Selections of an iteration semigroup of linear set-valued functions

Abstract

Let {F t : t ≥ 0} be an iteration semigroup of linear set-valued functions defined on a cone with a finite cone-basis. Under additional assumptions there exists a unique iteration semigroup {f t : t ≥ 0} of continuous linear selections f t of F t for every t ≥ 0. Mathematics Subject Classification (1991). 28B30, 54C60, 39B12.We begin with some notations and definitions. All vector spaces are over R, the set of all real numbers. Let n(X) denote the set of all nonempty subsets of a nonempty set X. If X is a normed space, then c(X) denotes the space of all compact elements of n(X) and cc(X) the family of all convex sets from c(X). Throughout this paper the space cc(X) is endowed with the Hausdorff metric h.We say that a set C ⊂ X is a cone if and only if tC ⊂ C for every t > 0. A linearly independed set E is said to be a basis of C if and only ifλ i e i , n ∈ N, e i ∈ E, λ i ≥ 0, i = 1, . . . , n .

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“…[12]), studied by J. Olko (see e.g. [9,10]) and by Zdun [15]. In [4,5] we introduced and studied a family of set-valued functions which now will be denoted by (A) (see Sect.…”

Recent results on iteration theory: iteration groups and semigroups in the real case

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“…[10]), studied by Olko (see e.g. [11]) and by Zdun in [13]. In [5] we introduced a family of set-valued functions which now will be denoted by (A) (see Sect.…”