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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 mappings in real Hilbert spaces. Novel contributions
are made on iterative algorithms for fixed points and solutions of the split
equality fixed point problems of some single-valued pseudocontractive-type
mappings in real Hilbert spaces. Interesting contributions are also made on iterative
algorithms for fixed points of a general class of multivalued strictly pseudocontractive
mappings in real Hilbert spaces using a new and novel approach
and the thorems were gradually extended to a countable family of multi-valued
mappings in real Hilbert spaces.It also contains contains original research and
important results on iterative approximations of fixed points of multi-valued
tempered Lipschitz pseudocontractive mappings in Hilbert spaces.
Apart from using some well known iteration methods and identities, some
very new and innovative iteration schemes and identities are constructed. The
thesis serves as a basis for unifying existing ideas in this area while also generalizing
many existing concepts. In order to demonstrate the wide applicability
of the theorems, there are given some nontrivial examples and the technique
is demonstrated to be more valuable than other methods currently in the literature.
The second part of the thesis focuses on some related optimization problems
in some Banach spaces. Some iterative algorithms are proposed for common
ii
solutions of zeroes of a monotone mapping and a finite family of nonexpansive
mappings in Lebesgue spaces.
The thesis presents in a unified manner, most of the recent works of this author
in this direction, namely:
Let H1;H2;H3 be real Hilbert spaces, S : H1 ! H1 and T : H2 ! H2 two
Lipschitz hemicontractive mappings, and A : H1 ! H3 and B : H2 ! H3
are two bounded linear mappings. Then the coupled sequence (xn; yn)
generated by the algorithm
8>>>>>>>>>>><
>>>>>>>>>>>:
(x1; y1) 2 H1 H2; chosen arbitarily;
(xn+1; yn+1) = (1 􀀀 )[(xn 􀀀 A(Axn 􀀀 Byn); yn + B(Axn 􀀀 Byn)]
+G(un; vn);
(un; vn) = (1 􀀀 )[(xn 􀀀 A(Axn 􀀀 Byn); yn + B(Axn 􀀀 Byn)]
+G(xn; yn);
2 (0; L􀀀2(
p
L2 + 1 􀀀 1))
2 (0; 2

(A;B) );
converges weakly to a solution (x; y) of the Split Equality Problem.
Let K be a nonempty, closed, convex subset of a real Hilbert space H.
Let T : K ! CB(K) be a mapping satisfying
D(Tx; Ty) kx 􀀀 yk2 + kD(Ax; Ay); k 2 (0; 1);A := I 􀀀 T:
Assume that F(T) 6= ; and Tp = fpg 8p 2 F(T): Then, the sequence
fxng generated by a certain Krasnolselskii type algorithm is an approximate
fixed point sequence of T and under appropriate mild conditions,
the sequence fxng converges strongly to a fixed point of T.
Let K be a nonempty, closed and convex subset of a real Hilbert space
H. For i = 1; 2; :::; m; let Ti : K ! CB(K) be a family of mappings
satisfying
D(Tix; Tiy) kx 􀀀 yk2 + kiD(Aix;Aiy); ki 2 (0; 1); Ai := I 􀀀 Ti;
for each i. Suppose that \mi
=1F(Ti) 6= ; and assume that for p 2
\mi
=1F(Ti); Tip = fpg. Then, the sequence fxng generated by the aliii
gorithm:
8>>>>>>><
>>>>>>>:
x0 2 K chosen arbitarily;
xn+1 = (0)xn +
mP
i=1
iyin
;
yin
2 Sin
:=
n
zin
2 Tixn : D2(fxng; Tixn) kxn 􀀀 zin
k2 + 1
n2
o
0 2 (k; 1);
mP
i=0
i = 1; and k := maxfki; i = 1; 2; :::; m; g:
is an approximate fixed point sequence for the finite family of mappings.
Let Ti : K ! CB(K) be a countably infinite family of mappings satisfying
D(Tix; Tiy) kx 􀀀 yk2 + kiD(Aix;Aiy); ki 2 (0; 1); Ai := I 􀀀 Ti:
Assume that := sup
i
ki 2 (0; 1), \1i
=1F(Ti) 6= ; and for p 2 \1i
=1F(Ti); Tip =
fpg. Then, the Krasnoselskii type sequence fxng generated by the algorithm:
8>>>>><
>>>>>:
x0 2 K; arbitrary;
in
2 􀀀i
n :=
n
zin
2 Tixn : D2(fxng; Tixn) kxn 􀀀 zin
k2 + 1
n2
o
xn+1 = 0xn +
1P
i=1
iin
;
0 2 (; 1);
P1
i=0 i = 1;
is an approximate fixed point sequence of the family Ti.
Let H be a real Hilbert space, K H be a nonempty, closed and convex.
Let T : K ! CB(K) be a multivalued mapping satisfying F(T) 6= ;,
diam(Tx [ Ty) Lkx 􀀀 yk for some L > 0, and
D2(Tx; Tp) kx 􀀀 pk2 + D2(x; Tx); 8x 2 H; p 2 F(T): (0.0.1)
Let fxng be a sequence defined by the algorithm:
8>>>>>>>>>><
>>>>>>>>>>:
x1 2 K
xn+1 = (1 􀀀 )xn + zn; 2 (0; L􀀀2[
p
1 + L2 􀀀 1])
zn 2 􀀀n := fun 2 Tyn : D(xn; Tyn) kxn 􀀀 unk2 + ng
yn = (1 􀀀 )xn + wn;
wn 2 n := fvn 2 Txn : D(xn; Txn) kxn 􀀀 vnk2 + ng
n 0;
1P
n=1
n < 1

 

 

CHAPTER ONE

General Introduction
Fixed Point Theory is concerned with solutions of the equation
x = Tx (1.0.1)
where T is a (possibly) nonlinear operator defined on a metric space. Any x
that solves (1.0.1) is called a fixed point of T and the collection of all such
elements is denoted by F(T). For a multi-valued mapping T : X ! 2X, a
fixed point of T is any x in X such that x 2 Tx:
Fixed Point Theory is inarguably the most powerful and effective tools
used in modern nonlinear analysis today. It is still an area of current intensive
research as it has vast applicability in establishing existence and uniqueness of
solutions of diverse mathematical models like solutions to optimization problems,
variational analysis, and ordinary differential equations. These models
represent various phenomena arising in different fields, such as steady state
temperature distribution, neutron transport theory, economic theories, chemical
equations, optimal control of systems, models for population, epidemics
and flow of fluids.
For example, given an initial value problem
dx(t)
dt = f(t; x(t));
x(t0) = x0:
(1.0.2)
This system is transformed into the functional equation
x(t) = x0 +
Z t
t0
f(s; x(s))ds:
1
To establish existence of solution to system (1.0.2), we consider the operator
T : X ! X(X = C([a; b])) defined by
Tx = x0 +
Z t
t0
f(s; x(s))ds:
Then finding a solution to the initial value problem (1.0.2) amounts to finding
a fixed point of T.
The existence(and uniqueness) of solution to equation (1.0.1), certainly, depends
on the geometry of the space and the nature of the mapping T. Existence
theorems are concerned with establishing sufficient conditions under
which the equation (1.0.1) will have a solution, but does not neccesarily show
how to find them. There are very many existence and uniqueness theorems in
the literature(see e.g. Kirk [67], Kato [62], Komura [68]).
Though existence theorems do not indicate how to construct a process starting
from a nonfixed point and convergent to a fixed point, they nevertheless
enhance understanding of conditions under which the existence of such fixed
points is guaranteed.
On the other hand, iterative methods of fixed points theory is concerned with
approximation or computation of sequences which converge to solutions of
(1.0.1). This is part of the problem that is being addressed in this thesis.
The pivot of the iterative methods of fixed point theory is the Banach contraction
mapping principle. It states that a self map T on a complete metric
space (X; d) satisfying
d(Tx; Ty) kd(x; y); 0 k < 1; 8x; y 2 X; (1.0.3)
neccesarily has a unique fixed point and for any starting point x1, the sequence
fTnx1g converges strongly to that fixed point.
Many authors, see for example Alber [7], Boyd and Wong [25], have now investigated
more general conditions under which a mapping will have a unique fixed
point and also developed iterative sequences that converge to such fixed points.
If k = 1 in the inequality (1.0.3) above, the mapping T is tagged nonexpansive.
There are many examples that show that xn+1 = Tn(x) need not converge to
a fixed point of a nonexpansive mapping T, even if it has a unique fixed point.
We then need to impose additional conditions on T (and/or the space X) and
also modify the sequence Tn(x) to ensure convergence to a fixed point of T .
2
These notable iterative algorithms were introduced for nonexpansive mappings,
namely, the Krasnosel’skii sequence presented in [69] as: x1 2 X and
xn+1 =
1
2
(xn + Txn);
the Krasnoselskii-Mann algorithm given by: x1 2 X,
xn+1 = (1 􀀀 )xn + Txn; 2 (0; 1);
the Halpern algorithm given in [59] as: u 2 X arbitrary and
xn+1 = nu + (1 􀀀 n)Txn;
and the more general Mann sequence presented in [72] as
xn+1 = (1 􀀀 n)xn + nTxn:
Diverse convergence theorems have been proved for these sequences, depending
on the smoothness of the underlying space and/or the compactness of the
mapping T:
Efforts to establish convergence theorems for nonexpansive mappings is likely
the most rewarding research venture in nonlinear analysis. It has helped in
the development of the geometry of Banach spaces and other related class of
mappings, namely, monotone and accretive operators.
A mapping M : X ! X is called 􀀀strongly monotone if
hx 􀀀 y;Mx 􀀀Myi kx 􀀀 yk2; 8x; y 2 X;
and A : X ! X is called 􀀀strongly accretive if
hAx 􀀀 Ay; j(x 􀀀 y)i kx 􀀀 yk2; 8x; y 2 X;
where h:; :i is the duality pairing between X and X; j(x􀀀y) 2 J(x􀀀y) where
J is the normalized duality mapping. When = 0, these mappings are called
monotone and accretive, respectively. If X is Hilbert space, these two notions
agree and they are simply refered to as monotone.
Accretive mappings have properties that are similar to those of monotone mappings.
However, the use of the strongly nonlinear mapping J make the study
of such mappings difficult. In a sense, the duality mapping on a Banach space
has all the properties of the Banach space that makes it differ from a Hilbert
space and the space can be characterized, almost, exclusive by the mapping.
3
These two ideas have proved to be very useful in many areas of interest. The
idea of accretive operators appear very often in partial differential equation,
in the existence theory of nonlinear evolution equations. On the other hand,
the idea of monotone operators appear in optimization theory and that, in
particular, include the increasingly important set-valued mapping called the
subdifferential. Given a convex, lower semicontinous function f, the subdifferential
is @f : X ! 2X given by
@f(x) := fx 2 X : f(y) 􀀀 f(x) hy 􀀀 x; xi; 8y 2 Xg:
The subdifferential is a monotone mapping and it is well known that 0 2 @f(x)
if and only if f(x) = inf
x2X
f(x). This motivates the study of the more general
problem of finding a zero, i.e x such that 0 2 Ax, of a monotone operator A.
The question on the existence of zeros is studied under the concept of maximal
monotone operators. A monotne mapping A is maximal monotone if the graph
G(A) is a maximal element when graphs of monotone operators in X X are
partially ordered by set inclusion. In that case, for any (x; y) 2 X X, the
inequality
hy1 􀀀 y2; x1 􀀀 x2i 0; 8×2 2 D(A); y2 2 Ax2
implies y1 2 Ax2: Maximal accretive mappings are defined accordingly.
The accretive operators are intimately connected with an important generalization
of nonexpansive mappings called the pseudocontractive mappings. A
mapping is pseudocontractive in the terminology of Browder and Petryshyn
[23] if for x; y in X, and for all r > 0,
kx 􀀀 yk k(x 􀀀 y) + r[(x 􀀀 Tx) 􀀀 (y 􀀀 Ty)]k; :
By a result of Kato [62], this is equivalent to
h(I 􀀀 T)x 􀀀 (I 􀀀 T)y; j(x 􀀀 y)i 0:
Thus, a mapping T is pseudocontractive if and only if the complementary operator
A := I 􀀀 T is accretive. Moreover, the zeros of A coincides with the
fixed points of T.
Another interesting relationship is that the resolvent of an accretive mapping
A always exists(i.e I +A is invertible ) and it is nonexpansive. The resolvent
of A is a set valued mapping J : X ! 2X defined by
J(x) = (I + A)􀀀1x; > 0:
4
In this case, A􀀀1(0) = Fix(J). More precisely, the mapping J is in fact
firmly nonexpansive, i.e
kJ(x) 􀀀 J(y)k2 hx 􀀀 y; J(x) 􀀀 J(y)i; 8x; y 2 X:
The existence and approximation algorithms for zeros of maximal monotone
operators are usually formulated in relation with the corresponding problem
for fixed points of firmly nonexpansive mappings. This makes the study of
firmly nonexpansive, and the more general pseudocontractive mappings, an
important tool for monotone operators and the theory of optimization.
The metric projection operator has become a veritable tool in dealing with variational
inequalities problem by iterative-projection method in Hilbert spaces.
Variational inequality problem V IP(A;C) involving an accretive operator A
and a convex set C can be proved to be equivalent to the fixed point problem
involving the nonexpansive mapping
T = PC(I 􀀀 A)
for arbitrary positive number . Conversely, given a differentiable functional
f, the V IP(rf;C) is simply the optimality condition for the minimization
problem
min
x2C
f(x):
Metric projection operators in Hilbert spaces are accretive and nonexpansive
and gives absolutely best approximations of any element of the closed convex
set. However, in the Banach space setting, this operator no longer possess
most of those properties that made them so effective in Hilbert spaces.
To study monotone-type mappings and the related pseudocontractive mappings
in Banach spaces, some analogues of the Hilbert space type projection
operators were introduced. These mappings are natural extentions of the classical
projection operators to Banach spaces. They have also helped in the
approximation of monotone operator in Banach spaces.
In the last five years or so, intensive effort are invested in developing feasible
iterative algorithm for approximating fixed points of multivalued pseudocontractive
type mappings and/or, correspondingly, zeros of monotone mappings
in Hilbert spaces and in the general Banach spaces. In each case, attempts
are made to recover Hilbert space type identities for these mappings. Most
of the study aim to derive a generalization of the multi-valued nonexpansive
mapping introduced in the classical work of Nadler [80]. Such method depends
heavily on the characterisation of the Hausdorf distance defined on closed and
bounded sets. The generalizations of existing ideas, on the other hand, should
5
be due to the generalization of some properties of the Hausdorff distance.
In this thesis, we first establish some new characterizations of the Hausdorf
metric and use the ideas thereby to define some more general class of multivalued
pseudocontractive mappings and prove convergent theorems for the
class of mappings defined. Attempts would be made to apply some of the
ideas obtained to real problems of interest. An example in this regard include
applications to split equality fixed point problems, introduced by Moudafi and
Al-Shemas[79] in (2013), which is formulated as finding a point x in a convex
set C and y in a convex set Q such that their images Ax and By under some
linear transformations A and B satisfy Ax = By. It serves as an inverse problem
model in which constraints are imposed on the solutions in the domain of
a linear mapping as well as in its range.
This thesis gives new insight and direction in the study of a general class of
multivalued pseudocontractive mappings. It also studies a new method for
finding a common solution of a monotone operator and family of a general
class of nonexpansive mappings in some classical Banach spaces using the idea
of generalized projections.
The rest of the thesis is organized as follows. Chapter 2 introduces some notions
and recalls some basic definitions and ideas which are the bedrocks for
the formulation of our theorems and for effective reading of the subsequent
chapters.Detailed literature review involving multi-valued nonexpansive and
pseudocontractive-type mappings are presented. In Chapter 3, convergence
of a coupled iterative algorithm to a solution of some split equality problem
is presented. Chapter 4, deals with some contributions to convergence theorems
for a general class of multivalued striclty pseudocontractive mappings
and Chapter 5 deals with the extension to finite and countable family. Chapter
6 is devoted to convergence theorems for a class of multivalued Lipschitz
pseudocontractive mappings. We finally present in Chapter 7, an iterative algorithm
for common element of zeros of a monotone mapping and fixed points
of a general class of nonexpansive mappings in real Banach spaces.
6

 

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