SavedAdd your journal

Journals / Turkish Journal of Mathematics and Computer Science / 2026 / Volume: 13 Issue: 1

Limit-point Classification for Singular Conformable Fractional Sturm-Liouville Operators

Yüksel Yalçınkaya(SULEYMAN DEMIREL UNIVERSITY, .)0000-0002-1633-8343
Bilender Paşaoğlu(Süleyman Demirel University)0000-0002-9315-4652
Hüseyin Tuna(MEHMET AKIF ERSOY UNIVERSITY)0000-0003-2483-1493
Pages
—
Publication Date
2021-06-30
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

We give a criterion for the limit point classification of conformable fractional Sturm-Liouville operators in singular case.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this work, we study the following conformable fractional Sturm--Liouville problem \[ l[y]=-T_{\alpha }(p(t)T_{\alpha }y(t))+q(t)y(t), \] where $t\in \lbrack 0,\infty ),$ the real-valued functions $p$ and $q$ satisfy the following conditions: \[ \begin{array}{cc} (i) & q\in L_{\alpha }^{2}[0,\infty ), \\ (ii) & p\ \text{is\ absolutely\ continuous\ on}\ [0,\infty ), \\ (iii) & p(t)>0\ \ \text{for\ all}\ t\in \lbrack 0,\infty ).% \end{array}% \] The conformable fractional Sturm--Liouville problem$\ $is of the limit-point case if the number of linearly independent $\alpha -$square integrable solutions of the equation$\ l[y]=\lambda y\ $is less than 2.
📊
Key Findings & Data:In this work, we study the following conformable fractional Sturm--Liouville problem \[ l[y]=-T_{\alpha }(p(t)T_{\alpha }y(t))+q(t)y(t), \] where $t\in \lbrack 0,\infty ),$ the real-valued functions $p$ and $q$ satisfy the following conditions: \[ \begin{array}{cc} (i) & q\in L_{\alpha }^{2}[0,\infty ), \\ (ii) & p\ \text{is\ absolutely\ continuous\ on}\ [0,\infty ), \\ (iii) & p(t)>0\ \ \text{for\ all}\ t\in \lbrack 0,\infty ).% \end{array}% \] The conformable fractional Sturm--Liouville problem$\ $is of the limit-point case if the number of linearly independent $\alpha -$square integrable solutions of the equation$\ l[y]=\lambda y\ $is less than 2.
💡
Conclusion & Impact:We give a criterion for the limit point classification of conformable fractional Sturm-Liouville operators in singular case.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Sturm-Liouville operator · conformable fractional Sturm-Liouville operator · limit-point case

Abstract

In this work, we study the following conformable fractional Sturm--Liouville problem \[ l[y]=-T_{\alpha }(p(t)T_{\alpha }y(t))+q(t)y(t), \] where $t\in \lbrack 0,\infty ),$ the real-valued functions $p$ and $q$ satisfy the following conditions: \[ \begin{array}{cc} (i) & q\in L_{\alpha }^{2}[0,\infty ), \\ (ii) & p\ \text{is\ absolutely\ continuous\ on}\ [0,\infty ), \\ (iii) & p(t)>0\ \ \text{for\ all}\ t\in \lbrack 0,\infty ).% \end{array}% \] The conformable fractional Sturm--Liouville problem$\ $is of the limit-point case if the number of linearly independent $\alpha -$square integrable solutions of the equation$\ l[y]=\lambda y\ $is less than 2. We give a criterion for the limit point classification of conformable fractional Sturm-Liouville operators in singular case.

🔖

Cite This Article

Yalçınkaya, Y., Paşaoğlu, B., Tuna, H. (2021). Limit-point Classification for Singular Conformable Fractional Sturm-Liouville Operators. Turkish Journal of Mathematics and Computer Science, Volume: 13 Issue: 1. https://doi.org/10.47000/tjmcs.823766

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (16)

  1. Abdeljawad T., On conformable fractional calculus, J. Comput. Appl. Math. 279(2015), 57-66.
  2. Allahverdiev, B. P., Tuna, H., Yalc¸ınkaya, Y., Conformable fractional Sturm-Liouville equation, Math. Meth. Appl. Sci., 42(10)(2019), 3508- 3526.
  3. Baleanu, D., Jarad, F., Ugurlu, E., Singular conformable fractional sequential differential equations with distributional potentials, Quaest. Math. 42(3)(2018), 277-287.
  4. Everitt W. N., On the limit-point classification of second-order differential expressions, J. London Math. Soc. 41(1966), 531-534.
  5. Everitt W. N., On the limit-circle classification of second order differential expressions, Quart. J. Math. (Oxford) 2(23)(1972), 193-6.
  6. Hardy G. H., Littlewood J. E., Polya G., Inequalities, Cambridge University Press, New York, 1934.
  7. Khalil R. M., Al Horani,Yousef A. and Sababheh M., A new definition of fractional derivative, J. Comput. Appl. Math. 264(2014), 65-70.
  8. Krein M. G., On the indeterminate case of the Sturm–Liouville boundary problem in the interval (0;1), Izv. Akad. Nauk SSSR Ser. Mat., 16(4)(1952), 293-324 (in Russian).
  9. Levinson N., Criteria for the limit-point case for second order linear differential operators. Pest. Mat. Fys. 74(1949), 17-20.
  10. Miller, K. S., Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equation, Wiley Interscience, New York, 1993.
  11. Oldham K. B., Spainer J., The Fractional Calculus, Academic Press, New York, 1974.
  12. Podlubny I., 1999, Fractional Differential Equations, Mathematics in Science and Enginering, vol. 198, Academic Press, New York, London, Tokyo and Toronto.
  13. Ross B., Fractional Calculus and Its Applications, Springer, New York, 1975.
  14. Titchmarsh E. C., Eigenfunction expansions associated with second-order differential equations, Part I (2nd edition, Oxford University Press, 1962).
  15. Zhaowen, Z., Huixin, L., Jinming, C., Yanwei, Z., Criteria of limit-point case for conformable fractional Sturm-Liouville operators, Math. Meth. Appl. Sci. 43(2020), 2548–2557.
  16. Khalil R. M., Al Horani,Yousef A. and Sababheh M., A new definition of fractional derivative, J. Comput. Appl. Math. 264(2014), 65-70.