Study Of Spectral Problems Of Second-Order Symmetric Linear Equations On Time Scales

This paper mainly deals with three problems: eigenvalues of second-order symmetric linear equations with periodic and antiperiodic boundary conditions on time scales, classification for singular second-order symmetric linear equations on time scales, and L'Ho|^spital rules on time scales.As a tool for establishing a unified framework for continuous and discrete analysis, a theory of dynamic equations on measure chains was introduced by S. Hilger in his Ph.D. thesis [1] in 1988. In many cases, it is necessary to study a special case of measure chains-time scales. In the last decade, the investigation of dynamic systems on time scales has involved much interest in quite a few fields, such as calculus, oscillation of dynamic systems. eigenvalue problems, boundary value problems, partial differential equations on time scales, and etc [2. 3, 4, 5]. The theory of dynamic systems on time scales is of very important theoretical significance and has a wide range of applications. It can not only reveal the similarity between the discrete case and the continuous case, but also explain the discrepancies that occur in parallel statements in continuous and discrete cases. In the real world, there are a lot of processes that depend on continuous time variable sometimes and discrete time variable sometimes, and there are many other processes that depend on piecewise continuous time variable. So we can work out more exactly mathematical models by using dynamic equations on time scales for these cases. For example, the time scales calculus can model insect populations that are continuous while in season, die out in winter, while their eggs are incubating or dormant, and then hatch in a new season, giving rise to a nonoverlapping population. An example of a simple electric circuit with resistance, inductance and capacitance is given in [3]. Recently, cobweb models on time scales are established and discussed.E. A. Coddingtong and N. Levinson, J. K. Hale, W. Magnus and S. Winkler [6, 7, 8] respectively studied properties of eigenvalues of second-order differential equations with periodic and antiperiodic boundary conditions and compared their eigenvalues.For eigenvalue problem of difference equations, F. V. Atkinson, M. Bohner, A. Jirari, Y. Shi, and S. Chen [9, 10, 11. 12, 13, 14] did a lot of profound and creative work. In 2005, Y. Wang and Y. Shi [15] made the comparison of eigenvalues of second-order difference equations with periodic and antiperiodic boundary conditions. In 2006, H. Sun and Y. Shi [16] extended these results to a coupled boundary condition. Although the numbers of eigenvalues of second-order differential and difference equations with periodic and antiperiodic boundary conditions are quite different, there comparison results are similar. So we wonder if the comparison results can be extended to time scales. This is one of the main aims of this paper.The spectral problems of symmetric linear differential operators and difference operators can both be divided into two cases. Those defined over finite closed intervals with well-behaved coefficients are called regular. Otherwise, they are called singular. In 1910, H. Weyl [17] gave a dichotomy of the limit-point and limit-circle cases for singular second-order symmetric linear differential equations. Later, many mathematicians, such as E. C. Titchmarsh, E. A. Coddington, N. Levinson [6, 18] developed his work and established the Weyl-Titchmarsh theory. Singular second-order formally self-adjoint scalar difference equations over infinite intervals were firstly studied by F. V. Atkinson [9]. His work was followed by D. B. Hinton, A. Jirari, etc.[10, 19]. The spectral problems of second-order and higher-order formally self-adjoint vector difference equations and discrete linear Hamiltonian systems were investigated systematically by Y. Shi, S. Chen. S. L. Clark, B. Beckermann, M. Bohner etc [11, 13, 14, 20, 21, 22]. In 2001. J. Chen and Y. Shi [23] obtained a sufficient and necessary condition and several criteria of limit-point and limit-circle cases for second-order formally self-adjoint linear difference equations with real coefficients. Recently, Y. Shi [24] established the Weyl-Titchmarsh theory of discrete linear Hamiltonian systems. More recently, S. Sun [25] extended Shi's work to Hamiltonian systems on time scales and established Weyl-Titchmarsh theory of Hamiltonian systems on time scales. Sun give the classification of singular Hamiltonian systems on time scales in terms of the defect indices of the minimal operator. In the present thesis, we employ Weyl's method to divide singular second-order symmetric linear differential equations on time scales into two cases: limit-point and limit-circle cases. This is another focus of this paper.M. Bohner and A. Peterson [3, 4] have made great progress on the basic calculus on time scales. But many results are not complete, such as L'Ho|^spital rules [3, 4, 26]. As we all know, the L'Ho|^spital rule plays an important role in classical calculus. It can help us deal with many problems. In this paper, we will give some revised L'Ho|^spital rules on time scales.This paper is divided into four chapters. In Chapter 1, the time scale calculus is introduced and some fundamental relative theories are given.In Chapter 2. we study eigenvalue problems of second-order symmetric linear equations with periodic and antiperiodic boundary conditions on time scales. We mainly employ the properties of eigenvalues of the Dirichlet boundary value problem and an os(?)illation result to compare eigenvalues of the periodic and antiperiodic boundary value problems on time scales. Finally, we will show our result not only covers those existing results in the differential and difference cases, which are studied by E. A. Cod-dington and X. Levinson [6] and Y. Wang and Y. Shi [15], but also covers other more complicated time scales.In Chapter 3. we focus on the classification of singular second-order symmetric linear differential equations on time scales. Firstly. L~2(I) is proved to be a Hilbert space. Secondly, we construct a family of nested circles. These circles converge to a limiting set. The dichotomy of the limit-point case and limit-circle case for singular second-order symmetric linear differential equations on time scales is given by geometric properties of the limiting set. Finally, several criteria of the limit-point case and limit-circle case are established, respectively.In Chapter 4. applying chain rule and the mean value theorems on time scales, we give two L'Ho|^spital rules on time scales under some weaker conditions.