Iterative equations in Banach spaces

AbstractLet 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 ∑iAifi(x)=F(x)or∑iAif(ϕi(x))=F(x) with operator coefficients in Banach spaces