Iterative equations in Banach spaces

Let X be a Banach space, let K(X) := f :X→X Lipschitz: f − id sup <∞ , and let P :K(X)→K(X), F ∈ K(X). By applying the Banach contraction principle we prove that if P is sufficiently close (in a certain sense) to the identity then the equation Pf = F has a unique solution f . As a corollary we obtain results on iterative equations of the types i Aif i(x) = F(x) or i Aif φi(x) = F(x) with operator coefficients in Banach spaces.  2004 Published by Elsevier Inc.