Global Structure Of Positive Solutions For Some Periodic Boundary Value Problems Of Second-order Ordinary Differential Equations

This thesis focuses on the global structure of positive solutions of some periodic boundary value problems for second order ordinary differential equations with a pa-rameter, via bifurcation techniques and the theory of spectrum of periodic boundary value problems of linear second order ordinary differential equations. It is divided into six chapters.In chapter one, The background material and some preliminaries are introduced. The main problems as well as the corresponding results are also presented.By using Rabinowitz global bifurcation theorems [48,50], chapter two is con-cerned with the global structure and the stability of positive solutions for periodic boundary value problems of second order ordinary differential equation whereλ∈R is a parameter. Different global structures of positive solutions for (0.0.1) are achieved with the condition that f satisfies asymptotically linear growth at the origin, according to whether f has zeros in (0,∞) and whether a is sign changing. Based on these results, the existence and multiplicity of positive solutions for (0.0.1) are obtained in an interval of A. Furthmore, by adding sign condition for the second derivative of f, we also discuss bifurcation direction and stability of the branch of positive solutions. The main results in this chapter improve and generalize many previous theroems on the existence of positive solutions.Chapter three continue to study the global structures of positive solutions for (0.0.1). In this chapter,f is required to satisfy sublinear growth at the origin, in the case that whether f has zeros in (0,∞) or not and whether a is sign changing or not. Such circumstances, Rabinowitz global bifurcation theorem can not applied directly to discuss the branch of positive solutions for (0.0.1) bifurcated from the trivial solution. To overcome this difficulty, a brand new approach by approximation is developed, and then the global structures of positive solutions for (0.0.1) can be also achieved. The results in this chapter improve and generalize some main results of [22] (J. R. Greaf, L. J. Kong, H. Y. Wang, J. Diff. Equ.,2008).In chapter four,∫is needed to satisfy suplinear growth condition at the origin. To achieve different global structures of positive solutions of (0.0.1) according to whether f has zeros in (0,∞) and whether a is sign changing, more difficulties is appearing compared to chapter three. The results in this chapter also improve and generalize some main results of [22](J. R. Greaf, L. J. Kong, H. Y. Wang, J. Diff. Equ.,2008).The fifth chapter is devoted to obtaining global structures of positive solutions for (0.0.1) under the conditions that the limits lims→0+(f(s))/s and limss→∞+(f(s))/sdon't exist and the weighted term a is nonnegative. Furthermore, the results are extended to a more generalized problem The main methods applied in the chapter include topological degree theory and the bifurcation theorems from an interval[41,42], which are brand new in the sence for periodic boundary value problems. Different from the first three chapter, the branch of positive solutions bifurcates from an interval in this chapter.Inspired by [74], in which B. P. Rynne studied the existence of infinitely many positive solutions for a semilinear elliptic problems, in the sixth chapter we pay our attention to the oscillation property of the branch of positive solutions and the existence of infinitely many positive solutions for the problem For the above problem, we show that with certain oscillation condition on the non-linearity g, the continuum of positive solutions of the problem (0.0.3) also begin to oscillate correspondingly,and the amplitude of these oscillations is bounded away from zero.The results in this chapter generalize some results of[74]to the periodic boundary value problems and improve the main conclusions of[24].