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Generalized $$I$$-nonexpansive selfmaps and invariant approximations. (English) Zbl 1175.41026
Let $$(X,d)$$ be a metric space and $$E$$ be a (closed) subset of $$X$$. Assume $$S,T:X\to X$$ be two maps such that $$E$$ is invariant under both maps. For $$x\in X$$, $$d(x,E)=\inf\{d(x,e): e\in E\}$$. Let $$A(x,E)=\{y\in E: d(x,E)=d(x,y)\}$$. Let $$F(S,T)=\{z\in E:S(z)=T(z)=z\}$$. The authors discuss two problems: (i) When $$F(S,T)$$ is nonempty? (ii) When $$F(S,T)\cap A(x,E)$$ is nonempty, where $$x\in X$$.

##### MSC:
 41A50 Best approximation, Chebyshev systems 47H10 Fixed-point theorems 54H25 Fixed-point and coincidence theorems (topological aspects)
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