VIETNAM NATIONAL UNIVERSITY
UNIVERSITY OF SCIENCE
FACULTY OF MATHEMATICS, MECHANICS AND INFORMATICS
Nguyen Ly Vinh Hanh
SOME METHODS FOR COMMON FIXED POINTS OF
A FAMILY OF NONEXPANSIVE MAPPINGS
Undergraduate Thesis
Advanced Undergraduate Program in Mathematics
Hanoi - 2012

VIETNAM NATIONAL UNIVERSITY
UNIVERSITY OF SCIENCE
FACULTY OF MATHEMATICS, MECHANICS AND INFORMATICS
Nguyen Ly Vinh Hanh
SOME METHODS FOR COMMON FIXED POINTS OF
A FAMILY OF NONEXPANSIVE MAPPINGS
Undergraduate Thesis
Advanced Undergraduate Program in Mathematics
Thesis Advisor: Prof. Dr. Sc. Pham Ky Anh
Hanoi - 2012

Acknowledgments
I express my sincere gratitude to my thesis advisor Prof. Pham Ky Anh, who has intro-
duced me to the field of Numerical Analysis. I am especially grateful for his patience and
ability of making abstract mathematics so easily to be perceived.
I also want to thank my family since they always motivate, encourage and create favor-
able conditions for my study and research.
Finally, I want to thank my friends in K53 Advanced Maths. They always stay by my
side to encourage and to help me.
Ha Noi, May, 2012.
Nguyen Ly Vinh Hanh

nach (1892 −1945) formulated his famous contraction mapping principle. Banach proved
that if (X,d) is a complete metric space and T : X → X is a given contraction, then T has a
unique fixed point p, i.e., T(p) = p and T
n
(x) → p (as n → ∞).
Many scientific problems in game theory, theory of phase transitions, optimization theory,
differential equations, differential geometry, image processing, etc. lead to a problem of
finding fixed points of nonexpansive mappings.
This minor thesis will attempt to highlight some achievements in the theory of nonexpansive
mappings and concentrate to the following problems.
• The existence of fixed points;
• The approximation of fixed points;
• Applications of the fixed point theory.
The thesis consists of four chapters.
The first chapter is devoted to some minimal functional analysis background.
The second chapter is devoted to Krasnoselskij and Mann iterations for nonexpansive
mappings and the third chapter deals with hybrid methods for relatively nonexpansive
mappings.
The last chapter provides a crucial application of nonexpansive mappings in image
processing.
5
Chapter 1
Preliminary
In this chapter we collect some facts on nonlinear operators and geometry of Banach
spaces.
Definition 1.1. Let C be a nonempty set of a metric space and T be a mapping from C into
itseft. An element x
∗
∈ C is called a fixed point of T if Tx = x and the set of all fixed points
of T is denoted by F(T).

x+ (1−
λ
)y :
λ
∈ [0,1)}
is contained in C.
Definition 1.5. A Banach space (X,||.||) is called strictly convex if, for all x,y ∈ X satisfying
||x|| ≤ 1, ||y|| ≤ 1 and x = y, we have ||x+ y|| < 2.
Example 1.2.
6
• All inner product spaces are strictly convex.
• Let X = R
2
. Then (X,||.||
2
) is strictly covex, (X,||.||
1
) and (X,||.||
∞
) are not strictly
convex.
Example 1.3. C[a,b] is not stritly convex.
Proof. Choose, x ≡ 1 and y =
t−a
b−a
. Clearly, x = y.
On the other hand,
||x|| = max
t∈[a,b]
x(t) = 1,

, we have
1
2
||x+ y|| < 1−
δ
.
Definition 1.7. Let X be a real Banach space. The space X
∗
of all linear continous functionals
on X is called the dual space of X. For f ∈ X
∗
and x ∈ X the value of f at x is denoted by
 f,x and is called the duality pairing.
• The dual X
∗
is a Banach space with respect to the norm
|| f ||
∗
= sup{ f, x : ||x|| ≤ 1}
usually denoted by ||.||;
• The dual space of X
∗
is X
∗∗
, the bidual space of X. Since, in general, X ⊆ X
∗∗
, we say
X is reflexive if X = X
∗∗
.

∗
.
Proof. For any fixed element x ∈ X, x = 0 consider the subspace X
0
and the linear func-
tional f defined as follows:
X
0
:= {z = tx : t ∈ R
1
}; X
0
⋐ X; f(z) = f(tx) := t||x||
2
.
Then f ∈ X
∗
0
and || f| |
X
∗
0
= ||x||
X
.
Applying Haln-Banach theorem, we obtain its linear extension f  Jx ∈ X
∗
such that || f|| =
||Jx||,
Jx,x =  f,x = ||x||

||
2
+ || f
2
||
2
= (|| f
1
|| + || f
2
||)||x| |
≥ || f
1
+ f
2
||||x||.
Therefore, || f
1
+ f
2
|| = || f
1
|| + || f
2
||. Since X
∗
is stritly convex, f
1
≡ f
2

τ
= 0
and, for q > 1, X is said to be q-uniformly smooth if there exists a constant c > 0 such that
ρ
X
(
τ
) ≤ c
τ
2
,
τ
∈ [0,∞).
Theorem 1.1. (Klee-Smulian)
A Banach space X is uniformly smooth iff the norm in X is uniformly Frechet differentiable,
i.e
lim
t→∞
||x+th|| − ||x| |
t
exists uniformly for x and h (||x|| = ||h|| = 1).
8
Example 1.4.
1. A Hilbert space is uniformly convex and uniformly smooth.
2. L
p
(G) is a uniformly convex and uniformly smooth B-space, where 1 < q < ∞; G ⊂ R
n
measurable set.
3. l

|| = 1, y
n
⇀
x
||x||
= x.
Since X is uniformly convex then
2(1−
δ
(||y
n
− x| |)) ≥ ||x+ y
n
|| ≥  f,y
n
+ x → 2 f,x,
where f ∈ X
∗
,|| f|| = 1,
δ
=
δ
(
ξ
) > 0. Therefore,
lim
2(1−
δ
(||y
n

Definition 1.10. Let C be a subset of a normed space (X,||.||). A mapping T : C → C is
called
(C
1
) Lipschitzian (or L-Lipschitzian) if there exists L > 0 such that
||Tx− Ty|| ≤ L||x− y||, ∀x,y ∈ C;
(C
2
) (strict) contraction ( or L-contraction) if T is L-Lipschitzian, with L ∈ [0,1);
(C
3
) nonexpansive if T is 1-Lipschitzian;
(C
4
) quasi-nonexpansive if F(T) = /0 and
||Tx− p|| ≤ ||x− p|| ∀x ∈ C, p ∈ F(T);
(C
5
) contractive if ||Tx− Ty|| < ||x− y||, for all x,y ∈ C, x = y;
(C
6
) isometry if ||Tx− Ty|| = ||x− y||, for all x,y ∈ C.
Definition 1.11. Let E be a smooth Banach space and let E
∗
be the dual of E. The funtion
φ
: E × E → R is defined by
φ
(y,x) = ||y||
2

A point p in C is said to be an asymptotic fixed point of T if C contains a sequence { x
n
}
which converges weakly to p such that lim
n→∞
||Tx
n
− x
n
|| = 0. The set of asymptotic fixed
point of T will be denoted by
ˆ
F(T).
Definition 1.13. Let C be a closed convex subset of E, and let T be a mapping from C into
itself. A mapping T is called relatively nonexpansive if
• F(T) is nonempty,
•
ˆ
F(T) = F(T),
•
φ
(p,Tx) ≤
φ
(p,x) for all x ∈ C and p ∈ F(T).
10
A point p in C is said to be a strong asymptotic fixed point of T if C contains a sequence
{x
n
} which converges strongly to p such that lim
n→∞

≤ r
2
||x− y||
2
+ ||Tx− Ty− r(x− y)||
2
.
It is equivalent to
Tx− Ty,x− y ≤ r||x− y||
2
or
(I − T)x− (I − T)y,x− y ≥ (1− r)||x− y||
2
,
where I is the identity mapping.
Definition 1.17. Let X be an arbitrary real Banach space. A mapping T with domain D(T)
and range R(T) in X is called
• pseudocontractive if for each x,y ∈ D(T) there exists j(x− y) ∈ J(x− y) such that
(I − T)x− (I −T)y, j(x− y) ≥ 0,
where J is the normalized duality mapping.
• strong pseudocontractive if the exists k > 0 such that for all x, y ∈ D(T) there exists
j(x,y) ∈ J(x− y) such that
(I − T)x− (I −T)y, j(x− y) ≥ k||x− y||
2
.
11
Definition 1.18. A mapping U with domain and range in X is called
• Accretive if for each x,y ∈ D(U), we have
Ux−Uy, j(x− y) ≥ 0.
• Strongly accretive if there exists a positive number k such that for each x,y ∈ D(U),

Definition 1.20. A set of points is defined to be convex if it contains the line segments
connecting each pair of its points. The convex hull of a given set X is defined as the (unique)
minimal convex set containing X, and denoted by coX.
Theorem 1.3. ( The Schauder Fixed Point Theorem ) Let E be a closed, bounded, convex
subset of a normed space X. If T : E → X is a compact map such that T(E) ⊆ E, then there
is an x ∈ E such that T(x) = x.
12
Theorem 1.4. ( Mazur Theorem ) If K is a compact subset of a Banach space X, then the
closed convex hull
co(K) is compact.
Definition 1.21. Let X be a normed linear space and f ∈ X
∗
.
• X
c
= {x : f(x) = c} is called a hyperplane.
• X
0
= {x ∈ X : f(x) = 0} is called a hyperspace.
•
X
c
= {x : f(x) ≤ c} or X
c
= {x : f(x) ≥ c} is called a halfspace.
Remark 1.2. Let X be a Hilbert space and y ∈ X be a fixed vector.
• X
c
= {x : x,y = c} is a hyperplane.
• X

need no longer converge (to a fixed point of T ). Then Krasnoselskij used the averaged map-
ping associated to T, namely T
λ
= (1−
λ
)I +
λ
T. Clearly, T
λ
has the same fixed point set
as T. Furthermore, it will be proved that T
n
λ
x
0
converges to a fixed point as n → ∞.
Theorem 2.1. Let C be a closed bounded convex subset of the Hilbert space H and T :C →C
be a nonexpansive operator. Then T has at least on fixed point.
Proof. For a fixed element v
0
in C and a number
λ
with 0 <
λ
< 1, we denote
T
λ
(x) = (1−
λ
)v

|||x− y||.
Therefore, T
λ
:C →C is a
λ
-contraction, then it has a unique fixed point, say x
λ
. On the other
hand, since C is closed, convex and bounded in the Hilbert space H, it is weakly compact.
14
Hence we may find a sequence

λ
j

in (0, 1) such that
λ
j
→ 1 (as j → ∞ ) and x
j
= x
λ
j
converges weakly to an element p of H.
Since C is weakly closed, p lies in C. We shall prove that p is a fixed point of T. If x is any
arbitrary point in H, we have
||x
j
− x| |
2

) = ||p− T p||
2
.
Moreover, since
λ
j
→ 1 and T
λ
j
x
j
= x
j
, we have
Tx
j
− x
j
= [
λ
j
Tx
j
+ (1−
λ
)v
0
] − x
j
+ (1−

Therefore,
lim
j→∞
||Tx
j
− x
j
|| = 0.
On the other hand, since T is nonexpansive, we have
||Tx
j
− T p|| ≤ ||x
j
− p||.
Hence,
||x
j
− T p|| ≤ ||x
j
− Tx
j
|| + ||Tx
j
− T p|| ≤ ||x
j
− Tx
j
|| + ||x
j
− p||.

which yields
lim
j→∞
(||x
j
− T p||
2
− ||x
j
− p||
2
) = 0.
Hence
||p− T p||
2
= 0,
i.e., p is a fixed point of T.
15
Lemma 2.1. Let C be a bounded closed convex subset of a Hilbert space H and T : C → C
be a nonexpansive and demicompact operator. Then the set F(T) of fixed points of T is a
nonempty convex set.
Proof. Since T is nonexpansive then T has fixed points in C. Therefore, F(T) = /0.
Let x,y ∈ F(T) and
λ
∈ [0,1]. We need to prove that
z = (1−
λ
)x+
λ
y ∈ F(T).

Tx
n
, n = 0,1,2, (2.3)
converges (strongly) to a fixed point of T.
Proof. The proof is divided into two steps.
1. F(T) is a nonempty convex set (Lemma 2.1).
2. The Krasnoselskij iteration converges .
For the second step, observe that for any fixed x
0
∈ C, the sequence {x
n
}
∞
n=0
given by (2.3)
lies in C and is bounded. Let p be a fixed point of T, and consider the averaged map T
λ
,
given by
T
λ
= (1−
λ
)I +
λ
T, (2.4)
where I is the identity map.
We first prove that the sequence {x
n
− Tx

n
− p) − a(Tx
n
− p).
Then
||x
n+1
− p||
2
= (1−
λ
)
2
||x
n
− p||
2
+
λ
2
||Tx
n
− p||
2
+ 2
λ
(1−
λ
)Tx
n

− p.
Hence, summing up both sides of ther preceding two inequalties and using the fact that T is
nonexpansive and T p = p, we get
||x
n+1
− p||
2
+ a
2
||x
n
− Tx
n
||
2
≤ [2a
2
+
λ
2
+ (1−
λ
)
2
]||x
n
− p||
2
+
+ 2[

2
+
λ
2
+(1−
λ
)
2
+2
λ
(1−
λ
)−2a
2
)||x
n
− p||
2
= ||x
n
− p||
2
.
Using the Cauchy-Schwarz inequalty,
Tx
n
− p,x
n
− p ≤ ||Tx
n

.
For n = 0 to n = N, we get
λ
(1−
λ
)
N
∑
n=0
||x
n
− Tx
n
||
2
≤
N
∑
n=0
[||x
n
− p||
2
− ||x
n+1
− p||
2
]
= ||x
0

x
n
i
→ p ∈ F(T).
Since T is nonexpansive, Tx
n
j
→ T p and T p = p.
The convergence of the entire sequence {x
n
}
∞
n=0
to p now follows from the inequality
||x
n+1
− p|| ≤ ||x
n
− p||,
which can be deduce from the nonexpansiveness of T and is valid for each n.
17
Algorithm 2.1. Let x
0
be an initial approximation, n be maximum number of iterations,
λ
be
a given number in [0, 1], and
ε
be max change of x value to allow abort. For k = 1,2, we
do

Corollary 2.1. [2] Let X be a uniformly convex Banach space, D a closed bounded convex
set in X, and T a nonexpansive mapping of D into D such that T satisfies any one of the
following two conditions.
1. (I − T) maps closed sets in D into closed sets in X;
2. T is demicompact at 0.
Then for any given x
0
inC and any fixed number
λ
with 0<
λ
< 1, the Krasnoselskij iteration
{x
n
}
∞
n=0
given by (2.3) coverges (strongly) to a fixed point of T.
Theorem 2.3. Suppose T is a nonexpansive operator that maps a bounded closed convex set
C of H into C and that F(T) = {p}. Then the Krasnoselskij iteration converges weakly to p,
T
n
λ
⇀ p, for any x
0
∈ C. (2.5)
Proof. It suffices to show that if {x
n
j
}

λ
p
0
|| ≤ ||T
λ
x
n
j
− T
λ
p
0
|| + ||x
n
j
− T
λ
x
n
j
≤ ||x
n
j
− p
0
|| + ||x
n
j
− T
λ

n
j
− T
λ
p
0
||
2
= ||(x
n
j
− p
0
) + (p
0
− T
λ
p
0
)||
2
= ||x
n
j
− p
0
||
2
+ ||p
0

p
0
||
2
− ||x
n
j
− p
0
||
2
] = ||p
0
− T
λ
p
0
||
2
. (2.7)
On the other hand, we have
||x
nj
− T
λ
p
0
||
2
− ||x

n
j
− T
λ
p
0
|| + ||x
nj
− p
0
||} is bounded, too, and by the
relations (2.6) − (2.8) we get
||p
0
− T
λ
p
0
|| ≤ 0, i.e. T
λ
p
0
= p
0
↔ p
0
∈ F(T) = {p}.
Theorem 2.4. Let C be a bounded closed convex subset of a Hilbert space and T :C → C be
a nonexpansive operator. Then, for any x
0

ε
>0
F
ε
= {y : g(y) = d
0
} ≡ F
0
.
19
Hence,

ε
>0
F
ε
= /0. Moreover, F
0
contains exactly one point. Indeed, since F
0
is convex
and closed, for p
0
, p
1
∈ F
0
, and p
λ
= (1−

)(p
0
− x
n
)||
2
)
= lim
n→∞
(
λ
2
||p
1
− x
n
||
2
+ (1−
λ
)
2
||p
0
− x
n
||
2
+ 2
λ

||
2
+ 2
λ
(1−
λ
)||p
1
− x
n
||||p
0
− x
n
||)
+ lim
n→∞
{2
λ
(1−
λ
)[p
1
− x
n
, p
0
− x
n
 − ||p

− x
n
||}.
Hence,
lim
n→∞
{2
λ
(1−
λ
)p
1
− x
n
, p
0
− x
n
 − ||p
1
− x
n
||||p
0
− x
n
||} = 0.
Since
||p
1

n
||
2
+ ||x
n
− p
0
||
2
− 2p
1
− x
n
, p
0
− x
n

→ d
2
0
− d
2
0
− 2d
2
0
= 0,
which leads to a contraction.
Now, in order to show that x

0
||
2
= ||x
n
j
− p||
2
+ ||p− p
0
||
2
− 2x
n
j
− p, p− p
0

→ g
2
(p) + ||p− p
0
||
2
= g
2
(p
0
) = d
2

1
) a
nj
≥ 0 for all n, j and a
nj
= 0 for j > n.
(A
2
)
∑
n
j=1
a
nj
= 1 for all n ≥ 1.
(A
3
) lim
n→∞
a
nj
= 0 for all j ≥ 1.
The sequence {x
n
}
∞
n=1
defined by x
n+1
= T(v

the Mann iteration by M(x
1
,A,T).
2. There exists a rich literature on the convergence of Mann iteration for different classes
of operators considered on various spaces. We will state without proof some results on
Mann iterations.
Theorem 2.5. [2] Suppose E is a locally convex Hausdorff linear topological space, C is a
closed convex subset of E, T : C → C is continuous, x
1
∈ C and A = [a
nj
] satisfies conditions
(A
1
),(A
2
) and (A
3
). If either of the sequences {x
n
} or {v
n
} in the Mann iterative process
M(x
1
,A,T) converges to a point p, then the other sequence also converges to p, and p is a
fixed point of T.
Definition 2.2. A Mann process M(x
1
,A,T) is said to be normal provided that A = [a

= (1− a
n+1,n+1
)a
nj
, j = 1,2, , n; n = 1,2, ,n.
(A
5
) Either a
nn
= 1 for all n, or a
nn
< 1 for all n > 1.
21
Theorem 2.6. [2]
• M(x
1
,A,T) is a normal Mann process if and only if A = [a
nj
] satisfies conditions
(A
1
),(A
2
),(A
4
),(A
5
) and (A
′
3









a
11
= 1,a
1j
= 0 for j > 1,
a
n+1,n+1
= c
n
, n = 1,2,3, ,
a
n+1, j
= a
j j
∏
n
j=1
(1− c
i
), for j = 1,2, , n
a
n+1, j

n+1
then the obtained matrix A is the Cesaro matrix.
Example 2.3. Let
λ
∈ (0, 1), and A
λ
= [a
nj
] be defined by
• a
n1
=
λ
n−1
, a
nj
=
λ
n− j
(1−
λ
), for j = 2,3, , n.
• a
nj
= 0 for j > n, n = 1,2,3,
then M(x
1
,A
λ
,T) is the normal Mann process. Indeed, it can be easy seen that the matrix A

} satisfies c
1
= 1, 0 < c
n
< 1, n ≥ 2 and
∑
∞
n=1
c
n
= ∞.
22