On maximum infinitesimal order solutions of nonlinear equations in sectorial neighbourhood of zero

A nonlinear operator equation $B(\lambda)x=R(x,\lambda)+b(\lambda)$ with conditions $R(0,0)\equiv0$, $b(0)=0$ is considered. The linear operator $B(\lambda)$ hasn't a continuous inverse operator at $\lambda=0,$ but it has a bounded inverse operator when $\lambda\in S$, where $S$ is a set named a sectorial neighbourhood of zero. The question of existence of infinitesimal continuous solutions $x(\lambda)\rightarrow0$ at $\lambda\in S$ when $\lambda\rightarrow0.$ The proved theorems propose a constructive way the solution of the maximum infinitesimal order.