Qualitative behaviour of local non-Lipschitz stochastic integrodifferential system with Rosenblatt process and infinite delay

Document Type : Research Paper

Authors

1 Numerical Analysis and Computer Science Laboratory, Department of Mathematics, Gaston Berger University of Saint-Louis, UFR SAT, B.P:234, Saint-Louis, Senegal

2 Department of Mathematics, PSG College of Arts and Science, Coimbatore, 641 046, India

3 Department of Mathematics, PSG college of Arts and Science, Coimbatore, 641 014, India

4 Universit´e Cadi Ayyad, Facult´e des Sciences Semlalia D´epartement de Math´ematiques B.P. 2390 Marrakech, Morocco

10.22075/ijnaa.2023.29778.4255

Abstract

The objective of this paper is to investigate the existence and uniqueness of mild solutions for stochastic integrodifferential evolution equations in Hilbert spaces with infinite delay and a Rosenblatt Process. The main results of this discussion are provided by Grimmer's resolvent operator theory and stochastic analysis. The theory is demonstrated with an example.

Keywords

[1] B. Bayour and D.F.M. Torres, Existence of solution to a local fractional nonlinear differential equation, J. Comput. Appl. Math. 312 (2017), 127–133.
[2] B. Boufoussi and S. Hajji, Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space, Statist. Probab. Lett. 82 (2012), 1549–1558.
[3] T. Caraballo, M.J. Garrido-Atienza, and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal.: Theory Meth. Appl. 74 (2011), no. 11, 3671–3684.
[4] 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 Continuous Dyn. Syst.-B 25 (2020), no. 2, 507.
[5] J. Deng and S. Wang, Existence of solutions of nonlocal Cauchy problem for some fractional abstract differential equation, Appl. Math. Lett. 55 (2016), 42–48.
[6] K. Dhanalakshmi and P. Balasubramaniam, Stability result of higher-order fractional neutral stochastic differential system with infinite delay driven by Poisson jumps and Rosenblatt process, Stochastic Anal. Appl. 38 (2020), no. 2, 352—372.
[7] M.A. Diop, K. Ezzinbi, and M. Lo, Existence and exponential stability for some stochastic neutral partial functional integrodifferential equations, Random Oper. Stoch. Equ. 22 (2014), 73–83.
[8] R. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Amer. Math. Soc. 273 (1982), no. 1, 333–349.
[9] E. Hernandez, Existence results for partial neutral functional integrodifferential equations with unbounded delay, J. Math. Anal. Appl. 292 (2004), 194–210.
[10] N.N. Leonenko and V.V. Anh, Rate of convergence to the Rosenblatt distribution for additive functionals of stochastic processes with long-range dependence, J. Appl. Math. Stoch. Anal. 14 (2001), no. 1, 27–46.
[11] M. Maejima and C.A. Tudor, Selfsimilar processes with stationary increments in the second wiener chaos, Probab. Math. Statist. 32 (2012), no. 1, 167–186.
[12] B. Maslowski and J. Pospisil Ergodicity and, and parameter estimates for infinite-dimensional fractional Ornstein-Uhlenbeck process, Appl. Math. Optim. 57 (2008), 401–429.
[13] B. Maslowski and D. Nualart, Evolution equations driven by a fractional Brownian motion, J. Funct. Anal. 202 (2003), 277-–305.
[14] J.Y. Park, K. Balachandran, and N. Annapoorani, Existence results for impulsive neutral functional integrodifferential equations with infinite delay, Nonlinear Anal. 71 (2009), 3152–3162.
[15] C. Parthasarathy, A. Vinodkumar, and M. Mallika Arjunan, and Uniqueness and existence, and stability of neutral stochastic functional integro-differential evolution with infinite delay, Int. J. Comput. Appl. 65 (2013), no. 15, 0975–8887.
[16] K. Ramkumar, A.A. Gbaguidi, C. Ogouyandjou, and M. A. Diop, Existence and uniqueness of mild solutions of stochastic partial integro-differential impulsive equations with infinite delay via resolvent operator, Adv. Dyn. Syst. Appl. 14 (2019), no. 1, 83–118.
[17] Y. Ren, X. Cheng, and R. Sakthivel, On time-dependent stochastic evolution equations driven by fractional Brownian motion in a Hilbert space with finite delay, Math. Meth. Appl. Sci. 37 (2014), no. 14, 2177–2184.
[18] M. Rosenblatt, Independence and dependence, Proc. 4th Berkeley Symp. Math. Statist. Prob., 1961, pp. 431–443.
[19] G.J. Shen and Y. Ren, Neutral stochastic partial differential equations with delay driven by Rosenblatt process in a Hilbert space, J. Korean Statist. Soc.44 (2015), 123–133.
[20] G.J. Shen, Y. Ren, and Li M. Controllability and, and stability of fractional stochastic functional systems driven by Rosenblatt process, Collect. Math. 71 (2020), 63–82.
[21] T. Taniguchi, Successive approximations to solutions of stochastic differential equations, J. Differ. Equ. 96 (1992), 152–169.
[22] T. Taniguchi, The existence and uniqueness of energy solutions to local non-Lipschitz stochastic evolution equations, J. Math. Anal. Appl. 360 (2009), 245–253.
[23] M. Taqqu, Weak convergence to fractional Brownian motion and to the Rosenblatt process, Adv. Appl. Probab. 7 (1975), no. 2, 249–249.
[24] C.A. Tudor, Analysis of the Rosenblatt process, ESAIM: Probab. Statist. 12 (2008), 230–257.

Articles in Press, Corrected Proof
Available Online from 15 January 2024
  • Receive Date: 30 January 2023
  • Accept Date: 26 November 2023