1,170
Views
19
CrossRef citations to date
0
Altmetric
Research Articles

On the asymptotic stability of solutions of stochastic differential delay equations of second order

& ORCID Icon
Pages 875-882 | Received 26 May 2019, Accepted 01 Aug 2019, Published online: 12 Aug 2019

References

  • Abou-El-Ela AMA, Sadek AI, Mahmoud AM. On the stability of solutions for certain second-order stochastic delay differential equations. Differ Uravn Protsessy Upr. 2015;2:1–13.
  • Abou-El-Ela AMA, Sadek AR, Mahmoud AM, et al. Asymptotic stability of solutions for a certain non-autonomous second-order stochastic delay differential equation. Turkish J Math. 2017;41(3):576–584.
  • Abou-El-Ela AMA, Sadek AI, Mahmoud AM, et al. On the stochastic stability and boundedness of solutions for stochastic delay differential equation of the second order. Chin J Math. 2015: 1–8. Art. ID 358936.
  • Ademola AT, Moyo S, Ogundare BS, et al. Stability and boundedness of solutions to a certain second order nonautonomous stochastic differential equation. Int J Anal. 2016: 1–11. Art. ID 2012315.
  • Bao J, Yin G, Yuan C. Asymptotic analysis for functional stochastic differential equations. Springer Briefs in Mathematics. Cham: Springer; 2016.
  • Burton TA. Stability and periodic solutions of ordinary and functional differential equations. Corrected version of the 1985 original. Mineola (NY): Dover Publications, Inc; 2005.
  • Burton TA. Stability by fixed point theory for functional differential equations. New York (NY): Dover Publications; 2006.
  • Hale JK, Verduyn Lunel SM. Introduction to functional-differential equations. applied Mathematical sciences, 99. New York (NY): Springer-Verlag; 1993.
  • Huang L, Deng F. Razumikhin-type theorems on stability of neutral stochastic functional differential equations. IEEE Trans Automat Control. 2008;53(7):1718–1723.
  • Krasovskii NN. Stability of motion. Applications of Lyapunov's second method to differential systems and equations with delay. Translated by J. L. Brenner. Stanford (CA): Stanford University Press; 1963.
  • Ladde AG, Ladde GS. An introduction to differential equations. Vol. 2. stochastic modeling, methods and analysis. Hackensack (NJ): World Scientific Publishing; 2013.
  • Ladde GS, Wu L. Nonlinear stochastic differential equations: ARIMA models and applications. Proc Neural Parallel Sci Comput. 2010;4:236–240.
  • Lei J, Mackey MC. Stochastic differential delay equation, moment stability, and application to hematopoietic stem cell regulation system. SIAM J Appl Math. 2006/07;67(2):387–407.
  • Li M, Deng F, Mao X. Basic theory and stability analysis for neutral stochastic functional differential equations with pure jumps. Sci China Inf Sci. 2019; 62 (1):1–15; 012204.
  • Liu K. Stability of infinite dimensional stochastic differential equations with applications. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 135. Boca Raton, FL: Chapman & Hall/CRC; 2006.
  • Mao X. LaSalle-type theorems for stochastic differential delay equations. J Math Anal Appl. 1999;236(2):350–369.
  • Mao X. Attraction, stability and boundedness for stochastic differential delay equations. Nonlinear Anal. 2001;47(7):4795–4806.
  • Mao X. Stochastic differential equations and applications. Second edition. Chichester: Horwood Publishing Limited; 2008.
  • Mao X, Shah A. Exponential stability of stochastic differential delay equations. Stochastics Stochastics Rep. 1997;60(1–2):135–153.
  • Mao X, Yuan C, Zou J. Stochastic differential delay equations of population dynamics. J Math Anal Appl. 2005;304(1):296–320.
  • Mohammed SEA. Stochastic functional differential equations. Research Notes in Mathematics, 99. Boston (MA): Pitman (Advanced Publishing Program); 1984.
  • Pedjeu JC, Ladde GS. Stochastic fractional differential equations: modeling, method and analysis. Chaos Solitons Fractals. 2012;45(3):279–293.
  • Rong HW, Fang T. Asymptotic stability of second-order linear stochastic differential equations. Chinese J Appl Mech. 1996;13(3):72–78. VI. Chinese.
  • Ruan D, Xu L, Luo J. Stability of hybrid stochastic functional differential equations. Appl Math Comput. 2019;346:832–841.
  • Shaikhet L. Lyapunov functionals and stability of stochastic functional differential equations. Cham: Springer; 2013.
  • Smith H. An introduction to delay differential equations with applications to the life sciences, Texts in Applied Mathematics, 57. New York (NY): Springer; 2011.
  • Sumafov MM. Construction of Ljapunov functions for some second-order nonlinear stochastic differential equations and questions of stability. Differentsial’nye Uravneniya. 1981;17(6):1143–1145. 1152. Russian.
  • Tunç C. On the stability and boundedness of solutions of a class of Lienard equations with multiple deviating arguments. Vietnam J Math. 2011;39(2):177–190.
  • Tunç C. Uniformly stability and boundedness of solutions of second order nonlinear delay differential equations. Appl Comput Math. 2011;10(3):449–462.
  • Tunç C. Stability and uniform boundedness results for non-autonomous Lienard-type equations with a variable deviating argument. Acta Math Vietnam. 2012;37(3):311–325.
  • Tunç C. Stability to vector Lienard equation with constant deviating argument. Nonlinear Dynam. 2013;73(3):1245–1251.
  • Tunç 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. 2006;30(2):213–221.
  • Tunç C. Stability and boundedness of solutions of non-autonomous differential equations of second order. J Comput Anal Appl. 2011;13(6):1067–1074.
  • Tunç C. Some new stability and boundedness results of solutions of Lienard type equations with deviating argument. Nonlinear Anal Hybrid Syst. 2010;4(1):85–91.
  • Tunç C. A note on boundedness of solutions to a class of non-autonomous differential equations of second order. Appl Anal Discrete Math. 2010;4(2):361–372.
  • Tunç C. New stability and boundedness results of Lienard type equations with multiple deviating arguments. Izv Nats Akad Nauk Armenii Mat. 2010;45(4):47–56.
  • Tunç C. Boundedness results for solutions of certain nonlinear differential equations of second order. J Indones Math Soc. 2010;16(2):115–128.
  • Tunç C. Stability and boundedness of solutions of non-autonomous differential equations of second order. J Comput Anal Appl. 2011;13(6):1067–1074.
  • Tunç C. A note on the bounded solutions. Appl Math Inf Sci. 2014;8(1):393–399.
  • Tunç C, Çinar I. On the existence of periodic solutions to nonlinear differential equations of second order. Differ Uravn Protsessy Upr. 2008;3:20–25.
  • Tunç C, Tunç O. On behaviors of functional Volterra integro-differential equations with multiple time-lags. J Taibah Univ Sci. 2018;12 (2):173–179.
  • Tunç C. On the qualitative behaviors of a functional differential equation of second order. Appl Appl Math. 2017;12(2):813–842.
  • Tunç C. On the properties of solutions for a system of non-linear differential equations of second order. Int J Math Comput Sci. 2019;14(2):519–534.
  • Tunç C. Qualitative properties in nonlinear Volterra integro-differential equations with delay. J Taibah Univ Sci. 2017;11(2):309–314.
  • Tunç C, Ayhan T. On the asymptotic behavior of solutions to nonlinear differential equations of the second order. Comment Math. 2015;55(1):1–8.
  • Tunc C, Dinç Y. Qualitative properties of certain non-linear differential systems of second order. J Taibah Univ Sci. 2017;11(2):359–366.
  • Tunç C, Tunç E. On the asymptotic behavior of solutions of certain second-order differential equations. J Franklin Inst. 2007;344(5):391–398.
  • Tunç C, Şevli H. Stability and boundedness properties of certain second-order differential equations. J Franklin Inst. 344 2007;344(5):399–405.
  • Tunç C, Erdur S. New qualitative results for solutions of functional differential equations of second order. Discrete Dyn Nat Soc. 2018; 1–13. Art. ID 3151742.
  • Tunç C, Tunç O. A note on certain qualitative properties of a second order linear differential system. Appl Math Inf Sci. 2015;9(2):953–956.
  • Tunç C, Tunç O. On the boundedness and integration of non-oscillatory solutions of certain linear differential equations of second order. J Adv Res. 2016;7(1):165–168.
  • Tunç C, Tunç O. A note on the stability and boundedness of solutions to non-linear differential systems of second order. J Assoc Arab Univ Basic Appl Sci. 2017;24:169–175.
  • Tunç C, Tunç O. Qualitative analysis for a variable delay system of differential equations of second order. J Taibah Univ Sci. 2019;13(1): 13 468–477.
  • Tunç C, Yazgan R. Existence of periodic solutions to multidelay functional differential equations of second order. Abstr Appl Anal. 2013: 1–5; Art. ID 968541.
  • Wu RQ, Mao X. Existence and uniqueness of the solutions of stochastic differential equations. Stochastics. 1983;11(1–2):19–32.
  • Golmankhaneh AK, Neda AP, Baleanu D. Mean square solutions of second-order random differential equations by using homotopy analysis method. Rom Rep Phys. 2013;65(2):350–362.
  • Sakthivel R, Ren Y, Kim H. Asymptotic stability of second-order neutral stochastic differential equations. J Math Phys. 2010;51(5):1–9; Art. ID 052701.
  • Verriest EI, Florchinger P. Stability of stochastic systems with uncertain time delays. Syst Control Lett. 1995;24(1):41–47.