Cyclic generalized iterated function systems

Abstract

In this article, we introduce the notion of cyclic generalized iterated function system (GIFS), which is a family of functions $f_1,f_2,\dots ,f_M:X^k\to X$, where each $f_i$ is a cyclic generalized $\phi$-contraction (contractive) map on a collection of subsets ${\left\{B_j\right\}}_{j=1}^p$ of a complete metric space $(X,d)$ respectively, and $k,M,p$ are natural numbers. When $B_j,j=1,2,\dots ,p$ are closed subsets of X, we show the existence of attractor of this cyclic GIFS, and investigate its properties. Further, we extend our ideas to cyclic countable GIFS.