On the Fu\v{c}ik spectrum of the wave operator and an asymptotically linear problem

We study generalized solutions of the nonlinear wave equation $$u_{tt}-u_{ss}=au^+-bu^-+p(s,t,u),$$ with periodic conditions in $t$ and homogeneous Dirichlet conditions in $s$, under the assumption that the ratio of the period to the length of the interval is two. When $p\equiv 0$ and $\lambda$ is a nonzero eigenvalue of the wave operator, we give a proof of the existence of two families of curves (which may coincide) in the Fu\v{c}ik spectrum intersecting at $(\lambda,\lambda)$. This result is known for some classes of self-adjoint operators (which does not cover the situation we consider here), but in a smaller region than ours. Our approach is based on a dual variational formulation and is also applicable to other operators, such as the Laplacian. In addition, we prove an existence result for the nonhomogeneous situation, when the pair $(a,b)$ is not `between' the Fu\v{c}ik curves passing through $(\lambda,\lambda)\neq(0,0)$ and $p$ is a continuous function, sublinear at infinity.