write the different application of contraction maps (i) and contraction maps (ii)
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In mathematics, a contraction mapping, or contraction or contractor, on a metric space (M, d) is a function f from M to itself, with the property that there is some nonnegative real number {\displaystyle 0\leq k<1} such that for all x and y in M,
{\displaystyle d(f(x),f(y))\leq k\,d(x,y).}
The smallest such value of k is called the Lipschitz constant of f. Contractive maps are sometimes called Lipschitzian maps. If the above condition is instead satisfied for k ≤ 1, then the mapping is said to be a non-expansive map.
More generally, the idea of a contractive mapping can be defined for maps between metric spaces. Thus, if (M, d) and (N, d') are two metric spaces, then {\displaystyle f:M\rightarrow N} is a contractive mapping if there is a constant {\displaystyle 0\leq k<1} such that
{\displaystyle d'(f(x),f(y))\leq k\,d(x,y)}
for all x and y in M.
Every contraction mapping is Lipschitz continuous and hence uniformly continuous (for a Lipschitz continuous function, the constant k is no longer necessarily less than 1).
A contraction mapping has at most one fixed point. Moreover, the Banach fixed-point theorem states that every contraction mapping on a non-empty 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 is used in one proof of the inverse function theorem.[1]
Contraction mappings play an important role in dynamic programming problems.
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