Lyapunov and Hartman–Wintner Inequalities for Caputo–Fabrizio Fractional Dirichlet Boundary Value Problems
Abstract
In this paper we establish Lyapunov- and Hartman–Wintner-type inequalities for a class of two-point Dirichlet boundary value problems involving the Caputo–Fabrizio fractional derivative. The analysis is based on a rigorous construction of the associated Caputo–Fabrizio Green’s function and explicit kernel-dependent bounds that reflect the non-singular exponential memory of the operator. As direct consequences, we obtain nonexistence criteria for nontrivial solutions and a lower bound for the first eigenvalue of the corresponding Caputo–Fabrizio spectral problem. A numerical illustration is provided to confirm the sharpness and qualitative
behavior of the derived inequalities.
2020 Mathematics Subject Classification:
26A33, 34A08, 34B15, 47H10Keywords
Caputo–Fabrizio fractional derivative, Dirichlet boundary value problem, Green’s function, Lyapunov-type inequality, Hartman–Wintner inequality, eigenvalue estimate
Author Details
Sukalwad Umesh Ramrao
Late Babasaheb Deshmukh Gorthekar College
Umri, India
e-mail: usukalwad@gmail.com
Rajkumar N. Ingle
Bahirji Smarak Mahavidyalaya
Basmathnagar, India
e-mail: ingleraju11@gmail.com
References
- A. Atangana, D. Baleanu. New fractional derivatives with nonlocal and non-singular kernel. Thermal Sci. 20, 2 (2016), 763–769, https://scispace.com/pdf/new-fractional-derivatives-with-nonlocal-and-non-singular-frab4xusw4.pdf.
- M. Caputo. Linear models of dissipation whose Q is almost frequency independent. Geophys. J. R. Astr. Soc. 13, 5 (1967), 529–539.
- M. Caputo, M. Fabrizio. A new definition of fractional derivative without singular kernel. Progr. Fract. Differ. Appl. 1, 2 (2015), 73–85, https://www.naturalspublishing.com/files/published/0gb83k287mo759.pdf.
- J. Harjani, K. Sadarangani, B. Samet. Hartman-Wintner type inequalities for a class of fractional BVPs with higher order. J. Inequal. Appl. (2019), Paper No. 278, 12 pp, https://doi.org/10.1186/s13660-019-2232-2.
- P. Hartman. On the zeros of solutions of second order linear differential equations. J. London Math. Soc. 27 (1952), 492–496, https://doi.org/10.1112/jlms/s1-27.4.492.
- M. Jleli, M. Kirane, B. Samet. Hartman-Wintner-type inequality for a fractional boundary value problem via a fractional derivative with respect to another function. Nature and Society (2017) Article ID 5123240, 8 pp, https://doi.org/10.1155/2017/5123240.
- M. Jleli, M. Kirane, B. Samet. Lyapunov-type inequalities for fractional boundary value problems. Handbook of fractional calculus with applications. Vol. 2, (eds A. Kochubei and Y. Luchko), 119–144. Berlin, De Gruyter, 2019, https://doi.org/10.1515/9783110571660-006.
- M. Jleli, B. Samet. Lyapunov-type inequalities for fractional boundary value problems. Electron. J. Differential Equations 2015 (2015), No. 88, 11 pp.
- A. A. Kilbas, H. M. Srivastava, J. J. Trujillo. Theory and applications of fractional differential equations. North-Holland Math. Stud., vol. 204. Amsterdam, Elsevier Science B.V., 2006.
- V. Lakshmikantham, S. Leela, J. Vasundhara Devi. Theory of Fractional Dynamic Systems. Cambridge, Cambridge Scientific Publishers, 2009.
- A. Liapounoff. Probléme général de la stabilité du mouvement. Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (2) 9 (1907), 203–474.
- J. Losada, J. J. Nieto. Properties of a new fractional derivative without singular kernel. Progr. Fract. Differ. Appl. 1, 2 (2015), 87–92, https://www.naturalspublishing.com/files/published/2j1ns3h8o2s789.pdf.
- I. Podlubny. Fractional Differential Equations. Math. Sci. Engrg., vol. 198. San Diego, CA, Academic Press, Inc., 1999.
- Y. Zhou. Fractional Evolution Equations and Inclusions. London, Elsevier/Academic Press, 2016.