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Common fixed points under contractive conditions in symmetric spaces. (English) Zbl 1056.47036

This article deals with fixed points of mappings in symmetric spaces, i.e., pairs \((X,d)\) where \(X\) is a set and \(d: \;X \times X \to [0,\infty)\) satisfies the properties (i) \(d(x,y) = 0\) if and only if \(x = y\) and \(d(x,y) = d(y,x)\). It is assumed that the following properties hold: (W3) \(d(x_n,x) \to 0\) and \(d(x_n,y) \to 0\) imply \(x = y\); (W4) \(d(x_n,x) \to 0\) and \(d(x_n,y_n) \to 0\) imply \(d(y_n,x) \to 0\); (\(H_E\)) \(d(x_n,x) \to 0\) and \(d(y_n,x) \to 0\) imply \(d(x_n,y_n) \to 0\). It is proved that each two selfmappings \(A\) and \(B\) of \((X,d)\) have a unique common fixed point provided that (0) \(A\) and \(B\) commute at their coincidence points and \(d(Ax_n,t), d(Bx_n,t) \to 0\) for some \(t \in X\) implies \(d(ABx_n,BAx_n) \to 0\), (1) \(d(Ax,Ay) \leq \phi(\max \{d(Bx,By),d(Bx,Ay),d(Ay,By)\})\) for all \(x, y \in X\) with some \(\phi: \;[0,\infty) \to [0,\infty)\) and such that \(0 < \phi(t) < t, \;0 < t < \infty\), (2) there exists a sequence \((x_n)\) such that \(d(Ax_n,t), d(Bx_n,t) \to 0\) for some \(t \in X\), (3) \(AX \subset BX\) and either \(AX\) or \(BX\) is a complete subspace of \(X\) (in a natural topology generated with \(d\)). A similar result for four selfmappings is also proved.

MSC:

47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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