Abstract
AbstractIn this paper we introduce the concept of iterated function system consisting of generalized convex contractions. More precisely, given n ∈ ℕ*, an iterated function system consisting of generalized convex contractions on a complete metric space (X; d) is given by a finite family of continuous functions (fi)i ∈I, fi: X → X, having the property that for every ω ∈ λn(I) there exists a family of positive numbers (aω;υ)υ∈Vn(I)such that:x; y ∈ X. Here λn(I) represents the family of words with n letters from I, Vn(I) designates the family of words having at most n - 1 letters from I, while, if ω1= ω1ω2... ωp, by fωwe mean fω1⃘fω2⃘... ⃘ fωp. Denoting such a system by S = ((X; d); n; (fi)i∈I), one can consider the function FS: K(X) → K(X) described by, for all B ∈ K(X), where K(X) means the set of non-empty compact subsets of X. Our main result states that FSis a Picard operator for every iterated function system consisting of generalized convex contractions S.
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