Boundedness and asymptotic Behaviour of Solutions of some second-order nonlinear stochastic differential equations with delay
DOI:
https://doi.org/10.22199/issn.0717-6279-5788Keywords:
stochastic delay differential equations, boundedness of solution, asymptotic behaviour of solutions, complete Lyapunov functionalAbstract
This paper considers a certain second-order nonlinear stochastic differential equation with delay. Novel conditions for the existence of solutions that are uniformly bounded and ultimately bounded are obtained. Moreover, we also study the asymptotic behaviour of solutions for the considered equation. We employ Lyapunov’s second method via an appropriate complete Lyapunov functional to achieve these. Obtained results are new, and they improve and complement some existing relatively recent results in the literature. Finally, an example is provided to illustrate the obtained results.
References
A. M. A. Abou-El-Ela, A. I. Sadek and A. M. Mahmoud, “On the stability of solutions for certain second-order stochastic delay differential equations”, Differential Equations and Control Processes, vol. 2, 2015.
A. M. A. Abou-El-Ela, A. I. Sadek, and A. M. Mahmoud and R. O. A Taie, “On the stochastic stability and boundedness of solutions for stochastic delay differential equation of the second-order”, Chinese Journal of Mathematics, vol. 2015, pp. 1-8, 2015. https://doi.org/10.1155/2015/358936
A. M. A. Abou-El-Ela, A. I. Sadek, A. M. Mahmoud, “Asymptotic stability of solutions for a certain nonautonomous second-order stochastic delay differential equation”, Turkish Journal of Mathematics, vol. 41, no. 3, pp. 576-584, 2017. https://doi.org/10.3906/mat-1508-62
A. T. Ademola, S. O. Akindeinde, B. S. Ogundare, M. O. Ogundiran and O. A. Adesina, “On the stability and boundedness of solutions to certain second-order nonlinear stochastic delay differential equations”, Journal of the Nigerian Mathematical Society, vol. 38, no. 2, pp. 185-209, 2019.
A. T. Ademola, S. Moyo, B. S. Ogundare, M. O. Ogundiran and O. A. Adesina, “Stability and boundedness of solutions to a certain second-order nonautonomous stochastic differential equation”, International Journal of Analysis, 2016, Artcle ID 2012315. https://doi.org/10.1155/2016/2012315
A. T. Ademola, B. S. Ogundare, M. O. Ogundiran and O. A. Adesina, “‘Periodicity, stability, and boundedness of solutions to certain second-order delay differential equations”, International Journal of Differential Equations, Article ID 2843709, 2016. https://doi.org/10.1155/2016/2843709
A. T. Ademola, P. O. Arawomo and A. S. Idowu, “Stability, boundedness and periodicity of solutions to certain second-order delay differential equations”, Proyecciones (Antofagasta, On line), vol. 36, no. 2, pp. 257-282, 2017. https://doi.org/10.4067/S0716-09172017000200257
A. T. Ademola, “Boundedness and stability of solutions to certain second-order differential equations”, Differential Equations and Control Processes, no. 3, pp. 38-50, 2015.
O. A. Adesina, A. T. Ademola, M. O. Ogundiran and B. S. Ogundare, “Stability, boundedness and unique global solutions to certain second-order nonlinear stochastic delay differential equations with multiple deviating arguments”, Nonlinear Studies, vol. 26, no. 1, pp. 71-94, 2019.
L. Arnold, Stochastic differential equations: Theory and applications. New York: Willey-Interscience, 1974.
A. V. Balakrishnam, Stochastic differential systems. I: Filtering and control, a function space approach. Lecture notes in economics and mathematical systems, vol. 84. Berlin: Springer, 1973.
T. A. Burton, Volterra Integral and Differential Equations. New York, NY: Academic Press, 1983.
T. A. Burton, Stability and periodic solutions of ordinary and functional differential equations. Mathematics in Science and Engineering, vol. 178. Orlando, FL: Academic Press, 1985.
T. A. Burton and L. Hatvani, “Asymptotic stability of second-order ordinary, functional, and partial differential equations”, Journal of Mathematical Analysis and Applications, vol. 176, no. 1, pp. 261-281, 1993. https://doi.org/10.1006/jmaa.1993.1212
J. K. Hale, Functional differential equations. New York: Springer, 1971.
J. K. Hale, Theory of functional differential equations. Applied Mathematical Science 3. New York: Springer, 1977.
J. K. Hale and L. S. M. Verduyn, Introduction to functional differential equations. New York: Springer, 1993.
L Berezansky and E Braverman, “Explicit conditions of exponential stability for a linear impulsive delay differential equation”, Journal of Mathematical Analysis and Applications, vol. 214, no. 2, pp. 439-458, 1997.
K. Itô, and M. N. Stratonovich, “On stationary solution of a stochastic differential equations”, Journal of Mathematics of Kyoto University, vol. 4, pp. 1-75, 1964. https://doi.org/10.1215/kjm/1250524705
A. F. Ivanov, V. I. Kazmerchuk and A. V. Swishchuk, “Theory stochastic stability and applications of stochastic delay differential equations: a survey of recent results”, Differential Equations and Dynamical Systems, vol. 11, no. 1, pp. 1-43, 2003.
R. I. Kadiev, “Sufficient conditions for stability of linear stochastic system with after effect with respect to a part variables”, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, vol. 44, no. 6, pp. 72-76, 2000.
G. Kellert, G. Kersting, and U. Rosler, “On the asymptotic behavior of solutions of stochastic differential equations”, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 68, pp. 163-189, 1984. https://doi.org/10.1007/BF00531776
P. E. Kloeden, and E. Platen, Numerical solutions of stochastic differential equations. Berlin: Springer, 1998.
E. Kolarova, “An application of stochastic integral equations to electrical networks”, Acta Electrotechnica et informatica, vol. 8, no. 3, pp. 14-17, 2008.
V. B. Kolmanovskii and A. D. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations. Dordrecht: Kluwer Academic Publishers, 1999.
V. B. Kolmanovskii and V. R. Nosov, Stability of Functional Differential Equations. Mathematics in Science and Engineering, vol. 180. Elsevier Science, 1986.
V. B. Kolmanovskii and L. Shaikhet, “Construction of Lyapunov Functionals for stochastic hereditary systems: a survey of some recent results”, Mathematical and Computer Modelling, vol. 36, no. 6, pp. 691-716, 2002. https://doi.org/10.1016/S0895-7177(02)00168-1
R. Liu, and Y. Raffoul, “Boundedness and exponential stability of highly nonlinear stochastic differential equations”, Electronic Journal of Differential Equations, vol. 143, pp. 1-10, 2009.
X. Mao, Stochastic differential equations and their applications. Chichester: Horwood Publishing, 1997.
X. Mao, “Attraction, stability and boundedness for stochastic delay differential equations”, Nonlinear Analysis, vol. 47, pp. 4795-4806, 2001. https://doi.org/10.1016/S0362-546X(01)00591-0
X. Mao, “Some contribution to stochastic asymptotic stability and boundedness via multiple Lyapunov functions”, Journal of Mathematical Analysis and Applications, vol. 260, no. 2, pp. 325-340, 2001. https://doi.org/10.1006/jmaa.2001.7451
B. S. Ogundare, A. T. Ademola, M. O. Ogundiran, and O. A. Adesina, “On the qualitative behaviour of solutions to certain second-order nonlinear differential equation with delay”, Annali dell Universita’ di Ferrara, vol. 2016, doi 10.1007/s11565-016-0262-y
A. Rodkina, and M. Basin, “On delay-dependent stability for a class of nonlinear stochastic delay-differential equations”, Mathematics of Control, Signals, and Systems, vol. 8, no. 2, pp. 187-197, 2006. https://doi.org/10.1007/s00498-006-0163-1
A. F. Yeniçerioğlu, “The behaviour of solutions of second-order delay differential equations”, Journal of Mathematical Analysis and Applications, vol. 332, pp. 1278-1290, 2006. https://doi.org/10.1016/j.jmaa.2006.10.069
A. F. Yeniçerioğlu, “Stability properties of second-order delay differential equations”, Computers and Mathematics with Applications, vol. 56, no. 12, pp. 3109-3117, 2008. https://doi.org/10.1016/j.camwa.2008.07.004
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