I am trying to understand the following theorem, which can be found in Kolmogorov and Fomin's (p. 509 here):
Let map $F$ [$:X\to Y$ where $X,Y$ are Banach spaces] be strongly differentiable in a sphere $B(x_0,r)$ having centre $x_0$ and radius $r$ and let the derivative $F'(x)$ satisfy Lipschitz condition$$\|F(x_1)-F(x_2)\|\le L\|x_1-x_2\|.$$Let $[F'(x_0)]^{-1}$ [in $\mathscr{L}(Y,X)$, I would say] exist and$$M=\|[F'(x_0)]^{-1}\|,\quad k=\|[F'(x_0)]^{-1}F(x_0)\|,\quad > h=MkL.$$Then if $h<1/4$, in sphere $\|x-x_0\|\le kt_0$, where $t_0$ is the smallest root of equation $ht^2-t+1=0$, equation $F(x)=0$ has the unique solution $x^\ast$ and sequence $\{x_n\}$ defined by recursive formula$$x_{n+1}=x_n-[F'(x_0)]^{-1}(F(x_n))$$converges to this solution.
The proof uses the fact that the derivative of $\Phi:x\mapsto F(x)-F(x_0)-F'(x_0)(x-x_0)$, which I guess we can be sure to exist only if $x\in B(x_0,r)$, is $F'(x)-F'(x_0)$ and that, if $\|x-x_0\|\le kt_0$, $\|\Phi'(x)\|\le Lt_0 k$. I wonder: is not it required that $kt_0<r$? Without such an assumption, I do not see how we can establish the existence of $\Phi'(x)$ for $\|x-x_0\|\le kt_0$. Does anybody know this theorem and what are the required assumptions? I have noticed that Kolmogorov-Fomin's sometimes states theorems valid only under unstated assumptions, and I fear this is one of those cases...
P.S.: I notice that the question has been downvoted: I would be grateful to the downvoter if (s)he made his/her vote more useful and suggest how to improve the question.