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1. Let (X, d) be a metric space and ƒ : X → X be a function. We say that ƒ is a weak contraction if d(f(x), f (y)) < d(x, y), for all x, y € X (a) Is a weak contraction always a contraction for X not compact? Give an example. (b) If X is a compact metric space, is a weak contraction always a contraction? Give an example. (c) Show that if X is compact and f is a weak contraction on X, then f has a unique fixed point.

1. Let (X, d) be a metric space and ƒ : X → X be a function. We say that ƒ is a weak contraction if d(f(x), f (y)) < d(x, y), for all x, y € X (a) Is a weak contraction always a contraction for X not compact? Give an example. (b) If X is a compact metric space, is a weak contraction always a contraction? Give an example. (c) Show that if X is compact and f is a weak contraction on X, then f has a unique fixed point.

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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1. Let (X, d) be a metric space and ƒ : X → X be a function. We say that f is a weak contraction if d(f(x), f(y)) < d(x, y), for all x, y ≤ X (a) Is a weak contraction always a contraction for X not compact? Give an example. (b) If X is a compact metric space, is a weak contraction always a contraction? Give an example. (c) Show that if X is compact and f is a weak contraction on X, then f has a unique fixed point.
Transcribed Image Text:1. Let (X, d) be a metric space and ƒ : X → X be a function. We say that f is a weak contraction if d(f(x), f(y)) < d(x, y), for all x, y ≤ X (a) Is a weak contraction always a contraction for X not compact? Give an example. (b) If X is a compact metric space, is a weak contraction always a contraction? Give an example. (c) Show that if X is compact and f is a weak contraction on X, then f has a unique fixed point.
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