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Common fixed point results for noncommuting mappings without continuity in generalized metric spaces. (English) Zbl 1185.54037
Let $X\ne\emptyset$. 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 $x\ne y$. (c) $G(x,x,y)\le G(x,y,z)$ for all $x,y\in X$, with $z\ne y$, (d) $G(x,y,z)= G(x,z,y)= G(y,z,x)=\cdots$ (symmetry in all three variables), (e) $G(x,y,z)\le G(x,a,a)+ G(a,y,z)$ 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: $$G(fx,fy,fy)\le k\max\{G(gx,fy,fy), G(gy,fx, fx), G(gy,fy,fy)\}$$ and $$G(fx,fy,fy)\le k\max\{G(gx,gx,fy), G(gy, gy,fx), G(gy, gy, fy)\}$$ for all $x,y\in X$, where $0\le k< 1$. 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 in topological spaces
Full Text:
References:
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