Existence Of Solutions For Boundary Value Problems Of Differential Equations

Nonlinear functional analysis originated in physics, mathematics and many other disciplines, has gradually developed into an important branch of modern mathematics. As an important theoretical tool, it has played a unique application value. Over the years, mathematicians conducted in-depth research to nonlinear functional analysis, and established a series of methods and theories for nonlinear problems. Those researches provide an efficient method for studying ordinary differential equation boundary value problem. Fixed point theorem of cone mapping plays an important role in solving these above problems. However, the limitation appears when the operator isn’t a cone mapping. Later, the fixed point theorem under the lattice structure, a new method for computing topological degree made up for the gap. Based on these fixed point theorem and predecessors’ researches, second order multipoint boundary value problem, existence of solutions for dynamic equations on the measure chain, existence of positive solutions for second order functional differential equation multipoint boundary value problems, and existence of solutions for integral boundary value problems of second order ordinary differential equation are studied in this thesis.There are five chapters in this thesis.Chapter 1 is the foreword of this thesis, and in this part, we mainly introduced the research background and contents of our research.Chapter 2, we study a second order multipoint boundary value problem, and analyze the existence of nontrivial solutions by the new method of fixed point theorem under the lattice structure. For this problem, we discuss corresponding result under the conditions that nonlinear term satisfies asymptotic linear, sublinear and superlinear respectively. The existence of sign-changing solution for this boundary value problem is obtained under the condition that the nonlinear term satisfies asymptotically linear condition, and the existence of positive,negative and sign-changing solutions are obtained under the condition that the nonlinear term satisfies sublinear condition, and the existence of negative and sign-changing solutions are obtained under the condition that the nonlinear term satisfies superlinear condition.Chapter 3,we study a dynamic equation on measure chain, and analyze the existence of nontrivial solutions by the new method of fixed point theorem under the lattice structure. Meanwhile, we discuss this problem under the conditions that nonlinear term satisfies asymptotic linear, sublinear and superlinear respectively. The existence of sign-changing solution for this boundary value problem is obtained under the condition that the nonlinear term satisfies asymptotically linear condition, and the existence of positive, negative and sign-changing solutions are obtained under the condition that the nonlinear term satisfies sublinear condition.Chapter 4, we study a multipoint boundary value problem of a second order functional differential equation by fixed point index method and the related properties of the first eigenvalue, and analyze the existence of solutions.Chapter 5, combined the Leray-Schauder degreees and cone theory,we study a nonlinear integral boundary value problem, and analyze the existence of sign-changing solutions and the multiplicity of solutions.