We give a criterion for the limit point classification of conformable fractional Sturm-Liouville operators in singular case.
📌 3 Maddede Bu Makale
🎯
Amaç & Hipotez: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.
📊
Temel Bulgular & Veri: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.
💡
Sonuç & Katkı:We give a criterion for the limit point classification of conformable fractional Sturm-Liouville operators in singular case.
🔬Tür:Araştırma Makalesi
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler: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.
Baleanu, D., Jarad, F., Ugurlu, E., Singular conformable fractional sequential differential equations with distributional potentials, Quaest. Math. 42(3)(2018), 277-287
Everitt W. N., On the limit-point classification of second-order differential expressions, J. London Math. Soc. 41(1966), 531-534
Everitt W. N., On the limit-circle classification of second order differential expressions, Quart. J. Math. (Oxford) 2(23)(1972), 193-6
Hardy G. H., Littlewood J. E., Polya G., Inequalities, Cambridge University Press, New York, 1934
Khalil R. M., Al Horani,Yousef A. and Sababheh M., A new definition of fractional derivative, J. Comput. Appl. Math. 264(2014), 65-70
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)
Levinson N., Criteria for the limit-point case for second order linear differential operators. Pest. Mat. Fys. 74(1949), 17-20
Miller, K. S., Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equation, Wiley Interscience, New York, 1993
Oldham K. B., Spainer J., The Fractional Calculus, Academic Press, New York, 1974
Podlubny I., 1999, Fractional Differential Equations, Mathematics in Science and Enginering, vol. 198, Academic Press, New York, London, Tokyo and Toronto
Ross B., Fractional Calculus and Its Applications, Springer, New York, 1975
Titchmarsh E. C., Eigenfunction expansions associated with second-order differential equations, Part I (2nd edition, Oxford University Press, 1962)
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.