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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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##### References:
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