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

Öz 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 $L_p(-1,1)$.
Anahtar Kelimeler:

## ODE with involution, nonlocal boundary-value problem, basicity, root functions

#### Kaynak Göster

 Bibtex @ { tbtkmath575270, journal = {Turkish Journal of Mathematics}, issn = {1300-0098}, eissn = {1303-6149}, address = {}, publisher = {TÜBİTAK}, year = {2019}, volume = {43}, pages = {393 - 401}, doi = {}, title = {Properties in \$L\_p\$ of root functions for a nonlocal problem with involution}, key = {cite}, author = {Krıtskov, Leonid and Sadybekov, Makhmud and Sarsenbı, Abdizhahan} } APA Krıtskov, L , Sadybekov, M , Sarsenbı, A . (2019). Properties in $L_p$ of root functions for a nonlocal problem with involution . Turkish Journal of Mathematics , 43 (1) , 393-401 . MLA Krıtskov, L , Sadybekov, M , Sarsenbı, A . "Properties in $L_p$ of root functions for a nonlocal problem with involution" . Turkish Journal of Mathematics 43 (2019 ): 393-401 < Chicago Krıtskov, L , Sadybekov, M , Sarsenbı, A . "Properties in $L_p$ of root functions for a nonlocal problem with involution". Turkish Journal of Mathematics 43 (2019 ): 393-401 RIS TY - JOUR T1 - Properties in $L_p$ of root functions for a nonlocal problem with involution AU - Leonid Krıtskov , Makhmud Sadybekov , Abdizhahan Sarsenbı Y1 - 2019 PY - 2019 N1 - DO - T2 - Turkish Journal of Mathematics JF - Journal JO - JOR SP - 393 EP - 401 VL - 43 IS - 1 SN - 1300-0098-1303-6149 M3 - UR - Y2 - 2021 ER - EndNote %0 Turkish Journal of Mathematics Properties in $L_p$ of root functions for a nonlocal problem with involution %A Leonid Krıtskov , Makhmud Sadybekov , Abdizhahan Sarsenbı %T Properties in $L_p$ of root functions for a nonlocal problem with involution %D 2019 %J Turkish Journal of Mathematics %P 1300-0098-1303-6149 %V 43 %N 1 %R %U ISNAD Krıtskov, Leonid , Sadybekov, Makhmud , Sarsenbı, Abdizhahan . "Properties in $L_p$ of root functions for a nonlocal problem with involution". Turkish Journal of Mathematics 43 / 1 (Şubat 2019): 393-401 . AMA Krıtskov L , Sadybekov M , Sarsenbı A . Properties in $L_p$ of root functions for a nonlocal problem with involution. Turkish Journal of Mathematics. 2019; 43(1): 393-401. Vancouver Krıtskov L , Sadybekov M , Sarsenbı A . Properties in $L_p$ of root functions for a nonlocal problem with involution. Turkish Journal of Mathematics. 2019; 43(1): 393-401. IEEE L. Krıtskov , M. Sadybekov ve A. Sarsenbı , "Properties in $L_p$ of root functions for a nonlocal problem with involution", Turkish Journal of Mathematics, c. 43, sayı. 1, ss. 393-401, Şub. 2019