Research On Some Periodic Boundary Value Problems For Difference Equations

This dissertation consists of six chapters. Its concern focuses on some type pe-riodic boundary value problems for difference equations. We investigate the spectral results of the linear periodic problem and obtain the existence results of solutions for some nonlinear periodic boundary value problems of difference equations.The first chapter is prolegomenon, in which the studying background, theoret-ical frame of this thesis are exhibited. And it displays the main problem we study and the main results we obtain.In the second chapter, applying Leray-Schauder principle, we study the follow-ing resonance periodic boundary value problems of difference equations with weight function a(·). The existence results of solution is achieved. Setλk the k-th eigenvalue of the linear problem corresponding to (0.0.1), and g is a continuous function. With some condition subjected to g and h, the existence result for solution of problem (0.0.1) is obtained.We pay our attention in the third chapter to the linear periodic boundary value problem of difference equation The distribution of the eigenvalues of the above problem are investigated systemati-cally and the properties of the eigenfunction are studied as well. The different parity of T decides the exact multiplicity of the eigenvalue of (0.0.2). At the same time, the multiplicity of sign-changing times of the eigenfunction are concluded. Wang and Shi[87] has obtained the eigenvalue's arrary of the general periodic boundary value problems of discrete case. However, the exact multiplicity of these eigenvalues is not quite clear, not mention to the properties of the corresponding eigenfunctions. Therefore, all content in this chapter is brand new in the sense for periodic boundary value problem in discrete case.The emphasis in the fourth chapter is to discuss the existence result of the symmetric solution for the periodic boundary value problem of difference equation With the help of the spectrum structure of the corresponding linear problem ob-tained in the third chapter, the existence result for symmetric solution of (0.0.3) is obtained, and these solutions's parity have been got. We get correlative existence results similar to that in continuous case.The existence result of the sigh-changing solution of the nonlinear periodic boundary value problem of difference equation is of our interest in the fifth chapter. The main tool we use in this chapter is the Rabinowitz bifurcation theorem. With the help of the spectrum structure of the corresponding linear problem obtained in the third chapter, it determines in this chapter the range of the parameter r to guarantee two sign-changing solutions. Specific sign-changing times of these solution are also expressed, and in addition, we can know the parity of these solutions. Many known results related to the positive solution of (0.0.4) has become individual cases.Both the Leray-Schauder principle and the bifurcation theorem are put to use to study the nonlinear periodic boundary value problem of difference equation Using the spectrum structure of the corresponding linear problem obtained in the third chapter, we got the multiplicity result of symmetric solution for (0.0.5) in the condition that the nonlinear term grows in sublinear state. Different range ofλcan determine different number of solutions of (0.0.5), and theses solutions has its specific parity.