Is There A Generalization Of Brouwer's Fixed Point Theorem?

Banach fixed point theorem?

• (scalar version) Let Ф: Ą ---> Ą be a contraction with constant k <1 then there exist a unique fixed point X* = Ф(x*) must be Ą show that the contraction constant k is given by k = max |Ф' (ε) | (max ε must be Ą) and show that the fixed point is unique. thanks

• Answer:

  I am sorry but such real analysis questions are not possible to answer on this site. This result is a stsndard theorem in contraction mapping theory and is readily available in most real analysis text books. JMM