The author considers a Noortype iterative scheme to approximate com mon fixed points of an infinite family of uniformly quasisup(
f_{n}
)Lipschitzian map pings and an infinite family of
g_{n}
expansive mappings in convex cone metric spaces. His results generalize, improve and unify some corresponding results in convex met ric spaces
[1
,
3
,
9
,
16
,
18
,
19]
and convex cone metric spaces
[8]
.
1. INTRODUCTION
Recently, a class of threestep approximation schemes, which includes Mann and Ishikawa iterative schemes for solving general variational inequalities and related problems in Hilbert spaces, was considered by Noor
[11]
. And then, threestep methods (named as Noor methods by some authors) for solving various classes of variational inequalities and related problems were extensively studied by the same author in
[12]
. Since then Noor iteration schemes have been applied to study strong and weak convergences of nonexpansive mappings
[2
,
10
,
13
,
14
,
16
,
20
,
21]
. In 2002, Xu and Noor
[20]
considered a threestep iterative scheme with fixedpoint iterations for asymptotically nonexpansive mappings in Banach spaces. In 2007, Noor and Huang
[13]
analyzed threestep iteration methods for finding the common element of the set of fixed points of nonexpansive mappings and studied the convergence criteria for threestep iterative methods. In 2007, Nammanee and Suanti
[8]
considered the weak and strong convergences for asymptotically nonexpansive mappings for the modified Noor iteration schemes with errors in a uniformly convex Banach spaces. In 2008, Khan et al.
[7]
generalized the Noortype iterative process considered in
[20]
to the case of a finite family of mappings.
On the other hand, there have been many researches
[1
,
3
,
5
,
6
,
9
,
16

19]
on iterative schemes for various kinds of nonexpansive mappings in convex metric spaces with convex structure
[15]
in the usual metric spaces. In 2010, Khan and Ahmed
[6]
introduced a generalized iterative scheme due to Khan et al.
[7]
in convex metric spaces and established a strong convergence to a unique common fixed point of a finite family of asymptotically quasinonexpansive mappings under the scheme. Very recently, Tian and Yang
[17]
gave some suffcient and necessary conditions for a new Noortype iteration with errors to approximate a common fixed point for a finite family of uniformly quasiLipschitzian mappings in convex metric spaces.
A cone metric
[4]
in Banach spaces, which is a coneversion of the usual metric in ℝ is very applicable in applied mathematics including nonlinear analysis by joining it with convex structures. Very recently, Lee
[8]
extended an Ishikawa type itera tive scheme with errors to approximate a common fixed point of two sequences of uniformly quasiLipschitzian mappings on convex cone metric spaces.
Inspired by the works mentioned above, the author recalls some generalized nonexpansive mappings on cone metric spaces and gives some sufficient and necessary conditions for a Noortype iteration to approximate a common fixed point of an infi nite family of uniformly quasisup(
fn
)Lipschitzian mappings and an infinite family of
gn
expansive mappings in convex cone metric spaces. His results generalize and improve many corresponding results in convex metric spaces
[1
,
3
,
9
,
16
,
18
,
19]
and convex cone metric spaces
[8]
.
2. PRELIMINARIES
Throughout this paper,
E
is a normed vector space with a normal solid cone
P
.
A nonempty subset
P
of
E
is called a cone if
P
is closed,
P
≠ {
θ
}, for
a
,
b
∈
and
x
,
y
∈
P
,
ax
+
by
∈
P
and
P
∩ {−
P
} = {
θ
}. We define a partial ordering
in
E
as
x
y
if
y
−
x
∈
P
.
x
<<
y
indicates that
y
−
x
∈
intP
and
x
≺
y
means that
x
y
but
x
≠
y
. A cone
P
is said to be solid if its interior
intP
is nonempty. A cone
P
is said to be normal if there exists a positive number
t
such that for
x
,
y
∈
P
, 0
x
y
implies ∥
x
∥ ≤
t
∥
y
∥. The least positive number
t
is called the normal constant of
P
.
Let
X
be a nonempty set. A mapping
d
:
X
×
X
→ (
E
,
P
) is called a cone metric if (i) for
x
,
y
∈
X
,
θ
d
(
x
,
y
) and
d
(
x
,
y
) =
θ
iff
x
=
y
, (ii) for
x
,
y
∈
X
,
d
(
x
,
y
) =
d
(
y
,
x
) and (iii) for
x
,
y
,
z
∈
X
,
d
(
x
,
y
)
d
(
x
,
z
) +
d
(
z
,
y
). A nonempty set
X
with a cone metric
d
:
X
×
X
→ (
E
,
P
) is called a cone metric space denoted by (
X
,
d
), where
P
is a solid normal cone.
A sequence {
x
_{n}
} in a cone metric space (
X
,
d
) is said to converge to
x
∈ (
X
,
d
) and denoted as
or
x
_{n}
→
x
(as
n
→ ∞) if for any
c
∈
intP
, there exists a natural number
N
such that for all
n
>
N
,
c
−
d
(
x
_{n}
,
x
) ∈
intP
. A sequence {
x
_{n}
} in (
X
,
d
) is called a Cauchy sequence if for any
c
∈
intP
, there exists a natural number
N
such that for all
n
,
m
>
N
,
c
−
d
(
x
_{n}
,
x
_{m}
) ∈
intP
. A cone metric space (
X
,
d
) is said to be complete if every Cauchy sequence converges.
Lemma 2.1
(
[4]
).
Let
{
x
_{n}
}
be a sequence in a cone metric space
(
X
,
d
)
and
P
be a normal cone with a normal constant t. Then
(i) {x_{n}} converges to x in X if and only if d(x_{n}, x) → θ (as n → ∞) in E.
(ii) {x_{n}} is a Cauchy sequence if and only if d(x_{n}, x_{m}) → θ (as n, m → ∞) in E.
We recall some generalized nonexpansive mappings and convex structures on cone metric spaces.
Definition 2.1
. Let
T
be a selfmapping on a cone metric space (
X
,
d
) and
f
:
X
→ (0, ∞) a function which is bounded above.
(i) T is fexpansive, if
In particular,
T
is said to be
nonexpansive
, if
and
T
is said to be
contractive
, if
(ii) T is asymptotically fexpansive, if there exists a sequence in X such that satisfying
d
(
T
^{n}
x
,
T
^{ n}
y
)
f
(
x
_{n}
)
d
(
x
,
y
) for
x
,
y
∈
X
,
in particular,
T
is
asymptotically nonexpansive
, if there exists a sequence
in [1, ∞) with
satisfying
d
(
T
^{n}
x
,
T
^{n}
y
)
k
_{n}
d
(
x
,
y
) for
x
,
y
∈
X
.
(iii) T is asymptotically quasifexpansive, if there exists a sequence in X such that satisfying
d
(
T
^{n}
x
,
^{p}
)
f
(
x
_{n}
)
d
(
x
,
p
) for
x
∈
X
and
p
∈
F
(
T
) a set of fixed points of
T
, in particular,
T
is
asymptotically quasinonexpansive
, if there exists a sequence
in [1, ∞) with
satisfying
d
(
T
^{n}
x
,
p
)
k
_{n}
d
(
x
,
p
) for
x
∈
X
and
p
∈
F
(
T
)
(iv) T is uniformly quasisup(f)Lipschitzian, if
for
x
∈
X
and
p
∈
F
(
T
),
in particular,
T
is
uniformly quasiLLipschitzian
, if there exists a constant
L
> 0 such that
d
(
T
^{n}
x
,
p
)
L
·
d
(
x
,
p
) for
x
∈
X
and
p
∈
F
(
T
)
.
Definition 2.2.
Let (
X
,
d
) be a cone metric space. A mapping
W
:
X
^{3}
×
I
^{3}
→
X
is a
convex structure
on
X
if
d
(
W
(
x, y, z, a_{n}, b_{n}, c_{n}), u
)
a
_{n}
·
d
(
x
,
u
) +
b
_{n}
·
d
(
y
,
u
) +
c
_{n}
·
d
(
z
,
u
) for real sequences {
a
_{n}
}, {
b
_{n}
} and {
c
_{n}
} in
I
= [0, 1] satisfying
a
_{n}
+
b
_{n}
+
c
_{n}
= 1
and
x
,
y
,
z
and
u
∈
X
. A cone metric space (
X
,
d
) with a convex structure
W
is called a
convex cone metric space
and denoted as (
X
,
d
,
W
). A nonempty subset
C
of a convex cone metric space (
X
,
d
,
W
) is said to be
convex
if
W
(
x
,
y
,
z
,
a
,
b
,
c
) ∈
C
for all
x
,
y
,
z
∈
C
and
a
,
b
,
c
∈
I
.
In this paper, we consider the following Noortype threestep iteration in convex cone metric spaces ;
For given
x
_{0}
∈
C
, define a sequence {
x
_{n}
} in
C
as follows;
where {
u
_{n}
}, {
v
_{n}
} and {
w
_{n}
} are any sequences in
X
,
C
is a nonempty convex subset of (
X
,
d
,
W
),
T
_{n}
:
C
→
C
is a uniformly quasisup(
fn
)Lipschitzian mapping,
Sn
:
C
→
C
is a
g_{n}
expansive mapping and {
α
_{n}
}, {
β
_{n}
}, {
γ
_{n}
}, {
a
_{n}
}, {
b
_{n}
}, {
c
_{n}
}, {
d
_{n}
}, {
e
_{n}
} and {
l
_{n}
} are sequences in
I
such that
a
_{n}
+
b
_{n}
+
c
_{n}
= 1,
α
_{n}
+
β
_{n}
+ ≥
n
= 1 and
d
_{n}
+
e
_{n}
+
l
_{n}
= 1
.
Remark 2.1.
(i) We have the following iteration by putting
S
_{n}
x
_{n}
=
x
_{n}
in
of (2.1)
(ii) By putting
S
_{n}
x
_{n}
=
x
_{n}
, we obtain the following iteration from (2.2) ;
which generalizes many kinds of Ishikawatype iterations.
(iii) The following Ishikawatype iteration is a special case of (2.1);
which was considered in
[1]
.
3. Main Results
Throughout this paper,
f
_{n}
,
g
_{n}
:
X
→ (0, ∞) are functions, which are bounded above, and whose least upper bounds are
M
_{n}
and
N
_{n}
, respectively
.
Lemma 3.1
(
[5
,
9]
).
Let
{
a
_{n}
}, {
b
_{n}
} and {
c
_{n}
}
be nonnegative real sequences satisfying the following conditions
;
(i) a_{n}+1 ≤ (1 + b_{n})a_{n} + c_{n} ,
(ii) and are finite, then the following hold;
(1) exists.
(2) If, then
The following lemma considered in convex cone metric space (
X
,
d
,
W
) is a result on the properties of an iteration sequence {
x
_{n}
} defined as (2.1) for an infinite family {
T
_{n}
} of uniformly quasisup(
f
_{n}
)Lipschitzian mappings and an infinite family {
S
_{n}
} of
g
_{n}
expansive mappings.
Lemma 3.2
.
Let d : X × X → (E, P) be a cone metric, where P is a solid normal cone with the normal constant t
.
Let C be a nonempty convex subset of a convex cone metric space
(
X
,
d
,
W
).
Let
T
_{n}
:
C
→
C
be uniformly quasisup
(
f
_{n}
)
Lipschitzian mappings and
S
_{n}
:
C
→
C
be
g
_{n}

expansive mappings
with a nonempty set
Let
{
u
_{n}
}, {
v
_{n}
}
and
{
w
_{n}
}
be bounded sequences in C and
{
α
_{n}
}, {
β
_{n}
}, {
γ
_{n}
}, {
a
_{n}
}, {
b
_{n}
}, {
c
_{n}
}, {
d
_{n}
}, {
e
_{n}
} and {
l
_{n}
}
be sequences in I such that
α
_{n}
+
β
_{n}
+
γ
_{n}
=
a
_{n}
+
b
_{n}
+
c
_{n}
=
d
_{n}
+
e
_{n}
+
l
_{n}
= 1
.
Assume that the following conditions hold;
(i) f_{n} > 1 and g_{n} > 1 ,
(ii) and are finite,
(iii) L = (1 + M + M^{2} ) · L_{0}, where
is finite
.
Let
{
x
_{n}
}
be the iteration defined as
(2.1),
then the following holds
.
for p
∈
D
,
where
η
n
= α
_{n}
+ β
_{n}
and
δ
_{n}
= β
_{n}
+ γ
_{n}
.
(2)
there exists a constant
K
> 0
such that
Proof.
For any
p
∈
D
, we have
and
From (3.1) to (3.3), by the fact that 1 +
x
≤
e
^{x}
for
x
≥ 0, we have
Moreover, we obtain the following result (2) from the result (1).
where
Remark 3.1
. If
f
_{n}
≥ 1 and
, then
Hence from (3.4)
and from (3.5)
where
Remark 3.2
. If
f
_{n}
≤ 1 and
, then
Hence from (3.4)
and from (3.5)
where
Remark 3.3
. If
f
_{n}
≤ 1 and
g
_{n}
≤ 1, then from (3.4) and (3.5), respectively,
and
Remark 3.4.
Lemma 2 in
[1]
dealt with the Ishikawatype iteration (2.4) for an infi nite family of uniformly quasi
L
_{n}
Lipschitzian mappings
T
_{n}
with
< ∞ and an infinite family of nonexpansive mappings
S
_{n}
, i.e.,
L
≥ 1 and
g
_{n}
= 1
. Hence Lemma 2 in
[1]
is a corollary of Lemma 3.1.
Now we introduce our main result for the iteration defined as (2.1) with an infinite family {
T
_{n}
} of uniformly quasisup(
f
_{n}
)Lipschitzian mappings and an infinite family {
S
_{n}
} of
g
_{n}
expansive mappings in convex cone metric spaces (
X, d, W
).
Theorem 3.1.
Let C be a nonempty closed convex subset of a convex cone metric space
(
X, d, W
)
with a solid normal cone P with a normal constant t
.
Let
T
_{n}
:
C
→
C
be uniformly quasisup
(
f
_{n}
)
Lipschitzian mappings and
S
_{n}
:
C
→
C
be
g
_{n}

expansive mappings with a nonempty set
Let
{
u
_{n}
}, {
v
_{n}
}
and
{
w
_{n}
}
be bounded sequences in C and
{
α
_{n}
}, {
β
_{n}
}, {
γ
_{n}
}, {
a
_{n}
}, {
b
_{n}
}, {
c
_{n}
}, {
d
_{n}
}, {
e
_{n}
} and {
l
_{n}
}
be real sequences in I such that
α
_{n}
+
β
_{n}
+
γ
_{n}
=
a
_{n}
+
b
_{n}
+
c
_{n}
=
d
_{n}
+
e
_{n}
+
l
_{n}
= 1
.
Assume that the following conditions hold;
(i) f_{n} > 1 and g_{n} > 1 ,
(ii) and are finite,
(iii) L = (1 + M + M^{2} ) · L_{0}, where
is finite,
(iv)
and
)
are finite.
Let
{
x
_{n}
}
be the iteration defined as
(2.1).
We have the following equivalent result;
(1) {x_{n}} converges to a common fixed point p ∈ D.
(2)
Proof
. We only consider the case of
f
_{n}
(
x
) > 1 and
g
_{n}
(
x
) > 1 for
x
∈
X
. Obviously the statement (1) implies the statement (2). Now we show that the statement (2) implies the statement (1). From (3.4), we have
where
η
_{n}
=
α
_{n}
+
β
_{n}
and
δ
_{n}
=
β
_{n}
+
γ
_{n}
for
.
Thus the normality of
P
implies that
where
t
is the normal constant of
P
.
By the condition (iv) that
and
are finite,
exists from Lemma 3.1. Since
by the hypothesis, we have
Now, we show that the sequence {
x
_{n}
} is convergent. For any given
ε
> 0, take a positive integer
N
_{0}
such that
where
for
n
≥
N
_{0}
.
By using a property of infimum for ∥
d
(
x
_{n}
,
D
)∥, we take a positive integer
N
_{1}
≥
N
_{0}
and
p
_{0}
∈
D
such that
Hence for any positive integer
n
>
N
_{1}
, from (3.5) by the normality of
P
we have
Hence
, which shows that
, i.e.,
by Lemma 2.1(i). Since
C
is closed,
.
Now, we show that the set
D
is closed. In fact, let {
p
_{n}
} a sequence in
D
converging to
p
in
C
. Then
Letting
n
→ ∞ in (3.6), we have
p
∈
F
(
T
_{i}
) for all
.
Similarly,
Letting also
n
→ ∞ in (3.7), we have
p
∈
F
(
S
_{i}
) for all
.
Hence
p
∈
D
, which implies that
D
is closed.
Moreover, we have
p
^{∗}
∈
D
. In fact, since
by Lemma 2.1 (i), we have
p
^{∗}
∈
D
, which says that {
x
_{n}
} converges to a common fixed point in
D
.
Remark 3.5
. In Theorem 3.1, the full space need not to be complete.
Remark 3.6.
We obtain the same results for iterations defined as (2.2) and (2.3) with an infinite family {
T
_{n}
} of uniformly quasisup(
f
_{n}
)Lipschitzian mappings and an infinite family {
S
_{n}
} of
g_{n}
expansive mappings in convex metric spaces (
X, d, W
).
Remark 3.7
. The main result of
[1]
considered the Ishikawatype iteration (2.4) for an infinite family of uniformly quasi
L
_{n}
Lipschitzian mappings
T
_{n}
with
L
=
and an infinite family of nonexpansive mappings
S
_{n}
, i.e.,
L
≥ 1 and
g
_{n}
= 1
in convex metric spaces. Hence Theorem 1 in
[1]
is a corollary of Theorem 3.1.
Remark 3.8.
By putting
f
_{n}
(
x
) = 1 and
g
_{n}
(
x
) = 1 for
x
∈
X
in Theorem 3.1, we have the corresponding results in convex metric spaces and convex cone metric spaces.
Remark 3.9.
We obtain the same results as Theorem 3.1 by replacing the uniform quasisup(
f
)Lipschitzian of
T
_{n}
with the asymptotical quasi
f
expansiveness of
T
_{n}
.
Remark 3.10.
Theorem 3.1 generalizes, improves and unifies the corresponding results in convex metric spaces
[1
,
3
,
9
,
16
,
18
,
19]
.
View Fulltext
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