Fixed point sets through iteration schemes
Let \((f_n)\) be a sequence of selfmaps on a Hausdorff space \(X\), \(s_n=\prod_{i=1}^nf_i\) and \(\mathcal S=(s_n)\). The sequence \(\mathcal S\) is called a fixed point iteration scheme (or simply a scheme) on \(X\) if \(\emptyset\neq F(\mathcal S)=r(\mathcal C(\mathcal S))\), where \(F(\mathcal S)=\bigcap_{f\in\mathcal S}F(f)\), \(F(f)\) is the fixed point set of \(f\), \(\mathcal C(\mathcal S)=\{x\in X:(s_n(x))\text{ converges}\}\), and \(r:\mathcal C(\mathcal S)\to F(\mathcal S)\) is defined by \(r(x)=\lim_{n\to\infty}s_n(x)\) for all \(x\in\mathcal C(\mathcal S)\). The scheme \(\mathcal S\) is said to have a continuous subsequence if there is a subsequence \((s_{c_n})\) of \(\mathcal S\) consisting of continuous maps. The authors call the scheme \(\mathcal S\) virtually stable if every point \(p\in F(\mathcal S)\) is virtually stable, i.e., for each neighborhood \(U\) of \(p\), there exist a neighborhood \(V\) of \(p\) and an increasing sequence \((k_n)\) of positive integers such that \(\prod_{i=j}^{k_n}f_i(V)\subseteq U\) for all \(n\in\mathbb{N}\) and \(j\leq k_n\). They prove the following: if \(X\) is a regular space and \(\mathcal S=(s_n)=(\prod_{i=1}^nf_i)\) is a virtually stable scheme having a continuous subsequence, then the map \(r:\mathcal C(\mathcal S)\to F(\mathcal S)\) defined above is continuous and hence \(F(\mathcal S)\) is a retract of \(\mathcal C(\mathcal S)\). The result is applied to obtain a new criterion for contractibility of the fixed point set of a Suzuki-type generalized nonexpansive selfmap on a compact convex space. Examples are provided to illustrate these results.
scientific article; zbMATH DE number 5960928
scientific article