Contraction Mapping Theorem Question

[nugget]

Homework Statement

Consider the function g : [0,∞) → R defined by g(x) = x + e^−2x.

Given |g(x₂) − g(x₁)| < |x₂ − x₁| for all x₁, x₂ ∈ [0,∞) with x₁ ≠ x₂.

Is g a contraction on [0,∞)? Why?

Homework Equations

I think we are intended to use the given equation and the CMT

CMT states that |f(x)-f(y)|=c|x-y|, where x,y∈R.

Which is similar to |g(x₂) − g(x₁)|<|x₂ − x₁|

The Attempt at a Solution

I really don't know where to begin with this; I'm not simply after the answer though, just a prod in the right direction would be great

 

[Tedjn]

On my computer, the formula for g(x) displays an unreadable symbol before the e^-2x. Also, I confess that I have only seen CMT from browsing my friend's analysis book, and it doesn't exactly say

CMT states that |f(x)-f(y)|=c|x-y|, where x,y∈R.

.
Are they equivalent?

EDIT: The actual statement I read is at http://en.wikipedia.org/wiki/Banach_fixed_point_theorem

 

[gauss^2]

g is a contraction precisely when there is some positive real number c < 1 with

|g(y) - g(x)| &lt; c |y-x|

for all x,y in the domain.

You have to either find a positive c < 1 that satisfies the inequality above for all x,y \in [0,\infty) or show no positive c < 1 exists which satisfies the inequality. Note that

|g(y) - g(x)| &lt; |y-x|

does not show g is a contraction. The constant 0 < c < 1 really is needed (check the proof of the contraction mapping theorem to see why if you're interested; it's a really cool piece of mathematics).

 

[HallsofIvy]

Homework Statement

Consider the function g : [0,∞) → R defined by g(x) = x + e^−2x.

Given |g(x₂) − g(x₁)| < |x₂ − x₁| for all x₁, x₂ ∈ [0,∞) with x₁ ≠ x₂.

Is g a contraction on [0,∞)? Why?

Homework Equations

I think we are intended to use the given equation and the CMT

CMT states that |f(x)-f(y)|=c|x-y|, where x,y∈R.

No, it doesn't. That is simply the definition of "contraction map". The CMT states that if f is a contraction map that take set S into itself, then it has exactly one fixed point in S.

Which is similar to |g(x₂) − g(x₁)|<|x₂ − x₁|

The Attempt at a Solution

I really don't know where to begin with this; I'm not simply after the answer though, just a prod in the right direction would be great