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Difference between revisions of "Banach fixed point theorem"
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Latest revision as of 21:26, 16 February 2011
Let $(X,d)$ be a metric space. A function $T:X \to X$ is said to be a contraction map if there is a constant $q$ with $0 \leq q < 1$ such that $$ d(Tx,Ty)\leq q\cdot d(x,y)$$ for all $x,y\in X$.
Banach Fixed Point Theorem. Every contraction map on a complete metric space has a unique fixed point.