Title of Thesis
SOME TYPES OF BEST APPROXIMATION AND THEIR APPLICATIONS
Author(s)
NAWAB HUSSAIN
Institute/ University/ Department Details
BAHAUDDIN ZAKARIYA UNIVERSITY/
CENTRE FOR ADVANCED STUDIES IN PURE AND
APPLIED MATHEMATICS
Status (Published/ Not Pu
blished/ In Press etc)
Published
Date of Publishing
August, 2001
Subject
MATHEMATICS
Number of Pages
123
Keywords (Extracted from title, table of contents and abstract of thesis)
Fixed Point, Best Approximation, Hyperconvex Spaces, Invariant approxima
t
ion, topological vector spaces
Abstract
The study of abstract fixed point theory of single valued maps has evolved as a natural
extension of the corresponding classical theory on Euclidean spaces. The celebrated
Banach contraction principle has become
a vigorous tool for studying nonlinear volterra
integral equations and nonlinear functional differential equations in Banach spaces. The
development of the geometric fixed poi
nt theory for multivalue
d maps was initiated by
Nadler [90] in 1969. Using the co
ncept of Hausdorff metric, he established the multi
valued contraction principle containing the Banach contraction principle as a sp
ecial case.
The theory of multi
valued maps
has applications in control
theory, convex optimization
and economics. The field
of approximation theory has become so vast that it intersects
with every other branch of analysis and plays an important role in applications in the
applied sciences and eng
ineering.
Fixed point theorems have been used in many instances
in approximation th
eory (e.g. Al

Thagafi [2]
, Brosowski [18], Habinaik [42], Prolla [10
5]
and Sehgal and Singh [116]).
In the subject of approximation theory one often wishes to
know whether s
ome useful properties of the f
unction being approximated are inherited by
the appro
ximating function. Meinardus [85] was the first who employed a fixed point
theorem to achieve this goal.
Random fixed points and random approximations are the stochastic generalizations
o
f
the
deterministic fixed point and approximation results. The stud
y of the random fixed points
and random best approximation of random operators of different
types is an active area of
investigation lying at the intersection of nonlinear analysis and probability theory. In the
setting of Banach spaces, Sehgal and Waters,
Sehgal and Singh and Papageorgiou
initiated the study of random approximations and obtained stochastic version of Ky Fan's
approximation theorem. More work and the inter

play between random fixed point theory
and random approximation theory
have
been carr
ied out by Beg and Shahzad, Khan, Lin
and Tan and Yaun.
The main purpose of this thesis is to esta
blish approximation theorems fo
r continuous and
non

continuous maps in the setting of
certain topological vector space
s and cite
applications of some types
of best approximation in fixed point theory.
We divide this
thesis into four chapters.
Chapter 1 is essentially introductory in nature. Here we fix our notations, recall some
basic definitions and summarize some of the familiar classical and recent
resul
ts about
fixed points and best approximations for our need in the sequel.
In chapter 2, common fixed point theorems for commuting as well as non

commuting
maps in the setup of Hausdorff locally convex spaces are proved. As applications, we
derive certain
Brosowski

Meinardus type results on invariant best approximation. Similar
results in the context
of convex metric spaces for R

sub commutative
maps are
established by using a recent common fixed point theorem of Pant [96]. The latter part of
this chapter
deals with results on invariant approximation in the setting of locally
bounded topological vector spaces where the set of best approximations need not be even
star shaped
. Finally, we approximate fixed point of a non

expansive map through the
iterates of
its bisection map in metrizable topological vector spaces. The results of section
2.2 will appear in [48] and [63]. The results of section 2.3 will appear in [61] and those of
section 2.4 have appeared in [58] and [64].
In chapte
r
3, we discuss mainly th
e inter

play between random fixed point theory and
random best approximation theory. In section 3.2, a number of random fixed point
theorems for single valued non

expansive random operators defined on a non

convex
domain (not even star
shaped) in locally b
ounded topological vector space are obtained
and the new results are applied to derive Brosowski

Meinardus type theorems regarding
inv
ariant random approximation. A
similar result for condensing and demicompact
random operators on a stars
s
haped domain is
also presented. In Theorem 3.2.18, a
multivalued extended analogue of a well

known result of Singh [131] in Frechet s
paces is
proved. The class of
*

nonexpansive multivalued mappings is different from that of
continuous maps. We utilize in section 3.3, the
best approximation operator as a selector
to find random fixed point and ran
dom approximation results for
*

non
expansive
multivalued random operators in various settings. The condition (A), introduced by
Shahzad and Latif [126] generalizes hemicompactness
and the concept of condensing
operators. In section 3.4 random fixed poin
t theorems for continuous and
*

nonexpansive
multivalued random maps satisfying condition (A) defined on unbounded domain in
Banach spaces and Frechet spaces are proved. Consequently
, a ra
ndom approximation
result for
*

nonexpansive maps on unbounded domain is obtained which in turn leads to a
random fixed point theorem under a number of boundary conditions. The results of
section 3.2 will appear in [65] and [68], while the results of
section 3.3 have appeared in
l59) and [60] and those of section 3.4 will appear in [7] and [57].
Fan's best
approximation theorem provides a connection between approximation theory and fixed
point theory. A number of deterministic and stochastic generaliz
ations of this result have
appeared in the literature; in particular contributions made by Beg and shahzad, Carbone,
Park, Prolla, Sehgal and Singh and Sine are worth mentioning. Section 4.2 concerns the
existence of deterministic and random best approxima
tion results for upper semi
continuous
maps through the notion of a
*

nonexpansive map in the setup of Banach
spaces and Frec
het spaces. In section 4.3, we f
ollow O’
Regan [95] to establish the
existence and approximation of solutions in the sense that the
r
e exists a bounded
sequence
X
n
and a point X
o
in C such that
X
n
X
o
and
X
o
is a solution to the nonlinear
inclusion
operator y E Ty where T is a
*

nonexpansive multivalued map defined on
suitable subset C of a Banach space. As a consequence we obtain rando
m approximation
results on unbounded domai
n in hyperconvex Banach spaces.
In section 4.4, we establish
Ky Fan type approximati
on results for continuous and
*

nonexpansive maps defined on a
compact and non

compact subset of a metrizable topological vector s
pace and
hyperconvex space. As an application, random approximation result for stochastic
domain is established.
The results of earlier part of
section 4.2 have appeared in [56] a
nd
the latter part may appear in [62].
The results of
section 4.3 may appear
i
n [6].
Table of Contents
List of Research Papers
vii
Abstract
viii
Chapter I
Fixed Point, Best Approximation and Hyperconvex
1
Spaces
1 .1
Introduction
1
1.2
Some classes of topological vector spaces
2
1.3
Random
fixed point theorems
7
1.4
Best approximation results
20
Chapter 2
Invariant Approximations
27
2.1
Introduction
27
2.2
Common fixed point as a best approximation
28
2.3
An extension of
a theorem of Sahab, Khan and Sessa
35
2.4
Invariant approximation in metrizable topological
42
vector spaces
Chapter 3
Random Approximations
56
3.1
Introduction
56
3.2
Invariant random best approximations
57
3.3
*

nonexpansive operators and random approxima
tion
65
3.4
Random fixed points and approximations on unbounded
77
Domain
Chapter 4
Ky Fan Type Approximation Theorems
86
4.1
Introduction
86
4.2
Approximation and fixed point theorems
87
4.3
Random approximation in Banach sp
aces
94
4.4
Generalizations of Ky Fan's approximation theorem
106
Bibliography
114
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