The Rotation Number Approach to the Periodic Fuc caron ik Spectrum

Multiplicity Results for Asymptotically Linear Equations, Using the Rotation Number Approach

“…We observe that Theorem 3.14 could be generalized, by replacing the linear operator u −→ u with the one-dimensional p-Laplacian operator φ p , defined by φ p (u) = |u| p−2 u, whenever p ∈ (1, +∞). Such an extension is guaranteed by the relations between eigenvalues and rotation numbers established by Zhang in [51] and [52].…”

Half-eigenvalues of periodic Sturm–Liouville problems

“…Again following the proof of Theorem 2.1, we find that y l is also continuous. Hence, % l k ða; a; bÞ is continuous in ða; a; bÞ; uniformly in a; and so the extreme values of % l k ðÁ; a; bÞ are continuous in ða; bÞ: & To conclude this section, we compare our results with those of [35], where Zhang defines a generalised Fucˇı´k spectrum as the set of ða; bÞAR 2 for which the problem…”

“…By (1.7), this is equivalent to whether 0 is to one side of, or between, the corresponding half-eigenvalues of ðL; a; bÞ: Usually, the known set S F ðL 0 Þ is used to give these conditions, in both the periodic and the separated cases, although more general operators L are considered in [26,30] and [31] in the separated case. In the periodic case the Fucˇı´k spectrum for more general operators L does not appear to have been discussed, although a different generalisation has been considered in [35] and will be compared with our approach in Section 2.…”

“…In this case the nonlinearity f is termed jumping. The solvability of (1.1)-(1.2) has been studied extensively, see for example, [8,9,11,[13][14][15][16][17]19,20,35], and the references therein. The corresponding problem with separated (e.g., Dirichlet or Neumann) boundary conditions has also been studied, see [3,9,17,26,[29][30][31], and the references therein.…”

Exponential Growth Rates of Periodic Asymmetric Oscillators

“…This will lead to a typical nonlinear spectrum problem, i.e., the Fučik spectrum of (1.3) which is a generalization of eigenvalues of Hill's equations [16] and the classical Fučik spectrum [6] as well. For the non-autonomous oscillators (1.3), a nice rotation number approach to a partial characterization of the Fučik spectrum of (1.3) has been given by Zhang [20]. Very recently, Binding and Rynne [2] have revealed some crucial difference between the spectra of (1.3) and that of the Hill's operators.…”