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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 \(k < \sqrt 2\).
47H10 Fixed-point theorems
46B20 Geometry and structure of normed linear spaces
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