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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