Fixed point properties of mappings satisfying local contractive conditions (Q2746996)

Let \(X\) be a Banach space, let \(G\) be a connected open subset of X, and let \(T: \overline{G} \to X\) be continuous on \(\overline{G}\) and locally nonexpansive on \(G\), with no fixed points on the boundary of \(G\). The following results are obtained. NEWLINENEWLINENEWLINE1. If \(X\) is strictly convex and \(\text{Fix}(T) \subset G\), then the components of \(\text{Fix}(T)\) are convex. NEWLINENEWLINENEWLINE2. Suppose that \(X\) is either uniformly convex or uniformly smooth, and that \(T\) is a local pointwise contraction on \(G\). Furthermore, suppose that there exists a \(z \in G\) such that \(\|z-T(z) \|< \|x-T(x)\|\) for all \(x \in \partial G\). Then \(T\) has a unique fixed point and \(\{T^n(x)\}\) converges to this fixed point for all \(x \in G\). NEWLINENEWLINENEWLINE3. Suppose that \(X\) is strictly convex whose nonempty closed subsets are densely proximinal. Suppose that \(\overline {\text{Fix}(T)} \subset G\) and \(int(\text{Fix}(T)) \neq \emptyset\). Then \(\text{Fix}(T)\) is a closed convex subset of \(G\) and \(\{T^n(x)\}\) converges to a point of \(\text{Fix}(T)\) for each \(x \in G\).