Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2011, Article ID 572156, 14 pages
doi:10.1155/2011/572156
Research Article
A New Strong Convergence Theorem for
Equilibrium Problems and Fixed Point Problems in
Banach Spaces
Weerayuth Nilsrakoo
Department of Mathematics, Statistics and Computer, Faculty of Science, Ubon Ratchathani University,
Ubon Ratchathani 34190, Thailand
Correspondence should be addressed to Weerayuth Nilsrakoo,
Received 5 June 2010; Revised 28 December 2010; Accepted 20 January 2011
Academic Editor: Fabio Zanolin
Copyright q 2011 Weerayuth Nilsrakoo. 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 w ork is properly cited.
We introduce a new iterative sequence for finding a common element of the set of fixed points of
a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in a Banach
space. Then, we study the strong convergence of the sequences. With an appropriate setting, we
obtain the corresponding results due to Takahashi-Takahashi and Takahashi-Zembayashi. Some of
our results are established with weaker assumptions.
1. Introduction
Throughout this paper, we denote by and the sets of positive integers a nd real numbers,
respectively. Let E be a Banach space, E
∗
the dual space of E and C a closed convex subsets
of E.LetF : C × C →
be a bifunction. The equilibrium problem is to find x ∈ C such that
F
x, y ∈ E
.
1.2
2 Fixed Point Theory and Applications
Using this functional, Matsushita and Takahashi 2, 3 studied and investigated the following
mappings in Banach spaces. A mapping S : C → E is relatively nonexpansive if the following
properties are satisfied:
R1 FS
/
,
R2 ϕp, Sx ≤ ϕp, x for all p ∈ FS and x ∈ C,
R3 FS
FS,
where FS and
FS denote the set of fixed points of S and the set of asymptotic fixed points
of S, respectively. It is known that S satisfies condition R3 if and only if I − S is demiclosed
at zero, where I is the identity mapping; that is, whenever a sequence {x
n
} in C converges
weakly to p and {x
n
−Sx
n
} converges strongly to 0, it follows that p ∈ FS. In a Hilbert space
H, the duality mapping J is an identity mapping and ϕx, yx − y
2
≥ 0, ∀y ∈ C,
x
n1
β
n
x
n
1 − β
n
S
α
n
u
1 − α
n
z
n
,
1.3
for every n ∈
,whereS is nonexpansive, {α
n
} and {β
n
− Ju
n
≥ 0, ∀y ∈ C,
u
n1
J
−1
α
n
Jx
n
1 − α
n
JSx
n
,
1.4
for every n ∈
, S is relatively nonexpansive, {α
n
} is an appropriate sequence in 0, 1,and
{r
n
< 1. 2.1
It is also said to be uniformly convex if for any ε>0, there exists δ>0suchthat
x
y
1,
x − y
≥ ε imply
x y
2
y
2
x, y ∈ E
,
2.3
where the duality mapping J : E → E
∗
is given by
x, Jx
x
2
Jx
2
x ∈ E
2
,
2.5
ϕ
x, J
−1
λJy
1 − λ
Jz
≤ λϕ
x, y
1 − λ
ϕ
x, z
, 2.6
for all λ ∈ 0, 1 and x, y, z ∈ E. The following lemma is an analogue of Xu’s inequality 22,
g
Jy − Jz
, 2.7
for all λ ∈ 0, 1, x ∈ E,andy, z ∈ B
r
.
It is also easy to see that if {x
n
} and {y
n
} are bounded sequences of a smooth Banach
space E,thenx
n
− y
n
→ 0 implies that ϕx
n
,y
n
→ 0.
4 Fixed Point Theory and Applications
Lemma 2.2 see 23,Proposition2. Let E be a uniformly convex and smooth Banach space, and
let {x
−→ 0 ⇐⇒ x
n
− y
n
−→ 0 ⇐⇒ Jx
n
− Jy
n
−→ 0. 2.8
Let C be a nonempty closed convex subset of a reflexive, strictly convex, and smooth
Banach space E.Itisknownthat1, 23 for any x ∈ E, there exists a unique point x ∈ C such
that
ϕ
x,x
min
y∈C
ϕ
y, x
.
2.9
Following Alber 1,wedenotesuchanelementx by Π
C
x. The mapping Π
C
is called the
generalized projection from E onto C. It is easy to see that in a Hilbert space, the mapping Π
,
that is, J
∗
J
−1
.WemakeuseofthefollowingmappingV : E × E
∗
→ studied in Alber 1
V
x, x
∗
x
2
− 2
x, x
∗
x
∗
2
,
∗
− x, y
∗
≤ V
x, x
∗
y
∗
, 2.11
for all x ∈ E and x
∗
,y
∗
∈ E
∗
.
Fixed Point Theory and Applications 5
Lemma 2.7 see 25, Lemma 2.1. Let {a
n
} be a sequence of nonnegative real numbers. Suppose
that
a
n1
≤
1 − γ
a
n
0.
Lemma 2.8 see 26, Lemma 3.1. Let {a
n
} be a sequence of real numbers such that there exists
a subsequence {n
i
} of {n} such that a
n
i
<a
n
i
1
for all i ∈ . Then, there exists a nondecreasing
sequence {m
k
}⊂ such that m
k
→∞,
a
m
k
≤ a
m
k
1
,a
k
r
x
z ∈ C : F
z, y
1
r
y − z, Jz − Jx
≥ 0 ∀y ∈ C
, 2.14
for all x ∈ E. Then, the following hold:
i T
r
is single-valued,
ii T
r
is a firmly nonexpansive-type mapping 27, that is, for all x, y ∈ E
T
r
x − T
≤ ϕ
z, Sx
ϕ
Sx, x
≤ ϕ
z, x
, 2.16
for all x ∈ C and z ∈ FS.Inparticular,S satisfies condition R2.
Lemma 2.12 see 3,Proposition2.4. Let C be a nonempty closed convex subset of a strictly
convex and smooth Banach space E and S : C → E a relatively nonexpansive mapping. Then, FS
is closed and convex.
3. Main Results
In this section, we prove a strong convergence theorem for finding a common element of
the fixed points set of a relatively nonexpansive mapping and the set of solutions of an
equilibrium problem in a uniformly convex and uniformly smooth Banach space.
Theorem 3.1. Let C be a nonempty closed convex subset of a uniformly convex and uniformly smooth
Banach space E and F : C × C →
a bifunction satisfying conditions A1–A4 and S : C → E
a relatively nonexpansive mapping such that FS ∩ EPF
/
.Let{u
n
} and {x
α
n
Ju
1 − α
n
Jx
n
,
u
n1
J
−1
β
n
Jx
n
1 − β
n
JSy
n
,
T
r
n
u
n
.SinceFS ∩ EPF is nonempty, closed,
and convex, we put u Π
FS∩EPF
u.SinceΠ
C
, T
r
n
,andS satisfy condition R2,by2.6,we
get
ϕ
u, y
n
≤ ϕ
u, J
−1
α
n
Ju
1 − α
1 − α
n
ϕ
u, u
n
,
3.2
Fixed Point Theory and Applications 7
and so
ϕ
u, u
n1
≤ β
n
ϕ
u, x
n
1 − β
n
ϕ
1 − α
n
1 − β
n
ϕ
u, u
n
≤ max
ϕ
u, u
,ϕ
u, u
n
.
3.3
By induction, we have
ϕ
−1
α
n
Ju
1 − α
n
Jx
n
.
3.5
Then, y
n
≡ Π
C
z
n
.UsingLemma 2.6 gives
ϕ
u, y
n
≤ ϕ
u,z
n
n
J u
1 − α
n
Jx
n
2α
n
z
n
− u, Ju − J u
≤ α
n
ϕ
u, u
1 − α
n
ϕ
sup{x
n
, Sy
n
: n ∈ }. Then, by Remark 2.11 and 3.6,weget
ϕ
u, u
n1
≤ β
n
ϕ
u, x
n
1 − β
n
ϕ
u, Sy
n
− β
n
1 − β
n
ϕ
u, y
n
− β
n
1 − β
n
g
Jx
n
− JSy
n
≤ β
n
ϕ
x
n
,u
n
− β
n
1 − β
n
g
Jx
n
− JSy
n
1 − γ
n
ϕ
u, u
n
− JSy
n
≤
1 − γ
n
ϕ
u, u
n
2γ
n
z
n
− u, Ju − J u
,
3.8
where γ
n
α
n
1 − β
u, u
n
− ϕ
u, u
n1
−→ 0. 3.9
It follows from 3.7 and γ
n
→ 0that
β
n
ϕ
x
n
,u
n
β
n
1 − β
n
g
−→ 0. 3.11
Consequently, by Remark 2.3,
x
n
− u
n
−→ 0,Jx
n
− JSy
n
−→ 0,x
n
− Sy
n
−→ 0. 3.12
From 2.6 and α
n
→ 0, we obtain
ϕ
x
n
,y
n
≤ ϕ
x
n
,z
−→ 0. 3.13
This implies that
x
n
− y
n
−→ 0,z
n
− y
n
−→ 0. 3.14
Therefore,
y
n
− Sy
n
−→ 0. 3.15
Since {y
n
} is bounded and E isreflexive,wechooseasubsequence{y
n
i
} of {y
n
} such that
y
n
i
zand
lim sup
n
Jx
n
− Ju
n
0.
3.17
Notice that
F
x
n
,y
1
r
n
y − x
n
,Jx
n
− Ju
n
≥ 0, ∀y ∈ C.
3.18
i
, ∀y ∈ C.
3.19
Letting i →∞,wehavefrom3.17 and A4 that
F
y, z
≤ 0, ∀y ∈ C. 3.20
From Lemma 2.10,wehavez ∈ EPF.SinceS satisfies condition R3 and 3.15, z ∈ FS.
It follows that z ∈ FS ∩ EPF.ByLemma 2.4a, we immediately obtain that
lim sup
n →∞
y
n
− u, Ju − J u z − u, Ju − J u≤0.
3.21
Since z
n
− y
n
→ 0,
lim sup
n →∞
z
n
− u, Ju − J u≤0.
3.22
It follows from Lemma 2.7 and 3.8 that ϕu, u
ϕ
u, u
m
k
≤ ϕ
u, u
m
k
1
,ϕ
u, u
k
≤ ϕ
u, u
m
k
1
3.24
for all k ∈
.From3.7 and γ
n
→ 0, we have
≤
ϕ
u, u
m
k
− ϕ
u, u
m
k
1
− γ
m
k
ϕ
u, u
m
k
2γ
m
z
m
k
− u, Ju − J u
≤ 0.
3.26
From 3.8,wehave
ϕ
u, u
m
k
1
≤
1 − γ
m
k
ϕ
u, u
m
k
2γ
m
k
− ϕ
u, u
m
k
1
2γ
m
k
z
m
k
− u, Ju − J u
≤ 2γ
m
k
y
m
k
− u, Ju − J u
.
3.28
In particular, since γ
k
≤ ϕu, u
m
k
1
for all k ∈ ,weconcludethatu
k
→ u,andx
k
→ u.
From two cases, we can conclude that {u
n
} and {x
n
} converge strongly to u and the
proof is finished.
Applying Theorem 3.1 and 28,Theorem3.2, we have the following result.
Theorem 3.2. Let C be a nonempty closed convex subset of a uniformly convex and uniformly smooth
Banach space E, F : C × C →
a bifunction satisfying conditions (A1)–(A4), and {T
i
: C → E}
∞
i1
a sequence of relatively nonexpansive mappings such that
∞
i1
FT
i
i
∩EPF
u.
Setting F ≡ 0andr
n
≡ 1inTheorem 3.1, we have the following result.
Corollary 3.3. Let C be a nonempty closed convex subset of a uniformly convex and uniformly smooth
Banach space E and S : C → E a relatively nonexpansive mapping. Let {u
n
} and {x
n
} be sequences
generated by u ∈ C, u
1
∈ E and
x
n
Π
C
u
n
,
y
n
Π
C
J
−1
α
,where{α
n
}⊂0, 1 satisfying lim
n →∞
α
n
0 and
∞
n1
α
n
∞, {β
n
}⊂a, b ⊂ 0, 1.
Then, {u
n
} and {x
n
} converge strongly to Π
FS
u.
Fixed Point Theory and Applications 11
Letting S : C → C in Corollary 3.3, we have the following result.
Corollary 3.4. Let C be a nonempty closed convex subset of a uniformly convex and uniformly smooth
Banach space E and S : C → C a relatively nonexpansive mapping. Let {x
n
} be a sequence in C
defined by u ∈ C, x
1
1 − β
n
JSy
n
,
3.33
for all n ∈
,where{α
n
}⊂0, 1 satisfying lim
n →∞
α
n
0 and
∞
n1
α
n
∞, {β
n
}⊂a, b ⊂ 0, 1.
Then {x
n
} converges strongly to Π
FS
u.
Let S be the identity mapping in Theorem 3.1, we also have the following result.
≥ 0, ∀y ∈ C,
y
n
Π
C
J
−1
α
n
Ju
1 − α
n
Jx
n
,
u
n1
J
−1
β
n
Jx
n
} converge strongly to Π
EPF
u.
4. Deduced Theorems in Hilbert Spaces
In Hilbert spaces, every nonexpansive mappings are relatively nonexpansive, and J is the
identity operator. We obtain the following result.
Theorem 4.1. Let C be a nonempty closed convex subset of a Hilbert space H, F : C × C →
a bifunction satisfying conditions (A1)–(A4), and S : C → H a nonexpansive mapping such that
FS ∩ EPF
/
.Let{x
n
} be a sequence in C defined by u ∈ C, x
1
∈ H and
x
n1
β
n
T
r
n
x
n
1 − β
n
α
n
∞, {β
n
}⊂a, b ⊂ 0, 1,and{r
n
}⊂c, ∞ ⊂ 0, ∞.Then,{x
n
} converges strongly to
P
FS∩EPF
u.
Remark 4.2. In Theorem 4.1,wehavethesameconclusionifthemappingS : C → H is only
quasinonexpansive i.e., FS
/
and p − Sx≤p − x for all x ∈ C and p ∈ FS such that
I − T is demiclosed at zero.
12 Fixed Point Theory and Applications
Letting F ≡ 0inTheorem 4.1, we have the following result.
Corollary 4.3. Let C be a nonempty closed convex subset of a Hilbert space H and S : C → H a
nonexpansive mapping such that FS
/
.Let{x
n
} be a sequence in C defined by u ∈ C, x
1
∈ H
and
n
}⊂0, 1 satisfying lim
n →∞
α
n
0,
∞
n1
α
n
∞,and{β
n
}⊂a, b ⊂
0, 1.Then,{x
n
} converges strongly to P
FS
u.
Let S be the identity mapping in Theorem 4.1, we have the following r esult.
Corollary 4.4. Let C be a nonempty closed convex subset of a Hilbert space H and F : C × C →
a
bifunction satisfying conditions (A1)–(A4). Let {x
n
} be a sequence in H defined by u, x
1
∈ H and
x
n1
γ
n
}⊂c, ∞ ⊂ 0, ∞.Then{x
n
} converges strongly to Π
EPF
u.
Proof. We may assume without loss of generality that γ
n
< 1/2foralln ∈ . Setting α
n
2γ
n
and β
n
1/2foralln ∈ ,weget
x
n1
1
2
T
r
n
x
n
1
2
I
Remark 4.5. Corollary 4.4 improves and extends 29, Corollary 5.3.Moreprecisely,the
conditions lim
n →∞
γ
n1
/γ
n
1and
∞
n1
|r
n1
− r
n
| < ∞ are removed.
Applying Corollary 4.4 and 30,Theorem8, we have the following result.
Corollary 4.6. Let C be a nonempty closed convex subset of a Hilbert space H, F : C × C →
a
bifunction satisfying conditions (A1)–(A4), and f : C → C a contraction of H into itself. Let {x
n
}
be a sequence in H defined by u, x
1
∈ H and
x
n1
γ
n
f
n
∞
and {r
n
}⊂c, ∞ ⊂ 0, ∞.Then,{x
n
} converges strongly to z P
EPF
fz.
Remark 4.7. Corollary 4.6 improves and extends 16, Corollary 3.4.Moreprecisely,the
conditions
∞
n1
|γ
n1
− γ
n
| < ∞ and
∞
n1
|r
n1
− r
n
| < ∞ are removed.
Fixed Point Theory and Applications 13
Acknowledgment
The author would like to thank the referees for their comments and helpful suggestions.
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