On systems of telegraph equations

Abstract: Let $\Omega=]0,2\pi[\times]0,\pi[$ and assume that $(t,x,s)\mapsto g(t,x,s)$ from $\Omega\times{\Bbb R}^n$ to ${\Bbb R}^n$ is a Carathéodory function which is $2\pi$-periodic in $t$. We shall discuss the nonresonance for the system of wave equations $$ \text{(WE)}\qquad \cases u_{tt}-u_{xx}-g(t,x,u)=h(t,x) \quad \text{in }\Omega
u(t,0)=u(t,\pi)=0,\quad t\in{\Bbb R}
u(t+2\pi,x)=u(t,x),\quad (t,x)\in{\Bbb R}_\times]0,\pi[, \endcases $$ i.e., the conditions for $g$ such that a solution $u\in L^2(\Omega,{\Bbb R}^n)$ exists for any given $h\in L^2(\Omega,{\Bbb R}^n)$.

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