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Flavian Georgescu

Abstract

The concept of generalized convex contraction was introduced and studied by V. Istrăţescu and the notion of b-metric space was introduced by I. A. Bakhtin and S. Czerwik. In this paper we combine these two elements by studying iterated function systems consisting of generalized convex contractions on the framework of b-metric spaces. More precisely we prove the existence and uniqueness of the attractor of such a system providing in this way a generalization of Istrăţescu’s convex contractions fixed point theorem in the setting of complete strong b-metric spaces.

Open access

Flavian Georgescu

Abstract

In 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 fω we mean fω1 ⃘fω2 ⃘... ⃘ 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 FS is a Picard operator for every iterated function system consisting of generalized convex contractions S.