Lyapunov-Krasovskii stability analysis of nonlinear integro-differential equation
DOI:
https://doi.org/10.14419/ijamr.v7i2.10168Keywords:
Lyapunov-Krasovskii Functional, Uniform Asymptotic Stability, Integro-Differential Equation.Abstract
The purpose of this paper is to develop a qualitative stability analysis of a class of nonlinear integro-differential equation within the framework of Lyapunov-Krasovskii. We show that the existence of a Lyapunov-Krasovskii functional is a necessary and sufficient condition for the uniform asymptotic stability of the nonlinear Volterra integro-differential equations.
References
[1] Adivar M. and Raffoul Y. N., 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] Becker L. C., Uniform continuous L1 – solution of Volterra equations and global asymptotic stability, Cubo 11(2009), 1-24.
[3] Burton T. A., Volterra integral and differential equations Second Edition, Mathematics in Science and Engineering, 202, Elsevier B. V. Amsterdam, 2005.
[4] Burton T. A., Stability theory for Volterra equations, J. Differential Equations 32(1979), 1, 101-118.
[5] Burton T. A. and Mahfoud W. E., Stability criteria for Volterra Equations, Trans. Amer. Soc. 279 (1983), 1, 143-174.
[6] Burton T. A. and Haddock J. R., Qualitative properties of solutions of integral equations, Nonlinear Anal. 71 (2009), No. 11, 5712-5723.
[7] Burton, T., Volterra integral and Differential Equations, Elsevier B. V., Amsterdam, 2005.
[8] Caraballo, T., Real, J. and Shaikhet, L., 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] Davies I., Nwoaburu A. O., Jackreece P. C.; Criteria for Asymptotic Stability for Linear Delay Systems of the Volterra Type, Journal of The Mathematical Association of Nigeria, (Abacus), Vol. 32, #2A, pp 101-109, 2005.
[10] Driver, R., Existence and stability of solution of a delay-Differential system, Arch. Rat. Mech. Anal., 10, (1962):401-426. https://doi.org/10.1007/BF00281203.
[11] Eloe P., Islam M. and Zhang B., Uniform asymptotic stability of Linear Volterra integro-differential equations with application to Delay systems, Dynam. Systems Appl. 9(200), No. 3, 331-344.
[12] Hale, J. K. and Lunel, S. M. V., Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. https://doi.org/10.1007/978-1-4612-4342-7.
[13] Jackreece, P. C., and Aniaku, S., Stability Results of Nonlinear Integro-differential Equations. Mathematical Theory and Modeling, 8(1), (2018), 27–33.
[14] Kuang, Y., Delay Differential Equations with Applications in Population Dynamics, Academic Press Inc., San Diego, 1993.
[15] La Salle, J. and Lefschetz S., Stability by Lyapunov’s direct method with applications, New York Academic Press, 1961.
[16] Lakshmikanthan, V. and Rao, M. R. M., Theory of Integro-differential Equations, Gordon and Breach Science Publishers, Amsterdam, 1995.
[17] Stamova, I. M., and Stamov, G. T., Analysis of differential equation with maximum, Math. Slovaca, 63(6), (2013): 1291-1302. https://doi.org/10.2478/s12175-013-0171-9.
[18] Stamov, I. M., and Stamov, G., T., Lyapunov-Razumikhim method for impulsive functional differential equations and application to population dynamics, J. compt. Appl. Math. 130, (2001):163-171
[19] Sergeev, V. S., 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.
[20] Tunc, C. A note on the qualitative behaviors of nonlinear Volterra integro-differential equation, Journal of the Egyptian Mathematical Society, 24, (2016), 187-192. https://doi.org/10.1016/j.joems.2014.12.010.
[21] Tunç, C., and Abid, S. (2017). A remark on the stability and boundedness criteria in retarded Volterra integro-differential equations, Journal of the Egyptian Mathematical Society, 25, 363–368. https://doi.org/10.1016/j.joems.2017.05.001.
[22] Tunc, C., 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.
[23] Tunç, C. (2016). Qualitative properties in nonlinear Volterra integro-differential equations with delay. Journal of Taibah University for Science. ps://doi.org/10.1016/j.jtusci.2015.12.009.
[24] Vanualailai, J. and Nakagiri, S., Stability of a System of Volterra integro-differential equations. J. Math. Anal. Appl., Vol. 281(2), 2003, 602-619. https://doi.org/10.1016/S0022-247X(03)00171-9.
[25] Wazwaz, A. M., 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.
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Received date: March 14, 2018
Accepted date: April 16, 2018
Published date: April 28, 2018