# Strong Convergence Theorems for Mixed Equilibrium Problem and AsymptoticallyI-Nonexpansive Mapping in Banach Spaces

Authors: Bin-Chao Deng; Tong Chen; Yi-Lin Yin
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

This paper aims to use a hybrid algorithm for finding a common element of a fixed point problem for a finite family of asymptotically nonexpansive mappings and the set solutions of mixed equilibrium problem in uniformly smooth and uniformly convex Banach space. Then, we prove some strong convergence theorems of the proposed hybrid algorithm to a common element of the above two sets under some suitable conditions.


---

## Body

## 1. Introduction

LetE be a Banach space with norm ∥
·
∥. Let C be a nonempty closed convex subset of E and E
* denoted the dual space of E. Let B
:
C
→
E
* be a nonlinear mapping and H a bifunction from C
×
C to R, where R denotes the set of numbers. The generalized equilibrium problem is to find x
∈
C such that
(1)
H
(
x
,
y
)
+
〈
B
x
,
y
-
x
〉
≥
0
,
∀
y
∈
C
.
The set of solution of (1) is denoted by GEP
⁡
(
H
,
B
), that is,
(2)
GEP
⁡
(
H
,
B
)
:
=
{
x
∈
C
,
H
(
x
,
y
)
+
〈
B
x
,
y
-
x
〉
≥
0
,
∀
y
∈
C
}
.In this paper, we are interested in solving the generalized equilibrium problem with thoseH given by
(3)
H
(
x
,
y
)
=
F
(
x
,
y
)
+
G
(
x
,
y
)
,
where F, G
:
C
×
C
→
R are two bifunctions satisfying the following special properties (
f
1
)
–
(
f
4
), (
g
1
)
–
(
g
3
) and (
H
):(f
1) F
(
x
,
x
)
=
0, for all x
∈
C;
(f
2) F is maximal monotone;
(f
3) for all x, y, z
∈
C, we have lim
⁡
⁡
sup
⁡
t
→
0
+
(
F
(
t
z
+
(
1
-
t
)
x
,
y
)
)
≤
F
(
x
,
y
);
(f
4) for all x
∈
C, the function y
↦
F
(
x
,
y
) is convex and weakly lower semicontinuous;
(g
1) G
(
x
,
x
)
=
0, for all x
∈
C;
(g
2) G is monotone and maximal monotone, and weakly upper semicontinuous in the first variable;
(g
3) G is convex in the second variable;
(H) for fixed λ
>
0 and x
∈
C, there exist a bounded set K
⊂
C and a
∈
K such that
(4)
-
F
(
a
,
z
)
+
G
(
z
,
a
)
+
1
λ
〈
a
-
z
,
z
-
x
〉
<
0
,
∀
z
∈
C
∖
K
.This is the well-know generalized mixed equilibrium problem, that is, to find an x in C such that
(5)
F
(
x
,
y
)
+
G
(
x
,
y
)
+
〈
B
x
,
y
-
x
〉
≥
0
,
∀
y
∈
C
.
The solution set of (5) is denoted by GMEP
⁡
(
F
,
G
,
B
), that is,
(6)
GMEP
⁡
(
F
,
G
,
B
)
:
=
{
x
∈
C
,
F
(
x
,
y
)
+
G
(
x
,
y
)
+
〈
B
x
,
y
-
x
〉
≥
0
,
∀
y
∈
C
}
.
If B
≡
0, problem (5) reduces into mixed equilibrium problem for F and G, denoted by MEP
⁡
(
F
,
G
), which is to find x
∈
C such that (3).IfG
=
0 and B
≡
0, reduces into equilibrium problem for F, denoted by EP
⁡
(
F
), which is to find x
∈
C such that
(7)
F
(
x
,
y
)
≥
0
,
∀
y
∈
C
.
Mixed equilibrium problems are suitable and common format for investigation of various applied problems arising in economics, mathematical physics, transportation, communication systems, engineering, and other fields. Moreover, equilibrium problems are closely related with other general problems in nonlinear analysis, such as fixed points, game theory, variational inequality, and optimization problems. Recently, many authors studied a great number of iterative methods for solving a common element of the set of fixed points for a nonexpansive mapping and the set of solutions to a mixed equilibrium problem in the setting of Hilbert space and uniformly smooth and uniformly convex Banach space, respectively (please see, e.g., [1–11] and the references therein).LetE be a real Banach space with norm ∥
·
∥, let C be a nonempty closed convex subset of E, and let J be the normalized duality mapping from E into E
* given by
(8)
J
x
=
{
x
*
∈
E
*
:
〈
x
,
x
*
〉
=
∥
x
∥
∥
x
*
∥
,
∥
x
∥
=
∥
x
*
∥
}
,
iiiiiiiiiiiiiiiiiiiiiiiiiii
kkkkkkkkkkkkkkki
∀
x
∈
E
,
where E
* denotes the dual space of E and 〈
·
,
·
〉 the generalized duality pairing between E and E
*. It is easily known that if E
* is uniformly convex, then J is uniformly continuous on bounded subsets of E.Consider the functional defined by(9)
ϕ
(
x
,
y
)
=
∥
x
∥
2
-
2
〈
x
,
J
y
〉
+
∥
y
∥
2
,
∀
x
,
y
∈
E
.
It is obvious from the definition of ϕ that
(10)
(
∥
x
∥
-
∥
y
∥
)
2
≤
ϕ
(
x
,
y
)
≤
(
∥
x
∥
+
∥
y
∥
)
2
,
∀
x
,
y
∈
E
.
On the other hand, in a Hilbert space H, (9) reduced to ϕ
(
x
,
y
)
=
∥
x
-
y
∥
2. Following Alber [12], the generalized projection Π
C
:
E
→
C is defined by
(11)
Π
C
(
x
)
=
inf
⁡
y
∈
C
ϕ
(
y
,
x
)
,
where is a map that assigns to an arbitrary point x
∈
E the minimum point of the functional ϕ
(
x
,
y
).In 2011, Kim [13] considered the following shrinking projection methods to obtain a convergence theorem, and these methods were introduced in [14] for quasi-φ-nonexpansive mappings in a uniformly convex and uniformly smooth Banach space.Theorem 1 (see [13]).
LetE be a uniformly smooth and strictly convex Banach space which has the Kadec-Klee property and C a nonempty closed convex subset of E. Let f be a bifunction from C
×
C to R satisfying (
f
1
)
–
(
f
4
) and T
:
C
→
C a closed and asymptotically quasi-φ-nonexpansive mapping. Assume that T is asymptotically regular on C and Ϝ
=
F
i
x
(
T
)
⋂
‍
E
F
(
f
) is nonempty and bounded. Let {
x
n
} be a sequence generated in the following manner:
(12)
∀
x
0
∈
E
,
C
1
=
C
,
x
1
=
∏
C
1
‍
x
0
,
y
n
=
J
-
1
(
α
n
J
x
n
+
(
1
-
α
n
)
J
T
n
x
n
)
,
u
n
∈
s
u
c
h
t
h
a
t
f
(
u
n
,
x
)
+
1
r
n
〈
x
-
u
n
,
J
u
n
-
J
y
n
〉
≥
0
,
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
i
∀
x
∈
C
,
C
n
+
1
=
{
z
∈
C
n
:
ϕ
(
z
,
u
n
)
≤
ϕ
(
z
,
x
n
)
+
(
k
n
-
1
)
M
n
}
,
x
n
+
1
=
∏
C
n
+
1
‍
x
0
,
where M
n
=
sup
⁡
{
ϕ
(
z
,
x
n
)
:
z
∈
Ϝ
} for each n
≥
1, {
α
n
} is a real sequence in [
0,1
] such that liminf
⁡
n
→
∞
α
n
(
1
-
α
n
)
>
0, and {
r
n
} is a real sequence in [
a
,
∞
), where a is some positive real number and J is the duality mapping on E. Then the sequence {
x
n
} converges strongly to ∏
Ϝ
‍
x
0, where ∏
Ϝ
‍ is the generalized projection from E onto Ϝ.Motivated and inspired by the researches going on in this direction (i.e., [4–11, 13–16]), the purpose of this paper is to use the following hybrid algorithm for finding a common element of the set of solutions to a mixed equilibrium problem and the set of the set of common fixed points for a finite family of asymptotically nonexpansive mappings in a uniformly smooth and uniformly convex Banach space.Algorithm 2.
Let(13)
u
n
∈
C
such
that
F
(
u
n
,
y
)
+
G
(
u
n
,
y
)
≤
1
r
n
〈
y
-
u
n
,
J
u
n
-
J
x
n
〉
,
∀
y
∈
C
,
y
n
=
β
n
x
n
+
(
1
-
β
n
)
T
n
u
n
,
x
n
+
1
=
α
n
(
x
n
)
+
(
1
-
α
n
)
I
n
y
n
,
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
∀
n
≥
1
.
Consequently, under suitable conditions, we show that iterative algorithms converge strongly to a solution of some optimization problem. Note that our methods do not use any projection.


## 2. Preliminaries

LetT
:
C
→
C be a mapping. Denote by F
i
x
(
T
) the set of fixed points of T, that is, F
i
x
(
T
)
=
{
x
∈
C
:
T
x
=
x
}. Throughout this paper, we always assume that F
i
x
(
T
)
≠
∅. Now we need the following known definitions.Definition 3.
A mappingT
:
C
→
C is said to be(1)
nonexpansive, if∥
T
x
-
T
y
∥
≤
∥
x
-
y
∥,  for all x, y
∈
C;
(2)
asymptotically nonexpansive, if there exists a sequence{
λ
n
}
⊂
[
1
,
∞
) with lim
⁡
n
→
∞
⁡
λ
n
=
1 such that ∥
T
n
x
-
T
n
y
∥
≤
λ
n
∥
x
-
y
∥,  for all x, y
∈
C and n
∈
N;
(3)
quasi-nonexpansive,∥
T
x
-
p
∥
≤
∥
x
-
p
∥, for all x
∈
C and p
∈
F
i
x
(
T
);
(4)
asymptotically quasi-nonexpansive, if there exists a sequence{
μ
n
}
⊂
[
1
,
∞
) with lim
⁡
n
→
∞
⁡
μ
n
=
1 such that ∥
T
n
x
-
p
∥
≤
μ
n
∥
x
-
p
∥, for all x, y
∈
C, p
∈
F
i
x
(
T
) and n
∈
N.There are many concepts which generalize a notion of nonexpansive mapping. In 2004, Shahzad [17] introduced the following concepts about I-nonexpansivity of a mapping T.Definition 4.
LetT
:
C
→
C and I
:
C
→
C be two mappings of a nonempty subset C, a real normal linear space E. Then T is said to be(i)
I-nonexpansive, if ∥
T
x
-
T
y
∥
≤
∥
I
x
-
I
y
∥, for all x, y
∈
C;
(ii)
asymptoticallyI-nonexpansive, if there exists a sequence {
λ
n
}
⊂
[
1
,
∞
) with lim
⁡
n
→
∞
⁡
λ
n
=
1 such that ∥
T
n
x
-
T
n
y
∥
≤
λ
n
∥
I
n
x
-
I
n
y
∥, for all x, y
∈
C and n
≥
1;
(iii)
asymptotically quasi-I-nonexpansive, if there exists a sequence {
μ
n
}
⊂
[
1
,
∞
) with lim
⁡
n
→
∞
⁡
μ
n
=
1 such that ∥
T
n
x
-
p
∥
≤
μ
n
∥
I
n
x
-
p
∥, for all x, y
∈
C, p
∈
F
i
x
(
T
)
⋂
‍
F
i
x
(
I
) and n
≥
1.Lemma 5 (see [4]).
Assume thatψ
:
K
→
R is convex, x
0
∈
core
K
C, ψ
(
x
0
)
≤
0, and ψ
(
y
)
≥
0, for all y
∈
C. Then ψ
(
y
)
≥
0, for all y
∈
K.Lemma 6 (see [18]).
LetC be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E, and let T be a relatively nonexpansive mapping from C into itself. Then F
i
x
(
T
) is closed and convex.Lemma 7 (see [19]).
Let{
a
n
}, {
b
n
} and {
σ
n
} be sequences of nonnegative real sequences satisfying the following conditions: for all n
≥
1
(1)
a
n
≤
a
n
+
b
n,
(2)
a
n
≤
(
1
+
σ
n
)
a
n
+
b
n,where ∑
n
=
1
∞
‍
σ
n
<
∞ and ∑
n
=
1
∞
‍
b
n
<
∞. Then lim
⁡
n
→
∞
⁡
a
n exists.Lemma 8 (see [20]).
LetE be a uniformly convex Banach space. Then, for each r
>
0, there exists a strictly increasing, continuous, and convex function h
:
[
0,2
r
]
→
R such that h
(
0
)
=
0 and
(14)
∥
t
x
+
(
1
-
t
)
y
∥
2
≤
t
∥
x
∥
2
+
(
1
-
t
)
∥
y
∥
2
-
t
(
1
-
t
)
h
(
∥
x
-
y
∥
2
)
,
for ∀
x, y
∈
B
r, t
∈
[
0,1
], where B
r
=
{
z
∈
E
:
∥
z
∥
≤
r
}.Lemma 9 (see [21]).
LetE be a uniformly convex Banach space and let b, c be two constants with 0
<
b
<
c
<
1. Suppose that {
t
n
} is a sequence in [
b
,
c
] and {
x
n
}, {
y
n
} are two sequence in E such that
(15)
lim
⁡
n
→
∞
∥
t
n
x
n
+
(
1
-
t
n
)
y
n
∥
=
d
,
limsup
⁡
n
→
∞
∥
x
n
∥
≤
d
,
limsup
⁡
n
→
∞
∥
y
n
∥
≤
d
holds some d
≥
0. Then lim
⁡
∥
x
n
-
y
n
∥
=
0.Definition 10 (see [22]).
The mappingsT, I
:
C
→
C are said to be satisfying condition (A) if there is a nondecreasing function f
:
[
0
,
∞
)
→
[
0
,
∞
) with f
(
0
)
=
0, f
(
r
)
>
0 for each r
∈
[
0
,
∞
) such that (
1
/
2
)
(
∥
x
-
T
x
∥
+
∥
x
-
I
x
∥
)
≥
f
(
d
(
x
,
Ω
)
) for all x
∈
C, where d
(
x
,
Ω
) = inf
⁡
{
∥
x
-
p
∥
:
p
∈
Ω = F
i
x
(
T
)
⋂
‍
F
i
x
(
I
)
}.Lemma 11 (see [23]).
LetE be a uniformly convex Banach space satisfying the Opial’s condition, C a nonempty closed subset of E, and T
:
C
→
C an asymptotically nonexpansive mapping. If the sequence {
x
n
}
⊂
C is a weakly convergent sequence with the weak limit p and if  lim
⁡
n
→
∞
⁡
∥
x
n
-
T
x
n
∥
=
0, then T
p
=
p.


## 3. Main Results

Theorem 12.
LetE be a smooth, strictly convex, and reflexive Banach space, and let C be a nonempty closed convex subset of E. Let F, G
:
C
×
C
→
R be two bifunctions which satisfy the conditions (
f
1
)
–
(
f
4
), (
g
1
)
–
(
g
3
), and (
H
). Then for every x
*
∈
E
*, there exists a unique point z
∈
C such that
(16)
0
≤
F
(
z
,
y
)
+
G
(
z
,
y
)
+
1
r
〈
y
-
z
,
J
z
-
J
x
*
〉
,
∀
y
∈
C
.The proof goes over the following three steps.Proof.
Step
1. There exists point z
∈
C such that
(17)
F
(
y
,
z
)
≤
G
(
z
,
y
)
+
1
r
〈
y
-
z
,
J
z
-
J
x
*
〉
,
∀
y
∈
C
.
Consider the closed sets(18)
T
r
(
y
)
=
{
+
1
r
〈
y
-
z
,
J
z
-
J
x
*
〉
,
y
∈
C
z
∈
C
∣
F
(
y
,
z
)
≤
G
(
z
,
y
)
+
1
r
〈
y
-
z
,
J
z
-
J
x
*
〉
,
y
∈
C
}
.
We will show that ⋂
y
∈
C
‍
T
r
(
y
)
≠
∅. Let y
i, i
∈
N, be a finite subset of C. Let I
⊂
N be nonempty. Let for all ξ
∈
conv
{
y
i
∣
i
∈
N
}. Then
(19)
ξ
=
∑
i
∈
I
‍
μ
i
y
i
with
μ
i
≥
0
(
i
∈
I
)
,
∑
i
∈
I
‍
μ
i
=
1
.
Assume, for contradiction, that
(20)
-
F
(
y
i
,
ξ
)
+
G
(
ξ
,
y
i
)
+
1
r
〈
y
i
-
ξ
,
J
ξ
-
J
x
*
〉
<
0
,
∀
i
∈
N
.
By the convexity of F and G and the monotonicity of F, we obtain that
(21)
0
=
F
(
ξ
,
ξ
)
+
G
(
ξ
,
ξ
)
+
1
r
〈
ξ
-
ξ
,
J
ξ
-
J
x
*
〉
≤
∑
i
∈
I
‍
μ
i
F
(
ξ
,
y
i
)
+
∑
i
∈
I
‍
μ
i
G
(
ξ
,
y
i
)
+
1
r
∑
i
∈
I
‍
μ
i
〈
y
i
-
ξ
,
J
ξ
-
J
x
*
〉
≤
-
∑
i
∈
I
‍
μ
i
F
(
y
i
,
ξ
)
+
∑
i
∈
I
‍
μ
i
G
(
ξ
,
y
i
)
+
1
r
∑
i
∈
I
‍
μ
i
〈
y
i
-
ξ
,
J
ξ
-
J
x
*
〉
=
∑
i
∈
I
‍
μ
i
[
+
1
r
〈
y
i
-
ξ
,
J
ξ
-
J
x
*
〉
-
F
(
y
i
,
ξ
)
+
G
(
ξ
,
y
i
)
+
1
r
〈
y
i
-
ξ
,
J
ξ
-
J
x
*
〉
]
<
0
,
and that is absurd. Hence (20) cannot be true. and we have F
(
y
i
,
ξ
)
≤
G
(
ξ
,
y
i
)
+
(
1
/
r
)
〈
y
i
-
ξ
,
J
ξ
-
J
x
*
〉 for some i
∈
I. Thus ξ
∈
⋂
y
∈
C
‍
T
r
(
y
i
) for some i
∈
N. Since for all ξ
∈
conv
{
y
i
∣
i
∈
N
}, it follows that
(22)
conv
{
y
i
∣
i
∈
N
}
⊂
{
T
r
(
y
i
)
∣
i
∈
N
}
.
By the sets T
r
(
y
i
) being closed, it follows form the standard version of the KKM-Theorem that
(23)
⋂
i
∈
N
‍
T
r
(
y
i
)
≠
∅
.
In other words, any finite subfamily of the family T
r
(
y
)
y
∈
C has nonempty intersection. Since these sets are closed subsets of the compact set C, it follows that the entire family has nonempty intersection. Hence(24)
⋂
y
∈
C
‍
T
r
(
y
)
≠
∅
.
Step
2. For every x
*
∈
E
*, the following statement are equivalent:(i)
z
∈
C, F
(
y
,
z
)
≤
G
(
z
,
y
)
+
〈
y
-
z
,
J
z
-
J
x
*
〉, for all y
∈
C,
(ii)
z
∈
C, 0
≤
F
(
z
,
y
)
+
G
(
z
,
y
)
+
〈
y
-
z
,
J
z
-
J
x
*
〉, for all y
∈
C.
Case
1. Let (ii) hold; since F is monotone, one has
(25)
F
(
z
,
y
)
≤
-
F
(
y
,
z
)
.
Hence (i) follows.
Case
2. Let (i) hold, for t with 0
<
t
≤
1 and y
∈
C, and let
(26)
x
t
=
t
y
+
(
1
-
t
)
z
.
Then x
t
∈
C, and from (i), F
(
x
t
,
z
)
≤
G
(
z
,
x
t
)
+
〈
x
t
-
z
,
J
z
-
J
x
*
〉. By the properties of F and G, it follows then, for all 0
<
t
≤
1,
(27)
0
=
F
(
x
t
,
x
t
)
+
G
(
x
t
,
x
t
)
+
〈
x
t
-
x
t
,
J
z
-
J
x
*
〉
≤
t
F
(
x
t
,
y
)
+
(
1
-
t
)
F
(
x
t
,
z
)
+
t
G
(
x
t
,
y
)
+
(
1
-
t
)
G
(
x
t
,
z
)
≤
F
(
x
t
,
y
)
+
G
(
x
t
,
y
)
.
Let t
→
0 and thereby x
t
→
z and using the hemicontinuity of F we obtain in the limit(28)
0
≤
F
(
z
,
y
)
+
G
(
z
,
y
)
+
〈
y
-
z
,
J
z
-
J
x
*
〉
.
Step
3. Take ψ
(
·
)
=
F
(
z
,
·
)
+
G
(
z
,
·
)
+
〈
·
-
z
,
J
z
-
J
x
*
〉. Then the function ψ
(
·
) is convex and ψ
(
y
)
≥
0, for all y
∈
C. If z
∈
core
K
C, then set x
0
=
z. If z
∈
C
∖
core
K
C, then set x
0
=
a, where a is as in assumption H for x
=
z. In both cases x
0
∈
core
K
C, and ψ
(
x
0
)
≤
0. Hence it follows from the Lemma 5 that
(29)
ψ
(
y
)
≥
0
∀
y
∈
C
,
that
is
,
F
(
z
,
y
)
+
G
(
z
,
y
)
+
〈
y
-
z
,
J
z
-
J
x
*
〉
≥
0
,
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
∀
y
∈
K
.Corollary 13.
LetE be a smooth, strictly convex, and reflexive Banach space, and let C be a nonempty closed convex subset of E. Let F, G
:
C
×
C
→
R be two bifunctions which satisfy the following conditions: (
f
1
)
–
(
f
4
), (
g
1
)
–
(
g
3
), and (
H
) in Theorem 12. There for every x
*
∈
E and r
>
0, there exists a unique point z
r
∈
C such that
(30)
0
≤
F
(
z
r
,
y
)
+
G
(
z
r
,
y
)
+
1
r
〈
y
-
z
r
,
J
z
r
-
J
x
*
〉
,
kkkkkkkkkkkkkkkkkkkkkkkkkkkkkk
∀
y
∈
C
.Proof.
Letx
∈
E and r
>
0 be given. Note that functions r
F and r
G also satisfy the conditions (
f
1
)
–
(
f
4
) and (
g
1
)
–
(
g
3
). Therefore, for J
x
*
∈
E
*, there exists a unique point z
r
∈
C such that
(31)
r
F
(
z
r
,
y
)
+
r
G
(
z
r
,
y
)
+
〈
y
-
z
r
,
J
z
r
-
J
x
*
〉
≥
0
,
kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk
∀
y
∈
C
.
This completes the proof.Under the same assumptions in Corollary13, for every r
>
0, we may define a single-valued mapping S
r
:
E
→
C as follows:
(32)
S
r
(
x
)
=
{
1
r
z
∈
C
∣
0
≤
F
(
z
,
y
)
+
G
(
z
,
y
)
+
1
r
〈
y
-
z
,
J
z
-
J
x
〉
,
∀
y
∈
C
}
,
for x
∈
E, which is called the resolvent of F and G for r.Theorem 14.
LetE be a smooth, strictly convex, and reflexive Banach space, and let C be a nonempty closed convex subset of E. Let F, G
:
C
×
C
→
R be two bifunctions which satisfy conditions (
f
1
)
–
(
f
4
), (
g
1
)
–
(
g
3
), and (
H
). For r
>
0 and x
∈
E, define a mapping S
r in (32). Then, the following hold:(a)
S
r is single-valued;
(b)
S
r is a firmly nonexpansive mapping, that is,
(33)
〈
S
r
x
-
S
r
y
,
J
S
r
x
-
J
S
r
y
〉
≤
〈
S
r
x
-
S
r
y
,
J
x
-
J
y
〉
,
iiiiiiiiiiiiiiiiiiii
∀
x
,
y
∈
E
;
(c)
F
i
x
(
S
r
)
=
MEP
⁡
(
F
,
G
);
(d)
MEP
⁡
(
F
,
G
) is closed and convex;
(e)
ϕ
(
p
,
S
r
x
)
+
ϕ
(
S
r
x
,
x
)
≤
ϕ
(
p
,
x
).Proof.
We divide the proof into several steps.
Step
1 (S
r is single-valued). Indeed, for x
∈
C and r
>
0, let z
1
,
z
2
∈
S
r
x. Then
(34)
F
(
z
1
,
z
2
)
+
G
(
z
1
,
z
2
)
+
1
r
〈
z
2
-
z
1
,
J
z
1
-
J
x
〉
≥
0
,
F
(
z
2
,
z
1
)
+
G
(
z
2
,
z
1
)
+
1
r
〈
z
1
-
z
2
,
J
z
1
-
J
x
〉
≥
0
.
Adding the two inequalities, we obtain
(35)
F
(
z
1
,
z
2
)
+
F
(
z
2
,
z
1
)
+
G
(
z
1
,
z
2
)
+
G
(
z
2
,
z
1
)
+
1
r
〈
z
1
-
z
2
,
J
z
1
-
J
z
2
〉
≥
0
.
From (
f
2
), (
g
2
), and r
>
0, we obtain
(36)
1
r
〈
z
1
-
z
2
,
J
z
1
-
J
z
2
〉
≥
0
.
Since E is strictly convex, we obtain(37)
z
1
=
z
2
.
Step
2 (S
r is a firmly nonexpansive mapping). For x, y
∈
C, we obtain
(38)
F
(
S
r
x
,
S
r
y
)
+
G
(
S
r
x
,
S
r
y
)
+
1
r
〈
S
r
y
-
S
r
x
,
J
S
r
x
-
J
x
〉
≥
0
,
F
(
S
r
y
,
S
r
x
)
+
G
(
S
r
y
,
S
r
x
)
+
1
r
〈
S
r
x
-
S
r
y
,
J
S
r
y
-
J
y
〉
≥
0
.
Adding the two inequalities, we obtain
(39)
F
(
S
r
x
,
S
r
y
)
+
F
(
S
r
y
,
S
r
x
)
+
G
(
S
r
x
,
S
r
y
)
+
G
(
S
r
y
,
S
r
x
)
+
1
r
〈
S
r
y
-
S
r
x
,
J
S
r
x
-
J
S
r
y
-
J
x
+
J
y
〉
≥
0
.
From (
f
2
), (
g
2
), and r
>
0, we obtain
(40)
〈
S
r
y
-
S
r
x
,
J
S
r
x
-
J
S
r
y
-
J
x
+
J
y
〉
≥
0
.
Therefore, we have(41)
〈
S
r
x
-
S
r
y
,
J
S
r
x
-
J
S
r
y
〉
≤
〈
S
r
x
-
S
r
y
,
J
x
-
J
y
〉
.
Step  3 (F
i
x
(
S
r
)
=
MEP
⁡
(
F
,
G
)). Indeed, we obtain the following equation:(42)
u
∈
F
i
x
(
S
r
)
⟺
u
=
S
r
u
⟺
F
(
u
,
y
)
+
G
(
u
,
y
)
+
1
r
〈
y
-
u
,
J
u
-
J
u
〉
≥
0
,
∀
y
∈
C
,
⟺
F
(
u
,
y
)
+
G
(
u
,
y
)
,
∀
y
∈
C
,
⟺
u
∈
MEP
⁡
(
F
,
G
)
.
Step
4 (MEP
⁡
(
F
,
G
) is closed and convex). From (c), we have MEP
⁡
(
F
,
G
)
=
F
i
x
(
S
r
), and from (b), we obtain
(43)
〈
S
r
x
-
S
r
y
,
J
S
r
x
-
J
S
r
y
〉
≤
〈
S
r
x
-
S
r
y
,
J
x
-
J
y
〉
,
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
i
x
,
y
∈
C
.
Moreover, we obtain
(44)
ϕ
(
S
r
x
,
S
r
y
)
+
ϕ
(
S
r
y
,
S
r
x
)
=
2
∥
S
r
x
∥
2
-
2
〈
S
r
x
,
J
S
r
y
〉
-
2
〈
S
r
y
,
J
S
r
x
〉
+
2
∥
S
r
y
∥
2
=
2
〈
S
r
x
,
S
r
x
-
J
S
r
y
〉
+
2
〈
S
r
y
,
S
r
y
-
J
S
r
x
〉
=
2
〈
S
r
x
-
S
r
y
,
S
r
x
-
J
S
r
y
〉
,
ϕ
(
S
r
x
,
y
)
+
ϕ
(
S
r
y
,
x
)
-
ϕ
(
S
r
x
,
x
)
-
ϕ
(
S
r
y
,
y
)
=
∥
S
r
x
∥
2
-
2
〈
S
r
x
,
J
y
〉
+
∥
y
∥
2
+
∥
S
r
y
∥
2
-
2
〈
S
r
y
,
J
x
〉
+
∥
x
∥
2
-
∥
S
r
x
∥
2
+
2
〈
S
r
x
,
J
x
〉
-
∥
y
∥
2
-
∥
S
r
y
∥
2
+
2
〈
S
r
y
,
J
y
〉
-
∥
x
∥
2
=
2
〈
S
r
x
,
J
x
-
J
y
〉
+
2
〈
S
r
y
,
J
y
-
J
x
〉
=
2
〈
S
r
x
-
S
r
y
,
J
x
-
J
y
〉
.
Hence, we obtain
(45)
ϕ
(
S
r
x
,
S
r
y
)
+
ϕ
(
S
r
y
,
S
r
x
)
≤
ϕ
(
S
r
x
,
y
)
+
ϕ
(
S
r
y
,
x
)
-
ϕ
(
S
r
x
,
x
)
-
ϕ
(
S
r
y
,
y
)
.
So we get
(46)
ϕ
(
S
r
x
,
S
r
y
)
+
ϕ
(
S
r
y
,
S
r
x
)
≤
ϕ
(
S
r
x
,
y
)
+
ϕ
(
S
r
y
,
x
)
.
Taking y
=
u
∈
F
i
x
(
S
r
), we obtain
(47)
ϕ
(
u
,
S
r
x
)
≤
ϕ
(
u
,
x
)
.
Next, we show thatF
^
i
x
(
S
r
)
=
MEP
⁡
(
F
,
G
). Let p
∈
F
^
i
x
(
S
r
). Then, there exists the sequence of {
z
n
∈
E
} such that z
n
⇀
p and lim
⁡
n
→
∞
⁡
(
z
n
-
S
r
z
n
)
=
0. Moreover, we obtain S
r
z
n
⇀
p. Hence we have p
∈
C. Since J is uniformly continuous on bounded sets, we obtain
(48)
lim
⁡
n
→
∞
∥
J
z
n
-
J
S
r
z
n
∥
=
0
.
Form the definition of S
r, we obtain
(49)
F
(
S
r
z
n
,
y
)
+
G
(
S
r
z
n
,
y
)
+
1
r
〈
y
-
S
r
z
n
,
J
S
r
z
n
-
J
z
n
〉
≥
0
.
Since the monotone of the F, we have
(50)
G
(
S
r
z
n
,
y
)
+
1
r
〈
y
-
S
r
z
n
,
J
S
r
z
n
-
J
z
n
〉
≥
-
F
(
S
r
z
n
,
y
)
=
F
(
y
,
S
r
z
n
)
.
According to (48) and z
n
⇀
p and form (
f
3
) and (
g
2
), we obtain
(51)
F
(
y
,
p
)
≤
G
(
p
,
y
)
,
∀
y
∈
C
.
For t with 0
<
t
≤
1 and y
∈
H, let x
t
=
t
y
+
(
1
-
t
)
p; then by the convexity of F and G we have
(52)
0
=
F
(
x
t
,
x
t
)
+
G
(
x
t
,
x
t
)
≤
t
F
(
x
t
,
y
)
+
(
1
-
t
)
F
(
x
t
,
p
)
+
t
G
(
x
t
,
y
)
+
(
1
-
t
)
G
(
x
t
,
ω
)
≤
t
F
(
x
t
,
y
)
+
t
G
(
x
t
,
y
)
.
Passing t
→
0
+ and by (
f
1
) and (
g
1
), we have 0
≤
F
(
p
,
y
)
+
G
(
p
,
y
) for all y
∈
H. Therefore, p
∈
MEP
⁡
(
F
,
G
). So, we get F
i
x
(
S
r
)
=
MEP
⁡
(
F
,
G
)
=
F
^
i
x
(
S
r
). Therefore, we have that S
r is a relatively nonexpansive mapping. From Lemma 6, then F
i
x
(
S
r
)
=
MEP
⁡
(
F
,
G
) is closed and convex.
Step
5 (ϕ
(
p
,
S
r
x
)
+
ϕ
(
S
r
x
,
x
)
≤
ϕ
(
p
,
x
)). From (b) and (45), for each x, y
∈
E, we obtain
(53)
ϕ
(
S
r
x
,
S
r
y
)
+
ϕ
(
S
r
y
,
S
r
x
)
≤
ϕ
(
S
r
x
,
y
)
+
ϕ
(
S
r
y
,
x
)
-
ϕ
(
S
r
x
,
x
)
-
ϕ
(
S
r
y
,
y
)
.
Letting y
=
p
∈
F
i
x
(
S
r
), we obtain
(54)
ϕ
(
p
,
S
r
x
)
+
ϕ
(
S
r
x
,
x
)
≤
ϕ
(
p
,
x
)
.IfG
(
x
,
y
)
=
ψ
(
x
)
-
ψ
(
y
) and form Theorems 12 and 14, we obtain the following corollary.Corollary 15 (see [24]).
LetE be a smooth, strictly convex, and reflexive Banach space, and C be a nonempty closed convex subset of E. Let F
:
C
×
C
→
R be a bifunctions which satisfy conditions (
f
1
)
–
(
f
4
). Let ψ
:
C
→
R be a lower semi-continuous and convex function. For r
>
0 and x
∈
E. Then, the following hold:(i)
0
≤
F
(
z
,
y
)
+
ψ
(
y
)
-
ψ
(
z
)
+
(
1
/
r
)
〈
y
-
z
,
J
z
-
J
x
*
〉, for all y
∈
C.
(ii)
If we define a mappingS
r
:
E
→
C as follows:
(55)
S
r
(
x
)
=
{
1
r
x
∈
C
∣
≤
0
≤
F
(
z
,
y
)
+
ψ
(
y
)
-
ψ
(
z
)
+
1
r
〈
y
-
z
,
J
z
-
J
x
〉
,
y
∈
C
}
,
and the mapping S
r has the following properties:
(a)
S
r is single-valued;
(b)
S
r is a firmly nonexpansive mapping, that is,
(56)
〈
S
r
z
-
S
r
y
,
J
S
r
z
-
J
S
r
y
〉
≤
〈
S
r
z
-
S
r
y
,
J
z
-
J
y
〉
,
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
∀
z
,
y
∈
E
;
(c)
F
i
x
(
S
r
)
=
MEP
⁡
(
F
,
ψ
);
(d)
MEP
⁡
(
F
,
ψ
) is closed and convex;
(e)
ϕ
(
p
,
S
r
z
)
+
ϕ
(
S
r
z
,
z
)
≤
ϕ
(
p
,
z
).


## 4. Strong Convergence Theorems

In this section, we introduce a new iterative scheme for finding a common element of the set of solutions of the mixed equilibrium problems and the set of fixed points forI-asymptotically nonexpansive mapping in Banach spaces.Theorem 16.
LetE be uniformly smooth and uniformly convex Banach space, and let C be a nonempty closed convex subset of E. Let F, G
:
C
×
C
→
R be two bifunctions which satisfy the conditions (
f
1
)
–
(
f
4
), (
g
1
)
–
(
g
3
), and (
H
), and let T be I-asymptotically nonexpansive self-mapping of C with sequences {
s
n
}
⊂
[
0
,
∞
} such that ∑
n
=
1
∞
‍
s
n
<
∞, and let I be asymptotically nonexpansive self-mapping of C with sequences {
t
n
}
⊂
[
0
,
∞
) such that ∑
n
=
1
∞
‍
t
n
<
∞, and Ω
=
F
i
x
(
I
)
⋂
‍
F
i
x
(
T
)
⋂
‍
MEP
⁡
(
F
,
G
)
≠
∅. For an initial point x
0
∈
C, generate a sequence {
x
n
} by
(57)
u
n
∈
C
s
u
c
h
t
h
a
t
F
(
u
n
,
y
)
+
G
(
u
n
,
y
)
≤
1
r
n
〈
y
-
u
n
,
J
u
n
-
J
x
n
〉
,
sh
that
F
(
u
n
,
y
)
+
G
(
u
n
,
y
)
≤
1
r
n
〈
y
-
u
n
,
J
u
n
〉
∀
y
∈
C
,
y
n
=
β
n
x
n
+
(
1
-
β
n
)
T
n
u
n
,
x
n
+
1
=
α
n
(
x
n
)
+
(
1
-
α
n
)
I
n
y
n
,
∀
n
≥
1
,
where {
α
n
} is a sequence in [
0,1
], {
β
n
}
⊂
[
a
,
b
] for some a, b
∈
(
0,1
) and {
r
n
}
⊂
[
d
,
+
∞
) for d
>
0. If the following conditions are satisfied:(i)
∑
n
=
0
∞
‍
α
n
<
∞;
(ii)
∑
n
=
0
∞
‍
β
n
<
∞;
(iii)
liminf
⁡
n
→
∞
⁡
r
n
>
0, then the sequence {
x
n
} generated by (57) converges strongly to a fixed point in Ω if and only if
(58)
liminf
⁡
n
→
∞
⁡
d
(
x
n
,
Ω
)
=
0
.Proof.
We divide the proof into several steps.
Step
1 (The sequence {
x
n
} is bounded). Let u
n
=
T
r
n
x
n. Since T is a I-asymptotically nonexpansive mapping, it follows from and Theorem 14 that Ω
:
=
F
i
x
(
T
)
⋂
‍
F
i
x
(
I
)
⋂
‍
MEP
⁡
(
F
,
G
) is nonempty closed convex subset E and for each p
∈
Ω. 							(59)
∥
x
n
+
1
-
p
∥
=
∥
α
n
(
x
n
)
+
(
1
-
α
n
)
I
n
y
n
-
p
∥
=
α
n
∥
x
n
-
p
∥
+
(
1
-
α
n
)
∥
I
n
y
n
-
p
∥
≤
α
n
∥
x
n
-
p
∥
+
(
1
-
α
n
)
(
1
+
t
n
)
∥
y
n
-
p
∥
.
Again from (57), we obtain that
(60)
∥
y
n
-
p
∥
=
∥
β
n
x
n
+
(
1
-
β
n
)
T
n
u
n
-
p
∥
≤
β
n
∥
x
n
-
p
∥
+
(
1
-
β
n
)
∥
T
n
u
n
-
p
∥
≤
β
n
∥
x
n
-
p
∥
+
(
1
-
β
n
)
×
(
1
+
s
n
)
∥
I
n
u
n
-
p
∥
≤
β
n
∥
x
n
-
p
∥
+
(
1
-
β
n
)
×
(
1
+
s
n
)
(
1
+
t
n
)
∥
T
r
x
n
-
p
∥
≤
β
n
∥
x
n
-
p
∥
+
(
1
-
β
n
)
×
(
1
+
s
n
)
(
1
+
t
n
)
∥
x
n
-
p
∥
=
[
1
+
(
1
-
β
n
)
(
s
n
+
t
n
+
s
n
t
n
)
]
∥
x
n
-
p
∥
.
From (59) and (60), we obtain
(61)
∥
x
n
+
1
-
p
∥
≤
α
n
∥
x
n
-
p
∥
+
(
1
-
α
n
)
(
1
+
t
n
)
×
[
1
+
(
1
-
β
n
)
(
s
n
+
t
n
+
s
n
t
n
)
]
∥
x
n
-
p
∥
≤
α
n
∥
x
n
-
p
∥
+
(
1
-
α
n
)
(
1
+
t
n
)
×
[
1
+
(
1
-
β
n
)
(
s
n
+
t
n
+
s
n
t
n
)
]
∥
x
n
-
p
∥
≤
(
1
+
ρ
n
)
∥
x
n
-
p
∥
,
where
(62)
ρ
n
=
(
1
-
α
n
)
(
1
-
β
n
)
(
s
n
+
t
n
+
s
n
t
n
)
+
(
1
-
α
n
)
t
n
+
(
1
-
α
n
)
(
1
-
β
n
)
×
t
n
(
s
n
+
t
n
+
s
n
t
n
)
.
Moreover since {
α
n
}
⊂
[
0,1
], {
β
n
}
⊂
[
a
,
b
] for some a, b
∈
(
0,1
), ∑
n
=
1
∞
‍
s
n
<
∞, and ∑
n
=
1
∞
‍
t
n
<
∞, it follow that ∑
n
=
1
∞
‍
ρ
n
<
∞. Form (60) and, by Lemma 7, we obtain that the limit of {
∥
x
n
-
p
∥
} exists for each p
∈
Ω. This implies that {
∥
x
n
-
p
∥
} is bounded and so are {
x
n
}, {
y
n
}, {
u
n
}, {
I
n
y
n
}, and {
T
n
u
n
}; on the other hand, we obtain that d
(
x
n
+
1
,
Ω
)
≤
(
1
+
ρ
n
)
d
(
x
n
,
Ω
). Then by Lemma 7, lim
⁡
n
→
∞
⁡
d
(
x
n
,
Ω
) exists and, by assumption liminf
⁡
n
→
∞
⁡
d
(
x
n
,
Ω
)
=
0, we obtain
(63)
lim
⁡
n
→
∞
d
(
x
n
,
Ω
)
=
0
.
Step
2 (lim
⁡
n
→
∞
⁡
∥
x
n
-
x
n
+
1
∥
=
0). Taking lim sup on both sides in the above inequality,
(64)
lim
⁡
n
→
∞
∥
y
n
-
p
∥
=
d
.
Since I
n is asymptotically nonexpansive self-mappings of C, we can get that ∥
I
n
y
n
-
p
∥
≤
(
1
+
t
n
)
∥
y
n
-
p
∥, which on taking limsup
⁡
n
→
∞
⁡ and using (64), we obtain
(65)
limsup
⁡
n
→
∞
∥
I
n
y
n
-
p
∥
≤
d
.
Further,
(66)
lim
⁡
n
→
∞
∥
x
n
+
1
-
p
∥
≤
d
.
That means that
(67)
lim
⁡
n
→
∞
∥
α
n
(
x
n
)
+
(
1
-
α
n
)
I
n
y
n
-
p
∥
≤
d
,
lim
⁡
n
→
∞
α
n
∥
(
x
n
)
-
p
∥
+
(
1
-
α
n
)
∥
I
n
y
n
-
p
∥
≤
d
.
It follows from Lemma 9 that
(68)
lim
⁡
n
→
∞
∥
I
n
y
n
-
x
n
∥
=
0
.
Moveover,
(69)
∥
x
n
+
1
-
x
n
∥
=
∥
(
1
-
α
n
)
(
I
n
y
n
-
x
n
)
∥
=
(
1
-
α
n
)
∥
I
n
y
n
-
x
n
∥
.
Thus, from (68), we have
(70)
lim
⁡
n
→
∞
∥
x
n
+
1
-
x
n
∥
=
0
.
Step
3 (lim
⁡
n
→
∞
⁡
∥
x
n
-
T
n
u
n
∥
=
0). Use (57) again, and Lemma 8 that for r
=
sup
⁡
n
≥
1
⁡
{
∥
x
n
∥
,
∥
u
n
∥
,
∥
T
n
u
n
∥
}, there exists a strictly increasing, continuous and convex function h
:
[
1,2
]
→
R that h
(
0
)
=
0 and
(71)
∥
x
n
+
1
-
p
∥
2
≤
α
n
∥
x
n
-
p
∥
2
+
(
1
-
α
n
)
∥
I
n
y
n
-
p
∥
2
-
α
n
(
1
-
α
n
)
h
(
∥
x
n
-
I
n
y
n
∥
2
)
≤
α
n
∥
x
n
-
p
∥
2
+
(
1
-
α
n
)
(
1
+
t
n
)
∥
y
n
-
p
∥
2
-
α
n
(
1
-
α
n
)
h
(
∥
x
n
-
I
n
y
n
∥
2
)
≤
α
n
∥
x
n
-
p
∥
2
+
(
1
-
α
n
)
(
1
+
t
n
)
×
(
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
∥
T
n
u
n
-
p
∥
2
-
β
n
(
1
-
β
n
)
h
(
∥
x
n
-
T
n
u
n
∥
2
)
)
-
α
n
(
1
-
α
n
)
h
(
∥
x
n
-
I
n
y
n
∥
2
)
≤
α
n
∥
x
n
-
p
∥
2
+
(
1
-
α
n
)
(
1
+
t
n
)
×
(
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
(
1
+
s
n
)
∥
u
n
-
p
∥
2
-
β
n
(
1
-
β
n
)
h
(
∥
x
n
-
T
n
u
n
∥
2
)
)
-
α
n
(
1
-
α
n
)
h
(
∥
x
n
-
I
n
y
n
∥
2
)
≤
α
n
∥
x
n
-
p
∥
2
+
(
1
-
α
n
)
×
(
1
+
t
n
)
(
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
(
1
+
s
n
)
∥
x
n
-
p
∥
2
-
β
n
(
1
-
β
n
)
h
(
∥
x
n
-
T
n
u
n
∥
2
)
)
-
α
n
(
1
-
α
n
)
h
(
∥
x
n
-
I
n
y
n
∥
2
)
≤
(
1
+
ρ
n
)
∥
x
n
-
p
∥
2
-
(
1
-
α
n
)
×
(
1
+
t
n
)
β
n
(
1
-
β
n
)
h
(
∥
x
n
-
T
n
u
n
∥
2
)
-
α
n
(
1
-
α
n
)
h
(
∥
x
n
-
I
n
y
n
∥
2
)
,
where
(72)
ρ
n
=
(
1
-
α
n
)
(
1
-
β
n
)
×
(
s
n
+
t
n
+
s
n
t
n
)
+
(
1
-
α
n
)
t
n
+
(
1
-
α
n
)
×
(
1
-
β
n
)
t
n
(
s
n
+
t
n
+
s
n
t
n
)
.
From the discuss of the Step 1, we can easily know that ∑
n
=
1
∞
‍
ρ
n
<
∞. On the other hand, by (71) and the bounded sequence of {
x
n
}, we obtain that
(73)
(
1
-
α
n
)
(
1
+
t
n
)
β
n
(
1
-
β
n
)
h
(
∥
x
n
-
T
n
u
n
∥
)
≤
ϕ
(
x
n
,
p
)
-
ϕ
(
x
n
+
1
,
p
)
+
ρ
n
ϕ
(
x
n
,
p
)
.
From lim
⁡
n
→
∞
⁡
h
(
∥
x
n
-
T
n
u
n
∥
)
=
0, (73) and the property of h, we have
(74)
lim
⁡
n
→
∞
∥
x
n
-
T
n
u
n
∥
=
0
.
The same as the proof of (74), we can easily obtain that
(75)
lim
⁡
n
→
∞
∥
x
n
-
I
n
y
n
∥
=
0
.
From (57), we obtain that
(76)
∥
y
n
-
x
n
∥
≤
(
1
-
β
n
)
∥
x
n
-
T
n
u
n
∥
.
It follows that
(77)
lim
⁡
n
→
∞
∥
x
n
-
y
n
∥
=
0
.
Step
4 (lim
⁡
n
→
∞
⁡
∥
x
n
-
u
n
∥
=
0). Let p
∈
Ω
=
F
i
x
(
I
)
⋂
‍
F
i
x
(
T
)
⋂
‍
MEP
⁡
(
F
,
G
). Then, from (59) and (60), it follows that
(78)
∥
u
n
+
1
-
p
∥
=
∥
T
r
n
+
1
x
n
+
1
-
p
∥
≤
∥
x
n
+
1
-
p
∥
≤
∥
α
n
x
n
+
(
1
-
α
n
)
I
n
y
n
-
p
∥
≤
α
n
∥
x
n
-
p
∥
+
(
1
-
α
n
)
×
(
1
+
t
n
)
∥
y
n
-
p
∥
≤
α
n
∥
x
n
-
p
∥
+
(
1
-
α
n
)
(
1
+
t
n
)
×
[
β
n
∥
x
n
-
p
∥
+
(
1
-
β
n
)
×
(
1
+
s
n
)
(
1
+
t
n
)
∥
u
n
-
p
∥
]
≤
[
α
n
+
(
1
-
α
n
)
(
1
+
t
n
)
β
n
]
×
∥
x
n
-
p
∥
+
(
1
-
α
n
)
(
1
-
β
n
)
×
(
1
+
s
n
)
(
1
+
t
n
)
2
∥
u
n
-
p
∥
≤
M
1
∥
x
n
-
p
∥
+
(
1
+
M
2
)
∥
u
n
-
p
∥
,
where
(79)
M
1
=
α
n
+
(
1
-
α
n
)
(
1
+
t
n
)
β
n
,
M
2
=
[
t
n
(
2
+
t
n
)
(
1
+
s
n
)
+
s
n
]
(
α
n
β
n
-
α
n
-
β
n
)
+
(
t
n
(
2
+
t
n
)
(
1
+
s
n
)
+
s
n
)
+
(
α
n
β
n
-
α
n
-
β
n
)
.
Moreover since {
α
n
}
⊂
[
0,1
], {
β
n
}
⊂
[
a
,
b
] for some a, b
∈
(
0,1
), ∑
n
=
1
∞
‍
s
n
<
∞ and ∑
n
=
1
∞
‍
t
n
<
∞, we can easily claim that ∑
n
=
1
∞
‍
M
1
<
∞ and ∑
n
=
1
∞
‍
M
2
<
∞. By Lemma 7, we obtain that lim
⁡
n
→
∞
⁡
∥
u
n
-
p
∥ exists and from Theorem 14(b) and (78), we have
(80)
∥
x
n
-
u
n
∥
≤
∥
x
n
-
p
∥
-
∥
T
r
n
x
n
-
p
∥
=
∥
x
n
-
p
∥
-
∥
u
n
-
p
∥
≤
M
1
∥
x
n
-
1
-
p
∥
+
(
1
+
M
2
)
×
∥
u
n
-
1
-
p
∥
-
∥
u
n
-
p
∥
≤
M
1
∥
x
n
-
1
-
p
∥
+
M
2
∥
u
n
-
1
-
p
∥
+
∥
u
n
-
1
-
p
∥
-
∥
u
n
-
p
∥
.
Thus, since {
u
n
} converges, ∑
n
=
1
∞
‍
M
1
<
∞ and ∑
n
=
1
∞
‍
M
2
<
∞ and {
x
n
} is bounded, it follows form Lemma 7 that
(81)
lim
⁡
n
→
∞
∥
x
n
-
u
n
∥
=
0
.
Step
5 (lim
⁡
n
→
∞
⁡
∥
u
n
-
T
n
u
n
∥
=
0). By using the triangle inequality, we have
(82)
∥
T
n
u
n
-
u
n
∥
≤
∥
T
n
u
n
-
x
n
∥
+
∥
x
n
-
u
n
∥
.
Thus, from (74) and (81), we obtain that
(83)
lim
⁡
n
→
∞
∥
u
n
-
T
n
u
n
∥
=
0
.
Step
6 (lim
⁡
n
→
∞
⁡
∥
x
n
-
T
n
x
n
∥
=
lim
⁡
n
→
∞
∥
x
n
-
I
n
x
n
∥
=
0). By using the triangle inequality again, we obtain
(84)
∥
x
n
-
T
n
x
n
∥
≤
∥
x
n
-
T
n
u
n
∥
+
∥
T
n
x
n
-
T
n
u
n
∥
≤
∥
x
n
-
T
n
u
n
∥
+
(
1
+
s
n
)
∥
x
n
-
u
n
∥
.
From (74) and (81), we have
(85)
lim
⁡
n
→
∞
∥
x
n
-
T
n
x
n
∥
=
0
.
From (57), we have
(86)
∥
x
n
-
I
n
x
n
∥
≤
∥
x
n
-
I
n
y
n
∥
+
∥
I
n
x
n
-
I
n
y
n
∥
≤
∥
x
n
-
I
n
y
n
∥
+
(
1
+
t
n
)
∥
x
n
-
y
n
∥
≤
∥
x
n
-
I
n
y
n
∥
+
(
1
+
t
n
)
×
(
1
-
α
n
)
∥
x
n
-
I
n
y
n
∥
.
From {
α
n
}
⊂
[
0,1
], ∑
n
=
1
∞
‍
t
n
<
∞, and (68), we obtain
(87)
lim
⁡
n
→
∞
∥
x
n
-
I
n
x
n
∥
=
0
.
Step
7 (x
*
∈
Ω
=
F
i
x
(
I
)
⋂
‍
F
i
x
(
T
)
⋂
‍
MEP
⁡
(
F
,
G
)). Since {
u
n
} is bounded, there exists a subsequence {
u
n
k
} of {
u
n
} such that {
u
n
k
} converges weakly to x
*
∈
C when x
*
=
J
-
1
p
* for some p
*
∈
J
(
C
). From (61), we have that {
x
n
k
} converges weakly to x
*
∈
C and, by (77), we also have that {
y
n
k
} converges weakly to x
*
∈
C. Also, by (85), (87), and Lemma 11, we obtain that x
*
∈
F
i
x
(
I
)
⋂
‍
F
i
x
(
T
).
Next, we show thatx
*
∈
MEP
⁡
(
F
,
G
); that is, J
x
*
=
p
∈
J
(
MEP
⁡
(
F
,
G
)
). Since J is uniformly norm-to-norm continuous on bounded subset of E, it follows from (61) that
(88)
lim
⁡
n
→
∞
∥
J
x
n
-
J
u
n
∥
=
0
.
From the assumption r
n
∈
[
d
,
∞
), one sees
(89)
lim
⁡
n
→
∞
∥
J
x
n
-
J
u
n
∥
r
n
=
0
.
Since {
x
n
} is bounded and so is {
J
x
n
}, there exists a subsequence {
J
x
n
k
} of {
J
x
n
} such that {
J
x
n
⇀
p
*
}. Since {
u
n
} is bounded, by (89), we also obtain {
J
u
n
⇀
p
*
}. Noticing that u
n
=
T
r
n
x
n, we obtain
(90)
F
(
u
n
,
y
)
≤
G
(
y
,
u
n
)
+
1
r
n
〈
y
-
u
n
,
J
u
n
-
J
x
n
〉
,
y
∈
C
,
F
(
u
n
k
,
y
)
≤
G
(
y
,
u
n
k
)
+
〈
y
-
u
n
k
,
J
u
n
k
-
J
x
n
k
r
n
〉
,
F
(
u
n
k
,
y
)
i
1
G
(
y
,
u
n
k
)
〈
y
u
n
k
,
J
u
n
k
-
J
x
n
k
r
n
〉
y
∈
C
.
According to (89), we obtain lim
⁡
k
→
∞
⁡
(
∥
J
x
n
k
-
J
u
n
k
∥
/
r
n
k
)
=
0. Then, by the conditions of (f
2) and (h
2), we obtain
(91)
1
r
n
∥
y
-
u
n
∥
∥
J
x
n
-
J
u
n
∥
≥
〈
y
-
u
n
,
J
u
n
-
J
x
n
〉
≥
-
F
(
u
n
,
y
)
+
G
(
y
,
u
n
)
≥
F
(
y
,
u
n
)
+
G
(
y
,
u
n
)
.
Since (
1
/
r
n
)
∥
J
x
n
-
J
u
n
∥
⇀
0 and {
J
u
n
⇀
p
*
}, we obtain
(92)
F
(
y
,
p
*
)
+
G
(
y
,
p
*
)
≤
0
.
For t with 0
<
t
≤
1 and y
∈
E, let y
t
=
t
y
+
(
1
-
t
)
p
*, we obtain
(93)
F
(
y
t
,
p
*
)
+
G
(
y
t
,
p
*
)
≤
0
.
So, from the conditions of (f
1), (f
3), (h
1), and (h
3), we have
(94)
0
=
F
(
y
t
,
y
t
)
+
G
(
y
t
,
y
t
)
≤
t
F
(
y
t
,
y
)
+
(
1
-
t
)
F
(
y
t
,
p
*
)
+
t
G
(
y
t
,
y
)
+
(
1
-
t
)
G
(
y
t
,
p
*
)
≤
F
(
y
t
,
y
)
+
G
(
y
t
,
y
)
.
Consequently
(95)
F
(
y
t
,
y
)
+
G
(
y
t
,
y
)
≥
0
by (f
2) and (h
2), as t
→
0, and we obtain p
*
∈
MEP
⁡
(
F
,
G
).
Step
8 (The sequence of {
x
n
} converges strongly to a common Ω). From Step 1 and (61), for all p
∈
Ω, ∥
x
n
+
1
-
p
∥
≤
(
1
+
ρ
n
)
∥
x
n
-
p
∥ for n
≥
1 with ∑
n
=
1
∞
‍
ρ
n
<
∞. This implies that d
(
x
n
+
1
-
Ω
)
≤
(
1
+
ρ
n
)
d
(
x
n
-
Ω
). Then by Lemma 7, lim
⁡
n
→
∞
⁡
d
(
x
n
+
1
-
Ω
) exists. Also by Step 6, lim
⁡
n
→
∞
∥
x
n
-
T
n
x
n
∥
=
∥
x
n
-
I
n
x
n
∥
=
0, and by the condition (A) in Definition 10 which guarantees that lim
⁡
n
→
∞
f
(
d
(
x
n
+
1
-
Ω
)
)
=
0. Since f is a nondecreasing function and f
(
0
)
=
0, it follows that lim
⁡
n
→
∞
d
(
x
n
-
Ω
). Form (81), we obtain
(96)
∥
x
n
-
x
n
+
m
∥
≤
∥
x
n
-
u
n
∥
+
∥
x
n
+
m
-
u
n
+
m
∥
.
We know that {
x
n
} is Cauchy sequence in C for all numbers m, n. This implies that {
x
n
} converges strongly to p
∈
Ω. This completes the proof.IfT is an asymptotically quasi-nonexpansive self-mapping in Theorem 16, we easily obtain the following corollary.Corollary 17.
LetE be uniformly smooth and uniformly convex Banach space, and let C be a nonempty closed convex subset of E. Let F, G
:
C
×
C
→
R be two bifunctions which satisfy the conditions (
f
1
)
–
(
f
4
), (
g
1
)
–
(
g
3
), and (
H
), and let T be asymptotically quasi-nonexpansive self-mapping of C with sequences {
s
n
}
⊂
[
0
,
∞
} such that ∑
n
=
1
∞
‍
s
n
<
∞, and let I be identity self-mapping of C, and Ω
=
F
i
x
(
T
)
⋂
‍
MEP
⁡
(
F
,
G
)
≠
∅. For an initial point x
0
∈
C, generate a sequence {
x
n
} by
(97)
u
n
∈
C
s
u
c
h
t
h
a
t
F
(
u
n
,
y
)
+
G
(
u
n
,
y
)
≤
1
r
n
〈
y
-
u
n
,
J
u
n
-
J
x
n
〉
,
∀
y
∈
C
,
y
n
=
β
n
x
n
+
(
1
-
β
n
)
T
n
u
n
,
x
n
+
1
=
α
n
x
n
+
(
1
-
α
n
)
y
n
,
∀
n
≥
1
,
where {
α
n
} is a sequence in [
0,1
], {
β
n
}
⊂
[
a
,
b
] for some a, b
∈
(
0,1
) and {
r
n
}
⊂
[
d
,
+
∞
) for d
>
0. If the following conditions are satisfied:(i)
∑
n
=
0
∞
‍
α
n
<
∞;
(ii)
∑
n
=
0
∞
‍
β
n
<
∞;
(iii)
liminf
⁡
n
→
∞
⁡
r
n
>
0,then the sequence {
x
n
} generated by (97) converges strongly to a fixed point in Ω if and only if liminf
⁡
n
→
∞
d
(
x
n
,
Ω
)
=
0.


## 5. Numerical Example

In this section, we introduce an example of numerical test to illustrate the algorithms given in Corollary17.Example 1.
LetE
=
R, C
=
[
-
2000,2000
]. The mixed equilibrium problem is to find x
∈
C such that
(98)
F
(
x
,
y
)
+
G
(
x
,
y
)
≥
0
,
∀
y
∈
C
,
where we define F
(
x
,
y
)
=
-
3
x
2
+
2
x
y
+
y
2 and G
(
x
,
y
)
=
x
2
+
3
x
y
-
4
y
2.
Now, we can easily know thatF and G satisfy the conditions (
f
1
)
–
(
f
4
), (
g
1
)
–
(
g
3
), and (
H
) as follows:(f
1) F
(
x
,
x
)
=
-
3
x
2
+
2
x
x
+
x
2
=
0 for all x
∈
[
-
2000,2000
];
(f
2) F
(
x
,
y
)
+
F
(
y
,
x
)
=
-
2
(
x
-
y
)
2
≤
0 for all x
,
y
∈
[
-
2000,2000
];
(f
3) for all x
,
y
,
z
∈
[
-
2000,2000
],
(99)
limsup
⁡
t
→
0
+
F
(
x
+
t
(
z
-
x
)
,
y
)
=
limsup
⁡
t
→
0
+
-
3
(
x
+
t
(
z
-
x
)
)
2
+
2
x
+
t
(
z
-
x
)
y
+
y
2
=
-
3
x
2
+
2
x
y
+
y
2
≤
F
(
x
,
y
)
;
(f
4) for each x
∈
[
-
2000,2000
], θ
(
y
)
=
F
(
x
,
y
)
=
-
3
x
2
+
2
x
y
+
y
2 is convex and weakly lower semicontinuous.
(g
1) G
(
x
,
x
)
=
x
2
+
3
x
x
-
4
x
2
=
0 for each x
∈
[
-
2000,2000
];
(g
2) G
(
x
,
y
)
+
G
(
y
,
x
)
=
-
3
(
x
-
y
)
2
≤
0 for all x
,
y
∈
[
-
2000,2000
], and weakly upper semicontinuous in first variable;
(g
3) for each x
∈
[
-
2000,2000
], θ
(
y
)
=
G
(
x
,
y
)
=
x
2
+
3
x
y
-
4
y
2 is convex.
Next, we find the formula ofS
r
x. From Theorem 14, we can claim that S
r
x is single-valued, for any y
∈
C, r
>
0,
(100)
F
(
x
,
y
)
+
G
(
x
,
y
)
+
1
r
〈
x
-
z
,
y
-
x
〉
⟺
-
3
r
y
2
+
(
5
r
x
+
x
-
z
)
y
+
x
z
-
2
r
x
2
-
x
2
≥
0
.
Let M
(
y
)
=
-
3
r
y
2
+
(
5
r
x
+
x
-
z
)
y
+
x
z
-
2
r
x
2
-
x
2. Then M
(
y
) is a quadratic function of y with coefficients a
=
-
3
r, b
=
5
r
x
+
x
-
z, and c
=
x
z
-
2
r
x
2
-
x
2. So its discriminant Δ
=
b
2
-
4
a
c is
(101)
Δ
=
(
5
r
x
+
x
-
z
)
2
-
4
(
-
3
r
)
(
x
z
-
2
r
x
2
-
x
2
)
=
(
(
r
+
1
)
x
-
z
)
2
.According to M
(
y
)
≥
0 for all y
∈
C, form Δ
≤
0, that is(102)
(
(
r
+
1
)
x
-
z
)
2
≤
0
.
Therefore, it follows that
(103)
x
=
z
r
+
1
and so
(104)
S
r
z
=
z
r
+
1
.
Now, letC
=
[
-
1
/
π
,
1
/
π
] and |
k
|
<
1, and define a mapping T
:
C
→
C by
(105)
T
(
x
)
=
{
k
x
sin
1
x
,
if
x
≠
0
,
0
if
x
=
0
,
for all x
∈
C. From the example in [25–27], we can easily know that T is an asymptotically quasi-nonexpansive mapping; furthermore F
i
x
(
T
)
=
{
0
}.
According to Theorem14, we obtain
(106)
F
i
x
(
S
r
)
=
MEP
⁡
(
F
,
G
)
=
0
,
F
i
x
(
T
)
=
0
,
and so Ω
=
0. Therefore, all the assumptions in Corollary 17 are satisfied. we can obtain the following numerical algorithms.Algorithm 18.
Letr
n
=
1, α
n
=
1
/
n
2, and β
n
=
1
/
2
n
2. It is claim to check that
(107)
∑
n
=
0
∞
‍
α
n
<
∞
,
∑
n
=
0
∞
‍
β
n
<
∞
,
liminf
⁡
n
→
∞
r
n
=
1
.
For an initial value x
0
=
0.2 and k
=
0.5, let the sequences {
u
n
} and {
x
n
} generate by
(108)
T
(
x
)
=
1
2
x
sin
1
x
,
u
n
=
S
r
(
x
n
)
=
1
2
x
n
,
x
n
+
1
=
1
+
n
2
2
n
4
x
n
+
(
1
-
n
)
(
1
-
2
n
2
)
4
n
4
T
n
x
n
,
kkkkkkkkkkkkkkkkkkkkkkkkkikk
∀
n
≥
1
.
Then, by the Corollary17, the sequence {
x
n
} converges to a solution of Example 1. Let ∥
x
n
+
1
-
x
n
∥
≤
1
0
-
5 and x
* be the fixed point of the Algorithm 18. Using the software of MATLAB, we generated a sequence {
x
n
} convergence to x
*
=
x
7
=
0 as shown in Figure 1.
Hence the sequencex
n converges strongly to solve Example 1.Figure 1
Convergence of iterative sequence{
x
n
}.



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*Source: 965737-2014-10-27.xml*