## Noncommuting selfmaps and invariant approximations.(English)Zbl 1108.41025

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.

### 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)
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