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The contraction principle for mappings on a metric space with a graph. (English) Zbl 1139.47040
Let \((X,d)\) be a metric space and \(\Delta\) denote the diagonal of \(X \times X\). Let \(G\) be a directed graph such that the set of its vertices coincides with \(X\) and the set \(E(G)\) of its edges contains all loops, i.e., \(E(G) \supseteq \Delta\). A map \(f: X \to X\) is called \(G\)-contraction if it preserves the edges of \(G\), i.e., \((x,y) \in E(G)\) implies \((fx,fy) \in E(G)\) and \(f\) decreases weights of edges of \(G\), i.e., \((x,y) \in E(G)\) implies \(d(fx,fy) \leq \alpha d(x,y)\) for some \(\alpha \in (0,1)\). The author presents some fixed point results for \(G\)-contractions, being a hybrid of the Banach and Knaster–Tarski theorems and generalizing a number of known assertions. As an application, the convergence of successive approximations for some linear operators on Banach spaces is considered.

MSC:
47H10 Fixed-point theorems
05C40 Connectivity
54H25 Fixed-point and coincidence theorems (topological aspects)
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