Abstract:
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 $a=a_1<a_2<\dots<a_m=b$, $m\geqslant2$,
$\sum\nu_i=n$, $p_i(\,\cdot\,)$ 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.
Fulltext:
PDF file (685 kB)
