On solutions with the maximal order of vanishing of nonlinear equations with a vector parameter in sectorial neighborhoods

The nonlinear operator equation $B(\lambda)x+R(x,\lambda)=0$ is considered. The linear operator $B(\lambda)$ has no bounded inverse operator for $\lambda=0$. The nonlinear operator $R(x,\lambda)$ is continuous in a neighborhood of zero and $R(0,0)=0$. Sufficient conditions for the existence of a continuous solution $x(\lambda)\to0$ as $\lambda\to0$ in some open set $S$ of a linear normed space $\Lambda$ are obtained. The zero of the space $\Lambda$ belongs to the boundary of the set $S$. A method of constructing a solution with the maximal order of vanishing in a neighborhood of the point $\lambda=0$ is suggested. The zero element is taken as the initial approximation.