Bohl Theorem for Volterra Equations
Main Article Content
Abstract
This work deals with the preservation of exponential stability under small perturbations for Volterra differential equations. The so-called Bohl-Perron type stability theorems for these systems are also studied.
Keywords:
Hardy inequality, time scales, exponential function.
References
[1] N. H. Du, Stability Radii of Differential-algebraic Equations with Structured Perturbations, Systems & Control Lett., Vol. 57, 2008, pp. 546-553, https://doi.org/10.1016/j.sysconle.2007.12.001.
[2] B. Jacob, A Formula for the Stability Radius of Time-varying Systems, J. Differential Equations, Vol. 142, 1998, pp. 167-187, https://doi.org/10.1006/jdeq.1997.3348.
[3] N. H. Du, V. H. Linh, N. T. T. Nga, On Stability and Bohl Exponent of Linear Singular Systems of Difference Equations with Variable Coefficients, Journal of Difference Equations and Applications., Vol. 22, 2016,
pp. 1350-1377, https://doi.org/10.1080/10236198.2016.1198341.
[4] M. Pituk, A Criterion for the Exponential Stability of Linear Difference Equations, Appl. Math. Lett., Vol. 17, 2004, pp. 779-783, https://doi.org/10.1016/j.aml.2004.06.005.
[5] E. Braverman, I. M. Karabash, Bohl-perron Type Stability Theorems for Linear Difference Equations with Infinite Delay, Journal of Difference Equations and Applications, Vol. 18, 2012, pp. 909-939, https://doi.org/10.48550/arXiv.1009.6163.
[6] M. R. Crisci, V. B. Kolimanovskll, E. Russo, A. Vecchio, on the Exponential Stability of Discrete Volterra Systems, Journal of Difference Equations and Applications, Vol. 6, 2000, pp. 667-480, https://doi.org/10.1080/10236190008808251.
[7] O. Perron, Die Stabilitatsfrage Bei Differentialgleichungen, Math. Z., Vol. 32, 1930, pp. 703-728, https://doi.org/10.1007/BF01194662.
[8] T. T. Anh, P. H. A. Ngoc, New Stability Criteria for Linear Volterra Time-varying Integro-differential Equations, Taiwanese Journal of Mathematics, Vol. 21, 2017, No. 4, pp. 841-863, https://doi.org/10.11650/tjm/7812.
[9] G. Seifert, Liapunov-razumikhin Conditions for Asymptotic Stability in Functional Differential Equations of Volterra Type, Journal of Differential Equations, Vol. 16, 1974, pp. 289-297, https://doi.org/10.1016/0022-0396(74)90016-3.
[10] A. Filatov, L. Sarova, Integralnye Neravenstva I Teorija Nelineinyh Kolebanii, Moskva, 1976.
[11] R. Grimmer, G. Seifert, Stability Properties of Integrodifferential Equations, J. Differential Equations, Vol. 19, 1975, pp. 142-166, https://doi.org/10.1016/0022-0396(75)90025-X.
[12] M. Pituk, A Perron Type Theorem for Functional Differential Equations, J. Math. Anal. Appl., Vol. 316, 2006,
pp. 24-41, https://doi.org/10.1016/j.jmaa.2005.04.027.
[2] B. Jacob, A Formula for the Stability Radius of Time-varying Systems, J. Differential Equations, Vol. 142, 1998, pp. 167-187, https://doi.org/10.1006/jdeq.1997.3348.
[3] N. H. Du, V. H. Linh, N. T. T. Nga, On Stability and Bohl Exponent of Linear Singular Systems of Difference Equations with Variable Coefficients, Journal of Difference Equations and Applications., Vol. 22, 2016,
pp. 1350-1377, https://doi.org/10.1080/10236198.2016.1198341.
[4] M. Pituk, A Criterion for the Exponential Stability of Linear Difference Equations, Appl. Math. Lett., Vol. 17, 2004, pp. 779-783, https://doi.org/10.1016/j.aml.2004.06.005.
[5] E. Braverman, I. M. Karabash, Bohl-perron Type Stability Theorems for Linear Difference Equations with Infinite Delay, Journal of Difference Equations and Applications, Vol. 18, 2012, pp. 909-939, https://doi.org/10.48550/arXiv.1009.6163.
[6] M. R. Crisci, V. B. Kolimanovskll, E. Russo, A. Vecchio, on the Exponential Stability of Discrete Volterra Systems, Journal of Difference Equations and Applications, Vol. 6, 2000, pp. 667-480, https://doi.org/10.1080/10236190008808251.
[7] O. Perron, Die Stabilitatsfrage Bei Differentialgleichungen, Math. Z., Vol. 32, 1930, pp. 703-728, https://doi.org/10.1007/BF01194662.
[8] T. T. Anh, P. H. A. Ngoc, New Stability Criteria for Linear Volterra Time-varying Integro-differential Equations, Taiwanese Journal of Mathematics, Vol. 21, 2017, No. 4, pp. 841-863, https://doi.org/10.11650/tjm/7812.
[9] G. Seifert, Liapunov-razumikhin Conditions for Asymptotic Stability in Functional Differential Equations of Volterra Type, Journal of Differential Equations, Vol. 16, 1974, pp. 289-297, https://doi.org/10.1016/0022-0396(74)90016-3.
[10] A. Filatov, L. Sarova, Integralnye Neravenstva I Teorija Nelineinyh Kolebanii, Moskva, 1976.
[11] R. Grimmer, G. Seifert, Stability Properties of Integrodifferential Equations, J. Differential Equations, Vol. 19, 1975, pp. 142-166, https://doi.org/10.1016/0022-0396(75)90025-X.
[12] M. Pituk, A Perron Type Theorem for Functional Differential Equations, J. Math. Anal. Appl., Vol. 316, 2006,
pp. 24-41, https://doi.org/10.1016/j.jmaa.2005.04.027.