On Nonlinear Fučik Type Spectra

Abstract

Eigenvalue problems of the form x" = -A/{x' *") + fi.g{x~), x(0) = 0, rE(l) = 0 are considered, where x"^ and x~ are respectively the positive and the negative parts of x. We are looking for (A,/x) such that the problem has a nontrivial solution. This problem generalizes the famous Fu(5ik problem for piece-wise linear equations. In order to show that nonlinear PuCik spectra may differ essentially from the classical ones, we consider piece-wise linear functions f(x) and ^(x). We show that the first branches of the Fucik spectrum may contain bounded components.

Two-Parameter Nonlinear Oscillations: The Neumann Problem

“…Besides it has the vertical asymptote at λ = λ * and the horizontal asymptote at µ = µ * , see points 4 and 5 of the theorem. Similar approach was realized to prove Theorem 2 in [5].…”

“…The centro-affine mapping Φ i,2j (i = 2j) preserves volumes with the coefficient J(Φ i,2j ) = 32 (j/i) 5 and has the inverse mapping N) is a cross point of the following subsets: 1) the solution surface F ± 2j−1 and the parabolic cylinders λ = λ1 α 2 1 α 2 and µ = µ1 α 2 1 α 2 , or equivalently 2) the solution surface F ± 2j−1 , the parabolic cylinder λ = λ1 α 2 1 α 2 and the plane µ = µ1 λ1 λ.…”

“…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.…”

“…Gritsans and F. Sadyrbaev associated with equation x ′′ = −λf (x + ) + µg(x − ), where f and g are nonlinear nonnegative functions. This equation with the Dirichlet boundary conditions was considered in [4,5,7]. In [4] the Dirichlet boundary value problem was investigated together with a normalization condition which is not needed in case of the Fučík equation since it fulfils automatically.…”

“…In [7] the Dirichlet problem was studied provided that one of the functions f and g is linear. In [5] some properties of the spectrum were analyzed. It was pointed out that branches of the spectrum may have separate components connected at infinity and even bounded separate components.…”