Abstract
In this chapter we give some proofs of Lyapunov’ inequality, in both the linear and nonlinear contexts.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Borg, G.: Uber die Stabilität gewisser Klassen von linearen Differentialgleichungen, Ark. for Matematik, Astronomi och Fysik 31, 1–31 (1945)
Borg, G.: On a Liapunoff criterion of stability. Amer. J. Math. 71, 67–70 (1949)
Brown, R. C., Hinton, D. B.: Lyapunov inequalities and their applications. In: Rassias, T. M. (ed.) Survey on Classical Inequalities, pp. 1–25. Springer (2000)
Cohn, J. H. E.: Consecutive zeroes of solutions of ordinary second order differential equations. J. London Math. Soc. 5, (1972) 465–468.
Courant, R., Hilbert, D,: Methods of Mathematical Physics, vol. I, Interscience Publishers, Inc., New York (1953)
Das, K. M., Vatsala, A. S.: Green’s function for n-n boundary value problem and an analogue of Hartman’s result. Journal of Mathematical Analysis and Applications 51, 670–677 (1975)
De Nápoli, P. L., Pinasco, J. P.: Lyapunov-type inequalities in R N. Preprint (2012)
Dosly, O., Rehak, P.: Half-Linear Differential Equations. Volume 202 North-Holland Mathematics Studies. North Holland (2005)
Egorov, Y. V., Kondriatev, V. A.: On Spectral Theory of Elliptic Operators (Operator Theory: Advances and Applications) Birkhäuser (1996)
Fernández Bonder, J., Pinasco, J. P., Salort, A. M.: A Lyapunov-type Inequality and Eigenvalue Homogenization with Indefinite Weights. Preprint (2013)
Harris, B. J., Kong, Q.: On the oscillation of differential equations with an oscillatory coefficient. Transactions of the American Mathematical Society 347, 1831–1839 (1995)
Hartman, P., Wintner, A.: On an Oscillation Criterion of Liapunoff. American Journal of Mathematics 73, 885–890 (1951)
Hong, H-L., Lian, W-C., Yeh, C.C.: The oscillation of half-linear differential equations with an oscillatory coefficient. Mathematical and Computer Modelling 24, 77–86 (1996)
Lee, C.-F., Yeh, C.-C., Hong, C.-H., Agarwal, R. P.: Lyapunov and Wirtinger Inequalities. Appl. Math. Letters 17, 847–853 (2004)
Levin, A.: A Comparison Principle for Second Order Differential Equations. Sov. Mat. Dokl. 1, 1313–1316 (1960)
Nehari, Z.: On the zeros of solutions of second-order linear differential equations. American Journal of Mathematics 76 689–697 (1954)
Nehari, Z.: Oscillation criteria for second-order linear differential equations. Transactions of the American Mathematical Society 85, 428–445 (1957)
Patula, W. T.: On the Distance between Zeroes. Proceedings of the American Mathematical Society 52, 247–251 (1975)
Pinasco, J. P.: Lower bounds for eigenvalues of the one-dimensional p-Laplacian. Abstract and Applied Analysis 2004, 147–153 (2004)
Pinasco, J. P.: Comparison of eigenvalues for the p-Laplacian with integral inequalities, Appl. Math. Comput. 182, 1399–1404 (2006)
Reid, W. T.: A matrix Liapunov inequality. J. of Math. Anal. and Appl. 32, 424–434 (1970)
Reid, W. T.: A Generalized Lyapunov Inequality. J. Differential Equations 13, 182–196 (1973)
Reid, W. T.: Interrelations between a trace formula and Liapunov type inequalities. J. of Differential Equations 23, 448–458 (1977)
St. Mary, D. F.: Some Oscillation and Comparison Theorems for \((r(t)y^{\prime})^{\prime} + p(t)y = 0\), J. of Differential Equations, 5, 314–323 (1969)
Watanabe, K.: Lyapunov-type inequality for the equation including 1-dim p-Laplacian. Mathematical Inequalities and Applications 15, 657–662 (2012)
Watanabe, K., Kametaka, Y., Yamagishi, H., Nagai, A., Takemura, K.: The best constant of Sobolev inequality corresponding to clamped boundary value problem. Boundary Value Problems 2011 Article ID 875057 (2011)
Wintner, A.: On the Non-Existence of Conjugate Points. American Journal of Mathematics 73, 368–380 (1951)
Yang, X.: On inequalities of Lyapunov type. Applied Math. Comp. 134, 293–300 (2003)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Juan Pablo Pinasco
About this chapter
Cite this chapter
Pinasco, J.P. (2013). Lyapunov’s Inequality. In: Lyapunov-type Inequalities. SpringerBriefs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8523-0_2
Download citation
DOI: https://doi.org/10.1007/978-1-4614-8523-0_2
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-8522-3
Online ISBN: 978-1-4614-8523-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)