Periodic solutions of semilinear higher dimensional wave equations (Q1862310)

Using an approach of the Castro-Lazer type, the author proves existence and uniqueness results for the weak periodic-Dirichlet problem on \([0,2\pi] \times [0,\pi]^n\) for semilinear wave equations of the type \[ \square u = p(t,x,u). \] When \[ \theta_- (\xi_1 - \xi_2)^2 \leq (\xi_1 - \xi_2)[p(t,x,\xi_1) - p(t,x,\xi_2)] \leq \theta_+ (\xi_1 - \xi_2)^, \] the existence and uniqueness of a solution is proved when \([\theta_-,\theta_+] \subset \rho(\square),\) where \(\rho(\square)\) denotes the resolvant of \(\square\) with the periodic-Dirichlet conditions. Existence is also guaranteed when \((\theta_-,\theta_+) \subset \rho(\square)\) and \[ \beta_- \leq \liminf_{| \xi| \to \infty} 2P(t,x,\xi)/\xi^2 \leq \limsup_{| \xi| \to \infty} 2P(t,x,\xi)/\xi^2 \leq \beta_+, \] where \(P(t,x)\xi) = \int_0^\xi p(t,x,s)\,ds,\) \(\theta_- \leq \beta_- \leq \beta_+ \leq \theta_+\) and \([\beta_-,\beta_+] \subset \rho(\square).\) Notice that the first result is an easy consequence of Lemma 9 of \textit{J. Mawhin} [Chaos Solitons Fractals 5, No. 9, 1651--1669 (1995; Zbl 1079.35532)].