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An implicit function implies several contraction conditions. (English) Zbl 1180.54052

The article deals with new generalization of coincidence point theorem. It is assumed that $A,B,C,D$ are self-mappings of a metric space $\left(X,d\right)$ satisfying, for all $x,y\in X$, the inequality

$F\left(d\left(Ax,By\right),d\left(Sx,Ty\right),d\left(Ax,Sx\right),d\left(By,Ty\right),d\left(Sx,By\right),d\left(Ty,Ax\right)\right)\le 0,$

where $F:{ℝ}_{+}^{6}\to ℝ$ is a function with properties: (F${}_{1}$) $F\left(t,0,t,0,0,t\right)>0$ for all $t>0$; (F${}_{2}$) $F\left(t,0,0,t,t,0\right)>0$ for all $t>0$; (F${}_{3}$) $F\left(t,t,0,0,t,t\right)>0$ for all $t>0$. Furthermore, it is assumed that there exist two sequences $\left\{{x}_{n}\right\},\left\{{y}_{n}\right\}$ such that

$\underset{n\to \infty }{lim}A{x}_{n}=\underset{n\to \infty }{lim}S{x}_{n}=\underset{n\to \infty }{lim}B{y}_{n}=\underset{n\to \infty }{lim}T{y}_{n}=t\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{some}\phantom{\rule{4pt}{0ex}}t\in X,$

and that $S\left(X\right)$ and $T\left(X\right)$ are closed. Under these conditions, it is stated that both pairs $\left(A,S\right)$ and $\left(B,T\right)$ have a coincidence point and, moreover, $A,B,C,T$ have a unique common fixed point, provided that both pairs $\left(A,S\right)$ and $\left(B,T\right)$ are weakly compatible ($\left(A,S\right)$ is called weakly compatible if $Ax=Sx$ for some $X$ implies $ASx=SAX$, and, similarly, for $\left(B,T\right)$). Some modifications of this statement are also given. A number of examples of functions $F$ are presented. Comparisons between old and new results are gathered at the end of the article.

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