ABSTRACT

 

Let H be a real Hilbert space and K a nonempty, closed and convex subset of H. Let
T : K ! K be an asymptotically nonexpansive map with a nonempty xed points set.
Let fng1n
=1 and ftng1 n=1 be real sequences in (0,1). Let fxng be a sequence generated
from an arbitrary x0 2 K by

yn = PK[(1 􀀀 tn)xn]; n 0
xn+1 = (1 􀀀 n)yn + nTnyn; n 0:
where PK : H ! K is the metric projection. Under some appropriate mild conditions
on fng1n
=1 and ftng1 n=1, we prove that fxng converges strongly to xed point of T. No
compactness assumption is imposed on T and or K and no further requirement is imposed
on the xed point set Fix(T) of T.
vii

TABLE OF CONTENTS

Certication ii
Dedication iii
Acknowledgement iv
Abstract vii
1 Introduction 1
1.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Some Banach Spaces and their Properties. . . . . . . . . . . . . . . . . . . 3
1.3 Iterative Algorithms for Asymptotically Nonexpansive Mappings . . . . . 8
1.3.1 Modied Mann Iterative Algorithm . . . . . . . . . . . . . . . . . . 8
1.3.2 Iterative method of Schu . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.3 Halpern-type process . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Organization of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Preliminaries 11
2.1 Some Classical Results on Sequences of Real Numbers . . . . . . . . . . . . 11
2.2 Some Denitions and Results Used in the Main Result . . . . . . . . . . . 23
v
2.3 Projection on convex subsets of a Hilbert Space . . . . . . . . . . . . . . . 27
3 Strong Convergence of Modied Averaging Iterative Algorithm for Asymp-
totically Nonexpansive Maps 32
3.1 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Refrences 43

 

CHAPTER ONE

Introduction
1.1 General Introduction
The theory of xed points of nonlinear operators has found many powerful and important
applications in diverse elds such as Dierential Equations,Topology, Economics, Biology,
Chemistry, Engineering, Game Theory, Physics, Dynamics, Optimal Control, and Func-
tional Analysis . Iterative algorithms for approximating xed points of some nonlinear
operators belonging to certain classes of mappings that generalize nonexpansive mappings
and dened in appropriate Banach spaces have been ourishing area of research for many
mathematicians.The class of nonlinear mappings we studied in this work, is the class of
asymptotically nonexpansive mappings. This class of asymptotically nonexpansive map-
pings which has engaged the interest of many researchers(for example see [18] and the
references there in ) was rst introduced by Goebel and Kirk [10] in the year 1972.
Denition 1.1.1: Let K be a nonempty subset of a normed linear space E. A mapping
T : K ! K is said to be nonexpansive if
kTx 􀀀 Tyk kx 􀀀 yk; 8x; y 2 K:
Denition 1.1.2: Let K be a nonempty subset of a normed linear space E. A mapping
T : K ! K is called asymptotically nonexpansive, if there exists a sequence fkng; kn 2
1
[1;1) such that limn!1 kn = 1, and
kTnx 􀀀 Tnyk knkx 􀀀 yk
holds for each x; y 2 K and for each integer n 1. It is clear that every nonexpansive
mapping is asymptotically nonexpansive with kn = 1 8n 1.
The following example reveals that the class of asymptotically nonexpansive mappings
properly contains the class of nonexpansive mappings.
Example 1 (Goebel and Kirk [10]). Let B denote the unit ball in the Hilbert space l2
and let T be dened as follows:
T : (x1; x2; x3; :::) ! (0; x21
; a2x2; a3x3; :::):
where faig is a sequence of numbers such that 0 < ai < 1 and
Q1
i=2 ai =
1
2
.
Then, T is Lipschitz and kTx 􀀀 Tyk 2kx 􀀀 yk; 8x; y 2 B.
Moreover,
kTix 􀀀 Tiyk 2
Yi
j=2
ajkx 􀀀 yk 8i = 2; 3; :::
Thus,
lim
i!1
ki = lim
i!1
2
Yi
j=2
aj = 1
But let,
x = (
2
3
; 0; 0; :::)
and
y = (
1
2
; 0; 0; :::);
clearly x; y are in B.
kTx 􀀀 Tyk = j
4
9
􀀀
1
4
j =
7
36
>
1
6
= kx 􀀀 yk:
therefore T is not nonexpasnsive.
2
1.2 Some Banach Spaces and their Properties.
In this section we give the denition of some special Banach spaces as well as some of their
geometric properties.
Denition 1.2.1 A Banach space E is said to be uniformly convex if for any 2 (0; 2],
there exists = () > 0 such that for all x; y 2 E with jjxjj 1, jjyjj 1 and jjx􀀀yjj > ,
then jj 1
2 (x + y)jj 1 􀀀 . Geometrically, a Banach space is uniformly convex if the unit
ball centred at the origin is uniformly round. Again, uniform convexity is a property of
the norm on E.
The modulus of convexity of E is dened by
E() = inf f1 􀀀 jj
x + y
2
jj : jjxjj = 1 = jjyjj; jjx 􀀀 yjj g;
0 < 2.
Denition 1.2.2 A Banach space E is said to be smooth if for all x 2 E, x 6= 0 with
jjxjj = 1, there exists f 2 E such that hx; fi = 1. Smoothness is a property of the norm.
In fact, E is smooth if and only if 8x; y 2 E, x 6= 0
lim
t!0
jjx + tyjj 􀀀 jjxjj
t
(1:1)
exists. The limit, when it exits, is of the form fx(y) with fx 2 E and is called the
Gateaux derivative of the norm in E. Thus T is smooth if and only if the norm is Gateaux
dierentiable.
Denition 1.2.3 A Banach space is uniformly smooth if and only if the limit (1.1) exists
uniformly on the set
U = f(x; y) 2 E E : jjxjj = jjyjj = 1g:
Denition 1.2.4 A Banach space E is said to be an Opial space (see for example [1],
[16]) if for each sequence fxng1 n=1 in E which converges weakly to a point x 2 E
lim inf jjxn 􀀀 xjj < lim inf jjxn 􀀀 yjj;
3
for all y 2 E, y 6= x. It is known (see [17]) that every Hilbert space and every lp(1 < p < 1)
space enjoy the property. Also in [9], D Van showed that any separable Banach space can
be equivalently re-normed so that it satises Opial condition. Indeed, for any normed
space E the existence of a weakly sequentially continuous duality map implies that E is
an Opial’s space, but the converse implication does not hold. Notably, the Lebesgue space
Lp is not an Opial space for p 6= 2.
Denition 1.2.5 A function : [0;1) ! [0;1) is said to be a guage function if is
continuous and strictly increasing with (0) = 0 and lim
t!1
(t) = 1.
Denition 1.2.6 Let E be a real Banach space. Let E denote the topological dual of E
and 2E be the collection of all subsets of E. Let : [0;1) ! [0;1) be a gauge function.
The mapping J:E ! 2E dened by
Jx = ff 2 E : hx; fi = kxkkfk; kfk = (kxk)g:
is called duality map with gauge function , where h:; :i denotes the generalized duality
paring between E and E. We note that if 1 < q < 1, then (t) = tq􀀀1 is a gauge
function. The duality mapping Jq : E ! 2E with gauge (t) = tq􀀀1 dened for each
x 2 E by
Jx = ff 2 E : hx; fi = kxkkfk; kfk = kxkq􀀀1g:
is called the generalised duality mapping. If q = 2, we obtain
J2 := J : E ! 2E
;
dened for all x 2 E by
J2x := J(x) = ff 2 E : hx; fi = kxk2 = kfk2g:
J2 is known as the normalised duality map. It is well known ( see for instance [2; 8]) that
for 1 < q < 1; Jq(x) = kxkq􀀀2J(x); for x 2 X; x 6= 0: The following theorem has been
proved for uniformly convex Banach space.
4
Theorem 1.2.7(Xu, [30]) Let p > 1 and r > 0 be two xed real numbers.Then a Banach
space X is uniformly convex if and only if there exists a continuous, strictly increasing
and convex function
g : R+ ! R+; g(0) = 0;
such that for all x; y 2 Br and 0 1,
kx + (1 􀀀 )ykp kxkp + (1 􀀀 )kykp 􀀀Wp()g(kx 􀀀 yk): (1:3)
where Wp() := p(1 􀀀 ) + (1 􀀀 )p and Br := fx 2 X : kxk rg:
The next result below establishes the existence of a xed point for asymptotically nonex-
pansive mapping in a nonempty, closed, convex and bounded subset of a uniformly convex
Banach space.
Theorem 1.2.8([10] Theorem 1) Let K be a nonemtpty, closed, convex and bounded
subset of a uniformly convex Banach space X, let F : K ! K be an asymptotically
nonexpansive mapping,then F has a xed point.
Proof For each x 2 K and r > 0, Let S(x; r) denote the spherical ball centred at x with
radius r. Let y 2 K be xed, and let the set Ry consist of those numbers for which there
exists an integer k such that
K (1
i=kS(Fiy; )) 6= ;:
If d is the diameter of K then d 2 Ry, so Ry 6= ;. Let 0 = g:l:b:Ry, and for each > 0,
dene C = [1k
=1(1i
=kS(Fiy; + )). Thus for each > 0 the set C K are nonempty
and convex, so re exivity of X implies that
C = >0( C K) 6= ;:
Note that for x 2 C and > 0 there exists an integer N such that if i N; kx 􀀀 Fiyk
0 + .
5
Now let x 2 C and suppose the sequence fFnxg does not converge to x (i.e., suppose
Fx 6= x). Then there exists > 0 and a subsequence fFnixg of fFnxg such that
kFnix 􀀀 xk ; i = 1; 2; ::::
For m > n,
kFnx 􀀀 Fmxk knkx 􀀀 Fm􀀀nxk;
where kn is the Lipschitz constant for Fn obtained from the denition of asymptotic
nonexpansiveness. Assume 0 > 0 and choose > 0 so that (1 􀀀 (
0+))(0 + ) < 0.
Select n so that kx􀀀Fnxk and also that kn(0 +

2
) 0 +. If N n is suciently
large, then m > N implies
kx 􀀀 Fm􀀀nyk 0 +

2
and we have
kFnx 􀀀 Fmyk knkx 􀀀 Fm􀀀nyk 0 + ;
kx 􀀀 Fmyk 0 + :
Thus by uniform convexity of X, if m > n,
k(
x + Fnx
2
) 􀀀 Fmyk (1 􀀀 (

0 +
))(0 + ) < 0;
and this contradicts the denition of 0. Hence we conclude 0 = 0 or Fx = x. But 0 = 0
implies fFnyg is a Cauchy sequence yielding Fny ! x = Fx as n ! 1: Therefore the set
consists of a single point which is a xed point under F.
Kirk, Yanez and Shin [13] improved the result of Goebel and Kirk [10]. They proved that
if a re exive Banach space E has the property that each of its closed, bounded and convex
subset has the xed point property for nonexpansive maps, then it will also have the
xed point property for asymptotically nonexpansive mapping which has a nonexpansive
iterate.
Theorem 1.2.9 Let H be a real Hilbert space and C a nonempty closed and convex
subset of H. Let T : C ! C be an asymptotically non expansive map. Then Fix(T) =
fx 2 C : Tx = xg is closed and convex.
6
Proof:
Convexity.
For any x; y 2 Fix(T) and 2 (0; 1). Let z := (1 􀀀 )x + y. Then,
kTnz 􀀀 zk2 = kTnz 􀀀 [(1 􀀀 )x + y]k2
= k(1 􀀀 )(Tnz 􀀀 x) + (Tnz 􀀀 y)k2
= (1 􀀀 )kTnz 􀀀 xk2 + kTnz 􀀀 yk2 􀀀 (1 􀀀 )kx 􀀀 yk2
(1 􀀀 )k2n
kz 􀀀 xk2 + k2n
kz 􀀀 yk2 􀀀 (1 􀀀 )kx 􀀀 yk2
= (1 􀀀 )k2n
k(1 􀀀 )x + y 􀀀 xk2 + k2n
k(1 􀀀 )x + y 􀀀 yk2 􀀀 (1 􀀀 )kx 􀀀 yk2
= 2(1 􀀀 )k2n
kx 􀀀 yk2 + k2n
k(1 􀀀 )x + (1 􀀀 )yk2 􀀀 (1 􀀀 )kx 􀀀 yk2
= 2(1 􀀀 )k2n
kx 􀀀 yk2 + (1 􀀀 )2k2n
kx 􀀀 yk2 􀀀 (1 􀀀 )kx 􀀀 yk2
= (1 􀀀 )[k2n
+ (1 􀀀 )k2n
􀀀 1]kx 􀀀 yk2
= (1 􀀀 )(k2n
􀀀 1)kx 􀀀 yk2 ! 0:
Therefore,
kTnz 􀀀 zk ! 0:
Now,
0 kTz 􀀀 zk
kTz 􀀀 Tnzk + kTnz 􀀀 zk
k1kz 􀀀 Tn􀀀1zk + kTnz 􀀀 zk ! 0:
Hence
Tz = z:
We now show that Fix(T) is closed. Let fxng Fix(T) be arbitrary and let xn ! x as
n ! 1, we show that x is in Fix(T):
Tx = T lim
n!1
xn = lim
n!1
Txn = lim
n!1
xn = x:
7
Two other denitions of asymptotically nonexpansive maps has also appeared in the lit-
erature. One of the denitions which is weaker than Denition 1.1.2 was introduced by
Kirk[12] and requires that
lim sup
n!1
sup
y2K
(kTnx 􀀀 Tnyk 􀀀 kx 􀀀 yk) 0:
for every x 2 K and that TN be continuous for some integer N > 1:
The other denition which has appeared require that
lim sup
n!1
(kTnx 􀀀 Tnyk 􀀀 kx 􀀀 yk) 0 8x; y 2 K:
This, however, has been shown to be unsatisfactory from the point of view of xed
point theory. Tingly [24] constructed an example of a closed convex set K in a Hilbert
space and a continuous map T : K ! K which actually satises the following condition
limn!1 kTnx 􀀀 Tnyk = 0 8x; y 2 K but has no xed point.
1.3 Iterative Algorithms for Asymptotically Nonex-
pansive Mappings
1.3.1 Modied Mann Iterative Algorithm
The averaging iteration process,
xn+1 = (1 􀀀 n)xn + nTnxn; n 1;
where T : K ! K is asymptotically nonexpansive in the sense of denition 1.1.2, K a
closed, convex and bounded subset of a Hilbert space was introduced by Schu[23].
In [21] Schu used the modied Mann iteration method,
xn+1 = (1 􀀀 n)xn + nTnxn; n 1
and proved the following theorem.
8
Theorem(1.3.1): Let H be a Hilbert space, K a nonempty closed convex and bounded
subset of H. Let T : K ! K be a completely continuous asymptotically nonexpansive map
with sequence fkng1 n=1 with kn 2 [1;1) for all n 1; limn!1 kn = 1 and
P1
n=1 (k2n
􀀀 1) <
1. Let fng1 n=1 be a sequence in [0,1] satifying the condition < n < 1 􀀀 for some
> 0. Then the sequence fxng generated from an arbitrary x1 2 K by
xn+1 = (1 􀀀 n) xn + nTnxn; n 1;
converges strongly to a xed point of T.
1.3.2 Iterative method of Schu
In this subsection, consider algorithm for approximating xed points of asymptotically
nonexpansive mappings which deals with almost xed points,
xn := nTnxn
of an asymptotically nonexpansive mappings T. Schu [24] proved the convergence of this
sequence fxng to some xed point of T under additional assumption that T is uniformly
asymptotically regular and (I 􀀀 T) is demiclosed. These assumptions had actually been
made by Vijayaraju[27] to ensure the existence of a xed point of T By strengthening
the asymptotic regularity of T, Schu established the convergence of an explicit iteration
scheme,
zn+1 := nTnzn
to some xed point of T.
1.3.3 Halpern-type process
One of the most useful results concerning algorithms for approximating xed points of non-
expansive mappings in real uniformly smooth Banach spaces is the celebrated convergence
theorem of Riech[29] who proved that the implicit sequence fxng dened as,
xn =
1
n
u + (1 􀀀
1
n
)Txn:
9
converges strongly to a xed point of T. Several authors have tried to obtain a result
analogous to that of Reich [19] for asymptotically nonexpansive mappings. Suppose K
is a nonempty bounded closed convex subset of a real uniformly smooth Banach space E
and T : K ! K is an asymptotically nonexpansive mapping with sequence kn 1 for all
n 1. Fix u 2 K and dene, for each integer n 1, the contraction mapping Sn : K ! K
by, (see [6]),
Sn(x) = (1 􀀀
tn
kn
)u +
tn
kn
Tnx;
where ftng [0; 1) is any sequence such that tn ! 1.Then by the Banach Contraction
Mapping Principle, there is a unique point xn xed by Sn, i.e. there is xn such that
xn = (1 􀀀
tn
kn
)u +
tn
kn
Tnxn:
For some existing results on asymptotically nonexpansive maps, an interested reader
should see [4,7,14,18,20,22] and the references there in.
1.4 Organization of Thesis
We have introduced in this chapter (Chapter One), various iteration methods for asymp-
totically nonexpansive maps and some existing results on them. We also studied some of
the important Banach spaces which are encountered in this work and their properties.
In Chapter Two (the Preliminaries), we presented most of the classical results on the se-
quences of real numbers encountered in Operator Theory. We also looked at projection
maps and some other results vital to this work.
In Chapter Three, we study certain averaging iterative algorithm for approximating the
xed point of asymptotically nonexpansive mappings introduced by Goebel and Kirk [10].
10

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