The Research Of (k,n-k) Conjugate Boundary Value Problems For Ordinary Differential Equations

This dissertation discusses (k, n-k) conjugate boundary value problems from various perspectives by different fixed-point theorem. The paper is divided into six chapters. Main contents as follows:The chapter one is the Introduction Part. In this part, we introduce the research background and the current state of the research on (k, n-k) conjugate boundary value problems, and the main content in this paper.In chapter two, by using the fixed point theorem on cone, we discuss the positive solutions for systems of two-point (k, n-k) conjugate boundary value problems. Using the properties of Green's function, the existence of one positive solution is obtained by constructing the integral operator.The chapter three is the extension of the chapter two. We discuss the positive solutions for systems of m-point (k, n-k) conjugate boundary value problems. From another perspective, the existence of one positive solution is obtained by constructing a different integral operator from chapter two.In chapter four, by using the fixed point theory in a cone for strict set contraction operators, the existence of positive solutions for m-point (k, n-k) conjugate boundary value problems in Banach spaces is obtained.In chapter five, we discuss the positive solution for m-point (k, n-k) nonhomogeneous boundary value problems. By using the Schauder fixed point theory, the existence of positive solutions has been established while the nonlinearity satisfying certain conditions.The chapter six discusses the impulsive two-point (k, n-k) conjugate boundary value problems. By using the Avery-Peterson theorem, we obtain the existence of at least three positive solutions of this problems through looking for the formula and the properties of Green's function.