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We study generalized solutions of the nonlinear wave equation$$u_{tt}-u_{ss}=au^+-bu^-+p(s,t,u),$$ with periodic conditions in $t$ andhomogeneous Dirichlet conditions in $s$, under the assumption that the ratio ofthe 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 existenceof two families of curves (which may coincide) in the Fu\v{c}ik spectrumintersecting at $(\lambda,\lambda)$. This result is known for some classes ofself-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 variationalformulation 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 atinfinity.

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