Contraction Mappings, Real Analysis II

We cover contraction mappings on a metric space (f: M to M). Beginning with the definition, that there exists some Lipschitz constant α∈[0,1) so that d(f(x),f(y) is less than or equal to α*d(x,y), we discuss the idea that such functions pull points closer together. Several examples are analyzed in detail (straight lines f(x) = mx+b, the square root function, trigonometric functions) with special attention paid to how the choice of domain affects whether the contraction condition is satisfied.

We define Lipschitz continuity, and prove that all contraction maps (as examples of Lipschitz continuous functions) are uniformly continuous, and hence continuous.

The central result is the Contraction Mapping Theorem (aka the Banach Fixed Point Theorem). We prove it carefully and constructively: beginning with bounds on adjacent terms in the iterated sequence {x1, f(x1), f(f(x1)), ...}, then showing it forms a Cauchy sequence, converges in a complete metric space, and trends to the unique fixed point.

Finally, we show how to use this result numerically: any starting input generates a sequence converging to the fixed point, providing an effective method of approximation.

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