Several Kinds Of Fixed-point Theorem And Its Application In Nonlinear Analysis

In this article, not only Banach's fixed-point theorem and its wide application,but also Krasno-selskii's fixed-point theorem and its application are studied.The whole article can divide into three parts. In the first part, Banach's fixed-point theorem that is compressing map theorem is investigated. Also, its important applications in solving the existence and uniqueness of solutions for differential equations and integral equations are obtained. The problems of the existence and uniqueness of solution in equations are discussed by applying instances.The theory value and application value of Banach's fixed-point theorem are both emphasized in the chapter. In the second part, the existence and uniqueness of random integral solutions is proved for a class of maccretive population dynamics with random migration perturbations in arbitrary finite interval of time by using Banach fixed point theorem of the nonlinear functional theory, which are the improvement of the results obtained by using Schauder fixed point theorem and Sadovskii fixed point theorem. In the last part, the existence of positive solution for a second-order three-point boundary value problem is investigated. Several sufficient conditions for the existence of positive solution are obtained under the condition that nonlinear item are all superlinear or all sublinear or one is superlinear,another is sublinear by using Krasnoselskii's fixed point theorem, which has improved and generalized the results of the the existence of positive solution for a second-order three-point boundary value problem under the condition that nonlinear item are only superlinear or only sublinear.