Having trouble with this one...
Question: (S,rho) a metric space. The function L: S --> S is a contraction with modulus A. L(S) belongs to K, a compact space with at least one element. I need to show that there's one fixed point in S.
Any ideas?
Having trouble with this one...
Question: (S,rho) a metric space. The function L: S --> S is a contraction with modulus A. L(S) belongs to K, a compact space with at least one element. I need to show that there's one fixed point in S.
Any ideas?
Hm, let me see...
If f(x0)=x1, f(x1)=x2 and so on...
d(x_n,x_n-1) <= A^n d(x1,x0)
For n high enough, we should have A^n-->0, so:
d(x_n,x_n-1) = 0
So x_n=x_n-1, so f^n(x0)=f^(n-1)(x0)
This implies that f(x_n-1)=x_n-1, so lim x_n-1 = v is a fixed point? Can I do that?