Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space

Controllability of time-dependent neutral stochastic functional differential equations driven by a fractional Brownian motion

“…Our results are inspired by the one in [8] where the existence and uniqueness of mild solutions to model (1.1) with B = 0, is studied.…”

“…Very recently, neutral stochastic functional differential equations driven by fractional Brownian motion have attracted the interest of many researchers. One can see [6,7,[12][13][14][15] and the references therein. The literature concerning the existence and qualitative properties of solutions of time-dependent functional stochastic differential equations is very restricted and limited to a very few articles.…”

Controllability of neutral stochastic functional integro-differential equations driven by fractional Brownian motion

“…Many studies of the solutions of stochastic equations in an infinite dimensional space with a fBm have been emerged recently, see [1,2,3,4,6,11,13]. The literature related to neutral stochastic partial functional integro-differential equations driven by a fBm is not vast.…”

Neutral stochastic delay partial functional integro-differential equations driven by a fractional Brownian motion

“…The literature related to neutral differential equations of the type (1.1) is not vast. Recently in [1], it was established the existence and the asymptotic behaviour of solutions of the following neutral stochastic differential equation with finite delay    d [u(t) + G(t, u(t − r(t)))] = [Au(t) + F (t, u(t − δ(t)))] dt + σ(t)dB H (t) for t ∈ [0, T ], u 0 (t) = ϕ(t), −τ ≤ t ≤ 0 (1.2) where A generates an analytic semigroup of bounded linear operators (S(t)) t≥0 in a Hilbert space X, and G, F : [0, +∞[×X → X. The goal of this work is to extend this problem to neutral integro-differential type equation and to discuss the existence of solutions for Eq.…”