Several Problems Of Multiplicity 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 bing one of the most important research fields in modern mathematics. It mainly includes partial ordering method, topological degree method 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 degree for 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. 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 Jingx-ian etc., had proud works in various fields of nonlinear functional analysis. (See [1-12]).The present paper mainly investigates existence of solutions, multiplicity, exact number of solutions for some boundary value problems of differential equations by using topological degree ,cone theory and monotone iterative technique. It is made up of six chapters and the main contents are as follows:Chapter 1 gives serval lemmas on fixed point index, which will be used in next chapters. The idea of Lemma 1.7 is the especial case of Dancer's conjecture (see paper[3,31,32]), the proof of this lemma is containing in the paper [32,33].Chapter 2 investigates the existence and multiplicity of nontrivial solutions for the fourth-order two point boundary value problemsWhere f is continuous. Making use of the theory of fixed point index in cone and Leray-Schauder degree, under general conditions on nonlinearity, we prove that there exist at least six different nontrivial solutions for the fourth-order two point boundary value problems. Furthermore, if the nonlinearity is odd, we obtain that there exist at least eight different nontrivial solutions. By the same way, we can study 2n order two point boundary value problem(see[84]).In Chapter 3, By means of calculation of the fixed point index in cone, we study the existence of one or two positive solutions of the fourth-order two point boundary value problems with two parametersIn Chapter 4, we study the expression and properties of Green's function for nth order m-point boundary value problem.where furthermore,we obtained the existence of positive solutions by meas of fixed point index theory.By using of the shooting method, Chapter 5 establishes the exact multiplicities of positive solutions for a class of differential equations involving p-Lapalacian operatorWhere p > 1.Chapter 6 investigates the existence of solutions of periodic boundary value problems for nonlinear second order functional differential equationsBy establishing comparison results, criteria on the existence of maximal and minimal solutions are obtained.