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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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##### References:
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