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Common fixed point theorems for commuting $$k$$-uniformly Lipschitzian mappings. (English) Zbl 0985.47045
Let $$(X, d)$$ be a metric space. A selfmap $$T$$ of $$X$$ is said to be k-uniformly Lipschitzian if there exists a constant $$k > 0$$ such that $$d(T^ix, T^iy) \leq k d(x, y)$$ for every natural number $$i$$ and every $$x, y,$$ in $$X$$. With $$(X, d)$$ a complete bounded metric space with a uniform normal structure with constant $$\beta < 1$$, the authors prove that any k-uniformly Lipschitzian selfmap of $$X$$ has a fixed point if $$k^2\beta < 1$$. If $$X$$ has a uniform convexity sructure with constant $$\beta < 1/k^2$$, and $$T_n$$ is a family of commuting k-uniformly Lipschitzian selfmaps of $$X$$, then the authors show that the intersection of the fixed point sets of $$T_n$$ is nonempty and is a k-local retract of $$X$$. A metric space $$(X, d)$$ is called hyperconvex if any family $$\{B(x_i, r_i), i \in I\}$$ of closed balls of $$(X, d)$$ such that $$d(x_i, x_j) \leq r_i + r_j$$, for every $$i, j \in I$$ has nonempty intersection. The final result of the paper is that, if $$(X, d)$$ is a bounded hyperconvex metric space, then any family of commuting k-uniformly Lipschitzian mappings defined on $$X$$ has a common fixed point if $$k < \sqrt 2$$.
##### MSC:
 47H10 Fixed-point theorems 46B20 Geometry and structure of normed linear spaces
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