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The contraction principle for mappings on a metric space with a graph. (English) Zbl 1139.47040
Let $\left(X,d\right)$ be a metric space and ${\Delta }$ denote the diagonal of $X×X$. Let $G$ be a directed graph such that the set of its vertices coincides with $X$ and the set $E\left(G\right)$ of its edges contains all loops, i.e., $E\left(G\right)\supseteq {\Delta }$. A map $f:X\to X$ is called $G$-contraction if it preserves the edges of $G$, i.e., $\left(x,y\right)\in E\left(G\right)$ implies $\left(fx,fy\right)\in E\left(G\right)$ and $f$ decreases weights of edges of $G$, i.e., $\left(x,y\right)\in E\left(G\right)$ implies $d\left(fx,fy\right)\le \alpha d\left(x,y\right)$ for some $\alpha \in \left(0,1\right)$. The author presents some fixed point results for $G$-contractions, being a hybrid of the Banach and Knasterâ€“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 for nonlinear operators on topological linear spaces 05C40 Connectivity 54H25 Fixed-point and coincidence theorems in topological spaces