Research Article
Yeni Tipten Sturm-Liouville Problemlerinin Ürettiği Diferansiyel Operatörlerin Kendine Eşlenikliği ve Pozitivliği
Abstract
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.
References
- [1]. C. T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. R. Soc. Edinburgh, A77(1977), 293-308.
- [2]. D. B. Hinton, An expansion theorem for an eigenvalue problem with eigenvalue parameter contained in the boundary condition, Quart. J. Math. Oxford, 30(1979), 33-42.
- [3]. A. Schneider, A Note on Eigenvalue Problems with Eigenvalue Parameter in the Boundary Conditions, Math. Z., 136(1974) 163-167.
- [4]. J. Walter, Regular eigenvalue problems with eigenvalue parameter in the boundary conditions. Math. Z. 133(1973), 301-312.
- [5]. B. Harmsen, A. Li, Discrete Sturm-Liouville problems with parameter in the boundary conditions, Journal of Difference Equations and Applications, 8(11)(2002), 969-981.
- [6]. F. V. Atkinson, Discrete and Continuous Boundary Problems, Academic Press, New York, 1964.
- [7]. P. A. Binding, P. J. Browne, Oscillation theory for indefinite Sturm-Liouville problems with eigenparameter dependent boundary conditions, Proc. Roy. Soc. Edinburgh Sect., A 127(1997), 1123-1136.
- [8]. P. A. Binding, P. J. Browne and K. Seddighi, Sturm-Liouville problems with eigenparameter dependent boundary conditions, Proc. Edinburgh Math. Soc., 37(2)(1993), 57-72.
- [9]. L. Greenberg, I. Babuska, A continuous analogue of Sturm sequences in the context of Sturm-Liouville problems, SIAM Journal on Numerical Analysis, 26(1989), 920-945.
- [10]. B. P. Belinskiy, J. P. Dauer, On a regular Sturm-Liouville problem on a finite interval with the eigenvalue parameter appearing linearly in the boundary conditions, Spectral theory and computational methods of Sturm-Liouville problem. Eds. D. Hinton and P. W. Schaefer, 1997.
- [11]. B. P. Belinskiy, J. P. Dauer, Eigenoscillations of mechanical systems with boundary conditions containing the frequency, Quarterly of Applied Math., 56(1998), 521-541.
- [12]. N. B. Kerimov, R. Kh. Mamedov, On a boundary value problem with Spectral parameter in the boundary conditions, (Russian) Sibirsk. Mat. Zh. 40, no 2, (1999) pp. 325-335.
- [13]. H. Olğar, O. Sh. Mukhtarov, Weak Eigenfunctions Of Two-Interval Sturm-Liouville Problems Together With Interaction Conditions, Journal of Mathematical Physics, 58, 042201 (2017), DOI: 10.1063/1.4979615.
- [14]. E. M. Russakovskii, Sturm-Liouville problem with parameter in the boundary conditions, Trudy Seminara imeni I. G. Petrovskogo, 18, 1993.
- [15]. A. M. Sarsenbi, A. A. Tengaeva, On the basis properties of root functions of two generalized eigenvalue problems, ISSN 0012-2661, Differential Equations, 2012, Vol. 48, No. 2, pp. 306-308.
- [16]. A. A. Shkalikov, On the Basis Problem of the Eigenfunctions of an Ordinary Differential Operators, Uspekhi Mat. Nauk, 34:5 (209)(1979), 235-236.
- [17]. R. Kh. Amirov, A. S. Ozkan and B. Keskin, Inverse problems for impulsive Sturm-Liouville operator with spectral parameter linearly contained in boundary conditions, Integral Transforms and Special Functions, 20(8)(2009), 607-618.
- [18]. R. Amirov, N. Topsakal, On inverse problem for singular Sturm-Liouville operator with discontinuity conditions, Bull. Iranian Math. Soc. Vol. 40(2014), No. 3, pp. 585–607.
- [19]. B. P. Allahverdiev, E. Bairamov and E. Ugurlu, Eigenparameter dependent Sturm Liouville problems in boudary conditions with transmission conditions, J. Math. Anal. Appl. 401(2013), 388-396.
- [20]. K. Aydemir, H. Olğar, O. Sh. Mukhtarov and F. S. Muhtarov, Differential operator equations with interface conditions in modified direct sum spaces, Filomat, 32:3(2018), 921-931.
- [21]. K. Aydemir, O. Mukhtarov, A Class of Sturm-Liouville Problems with Eigenparameter Dependent Transmission Conditions, Numerical Functional Analysis and Optimization, (2017), Doi:10.1080/01630563.2017.1316995.
- [22]. K. Aydemir, O. Sh. Mukhtarov, Second-order differential operators with interior singularity, Advances in Difference Equations, (2015), 2015:26 DOI 10.1186/s13662-015-0360-7.
- [23]. Y. Güldü, R. Kh. Amirov and N. Topsakal, On impulsive Sturm–Liouville operators with singularity and spectral parameter in boundary conditions , Ukrainian Mathematical Journal, Vol. 64, No. 12, May,2013 (Ukrainian Original Vol. 64, No. 12, December, 2012), 1816-1838.
- [24]. M. Kandemir, O. Sh. Mukhtarov, Nonlocal Sturm-Liouville Problems with Integral Terms in the Boundary Conditions, Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 11, pp. 1-12.
- [25]. O. Sh. Mukhtarov, K. Aydemir, Eigenfunction expansion for Sturm-Liouville problems with transmission conditions at one interior point, Acta Mathematica Scientia, 35B(3)(2015), 639-649.
- [26]. O. Sh. Mukhtarov, K. Aydemir, The Eigenvalue Problem with Interaction Conditions at One Interior Singular Point, Filomat 31:17(2017), 5411-5420.
- [27]. O. Sh. Mukhtarov, H. Olğar and K. Aydemir, Resolvent Operator and Spectrum of New Type Boundary Value Problems, Filomat, 29:7(2015), 1671-1680.
- [28]. O. Sh. Mukhtarov, M. Kandemir, Asymptotic behaviour of eigenvalues for the discontinuous boundary-value problem with Functional-Transmissin conditions, Acta Mathematica Scientia, 22B(3)(2002), pp.335-345.
- [29]. A. S. Ozkan, B. Keskin and Y. Cakmak, Double discontinuous inverse problems for Sturm-Liouville operator with parameter-dependent conditions, Abstract and Applied Analysis, 2013, Article ID 794262, P.7.
- [30]. E. S. Panakhov, T. Gulsen, On discontinuous Dirac systems with eigenvalue dependent boundary conditions, AIP Conference Proceeding, 1648, 260003(2015), 1-4, https://doi.org/10.1063/1.4912520.
- [31]. A. V. Likov, Yu. A. Mikhailov, The Theory Of Heat And Mass Transfer, Qosenergaizdat (Russian), 1963.
- [32]. A. N. Tikhonov, A. A. Samarskii, Equations Of Mathematical Physics, Oxford and New York, Pergamon, 1963.
- [33]. N. N. Voitovich, B. Z. Katsenelbaum and A. N. Sivov, Generalized Method Of Eigen-Vibration In The Theory Of Diffraction, Nakua, Moskow (Russian), 1997.
- [34]. O. A. Ladyzhenskaia, The Boundary Value Problems of Mathematical Physics, Springer-Verlag, New York, 1985.
- [35]. I. C. Gohberg, M. G. Krein, Introduction to The Theory of Linear Non-Selfadjoint Operators, Translation of Mathematical Monographs, vol. 18, Amer. Math. Soc., Providence, Rhode Island, 1969.
- [36]. D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, 2nd ed. Springer-Verlag, Berlin, 1983.
- [37]. E. Kreyszig, Introductory Functional Analysis With Application, New-York, 1978.
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.
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
- [1]. C. T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. R. Soc. Edinburgh, A77(1977), 293-308.
- [2]. D. B. Hinton, An expansion theorem for an eigenvalue problem with eigenvalue parameter contained in the boundary condition, Quart. J. Math. Oxford, 30(1979), 33-42.
- [3]. A. Schneider, A Note on Eigenvalue Problems with Eigenvalue Parameter in the Boundary Conditions, Math. Z., 136(1974) 163-167.
- [4]. J. Walter, Regular eigenvalue problems with eigenvalue parameter in the boundary conditions. Math. Z. 133(1973), 301-312.
- [5]. B. Harmsen, A. Li, Discrete Sturm-Liouville problems with parameter in the boundary conditions, Journal of Difference Equations and Applications, 8(11)(2002), 969-981.
- [6]. F. V. Atkinson, Discrete and Continuous Boundary Problems, Academic Press, New York, 1964.
- [7]. P. A. Binding, P. J. Browne, Oscillation theory for indefinite Sturm-Liouville problems with eigenparameter dependent boundary conditions, Proc. Roy. Soc. Edinburgh Sect., A 127(1997), 1123-1136.
- [8]. P. A. Binding, P. J. Browne and K. Seddighi, Sturm-Liouville problems with eigenparameter dependent boundary conditions, Proc. Edinburgh Math. Soc., 37(2)(1993), 57-72.
- [9]. L. Greenberg, I. Babuska, A continuous analogue of Sturm sequences in the context of Sturm-Liouville problems, SIAM Journal on Numerical Analysis, 26(1989), 920-945.
- [10]. B. P. Belinskiy, J. P. Dauer, On a regular Sturm-Liouville problem on a finite interval with the eigenvalue parameter appearing linearly in the boundary conditions, Spectral theory and computational methods of Sturm-Liouville problem. Eds. D. Hinton and P. W. Schaefer, 1997.
- [11]. B. P. Belinskiy, J. P. Dauer, Eigenoscillations of mechanical systems with boundary conditions containing the frequency, Quarterly of Applied Math., 56(1998), 521-541.
- [12]. N. B. Kerimov, R. Kh. Mamedov, On a boundary value problem with Spectral parameter in the boundary conditions, (Russian) Sibirsk. Mat. Zh. 40, no 2, (1999) pp. 325-335.
- [13]. H. Olğar, O. Sh. Mukhtarov, Weak Eigenfunctions Of Two-Interval Sturm-Liouville Problems Together With Interaction Conditions, Journal of Mathematical Physics, 58, 042201 (2017), DOI: 10.1063/1.4979615.
- [14]. E. M. Russakovskii, Sturm-Liouville problem with parameter in the boundary conditions, Trudy Seminara imeni I. G. Petrovskogo, 18, 1993.
- [15]. A. M. Sarsenbi, A. A. Tengaeva, On the basis properties of root functions of two generalized eigenvalue problems, ISSN 0012-2661, Differential Equations, 2012, Vol. 48, No. 2, pp. 306-308.
- [16]. A. A. Shkalikov, On the Basis Problem of the Eigenfunctions of an Ordinary Differential Operators, Uspekhi Mat. Nauk, 34:5 (209)(1979), 235-236.
- [17]. R. Kh. Amirov, A. S. Ozkan and B. Keskin, Inverse problems for impulsive Sturm-Liouville operator with spectral parameter linearly contained in boundary conditions, Integral Transforms and Special Functions, 20(8)(2009), 607-618.
- [18]. R. Amirov, N. Topsakal, On inverse problem for singular Sturm-Liouville operator with discontinuity conditions, Bull. Iranian Math. Soc. Vol. 40(2014), No. 3, pp. 585–607.
- [19]. B. P. Allahverdiev, E. Bairamov and E. Ugurlu, Eigenparameter dependent Sturm Liouville problems in boudary conditions with transmission conditions, J. Math. Anal. Appl. 401(2013), 388-396.
- [20]. K. Aydemir, H. Olğar, O. Sh. Mukhtarov and F. S. Muhtarov, Differential operator equations with interface conditions in modified direct sum spaces, Filomat, 32:3(2018), 921-931.
- [21]. K. Aydemir, O. Mukhtarov, A Class of Sturm-Liouville Problems with Eigenparameter Dependent Transmission Conditions, Numerical Functional Analysis and Optimization, (2017), Doi:10.1080/01630563.2017.1316995.
- [22]. K. Aydemir, O. Sh. Mukhtarov, Second-order differential operators with interior singularity, Advances in Difference Equations, (2015), 2015:26 DOI 10.1186/s13662-015-0360-7.
- [23]. Y. Güldü, R. Kh. Amirov and N. Topsakal, On impulsive Sturm–Liouville operators with singularity and spectral parameter in boundary conditions , Ukrainian Mathematical Journal, Vol. 64, No. 12, May,2013 (Ukrainian Original Vol. 64, No. 12, December, 2012), 1816-1838.
- [24]. M. Kandemir, O. Sh. Mukhtarov, Nonlocal Sturm-Liouville Problems with Integral Terms in the Boundary Conditions, Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 11, pp. 1-12.
- [25]. O. Sh. Mukhtarov, K. Aydemir, Eigenfunction expansion for Sturm-Liouville problems with transmission conditions at one interior point, Acta Mathematica Scientia, 35B(3)(2015), 639-649.
- [26]. O. Sh. Mukhtarov, K. Aydemir, The Eigenvalue Problem with Interaction Conditions at One Interior Singular Point, Filomat 31:17(2017), 5411-5420.
- [27]. O. Sh. Mukhtarov, H. Olğar and K. Aydemir, Resolvent Operator and Spectrum of New Type Boundary Value Problems, Filomat, 29:7(2015), 1671-1680.
- [28]. O. Sh. Mukhtarov, M. Kandemir, Asymptotic behaviour of eigenvalues for the discontinuous boundary-value problem with Functional-Transmissin conditions, Acta Mathematica Scientia, 22B(3)(2002), pp.335-345.
- [29]. A. S. Ozkan, B. Keskin and Y. Cakmak, Double discontinuous inverse problems for Sturm-Liouville operator with parameter-dependent conditions, Abstract and Applied Analysis, 2013, Article ID 794262, P.7.
- [30]. E. S. Panakhov, T. Gulsen, On discontinuous Dirac systems with eigenvalue dependent boundary conditions, AIP Conference Proceeding, 1648, 260003(2015), 1-4, https://doi.org/10.1063/1.4912520.
- [31]. A. V. Likov, Yu. A. Mikhailov, The Theory Of Heat And Mass Transfer, Qosenergaizdat (Russian), 1963.
- [32]. A. N. Tikhonov, A. A. Samarskii, Equations Of Mathematical Physics, Oxford and New York, Pergamon, 1963.
- [33]. N. N. Voitovich, B. Z. Katsenelbaum and A. N. Sivov, Generalized Method Of Eigen-Vibration In The Theory Of Diffraction, Nakua, Moskow (Russian), 1997.
- [34]. O. A. Ladyzhenskaia, The Boundary Value Problems of Mathematical Physics, Springer-Verlag, New York, 1985.
- [35]. I. C. Gohberg, M. G. Krein, Introduction to The Theory of Linear Non-Selfadjoint Operators, Translation of Mathematical Monographs, vol. 18, Amer. Math. Soc., Providence, Rhode Island, 1969.
- [36]. D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, 2nd ed. Springer-Verlag, Berlin, 1983.
- [37]. E. Kreyszig, Introductory Functional Analysis With Application, New-York, 1978.
There are 37 citations in total.
Details
| Primary Language | English |
|---|---|
| Journal Section | Natural Sciences |
| Authors | |
| Publication Date | March 22, 2019 |
| Submission Date | August 6, 2018 |
| Acceptance Date | December 27, 2018 |
| Published in Issue | Year 2019Volume: 40 Issue: 1 |