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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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