Properties in $L_p$ of root functions for a nonlocal problem with involution

Abstract

The spectral problem $-u''(x)+\alpha u''(-x)=\lambda u(x)$, $-1$%lt; $x$ < $1$, with nonlocal boundary conditions $u(-1)=\beta u(1)$, $u'(-1)=u'(1)$, is studied in the spaces $L_p(-1,1)$ for any $\alpha\in (-1,1)$ and $\beta\ne\pm 1$. It is proved that if $r=\sqrt{(1-\alpha)/(1+\alpha)}$ is irrational then the system of its eigenfunctions is complete and minimal in $L_p(-1,1)$ for any $p>1$, but does not form a basis. In the case of a rational value of $r$, the way of supplying this system with associated functions is specified to make all the root functions a basis in $L_p(-1,1)$.