Differential Equations Of Periodic Boundary Value Problem

Because of the important physical background, the existence problem of periodic solutions fordifferential equations has been studied for some good results. Many people pay attention to theexistence of periodic solutions for the telegraph systems and do much study work, such as theexistence of double periodic solutions for the nonlinear telegraph systems, the maximum principle oftelegraph systems, and the other related study. In this paper, we mainly consider the existence ofdouble periodic solution for a class of coupled differential systems and some other similar problems.First of all, different from the past, we will show the maximum principle of the telegraph systemsusing a abstract results by [23] and [32] though we relax the restrictions on a (t, x). In addition, usingthe strongly positive estimate and the fixed point theorem of cone mapping, we obtain the existence ofpositive doubly solutions of nonlinear telegraph system. While the nonlinear term may be negative,we will discuss the existence of double periodic positive solutions. In the end of this chapter, weestablish the existence and multiplicity results of positive doubly periodic solutions for a nonlinearsingular telegraph system with twin-parameters. The main method using here is a well-known fixedpoint theorem in a cone.Second, we study a coupled of second-order damped differential equations. One is that thenonlinear term be positive, we will discuss the existence of double periodic positive solutions.Another is that the nonlinear term may be negative, we can also get the existence of double periodicpositive solutions.Finally, we give some summary and outlook.