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Invariant approximations for generalized \(I\)-contractions. (English) Zbl 1084.41023
Let \(X\) be a normed vector space. Let \(S\) be a subset of \(X\) and let \({T, I}\) be self-mappings of \(X\). Then \(T\) is called \(I\)-contraction on \(S\) if there exists \({k\in [0,1)}\) such that \({\| Tx-Ty\| \leq k \| Ix-Iy\| }\) for all \({x,y\in S}\). By \({[a,b]}\) we denote a linear segment in \(X\) joining points \({a,b\in X}\). Suppose \(p\) is a point in \(S\) and let \(S\) be \(p\)-star-shaped. The mappings \(T\) and \(I\) are called \(R\)-subweakly commuting on \(S\) if there exists \({R\in (0,\infty)}\) such that \({\| TIx-ITx\| \leq R\cdot dist(Ix,[Tx,p])}\) for all \({x\in S}\). The purpose of the paper under review is to obtain some results on common fixed points for generalized \(I\)-contractions and \(R\)-subweakly commuting maps. As applications, various invariant approximation results are derived.

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