KaydedilenlerDerginizi ekleyin

Dergiler / Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics / 2021 / Volume: 70 Issue: 2

On the resolvent of singular q-Sturm-Liouville operators

Bilender Paşaoğlu(SULEYMAN DEMIREL UNIVERSITY)0000-0002-9315-4652
Hüseyin Tuna(MEHMET AKIF ERSOY UNIVERSITY)0000-0001-7240-8687
Sayfa
—
Yayın Tarihi
2021-12-31
🏛️Bu makalenin bibliyografik verileri ve tam metin bağlantısı DergiPark üzerinden sağlanmıştır.
DergiPark Kaydı ↗
✨

Yapay Zeka Hızlı Bakış & Çıkarımlar

AI Insight
⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

Furthermore, we obtain a formula for the Titchmarsh-Weyl function of the singular $q$-Sturm-Liouville problem.

📌 3 Maddede Bu Makale

🎯
Amaç & Hipotez: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 −(1/q)Dq⁻¹Dqy(x)+r(x)y(x)=λy(x) , with the boundary condition y ( 0 , λ ) c o s β + D q ⁻ ¹ y ( 0 , λ ) s i n β = 0 y(0,λ)cosβ+Dq⁻¹y(0,λ)sinβ=0 , where λ ∈ C λ∈C , $r$ is a real function defined on $[0,∞)$, continuous at zero and r ∈ L q , l o c ¹ ( 0 , ∞ ) r∈Lq,loc¹(0,∞) .
📊
Temel Bulgular & Veri:We give an integral representation for the resolvent operator and investigate some properties of this operator.
💡
Sonuç & Katkı:Furthermore, we obtain a formula for the Titchmarsh-Weyl function of the singular $q$-Sturm-Liouville problem.
🔬Tür:Research Article
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:q-Sturm-Liouville operator · spectral function · resolvent operator

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 −(1/q)Dq⁻¹Dqy(x)+r(x)y(x)=λy(x) , with the boundary condition y ( 0 , λ ) c o s β + D q ⁻ ¹ y ( 0 , λ ) s i n β = 0 y(0,λ)cosβ+Dq⁻¹y(0,λ)sinβ=0 , where λ ∈ C λ∈C , $r$ is a real function defined on $[0,∞)$, continuous at zero and r ∈ L q , l o c ¹ ( 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.

🔖

Bu Makaleye Atıf Yap

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, Volume: 70 Issue: 2. https://doi.org/10.31801/cfsuasmas.866753

📄 Tam Metin PDF Okuyucu

📄

PDF görüntüleyici hazırlanıyor...

Kaynakça (25)

  1. 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.
  2. 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
  3. 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
  4. 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
  5. 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
  6. 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
  7. 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
  8. 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
  9. 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
  10. 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
  11. 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
  12. 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.
  13. Hahn, W., Beitraäge zur Theorie der Heineschen Reihen, Math. Nachr. 2 (1949), 340-379 (in German). doi:10.1002/mana.19490020604
  14. 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
  15. Hamza, A. E., Ahmed, S. M., Theory of linear Hahn difference equations, J. Adv. Math., 4(2) (2013), 441-461.
  16. Jackson, F. H., On q-de…nite integrals, Quart. J. Pure Appl. Math., 41 (1910), 193-203.
  17. Kac, V., Cheung, P., Quantum Calculus, Springer-Verlag, Berlin Heidelberg, 2002. doi:10.1007/978-1-4613-0071-7
  18. 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.
  19. Kolmogorov, A. N., Fomin, S. V., Introductory Real Analysis, Translated by R. A. Silverman, Dover Publications, New York, 1970.
  20. 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
  21. 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
  22. Naimark, M. A., Linear Di¤erential Operators, 2nd edn.,1969, Nauka, Moscow; English transl. of 1st. edn., 1, 2, New York, 1968.
  23. 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
  24. 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
  25. Titchmarsh, E. C., Eigenfunction Expansions Associated with Second-Order Differential Equations, Part I. Second Edition, Clarendon Press, Oxford, 1962.