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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 $F:(\Bbb{R}^{+})^6\rightarrow\Bbb{R}$ satisfying the following conditions: (i) $F$ is non-increasing in $t_5$ and $t_6$; (ii) there exists $h\in (0,1)$ such that if $u,v\geq 0$ each of the relations $F(u,v,v,u,u+v,0)\leq 0$ or $F(u,v,u,v,0,u+v)\leq 0$ imply $u\leq hv$; (iii) $F(u,u,0,0,v,v)>0$ for all $u>0$. The following theorem is the main result of the paper. Let $(X,d)$ be a metric space and $S,T,I,J:X\rightarrow X$ four mappings satisfying the following conditions: (a) $S(X)\subset J(X)$, $T(X)\subset I(X)$ and one of the sets $S(X),T(X),I(X)$ and $J(X)$ is complete; (b) for all $x,y\in X$ and $F\in J$ $$ F(d(Sx,Ty),d(Ix,Jy),d(Ix,Sx),d(Jy,Ty),d(Ix,Ty),d(Jy,Sx))\leq 0. $$ Then each of the pairs of mappings $(S,I)$ and $(T,J)$ has a coincidence point. Moreover, if each of the pairs of mappings $(S,I)$ and $(T,J)$ 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 {\it 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.

54H25Fixed-point and coincidence theorems in topological spaces
47H10Fixed-point theorems for nonlinear operators on topological linear spaces