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Contributions to Degree theory, and Applications

 Author / Creator
 Niksirat, Mohammad Ali

This thesis is dedicated to the study of topological degree for different classes of monotone maps and applications to nonlinear problems in mathematical analysis and the applied mathematics. Employing topological methods for nonlinear problems in mathematics goes back to the pioneering work of H. Poincaré on the three body problem. The generalization of Brouwer degree for mappings in finite dimensional spaces to compact perturbations of the identity in arbitrary Banach spaces, by J. Schauder and J. Leray, opened up a way to apply such a powerful method to a broad class of complicated nonlinear problems. Further generalizations, including the degree for classes of monotone maps as well as the degree for multivalued maps have been carried out by several authors in recent decades. This thesis is divided into two parts. The first part, consisting of Chapters 1 and 2, is about the theoretical aspects of topological degree. The second part, Chapters 35, is devoted to the applications of the topological degree in three fields: an integral equation coming from the DoiOnsager model for liquid crystals; dynamical systems governed by nonlinear ordinary differential equations and finally fully nonlinear elliptic and parabolic partial differential equations. After a detailed introduction on various topological methods for linear and quasilinear elliptic problems and presentation of some of its implications for monotone maps and variational problems in Chapter 1, we systematically introduce the concept of finite rank approximation of a map in Chapter 2. This concept enables us to prove the stability of the homotopy class of finite rank approximations for different types of monotone maps including (S)+, pseudomonotone and maximal monotone maps in separable, locally uniformly convex Banach spaces. Furthermore, we generalize the degree for mappings that are only demicontinuous in a subspace, not necessarily dense, of the focal space. We use this generalization for the DoiOnsager problem presented in Chapter 3. The DoiOnsager problem is a mathematical formulation to model the behaviour of the liquid crystals in terms of the interaction potential field and the temperature of the liquid. In our work on this problem, we solve the problem in dimension D=2 and also prove the uniqueness of the isotropic solution for high temperature and the bifurcation of nematic solutions for low temperature in general dimension. For a classical application of degree theory, we return in Chapter 4 to the problem of periodic solutions for dynamical systems described in ordinary differential equations. The method that we employ in this chapter is based on the continuation method. For this, we consider a oneparameter family of dynamical systems and then prove (under certain conditions) that periodic orbits survive when the parameter increases. The last chapter of the thesis, Chapter 5 is dedicated to defining a degree for fully nonlinear elliptic and parabolic equations. Even though it is not novel to define a degree for fully nonlinear elliptic equations, our construction can be employed to define a degree for fully nonlinear parabolic equations which, to our knowledge is new.

 Graduation date
 Fall 2014

 Type of Item
 Thesis

 Degree
 Doctor of Philosophy

 License
 This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for noncommercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.