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Green’s Function and a Lyapunov-Type Inequality for a Second-Order Boundary Value Problem with Robin Conditions
Subject area: Science,Engineering and Technology · Area of research: Fractional Inequalities
Abstract
We study the second-order differential equation x′′(t) + q(t)x(t) = 0, a < t < b, subject to separated Robin boundary conditions. Under explicit assumptions on the boundary coefficients, we construct the Green’s function for the auxiliary operator −d2/dt2, establish its positivity and symmetry, and derive an integral representation for solutions. This representation yields a Lyapunov-type necessary condition, a weighted eigenvalue estimate, and an Lp-version. Dirichlet and mixed boundary conditions are recovered as special cases. The results provide a self-contained baseline for further work on optimal constants and more general differential operators.
Keywords
Green’s function; Lyapunov inequality; Robin boundary conditions; mixed boundary conditions; second-order differential equation; eigenvalue estimate
References
[1] Hashizume M, Takahashi F. Lyapunov inequality for an elliptic problem with the Robin boundary condition. Nonlinear Analysis: Theory, Methods & Applications. 2015;129:189–197. https://doi.org/10.1016/j.na.2015.08.006
[2] Kantar ÜD, Özden T. Lyapunov-type inequalities for quasilinear elliptic equations with Robin boundary condition. Journal of Inequalities and Applications. 2017;2017:45. https://doi.org/10.1186/s13660-017-1320-4
[3] Jonnalagadda JM, Nápoles Valdés JE. Green’s function and a Lyapunov-type inequality for a generalized boundary value problem. Open Journal of Mathematical Analysis. 2026;10(1):176–185. https://doi.org/10.30538/psrp-oma2026.0195
[4] Oza P. Lyapunov-type inequality to general second-order elliptic equations. Mathematical Methods in the Applied Sciences. 2024. https://doi.org/10.1002/mma.10297
How to cite this paper
@article{1724444,
author = {Machhindra Rambhau Khandare, Jayashri Patil},
title = {Green’s Function and a Lyapunov-Type Inequality for a Second-Order Boundary Value Problem with Robin Conditions},
journal = {Iconic Research And Engineering Journals},
year = {2026},
volume = {10},
number = {4},
pages = {4975-4981},
issn = {2456-8880},
url = {https://www.irejournals.com/formatedpaper/1724444.pdf},
abstract = {We study the second-order differential equation x′′(t) + q(t)x(t) = 0, a < t < b, subject to separated Robin boundary conditions. Under explicit assumptions on the boundary coefficients, we construct the Green’s function for the auxiliary operator −d2/dt2, establish its positivity and symmetry, and derive an integral representation for solutions. This representation yields a Lyapunov-type necessary condition, a weighted eigenvalue estimate, and an Lp-version. Dirichlet and mixed boundary conditions are recovered as special cases. The results provide a self-contained baseline for further work on optimal constants and more general differential operators.},
keywords = {Green’s function; Lyapunov inequality; Robin boundary conditions; mixed boundary conditions; second-order differential equation; eigenvalue estimate},
month = {October},
}