الفهرس 
COVEROnly 14 pages are availabe for public view
 |  | from | 122 |  |  |
| | | from | 122 | | |
Abstract
In this thesis some fixed point theorems are given. The
importance of metrical fixed point theory and its application to
differential equations are studied. The fixed point theory for solving
the functional equations depends on reformulating the functional
equation:
= 0
to the equivalent fixed point problem
=
from which, we can write the iteration form
= , = 0, 1, 2,⋯
from this iteration form, we construct the sequence
which in general and subject to some conditions approximates the
solution. There is some questions about the convergence of this
sequence. If this sequence is convergent what about its limit, is this
limit satisfies the original equation what about the conditions which
must be imposed and what we can say about the rate of convergence,
is it possible to accelerate the convergence process!
The difficulty in applying Picard fixed point depends on the
choice of appropriate iteration form, the situation has more
complications in differential equations because of the included
integrations! These difficulties has restricted the use of Picard fixed
point theorem as a method of solution. For a long time Picard method
was used as a method for proving the existence or existence and
uniqueness of a solution. In comparing the series solution methods
with Picard method, we can treat equations with more general forms
not only polynomial functions by Picard method. It is important to
note that the point of solution in this case represent a function ”a point
in the solution space”. In this work, we tried to introduce the basic
concepts that help in the use of fixed point theorem as a method of
solution as well as accelerate the convergence of series of
approximations.
The thesis contains summary, motivations and four chapters.
Chapter One: ”Definitions and basic theorems for Contractions”, this
chapter contains four sections. In section 1: introduction. In section 2:
iii
contraction mapping principle. In section 3: the convergence and the
stability of a fixed point of a contraction operator. In section 4: the
relation between Picard iteration and other iterations ”Mann and
Ishikawa iterations”.
Chapter two: ”Successive approximations of differential and integral
equations” the chapter contains three sections. In section 1:
Introduction. In section 2: the fundamental theorems of existence and
uniqueness. In section 3: Construction of Green functions.
Chapter three: ”On the Effect of Using Integrating Factors
Approach with Picard Iteration”, the chapter contains five sections.
In section 1: Introduction. In section 2: the Gauss-Seidel modification
of Picard iterations. In section 3: the integrating factor modification of
Picard iteration. In section 4: the general linear two dimensional
systems In section 5: differential equations with critical points.
Chapter four: ”Accelerating the Modified Picard Iteration by using
Green’s Function Approach”, the chapter contains four sections. In
section 1: Introduction. In section 2: modification of Picard iteration
with Green’s function integral, In section 3: second order non-linear
equation of Bernoulli type. In section 4: Application to the Lotka-
Volterra model.
It is worth to mention that:
- all calculations were done by the use Mathematica 6.
- the results of chapter three has been accepted for publication in
the Egyptian Journal of Pure and Applied Science 2011 with
title ”Picard Fixed Point Iteration Combined with Integrating
Factor Approach”
- the content of chapter four is in the preparation for submission.