Some existence results for systems of impulsive stochastic differential equations
Abstract
In this paper we study a class of impulsive systems of stochastic differential equations with infinite Brownian motions. Sufficient conditions for the existence and uniqueness of solutions are established by mean of some fixed point theorems in vector Banach spaces. An example is provided to illustrate the theory.
Keywords
stochastic differential equation; Wiener process; impulsive differential equations; matrix convergent to zero; generalized Banach space; fixed point
References
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32. J.J. Nieto, A. Ouahab, and M.A. Slimani, Existence and boundedness of solutions for systems of difference equations with infinite delay, Glas. Mat. Ser. III 53(73) (2018), no. 1, 123–141.
33. A.A. Novikov, The moment inequalities for stochastic integrals, (in Russian), Teor. Verojatnost. i Primenen. 16 (1971), 548–551.
34. A.I. Perov, On the Cauchy problem for a system of ordinary differential equations, (in Russian), Približ. Metod. Rešen. Differencial’. Uravnen. Vyp. 2, (1964), 115–134.
35. R. Precup and A. Viorel, Existence results for systems of nonlinear evolution equations, Int. J. Pure Appl. Math. 47 (2008), no. 2, 199–206.
36. D. Revuz and M. Yor, Continuous Martingales and Brownian Motion. Third edition, Grundlehren der Mathematischen Wissenschaften, 293, Springer-Verlag, Berlin, 1999.
37. I.A. Rus, The theory of a metrical fixed point theorem: theoretical and applicative relevances, Fixed Point Theory 9 (2008), no. 2, 541–559.
38. R. Sakthivel and J. Luo, Asymptotic stability of nonlinear impulsive stochastic differential equations, Statist. Probab. Lett. 79 (2009), no. 9, 1219–1223.
39. A.M. Samoilenko and N.A. Perestyuk, Impulsive Differential Equations, World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, 14, World Scientific Publishing Co., Singapore, 1995.
40. K. Sobczyk, Stochastic Differential Equations. With Applications to Physics and Engineering, Mathematics and its Applications (East European Series), 40, Kluwer Academic Publishers Group, London, 1991.
41. C.P. Tsokos and W.J. Padgett, Random Integral Equations with Applications to Life Sciences and Engineering, Mathematics in Science and Engineering, 108, Academic Press, New York, 1974.
42. A. Viorel, Contributions to the Study of Nonlinear Evolution Equations, Ph.D. Thesis, Babeş-Bolyai University, Department of Mathematics, Cluj-Napoca, 2011.
2. M. Benchohra, J. Henderson, and S.K. Ntouyas, Impulsive Differential Equations and Inclusions, Contemporary Mathematics and Its Applications, 2, Hindawi Publishing Corporation, New York, 2006.
3. H. Berrezoug, J. Henderson, and A. Ouahab, Existence and uniqueness of solutions for a system of impulsive differential equations on the half-line, J. Nonlinear Funct. Anal. 2017 (2017), Article ID 38, 16 pp.
4. A.T. Bharucha-Reid, Random Integral Equations, Mathematics in Science and Engineering, 96, Academic Press, New York, 1972.
5. O. Bolojan-Nica, G. Infante. and R. Precup, Existence results for systems with coupled nonlocal initial conditions, Nonlinear Anal. 94 (2014), 231–242.
6. A. Boudaoui, T. Caraballo, and A. Ouahab, Stochastic differential equations with noninstantaneous impulses driven by a fractional Brownian motion, Discrete Contin. Dyn. Syst. Ser. B 22 (2017), no. 7, 2521–2541.
7. D.L. Burkholder, Martingale transforms, Ann. Math. Statist. 37 (1966), 1494–1504.
8. D.L. Burkholder and R.F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 124 (1970), 249–304.
9. T. Caraballo, C. Ogouyandjou, F.K. Allognissode, and M.A. Diop, Existence and exponential stability for neutral stochastic integro-differential equations with impulses driven by a Rosenblatt process, Discrete Contin. Dyn. Syst. Ser. B 25, (2020), no. 2, 507–528.
10. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1992.
11. B. Davis, On the integrability of the martingale square function, Israel J. Math. 8 (1970), 187–190.
12. D. Gao and J. Li, Existence and mean-square exponential stability of mild solutions for impulsive stochastic partial differential equations with noncompact semigroup, J. Math. Anal. Appl. 484 (2020), no. 1, 123717, 15 pp.
13. T.C. Gard, Introduction to Stochastic Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics, 114, Marcel Dekker, New York, 1988.
14. I.I. Gikhman and A.V. Skorokhod, Stochastic Differential Equations, Ergebnisse der Mathematik und ihrer Grenzgebiete, 72, Springer-Velag, New York, 1972.
15. J.R. Graef, J. Henderson, and A. Ouahab, Impulsive Differential Inclusions. A Fixed Point Approach, De Gruyter Series in Nonlinear Analysis and Applications, 20, De Gruyter, Berlin, 2013.
16. J.R. Graef, J. Henderson, and A. Ouahab, Topological Methods for Differential Equations and Inclusions, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2019.
17. J.R. Graef, H. Kadari, A. Ouahab, and A. Oumansour, Existence results for systems of second-order impulsive differential equations, Acta Math. Univ. Comenian. (N.S.) 88 (2019), no. 1, 51–66.
18. Y. Guo, Q. Zhu, and F. Wang, Stability analysis of impulsive stochastic functional differential equations, Commun. Nonlinear Sci. Numer. Simul. 82 (2020), 105013, 12 pp.
19. A. Halanay and D. Wexler, Teoria Calitativa a Sistemelor Cu Impulsuri, (in Romanian), Editura Academiei Republicii Socialiste România, Bucharest, 1968.
20. J. Henderson, A. Ouahab, and M. Slimani, Existence results for a semilinear system of discrete equations, Int. J. Difference Equ. 12 (2017), no. 2, 235–253.
21. W. Hu and Q. Zhu, Stability analysis of impulsive stochastic delayed differential systems with unbounded delays, Systems Control Lett. 136, (2020), 104606, 7 pp.
22. H. Kadari, J.J. Nieto, A. Ouahab, and A. Oumansour, Existence of solutions for implicit impulsive differential systems with coupled nonlocal conditions, Int. J. Difference Equ. 15 (2020), no. 2, 429–451.
23. V. Lakshmikantham, D.D. Bainov, and P.S. Simeonov, Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics, 6, World Scientific Publishing Co., Singapore, 1989.
24. C. Li, J. Shi, and J. Sun, Stability of impulsive stochastic differential delay systems and its application to impulsive stochastic neural networks, Nonlinear Anal. 74 (2011), no. 10, 3099–3111.
25. J. Liu, X. Liu, and W.C. Xie, Existence and uniqueness results for impulsive hybrid stochastic delay systems, Comm. Appl. Nonlinear Anal. 17 (2010), no. 3, 37–54.
26. M. Liu and K. Wang, On a stochastic logistic equation with impulsive perturbations, Comput. Math. Appl. 63 (2012), no. 5, 871–886.
27. X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Series in Mathematics & Applications, Horwood Publishing Limited, Chichester, 1997.
28. P.W. Millar, Martingale integrals, Trans. Amer. Math. Soc. 133 (1968), 145–166.
29. V.D. Milman and A.D. Myshkis, On the stability of motion in the presence of impulses, (in Russian), Sibirsk. Math. Ž. 1 (1960), 233–237.
30. O. Nica, Initial-value problems for first-order differential systems with general nonlocal conditions, Electron. J. Differential Equations 2012, No. 74, 15 pp.
31. O. Nica and R. Precup, On the nonlocal initial value problem for first order differential systems, Stud. Univ. Babeş-Bolyai Math. 56 (2011), no. 3, 113–125.
32. J.J. Nieto, A. Ouahab, and M.A. Slimani, Existence and boundedness of solutions for systems of difference equations with infinite delay, Glas. Mat. Ser. III 53(73) (2018), no. 1, 123–141.
33. A.A. Novikov, The moment inequalities for stochastic integrals, (in Russian), Teor. Verojatnost. i Primenen. 16 (1971), 548–551.
34. A.I. Perov, On the Cauchy problem for a system of ordinary differential equations, (in Russian), Približ. Metod. Rešen. Differencial’. Uravnen. Vyp. 2, (1964), 115–134.
35. R. Precup and A. Viorel, Existence results for systems of nonlinear evolution equations, Int. J. Pure Appl. Math. 47 (2008), no. 2, 199–206.
36. D. Revuz and M. Yor, Continuous Martingales and Brownian Motion. Third edition, Grundlehren der Mathematischen Wissenschaften, 293, Springer-Verlag, Berlin, 1999.
37. I.A. Rus, The theory of a metrical fixed point theorem: theoretical and applicative relevances, Fixed Point Theory 9 (2008), no. 2, 541–559.
38. R. Sakthivel and J. Luo, Asymptotic stability of nonlinear impulsive stochastic differential equations, Statist. Probab. Lett. 79 (2009), no. 9, 1219–1223.
39. A.M. Samoilenko and N.A. Perestyuk, Impulsive Differential Equations, World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, 14, World Scientific Publishing Co., Singapore, 1995.
40. K. Sobczyk, Stochastic Differential Equations. With Applications to Physics and Engineering, Mathematics and its Applications (East European Series), 40, Kluwer Academic Publishers Group, London, 1991.
41. C.P. Tsokos and W.J. Padgett, Random Integral Equations with Applications to Life Sciences and Engineering, Mathematics in Science and Engineering, 108, Academic Press, New York, 1974.
42. A. Viorel, Contributions to the Study of Nonlinear Evolution Equations, Ph.D. Thesis, Babeş-Bolyai University, Department of Mathematics, Cluj-Napoca, 2011.
MekkiS., BlouhiT., NietoJ. J., & OuahabA. (2021). Some existence results for systems of impulsive stochastic differential equations. Annales Mathematicae Silesianae, 35(2), 260-281. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/13451
Sliman Mekki
Laboratory of Mathematics, Univ Sidi Bel Abbes, Algeria Algeria
Laboratory of Mathematics, Univ Sidi Bel Abbes, Algeria Algeria
Tayeb Blouhi
Laboratory of Mathematics, Univ Sidi Bel Abbes, Algeria Algeria
Laboratory of Mathematics, Univ Sidi Bel Abbes, Algeria Algeria
Juan J. Nieto
Departamento de Análisis Matemático, Instituto de Matemáticas, Universidad de Santiago de Compostela, Spain Spain
Departamento de Análisis Matemático, Instituto de Matemáticas, Universidad de Santiago de Compostela, Spain Spain
Abdelghani Ouahab
agh_ouahab@yahoo.fr
Laboratory of Mathematics, Univ Sidi Bel Abbes, Algeria Algeria
https://orcid.org/0000-0002-4639-2092
Laboratory of Mathematics, Univ Sidi Bel Abbes, Algeria Algeria
https://orcid.org/0000-0002-4639-2092
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