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Q. Provide an example of contraction with exactly two fixed points.

My approach: Let T: R->R such that $T(x)=x^2$. Suppose $x$ is a fixed point of $T$ then $x^2 = x$. This implies $x=0$ or $x=1$. Thus, contraction T has two fixed points.

However, it doesn't hold true for suppose T(2) and T(3) as 5 is not less than or equal to 1.

Where am I going wrong?

Sepia Glen
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  • Banach contraction principle guarantees a unique fixed point. Meaning that there cannot be two different fixed points for a contraction. For a non-expansive map, though, you can have more than one fixed point. – Aniruddha Deshmukh Nov 10 '21 at 13:25
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    $T$ is not a contraction in $\mathbb R$, not even in $[0,1]$. But it is in say $[-0.4,0.4]$. – lhf Nov 10 '21 at 13:27
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    How do you define a "contraction"? –  Nov 10 '21 at 13:31

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