BOHL THEOREM FOR VOLTERRA EQUATION ON TIME SCALES
DOI:
https://doi.org/10.51453/2354-1431/2021/630Keywords:
Volterra differential equations, Boundedness of solutions, Exponential stability, Bohl-Perron theorem.Abstract
This paper is concerned with the Bohl-Perron theorem for Volterra in the form equations
\(x^\Delta (t) = A(t) x(t)+\int_{t_0}^t K(t,s)x(s)\Delta s + f(t). \)
on time scale T. We will show a relationship between the boundedness of the solution of Volterra equation and the stability of the corresponding homogeneous equation.
Downloads
References
[1] O. Perron, Die Stabilitatsfrage bei Differentialgleichungen, Math. Z., 32 (1930), pp. 703-728.
[2] E. Akin-Bohner, M. Bohner and F. Akin, Pachpatte inequalities on time scales, J. Inequal. Pure Appl. Math. 6 (2005) no. 1, 23 pp.
[3] M. Bohner and A. Peterson, Dynamic equations on time scales: An Introduction with Applications, Birkh¨auser, Boston, 2001.
[4] Gusein Sh. Guseinov, Integration on time scales, J. Math. Anal. Appl., 285(2003), 107{127.
[5] E. Bravyi, R. Hakl, A. Lomtatidze, On Cauchy problem for first order nonlinear functional differential equations of non-Volterra’s type. (English), Czechoslovak Mathematical Journal, 52(2002), issue 4, pp. 673-690.
[6] E. Braverman, I.M. Karabash, Bohl-Perron type stability theorems for linear difference equations with infinite delay, J. Differ. Equ. Appl., 18(2012), pp. 909-939.
[7] M.R. Crisci, V.B. Kolimanovskll, E. Russo, A. Vecchio, On the exponential stability of discrete volterra systems, Journal of Difference Equations and Applications, 6(2000), pp. 667-480.
[8] N.H. Du, V.H. Linh and N.T.T. Nga, On stability and Bohl exponent of linear singular systems of difference equations with variable coefficients, J. Differ. Equ. Appl., 22 (2016), pp. 1350-1377.
[9] N.H. Du, L.H. Tien, On the exponential stability of dynamic equations on time scales. J. Math. Anal. Appl., 331(2007), pp. 1159-1174.
[10] A. Filatov and L. Sarova, Integral’nye neravenstva i teorija nelineinyh kolebanii. Moskva, 1976. 19(1975), pp. 142-166.
[11] S. Grossman and R.K. Miller, Perturbation Theory for Volterra Integrodiffererential Systems, J. Differential Equations, 8(1970), pp. 457-474.
[12] M. Pituk. A Perron type theorem for functional differential equations, J. Math. Anal. Appl., 316(2006), pp 24-41.
Downloads
Published
How to Cite
Issue
Section
License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
All articles published in SJTTU are licensed under a Creative Commons Attribution-ShareAlike 4.0 International (CC BY-SA) license. This means anyone is free to copy, transform, or redistribute articles for any lawful purpose in any medium, provided they give appropriate attribution to the original author(s) and SJTTU, link to the license, indicate if changes were made, and redistribute any derivative work under the same license.
Copyright on articles is retained by the respective author(s), without restrictions. A non-exclusive license is granted to SJTTU to publish the article and identify itself as its original publisher, along with the commercial right to include the article in a hardcopy issue for sale to libraries and individuals.
Although the conditions of the CC BY-SA license don't apply to authors (as the copyright holder of your article, you have no restrictions on your rights), by submitting to SJTTU, authors recognize the rights of readers, and must grant any third party the right to use their article to the extent provided by the license.
