Resonant Problems by Quasilinearization

Abstract

The Dirichlet resonant boundary value problems are considered. If the respective nonlinear equation can be reduced to a quasilinear one with a nonresonant linear part and both equations are equivalent in some domainΩand if solutions of the quasilinear problem are inΩ, then the original problem has a solution. We say then that the original problem allows for quasilinearization. If quasilinearization is possible for essentially different linear parts, then the original problem has multiple solutions. We give conditions for Emden-Fowler type resonant boundary value problem solvability and consider examples.

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]). …”

“…Usually the appropriate conditions on the linear part and on the nonlinearity are imposed in order the problem to be solvable. One may consult recent articles [1,5,12] and references therein for the respective bibliography.…”

“…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].…”

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

“…There are numerous papers devoted to resonant boundary value problems. Some of them can be found in the bibliography of papers [25][26][27].…”

Uniqueness of solutions of boundary value problems at resonance

“…The application of the method of quasilinearization to boundary value problems at resonance is not new; see [27,28]. The motivation and development here is different than that in [27] or [28], since uniqueness of solutions is a key feature in this work and multiplicity of solutions is key in [27] or [28].…”