (2008) no. 6, pp. 891-921

Résumé

The Green's function for the de la Vallée-Poussin problem \begin{gather*} Lx\equiv x^{(n)}+p_1(t)x^{(n-1)}+\dots+p_n(t)x=f, \\ x(a_i)=A_i^{(0)}, \ \ x'(a_i)=A_i^{(1)}, \ \ \dots, \ \ x^{(\nu_i-1)}(a_i)=A_i^{(\nu_i-1)}, \ \ i= {1,\dots,m}, \end{gather*} where $a=a_1$, $m\geqslant2$, $\sum\nu_i=n$, $p_i(\,\cdot\,)$ and $f(\,\cdot\,)\in L_1[a,b]$, is investigated. It is defined in the square $a\leqslant t,s\leqslant b$, and vanishes at the lines $t=a_i$, $i={1,\dots,m}$, $s=a$, $s=b$; it is proved that the orders of its zeros have uniform bounds. Bibliography: 27 titles.