Introduction

Tha goal of this section is to investigate the structure and qualitative properties of the Fucik spectrum for several differential operators. The Fucik spectrum for a linear operator L is the set

where u+ and u- stand for the positive and the negative parts of u, respectively. S. Fucik [F76] and E. N. Dancer [D77] firstly recognized the importance of this set in the case of semilinear boundary value problems. Nowadays, there exists several papers such as [NFS00] and [NFS00] that deal with the Fucik spectrum for the general operator L. But, there are still many open problems, especially in applying these general results in particular cases of differential operators for partial differential equations such as wave or beam operators.

References

[HN10]	G. Holubova and P. Necesal, The Fučík Spectra for multi-point boundary value problems, Electronic Journal of Differential Equations Conference 18 (2010), pp. 33--44.
[DR10]	P. Drabek and S.B. Robinson, On the Fredholm alternative for the Fucik spectrum, Abstr. Appl. Anal. (2010), pp. 20.
[BDG11]	J. Benedikt and P. Drabek and P. Girg, The first nontrivial curve in the Fucik spectrum of the Dirichlet Laplacian on the ball consists of nonradial eigenvalues, Boundary Value Problems (2011), pp. 27.
[NFS00]	K. Ben-Naoum and C. Fabry and D. Smets, Resonance with respect to the Fucik spectrum, Electron. J. Differential Equations 2000 (2000), No. 37, pp. 1--21.
[D77]	E. N. Dancer, On the Dirichlet problem for weakly nonlinear elliptic partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 76A (1977), pp. 283--300.
[F00]	C. Fabry, Fucik spectra for vector equations, Electron. J. Qual. Theory Differ. Equ. (2000), No. 7, pp. 1--24.
[F76]	S. Fucik, Boundary value problems with jumping nonlinearities, Casopis pro pest. mat. (1976), pp. 69--87.
[HN08]	G. Holubova and P. Necesal, Nontrivial Fucik spectrum of one non-selfadjoint operator, Nonlinear Anal. 69 (2008), pp. 2930--2941.
[HN09]	G. Holubova and P. Necesal, Nonlinear four-point problem: Non-resonance with respect to the Fucik spectrum, Nonlinear Anal. 71 (2009), pp. 4559--4567.
[K83]	P. Krejci, On solvability of equations of the 4th order with jumping nonlinearities, Casopis pro pest. mat. 108 (1983), pp. 29--39.
[MM97]	C. A. Margulies and W. Margulies, An example of the Fucik spectrum, Nonlinear Anal. 29 (1997), No. 12, pp. 1373--1378.
[MM99]	C. A. Margulies and W. Margulies, Nonlinear resonance set for nonlinear matrix equations, Linear Algebra and its Appl. 293 (1999), No. 1--3, pp. 187--197.
[N01]	P. Necesal, On the Resonance Problem for the 4th Order Ordinary Differential Equations, Fucik's Spectrum, Archivum Mathematicum, CDDE 2000 issue 36 (2000), pp. 531--542.
[N04]	P. Necesal, The Fucik Spectrum in Models of Suspension Bridges, Proc. of Dynamic Systems and Applications 4 (2004), pp. 320--327.
[S94]	M. Schechter, The Fucik spectrum, Indiana Univ. Math. J. 43 (1994), pp. 1139--1157.