Global attractors for multivalued random dynamical systems

On the Relationship Between Solutions of Stochastic and Random Differential Inclusions

“…As a consequence, all solutions of the stochastic inclusion with the initial condition in a bounded set of the phase space converge uniformly (in the sense of pullback attraction) to this attractor in the Hausdorff semidistance. This justifies the interest of the results of [4,5,6], since they give some information on the asymptotic behaviour of the solutions of the stochastic inclusion.…”

“…In our previous works Caraballo et al [4,5,6], the concept of multivalued random dynamical system (MRDS) has been introduced and some applications to stochastic differential inclusions have been considered. In fact, a reaction-diffusion inclusion perturbed by additive and multiplicative noise is considered.…”

“…However, there exits a result due to Da Prato and Frankowska [7] which guarantees the existence of stochastic solutions for a more general noise than the ones considered in [4,5,6].…”

“…It is shown in [4,5,6] that the random inclusion generates a perfect cocycle having a compact global random attractor. As a consequence, all solutions of the stochastic inclusion with the initial condition in a bounded set of the phase space converge uniformly (in the sense of pullback attraction) to this attractor in the Hausdorff semidistance.…”

Global attractor for a non-autonomous integro-differential equation in materials with memory

“…The next lemma can be proved in a similar way as in [6,Proposition 4] Lemma 14 U defined by (26) satisfies the strict process property U(t, τ, ψ) = U(t, s, U(s, τ, ψ)) for all τ ≤ s ≤ t and ψ ∈ H. Hence, U is a strict MNDS.…”