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

Bu makalenin amacı 1 1 2 2 [ , ) ( , ) ( , ]       çok-aralığında tanımlı olan, özdeğer parametresini doğrusal olarak sınır şartlarında bulunduran ve iki tane ek geçiş şartı içeren u(x)q(x)u(x)  u(x) 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 bilinen 2L [ , ] Hilbert uzayında ortogonal baz oluşturma özelliği sağlanmaktadır. Genellikle sınır-değer-geçiş problemleri kendine-eşlenik değildir ve özfonksiyonlar sistemi klasik 2L [ , ] 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, eğer ()qx potansiyeli sürekli ise, o halde genelleşmiş özfonksiyonlar incelediğimiz problemi klasik anlamda da sağlıyor. Daha sonra bazı kompakt operatörleri öyle tanımladık ki araştırılan sınır-değergeçiş 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.

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

It is purpose of this paper to investigate Sturm-Liouville equation u(x)q(x)u(x)  u(x)on many-interval 1 1 2 2 [ , ) ( , ) ( , ]       with the eigenvalue parameter appearing linearly in theboundary conditions and with two supplementary transmission conditions. The classical Sturmian theory didnot cover such type of many-interval boundary value transmission problems. For the classical Sturm-Liouvilleproblems it is guaranteed that the problem is self-adjoint with compact resolvent, the spectrum is disctrete andconsist of eigenvalues and the corresponding eigenfunctions form an orthogonal basis in the well-known Hilbertspace 2[ , ] L . But the boundary-value-transmission problems are not self-adjoint and the system ofeigenfunctions did not form a basis in the classical Hilbert space 2[ , ] L  in general. Taking in view this factwe suggest a new approach for self-adjoint realization of such type transmission problems. Moreover, we definesome new Hilbert spaces to establish positiveness of corresponding operator-pencil. At first we define a conceptof generalized eigenfunctions for this kind of spectral problems. In particular it is shown that if the potential()qx 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-valuetransmissionproblem can be reduced to the appropriate operator-pencil equation. Finally, we prove that thisoperator-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-Liouvilleproblems.

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C. T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. R. Soc. Edinburgh, A77(1977), 293-308.
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.
A. Schneider, A Note on Eigenvalue Problems with Eigenvalue Parameter in the Boundary Conditions, Math. Z., 136(1974) 163-167.
J. Walter, Regular eigenvalue problems with eigenvalue parameter in the boundary conditions. Math. Z. 133(1973), 301-312.
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.
F. V. Atkinson, Discrete and Continuous Boundary Problems, Academic Press, New York, 1964.
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.
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.
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.
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.
B. P. Belinskiy, J. P. Dauer, Eigenoscillations of mechanical systems with boundary conditions containing the frequency, Quarterly of Applied Math., 56(1998), 521-541.
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.
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.
E. M. Russakovskii, Sturm-Liouville problem with parameter in the boundary conditions, Trudy Seminara imeni I. G. Petrovskogo, 18, 1993.
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.
A. A. Shkalikov, On the Basis Problem of the Eigenfunctions of an Ordinary Differential Operators, Uspekhi Mat. Nauk, 34:5 (209)(1979), 235-236.
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.
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.
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.
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.
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.
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.
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.
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.
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.
O. Sh. Mukhtarov, K. Aydemir, The Eigenvalue Problem with Interaction Conditions at One Interior Singular Point, Filomat 31:17(2017), 5411-5420.
O. Sh. Mukhtarov, H. Olğar and K. Aydemir, Resolvent Operator and Spectrum of New Type Boundary Value Problems, Filomat, 29:7(2015), 1671-1680.
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.
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.
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.
A. V. Likov, Yu. A. Mikhailov, The Theory Of Heat And Mass Transfer, Qosenergaizdat (Russian), 1963.
A. N. Tikhonov, A. A. Samarskii, Equations Of Mathematical Physics, Oxford and New York, Pergamon, 1963.
N. N. Voitovich, B. Z. Katsenelbaum and A. N. Sivov, Generalized Method Of Eigen-Vibration In The Theory Of Diffraction, Nakua, Moskow (Russian), 1997.
O. A. Ladyzhenskaia, The Boundary Value Problems of Mathematical Physics, Springer-Verlag, New York, 1985.
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.
D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, 2nd ed. Springer-Verlag, Berlin, 1983.
E. Kreyszig, Introductory Functional Analysis With Application, New-York, 1978.