Equivalence of Metrics
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, a contraction mapping, or contraction, on a metric space (M,d) is a function f from M to itself, with the property that there is some real number k 1 such that for all x and y in M,The smallest such value of k is called the Lipschitz constant of f. Contractive maps are sometimes called Lipschitzian maps.A contraction mapping has at most one fixed point. Moreover, the Banach fixed point theorem states that every contraction mapping on a nonempty…mehr

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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, a contraction mapping, or contraction, on a metric space (M,d) is a function f from M to itself, with the property that there is some real number k 1 such that for all x and y in M,The smallest such value of k is called the Lipschitz constant of f. Contractive maps are sometimes called Lipschitzian maps.A contraction mapping has at most one fixed point. Moreover, the Banach fixed point theorem states that every contraction mapping on a nonempty complete metric space has a unique fixed point, and that for any x in M the iterated function sequence x, f (x), f (f (x)), f (f (f (x))), ... converges to the fixed point. This concept is very useful for iterated function systems where contraction mappings are often used. Banach''s fixed point theorem is also applied in proving the existence of solutions of ordinary differential equations, and isused in one proof of the inverse function theorem.