Research On The Solutions Of Differential Equations In Banach Spaces

Nonlinear functional analysis is a research discipline both to have the profound theory and to have the widespread application. It is by the nonlinear problems appearing in mathematics and the natural sciences for a background, to establish some general theories and method to handle nonlinear problem. Because it can commendably explain all kinds of natural phenomena, its rich theory and advanced method have provided the effective theory tool for solving the nonlinear problem which emerges in the technical domain one after another incessantly. To solve all kinds of actual problems which correspond nonlinear integral equation, the differential equation and the partial differential equation, it plays role which can not be substituted.The main questions to research in this thesis are the existence of solution for nonlinear operator equation, uniqueness of solution, multiple solution, structuring the iteration algorithm converging to solution, and applying the fixed point method, half ordering method, the upper and lower solution method, topology degree method and so on to study initial value problems for integro-differential equations in Banach spaces or singular nonlinear boundary value problems, we obtained many new results, and published (include preprint) more than forty dissertations, the main publication of the paper publishing is: 《Nonlinear Analysis》,《J. Math. Anal. Appl.》 ,《Comput. Math. Appl.》 ,《Applied Math. Letters》, 《Applied Math. Comput.》 , 《Dynamic Sys. Appl.》,《Dynamic of Continuous, Discrete and Impulsive Systems》 etc. Because the length is limited, this thesis only selects twelve papers to come the key introduction.The whole paper is divided into six chapters.In the chapter one, we mainly introduce the developing history and background of nonlinear analysis, and main results of this thesis.The chapter two, we firstly generalize the famous strict set compression fixed point theorem, and then by using fixed point theorem we obtained and Monch fixed point theorem, under the weaker conditions, consider the existence of global solutions for nonlinear integral equation of Volterra type in Banach spaces and...