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## entrSelfadjointness and Positiveness of the Differential Operators Generated by New Type Sturm-Liouville ProblemsYeni Tipten Sturm-Liouville Problemlerinin Ürettiği Diferansiyel Operatörlerin Kendine Eşlenikliği ve Pozitivliği

#### Hayati OLĞAR [1]

##### 32 34

It is purpose of this paper to investigate Sturm-Liouville equation on many-interval with the eigenvalue parameter appearing linearly in the boundary conditions and with two supplementary transmission conditions. The classical Sturmian theory did not cover such type of many-interval boundary value transmission problems. For the classical Sturm-Liouville problems it is guaranteed that the problem is self-adjoint with compact resolvent, the spectrum is disctrete and consist of eigenvalues and the corresponding eigenfunctions form an orthogonal basis in the well-known Hilbert space . But the boundary-value-transmission problems are not self-adjoint and the system of eigenfunctions did not form a basis in the classical Hilbert space in general. Taking in view this fact we suggest a new approach for self-adjoint realization of such type transmission problems. Moreover, we define some new Hilbert spaces to establish positiveness of corresponding operator-pencil. At first we define a concept of generalized eigenfunctions for this kind of spectral problems. In particular it is shown that if the potential  is continuous then the generalized eigenfunctions satisfies the considered problem is the classical sense. Then we introduce to the consideration some compact operators such a way that the considered boundary-value-transmission problem can be reduced to the appropriate operator-pencil equation. Finally, we prove that this operator-pencil is self-adjoint and positive definite for sufficiently large negative values of the eigenparameter. It is important to note that the obtained results extends classical results associated with regular Sturm-Liouville problems.

Bu makalenin amacı çok-aralığında tanımlı olan, özdeğer parametresini doğrusal olarak sınır şartlarında bulunduran ve iki tane ek geçiş şartı içeren Sturm-Liouville problemini araştırmaktır. Klasik Sturm-Liouville teorisi bu tipten çok-aralıklı sınır-değer-geçiş problemlerini kapsamaktadır. Klasik Sturm-Liouville problemleri için kendine-eşleniklik, rezolventin kompaktlığı, spektrumun diskretliği ve uygun özfonksiyonların iyi bilinenHilbert uzayında ortogonal baz oluşturma özelliği sağlanmaktadır. Genellikle sınır-değer-geçiş problemleri kendine-eşlenik değildir ve özfonksiyonlar sistemi klasik Hilbert uzayında baz oluşturmuyor. Bunu dikkate alarak, bu tipten geçiş problemlerinin kendine-eşlenik biçimde sonuçlanabilmesi için yeni bir yaklaşım önermişiz. Bunun dışında uygun operatör-demetinin pozitivliğini gösterebilmek için bazı yeni Hilbert uzayları tanımladık. İlk olarak bu türden spektral problemlerin genelleştirilmiş özfonksiyonları kavramını tanımladık. Özel olarak gösterdik ki, eğerpotansiyeli sürekli ise, o halde genelleşmiş özfonksiyonlar incelediğimiz problemi klasik anlamda da sağlıyor. Daha sonra bazı kompakt operatörleri öyle tanımladık ki araştırılan sınır-değer-geçiş problemlerini uygun operatör demetine dönüştürmek mümkün olsun. Son olarak özdeğer parametresinin mutlak değerce yeteri kadar büyük negativ değerleri için bu operatör demetinin kendine eşlenik ve pozitiv olduğunu ispat ettik. Elde edilen sonuçların düzgün Sturm-Liouville problemlerinin sağladığı klasik sonuçları genelleştirmesi önem arz etmektedir.

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Primary Language en Basic Sciences Natural Sciences Orcid: 0000-0003-4732-1605Author: Hayati OLĞAR (Primary Author)Institution: GAZIOSMANPASA UNIVERSITYCountry: Turkey
 Bibtex @research article { csj451174, journal = {Cumhuriyet Science Journal}, issn = {2587-2680}, eissn = {2587-246X}, address = {Cumhuriyet University}, year = {2019}, volume = {40}, pages = {24 - 34}, doi = {10.17776/csj.451174}, title = {Selfadjointness and Positiveness of the Differential Operators Generated by New Type Sturm-Liouville Problems}, key = {cite}, author = {OLĞAR, Hayati} } APA OLĞAR, H . (2019). Selfadjointness and Positiveness of the Differential Operators Generated by New Type Sturm-Liouville Problems. Cumhuriyet Science Journal, 40 (1), 24-34. DOI: 10.17776/csj.451174 MLA OLĞAR, H . "Selfadjointness and Positiveness of the Differential Operators Generated by New Type Sturm-Liouville Problems". Cumhuriyet Science Journal 40 (2019): 24-34 Chicago OLĞAR, H . "Selfadjointness and Positiveness of the Differential Operators Generated by New Type Sturm-Liouville Problems". Cumhuriyet Science Journal 40 (2019): 24-34 RIS TY - JOUR T1 - Selfadjointness and Positiveness of the Differential Operators Generated by New Type Sturm-Liouville Problems AU - Hayati OLĞAR Y1 - 2019 PY - 2019 N1 - doi: 10.17776/csj.451174 DO - 10.17776/csj.451174 T2 - Cumhuriyet Science Journal JF - Journal JO - JOR SP - 24 EP - 34 VL - 40 IS - 1 SN - 2587-2680-2587-246X M3 - doi: 10.17776/csj.451174 UR - https://doi.org/10.17776/csj.451174 Y2 - 2018 ER - EndNote %0 Cumhuriyet Science Journal Selfadjointness and Positiveness of the Differential Operators Generated by New Type Sturm-Liouville Problems %A Hayati OLĞAR %T Selfadjointness and Positiveness of the Differential Operators Generated by New Type Sturm-Liouville Problems %D 2019 %J Cumhuriyet Science Journal %P 2587-2680-2587-246X %V 40 %N 1 %R doi: 10.17776/csj.451174 %U 10.17776/csj.451174 ISNAD OLĞAR, Hayati . "Selfadjointness and Positiveness of the Differential Operators Generated by New Type Sturm-Liouville Problems". Cumhuriyet Science Journal 40 / 1 (March 2019): 24-34. https://doi.org/10.17776/csj.451174