Study On Existence Of Positive Periodic Solutions For Two Kinds Of Singular Differential Equations

Singular differential equations in the physical,biological,engineering,economic and other fields are used widely,and they have attracted many scholars.In recent years,using the methods of continuation theorem,fixed point theorem,topological degree theory,and so on,scholars got a lot of results of the existence of positive periodic solutions for singular differential equations,and most of the results is the existence of positive periodic solutions of singular differential equations with an attractive type or a repulsive type.However,indefinite singular differential equations have received little attention for a long time.Indefinite singularity as the combination of attractive singularity,repulsive singularity and non-singularity,it is very complicated to study,and there are very few works about it.In this paper,at first,we introduce the application background of singular differential equations,and we study sufficient conditions for positivity and negativity of Green's function of second-order linear differential equations in three cases,the result improve and supplement conclusions of Green's function in the existing literature.Next,we study the existence of positive periodic solutions of a kind of indefinite singular differential equations,the difficulty lies in dealing with the indefinite singular term.In this paper,we transform the indefinite singular differential equations into normal singular differential equations with a strong repulsive type by means of LeraySchauder alternative principle and a transformation of variable,overcoming the difficulty that indefinite singular can't confirm its property.And we discuss the existence of positive periodic solutions of the equations at different cases.Furthermore,we overcome the difficulty that upper and lower bound of positive periodic solutions of differential equations is difficult to estimate,and give the existing areas of positive periodic solutions.Lastly,as an application of singular differential equations in biology,we study a resources and population model in an isolated island(Basener-Ross model).We first transform the model into a second-order differential equation.Depending on the parameters,the equation may have a singularity or not.We give existence conditions of positive periodic solutions of the model by using positivity and negativity of Green's function,Man?asevich-Mawhin continuation theorem,Leray-Schauder alternative principle and fixed point theorem in cones in different cases.