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

Authors : Leonid KRITSKOV, Makhmud SADYBEKOV, Abdizhahan SARSENBI

Pages : 393-401

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Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :The spectral problem $-u``insert ignore into journalissuearticles values(x);+\alpha u``insert ignore into journalissuearticles values(-x);=\lambda uinsert ignore into journalissuearticles values(x);$, $-1$%lt; $x$ < $1$, with nonlocal boundary conditions $uinsert ignore into journalissuearticles values(-1);=\beta uinsert ignore into journalissuearticles values(1);$, $u`insert ignore into journalissuearticles values(-1);=u`insert ignore into journalissuearticles values(1);$, is studied in the spaces $L_pinsert ignore into journalissuearticles values(-1,1);$ for any $\alpha\in insert ignore into journalissuearticles values(-1,1);$ and $\beta\ne\pm 1$. It is proved that if $r=\sqrt{insert ignore into journalissuearticles values(1-\alpha);/insert ignore into journalissuearticles values(1+\alpha);}$ is irrational then the system of its eigenfunctions is complete and minimal in $L_pinsert ignore into journalissuearticles values(-1,1);$ 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 $L_pinsert ignore into journalissuearticles values(-1,1);$.
Keywords : ODE with involution, nonlocal boundary value problem, basicity, root functions