A classical result of J. E. Hutchinson [Indiana Univ. Math. J. 30, 713–747 (1981; Zbl 0598.28011)] states that given a complete space \(X\) and a finite collection of strict contractions \(f_i:X\to X\) \((1\leq i \leq N)\), there exists a unique, non-empty subset \(F\subseteq X\) such that \[ F = \bigcup_{i=1}^N f_i(F). \] This set is known as the attractor of the iterated function system (IFS) \(\left\{ f_i \right\}\).
Many generalisations have been made to this result, often at the expense of the uniqueness of the attractor. One such result is that of R. Miculescu et al. [An. Univ. Vest Timiș., Ser. Mat.-Inform. 56, No. 2, 71–80 (2018; Zbl 1513.28014)] of which the present article is a generalization.
Let \(\Phi: \mathbb{R}^+\times\mathbb{R}^+ \to \mathbb{R}^+\) be a function satisfying

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\(\Phi(r,s)\) is non-decreasing in \(r\) and non-increasing in \(s\)

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\(\sum_{i=0}^\infty \Phi(i,t)<\infty\) for all \(t\geq 0\).

Let \(\mathcal{I}\) be a finite index set and write \(\mathcal{I}^*\) for the set of all finite sequences over \(\mathcal{I}\). Let \(f_i\) be a collection of self-maps on a complete metric space \(X\). Write \(f_{\omega}(x) = f_{\omega_1}\circ f_{\omega_2}\circ \dots \circ f_{\omega_n}(x)\), where \(\omega\) is a word of length \(n=|\omega|\). Assume that the IFS \(\left\{ f_i \right\}\) satisfies \[ d\left( f_\omega(x), f_{\omega}\circ f_i(x) \right) \leq \Phi(\gamma |\omega|, d(x, f_i(x)) \] for every \(x\in X\), \(\omega\in\mathcal{I}^*\) and some \(\gamma>0\).
The main result of the article is arguably Theorem 2.1, which states that under these conditions the IFS \(\left\{ f_i \right\}\) has an attractor. The authors investigate properties of the attractor and the induced Hutchinson operator under these and additional conditions. They further provide examples of how these conditions extend on previous assumptions.