The Existence Of Solutions For Functional Boundary Value Problems Of Differential Equations At Resonance

The differential boundary value problem is an important branch of differential equation research.It is also widely used.The problems in the disciplines related to solving derivatives such as physics,engineering,biology,medicine and materials science can be transformed into the solution of the differential equation(group)boundary value problem,so the solution of the differential boundary value problem has aroused the concern of many scholars.In the past,the boundary conditions of the boundary value problem of the differential equation are generally specific,such as the boundary condition of two points,the boundary condition of multipoints,and the boundary condition of integral and differential.The boundary condition of the differential equation can be generalized by the functional boundary condition of the differential equation,and the boundary condition can be abstracted so that the abstract boundary condition contain the specific boundary condition of each class.As long as the boundary conditions satisfies certain linear conditions,they can be replaced by abstract boundary conditions.Because it can rich boundary value theory of of the study,the study of practical problems is of great significance.Compared with the boundary value problem of the nonresonant differential equation,it is more difficult to solve the existence of solution of the boundary value problem at resonance.Based on these considerations,this paper will study two boundary value problems at resonance:1)The existence of solutions for boundary value problems of second order nonlinear differential equations with functional boundary conditions;2)The existence of solutions for (k,n-k) conjugate boundary value problems with functional boundary conditions;In this paper,by the Mawhin continuous theorem,defining the appropriate Banach space and norm,giving the appropriate projection operators P and Q,we give the detailed existence theorems of boundary value problems and examples to illustrate the results.