Magister Scientiae - MSc / Although the interior operators correspond to a special class of neighbourhood operators, the closure operators are not nicely related to the latter. We introduce and study the notion of topogenous orders on a category which provides a basis for categorical study of topology. We show that they are equivalent to the categorical neighbourhood operators and house the closure and interior operators. The natural notion of strict morphism with respect to a topogenous order is shown to capture the known ones in the settings of closure, interior and neighbourhood operators.

No description available.

No description available.

Our aim in this research was to study monotone operators in Banach spaces. In particular, the most important concept in this theory, the maximal monotone operators. Here we make an attempt to describe most of the important results and concepts on maximal monotone operators and how they all tie together. We will take a brief look at subdifferentials, which generalize the notion of a derivative. The subdifferential is a maximal monotone operator and it has proved to be of fundamental importance for the study of maximal monotone operators. The theory of maximal monotone operators is somewhat complete in reflexive Banach spaces. However, in nonreflexive Banach spaces it is still to be developed fully. As such, here we will describe most of the important results about maximal monotone operators in Banach spaces and we will distinguish between the reflexive Banach spaces and nonreflexive Banach spaces when a property is known to hold only in reflexive Banach spaces. In the latter case, we will state what the corresponding situation is in nonreflexive Banach spaces and we will give counter examples whenever such a result is known to fail in nonreflexive Banach spaces. The representations of monotone operators by convex functions have found to be extremely useful for the study of maximal monotone operators and it has generated a lot of interest of late. We will discuss some of those key representations and their properties. We will also demonstrate how these representations could be utilized to obtain results about maximal monotone operators. We have included a discussion about the very important Rockafellar sum theorem and some its generalizations. This key result and its generalizations have only been proved in reflexive Banach spaces. We will also discuss several special cases where the Rockafellar sum theorem is known to be true in nonreflexive Banach spaces. The subclasses which provide a basis for the study of monotone operators in nonreflexive Banach spaces are also discussed here

Thesis--Bonn. / Includes bibliographical references (p. 58-60).

Mean-field games (MFGs) model the behavior of large populations of rational agents. Each agent seeks to minimize an individual cost that depends on the statistical distribution of the population. Roughly speaking, MFGs are given by the limit of differential games with N agents as N goes to infinity. This limit describes an average effect of the population’s behavior. Instead of modeling large systems for all agents, we consider two coupled equations: the Hamilton–Jacobi equation and the Fokker–Planck equation. A solution to MFGs is given by two functions: a value function and a population density. From the point of view of mathematics, monotonicity conditions for MFGs are a natural way to obtain the uniqueness of solutions and the stability of systems. In this thesis, we develop a new framework to establish the existence of solutions to MFGs through monotonicity. First, we study first-order stationary monotone MFGs with Dirichlet boundary conditions. In MFGs, boundary conditions arise when agents can leave the domain. There are exit costs for agents given by Dirichlet boundary conditions. Here, we establish the existence of solutions to MFGs that fulfill those boundary conditions in the trace sense. In particular, our solution is continuous up to the boundary in the one-dimensional case. Second, we consider time-dependent monotone MFGs with space-periodic boundary conditions. To solve the time-dependent monotone MFG, we introduce a mono- tone high-order regularized elliptic problem in Rn+1, although the original MFG is a parabolic type. To preserve monotonicity, we need to determine the specific boundary conditions for the time variable. Then, we can apply our method of stationary MFGs to this regularization. In particular, we prove that a solution to the problem exists for any terminal time. Third, we investigate stationary MFGs with hypoelliptic operators that are degenerate differential operators. Those models arise from stochastic control problems with the Stratonovich integration. We study a hypoelliptic MFG with the standard quadratic Hamiltonian. Under standard assumptions, although there is no uniform elliptic condition in hypoelliptic operators, we verify that there is a unique solution to our hypoelliptic MFG.

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Submitted in fulfillment of the requirements for the Degree of Masters in Business Administration, Durban University of Technology, 2014. / This study is an attempt to shed light on how tour operators can help to promote tourism in the south coast region of KZN. Even though tourism claims to bring economic development to the tourist destinations, in some cases, the development of tourism has created important social transformation in the other sectors of the tourism destinations, where the relationship between other sectors of the tourism industry and tour operators was weak or even negative. For some people, this situation is no longer considered a sustainable way to travel and do business. Therefore, the tour operator’s business does not end at picking up and dropping off tourists at their booked accommodation. The aim is to explore the role of tour operators in promoting tourism in the south coast region of KZN. The objectives of the study are: to identify and examine the nature of the tour operators’ business; to identify and explore the role of tour operators in the tourism industry in sustaining their businesses; to identify the changes needed to improve the promotion and sustainability of both tourism in the south coast region of KZN and that of the tour operator; and to explore the challenges encountered by tour operators which hinder effective contribution to the economy of South Africa. This research explored and investigated the perceptions of the tour operators in the south coast region in KZN in terms of their business growth, their role as tour operators and the problems they may encounter in growing their small businesses. A quantitative research methodology was utilised to collect the primary data for this study. Data was collected through the use of a structured questionnaire. The sample size for the study was 50 tour operators. The results from the literature and primary study revealed some interesting findings from the objectives and each set of questions. The highlighted finding in the study revealed the need for tour operators to provide the best possible service as the tourists are important for the destination’s economy (TOI, 2008:108). The study revealed that tour operators have direct contact with clients when they take them on tours; hence, a personal one-on-one safe relationship is established with clients (Moutinho, 2011:161). The conclusions drawn from this study point to the lack of access to information that would enable the majority of tour operators to understand the tourism sector holistically. Furthermore, information available to tour operators owing to technological advancements has increased exponentially (e.g. internet access).

published_or_final_version / abstract / Mathematics / Doctoral / Doctor of Philosophy

published_or_final_version / Mathematics / Master / Master of Philosophy