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 equationutt−uxx+cut=F(t,x,u),(t,x)∈R2, where c>0c>0 is a constant, F∈C(R3,R)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.