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The Fucik spectrum of general Sturm-Liouville problems. (English) Zbl 0976.34024
The author studies the boundary value problem consisting of the Sturm-Liouville-like equation \[ -(pu')'+ qu= \alpha u^+ -\beta u^-\text{ on }[0,\pi] \] and a separated real boundary condition such as the Dirichlet boundary condition \(u(0)=0 = u(\pi)\), with \(p \in C^1([0,\pi],\mathbb{R})\), \(p > 0\) on \([0,\pi]\), \(q\in C^0([0,\pi],\mathbb{R})\), \(\alpha,\beta\in \mathbb{R}\), and \(u^\pm(x) = \max\{\pm u(x),0\}\). The set of points \((\alpha,\beta)\in\mathbb{R}^2\) for which this problem has a nontrivial solution is called the Fucik spectrum. This set is of importance in the study of nonlinear boundary value problems with jumping nonlinearities.
When \(p\equiv 1\), \(q\equiv 0,\) and either the Dirichlet or the periodic boundary condition is imposed, the Fucik spectrum is explicitly known, and consists of a countable collection of curves in \(\mathbb{R}^2\). Here the author obtains various geometric properties of the general problem above. In particular, it is shown that the spectrum is always a countable collection of curves in \(\mathbb{R}^2\), and the asymptotic behaviour of these curves is determined using certain eigenvalues for the associated usual Sturm-Liouville equation \[ -(pv')'+qv=\lambda v\text{ on }[0,\pi]. \] A generalized Fucik spectrum is considered and is shown to have similar geometric properties.
Some spectral-type properties of a positively homogeneous “half-linear” problem are also discussed via similar methods. These results are then used to give necessary and sufficient conditions for the solvability of a nonlinear problem, with jumping nonlinearity, in terms of the location of the “half-eigenvalues” of the associated half-linear problem.

MSC:
34B24 Sturm-Liouville theory
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[1] Ambrosetti, A.; Prodi, G., A primer of nonlinear analysis, (1993), Cambridge Univ. Press Cambridge · Zbl 0781.47046
[2] Berestycki, H., On some nonlinear sturm – liouville problems, J. differential equations, 26, 375-390, (1977) · Zbl 0331.34020
[3] Browne, P.J., A Prüfer approach to half-linear sturm – liouville problems, Proc. Edinburgh math. soc., 41, 573-583, (1998) · Zbl 0918.34031
[4] Coddington, E.A.; Levinson, N., Theory of ordinary differential equations, (1955), McGraw-Hill New York · Zbl 0042.32602
[5] Dancer, E.N., On the Dirichlet problem for weakly non-linear elliptic partial differential equations, Proc. roy. soc. Edinburgh sect. A, 76, 283-300, (1977) · Zbl 0351.35037
[6] Dancer, E.N., Generic domain dependence for non-smooth equations and the open set problem for jumping nonlinearities, Topol. methods nonlinear anal., 1, 139-150, (1993) · Zbl 0817.35026
[7] Deimling, K., Nonlinear functional analysis, (1985), Springer-Verlag Berlin · Zbl 0559.47040
[8] Fabry, C., Landesman – lazer conditions for periodic boundary value problems with asymmetric nonlinearities, J. differential equations, 116, 405-418, (1995) · Zbl 0816.34014
[9] Fucik, S., Solvability of nonlinear equations and boundary value problems, (1980), Reidel Dordrecht · Zbl 0453.47035
[10] Invernizzi, S., A note on nonuniform nonresonance for jumping nonlinearities, Comment. math. univ. carolin., 27, 285-291, (1986) · Zbl 0603.34016
[11] Pistoia, A., A generic property of the resonance set of an elliptic operator with respect to the domain, Proc. roy. soc. Edinburgh sect. A, 127, 1301-1310, (1997) · Zbl 0887.35060
[12] Ruf, B., A non-linear Fredholm alternative for second order ordinary differential equations, Math. nachr., 127, 299-308, (1986) · Zbl 0605.34020
[13] Schechter, M., The Fucik spectrum, Indiana univ. math. J., 43, 1139-1157, (1994) · Zbl 0833.35050
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