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4. Let f be differentiable on R with a = sup{|f'(x)| : x € R} < 1. (i) Pick a number so € R and inductively define sn = f(sn-1) for n ≥ 1. Prove that {n} is a convergent sequence in R. (ii) Show that f has a fixed point, that is, f(s) = s for some s € R.

4. Let f be differentiable on R with a = sup{|f'(x)| : x € R} < 1. (i) Pick a number so € R and inductively define sn = f(sn-1) for n ≥ 1. Prove that {n} is a convergent sequence in R. (ii) Show that f has a fixed point, that is, f(s) = s for some s € R.

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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**Problem 4:** Let \( f \) be differentiable on \( \mathbb{R} \) with \( a = \sup\{|f'(x)| : x \in \mathbb{R}\} < 1 \). (i) Pick a number \( s_0 \in \mathbb{R} \) and inductively define \( s_n = f(s_{n-1}) \) for \( n \geq 1 \). Prove that \(\{s_n\}\) is a convergent sequence in \( \mathbb{R} \). (ii) Show that \( f \) has a fixed point, that is, \( f(s) = s \) for some \( s \in \mathbb{R} \).
Transcribed Image Text:**Problem 4:** Let \( f \) be differentiable on \( \mathbb{R} \) with \( a = \sup\{|f'(x)| : x \in \mathbb{R}\} < 1 \). (i) Pick a number \( s_0 \in \mathbb{R} \) and inductively define \( s_n = f(s_{n-1}) \) for \( n \geq 1 \). Prove that \(\{s_n\}\) is a convergent sequence in \( \mathbb{R} \). (ii) Show that \( f \) has a fixed point, that is, \( f(s) = s \) for some \( s \in \mathbb{R} \).
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