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Common fixed point results for noncommuting mappings without continuity in generalized metric spaces. (English) Zbl 1185.54037
Let $$X\neq\emptyset$$. Suppose that a mapping $$G: X\times X\times X\to[0,\infty)$$ satisfies:
(a) $$G(x,y,z)= 0$$ if and only if $$x= y= z$$,
(b) $$0< G(x,y,z)$$ for all $$x,y\in X$$, with $$x\neq y$$.
(c) $$G(x,x,y)\leq G(x,y,z)$$ for all $$x,y\in X$$, with $$z\neq y$$,
(d) $$G(x,y,z)= G(x,z,y)= G(y,z,x)=\cdots$$ (symmetry in all three variables),
(e) $$G(x,y,z)\leq G(x,a,a)+ G(a,y,z)$$ for all $$x,y,z,a\in X$$.
Then $$G$$ is called a $$G$$-metric on $$X$$ and $$(X,G)$$ is called a $$G$$-metric space.
In the present paper the authors, using the setting of $$G$$-metric space, prove a fixed point theorem for one map, and several fixed point theorems for two maps. They prove, for example:
Theorem 2.5. Let $$(X, G)$$ be a $$G$$-metric space. Suppose that $$f,g: X\to X$$ satisfy one of the following conditions:
$G(fx,fy,fy)\leq k\max\{G(gx,fy,fy), G(gy,fx, fx), G(gy,fy,fy)\}$ and
$G(fx,fy,fy)\leq k\max\{G(gx,gx,fy), G(gy, gy,fx), G(gy, gy, fy)\}$ for all $$x,y\in X$$, where $$0\leq k< 1$$. If the range of $$g$$ contains the range of $$f$$ and $$g(X)$$ is a complete subspace of $$X$$, then $$f$$ and $$g$$ have a unique point of coincidence in $$X$$. Moreover, if $$f$$ and $$g$$ are weakly compatible, then $$f$$ and $$g$$ have a unique common fixed point.

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