Download Math 2030 Lecture: Attracting Fixed Points

Fixed Point Behavior
Contraction Mapping Principle
Math 2030 Lecture: Attracting Fixed Points
Neal Stoltzfus3
3 Louisiana
State University
Lectures: Math 2030
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Fixed Point Behavior
Contraction Mapping Principle
Outline
1
Fixed Point Behavior
Definitions
Theorem
Lipschitz Condition
2
Contraction Mapping Principle
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Fixed Point Behavior
Contraction Mapping Principle
Attracting Fixed Points
Definition
A function F has an attracting fixed point at x0 provided
F (x0 ) = x0 (i.e. fixed) and |F 0 (x0 )| < 1.
Example: F [x] = 2x(1 − x) has an attracting fixed points at
x0 = 1/2. F 0 [1/2] = 0!
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Fixed Point Behavior
Contraction Mapping Principle
Theorem
Theorem
Let F be a continuously differentiable function (i.e. the
derivative exists and is continuous every where). Suppose that
F has an attracting fixed point at x0 , |F 0 (x0 )| < 1. Then there is
a number δ > 0 such that:
1
F maps the interval (x0 − δ, x0 + δ) = I to itself.
2
limn→∞ F n [p] → x0 for all numbers p in the interval I.
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Fixed Point Behavior
Contraction Mapping Principle
Proof
F is differentiable everywhere, hence continuous as well and
the Mean Value Theorem from calculus holds. By the
hypothesis, |F 0 (x0 )| < λ < 1 for some λ. By the continuity of the
derivative at x0 , there is a δ > 0 such that F 0 [x] < λ on the
interval I = (x0 − δ, x0 + δ).
Applying the conclusion of the Mean Value Theorem, the
[x0 ]
= F 0 [c] for some c in I. Hence the
secant slope F [p]−F
p−x0
distance from F [p] to x0 decreases by a factor of λ:
|F [p] − F [x0 ]| = |F [p] − x0 | = |p − x0 ||F 0 [c]| < λ|p − x0 |
Therefore F maps the interval (x0 − δ, x0 + δ) = I to itself since
the distances to x0 decreases.
Continued on the next slide
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Fixed Point Behavior
Contraction Mapping Principle
Proof
Next, we observe:
|F 2 [p] − x0 | = |F 2 [p] − F 2 [x0 ]| < λ|F [p] − x0 | < λ2 |p − x0 |
By induction, |F n [p] − x0 | < λn |p − x0|. For λ < 1, λn
converges to zero hence
lim F n [p] → x0 .
n→∞
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Fixed Point Behavior
Contraction Mapping Principle
Lipschitz Condition
Definition
A function F satisfies a Lipschitz condition with Lipschitz
parameter λ > 0 provided |F [x] − F [y ]| < λ|x − y | for all points
x, y .
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Fixed Point Behavior
Contraction Mapping Principle
Contraction Mapping Principle: Fixed Points
Theorem
Let U be a subset of Rn and F : U 7→ U such that
|F [x] − F [y ]| < λ|x − y | for some number λ < 1 (i.e. F contracts
distances.) Then F has a unique fixed point and for any u ∈ U,
the iterates, F n [u] converge to the fixed point.
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