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Strong convergence theorems and rate of convergence of multi-step iterative
methods for continuous mappings on an arbitrary interval
Fixed Point Theory and Applications 2012, 2012:9 doi:10.1186/1687-1812-2012-9
Withun Phuengrattana ()
Suthep Suantai ()
ISSN 1687-1812
Article type Research
Submission date 6 October 2011
Acceptance date 31 January 2012
Publication date 31 January 2012
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Strong convergence theorems and rate of
convergence of multi-step iterative methods for
continuous mappings on an arbitrary interval
Withun Phuengrattana
1,2
and Suthep Suantai
1,2,∗
1
Department of Mathematics, Faculty of Science,
Chiang Mai University, Chiang Mai 50200, Thailand

2
Centre of Excellence in Mathematics,
CHE, Si Ayutthaya Road, Bangkok 10400, Thailand
∗
Corresponding author:
E-mail address:
WP: phun26 , withun
Abstract
In this article, by using the concept of W -mapping introduced by Atsushiba
and Takahashi and K-mapping introduced by Kangtunyakarn and Suantai, we
1
define W
(T,N )
-iteration and K
(T,N )
-iteration for finding a fixed point of continu-
ous mappings on an arbitrary interval. Then, a necessary and sufficient condition
for the strong convergence of the proposed iterative methods for continuous map-
pings on an arbitrary interval is given. We also compare the rate of convergence of
those iterations. It is proved that the W
(T,N )
-iteration and K
(T,N )
-iteration are
equivalent and the K
(T,N )
-iteration converges faster than the W
(T,N )
-iteration.
Moreover, we also present numerical examples for comparing the rate of conver-

gence between W
(T,N )
-iteration and K
(T,N )
-iteration.
MSC: 26A18; 47H10; 54C05.
Keywords: fixed point; continuous mapping; W-mapping; K-mapping; rate of
convergence.
1 Introduction
There are several classical methods for approximation of solutions of nonlinear equation
of one variable
f(x) = 0 (1.1)
where f : E → E is a continuous function and E is a closed interval on the real line.
Classical fixed point iteration method is one of the methods used for this problem. To
use this method, we have to transform (1.1) to the following equation:
g(x) = x (1.2)
2
where g : E → E is a contraction. Then, Picard’s iteration can be applied for finding
a solution of (1.2).
Question: If g : E → E is continuous but not contraction, what iteration methods
can be used for finding a solution of (1.2) (that is a fixed point of g) and how about
the rate of convergence of those methods.
There are many iterative methods for finding a fixed point of g. For example, the
Mann iteration (see [1]) is defined by x
1
∈ E and
x
n+1
= (1 − α
n

)x
n
+ α
n
g(x
n
) (1.3)
for all n ≥ 1, where {α
n
}
∞
n=1
is sequences in [0, 1]. The Ishikawa iteration (see [2]) is
defined by x
1
∈ E and







y
n
= (1 − β
n
)x
n
+ β

n
g(x
n
)
x
n+1
= (1 − α
n
)x
n
+ α
n
g(y
n
)
(1.4)
for all n ≥ 1, where {α
n
}
∞
n=1
, {β
n
}
∞
n=1
are sequences in [0, 1]. The Noor iteration
(see [3]) is defined by x
1
∈ E and


















z
n
= (1 − γ
n
)x
n
+ γ
n
g(x
n
)
y
n

= (1 − β
n
)x
n
+ β
n
g(z
n
)
x
n+1
= (1 − α
n
)x
n
+ α
n
g(y
n
)
(1.5)
for all n ≥ 1, where {α
n
}
∞
n=1
, {β
n
}
∞

n=1
, and {γ
n
}
∞
n=1
are sequences in [0, 1]. Clearly
Mann and Ishikawa iterations are special cases of Noor iteration. The SP-iteration
3
(see [4]) is defined by x
1
∈ E and

















z

n
= (1 − γ
n
)x
n
+ γ
n
g(x
n
)
y
n
= (1 − β
n
)z
n
+ β
n
g(z
n
)
x
n+1
= (1 − α
n
)y
n
+ α
n
g(y

n
)
(1.6)
for all n ≥ 1, where {α
n
}
∞
n=1
, {β
n
}
∞
n=1
, and {γ
n
}
∞
n=1
are sequences in [0, 1]. Clearly
Mann iteration is special cases of SP-iteration.
In 1976, Rhoades [5] proved the convergence of the Mann and Ishikawa iterations
to a solution of (1.2) when E = [0, 1]. He also proved the Ishikawa iteration converges
faster than the Mann iteration for the class of continuous and nondecreasing functions.
Later in 1991, Borwein and Borwein [6] proved the convergence of the Mann iteration
of continuous functions on a bounded closed interval. In 2006, Qing and Qihou [7]
extended their results to an arbitrary interval and to the Ishikawa iteration and gave
some control conditions for the convergence of Ishikawa iteration on an arbitrary in-
terval. Recently, Phuengrattana and Suantai [4] obtained a similar result for the new
iteration, called the SP-iteration, and they proved the Mann, Ishikawa, Noor and SP-
iterations are equivalent and the SP-iteration converges faster than the others for the

class of continuous and nondecreasing functions.
In this article, we are interested to employ the concept of W -mappings and K-
mappings for approximation of a solution of (1.2) for a continuous function on an
arbitrary interval and compare which one converges faster. The concept of W -mapping
was first introduced by Atsushiba and Takahashi [8]. They defined W -mapping as
4
follows. Let C be a subset of a Banach space X and T : C → C be a mapping. A
point x ∈ C is a fixed point of T if T x = x. The set of all fixed points of T is denoted
by F(T ). Let {T
i
}
N
i=1
be a finite family of mappings of C into itself. Let W
n
: C → C
be a mapping defined by
S
n,0
= I,
S
n,1
= λ
n,1
T
1
S
n,0
+ (1 − λ
n,1

)I,
S
n,2
= λ
n,2
T
2
S
n,1
+ (1 − λ
n,2
)I, (1.7)
.
.
.
S
n,N−1
= λ
n,N−1
T
N−1
S
n,N−2
+ (1 − λ
n,N−1
)I,
W
n
= S
n,N

= λ
n,N
T
N
S
n,N−1
+ (1 − λ
n,N
)I,
where I is the identity mapping of C and λ
n,i
∈ [0, 1] for all i = 1, 2, . . . , N . Such a map-
ping W
n
is called the W-mapping generated by T
1
, T
2
, . . . , T
N
and λ
n,1
, λ
n,2
, . . . , λ
n,N
.
Many researchers have studied and applied this mapping for finding a common fixed
point of nonexpansive mappings, for instance, see [8–23].
In 2009, Kangtunyakarn and Suantai [24] introduced a new concept of the K-

5
mapping in a Banach space as follows. Let K
n
: C → C be a mapping defined by
U
n,0
= I,
U
n,1
= λ
n,1
T
1
U
n,0
+ (1 − λ
n,1
)U
n,0
,
U
n,2
= λ
n,2
T
2
U
n,1
+ (1 − λ
n,2

)U
n,1
, (1.8)
.
.
.
U
n,N−1
= λ
n,N−1
T
N−1
U
n,N−2
+ (1 − λ
n,N−1
)U
n,N−2
,
K
n
= U
n,N
= λ
n,N
T
N
U
n,N−1
+ (1 − λ

n,N
)U
n,N−1
,
where I is the identity mapping of C and λ
n,i
∈ [0, 1] for all i = 1, 2, . . . , N . Such a map-
ping K
n
is called the K-mapping generated by T
1
, T
2
, . . . , T
N
and λ
n,1
, λ
n,2
, . . . , λ
n,N
.
They showed that if C is a nonempty closed convex subset of a strictly convex Banach
space X and {T
i
}
N
i=1
is a finite family of nonexpansive mappings of C into itself, then
F (K

n
) =

N
i=1
F (T
i
) and they also introduced an iterative method by using the con-
cept of K-mapping for finding a common fixed point of a finite family of nonexpansive
mappings and a solution of an equilibrium problem. Applications of K-mappings for
fixed point problems and equilibrium problems can be found in [23–26].
By using the concept of W -mappings and K-mappings, we introduce two new iter-
ations for finding a fixed point of a mapping T : E → E on an arbitrary interval E as
follows.
The W
(T,N )
-iteration is defined by u
1
∈ E and
u
n+1
= W
(T,N )
n
u
n
∀n ≥ 1, (1.9)
6
where N ≥ 1 and W
(T,N )

n
is a mapping of E into itself generated by
S
n,0
= I,
S
n,1
= λ
n,1
T S
n,0
+ (1 − λ
n,1
)I,
S
n,2
= λ
n,2
T S
n,1
+ (1 − λ
n,2
)I, (1.10)
.
.
.
S
n,N−1
= λ
n,N−1

T S
n,N−2
+ (1 − λ
n,N−1
)I,
W
(T,N )
n
= S
n,N
= λ
n,N
T S
n,N−1
+ (1 − λ
n,N
)I,
where I is the identity mapping of E and λ
n,i
∈ [0, 1] for all i = 1, 2, . . . , N. We call
a mapping W
(T,N )
n
as the W -mapping generated by T and λ
n,1
, λ
n,2
, . . . , λ
n,N
. Clearly,

W
(T,1)
-iteration is Mann iteration, W
(T,2)
-iteration is Ishikawa iteration and W
(T,3)
-
iteration is Noor iteration.
The K
(T,N )
-iteration is defined by x
1
∈ E and
x
n+1
= K
(T,N )
n
x
n
∀n ≥ 1, (1.11)
7
where N ≥ 1 and K
(T,N )
n
is a mapping of E into itself generated by
U
n,0
= I,
U

n,1
= λ
n,1
T U
n,0
+ (1 − λ
n,1
)U
n,0
,
U
n,2
= λ
n,2
T U
n,1
+ (1 − λ
n,2
)U
n,1
, (1.12)
.
.
.
U
n,N−1
= λ
n,N−1
T U
n,N−2

+ (1 − λ
n,N−1
)U
n,N−2
,
K
(T,N )
n
= U
n,N
= λ
n,N
T U
n,N−1
+ (1 − λ
n,N
)U
n,N−1
,
where I is the identity mapping of E and λ
n,i
∈ [0, 1] for all i = 1, 2, . . . , N. We call
a mapping K
(T,N )
n
as the K-mapping generated by T and λ
n,1
, λ
n,2
, . . . , λ

n,N
. Clearly,
K
(T,1)
-iteration is Mann iteration and K
(T,3)
-iteration is SP-iteration.
Obviously, the mappings (1.10) and (1.12) are special cases of the W -mapping and
K-mapping, respectively.
The purpose of this article is to give a necessary and sufficient condition for the
strong convergence of the W
(T,N )
-iteration and K
(T,N )
-iteration of continuous mappings
on an arbitrary interval. We also prove that the K
(T,N )
-iteration and W
(T,N )
-iteration
are equivalent and the K
(T,N )
-iteration converges faster than the W
(T,N )
-iteration for
the class of continuous and nondecreasing mappings. Moreover, we present numerical
examples for the K
(T,N )
-iteration to compare with the W
(T,N )

-iteration. Our results
extend and improve the corresponding results of Rhoades [5], Borwein and Borwein [6],
Qing and Qihou [7], Phuengrattana and Suantai [4], and many others.
8
2 Convergence theorems
We first give a convergence theorem for the K
(
T,N
)
-iteration for continuous mappings
on an arbitrary interval.
Theorem 2.1 Let E be a closed interval on the real line and T : E → E be a contin-
uous mapping. For x
1
∈ E, let the K
(T,N )
-iteration {x
n
}
∞
n=1
defined by (1.11), where
{λ
n,i
}
∞
n=1
(i = 1, 2, . . . , N ) are sequences in [0, 1] satisfying the following conditions:
(C1)


∞
n=1
λ
n,i
< ∞ for all i = 1, 2, . . . , N − 1;
(C2) lim
n→∞
λ
n,N
= 0 and

∞
n=1
λ
n,N
= ∞.
Then {x
n
}
∞
n=1
is bounded if and only if {x
n
}
∞
n=1
converges to a fixed point of T .
Proof. It is obvious that if {x
n
}

∞
n=1
converges to a fixed point of T, then it is bounded.
Now, assume that {x
n
}
∞
n=1
is bounded. We will show that {x
n
}
∞
n=1
converges to a fixed
point of T . First, we show that {x
n
}
∞
n=1
is convergent. To show this, we suppose not.
Then there exist a, b ∈ R, a = lim inf
n→∞
x
n
, b = lim sup
n→∞
x
n
and a < b.
Next, we show that

if m ∈ (a, b), then T m = m. (2.1)
To show this, suppose that T m = m for some m ∈ (a, b). Without loss of generality,
we may assume that Tm − m > 0. By continuity of T, there exists δ ∈ (0, b − a) such
that
T x − x > 0 for |x − m| ≤ δ. (2.2)
9
By boundedness of {x
n
}
∞
n=1
, we have {x
n
}
∞
n=1
belongs to a bounded closed interval.
Continuity of T implies that {T x
n
}
∞
n=1
belongs to another bounded closed interval,
so {T x
n
}
∞
n=1
is bounded. Since U
n,1

x
n
= λ
n,1
T x
n
+ (1 − λ
n,1
)x
n
, we get {U
n,1
x
n
}
∞
n=1
is bounded, and thus {T U
n,1
x
n
}
∞
n=1
is bounded. Similarly, by using (1.11), we have
{U
n,i
x
n
}

∞
n=1
and {T U
n,i
x
n
}
∞
n=1
are bounded for all i = 2, 3, . . . , N − 1. It follows by
(1.11) that U
n,i
x
n
− U
n,i−1
x
n
= λ
n,i
(T U
n,i−1
x
n
− U
n,i−1
x
n
) for all i = 1, 2, . . . , N. By
condition (C1) and (C2), we get lim

n→∞
|U
n,i
x
n
− U
n,i−1
x
n
| = 0 for all i = 1, 2, . . . , N .
Since
|x
n+1
− x
n
| = |U
n,N
x
n
− U
n,0
x
n
|
≤ |x
n+1
− U
n,N−1
x
n

| + |U
n,N−1
x
n
− U
n,N−2
x
n
| + · · · + |U
n,1
x
n
− U
n,0
x
n
|,
it implies that lim
n→∞
|x
n+1
− x
n
| = 0. Thus, there exists M
0
such that
|x
n+1
− x
n

| <
δ
N
and |U
n,i
x
n
− U
n,i−1
x
n
| <
δ
N
(i = 1, 2, . . . , N − 1), (2.3)
for all n > M
0
. Since b = lim sup
n→∞
x
n
> m, there exists k
1
> M
0
such that x
k
1
> m.
Let k = k

1
, then x
k
> m. If x
k
≥ m +
δ
N
, then by (2.3), we have x
k+1
> x
k
−
δ
N
≥ m,
so x
k+1
> m. If x
k
∈ (m, m +
δ
N
), then by (2.3), we have
m −
δ
N
i < U
k,i
x

k
< m +
δ
N
(i + 1) for all i = 1, 2, . . . , N − 1.
So we have
|x
k
− m| < δ and |U
k,i
x
k
− m| < δ for all i = 1, 2, . . . , N − 1.
10
This implies by (2.2) that
T x
k
− x
k
> 0 and T U
k,i
x
k
− U
k,i
x
k
> 0 for all i = 1, 2, . . . , N − 1. (2.4)
Using (1.11), we obtain
x

k+1
= λ
k,N
T U
k,N −1
x
k
+ (1 − λ
k,N
)U
k,N −1
x
k
= U
k,N −1
x
k
+ λ
k,N
(T U
k,N −1
x
k
− U
k,N −1
x
k
)
= U
k,N −2

x
k
+ λ
k,N −1
(T U
k,N −2
x
k
− U
k,N −2
x
k
) + λ
k,N
(T U
k,N −1
x
k
− U
k,N −1
x
k
)
.
.
.
= x
k
+
N


i=1
λ
k,i
(T U
k,i−1
x
k
− U
k,i−1
x
k
) . (2.5)
By (2.4), we have x
k+1
> x
k
. Thus, x
k+1
> m.
By using the above argument, we obtain x
k+j
> m for all j ≥ 2. Thus we get
x
n
> m for all n > k. So a = lim inf
n→∞
x
n
≥ m, which is a contradiction with a < m.

Thus Tm = m. Therefore, we obtain (2.1).
For the sequence {x
n
}
∞
n=1
, we consider the following two cases:
Case 1: There exists x
¯
M
such that a < x
¯
M
< b. Then T x
¯
M
= x
¯
M
. By using (1.11),
we obtain that U
¯
M,i
x
¯
M
= x
¯
M
for all i = 1, 2, . . . , N. Thus, we have x

¯
M+1
= x
¯
M
. By
induction, we obtain x
¯
M
= x
¯
M+1
= x
¯
M+2
= · · · , so x
n
→ x
¯
M
. This implies that
x
¯
M
= a and x
n
→ a, which contradicts with our assumption.
Case 2: For all n, x
n
≤ a or x

n
≥ b. Because b − a > 0 and lim
n→∞
|x
n+1
− x
n
| = 0,
there exists M
1
such that |x
n+1
− x
n
| <
b−a
N
for all n > M
1
. It implies that either
x
n
≤ a for all n > M
1
or x
n
≥ b for all n > M
1
. If x
n

≤ a for n > M
1
, then
11
b = lim sup
n→∞
x
n
≤ a, which is a contradiction with a < b. If x
n
≥ b for n > M
1
, so
we have a = lim inf
n→∞
x
n
≥ b, which is a contradiction with a < b.
Hence, we have {x
n
}
∞
n=1
is convergent.
Finally, we show that {x
n
}
∞
n=1
converges to a fixed point of T . Let lim

n→∞
x
n
= p
and suppose T p = p. Since {U
n,i
x
n
}
∞
n=1
is bounded for all i = 1, 2, . . . , N −1, it implies
by (1.11), condition (C1) and (C2) that lim
n→∞
U
n,i
x
n
= p for all i = 1, 2, . . . , N − 1.
Let h
k,i
= T U
k,i−1
x
k
− U
k,i−1
x
k
for all i = 1, 2, . . . , N. Continuity of T implies that

lim
k→∞
h
k,i
= Tp − p = 0 for all i = 1, 2, . . . , N. Put w = Tp − p. Then w = 0. By
(2.5), we have
n−1

k=1
(x
k+1
− x
k
) =
n−1

k=1
(λ
k,1
h
k,1
+ λ
k,2
h
k,2
+ · · · + λ
k,N
h
k,N
).

This implies that
x
n
= x
1
+
n−1

k=1
(λ
k,1
h
k,1
+ λ
k,2
h
k,2
+ · · · + λ
k,N
h
k,N
). (2.6)
By condition (C1), (C2), and lim
k→∞
h
k,i
= w = 0 for all i = 1, 2, . . . , N, we get that

∞
k=1

λ
k,i
h
k,i
is convergent for all i = 1, 2, . . . , N − 1 and

∞
k=1
λ
k,N
h
k,N
is divergent.
It follows by (2.6) that {x
n
}
∞
n=1
is divergent, which is a contradiction. Hence, {x
n
}
∞
n=1
converges to a fixed point of T.
We now obtain the convergence theorem of W
(T,N )
-iteration. The proof is omitted
because it is similar as above theorem and Theorem 2.2 of [4].
Theorem 2.2 Let E be a closed interval on the real line and T : E → E be a contin-
uous mapping. For x

1
∈ E, let the W
(T,N )
-iteration {x
n
}
∞
n=1
defined by (1.9), where
12
{λ
n,i
}
∞
n=1
(i = 1, 2, . . . , N ) are sequences in [0, 1] satisfying the following conditions:
(C1) lim
n→∞
λ
n,i
= 0 for all i = 1, 2, . . . , N;
(C2)

∞
n=1
λ
n,N
= ∞.
Then {x
n

}
∞
n=1
is bounded if and only if {x
n
}
∞
n=1
converges to a fixed point of T .
The following results are obtained direclty from Theorem 2.1.
Corollary 2.3 ([4, Theorem 2.1]) Let E be a closed interval on the real line and
T : E → E be a continuous mapping. For x
1
∈ E, let the SP-iteration {x
n
}
∞
n=1
defined
by (1.6), where {λ
n,1
}
∞
n=1
, {λ
n,2
}
∞
n=1
, and {λ

n,3
}
∞
n=1
are sequences in [0, 1] satisfying
the following conditions:
(C1)

∞
n=1
λ
n,1
< ∞ and

∞
n=1
λ
n,2
< ∞;
(C2) lim
n→∞
λ
n,3
= 0 and

∞
n=1
λ
n,3
= ∞.

Then {x
n
}
∞
n=1
is bounded if and only if {x
n
}
∞
n=1
converges to a fixed point of T .
Corollary 2.4 ([7, Theorem 3]) Let E be a closed interval on the real line and
T : E → E be a continuous mapping. For x
1
∈ E, let the Mann iteration {x
n
}
∞
n=1
defined by (1.3), where {λ
n,1
}
∞
n=1
is a sequence in [0, 1] satisfying lim
n→∞
λ
n,1
= 0 and


∞
n=1
λ
n,1
= ∞. Then {x
n
}
∞
n=1
is bounded if and only if {x
n
}
∞
n=1
converges to a fixed
point of T .
The following results are obtained directly from Theorem 2.2.
13
Corollary 2.5 ([4, Theorem 2.2]) Let E be a closed interval on the real line and
T : E → E be a continuous mapping. For x
1
∈ E, let the Noor iteration {x
n
}
∞
n=1
defined by (1.5), where {λ
n,1
}
∞

n=1
, {λ
n,2
}
∞
n=1
, {λ
n,3
}
∞
n=1
are sequences in [0, 1] satisfying
the following conditions:
(C1) lim
n→∞
λ
n,1
= 0, lim
n→∞
λ
n,2
= 0 and lim
n→∞
λ
n,3
= 0;
(C2)

∞
n=1

λ
n,3
= ∞.
Then {x
n
}
∞
n=1
is bounded if and only if {x
n
}
∞
n=1
converges to a fixed point of T .
Corollary 2.6 ([7]) Let E be a closed interval on the real line and T : E → E be a
continuous mapping. For x
1
∈ E, let the Ishikawa iteration {x
n
}
∞
n=1
defined by (1.4),
where {λ
n,1
}
∞
n=1
, {λ
n,2

}
∞
n=1
are sequences in [0, 1] satisfying the following conditions:
(C1) lim
n→∞
λ
n,1
= 0 and lim
n→∞
λ
n,2
= 0;
(C2)

∞
n=1
λ
n,2
= ∞.
Then {x
n
}
∞
n=1
is bounded if and only if {x
n
}
∞
n=1

converges to a fixed point of T .
3 Rate of convergence and numerical examples
There are many articles have been published on the iterative methods using for ap-
proximation of fixed points of nonlinear mappings, see for instance [1–7]. However,
there are only a few articles concerning comparison of those iterative methods in order
to establish which one converges faster. As far as we know, there are two ways for
14
comparison of the rate of convergence. The first one was introduced by Berinde [27].
He used this idea to compare the rate of convergence of Picard and Mann iterations for
a class of Zamfirescu operators in arbitrary Banach spaces. Popescu [28] also used this
concept to compare the rate of convergence of Picard and Mann iterations for a class of
quasi-contractive operators. It was shown in [29] that the Mann and Ishikawa iterations
are equivalent for the class of Zamfirescu operators. In 2006, Babu and Prasad [30]
showed that the Mann iteration converges faster than the Ishikawa iteration for this
class of operators. Two years later, Qing and Rhoades [31] provided an example to
show that the claim of Babu and Prasad [30] is false.
However, this concept is not suitable or cannot be applied to a class of continuous
self-mappings defined on a closed interval. In order to compare the rate of convergence
of continuous self-mappings defined on a closed interval, Rhoades [5] introduced the
other concept which is slightly different from that of Berinde to compare iterative
methods which one converges faster as follows.
Definition 3.1 Let E be a closed interval on the real line and T : E → E be a
continuous mapping. Suppose that {x
n
}
∞
n=1
and {u
n
}

∞
n=1
are two iterations which
converge to the fixed point p of T . We say that {x
n
}
∞
n=1
converges faster than {u
n
}
∞
n=1
if
|x
n
− p| ≤ |u
n
− p| for all n ≥ 1.
In this section, we study the rate of convergence of W
(T,N )
-iteration and K
(T,N )
-
iteration for continuous and nondecreasing mappings on an arbitrary interval in the
15
sense of Rhoades. The following lemmas are useful and crucial for our following results.
Lemma 3.2 Let E be a closed interval on the real line and T : E → E be a con-
tinuous and nondecreasing mapping such that F(T ) is nonempty and bounded with
x

1
> sup{p ∈ E : p = T p}. Let {x
n
}
∞
n=1
be defined by W
(T,N )
-iteration or K
(T,N )
-
iteration. If T x
1
> x
1
, then {x
n
}
∞
n=1
does not converge to a fixed point of T.
Proof. We prove only the case that {x
n
}
∞
n=1
is defined by K
(T,N )
-iteration because the
other case can be proved similarly.

Let Tx
1
> x
1
. Since x
1
> sup{p ∈ E : p = T p} and by using (1.11) and mathemat-
ical induction, we can show that x
n
≥ sup{p ∈ E : p = T p} for all n ≥ 1. It is clear
that T x
n
≥ x
n
for all n ≥ 1. Using (1.11), we have
U
n,1
x
n
= λ
n,1
T x
n
+ (1 − λ
n,1
)x
n
≥ x
n
for all n ≥ 1.

Since T is nondecreasing, we have T U
n,1
x
n
≥ T x
n
≥ x
n
. Using (1.11) again, we have
U
n,2
x
n
= λ
n,2
T U
n,1
x
n
+ (1 − λ
n,2
)U
n,1
x
n
≥ x
n
for all n ≥ 1.
This implies that T U
n,2

x
n
≥ T x
n
≥ x
n
. By continuity in this way, we can show that
x
n+1
= K
(T,N )
n
x
n
= U
n,N
x
n
≥ x
n
for all n ≥ 1. Thus {x
n
}
∞
n=1
is nondecreasing. But
x
1
> sup{p ∈ E : p = T p}, it implies that {x
n

}
∞
n=1
does not converges to a fixed point
of T .
By using the same argument of proof as in above lemma, we get the following result.
16
Lemma 3.3 Let E be a closed interval on the real line and T : E → E be a continuous
and nondecreasing mapping such that F (T ) is nonempty and bounded with x
1
< inf{p ∈
E : p = T p}. Let {x
n
}
∞
n=1
be defined by W
(T,N )
-iteration or K
(T,N )
-iteration. If
T x
1
< x
1
, then {x
n
}
∞
n=1

does not converge to a fixed point of T.
We now get the following theorem for compare rate of convergence between W
(T,N )
-
iteration and K
(T,N )
-iteration.
Theorem 3.4 Let E be a closed interval on the real line and T : E → E be a con-
tinuous and nondecreasing mapping such that F (T ) is nonempty and bounded. For
u
1
= x
1
∈ E, let {u
n
}
∞
n=1
and {x
n
}
∞
n=1
are the sequences defined by (1.9) and (1.11),
respectively. Let {λ
n,i
}
∞
n=1
be sequences in [0, 1) for all i = 1, 2, . . . , N. Then, the

W
(T,N )
-iteration {u
n
}
∞
n=1
converges to the fixed point p of T if and only if the K
(T,N )
-
iteration {x
n
}
∞
n=1
converges to p. Moreover, the K
(T,N )
-iteration converges faster than
the W
(T,N )
-iteration.
Proof. Put L = inf{p ∈ E : p = T p} and U = sup{p ∈ E : p = T p}.
(⇒) Suppose that the W
(T,N )
-iteration {u
n
}
∞
n=1
converges to the fixed point p of T .

We divide our proof into the following three cases:
Case 1: u
1
= x
1
> U . By Lemma 3.2, we have T u
1
< u
1
and T x
1
< x
1
. We now
show that x
n
≤ u
n
for all n ≥ 1. Assume that x
k
≤ u
k
. Thus, T x
k
≤ T u
k
. Since
x
1
> U and by using (1.11) and mathematical induction, we can show that x

n
≥ U for
all n ≥ 1. It is clear that T x
k
≤ x
k
. This implies that T x
k
≤ U
k,1
x
k
≤ x
k
. Since T is
17
nondecreasing, T U
k,1
x
k
≤ T x
k
. Thus, we have
T U
k,1
x
k
≤ U
k,2
x

k
≤ U
k,1
x
k
. (3.1)
It follows that U
k,2
x
k
≤ x
k
. By (3.1) and T is nondecreasing, we have T U
k,2
x
k
≤
T U
k,1
x
k
≤ U
k,2
x
k
. This implies that
T U
k,2
x
k

≤ U
k,3
x
k
≤ U
k,2
x
k
.
Thus, we have U
k,3
x
k
≤ x
k
. By continuity in this way, we can show that
U
k,i
x
k
≤ x
k
for all i = 1, 2, . . . , N.
Using (1.9) and (1.11), we get
U
k,1
x
k
− S
k,1

u
k
= λ
k,1
(x
k
− u
k
) + (1 − λ
k,1
)(T x
k
− Tu
k
) ≤ 0.
Since T is nondecreasing, we have T U
k,1
x
k
≤ T S
k,1
u
k
. It follows that
U
k,2
x
k
− S
k,2

u
k
= λ
k,2
(U
k,1
x
k
− u
k
) + (1 − λ
k,2
)(T U
k,1
x
k
− TS
k,1
u
k
)
≤ λ
k,2
(U
k,1
x
k
− x
k
) + (1 − λ

k,2
)(T U
k,1
x
k
− TS
k,1
u
k
)
≤ 0.
That is U
k,2
x
k
≤ S
k,2
u
k
. Since T is nondecreasing, we have T U
k,2
x
k
≤ T S
k,2
u
k
. This
implies that
U

k,3
x
k
− S
k,3
u
k
= λ
k,3
(U
k,2
x
k
− u
k
) + (1 − λ
k,3
)(T U
k,2
x
k
− TS
k,2
u
k
)
≤ λ
k,3
(U
k,2

x
k
− x
k
) + (1 − λ
k,3
)(T U
k,2
x
k
− TS
k,2
u
k
)
≤ 0.
18
That is U
k,3
x
k
≤ S
k,3
u
k
. By continuity in this way, we can show that U
k,N
x
k
≤ S

k,N
u
k
.
Thus, x
k+1
≤ u
k+1
. Hence, by mathematical induction, we obtain x
n
≤ u
n
for all
n ≥ 1. By x
n
≥ U for all n ≥ 1, we get 0 ≤ x
n
− p ≤ u
n
− p, so
|x
n
− p| ≤ |u
n
− p| for all n ≥ 1. (3.2)
Since lim
n→∞
u
n
= p, it implies that lim

n→∞
x
n
= p. That is, the K
(T,N )
-iteration
{x
n
}
∞
n=1
converges to the same fixed point p. Moreover, by (3.2), we see that the
K
(T,N )
-iteration {x
n
}
∞
n=1
converges faster than the W
(T,N )
-iteration {u
n
}
∞
n=1
.
Case 2: u
1
= x

1
< L. By Lemma 3.3, we have T u
1
> u
1
and Tx
1
> x
1
. By using
(1.9), (1.11) and the same argument as in Case 1, we can show that x
n
≥ u
n
for all
n ≥ 1. We note that x
1
< L and by using (1.11) and mathematical induction, we can
show that x
n
≤ L for all n ≥ 1. Thus, we have |x
n
− p| ≤ |u
n
− p| for all n ≥ 1. It
follows that lim
n→∞
x
n
= p and the K

(T,N )
-iteration {x
n
}
∞
n=1
converges faster than the
W
(T,N )
-iteration {u
n
}
∞
n=1
.
Case 3: L ≤ u
1
= x
1
≤ U. Suppose that T u
1
= u
1
. Without loss of generality, we
suppose T u
1
< u
1
. It follows by (1.9) that u
n

≤ u
1
for all n ≥ 1. Since lim
n→∞
u
n
= p,
we must get p < u
1
= x
1
. By the same argument as in Case 1, we have p ≤ x
n
≤ u
n
for all n ≥ 1. It follows that |x
n
− p| ≤ |u
n
− p| for all n ≥ 1. Hence, lim
n→∞
x
n
= p
and the K
(T,N )
-iteration {x
n
}
∞

n=1
converges faster than the W
(T,N )
-iteration {u
n
}
∞
n=1
.
(⇐) Suppose that the K
(T,N )
-iteration {x
n
}
∞
n=1
converges to the fixed point p of
T . Put λ
n,i
= 0 for all i = 1, 2, . . . , N − 1 and n ≥ 1, we get the sequence {x
n
}
∞
n=1
19
generated by
x
n+1
= λ
n,N

T x
n
+ (1 − λ
n,N
)x
n
for all n ≥ 1 (3.3)
that converges to p. We will show that W
(T,N )
-iteration {u
n
}
∞
n=1
converges to p. We
shall prove only the case x
1
= u
1
> U, because other cases can be proved similarly
as the first part. By Proposition 3.5 in [4], we get T x
1
< x
1
and T u
1
< u
1
. Assume
that u

k
≤ x
k
. Thus T u
k
≤ Tx
k
. Since u
1
> U and by using (1.9) and mathematical
induction, we can show that u
n
≥ U for all n ≥ 1. It is clear that T u
k
≤ u
k
.
This implies that T u
k
≤ S
k,1
u
k
≤ u
k
. Since T is nondecreasing, T S
k,1
u
k
≤ T u

k
≤
S
k,1
u
k
. Thus, T S
k,1
u
k
≤ u
k
≤ x
k
. It follows that T S
k,1
u
k
≤ S
k,2
u
k
≤ u
k
. Since T is
nondecreasing, T S
k,2
u
k
≤ Tu

k
≤ S
k,1
u
k
. Thus, T S
k,2
u
k
≤ u
k
≤ x
k
. By continuity in
this way, we have T S
k,i
u
k
≤ x
k
for all i = 1, 2, . . . , N. By (1.9) and (3.3), we obtain
S
k,i
u
k
− x
k
= λ
k,i
(u

k
− x
k
) + (1 − λ
k,i
)(T S
k,i−1
u
k
− x
k
) ≤ 0,
for all i = 2, 3, . . . , N − 1. Since T is nondecreasing, we have
T S
k,i
u
k
≤ T x
k
for all i = 2, 3, . . . , N − 1.
It follows by (1.9) and (3.3) that
u
k+1
− x
k+1
= λ
k,N
(u
k
− x

k
) + (1 − λ
k,N
)(T S
k,N −1
u
k
− Tx
k
) ≤ 0.
By mathematical induction, we have u
n
≤ x
n
for all n ≥ 1. We note that x
1
> U and
by using (3.3) and mathematical induction, we can show that x
n
≥ U for all n ≥ 1.
Thus, we have 0 ≤ u
n
− p ≤ x
n
− p for all n ≥ 1. Since lim
n→∞
x
n
= p, it follows
20

that lim
n→∞
u
n
= p. That is, the W
(T,N )
-iteration {u
n
}
∞
n=1
converges to the same fixed
point p.
We also consider the speed of convergence of the K
(T,N )
-iteration which depends
on the choice of control sequences {λ
n,i
}
∞
n=1
(i = 1, 2, . . . , N) as the following theorem.
Theorem 3.5 Let E be a closed interval on the real line and T : E → E be a
continuous and nondecreasing mapping such that F (T ) is nonempty and bounded.
Let {λ
n,i
}
∞
n=1
, {λ

∗
n,i
}
∞
n=1
are the sequences in [0, 1) such that λ
n,i
≤ λ
∗
n,i
for all i =
1, 2, . . . , N. Let {x
n
}
∞
n=1
be a sequence defined by x
1
∈ E and
x
n+1
= K
(T,N )
n
x
n
∀n ≥ 1, (3.4)
where K
(T,N )
n

is the K-mapping generated by T and λ
n,1
, λ
n,2
, . . . , λ
n,N
, and {x
∗
n
}
∞
n=1
be
a sequence defined by x
∗
1
= x
1
∈ E and
x
∗
n+1
=
¯
K
(T,N )
n
x
∗
n

∀n ≥ 1, (3.5)
where
¯
K
(T,N )
n
is the K-mapping generated by T and λ
∗
n,1
, λ
∗
n,2
, . . . , λ
∗
n,N
.
If {x
n
}
∞
n=1
converges to the fixed point p of T , then {x
∗
n
}
∞
n=1
converges to p. More-
over, {x
∗

n
}
∞
n=1
converges faster than {x
n
}
∞
n=1
.
Proof. Put L = inf{p ∈ E : p = Tp} and U = sup{p ∈ E : p = Tp}. Suppose that
{x
n
}
∞
n=1
converges to a fixed point p of T . We divide our proof into the following three
cases:
Case 1: x
∗
1
= x
1
> U. By Lemma 3.2, we have T x
∗
1
< x
∗
1
and T x

1
< x
1
. Assume
that x
∗
k
≤ x
k
. Thus, T x
∗
k
≤ T x
∗
k
. Since x
∗
1
> U and by using (3.5) and mathematical
21
induction, we can show that x
∗
n
≥ U for all n ≥ 1. It is clear that T x
∗
k
≤ x
∗
k
. This

implies that T x
∗
k
≤ U
k,1
x
∗
k
≤ x
∗
k
. Since T is nondecreasing, T U
k,1
x
∗
k
≤ T x
∗
k
. Thus, we
have
T U
k,1
x
∗
k
≤ U
k,2
x
∗

k
≤ U
k,1
x
∗
k
.
It follows that TU
k,2
x
∗
k
≤ T U
k,1
x
∗
k
≤ U
k,2
x
∗
k
. This implies that
T U
k,2
x
∗
k
≤ U
k,3

x
∗
k
≤ U
k,2
x
∗
k
.
By continuity in this way, we can show that
T U
k,i
x
∗
k
≤ U
k,i
x
∗
k
for all i = 0, 1, . . . , N. (3.6)
Using (3.4), (3.5), and (3.6), we have
U
k,1
x
∗
k
− U
k,1
x

k
= (U
k,0
x
∗
k
− U
k,0
x
k
) + λ
∗
k,1
(T U
k,0
x
∗
k
− U
k,0
x
∗
k
) + λ
k,1
(U
k,0
x
k
− TU

k,0
x
k
)
≤ (U
k,0
x
∗
k
− U
k,0
x
k
) + λ
∗
k,1
(T U
k,0
x
∗
k
− U
k,0
x
∗
k
) + λ
∗
k,1
(U

k,0
x
k
− TU
k,0
x
k
)
= (1 − λ
∗
k,1
)(U
k,0
x
∗
k
− U
k,0
x
k
) + λ
∗
k,1
(T U
k,0
x
∗
k
− TU
k,0

x
k
)
≤ 0.
This implies TU
k,1
x
∗
k
≤ T U
k,1
x
k
. It follows that
U
k,2
x
∗
k
− U
k,2
x
k
= (U
k,1
x
∗
k
− U
k,1

x
k
) + λ
∗
k,2
(T U
k,1
x
∗
k
− U
k,1
x
∗
k
) + λ
k,2
(U
k,1
x
k
− TU
k,1
x
k
)
≤ (U
k,1
x
∗

k
− U
k,1
x
k
) + λ
∗
k,2
(T U
k,1
x
∗
k
− U
k,1
x
∗
k
) + λ
∗
k,2
(U
k,1
x
k
− TU
k,1
x
k
)

= (1 − λ
∗
k,2
)(U
k,1
x
∗
k
− U
k,1
x
k
) + λ
∗
k,2
(T U
k,1
x
∗
k
− TU
k,1
x
k
)
≤ 0.
22
By continuity in this way, we can show that
¯
K

(T,N )
k
x
∗
k
− K
(T,N )
k
x
k
= U
k,N
x
∗
k
− U
k,N
x
k
≤ 0.
That is, x
∗
k+1
≤ x
k+1
. By mathematical induction, we obtain x
∗
n
≤ x
n

for all n ≥ 1.
Since x
∗
n
≥ U for all n ≥ 1, we get 0 ≤ x
∗
n
− p ≤ x
n
− p, so |x
∗
n
− p| ≤ |x
n
− p| for all
n ≥ 1. It follows that lim
n→∞
x
∗
n
= p and {x
∗
n
}
∞
n=1
converges faster than {x
n
}
∞

n=1
.
Case 2: x
∗
1
= x
1
< L. By Lemma 3.3, we have T x
∗
1
> x
∗
1
and Tx
1
> x
1
. By using
(3.4), (3.5) and the same argument as in Case 1, we can show that x
∗
n
≥ x
n
for all
n ≥ 1. We note that x
∗
1
< L and by using (3.5) and mathematical induction, we can
show that x
∗

n
≤ L for all n ≥ 1. Thus, we have |x
∗
n
− p| ≤ |x
n
− p| for all n ≥ 1. It
follows that lim
n→∞
x
∗
n
= p and {x
∗
n
}
∞
n=1
converges faster than {x
n
}
∞
n=1
.
Case 3: L ≤ x
∗
1
= x
1
≤ U . Suppose that T x

∗
1
= x
∗
1
. Without loss of generality,
we suppose T x
∗
1
< x
∗
1
. It follows by (3.5) that x
n+1
≤ x
n
for all n ≥ 1. Since
lim
n→∞
x
n
= p, we must get p < x
∗
1
= x
1
. By the same argument as in Case 1, we
have p ≤ x
∗
n

≤ x
n
for all n ≥ 1. It follows that |x
∗
n
− p| ≤ |x
n
− p| for all n ≥ 1. Hence,
lim
n→∞
x
∗
n
= p and {x
∗
n
}
∞
n=1
converges faster than {x
n
}
∞
n=1
.
Finally, we present two numerical examples for comparing rate of convergence be-
tween W
(T,N )
-iteration and K
(T,N )

-iteration.
Example 3.6 Let T : [0, 8] → [0, 8] be defined by T x = − sin(
x−3
2
) + x +
1
2
. Then T is
a continuous and nondecreasing mapping. The comparison of the rate of convergence
of the W
(T,N )
-iteration {u
n
}
∞
n=1
and K
(T,N )
-iteration {x
n
}
∞
n=1
to a fixed point of T are
23
given in Table 1, with the initial point u
1
= x
1
= 1 when N = 10.

From Table 1, we see that the K
(T,10)
-iteration converges faster than the W
(T,10)
-
iteration under the same control conditions. We also observe that x
45
= 4.047155172
is an approximation of the fixed point of T with accuracy at 6 significant digits.
Example 3.7 Let T : [−7, 7] → [−7, 7] be defined by
T x =















0.7x + e
−0.8
+ 0.8, if x ∈ [−7, −4)
e

x
5
− 2, if x ∈ [−4, 5)
(x − 5)
2
+ e − 2, if x ∈ [5, 7].
Then T is a continuous and nondecreasing mapping. The comparison of the rate of
convergence of the W
(T,N )
-iteration {u
n
}
∞
n=1
and K
(T,N )
-iteration {x
n
}
∞
n=1
to a fixed
point of T are given in Table 2, when N = 12.
In Example 3.7, the mapping T is continuous on [−7, 7] but it not differentiable
at x = −4 and x = 5. In Table 2, we observe that the K
(T,12)
-iteration and W
(T,12)
-
iteration with the initial point is x = 5 converge to a fixed p oint p ≈ −1.215863862 of

T . Moreover, the K
(T,12)
-iteration converges faster than the W
(T,12)
-iteration.
Open Problem: Is it possible to prove the convergence theorem of a finite family of
continuous mappings on an arbitrary interval by using W-mappings and K-mappings
and how about the rate of convergence of those methods?
24