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 .
Abstract. We prove, using the fixed point approach, some stability results for the general linear functional equation. Namely we obtain sufficient conditions for the stability of a wide class of functional equations and control functions. Our results generalize a lot of the well known and recent outcomes concerning stability. In some examples we indicate how our method may be used to check if the particular functional equation is stable and we discuss the optimality of obtained bounding constants.Mathematics Subject Classification. Primary 39B82; Secondary 39B52.
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