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Let (X, d) be a compact metric space. Let f : (X, d) ? (X, d) be a function. Show that if d(f(a), f(b)) < d(a, b) for all a, b ? X, a ? b, then f has a unique fixed point (that is, there exists a unique c ? X such that f(c) = c). (Hint: define g(x) = d(x, f(x)) for all x ? X. Show that g : (X, d) ? (?, d_1) is continuous (use continuity of d and f (why is f continuous (< ? ?)?)). Thus g(X) is compact (why?). Next, explain why there exists c ? X such that inf_{x ? X} g(x) = g(c). Show g(c) = 0. This gets you a fixed point for f (why?). Show it is the unique fixed point.)

          Let (X, d) be a compact metric space. Let f : (X, d) ? (X, d) be a function. Show that if d(f(a), f(b)) < d(a, b) for all a, b ? X, a ? b, then f has a unique fixed point (that is, there exists a unique c ? X such that f(c) = c). (Hint: define g(x) = d(x, f(x)) for all x ? X. Show that g : (X, d) ? (?, d_1) is continuous (use continuity of d and f (why is f continuous (< ? ?)?)). Thus g(X) is compact (why?). Next, explain why there exists c ? X such that inf_{x ? X} g(x) = g(c). Show g(c) = 0. This gets you a fixed point for f (why?). Show it is the unique fixed point.)

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Let (X, d) be a compact metric space. Let f : (X, d) ? (X, d) be a function. Show that if d(f(a), f(b)) < d(a, b) for all a, b ? X, a ? b, then f has a unique fixed point (that is, there exists a unique c ? X such that f(c) = c). (Hint: define g(x) = d(x, f(x)) for all x ? X. Show that g : (X, d) ? (?, d1) is continuous (use continuity of d and f (why is f continuous (< ? ?)?)). Thus g(X) is compact (why?). Next, explain why there exists c ? X such that infx ? X g(x) = g(c). Show g(c) = 0. This gets you a fixed point for f (why?). Show it is the unique fixed point.)

Added by Julio C.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Madhur L

08/15/2023

Step 1

Define g(x) = d(x, f(x)) for all x ∈ X. We want to show that g: (X, d) → (R, d') is continuous. To show this, let ε > 0 be given. We want to find δ > 0 such that for all x, y ∈ X, if d(x, y) < δ, then |g(x) - g(y)| < ε. Now, consider d(x, f(x)) and d(y, f(y)). Show more…

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Let (X, d) be a compact metric space. Let f: X → X be a function. Show that if d(f(a), f(b)) < d(a, b) for all a, b ∈ X, a ≠ b, then f has a unique fixed point (that is, there exists a unique c ∈ X such that f(c) = c). Hint: define g(x) = d(x, f(x)) for all x ∈ X. Show that g: X → ℝ is continuous (use continuity of d and f). Thus, X is compact. Next, explain why there exists c ∈ X such that inf g(x) = g(c). Show that g(c) = 0, which gives you the fixed point for f. Show that it is the unique fixed point.

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