Existence Of Solutions For Boundary Value Problems Of Several Classes Of Second Order Nonlinear Impulsive Differential Equations

Nonlinear functional analysis is an important branch of morden analysis mathe-matics, because it can explain all kinds of natural phenomena, more and more math-ematicians are devoting their time to it. It mainly includes partial ordering method, topological degree theory, cone theory and the variational method. Also it provides an effect theoretical tool for solving many nonlinear problems in the fields of science and technology. And what is more, it is an important approach for studying nonlin-ear differential equations arising from many applied mathematics. L.E.J. Brouwer had established the conception of topological degree in finite dimensional spaces in 1912. J. Leray and J. Schauder had extended the conception to completely continuous oper-ators in Banach spaces in 1934, afterwards M. A. Krasnosel'skii [1-2], H. Amann [3], K. Deimling [4], etc. had carried on intensive research on topological degree and cone theory. Many well known mathematicians in China, say Zhang Gongqing, Guo Dajun, Chen Wenyuan and Sun Jingxian etc., had excellent works in various fields of nonlinear functional analysis (see [5-13]).The present paper mainly investigates the existence of solutions for boundary value problems of several classes of second order nonlinear impulsive differential equations in Banach spaces by using cone theorey, the method of lower and upper solutions, the fixed point theory and the iterative technique. It is composed of four chapters and the main contents are as follows:In Chapter 1, we first give several definitions and theorems, which will play an important role in next chapters.In Chapter 2, we consider the existence of solutions for boundary value problems of second order nonlinear impulsive differential equations in Banach spaces By using a new comparison result and the Monch fixed theorem, the existence theorem is established.In Chapter 3, we study the unique positive solutions for boundary value problems of second order nonlinear impulsive singular differential equations in Banach spaces By applying the Banach fixed point theory and the iterative technique, we establish the unique positive solution for the boundary value problems. In addition, an explicit iterative sequence for approximating the positive solution is derived together with an error estimate.In Chapter 4, we investigate the existence of solutions for multipoint infinte bound-ary value problems of second order nonlinear impulsive singular differential equations in Banach spaces By constructing a special cone and using the Monch fixed point theory with the mono-tone iterative technique, we establish the existence of positive solutions and the iterative sequence for approximating the positive solution of boundary value problems in Banach spaces.