Minimal branches of solutions of nonlinear operator equations in Banach spaces

We consider the nonlinear equation $B(\lambda)x=R(x,\lambda)+b(\lambda)$, where $R(0,0)=0$, $b(0)=0$, the linear operator $B(\lambda)$ has a bounded inverse operator for $S\ni\lambda\rightarrow0$, and $S$ is an open set, $0\in\partial S$. We examine the problem on the existence of a small continuous solution of the maximal order od smallness $x(\lambda)\rightarrow0$ as $S\ni\lambda\rightarrow0$. A constructive method way of constructing this solution is presented.