Selfadjointness and Positiveness of the Differential Operators Generated by New Type Sturm-Liouville Problems

Hayati Olğar

doi:10.17776/csj.451174

TR EN

Yeni Tipten Sturm-Liouville Problemlerinin Ürettiği Diferansiyel Operatörlerin Kendine Eşlenikliği ve Pozitivliği

Öz

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.

Anahtar Kelimeler

Sınır değer problemleri,özfonksiyonlar,sınır ve geçiş şartları,pozitiv operatörler

Selfadjointness and Positiveness of the Differential Operators Generated by New Type Sturm-Liouville Problems

Abstract

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.

Keywords

Boundary value problems,eigenfunctions,boundary and transmission conditions,positive operators

References

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Details

Primary Language

English

Subjects

-

Journal Section

Research Article

Authors

Hayati Olğar ^*
0000-0003-4732-1605
Türkiye

Publication Date

March 22, 2019

Submission Date

August 6, 2018

Acceptance Date

December 27, 2018

Published in Issue

Year 2019 Volume: 40 Number: 1

DOI

https://doi.org/10.17776/csj.451174

IZ

https://izlik.org/JA54AB25GM

Cite

RIS / Bibtex

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. https://doi.org/10.17776/csj.451174

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