Nonlinear Differential Equations And Their Applications

Nonlinear functional analysis is an important branch of mordern analysis mathematics. It can explain a lot of natural phenomena clearly,so more and more mathematicans and mathematical researchers are devoting their time to it. Among them, the nonlinear initial value problem and boundary value problem comes from a lot of branches of applied mathematics and physics, it is at present one of the most active fields that is studied in analyse mathematics. The present paper employs the cone theory, fixed point theory, fixed point index methord and upper and lower methord and so on, to investigate the existence of solutions to initial value problem and boundary value problem of several kinds of nonlinear systems of differential equations. The obtained results are either new or intrinsically generalize and improve the previous relevant ones under weaker conditions.The thesis is divided into four sections according to contents.In chapter 1, it is the introduction of this paper, which introduces the main contents of this paper.In chapter 2, Allowing h(t) is singular at t=0 and t=1,the existence of multiple positive solutions of second-order three-point boundary value problems is attained.The existence results are given by using Krasnoselskii fixed point theorem and Leggett-Williams fixed point theorem.In Chapter 3, Allowing h(t) is singular at t=0 and t=1,the existence of positive solutions of sublinear impulsive Sturm-Liouville boundary value problems is attained.The existence results are given by means of the fixed point index under some conditions concerning the first eigenvalues corresponding to the relevant linear operator .In Chapter 4, In this paper,the initial value problem of the nonlinear integro-differential equations in Banach spaces is investigated under the condi- tion weaker than the continuity of f.The existence and uniqueness and iterative approximation of solution for the initial value problem is abtained.The results presented here not only improve and extend many results,but also apply to other aspects.