On Types of Solutions of the Second Order Nonlinear Boundary Value Problems

Abstract

We review the results concerning types of solutions of boundary value problems for the second order nonlinear equation(l2x)(t)=f(t,x,x′),where(l2x)(t)is the second order linear differential form. The existence results and the multiplicity results are stated in terms of types of solutions.

On conversion of resonant problem to non-resonant one

“…Therefore multiple application of this scheme using multiple different linear parts can prove the existence of multiple solutions of the (resonant) problem (4.1). This scheme was tested on equations of the EmdenFowler type in [12] (see also [3]). …”

“…It is to be mentioned that the natural idea of "a shift" from a resonant linear part to a non-resonant one was employed in the papers [4,5,13]. "Shift" arguments in a broad sense (replacing the linear part of a given equation multiply by different non-resonant linear parts and proving the existence of multiple solutions to a given boundary value problem) were applied for the study of resonant problems in [12] in context of the quasi-linearization approach [3]. In Section 4 of this paper we consider two ways of relaxing the quasilinearization arguments used in [12].…”