Nonlinear Spectra: The Neumann Problem

Abstract

Eigenvalue problems of the form x” = −λf(x+ ) + μg(x− ), x‘(a) = 0, x' (b) = 0 are considered. We are looking for (λ,μ) such that the problem (i), (ii) has a nontrivial solution. This problem generalizes the famous Fučík problem for piece‐wise linear equations. In our considerations functions f and g may be nonlinear. Consequently spectra may differ essentially from those for the Fučík equation.

Two-Parameter Nonlinear Oscillations: The Neumann Problem

“…The proof is similar to that of Theorem 1 in [8]. ⊓ ⊔ Now let us consider properties of an α-spectrum.…”

“…The conditions (A1) and (A2) together are more general than those imposed on f in [8]. The conditions (A3) and (A4) together ensure that the relations (2.3) are valid.…”

“…We will call α-spectrum F (α) of α-normalized problem a set of all pairs (λ, µ) such that the α-normalized problem is nontrivially solvable. Description of α-spectrum is similar to that given in [8], where a normalization condition |x ′ (z)| = 1 was used: the αspectrum is a union…”

“…The main result in [8] is an analytical description of the spectrum provided in Theorem 1. In Definition 1 of the present article we substantially expand notion of a spectrum (a solution set is introduced) for the problem (1.1) and give full description of a spectrum in Theorem 4, thereby generalizing the main result in [8].…”

“…Two-parameter nonlinear boundary value problems have been extensively studied in the literature, see for example [1,2,3,5,8,11,14,16]. Some of the mentioned references deal with the so called asymmetric oscillators.…”