Small solutions of nonlinear equations in sectorial neighbourhoods

We consider nonlinear operator equation $B(\lambda )x+R(x,\lambda )=0$. Linear operator $B(\lambda )$ does not have bounded inverse operator at $\lambda=0$. Nonlinear operator $R(x,\lambda)$ is continuous in neighborhood of zero, $R(0,0)=0$. We have deduced sufficient conditions of existence of the continuous solution $x(\lambda)\rightarrow0$ as $\lambda\rightarrow0$ in some open set $S$ of linear normalized space $\Lambda$. Zero belongs to frontier of set $\Lambda$. We have proposed way of construction the solution of maximum infinitesimal order in neighborhood of zero. The initial estimate is null element.