Lyapunov-type inequalities for systems of second order quasilinear differential equations

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

We establish Lyapunov-type inequalities for systems of second order quasilinear differential equations with Dirichlet boundary conditions. By incorporating the positive parts of the coefficient functions, we correct and extend several known inequalities for quasilinear systems. Our results cover constant and variable coefficient cases and lead to explicit lower bounds for generalized eigenvalues. These inequalities generalize classical scalar results and improve the existing estimates within the literature. As an application, we obtain explicit and computable lower bounds for generalized eigenvalues of coupled quasilinear operators.

Keywords

systems of nonlinear differential equations / Lyapunov-type inequalities / eigenvalue estimation

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] A. M. Liapunov, Probleme general de la stabilite du mouvement, Ann. Math. Stud., 17 (1947), 203-474.

[2] G. Borg, On a Liapunoff criterion of stability, Am. J. Math., 71 (1949), 67-70. https://doi.org/10.2307/2372093

[3] A. Wintner, On the non-existence of conjugate points, Am. J. Math., 73 (1951), 368-380. https://doi.org/10.2307/2372182

[4] S. Dhar, Q. Kong, Liapunov-type inequalities for third-order half-linear equations and applications to boundary value problems, Nonlin. Anal. TMA, 110 (2014), 170-181. https://doi.org/10.1016/j.na.2014.07.020

[5] S. Dhar, Q. Kong, Lyapunov-type inequalities for higher order half-linear differential equations, Appl. Math. Comput., 273 (2016), 114-124. https://doi.org/10.1016/j.amc.2015.09.090

[6] S. Dhar, Q. Kong, Lyapunov-type inequalities for third-order linear differential equations, Math. Inequal. Appl., 19 (2016), 297-312. https://doi.org/10.7153/mia-19-22

[7] S. Dhar, Q. Kong, Lyapunov-type inequalities for odd order linear differential equations, Electr. J. Differ. Equation, 2016 (2016), 1-10.

[8] S. Dhar, J. S. Kelly, Lower bounds for eigenvalues of even ordered quasilinear differential equations, Proc. Am. Math. Soc., 151 (2023), 647-661. https://doi.org/10.1090/proc/16122

[9] J. P. Pinasco, Lower bounds for eigenvalues of the one-dimensional p-Laplacian, Abstr. Appl. Anal., 2 (2004), 147-153. https://doi.org/10.1155/S108533750431002X

[10] J. P. Pinasco, Lyapunov-type Inequalities, with Applications to Eigenvalue Problems, Springer Briefs in Mathematics, Springer, New York, 2013.

[11] K. Watanabe, Lyapunov type inequality for the equation including 1-dim p-Laplacian, Math. Inequal. Appl., 15 (2012), 657-662. https://doi.org/10.7153/mia-15-58

[12] P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964, and Birkhauser, Boston, 1982.

[13] X. Yang, On inequalities of Lyapunov type, Appl. Math. Comput., 134 (2003), 293-300. https://doi.org/10.1016/S0096-3003(01)00283-1

[14] P. L. Napoli, J. P. Pinasco, Estimates for eigenvalues of quasilinear elliptic systems, J. Differ. Equation, 227 (2006), 102-115. https://doi.org/10.1016/j.jde.2006.01.004

[15] D. Ç akmak, A. Tiryaki, Lyapunov-type inequality for a class of Dirichlet quasilinear systems involving the (p₁, p₂, · · · , p_n)-Laplacian , J. Math. Anal. Appl., 369 (2010), 76-81. https://doi.org/10.1016/j.jmaa.2010.02.043

[16] X. H. Tang, X. He, Lower bounds for generalized values of the quasilinear systems, J. Math. Anal. Appl., 385 (2012), 72-85. https://doi.org/10.1016/j.jmaa.2011.06.026

[17] M. F. Aktas, D. Çakmak, A. Tiryaki, A note on Tang and He’s paper, Appl. Math. Comput., 218 (2012), 4867-4871. https://doi.org/10.1016/j.amc.2011.10.050

[18] D. Ç akmak, A. Tiryaki, On Lyapunov-type inequality for quasilinear systems, Appl. Math. Comput., 216 (2010), 3584-3591. https://doi.org/10.1016/j.amc.2010.05.004

[19] G. Guseinov, B. Kaymakcalan, Lyapunov inequalities for discrete linear Hamiltonian system, Comput. Math. Appl., 45 (2003), 1399-1416. https://doi.org/10.1016/S0898-1221(03)00095-6

[20] I. Sim, Y. Lee, Lyapunov inequalities for one-dimensional p-Laplacian problems with singular weight function, J. Inequal. Appl., 2010 (2010), 1-9. https://doi.org/10.1155/2010/865096