Two Kinds Of Problems About Q-difference Equations

The main object of this paper is to study the spectral properties of high order linear q-difference equations and Floquet theory of the 2nd order q-difference equation.The paper is mainly divided into five chapters.In the first chapter, we introduce the background and the development of q-fractional calculus, and the main results in this paper are presented.In the the second chapter, some definitions and basic properties are presented. In the first section, we introduce the q-difference operator, present its concept and relative properties. In the second section, we discuss the q-exponential function which is very important to the proof in the third chapter.The third chapter is the main part of the paper. We investigate the singular boundary value problems of 2n-th order liner q-difference equations. It is proved that the corresponding resolvent operator of the q-difference operator is linear self-adjoint and completely continuous. The spectral properties of the singular boundary value problems of the 2n-th order liner q-difference equations axe obtained by means of the spectral theory of linear self-adjoint completely continuous operator.In the fourth chapter, we study the Floquet theory of the 2nd order q-difference equation in which the coefficients are q-m periodic. Firstly, we define the q-m periodic functions. Then we obtain a linear relationship between the non-trivial solutions y and gmy(qmx), i.e. qmy(qmx) = py. Finally, we get the expression of the solution by discussing the value of p.The last chapter is the summation of this paper, we point out some unsolved problems and the direction for the further research.