On Small Solutions of Nonlinear Equations with Vector Parameter in Sectorial Neighborhoods

We consider the nonlinear operator equation $B(\lambda)x+R(x,\lambda)=0$ with parameter $\lambda$, which is an element of a linear normed space $\Lambda$. The linear operator $B(\lambda)$ has no bounded inverse for $\lambda=0$. The range of the operator $B(0)$ can be nonclosed. The nonlinear operator $R(x,\lambda)$ is continuous in a neighborhood of zero and $R(0,0)=0$. We obtain sufficient conditions for the existence of a continuous solution $x(\lambda)\to 0$ as $\lambda\to 0$ with maximal order of smallness in an open set $S$ of the space $\Lambda$. The zero of the space $\Lambda$ belongs to the boundary of the set $S$. The solutions are constructed by the method of successive approximations.