Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2011, Article ID 372975, 18 pages
doi:10.1155/2011/372975
Research Article
New Iterative Scheme for Finite Families of
Equilibrium, Variational Inequality, and Fixed Point
Problems in Banach Spaces
Shenghua Wang
1, 2
and Caili Zhou
3
1
School of Applied Mathematics and Physics, North China Electric Power University,
Baoding 071003, China
2
Department of Mathematics, Gyeongsang National University, Jinju 660-714, Republic of Korea
3
College of Mathematics and Computer, Hebei University, Baoding 071002, China
Correspondence should be addressed to Shenghua Wang,
Received 6 December 2010; Accepted 30 January 2011
Academic Editor: S. Al-Homidan
Copyright q 2011 S. Wang and C. Zhou. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We introduced a new iterative scheme for finding a common element in the set of common fixed
points of a finite family of quasi-φ-nonexpansive mappings, the set of common solutions of a finite
family of equilibrium problems, and the set of common solutions of a finite family of variational
inequality problems in Banach spaces. The proof method for the main result is simplified under
some new assumptions on the bifunctions.
1. Introduction
x
x
∗
∈ E
∗
:
x, x
∗
x
2
x
∗
2
, ∀x ∈ E. 1.2
2 Fixed Point Theory and Applications
Let C be a nonempty closed and convex subset of E. The generalized projection Π : E → C is
a mapping that assigns to an arbitrary point x ∈ E the minimum point of the function φx, y,
−
x
2
≤ φ
y, x
≤
y
x
2
, ∀x, y ∈ E.
1.4
We remark that if E is a reflexive, strictly convex and smooth Banach space, then for x, y ∈ E,
φx, y0 if and only if x y. For more details on φ and Π, the readers are referred to 1–4.
Let T be a mapping from C into itself. We denote the set of fixed points of T by FT. T
is called to be nonexpansive if Tx − Ty≤x − y for all x, y ∈ C and quasi-nonexpansive
n
J
−1
α
n
Jx
n
1 − α
n
JTx
n
,
C
n
z ∈ C : φ
z, y
n
≤ φ
z, x
n
n
< 1 and lim sup
n →∞
α
n
< 1and
proved that the sequence {x
n
} generated by 1.5 converges strongly to P
FT
x, where P
FT
is
the generalized projection from C onto FT.
Let f be a bifunction from C × C to R. The equilibrium problem for f is to find p ∈ C
such that
f
p, y
≥ 0, ∀y ∈ C. 1.6
We use EPf to denote the solution set of the equilibrium problem 1.6.Thatis,
EP
f
p ∈ C : f
,
y
n
J
−1
α
n,0
Jx
n
N
i1
α
n,i
JT
i
x
n
,
u
n
∈ C such that f
u
n
,y
,
x
n1
Π
C
n1
x
0
,
1.8
where T
i
: C → C is a closed quasi-φ-nonexpansive mapping for each i ∈{1, 2, ,N},
α
n,0
, {α
n,1
}, ,{α
n,N
} are real sequences in 0, 1 satisfying
N
j0
α
n,j
1 for each n ≥ 1and
lim inf
n →∞
α
n,0
:C → E
∗
, i 1, 2, ,N, a finite family of continuous monotone mappings. For x ∈ E,
define the mappings F
r
n
, T
r
n
: E → C by
F
r
n
x
z ∈ C :
y − z, A
n
z
1
r
n
y − z, Jz − Jx
≥ 0, ∀y ∈ C
nmod L
and r
n
⊂ c
1
, ∞ for some c
1
> 0. Zegeye and Shahzad
25 introduced the following scheme:
x
0
∈ C
0
C chosen arbitrarily,
z
n
F
r
n
x
n
,
u
n
T
r
n
x
n
,
n
≤ φ
z, x
n
,
x
n1
Π
C
n1
x
0
,
1.10
where S
n
S
nmod D
, α
0
,α
1
,α
2
∈ 0, 1 such that α
0
Banach space. More precisely, let {S
i
}
N
1
i1
: C → C be a family of quasi-φ-nonexpansive
mappings, {f
i
}
N
2
i1
: C × C → R a finite family of bifunctions, and {A
i
}
N
3
i1
: C → E
∗
a
finite family of continuous monotone mappings such that F
N
1
i1
FS
i
∩
1,i
,
F
r
2,i
: E → C by
T
r
1,i
x
z ∈ C : f
i
z, y
1
r
1,i
y − z, Jz − Jx
≥ 0, ∀y ∈ C
,i 1, ,N
2
, 1.11
F
r
α
0
Jx
n
α
1
N
1
i1
λ
1,i
JS
i
x
n
α
2
N
2
i1
λ
2,i
JT
r
1,i
x
n
,
D
n
n
i1
C
i
,
x
n1
Π
D
n
x
1
,n≥ 1,
1.13
where α
0
,α
1
,α
2
,α
3
are the real numbers in 0, 1 satisfying α
0
α
r
2,j
, i 1, ,N
2
, j 1, ,N
3
is closed which is such that the proof
for the main result of this paper is simplified.
2. Preliminaries
The modulus of smoothness of a Banach space E is the function ρ
E
: 0, ∞ → 0, ∞ defined
by
ρ
E
τ
sup
x y
x − y
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.
It is known that if a Banach space E is uniformly smooth, then its dual space E
∗
is uniformly
convex.
A Banach space E is called to have the Kadec-Klee property if for any sequence {x
n
}⊂
E and x ∈ E with x
n
x, where denotes the weak convergence, and x
n
the following inequality holds:
x − y, Ax − Ay
≥ 0. 2.2
Let A be a monotone mapping from C into E
∗
. The variational inequality problem on A is
formulated as follows:
find a point u ∈ C such that
v − u, Au
≥ 0, ∀v ∈ C. 2.3
The solution set of the above variational inequality problem is denoted by VIC, A.
6 Fixed Point Theory and Applications
Next we state some lemmas which will be used later.
Lemma 2.1 see 1. Let C be a nonempty closed and convex subset of a smooth Banach space E and
x ∈ E. Then, x
0
Π
C
x if and only if
x
0
− y, Jx − Jx
0
≥ 0 ∀y ∈ C. 2.4
z, y
1
r
y − z, Jz − Jx
≥ 0, ∀y ∈ C;
2.6
b define a mapping T
r
: E → Cby
T
r
x
z ∈ C : f
z, y
1
r
y − z, Jz − Jx
≥ 0, ∀y ∈ C
. 2.7
4 T
r
is quasi-φ-nonexpansive;
5 EPf is closed and convex;
6 φp, T
r
xφT
r
x, x ≤ φp, x, for all p ∈ FT
r
.
Fixed Point Theory and Applications 7
Remark 2.5. Let A : C → E
∗
be a continuous monotone mapping and define fx, yy −
x, Ax for all x, y ∈ C. It is easy to see that f satisfies the conditions A1, A2, A3’,and
A4 and EPfVIC, A. Hence, for every real number r>0, if defining a mapping F
r
:
E → C by
F
r
x
z ∈ C :
y − z, Az
1
1 − λ
y
2
− w
p
λ
g
x − y
2.10
for all x, y ∈ B
s
0{x ∈ E : x≤s} and λ ∈ 0, 1,wherew
p
λλ
p
1 − λλ1 − λ
p
α
i
x
i
2
− α
j
α
k
g
x
j
− x
k
,j,k∈
{
1, 2, ,N
}
with j
/
k
2.11
}⊂E converge to x
and {T
r
x
n
} converge to x. To end the conclusion, we need
to prove that T
r
x
x. Indeed, for each x
n
, Lemma 2.4 shows that there exists a unique z
n
∈ C
such that z
n
T
r
x
n
,thatis,
f
z
n
,y
≥ 0, ∀y ∈ C,
2.13
which implies that T
r
x
x. This completes t he proof.
3. Main Results
Theorem 3.1. Let C be a nonempty closed and convex subset of a uniformly smooth and strictly
convex Banach space E which has the Kadec-Klee property. Let {S
i
}
N
1
i1
: C → C be a family of closed
quasi-φ-nonexpansive mappings, {f
i
}
N
2
i1
: C × C → R a finite family of bifunctions satisfying the
conditions (A1), (A2), (A3’), and (A4), and {A
i
}
N
3
i1
: C → E
i1
, {r
2,i
}
N
3
i1
⊂
0, ∞.Let{x
n
} be a sequence generated by the following manner:
x
1
∈ C chosen arbitrarily,
z
n
N
1
i1
λ
1,i
JS
i
x
n
,
u
n
J
−1
α
0
Jx
n
α
1
z
n
α
2
u
n
α
3
w
n
,
C
n
z ∈ C : φ
v, y
n
and F
r
2,j
j 1, 2, ,N
3
are defined by 1.11 and 1.12, α
0
,α
1
,α
2
,α
3
are the real numbers in 0, 1 satisfying α
0
α
1
α
2
α
3
1 and for each j 1, 2, 3, λ
j,1
, ,λ
j,N
j
are the real numbers in 0, 1 satisfying
N
j
n
2
y
n
2
≥ 0, it follows that C
n
is convex for each n ≥ 1. By the definition of D
n
,
we can conclude that D
n
is closed and convex for each n ≥ 1.
Fixed Point Theory and Applications 9
Next, we prove that F⊂D
n
for each n ≥ 1. From Lemma 2.4 and Remark 2.5,wesee
that each T
r
1,i
i 1, 2, ,N
2
and F
r
2,j
j 1, 2, ,N
3
are quasi-φ-nonexpansive. Hence, for
p
2
− 2
p, α
0
Jx
n
α
1
z
n
α
2
u
n
α
3
w
n
α
0
Jx
p, z
n
− 2α
2
p, u
n
− 2α
3
p, w
n
α
0
x
n
2
α
1
z
n
2
1
i1
λ
1,i
p, JS
i
x
n
− 2α
2
N
2
i1
λ
2,i
p, JT
r
1,i
x
n
− 2α
3
N
3
n
2
α
2
N
2
i1
λ
2,i
JT
r
1,i
x
n
2
α
3
N
3
i1
λ
3,i
α
2
N
2
i1
λ
2,i
φ
p, T
r
1,i
x
n
α
3
N
3
i1
λ
3,i
φ
p, F
r
2,i
2,i
φ
p, x
n
α
3
N
3
i1
λ
3,i
φ
p, x
n
φ
p, x
n
,
3.2
which implies that F⊂C
n
for each n ≥ 1. So, it follows from the definition of D
n
p, x
n1
≤ φ
p, x
1
, 3.3
for each p ∈F. This shows that the sequence {φx
n
,x
1
} is bounded. It follows from 1.4 that
the sequence {x
n
} is also bounded.
10 Fixed Point Theory and Applications
Since E is reflexive, we may, without loss of generality, assume that x
n
x
∗
. Since D
n
is closed and convex for each n ≥ 1, we can conclude that x
∗
∈ D
n
for each n ≥ 1. By the
definition of {x
n
,x
1
≤ lim sup
n →∞
φ
x
n
,x
1
≤ φ
x
∗
,x
1
.
3.5
This implies that
lim
n →∞
φ
x
n
,x
n
and x
n2
Π
D
n1
x
1
⊂ D
n
. It follows
from Lemma 2.2 that
φ
x
n2
,x
n1
φ
x
n2
, Π
D
n
x
1
≤ φ
.
3.8
Letting n →∞,weobtainthatφx
n2
,x
n1
→ 0. In view of x
n1
∈ D
n
n
i1
C
n
, we have
x
n1
∈ C
n
and hence
φ
x
n1
,y
n
≤ φ
as n →∞. 3.11
Fixed Point Theory and Applications 11
Hence,
Jy
n
−→
Jx
∗
as n →∞. 3.12
This implies that the sequence {Jy
n
} is bounded. Note that reflexivity of E implies reflexivity
of E
∗
. Thus, we may assume that Jy
n
y∈ E
∗
. Furthermore, reflexivity of E implies that
there exists x ∈ E such that y Jx. Then, it follows that
φ
x
2
− 2
x
n1
,Jy
n
Jy
n
2
.
3.13
Take lim inf on both sides of 3.13 over n and use weak lower semicontinuity of norm to get
that
0 ≥
x
∗
2
− 2
x
∗
2
− 2
x
∗
,Jx
x
2
φ
x
∗
,x
,
3.14
which implies that x
∗
x. Hence, y Jx
∗
. It follows that Jy
n
Jx
∗
.Now,from3.12 and
x
n
− x
∗
x
∗
− y
n
. 3.15
It follows that
lim
n →∞
x
n
− y
n
0.
3.16
Since J is uniformly norm-to-norm continuous on any bounded sets, we have
n
α
1
z
n
α
2
u
n
α
3
w
n
p
2
−2
p, α
0
Jx
n
α
1
2
≤
p
2
− 2
p, α
0
Jx
n
α
1
z
n
α
2
u
n
α
3
w
n
α
0
r
1,i
x
n
2
α
3
N
3
i1
λ
3,i
F
r
2,i
x
n
2
− α
0
α
1
λ
i
x
n
α
2
N
2
i1
λ
2,i
φ
p, T
r
1,i
x
n
α
3
N
3
i1
λ
3,i
φ
α
1
N
1
i1
λ
1,i
φ
p, x
n
α
2
N
2
i1
λ
2,i
φ
p, x
n
α
3
N
3
− α
0
α
1
λ
1,1
g
Jx
n
− JS
1
x
n
.
3.18
It follows that
α
0
α
1
λ
1,1
g
x
n
2
−
y
n
2
− 2
p, Jx
n
− Jy
n
≤
x
n
− y
n
φ
p, x
n
− φ
p, y
n
−→ 0asn −→ ∞ . 3.21
By 3.19, 3.21,andα
0
α
1
λ
1,1
> 0, we have
g
Jx
n
− JS
1
x
n
−→ 0asn −→ ∞ . 3.22
−
Jx
∗
|
|
x
n
−
x
∗
|
≤
x
n
− x
∗
. 3.24
This implies that
lim
n →∞
∗
≤
JS
1
x
n
− Jx
n
Jx
n
− Jx
∗
. 3.27
From 3.23 and 3.26, we arrive at
lim
n →∞
JS
1
x
n
− Jx
∗
JS
1
x
n
−
Jx
∗
|
≤
JS
1
x
n
− Jx
∗
, 3.29
by 3.28 we conclude that S
1
x
n
→x
∗
as n →∞. Since E enjoys the Kadec-Klee prop-
erty, we obtain that
lim
T
r
1,i
x
n
− x
∗
0,i 1, ,N
2
, 3.32
lim
n →∞
F
r
2,i
x
n
− x
∗
0,i 1, ,N
3
.
3.33
14 Fixed Point Theory and Applications
x
∗
x
∗
i 1, 2, ,N
2
and F
r
2,i
x
∗
x
∗
i 1, 2, ,N
3
. Now, it follows from
Lemma 2.4 and Remark 2.5 that FT
r
1,i
EPf
i
i 1, 2, ,N
2
and FF
r
2,i
VIC, A
i
i 1, 2, ,N
,byLemma 2.1,weseethat
x
n1
− p, Jx
1
− Jx
n1
≥ 0, ∀p ∈ D
n
. 3.34
Since F⊂D
n
for each n ≥ 1, we have
x
n1
− p, Jx
1
− Jx
n1
≥ 0, ∀p ∈F. 3.35
Letting n →∞in 3.35,weseethat
x
∗
− p, Jx
1
and F
r
2,j
i 1, 2, ,N
2
,j
1, 2, ,N
3
are closed. In fact, although the condition A3’ is stronger than A3,itisnot
easier to verify the condition A3 than verify the condition A3’. Hence, from this point, the
condition A3’ is acceptable. On the other hand, the definition of D
n
is of some interest.
If S
i
S for each i 1, 2, ,N
1
, f
i
f for each i 1, 2, ,N
2
and A
i
A for each
i 1, 2, ,N
3
, then Theorem 3.1 reduces to the following result.
Corollary 3.3. Let C be a nonempty closed and convex subset of a uniformly smooth and strictly
convex Banach space E which has the Kadec-Klee property. Let S : C → C be a closed quasi-φ-
nonexpansive mapping, f : C × C → R a bifunction satisfying the conditions (A1), (A2), (A3’), and
JT
r
1
x
n
α
3
JF
r
2
x
n
,
C
n
z ∈ C : φ
v, y
n
≤ φ
v, x
n
,
2
and r
2,j
r
2
j
1, 2, ,N
3
, α
0
,α
1
,α
2
,α
3
are the real numbers in 0, 1 satisfying α
0
α
1
α
2
α
3
1. Then the
sequence {x
n
} converges strongly to P
F
x
FS
i
∩
N
2
i1
EPf
i
∩
N
3
i1
VIC, A
i
/
∅.Let{r
1,i
}
N
2
i1
, {r
2,i
}
N
3
i1
T
r
1,i
x
n
,
w
n
N
3
i1
λ
3,i
F
r
2,i
x
n
,
y
n
α
0
x
n
α
,
D
n
n
i1
C
i
,
x
n1
P
D
n
x
1
,n≥ 1,
3.38
where {T
r
1,i
}
N
2
i1
and {F
r
1,i
}
i1
λ
j,i
1. Then the sequence {x
n
} converges
strongly to P
F
x
1
,whereP
F
is the projection from H onto F.
Proof. By the proof of Theorem 3.1, we have x
n
→ x
∗
as n →∞,
lim
n →∞
S
i
x
n
− x
n
0,i 1, 2, ,N
1
0,i 1, 2, ,N
3
.
3.39
Since each S
i
is closed, we can conclude that x
∗
∈ FS
i
, i 1, 2, ,N
1
.Notethatina
Hilbert space, a firmly-nonexpansive mapping is also nonexpansive. Hence, T
r
1
,i
and F
r
2,j
are
nonexpansive for each i 1, 2, ,N
2
and j 1, 2, ,N
3
. By demiclosed principle, we can
conclude that x
∗
∈ FT
r1,i
2
≤
x − y
2
I − T
x −
I − T
y
2
,
3.40
or equivalently,
I − T
Corollary 3.5. Let C be a nonempty closed and convex subset of a Hilbert space H. Let {S
i
}
N
1
i1
:
C → C be a family of closed quasi-nonexpansive mappings, {f
i
}
N
2
i1
: C × C → R a finite
family of bifunctions satisfying the conditions (A1)–(A4), and {T
i
}
N
3
i1
: C → H a finite family of
continuous pseudocontractions such that F
N
1
i1
FS
i
∩
x
1
∈ C chosen arbitrarily,
z
n
N
1
i1
λ
1,i
S
i
x
n
,
u
n
N
2
i1
λ
2,i
T
r
1,i
x
2
u
n
α
3
w
n
,
C
n
z ∈ C :
v − y
n
≤
v − x
n
,
D
n
x
z ∈ C :
y − x,
I − T
i
x
1
r
2,i
y − z, z − x
≥ 0 ∀y ∈ C
,i 1, 2, ,N
3
,
3.44
α
0
,α
F
x
1
,whereP
F
is the projection from H onto F.
If S
i
S, f
j
f,andT
k
T for each i 1, 2, ,N
1
, j 1, 2, ,N
2
,and
k 1, 2, ,N
3
, then Corollary 3.5 reduced the following result.
Corollary 3.6. Let C be a nonempty closed and convex subset of a Hilbert space H. Let S : C → C
be a closed quasi-nonexpansive mapping, f : C × C → R a bifunction satisfying the conditions (A1)–
(A4), and T : C → H a continuous pseudocontraction such that F FS ∩ EPf ∩ FT
/
∅.Let
r
1
, r
2
⊂ 0, ∞. Define a sequence {x
x
n
,
C
n
z ∈ C :
v − y
n
≤
v − x
n
,
D
n
n
i1
C
i
0
,α
1
,α
2
,α
3
are the real numbers in 0, 1 satisfying α
0
α
1
α
2
α
3
1.
Then the sequence {x
n
} converges strongly to P
F
x
1
,whereP
F
is the projection from H onto F.
Acknowledgment
This work was supported by the Natural Science Foundation of Hebei Province
A2010001482.
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