Research On Existence Of Multiplicity Solutions For Nonlinear Differential Equations

In this thesis, we mainly consider the existence and multiplicity of positive solutions for multi-point boundary value problems in abstract space, the existence of multiple solutions for some kinds of differential equation with impulses, and the existence and multiplicity of positive solutions for singular functional differential equation with delay and for periodic boundary value problems of impulsive functional differential equation. The thesis contains five chapters.In Chapter 1, the background and history of the existence and multiplicity of solutions for boundary value problems of differential equation in abstract space, boundary value problems of impulsive differential equation, and (periodic) boundary value problems of functional differential equation are briefly addressed, and the main works of the thesis are given.Chapter 2 investigates the existence of positive solution for two kinds of multi-point boundary value problems in abstract space. In Section 2.2, by using the fixed point theorem of strict-set-contractions and theory of Kuratowski measures noncompactness, we prove the existence theorem of multiplicity positive solutions for second order m- point boundary value problems. In Section 2.3, by using the method of upper and lower solutions coupled with the properties of weak topological and the modified functions, we obtain the existence of at least three positive solutions for second order m-point boundary value problems with nonlinear terms depending on the first derivate order in abstract space.In Chapter 3, we study the existence of multiple solutions for some kinds of impulsive differential equation that the first order derivate appears in the nonlinear term. Making use of the functionals fixed point theorem of Avery, Section 3.2 gives the existence of at least three positive solutions of second order differential equation with impulses in infinte intervals. By means of the theorem of Leray-Schauder degree and the method of upper and lower solutions, Section 3.3 investigates the existence of at least three solutions of second order two-point boundary value problems for impulsive differential equation when the nonlinearity satisfies the Nagumo's condition. In Section 3.4, using the theory of the coincidence degree of Mawhin and the method of the two pairs of upper and lower solutions, the existence theorem of at least three and 2n-1 solutions for three point boundary value problem of second order impulsive differential equation at resonance is established.Chapter 4 establishes the existence theorem of multiplicity positive solutions of singular second-order functional differential equation with delay. In Section 4.2, by applying the fixed point index theorem and the changing technique of translation, we obtain the conditions for the existence of at least two, three and 2n + 1 positive solutions respectively of boundary value problem for second order two-point singular functional differential equation with delay that the nonlinearity may take negative value and has no lower bound.Chapter 5 studies the existence and multiple positive solutions for two kinds of periodic boundary value problems for impulsive functional differential equations. In Section 5.2 and 5.3, by using the fixed point theorem in a cone, we establish the existence of at least one and multiple positive solutions of first-order and second-order periodic boundary value problems for impulsive functional differential equation. In particular, our result does not assume any monotonicity condition on the nonlinearity which is usually needed in literatures.