Contraction mapping on metric space

Question

We know the definition of contraction mapping. But it is unkown to me the definition of weak contraction mapping. Help me

May be this is helpful. – Peter Melech Jun 06 '19 at 12:19 — Peter Melech, Jun 06 '19 at 12:19

Peter Melech · Answer 1 · 2019-06-07T13:15:06.573

0

A mapping $f:X\rightarrow X $ on a metric space is a $\bf{weak} $ contraction if for all $x\neq y\in X$ you have $$d(f(x),f(y))<d(x,y)$$ whereas a contraction is such a mapping where there is a constant $0<C<1$ so that $$d(f(x),f(y))\leq Cd(x,y)$$ for $\bf{all}$ $x,y\in X$. The answer I cited in the comment shows not every weak contraction is a contraction.

edited Jun 07 '19 at 13:15

answered Jun 06 '19 at 12:26

Peter Melech

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Contraction mapping on metric space

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