Abstract
In this paper, we investigate a class of time-dependent neutral stochastic functional differential equations with finite delay driven by Rosenblatt process in a real separable Hilbert space. We prove the existence of unique mild solution by the well-known Banach fixed point principle. At the end we provide a practical example in order to illustrate the viability of our result.
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References
P. Acquistapace, B. Terreni, A unified approach to abstract linear parabolic equations. Tend. Sem. Mat. Univ. Padova 78, 47–107 (1987)
D. Aoued, S. Baghli, Mild solutions for Perturbed evolution equations with infinite state-dependent delay. Electron. J. Qual. Theory Differ. Equ. 59, 1–24 (2013)
B. Boufoussi, S. Hajji, E. Lakhel, Functional differential equations in Hilbert spaces driven by a fractional Brownian motion. Afrika Matematika 23(2), 173–194 (2011)
T. Caraballo, M.J. Garrido-Atienza, T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion. Nonlinear Anal. 74, 3671–3684 (2011)
G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimension (Cambridge University Press, Cambridge, 1992)
S. Hajji, E. Lakhel, Existence and uniqueness of mild solutions to neutral SFDEs driven by a fractional Brownian motion with non-Lipschitz coefficients. J. Numer. Math. Stochast. 7(1), 14–29 (2015)
E. Lakhel, Exponential stability for stochastic neutral functional differential equations driven by Rosenblatt process with delay and Poisson jumps. Random Oper. Stoch. Equ. 24(2), 113–127 (2016)
N.N. Leonenko, V.V. Ahn, Rate of convergence to the Rosenblatt distribution for additive functionals of stochastic processes with long-range dependence. J. Appl. Math. Stoch. Anal. 14, 27–46 (2001)
M. Maejima, C.A. Tudor, On the distribution of the Rosenblatt process. Statist. Probab. Lett. 83, 1490–1495 (2013)
M. Maejima, C.A. Tudor, Wiener integrals with respect to the Hermite process and a non central limit theorem. Stoch. Anal. Appl. 25, 1043–1056 (2007)
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences (Springer, New York, 1983)
V. Pipiras, M.S. Taqqu, Integration questions related to the fractional Brownian motion. Probab. Theory Relat. Fields 118, 251–281 (2001)
M. Rosenblatt, Independence and dependence, in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 2: Contributions to Probability Theory, pp. 431-443. (University of California Press, Berkeley, CA, 1961)
M.S. Taqqu, Weak convergence to fractional Brownian motion and the Rosenblatt Process. Z. Wahr. Geb. 31, 287–302 (1975)
M. Taqqu, Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrscheinlichkeitstheor, Verw. Geb., 50, 53–83 (1979)
C.A. Tudor, Analysis of the Rosenblatt process. ESAIM Probab. Statist. 12, 230–257 (2008)
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Lakhel, E., Tlidi, A. (2019). Time-Dependent Neutral Stochastic Delay Partial Differential Equations Driven by Rosenblatt Process in Hilbert Space. In: Melliani, S., Castillo, O. (eds) Recent Advances in Intuitionistic Fuzzy Logic Systems. Studies in Fuzziness and Soft Computing, vol 372. Springer, Cham. https://doi.org/10.1007/978-3-030-02155-9_24
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DOI: https://doi.org/10.1007/978-3-030-02155-9_24
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