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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com


Neutral Stochastic Impulsive Integro-Differential Equations Driven by Fractional Brownian Motion and Brownian Motion with Nonlocal Condition

Discontinuity, Nonlinearity, and Complexity 11(3) (2022) 487--500 | DOI:10.5890/DNC.2022.09.010

S. Abinaya$^1$, Sayooj Aby Jose$^{2,3}$, Weerawat Sudsutad$^{4,5}$

$^1$ Department of Mathematics, Rathinam College of Arts and Science, Coimbatore, Tamil Nadu, India

$^2$ Ramanujan Centre for Higher Mathematics, Alagappa University, Karaikudi-630 004, Tamil Nadu, India

$^3$ Department of Mathematics, Alagappa University, Karaikudi-630 004, Tamil Nadu, India

$^4$ Department of General Education, Navamindradhiraj University, Bangkok, Thailand

$^5$ Department of Statistics, Faculty of Science, Ramkhamhaeng University, Bangkok 10240, Thailand

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Abstract

In this paper, we present the existence, uniqueness and asymptotic behaviour of mild solution for neutral stochastic impulsive integro-differential equations driven by fractional Brownian motion and Brownian motion with the Hurst index $H>\frac{1}{2}$ with nonlocal condition. The results are obtained by using Banach fixed point principle in a Hilbert space and the theory of resolvent operator.

References

  1. [1]  Dung, N.T. (2014), Neutral stochastic differential equations driven by a fractional Brownian motion with impulsive effects and varying-time delays, Journal of the Korean Statistical Society, http://dx.doi.org/10.1016/j.jkss.2014.02.003
  2. [2]  Duan, P.J. and Ren, Y. (2018), Solvability and stability for neutral stochastic integro-differential equations driven by fractional Brownian motion with impulses, Mediterranean Journal of Mathematics, 15(207), https://doi.org/ 10.1007/s00009-018-1253-2.
  3. [3]  Abinaya, S. and Jose, S.A. (2019), Neutral impulsive stochastic differential equations driven by fractional Brownian motion with Poisson jumps and nonlocal conditions, Global Journal of Pure and Applied Mathematics, 15(3), 305-321.
  4. [4]  Balachandran, K. and Chandrasekaran, M. (1996), Existence of solutions of a delay differential equation wih nonlocal condition, Indian Journal of Pure and Applied Mathematics, 27, 443-449.
  5. [5]  Balasubramaniam, P., Park, J.Y., and Kumar, A.V.A. (2009), Existence of solutions for semilinear neutral stochastic functional differential equations with nonlocal conditions, Nonlinear Analysis, 71, 1049-1058.
  6. [6] Byszewski, L. (1991), Theorems about the existence and uniqueness of a solution of a semilinear evolution nonlocal cauchy problem, Journal of Mathematical Analysis and Applications, 162, 496-505.
  7. [7] Fu, X. and Ezzinbi, K. (2003), Existence of solutions for neutral functional differential evolution equations with nonlocal conditions, Nonlinear Analysis, 54, 215-227.
  8. [8]  Lv, J.Y. and Yang, X.Y. (2017), Nonlocal fractional stochastic differential equations driven by fractional Brownian motion, Advances in Difference Equations, 198.
  9. [9] Ntouyas, S.K. and Tsamaos, P.Ch. (1997), Global existence for semilinear evolution equations with nonlocal conditions, Journal of Mathematical Analysis and Applications, 210, 679-687.
  10. [10]  Jose, S.A. and Usha, V. (2018), Existence of solutions for random impulsive differential equation with nonlocal conditions, International Journal for Computer Sciences and Engineering, 6(10), 549-554.
  11. [11]  Jose, S.A., Yukunthorn, W., Napoles, J.E., and Leiva, H. (2020), Some existence, uniqueness and stability results of nonlocal random impulsive integro-differential equations, Applied Mathematics-E Notes, (20), 481-492.
  12. [12] Jose, S.A., Tom, A., Abinaya, S., and Yukunthorn, W. (2021), Some characterization results of nonlocal special random impulsive differential evolution equation, Journal of Applied Nonlinear Dynamics, 10(4), 711-723
  13. [13] Ferrante, M. and Rovira, C. (2006), Stochastic delay diferential equations driven by fractional Brownian motion with Hurst parameter $H>\frac{1}{2}$, Bernoulli, 12, 85-100.
  14. [14]  Boufoussi, B. and Hajji, S. (2011), Functional differential equations driven by a fractional Brownian motion, Computers and Mathematics with Applications, 62, 746-754.
  15. [15] Boufoussi, B., Hajji, S., and Lakhel, E. (2011), Functional differential equations in Hilbert spaces driven by a fractional Brownian motion, Afrika Matematika, DOI:10.1007/s13370-011-0028-8.
  16. [16] Caraballo, T., Garrido-Atienza, M.J., and Taniguchi, T. (2011), The existence and exponential behaviour of solutions to stochastic delay evolution equations with frational Brownian motion, Nonlinear Analysis, 74(11), 3611-3684.
  17. [17] Ferrante, M. and Rovira, C. (2010), Convergence of delay differential equations driven by fractional Brownian motion, Journal of Evolution Equations, 10(4), 761-783.
  18. [18]  Jumarie, G. (2005), On the solutions of the stochastic differential equation of exponential growth driven by fractional Brownian motion, Applied Mathematics Letters, 18, 817-826.
  19. [19]  Arthi, G., Ju, H., and Jung, H.(2016), Existence and exponential stability for neutral stochastic integro-differential equations with impulses driven by a fractional Brownian motion, Communications in Nonlinear Science Numerical Simulation, 32, 145-157.
  20. [20] Diop, M., Sakthivel, R., and Ndiaye, A. (2016), Neutral stochastic integro-differential equations driven by a fractional Brownian motion with impulsive effects and time varying Delays, Mediterranean Journal of Mathematics, 13(5), 2425-2442.
  21. [21] Da Prato, G. and Zabckzyk, J. (1992), Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge.
  22. [22]  Mao, X. (1997), Stochastic Differential Equations and their Applications, Horwood Publishing, Chichester.
  23. [23] Pazy, A. (1983), Semigroups of linear operators and applications to partial differential equations, In Applied Mathematical Sciences, 44, New York: Springer-Verlag.
  24. [24]  Grimmer, R.(1982), Resolvent operators for integral equations in a Banach space, Transactions of the American Mathematical Society, 273, 333-349.