Theorem 3.18: The Contraction Lemma Let (X, d) be a complete metric space and f: X→ X a contractive function. Then there is exactly one point x in X for which f(x) = x. x is called a fixed point of f Cany). Proof: To show the existence of such a point, choose a point x; in X and define x2 = f(x), x; = f(x2), ..., X, = f(x-1),. n2 2. The fact that f is a contractive function insures that {xn}%-1 is a Cauchy sequence. By completeness, this sequence has a limit x in X. v - Use Thus f(x) = x. To show the required uniqueness property, suppose that y is a second point satisfying f(y) = y. Then d(x, y) = d(f(x), f(y)) s ad(x, y). Since a < 1, this relation cannot hold unless d(x, y) = 0 and x = y.

Theorem 3.18: The Contraction Lemma Let (X, d) be a complete metric space and f: X→ X a contractive function. Then there is exactly one point x in X for which f(x) = x. x is called a fixed point of f Cany). Proof: To show the existence of such a point, choose a point x; in X and define x2 = f(x), x; = f(x2), ..., X, = f(x-1),. n2 2. The fact that f is a contractive function insures that {xn}%-1 is a Cauchy sequence. By completeness, this sequence has a limit x in X. v - Use Thus f(x) = x. To show the required uniqueness property, suppose that y is a second point satisfying f(y) = y. Then d(x, y) = d(f(x), f(y)) s ad(x, y). Since a < 1, this relation cannot hold unless d(x, y) = 0 and x = y.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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