On the Fredholm Alternative for the Fucيk Spectrum

and Applied Analysis 3 which is extended to 0, πp and then 0, 2πp by symmetry, and then to all of as a 2πp periodic function. See, for example, 3, 4 . Note that, we have u1 > 0 in 0, T , and u1 is a nontrivial solution of 1.5 for α, β λ1, β , with arbitrary β ∈ . Obviously, this implies that λ1 × ⊂ Σp. Similarly, × λ1 ⊂ Σp with a corresponding nontrivial solution, −u1 < 0 in 0, T . It is helpful to separate this so-called trivial part of the Fučı́k Spectrum into C1 : {( λ1, β ) ∈ 2 : β ≤ λ1 } ∪ { α, λ1 ∈ 2 : α ≥ λ1 } , C 1 : {( λ1, β ) ∈ 2 : β ≥ λ1 } ∪ { α, λ1 ∈ 2 : α ≤ λ1 } . 1.10 We set C1 C1 ∪ C 1 . The set C1 is the component, that is, maximal connected subset, of Σp which contains λ1, λ1 . The other components of Σp lie in the first quadrant. Hence, from now on we assume α > 0, β > 0. The next component of Σp, which contains λ2, λ2 , is called C2. It is a curve p-hyperbola which passes through λ2, λ2 and has the asymptotes α λ1 and β λ1. More precisely, C2 : { ( α, β ) ∈ 2 : 1 α1/p 1 β1/p ( p − 1)−1/p T πp } . 1.11 For α, β ∈ C2, a corresponding nontrivial solution, uαβ, of 1.5 is a two-bump function in 0, T which can be constructed in a piecewise fashion using appropriately shifted and scaled sinp functions as the pieces. In particular, for α > β, uαβ is a positive multiple of either φ21 or φ22, both L normalized, as shown in Figure 1. We note that φ22 t φ21 T − t , t ∈ 0, T . For α < β, the situation is similar but the positive bump is now bigger than the negative one. For further reference, we denote C2 : {( α, β ) ∈ C2 : α ≥ β } , C 2 : {( α, β ) ∈ C2 : α ≤ β } . 1.12 In the special case where α β λ2, we have φ21 cu2 and φ22 −cu2, where c > 0 is an L normalizing constant. C2 is called the first nontrivial part of the Fučı́k Spectrum. The component of Σp containing λ3, λ3 is denoted by C3 and consists of two p-hyperbolas which intersect at λ3, λ3 . C3 : { ( α, β ) ∈ 2 : 2 α1/p 1 β1/p ( p − 1)−1/p T πp } , C3 : { ( α, β ) ∈ 2 : 1 α1/p 2 β1/p ( p − 1)−1/p T πp } . 1.13 Note that the asymptote for C3 as α → ∞ is β λ1, while as β → ∞ the asymptote is α λ2. The set C3 is just the reflection of C3 with respect to the diagonal line α β. In Theorem 1.4 below we will consider α, β ∈ C3 with α > β, with a corresponding normalized eigenfunction, φ31, as depicted in Figure 2. 4 Abstract and Applied Analysis a φ21 b φ22 Figure 1: Eigenfunctions for α, β ∈ C2 with α > β. Figure 2: Eigenfunction for α, β ∈ C1 3 with α > β, that is, φ31. Components Cn of Σp containing the points λn, λn , n > 3, are obtained similarly see 2 . Note that a nontrivial solution of 1.5 associated with α, β ∈ Cn consists of n bumps in the interval 0, T , and can be expressed explicitly in terms of the sinp function. We adopt a convention used in 5 to identify the Fučı́k eigenvalues and eigenfunctions referred to in our theorems and proofs. Fix a positive constant s and consider the intersection of the line λ s, λ with Σp. This produces a sequence 0 < λ11 < λ12 < λ21 λ22 < λ31 < λ32 < · · · with associated normalized eigenfunctions φ11, φ12, φ21, φ22, φ31, φ32, . . .. See Figure 3. There is an interesting literature developing for problems such as 1.1 . There are two different cases which have to be distinguished: α, β /∈Σp, the so-called nonresonance case; α, β ∈ Σp, the so-called resonance case. The nonresonance case is well understood. If α, β belongs to a component of 2 \Σp containing a point λ, λ , then 1.1 has a solution for arbitrary f ∈ L1 0, T . The proof is based on the homotopy invariance property of Leray-Schauder degree. On the other hand, if the component of 2 \Σp does not contain a point λ, λ , then there exists f ∈ L1 0, T for which 1.1 does not have a solution see, e.g., 2 . Abstract and Applied Analysis 5and Applied Analysis 5