Asymptotic stability of nonlinear integrodifferential equation
DOI:
https://doi.org/10.14419/ijamr.v7i2.9592Published:
2018-04-19Keywords:
Asymptotic Stability, Lyapunov Functional, Integro-Differential Equations.Abstract
The investigation of stability for nonlinear dynamical system often related to the construction of Lyapunov functionals. We employ Lyapunov functionals to the system of nonlinear Volterra integro-differential equations of the form and obtain conditions for the asymptotic stability of the zero solution. Also, we give examples to illustrate the obtained results.
References
[1] M. Adivar and Y. N. Raffoul., Inequalities and exponential stability in finite delay Volterra integro-differential equations, Rend. Circ Mat. Palermo (2) 61, (2012), 321-330. https://doi.org/10.1007/s12215-012-0092-4.
[2] L. C. Becker, Uniform continuous L1 – solution of Volterra equations and global asymptotic stability, Cubo 11, (2009): 1-24.
[3] T. A. Burton, Volterra integral and differential equations Second edition, Mathematics in Science and Engineering, 202, Elsevier B. V. Amsterdam, 2005.
[4] T. A. Burton, Stability theory for Volterra equations, J. Differential equations 32, 1, (1979): 101-118.
[5] T. A. Burton and W. E. Mahfoud, Stability criteria for Volterra equations, Trans. Amer. Soc. 279, 1, (1983):143-174.
[6] T. A. Burton and J. R. Haddock, Qualitative properties of solutions of integral equations, Nonlinear Anal. 71, No. 11, (2009):5712- 5723. https://doi.org/10.1016/j.na.2009.04.047.
[7] T. A. Burton, Q. C. Huang and W. E. Mahfoud, Rate of decay of solutions of Volterra equations, Nonlinear Anal. 9(7), (1985): 651- 663. https://doi.org/10.1016/0362-546X(85)90011-2.
[8] T. Caraballo, J. Real. and L. Shaikhet, Method of Lyapunov functional construction in stability of delay evolution equation. J. Math. Anal. 334. (2007):1130-1145 https://doi.org/10.1016/j.jmaa.2007.01.038.
[9] R. Driver, Existence and stability of solution of a delay-differential system, Arch. Rat. Mech. Anal., 10, (1962):401-426. https://doi.org/10.1007/BF00281203.
[10] P. Eloe , M. Islam and B. Zhang, Uniform asymptotic stability of linear Volterra integro-differential equations with application to delay systems, Dynam. Systems Appl. 9, No. 3, (2000)331-344.
[11] P. C. Jackreece and S. Aniaku, Stability Results of Nonlinear Integro-differential Equations. Mathematical Theory and Modeling, 8(1), (2018): 27–33.
[12] K. Magnus, Development of the stability concept in mechanics. Naturwissenshaften. 46, (1959):590-595. https://doi.org/10.1007/BF00684205.
[13] V. S. Sergeev, Stability of solutions of Voterra integro-differential equations. Mathematical and computer modeling. 45, (2007):1376- 1394. https://doi.org/10.1016/j.mcm.2006.09.023.
[14] I. M. Stamova and G. T. Stamov, Analysis of differential equation with maximum, Math. Slovaca, 63, (6)(2013): 1291-1302. https://doi.org/10.2478/s12175-013-0171-9.
[15] I. M. Stamova, and G. T. Stamov, Lyapunov-Razumikhim method for impulsive functional differential equations and application to population dynamics, J. compt. Appl. Math. 130, (2001):163-171
[16] C. Tunç and S. Abid, Journal of the Egyptian Mathematical Society A remark on the stability and boundedness criteria in retarded Volterra integro-differential equations, 25, (2017), 363– 368. https://doi.org/10.1016/j.joems.2017.05.001.
[17] C. Tunc., New Stability and boundedness results to Volterra integro-differential equations with delay, Journal of the Egyptian Mathematical Society, 24(2016): 210-213. https://doi.org/10.1016/j.joems.2015.08.001.
[18] C. Tunc, Qualitative properties in nonlinear Volterra integro- differential equation with delay, J. Taibah Univ. Sci. (2016), https://doi.org/10.1016/j.jtusci.2015.12.009.
[19] J. Vanualailai, and S. Nakagiri,., Stability of a System of Volterra integro-differential equations. J. Math. Anal. Appl., Vol. 281(2003): 602-619. https://doi.org/10.1016/S0022-247X(03)00171-9.
[20] A. M. Wazwaz,, Linear and nonlinear integral equations, methods and applications, Higher Education Press, Beijing, Springer, Heidelberg, 2011. https://doi.org/10.1007/978-3-642-21449-3.
License
Authors who publish with this journal agree to the following terms:
                          [1]           Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
                          [2]           Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
                          [3]           Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).