On the Fredholm Alternative for the Fu？ík Spectrum

We consider resonance problems for the one-dimensional -Laplacian assuming Dirichlet boundary conditions. In particular, we consider resonance problems associated with the first three curves of the Fu？ík Spectrum. Using variational arguments based on linking theorems, we prove sufficient conditions for the existence of at least one solution. Our results are related to the classical Fredholm Alternative for linear operators. We also provide a new variational characterization for points on the third Fu？ík curve. 1. Introduction In this paper, we study the solvability of the problem where , , , and . A solution of (1.1) is defined as a function , such that is absolutely continuous and satisfies (1.1) a.e. in . It is helpful to restate the given problem as an operator equation. Let and let represent the duality pairing between and . Define the operators by In [1, page 306], it is proved that is an isomorphism (in particular, exists and is continuous) and that is continuous and compact. Using an integration by parts argument, it is easy to verify [2, page 120] that solutions of (1.1) are in a one-to-one correspondence with solutions of the operator equation where is defined by We are interested in the case where which represents the Fu？ík Spectrum associated with (1.1), that is, the set of all such that has a nontrivial solution. An explicit form of is given in [2, page 132]. For convenience, we recall the first parts of it. First we note that for , we have if and only if is an eigenvalue of These eigenvalues can be expressed explicitly as where . The associated eigenfunctions are scalar multiples of where is defined by the implicit formula which is extended to and then by symmetry, and then to all of as a periodic function. See, for example, [3, 4]. Note that, we have in , and is a nontrivial solution of (1.5) for , with arbitrary . Obviously, this implies that . Similarly, with a corresponding nontrivial solution, in . It is helpful to separate this so-called trivial part of the Fu？ík Spectrum into We set . The set is the component, that is, maximal connected subset, of which contains . The other components of lie in the first quadrant. Hence, from now on we assume , . The next component of , which contains , is called . It is a curve ( -hyperbola) which passes through and has the asymptotes and . More precisely, For , a corresponding nontrivial solution, , of (1.5) is a two-bump function in which can be constructed in a piecewise fashion using appropriately shifted and scaled functions as the pieces. In particular, for , is a positive multiple of either