Project Materials

GENERAL

Contributions to Operator Theory and Applications

click here to get this project topic material with complete chapters 1-5 for just ₦3000 flat rate.

Do You Have New or Fresh Topic? Send Us Your Topic

Contributions to Operator Theory and

ABSTRACT

This thesis consists of two parts. The first part deals with existence and approximation techniques for finding solutions of operator equations or fixed points of operators belonging to certain classes of mappings. The classes of mappings studied include the K-posztz~~dee finzte operators, the suppressive mappings and accretive-type rntippings. In particular, it is proved that for a real space X, the equation Au = f , f E X, where A is a Kpd operator with the same domain as A’, has a unique solution. An process is constructed ant1 shown to converge strongly to the unique solution of this equation. Furthermore, an asyrnptotzc version of Kpd operators is introduced and studied and a convergence result is proved. Drawing from the ideas of Alber [I], Alber and Guerre-Delabriere [2, 31, suppressive and accretive-type mappings are studied in more general settings. In particular, it is proved that if I

The second part of the thesis deals with ling of infectious diseases. Models for drug-resistant malaria parasites are presented both for single populations of humans and vectors and also for multi-group populations. Eacll’ of the models results in a system of nonlinear ordinary differential equations, which under suitable conditions leads to a ly stable equilibrium. The ecological significance of these ecluilibriunl poirit s emerges as a by-product. For the compartmental models, attention is devoted to the question of quantitative agreement with published field s by the application of new nonlinear least squares techniques. A time dependent immunity model is formulated arid used :is a baseline study to investigate parameter behaviour.
Furthermore, the multi-group models are studied in Rn. The ultimate intention is to extend to infinite dimension, thereby providing a link between the analysis of these models and some well known and developed Hilbert space theory.

OF CONTENTS

1 GENERAL INTRODUCTION AND PRELIMINARIES
2 Existence, Uniqueness and Approximation of a Solution for a K-Positive Definite
Operator Equation 17
2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Main s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 A Local Approximation Methods for the Solution of K-Positive Definite Operator
Equations 2 6
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Main s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
,.,.. :..’.”‘ ,..a Pa ‘
4 Approximations of points of weakly contractive Non-self Maps in
Spaces 3 2
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2 Preliminaries . . . . . . . . . . . .,I . . . . … . . . . . . . . . . . . . . . . . . . . . . . 34
4.3 Main s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5 Iterative Methods for points of Asymptotically weakly contractive Maps 43
5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.3 Main s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6 Mathematical Modelling of Drug Resistant Malaria Parasites and Vector Populat
ions 5 6
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6.2 Simple host-vector model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.3 Resistant parasites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
7 Some Malaria Models Treating both Sensitive and Resistant Strains in Single
and Multigroup Populations 69
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7.2 Single population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
7.3 Multigroup population …………………………… 74
8 A Mathematical Model for Malaria Treating both Sensitive and Resistant
Strains in a Spatially Distributed Population 79
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
8.2 Spatially Distributed Population Model . . . . . . . . . . . . . . . . . . . . . . . . 80
8.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

CHAPTER ONE

GENERAL INTRODUCTION

PRELIMINARIES

This thesis is divided into two parts. The contributions of the first part fall within the general area of operator theory while the second part is concerned with applications to ling of communicable diseases, in particular, malaria models. For the first part, we shall, ‘ in particular, devote attention to existence and approximation methods for finding solutions of operator equations or fixed points of certain nonlinear mappings defined in a subset of a space. The classes of mappings st~di&l’k’cl;ide: the K-positive definite operators, the suppressive operators and certain accretive-type operators.

It is well known that many physically significant problems can be modelled in terms of an initial value problem of the form where A is an acceptive-type operator defined in an appropriate space. It is clear that if the solution of equation (1.0) is independent oft, then Au = 0 and the solutions of this equation correspond to the equilibrium of the system (1.0). equently, considerable research efforts have been devoted within the past half a century to finding techniques for the determination of zeros of accretive-type operators (see, for e.g., [17, 18, 19, 22, 52, 54, 581). The study of operator equations is partly linked with fixed point theory for; u is a fixed point of the operator A if and only if u is the solution of the operator equation ( I – A)u = 0, where I is the identity operator. The classical importance and application of fixed point theory can be seen largely in the theory of ordinary differential equations. The existence or construction of a solution to a differential equation is often reduced to the existence or location of a fixed point for an operator defined on a subset of a space of functions. In this thesis we shall.employ fixed point techniques where appropriate.

Direct and iterative methods for finding solutions of operator equations or fixed points of an operator defined in an appropriate space have been studied by many authors. These studies have given rise to the development of results and techniques which are now widely available in the literature.

Petryshyn [54] considered the operator equation .4u = f , in a Hilbert space, when A is K positive definite. Let HI be a dense subspace of a Hilbert space, H. An operator T with domain ‘D(T) 2 HI is called continuously HI invertzble if the range of T, R(T), with T considered as an operator restricted to HI is dense in H and T has a bounded inverse on R(T). Let H be a complex and separable HilberC space%ndaA ‘be a linear unbounded operator defined on a dense domain D(A) in H with the property that there exist a continuously D(A)-invertible closed linear operator K with D(A) c D(K), and a constant c > 0 such that (1.0) (Au, Ku) 12 c(. ~.. Ku~(~u ,E D(A), then A is called I 0 such that for all u E D(K), ll.4ull the equation Au = f , f E X, where A is a Kpd operator with the same domain as K has a unique solution. Furthermore, if X = Lp (or lp), p > 2, and is separable, we construct an process which converges strongly to this solution.

Not What You Were Looking For? Send Us Your Topic

click here to get this project topic material with complete chapters 1-5 for just ₦3000 flat rate.

TRUCTIONS AFTER PAYMENT

After making payment, kindly send the following:
  • 1.Your Full name
  • 2. Your Active Email Address
  • 3. Your Phone Number
  • 4. Amount Paid
  • 5. Project Topic
  • 6. Location you made payment from

» Send the above details to our email; contact@premiumresearchers.com or to our support phone number; (+234) 0813 2546 417 . As soon as details are sent and payment is confirmed, your project will be delivered to you within minutes.

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Advertisements