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Some fixed point theorems for compatible mappings satisfying an implicit relation. (English) Zbl 0926.54030
Let \(S,T,I,J\) be selfmaps of a metric space \((X,d)\) such that \[ F(d(Sx,Ty), d(Ix,Jy), d(Ix,Sx), d(Jy,Ty), d(Ix,Ty), d(Jy,Sx))\leq 0, \] for all \(x,y\) in \(X\), where \(F: [0,+\infty)^6\to \mathbb{R}\) is a function having the following properties:
(i) \(F\) is non-increasing in the 5th and 6th variables,
(ii) \(F(u,v,v,u,u+v,0)\leq 0\) or \(F(u,v, u,v, 0,u+v)\leq 0\) implies \(u\leq hv\) for some \(h\in (0,1)\),
(iii) \(F(u,u, 0,0, u,u)> 0\) for all \(u>0\).
The author gives suitable examples of such functions. Main result: The above maps have a unique common fixed point provided that the ranges of \(J\) and \(I\) contain the range of \(S\) and \(T\) respectively, one of them is continuous and the pairs \((S,I)\) and \((T,J)\) are compatible.
Reviewer: S.Sessa (Napoli)

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