Positive Solutions Of Periodic Boundary Value Problems For Third-order Differential Equations

Differential equation is an important bridge connecting mathematic to the real world,which is mainly divided into ordinary differential equation and partial dif-ferential equation.As the basic theory of ordinary differential equation,the study of integer order differential equation has been applied to many fields to solve practi-cal problems,and fractional differential equation is an extension based on integer order.In the past decades,there have been many achievements in the study of third-order differential equations,such as boundary value problems,periodic boundary value problems,and the insertion of points,the exploration of solutions and posi-tive solutions.Periodic boundary conditions are widely used in computer simulation,metamaterial band structure and molecular dynamics simulation.Therefore,the study of periodic boundary value problems of third-order differential equations has attracted more and more attention of experts and scholars.This paper will study the existence of positive solutions for periodic boundary value problems of nonlinear third-order differential equations(?) where b-a2/3<1/16,|c+1/24a3-ab/3|<2(1/16-b+a2/3)1/2(4-b+a2/3)/3(?),c>0,??C([0,27?]×[0,+?),[0,+?)),a,b,c ?R.In the first chapter,some research background and basic theoretical knowl-edge about boundary value problems of differential equations are introduced.These basic contents provide a theoretical basis for the follow-up study on the existence of positive solutions of periodic boundary value problems of third-order differential equations.In the second chapter,we first introduce the application and development of Green's function in various disciplines and research fields,and indicate the impor-tance of Green's function.Secondly,the basic theory of differential equations is used to obtain the characteristic function of the problem to be studied,and the Car-dano formula is used to convert the specific problem into four cases,and the specific form of eigenvalues in different cases is solved.Finally,the Green's function in each case is obtained.In the third chapter,we explore the properties of Green's function,and make preparations for the follow-up study.According to the Green's function and co-efficient conditions in four different cases solved in the second chapter,we explore the properties of the corresponding Green's function,and prepare for proving the existence of positive solutions for periodic boundary value problems of third-order differential equations.In the fourth chapter,we introduce the fixed point index theory on cone.The positive solution of the periodic boundary value problem of the differential equation is transformed into the fixed point of the corresponding operator.The non-zero fixed point of the operator is found by using the fixed point index theory on cone,and it is proved that the operator is completely continuous.Finally,we prove the existence of positive solutions of the periodic boundary value problem of the third-order differential equation under four different conditions.