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Let f: RR be an increasing function. Suppose that there exist a, b € R satisfy b>a and a ≤ f(a) and f(b) < b. Show that f has at least one fixed point., that is, there is some x ER such that f(x) = x. (Hint: Consider z := sup{y € R: a ≤ y ≤by ≤ f(y)} and z)

Let f: RR be an increasing function. Suppose that there exist a, b € R satisfy b>a and a ≤ f(a) and f(b) < b. Show that f has at least one fixed point., that is, there is some x ER such that f(x) = x. (Hint: Consider z := sup{y € R: a ≤ y ≤by ≤ f(y)} and z)

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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Explain the basics: Explain the concepts as well Let f RR be an increasing function. Suppose that there exist a, b € R satisfy b> a and a ≤ f(a) and f(b) < b. Show that f has at least one fixed point., that is, there is some x ER such that f(x) = x. (Hint: Consider z = sup{y € R: a ≤ y ≤by≤ f(y)} and z) | Also explain sup
Transcribed Image Text:Explain the basics: Explain the concepts as well Let f RR be an increasing function. Suppose that there exist a, b € R satisfy b> a and a ≤ f(a) and f(b) < b. Show that f has at least one fixed point., that is, there is some x ER such that f(x) = x. (Hint: Consider z = sup{y € R: a ≤ y ≤by≤ f(y)} and z) | Also explain sup
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