In this note we extend a previous result concerning the existence of global compact attractors for multivalued semiflows in metric spaces.Keywords Multivalued semiflow · Global attractor · Multivalued dynamical systemThe aim of this note is to extend the result proved in [1, Theorem 3] to the case where the semiflow G does not satisfy a continuity property.The statement of Theorem 3 in Melnik and Valero [1] is the following:Theorem 1 Let us consider a complete metric space X. Let G be a pointwise dissipative and asymptotically upper semicompact multivalued semiflow. Suppose that G (t, ·) : X → C (X) is upper semicontinuous for any t ∈ + . If for any B ∈ B (X) there exists T (B) ∈ + such that γ + T(B) (B) ∈ B(X), then G has the compact global attractor . It is minimal among all closed sets attracting each B ∈ B(X).In Remark 7 in Melnik and Valero [1] it is stated that the result remains valid if the map G (t, ·) : X → C (X) is not upper semicontinuous but has closed graph. This statement is misleading, as in such a case we need an extra assumption, namely, that the semiflow G is strict.

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The asymptotic behaviour of some types of retarded differential equations, with both variable and distributed delays, is analysed. In fact, the existence of global attractors is established for different situations: with and without uniqueness, and for both autonomous and non-autonomous cases, using the classical notion of attractor and the recently new concept of pullback one, respectively.

Abstract. We first prove the existence and uniqueness of pullback and random attractors for abstract multi-valued non-autonomous and random dynamical systems. The standard assumption of compactness of these systems can be replaced by the assumption of asymptotic compactness. Then, we apply the abstract theory to handle a random reaction-diffusion equation with memory or delay terms which can be considered on the complete past defined by R − . In particular, we do not assume the uniqueness of solutions of these equations.1. Introduction. The intention of this article is to study the asymptotic behaviour of multi-valued non-autonomous and random dynamical systems. The long-time behaviour of these systems can be expressed by terms like pullback attractor and random attractor. The theories of these attractors are now well established as have been extensively developed over the last one and a half decades (see, e.g. Caraballo et al.

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In this paper we study the structure of the global attractor for a reactiondiffusion equation in which uniqueness of the Cauchy problem is not guarantied. We prove that the global attractor can be characterized using either the unstable manifold of the set of stationary points or the stable one but considering in this last case only solutions in the set of bounded complete trajectories.

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