ABSTRACT
This thesis examines a general iterative scheme for fixed points of nonlinear operators using the
special cases of the Krasnoselskii-Mann (KM) iterative procedure for particular choices of the
nonexpansive operator N. We prove many theorems and lemmas that help to see the relationship
between some iterative algorithms. The iterative scheme xk+1 =Txk =
(1- )xk+ Nxk has been shown to be one of the schemes that can be generally used to represent
some iterative schemes for finding fixed points of nonlinear operators.
TABLE OF CONTENTS
Title page..……………………………………………………………………….……………. i
Declaration page ………………………………………………………………………………ii
Certification………………………………………………………………..………………… iii
Dedication…………………………………………………………………………………….. iv
Acknowledgement……………………………………………………………………….……. v
Abstract………………………………………………………………………….…………… vi
Content……………………………………………………………………………………… vii
1 .0 INTRODUCTION AND PRELIMINARIES ……………… .………………………1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . ………………………………………… . …. 1
1.2 Research questions . . . . . . . . . . . . . . . . . …………………………………… . . . . . ……1
1.3 Aim and Objectives of the Study . . . . ………………………..………… . . . . . . . . . . . 2
1.4 Statement of the Research Problem . . ………………………………… . . . . . . . . . . . . . 2
1.5 Research Methodology . . . . . . . . . . . . . . ……………………………….. . . . . . . ……. 3
1.6 Convergence in the normed space . . . ….……………………………………………… 3
1.7 Nonexpansive operator . . . . . . . . . . …………………………………………………… 3
1.8 Hilbert space . . . . . . . . . . . . . . . . . …………………………………………………….. 3
1.9 Fixed Point . . . . . . . . . . . . . . . . . ………………………………………..……………….3
1.10 Operators on Hilbert Space . . . . . . . . . . . . . . . . . . ……………………………………. 4
1.11 Firmly nonexpansive operator . . . . . . . . . . . . . . . . . …………………………………… 5
1.12 Monotone Operator . . . . . . . . . . . . . . . . . . . . . . ………………………………………… 5
1.13 Strict contraction . . . . . . . . . . . . . . . . . . ……………………………………… . . . . . …. 5
1.14 Average operator . . . . . . . . . . . . . . . . . . . . . . . ………………………………………. . ..5
8
1.15 V-ism operator . . . . . . . . . . . . . . . . . . . . . . . . . ………………………………………… 6
1.16 Sequences in a Metric Space . . . . . . . . . . . . . . . . . . ……………………………………. 6
1.17 Subsequences in a Metric Space . . . . . . . . . . . . . . . . …………………………………… 6
1.18 Convergence of a Sequence in a Metric Space . . . . . . . . ………………………………. 7
1.19 Bounded Sequences . . . . . . . . . . . . . . . . . . . . . . . …………………………………………7
1.20 Attracting mapping . . . . . . . . . . . . . . . . . . . . . . . ………………………………………..8
1.21 Outline of the thesis . . . . . . . . . . . . . . . . . ……………………………………………….9
2.0 Literature Review ……………………………………………………………..…..……10
2.1 A Superior Implementation of the Algebraic Reconstruction Technique (ART) Algorithm . .
. . . . . . . . . . . . . . . . . . …………………………………………………………………. . . .10
2.2 The Multiple-sets Split Feasibility Problem and its Applications for Inverse Problems
………………………………………………………… . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Projection Methods and their advantages . . . . . …………………………………. . . . . . 12
2.4 The Split Feasibility Problem . . . . . . . . . . . . . . . . . ……………………………………. 13
2.5 Cimmino’s Method and the Algebraic Reconstruction Technique …. . . . . . . . . . . . . . . . 14
2.6 Bandlimited Extrapolation Methods . . . . . . . . . . . . …………………………………… . 15
2.7 The Landweber algorithms . . . . . . . . . . . . . . . . . ……………………………………… . 19
2.8 The Geometric Properties of Banach Spaces and Nonlinear Iterations . . . . . . . . . . . . . . 21
2.9 Interior point optimization algorithms . . . . . . . ………………………………….. . . . . . 24
3.0 SOME NONLINEAR OPERATORS AND THEIR RELATIONS…………………..25
3.1 Introduction . . . . …………………………………….. . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Firmly nonexpansive operator . . . ……………………………………………….. . . . . . 31
3.3 Prototype of a strongly attracting mapping . . . . . . …………….…………………… . . . 35
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3.4 A system of generalized equilibrium problems . . . ……………………………… . . . . . 36
4.0 SOME ITERATIVE ALGORITHMS FOR FIXED POINT OF NONLINEAR
OPERATORS………………………………………………………………………… 40
4.1 Introduction . . . . ………………………………………… . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 Constrained Optimization Algorithms . . . . . . . . . . . ………………………………….. . 40
4.3 Orthogonal Projection onto Sets C and Q . . . . . . . . …………………………………… . 41
4.4 The convex feasibility problem . . . . . . . . . . . . . . . . ……………………………………. 43
4.5 The split feasibility problem . . . . . . . . . . . . . . . . . . ……………………………………. 44
5.0 THE MAIN RESULT …………………………………………………………………..48
5.1 Introduction . . . . . . . . . ……………………………………….. . . . . . . . . . . . ……….. . 48
5.2 The Main result……………………………………………………………………….…..48
6.0 SUMMARY, CONCLUSION AND RECOMMENDATIONS……………….….….. 51
6.1 Summary . . . . . …………………………………………. . . . . . . . . . . . . . . . . . . . . . . . ..51
6.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . ………………………………………….. 52
6.3 Recommendations………………………………………………………………………..52
REFERENCES………………………………………………………………………………..53
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CHAPTER ONE
Introduction and Preliminaries
1.1 Introduction
Many problems in Mathematics and related field can be solved by finding fixed point of a
particular operator, and algorithms for finding such points play a prominent role in a number of
applications.
The article by Bauschke (1996) is fundamental to this work. This section deals with definitions
of some basic terms used in the subsequent discussions while some examples are also given to
make these definitions clearer.
1.2 Research questions
There are many open challenging questions on fixed point theorem that have still not been
answered which include the following;
i. Is there a general algorithm for some iterative schemes finding fixed point of nonlinear
operator?
ii. Are there significant relationships between nonexpansive ne operator N, firmly nonexpansive
fne operator F and average av operator T?
iii. Are the iterative schemes of the complements of nonexpansive ne, firmly nonexpansive fne
and average operator av T converge to fixed points of T as the main operators?
iv. Which of the properties of the operator T is sufficient to guarantee convergence of the
sequence {Tkx} to fixed point of T, whenever such fixed points exist?
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1.3 Aim and Objectives of the Study
The aim of this thesis is to unify the several iterative algorithms for finding the fixed points of
some nonlinear operators. The iterative methods considered have the form,
x k+1 = T x k. for k = 0,1… (1.3.1)
Where T is a linear or nonlinear continuous operator on a real (possibly infinite dimensional)
Hilbert space H and x0 is an arbitrary starting vector. For any operator T on H the fixed point
set of T is denoted by Fix(T ) = {z/ Tz = z}.
The main objectives here are;
i. To find a common iterative scheme for some iterative algorithms for finding the fixed points of
some nonlinear operators.
ii. To find the property of the operator T which is sufficient to guarantee convergence of the
sequence {Tkx} to a fixed point of T whenever such fixed points exist.
1.4 Statement of the Research Problem
In the algorithms of interest here, the operator T is selected so that the set Fix(T) contains those
vectors z that possess the properties we desire that is ‘find a general iterative scheme for some
iterative algorithms for nonlinear operators’ finding a fixed point of the some iterative schemes
leads to a solution of the problem.
Some applications involve constrained optimization, in which we seek a vector x in a given
convex set C that minimizes a certain function f. For suitable > 0 the fixed points of the
operator T = PC(I – ∇f ) will solve the problem under conditions to be discussed below.
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1.5 Research Methodology
This thesis is a review of the paper of Byrne (2004). Many related and necessary papers were
consulted for effective outcome. These papers are thoroughly reviewed to cover a major part of
this work. Lemmas, propositions and Theorems are stated and proved in order to ascertain the
properties, relationship and applications of the operators used in the thesis while conclusion are
drawn based on the theorems, propositions and lemmas proved.
1.6 Convergence in the normed space
Let (X, .) be a normed space and let {xn} be a sequence in X, then {xn} is said to
(i) converge strongly, if ∃ x ∈ X ∋ xn- x→ 0 as n→ ∞ ,
(ii) converge strongly if ∃ x ∈ X ∋ |f( xn) – f(x)|→ 0 as n→ ∞ ∀ f ∈ x*
1.7 Nonexpansive operator
An operator N on H is called nonexpansive ne if ∀ x, y ∈ H, we have
Nx – Ny ≤ x – y
1.8 Hilbert space
An inner product space (or a pre-Hilbert space) is said to be complete if every Cauchy sequence
{x n}∈ X converges to a point x ∈ X. A complete pre-Hilbert space is called a Hilbert space.
1.9 Fixed Point
Let X be a nonempty set and T: X → X be a map. We say that x ∈ X is a fixed point of T if
T(x) = x and we denote the set of all fixed points of T by F(T) = {x ∈ X : T x = x}.
Many problems of nonlinear analysis could be solved by the use of various fixed points
theorems.
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Example 1.9.1
1. If X = ℝ and T(x) = x2+ 5x + 4, then F (T ) ={-2};
2. If X = R and T (x) = x 2 – x, then F(T) = {0, 2}.
Let X be any set and T: X → X be a map. For any given x ∈ X, we denote Tn(x) inductively by
T0(x) = x and Tn+1(x) = T(Tn(x)); we call Tn (x) the nth iterate of x under T, For simplicity we use
Tx instead of T (x).
The mapping Tn (n ≥ 0) is called the nth iterate of T. For any x ∈ X; the sequence {xn} n ≥ 0⊂ X
given by xn = Txn-1, n = 1, 2, 3,… is known as the Picard’s iterations. For any given self-mapping
T the following properties obviously hold:
i. F (T) ⊂ F(Tn) for each n ∈ N,
ii. F(Tn) = {x}, ⇒ F(T) = {x}.
But the reverse in (i) is not true in general, as shown by the next example.
Example 1.9.2 Let T: {1, 2, 3} → {1, 2, 3} be denoted by T(1) = 3,T (2) = 2 and T (3) = 1.
Then F (T) = {2} and FT(2) = {1, 2, 3}.
The fixed point theorems are concerned with finding conditions on the structure that the set X
must be endowed as well as on the properties of the operator T: X → X, in order to obtain result
on,
a. existence (and possibly uniqueness) of fixed point .
b. the data dependence of fixed points.
c. the construction of fixed point.
1.10 Operators on Hilbert Space
(i) Adjoint Operator
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Let y be a fixed element in H and let T (y) be defined on H by T(y)x =⟨T(x), y⟩, ∀ x ∈ H then
T(y) is a bounded linear functional on H.
Therefore, by Riez representation Theorem ∃ a unique vector z ∈ H ∋ T (y) =⟨ x, z⟩, ∀ x ∈ H that
is ⟨x, z⟩ = ⟨T(x), y⟩ ∀ x ∈ H putting z = T* y we see that ⟨x, Ty⟩ = ⟨T (x), y)⟩ ∀ x, y ∈ H.
The mapping T* is called the adjoint of T
(ii) Self-adjoint Operator
Let T be a bounded linear operator on a Hilbert space H. Then T is said to be self-adjoint if T =
T*. Note that 0 = 0* and I = I*, that is zero and identity operators are self-adjoints.
(iii) Normal Operator
A bounded linear operator T on a Hilbert space H is called a Normal operator if T* T= TT*
(iv) Unitary Operator
A bounded linear operator T on a Hilbert space H is called unitary, if T* T= TT*= I
1.11 Firmly nonexpansive operator
An operator F on H is called firmly nonexpansive fne if F is defined as
⟨Fx – Fy, x – y⟩ ≥ Fx – Fy2 ∀ x,y ∈ H
1.12 Monotone Operator
An operator G: H → H is monotone if, for all x, y ∈ H, we have
⟨Gx – Gy, x – y⟩ ≥ 0
1.13 Strict contraction
An operator S on H is said to be a strict contraction (sc) if ∃ ∈ [0, 1), such that ∀ x, y ∈ H,
Sx – Sy ≤ x – y
1.14 Average operator
An operator A is called average if ∃ ∈ (0; 1) and ne operator N, ∋ we have A = (1- )I + N
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1.15 V-ism operator
An operator G on H is called v-inverse strongly monotone (v- ism), v > 0, ∋
⟨Gx – Gy, x – y⟩ ≥ v Gx – Gy 2
1.16 Sequences in a Metric Space
Let X be a nonempty set, a sequence in X is a function from ℕ (the set of positive integers) into
X. If x is a sequence in X, the image x(n) of n ∈ ℕ is usually denoted by { xn}. It is customary
to denote the sequence x by the classical symbol {xn}.
Some time we write it as {x1, x2, x3 …, xn …}.
The following are examples of real sequences: {
}, {
}, { cos
}.
A sequence {xn} of real number is called
(a) strictly monotone decreasing if xn+1 < xn ∀ n ∈ ℕ;
(b) Monotone non-decreasing if xn+1 ≥ xn ∀ n ∈ ℕ;
(c) Strictly monotone increasing if xn+1 < xn ∀ n ∈ ℕ;
(d) Monotone non-increasing if xn+1 ≤ xn ∀ n ∈ ℕ;
1.17 Subsequences in a Metric Space
Let x: ℕ → X is a sequence on X. Let n: ℕ → ℕ be a strictly increasing function, then the
function x n: ℕ →X is called a subsequence of the original sequence x: ℕ → X.
Example 1.17.1 The sequence of prime numbers {2, 3, 5, 7, 11,} is a subsequence of the
sequence of natural numbers.
The proof of this is shown below:
Let {xn} = {2, 3, 5, 7, 11, …}.
Then x1 = 2, x2 = 3, x3 = 5, x4 = 7, e t c.
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Take n1= 2, n2= 3, n3= 5, n4= 7, … . Then {nk} is a strictly increasing sequence of positive
integers. Therefore,
x(n(1)) = 2
x(n(2)) =3
x(n(3)) = 5 etc.:
Hence, {2, 3, 5, 7, 11, …}. Is a subsequence of {xn}.
1.18 Convergence of a Sequence in a Metric Space
Let {xn} be a sequence in a metric space (X; d), this sequence is said to converge to a point x ∈
X if given > 0, ∃ n0 ∈ N; such that d (xn, x) < ∀ n ≥ n0
The fact that {xn} converges to x is expressed by writing xn → x as n → 1 or lim x n = x
A sequence {xn} in a metric space (X, d) is called a convergent sequence if the sequence
converges to a point say x ∈ X.
1.19 Bounded Sequences
A sequence {xn} is said to be bounded below if there exists a real number such that xn ≥
∀ n ∈ N.
The number is called a lower bound of the sequence {xn}.
Example 1.19.1
The sequence {xn} = {1, 2, 3, 4, …} is bounded below as there exists a number 1 which is less
than each term of the sequence i.e.
xn ≥ 1, ∀ n ∈ N.
A sequence {xn} is said to be bounded above if there exists a real number such that xn ≤
∀ n ∈ N.
The number is called an upper bound of the sequence {xn}.
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Example 1.19.2
For example, the sequence {xn} = {-1, -2, -3, -4 …} is bounded above as there exists a number –
1 which is greater than each term of the sequence i.e. xn ≤ −1, ∀ n ∈ N.
A sequence {xn} is said to be bounded if it is bounded both above and below. i.e. a sequence
{xn} is bounded if there exist two real numbers and such that ≤ xn ≤ ∀ n ∈ N.
If we choose M = max {||||}, then the sequence {xn}is bounded provided |xn| ≤ M
∀ n ∈ N.
Example 1.19.3
if x n =(
), then {xn} = {
,
,
,
,…}, where xn ≤ 0 and xn ≥ 1, ∀ n ∈ N. i.e. 0 ≤ xn ≤ 1.
∀ n ∈ N.
Hence, the sequence {xn} is bounded.
1.20 Attracting mapping
Definition 1.21.1 Suppose D is a closed convex nonempty set, T: D→D is nonexpansive, and F is
a closed convex nonempty subset of D. we say that T is attracting w:r:t: F if for every x ∈ D|F,
f ∈ F, then we have T x – f < x – f ,
In other words, every point in F attracts every point outside F. A more quantitative and stronger
version is the following. We say that T is strongly attracting w,r,t, F if there is some k > 0 s.t for
every x ∈ D, f ∈ F
k x – Tx 2 ≤ x – f 2 – Tx – f 2,
Alternatively, if emphasis is laid on k explicitly, we say that T is k-attracting w.r.t F. In several
instances, F is Fix T, in this case, we simply speak of attracting, strongly attracting, or
k – attracting mapping.
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1.21 Outline of the thesis
This thesis contains five other chapters after this chapter. The outlines of chapter two to
six are as follows:
Chapter 2: This contains the reviewed relevant literature that has direct bearing with the thesis
‘a general iterative scheme for signal enhancement and image modification’
Chapter 3: In this chapter, we proved lemmas, theorems, definitions and propositions; we also
compare them in order to see their significant relationship
Chapter 4: This chapter contains the analysis of some of the algorithms that are popularly used in
signal enhancement and image modification which are relevant to the work.
Chapter 5: Here, we present the main result of the thesis through a theorem which is proved to
see how the objectives are accomplished.
Chapter 6: We hereby give the conclusion based on the main result presented in chapter five and
also present the summary of the entire work.
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