In almost all works in the literature there are several results showing asymptotic relationships between the solutions of x&prime / &prime / = f (t, x) (0.1) and the solutions 1 and t of x&prime / &prime / = 0. More specifically, the existence of a solution of (0.1) asymptotic to x(t) = at + b, a, b &isin / R has been obtained. In this thesis we investigate in a systematic way the asymptotic behavior as t &rarr / &infin / of solutions of a class of differential equations of the form (p(t)x&prime / )&prime / + q(t)x = f (t, x), t &ge / t_0 (0.2) and (p(t)x&prime / )&prime / + q(t)x = g(t, x, x&prime / ), t &ge / t_0 (0.3) by the help of principal u(t) and nonprincipal v(t) solutions of the corresponding homogeneous equation (p(t)x&prime / )&prime / + q(t)x = 0, t &ge / t_0. (0.4) Here, t_0 &ge / 0 is a real number, p &isin / C([t_0,&infin / ), (0,&infin / )), q &isin / C([t_0,&infin / ),R), f &isin / C([t_0,&infin / ) &times / R,R) and g &isin / C([t0,&infin / ) &times / R &times / R,R). Our argument is based on the idea of writing the solution of x&prime / &prime / = 0 in terms of principal and nonprincipal solutions as x(t) = av(t) + bu(t), where v(t) = t and u(t) = 1. In the proofs, Banach and Schauder&rsquo / s fixed point theorems are used. The compactness of the operator is obtained by employing the compactness criteria of Riesz and Avramescu. The thesis consists of three chapters. Chapter 1 is introductory and provides statement of the problem, literature review, and basic definitions and theorems. In Chapter 2 first we deal with some asymptotic relationships between the solutions of (0.2) and the principal u(t) and nonprincipal v(t) solutions of (0.4). Then we present existence of a monotone positive solution of (0.3) with prescribed asimptotic behavior. In Chapter 3 we introduce the existence of solution of a singular boundary value problem to the Equation (0.2).

In the first part of this thesis the spectrum of a matrix operator is determined. For this the coefficients of the matrix operator are assumed to satisfy rather general properties which combine smoothness and decay. With this the asymptotics of the eigenfunctions can be determined. This in turn leads to properties of the spectra with the aid of the M-matrix. In the second part it will be shown that if a discrete canonical system has absolutely continuous spectrum of a certain multiplicity, then there is a corresponding number of linearly independent solutions y which are bounded in a weak sense.

In this thesis the spectral properties of differential operators generated by the formally self-adjoint differential expression Τy = w⁻₁[(ry″)″ - (py′)′ + qy] are investigated. The main tools to be used are the theory of asymptotic integration and the Titchmarsh--Weyl M-matrix. Subject to certain regularity conditions on the coefficients asymptotic integration leads to estimates for the eigenfunctions of the corresponding differential equation Τy = zy. According to the theory of asymptotic integration the regularity conditions combine smoothness with decay, i.e. admissible coefficients are (in an appropriate sense) either short range or slowly varying. Knowledge of the asymptotics (x → ∞) of the solutions will then be used to determine the deficiency index and to derive properties of the M-matrix which is closely related to the spectral measure of an associated self-adjoint realization Τ. Consequently we can compute the multiplicity of the spectrum, locate the absolutely continuous spectrum and give conditions for the singular continuous spectrum to be empty. This generalizes classical results on second order operators.