# Invariant Means and Reversible Semigroup of Relatively Nonexpansive Mappings in Banach Spaces

Authors: Kyung Soo Kim
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

The purpose of this paper is to study modified Halpern type and Ishikawa type iteration for a semigroup of relatively nonexpansive mappingsI
=
{
T
(
s
)
:
s
∈
S
} on a nonempty closed convex subset C of a Banach space with respect to a sequence of asymptotically left invariant means {
μ
n
} defined on an appropriate invariant subspace of l
∞
(
S
), where S is a semigroup. We prove that, given some mild conditions, we can generate iterative sequences which converge strongly to a common element of the set of fixed points F
(
I
), where F
(
I
)
=
⋂
{
F
(
T
(
s
)
)
:
s
∈
S
}.


---

## Body

## 1. Introduction

LetE be a real Banach space with the topological dual E
* and let C be a closed and convex subset of E. A mapping T of C into itself is callednonexpansive if ∥
T
x
-
T
y
∥
≤
∥
x
-
y
∥ for each x
,
y
∈
C.Three classical iteration processes are often used to approximate a fixed point of a nonexpansive mapping. The first one is introduced by Halpern [1] and is defined as follows:
(1)
x
0
=
u
∈
C
,
chosen
arbitrarily
,
x
n
+
1
=
α
n
u
+
(
1
-
α
n
)
T
x
n
,
∀
n
≥
1
,
where {
α
n
} is a sequence in [
0,1
]. He pointed out that the conditions lim
⁡
n
→
∞
α
n
=
0 and ∑
n
=
1
∞
‍
α
n
=
∞ are necessary in the sense that if the iteration (1) converges to a fixed point of T, then these conditions must be satisfied. The second iteration process is known as Mann’s iteration process [2] which is defined as follows:
(2)
x
n
+
1
=
α
n
x
n
+
(
1
-
α
n
)
T
x
n
,
∀
n
≥
1
,
where the initial x
1 is taken in C arbitrary and the sequence {
α
n
} is in [
0,1
].The third iteration process is referred to as Ishikawa’s iteration process [3] which is defined as follows:
(3)
y
n
=
β
n
x
n
+
(
1
-
β
n
)
T
x
n
,
x
n
+
1
=
α
n
x
n
+
(
1
-
α
n
)
T
y
n
,
∀
n
≥
1
,
where the initial x
1 is taken in C arbitrary and {
α
n
} and {
β
n
} are sequences in [
0,1
].In 2007, Lau et al. [4] proposed the following modification of Halpern’s iteration (1) for amenable semigroups of nonexpansive mappings in a Banach space.Theorem 1.
LetS be a left reversible semigroup and let I
=
{
T
(
s
)
:
s
∈
S
} be a representation of S as nonexpansive mappings from a compact convex subset C of a strictly convex and smooth Banach space E into C, let X be an amenable and I-stable subspace of l
∞
(
S
), and let {
μ
n
} be a strongly left regular sequence of means on X. Let {
α
n
} be a sequence in [
0,1
] such that lim
⁡
n
→
∞
α
n
=
0 and ∑
n
=
1
∞
‍
α
n
=
∞. Let x
1
=
x
∈
C and let {
x
n
} be the sequence defined by(4)
x
n
+
1
=
α
n
x
+
(
1
-
α
n
)
T
(
μ
n
)
x
n
,
n
≥
2
.
Then {
x
n
} converges strongly to P
x, where P denotes the unique sunny nonexpansive retraction of C onto F
(
I
).LetC be a closed and convex subset of E and let T be a mapping from C into itself. We denote by F
(
T
) the set of fixed points of T. Point p in C is said to be anasymptotic fixed point of T [5] if C contains a sequence {
x
n
} which converges weakly to p such that the strong lim
⁡
n
→
∞
(
T
x
n
-
x
n
)
=
0. The set of asymptotic fixed points of T will be denoted by F
^
(
T
). A mapping T from C into itself is calledrelatively nonexpansive [6–8], if F
^
(
T
)
=
F
(
T
) and ϕ
(
p
,
T
x
)
≤
ϕ
(
p
,
x
) for all x
∈
C and p
∈
F
(
T
). The asymptotic behavior of relatively nonexpansive mappings was studied in [6, 7, 9].Recently, Kim [10] proved a strong convergence theorem for relatively nonexpansive mappings in a Banach space by using the shrinking method.Theorem 2.
LetS be a left reversible semigroup and let I
=
{
T
(
s
)
:
s
∈
S
} be a representation of S as relatively nonexpansive mappings from a nonempty, closed, and convex subset C of a uniformly convex and uniformly smooth Banach space E into C with F
(
I
)
≠
∅. Let X be a subspace of l
∞
(
S
) and let {
μ
n
} be a asymptotically left invariant sequence of means on X. Let {
α
n
} be a sequence in [
0,1
] such that 0
<
α
n
<
1 and lim
⁡
n
→
∞
α
n
=
0. Let {
x
n
} be a sequence generated by the following algorithm:
(5)
x
0
∈
C
,
c
h
o
s
e
n
a
r
b
i
t
r
a
r
i
l
y
,
C
1
=
C
,
x
1
=
Π
C
1
x
0
,
y
n
=
J
-
1
(
α
n
J
x
1
+
(
1
-
α
n
)
J
T
μ
n
x
n
)
,
C
n
+
1
=
{
z
∈
C
n
:
ϕ
(
z
,
y
n
)
≤
α
n
ϕ
(
z
,
x
1
)
+
(
1
-
α
n
)
ϕ
(
z
,
x
n
)
}
,
x
n
+
1
=
Π
C
n
+
1
x
1
,
∀
n
≥
1
.
Then {
x
n
} converges strongly to Π
F
(
I
)
x
1, where Π
F
(
I
) is the generalized projection from C onto F
(
I
).LetS be a semigroup. The purpose of this paper is to study modified Halpern type and Ishikawa type iterations for a semigroup of relatively nonexpansive mappings I
=
{
T
(
s
)
:
s
∈
S
} on a nonempty closed convex subset C of a Banach space with respect to a sequence of asymptotically left invariant means {
μ
n
} defined on an appropriate invariant subspace of l
∞
(
S
). We prove that, given some mild conditions, we can generate iterative sequences which converge strongly to a common element of the set of fixed points F
(
I
), where F
(
I
)
=
⋂
‍
{
F
(
T
(
s
)
)
:
s
∈
S
}.


## 2. Preliminaries

A real Banach spaceE is said to bestrictly convex if ∥
(
x
+
y
)
/
2
∥
<
1 for all x
,
y
∈
E with ∥
x
∥
=
∥
y
∥
=
1 and x
≠
y. It is said to beuniformly convex if lim
⁡
n
→
∞
∥
x
n
-
y
n
∥
=
0 for any two sequences {
x
n
} and {
y
n
} in E such that ∥
x
n
∥
=
∥
y
n
∥
=
1 and lim
⁡
n
→
∞
∥
(
x
n
+
y
n
)
/
2
∥
=
1. Let U
=
{
x
∈
E
:
∥
x
∥
=
1
} be the unit sphere of E. Then the Banach space E is said to besmooth if
(6)
lim
⁡
t
→
0
∥
x
+
t
y
∥
-
∥
x
∥
t
exists for each x
,
y
∈
U. It is said to beuniformly smooth if the limit is attained uniformly for x
,
y
∈
E.LetE be a real Banach space with norm ∥
·
∥ and let E
* be the dual space of E. Denote by 〈
·
,
·
〉 the duality product. We denote by J the normalized duality mapping from E to 2
E
* defined by
(7)
J
x
=
{
f
*
∈
E
*
:
〈
x
,
f
*
〉
=
∥
x
∥
2
=
∥
f
*
∥
2
}
,
for x
∈
E. A Banach space E is said to have the Kadec-Klee property if a sequence {
x
n
} of E satisfies that x
n
⇀
x and ∥
x
n
∥
→
∥
x
∥ and then x
n
→
x, where ⇀ and → denote the weak convergence and the strong convergence, respectively.We know the following:(1)
the duality mappingJ is monotone, that is, 〈
x
-
y
,
x
*
-
y
*
〉
≥
0 whenever x
*
∈
J
x and y
*
∈
J
y;
(2)
ifE is strictly convex, then J is one-to-one; that is, if J
x
∩
J
y is nonempty, then x
=
y;
(3)
ifE is strictly convex, then J is strictly monotone; that is,  x
=
y whenever 〈
x
-
y
,
x
*
-
y
*
〉
=
0, x
*
∈
J
x and y
*
∈
J
y;
(4)
ifE is uniformly convex, then E has the Kadec-Klee property;
(5)
ifE is uniformly convex, then E is reflexive and strictly convex;
(6)
ifE is smooth, then J is single-valued and norm-to-weak
* continuous;
(7)
ifE is uniformly smooth, then J is uniformly norm-to-norm continuous on bounded subsets of E;
(8)
ifE is reflexive, then J is onto;
(9)
ifE is smooth and reflexive, then J is norm-to-weak continuous; that is, J
x
n
⇀
J
x whenever x
n
→
x;
(10)
ifE is smooth, strictly convex, and reflexive, then J is single-valued, one-to-one and onto; in this case, the inverse mapping J
-
1 coincides with the duality mapping on E;
(11)
ifE
* is strictly convex, then J is single-valued;
(12)
the norm ofE
* is Fréchet differentiable if and only if E is strictly convex and reflexive Banach space which has the Kadec-Klee property.For more details, see [11].As well known, ifC is a nonempty, closed, and convex subset of a Hilbert space H and P
C
:
H
→
C is the metric projection of H onto C, then P
C is nonexpansive (see, the reference therein). This fact actually characterizes Hilbert spaces. Consequently, it is not true to more general Banach spaces. In this connection, Alber [12] introduced a generalized projection operator Π
C in a Banach space E which is an analogue of the metric projection in Hilbert spaces. Consider the function defined by
(8)
ϕ
(
x
,
y
)
=
∥
x
∥
2
-
2
〈
x
,
J
y
〉
+
∥
y
∥
2
,
for x
,
y
∈
E. Observe that, in a Hilbert space H, (8) reduces to
(9)
ϕ
(
x
,
y
)
=
∥
x
-
y
∥
2
,
for x
,
y
∈
H. The generalized projection Π
C
:
E
→
C is a mapping that assigns an arbitrary point x
∈
E to the minimum point of the functional ϕ
(
x
,
y
); that is, Π
C
x
=
x
¯, where x
¯ is the solution to the minimization problem:
(10)
ϕ
(
x
¯
,
x
)
=
inf
⁡
y
∈
C
ϕ
(
y
,
x
)
.
The existence and uniqueness of the operator Π
C follow from the properties of the functional ϕ
(
x
,
y
) and strict monotonicity of the mapping J (see, e.g., [12, 13]). In a Hilbert space, Π
C
=
P
C. It is obvious from the definition of the function ϕ that (ϕ
1)
(
∥
x
∥
-
∥
y
∥
)
2
≤
ϕ
(
x
,
y
)
≤
(
∥
x
∥
+
∥
y
∥
)
2 for all x
,
y
∈
E,
(ϕ
2)
ϕ
(
x
,
y
)
=
ϕ
(
x
,
z
)
+
ϕ
(
z
,
y
)
+
2
〈
x
-
z
,
J
z
-
J
y
〉 for all x
,
y
,
z
∈
E,
(ϕ
3)
ϕ
(
x
,
y
)
=
〈
x
,
J
x
-
J
y
〉
+
〈
y
-
x
,
J
y
〉
≤
∥
x
∥
∥
J
x
-
J
y
∥
+
∥
y
-
x
∥
∥
y
∥ for all x
,
y
∈
E,
(ϕ
4)
ifE is a reflexive, strictly convex, and smooth Banach space, then, for all x
,
y
∈
E,
(11)
ϕ
(
x
,
y
)
=
0
iff
x
=
y
.
For more details see [14].LetS be a semigroup. We denote by l
∞
(
S
) the Banach space of all bounded real-valued functionals on S with supremum norm. For each s
∈
S, we define the left and right translation operators l
(
s
) and r
(
s
) on l
∞
(
S
) by
(12)
(
l
(
s
)
f
)
(
t
)
=
f
(
s
t
)
,
(
r
(
s
)
f
)
(
t
)
=
f
(
t
s
)
,
for each t
∈
S and f
∈
l
∞
(
S
), respectively. Let X be a subspace of l
∞
(
S
) containing 1. An element μ in the dual space X
* of X is said to be amean on X if ∥
μ
∥
=
μ
(
1
)
=
1. For s
∈
S, we can define a point evaluation δ
s by δ
s
(
f
)
=
f
(
s
) for each f
∈
X. It is well known that μ is mean on X if and only if
(13)
inf
⁡
s
∈
S
f
(
s
)
≤
μ
(
f
)
≤
sup
⁡
s
∈
S
f
(
s
)
,
for each f
∈
X.LetX be a translation invariant subspace of l
∞
(
S
) (i.e., l
(
s
)
X
⊂
X and r
(
s
)
X
⊂
X for each s
∈
S) containing 1. Then a mean μ on X is said to beleft invariant (resp.,right invariant) if
(14)
μ
(
l
(
s
)
f
)
=
μ
(
f
)
,
(
resp
.
,
μ
(
r
(
s
)
f
)
=
μ
(
f
)
)
for each s
∈
S and f
∈
X. A mean μ on X is said to beinvariant if μ is both left and right invariant [15–19]. X is said to beleft (resp.,right)amenable if X has a left (resp., right) invariant mean. X is amenable if X is left and right amenable. We call a semigroup S
amenable if X is amenable. Further, amenable semigroups include all commutative semigroups and solvable groups. However, the free group or semigroup of two generators is not left or right amenable (see [20–22]).A net{
μ
α
} of means on X is said to beasymptotically left (resp.,right)invariant if
(15)
lim
⁡
α
(
μ
α
(
l
(
s
)
f
)
-
μ
α
(
f
)
)
=
0
,
(
resp
.
,
lim
⁡
α
(
μ
α
(
r
(
s
)
f
)
-
μ
α
(
f
)
)
=
0
)
,
for each f
∈
X and s
∈
S, and it is said to beleft (resp.,right)strongly asymptotically invariant (orstrong regular) if(16)
lim
⁡
α
∥
l
*
(
s
)
μ
α
-
μ
α
∥
=
0
,
(
resp
.
,
lim
⁡
α
∥
r
*
(
s
)
μ
α
-
μ
α
∥
=
0
)
,
for each s
∈
S, where l
*
(
s
) and r
*
(
s
) are the adjoint operators of l
(
s
) and r
(
s
), respectively. Such nets were first studied by Day in [20] where they were called w
e
a
k
*
invariant andnorm invariant, respectively.It is easy to see that if a semigroupS is left (resp., right) amenable, then the semigroup S
′
=
S
∪
{
e
}, where e
s
′
=
s
′
e
=
s
′ for all s
′
∈
S, is also left (resp., right) amenable and converse.From now onS denotes a semigroup with an identity e. S is calledleft reversible if any two right ideals of S have nonvoid intersection; that is, a
S
∩
b
S
≠
∅ for a
,
b
∈
S. In this case, (
S
,
⪯
) is a directed system when the binary relation “⪯” on S is defined by a
⪯
b if and only if {
a
}
∪
a
S
⊇
{
b
}
∪
b
S for a
,
b
∈
S. It is easy to see that t
⪯
t
s for all t
,
s
∈
S. Further, if t
⪯
s then p
t
⪯
p
s for all p
∈
S. The class of left reversible semigroup includes all groups and commutative semigroups. If a semigroup S is left amenable, then S is left reversible. But the converse is not true [23–28].LetS be a semigroup and let C be a closed and convex subset of E. Let F
(
T
) denote the fixed point set of T. Then I
=
{
T
(
s
)
:
s
∈
S
} is called arepresentation of
S
as relatively nonexpansive mappings on
C if T
(
s
) is relatively nonexpansive with T
(
e
)
=
I and T
(
s
t
)
=
T
(
s
)
T
(
t
) for each s
,
t
∈
S. We denote by F
(
I
) the set of common fixed points of {
T
(
s
)
:
s
∈
S
}; that is,
(17)
F
(
I
)
=
⋂
s
∈
S
F
(
T
(
s
)
)
=
⋂
s
∈
S
{
x
∈
C
:
T
(
s
)
x
=
x
}
.We know that ifμ is a mean on X and if for each x
*
∈
E
* the function s
↦
〈
T
(
s
)
x
,
x
*
〉 is contained in X and C is weakly compact, then there exists a unique point x
0 of E such that μ
〈
T
(
·
)
x
,
x
*
〉
=
〈
x
0
,
x
*
〉 for each x
*
∈
E
*. We denote such a point x
0 by T
μ
x. Note that T
μ
x is contained in the closure of the convex hull of {
T
(
s
)
x
:
s
∈
S
} for each x
∈
C. Note that T
μ
z
=
z for each z
∈
F
(
I
); see [29–31].


## 3. Lemmas

We need the following lemmas for the proof of our main results.Lemma 3 (see [9]).
LetE be a strictly convex and smooth Banach space, let C be a closed convex subset of E, and let T be a relatively nonexpansive mapping from C into itself. Then F
(
T
) is closed and convex.Lemma 4 (see [12, 32]).
LetE be a reflexive, strictly convex, and smooth Banach space and let C be a nonempty, closed, and convex subset of E and x
∈
E. Then
(18)
ϕ
(
y
,
Π
C
x
)
+
ϕ
(
Π
C
x
,
x
)
≤
ϕ
(
y
,
x
)
,
for all y
∈
C.Lemma 5 (see [32]).
LetE be a uniformly convex and smooth Banach space and let {
x
n
}, {
y
n
} be two sequences of E. If lim
⁡
n
→
∞
ϕ
(
x
n
,
y
n
)
=
0 and either {
x
n
} or {
y
n
} is bounded, then lim
⁡
n
→
∞
∥
x
n
-
y
n
∥
=
0.Lemma 6 (see [4, 33]).
Letμ be a left invariant mean on X. Then F
(
I
)
=
F
(
T
μ
)
∩
C
a, where C
a denotes the set of almost periodic elements in C; that is, all x
∈
C such that {
T
(
s
)
x
:
s
∈
S
} is relatively compact in the norm topology of E.Lemma 7 (cf. [4, 10]).
Let{
μ
n
} be an asymptotically left invariant sequence of means on X. If z
∈
C
a and liminf
⁡
n
→
∞
∥
T
μ
n
z
-
z
∥
=
0, then z is a common fixed point of I.


## 4. Strong Convergence Theorems

In this section, we will establish two strong convergence theorems of various iterative sequences for finding common fixed point of relatively nonexpansive mappings in a uniformly convex and uniformly smooth Banach spaces (cf. [34–36]).We begin with a strong convergence theorem of modified Halpern’s type.Theorem 8.
LetS be a left reversible semigroup and let I
=
{
T
(
s
)
:
s
∈
S
} be a representation of S as relatively nonexpansive mappings from a nonempty, closed, and convex subset C of a uniformly convex and uniformly smooth Banach space E into itself. Let X be a subspace of l
∞
(
S
) and let {
μ
n
} be an asymptotically left invariant sequence of means on X. Let {
α
n
} be a sequence in (
0,1
) such that lim
⁡
n
→
∞
α
n
=
0. Let {
x
n
} be a sequence generated by the following algorithm:
(19)
x
0
∈
C
,
c
h
o
s
e
n
a
r
b
i
t
r
a
r
i
l
y
,
x
n
+
1
=
Π
C
J
-
1
(
α
n
J
x
0
+
(
1
-
α
n
)
J
T
μ
n
x
n
)
,
∀
n
≥
0
.
If the interior ofF
(
I
) is nonempty, then {
x
n
} converges strongly to some common fixed point F
(
I
).Proof.
We show first that the sequence{
x
n
} converges strongly in C.
From Lemma3, we know F
(
T
) is closed and convex. So, we can define the generalized projection Π
C onto F
(
I
). Most of all, we have
(20)
∥
T
μ
n
x
n
∥
=
sup
⁡
{
|
〈
T
μ
n
x
n
,
x
*
〉
|
:
x
*
∈
E
*
,
∥
x
*
∥
=
1
}
=
sup
⁡
{
|
(
μ
n
)
s
〈
T
(
s
)
x
n
,
x
*
〉
|
:
x
*
∈
E
*
,
∥
x
*
∥
=
1
}
≤
sup
⁡
{
(
μ
n
)
s
(
∥
T
(
s
)
x
n
∥
∥
x
*
∥
)
:
x
*
∈
E
*
,
∥
x
*
∥
=
1
}
=
(
μ
n
)
s
∥
T
(
s
)
x
n
∥
.
Then, from the definition of relatively nonexpansive, we have(21)
ϕ
(
u
,
T
μ
n
x
n
)
=
∥
u
∥
2
-
2
〈
u
,
J
T
μ
n
x
n
〉
+
∥
T
μ
n
x
n
∥
2
=
∥
u
∥
2
-
2
(
μ
n
)
s
〈
u
,
J
T
(
s
)
x
n
〉
+
(
μ
n
)
s
∥
T
(
s
)
x
n
∥
2
=
(
μ
n
)
s
ϕ
(
u
,
T
(
s
)
x
n
)
≤
(
μ
n
)
s
ϕ
(
u
,
x
n
)
=
ϕ
(
u
,
x
n
)
,
for all u
∈
F
(
I
). From the convexity of ∥
·
∥
2 and (21), we get
(22)
ϕ
(
u
,
x
n
+
1
)
=
ϕ
(
u
,
Π
C
J
-
1
(
α
n
J
x
0
+
(
1
-
α
n
)
J
T
μ
n
x
n
)
)
≤
ϕ
(
u
,
J
-
1
(
α
n
J
x
0
+
(
1
-
α
n
)
J
T
μ
n
x
n
)
)
=
∥
u
∥
2
-
2
〈
u
,
α
n
J
x
0
+
(
1
-
α
n
)
J
T
μ
n
x
n
〉
+
∥
α
n
J
x
0
+
(
1
-
α
n
)
J
T
μ
n
x
n
∥
2
≤
∥
u
∥
2
-
2
α
n
〈
u
,
J
x
0
〉
-
2
(
1
-
α
n
)
〈
u
,
J
T
μ
n
x
n
〉
+
α
n
∥
x
0
∥
2
+
(
1
-
α
n
)
∥
T
μ
n
x
n
∥
2
=
α
n
ϕ
(
u
,
x
0
)
+
(
1
-
α
n
)
ϕ
(
u
,
T
μ
n
x
n
)
≤
α
n
ϕ
(
u
,
x
0
)
+
(
1
-
α
n
)
ϕ
(
u
,
x
n
)
.
So, we have
(23)
(
1
-
α
n
)
{
ϕ
(
u
,
x
n
+
1
)
-
ϕ
(
u
,
x
n
)
}
≤
α
n
{
ϕ
(
u
,
x
0
)
-
ϕ
(
u
,
x
n
+
1
)
}
≤
α
n
ϕ
(
u
,
x
0
)
.
Since lim
⁡
n
→
∞
α
n
=
0, we obtain
(24)
lim
⁡
n
→
∞
{
ϕ
(
u
,
x
n
+
1
)
-
ϕ
(
u
,
x
n
)
}
≤
0
.
Therefore {
ϕ
(
u
,
x
n
)
} is bounded and lim
⁡
n
→
∞
ϕ
(
u
,
x
n
) exists. Then {
x
n
} is also bounded. This implies that {
T
μ
n
x
n
} is bounded. Since the interior of F
(
I
) is nonempty, there exist p
∈
F
(
I
) and r
>
0 such that
(25)
p
+
r
q
∈
F
(
I
)
,
whenever ∥
q
∥
≤
1. By (
ϕ
2
), we have
(26)
ϕ
(
u
,
x
n
)
=
ϕ
(
u
,
x
n
+
1
)
+
ϕ
(
x
n
+
1
,
x
n
)
+
2
〈
u
-
x
n
+
1
,
J
x
n
+
1
-
J
x
n
〉
,
for any u
∈
F
(
I
). This implies
(27)
〈
x
n
+
1
-
u
,
J
x
n
-
J
x
n
+
1
〉
+
1
2
ϕ
(
x
n
+
1
,
x
n
)
=
1
2
(
ϕ
(
u
,
x
n
)
-
ϕ
(
u
,
x
n
+
1
)
)
.
Also, we have
(28)
〈
x
n
+
1
-
p
,
J
x
n
-
J
x
n
+
1
〉
=
〈
x
n
+
1
-
(
p
+
r
q
)
+
r
q
,
J
x
n
-
J
x
n
+
1
〉
=
〈
x
n
+
1
-
(
p
+
r
q
)
,
J
x
n
-
J
x
n
+
1
〉
+
r
〈
q
,
J
x
n
-
J
x
n
+
1
〉
.
On the other hand, by (24) and (25), we have that
(29)
ϕ
(
p
+
r
q
,
x
n
+
1
)
≤
ϕ
(
p
+
r
q
,
x
n
)
.
From (27), we get
(30)
0
≤
1
2
(
ϕ
(
p
+
r
q
,
x
n
)
-
ϕ
(
p
+
r
q
,
x
n
+
1
)
)
=
〈
x
n
+
1
-
(
p
+
r
q
)
,
J
x
n
-
J
x
n
+
1
〉
+
1
2
ϕ
(
x
n
+
1
,
x
n
)
=
〈
x
n
+
1
-
p
,
J
x
n
-
J
x
n
+
1
〉
-
r
〈
q
,
J
x
n
-
J
x
n
+
1
〉
+
1
2
ϕ
(
x
n
+
1
,
x
n
)
.
Then, by (27), we have
(31)
r
〈
q
,
J
x
n
-
J
x
n
+
1
〉
≤
〈
x
n
+
1
-
p
,
J
x
n
-
J
x
n
+
1
〉
+
1
2
ϕ
(
x
n
+
1
,
x
n
)
=
1
2
(
ϕ
(
p
,
x
n
)
-
ϕ
(
p
,
x
n
+
1
)
)
,
for p
∈
F
(
I
). Hence
(32)
〈
q
,
J
x
n
-
J
x
n
+
1
〉
≤
1
2
r
(
ϕ
(
p
,
x
n
)
-
ϕ
(
p
,
x
n
+
1
)
)
.
Sinceq with ∥
q
∥
≤
1 is arbitrary, by (24), we have
(33)
∥
J
x
n
-
J
x
n
+
1
∥
≤
1
2
r
(
ϕ
(
p
,
x
n
)
-
ϕ
(
p
,
x
n
+
1
)
)
.
So, we have(34)
∥
J
x
n
+
m
-
J
x
n
∥
=
∥
J
x
n
+
m
-
J
x
n
+
m
-
1
+
J
x
n
+
m
-
1
-
⋯
-
J
x
n
+
1
+
J
x
n
+
1
-
J
x
n
∥
≤
∑
i
=
n
n
+
m
-
1
‍
∥
J
x
i
-
J
x
i
+
1
∥
≤
1
2
r
∑
i
=
n
n
+
m
-
1
‍
(
ϕ
(
p
,
x
i
)
-
ϕ
(
p
,
x
i
+
1
)
)
=
1
2
r
(
ϕ
(
p
,
x
n
)
-
ϕ
(
p
,
x
n
+
1
)
)
.
We know that {
ϕ
(
p
,
x
n
)
} converges. Hence, {
J
x
n
} is a Cauchy sequence. Since E
* is complete, {
J
x
n
} converges strongly to some point in E
*. Since E is uniformly convex, E
* has a Fréchet differentiable norm. Then J
-
1 is continuous on E
*. Hence {
x
n
} converges strongly to some point v in C.
Now, we show thatv
∈
F
(
I
), where v
=
lim
⁡
n
→
∞
Π
F
(
I
)
x
n.
By (33) and the convergence of {
ϕ
(
p
,
x
n
)
}, we have
(35)
lim
⁡
n
→
∞
∥
J
x
n
-
J
x
n
+
1
∥
=
0
.
Since J
-
1 is uniformly norm-to-norm continuous on bounded sets, it follows that
(36)
lim
⁡
n
→
∞
∥
x
n
-
x
n
+
1
∥
=
0
.
Let z
n
=
J
-
1
(
α
n
J
x
0
+
(
1
-
α
n
)
J
T
μ
n
x
n
). Then, we have
(37)
∥
J
z
n
-
J
T
μ
n
x
n
∥
=
∥
α
n
J
x
0
+
(
1
-
α
n
)
J
T
μ
n
x
n
-
J
T
μ
n
x
n
∥
=
α
n
∥
J
x
0
-
J
T
μ
n
x
n
∥
.
Since lim
⁡
n
→
∞
α
n
=
0, we have
(38)
lim
⁡
n
→
∞
∥
J
z
n
-
J
T
μ
n
x
n
∥
=
0
.
Since J
-
1 is uniformly norm-to-norm continuous on bounded sets, we get
(39)
lim
⁡
n
→
∞
∥
z
n
-
T
μ
n
x
n
∥
=
0
.
From x
n
+
1
=
Π
C
z
n and Lemma 4, we have
(40)
ϕ
(
T
μ
n
x
n
,
x
n
+
1
)
+
ϕ
(
x
n
+
1
,
z
n
)
=
ϕ
(
T
μ
n
x
n
,
Π
C
z
n
)
+
ϕ
(
Π
C
z
n
,
z
n
)
≤
ϕ
(
T
μ
n
x
n
,
z
n
)
.
Since
(41)
ϕ
(
T
μ
n
x
n
,
z
n
)
=
ϕ
(
T
μ
n
x
n
,
J
-
1
(
α
n
J
x
0
+
(
1
-
α
n
)
J
T
μ
n
x
n
)
)
=
∥
T
μ
n
x
n
∥
2
-
2
〈
T
μ
n
x
n
,
α
n
J
x
0
+
(
1
-
α
n
)
J
T
μ
n
x
n
〉
+
∥
α
n
J
x
0
+
(
1
-
α
n
)
J
T
μ
n
x
n
∥
2
≤
∥
T
μ
n
x
n
∥
2
-
2
α
n
〈
T
μ
n
x
n
,
J
x
0
〉
-
2
(
1
-
α
n
)
〈
T
μ
n
x
n
,
J
T
μ
n
x
n
〉
+
α
n
∥
x
0
∥
2
+
(
1
-
α
n
)
∥
T
μ
n
x
n
∥
2
=
α
n
ϕ
(
T
μ
n
x
n
,
x
0
)
+
(
1
-
α
n
)
ϕ
(
T
μ
n
x
n
,
T
μ
n
x
n
)
=
α
n
ϕ
(
T
μ
n
x
n
,
x
0
)
and lim
⁡
n
→
∞
α
n
=
0, we have
(42)
lim
⁡
n
→
∞
ϕ
(
T
μ
n
x
n
,
z
n
)
=
0
.
From (40), we get
(43)
Lim
n
→
∞
ϕ
(
T
μ
n
x
n
,
x
n
+
1
)
=
lim
⁡
n
→
∞
ϕ
(
x
n
+
1
,
z
n
)
=
0
.
By Lemma 5, we obtain(44)
lim
⁡
n
→
∞
∥
T
μ
n
x
n
-
x
n
+
1
∥
=
lim
⁡
n
→
∞
∥
x
n
+
1
-
z
n
∥
=
0
.
Since ∥
x
n
-
T
μ
n
x
n
∥
≤
∥
x
n
-
x
n
+
1
∥
+
∥
x
n
+
1
-
z
n
∥
+
∥
z
n
-
T
μ
n
x
n
∥, from (36), (39), and (44), we have
(45)
lim
⁡
n
→
∞
∥
x
n
-
T
μ
n
x
n
∥
=
0
.
From Lemma 7, we have x
n
∈
F
(
I
). Since F
(
I
) is closed and lim
⁡
n
→
∞
x
n
=
v, we have v
∈
F
(
I
), where v
=
lim
⁡
n
→
∞
Π
F
(
I
)
x
n.We now establish a convergence theorem of modified Ishikawa type.Theorem 9.
LetS be a left reversible semigroup and let I
=
{
T
(
s
)
:
s
∈
S
} be a representation of S as relatively nonexpansive mappings from a nonempty, closed, and convex subset C of a uniformly convex and uniformly smooth Banach space E into itself. Let X be a subspace of l
∞
(
S
) and let {
μ
n
} be an asymptotically left invariant sequence of means on X. Let {
α
n
} and {
β
n
} be sequences of real numbers such that α
n
,
β
n
∈
(
0,1
) and lim
⁡
n
→
∞
α
n
=
0, lim
⁡
n
→
∞
β
n
=
1. Let {
x
n
} be a sequence generated by the following algorithm:
(46)
x
0
∈
C
,
c
h
o
s
e
n
a
r
b
i
t
r
a
r
i
l
y
,
y
n
=
J
-
1
(
β
n
J
x
n
+
(
1
-
β
n
)
J
T
μ
n
x
n
)
,
x
n
+
1
=
Π
C
J
-
1
(
α
n
J
x
n
+
(
1
-
α
n
)
J
T
μ
n
y
n
)
,
∀
n
≥
0
.
If the interior of F
(
I
) is nonempty, then {
x
n
} converges strongly to some common fixed point F
(
I
).Proof.
Firstly, we show that{
x
n
} converges strongly in C.
From Lemma3, we know F
(
T
) is closed and convex. So, we can define the generalized projection Π
C onto F
(
I
). Let u
∈
F
(
I
). From the definition of relatively nonexpansive and the convexity of ∥
·
∥
2, from (21), we have
(47)
ϕ
(
u
,
y
n
)
=
ϕ
(
u
,
J
-
1
(
β
n
J
x
n
+
(
1
-
β
n
)
J
T
μ
n
x
n
)
)
≤
β
n
ϕ
(
u
,
x
n
)
+
(
1
-
β
n
)
ϕ
(
u
,
T
μ
n
x
n
)
≤
ϕ
(
u
,
x
n
)
,
for allu
∈
F
(
I
). From (47), we obtain
(48)
ϕ
(
u
,
x
n
+
1
)
=
ϕ
(
u
,
Π
C
J
-
1
(
α
n
J
x
n
+
(
1
-
α
n
)
J
T
μ
n
y
n
)
)
≤
ϕ
(
u
,
J
-
1
(
α
n
J
x
n
+
(
1
-
α
n
)
J
T
μ
n
y
n
)
)
≤
α
n
ϕ
(
u
,
x
n
)
+
(
1
-
α
n
)
ϕ
(
u
,
T
μ
n
y
n
)
≤
α
n
ϕ
(
u
,
x
n
)
+
(
1
-
α
n
)
ϕ
(
u
,
y
n
)
≤
ϕ
(
u
,
x
n
)
.
Hence, {
ϕ
(
u
,
x
n
)
} is bounded and lim
⁡
n
→
∞
ϕ
(
u
,
x
n
) exists. This implies that {
x
n
}, {
T
μ
n
x
n
}, and {
y
n
} are bounded. Since the interior of F
(
I
) is nonempty, similar to the proof of Theorem 8, we obtain that {
x
n
} converges strongly to v in C.
Next, we show thatv
∈
F
(
I
), where v
=
lim
⁡
n
→
∞
Π
F
(
I
)
x
n.
Let(49)
z
n
=
J
-
1
(
α
n
J
x
n
+
(
1
-
α
n
)
J
T
μ
n
y
n
)
.
From Lemma 4, we have
(50)
ϕ
(
x
n
,
x
n
+
1
)
+
ϕ
(
x
n
+
1
,
z
n
)
=
ϕ
(
x
n
,
Π
C
z
n
)
+
ϕ
(
Π
C
z
n
,
z
n
)
≤
ϕ
(
x
n
,
z
n
)
.
Also,
(51)
ϕ
(
x
n
,
z
n
)
=
ϕ
(
x
n
,
J
-
1
(
α
n
J
x
n
+
(
1
-
α
n
)
J
T
μ
n
y
n
)
)
≤
α
n
ϕ
(
x
n
,
x
n
)
+
(
1
-
α
n
)
ϕ
(
x
n
,
T
μ
n
y
n
)
≤
α
n
ϕ
(
x
n
,
x
n
)
+
(
1
-
α
n
)
ϕ
(
x
n
,
y
n
)
≤
ϕ
(
x
n
,
y
n
)
,
(52)
∥
J
x
n
-
J
y
n
∥
=
∥
J
x
n
-
(
β
n
J
x
n
-
(
1
-
β
n
)
J
T
μ
n
x
n
)
∥
=
(
1
-
β
n
)
∥
J
x
n
-
J
T
μ
n
x
n
∥
.
From lim
⁡
n
→
∞
β
n
=
1 and (52), we have
(53)
lim
⁡
n
→
∞
∥
J
x
n
-
J
y
n
∥
=
0
.
Since J
-
1 is uniformly norm-to-norm continuous, we obtain
(54)
lim
⁡
n
→
∞
∥
x
n
-
y
n
∥
=
0
.
Hence,
(55)
ϕ
(
x
n
,
y
n
)
=
∥
x
n
∥
2
-
2
〈
x
n
,
J
y
n
〉
+
∥
y
n
∥
2
=
∥
x
n
∥
2
-
2
〈
x
n
,
J
y
n
-
J
x
n
〉
-
2
〈
x
n
,
J
x
n
〉
+
∥
y
n
∥
2
≤
∥
y
n
∥
2
-
∥
x
n
∥
2
+
2
∥
x
n
∥
∥
J
y
n
-
J
x
n
∥
≤
∥
y
n
-
x
n
∥
(
∥
y
n
∥
+
∥
x
n
∥
)
+
2
∥
x
n
∥
∥
J
y
n
-
J
x
n
∥
.
By (53) and (54), we have
(56)
lim
⁡
n
→
∞
ϕ
(
x
n
,
y
n
)
=
0
.
From (50) and (51), we obtain
(57)
lim
⁡
n
→
∞
ϕ
(
x
n
,
x
n
+
1
)
=
lim
⁡
n
→
∞
ϕ
(
x
n
,
z
n
)
=
0
.
From Lemma 5, we get
(58)
lim
⁡
n
→
∞
∥
x
n
-
x
n
+
1
∥
=
lim
⁡
n
→
∞
∥
x
n
-
z
n
∥
=
0
.
Since
(59)
∥
J
z
n
-
J
T
μ
n
y
n
∥
=
∥
α
n
J
x
n
+
(
1
-
α
n
)
J
T
μ
n
y
n
-
J
T
μ
n
y
n
∥
=
α
n
∥
J
x
n
-
J
T
μ
n
x
n
∥
and lim
⁡
n
→
∞
α
n
=
0, we have
(60)
lim
⁡
n
→
∞
∥
J
z
n
-
J
T
μ
n
y
n
∥
=
0
.
Since J
-
1 is uniformly norm-to-norm continuous, we obtain
(61)
lim
⁡
n
→
∞
∥
z
n
-
T
μ
n
y
n
∥
=
0
.
Since lim
⁡
n
→
∞
∥
x
n
-
z
n
∥
=
0 and J is uniformly norm-to-norm continuous,
(62)
lim
⁡
n
→
∞
∥
J
x
n
-
J
z
n
∥
=
0
.
By (46) and (49), we have
(63)
J
T
μ
n
x
n
=
1
1
-
β
n
(
J
y
n
-
β
n
J
x
n
)
,
J
T
μ
n
y
n
=
1
1
-
α
n
(
J
z
n
-
α
n
J
x
n
)
.
From (63), we obtain
(64)
∥
J
T
μ
n
x
n
-
J
T
μ
n
y
n
∥
=
∥
1
1
-
β
n
(
J
y
n
-
β
n
J
x
n
)
-
1
1
-
α
n
(
J
z
n
-
α
n
J
x
n
)
∥
=
∥
(
J
z
n
+
α
n
1
-
α
n
(
J
z
n
-
J
x
n
)
)
J
y
n
+
β
n
1
-
β
n
(
J
y
n
-
J
x
n
)
-
(
J
z
n
+
α
n
1
-
α
n
(
J
z
n
-
J
x
n
)
)
β
n
1
-
β
n
∥
≤
∥
J
y
n
-
J
x
n
∥
+
∥
J
x
n
-
J
z
n
∥
+
β
n
1
-
β
n
∥
J
y
n
-
J
x
n
∥
+
α
n
1
-
α
n
∥
J
z
n
-
J
x
n
∥
=
1
1
-
α
n
∥
J
z
n
-
J
x
n
∥
+
1
1
-
β
n
∥
J
y
n
-
J
x
n
∥
.
Combining (53), (62), and (64), we get
(65)
lim
⁡
n
→
∞
∥
J
T
μ
n
x
n
-
J
T
μ
n
y
n
∥
=
0
.
Since J
-
1 is uniformly norm-to-norm continuous, we have
(66)
lim
⁡
n
→
∞
∥
T
μ
n
x
n
-
T
μ
n
y
n
∥
=
0
.
Since
(67)
∥
x
n
-
T
μ
n
x
n
∥
≤
∥
x
n
-
z
n
∥
+
∥
z
n
-
T
μ
n
y
n
∥
+
∥
T
μ
n
y
n
-
T
μ
n
x
n
∥
,
therefore, by (58), (61), (66), and (67), we obtain
(68)
lim
⁡
n
→
∞
∥
x
n
-
T
μ
n
x
n
∥
=
0
.
From Lemma 7, we have x
n
∈
F
(
I
). Since F
(
I
) is closed and lim
⁡
n
→
∞
x
n
=
v, we have v
∈
F
(
I
), where v
=
lim
⁡
n
→
∞
Π
F
(
I
)
x
n.If we setβ
n
=
1, then the iteration (46) reduces modified Mann type. Hence we obtain the following corollary.Corollary 10.
LetS be a left reversible semigroup and let I
=
{
T
(
s
)
:
s
∈
S
} be a representation of S as relatively nonexpansive mappings from a nonempty, closed, and convex subset C of a uniformly convex and uniformly smooth Banach space E into itself. Let X be a subspace of l
∞
(
S
) and let {
μ
n
} be an asymptotically left invariant sequence of means on X. Let {
α
n
} be a sequence of real number such that α
n
∈
(
0,1
) and lim
⁡
n
→
∞
α
n
=
0. Let {
x
n
} be a sequence generated by the following algorithm:
(69)
x
0
∈
C
,
c
h
o
s
e
n
a
r
b
i
t
r
a
r
i
l
y
,
x
n
+
1
=
Π
C
J
-
1
(
α
n
J
x
n
+
(
1
-
α
n
)
J
T
μ
n
x
n
)
,
∀
n
≥
0
.
If the interior of F
(
I
) is nonempty, then {
x
n
} converges strongly to some common fixed point F
(
I
).In a Hilbert space,J is the identity operator. Theorems 8 and 9 reduce to the following.Corollary 11.
LetS be a left reversible semigroup and let I
=
{
T
(
s
)
:
s
∈
S
} be a representation of  S as relatively nonexpansive mappings from a nonempty, closed, and convex subset C of a Hilbert space H into itself. Let X be a subspace of  l
∞
(
S
) and let {
μ
n
} be an asymptotically left invariant sequence of means on X. Let {
α
n
} be a sequence in (
0,1
) such that lim
⁡
n
→
∞
α
n
=
0. Let {
x
n
} be a sequence generated by the following algorithm:
(70)
x
0
∈
C
,
c
h
o
s
e
n
a
r
b
i
t
r
a
r
i
l
y
,
x
n
+
1
=
P
C
(
α
n
x
0
+
(
1
-
α
n
)
T
μ
n
x
n
)
,
∀
n
≥
0
.
If the interior of F
(
I
) is nonempty, then {
x
n
} converges strongly to some common fixed point F
(
I
), where P
C is a metric projection.Corollary 12.
LetS be a left reversible semigroup and let I
=
{
T
(
s
)
:
s
∈
S
} be a representation of  S as relatively nonexpansive mappings from a nonempty, closed, and convex subset C of a Hilbert space H into itself. Let X be a subspace of l
∞
(
S
) and let {
μ
n
} be an asymptotically left invariant sequence of means on X. Let {
α
n
} and {
β
n
} be sequences of real numbers such that α
n
,
β
n
∈
(
0,1
) and lim
⁡
n
→
∞
α
n
=
0, lim
⁡
n
→
∞
β
n
=
1. Let {
x
n
} be a sequence generated by the following algorithm:
(71)
x
0
∈
C
,
c
h
o
s
e
n
a
r
b
i
t
r
a
r
i
l
y
,
y
n
=
β
n
x
n
+
(
1
-
β
n
)
T
μ
n
x
n
,
x
n
+
1
=
P
C
(
α
n
x
n
+
(
1
-
α
n
)
T
μ
n
y
n
)
,
∀
n
≥
0
.
If the interior of F
(
I
) is nonempty, then {
x
n
} converges strongly to some common fixed point F
(
I
), where P
C is a metric projection.



---

*Source: 694783-2014-07-20.xml*