The Study Of The Existence Solutions For Two Kinds Of Differential Equations

This paper mainly studies the behavior of two kinds of differential equations-nonlinear differential equation and kirchhoff-type equation.The main contents are studied in teo parts.In the first part,a new fixed point theory of mixed monotone operators is presented,and the conclusion is applied to boundary value problems,and the existence and uniqueness of its non-trivial solutions are obtained.The second part focus on the existence of solutions of two different types of kirchhoff type equations.The specific contents are arranged as follows.The first chapter expounds the historical background and significance of the corresponding differential equation,the research status at home and abroad,and describes the main research content of this paper in detail.The second chapter introduces the first main result of this paper,that is,the existence and uniqueness of the fixed point are obtained by introducing mixed monotone operator,using cone theory and monotone iteration method.Then the existence and uniqueness of non-trivial solutions are obtained for the specific boundary value problem of Sturm-Liouville.Some existing results are improved and generalized,and a new method for studying nonlinear differential equations is presented.In the third chapter,the second problem of this paper is expounded.The equation of type with general singular terms is studied by using the embedding theorem,Fatou lemma and the weak lower semi-continuity of norm.Firstly,the existence of global minimum of energy functional is proved and obtained.Then the variational method is used to obtain the existence result of the positive solution.In the fourth chapter,the kirchhoff type equations with critical exponent are studied.In this paper,the existence of the global minimum value m of energy functional is first obtained.and then the variational method is adopted to obtain a positive partial minimal solution of the problem.