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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 $(X,d)$ into itself satisfying the conditions: (i) $A(X)\subset T(X)$ and $B(X)\subset S(X)$, $$d(Ax,By)\le h\cdot\max\bigl\{d(Ax,Sx),\ d(By,Ty),\ \textstyle{{1\over 2}}[d(Ax,Ty)+d(By,Sx)],\ d(Sx,Ty)\bigl\}\tag ii$$ 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.

54H25Fixed-point and coincidence theorems in topological spaces