Let f: R→R be a function with the property that there exists K € (0, 1) such that, for all x, y = R, f(x)-f(y)| ≤ Kx-y. Such a function is called a contraction of contraction factor K. (a) Prove that f is continuous. (b) Let e ER and (₂) be the sequence defined by 21 = C, In+1 = f(n) for all n ≥ 1. 0 i. Prove, by induction, that n+1-n| ≤ K-¹2-21 for all n ≥ 1. Kn-1 ii. Prove that, for all m≥n ≥ 1, xm-n ≤ 1- K iii. Deduce that (r) is Cauchy, and hence converges. = |x₂ - 11. iv. Hence prove that f has a fixed point. v. Prove that the fixed point of f is unique. (You have just proved a special case of the Contraction Mapping Theorem.) (c) Write down a continuous function g: R → R which has no fixed points. Verify that it is not a contraction.
Let f: R→R be a function with the property that there exists K € (0, 1) such that, for all x, y = R, f(x)-f(y)| ≤ Kx-y. Such a function is called a contraction of contraction factor K. (a) Prove that f is continuous. (b) Let e ER and (₂) be the sequence defined by 21 = C, In+1 = f(n) for all n ≥ 1. 0 i. Prove, by induction, that n+1-n| ≤ K-¹2-21 for all n ≥ 1. Kn-1 ii. Prove that, for all m≥n ≥ 1, xm-n ≤ 1- K iii. Deduce that (r) is Cauchy, and hence converges. = |x₂ - 11. iv. Hence prove that f has a fixed point. v. Prove that the fixed point of f is unique. (You have just proved a special case of the Contraction Mapping Theorem.) (c) Write down a continuous function g: R → R which has no fixed points. Verify that it is not a contraction.
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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Transcribed Image Text:2. Let f: R → R be a function with the property that there exists K € (0, 1) such that,
for all r, y € R, f(x) − f(y)| ≤ K|x − y. Such a function is called a contraction of
contraction factor K.
(a) Prove that f is continuous.
(b) Let c ER and (zn) be the sequence defined by
x1 = C,
In+1 =
0
f(n) for all n ≥ 1.
i. Prove, by induction, that n+1-n| ≤ K-¹|x₂ - x₁ for all n ≥ 1.
Kn-1
ii. Prove that, for all m≥ n ≥ 1, xm - In| ≤ 7|₂ - 01|-
1- K
iii. Deduce that (n) is Cauchy, and hence converges.
iv. Hence prove that f has a fixed point.
v. Prove that the fixed point of f is unique.
(You have just proved a special case of the Contraction Mapping Theorem.)
(c) Write down a continuous function g: R → R which has no fixed points. Verify
that it is not a contraction.
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