Maximum principles and the method of upper and lower solutions for time-periodic problems of the telegraph equations ✩

This paper deals with the existence of bounded time-periodic solutions for the nonlinear telegraph equation utt −uxx + cut = F(t,x,u), (t,x) ∈ R2, where c > 0 is a constant, F ∈ C(R3,R) is 2π-periodic in t . We build a maximum principle for the time-periodic solutions of the corresponding linear telegraph equation. Using this maximum principle, we develop a method of the upper and lower solutions for the time-periodic problem of the nonlinear telegraph equation and obtain some existence and uniqueness results. © 2006 Elsevier Inc. All rights reserved.