| In this paper, we will study the optimal bounds of arbitrary eigenvalue ratios for one-dimensional p-Laplacian with separated endpoint boundary conditions using a com-bination of the Prufer transformation and comparision theorem.The paper is mainly divided into four chapters.The first chapter is the introduction of the whole paper.We talk about the background and significance of the problem, propose the problems to be studied and the main results in this paper.In the second chapter of the paper, we introduce the basic concepts and theories. This chapter is divided into two sections, we introduce the generalized trigonometric functions sinp (x), cosp (x) and its properties in the first section, and describes the generalized Prufer transformation in the second section, in order to meet the needs of this paper, two different forms of the generalized Prufer transform are given in this paper, they will be the key elements of the proof.The third chapter is the main part of the paper, consisting of four parts. Firstly, we consider some functions containing the generalized trigonometric functions tanp (x), and discuss its monotony. On this basis, we study the ratios of any two positive eigenvalues of the one-dimensional p-Laplace equation with potential function q= 0 and q? 0 using a combination of the Prufer transformation and comparision theorem. For q= 0, we first consider the ratios of any two positive eigenvalues of the one-dimensional p-Laplace equation with Dirichlet boundary conditions, and further extended to the general separate boundary conditions. For q? 0, the last two section get the bounds of the ratios of any positive eigenvalues with the first eigenvalue, further, we get the upper and lower bounds of any two positive eigenvalues with separated boundary conditions by comparing the eigenvalue with the same interval and different boundary conditions.The last chapter is the summation of this paper, we point out some unsolved prob-lems and direction for further research. |
PDF Full Text Request