On the resolvent of singular q-Sturm-Liouville operators

Bilender Paşaoğlu; Hüseyin Tuna

doi:10.31801/cfsuasmas.866753

Research Article

Year 2021, Volume: 70 Issue: 2, 702 - 718, 31.12.2021

Bilender Paşaoğlu Hüseyin Tuna

https://doi.org/10.31801/cfsuasmas.866753

Cited By: 1

Abstract

References

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.

On the resolvent of singular q-Sturm-Liouville operators

Year 2021, Volume: 70 Issue: 2, 702 - 718, 31.12.2021

Bilender Paşaoğlu Hüseyin Tuna

https://doi.org/10.31801/cfsuasmas.866753

Cited By: 1

Abstract

In this paper, we investigate the resolvent operator of the singular q-Sturm-Liouville problem defined as
$-$ $($ $1$ $/$ $q$ $)$ $D$ $_{q}$ $_{⁻}$ $_{¹}$ ${[D}$ $_{q}$ $y$ $($ $x$ $)]$ $+$ $[r$ $($ $x$ $)$ $-λ$ $]y$ $($ $x$ $)=0$ ,

with the boundary condition $y (0, λ) c o s β + D_{q ⁻ ¹} y (0, λ) s i n β = 0$ ,

where $λ \in C$ , $r$ is a real function defined on $[0,∞)$, continuous at zero and $r \in L_{q, l o c} ¹ (0, \infty)$ . 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

References

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.

There are 25 citations in total.

Details

Primary Language	English
Subjects	Applied Mathematics
Journal Section	Research Articles
Authors	Bilender Paşaoğlu 0000-0002-9315-4652 Hüseyin Tuna 0000-0001-7240-8687
Publication Date	December 31, 2021
Submission Date	January 22, 2021
Acceptance Date	April 3, 2021
Published in Issue	Year 2021 Volume: 70 Issue: 2

Cite

APA	Paşaoğlu, B., & Tuna, H. (2021). On the resolvent of singular q-Sturm-Liouville operators. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 70(2), 702-718. https://doi.org/10.31801/cfsuasmas.866753
AMA	Paşaoğlu B, Tuna H. On the resolvent of singular q-Sturm-Liouville operators. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. December 2021;70(2):702-718. doi:10.31801/cfsuasmas.866753
Chicago	Paşaoğlu, Bilender, and Hüseyin Tuna. “On the Resolvent of Singular Q-Sturm-Liouville Operators”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70, no. 2 (December 2021): 702-18. https://doi.org/10.31801/cfsuasmas.866753.
EndNote	Paşaoğlu B, Tuna H (December 1, 2021) On the resolvent of singular q-Sturm-Liouville operators. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70 2 702–718.
IEEE	B. Paşaoğlu and H. Tuna, “On the resolvent of singular q-Sturm-Liouville operators”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 70, no. 2, pp. 702–718, 2021, doi: 10.31801/cfsuasmas.866753.
ISNAD	Paşaoğlu, Bilender - Tuna, Hüseyin. “On the Resolvent of Singular Q-Sturm-Liouville Operators”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70/2 (December 2021), 702-718. https://doi.org/10.31801/cfsuasmas.866753.
JAMA	Paşaoğlu B, Tuna H. On the resolvent of singular q-Sturm-Liouville operators. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2021;70:702–718.
MLA	Paşaoğlu, Bilender and Hüseyin Tuna. “On the Resolvent of Singular Q-Sturm-Liouville Operators”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 70, no. 2, 2021, pp. 702-18, doi:10.31801/cfsuasmas.866753.
Vancouver	Paşaoğlu B, Tuna H. On the resolvent of singular q-Sturm-Liouville operators. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2021;70(2):702-18.

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

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On the resolvent of singular q-Sturm-Liouville operators

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Multiplicative Laplace transform in q−calculus

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https://doi.org/10.2298/FIL2318859Y