The pervading theme of this thesis is the development of insights that contribute to the understanding of whether certain classes of functional differential equation have solutions that are all oscillatory. The starting point for the work is the analysis of simple (linear autonomous) ordinary differential equations where existing results allow a full explanation of the phenomena. The Laplace transform features as a key tool in developing a theoretical background. The thesis goes on to explore the corresponding theory for delay equations, advanced equations and functional di erential equations of mixed type. The focus is on understanding the links between the characteristic roots of the underlying equation, and the presence or otherwise of oscillatory solutions. The linear methods are used as a class of numerical schemes which lead to discrete problems analogous to each of the classes of functional differential equation under consideration. The thesis goes on to discuss the insights that can be obtained for discrete problems in their own right, and then considers those new insights that can be obtained about the underlying continuous problem from analysis of the oscillatory behaviour of the analogous discrete problem. The main conclusions of the work are some semi-automated computational approaches (based upon the Principle of the Argument) which allow the prediction of oscillatory solutions to be made. Examples of the effectiveness of the approach are provided, and there is some discussion of its theoretical basis. The thesis concludes with some observations about further work and some of the limitations of existing analytical insights which restrict the reliability with which the approach developed can be applied to wider classes of problem.

Wudu Lu. / Thesis (Ph.D.)--Chinese University of Hong Kong, 1995. / Includes bibliographical references (leaves 114-116). / Abstract --- p.ii / Introduction --- p.iv / Chapter 1 --- The Fundamental Theory of NFDEs with Infinite Delay --- p.1 / Chapter 1.1 --- Introduction --- p.1 / Chapter 1.2 --- Phase spaces and NFDEs with infinite delay --- p.2 / Chapter 1.3 --- Local theory --- p.4 / Chapter 2 --- Periodicity and Bp -Boundedness in Neutral Systems of Non- linear D-operator with Infinite Delay --- p.12 / Chapter 2.1 --- Introduction --- p.12 / Chapter 2.2 --- Preliminaries --- p.15 / Chapter 2.3 --- Existence of periodic solutions --- p.22 / Chapter 2.4 --- BP-U.B and Bp -U.U.B of solutions --- p.29 / Chapter 2.5 --- Applications --- p.42 / Chapter 3 --- Stability in Neutral Differential Equations of Nonlinear D- operator with Infinite Delay --- p.47 / Chapter 3.1 --- Introduction --- p.47 / Chapter 3.2 --- Preliminaries --- p.49 / Chapter 3.3 --- Uniformly Asymptotic Stability --- p.57 / Chapter 3.4 --- Applications --- p.74 / Chapter 4 --- Nonoscillation and Oscillation of First Order Linear Neutral Equations --- p.79 / Chapter 4.1 --- Introduction --- p.79 / Chapter 4.2 --- Existence of Nonoscillatory Solutions --- p.80 / Chapter 4.3 --- Oscillation --- p.90 / Chapter 5 --- Nonoscillation and Oscillation of First Order Nonlinear Neu- tral Equations --- p.94 / Chapter 5.1 --- Introduction --- p.94 / Chapter 5.2 --- Existence of Nonoscillatory Solutions --- p.95 / Chapter 5.3 --- Oscillation --- p.102 / Bibliography --- p.108 / List of Author's Publications --- p.114

Thesis (Ph.D.)--University of Illinois, 1917. / Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

published_or_final_version / Mathematics / Master / Master of Philosophy

Thesis (M. Phil.)--University of Hong Kong, 2010. / Includes bibliographical references (leaves 55-59). Also available in print.

Estudamos a representação das equações diferenciais funcionais (EDF) lineares autônomas do tipo neutro como uma classe de equações de renovação, isto é, equações do tipo convolução. Utilizando ferramentas como a transformada de Laplace-Stieltjes, estudamos o comportamento assintótico das soluções desta equações quando t \'SETA\' \' INFINITO\' / We studied the representation of linear autonomous functional differential equations (FDE) as a class of renewall equations, that is, convolution-type equations. Using tools like the Laplace-Stieltjes trnsform, we obtained the asymptotic behaviour of those solutions as t \'ARROW\' \'INFINITY

published_or_final_version / Mathematics / Master / Master of Philosophy

Thesis (M.S.)--Virginia Polytechnic Institute and State University, 1994. / Vita. Abstract. Includes bibliographical references (leaves 41-42). Also available via the Internet.

Thesis (Ph. D.)--Rutgers University, 2007. / "Graduate Program in Mathematics." Includes bibliographical references (p. 174-176).

Estudamos a representação das equações diferenciais funcionais (EDF) lineares autônomas do tipo neutro como uma classe de equações de renovação, isto é, equações do tipo convolução. Utilizando ferramentas como a transformada de Laplace-Stieltjes, estudamos o comportamento assintótico das soluções desta equações quando t \'SETA\' \' INFINITO\' / We studied the representation of linear autonomous functional differential equations (FDE) as a class of renewall equations, that is, convolution-type equations. Using tools like the Laplace-Stieltjes trnsform, we obtained the asymptotic behaviour of those solutions as t \'ARROW\' \'INFINITY