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

Sougata Dhar; Jessica Stewart Kelly; Sougata Dhar; Jessica Stewart Kelly

doi:10.3934/era.2026173

Electronic Research Archive

2026, Volume 34, Issue 6: 3843-3857. doi: 10.3934/era.2026173

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

Sougata Dhar ¹,
Jessica Stewart Kelly ^{2
,
,}

1.
Department of Mathematics, Fairfield University, Fairfield, CT 06824, USA
2.
Department of Mathematics, Christopher Newport University, Newport News, VA 23606, USA

Received: 12 February 2026 Revised: 14 April 2026 Accepted: 23 April 2026 Published: 11 May 2026

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.
- systems of nonlinear differential equations,
- Lyapunov-type inequalities,
- eigenvalue estimation
Citation: Sougata Dhar, Jessica Stewart Kelly. Lyapunov-type inequalities for systems of second order quasilinear differential equations[J]. Electronic Research Archive, 2026, 34(6): 3843-3857. doi: 10.3934/era.2026173

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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.

References

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