Existence Of Solutions For Ordinary Differential Equation System And Fractional Differential Equation

Since the 20th century, all sorts of nonlinear problems have resulted from mathematics, physics, chemistry, biology, medicine, economics, engineering, cy-bernetics and so on. During the development of solving such problems, nonlinear functional analysis has been one of the most important research fields in modern mathematics. L. E.J. Brouwer had established the conception of topological de-gree for finite dimensional space in 1912.J. Leray and.1. Schauder had extended the conception to completely continuous field of Banach space in 1934, after-ward E. Rothe, M. A. Krasnosel'skii, P. H. Rabinowitz, H. Amann, K. Deimling had carried on embedded research on topological degree and cone theory. Many well known mathematicians in China, say Zhang Gongqing, Guo Dajun, Chen Wenyuan, Ding Guanggui and Sun Jingxian etc., had proud works in various fields of nonlinear functional analysis. (Sce[1-9]).The method to research nonlinear problem mainly has topological degree method, partial order method, lower and upper solution method, fixed point theory, monotone iterative technique and so on. Also it provides a much effect theoretical tool for solving many nonlinear problems in the fields of the science and technology. And what is more, it is an important approach for studying nonlinear differential equations arising from many applied mathematics.The present paper mainly investigates existence of solutions for some dif-ferential equation system and fractional differential equation by using lower and upper solution method and monotone iterative technique. And the main contents are as follows:Chapter 1 gives some preliminary definitions and properties of nonlinear functional analysis, and gives several lemmas on the existence of fixed point, which play an important role in next chapters. Chapter 2 considers the following fourth-order and second-order differential equation system of singular Sturm-Liouville boundary value problem where f, g, ai,bi, ci,di,(i=1,2) satisfy the following hypothesis(Hi):ai>≥0, bi≥0, ci≥0, di≥0, ai+bi>0, ci+di>0, pi=aici+aidi+bici>0,i=1.2.(H2):f∈C((0,1)x(0,∞)3.[0,∞)). g∈C((0,1)×(0,∞)2,[0,∞)). f(t,1,1,1),g(t,1,1)∈C(0,1),f(t,1,1,1)> 0, g(t,1,1)> 0, t∈(0,1), that is,f(t,x1,x2,x3),g(t,x1,x2) may be singular at x1=0,x2=0,x3=0,t= 0 and or t=1.Chapter 3 by means of the lower and upper solution method and monotone iterative technique, investigate initial value problem involving Riemann-Liouville fractional derivatives where 0