On the resolvent of singular q-Sturm-Liouville operators

In this paper, we investigate the resolvent operator of the singular q-Sturm-Liouville problem defined as −(1/q)Dq⁻¹[Dqy(x)]+[r(x)-λ]y(x)=0−(1/q)Dq⁻¹Dqy(x)+r(x)y(x)=λy(x), with the boundary condition y(0,λ)cosβ+Dq⁻¹y(0,λ)sinβ=0y(0,λ)cosβ+Dq⁻¹y(0,λ)sinβ=0, where λ∈Cλ∈C, $r$ is a real function defined on $[0,∞)$, continuous at zero and r∈Lq,loc¹(0,∞)r∈Lq,loc¹(0,∞). We give an integral representation for the resolvent operator and investigate some properties of this operator. Furthermore, we obtain a formula for the Titchmarsh-Weyl function of the singular $q$-Sturm-Liouville problem.

Keywords:

q-Sturm-Liouville operator, spectral function resolvent operator, Titchmarsh-Weyl function,

PDF

___

Aldwoah, K. A., Malinowska, A. B., Torres, D. F. M., The power quantum calculus and variational problems, Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms, 19 (2012), 93-116.
Allahverdiev, B. P., Tuna, H., A representation of the resolvent operator of singular Hahn-Sturm-Liouville problem, Numer. Funct. Anal. Optimiz., 41(4) (2020), 413-431. doi:10.1080/01630563.2019.1658604
Allahverdiev, B. P., Tuna, H., An expansion theorem for q-Sturm-Liouville operators on the whole line, Turk. J. Math., 42 (2018), 1060-1071. doi:10.3906/mat-1705-22
Allahverdiev, B. P., Tuna, H., Eigenfunction expansion in the singular case for q-Sturm-Liouville operators, Caspian J. Math. Sci., 8(2) (2019), 91-102. doi:10.22080/CJMS.2018.13943.1339
Allahverdiev, B. P., Tuna, H., Some properties of the resolvent of Sturm-Liouville operators on unbounded time scales, Mathematica, 61 (84) No. 1 (2019), 3-21. doi:10.24193/mathcluj.2019.1.01
Allahverdiev, B. P., Tuna, H., Spectral theory of singular Hahn difference equation of the Sturm-Liouville type, Commun. Math., 28(1) (2020), 13-25. doi:10.2478/cm-2020-0002
Allahverdiev, B. P., Tuna, H., On the resolvent of singular Sturm-Liouville operators with transmission conditions, Math. Meth. Appl. Sci., 43 (2020), 4286-4302. doi:10.1002/mma.6193
Annaby, M. H., Mansour, Z. S., q-Fractional calculus and equations. Lecture Notes in Mathematics, vol. 2056, Springer, Berlin, 2012. doi:10.1007/978-3-642-30898-7
Annaby, M. H., Mansour, Z. S., Soliman, I. A., q-Titchmarsh-Weyl theory: series expansion, Nagoya Math. J., 205 (2012), 67-118. doi:10.1215/00277630-1543787
Annaby, M. H., Mansour, Z. S., Basic Sturm-Liouville problems, J. Phys. A, Math. Gen., 38(17) (2005), 3775-3797. doi:10.1088/0305-4470/38/17/005
Annaby, M. H., Hamza, A. E., Aldwoah, K. A., Hahn di¤erence operator and associated Jackson-Nörlund integrals, J. Optim. Theory Appl., 154 (2012), 133-153. doi:10.1007/s10957-012-9987-7
Ernst, T., The History of q-Calculus and a New Method, U. U. D. M. Report (2000): 16, ISSN 1101-3591, Department of Mathematics, Uppsala University, 2000.
Hahn, W., Beitraäge zur Theorie der Heineschen Reihen, Math. Nachr. 2 (1949), 340-379 (in German). doi:10.1002/mana.19490020604
Hamza, A. E., Ahmed, S. M., Existence and uniqueness of solutions of Hahn difference equations, Adv. Differ. Equ., 316 (2013), 1-15. doi:10.1186/1687-1847-2013-316
Hamza, A. E., Ahmed, S. M., Theory of linear Hahn difference equations, J. Adv. Math., 4(2) (2013), 441-461.
Jackson, F. H., On q-denite integrals, Quart. J. Pure Appl. Math., 41 (1910), 193-203.
Kac, V., Cheung, P., Quantum Calculus, Springer-Verlag, Berlin Heidelberg, 2002. doi:10.1007/978-1-4613-0071-7
Karahan, D., Mamedov, Kh. R., Sampling theory associated with q-Sturm-Liouville operator with discontinuity conditions, Journal of Contemporary Applied Mathematics, 10(2) ( 2020), 1-9.
Kolmogorov, A. N., Fomin, S. V., Introductory Real Analysis, Translated by R. A. Silverman, Dover Publications, New York, 1970.
Levitan, B. M., Sargsjan, I. S., Sturm-Liouville and Dirac Operators. Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1991 (translated from the Russian). doi:10.1007/978-94-011-3748-5
Malinowska, A. B., Torres, D. F. M., The Hahn quantum variational calculus, J. Optim. Theory Appl., 147 (2010), 419-442. doi:10.1007/s10957-010-9730-1
Naimark, M. A., Linear Di¤erential Operators, 2nd edn.,1969, Nauka, Moscow; English transl. of 1st. edn., 1, 2, New York, 1968.
Swamy, P. N., Deformed Heisenberg algebra:origin of q-calculus, Physica A: Statistical Mechanics and its Applications, 328, 1-2 (2003), 145-153. doi:10.1016/S0378-4371(03)00518-1
Tariboon, J., Ntouyas, S. K., Quantum calculus on finite intervals and applications to impulsive difference equations, Adv. Differ. Equ., 282 (2013), 1-19. doi:10.1186/1687-1847-2013-282
Titchmarsh, E. C., Eigenfunction Expansions Associated with Second-Order Differential Equations, Part I. Second Edition, Clarendon Press, Oxford, 1962.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics-Cover

ISSN: 1303-5991
Yayın Aralığı: Yılda 4 Sayı
Başlangıç: 1948
Yayıncı: Ankara Üniversitesi

Arşiv