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Function contraction theorem Let (X, d) be a complete metric space. Suppose that f: XX satisfies d(f(x), f(y)) ≤ 2-¹d(x, y) for all x, y ≤ X. Show that there exists a unique point x* EX such that f(x*) = x*.

Function contraction theorem Let (X, d) be a complete metric space. Suppose that f: XX satisfies d(f(x), f(y)) ≤ 2-¹d(x, y) for all x, y ≤ X. Show that there exists a unique point x* EX such that f(x*) = x*.

Advanced Engineering Mathematics
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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Function contraction theorem Let (X, d) be a complete metric space. Suppose that f: XX satisfies d(f(x), f(y)) ≤ 2-¹d(x, y) for all x, y E X. Show that there exists a unique point x* EX such that f(x*) = x*. Problem 5 Hint: i=0 Step 1. Let xo E X. Define a sequence {x} by the recurrence relation xi = f(xi-1) for i=1,2,.... Step 2. Show that {x} Step 3. Show that x* is the point of interest. converges to a point x* by showing that it is Cauchy.
Transcribed Image Text:Function contraction theorem Let (X, d) be a complete metric space. Suppose that f: XX satisfies d(f(x), f(y)) ≤ 2-¹d(x, y) for all x, y E X. Show that there exists a unique point x* EX such that f(x*) = x*. Problem 5 Hint: i=0 Step 1. Let xo E X. Define a sequence {x} by the recurrence relation xi = f(xi-1) for i=1,2,.... Step 2. Show that {x} Step 3. Show that x* is the point of interest. converges to a point x* by showing that it is Cauchy.
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