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Let $(X,d)$ be a complete metric space and $T:X\to X$ be a mapping such that $T^m= T\circ T\circ T\circ\dots\circ T$ ($m$ times) is a contraction for some fixed $m$, then show $T$ has an unique fixed point.

egreg
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Amit Vishwakarma
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1 Answers1

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The Contraction Mapping Principle gives a unique fixed point of the map $T^m$, call it $x$. Now we only have one candidate for the fixed point of $T$, namely $x$. What happens if x is not a fixed point of $T$? Hint: Suppose $T(x)=y$ and apply $T^m$. Remember that the fixed point is unique.

phunfd
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