Lyapunov-type inequalities for higher-order Caputo fractional differential equations with general two-point boundary conditions

Satyam Narayan Srivastava; Smita Pati; John R. Graef; Alexander Domoshnitsky; Seshadev Padhi

doi:10.56754/0719-0646.2602.259

DOI:

https://doi.org/10.56754/0719-0646.2602.259

Abstract

In this paper the authors present three different Lyapunov-type inequalities for a higher-order Caputo fractional differential equation with identical boundary conditions marking the inaugural instance of such an approach in the existing literature. Their findings extend and complement certain prior results in the literature.

Keywords

Fractional integral , Caputo fractional derivative , boundary value problem , existence of solution , Lyapunov inequality , Green’s function

Mathematics Subject Classification:

26A33 , 34A08 , 34B2 , 26D10 , 34C10

Satyam Narayan Srivastava Department of Mathematics, Ariel University, Ariel-40700, Israel. https://orcid.org/0000-0002-4761-3109
Smita Pati Amity University Jharkhand, Ranchi, 834001, India. https://orcid.org/0000-0002-1371-7591
John R. Graef Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37401 USA. https://orcid.org/0000-0002-8149-4633
Alexander Domoshnitsky Department of Mathematics, Ariel University, Ariel-40700, Israel. https://orcid.org/0000-0002-8114-7084
Seshadev Padhi Department of Mathematics, Birla Institute of Technology, Ranchi-835215, India. https://orcid.org/0000-0002-0282-2791

Pages: 259–277
Date Published: 2024-07-11
Vol. 26 No. 2 (2024)

R. P. Agarwal, Y. Liu, D. O’Regan, and C. Tian, “Positive solutions of two-point boundary value problems for fractional singular differential equations,” Differ. Equ., vol. 48, no. 5, pp. 619–629, 2012, translation of Differ. Uravn. 48 (2012), no. 5, 611–621.

R. P. Agarwal, M. Bohner, and A. Özbekler, Lyapunov inequalities and applications. Springer, Cham, 2021, doi: 10.1007/978-3-030-69029-8.

M. F. Aktaş and D. Çakmak, “Lyapunov-type inequalities for third-order linear differential equations,” Electron. J. Differential Equations, 2017, Art. ID 139.

M. F. Aktaş and D. Çakmak, “Lyapunov-type inequalities for third-order linear differential equations under the non-conjugate boundary conditions,” Differ. Equ. Appl., vol. 10, no. 2, pp. 219–226, 2018, doi: 10.7153/dea-2018-10-14.

M. F. Aktaş and D. Çakmak, “Lyapunov-type inequalities for third order linear differential equations with two points boundary conditions,” Hacet. J. Math. Stat., vol. 48, no. 1, pp. 59–66, 2019, doi: 10.15672/hjms.2017.514.

M. Bohner, A. Domoshnitsky, S. Padhi, and S. N. Srivastava, “Vallée-Poussin theorem for equations with Caputo fractional derivative,” Math. Slovaca, vol. 73, no. 3, pp. 713–728, 2023, doi: 10.1515/ms-2023-0052.

I. Cabrera, B. Lopez, and K. Sadarangani, “Lyapunov type inequalities for a fractional two-point boundary value problem,” Math. Methods Appl. Sci., vol. 40, no. 10, pp. 3409–3414, 2017, doi: 10.1002/mma.4232.

S. Dhar and Q. Kong, “Lyapunov-type inequalities for third-order linear differential equations,” Math. Inequal. Appl., vol. 19, no. 1, pp. 297–312, 2016, doi: 10.7153/mia-19-22.

S. Dhar and Q. Kong, “Fractional Lyapunov-type inequalities with mixed boundary conditions on univariate and multivariate domains,” J. Fract. Calc. Appl., vol. 11, no. 2, pp. 148–159, 2020.

A. Domoshnitsky, S. Padhi, and S. N. Srivastava, “Vallée-Poussin theorem for fractional functional differential equations,” Fract. Calc. Appl. Anal., vol. 25, no. 4, pp. 1630–1650, 2022, doi: 10.1007/s13540-022-00061-z.

J. R. Graef, K. Maazouz, and M. D. A. Zaak, “A generalized Lyapunov inequality for a pantograph boundary value problem involving a variable order Hadamard fractional derivative,” Mathematics, vol. 11, no. 13, 2023, Art. ID 2984, doi: 10.3390/math11132984.

J. R. Graef, R. Mahmoud, S. H. Saker, and E. Tunç, “Some new Lyapunov-type inequalities for third order differential equations,” Comm. Appl. Nonlinear Anal., vol. 22, no. 2, pp. 1–16, 2015.

A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, ser. North Holland Mathematics Studies. Elsevier Science B.V., Amsterdam, 2006, vol. 204.

A. M. Lyapunov, “Probléme général de la stabilité du mouvement,” Ann. Fac. Sci. Toulouse Math., vol. 9, pp. 203–474, 1907.

Q. Ma, C. Ma, and J. Wang, “A Lyapunov-type inequality for a fractional differential equation with Hadamard derivative,” J. Math. Inequal., vol. 11, no. 1, pp. 135–141, 2017, doi: 10.7153/jmi-11-13.

A. D. Oğuz, J. Alzabut, A. Özbekler, and J. M. Jonnalagadda, “Lyapunov and Hartman-type inequalities for higher-order discrete fractional boundary value problems,” Miskolc Math. Notes, vol. 24, no. 2, pp. 953–963, 2023, doi: 10.18514/mmn.2023.3931.

N. Parhi and S. Panigrahi, “ On Liapunov-type inequality for third-order differential equations,” J. Math. Anal. Appl., vol. 233, no. 2, pp. 445–460, 1999, doi: 10.1006/jmaa.1999.6265.

I. Podlubny, Fractional differential equations, ser. Mathematics in Science and Engineering. Academic Press, Inc., San Diego, CA, 1999, vol. 198.

T. Qiu and Z. Bai, “Existence of positive solutions for singular fractional differential equations,” Electron. J. Differential Equations, 2008, Art. ID 146.

T. Qiu and Z. Bai, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” J. Nonlinear Sci. Appl., vol. 1, no. 3, pp. 123–131, 2008, doi: 10.22436/jnsa.001.03.01.

J. Rong and C. Bai, “Lyapunov-type inequality for a fractional differential equation with fractional boundary conditions,” Adv. Difference Equ., 2015, Art. ID 82, doi: 10.1186/s13662- 015-0430-x.

S. N. Srivastava, S. Pati, S. Padhi, and A. Domoshnitsky, “Lyapunov inequality for a Caputo fractional differential equation with Riemann-Stieltjes integral boundary conditions,” Math. Methods Appl. Sci., vol. 46, no. 12, pp. 13 110–13 123, 2023, doi: 10.1002/mma.9238.

Y. Sun and X. Zhang, “Existence and nonexistence of positive solutions for fractional-order two-point boundary value problems,” Adv. Difference Equ., 2014, Art. ID 53, doi: 10.1186/1687-1847-2014-53.

C. Tian and Y. Liu, “Multiple positive solutions for a class of fractional singular boundary value problems,” Mem. Differ. Equ. Math. Phys., vol. 56, pp. 115–131, 2012.

National Board for Higher Mathematics of the Department of Atomic Energy of the Government of India in the research grant No 02011/17/2021 NBHM(R.P)/R&D II/9294 Dated 11.10.2021