2. Let f: R R be a function with the property that there exists K € (0, 1) such that, for all x, y ER, 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 (xn) be the sequence defined by x1 = C, n+1 = f(n) for all n ≥ 1. 0 i. Prove, by induction, that n+1- xn| ≤ Kn-¹|x2-x1 for all n ≥ 1. Kn-1 ii. Prove that, for all m n ≥ 1, xm- xn| ≤ = |x₂ = x11. - 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: RR which has no fixed points. Verify that it is not a contraction.
2. Let f: R R be a function with the property that there exists K € (0, 1) such that, for all x, y ER, 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 (xn) be the sequence defined by x1 = C, n+1 = f(n) for all n ≥ 1. 0 i. Prove, by induction, that n+1- xn| ≤ Kn-¹|x2-x1 for all n ≥ 1. Kn-1 ii. Prove that, for all m n ≥ 1, xm- xn| ≤ = |x₂ = x11. - 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: RR 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 ƒ : R → R be a function with the property that there exists K € (0, 1) such that,
for all x, y ≤ R, [ƒ(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 € R and (xn) be the sequence defined by
x1 = C,
xn+1 = f(xn) for all n ≥ 1.
0
i. Prove, by induction, that n+1 − Xn| ≤ Kn−¹|x2 − x₁| for all n ≥ 1.
Kn-1
ii. Prove that, for all m ≥ n ≥ 1, |xm − xn| ≤ -|X2 — X1|.
1- K
iii. Deduce that (n) is Cauchy, and hence converges.
iv. Hence prove that ƒ 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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