Asymptotic behavior of neutral stochastic partial functional integro-differential equations driven by a fractional Brownian motion. (English) Zbl 1322.60103

Caraballo, Tomás; Diop, Mamadou Abdoul; Ndiaye, Abdoul Aziz

J. Nonlinear Sci. Appl. 7, No. 6, 407-421 (2014).

Summary: This paper deals with the existence, uniqueness and asymptotic behavior of mild solutions to neutral stochastic delay functional integro-differential equations perturbed by a fractional Brownian motion \(B^H\) with Hurst parameter \(H \in (\frac12, 1)\). The main tools for the existence of a solution is a fixed point theorem and the theory of resolvent operators developed in [R. C. Grimmer, Trans. Am. Math. Soc. 273, 333–349 (1982; Zbl 0493.45015)], while a Gronwall-type lemma plays the key role for the asymptotic behavior. An example is provided to illustrate the results of this work.

MSC:

60H15	Stochastic partial differential equations (aspects of stochastic analysis)
60H20	Stochastic integral equations
60G22	Fractional processes, including fractional Brownian motion
60G15	Gaussian processes

Keywords:

neutral stochastic partial functional integro-differential equations; fractional Brownian motion; mild solutions; resolvent operators; \(C_0\)-semigroup; Wiener process; exponential decay of solutions