Nonlinear Oscillations in the Frame of Alternative Methods

Abstract

This chapter discusses the nonlinear oscillations in the frame of alternative methods. The injection of methods of functional analysis in the classical bifurcation process of Poincaré, Lyapunov, and Schmidt, and extensive subsequent work have made this process a fine tool in nonlinear analysis, particularly in the difficult problems at resonance in the usual terminology. The general theory that has ensued, with all its variants and ramifications, is often referred to as bifurcation theory, or alternative methods. There are many ideas that have been brought to bear in this theory in the last few years, such as Schauder's fixed point theorem, Banach's fixed point theorem, invariance properties of topological degree, Brouwer's fixed point theorem; the theory of monotone and maximal monotone operators in Banach spaces; Schauder's principle of invariance of domain; implicit function theorem; and Newton's polygon method in Banach spaces. The chapter discusses the recent aspects of the theory, mainly for non-self-adjoint problems. It presents the modification of Cesari's alternative scheme proposed by Hale, Bancroft, and Sweet for non-self-adjoint problems.