Numerical theory and Lipschitz function

  • #1
so I have this homework as I said and marks will be added on my total, so if anyone could help you will be a lifesaver, you dont have to answer the whole thing , just help me with the part you know,

here it is :
A function g (x) is called Lipschitz function on the interval [a,b] if there exists a constant L > 0, such that absolute(g(y) – g(x)) <= L *absolute(y-x).the constant L is called the Lipschitz constant.
1- Show that if g(x) is Lipschitz function on [a,b] with a Lipschitz constant L > 0, then g(x) is continuous function on [a,b].
2- Show that if g(x) is differentiable on [a,b], then g(x) is Lipschitz.
3- Show that if g(x) >= 0 is a Lipschitz function on [a,b], b > a >= 0 with a Lipschitz constant 0 < L =< 1, then g(x) maps the interval [a,b] into itself.
4- From the parts 1 and 2 , we deduct the existence of a fixed point P of g(x). show that P (the fixed point) is unique provided that g(x) is contraction function. A function g(x) is called a contraction function if g(x) is a Lipschitz function on [a,b] with a Lipschitz constant 0 < L < 1.
5- Assume that g(x) satisfies the condition in part 3 and 4. Show that the sequence of fixed point iterations defined by xn = g(xn-1) with any initial guess x0 converges to the unique fixed point.
 

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
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If this is "for points" then you are expected to turn in your own work, not someone elses. We may be able to help if you show what you have done. The first two, at least, are pretty straight forward using the definitions of "continuous" and "differentiable".
 

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