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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) $d(Ax,Ay) \le \phi(\max \{d(Bx,By),d(Bx,Ay),d(Ay,By)\})$ 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 for nonlinear operators on topological linear spaces 47H09 Mappings defined by “shrinking” properties
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