Asymptotic stability of nonlinear integrodifferential equation

Prebo Jackreece; Stephen Aniaku

doi:10.14419/ijamr.v7i2.9592

Authors

Prebo Jackreece University of Port Harcourt
Stephen Aniaku University of Nigeria, Nsukka

DOI:

https://doi.org/10.14419/ijamr.v7i2.9592

Published:

2018-04-19

Keywords:

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 L¹â€“ 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): 27â€“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. TunÃ§ 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), 363â€“ 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.