$(\sigma,\tau,\varphi)$ -cyclic contractions and related iterated function systems

In this paper, we generalize the concept of synchronous/asynchro-nous $ r $-cyclic contraction. More precisely, given a metric space $ (X,d) $, a covering of it consisting of $ p $ closed subsets, a comparison function $ \varphi $, and two permutations of $ \{1,...,p\} $, we consider the so-called $ (\sigma ,\tau ,\varphi ) $-cyclic contractions. For $ p = 2 $, a complete investigation of Picard's iteration method is presented. Based on this particular case, we present adequate requirements for such a contraction to be a (weakly) Picard operator when there exists $ n\in \mathbb{N} $ such that $ \tau = \sigma ^{n} $ or $ \sigma = \tau ^{n} $. Finally, we consider IFSs consisting of this kind of contractions and provide sufficient conditions for the existence and uniqueness of the attractor.