*(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 $(X,d)$ satisfying, for all $x,y\in X$, the inequality

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

and that $S\left(X\right)$ and $T\left(X\right)$ are closed. Under these conditions, it is stated that both pairs $(A,S)$ and $(B,T)$ have a coincidence point and, moreover, $A,B,C,T$ have a unique common fixed point, provided that both pairs $(A,S)$ and $(B,T)$ are weakly compatible ($(A,S)$ is called weakly compatible if $Ax=Sx$ for some $X$ implies $ASx=SAX$, and, similarly, for $(B,T)$). 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 |