This paper deals with a class of stochastic delay differential equations (SDDEs) of second order with multiple delays. Here, two main and novel results are proved on stochastic asymptotic stability and stochastic boundedness of solutions of the considered SDDEs. In the proofs of results, the Lyapunov-Krasovskii functional (LKF) method is used as the main tool. A comparison between our results and those are available in the literature shows that the main results of this paper have new contributions to the related ones in the current literature. Two numerical examples are given to show the applications of the given results.
[1] 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, Differ. Uravn. Protsessy Upr., 2 (2015), 1-13.
[2] A. M. A. Abou-El-Ela, A. I. Sadek, 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, Chin.J. Math. (N.Y.) 2015, Art. ID 358936, 8 PP.
[3] A. M. A. Abou-El-Ela, A. R. Sadek, A. M. Mahmoud, and E.S. Farghaly, Asymptotic stability of solutions for a certain non-autonomous second-order stochastic delay differential equation, Turkish J. Math., 41(3) (2017), 576-584.
[4] A.T. Ademola, S. O. Akindeinde, M. O. Ogundiran, and O. A. Adesina, On the stability and boundedness of solutions to certain second order nonlinear stochastic delay differential equations, J. Nigerian Math. Soc., 38(2) (2019), 185-209.
[5] 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, Int. J. Anal. 2016, Art. ID 2012315, 11 PP.
[6] O. A. Adesina, A. T. Ademola, M. O. Ogundiran, and B. S. Ogundare, Stability, boundedness and unique globel so- lutions to certain second order nonlinear stochastic delay differential equations with multiple deviating arguments, Nonlinear Stud. 26(1) (2019), 71-94.
[7] L. Arnold, Stochastic differential equations: Theory and applications, Jhon Wiley & Sons, 1974.
[8] J. Bao, G. Yin, and C. Yuan, Asymptotic analysis for functional stochastic differential equation, Springer Bringer in Mathematics. Springer, Cham, 2016.
[9] L. Huang and F. Deng, Razumikhin-type theorems on stability of neutral stochascit funcctional differential equa- tions. IEEE Trans. Automat. Control, 7 (2008), 1718-1723.
[10] A.G. Ladde and G. S. Ladde, An introduction to differentianl equations.Vol.2.Stochastic modeling, methods and analysis, World Scientific Publishing Co. Pte.Ltd., Hackensack, NJ, 2013.
[11] J. Lei and M.C. Mackey, Stochastic differential delay equation, moment stability, and application to hematopoietic stem cell regulation system. SIAM J.Appl.Math., 2 (2006/07), 387-407.
[12] K. Liu, Stability of infinite dimensional stochastic differential equations with applications, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 135. Chapman & Hall/CRC, Boca Raton, FL, 2006.
[13] A. M. Mahmoud and C. Tun¸c, Stability and ultimate boundedness for non-autonomous system with variable delay, Appl. Math. Inf. Sci., 3 (2020), 431-440.
[14] X. Mao, Some contributions to stochastic asymptotic stability and boundedness via multiple Lyapunov functions. J. Math. Anal. Appl., 2 (2001), 325-340.
[15] X. Mao, Attraction, stability and boundedness for stochastic differential delay equations. Proceedings of the Third World Congress of Nonlinear Analysts, Part 7 (Catania, 2000). Nonlinear Anal., 7 (2001), 4795-4806.
[16] X. Mao, Stochastic differential equations and applications, Second edition. Horwood Publishing Limited, Chich-ester, 2008.
[17] S. E. A. Mohammed, Stochastic functional differential equations, Research Notes in Mathematics, 99. Pitman (Advanced Publishing Program), Boston, MA, 1984.
[18] A. M. Mahmoud and C. Tun¸c, Asymptotic stability of solutions for a kind of third-order stochastic differential equations with delays, Miscolc Math. Notes, 1(2019), 381-393.
[19] P. H. A. Ngoc, Explicit criteria for mean square exponential stability of stochastic linear differential equations with distributed delays. Wietnam J. Math., 1 (2020), 159-169.
[20] H. W. Rong and T. Fang, Asymptotic stability of second-order linear stochastic differential equations, Chinese J. Appl. Mech., 3 (1996), 72-78.
[21] L. Shaikhet, Lyapunov functionals and stability of stochastic functional differential equations, Springer, Cham, 2013.
[22] C. Tun¸c, Some new stability and boundedness results on the solutions of the nonlinear vector differential equations of second order, Iran. J. Sci. Technol. Trans. A Sci., 2 (2006), 213-221.
[23] C. Tun¸c,Boundedness results for solutions of certain nonlinear differential equations of second order, J. Indones. Math. Soc., 2 (2010), 115-126.
[24] C. Tun¸c, Stability and boundedness of solutions of nonautonomous differential equations of second order, J. Com- put. Anal. Appl., 6 (2011), 1067-1074.
[25] C. Tun¸c, Stability and uniform boundedness results for non-autonomous Lineard-type equations with a variable deviating argument, Acta Math. Vietnam., 3 (2012), 311-325.
[26] C. Tun¸c, On the stability and boundedness of solutions of a class of Lienard equations with multiple deviating arguments, Vietnam J. Math., 2 (2011), 177-190.
[27] C. Tun¸c, A note on boudedness of solutions to a class of non-autonomous differential equations of second order, Appl. Anal. Discrete Math. , 4(2) (2010), 361-372.
[28] C. Tun¸c, New stability and boundedness results of Lienard type equations with multiple deviating arguments, Izv. Nats. Akad. Nauk Armenii Mat. 45 (2010), no. 4, 47-56; reprinted in J. Contemp. Math. Anal., 4 (2010), 214-220.
[29] C. Tun¸c, Stability to vector Lienard equation with constand deviating argument, Nonlinear Dynam., 3 (2013), 1245-1251.
[30] C. Tun¸c, A note on the bounded solutions to xII + c(t, x, xI) + b(t)f (x) = q(t), Appl. Math. Inf. Sci., 8(1) (2014), 393-399.
[31] C. Tun¸c, Some new stability and boundedness results of solutions of Lienard type equations with deviating argu- ment, Nonlinear Anal. Hybrid Syst., 4(1) (2010), 85-91.
[32] C. Tun¸c, Stability and boundedness of solutions of nonautonomous differential equations of second order, J. Com- put. Anal. Appl. 13, 6 (2011), 1067-1074.
[33] C. Tun¸c, Uniformly stability and boundedness of solutions of second order nonlinear delay differential equations, Appl. Comput. Math. 10, 3 (2011), 449-462.
[34] C. Tun¸c, On the properties of solutions for a system of non-linear differential equations of second order, Int. J. Math. Comput. Sci.,2 (2019), 519-534.
[35] C. Tun¸c, On the qualitative behaviors of a functional differential equation of second order, Appl. Appl. Math., 2 (2017), 813-842.
[36] C. Tun¸c and T. Ayhan, On the asymptotic behavior of solutions to nonlinear differential equations of the second order, Comment. Math., 1 (2015), 1-8.
[37] C. Tun¸c and E. Tun¸c, On the asymptotic behavior of solutions of certain second-order differential equations, J. Franklin Inst., 5 (2007), 391-398.
[38] C. Tun¸c and Y. Din¸c, Qualitative properties of certain nonlinear differential systems of second order, J. Taibah Univ. Sci., 11(2) (2017), 359-366.
[39] C. Tun¸c, H. S¸evli, Stability and boudedness properties of certain second-order differential equations, J. Franklin Inst. 344, 5 (2007), 399-405.
[40] C. Tun¸c and S. Erdur, New qualitative results for solutions of functional differential equations of second order, Discrete Dyn. Nat. Soc., (2018), Art. ID 3151742, 13 pp.
[41] C. Tun¸c and O. Tun¸c, On the asymptotic stability of solutions of stochastic differential delay equations of second order, J. Taibah Univ. Sci. 13, 1 (2019), 875-882.
[42] C. Tun¸c and O. Tun¸c, On the boundedness and integration of non-oscillatory solutions of certain linear differential equations of second order, J. Adv. Res., 7(1) (2016), 165-168.
[43] C. Tun¸c and O. Tun¸c, A note on the stability and boundedness of solutions to non-linear differential systems of second order, J. Assoc. Arab. Univ. Basic. Appl. Sci., 24 (2017), 169-175.
[44] C. Tun¸c and O. Tun¸c, A note on certain qualitative properties of a second order linear differential system, Appl. Math. Inf. Sci., 9(2) (2015), 953-956.
[45] R. Q. Wu and X. Mao, Existence and uniqueness of the solutions of stochastic differential equations, Stochastics, 11(1-2) (1983), 19-32.
[46] E. I. Verriest and P. Florchinger, Stability of stochastic systems with uncertain time delays, Syst Control Lett., 24(1) (1995), 41-47.
Tunc, C., & Oktan, Z. (2024). Improved new qualitative results on stochastic delay differential equations of second order. Computational Methods for Differential Equations, 12(1), 67-76. doi: 10.22034/cmde.2022.52821.2229
MLA
Cemil Tunc; Zozan Oktan. "Improved new qualitative results on stochastic delay differential equations of second order". Computational Methods for Differential Equations, 12, 1, 2024, 67-76. doi: 10.22034/cmde.2022.52821.2229
HARVARD
Tunc, C., Oktan, Z. (2024). 'Improved new qualitative results on stochastic delay differential equations of second order', Computational Methods for Differential Equations, 12(1), pp. 67-76. doi: 10.22034/cmde.2022.52821.2229
VANCOUVER
Tunc, C., Oktan, Z. Improved new qualitative results on stochastic delay differential equations of second order. Computational Methods for Differential Equations, 2024; 12(1): 67-76. doi: 10.22034/cmde.2022.52821.2229
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