Contributions to Operator Theory and Applications

Abstract

This thesis consists of two parts. The first part deals with existence and approximation techniquesfor finding solutions of operator equations or fixed points of operators belonging to certain classes of mappings. The classes of mappings studied include the K-positive definite operators, the suppressive mappings and accretive-type mappings. In particular, it is proved that for a real Banach space X, the equation Au = f , f € X, where A is a Kpd operator with the same domain as K, has a unique solution. An iteration process is constructed and shown to converge strongly to the unique solution of this equation. Furtherniore, an asymptotic 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, 3], suppressive and accretive-type mappings are studied in more general settings. In particular, it is proved that if K is a closed convex nonexpansive retract of a real uniformly smooth Banach space E, T : K > 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 x*. A related result deals with the approximation of a fixed point of T, when K is a subset of an arbitrary real Banach space E. .F(T) := {.x€ E : Tx + x} ≠ 0. Moreover, asymptotically d-weakly contractive mappings are introduced and studied and convergence result,s 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 multigroup populations. Each 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 quailtitative 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 behaviour. Furthermore, the multigroup 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.