## PROJECT TOPIC ON A STUDY OF COMMON FIXED POINT APPROXIMATIONS FOR FINITE FAMILIES OF TOTAL ASYMPTOTICALLY NONEXPANSIVE SEMIGROUP IN HYPERBOLIC SPACES

ABSTRACT

In this dissertation, a multi-step iterative scheme was used to establish the strong and convergence theorems for finite families of uniformly asymp-totic regular, total asymptotically nonexpansive semigroup in a uniformly convex hyperbolic space. We also used a diﬀerent method of proof, to es-tablish the polar and convergence theorems for finite families of uniformly asymptotic regular, uniformly L-Lipschitzian and total asymptotically nonex-pansive 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 uni-formly asymptotic regular and total asymptotically nonexpansive semigroup in hyperbolic spaces.

CHAPTER ONE

GENERAL INTRODUCTION

1.1 Introduction

The study of non-linear operators had its beginning about the start of the twen-tieth 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 T x = 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 nonexpan-sive mappings as a generalization of nonexpansive mappings. Alber et al.

(2006), introduced the class of total asymptotically nonexpansive mappings, which generalizes several classes of maps, which are extensions of asymptoti-cally 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 im-mediately apply to any CAT (k) space with k 0. As it is well known, the construction of common fixed points of nonexpansive semigroup and asymp-totically 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 non-commutative semigroup of nonexpansive mappings which generalises DeMarr’s (1963) result.

The concept of approximation of common fixed point of asymptotically nonex-pansive mappings in CAT (0) space and convergence theorems for finite families of total asymptotically nonexpansive mappings in hyperbolic space, were dis-cussed in Ugwunnadi and Ali (2016) and in Ali (2016). In this dissertation, using the definitions of the semigroup of the one parameter family of non-expansive mappings and the uniformly asymptotic regularity of a map, their results were a bit modified and presented in the same complete CAT (0) space.

### PROJECT TOPIC ON A STUDY OF COMMON FIXED POINT APPROXIMATIONS FOR FINITE FAMILIES OF TOTAL ASYMPTOTICALLY NON EXPANSIVE SEMI GROUP IN HYPERBOLIC SPACES

1.2 Statement of the Problem

Ali (2016) proved the strong and cconvergene 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 the-orems 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:

- establish the strong and convergence theorems, using the uniformly asymptotic regularity of self mappings on the total asymptotically quasi-nonexpansive semigroup in a uniformly convex hyperbolic space.
- prove (1) above, using a diﬀerent method of proof to show polar a

convergence theorems, using the uniformly asymptotic regularity of self mappings on the total asymptotically quasi-nonexpansive semigroup in a complete CAT (0) space.=

- 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

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 chap-ter. 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 nonexpan-sive mappings.

Chapter III: In this chapter, we establish the strong and convergence the-orems 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.

1.6.1 Metric Space

Definition 1.6.1 Let X be a nonempty set and R denote the set of real num-bers. 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;

- 1; x 6= y;

>

>

<

d(x; y) =

>

>

>

- 0; x = y:

for all x; y 2 X. Then (X; d) is a metric space called the trivial or discrete metric.

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: T x = x.

The set of all fixed points of T is represented by: F (T ) := fx 2 X : T x = xg.

The set of all common fixed points of T is represented by: F := ^{T}^{n}_{i=1} F (T_{i}) 6= ;, 8i 2 N. (see; Alber et al. (2006))

Let (X; d) be a metric space. A self mappings T : X ! X is called non-expansive if: d(T x; T y) d(x; y), for every x; y 2 X. A map T is called quasi-nonexpansive if: F (T ) := fx 2 X : T x = xg 6= ; and d(T x; 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 se-quence fk_{n}g [1; 1) with k_{n} ! 1 as n ! 1 such that: for every n 2 N, d(T ^{n}x; T ^{n}y) k_{n}d(x; y), for all x; y 2 X.

If F (T ) 6= ; and there exists a sequence fk_{n}g [1; 1) with k_{n} ! 1 as n ! 1 such that: for every n 2 N, d(T ^{n}x; p) k_{n}d(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 fu_{n}g and fv_{n}g of nonnegetive numbers (i.e u_{n}; v_{n} !

0 as n ! 1) and a strictly increasing function : [0; 1) ! [0; 1) with

(0) = 0 such that: d(T ^{n}x; T ^{n}y) d(x; y) + u_{n} (d(x; y)) + v_{n}, for every x; y 2 X.

And T is called total asymptotically quasi-nonexpansive, if: F (T ) 6= ; and there exists infinitesimal real sequences fu_{n}g and fv_{n}g of nonnegative numbers

(i.e u_{n}; v_{n} ! 0 as n ! 1) and a strictly increasing function : [0; 1) !

[0; 1) with (0) = 0 such that: d(T ^{n}x; p) d(x; p) + u_{n} (d(x; p)) + v_{n}, 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.

- 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(T ^{n}x; T ^{n}y) L^{n}d(x; y) Ld(x; y), for every x; y 2 X.

Example 1.6.3 The map T : R ! R defined by:

- (i) Tx = tanx (ii) Tx = sinx; are non-linear.
- (i) Tx = 5x + 10 (ii) Tx = ax, for any constant a; are linear.
- 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 defini-tions:

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(T x; T y) = 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] := f x + (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 (R^{2}) such that: d(x; y) = d_{R}2 (x; y),

d(y; z) = d_{R}2 (y; z) and d(z; x) = d_{R}2 (z; x).

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) d_{R}2 (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 (R^{2}). (see; Kirk and Panyanak (2008))

A CAT (0) space X, is said to be complete if every Cauchy sequence fx_{n}g in X, converges to a point x 2 X. A complete CAT (0) space is often called the Hadamard space.

CAT means Cartan Alexandrov T opogonov. Examples of the CAT (0) spaces includes; R tree, Hadamard space, Hilbert ball equipped with hyperbolic met-ric 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

- ,

1 | 1 | 1 | 1 | 1 | ||||||||||

d( | x | z; | z | y) | d(x; y): | |||||||||

2 | 2 | 2 | 2 | 2 |

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 : D D !

- by:

1 | + j | z w | j | |||||

d(z; w) = log( | 1 zw | ) | ||||||

1 | j | z w | j | |||||

1 zw | ||||||||

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:

1 | 1 | 1 | ||||

_{a}(r; ) = inff1 |
d( | x | y; a) : d(x; a) r; d(y; a) r; d(x; y) g > 0: | |||

r | 2 | 2 |

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:

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, (lim_{t!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(T ^{n}(t)x; T ^{n}(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 asymp-totically 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: lim_{t!1} u(t) = lim_{t!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 Semigr

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 in-creasing and continuous functions : R^{+} ! R^{+} such that: lim_{t!1} u(t) = lim_{t!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 d(T (s + t)x; T (t)x) = 0; 8 t 2 [0; 1) and x 2 K

t!1

It is also said to be uniformly asymptotic regular if: for any t 0 and for any bounded subset C of K,

lim sup d(T (s + t)x; T (t)x) = 0:

^{t!1} x2C

1.6.7 Polar and Convergence

Definition 1.6.6 (Ali (2016)) A sequence fx_{n}g in a complete CAT (0) space

- is said to converge to a point x, if x is a unique asymptotic centre of fu
_{n}g for every subsequence fu_{n}g of fx_{n}g. This is written as lim_{n!1}x_{n}= x.

A sequence fx_{n}g 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 N_{y} 2 N such that; d(x_{n}; x) < d(x_{n}; y), 8n N_{y}.

The sequence fx_{n}g converge strongly to a point x, if the limit lim_{n!1} d(x_{n}; x) exists and for any y 6= x, lim_{n!1} d(x_{n}; x) lim inf_{n!1} d(x_{n}; y).

Let fx_{n}g be a bounded sequence in a complete CAT (0) space X. For x 2 X, we set: r(x; fx_{n}g) = lim sup_{n!1} d(x; x_{n}).

The asymptotic radius r(fx_{n}g) of fx_{n}g is given by :

r(fx_{n}g) = inffr(x; fx_{n}g) : x 2 Xg,

and the asymptotic center A(fx_{n}g) of fx_{n}g is the set:

A(fx_{n}g) = fx 2 X : r(x; fx_{n}g) = r(fx_{n}g)g.

It is well known that in a CAT (0) space, A(fx_{n}g) 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 fx_{n}g in X which converges weakly to a point x 2 X, with d(x_{n}; T x_{n}) ! 0, as n ! 1, then T x = x, for all x 2 X.

(see; Chang et al. (2012)

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