Determining equations and the relatedness principle (Q795316)

This article is devoted to a very common method in nonlinear analysis of studying the operator equation \((1)\quad Fx=0\) with a nonlinear operator F mapping one Banach space X into another Y. If the operator U maps X into a finite dimensional subspace \(L\subseteq Y\) and the operator \(F+U\) has an inverse then the equation (1) is equivalent to the equation \((2)\quad z=U(F+U)^{-1}z=0\) with unknown \(z\in L\); the equation (2) is called determination equation for (1). The Lyapunov and Schmidt bifurcation equations, the determination equations by Cesari and the Poincaré and A. M. Samoilenko equations in the oscillation theory are particular cases of determination equations for suitable U. In the article the general relatedness principle binding the rotations of \(F:X\to Y\) and \(I-U(F+U)^{-1}:L\to L\) on the boundaries of corresponding domains in X and L are given; this principle generalized known relatedness principles stated by M. A. Krasnoselskij, V. V. Strygin, E. A. Lifsic and some new analoguous results.