#### Project File Details

3,000.00

File Type: MS Word (DOC) & PDF

File Size: 331 KB

Number of Pages:65

## CHAPTER ONE

Introduction
1.1 General Introduction
The contribution of this thesis falls under a branch of mathematics called Functional
Analysis. Functional Analysis as an independent mathematical discipline started at the
turn of the 19th century and was nally established in 1920’s and 1930’s, on one hand
under the in uence of the study of specic classes of linear operators-integral operators
and integral equations connected with them-and on the other hand under the in uence of
the purely intrinsic development of modern mathematics with its desire to generalize and
thus to clarify the true nature of some regular behaviour. Quantum Mechanics also had
a great in uence on the development of Functional Analysis, since its basic concepts, for
example energy, turned out to be linear operators (which physicists at rst rather loosely
interpreted as innite dimensional matrices) on innite dimensional spaces. In the early
stages of the development of Functional Analysis the problems studied were those that
could be stated and solved in terms of linear operators on elements of the space alone. But
as the concept of a space was being developed and deepened, the concept of a function was
being developed and generalized. In the end, it became necessary to consider mapping
(not necessary linear) from one space into another. One of the central problems in non-
linear Functional Analysis is the study of such mappings. In the modern view, Functional
Analysis is seen as the study of complete normed vector spaces over the real or complex
1
numbers. Such studies are narrowed to the study of Banach spaces. An important example
is a Hilbert space, where the norm arises from an inner product.
This project sets to solve the problem of constructing an iterative scheme for approximat-
ing xed points of Lipschitz Pseudo-contractive Maps in Hilbert spaces. We introduced a
modied Ishikawa iterative algorithm and prove that if
F(T) = fx 2 H : Tx = xg 6= ;, then our proposed iterative algorithm converges strongly
to a xed point of T. No compactness assumption is imposed on T and no further require-
ment is imposed on F(T).
We proceed with the denitions of some basic terms, and the introduction of various non
linear operators studied in this project.
Denition 1.1 : Let K be a non empty subset of a real normed space E and let T : K ! K
be a map. A point x 2 K is said to be a xed point of T if Tx = x. We shall denote the
set of xed points of T by F(T).
Denition 1.2 (Convex Set) : The set C of a real vector space X is called convex if,
for any pair of points x; y 2 C, the closed segment with extremities x; y 2 C that is, the
set fx + (1 􀀀 )y : 2 [0; 1]g is contained in C. A subset C of a real normed space is
called bounded if there exists M > 0 such that kxk M 8x 2 C.
Denition 1.3 : Let K be a non-empty closed convex subset of a Hilbert space H. The
(metric or nearest point) projection onto K is the mapping Pk : H ! K which assigns to
each x 2 H the unique point Pkx in K with the property
kx 􀀀 Pkxk = minfkx 􀀀 yk : y 2 Kg.
Lemma 1.1: Given x 2 H and z 2 K. Then z = Pkx if and only if
hx 􀀀 z; y 􀀀 zi 0 for all y 2 K.
As a consequence we have that
(i) kPkx 􀀀 Pkyk2 hx 􀀀 y; Pkx 􀀀 Pkyi for all x; y 2 H; that is, the projection is non
expansive;
(ii) kx 􀀀 Pkxk2 kx 􀀀 yk2 􀀀 ky 􀀀 Pkxk2 8x 2 H and y 2 K
2
(iii) If K is a closed subspace, then Pk coincides with the orthogonal projection from H
onto K; that is, for x 2 H; x 􀀀 Pkx is orthogonal to K (i.e. hx 􀀀 Pkx; yi = 0 for y 2 K).
If K is a closed convex subset with a particularly simple structure, then the projection Pk
has a closed form expression as described below:
(a.) If K = fx 2 H : kx 􀀀 uk rg is a closed ball centred at u 2 H with radius r > 0,
then
Pkx =
(
u + r (x􀀀u)
kx􀀀uk ; ifx =2 K
x; ifx 2 K:
(b.) If K = [a; b] is a closed rectangle in <n, where a = (a1; a2; :::; an)T and b =
(b1; b2; :::; bn)T where T is the transpose, then, for 1 i n; Pkx has the ith coordinate
given by
(Pkx)i =
8<
:
ai; if xi < ai;
xi; if xi 2 [ai; bi];
bi; if xi > bi:
(c.) If K = fy 2 H : ha; yi = g is a hyperplane, with a 6= 0 and 2 <, then
Pkx = x 􀀀 ha;xi􀀀
kak2 a.
(d.) If K = fy 2 H : ha; yi g is a closed half space, with a 6= o and 2 <, then
Pkx =
(
x 􀀀 ha;xi􀀀
kak2 a; if ha; xi >
x; if ha; xi :
(e) If K is the range of an m n matrix A with full column rank, then
Pkx = A(AA)􀀀1Ax
where A is the adjoint of A.
1.2 Demiclosedness Principles
A fundamental result in the theory of nonexpansive mappings is Browder’s demiclosedness
principle.
Denition 1.2.1 : A mapping T : K ! H is said to be demiclosed (at y) if the conditions
that fxng converges weakly to x and that fTxng converges strongly to y imply that x 2 K
3
and Tx = y. Moreover, we say that H satises the demiclosedness principle if for any
closed convex subset K of H and any nonexpansive mapping T : K ! H, the mapping
I 􀀀 T is demiclosed.
The demiclosedness principle plays an important role in the theory of non expansive map-
pings (and other classes of non linear mappings as well). In 1965, Browder [9] gave the
following demiclosed principle for non expansive mappings in Hilbert spaces.
Theorem 1.1(Browder [9]) Let K be a non empty closed convex subset of a real Hilbert
space H. Let T be a non expansive mapping on K into itself, and let fxng be a sequence
in K. If xn * w and limn!1 kxn 􀀀 Txnk = 0, then Tw = w.
1.3 Nonlinear Mappings
The following denitions contains the nonlinear mappings we are working with and that
will appear throughout the entire chapters.
Denition 1.3.1 Let K be a nonempty subset of a real Hilbert space H. The mapping
T : K ! H is called
(i) Lipschitz or Lipschitz continuous if there exists a constant L 0 such that
kTx 􀀀 Tyk Lkx 􀀀 yk 8 x; y 2 K (1:1)
If L = 1, then T is called nonexpansive; and if L < 1, then T is called a contraction.
It is easy to see from (1.1) that every contraction mapping is nonexpansive and every
nonexpansive mapping is Lipschitz.
Denition 1.3.2: Let K be a non empty subset of a real Hilbert space H.
(i) A mapping T : K ! H is called pseudo-contractive if
hTx 􀀀 Ty; x 􀀀 yi kx 􀀀 yk2 8 x; y 2 K (1:2)
(ii) A mapping T : K ! H is called a strict pseudo-contraction if for all x; y 2 K there
exists a constant 2 [0; 1) such that
kTx 􀀀 Tyk2 6 kx 􀀀 yk2 + k(I 􀀀 T)x 􀀀 (I 􀀀 T)yk2; 8 x; y 2 K (1:3)
4
Inequality (1.2) can be equivalently written as:
kTx 􀀀 Tyk2 kx 􀀀 yk2 + k(I 􀀀 T)x 􀀀 (I 􀀀 T)yk28x; y 2 K (1:4)
i.e. set A = I 􀀀 T
) k(I 􀀀 A)x 􀀀 (I 􀀀 A)yk2 kx 􀀀 yk2 + kAx 􀀀 Ayk2
) hx 􀀀 y 􀀀 (Ax 􀀀 Ay); x 􀀀 y 􀀀 (Ax 􀀀 Ay)i kx 􀀀 yk2 + kAx 􀀀 Ayk2
) kx 􀀀 yk2 + kAx 􀀀 Ayk2 􀀀 2hAx 􀀀 Ay; x 􀀀 yi kx 􀀀 yk2 + kAx 􀀀 Ayk2
) 􀀀2hAx 􀀀 Ay; x 􀀀 yi 0
) hAx 􀀀 Ay; x 􀀀 yi 0
) h(I 􀀀 T)x 􀀀 (I 􀀀 T)y; x 􀀀 yi 0
) kx 􀀀 yk2 􀀀 hTx 􀀀 Ty; x 􀀀 yi 0
) hTx 􀀀 Ty; x 􀀀 yi kx 􀀀 yk2
Nonexpansive ) strict pseudo-contraction ) pseudo-contraction. However, the following
examples show that the converse is not true.
Example 1.1: Take X = <2; B = fx 2 <2 : kxk 6 1g; B1 = fx 2 B : kxk 6 1
2g;
B2 = fx 2 B : 1
2 6 kxk 6 1g . If x = (a; b) 2 X we dened x? to be (b;􀀀a) 2 X.
Dene T : B ! B by
Tx =

x + x?; x 2 B1
x
kxk 􀀀 x + x?; x 2 B2
Then, T is Lipschitz and pseudo-contractive but not strictly pseudo-contractive. Example
(1.1) is due to Chidume and Mutangadura [12].
Proof
Trivially hx; x?i = 0; kx?k = kxk; hx?; yi + hx; y?i = 0; hx?; y?i = hx; yi; kx? 􀀀 y?k =
kx 􀀀 yk 8 x; y 2 H:
We now show that T is imfact Lipschitz. For x; y 2 B1 we have
kTx 􀀀 Tyk2 = kx + x? 􀀀 y 􀀀 y?k2 = hx 􀀀 y + x? 􀀀 y?; x 􀀀 y + x? 􀀀 y?i
= hx 􀀀 y; x 􀀀 y + x? 􀀀 y?i + hx? 􀀀 y?; x 􀀀 y + x? 􀀀 y?i
5
= hx 􀀀 y; x 􀀀 yi + hx 􀀀 y; x? 􀀀 y?i + hx? 􀀀 y?; x 􀀀 yi + hx? 􀀀 y?; x? 􀀀 y?i
= kx 􀀀 yk2 + kx? 􀀀 y?k2 + hx; x?i 􀀀 hx; y?i 􀀀 hy; x?i + hy; y?i = 2kx 􀀀 yk2 + 0
) kTx 􀀀 Tyk =
p
2kx 􀀀 yk
Next, for x; y 2 B2 we have
k(
x
kxk
􀀀
y
kyk
)k2 = h
x
kxk
􀀀
y
kyk
;
x
kxk
􀀀
y
kyk
i
= h
x
kxk
;
x
kxk
􀀀
y
kyk
i 􀀀 h
y
kyk
;
x
kxk
􀀀
y
kyk
i
= h
x
kxk
;
x
kxk
i 􀀀 h
x
kxk
;
y
kyk
i 􀀀 h
y
kyk
;
x
kxk
i + h
y
kyk
;
y
kyk
i
=
kxk2
kxk2
􀀀 2h
x
kxk
;
y
kyk
i +
kyk2
kyk2 = 2 􀀀 2h
x
kxk
;
y
kyk
i = 2 􀀀
2hx; yi
kxkkyk
=
2
kxkkyk
fkxkkyk 􀀀 hx; yig =
1
kxkkyk
f2kxkkyk 􀀀 2hx; yig
=
1
kxkkyk
f2kxkkyk + kx 􀀀 yk2 􀀀 (kxk2 + kyk2)g
=
1
kxkkyk
fkx 􀀀 yk2 􀀀 (kxk 􀀀 kyk)2g
) k
x
kxk
􀀀
y
kyk
k2 6 8kx 􀀀 yk2
) k
x
kxk
􀀀
y
kyk
k 6
p
8kx 􀀀 yk
Hence, for x; y 2 B2 we have
kTx 􀀀 Tyk = k
x
kxk
􀀀 x + x? 􀀀
y
kyk
+ y 􀀀 y?k
6 k
x
kxk
􀀀
y
kyk
k + kx 􀀀 yk + kx? 􀀀 y?k
=
p
8kx 􀀀 yk + kx 􀀀 yk + kx 􀀀 yk =
p
8kx 􀀀 yk + 2kx 􀀀 yk
6 3kx 􀀀 yk + 2kx 􀀀 yk = 5kx 􀀀 yk
So that T is Lipschitz on B2. Now let x and y be in the interiors of B1 and B2 respectively.
Then there exists 2 (0; 1) for which Z = x + (1 􀀀 )y. Hence,
kTx 􀀀 Tyk 6 kTx 􀀀 Tzk + kTz 􀀀 Tyk 6
p
2kx 􀀀 zk + 5kz 􀀀 yk
6 5kx 􀀀 yk + 5kz 􀀀 yk = 5kx 􀀀 yk
6
Thus kTx 􀀀 Tyk 6 5kx 􀀀 yk 8 x; y 2 B as required.
We now show that T is a pseudo-contraction. First, we note that we may put j(x) = x,
since H is Hilbert. For x; y 2 B put 􀀀(x; y) = kx 􀀀 yk2 􀀀 hTx 􀀀 Ty; x 􀀀 yiand, if x and
y are both non zero, put (x; y) = hx;yi
kxkkyk . Hence to show that T is a pseudo-contraction,
we need to proof that 􀀀(x; y) > 0 8 x; y 2 B. We only need examine the following three
cases
(i) for x; y 2 B1, we have
hTx 􀀀 Ty; x 􀀀 yi = hx + x? 􀀀 y 􀀀 y?; x 􀀀 yi
= hx 􀀀 y; x 􀀀 yi + hx? 􀀀 y?; x 􀀀 yi
= kx 􀀀 yk2 + hx?; xi 􀀀 hx?; yi 􀀀 hy?; xi + hy?; yi
= kx 􀀀 yk2 + 0 􀀀 fhx?; yi + hy?; xig + 0 = kx 􀀀 yk2
) hTx 􀀀 Ty; x 􀀀 yi = kx 􀀀 yk2
so that 􀀀(x; y) = 0;
(ii)For x; y 2 B2 we have
hTx 􀀀 Ty; x 􀀀 yi = h
x
kxk
􀀀 x + x? 􀀀
y
kyk
+ y 􀀀 y?; x 􀀀 yi
= h
x
kxk
; xi 􀀀 h
x
kxk
; yi 􀀀 hx; xi + hx; yi + hx?; xi 􀀀 hx?; yi 􀀀 h
y
kyk
; xi
+h
y
kyk
; yi + hx; yi 􀀀 hy; yi 􀀀 hy?; xi + hy?; yi
= kxk 􀀀
1
kxk
hx; yi 􀀀 kxk2 + hx; yi + 0 􀀀 hx?; yi
􀀀
1
kyk
hx; yi + kyk + hx; yi 􀀀 kyk2 􀀀 hy?; xi + 0
= kxk + kyk 􀀀 kxk2 􀀀 kyk2 􀀀
1
kxk
hx; yi + 2hx; yi
􀀀fhx?; yi + hy?; xig 􀀀
1
kyk
hx; yi
= kxk + kyk 􀀀 kxk2 􀀀 kyk2 􀀀
1
kxk
hx; yi + 2hx; yi 􀀀 0 􀀀
1
kyk
hx; yi
= kxk 􀀀 kxk2 + kyk 􀀀 kyk2 + hx; yi(2 􀀀
1
kxk
􀀀
1
kyk
)
= kxk 􀀀 kxk2 + kyk 􀀀 kyk2 +
hx; yi
kxkkyk
f2kxkkyk 􀀀 kyk 􀀀 kxkg
7
= kxk 􀀀 kxk2 + kyk 􀀀 kyk2 + (x; y)f2kxkkyk 􀀀 kyk 􀀀 kxkg
Hence
􀀀(x; y) = 2kxk2 + 2kyk2 􀀀 kxk 􀀀 kyk 􀀀 (x; y)f4kxkkyk 􀀀 kxk 􀀀 kykg
Since
f4kxkkyk 􀀀 kxk 􀀀 kykg > 0 8 x; y 2 B2
We have, for xed kxk and kyk; 􀀀(x; y) has a minimum when (x; y) = 1.
) 􀀀(x; y) 6 2kxk2 + 2kyk2 􀀀 4kxkkyk = 2(kxk 􀀀 kyk)2
Again, we have 􀀀(x; y) > 0 8 x; y 2 B2 as required.
(iii) For x 2 B1; y 2 B2 we have
hTx 􀀀 Ty; x 􀀀 yi = hx + x? 􀀀
y
kyk
+ y + y?; x 􀀀 yi
= hx; yi 􀀀 hx; yi + hx?; xi 􀀀 hx?; yi 􀀀
1
kyk
hy; xi
+
1
kyk
hy; yi + hy; xi 􀀀 hy; yi 􀀀 hy?; xi + hy?; yi
= kxk2 + 0 􀀀 hx?; yi 􀀀
1
kyk
hx; yi + kyk
􀀀kyk2 􀀀 hy?; xi + 0
= kxk2 􀀀 kyk2 + kyk 􀀀 [hx?; yi + hy?; xi]
􀀀
1
kyk
hx; yi = kxk2 􀀀 kyk2 + kyk 􀀀
1
kyk
hx; yi
= kxk2 + kyk 􀀀 kyk2 􀀀
1
kyk
hx; yikxkkyk
kxkkyk
= kxk2 + kyk 􀀀 kyk2 􀀀 (x; y)kxk
Hence
􀀀(x; y) = 2kxk2 􀀀 kyk + [kxk 􀀀 2kxkkyk](x; y)
Since (kxk 􀀀 2kxkkyk) 6 0 for x 2 B1 and y 2 B2; 􀀀(x; y) has its minimum, for xed
kxk and kyk when (x; y) = 1. We conclude that
􀀀(x; y) > 2kxk2 􀀀 kyk + kxk 􀀀 2kxkkyk = (kyk 􀀀 kxk)(2kyk 􀀀 1) > 0 8x 2 B1; y 2 B2
8
Therefore, T is a pseudo-contraction.
Next, we show that T is not a strictly pseudo-contraction
Let x; y 2 B1 be such that kx 􀀀 yk 6= 0 f for example x = ( 1
4 ; 0); y = (0; 0)g and let
2 [0; 1) be arbitrary. Then kx 􀀀 yk 6= 0 and
kTx 􀀀 Tyk2 = kx + x? 􀀀 (y 􀀀 y?)k2 = kx 􀀀 y + x? 􀀀 y?k2
= kx 􀀀 yk2 + kx? 􀀀 y?k2 + 2hx 􀀀 y; x? 􀀀 y?i = 2kx 􀀀 yk2 (1:5)
Furthermore,
kx 􀀀 Tx 􀀀 (y 􀀀 Ty)k2 = kx? 􀀀 y?k2 = kx 􀀀 yk2 (1:6)
(1.5) and (1.6) imply:
kTx 􀀀 Tyk2 = kx 􀀀 yk2 + kx 􀀀 Tx 􀀀 (y 􀀀 Ty)k2
> kx 􀀀 yk2 + kx 􀀀 Tx 􀀀 (y 􀀀 Ty)k2 8 2 [0; 1):
Therefore, T is not strictly pseudocontractive.
Example 1.2 Take X = <1 and dene T : X ! X by Tx = 􀀀3x: Then, T is a strict
pseudocontraction but not nonexpansive mapping.
Proof
jx 􀀀 Tx 􀀀 (y 􀀀 Ty)j2 = 16jx 􀀀 yj2:
Hence,
jTx 􀀀 Tyj2 = 9jx 􀀀 yj2 = jx 􀀀 yj2 + 8jx 􀀀 yj2
= jx 􀀀 yj2 +
1
2
jx 􀀀 Tx 􀀀 (y 􀀀 Ty)j2;
so that T is 􀀀strictly pseudocontractive with = 1
2 :
Next, we show that T is not non-expansive.
jTx 􀀀 Tyj = j 􀀀 3x + 3yj = j 􀀀 3(x 􀀀 y)j = 3jx 􀀀 yj > jx 􀀀 yj; 8 x 6= y
Hence T is not non-expansive.
Remark 1.3 The following example shows that the class of pseudocontractive maps prop-
erly contains the class of nonexpansive maps.
9
It also shows that the class of pseudocontractive maps is more general than the class
of strictly pseudocontractive maps.
Example 1.4: Let < be the reals with the usual norm and K = [0; 1]. Let T : [0; 1] ! [0; 1]
be dened by
Tx = 1 􀀀 x
2
3 (1:7):
We assert that T is not Lipschitz. To see this, let L > 0 be arbitrary, r = minf1; 1
L3 g, and
consider x 2 (0; r]; y = 0: Then jx 􀀀 yj = jxj = x and
jTx 􀀀 Tyj = j1 􀀀 x
2
3 􀀀 1j = x
2
3 = x
􀀀1
3 (x)
Since x < 1
L3 , then x
1
3 < 1
L. Thus 1
x
1
3
> L (i.e. x
􀀀1
3 > L). Thus
jTx 􀀀 Tyj = x
􀀀1
3 (x) > Lx = Ljxj = Ljx 􀀀 yj:
Since T is not Lipschitz, it is not nonexpansive. Let x; y;2 [0; 1], and let x y. Then
x
2
3 y
2
3 and 􀀀x
2
3 􀀀y
2
3 . Thus 1 􀀀 x
2
3 1 􀀀 y
2
3 , and this yields Tx Ty.
It follows that if
x 􀀀 y 0; then Tx 􀀀 Ty 0:
Hence
hTx 􀀀 Ty; x 􀀀 yi 0 tjx 􀀀 yj2 8 t > 0
Thus T is pseudocontractive. T is not strictly pseudocontractive because it is not
Lipschitz.
Next, we show that the mapping dened in (1.7) has a xed point. To see this, let x be
the xed point of the given map. Then we have
1 􀀀 x
2
3 = x
) (1 􀀀 x)3 = x2
) 1 􀀀 3x + 3×2 􀀀 x3 = x2
) x3 􀀀 2×2 + 3x 􀀀 1 = 0 (1:8)
10
We want to solve a cubic equation of the form x3 + ax2 + bx + c = 0
Using Cardano’s method of solving a cubic equation, we have x = m 􀀀 a
3 where a is a
coecient of x2
x = m +
2
3
(1:9)
Then (1.8) becomes (m + 2
3 )3 􀀀 2(m + 2
3 )2 + 3(m + 2
3 ) 􀀀 1 = 0
) m3 + 3m2
2
3
+ 3m
4
9
+
8
27
􀀀 2(m2 +
4m
3
+
4
9
) + 3m + 2 􀀀 1 = 0
) m3 +
5m
3
+
11
27
= 0 (1:10)
Equation (1.10) can be re-written as
m3 + pm + q = 0 (1:11)
where
p =
5
3
; q =
11
27
(1:12)
letting m = u + v, we re-write the above equation (1.12) as
u3 + v3 + (u + v)(3uv + p) + q = 0 (1:13)
Next, we set 3uv + p = 0, and equation (1.13) becomes
u3 + v3 = 􀀀q
Hence we are left with these equations
u3 + v3 = 􀀀q and u3v3 = 􀀀p3
27
Since the above equations are the product and sum of u3 and v3, then there is a quadratic
equation with roots u3 and v3. This quadratic equation is t2 + qt 􀀀 p3
27 = 0 with solutions
u3 =
􀀀q+
q
q2+4p3
27
2 and v3 =
􀀀q􀀀
q
q2+4p3
27
2
11
But from equation (1.12) q = 11
27 ; p = 5
3
) u3 =
􀀀11
27 +
q
( 11
27 )2 + 4( 5
3 )3
27
2
=
􀀀11
27 +
q
121
729 + 500
729
2
) u3 =
􀀀11 + 3
p
69
54
) u = (
􀀀11 + 3
p
69
54
)
1
3
and
v3 =
􀀀11 􀀀 3
p
69
54
) v = (
􀀀11 􀀀 3
p
69
54
)
1
3
Therefore,
m = (
􀀀11 + 3
p
69
54
)
1
3 + (
􀀀11 􀀀 3
p
69
54
)
1
3
But from equation (1.9) x = m + 2
3
) x = (
􀀀11 + 3
p
69
54
)
1
3 + (
􀀀11 􀀀 3
p
69
54
)
1
3 +
2
3
a xed point.
1.4 Iterative Algorithms
In this section, we will present several methods for solving xed point problems. We will
focus on iterative methods (we also call them iterative procedures or algorithms) which
are given in the form of the following recurrences:
1.4.1 The Picard iteration Method
Let X be any set and T : X ! X a self map. For any x0 2 X, the sequence fxngn>0 X
given by
xn = Txn􀀀1 = Tnx0; n = 1; 2; :::
12
is called the sequence of successive approximations with the initial value x0. It is also
known as the Picard iteration.
The theorem below, called the Banach xed point or the Banach theorem on contractions,
is widely applied in various areas of mathematics. The theorem holds for any complete
metric space, and hence, in particular, for every closed subset of a Hilbert space.
Theorem 1.2 (Banach, 1922) Let X be a complete metric space and T : X ! X be
a contraction. Then T has exactly one xed point x 2 X. Furthermore, for any x 2 X,
the orbit fTnxg1n
=0 converges to x at a rate of geometric progression.
The Banach xed point theorem is a widely applied tool for an iterative approximation
of xed points. Unfortunately, its application is restricted to contractions. We will need,
however, appropriate tools for an iterative approximation of xed points of non-expansive
operators T with F(T) 6= ?.
Below, we present several classical xed points theorems.
Theorem 1.3 (Brouwer, [8]) Let X < be non empty compact and convex and
T : X ! X be continuous. Then T has a xed point.
The Brouwer xed point theorem was generalized by Juliusz Schauder.
Theorem 1.4 (Schauder, [8]) Let X be a non empty compact and convex subset of a
Banach space and T : X ! X be continuous. Then T has a xed point.
For non-expansive operators in Hilbert space H the compactness of X H in the Schauder
Theorem (1.4) can be replaced by the boundedness and closeness of X.
1.4.2 Krasnoselskii Iteration Method [6]
For x0 2 K and 2 [0; 1] the sequence fxng1 n=0, dened by
xn+1 = (1 􀀀 )xn + Txn; n = 0; 1; 2; :::
is called Krasnoselkii iterative method and is denoted by Kn(x0; ; T).
13
1.4.3 The Mann Iteration Process [21]
Mann iteration process is essentially an averaged algorithm which generates a sequence
recursively by
xn+1 = (1 􀀀 n)xn + nTxn; n > 0 (1:14)
where the initial guess x0 2 K and fng is a sequence in (0; 1).
The Mann iteration method has been successfully employed in approximating xed points
(when they exist) of nonexpansive mappings. This success has not carried over to the
more general class of pseudo-contractions. If K is a compact convex subset of a Hilbert
space and T : K ! K is Lipschitz , then, by Schauder xed point theorem, T has a xed
point in K. All eorts to approximate such a xed point by means of the Mann sequence
when T is also assumed to be pseudo-contractive proved to be abortive.
Hicks and Kubicek [18], gave an example of a discontinuous pseudo-contraction with unique
xed point for which the Mann iteration does not always converge. Borwein and Borwein
[7] (proposition 8), gave an example of a Lipschitz map (which is not pseudo-contractive)
with a unique xed point for which the Mann sequence fails to converge. The problem
for Lipschitz pseudo-contraction still remained open. This was eventually resolved in the
negative by Chidume and Mutangadura [12] in the following example.
Example 1.4.3.1
Let H be a real Hilbert space <2 under the the usual Euclidean inner product.
If x = (a; b) 2 H we dene by x? 2 H to be (b;􀀀a). Trivially, we have
hx; x?i = 0; kx?k = kxk;
hx?; y?i = hx; yi; kx? 􀀀 y?k = kx 􀀀 yk
and hx?; yi + hx; y?i = 0 for all x; y 2 H.
We take our closed and bounded convex set K to be the closed unit ball in H and putK1 =
fx 2 H : kxk 1
2g, K2 = fx 2 H : 1
2 kxk 1g. We dene the map T : K ! K as
14
follows:
Tx =

x + x?; ifx 2 K1;
x
kxk 􀀀 x + x?; ifx 2 K2
Then T is a Lipschitz pseudo-contractive map of a compact convex set into itself with a
unique xed point for which no Mann sequence converges.
We notice that, for x 2 K1 \ K2, The two possible expressions for Tx coincide and that
T is continuous on both of K1 and K2. Hence T is continuous on all of K. We now show
that T is in fact, Lipschitz. One easily shows that
kTx 􀀀 Tyk =
p
2kx 􀀀 yk for x; y 2 K1. For x; y 2 K2, we have
k(
x
kxk
􀀀
y
kyk
)k2 =
2
kxkkyk
(kxkkyk 􀀀 hx; yi)
=
1
kxkkyk
fkx 􀀀 yk2 􀀀 (kxk 􀀀 kyk)2g

1
kxkkyk
2kx 􀀀 yk2 8kx 􀀀 yk2
Hence, for x; y 2 K2, we have
kTx 􀀀 Tyk k
x
kxk
􀀀
y
kyk
k
+kx 􀀀 yk + kx? 􀀀 y?k 5kx 􀀀 yk;
So that T is Lipschitz on K2. Now let x and y be in the interiors of K1 and K2 respectively.
Then there exist 2 (0; 1) and z 2 K1 \ K2 for which z = x + (1 􀀀 )y. Hence
kTx 􀀀 Tyk kTx 􀀀 Tzk + kTz 􀀀 Tyk

p
2kx 􀀀 zk + 5kz 􀀀 yk
5kx 􀀀 zk + 5kz 􀀀 yk = 5kx 􀀀 yk
Thus kTx 􀀀 Tyk 5kx 􀀀 yk8x; y 2 K, as required. The origin is clearly a xed point of
T. For x 2 K1; kTxk2 = 2kxk2, and for x 2 K2; kTxk2 = 1 + 2kxk2 􀀀 2kxk. From these
expressions and from the fact that Tx = x? 6= x if kxk = 1, it is easy to show that the
origin is the only xed point of T. We now show that no Mann iteration sequence for T
15
is convergent for any non zero starting point.
First, we show that no Mann sequence converges to the xed point. Let x 2 K be such
that x 6= 0. Then, in case x 2 K1, any Mann iterate of x is actually further away from
the xed point of T than x is. This is because
k(1􀀀)x+Txk2 = (1+2)kxk2 > kxk2 for 2 (0; 1). If x 2 K2 then, for any 2 (0; 1)
k(1 􀀀 )x + Txk2 = k(

kxk
+ 1 􀀀 2)x + x?k2
= [(

kxk
+ 1 􀀀 2)2 + 2]kxk2 > 0:
more generally, it is easy to see that for the recursion formula (1.2.8), if x0 2 K1 then
kxn+1k > kxnk for all integers n > 0, and if x0 2 K2, then kxn+1k >
p
2
2 kxnk for all
integers n > 0. We therefore conclude that, in addition, any Mann iterate of any non zero
vector in K is itself non zero. Thus any Mann sequence fxng, starting from a non zero
vector, must be innite. For such a sequence to converge to the origin, xn would have to
lie in the neighbourhood K1 of the origin for all n > N0, for some real N0. This is not
possible because, as already established for K1, kxnk < kxn+1k for all n > N0.
We now show that no Mann sequence converges to x 6= 0. We do this in the form of a
general lemma.
Lemma 1.2 Let M be a non empty closed and convex subset of a real Banach space
E and let S : M ! M be any continuous function. If a Mann sequence for S is norm
convergent, then the corresponding limit is a xed point for S.
proof. Let fxng be a Mann sequence in M for S, as dened in the recursion formula
(1.2.8). Assume, for proof by contradiction, that the sequence converges, in norm, to
x in M, where Sx 6= x. For each n 2 N put “n = xn 􀀀 Sxn 􀀀 x + Sx. Since S is
continuous, the sequence “n converges to 0. Pick p 2 N such that, if m > p and n > p,
then k”nk < 1
3kx 􀀀 Sxk kxn 􀀀 xmk < 1
3kx 􀀀 Sxk. Pick any positive integer q such that
Pp+q
n=p n > 1. We have that
kxp 􀀀 xp+q+1k = k
Xp+q
n=p
(xn 􀀀 xn+1)k = k
Xp+q
n=p
n(xn 􀀀 Sx + “n)k
16
> k
Xp+q
n=p
n(x 􀀀 Sx)k 􀀀 k
Xp+q
n=p
n”nk
>
Xp+q
n=p
n(kx 􀀀 Sxk) 􀀀
1
3
kx 􀀀 Sxk) > 2
3
kx 􀀀 Sxk
We now show that T is a peudo-contraction. First, we note that we may put j(x) = x,
since H is Hilbert. For x; y 2 K, put 􀀀(x; y) = kx 􀀀 yk2 􀀀 hTx 􀀀 Ty; x 􀀀 yi and, if x and
y are both non zero, put (x; y) = hx;yi
kxkkyk . Hence to show that T is a pseudo-contraction,
we need to proof that 􀀀(x; y) > 0 for all x; y 2 K. We only need examine the following
three cases:
(i) x; y 2 K1: An easy computation shows that hTx 􀀀 Ty; x 􀀀 yi = kx 􀀀 yk2 so that
􀀀(x; y) = 0; thus we are home and dry for this case.
(ii)x; y 2 K2: Again, a straight forward calculation shows that
hTx 􀀀 Ty; x 􀀀 yi = kxk 􀀀 kxk2 + kyk 􀀀 kyk2
+hx; yi(2 􀀀
1
kxk
􀀀
1
kyk
)
= kxk 􀀀 kxk2 + kyk 􀀀 kyk2
+(x; y)(2kxkkyk 􀀀 kxk 􀀀 kyk):
Hence 􀀀(x; y) = 2kxk2 + 2kyk2 􀀀 kxk 􀀀 kyk 􀀀 (x; y)(4kxkkyk 􀀀 kxk 􀀀 kyk)
It is not hard to establish that (4kxkkyk􀀀kxk􀀀kyk) > 08x; y 2 K2. Hence, for xed kxk
and kyk; 􀀀(x; y) has a minimum when (x; y) = 1. This minimum is therefore 2kxk2 +
2kyk2 􀀀 4kxkkyk = 2(kxk 􀀀 kyk)2.
Again, we have that 􀀀(x; y) > 0 8 x; y 2 K2 as required.
(iii) x 2 K1; y 2 K2: we have
hTx 􀀀 Ty; x 􀀀 yi = kxk2 + kyk 􀀀 kyk2 􀀀 (x; y)kxk. Hence 􀀀(x; y) = 2kyk2 􀀀 kyk +
(kxk 􀀀 2kxkkyk)(x; y). Since kxk 􀀀 2kxkkyk 0 for x 2 K1 and y 2 K2; 􀀀(x; y) has its
17
minimum, for xed kxk and kyk when (x; y) = 1. We conclude that
􀀀(x; y) > 2kyk2 􀀀 kyk + kxk 􀀀 2kxkkyk
= (kyk 􀀀 kxk)(2kyk 􀀀 1) > 08x 2 K1; y 2 K2:
This complete the proof.
1.4.4 The Ishikawa Iteration Process [19]
In 1974, Ishikawa introduced an iteration scheme which in some sense, is more general
than that of Mann and which converges under this setting, to a xed point of T. He
proved the following result.
Theorem 1.4.4.1 (Ishikawa [19]): If K is a compact convex subset of a Hilbert space
H, T : K ! K is a Lipschitzian pseudo-contractive map and x0 is any point of K, then the
sequence fxngn>0 converges strongly to a xed point of T, where xn is dened iteratively
for each integer n > 0 by
xn+1 = (1 􀀀 n)xn + nTyn; yn = (1 􀀀 n)xn + nTxn (1:15)
where fng; fng are sequences of positive numbers satisfying the conditions
(i) 0 n n < 1; (ii) limn!1 n = 0; (iii)
P
n>0 nn = 1.
The iteration method of Theorem 1.4.4.1 which is now referred to as the Ishikawa iterative
method has been studied extensively by various authors.The important thing here is that
Ishikawa yielded strong convergence (Normal Mann only yields weak convergence). But it
is still an open question whether or not this method can be employed to approximate xed
points of Lipschitz pseudo-contractive mapping without the compactness assumption on
K or T (see, e.g., [13,22,21]).
In order to obtain a strong convergence theorem for pseudo-contractive mappings without
the compactness assumption, Zhou [43] established the hybrid Ishikawa algorithm for
Lipschitz pseudo-contractive mappings as follows:
18
8>>>><
>>>>:
yn = (1 􀀀 n)xn + nTxn;
Zn = (1 􀀀 n)xn + nTyn;
Kn = fz 2 K : kzn 􀀀 zk2 kxn 􀀀 zk2 􀀀 nn(1 􀀀 2n 􀀀 L22n
)kxn 􀀀 Txnk2g Qn = fz 2 K : hxn 􀀀 z; x0 􀀀 xni > 0g;
xn+1 = Pkn
T
Qnx0; n > 1:
He proved that the sequence fxng dened by (1.3.0) converges strongly to Pkx0, where Pk
is the metric projection from H into K.
In [31] Rhoades compared the performance of these two iteration processes and showed
that despite they are similar, they may exhibit dierent behaviours for dierent classes of
nonlinear mappings. In its original form the Ishikawa iteration process does not include
Mann process as a special case because of the condition 0 n n 1. In an eort to
have an Ishikawa type iteration process which does include the Mann process as a special
case, several authors have modied the inequality condition to read 0 n,n 1.
In 1995, Liu [23] introduced the Ishikawa and Mann iteration processes with “errors” as
follows:
1.4.5 Mann Iteration Process with Errors in the Sense of Liu
Let T : E ! E be a given map. Then the Mann iteration process with errors in the sense
of Liu is given by
xn+1 = (1 􀀀 n)xn + nTxn + un; n 0
where fng1 n=1 is a suitable real sequence in [0; 1] and fung1 n=1 is a summable sequence in
E (i:e:;
1P
n=1
jjunjj < 1).
1.4.6 Ishikawa Iteration Process with Errors in the Sense of Liu
LetE be a real Banach Space, T : D(T) E ! E and x0 2 D(T). The Ishikawa iteration
process with errors is given by
yn = (1 􀀀 n)xn + nTxn + un; n 0
19
xn+1 = (1 􀀀 n)xn + nTyn + vn; n 0
where fng1 n=0, fng1 n=0 are suitable sequences in [0; 1] and fung1 n=0, fvng1 n=0 are summable
sequences in E.
Several authors have studied these iteration processes with errors and observed that it is
unsatisfactory because the conditions
X1
n=0
jjunjj < 1; and
X1
n=0
jjvnjj < 1
imply that error terms tend to zero. This is incompatible with the randomness of the
occurence of errors. Furthermore, if the operator T is dened on a nonempty convex
subset K of a normed linear space E, then the Mann and Ishikawa iteration process with
errors are not in general well dened.
1.4.7 The Agarwal-O’Regan-Sahu Iteration Process
The sequence fxng is dened by
8<
:
x1 2 K;
yn = nxn + (1 􀀀 n)Txn;
xn+1 = nTxn + (1 􀀀 n)Tyn; n > 1
where fng and fng are sequence in [0,1] is known as Agar-O’Regan-Sahu [19] iteration
scheme.
Theorem 1.4.7.1 [37] Let K be a non empty closed convex subset of a real Hilbert space
H, S : K ! K be non-expansive and T : K ! K be Lipschitz strongly pseudo-contractive
mappings respectively such that
F(S) \ F(T) = fx 2 K : Sx = x = Txg 6= ? and satisfying
kx 􀀀 Syk kSx 􀀀 Syk,
kx 􀀀 Tyk kTx 􀀀 Tyk for all x; y 2 K:
Let fngn>1; fngn>1 2 [0; 1] be such that
(i) n n
(ii)
P
n>1 n = 1 and
20
(iii)
P
n>1 n < 1:
For arbitrary x1 2 K let fxngn>1 be iteratively dened by

yn = (1 􀀀 n)xn + nTxn;
xn+1 = (1 􀀀 n)Syn + nTxn; n > 1:
Then fxngn > 1 converges strongly to the common xed point p of S and T.
corollary 1.4.7.2 [37] Let K be a nonempty closed convex subset of a real Hilbert
space H and T; S : K ! K be two Lipschitz pseudo-contractive mappings such that
p 2 F(T) \ F(S) = fx 2 K : Tx = x = Sxg. Let fng and fng be sequences in [0,1]
satisfying limn!1(1􀀀n) = 0: For arbitrary x1 2 K; let fxng1 n=1 be the sequence dened
iteratively by

yn = nxn + (1 􀀀 n)Txn; n > 1
xn+1 = nTxn + (1 􀀀 n)Syn;
Then the following are equivalent:
(a) fxng1 n=1 converges strongly to the common xed point q of T and S.
(b) fTxng1n
=1 and fSyng1 n=1 are bounded.
1.5 ORGANIZATION OF THESIS
We have introduced in Chapter one, various iterations methods for Lipschitz pseudo-
contractive maps and some existing results on them. We have also considered various
non-linear operators studied in this project.
In Chapter Two (the preliminary section), we presented most of the technical results about
convergent sequences of real numbers encountered in Operator Theory. We also considered
projection maps and some other results vital to this work.
Chapter Three featured the main results of this project. Using a modied Ishikawa it-
eration method we proved strong convergence theorem for Lipschitz pseudo-contractive
maps in Hilbert spaces. Our result extended the work of Yao and Li [20] from the class
of – strictly pseudo-contractive operators to the more general class of Lipschitz pseudo-
contractive operators.
21

GET THE FULL WORK