Two Classes Of Existence Of Solutions Of Nonlinear Boundary Value Problems

With the great development of science and technology, all sorts of nonlinear problems have resulted from mathematics, physics, chemistry, biology, medicine, economics, engineering, cybernetics and so on. During the development of solving such problems, nonlinear functional analysis has been being one of the most importantresearch fields in modern mathematics. It mainly includes partial ordering method, topological degree theory, cone theory and the variational method. 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. L. E. J. Brouwer had established the conception of topological degreefor finite dimensional space in 1912. J. Leray and J. Schauder had extend the conception to completely continuous field of Banach space in 1934. afterward E. H. Rothe [1-2], M. A. Krasnosel'skii [3-4], H. Amann [5], K. Deimling [6], M. S. Berger [7], etc. had carried on embedded 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 proud works in various fields of nonlinear functional analysis (see [8-16]).The present paper mainly investigates existence of position solutions for some boundary value problems of ordinary differential equations with p-Laplacian and singular Sturm-Liouville boundary value problems for a second-order differential system by using topological degree theory, cone theory or the method of lower and upper solutions. It is made up of five chapters and the main contents are as follows:Chapter 1 gives serval difinitions and fixed-points theorems, which will play an important role in next chapters.Chapter 2 considers existence of solution for the nonlinear two-point boundaryvalue problems with one-dimensional p-Laplacian operatorBy using Leray-Schauder fixed-point theorem, several existence theorems of solutionsare established for the class of equations. The man, results show that the class of equations has at least one solution or positive solutions if the "height" of nonlinear terra is appropriate on a bounded subset of its domain. By the same way, we can study the boundary value problemIn Chapter 3, we study the following nonlinear four-point boundary value problems with a p-Laplacian operatorSeveral sufficient conditions for the existence of positive solutions are obtained by constructing a completely continuous operator and combing fixed-point theorem of cone expansion and compression of norm type.In Chapter 4, we consider the following nonlinear singular four-point boundaryvalue problems with p-Laplacian operatorBy applying the fixed-point theorem of cone expansion and compression of norm type, several sufficient conditions for existence of positive solutions are established.Chapter 5 investigates the existence of positive solutions of singular SturmLiouvilleboundary value problems for a second-order differential system. A necessaryand sufficient condition for the existence of C[0,1]×C[0,1] positive solutions as well as C¹[0.1]×C¹[0, 1] positive solutions is given by means of the method of lower and upper solutions. Our nonlinearity f_i(t,x₁,x₂) may be singular at x₁ = 0. x₂ = 0, t = 0 and/or t = 1, i = 1, 2.