Iterative Algorithms for Single-valued and Multi-valued Non-expansive Type Mappings in Real Lebesgue Spaces

Iterative Algorithms for Single-valued and Multi-valued Non-expansive Type Mappings in Real Lebesgue Spaces

ABSTRACT

Algorithms for single-valued and multi-valued non-expansive-type mappings have continued to attract a lot of attentions because of their remarkable utility and wide applicability in modern mathematics and other research 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 single valued 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 pseudo-contractive-type mappings in real Hilbert spaces. Interesting contributions are also made on iterative algorithms for fixed points of a general class of multivalued strictly pseudo-contractive mappings in real Hilbert spaces using a new and novel approach and the theorems 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 pseudo-contractive 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 solutions of zeroes of a monotone mapping and a finite family of non-expansive
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 algorithm:
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 necessarily 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 non-fixed 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) necessarily 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 non-expansive. There are many examples that show that xn+1 = Tn(x) need not converge to a fixed point of a non-expansive 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. These notable iterative algorithms were introduced for non-expansive 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 non-expansive 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 referred 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. 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 sub-differential. Given a convex, lower semicontinuous function f, the sub-differential is @f : X ! 2X given by @f(x) := fx 2 X : f(y) ? f(x) hy ? x; xi; 8y 2 Xg: The sub-differential 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 non-expansive mappings called the pseudo-contractive mappings. A mapping is pseudo-contractive 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 pseudo contractive 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 non-expansive. 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 non-expansive, 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 non-expansive mappings. This makes the study of
firmly non-expansive, and the more general pseudo contractive 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 non-expansive 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 non-expansive 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 pseudo contractive mappings in Banach spaces, some analogues of the Hilbert space type projection operators were introduced. These mappings are natural extensions 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 pseudo contractive 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 non-expansive mapping introduced in the classical work of Nadler [80]. Such method depends heavily on the characterization of the Hausdorf distance defined on closed and bounded sets. The generalizations of existing ideas, on the other hand, should 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 pseudo contractive 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 pseudo contractive mappings. It also studies a new method for finding a common solution of a monotone operator and family of a general
class of non-expansive 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 non-expansive and pseudo-contractive-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 strictly pseudo-contractive 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 pseudo-contractive 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 non-expansive mappings in real Banach spaces.