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Common fixed point theorems. (English) Zbl 0767.54038

This paper contains some common fixed point theorems in metric spaces. The main result can be stated as follows: Theorem: Let $A$, $B$, $S$ and $T$ be mappings from a complete metric space $\left(X,d\right)$ into itself satisfying the conditions: (i) $A\left(X\right)\subset T\left(X\right)$ and $B\left(X\right)\subset S\left(X\right)$,

$d\left(Ax,By\right)\le h·max\left\{d\left(Ax,Sx\right),\phantom{\rule{4pt}{0ex}}d\left(By,Ty\right),\phantom{\rule{4pt}{0ex}}\frac{1}{2}\left[d\left(Ax,Ty\right)+d\left(By,Sx\right)\right],\phantom{\rule{4pt}{0ex}}d\left(Sx,Ty\right)\right\}\phantom{\rule{2.em}{0ex}}\left(\mathrm{ii}\right)$

for all $x$, $y$ in $X$, where $0\le h<1$. Further, suppose that (iii) one of $A$, $B$, $S$ and $T$ is continuous, (iv) pairs $A$, $S$ and $B$, $T$ are compatible on $X$. Then $A$, $B$, $S$ and $T$ have a unique common fixed point in $X$. The above theorem generalizes several known results due to B. Fisher, G. Jungck, M. S. Khan and M. Imdad. A result for compact metric space is also proved. Examples are given to illustrate all the results.

##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces
##### Keywords:
common fixed point