Multiple doubly periodic solutions of semilinear dissipative hyperbolic equations (Q1916747)

The author extends some generalized Ambrosetti-Prodi-type conditions obtained for ordinary differential equations by Fabry, Mawhin, Nkashama and Ding, to the weak doubly periodic solutions of semilinear hyperbolic equations of the form \[ \beta u_t+ u_{tt}- u_{xx}+ g(t,x,u)= s+h(t,x), \] with \(\beta\neq 0\), \(\iint h(t,x)dt dx=0\), \(g\) has at most linear growth in \(|u|\), \(g\) is nonnegative and \(\lim_{|u|\to\infty} g(t,x,u)= +\infty\) uniformly in \((t,x)\). The proofs use topological degree arguments and delicate estimates.