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A nonlinear Fredholm alternative for second order ordinary differential equations. (English) Zbl 0605.34020
Existence, nonexistence, and the existence of multiple solutions are discussed for a class of nonlinear Sturm-Liouville equations with Dirichlet boundary conditions. The nonlinearities considered are so- called ”jumping-nonlinearities” (S. Fučik), which cross asymptotically some eigenvalues of the linear differential operator. It is shown that the (nonlinear) eigenvalues of an associated positive- homogeneous equation determine two types of regions of parameters: For the first type the given problem has for any given data function exactly one solution, and for the second type, there exist data functions for which there exist at least two solutions while for other data functions there exist no solutions. This behaviour can be interpreted as a nonlinear Fredholm alternative. The proofs are based on the topological degree of Leray Schauder, and the so-called Lyapunov-Schmidt reduction.

34B15 Nonlinear boundary value problems for ordinary differential equations
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