[Aspects of Combinatorial Convexity]

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ACKNOWLEDGEMENT

I would take this opportunity to thank my research supervisor, family and friends for their support and guidance without which this research would not have been possible.

DECLARATION

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TABLE OF CONTENTS

ACKNOWLEDGEMENT2

DECLARATION3

ABSTRACT5

CHAPTER 1: INTRODUCTION6

CHAPTER 2: LITERTATURE REIVEW13

The theorems of Radon, Helly, and Caratheodory.13

Radon's theorem13

Helly's theorem13

Caratheodory's theorem14

Tverberg's Theorem15

Colorful Caratheodory16

CHAPTER 3: NOTATIONS AND FACTS18

CHAPTER 4: RESULTS AND DISCUSSION21

The existence of continuous Lyapunov function21

Proof of the main result29

An application to random operators33

Analysis of convergence36

Numerical examples39

CHAPTER 5: CONCLUSIONS50

CHAPTER 5: CONCLUSIONS50

REFERENCES51

ABSTRACT

We prove the existence of a Carathéodory selection for a set-valued mapping satisfying standard conditions. Also, an application to random fixed point for random operators is established. The converse problem of Lyapunov stability for systems of ODEs of Caratheodory type is considered. It is proved that if the right hand side of an ODE satisfies only the Osgood condition, the uniform stability of the origin is sufficient to the existence of a locally Lipschitz continuous Lyapunov function. Actually, the uniform stability is equivalent to the robust stability in this case. Moreover, as an auxiliary result, the generalization of the famous Gronwall-Bellman-Bihari inequality is also proved. In this paper, we relax the convergence conditions required in Ezquerro and Hernández (Int. J. Pure Appl. Math. 6(1) (2003) 103) for a multipoint third-order iteration of Halley type, where the conditions provided are the known ones for methods of order three. In this paper, we derive a class of new third-order methods free from second derivative from Halley's method. Per iteration the methods require two evaluations of the function and one evaluation of its first derivative. Analysis of efficiency, in term of function evaluations, of this class of methods shows that they have definite practical utility, which is also demonstrated by numerical examples.

CHAPTER 1: INTRODUCTION

We investigate the converse problem of Lyapunov stability for systems of ODEs with the right hand side being only Lebesgue measurable with respect to t

(1)

Problems described by such systems can be found in controllability problems, stabilization as well as in optimal control problems and neural networks. The existence of a Lyapunov function is a test for stability of solutions of ODEs. There are well known different kinds of stability for time-dependent systems (1) like uniform, asymptotic, robust and others. Particular features of Lyapunov function and of the right hand side of (1) imply the appropriate kind of stability. Many investigations on this topic can be found in e.g. Carathéodory systems in [11].

In this paper, we are interested in the converse problem, i.e. the existence of a Lyapunov function for uniformly stable Carathéodory systems. For such systems we cannot expect the existence of a classical C1Lyapunov function. Therefore, it is reasonable to consider a generalized Lyapunov function (for a number of such generalizations, which need not be even continuous. Kurzweil and Vrkoc in [9] give an example ...