On asymptotically stability, uniformly stability and boundedness of solutions of nonlinear Volterra integro-differential equations
In this paper, two new Lyapunov functionals are defined.
We apply these functionals to get sufficient conditions guaranteeing the asymptotic stability, uniform stability, and boundedness of solutions of certain nonlinear Volterra integro-differential equations of the first order.
The results obtained are improvements and extensions of known results that can be found in literature.
We also suggest examples to show the applicability of our results and for the sake of illustrations.
Using MATLAB-Simulink, in particular cases we clearly show the behavior of orbits of Volterra integro-differential equations under consideration.
L. C. Becker, Uniformly continuous $L^1$ solutions of Volterra equations and global asymptotic stability, Cubo, 11, № 3, 1 – 24 (2009).
T. A. Burton, Volterra integral and differential equations, Second ed. Math. Sci. and Eng., Vol. 202, Elsevier B. V., Amsterdam (2005).
T. Furumochi, S. Matsuoka, Stability and boundedness in Volterra integro-differential equations, Mem. Fac. Sci. Eng. Shimane Univ. Ser. B, Math. Sci., 32, 25 – 40 (1999).
J. R. Graef, C. Tun¸c, S. ¸Sevgin, Behavior of solutions of non-linear functional Voltera integro-differential equations with multiple delays, Dynam. Systems and Appl., 25, № 1-2, 39 – 46 (2016).
Y. Raffoul, Boundedness in nonlinear functional differential equations with applications to Volterra integro-differential equations, J. Integral Equat. and Appl., 16, № 4, 375 – 388 (2004), https://doi.org/10.1216/jiea/1181075297 DOI: https://doi.org/10.1216/jiea/1181075297
M. Rahman, Integral equations and their applications, WIT Press, Southampton, (2007).
M. Rama Mohana Rao, P. Srinivas, Asymptotic behavior of solutions of Volterra integro-differential equations, Proc. Amer. Math. Soc., 94, № 1, 55 – 60 (1985), https://doi.org/10.2307/2044951 DOI: https://doi.org/10.2307/2044951
C. Tun¸c, A note on the qualitative behaviors of nonlinear Volterra integro-differential equation, J. Egyptian Math. Soc., 24, № 2, 187 – 192 (2016), https://doi.org/10.1016/j.joems.2014.12.010 DOI: https://doi.org/10.1016/j.joems.2014.12.010
C. Tun¸c, New stability and boundedness results to Volterra integro-differential equations with delay, J. Egyptian Math. Soc., 24, № 2, 210 – 213 (2016), https://doi.org/10.1016/j.joems.2015.08.001 DOI: https://doi.org/10.1016/j.joems.2015.08.001
C. Tun¸c, Properties of solutions to Volterra integro-differential equations with delay, Appl. Math. Inf. Sci., 10, № 5, 1775 – 1780 (2016), https://doi.org/10.18576/amis/100518 DOI: https://doi.org/10.18576/amis/100518
C. Tun¸c, Qualitative properties in nonlinear Volterra integro-differential equations with delay, J. Taibah Univ. Sci., 11, № 2, 309 – 314 (2017), , http://dx.doi.org/10.1016/j.jtusci.2015.12.009 DOI: https://doi.org/10.1016/j.jtusci.2015.12.009
C. Tun¸c, On the qualitative behaviors of a functional differential equation of second order, Appl. Appl. Math., 12, № 2, 813 – 842 (2017).
C. Tun¸c, S. A. Mohammed, A remark on the stability and boundedness criteria in retarded Volterra integro-differential equations, J. Egyptian Math. Soc., 25, № 4, 363 – 368 (2017), https://doi.org/10.1016/j.joems.2017.05.001 DOI: https://doi.org/10.1016/j.joems.2017.05.001
C. Tun¸c, O. Tun¸c, On the exponential study of solutions of Volterra integro-differential equations with time lag, Electron. J. Math. Anal. and Appl., 6, № 1, 253 – 265 (2018).
C. Tun¸c, O. Tun¸c, New results on the stability, integrability and boundedness in Volterra integro-differential equations, Bull. Comput. Appl. Math., 6, № 1, 41 – 58 (2018).
C. Tun¸c, O. Tun¸c, On behaviors of functional Volterra integro-differential equations with multiple time-lags, J. Taibah Univ. Sci., 11, № 2, 173 – 179 (2018).
Ke Wang, Uniform asymptotic stability in functional-differential equations with infinite delay, Ann. Different. Equat., 9, № 3, 325 – 335 (1993).
Q. Wang, The stability of a class of functional differential equations with infinite delays, Ann. Different. Equat., 16, № 1, 89 – 97 (2000).
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 DOI: https://doi.org/10.1007/978-3-642-21449-3
This work is licensed under a Creative Commons Attribution 4.0 International License.