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Some remarks for improving fixed point theorems for more than two maps. (English) Zbl 0912.47031

Principles for the existence of common fixed points for two functions were established by Sehie Park. Such principles cannot be generalized for more than two functions, but the authors make some progress in improving fixed point theorems involving four maps by weakening the condition of compatibility of two of the maps. Two selfmaps $S$ and $T$ of a metric space $\left(X,d\right)$ are said to be compatible if for every sequence $\left\{{x}_{n}\right\}$ in $X$ such that $limT{x}_{n}=limS{x}_{n}=t\in X$, $limd\left(ST{x}_{n},TS{x}_{n}\right)=0$. Two maps $S$ and $T$ are said to be weakly compatible if they commute at coincidence points. The following theorem is typical for the results of the paper.

Theorem 1. Let $A,B,S,T$ be selfmaps of a complete metric space $X$ satisfying

(i) $A\left(X\right)\subset T\left(X\right)$ and $B\left(X\right)\subset S\left(X\right)$.

(ii) $d\left(Ax,Bx\right)\le {\Phi }\left(M\left(x,y\right)\right)$ for each $x,y$ in $X$, where

$M\left(x,y\right)=max\left\{d\left(Sx,Ty\right),d\left(Sx,Ax\right),d\left(Ty,By\right),\left(d\left(Sx,By\right)+d\left(Ax,Ty\right)\right)/2\right\}$

and ${\Phi }:\left[0,\infty \right)\to \left[0,\infty \right)$ is upper semicontinuous and such that ${\Phi }\left(t\right) for each $t>0$. If either

(iii) $\left\{A,S\right\}$ are compatible, $A$ or $S$ is continuous and $\left(B,T\right)$ are weakly compatible, or

(iv) $\left\{B,T\right\}$ are compatible, $B$ or $T$ is continuous and $\left(A,S\right)$ are weakly compatible

then $A,B,S,T$ have a unique common fixed point $z$.

Moreover, the theorem contains an assertion about sequences arising from the maps and converging to the common fixed point. The authors also point out that some fixed point theorems in the literature for three or four functions have incorrect or incomplete proofs.

##### MSC:
 47H10 Fixed point theorems for nonlinear operators on topological linear spaces
##### Keywords:
common fixed point; weakly compatible maps