IFSs consisting of generalized convex contractions

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 (f_i)_{i ∈I} , f_i : X → X, having the property that for every ω ∈ λ_n(I) there exists a family of positive numbers (a_ω;υ)_υ∈Vn(I) such that:

x; y ∈ X. Here λ_n(I) represents the family of words with n letters from I, V_n(I) designates the family of words having at most n - 1 letters from I, while, if ω₁ = ω₁ω₂ ... ω_p, by f_ω we mean f_ω1 ⃘f_ω2 ⃘... ⃘ f_ωp. Denoting such a system by S = ((X; d); n; (f_i)_i∈I), one can consider the function F_S : 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 F_S is a Picard operator for every iterated function system consisting of generalized convex contractions S.