5. Let X and A be Banach spaces, let ? be an open set of A, and let F : Q x X - X be a uniform contraction map on X, i.e., there exists a constant ke 0, 1) s.t. IF(X, 2) - F(X, y)| Skix - yll for all (X, x), (), y) 6 0x X. It then follows from the contraction mapping principle that for each A e ? there exists a unique r(A) s.t. F(X, r(1)) = I (X). Assume that there exists a positive constant L such that IF(X, 2) - F(A, )II SIX - All for all (X, r), (u, r) En x X. Prove that the following inequality holds I (X) - I()ISA- All. L for all A, M En. Assume, in addition, that Fe C'(0 x X, X). Show that r(.) ( C'(0)