IMPLICIT FUNCTION THEOREM IN SECTORIAL QUASI-NEIGHBORHOODS

We consider nonlinear operational equation $F(x,lambda)=0$ with condition $F(0,0)equiv0$. Operator $F_x(0,0)$ is not continuously inversible. We costruct continuous solutions $x(lambda)
ightarrow0$ as $lambda
ightarrow0$ in open set $S$ of linear normalized space $Lambda$. Zero belongs to frontier of set $S$. Solution existence theorems we have illustrated by examples.

Keywords

Banach space, implicit function theorem, sectorial quasi-neighborhoods, nonlinear operator equation, linear normalized space, two-point boundary problem