Double Periodic Positive Solutions For Nonlinear Telegraph Systems

Because of the important physical background, many people pay attention to the existence of periodic solution for the telegraph equation and do much study work, such as the existence of double periodic solution on time and space, the existence of periodic solution on time with Dirichelet boundary condition, and the other related study. In this paper, we mainly consider the existence of wealy double periodic solution for a class of coupled telegraph systems.First of all, we obtain the sufficient conditions of the existence of at least one, two, or three double periodic positive solutions for the nonlinear telegraph system using Fixed point theorem of cone mapping (Fixed theorem of cone expansion and comprehension) and topological degree approaches (Leggett-Williams multiple fixed theorem).Second, we obtain a maximum principle for the linear doubly periodic telegraph system using an abstract result in a recent paper by Correa and Souto[5]. We discuss the existence of double periodic positive solutions for the nonlinear telegraph system on the basis of the above maximum using the method of upper and lower solution and thereom of fixed point index.Finally, we study two telegraph systems with more general nonlinear terms. One is that the nonlinear term may be negative, we call such problem as semipositone problem, so we discuss the existence of double periodic positive solution for the semipositone telegraph system. Another is that the nonlinear term may be singular, we also consider the existence of double periodic positive solution of the singular telegraph system.