Long time behavior of fractional impulsive stochastic differential equations with infinite delay

Abstract

This paper is first devoted to the local and global existence of mild solutions for a class of fractional impulsive stochastic differential equations with infinite delay driven by both K-valued Q-cylindrical Brownian motion and fractional Brownian motion with Hurst parameter H ∈ (1/2, 1). A general framework which provides an effective way to prove the continuous dependence of mild solutions on initial value is established under some appropriate assumptions. Furthermore, it is also proved the exponential decay to zero of solutions to fractional stochastic impulsive differential equations with infinite delay.

Well-posedness and dynamics of impulsive fractional stochastic evolution equations with unbounded delay

“…Although the content of this section can be found in several published works (see, e.g. [28]), we prefer to include it in our paper to make it more readable and as much as self-contained as possible.…”

“…Xu and Caraballo analyzed in [28] these previously mentioned issues for the following impulsive fractional stochastic delay differential equation (see [28] for the details on the operators and noise)…”

“…), t = t k , k = 1, 2, · · · , x(t) = φ(t), t ∈ (−∞, 0], (1.2) where D α t is the Caputo fractional derivative of order 0 < α < 1, I 1−α t is the (1−α)-order fractional integral operator, x(·) takes the value in the separable Hilbert sapce H. A : D(A) ⊆ H → H is the infinitesimal generator of an α-order fractional compact and analytic operator T α (t)(t ≥ 0) (the same as the operator T α (t) in [28]). As usual, B(t) and B H Q (t) denote, respectively, a K-valued Q-cylindrical Brownian motion and fractional Brownian motion defined on a filtered complete probability space (Ω, F, {F t } t≥0 , P).…”

“…It is worth noticing that, the nonlinear terms of the right hand side of problem (1.2) have higher regularity because of the integral operator I 1−α t . For this model, by a fractional variation of constants formula (see Definition 3.1), the mild solution to problem (1.2) is only involving the α-order fractional solution operator T α (t) (t ≥ 0) which is compact, so that we are able to answer the open problem given in [28] but for our new model (1.2). We emphasize that the main advantage of our model is that we can extend the results obtained in [28] for the model (1.1) when α ∈ ( 1 2 , 1) to our model (1.2) for α ∈ (0, 1).…”

“…For this model, by a fractional variation of constants formula (see Definition 3.1), the mild solution to problem (1.2) is only involving the α-order fractional solution operator T α (t) (t ≥ 0) which is compact, so that we are able to answer the open problem given in [28] but for our new model (1.2). We emphasize that the main advantage of our model is that we can extend the results obtained in [28] for the model (1.1) when α ∈ ( 1 2 , 1) to our model (1.2) for α ∈ (0, 1). Also, thanks to the good properties of the α-order fractional solution operator (see Lemma 2.11) we are able to prove the existence of attracting sets and provide interesting information about the dynamics of the problem.…”