Abstract
Properties of asymmetric oscillator described by the equation (i), where and , are studied. A set of such that the problem (i), (ii), and (iii) have a nontrivial solution, is called α-spectrum. We give full description of α-spectra in terms of solution sets and solution surfaces. The exact number of nontrivial solutions of the two-parameter Dirichlet boundary value problem (i), and (ii) is given.
Highlights
Asymmetric oscillators were studied intensively starting from the works by Kufner and Fucık; see [1] and references therein
Simple equations like (2) given with the boundary conditions allow for complete investigation of spectra
There is a plenty of works devoted to one-parameter case of equations x + λf(x) = 0 given together with the two-point boundary conditions
Summary
Asymmetric oscillators were studied intensively starting from the works by Kufner and Fucık; see [1] and references therein. There is a plenty of works devoted to one-parameter case of equations x + λf(x) = 0 given together with the two-point boundary conditions. Λ and μ are nonnegative parameters, x+ = max{x, 0}, x− = max{−x, 0} This equation describes asymmetric oscillator with different nonlinear restoring forces on both sides of x = 0. Properties of the Fucık spectrum are well known (the Fucık spectrum is a set of all pairs (λ, μ) where λ, μ ≥ 0, such that the Dirichlet problem—(2) with boundary conditions x(0) = 0 = x(1)—has a non-trivial solution). The aim of our study in this paper is to describe properties of the spectrum of the problem x = −λ(x+)p + μ(x−)q, 0 < q ≤ 1, 1 ≤ p, (3). This paper continues series of publications by the authors devoted to nonlinear asymmetric oscillations [5–8]
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