ABSTRACT
Let E be a real normed space with dual space E and let A : E ! 2E be any map. Let J : E ! 2E be
the normalized duality map on E. A new class of mappings, J-pseudocontractive maps, is introduced
and the notion of J-fixed points is used to prove that T := (J A) is J-pseudocontractive if and
only if A is monotone. In the case that E is a uniformly convex and uniformly smooth real Banach
space with dual E, T : E ! 2E is a bounded J-pseudocontractive map with a nonempty J-fixed
point set, and J T : E ! 2E is maximal monotone, a sequence is constructed which converges
strongly to a J-fixed point of T. As an immediate consequence of this result, an analogue of a recent
important result of Chidume for bounded m-accretive maps is obtained in the case that A : E ! 2E is
bounded maximal monotone, a result which complements the proximal point algorithm of Martinet and
Rockafellar. Furthermore, this analogue is applied to approximate solutions of Hammerstein integral
equations and is also applied to convex optimization problems.
viii
TABLE OF CONTENTS
Title Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Certification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
1 INTRODUCTION 1
1.1 Background of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Zeros of Monotone operators on Hilbert spaces . . . . . . . . . . . . . . . . . . 1
1.1.2 Extension of Hilbert space Monotonicity to arbitrary normed spaces . . . . . . . 4
1.1.3 Application of Fixed Point Techniques . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Statement of Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Aim and Objectives of Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 LITERATURE REVIEW 8
2.0.1 Accretive-type mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.0.2 Monotone-type mappings in arbitrary normed spaces . . . . . . . . . . . . . . . 9
3 PRELIMINARY CONCEPTS AND RESULTS 12
3.1 Geometry of Some Banach spaces. Duality Mappings . . . . . . . . . . . . . . . . . . . 12
3.1.1 Strictly Convex and Uniformly Convex Spaces . . . . . . . . . . . . . . . . . . 13
3.1.2 Smooth and Uniformly smooth spaces . . . . . . . . . . . . . . . . . . . . . . . 15
3.1.3 Classical Banach spaces: Lp; 1 p 1 . . . . . . . . . . . . . . . . . . . . . 16
3.1.4 Moduli. p-uniformly convex and q-uniformly smooth spaces . . . . . . . . . . . 17
3.1.5 Duality Mapping of Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.6 Important Banach space Identities and Characterizations . . . . . . . . . . . . . 21
3.2 Nonlinear Operators. Maximal Monotone Mappings . . . . . . . . . . . . . . . . . . . 24
3.2.1 Topological Properties of Nonlinear Operators . . . . . . . . . . . . . . . . . . 24
3.2.2 Accretive Operators and Pseudocontractive Mappings . . . . . . . . . . . . . . 25
3.2.3 Monotone and Maximal monotone Operators . . . . . . . . . . . . . . . . . . . 26
3.2.4 Some Characterizations and Properties of Maximal Operators . . . . . . . . . . 28
3.2.5 Semigroup of Operators. Resolvents . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.6 Approximation of the Nonlinear Equation Au = 0 . . . . . . . . . . . . . . . . 30
3.3 Convex Analysis: Subdifferential and Optimization . . . . . . . . . . . . . . . . . . . . 31
3.3.1 Basic Definitions and Results in Convex Analysis . . . . . . . . . . . . . . . . . 31
3.3.2 Subdifferential and Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Fixed Point Theory: Approximate Fixed Points . . . . . . . . . . . . . . . . . . . . . . 35
v
3.4.1 Approximation and Iterative Algorithm . . . . . . . . . . . . . . . . . . . . . . 35
3.4.2 Important Recurrent Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4 MAIN RESULTS AND APPLICATIONS 40
4.1 Application to zeros of maximal monotone maps . . . . . . . . . . . . . . . . . . . . . 51
4.2 Complement to proximal point algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3 Application to solutions of Hammerstein integral equations . . . . . . . . . . . . . . . . 52
4.4 Application to convex optimization problem . . . . . . . . . . . . . . . . . . . . . . . . 57
CHAPTER ONE
INTRODUCTION
1.1 Background of study
The contributions of this thesis work fall within the general area of nonlinear functional analysis and
applications, in particular, nonlinear operator theory. We are interested in the solution or approximation
of solutions of nonlinear equations or inclusions (i.e., equations or inclusions defined by nonlinear operators)
in Banach spaces.
Problems in the area involve methods of fixed point theory and application of iterative algorithms to
approximate zeros or fixed points of nonlinear mappings. Research in the area is enormous due to varied
classification of Banach spaces, operators and topological assumptions on them (e.g., continuity, boundedness,
compactness, closedness e.t.c). The literature of the last four decades abounds with papers which
establish fixed point theorems for selfmaps or nonselfmaps satisfying a variety of contractive type conditions
on several ambient spaces. See figures 1.1, 1.2 and 3.1.
(1 < p 2) Lp (2 p < 1)
[
H
[
Rn
p-Uniformly convex q-Uniformly smooth
Uniformly convex Uniformly smooth
Reflexive Smooth
Unif. Gat. Diff. norm
Strictly convex
(a) Lattice of Banach spaces
k-Contractive maps
Nonexpansive maps
Strictly
pseudocontractive
Lipschitz
pseudocontractive
(b) Metric Fixed-point Operator lattice
Figure 1.1: Lattice of Spaces and Metric fixed-point Operator
1
k-Contractive maps
Nonexpansive maps
Strictly
pseudocontractive
Asymptotically
nonexpansve
Asymp. Nonexp. in the
Intermediate sense Pseudocontractive
Asymp. Strictly
pseudocontractive
Asymp. Strictly
pseudocontr. in the
Interm. Sense
Asymp.
pseudocontractive
Asymp. pseudocontr. in
the Interm. sense
Firmly
Quasi-nonexpansive
Quasi-nonexpansive
Demi-contractive
(a) Contractive-type self map Operator lattice
Figure 1.2: Lattice of Operators
Let H be a real inner product space. A map A : H ! 2H is called monotone if for each x; y 2 H,
; x y
0 8 2 Ax; 2 Ay: (1.1.1)
Monotone mappings were first studied in Hilbert spaces by Zarantonello [120], Minty [84], Kaˇcurovskii
[64] and a host of other authors. Interest in such mappings stems mainly from their usefulness in applications.
1.1.1 Zeros of Monotone operators on Hilbert spaces
We consider the problem given by
Au 3 0 (1.1.2)
where A : H ! 2H is a monotone map on a Hilbert space. Problems of this kind find relevance in
several areas of applications. In particular, we have the following examples:
Convex optimisation problems
Let g : H ! R [ f1g be a proper convex function. The subdifferential of g, @g : H ! 2H, is defined
for each x 2 H by
@g(x) =
x 2 H : g(y) g(x)
y x; x
8 y 2 H
:
It is easy to check that @g is a monotone operator on H, and that 0 2 @g(u) if and only if u is a minimizer
of g. Setting @g A, it follows that solving the inclusion 0 2 Au, in this case, is solving for a minimizer
of g.
2
In particular, as an example of the above, where g(x) = jxj, the subdifferential of g at zero, @g(0) =
[1; 1], which trivially contains zero. Hence, zero is the minimizer of g.
Equilibrium problem of dynamical systems: Evolution equation
The equation 0 2 Au when A is a monotone map from a real Hilbert space to itself also appears in
evolution systems. Consider the evolution equation for a single-valued operator,
du
dt
+ Au = 0
where A is a monotone map from a real Hilbert space to itself. At equilibrium state, du
dt = 0 so that
Au = 0, whose solutions correspond to the equilibrium state of the dynamical system.
In particular, consider the following diffusion equation
8<
:
@u
@t (t; x) = 4u(t; x) + g(u(t; x)); t 0; x 2
;
u(t; x) = 0; t 0; x 2 @
;
u(0; x) = u0(x); u0 2 L2(
);
(1.1.3)
where
is an open subset of Rn.
By simple transformation i.e., by setting v(t) = u(t; :); where v : [0;1) ! L2(
) is defined by
v(t)(x) = u(t; x) and f(‘)(x) = g(‘(x)); such that f : L2(
) ! L2(
); we see that equation (1.1.3)
is equivalent to
v0(t) = Av(t) + f(v(t)); t 0;
v(0) = u0;
(1.1.4)
where A is a nonlinear monotone-type mapping defined on L2(
).
Setting f to be identically zero, at an equilibrium state (i.e., when the system becomes independent of
time) we see that equation (1.1.4) reduces to
Au = 0: (1.1.5)
Thus, approximating zeros of equation (1.1.5) is equivalent to the approximation of solutions of the
diffusion equation (1.1.3) at equilibrium state.
Hammerstein integral equations
Definition 1.1.1. Let
Rn be bounded. Let k :
! R and f :
R ! R be measurable
real-valued functions. An integral equation (generally nonlinear) of Hammerstein-type has the form
u(x) +
Z
k(x; y)f(y; u(y))dy = w(x); (1.1.6)
where the unknown function u and inhomogeneous function w lie in a Banach space E of measurable
real-valued functions.
By simple transformation (1.1.6) can put in the abstract form
u + KFu = 0; (1.1.7)
Interest in Hammerstein integral equations stems mainly from the fact that several problems that arise in
differential equations, for instance, elliptic boundary value problems whose linear part possesses Green’s
function can, as a rule, be transformed into the form (1.1.6) (see e.g., Pascali and Sburian [88], p. 164).
3
1.1.2 Extension of Hilbert space Monotonicity to arbitrary normed spaces
We recall that for a Hilbert space H, H = H. So, in the definition of a monotone operator in a Hilbert
space, the map A : H ! H could have been A : H ! H. Thus, the notion of monotone mappings has
been extended to real normed spaces. We now briefly examine two well-studied extensions of Hilbert
space monotonicity to arbitrary normed spaces, say , E.
A
A : E ! E A : E ! E
Accretive Monotone
Figure 1.3: Extension of Hilbert space monotonicity
Accretive-type mappings
Let E be a real normed space with dual space E. A map
J : E ! 2E defined by
Jx :=
x 2 E :
x; x
= kxk:kxk; kxk = kxk
is called the normalized duality map on E. We denote J1 by J
A map A : D(A) ! 2E is called accretive if for each x; y 2 D(A), there exists j(x y) 2 J(x y)
such that
; j(x y)
0 8 2 Ax; 2 Ay: (1.1.8)
Roughly speaking, accretive mappings acting in a space E are generalizations of non-decreasing realvalued
functions. More precisely, A is said to be accretive if for all x1; x2 2 D(A), y1 2 Ax1, y2 2 Ax2
and 0,
kx1 x2k kx1 x2 + (y1 y2)k:
A is called maximal accretive if, in addition, the graph of A is not properly contained in the graph of any
other accretive operator. It is m-accretive if and only if A is accretive and R(I + tA) = E for all t > 0.
In a normed space, “m-accretive” implies “maximal accretive” . The converse assertion need not be
true. The first counterexample was constructed in lp by B.D. Calvert (1970). Moreover, A. Cernes
(1974) showed that even if both E and E are uniformly convex, but E is not a Hilbert space, then there
are maximal accretive mappings which are not m-accretive. However, it was proved by G. Minty (1962)
that in Hilbert spaces, the notions of ”m-accretive” and ”maximal accretive” are equivalent (see e.g.,
[80]) In a Hilbert space, the normalized duality map is the identity map, and so, in this case, inequality
(1.1.8) and inequality (1.1.1) coincide. Hence, accretivity is one extension of Hilbert space monotonicity
to general normed spaces.
Monotone-type mappings in arbitrary normed spaces
Let E be a real normed space with dual E. A map A : E ! 2E is called monotone if for each x; y 2 E,
the following inequality holds:
; x y
0 8 2 Ax; 2 Ay: (1.1.9)
4
It is called maximal monotone if, in addition, the graph of A is not properly contained in the graph of
any other monotone operator. Also, A is m-monotone if and only if it is monotone and R(J +tA) = E
for all t > 0. When E is a strictly convex Banach space with a Fr´echet differentiable norm, a maximal
monotone operator from E into E is m-monotone (see e.g., Kido [71]).
It is obvious that monotonicity of a map defined from a normed space to its dual is another extension of
Hilbert space monotonicity to general normed spaces.
The extension of the monotonicity condition from a Banach space into its dual has been
the starting point for the development of nonlinear functional analysis…: The monotone
mappings appear in a rather wide variety of contexts, since they can be found in many
functional equations. Many of them appear also in calculus of variations, as subdifferential
of convex functions (Pascali and Sburian [88], p. 101).
1.1.3 Application of Fixed Point Techniques
The theory of fixed point proves to be one of the most useful tools of modern mathematics. This comes
from earlier development of the theory and the fact that most important nonlinear problems in applications
can be transformed to fixed point problems.
Definition 1.1.2. Let X be a non-empty set and f be a self-map on X. A fixed point of f is a point
x 2 X such that f(x) = x. If f is a multivalued then a point p in X is called a fixed point of f if p 2 fp.
Theorems concerning the existence and properties of fixed points are known as fixed point theorems.
Several fixed point theorems include the Banach contraction mapping principle, Brouwer fixed point
theorem, Schauder fixed point theorem and a host of others (see e.g., Asati et al. [5], Khamsi [67], Smith
[107], Lee [74]).
Let E be a real normed space and A : E ! E be an accretive operator. Assume that Au = 0 has
a solution. Browder [14] introduced an operator T : E ! E by T = I A and called the map T,
pseudo-contractive. It is clear that zeros of A correspond to fixed points of T (i.e., Au = 0 if and
only if Tu = u). The class of pseudocontractive maps properly contains the class of nonexpansive maps
which are a generalisation of contraction maps. A map T : E ! E is called nonexpansive if for each
x; y 2 E, the inequality kTx Tyk kx yk is true.
Several existence theorems have been proved for the equation Au = 0; where A is of the monotone-type
(or accretive-type) (see e.g., Brezis [11], Browder [14], Deimling [50], Pascali and Sburian [88], e.t.c.).
Likewise, several results have appeared in the literature for approximating zeros of accretive-type (or
fixed points of pseudo-contractive) mappings in certain Banach spaces (see e.g., Chidume et al.[20],
Takahashi [113], Bruck [17], and host of other authors).
Let E be a real normed space and T := I A : E ! E a pseudocontractive mapping. If K is a
nonempty convex subset of E and F(T) := fx 2 K : Tx = xg 6= ;, the following recursion formula
has been used to approximate fixed points of T, x0 2 K,
xn+1 = (1 n)xn + nTxn; n 0;
where fng is a real sequence satisfying appropriate conditions. The most general iterative scheme for
bounded pseudocontractive maps seems to be that obtained from the following
Theorem 1.1.3 (C. E. Chidume [23]). Let E be a uniformly smooth real Banach space with modulus of
smoothness E, and let A : E ! 2E be a multi-valued bounded maccretive operator with D(A) = E
such that the inclusion 0 2 Au has a solution. For arbitrary x1 2 E, define a sequence fxng by,
xn+1 = xn nun nn(xn x1); un 2 Axn; n 1;
5
where fng and fng are sequences in (0; 1) satisfying the following conditions:
(i) limn!1 n = 0; fng is decreasing; (ii)
P
nn = 1;
P
E(nM1) < 1, for some
constantM1 > 0; (iii) limn!1
h
n1
n
1
i
nn
= 0. There exists a constant 0 > 0 such that E(n)
n
0n.
Then, the sequence fxng converges strongly to a zero of A.
1.2 Statement of Problem
In studying the inclusion (1.1.2) on real Banach spaces more general than Hilbert spaces when A is
of accretive-type mapping, several iterative algorithms have been constructed and results obtained for
approximating solutions of problems of the equation (see e.g., the following monographs: Berinde [9],
Browder [14], Chidume [22], Reich [90], and the references contained in them). Consequently, this has
generated interests and the question asked if similar results for the case of monotone-type mappings in
arbitrary Banach spaces can be obtained, where A maps a space into its dual.
Regrettably, the pursuit of analogous results has only been greeted with very little progress and seemingly
unpropitious prospects as the success for the accretive-type case doesn’t quite easily carry over to
the case of monotone-type mappings. The difficulty, for the most part, seems to be that all efforts made
to apply directly known geometric properties of Banach spaces proved abortive; also developing and
understanding concepts with applying knowledge of the structure and geometry of the dual space, existence
and uniqueness theorems for monotone-type mappings in arbitrary Banach spaces, weak topology
and relevant tools of functional analysis, and other notions of operator theory were rather too slow for
the ambitious researcher. Also, defining the iterative sequence to make sense posed a challenge.
Furthermore, the technique of converting the inclusion (1.1.2) into a fixed point problem of defining the
map T := I A is not applicable since, in this case when A is monotone, A maps E into E and such
T is never well-defined as the identity map does not make sense.
1.3 Aim and Objectives of Study
The aim of this work is to contribute to the efforts being made to approximate solutions of inclusion
(1.1.2) where A is of monotone-type. We consider the problem of solving zeros of nonlinear equations
of maximal monotone-type mappings with no continuity assumption. We proceed thus.
1. We introduce, as far as we know, a class of mappings called J-Pseudocontractive mappings and
study the concept of J-fixed points. We establish the relationship between monotone mappings
and J-pseudocontractive mappings and between J-fixed points and zeros of operators.
2. We construct an iterative algorithm which converges to a J-fixed point of a J-Pseudocontractive
mappings and hence, by extension, to a zero of a monotone mapping.
3. We apply our results to:
Zeros of maximal monotone mappings. ( This corresponds, as noted earlier, to the equilibrium
state of some dynamical system)
Proximal point algorithm
Solutions Hammerstein integral equations
Convex minimization problems
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