Implicit function theorem in sectorial quasi-neighborhoods

We consider nonlinear operational equation F(x, λ) = 0 with condition F(0,0) = 0. Operator Fx(0,0) is not continuously inversible. We costruct continuous solutions х(λ) → 0 as λ → 0 in open set S of linear normalized space A. Zero belongs to frontier of set S. Solution existence theorems we have illustrated by examples.