SPECTRAL PROPERTIES OF A CONFORMABLE BOUNDARY VALUE PROBLEM ON TIME SCALES

Year 2021, , 73 - 80, 31.01.2021

Zeki Ceylan

https://doi.org/10.33773/jum.695777

Abstract

We study a self-adjoint conformable dynamic equation of second order on an arbitrary time scale $\mathbb{T}$. We state an existence and uniqueness theorem for the solutions of this equation. We prove the conformable Lagrange identity on time scales. Then, we consider a conformable eigenvalue problem generated by the above-mentioned dynamic equation of second order and we examine some of the spectral properties of this boundary value problem.

Keywords

Time scales, Conformable derivative

References

• Referans1 T. Abdeljawad, On conformable fractional calculus, Journal of Computational and Applied Mathematics, 279(2015), 57-66.
• Referans2 M. Abu Hammad, R. Khalil, Abel's formula and Wronskian for conformable fractional differential equations, Internat. J. Diff. Equ. Appl., 13(2014), No. 3, 177-183.
• Referans3 M. Abu Hammad, R. Khalil, Conformable fractional heat differential equations, Internat. J. Pure Appl. Math., 94(2014), No. 2, 215-221.
• Referans4 H. Abu-Shaab, R. Khalil, Solution of some conformable fractional differential equations, Int. J. Pure Appl. Math., 103(2015), No. 4, 667-673.
• Referans5 M. J. Lazo, D. F. M. Torres, Variational calculus with conformable fractional derivatives, IEEE/CAA Journal of Automatica Sinica, 4(April 2017), No. 2.
• Referans6 W. Rosa, J. Weberspil, Dual conformable derivative:Definition, simple properties and perspectives for applications, Chaos, Solitons and Fractals, 117(2018), 137-141.
• Referans7 D. Anderson, R. I. Avery, Fractional-order boundary value problem with Sturm-Liouville boundary conditions, Electronic Journal of Differential Equations, 29(2015), 1-10.
• Referans8 H. Batarfi, J. Losada, J. J. Nieto, W. Shammakh, Three-point boundary value problems for conformable fractional differential equations, Journal of Function Spaces, Volume 2015, Article ID 706383, 6 pages, doi:10.1155/2015/706383.
• Referans9 B. P. Allahverdiev, H. Tuna, Y. Yalcinkaya, Conformable fractional Sturm-Liouville equation, Math. Meth. Appl. Sci., 42(2019), 3508-3526.
• Referans10 N. Benkhettou, S. Hassani, D. F. M. Torres, A conformable fractional calculus on arbitrary time scales, Journal of King Saud University-Science, 28(2016), 93-98.
• Referans11 M. Bohner, V. F. Hatipoğlu, Dynamic Cobweb models with conformable fractional derivatives, Nonlinear Anal., Hybrid Syst. 32(2019), 157-167.
• Referans12 T. Gulsen, E. Yilmaz, S. Goktas, Conformable fractional Dirac system on time scales, J. Inequal. Appl., 2017:10, 2017.
• Referans13 T. Gulsen, E. Yilmaz, H. Kemaloglu, Conformable fractional Sturm-Liouville equation and some existence results on time scales, Turk. J. Math. 42(2018), No. 3, 1348-1360.
• Referans14 S. Rahmat, M. Rafi, A new definition of conformable fractional derivative on arbitrary time scales, Adv. Difference Equ., 2019:354, 2019.
• Referans15 D. F. Zhou, X. X. You, A new fractional derivative on time scales, Adv. Appl. Math. Anal., 11(2016), No. 1, 1-9.
• Referans16 C. Zhang, S. Sun, Sturm-Picone comparison theorem of a kind of conformable fractional differential equations on time scales, J. Appl. math. Comput., 55(2017), 191-203, doi:10.1007/s12190-016-1032-9.
• Referans17 M. Bohner, A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications, Birkhauser, Boston Inc. Boston, MA, 2001.