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

A nonlinear operator equation В(λ)х = R(x, λ) + b(λ) with conditions Д(0,0) = 0, b(0) = 0 is considered. The linear operator B(λ) hasn't a continuous inverse operator at λ = 0, but it has a bounded inverse operator when λ Є S, where 5 is a set named a sectorial neighbourhood of zero. The question of existence of infinitesimal continuous solutions x(λ) → 0 at λ Є S when λ→ 0. The proved theorems propose a constructive way the solution of the maximum infinitesimal order.