Thesis (Ph. D.)--University of Wisconsin--Madison, 1985. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 127-130).

Thesis--University of Florida. / Description based on print version record. Typescript. Vita. Includes bibliographical references (leaves 90-91).

Let f be a point-to-set mapping from a topological X space X into the family 2(X) of nonempty closed subsets of X . K. Fan [13] proved that if X is a Hausdorff locally convex linear topological space and K is a nonempty compact convex subset of X , then an upper semicontinuous mapping (abbreviated by u.s.c.) f from K into k(K), the family of nonempty closed convex subsets of K, has a fixed point in K . Our main object in this work is to weaken "compactness" of K to "weak compactness" and prove a fixed point theorem for a mapping f on K into certain subfamily of 2(K). The definition of convex function has been extended to point-to-set mappings in Chapter I. Let I denote the identity mapping on a Banach space X. Assume that I-f is a convex mapping on a weakly compact closed convex subset K of X. Then any of the following conditions implies the existence of the fixed point of f on K: (1) f : K → 2(K) is u.s.c. and [formula omitted] d(x,f(x)) = 0. (2) f : K → 2(K) is u.s.c. and is asymptotically regular (see definition 1.3) at some point in K . (3) f : K → cc(K) is nonexpansive and the Banach space X has a strictly convex norm. Moreover, it has been shown that if f : K → cpt(K) (see definition 0.3) is nonexpansive and I-f is strictly convex (see definition 1.5) on K, then K has a fixed point on K . Finally, an effort has been made to investigate the properties of the set of fixed points of a point-to-set mappings. In Chapter II, we have confined ourselves to a reflexive Banach space X which has a weakly continuous duality map J (see definition 2.3) and X has a strictly convex norm. On such a special space we are able to prove that a nonexpansive mapping f : X → cc(X) such that f(x)ʗ K, for any x in a closed convex bounded subset K of X , has a fixed point. As an application of this result we prove a fixed point theorem for semicontractive mappings (see definition 2.7). F : X → cc(X) such that F(x)ʗK for any x ε K , where K and X are the same as above. . In the last Chapter, we have proved that if f is strictly nonexpansive on a.Banach space X into cpt(X) and if there is x(o) ε X such that [formula omitted] has a subsequence convergent to a set A ε cpt(X) under the Hausdorff metric D on cpt(X), then f has a fixed point in A . Furthermore we prove that a nonexpansive mapping f : K → cpt(K), where K is a weakly compact convex subset of a metrizable locally convex linear topological space X, has a fixed point in K, provided that a constant k > 0 exists such that the set E(x) = {y ε K ; d(x,y) ≥ kd(y,f(y))} is nonempty and convex and the mapping E : K → k(K), with E(x) defined above, is weakly locally closed (see definition 3.1). Finally the comparisons of the continuities of a point-to-set mapping have been made. / Science, Faculty of / Mathematics, Department of / Graduate

No description available.

In the study of differential equations there are two fundamental questions: is there a solution? and what is it? One of the most elegant ways to prove that an equation has a solution is to pose it as a fixed point problem, that is, to find a function f such that x is a solution if and only if f (x) = x. Results from fixed point theory can then be employed to show that f has a fixed point. However, the results of fixed point theory are often nonconstructive: they guarantee that a fixed point exists but do not help in finding the fixed point. Thus these methods tend to answer the first question, but not the second. One such result is Schauder’s fixed point theorem. This theorem is broadly applicable in proving the existence of solutions to differential equations, including the Navier-Stokes equations under certain conditions. Recently a semi-constructive proof of Schauder’s theorem was developed in Rizzolo and Su (2007). In this thesis we go through the construction in detail and show how it can be used to search for multiple solutions. We then apply the method to a selection of differential equations.

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2004.\ / Includes bibliographical references (p. 91-92).

No description available.

by Ip Che-ho. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1997. / Includes bibliographical references (leaf 136). / Covering Page --- p.i / Acknowledgement --- p.ii / Abstract --- p.iii / Table of Content --- p.v / Chapter 1. --- Fundamentals of Complex Analys --- p.is / Chapter 1.1 --- The extended complex plane --- p.1 / Chapter 1.2 --- Stereographic projection --- p.2 / Chapter 1.3 --- Analytic functions --- p.3 / Chapter 1.4 --- Rational functions --- p.5 / Chapter 1.5 --- Mobius transformation --- p.6 / Chapter 2. --- The Topology of the Extended Plane / Chapter 2.1 --- The topology of S2 and C ∞ --- p.9 / Chapter 2.2 --- Smooth map and manifolds --- p.10 / Chapter 2.3 --- Regular points --- p.11 / Chapter 2.4 --- Degree of maps --- p.13 / Chapter 2.5 --- Euler characteristics --- p.14 / Chapter 2.6 --- Covering space --- p.16 / Chapter 2.7 --- Riemann-Hurwritz formula --- p.17 / Chapter 3 --- The Montel Theorem / Chapter 3.1 --- Introduction --- p.21 / Chapter 3.2 --- Normality and Equicontinuous --- p.21 / Chapter 3.3 --- Local boundedness --- p.23 / Chapter 3.4 --- Covering and uniformization --- p.26 / Chapter 3.5 --- Montel's theorem --- p.28 / Chapter 4 --- Fatou Set and Julia Set / Chapter 4.1 --- Iteration of functions --- p.31 / Chapter 4.2 --- Fatou set and Julia set --- p.35 / Chapter 4.3 --- Iteration of Mobius transformtion --- p.39 / Chapter 4.4 --- Fixed points and their classification --- p.44 / Chapter 4.5 --- Periodic points and cycles --- p.45 / Chapter 4.6 --- Critical points --- p.47 / Chapter 4.7 --- Dlustractions of local behaviour of map near periodic points --- p.48 / Chapter 5 --- More about Julia Set / Chapter 5.1 --- Some examples of Julia set --- p.57 / Chapter 5.2 --- Completely invariant set --- p.58 / Chapter 5.3 --- Exceptional set --- p.61 / Chapter 5.4 --- Properties of Julia set --- p.63 / Chapter 5.5 --- Forward and backward convergence of sets --- p.66 / Chapter 6 --- More about Fatou Set / Chapter 6.1 --- Components of Fatou set --- p.97 / Chapter 6.2 --- Simply connected Fatou components --- p.98 / Chapter 6.3 --- Number of components in Fatou set --- p.100 / Chapter 6.4 --- Classification of forward invariant components of the Fatou set --- p.102 / Chapter 6.5 --- Examples illustrating the five possible forward invariant components --- p.104 / Chapter 7 --- Critical Points / Chapter 7.1 --- Introduction --- p.108 / Chapter 7.2 --- Some interesting results --- p.110 / Chapter 7.3 --- The Fatou set of polynomials --- p.114 / Chapter 7.4 --- Quadratic polynomial and Mandelbrot set --- p.116 / Appendix --- p.125 / Reference --- p.136

Kakutani’s fixed point theorem has many applications in economics and game theory. One of its most well known applications is in John Nash’s paper [8], where the theorem is used to prove the existence of an equilibrium strategy in n-person games. Sperner’s lemma, on the other hand, is a combinatorial result concerning the labelling of the vertices of simplices and their triangulations. It is known that Sperner’s lemma is equivalent to a result called Brouwer’s fixed point theorem, of which Kakutani’s theorem is a generalization. A natural question that arises is whether we can prove Kakutani’s fixed point theorem directly using Sperner’s lemma without going through Brouwer’s theorem. The objective of this thesis to understand Kakutani’s theorem, Sperner’s lemma, and how they are related. In particular, I explore ways in which Sperner’s lemma can be used to prove Kakutani’s theorem and related results.

Results from combinatorial topology have shown that certain combinatorial lemmas are equivalent to certain topologocal fixed point theorems. For example, Sperner’s lemma about labelings of triangulated simplices is equivalent to the fixed point theorem of Brouwer. Moreover, since Sperner’s lemma has a constructive proof, its equivalence to the Brouwer fixed point theorem provides a constructive method for actually finding the fixed points rather than just stating their existence. The goal of this research project is to develop a combinatorial analogue for the Poincare ́-Birkhoff fixed point theorem.