Positive Solutions For Several Kinds Of Multipoint Boundary Value Problems Of Differential Equations

In later years, 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 one of the most important research fields in modern mathematics. It mainly includes partial ordering mathod, topological degree method and the variational method. Also it provides a much effect theoreretical 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 degree for finite dimensional space in 1912. J. Leray and J. Schauder had extended the conception to completely continuous field of Banach space in 1934, afterward E. Rothe, M. A. Kras-nosel'skii, P. H. Rabinowitz, H. Amann, K. Derailing had caried 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 ficlds of nonlinear functional analysis. (See [1-12]).The present paper mainly investigates existence of solutions, multiplicity for some boundary value problems of differential equations by using topological degree, cone theory and monotone iterative technique. And the main contents are as follows:Chapter 1 gives several lemms on fixed point index, which will be used in next chapters, which play an important role in next chapters.Chapter 2 considers the following semipositone singular multipoint boundary vaue problem: where a_i,b_i,ξ_i>0(i=1,2…,m-2) are constants with 0 <(?)< 1, 0 <(?)<1,f(t,u)∈C((0,1)×[0,+∞),[0+∞))(f(t,u) may be singular at t = 0, 1),p(t)∈C([0,1],[0,+∞)),q(t):(0,1)→(-∞,+∞)is continuous and the limit of q(t)may be-∞. We obtain the sufficient conditions of the existence of at least one positive solution by theory of the fixed point index.Chapter 3 considers the following generalized Sturm-Liouvlle boundary value prob-lem with two parameters:where f∈C([0,1]×[0,+∞),[0,+∞))satisfying f(t,u)(?)0andα,β≥0,a,b,c,d∈[0,+∞)andρ:=ac+bc+ad>0,ξ_i∈(0,1),α_i,β_i∈[0,+∞)(i=1,2,…,m-2)are constants.Chapter 4 investigates the existence of multiplicity of the multipoint impulsive boundary value problem with p-Laplacian operator:whereφ_p(s) is p-Laplacian operator, i.e.,φ_p(s)φ_p(s)=|s|^p-2s,p>1,(φ_p)^-1=φ_q,(?)+(?)=1, t_k(k=1,2,…,m, where m is a fixed positive integer) is fixed points with 0 1< t₂<…k<…m<1,ξ_i(i=1,2,…,l,where l is a positive integer)∈(0,1) 0<ξ₁<ξ₂<…<ξ_l<1andξ_i≠l_k,i=1,2,….l.k=1,2,…,m,â–³u|_t=tk denotes the jump of u(t) at t = t_k, i.e.,Where u(t_k⁺),u(t_k^-) represent the right-hand limit and left-hand limit of u(t) at t =t_k respectively.