The Study On Positive Periodic And Homoclinic Solutions For Some Singular Functiolnal Differential Equations

The existence of periodic solutions for differential equations witli singularities has attracted many researchers because singular equations describe many problems in the applied sciences,such as Brillouin focusing system,nonlinear elasticity,and Bose-Einstein condensate.Besides these applications,it has been found that the theory of the singular functional differential equations play a significant role in studying of the Lyapunov stability of periodic solutions of Lagrangian equations.Therefore,the studies on the existence of positive periodic and homoclinic solutions for singular functional differential equations have practical significance.Inspired by the above facts,in this paper,by means of Mawhin's contianuation theorem,the problems of existence of positive periodic solutions are studied for a kind of preseribed mean curvature Lienard equation with a singularity and the neutral Lienard differential equations with a singularity,some sufficient conditions for the existence of positive periodic solutions are given.On the basis work of positive periodic solutions,the existence results of homoclinic solutions for n-dimensional prescribedmean curvature p-Laplacian equations are established.Our work generalizes and improves the corresponding results in the related lit-erature.Especially,it was the first time to consider the existence problem of positive periodic solutions for the neutral functional differential equations.