Fix α, a positive real number, and let X=[0,1] be a subset of R. Define f:X->R by f(x)=αx(1-x) for x in X. For what values of α does the mapping f:X->R have the property that f(X) is a subset of X and f:X->X is a contraction? A contraction is defined for me as when a Lipschitz function has a Lipschitz constant c<1. I already know that for α in [0,4], the function f(X) is a subset of X, now I need to find the values for α that make it a contraction.
Copyright © 2024 1QUIZZ.COM - All rights reserved.
Answers & Comments
Verified answer
|f(x) - f(y)| = α|x-y -(x^2-y^2)| = α|x-y||1-(x+y)|, and α|1-(x+y)| < 1 when α< 1/|1-(x+y)|, which is smallest only when |1-(x+y)| is maximum. |1-(x+y)| is easily seen to take max value 1 on [0,1], so 0 <= α < 1.
PUNCCCCCCCCCCHHHHHHHHHH satisfied Belated Birthday to you and it is one for you. The Beatles~ immediately is your birthday, satisfied birthday to you. so, on etc. lol it is my reviews with y/a. they type of direction from satisfied to loopy. LOL I stand by using you~ The Pretenders brilliant satisfied human beings~ R.E.M. My technology ~ The Who ought to I stay or ought to i bypass now~ The conflict by no skill say by no skill~ Romeo Void I wanna be sedated ~ The Ramones Breaking the regulation~ Judas Priest Stand or Fall~ The Fixx Firestarter~ Prodigy Blue Monday~ New Order quite everyone appears human beings, coverage of truth, ~ Depeche Mode.