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An Ambrosetti-Prodi-type result for periodic solutions of the telegraph equation. (English) Zbl 1149.35059

Summary: Using Leray-Schauder degree theory, a theorem of upper and lower solutions and a strong maximum principle for the telegraph equation
\[ \begin{aligned} &u_{tt}-u_{xx}+ cu_t+f(t,x,u)=s,\\ &u(t+2\pi,x)= u(t,x+2\pi)= u(t,x), \quad (t,x)\in\mathbb R^2, \end{aligned} \]
where \(c>0\), \(f:\mathbb R^3\to\mathbb R\) is a continuous function \(2\pi\)-periodic in \(t\) and \(x\), \(s\) is a real parameter. We prove an Ambrosetti-Prodi-type result for periodic solutions of the telegraph equation.

MSC:

35L70 Second-order nonlinear hyperbolic equations
35B10 Periodic solutions to PDEs
35B50 Maximum principles in context of PDEs
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