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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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