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

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.

Keywords

• Sturm-Liouville operator
• conformable fractional Sturm-Liouville operator
• limit-point case

References

1. [1] Abdeljawad T., On conformable fractional calculus, J. Comput. Appl. Math. 279(2015), 57-66.
2. [2] Allahverdiev, B. P., Tuna, H., Yalc¸ınkaya, Y., Conformable fractional Sturm-Liouville equation, Math. Meth. Appl. Sci., 42(10)(2019), 3508- 3526.
3. [3] Baleanu, D., Jarad, F., Ugurlu, E., Singular conformable fractional sequential differential equations with distributional potentials, Quaest. Math. 42(3)(2018), 277-287.
4. [4] Everitt W. N., On the limit-point classification of second-order differential expressions, J. London Math. Soc. 41(1966), 531-534.
5. [5] Everitt W. N., On the limit-circle classification of second order differential expressions, Quart. J. Math. (Oxford) 2(23)(1972), 193-6.
6. [6] Hardy G. H., Littlewood J. E., Polya G., Inequalities, Cambridge University Press, New York, 1934.
7. [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. [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. [9] Levinson N., Criteria for the limit-point case for second order linear differential operators. Pest. Mat. Fys. 74(1949), 17-20.
10. [10] Miller, K. S., Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equation, Wiley Interscience, New York, 1993.
11. [11] Oldham K. B., Spainer J., The Fractional Calculus, Academic Press, New York, 1974.
12. [12] Podlubny I., 1999, Fractional Differential Equations, Mathematics in Science and Enginering, vol. 198, Academic Press, New York, London, Tokyo and Toronto.
13. [13] Ross B., Fractional Calculus and Its Applications, Springer, New York, 1975.
14. [14] Titchmarsh E. C., Eigenfunction expansions associated with second-order differential equations, Part I (2nd edition, Oxford University Press, 1962).
15. [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.