Can you recall the definition of Lipschitz characteristic, in order to make your post more self-contained?
i'm beginner in metric fixed point theory. So i meet a problem to answer the proof of lipschitz characteristic for hilbert spaces is sqrt2. Can you help me?please. i can't find literatures that answer my question. Thanks for your attention.
c' is called lipschitz characteristic for metric space (X,d) if and only if equal to sup{c>=1 : balls in X is c-reguler}.
Balls in X is c-reguler if and only if for all k<c there exist a,b in (0,1) such that for all x,y elements in X and r>0 with d(x,y)>=(1-a)r,
there exist z element in X such that (B(x,(1+a)r) meet B(y,k(1+a)r)) in B(z,br).
Note: B is closed ball in X.