On an Application of a Modification of the Zincenko Method to the Approximation of Implicit Functions

Let E,A be Banach spaces and denote by U(x0 ,R) the closed ball with center x0 € E and of radius R in E. We will use the same symbol for the norm 11 11 in both spaces. Let Pbe a linear projection operator (P 2 = P) which projects Eon its subspace Ep and set Q I P. Suppose that the non-linear operators F(x, A) and G(x, A) with values in E are defined for N D, where D is some convex subset of E containing U(x0 ,R) and A € U(XO , S). For each fixed A€ U(X0 ,S) the operator PF(z, A) will be assumed to be Fréchet differentiable for all z ED. Then PF'(x,A) will denote the Frechet derivative of the operator PF(z,A) with respect to the argument z at z = x. Moreover, we assume that (PF'(x0 ,X0 )Y 1 exists and