Multiplicity Results For Periodic Solutions To A Class Of Second Order Delay Differential Equations

This paper mainly concerns with the existence and multiplicity of periodic solu-tions to the following second order delay differential equationwhere f∈C1(R2, R) andτ> 0 is a given constant. The main idea is to estab-lish the corresponding variational functional and reduce the problem of finding peri-odic solutions of delay differential equation to an existence problem for an associatedHamiltonian system and then find critical points of the corresponding functional.The historical background and the recent development of problems under con-sideration is introduced in Chapter 1 on the periodic solutions to delay differentialequations and some preparations are also given.Chapter 2 is focused on the studies in the case that the nonlinearity f possessesa symmetric property and is an odd function. When f grows asymptotically linearboth at zero and at infinity, some interesting results for the existence and multiplicityof periodic solutions are obtained by using the critical point theory and Z2 pseudogeometrical index theory.In chapter 3, by making use of the critical point theory and S1 geometrical indextheory, some new results for the existence and multiplicity of periodic solutions areobtained without the oddness assumption on the nonlinearity f.