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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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