ABSTRACT Compact operators are linear operators on Banach spaces that maps bounded set to relatively compact sets. In the case of Hilbert space H it is an extension of the concept of matrix acting on a finite dimensional vector space. In Hilbert space, compact operators are the closure of the finite rank operators in the topology induced by the operator norm. In general, operators on infinite dimensional spaces feature properties that do not appear in the finite dimension case; i.e matrices. ...
The scope of Quadratic Form Theory is historically wide although it usually appears almost as an afterthought when needed to solve a variety of problems such as the classification of Hessian matrices in finite dimensional Calculus [1], [2], [3], the finding of invariants that fully describe the equivalence class of a given form in Algebraic Geometry and Number Theory [4], the use of Rayleigh-Ristz methods for finding eigenvalues of real symmetric matrices in Linear Algebra [5], [6], the secon...
ABSTRACT Algorithms for single-valued and multi-valued nonexpansive-type mappings have continued to attract a lot of attentions because of their remarkable utility and wide applicability in modern mathematics and other reasearch areas,(most notably medical image reconstruction, game theory and market economy). The first part of this thesis presents contributions to some crucial new concepts and techniques for a systematic discussion of questions on algorithms for singlevalued and multi-valued...
The most popular method for studying stability of nonlinear systems is introduced by a Russian Mathematician named Alexander Mikhailovich Lyapunov. His work ”The General Problem of Motion Stability ” published in 1892 includes two methods: Linearization Method, and Direct Method. His work was then introduced by other scientists like Poincare and LaSalle . In chapter one of this work, we focussed on the basic concepts of the ordinary differential equations. Also, we emphasized on relevant ...
ABSTRACT In this thesis, a hybrid extragradient-like iteration algorithm for approximating a common element of the set of solutions of a variational inequality problem for a monotone, k-Lipschitz map and common fixed points of a countable family of relatively nonexpansive maps in a uniformly smooth and 2-uniformly convex real Banach space is introduced. A strong convergence theorem for the sequence generated by this algorithm is proved. The theorem obtained is a significant improvement of the...
ABSTRACT In this thesis, we consider the problem of approximating solution of generalized equilibrium problems and common fixed point of finite family of strict pseudocontractions. The result obtained is applied in approximation of solution of generalized mixed equilibrium problems and common fixed point of finite family of strict pseudocontractions. Our theorems improve and unify some existing results that were recently announced by several authors. Corollaries obtained and our method of pr...
ABSTRACT Let H be a real Hilbert space and A : D(A) ⊂ H → H be an unbounded, linear, self-adjoint, and maximal monotone operator. The aim of this thesis is to solve u 0 (t) + Au(t) = 0, when A is linear but not bounded. The classical theory of differential linear systems cannot be applied here because the exponential formula exp(tA) does not make sense, since A is not continuous. Here we assume A is maximal monotone on a real Hilbert space, then we use the Yosida approximation to solve. A...
INTRODUCTION Variational methods have proved to be very important in the study of optimal shape, time, velocity, volume or energy. Laws existing in mechanics, physics, astronomy, economics and other fields of natural sciences and engineering obey variational principles. The main objective of variational method is to obtain the solutions governed by these principles. Fermat postulated that light follows a part of least possible time, this is a subject in finding minimizers of a given functiona...
Introduction In physical sciences (e.g, elasticity, astronomy) and natural sciences (e.g, ecology) among others, the consideration of periodic environmental factor in the dynamics of multi-phenomena interractions (or multi-species interractions) leads to the study of differential systems with periodic data. Therefore, it is worth investigating, the fundamental questions inherent in systems of periodic (ordinary) differential equations such as: existence, uniqueness and stability.
Preface This Project is at the interface between Optimization, Functional analysis and Dierential equation. It concerns one of the powerful methods often used to solve optimization problems with constraints; namely Minimum Pontryagin Method. It is more precisely an optimization problem with constrain, an ordinary dierential equation. Their applications cover variational calculos as well as applied areas including optimization, economics, control theory and Game theory. But we shall focus on ...
ABSTRACT In this work a deterministic and stochastic model is developed to investigate the deterministic and stochastic model of dynamics of Ebola virus. The model includes susceptible, exposed, infected, quarantined and removed or recovered individuals. The model used in this work is based on a deterministic model. The Chowell (2015) work on early detection of Ebola is modified by introducing an assumption that the quarantined class is totally successful and cannot infect the susc...
This Project primarily falls into the field of Linear Functional Analysis and its Applications to Eigenvalue problems. It concerns the study of Compact Linear Operators (i.e., bounded linear operators which map the closed unit ball onto a relatively compact set) and their spectral analysis applicable to Fourier Analysis and to the solvability of Fredholm Integral Equations, linear elliptic Partial Differential Equations (PDEs) with the Dirichlet boundary condition, Sturm-Liouville problems, a...
ABSTRACT We consider classical Finite Difference Scheme for a system of singularly perturbed convection-diffusion equations coupled in their reactive terms, we prove that the classical SFD scheme is not a robust technique for solving such problem with singularities. First we prove that the discrete operator satisfies a stability property in the l2-norm which is not uniform with respect to the perturbation parameters, as the solution blows up when the perturbation parameters goes to zero. An e...
ABSTRACT In this thesis, we treated a Standard Finite Difference Scheme for a singularly perturbed parabolic reaction-diffusion equation. We proved that the Standard Finite Difference Scheme is not a robust technique for solving such problems with singularities. First we discretized the continuous problem in time using the forward Euler method. We proved that the discrete problem satisfied a stability property in the l∞ − norm and l2 − norm which is not uniform with respect to the pertu...
This project concerns Evolution Equations in Banach spaces and lies at the interface between Functional Analysis, Dynamical Systems, Modeling Theory and Natural Sciences. Evolution Equations include Partial Dierential Equations (PDEs) with time t as one of the independent variables and arise from many elds of Mathematics as well as Physics, Mechanics and Material Sciences (e.g., Systems of Conservation Law from Dynamics, Navier-Stokes and Euler equations from Fluid Mechanics, Diusion equation...