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.
In this paper we define and study multivalued dynamical processes in Hausdorff topological spaces. Existence theorems for attractors of multivalued processes are proved, their topological properties are studied. The abstract results are applied to a system of phase-field equations without conditions providing uniqueness of solutions and to nonautonomous differential inclusions.
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