中文摘要：

本文讨论非线性电报方程u（tt）-u（xx）＋cut=F（t,x,u）,（t,x）∈R^2时空双2π周期解的存在性。改进了Ortega与Robles-Perez关于线性电报方程双周期解的极大值原理,应用新获得的极大值原理,推广了相应的上下解定理,并且加强了极大值原理的结论,建立了线性方程解的强正性估计,利用这个强正性估计及锥上的不动点定理获得了超线性电报方程及奇异电报方程正双周期解的存在性。

英文摘要：

This paper deals with the existence of doubly periodic solutions for nonlinear telegraph equation u（tt）-u（xx）＋cut=F（t,x,u）,（t,x）∈R^2where c 〉 0 is a constant, F ∈C（R^3, R） is 2π-periodic in t and x. We improve the maximum principle built by Ortega and Robles-Perez for linear telegraph equation. Using our new maximum principle, we extend Ortega and Robles-Perez＇s theorem of upper and lower solutions. Moreover, we obtain a strongly positive estimate for the solutions of linear telegraph equations, which is the improvement of the maximum principle. Using this strongly positive estimate and the fixed point theorem of cone mapping, we obtain the existence of positive doubly periodic weak solutions of superlinear and singular telegraph equations.