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Assume that f : R → R is such that |f(x) – f(y)| < \|x – y| for all x, y E R and some A E (0, 1). (a) Prove that for every r > 0 one has that f(r) – r< f(0) – (1 – A)r and f(-r)+r> f(0) + (1 – A)r (b) Consider g(x) g(r*) < 0 and g(-r*) > 0. (c) Assume that r* is the number from part (b). Prove that there exists x* € (-r*, r*) such that g(x*) = 0 or, equivalently, f(x*)= x*. f (x) – x. Prove that there exists r* > 0 such that

Assume that f : R → R is such that |f(x) – f(y)| < \|x – y| for all x, y E R and some A E (0, 1). (a) Prove that for every r > 0 one has that f(r) – r< f(0) – (1 – A)r and f(-r)+r> f(0) + (1 – A)r (b) Consider g(x) g(r*) < 0 and g(-r*) > 0. (c) Assume that r* is the number from part (b). Prove that there exists x* € (-r*, r*) such that g(x*) = 0 or, equivalently, f(x*)= x*. f (x) – x. Prove that there exists r* > 0 such that

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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2. Assume that f : R → R is such that |f(x) – f(y)|< \/x – y| for all x, y E R and some A E (0, 1). (a) Prove that for every r > 0 one has that f (r) – r < f(0) – (1 – A)r and f(-r)+r > f(0) + (1 – A)r (b) Consider g(x) = f(x) – g(r*) < 0 and g(-r*) > 0. x. Prove that there exists r* > 0 such that (c) Assume that r* is the number from part (b). Prove that there exists x* € (-r*, r*) such that g(x*) = 0 or, equivalently, f(x*) = x*.
Transcribed Image Text:2. Assume that f : R → R is such that |f(x) – f(y)|< \/x – y| for all x, y E R and some A E (0, 1). (a) Prove that for every r > 0 one has that f (r) – r < f(0) – (1 – A)r and f(-r)+r > f(0) + (1 – A)r (b) Consider g(x) = f(x) – g(r*) < 0 and g(-r*) > 0. x. Prove that there exists r* > 0 such that (c) Assume that r* is the number from part (b). Prove that there exists x* € (-r*, r*) such that g(x*) = 0 or, equivalently, f(x*) = x*.
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Step 1

Explanation

Given - assume that f:ℝ→ℝ such that f(x)-f(y)≤λx-y for all x, y ∈ℝ and some λ∈(0, 1)

To prove - 

(a) prove that for every r>0 one has that

f(r)-r≤f(0)-(1-λ)r       &    f(-r)+r≥f(0)+(1-λ)r

(b) consider g(x)=f(x)-x prove that there exist r*>0 such that g(r*)<0 & g(-r*)>0

(c) assume that r* is the number from part (b) . prove that there exist x*∈(-r*,r*) such that g(x*)=0 or, equivalently , f(x*)=x*

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