Project Materials

Contributions to Operator Theory and Applications

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 Banach 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 iteration 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< is a closed convex non-expansive retract of a real uniformly smooth Banach space E, T :

I< + E, a d-weakly contractive map such that a fixed point x* E intK of T exists then a descent-like approximation sequence converges strongly to z*. A related result deals with the approximation of a fixed point of T, when K is a subset of an arbitrary real Banach space &, aiacl. R(T) := {.c E E : Tx = z} # 0. Moreover, asymptotically d-weakly contractive mappings are introduced and studied and convergence results are proved.

The second part of the thesis deals with mathematical modelling 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 locally 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 observations 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 behavior.

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 Banach space theory.

TABLE 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 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 A Local Approximation Methods for the Solution of K-Positive Definite Operator
Equations 2 6
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
,.,.. :..’.”‘ ,..a Pa ‘
4 Approximations of Fixed points of weakly contractive Non-self Maps in Banach
Spaces 3 2
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2 Preliminaries . . . . . . . . . . . .,I . . . . … . . . . . . . . . . . . . . . . . . . . . . . 34
4.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5 Iterative Methods for Fixed points of Asymptotically weakly contractive Maps 43
5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 mathematical modelling 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 Banach 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 Banach 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). Consequently, 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 Banach 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<-positive definite (Kpd) (see e.g., Petryshyn [54]). If K = I (the identity operator) inequality (1.0) reduces to (Au, u) 2 cJu(a~n~d ,in this case,

A is called positive definite. If in addition c = 0, A is a positive operator (or accretive operator). Positive definite operators have been studied by several authors (see e.g.[15, 17, 18, 19, 28, 38, 581). It is clear that the class of K-pd operators contains among others, the class of positive definite operators, and also contains the class of invertible operators (when K = A) as its subclasses.

Furthermore, Petryshyn [54] remarked that for a proper choice of K, the ordinary differential operators of odd order, the weakly elliptic partial differential operators of odd order, are members of the class of K-pd operators. Moreover, if the operators are bounded, the class of Kpd operators forms a subclass of symmetrical operators studied by Reid [58]. In [54], Petryshyri proved the following theorem.

Theorem P If A is a Kpd operator and D(A) = D(K), then there exists a constant a! > 0 such that for all u E D(K), ll.4ull < allKuII. Furthermore, the operator A is closed, R(A) = H and the equation Au = f , f E H, has a unique solution. In the case that K is bounded and A is closed, F. E. Browder[l9] obtained a result similar to the second part of Theorem P.

In chapter two of this thesis, we extend the definition of a Kpd operator to real Banach spaces, X . In particular, if X is a real – separable Banach space with a strictly convex dual, we prove that ,, . a*,. +. ‘,> 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 iteration process which converges strongly to this solution.