Generalized \(I\)-nonexpansive selfmaps and invariant approximations. (English) Zbl 1175.41026

Al-Thagafi, M. A.; Shahzad, Naseer

Acta Math. Sin., Engl. Ser. 24, No. 5, 867-876 (2008).

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 \(x\in X\).

Reviewer: Roshdi Khalil (Amman)

MSC:

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

Keywords:

common fixed point; occasionally weakly compatible map; ultraoccasionally weakly compatible map; generalized \(I\)-nonexpansive map; invariant approximation

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