| Study on asymptotic behavior of solutions of perturbed linear systems has a long time. In 1948, N. Levinson studied asymptotic behavior of solutions of the perturbed differential systemy'(x) = ((?)(x) + R(x))y(x)and obtained an important asymptotic result, called the Levinson theorem (see [12, Theorem 8.1 in Chapter 3] or [15, Theorem 1.3.1]), which played an important role in study of asymptotic problems of perturbed differential systems. Hartman and Wintner [19] got another important result, called the Hartman-Wintner theorem, in 1955. Later, their works were followed by Harris, Lutz [17, 18], Eastham [15], et al. Many excellent asymptotic results for differential systems were summarized in the monograph of Eastham [15] and many references were cited therein.At the beginning of the twenty century, Birkhoff [5] and Coffman [13] began to study asymptotic behavior of solutions of difference equations. In 1987, Benzaid and Lutz investigated asymptotic behavior of solutions of perturbed linear difference systemsy(t+1) = ((?)(t) + R(t))y(t)and got several asymptotic results [4], one of which is a discrete analog of the Levinson theorem, which plays an important role in our paper. Similarly to the case of differential systems, two types of conditions are crucial in studying asymptotic representations of solutions: the first is a dichotomy condition on the diagonal matrix A(t), and the second is a growth condition on the perturbation term R(t). These two conditions are interrelated, and so we can obtain asymptotic representations of solutions in variety of ways by strengthening one condition while weakening the other one. The asymptotic behavior of solutions of perturbed linear systems has very important theoretical value and wide applied perspective. It can be applied to investigate the stability of systems and spectral problems of operators.Since the beginning of the twenty-first century the investigation of dynamic systems on time scales has involved much interesting, including quite a few fields, such as the notion and theory of calculus, the oscillation of the dynamic systems, the eigenvalue problems and boundary value problems, partitial differential equations and functional differential equations on time scales, and etc (cf. [1, 3, 8]). The theory of dynamic systems on time scales has very important theoretical significance and wide applied perspective. It can not only reveal the similarity between differential equations and difference equations, but also unify the theory of differential equations and difference equations. In nature, there are a lot of processes which depend on continuous variables sometimes and depend on discrete variables sometimes. So we can give more exactly mathematical models by using dynamic equations on time scales. Recently, Bohner, Lutz et al investigated asymptotic behavior of dynamic equations on time scales and gave several beautiful results [6, 7], one of which is a generalization of Levinson theorem on time scales.Limit point and limit circle criteria for formally self-adjoint differential operators or difference operators are one of important parts of the spectral theory. The spectral problems of differential operators and difference operators are both divided into two cases. Those defined over finite intervals with well-behaved coefficients are called regular. Otherwise, they are called singular. H. Weyl [36] first found in his dissertation that singular second-order formally self-adjoint differential operators can be divided into two cases: limit point and limit circle cases. Later, E. C. Titchmarsh, E. A. Coddington, N. Levinson et al deepened and improved his results and established the theory of Titchmarsh-Weyl (cf. [12, 35] and their references). S. L. Clark, D. B. Hinton, C. Remling. J. K. Shaw. W. Krall, R. M. Kauffman, T. T. Read, A. Zettl, et al generalized the Titchmarsh-Weyl theory to Hamiltonian systems, and obtained many limit point and limit circle criteria for differential equations and Hamiltonian systems (cf. [14, 21, 25, 27] and their references).Spectral problems of second-order self-adjoint scalar difference operators over infinite intervals were firstly studied by F. V. Atkinson [2]. He was followed by D. B. Hinton, R. T. Lewis, S. L. Clark and A. Jirari (cf. [11, 20, 23] and their references). In 2001, J. Chen and Y. Shi [10] obtained a sufficient and necessary conditions and several criteria of limit point and limit circle cases for real coefficient second-order formally self-adjoint linear difference operators. Recently, spectral theory of discret linear Hamiltonian systems has been interested. Y. Shi [31-33] established Titchmarsh-Weyl theory of discrete linear Hamiltonian systems, and several equivalent conditions to limit circle and limit point cases which has very important theoretical value. But these equivalent conditions are notexpressed with the coefficients of the operators. So they can not be applied directly.In Chapter 1 we study the linear perturbed difference systems and give several asymptotic results. They can be regarded as discrete analogs of the well-known Hartman-Wintner theorem, the Harris-Lutz theorem and the Eastham theorem. At the end of this chapter, some examples are presented to demonstrate how the theorems can be applied and to compare conditions of our theorems with those of some existing results.In Chapter 2 we discuss the asymptotic behavior of dynamic systems on time scales using the basic theory of time scales. We obtain two results which can be regarded as generalizations of Hartman-Wintner theorem and Harris-Lutz theorem on time scales, respectively.In Chapter 3 we study fourth-order singular formally self-adjoint difference operators. Applying the results obrained in Chapter 1, we establish one criteria for limit circle and limit point cases which are concerning with the coefficients of the operators.
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