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Compatible maps and invariant approximations. (English) Zbl 1110.54024
The authors prove existence theorems for invariant best approximation of compatible maps which are based on common fixed point theorems for noncommuting maps. The basic result can be described as follows: Let $$M$$ be a nonempty subset in a metric space $$(X,d)$$ and $$f,g:M\to M$$ maps such that $$gf(x)=fg(x)$$ whenever $$f(x)=g(x)$$. Assume that $$\overline{g(M)}$$ is complete and contained in $$f(M)$$ and that there exists a $$k\in[0,1)$$ such that $$d(gx,gy)\leq k\max\{d(fx,fy), d(fx,gx), d(fy,gy), d(fx,gy), d(fy,gx)\}$$. Then there is a point which is the unique coincidence point of $$f$$ and $$g$$ and this point is the unique common fixed point of both maps.

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects) 47H10 Fixed-point theorems
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