**Completeness of the Weak Eigenfunctions of one Boundary-Value-Transmission Problem**

Hayati Olgar

^{1}

^{*}, O. Sh. Mukhtarov

^{2}, F. S. Muhtarov

^{3}, Kadriye Aydemir

^{4}

^{1}Gaziosmanpaşa University, Tokat, Turkey

^{2}AzerbaijanNational Academy of Sciences , Bakü, Azerbaijan

^{3}AzerbaijanNational Academy of Sciences , Bakü, Azerbaijan

^{4}Amasya University, Amasya, Turkey

** Corresponding author: holgar@gmail.com*

Presented at the 2nd International Symposium on Innovative Approaches in Scientific Studies (ISAS2018-Winter), Samsun, Turkey, Nov 30, 2018

SETSCI Conference Proceedings, 2018, 3, Page (s): 1168-1172 , https://doi.org/

**Published Date: **31 December 2018 |
1266 10

**Abstract**

Recently, the basis properties and eigenfunction expansions in various function spaces of the eigenfunction of the regular

boundary value problems with spectral parameter in the boundary conditions have been investigated by many mathematicans. However in different areas of applied mathematics and physics many problems arise in the form of singular boundary value problems

involving transmission conditions at the interior singular points. Such problems are called boundary-value-transmission problems.

For example, some boundary value problems with transmission conditions arise in heat and mass transfer problems, in vibrating string problems when the string loaded with additional point masses, in diffraction problems. It is not clear how the extend

the classical methods to a problem with additional transmission conditions. Another major diffuculty lies in the completeness of

the eigenfunctions, since the eigenvalues of a Sturm-Liouville problems with transmission conditions may not have an asymptotic expansion. This study devoted to the investigation of the Sturm-Liouville type boundary-value problems with supplementary

transmission conditions. We introduce a new concept, so-called generalized eigenfunctions and study the completeness and basis

properties of systems of generalized eigenfunctions for the problem under consideration.

**Keywords - **Boundary value problems, boundary conditions, weak eigenfunctions, eigenvalue, completeness

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