Lyapunov-type inequalities for self-adjoint differential equations of second order

Mohamed Jleli; Bessem Samet; Mohamed Jleli; Bessem Samet

doi:10.3934/math.2025963

AIMS Mathematics

2025, Volume 10, Issue 9: 21675-21692. doi: 10.3934/math.2025963

Previous Article Next Article

Research article Special Issues

Lyapunov-type inequalities for self-adjoint differential equations of second order

Mohamed Jleli ,
Bessem Samet ^,

Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia

Received: 01 July 2025 Revised: 09 September 2025 Accepted: 11 September 2025 Published: 18 September 2025
MSC : 34B05, 34B30, 34C10, 26D15

We present a general method that yields Patula, Hartman-Wintner, and Lyapunov-type inequalities for the second-order differential equation
$ -(\alpha(x)u'(x))'+\beta(x)u(x) = \gamma(x)u(x), \quad x\in I, $
where $ I $ is an interval of $ \mathbb{R} $, $ \alpha\in C^1(I) $, $ \alpha > 0 $, $ \beta, \gamma\in C(I) $, and $ \beta\geq 0 $. Applications cover the generalized radial Schrödinger equation and the modified Bessel equation and include an improved (sharpened) form of Bargmann's inequality.
Citation: Mohamed Jleli, Bessem Samet. Lyapunov-type inequalities for self-adjoint differential equations of second order[J]. AIMS Mathematics, 2025, 10(9): 21675-21692. doi: 10.3934/math.2025963

Related Papers:

Abstract

We present a general method that yields Patula, Hartman-Wintner, and Lyapunov-type inequalities for the second-order differential equation

$ -(\alpha(x)u'(x))'+\beta(x)u(x) = \gamma(x)u(x), \quad x\in I, $

where $ I $ is an interval of $ \mathbb{R} $, $ \alpha\in C^1(I) $, $ \alpha > 0 $, $ \beta, \gamma\in C(I) $, and $ \beta\geq 0 $. Applications cover the generalized radial Schrödinger equation and the modified Bessel equation and include an improved (sharpened) form of Bargmann's inequality.

References

[1]	A. Lyapunov, Problème général de la stabilité du mouvement, Ann. Fac. Sci. Toulouse, 9 (1907), 203–474. https://doi.org/10.5802/afst.246 doi: 10.5802/afst.246
[2]	A. Wintner, On the non-existence of conjugate points, Am. J. Math., 73 (1951), 368–380. https://doi.org/10.2307/2372182 doi: 10.2307/2372182
[3]	P. Hartman, A. Wintner, On an oscillation criterion of Liapunoff, Am. J. Math., 73 (1951), 885–890. https://doi.org/10.2307/2372122 doi: 10.2307/2372122
[4]	W. T. Patula, On the distance between zeroes, P. Am. Math. Soc., 52 (1975), 247–251. https://doi.org/10.1090/S0002-9939-1975-0379986-5 doi: 10.1090/S0002-9939-1975-0379986-5
[5]	P. Hartman, Ordinary differential equations, New York: Wiley, 1964.
[6]	B. Behrens, S. Dhar, Lyapunov-type inequalities for third-order nonlinear equations, Differ. Equat. Appl., 14 (2022), 265–277. https://doi.org/10.7153/dea-2022-14-18 doi: 10.7153/dea-2022-14-18
[7]	K. M. Das, A. S. Vatsala, Green's function for $n$-$n$ boundary value problem and an analogue of Hartman's result, J. Math. Anal. Appl., 51 (1975), 670–677. https://doi.org/10.1016/0022-247X(75)90117-1 doi: 10.1016/0022-247X(75)90117-1
[8]	J. Sánchez, V. Vergara, A Lyapunov-type inequality for a $\psi$-Laplacian operator, Nonlinear Anal.-Theor., 74 (2011), 7071–7077. https://doi.org/10.1016/j.na.2011.07.027 doi: 10.1016/j.na.2011.07.027
[9]	R. Yang, I. Sim, Y. H. Lee, Lyapunov-type inequalities for one-dimensional Minkowski-curvature problems, Appl. Math. Lett., 91 (2019), 188–193. https://doi.org/10.1016/j.aml.2018.11.006 doi: 10.1016/j.aml.2018.11.006
[10]	A. Cañada, S. Villegas, Lyapunov inequalities for partial differential equations at radial higher eigenvalues, Discrete Cont. Dyn., 33 (2013), 111–122. https://doi.org/10.3934/dcds.2013.33.111 doi: 10.3934/dcds.2013.33.111
[11]	M. Hashizume, F. Takahashi, Lyapunov inequality for an elliptic problem with the Robin boundary condition, Nonlinear Anal.-Theor., 129 (2015), 189–197. https://doi.org/10.1016/j.na.2015.08.006 doi: 10.1016/j.na.2015.08.006
[12]	M. Alotaibi, M. Jleli, M. A. Ragusa, B. Samet, On the absence of global weak solutions for a nonlinear time-fractional Schrödinger equation, Appl. Anal., 103 (2024), 1–15. https://doi.org/10.1080/00036811.2022.2036335 doi: 10.1080/00036811.2022.2036335
[13]	R. A. C. Ferreira, On a Lyapunov-type inequality and the zeros of a certain Mittag-Leffler function, J. Math. Anal. Appl., 412 (2014), 1058–1063.
[14]	Q. Li, V. D. Rǎdulescu, J. Zhang, W. Zhang, Concentration of normalized solutions for non-autonomous fractional Schrödinger equations, Z. Angew. Math. Phys., 76 (2025), 132. https://doi.org/10.1007/s00033-025-02510-0 doi: 10.1007/s00033-025-02510-0
[15]	R. P. Agarwal, M. Bohner, A. Özbekler, Lyapunov inequalities and applications, Berlin: Springer, 2021. https://doi.org/10.1007/978-3-030-69029-8
[16]	M. H. Protter, H. F. Weinberger, Maximum principles in differential equations, Springer Science & Business Media, 2012.
[17]	V. Bargmann, On the number of bound states in a central field of force, P. Natl. Acad. Sci. USA, 38 (1952), 961–966. https://doi.org/10.1073/pnas.38.11.961 doi: 10.1073/pnas.38.11.961
[18]	M. Abramowitz, I. A. Stegun, Handbook of mathematical functions: With formulas, graphs, and mathematical tables, Courier Corporation, 55 (1965).
[19]	F. W. J. Olver, A. B. O. Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, et al., NIST digital library of mathematical functions, Release, 1 (2016), 22.

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)