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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

2
Contents
Introduction 5
1 Preliminary 6
2 Iterations for nonexpansive mappings 14
2.1 Krasnoselskij iteration for nonexpansive mappings . . . . . . . . . . . . . 14
2.2 Mann iterations for nonexpansive mappings . . . . . . . . . . . . . . . . . 20
2.2.1 Strongly Pseudocontractive Operators . . . . . . . . . . . . . . . . 23
2.2.2 Nonexpansive and quasi-nonexpansive operators . . . . . . . . . . 26
3 Iterations for relatively nonexpansive mappings 34
3.1 A hybrid method for relatively nonexpansive mappings . . . . . . . . . . . 34
3.2 Strong Convergence theorems for a Finite Family of Relatively Nonexpan-
sive Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Parallel hybrid methods for a finite family of relatively nonexpansive mappings 44
4 Applications 52
4.1 Some basic facts about projections onto hyperplanes . . . . . . . . . . . . 52
4.2 The algebraic reconstruction technique . . . . . . . . . . . . . . . . . . . . 54
4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.2 The convergence of the ART . . . . . . . . . . . . . . . . . . . . . 55
4.3 Reconstruction by successive approximation . . . . . . . . . . . . . . . . . 57
4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3
4.3.2 The convergence of the reconstruction by successive approximation 58
4.4 A numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4.1 Applying the ART method . . . . . . . . . . . . . . . . . . . . . . 62
4.4.2 Applying Liu’s method . . . . . . . . . . . . . . . . . . . . . . . . 64
4
Introduction
In this thesis, we deal with iteration methods for finding fixed points of nonexpansive map-
pings on normed spaces. The origin of these methods dates back to 1920, when Stefan Ba-

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).

Example 1.1.
1) Let X = R Tx = x
2
− 3x . Then F(T) = {0,4}.
2) Let X = R Tx = x+ 5 . Then F(T) = { /0}.
Definition 1.2. The sequence {x
n
}
∞
n=0
in a normed space X is said to be strongly convergent
to a if
||x
n
− a|| → 0, as n → ∞.
Further, {x
n
}
∞
n=0
⊂ X coverges weakly to a if for any f ∈ X
∗
 f,x
n
 →  f,a, as n → ∞.
Definition 1.3. A subset C of a real normed space is called bounded if there exists M > 0
such that ||x|| ≤ M, for all x ∈ C.
Definition 1.4. A subset C of a real vector space X is called covex if, for any pair of points
x,y in C, the closed segment with the endpoints x, y, i.e., the set {
λ

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,

||y|| = max
t∈[a,b]
t −a
b− a
= 1,
and
(x+ y)(t) = 1+
t −a
b− a
=
t +b− 2a
b− a
.
Thus,
||x+ y|| = max
t∈[a,b]
(x+ y)(t) =
b+ b− 2a
b− a
=
2(b− a)
b− a
= 2 = x + y.
Definition 1.6. A Banach space (X, ||.||) is called uniformly convex if, given any
ε
> 0, there
exists
δ
> 0 such that for all x,y ∈ X satisfying ||x|| ≤ 1,||y|| ≤ 1, and ||x− y|| ≥
ε

, 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
∗∗
.

Definition 1.8. Let X
∗
be the dual space of a real Banach space. The multivalued mapping
J defined by
Jx = { f
∗
∈ X
∗
:< f
∗
,x >= ||x||
2
= || f
∗
||
2
}
is called the normalized duality mapping of E.
7
Lemma 1.1. If X
∗
is strictly convex then
∀x ∈ X ∃!Jx ∈ X
∗
: Jx,x = ||x| |
2
X
= ||Jx||
2
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
= ||Jx||
2
.
Now we prove that Jx is unique.
Assuming that there exist f
i
∈ X
∗
:  f
i
,x = ||x||
2
= || f
i
||
2
(i = 1,2) we get
|| f
1
+ f
2
||||x|| ≥  f
1
+ f
2
,x = || f
1

||
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

.
Definition 1.9. A Banach space X is called smooth if, for every x ∈ X with ||x|| = 1, there
exists a unique f ∈ X
∗
such that || f|| =  f,x = 1. The modulus of smoothness of X is the
fuction
ρ
X
: [0,∞) → [0,∞), defined by
ρ
X
(
τ
) = sup

1
2
(||x+ y|| + ||x− y||) − 1 : x,y ∈ X,||x|| = 1,||y|| =
τ

.
The Banach space X is called uniformly smooth if
lim
τ
→∞
ρ
X
(
τ
)

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

p
is uniformly convex and uniformly smooth for 1 < p < ∞ .
4. W
p
m
(G) is uniformly convex and uniformly smooth for 1 < p < ∞.
Proposition 1.1. If X is uniformly smooth, then J is uniformly norm-to-norm continous on
each bounded subset of X.
Theorem 1.2. A uniformly convex Banach space X has the Klee-Kadec (E fimov−Stechkin)
property, i.e., from x
n
⇀ x and ||x
n
|| → ||x|| it follows x
n
→ x.
Proof.
The case, when x = 0 is trivial. Let x = 0. Without loss of generality assuming that ||x|| = 1
and ||x
n
|| = 0, then we have y
n
:=
x
n
||x
n
||
, ||y
n

|| = 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

− x| |)) ≥ sup
|| f||=1
2 f,x = 2||x|| = 2.
Hence,
y
n
→ x, x
n
= y
n
||x
n
|| → x||x|| = x.
Remark 1.1.
• E(E
∗
) is uniformly convex if and only if E
∗
(E) is uniformly smooth.
• If E is uniformly convex, then it is reflexive and strictly convex and it satisfies the
Kadec-Klee property.
• If E is uniformly smooth and uniformly convex then J and J
−1
= J
∗
are single-valued
and uniformly norm-to-norm continuous on bounded subsets of E and E
∗
, respectively.
9

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

− 2y,Jx + ||x||
2
,
for all x,y ∈ X, where J is the normalized duality mapping from X → X
∗
.
Therefore, we have
(||x|| − ||y||)
2
≤
φ
(y,x) ≤ (|| y| |
2
+ ||x||
2
), for allx,y ∈ E. (1.1)
Definition 1.12. The generalized projection Π
C
: E → C is a map that assigns to an arbitrary
point x ∈ E the minimum point of the funtional
φ
(x,y), that is Π
C
x = ¯x, where ¯x is the
solution to the minimization problem
φ
( ¯x,x) = min
y∈C
φ
(y,x).

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→∞

||Tx
n
− x
n
|| = 0. The set of strong
asymptotic fixed points of T will be denoted by
¯
F(T).
Definition 1.14. A mapping T from C into itself is called weakly relatively nonexpansive if
• F(T) is nonempty,
•
φ
(p,Tx) ≤
φ
(p,x), ∀p ∈ F(T),x ∈ C,
•
¯
F(T) = F(T).
Definition 1.15. A mapping T from C into itself is called hemi-relatively nonexpansive if
• F(T) is nonempty;
•
φ
(p,Tx) ≤
φ
(p,x), ∀p ∈ F(T),x ∈ C.
Definition 1.16. Let H be a real Hilbert space with norm ||.|| and an inner product .,., and
K be a nonempty subset of H. An operator T : K → K is said to be a generalized pseudo-
contraction if for all x,y ∈ K, there exists a constant r > 0 such that
||Tx− Ty||
2

≤ 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),

there exists a j(x− y) ∈ J(x− y) such that
Ux−Uy, j(x− y) ≥ k||x− y||
2
.
Proposition 1.2.
1. An operator T is (strongly) pseudo-contractive if and only if (I − T) is (strongly) ac-
cretive.
2. T is strongly pseudo-contractive if there exists t > 1 such that, for all x,y ∈ D(T) and
k > 0, the following inequality holds
||x− y|| ≤ ||(1 + r)(x+ y) − rt(Tx− Ty)||. (1.2)
3. T is pseudo-contractive if t = 1 in inequality (1.2).
4. T is strongly accretive if there exists k > 0 such that the inequality
||x− y|| ≤ ||(x− y) + r[(T − kI)x− (T − kI)y]|| (1.3)
holds for all x,y ∈ D(U) and r > 0.
5. T is accretive if k = 0 in the inequality (1.3)
Definition 1.19. Let H be a Hilbert space and C a subset of H. A mapping T : C → H is
called demicompact if it has the property that whenever {u
n
} is a bounded sequence in H
and {Tu
n
− u
n
} is strongly convergent, then there exists a subsequence {u
n
k
} of {u
n
} which
is strongly convergent.

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

0
= {x : x,y = 0} is a hyperspace.
•
X
c
= {x : x,y ≤ c} or X
c
= {x : x,y ≥ c} is a halfspace.
13
Chapter 2
Iterations for nonexpansive mappings
2.1 Krasnoselskij iteration for nonexpansive mappings
Based probably on ideas of Cauchy and Liouville, Picard developed the following method
of successive approximations,
x
n
= T(x
n−1
) = T
n
(x
0
) n = 1,2, (2.1)
where T is an L-contraction.
However, if T is assumed to be only a nonexpansivemap, then the Picard iterations {T
n
x
0
}
n≥0

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

0
+
λ
Tx, x ∈ C. (2.2)
Since C is convex and closed, T
λ
maps C into itself. Moreover, we obtain
||T
λ
(x) − T
λ
(y)|| = ||(1−
λ
)v
0
+
λ
Tx− (1−
λ
)v
0
−
λ
Ty|| = ||
λ
(Tx− Ty)||
≤ |
λ
|||Tx− Ty|| ≤ |
λ

|||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

= ||(x
j
− p) + (p− x)||
2
= ||x
j
− p||
2
+ ||p− x||
2
+ 2x
j
− p, p− x,
where
2x
j
− p, p− x → 0 as j → ∞,
since x
j
− p converges weakly to zero in H. Setting x = T p, we obtain
lim
j→∞
(||x
j
− T p||
2
− ||x
j
− p||
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−

λ
j
)[Tx
j
− v
0
]
= (T
λ
j
x
j
− x
j
) + (1−
λ
j
)(Tx
j
− v
0
)
= (1−
λ
j
)(Tx
j
− v
0
) → 0, asj → ∞.

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||.

Thus
limsup(||x
j
− T p|| − ||x
j
− p||) ≤ lim
j→∞
||x
j
− Tx
j
|| = 0,
and due to the boundedness of C, we have also
limsup(||x
j
−T p||
2
−||x
j
− p||
2
) = limsup(||x
j
−T p||−||x
j
− p||)(||x
j
−T p||+||x
j
− p||) ≤ 0,

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).

Indeed, since x,y ∈ F(T) then
||x− Tz|| = ||Tx− Tz| | ≤ ||x− z||, and ||Tz− y|| = ||Tz − Ty|| ≤ ||z− y||.
Therefore,
||x− y|| ≤ ||x− Tz|| + ||Tz − y|| ≤ ||x− y||.
So
x− Tz = a(x− z) and y− Tz = b(y− z), for some a,b ∈ [0,1].
Hence, Tz = z.
Theorem 2.2. 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 and for any given x
0
in C and any fixed number
λ
with 0 <
λ
< 1, the
Krasnoselskij iteration {x
n
}
∞
n=0
given by
x
n+1
= (1−
λ
)x
n
+
λ

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
}
n∈N
converges strongly to zero. Indeed,
x
n+1
− p = (1−
λ
)x
n
+
λ
Tx
n
− p = (1−
λ
)(x
n
− p) +
λ
(Tx
n
− p).
16
On the other hand, for any constant a,
a(x
n
− Tx
n
) = a(x

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,x
n
− p,
and
a
2
||x
n
− Tx
n
||
2
= a
2
||x
n
− p||
2
= a
2
||Tx
n
− p||
2
− 2a
2
Tx
n
− p,x
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[

λ
(1−
λ
) − a
2
]Tx
n
− p,x
n
− p.
If we choose now an a such that a
2
≤
λ
(1−
λ
), then from the last inequality we obtain
||x
n+1
− p||
2
+a
2
||x
n
−Tx
n
||
2
≤ (2a

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

− P||.||x
n
− p|| ≤ ||x
n
− p||
2
.
Letting a
2
=
λ
(1−
λ
) > 0 and summing up the obtained inequality
a
2
||x
n
− Tx
n
||
2
≤ ||x
n
− p||
2
− ||x
n+1
− p||
2

.
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

− p||
2
− ||x
N+1
− p||
2
≤ ||x
0
− p||
2
,
which shows that
∑
∞
n=0
||x
n
− Tx
n
||
2
< ∞ and hence ||x
n
− Tx
n
|| → 0, as n → ∞.
Since T is demicompact, there exists a strongly convergent subsequence {x
n
i
} such that

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

• Calculate
x
new
:= (1−
λ
)x
0
+
λ
Tx
0
.
• Compute
err := ||x
new
− x
0
||.
If err ≤
ε
then print x
new
, else x
0
= x
new
and repeat.
Remark 2.1. In general, there is no estimation for the convergence rate of the Krasnoselskij
method. In fact, it is typical that the convergence of iteration methods involving nonexpan-
sive mappings may be arbitrary slow.

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
}

∞
j=0
, x
n
j
= T
n
j
λ
x converges weakly to a certain p
0
, then
p
0
is a fixed point of T or of T
λ
and therefore p
0
= p.
Suppose that {x
n
j
}
∞
j=0
does not converge weakly to p. Then
||x
n
j
− T

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

x
n
j
||
and, using the arguments in the proof of Theorem 2.2, it results
||x
n
j
− T
λ
x
n
j
|| → 0, as n → ∞.
18
The last inequality implies that
limsup(||x
nj
− T
λ
p
0
|| − ||x
n
j
− p
0
||) ≤ 0. (2.6)
But, like in the proof of Theorem 2.2, we have
||x

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

− T
λ
p
0
||
2
+ 2x
n
j
− p
0
, p
0
− T
λ
p
0
,
which shows, together with x
nj
⇀ p
0
(as j → ∞), that
lim
n→∞
[||x
n
j
− T
λ

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
− p
0
||
2
= (||x
n
j
− T
λ
p
0
|| − ||x
n
j
− p||)(||x
n
j
− T
λ
p
0
|| + ||x
n
j
− p
0
||). (2.8)
Since C is bounded, the sequence {||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

in C, the Krasnoselskij iteration converges weakly
to a fixed point of T.
Proof. Let F(T) be the set of all fixed points of T in C (which is nonempty, by Theorem
2.1). As T is nonexpansive, for each p ∈ F(T) and each n we have
||x
n+1
− p|| ≤ ||x
n
− p||,
which shows that the function g(p) = lim
n→∞
||x
n
− p|| is well defined and is a lower semi-
continuous covex function on F(T). Let
d
0
= inf{g(p) : p ∈ F(T)}.
For each
ε
> 0, the set
F
ε
= {y : g(y) ≤ d
0
+
ε
}
is closed, convex, nonempty and bounded. Hence the set is weakly compact. Therefore,


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−

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

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

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

||
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

1
− x
n
||||p
0
− x
n
||]}
= g
2
(p) + lim
n→∞
{2
λ
(1−
λ
)p
1
− x
n
, p
0
− x
n
 − ||p
1
− x
n
||||p
0

− 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

− x
n
|| → d
0
and ||p
0
− x
n
|| → d
0
,
||p
1
− p
0
||
2
= ||(p
1
− x
n
) − (x
n
− p
0
)||
2
= ||p
1
− x

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

n
= T
n
λ
x
0
⇀ p
0
, is suffices to assume that x
n
j
⇀ p for an infinite
subsequence and then prove that p = p
0
. By the arguments in Theorem 2.3, p ∈ F(T).
Taking into account the definition of g and the fact that x
n
j
→ p, we have
||x
n
j
− p
0
||
2
= ||x
n
j
− p+ p− p

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

0
.
Since g
2
(p) ≥ d
2
0
, the last inequality implies that
||p− p
0
|| ≤ 0,
which means p = p
0
.
2.2 Mann iterations for nonexpansive mappings
The Mann iteration was chronologically introduced two years earlier than the Krasnoselskij
iteration, even so it is generalization of the latter and in its normal form is obtained by
replacing the parameter
λ
in the Krasnoselskij iteration formula by a sequence {a
n
}.
20
Definition 2.1. Let E be a linear space, C a convex subset of E and let T : C → C be a
mapping and x
1
∈ C, arbitrary. Let A = [a
nj
] be an infinite real matrix satisfying
(A

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

n
), where
v
n
=
n
∑
j=1
a
nj
x
j
,
is called the Mann iterative process or the Mann iteration, denoted by M(x
1
,A,T).
Remark 2.2.
1. The Mann iterative process {x
n
}
∞
n=1
can be constructed by choosing the initial guess
x
1
, the matrix A and the operator T, then computing x
n
and {v
n
}. So, we will denote

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

nj
]
satisfies the following conditions.
(A
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.
(A
4
) a
n+1, j

= (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

)
∞
∑
n=1
a
nn
= ∞.
• The matrices A = [a
nj
] (other than the infinite identity matrix) in all normal Mann
process M(x
1
,A,T) are constructed as follows.
Choose {c
n
} such that 0 ≤ c
n
< 1 for all n and the series
∑
∞
n=1
c
n
diverges, and define
A = [a
nj
] by












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

= 0, for j > n+ 1, n = 1,2,3,
• The sequence {v
n
} in a normal Mann process M(x
1
,A,T) satisfies
v
n+1
= (1− c
n
)v
n
+ c
n
Tv
n
, for all n = 1,2,3, , (2.9)
where
c
n
= a
n+1,n+1
. (2.10)
Example 2.1. Let c
n
= 1 ∀n ≥ 1 then Mann iteration corresponds to Picard iteration.
Example 2.2. Let c
n
=
1

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

λ
satisfies all of five above conditions.
Remark 2.3. If we consider
T
n
= (1− c
n
)I + c
n
T,
then we have F(T) = F(T
n
), for all c
n
∈ (0, 1].
If the sequence c
n
=
λ
(constant), then the Mann iteration obviously reduces to the Kras-
noselskij iteration. So we can denote Krasnoselskij iteration by K(v
0
,
λ
,T).
Moreover, most of the literature deals with the specialized Mann iteration method defined by
x
1
∈ E and (2.9), where {c
n

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