# Strong Convergence on Iterative Methods of Cesàro Means for Nonexpansive Mapping in Banach Space

Authors: Zhichuan Zhu; Rudong Chen
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

Two new iterations with Cesàro's means for nonexpansive mappings are proposed and the strong convergence is obtained asn
→
∞. Our main results extend and improve the corresponding results of Xu (2004), Song and Chen (2007), and Yao et al. (2009).


---

## Body

## 1. Introduction

LetC be a nonempty closed convex subset of a real Banach space E and let T be nonexpansive mapping from C into itself (recall that a mapping T
:
C
→
C is nonexpansive if ∥
T
x
-
T
y
∥
≤
∥
x
-
y
∥, ∀
x
,
y
∈
C). We denote fixed points of T as F
(
T
); that is, F
(
T
)
=
{
x
∈
C
:
T
x
=
x
}.Recall that a mappingf
:
C
→
C is contractive if there exists a constant β
∈
(
0,1
) such that ∥
f
(
x
)
-
f
(
y
)
∥
≤
β
∥
x
-
y
∥, ∀
x
,
y
∈
C.In 1975, Baillon [1] proved the first nonlinear ergodic theorem.Theorem 1.
Suppose thatC is a nonempty closed convex subset of Hilbert space E and T
:
C
→
C mapping such that F
(
T
)
≠
ϕ; then ∀
x
∈
C, and the Cesàro means
(1)
T
n
x
=
1
n
+
1
∑
i
=
0
n
‍
T
i
x
weakly converges to a fixed point of T.In 1979, Bruck [2] showed the nonlinear ergodic theorem for nonexpansive mapping in uniformly convex Banach space withFréchet differentiable norms.In 2004, Xu [3] introduced the following viscosity iterative scheme {
x
n
} given by
(2)
x
n
+
1
=
α
n
f
(
x
n
)
+
(
1
-
α
n
)
T
x
n
,
where parameter {
α
n
}
⊂
[
0,1
], satisfying(X1)
lim
⁡
n
→
∞
α
n
=
0;
(X2)
∑
n
=
0
∞
α
n
=
∞;
(X3)
either∑
n
=
0
∞
|
α
n
+
1
-
α
n
|
<
∞, or lim
⁡
n
→
∞
(
α
n
+
1
/
α
n
)
=
1.He proved that the explicit iterative scheme {
x
n
} converges strongly to a fixed point p of T in uniformly smooth Banach space.In 2007, Song and Chen [4] defined the following viscosity iteration {
x
n
} of Cesàro means for nonexpansive mapping T:
(3)
x
n
+
1
=
α
n
f
(
x
n
)
+
(
1
-
α
n
)
1
n
+
1
∑
i
=
0
n
‍
T
i
x
n
,
and they proved that the sequence {
x
n
} converges strongly to some point in F
(
T
) in a uniformly convex Banach space with weakly sequentially continuous duality mapping.In 2009, Yao et al. [5] introduced the following process {
x
n
}:
(4)
x
n
+
1
=
α
n
u
+
β
n
x
n
+
γ
n
T
x
n
,
n
≥
0
.
They proved that the sequence {
x
n
} converges strongly to a fixed point of T under the following control conditions of parameters:(YLZ1)
α
n
+
β
n
+
γ
n
=
1,  for all n
≥
0;
(YLZ2)
lim
⁡
n
→
∞
α
n
=
0 and ∑
n
=
0
∞
α
n
=
∞;
(YLZ3)
lim
⁡
n
→
∞
γ
n
=
0.Motivated by the above results, we propose the following new iterations with Cesàro’s means for nonexpansive mappings:(5)
x
n
+
1
=
α
n
u
+
β
n
x
n
+
γ
n
1
n
+
1
∑
i
=
0
n
‍
T
i
x
n
,
n
≥
0
,
and viscosity iteration:
(6)
x
n
+
1
=
α
n
f
(
x
n
)
+
β
n
x
n
+
γ
n
1
n
+
1
∑
i
=
0
n
‍
T
i
x
n
,
n
≥
0
.Some examples are given to show the generation of our new iterations with Cesàro’s means as follows.Example 2.
Ifn
=
1, iteration (5) with Cesàro’s means for nonexpansive mappings is x
2
=
α
1
u
+
β
1
x
1
+
γ
1
·
(
1
/
2
)
(
x
1
+
T
x
1
), which is reduced as the same iteration of Yao et al. [5]. If n
≥
2, iteration (5) can be written as follows:
(7)
x
n
+
1
=
α
n
u
+
β
n
x
n
+
γ
n
·
1
n
+
1
(
x
n
+
T
x
n
+
T
2
x
n
+
⋯
+
T
n
x
n
)
,
which is a generation of Yao et al. [5].Example 3.
LetE
=
R with the usual metric, nonexpansive mapping defined by T
x
=
sin
x, the fixed contractive mapping f
(
x
)
=
(
1
/
2
)
x, and the parameters are defined as α
n
=
1
/
2
n, γ
n
=
1
/
4
n, and β
n
=
(
4
n
-
3
)
/
4
n. The new iterations with Cesa
`ro’s means which is related to iterative step n can be written as follows:
(8)
x
n
+
1
=
1
2
n
u
+
4
n
-
3
4
n
x
n
mmmn
+
1
4
n
(
n
+
1
)
(
x
n
+
T
x
n
+
T
2
x
n
+
⋯
+
T
n
x
n
)
,
x
n
+
1
=
1
2
n
·
1
2
x
n
+
4
n
-
3
4
n
x
n
mmmn
+
1
4
n
(
n
+
1
)
(
x
n
+
T
x
n
+
T
2
x
n
+
⋯
+
T
n
x
n
)
.


## 2. Preliminaries

Throughout the paper, letE be a real Banach space with norm ∥
·
∥. The normalized duality mapping J
:
E
→
2
E
* is defined by
(9)
J
(
x
)
=
{
f
∈
E
*
,
〈
x
,
f
〉
=
∥
x
∥
∥
f
∥
,
∥
x
∥
=
∥
f
∥
}
,
hhhhhhhhhhhhhhhhhhhhhhhhhhhhii
∀
x
∈
E
,
where E
* denotes the dual space of E and 〈
·
,
·
〉 denotes the generalized duality pairing. We will denote the single-valued normalized duality mapping by j.LetS
:
=
{
x
∈
E
:
∥
x
∥
=
1
} be the unit sphere of a Banach space. The space is said to have a Gâteaux differentiable norm (or E is said to be smooth), if the limit
(10)
lim
⁡
t
→
0
∥
x
+
t
y
∥
-
∥
x
∥
t
exists for every x
,
y
∈
S, and E is said to have a uniformly Gâteaux differentiable norm if for each y
∈
S the limit (10) is attained uniformly for x
∈
S. Further, E is said to be uniformly smooth if the limit (10) exists uniformly for (
x
,
y
)
∈
S
×
S.The following two results can be found in [6].IfE is smooth the duality mapping J is single-valued and strong-
weak
* continuous.IfE is Banach space with uniformly Gâteaux differentiable norm, then duality mapping J
:
E
→
E
* is single-valued and norm to weak star uniformly continuous on bounded sets of E.In order to prove our main results, the following lemmas will be used.Lemma 4 (see [7]).
LetC be a nonempty closed convex subset of a uniformly smooth Banach space X. Let T
:
C
→
C be a nonexpansive mapping with F
(
T
)
≠
∅. For each fixed u
∈
C and every t
∈
(
0,1
), the unique fixed point z
t
∈
C of the contraction C
∋
x
⊢
t
u
+
(
1
-
t
)
T
x as t
→
0 converges strongly to x
*
∈
F
(
T
) which is the nearest to u.Lemma 5 (see [1]).
LetX be a uniformly smooth Banach space, C a closed convex subset of X, T
:
C
→
C a nonexpansive mapping with F
(
T
)
≠
ϕ, and f
:
C
→
C a fixed contraction. Then x
t defined by x
t
=
t
f
(
x
t
)
+
(
1
-
t
)
T
x
t converges strongly to a unique fixed point in F
(
T
) as t
→
0.Lemma 6 (see [3]).
LetE be a real Banach space and let J be the normalized duality mapping. Then for any given x
,
y
∈
E, one has
(11)
∥
x
+
y
∥
2
≤
∥
x
∥
2
+
2
〈
y
,
j
(
x
+
y
)
〉
,
hhhhhhhhh
∀
j
(
x
+
y
)
∈
J
(
x
+
y
)
.Lemma 7 (see [8]).
Assume{
a
n
} is a sequence of nonnegative real numbers such that
(12)
a
n
+
1
≤
(
1
-
γ
n
)
a
n
+
γ
n
δ
n
,
n
≥
0
,
where {
γ
n
} is a sequence in (0,1) and {
δ
n
} is a sequence such that(1)
∑
n
=
0
∞
γ
n
=
∞;
(2)
limsup
⁡
n
→
∞
δ
n
≤
0.
Thenlim
⁡
n
→
∞
a
n
=
0.


## 3. Main Results

LetC be a nonempty closed convex subset of a uniformly smooth Banach space. Let T
:
C
→
C be a nonexpansive mapping such that F
(
T
)
≠
ϕ. Let {
α
n
}, {
β
n
}, and {
γ
n
} be three real sequences in (0,1) satisfying(i)
α
n
+
β
n
+
γ
n
=
1, for all n
≥
0;
(ii)
lim
⁡
n
→
∞
α
n
=
0 and ∑
n
=
0
∞
α
n
=
∞;
(iii)
lim
⁡
n
→
∞
γ
n
=
0.In the following, we will present the first main result. For eachn
≥
0 and t
∈
(
0,1
), let x
t
,
n be the unique fixed point of the contractive mapping T
t
,
n given by
(13)
T
t
,
n
x
=
(
1
-
α
n
)
t
γ
n
+
t
β
n
u
+
(
1
-
t
)
γ
n
γ
n
+
t
β
n
1
n
+
1
∑
i
=
0
n
‍
T
i
x
.
That is,
(14)
x
t
,
n
=
(
1
-
α
n
)
t
γ
n
+
t
β
n
u
+
(
1
-
t
)
γ
n
γ
n
+
t
β
n
1
n
+
1
∑
i
=
0
n
‍
T
i
x
t
,
n
.
From Lemma 4, for fixed n, we have
(15)
lim
⁡
t
→
0
x
t
,
n
=
p
∈
F
(
T
)
,
which is the unique fixed point.Theorem 8.
LetC be a nonempty closed convex subset of a uniformly smooth Banach space X. Let T
:
C
→
C be a nonexpansive mapping such that F
(
T
)
≠
ϕ. Let {
α
n
}, {
β
n
}, and {
γ
n
} be three real sequences in (0,1) satisfying conditions (i)–(iii). Then, for given x
0
∈
C arbitrarily, let the sequence {
x
n
} be generated iteratively by (5). Then the sequence {
x
n
} defined by (5) converges strongly to a fixed point of T.Proof.
Taking a fixed pointp
∈
F
(
T
), we have
(16)
∥
x
n
+
1
-
p
∥
=
α
n
∥
u
-
p
∥
+
β
n
∥
x
n
-
p
∥
+
γ
n
∥
1
n
+
1
∑
i
=
0
n
‍
(
T
i
x
n
-
T
i
p
)
∥
≤
α
n
∥
u
-
p
∥
+
β
n
∥
x
n
-
p
∥
+
γ
n
∥
x
n
-
p
∥
=
α
n
∥
u
-
p
∥
+
(
1
-
α
n
)
∥
x
n
-
p
∥
≤
max
⁡
{
∥
u
-
p
∥
,
∥
x
n
-
p
∥
}
.
By induction, we get that {
x
n
} is bounded. We observe that (14) can be rewritten as follows:
(17)
x
t
,
n
=
t
u
+
(
1
-
t
)
[
β
n
1
-
α
n
x
t
,
n
+
γ
n
1
-
α
n
1
1
+
n
∑
i
=
0
n
‍
T
i
x
t
,
n
]
,
lim
⁡
t
→
0
x
t
,
n
=
p
∈
F
(
T
)
,
∀
n
≥
0
.
By (14), we have
(18)
x
t
,
n
-
x
n
=
t
(
u
-
x
n
)
+
(
1
-
t
)
[
∑
i
=
0
n
‍
(
T
i
x
t
,
n
-
x
n
)
β
n
1
-
α
n
(
x
t
,
n
-
x
n
)
m
m
m
m
m
+
γ
n
1
-
α
n
1
1
+
n
∑
i
=
0
n
‍
(
T
i
x
t
,
n
-
x
n
)
]
.
Applying Lemma 6 to (18), we have
(19)
∥
x
t
,
n
-
x
n
∥
≤
(
1
-
t
)
2
∥
∑
i
=
0
n
‍
(
T
i
x
t
,
n
-
x
n
)
β
n
1
-
α
n
(
x
t
,
n
-
x
n
)
m
m
m
m
m
m
m
i
+
γ
n
1
-
α
n
1
1
+
n
∑
i
=
0
n
‍
(
T
i
x
t
,
n
-
x
n
)
∥
2
+
2
t
〈
u
-
x
n
,
j
(
x
t
,
n
-
x
n
)
〉
≤
(
1
-
t
)
2
[
∑
i
=
0
n
‍
∥
T
i
x
t
,
n
-
T
i
x
n
∥
β
n
1
-
α
n
∥
x
t
,
n
-
x
n
∥
mmmmmmm
+
γ
n
1
-
α
n
1
1
+
n
∑
i
=
0
n
‍
∥
T
i
x
t
,
n
-
T
i
x
n
∥
m
m
m
m
m
m
m
+
γ
n
1
-
α
n
1
1
+
n
∑
i
=
0
n
‍
∥
T
i
x
n
-
x
n
∥
]
2
+
2
t
∥
x
t
,
n
-
x
n
∥
2
+
2
t
〈
u
-
x
t
,
n
,
j
(
x
t
,
n
-
x
n
)
〉
≤
(
1
-
t
)
2
[
∥
x
t
,
n
-
x
n
∥
+
γ
n
1
-
α
n
1
1
+
n
∑
i
=
0
n
‍
∥
T
i
x
n
-
x
n
∥
]
2
+
2
t
∥
x
t
,
n
-
x
n
∥
2
+
2
t
〈
u
-
x
t
,
n
,
j
(
x
t
,
n
-
x
n
)
〉
=
(
1
+
t
2
)
∥
x
t
,
n
-
x
n
∥
2
+
(
1
-
t
)
2
(
γ
n
1
-
α
n
1
1
+
n
∑
i
=
0
n
‍
∥
T
i
x
n
-
x
n
∥
)
2
+
2
(
1
-
t
)
2
γ
n
1
-
α
n
∥
x
t
,
n
-
x
n
∥
(
1
1
+
n
∑
i
=
0
n
‍
∥
T
i
x
n
-
x
n
∥
)
+
2
t
〈
u
-
x
t
,
n
,
j
(
x
t
,
n
-
x
n
)
〉
≤
(
1
+
t
2
)
∥
x
t
,
n
-
x
n
∥
2
+
γ
n
1
-
α
n
Q
+
2
t
〈
u
-
x
t
,
n
,
j
(
x
t
,
n
-
x
n
)
〉
,
where Q is some constant such that
(20)
sup
⁡
{
1
1
+
n
∑
i
=
0
n
‍
∥
T
i
x
n
-
x
n
∥
m
n
m
+
2
∥
x
t
,
n
-
x
n
∥
1
1
+
n
∑
i
=
0
n
‍
∥
T
i
x
n
-
x
n
∥
,
n
≥
0
}
≤
Q
.
Hence, we get
(21)
〈
u
-
x
t
,
n
,
j
(
x
n
-
x
t
,
n
)
〉
≤
t
2
Q
1
+
1
2
t
(
γ
n
1
-
α
n
Q
)
,
where Q
1 is a constant such that
(22)
sup
⁡
{
∥
x
t
,
n
-
x
n
∥
,
n
≥
0
,
0
<
t
<
1
}
≤
Q
1
.
It follows that
(23)
limsup
⁡
t
→
0
limsup
⁡
n
→
∞
〈
u
-
x
t
,
n
,
j
(
x
n
-
x
t
,
n
)
〉
≤
0
.
By the fact that the order of limsup
⁡
t
→
0 and limsup
⁡
n
→
∞ is changeable, we have
(24)
limsup
⁡
n
→
∞
〈
u
-
p
,
j
(
x
n
-
p
)
〉
≤
0
.
Finally, we prove x
n
→
p. Indeed, applying Lemma 6 to (5), we obtain
(25)
∥
x
n
+
1
-
p
∥
2
=
∥
α
n
(
u
-
p
)
+
β
n
(
x
n
-
p
)
+
γ
n
(
1
n
+
1
∑
i
=
0
n
‍
T
i
x
n
-
p
)
∥
2
≤
∥
β
n
(
x
n
-
p
)
+
γ
n
1
n
+
1
∑
i
=
0
n
‍
(
T
i
x
n
-
T
i
p
)
∥
2
+
2
α
n
〈
u
-
p
,
j
(
x
n
+
1
-
p
)
〉
≤
(
β
n
∥
x
n
-
p
∥
+
γ
n
∥
x
n
-
p
∥
)
2
+
2
α
n
〈
u
-
p
,
j
(
x
n
+
1
-
p
)
〉
≤
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
2
α
n
〈
u
-
p
,
j
(
x
n
+
1
-
p
)
〉
.
Hence, by Lemma 7, we have that x
n
→
p as n
→
∞.
The proof is complete.Now we will give the second main result. In order to prove the strong convergence of viscosity iterative (6), we assume that z
t
,
n is the unique fixed point of the following contractive mapping T
t
,
n given by
(26)
T
t
,
n
x
=
(
1
-
α
n
)
t
γ
n
+
t
β
n
f
(
x
)
+
(
1
-
t
)
γ
n
γ
n
+
t
β
n
1
n
+
1
∑
i
=
0
n
‍
T
i
x
.
That is,
(27)
z
t
,
n
=
(
1
-
α
n
)
t
γ
n
+
t
β
n
f
(
z
t
,
n
)
+
(
1
-
t
)
γ
n
γ
n
+
t
β
n
1
n
+
1
∑
i
=
0
n
‍
T
i
z
t
,
n
.
From Lemma 5, for fixed n, we have
(28)
lim
⁡
t
→
0
z
t
,
n
=
p
∈
F
(
T
)
,
which is the unique fixed point.Theorem 9.
LetC be a nonempty closed convex subset of a uniformly smooth Banach space X. Let T
:
C
→
C be a nonexpansive mapping such that F
(
T
)
≠
ϕ. Let {
α
n
}, {
β
n
}, and {
γ
n
} be three real sequences in (0,1) satisfying the following control conditions (i–iii). Then, for given x
0
∈
C arbitrarily, the sequence {
x
n
} defined by (6) converges strongly to a fixed point of T.Proof.
Taking a fixed pointq
∈
F
(
T
), we have
(29)
∥
x
n
+
1
-
q
∥
≤
α
n
∥
f
(
x
n
)
-
q
∥
+
β
n
∥
x
n
-
q
∥
+
γ
n
∥
1
1
+
n
∑
i
=
0
n
‍
(
T
i
x
n
-
T
i
q
)
∥
≤
α
n
(
∥
f
(
x
n
)
-
f
(
q
)
∥
+
∥
f
(
q
)
-
q
∥
)
+
β
n
∥
x
n
-
q
∥
+
γ
n
∥
x
n
-
q
∥
≤
α
n
β
∥
x
n
-
q
∥
+
α
n
∥
f
(
q
)
-
q
∥
+
(
1
-
α
n
)
∥
x
n
-
q
∥
=
α
n
∥
f
(
q
)
-
q
∥
+
(
1
-
(
1
-
β
)
α
n
)
∥
x
n
-
q
∥
≤
max
⁡
{
1
1
-
β
∥
f
(
q
)
-
q
∥
,
∥
x
n
-
q
∥
}
.
By induction,
(30)
∥
x
n
-
q
∥
≤
max
⁡
{
1
1
-
β
∥
f
(
q
)
-
q
∥
,
∥
x
0
-
q
∥
}
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
n
≥
0
and {
x
n
} is bounded so are {
T
i
x
n
} and {
f
(
x
n
)
}.
We observe that (27) can be rewritten as
(31)
z
t
,
n
=
t
f
(
z
t
,
n
)
+
(
1
-
t
)
[
β
n
1
-
α
n
z
t
,
n
+
γ
n
1
-
α
n
1
1
+
n
∑
i
=
0
n
‍
T
i
z
t
,
n
]
,
and lim
⁡
t
→
0
z
t
,
n
=
q
∈
F
(
T
), for all n
≥
0.
From (31), we have
(32)
z
t
,
n
-
x
n
=
t
(
f
(
z
t
,
n
-
x
n
)
)
+
(
1
-
t
)
[
∑
i
=
0
n
‍
(
T
i
z
t
,
n
-
x
n
)
β
n
1
-
α
n
(
z
t
,
n
-
x
n
)
m
m
m
m
m
+
γ
n
1
-
α
n
1
1
+
n
∑
i
=
0
n
‍
(
T
i
z
t
,
n
-
x
n
)
]
.
Applying Lemma 6 to (32), we get
(33)
∥
z
t
,
n
-
x
n
∥
2
≤
(
1
-
t
)
2
∥
∑
i
=
0
n
‍
(
T
i
z
t
,
n
-
x
n
)
β
n
1
-
α
n
(
z
t
,
n
-
x
n
)
m
m
m
m
m
m
+
γ
n
1
-
α
n
1
1
+
n
∑
i
=
0
n
‍
(
T
i
z
t
,
n
-
x
n
)
∥
2
+
2
t
〈
f
(
z
t
,
n
)
-
x
n
,
j
(
z
t
,
n
-
x
n
)
〉
≤
(
1
-
t
)
2
∥
∑
i
=
0
n
‍
(
T
i
z
t
,
n
-
x
n
)
β
n
1
-
α
n
(
z
t
,
n
-
x
n
)
mmmmmmmn
+
γ
n
1
-
α
n
1
1
+
n
∑
i
=
0
n
‍
(
T
i
z
t
,
n
-
T
i
x
n
)
mmmmmmmn
+
γ
n
1
-
α
n
1
1
+
n
∑
i
=
0
n
‍
(
T
i
x
n
-
x
n
)
∥
2
+
2
t
∥
z
t
,
n
-
x
n
∥
2
+
2
t
〈
f
(
z
t
,
n
)
-
z
t
,
n
,
j
(
z
t
,
n
-
x
n
)
〉
≤
(
1
-
t
)
2
[
∥
z
t
,
n
-
x
n
∥
+
γ
n
1
-
α
n
1
1
+
n
∑
i
=
0
n
‍
∥
T
i
x
n
-
x
n
∥
]
2
+
2
t
∥
z
t
,
n
-
x
n
∥
2
+
2
t
〈
f
(
z
t
,
n
)
-
z
t
,
n
,
j
(
z
t
,
n
-
x
n
)
〉
=
(
1
+
t
2
)
∥
z
t
,
n
-
x
n
∥
2
+
(
1
-
t
)
2
(
γ
n
1
-
α
n
1
1
+
n
∑
i
=
0
n
‍
∥
T
i
x
n
-
x
n
∥
)
2
+
2
(
1
-
t
)
2
γ
n
1
-
α
n
1
1
+
n
∑
i
=
0
n
‍
∥
T
i
x
n
-
x
n
∥
∥
z
t
,
n
-
x
n
∥
+
2
t
〈
f
(
z
t
,
n
)
-
z
t
,
n
,
j
(
z
t
,
n
-
x
n
)
〉
≤
(
1
+
t
2
)
∥
z
t
,
n
-
x
n
∥
2
+
γ
n
1
-
α
n
M
+
2
t
〈
f
(
z
t
,
n
)
-
z
t
,
n
,
j
(
z
t
,
n
-
x
n
)
〉
,
where M is some constant such that
(34)
sup
⁡
{
1
1
+
n
∑
i
=
0
n
‍
(
∥
T
i
x
n
-
x
n
∥
2
m
m
m
m
m
m
m
m
+
2
∥
T
i
x
n
-
x
n
∥
∥
z
t
,
n
-
x
n
∥
∥
T
i
x
n
-
x
n
∥
2
)
,
n
≥
0
∑
i
=
0
n
‍
(
∥
T
i
x
n
-
x
n
∥
2
}
≤
M
.
Hence, we have
(35)
〈
f
(
z
t
,
n
)
-
z
t
,
n
,
j
(
x
n
-
z
t
,
n
)
〉
≤
t
2
M
1
+
1
2
t
(
γ
n
1
-
α
n
M
)
,
where M
1 is also a constant such that
(36)
M
1
≥
sup
⁡
{
∥
z
t
,
n
-
x
n
∥
2
,
n
≥
0
,
0
<
t
<
1
}
.
It follows that
(37)
limsup
⁡
t
→
0
limsup
⁡
n
→
∞
〈
f
(
z
t
,
n
)
-
z
t
,
n
,
j
(
x
n
-
z
t
,
n
)
〉
≤
0
.
Since the order of limsup
⁡
t
→
0 and limsup
⁡
n
→
∞ is exchangeable, hence
(38)
limsup
⁡
n
→
∞
〈
f
(
q
)
-
q
,
j
(
x
n
-
q
)
〉
≤
0
.
Finally, we prove thatx
n
→
q. Indeed, applying Lemma 6 to (6), we obtain
(39)
∥
x
n
+
1
-
q
∥
2
=
∥
∑
i
=
0
n
‍
(
T
i
x
n
-
T
i
q
)
α
n
(
f
(
x
n
)
-
q
)
+
β
n
(
x
n
-
q
)
m
m
m
+
γ
n
1
1
+
n
∑
i
=
0
n
‍
(
T
i
x
n
-
T
i
q
)
∥
2
≤
∥
β
n
(
x
n
-
q
)
+
γ
n
1
1
+
n
∑
i
=
0
n
‍
(
T
i
x
n
-
T
i
q
)
∥
2
+
2
α
n
〈
f
(
x
n
)
-
q
,
j
(
x
n
+
1
-
q
)
〉
≤
[
β
n
∥
x
n
-
q
∥
+
γ
n
∥
x
n
-
q
∥
]
2
+
2
α
n
〈
f
(
x
n
)
-
f
(
q
)
,
j
(
x
n
+
1
-
q
)
〉
+
2
α
n
〈
f
(
q
)
-
q
,
j
(
x
n
+
1
-
q
)
〉
≤
(
1
-
α
n
)
2
∥
x
n
-
q
∥
2
+
α
n
(
∥
f
(
x
n
)
-
f
(
q
)
∥
2
+
∥
x
n
+
1
-
q
∥
2
)
+
2
α
n
〈
f
(
q
)
-
q
,
j
(
x
n
+
1
-
q
)
〉
.
Therefore, we have
(40)
∥
x
n
+
1
-
q
∥
2
≤
1
-
2
α
n
+
β
2
α
n
+
α
n
2
1
-
α
n
∥
x
n
-
q
∥
2
+
2
α
n
1
-
α
n
〈
f
(
q
)
-
q
,
j
(
x
n
+
1
-
q
)
〉
≤
(
1
-
1
-
β
2
1
-
α
n
α
n
)
∥
x
n
-
q
∥
2
+
α
n
2
1
-
α
n
∥
x
n
-
q
∥
2
+
2
α
n
1
-
α
n
〈
f
(
q
)
-
q
,
j
(
x
n
+
1
-
q
)
〉
≤
(
1
-
α
n
~
)
∥
x
n
-
q
∥
2
+
α
n
~
β
n
~
.
Put
(41)
α
n
~
=
1
-
β
2
1
-
α
n
α
n
,
β
n
~
=
M
1
1
-
β
2
α
n
+
2
1
-
β
2
〈
f
(
q
)
-
q
,
j
(
x
n
+
1
-
q
)
〉
.
It follows that
(42)
∥
x
n
+
1
-
q
∥
2
≤
(
1
-
α
n
~
)
∥
x
n
-
q
∥
2
+
α
n
~
β
n
~
.
It is easily seen from (ii) and (38) that
(43)
α
n
~
⟶
0
,
∑
n
=
0
∞
‍
α
n
~
=
∞
,
limsup
⁡
n
→
∞
β
n
~
≤
0
.
Hence, applying Lemma 7 to (42), we have that x
n
→
q as n
→
∞.
The proof is complete.Remark 10.
Our main result extends the main result of Yao et al. to Cesàro means and viscosity iteration method. Our results are new and the proofs are simple and different from many others.



---

*Source: 205875-2014-05-12.xml*