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Common fixed point results for noncommuting mappings without continuity in generalized metric spaces. (English) Zbl 1185.54037

Let X. Suppose that a mapping G:X×X×X[0,) satisfies:

(a) G(x,y,z)=0 if and only if x=y=z,

(b) 0<G(x,y,z) for all x,yX, with xy.

(c) G(x,x,y)G(x,y,z) for all x,yX, with zy,

(d) G(x,y,z)=G(x,z,y)=G(y,z,x)= (symmetry in all three variables),

(e) G(x,y,z)G(x,a,a)+G(a,y,z) for all x,y,z,aX.

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:XX satisfy one of the following conditions:

G(fx,fy,fy)kmax{G(gx,fy,fy),G(gy,fx,fx),G(gy,fy,fy)}

and

G(fx,fy,fy)kmax{G(gx,gx,fy),G(gy,gy,fx),G(gy,gy,fy)}

for all x,yX, where 0k<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:
54H25Fixed-point and coincidence theorems in topological spaces
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