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Remarks on some fixed point theorems satisfying implicit relations. (English) Zbl 1033.54025

Let $I$ be the set of all continuous functions $F:{\left({ℝ}^{+}\right)}^{6}\to ℝ$ satisfying the following conditions: (i) $F$ is non-increasing in ${t}_{5}$ and ${t}_{6}$; (ii) there exists $h\in \left(0,1\right)$ such that if $u,v\ge 0$ each of the relations $F\left(u,v,v,u,u+v,0\right)\le 0$ or $F\left(u,v,u,v,0,u+v\right)\le 0$ imply $u\le hv$; (iii) $F\left(u,u,0,0,v,v\right)>0$ for all $u>0$.

The following theorem is the main result of the paper. Let $\left(X,d\right)$ be a metric space and $S,T,I,J:X\to X$ four mappings satisfying the following conditions: (a) $S\left(X\right)\subset J\left(X\right)$, $T\left(X\right)\subset I\left(X\right)$ and one of the sets $S\left(X\right),T\left(X\right),I\left(X\right)$ and $J\left(X\right)$ is complete; (b) for all $x,y\in X$ and $F\in J$

$F\left(d\left(Sx,Ty\right),d\left(Ix,Jy\right),d\left(Ix,Sx\right),d\left(Jy,Ty\right),d\left(Ix,Ty\right),d\left(Jy,Sx\right)\right)\le 0·$

Then each of the pairs of mappings $\left(S,I\right)$ and $\left(T,J\right)$ has a coincidence point. Moreover, if each of the pairs of mappings $\left(S,I\right)$ and $\left(T,J\right)$ commute at their coincidence points, then $S,T,I$ and $J$ have a unique common fixed point.

The previous theorem improves under many aspects an earlier result of V. Popa [Demonstr. Math. 32, 157–163 (1999; Zbl 0926.54030]. Next the author establishes related results and give illustrative examples which demonstrate the utility of the proved results.

##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 47H10 Fixed point theorems for nonlinear operators on topological linear spaces