an:05612897 Zbl 1185.54037 Abbas, M.; Rhoades, B. E. Common fixed point results for noncommuting mappings without continuity in generalized metric spaces EN Appl. Math. Comput. 215, No. 1, 262-269 (2009). 00253921 2009
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54H25 weakly compatible maps; common fixed point; generalized metric space 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. Jaros??aw G??rnicki (Rzesz??w)