Noncommuting selfmaps and invariant approximations. (English) Zbl 1108.41025

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

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 64, No. 12, 2778-2786 (2006).

Let \(D\) be a nonvoid set of a normed space \(X\), \(I\) and \(T\) self maps of \(D\) and suppose that \(q\in\text{Fix}(I)\). If \(D\) is \(q\)-starshaped, let \(T_{k}\) be the operator defined by \(T_{k}(x):=kT(x)+(1-k)q,\) \(k\in [ 0,1]\), let \(C(I,T_{k})\) be the set of coincidence points of \(I\) and \(T_{k}\), and \(C(I,T):=\cup \{C(I,T_{k}):0\leq k\leq 1\}.\) The operators \(I,\) \(T\) are called \(C_{q}\)-commuting if \(ITx=TIx\), for all \(x\in C(I,T)\). Firstly, the authors discuss the generality of this class of operators and give some results concerning the existence of common fixed points of \(C_{q}\)-commuting operators. As application, in sections \(3\) and \(4\), one obtains several invariant approximation results for starshaped, and respectively convex sets.

Reviewer: Costica Mustăţa (Cluj-Napoca)

MSC:

41A65	Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
47H10	Fixed-point theorems
54H25	Fixed-point and coincidence theorems (topological aspects)

Keywords:

best approximation; common fixed point; \(C_q\)-commuting map

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