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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.

54H25 Fixed-point and coincidence theorems (topological aspects)
Full Text: DOI
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