ABSTRACT

In this dissertation, a multi-step iterative scheme was used to establish the
strong and 􀀀convergence theorems for finite families of uniformly asymptotic
regular, total asymptotically nonexpansive semigroup in a uniformly
convex hyperbolic space. We also used a different method of proof, to establish
the polar and 􀀀convergence theorems for finite families of uniformly
asymptotic regular, uniformly L-Lipschitzian and total asymptotically nonexpansive
semigroup in a complete CAT(0) space. We then studied the modified
Mann iteration scheme for approximating common fixed point of a uniformly
asymptotic regular family of total asymptotically nonexpansive semigroup in
a complete CAT(0) space. We proved that the sequences generated by the
iterative schemes converges to a common fixed point of a finite family of uniformly
asymptotic regular and total asymptotically nonexpansive semigroup
in hyperbolic spaces.

 

 

TABLE OF CONTENTS

Flyleaf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Title Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Certification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
CHAPTER ONE
GENERAL INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . 3
1.3 Aim and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Research Methodology . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 Outline of the Dissertation . . . . . . . . . . . . . . . . . . . . . 5
1.6 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.6.1 Metric Space . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6.2 Fixed Point . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.6.3 CAT(0) Space . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6.4 Hyperbolic Space . . . . . . . . . . . . . . . . . . . . . . 10
1.6.5 Uniformly Convex Hyperbolic Space . . . . . . . . . . . 11
1.6.6 Semigroup of The One Parameter Family of Non-Expansive
Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6.7 Polar and 􀀀Convergence . . . . . . . . . . . . . . . . . 14
ix
1.6.8 Demiclosedness Property . . . . . . . . . . . . . . . . . . 14
CHAPTER TWO
LITERATURE REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 Fixed Point Theorems . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Semigroup of The One Parameter Family of Nonexpansive Mappings
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Fixed Point Theorems and Semigroup of The One Parameter
Family of Nonexpansive Mappings in Hyperbolic spaces (CAT(0)
Spaces) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
CHAPTER THREE
CONVERGENCE THEOREMS FOR FINITE FAMILIES OF
TOTAL ASYMPTOTICALLY NONEXPANSIVE
SEMIGROUP IN HYPERBOLIC SPACES . . . . . . . . . . . . . . . . . . . . . . . 21
3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
CHAPTER FOUR
COMMON FIXED POINT APPROXIMATION OF TOTAL
ASYMPTOTICALLY NONEXPANSIVE SEMIGROUP IN
CAT(0) SPACES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
x
CHAPTER FIVE
SUMMARY, CONCLUSION AND RECOMMENDATIONS. . . . 63
5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.3 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . 64
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

 

 

CHAPTER ONE

GENERAL INTRODUCTION
1.1 Introduction
The study of non-linear operators had its beginning about the start of the twentieth
century, with the investigation into the existence properties of solutions
to certain boundary valued problems arising in ordinary and partial differential
equations. The earliest techniques, largely devised by Picard (1893), involved
the iteration of an integral operator to obtain solutions to such problems. In
1922, these techniques of Picard were given precise abstract formulation by
Cacciopoli (1931) and Banach (1932), a powerful tool in analysis for establishing
existence and uniqueness of solution of problems of different kinds. The
fact that, fixed point theorem is an important tool for solving equation of the
form Tx = x; where T is a self-map defined on a subset of some suitable
space, leads to the significance of this area. And it is useful in the theory of
Newtonian and non-Newtonian calculus. The fixed point of a map plays the
role of an equilibrium or stable point of the body of a system defined in terms
of the operator. It is also known that the concept of equilibrium system is
very crucial in other scientific areas that includes: biology, ecology, economics,
physics, medicine, chemistry and also in engineering among others. Various
authors have generalized Banach-Cacciopoli contraction mapping principle in
different spaces which includes: Banach space, Hilbert space, Metric space,
Hyperbolic space and even the CAT(0) space.
Goebel and Kirk (1972) introduced the concept of asymptotically nonexpansive
mappings as a generalization of nonexpansive mappings. Alber et al.
1
(2006), introduced the class of total asymptotically nonexpansive mappings,
which generalizes several classes of maps, which are extensions of asymptotically
nonexpansive mappings. The concept of fixed point theory in CAT(k)
space was first introduced by Kirk (2003, 2004). His work was followed by a
series of new works by many authors, mainly focusing on the CAT(0) space,
which is a special case of the CAT(k) space, all results in CAT(0) space immediately
apply to any CAT(k) space with k 0. As it is well known, the
construction of common fixed points of nonexpansive semigroup and asymptotically
nonexpansive semigroup is an important problem in the theory of
nonexpansive mappings in nonlinear operator theory and applications. These
has applications in: image recovery, signal processing problems and convex
feasibility problems. (see; Yao and Shahzad (2011), Chidume and Chidume
(2006), Marino and Xu (2006), Xu (2004), Shimoji and Takahashi (2001),
among others). Takahashi (1969) proved the fixed point theorem for a noncommutative
semigroup of nonexpansive mappings which generalises DeMarr’s
(1963) result.
The concept of approximation of common fixed point of asymptotically nonexpansive
mappings in CAT(0) space and convergence theorems for finite families
of total asymptotically nonexpansive mappings in hyperbolic space, were discussed
in Ugwunnadi and Ali (2016) and in Ali (2016). In this dissertation,
using the definitions of the semigroup of the one parameter family of nonexpansive
mappings and the uniformly asymptotic regularity of a map, their
results were a bit modified and presented in the same complete CAT(0) space.
2
1.2 Statement of the Problem
Ali (2016) proved the strong and 􀀀convergence for finite families of total
asymptotically nonexpansive mappings in hyperbolic spaces, while Ugwunnadi
and Ali (2016) studied the modified Mann iterative scheme and proved that the
proposed sequence in the iterative scheme converges to a common fixed point
of total asymptotically nonexpansive mappings in a complete CAT(0) space.
In this dissertation, using the concept of uniformly asymptotic regularity of
self mappings and the semigroup of the one parameter family of nonexpansive
mappings, we shall study and prove the strong, polar and 􀀀convergence theorems
in a uniformly convex hyperbolic space and also in a complete CAT(0)
space.
1.3 Aim and Objectives
This dissertation is aimed at establishing new results in the field of non-linear
operator theory. The primary objectives of this study are to:
1. establish the strong and 􀀀convergence theorems, using the uniformly
asymptotic regularity of self mappings on the total asymptotically quasinonexpansive
semigroup in a uniformly convex hyperbolic space.
2. prove (1) above, using a different method of proof to show polar and
􀀀convergence theorems, using the uniformly asymptotic regularity of
self mappings on the total asymptotically quasi-nonexpansive semigroup
in a complete CAT(0) space.
3
3. establish the notion of convergence theorems in hyperbolic space and also
in a complete CAT(0) space, using the concept of the semigroup of the
one parameter family of nonexpansive mappings.
1.4 Research Methodology
The method used in this dissertation is by consulting necessary and relevant
books and articles, in literature; on the fixed point theorem, approximation
of common fixed points, convergence for finite families, uniformly asymptotic
regular family, total asymptotically nonexpansive semigroup, CAT(0) space,
hyperbolic space and so on. These articles were reviewed thoroughly to cover
a major part of the work done on polar and 􀀀 convergence in hyperbolic and
CAT(0) spaces. The work done on the common fixed point approximations
for finite families of total asymptotically nonexpansive semigroup in hyperbolic
space are then taken in the settings of a complete CAT(0) space.
In the theorems, we assumed that the finite families of mappings are uniformly
asymptotic regular and are semigroup of nonexpansive mappings, if they satisfy
some restricted and appropriate conditions, then the sequence either polar,
strongly or 􀀀converge to a point in the set of all common fixed points in the
space. And, in the proofs, we considered a sequence from the proposed iterative
schemes and applied the concepts of: a metric space, the uniformly asymptotic
regularity of a map, the total asymptotically quasi-nonexpansive semigroup of
self mappings, the definitions of polar convergence, strong convergence and
􀀀convergence. Using these concepts, we successfully showed that the limit
of the sequence exists in the set of all common fixed points in a complete
4
CAT(0) space and also in a uniformly convex hyperbolic space.
1.5 Outline of the Dissertation
The dissertation contains four other chapters apart from the introductory chapter.
The outline of the remaining chapters are as follows:
Chapter II: In this chapter, we present a survey of the necessary and relevant
literature for fixed point theorem, CAT(0) space and semigroup of nonexpansive
mappings.
Chapter III: In this chapter, we establish the strong and 􀀀convergence theorems
for finite families of total asymptotically nonexpansive semigroup of
mappings in both the hyperbolic space and the CAT(0) space.
Chapter IV: In this chapter, we introduce the approximation of common fixed
point of a family of uniformly asymptotic regular semigroup of mappings in a
complete CAT(0) space, using the modified Mann iterative scheme.
Chapter V: In the final chapter, we present the summary and conclusion of the
results obtained in this dissertation, along with some directions/recommendations
for further research.
1.6 Preliminaries
In this section we give some basic and important definitions and concepts which
are useful and related to the context of this dissertation.
5
1.6.1 Metric Space
Definition 1.6.1 Let X be a nonempty set and R denote the set of real numbers.
A metric d on X is a real-valued function d : X X ! R which satisfies
the following conditions: For any x; y; z 2 X;
M1: d(x; y) 0
M2: d(x; y) = 0 if and only if x = y;
M3: d(x; y) = d(y; x); and
M4: d(x; y) d(x; z) + d(z; y)
The pair (X; d) is called a metric space.
Example 1.6.1 Let d : R R ! R defined by d(x; y) = jx 􀀀 yj; 8x; y 2 R.
Then (R; d) is a metric space, called the usual metric on R.
Example 1.6.2 Let X be an arbitrary nonempty set. Define d : X X ! R
by;
d(x; y) =
8>>>><
>>>>:
1; x 6= y;
0; x = y:
for all x; y 2 X. Then (X; d) is a metric space called the trivial or discrete
metric.
6
1.6.2 Fixed Point
Definition 1.6.2 Let X be a nonempty set and T : X ! X be self mappings.
Then a point x 2 X is called a fixed point of T if: Tx = x.
The set of all fixed points of T is represented by: F(T) := fx 2 X : Tx = xg.
The set of all common fixed points of T is represented by: F :=
Tn
i=1 F(Ti) 6= ;,
8i 2 N. (see; Alber et al. (2006))
Let (X; d) be a metric space. A self mappings T : X ! X is called nonexpansive
if: d(Tx; Ty) d(x; y), for every x; y 2 X. A map T is called
quasi-nonexpansive if: F(T) := fx 2 X : Tx = xg 6= ; and d(Tx; p) d(x; p),
for every x 2 X and p 2 F(T). The class of quasi-nonexpansive mappings
properly contained the class of nonexpansive mappings with fixed points.
The mappings T is called asymptotically nonexpansive if there exists a sequence
fkng [1;1) with kn ! 1 as n ! 1 such that: for every n 2 N,
d(Tnx; Tny) knd(x; y), for all x; y 2 X.
If F(T) 6= ; and there exists a sequence fkng [1;1) with kn ! 1 as n ! 1
such that: for every n 2 N, d(Tnx; p) knd(x; p), for all x 2 X and p 2 F(T),
then T is called asymptotically quasi-nonexpansive.
The mappings T is called total asymptotically nonexpansive, if there exists
infinitesimal real sequences fung and fvng of nonnegetive numbers (i.e un; vn !
0 as n ! 1) and a strictly increasing function : [0;1) ! [0;1) with
(0) = 0 such that: d(Tnx; Tny) d(x; y) + un (d(x; y)) + vn, for every
x; y 2 X.
And T is called total asymptotically quasi-nonexpansive, if: F(T) 6= ; and
there exists infinitesimal real sequences fung and fvng of nonnegative numbers
7
(i.e un; vn ! 0 as n ! 1) and a strictly increasing function : [0;1) !
[0;1) with (0) = 0 such that: d(Tnx; p) d(x; p) + un (d(x; p)) + vn, for
every x;2 X and p 2 F(T). (see: Ali (2016))
A self mappings T : X ! X is called L-Lipschitz (or L-Lipschitzian), if there
exists a constant L > 0 such that: d(Tx; Ty) Ld(x; y), for every x; y 2 X.
T is called a contraction (or strict contraction), if L < 1 and its nonexpansive
if L = 1.
The mappings T is called uniformly L-Lipschitz (or uniformly L-Lipschitzian),
if for every constant L > 0, there exists n 2 N such that:
d(Tnx; Tny) Lnd(x; y) Ld(x; y), for every x; y 2 X.
Example 1.6.3 The map T : R ! R defined by:
1. (i) Tx = tanx (ii) Tx = sinx; are non-linear.
2. (i) Tx = 5x + 10 (ii) Tx = ax, for any constant a; are linear.
3. Tx = x + 1: T is nonexpansive and Lipschitz.
Since, d(Tx; Ty) = jTx􀀀Tyj = j(x+1)􀀀(y+1)j Ljx􀀀yj = Ld(x; y).
Clearly, T is Lipschitz with L > 0, its a contraction with L < 1 and its
nonexpansive with L = 1.
1.6.3 CAT(0) Space
Let (X; d) be a metric space, 8x; y 2 X. Then, we have the following definitions:
8
Geodesic Path
A geodesic path from x to y is an isometry. A map T : X ! X is an isometry
(distance preserving) if for any x; y 2 X, d(Tx; Ty) = d(x; y).
Geodesic Segment
The image of a geodesic path is called a geodesic segment. A geodesic segment
joining two points x; y in a metric space X is represented by [x; y], where
[x; y] := fx+(1􀀀)y : 2 [0; 1]g. A subset K of a metric space X is convex
if 8x; y 2 K, [x; y] K.
Geodesic Space
A metric space (X; d), is called a geodesic space if any two distinct points of
X are joined by the geodesic segment.
Geodesic Triangle
A geodesic triangle (4(x; y; z)), consist of three distinct points x; y; z 2 X
joined by three geodesic segments in a geodesic space.
Comparison Triangle
A comparison triangle (4(x; y; z)) or (4(x; y; z)) of a geodesic triangle (4(x; y; z))
is a triangle in the Euclidean space (R2) such that: d(x; y) = dR2(x; y),
d(y; z) = dR2(y; z) and d(z; x) = dR2(z; x).
9
A geodesic space is called a CAT(0) space if for every geodesic triangle (4)
and its comparison triangle (4), the following inequality holds:
d(x; y) dR2(x; y);
where; x; y 2 4 and x; y 2 4:
Also,
A metric space (X; d) is a CAT(0) space if it is geodesically connected and if
every geodesic triangle (4) in X, is at least as thin as its comparison triangle
(4) in the Euclidean space (R2). (see; Kirk and Panyanak (2008))
A CAT(0) space X, is said to be complete if every Cauchy sequence fxng in
X, converges to a point x 2 X. A complete CAT(0) space is often called the
Hadamard space.
CAT means Cartan Alexandrov Topogonov. Examples of the CAT(0) spaces
includes; R􀀀tree, Hadamard space, Hilbert ball equipped with hyperbolic metric
and so on. For details on these spaces, (see; Abramenco and Brown (2008),
Dhompongsa and Panyanak (2008), Burago et al. (2001), Bridson and Haefliger
(1999), Ballmann (1995) and Brown (1989)).
1.6.4 Hyperbolic Space
Definition 1.6.3 A geodesic space (X; d) is called hyperbolic, if for any x; y; z 2
X,
d(
1
2
x
1
2
z;
1
2
z
1
2
y)
1
2
d(x; y):
10
The class of hyperbolic spaces includes: normed spaces, CAT(0) spaces amongst
others. The following is an example of a hyperbolic space which is not a normed
space. (see; Reich and Shafrir (1990))
Example 1.6.4 Let D be a unit disc in a complex plane C. Define d : DD !
R by:
d(z;w) = log(
1 + j z􀀀w
1􀀀zwj
1 􀀀 j z􀀀w
1􀀀zwj
)
Then, (D; d) is a complete hyperbolic metric space.
From the example above, we have that the class of hyperbolic spaces are more
general than the class of normed spaces.
1.6.5 Uniformly Convex Hyperbolic Space
Definition 1.6.4 Let (X; d) be a hyperbolic space. Then X is called uniformly
convex if for any a 2 X, for every r > 0 and for each > 0:
a(r; ) = inff1 􀀀
1
r
d(
1
2
x
1
2
y; a) : d(x; a) r; d(y; a) r; d(x; y) g > 0:
1.6.6 Semigroup of The One Parameter Family of Non-
Expansive Mappings
Definition 1.6.5 Let K be a nonempty subset of a metric space X. A one
parameter family @ := fT(t) : K ! K; t 2 R+g, where R+ denotes the set of
non-negative real numbers of maps is called a semigroup of self mappings from
K into K satisfying:
11
S1: T(0)x = x, for all x 2 K;
S2: T(s + t)x = T(s)T(t)x, for all s; t 2 R+;
S3: For each x 2 K, the mapping t 7􀀀! T(t)x is continuous, (limt!0 T(t)x =
x)
Let X be a nonempty set together with the binary operation (+; :), then X is
said to be a semigroup if it satisfies the closure and associative properties under
the binary operators. A semigroup is also known as an associative magma.
Uniformly L-Lipschitzian Nonexpansive Semigroup
A one parameter family @ := fT(t) : K ! K; t 2 R+g, is said to be uniformly
L-Lipschitzian nonexpansive semigroup if conditions S1􀀀S3 above are satisfied
and in addition:
S4: for each t > 0, there exists a bounded function L(t) : (0;1) ! [0;1)
such that; d(Tn(t)x; Tn(t)y) L(t)d(x; y); 8 x; y 2 K.
A uniformly L-Lipschitzian semigroup of a one parameter family @ is called
nonexpansive (or contraction) if: L(t) = 1 (orL(t) < 1), for all t > 0.
Total Asymptotically Nonexpansive Semigroup
A one parameter family @ := fT(t) : K ! K; t 0g, is said to be total asymptotically
nonexpansive semigroup, if conditions S1􀀀S3 above are satisfied and
in addition:
S4: for each t 0, there exists functions u; v : [0:1) ! [0;1) and strictly in-
12
creasing and continuous functions : R+ ! R+ such that: limt!1 u(t) =
limt!1 v(t) = 0 and (0) = 0, then;
d(T(t)x; T(t)y) d(x; y) + u(t) (d(x; y)) + v(t); 8x; y 2 K.
Total Asymptotically Quasi-Nonexpansive Semigroup
A one parameter family @ := fT(t) : K ! K; t 0g, is said to be total
asymptotically quasi-nonexpansive semigroup, if conditions S1􀀀S3 above are
satisfied and in addition:
S4: for each t 0, there exists functions u; v : [0:1) ! [0;1) and strictly increasing
and continuous functions : R+ ! R+ such that: limt!1 u(t) =
limt!1 v(t) = 0 and (0) = 0. If F(T) := fp 2 X : Tp = pg 6= 0, then:
d(T(t)x; p) d(x; p) + u(t) (d(x; p)) + v(t), 8x 2 K and p 2 F(T)
.
Asymptotically Regular and Uniformly Asymptotically Regular
A one parameter family @ := fT(t) : K ! K; t 0g, is said to be asymptotic
regular if;
lim
t!1
d(T(s + t)x; T(t)x) = 0; 8 t 2 [0;1) and x 2 K
It is also said to be uniformly asymptotic regular if: for any t 0 and for any
bounded subset C of K,
lim
t!1
sup
x2C
d(T(s + t)x; T(t)x) = 0:
13
1.6.7 Polar and 􀀀Convergence
Definition 1.6.6 (Ali (2016)) A sequence fxng in a complete CAT(0) space
X is said to 􀀀converge to a point x, if x is a unique asymptotic centre of fung
for every subsequence fung of fxng. This is written as 􀀀 limn!1 xn = x.
A sequence fxng in a complete CAT(0) space X is said to polar converge to
a point x 2 X, if for every y 2 X, y 6= x, there exists Ny 2 N such that;
d(xn; x) < d(xn; y), 8n Ny.
The sequence fxng converge 􀀀strongly to a point x, if the limit limn!1 d(xn; x)
exists and for any y 6= x, limn!1 d(xn; x) lim infn!1 d(xn; y).
Let fxng be a bounded sequence in a complete CAT(0) space X. For x 2 X,
we set: r(x; fxng) = lim supn!1 d(x; xn).
The asymptotic radius r(fxng) of fxng is given by :
r(fxng) = inffr(x; fxng) : x 2 Xg,
and the asymptotic center A(fxng) of fxng is the set:
A(fxng) = fx 2 X : r(x; fxng) = r(fxng)g.
It is well known that in a CAT(0) space, A(fxng) consist of exactly one point,
see (Proposition 7 of Dhompongsa et al. (2006))
1.6.8 Demiclosedness Property
Definition 1.6.7 The mappings T : X ! X is said to be demiclosed at a
point, if for any sequence fxng in X which converges weakly to a point x 2 X,
with d(xn; Txn) ! 0, as n ! 1, then Tx = x, for all x 2 X.
(see; Chang et al. (2012))

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