Basicity in L_p of root functions for differential equations with involution

Abstract

We consider the differential equation $$ \alpha u''(-x)-u''(x)=\lambda u(x), \quad -1<x<1, $$ with the nonlocal boundary conditions $u(-1)=0$, $u'(-1)=u'(1)$ where $\alpha\in (-1,1)$. We prove 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 constitute a basis. In the case of a rational value of r we specify the way of choosing the associated functions which provides the system of all root functions of the problem forms a basis in $L_p(-1,1)$.