Year 2020, Volume 49 , Issue 1, Pages 338 - 351 2020-02-06

Basis properties of root functions of a regular fourth order boundary value problem

Ufuk KAYA [1] , Esma KARA KUZU [2]


In this paper, we consider the following boundary value problem

\[ y^{(4)}+q(x) y=\lambda y,~\ \ \ 0<x<1, \]

\[ y^{\prime\prime\prime}\left(1\right)-\left(-1\right)^{\sigma}y^{\prime\prime\prime}\left(0\right)+\alpha y\left(0\right) =0, \]

\[ y^{(s)}(1) -( -1) ^{\sigma}y^{(s) }( 0) =0,\ \ \ s=\overline{0,2}, \]

where $\lambda $ is a spectral parameter, $q( x)\in L_{1}(0,1)$ is complex-valued function and $\sigma =0,1$. The boundary conditions of this problem are regular but not strongly regular. Asymptotic formulae for eigenvalues and eigenfunctions of the considered boundary value problem are established. When $\alpha\ne 0$, we proved that all the eigenvalues, except for finite number, are simple and the system of root functions of this spectral problem forms a Riesz basis in the space $L_{2}( 0,1)$. Furthermore, we show that the system of root functions forms a basis in the space $L_{p}( 0,1)$, $1<p<\infty$ $(p\neq 2)$, under the conditions $\alpha\ne 0$ and $q( x) \in W_{1}^{1}( 0,1)$.

fourth order eigenvalue problem, not strongly regular boundary conditions, asymptotic behavior of eigenvalues and eigenfunctions, basis properties of the system of root functions
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Primary Language en
Subjects Mathematics
Journal Section Mathematics
Authors

Orcid: 0000-0003-1278-997X
Author: Ufuk KAYA
Institution: BITLIS EREN UNIVERSITY
Country: Turkey


Orcid: 0000-0003-2692-1070
Author: Esma KARA KUZU
Institution: BITLIS EREN UNIVERSITY
Country: Turkey


Dates

Publication Date : February 6, 2020

Bibtex @research article { hujms552213, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2020}, volume = {49}, pages = {338 - 351}, doi = {10.15672/hujms.552213}, title = {Basis properties of root functions of a regular fourth order boundary value problem}, key = {cite}, author = {KAYA, Ufuk and KARA KUZU, Esma} }
APA KAYA, U , KARA KUZU, E . (2020). Basis properties of root functions of a regular fourth order boundary value problem. Hacettepe Journal of Mathematics and Statistics , 49 (1) , 338-351 . DOI: 10.15672/hujms.552213
MLA KAYA, U , KARA KUZU, E . "Basis properties of root functions of a regular fourth order boundary value problem". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 338-351 <https://dergipark.org.tr/en/pub/hujms/issue/52287/552213>
Chicago KAYA, U , KARA KUZU, E . "Basis properties of root functions of a regular fourth order boundary value problem". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 338-351
RIS TY - JOUR T1 - Basis properties of root functions of a regular fourth order boundary value problem AU - Ufuk KAYA , Esma KARA KUZU Y1 - 2020 PY - 2020 N1 - doi: 10.15672/hujms.552213 DO - 10.15672/hujms.552213 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 338 EP - 351 VL - 49 IS - 1 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.552213 UR - https://doi.org/10.15672/hujms.552213 Y2 - 2018 ER -
EndNote %0 Hacettepe Journal of Mathematics and Statistics Basis properties of root functions of a regular fourth order boundary value problem %A Ufuk KAYA , Esma KARA KUZU %T Basis properties of root functions of a regular fourth order boundary value problem %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 1 %R doi: 10.15672/hujms.552213 %U 10.15672/hujms.552213
ISNAD KAYA, Ufuk , KARA KUZU, Esma . "Basis properties of root functions of a regular fourth order boundary value problem". Hacettepe Journal of Mathematics and Statistics 49 / 1 (February 2020): 338-351 . https://doi.org/10.15672/hujms.552213
AMA KAYA U , KARA KUZU E . Basis properties of root functions of a regular fourth order boundary value problem. Hacettepe Journal of Mathematics and Statistics. 2020; 49(1): 338-351.
Vancouver KAYA U , KARA KUZU E . Basis properties of root functions of a regular fourth order boundary value problem. Hacettepe Journal of Mathematics and Statistics. 2020; 49(1): 351-338.