Multi-point Boundary Value Problems And Functional Differential Equations With Positive Solutions For A Number Of Studies

In this thesis, we study mainly the existence of positive solutions for multi-point boundary value problems and boundary value problems for functional differential equations. The main results is summarized as follows:1. By means of index theory of fixed points in a cone, we prove the existence theorems of multiple positive solutions for two kind of nonlinear m-point boundary value problems for the one-dimensional p-Laplacian.2. Using the index theory of fixed points for A-proper semilinear operators, we study the existence of positive solutions for the multiple point boundary value problems at rersonance. The existence theorems of positive solutions for three-point boundary value problems at rersonance for a kind of nonlinear second-order differential equation are obtained.3. We use the Leggett and Williams fixed point theorem to discuss the existence of three positive solutions for the boundary value problem of second-order nonlinear functional differential equation. Further, we introduce the concept of partially symmetric positive solution and discuss the existence of three partially symmetric positive solutions for the boundary value problem of second-order functional differential equation. Lastly, the existence theorem of three positive solutions for the boundary value problem of second-order singular functional differential equation be established by applying the index theory of fixed points in a cone.4. Using the Bielecki's norm and the index theory of fixed points in a cone, we study the existence of positive solutions of boundary value problems for second order functional differential equations on infinite intervals.5. By means of a nonlinear alternative theorem of Leray-Schauder type and Guo-Krasnoselskii's fixed point theorem in a cone, we establish the existence theorem of a positive solution to a singular coupled system of nonlinear fractional differential equations.