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(a) Suppose that f(a) = b and f(6) = a with a # b. Prove that f has a fixed point in (a, b). In other words, every continuous, real-valued function that has a prime period-2 point has a fixed point as well. (Hint: Problem 5 may prove useful.) %3D

(a) Suppose that f(a) = b and f(6) = a with a # b. Prove that f has a fixed point in (a, b). In other words, every continuous, real-valued function that has a prime period-2 point has a fixed point as well. (Hint: Problem 5 may prove useful.) %3D

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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(Cury (c) In view of the fact provea tu pon- sense to call p "super"-attracting. 12. Period Doubling. We start with a definition: a. fn-1(a) + a, then we say that a but f(a) # a, f (a) a, ..., has prime period-n. Note that if k divides n, any period-k point is also a period-n point. bu not a prime period-n point. Note also that if a has period n, then it h prime period-n if and only if a, f(a), f(a), ..., f"¬(a) are distinet Let I be a closed interval in R, and let f : I I be a continuous function (a) Suppose that f(a) a fixed point in (a, b). In other words, every continuous, real-valued function that has a prime period-2 point has a fixed point as wel. (Hint: Problem 5 may prove useful.) (b) Let n e N. Prove that if f has a prime period-2" point, then it must have prime periodic points of periods {1,2, 4, 8, . , 2" }. (Hint: If r is a prime period-2n+1 point for f, then it is a prime period 2" point for f. = b and f(b) = a with a b. Prove that f has
Transcribed Image Text:(Cury (c) In view of the fact provea tu pon- sense to call p "super"-attracting. 12. Period Doubling. We start with a definition: a. fn-1(a) + a, then we say that a but f(a) # a, f (a) a, ..., has prime period-n. Note that if k divides n, any period-k point is also a period-n point. bu not a prime period-n point. Note also that if a has period n, then it h prime period-n if and only if a, f(a), f(a), ..., f"¬(a) are distinet Let I be a closed interval in R, and let f : I I be a continuous function (a) Suppose that f(a) a fixed point in (a, b). In other words, every continuous, real-valued function that has a prime period-2 point has a fixed point as wel. (Hint: Problem 5 may prove useful.) (b) Let n e N. Prove that if f has a prime period-2" point, then it must have prime periodic points of periods {1,2, 4, 8, . , 2" }. (Hint: If r is a prime period-2n+1 point for f, then it is a prime period 2" point for f. = b and f(b) = a with a b. Prove that f has
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