Thread: Calculus Contraction Mapping

T(x) = X+ 1/x X=[1,INFINITY]

|T(X) - T(Y)| < |X-Y|

|(x+ 1/x) - (y+ 1/y)| < |x-y|

Can i solve the left side further?

Originally Posted by littlefire

T(x) = X+ 1/x X=[1,INFINITY]

|T(X) - T(Y)| < |X-Y|

|(x+ 1/x) - (y+ 1/y)| < |x-y|

Can i solve the left side further?

I am not sure what you want.

This is bounded by, .

Thanks a million. Still now too sure on your answer after the bounded part.

This is my full question that I have to answer -

Let X=[1,infinity) with the metric d(x,y) = |x-y|. Define T:X->X by T(x) = x+1/x. Prove T satisfies

d(T(x),T(y)) < d(x,y) for all x,y E X

but T has no fixed points in X. Does this contradict Banach's fixed point theorem.

Originally Posted by littlefire

Let X=[1,infinity) with the metric d(x,y) = |x-y|. Define T:X->X by T(x) = x+1/x. Prove T satisfies
d(T(x),T(y)) < d(x,y) for all x,y E X

By simple differentiation we see the function is increasing on its domain.
.
So, .
.

By symmetry this holds if .