homework6 2010 - be a complete metric space and T be a...

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ECN/APEC 7240 Spring 2010 Homework 6 Due 3/31/10 1) Let f : R R be a continuously differentiable function with a fixed point x *. Suppose that | f ( x *)| < β < 1. a) Use Taylor’s Theorem to show there exists a neighborhood U of x * such that if we restrict f to U then f will be a contraction of modulus β . b) Let x 0 U and define the dynamical sequence { x n } by x n +1 = f ( x n ). Show that { x n } U and that x n converges to x *. 2) Let ( X
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Unformatted text preview: ) be a complete metric space and T be a contraction mapping on X with modulus ∈ (0, 1). Let x * be the unique fixed point of T . Show that for any x ∈ X , ). , ( 1 1 *) , ( 1 x T x T d x x T d n n n +-≤ This provides a lower bound on the speed at which the sequence { x n } is converging to x * that does not require prior knowledge of x *....
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