The Research Of Three Classes Differential Equation’ Solutions In Banach Space

According to contents.the whole paper is consisted of four chapters.Chapter1:It focuses on the introduction, and mainly introduces the research of back-ground and situation domestic and overseas, of nonlinear functional analysis. And then the introduction shows some related stages, which nonlinear functional analysis’s development ex-periences, as well as those representatives and their main achievements in this stage. Finally, it summarizes that the text researches the existence and uniqueness of three type of Banach spaces differential equation’ solution, based on the theory of fixed point theorem, non-compactness measurement, and condensing mapping.chapter2:Not all right-hand-side functions of differential equation are continuous. How-ever, the properties of solution have been researched, in the situation of that the right-hand-side functions are continuous in many articles. Besides, to make sure the existence of solution, it is commonly with the limit of compactness condition and that the right-hand-side functions is essential. This chapter is based on the limitation that, right-hand-side is not only continuous, but also without the limit of compactness condition, use the knowed lemma first, to derivate a new comparison theorem.We apply comparison theorem and monotone iterative techniques, to research Banach space’s differential equation with discontinuous terms: On the premise of satisfying related conditions, we get the result of initial value problem(2.1)’s equation, which are the existence and uniqueness.Chapter3:In this chapter, we mainly focus on the initial value of first-order impulsive differential equations: We have used the basic methods to research nonlinear functional analysis. On the condition of satisfying basic assumptions in this chapter, it mainly uses monotone iteratiive techniques. Firstly, it has proved the existing and unique result of the initial value’s solution(3.1), presented the approximation sequence of related solutions, and made the error estimation of solution-s. After that, we have discussed following problem of second-order impulsive differential equations: We apply the solution of first-order impulsive differential equation to study the second-order impulsive differential equation(3.10), and gain the existing result similar to the solution of first-order impulsive differential equation.Chapter4:In this chapter, we mainly discuss the existence of initial value of first-order mixed nonlinear differential equation: We understand related properties of condensed image and use the fixed point of Banach spaces’ condense mapping, and by researching solution’s existence of Banach spaces’ first-order mixed differential equation(4.1), we have obtained relatively reasonable results and applied these re-sults to second-order mixed differential equation: On the relatively looser condition, the existence result of second-order differential equation(4.6)’s solution is obtained, which is simplifying the common methods.