Minimal Branches of Solutions of Nonlinear Operator Equations in Banach Spaces.

We consider the nonlinear equation B(λ)x = R(x, λ) + b(λ), where R(0, 0) = 0, b(0) = 0, the linear operator B(λ) has a bounded inverse operator for S ∋ λ → 0, and S is an open set, 0 ∈ ∂S. We examine the existence of a small continuous solution of the maximal order of smallness x(λ) → 0 as S ∋ λ → 0. A constructive method of constructing this solution is presented. [ABSTRACT FROM AUTHOR]