Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2011, Article ID 173430, 15 pages
doi:10.1155/2011/173430
Research Article
An Application of Hybrid Steepest Descent
Methods for Equilibrium Problems and Strict
Pseudocontractions in Hilbert Spaces
Ming Tian
College of Science, Civil Aviation University of China, Tianjin 300300, China
Correspondence should be addressed to Ming Tian, [email protected]
Received 9 December 2010; Accepted 13 February 2011
Academic Editor: Shusen Ding
Copyright q 2011 Ming Tian. 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 use the hybrid steepest descent methods for finding a common element of the set of solutions
of an equilibrium problem and the set of fixed points of a strict pseudocontraction mapping in the
setting of real Hilbert spaces. We proved strong convergence theorems of the sequence generated
by our proposed schemes.
1. Introduction
Let H be a real Hilbert space and C a closed convex subset of H,andletφ be a bifunction of
C × C into R,whereR is the set of real numbers. The equilibrium problem for φ : C × C → R
is to find x ∈ C such that
EP : φ
x, y
≥ 0 ∀y ∈ C 1.1
denoted the set of solution by EPφ. Given a mapping T : C → H,letφx, yTx,y − x
for a ll x, y ∈ C,thenz ∈ EPφ if and only i f Tz,y − z≥0forally ∈ C,thatis,z
α
n
γf
x
n
I − α
n
A
Tx
n
,n≥ 0, 1.3
where T is a self-nonexpansive mapping on H, f is a contraction of H into itself with β ∈ 0, 1
and {α
n
}⊂0, 1 satisfies certain conditions, and A is a strongly positive bounded linear
operator on H and converges strongly to a fixed-point x
∗
of T which is the unique solution
to the following variational inequality:
γf − Ax
∗
,x− x
∗
≤0, for x ∈ FT, and is also the optimality condition for some
minimization problem. A mapping S : C → H is said to be k-strictly pseudocontractive if
Note that the class of k-strict pseudo-contraction strictly includes the class of nonex-
pansive mapping, that is, S is nonexpansive if and only if S is 0-srictly pseudocontractive; it
is also said to be pseudocontractive if k 1. Clearly, the class of k-strict pseudo-contractions
falls into the one between classes of nonexpansive mappings and pseudo-contractions.
The set of fixed points of S is denoted by FS. Very recently, by using the general
approximation method, Qin et al. 7 obtained a strong convergence theorem for finding an
element of FS. On the other hand, Ceng et al. 8 proposed an iterative scheme for finding
an element of EPφ ∩ FS and then obtained some weak and strong convergence theorems.
Based on the above work, Y. Liu 9 introduced two iteration schemes by the general iterative
method for finding an element of EPφ ∩ FS.
In 2001, Yamada 10 introduced the following hybrid iterative method for solving the
variational inequality:
x
n1
Tx
n
− μλ
n
F
Tx
n
,n≥ 0, 1.5
where F is k-Lipschitzian and η-strongly monotone operator with k>0, η>0, 0 <μ<2η/k
2
,
then he proved that if {λ
n
} satisfyies appropriate conditions, the {x
n
→ x implies that {x
n
} converges strongly to x.Foranyx ∈ H, there exists a unique
nearest point in C, denoted by P
C
x,suchthat
x − P
C
x
≤
x − y
, ∀y ∈ C. 2.1
Such a P
C
x is called the metric projection of H onto C.ItisknownthatP
C
is nonexpansive.
Furthermore, for x ∈ H and u ∈ C, u p
c
x, ⇔x − u, u − y≥0, for all y ∈ C.
It is widely known that H satisfies Opial’s condition 11, that is, for any sequence
{x
n
φ
tz
1 − t
x, y
≤ φ
x, y
;
2.3
A4 For each fixed x ∈ C, the function y → φx, y is convex and lower semicontinuous.
Let us recall the following lemmas which will be useful for our paper.
Lemma 2.1 see 12. Let φ be a bifunction from C × C into R satisfying (A1), (A2),(A3) and (A4)
then, for any r>0 and x ∈ H,thereexistsz ∈ C such that
φ
z, y
1
r
y − z, z − x≥0, ∀y ∈ C.
2.4
Further, if T
r
x {z ∈ C : φz, y1/ry − z, z − x≥0, ∀y ∈ C}, then the following hold:
4 Journal of Inequalities and Applications
Lemma 2.2 see 13. If S : C → H is a k-strict pseudo-contraction, then the fixed-point set FS
is closed convex, so that the projection P
FS
is well difened.
Lemma 2.3 see 14. Let S : C → H be a k-strict pseudo-contraction. Define T : C → H by
Tx λx 1 − λSx for each x ∈ C,then,asλ ∈ k, 1, T is nonexpansive mapping such that
FTFS.
Lemma 2.4 see 15. In a Hilbert space H, there holds the inequality
x y
2
≤
x
2
2
y,
x y
, ∀x, y ∈ H.
2.6
Lemma 2.5 see 16. Assume that {a
n
n
≤ 0 or
∞
n1
|δ
n
γ
n
| < ∞.
Then lim
n →∞
a
n
0.
3. Main Results
Throughout the rest of this paper, we always assume that F is a L-lipschitzian continuous and
η-strongly monotone operator with L, η > 0 and assume that 0 <μ<2η/L
2
. τ μη −μL
2
/2.
Let {T
λ
n
} be mappings defined as Lemma 2.1. Define a mapping S
n
: C → H by S
n
x
G
n
x − G
n
y
≤
1 − α
n
τ
T
λ
n
x − T
λ
n
y
≤
1 − α
n
τ
n
x
F
n
.
3.3
Journal of Inequalities and Applications 5
For simplicity, we will write x
n
for x
F
n
provided no confusion occurs. Next, we prove
that the sequence {x
n
} converges strongly to a q ∈ FS ∩ EPφ which solves the variational
inequality
Fq,p − q
≥ 0, ∀p ∈ F
S
∩ EP
φ
. 3.4
Equivalently, q P
n
≥0, ∀y ∈ C,
y
n
β
n
u
n
1 − β
n
Su
n
,
x
n
I − α
n
μF
y
n
, ∀n ∈ N,
3.5
where u
n
ii 0 ≤ k ≤ β
n
≤ λ<1 and lim
n →∞
β
n
λ,
then {x
n
} converges strongly to a point q ∈ FS ∩ EPφ which solves the variational inequality
3.4.
Proof. First, take p ∈ FS ∩ EPφ.Sinceu
n
T
λ
n
x
n
and p T
λ
n
p,fromLemma 2.1,forany
n ∈ N,wehave
u
n
− p
− p
S
n
u
n
− S
n
p
≤
u
n
− p
≤
x
n
− p
≤ α
n
−μF
p
1 − α
n
τ
y
n
− p
.
3.8
It follows that x
n
− p≤μFp/τ.
n
u
n
− x
n
α
n
−μFy
n
.
3.9
By Lemma 2.1,wehave
u
n
− p
2
− p
2
u
n
− p
2
−
u
n
− x
n
2
.
3.10
It follows that
u
n
− p
2
α
n
−μFp
I − μα
n
F
y
n
−
I − μα
n
F
p
2
≤
1 − α
− p
2
2α
n
−μFp, x
n
− p
≤
1 − α
n
τ
2
x
n
− p
2
−
x
2
x
n
− p
2
−
1 − α
n
τ
2
x
n
− u
n
2
2α
n
− μFp
2
−
1 − α
n
τ
2
x
n
− u
n
2
2α
n
−μFp
x
n
− p
.
2α
n
μFp
x
n
− p
.
3.13
Since α
n
→ 0, therefore
lim
n →∞
x
n
− u
n
0.
3.14
From 3.9,wederivethat
y
n
− u
n
≤
λ − β
n
u
n
− Su
n
y
n
− u
n
lim inf
n →∞
u
n
i
− q
< lim inf
n →∞
u
n
i
− Tq
≤ lim inf
n →∞
u
n
i
− Tu
n
i
n
,foranyy ∈ C,weobtain
φ
u
n
,y
1
λ
n
y − u
n
,u
n
− x
n
≥ 0.
3.19
From A2,wehave
1
λ
n
y − u
n
,u
y, u
n
i
.
3.21
Since u
n
i
− x
n
i
/λ
n
i
→ 0andu
n
i
q, it follows from A4 that 0 ≥ φy, q,forall
y ∈ C.Letz
t
ty 1 − tq for all t ∈ 0, 1 and y ∈ C,thenwehavez
t
∈ C and hence
φz
t
,q ≤ 0. Thus, from A1 and A4,wehave
0 φ
t
,y.FromA3,wehave0≤ φq, y for all y ∈ C and hence q ∈ EPφ.
Therefore, q ∈ FS ∩ EPφ. On the other hand, we note that
x
n
− q −α
n
μFq
I − μα
n
F
y
n
−
I − μα
n
F
q. 3.23
Hence, we obtain
x
n
− q
−μFq, x
n
− q
1 − α
n
τ
x
n
− q
2
.
3.24
It follows that
x
n
− q
2
≤
x
n
i
− q
2
≤
−μFq, x
n
i
− q
τ
.
3.27
Since x
n
i
q, it follows from 3.27 that x
n
i
→ q as i →∞. Next, we show that q
solves the variational inequality 3.4.
As a matter of fact, we have
x
n
I − S
n
T
λ
n
x
n
− μα
n
Fx
n
− FS
n
T
λ
n
x
n
.
3.29
Hence, for p ∈ FS ∩ EPφ,
μF
x
n
,x
n
− p
−
1
α
n
I − S
n
T
λ
n
x
n
−
I − S
n
T
λ
n
p, x
n
λ
n
x − I − S
n
T
λ
n
y≥0, for all x, y ∈ H.This
is due to the nonexpansivity of S
n
T
λ
n
.
Now replacing n in 3.30 with n
i
and letting i →∞,weobtain
μF
q, q − p
lim
i →∞
μFx
n
i
,x
n
k
→ x. Similiary to the proof above, we derive x ∈ FS ∩
EPφ. Moreover, it follows from the inequality 3.31 that
μF
q, q − x
≤ 0. 3.32
Interchange q and x to obtain
μF
x, x − q
≤ 0. 3.33
Adding up 3.32 and 3.33 yields
μη
q − x
2
≤
q − x,
I − μF
q q. 3.36
Theorem 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H and φ a bifunction
from C × C into R satisfying (A1), (A2), (A3) and (A4). Let S : C → H be a k-strictly
pseudocontractive nonself mapping such that FS∩EP φ
/
φ.LetF : H → H be an L-Lipschitzian
continuous and η-strongly monotone operator on H with L, η > 0 . Suppose that 0 <μ<2η/L
2
,
τ μη − μL
2
/2.Let{x
n
} and {u
n
} be sequences generated by x
1
∈ H and
φ
u
n
,y
1
λ
μF
y
n
, ∀n ∈ N,
3.37
10 Journal of Inequalities and Applications
where u
n
T
λ
n
x
n
, y
n
S
n
u
n
if {α
n
},{β
n
},and{λ
n
} satisfy the following conditions:
i {α
n
}⊂0, 1, lim
− β
n
| < ∞,
iii {λ
n
}∈0, ∞, lim
n →∞
λ
n
> 0 and
∞
n1
|λ
n1
− λ
n
| < ∞,
then {x
n
} and {u
n
} converge strongly to a point q ∈ FS∩EPφ which solves the variational
inequality3.4.
Proof. We first show that {x
n
} is bounded. Indeed, pick any p ∈ FS ∩ EPφ to derive that
x
−μF
p
1 − α
n
τ
x
n
− p
≤
1 − α
n
τ
x
n
− p
, ∀n ∈ N, 3.39
and hence {x
n
} is bounded. From 3.6 and 3.7, we also derive that {u
n
} and {y
n
} are
bounded. Next, we show that x
n1
− x
n
→0. We have
x
n1
− x
n
I − α
n
μF
y
n
−
n
μF
y
n−1
−
I − α
n−1
μF
y
n−1
≤
1 − α
n
τ
y
n
− y
n−1
− α
n−1
|
,
3.40
where
K sup
μFy
n
: n ∈ N
< ∞. 3.41
On the other hand, we have
y
n
− y
n−1
S
n
u
n
− u
n−1
S
n
u
n−1
− S
n−1
u
n−1
.
3.42
Journal of Inequalities and Applications 11
From u
n1
T
λ
n1
x
n1
and u
n
T
1
λ
n
y − u
n
,u
n
− x
n
≥ 0, ∀y ∈ C.
3.44
Putting y u
n
in 3.43 and y u
n1
in 3.44,wehave
φ
u
n1
,u
n
1
λ
− x
n
≥ 0.
3.45
So, from A2,wehave
u
n1
− u
n
,
u
n
− x
n
λ
n
−
u
n1
− x
n1
λ
n1
≥ 0 , 3.46
and hence
u
n
>a>0foralln ∈ N. Thus, we have
u
n1
− u
n
2
≤
u
n1
− u
n
,x
n1
− x
n
1 −
λ
n
λ
n1
u
n1
u
n1
− x
n1
u
n1
− u
n
≤
x
n1
− x
n
1
a
|
λ
n1
− λ
n
β
n
u
n−1
1 − β
n
Su
n−1
−
β
n−1
u
n−1
1 − β
n−1
Su
n−1
x
n
− x
n−1
M
0
a
|
λ
n
− λ
n−1
|
β
n
− β
n−1
u
n−1
− Su
n−1
where M
1
is an appropriate constant such that
M
1
≥
M
0
a
u
n−1
− Su
n−1
, ∀n ∈ N.
3.51
From 3.41 and 3.50,weobtain
x
n1
− x
n
≤ K
|
α
n
− α
n−1
M
1
≤
1 − α
n
τ
x
n
− x
n−1
M
|
α
n
− α
n−1
|
|
λ
n
− λ
n−1
|→0and|β
n
− β
n−1
|→0, we have
lim
n →∞
u
n1
− u
n
0, lim
n →∞
y
n1
− y
n
0.
3.54
Since
x
n1
n
x
n
− x
n1
α
n
−μFy
n
.
3.56
From α
n
→ 0and3.53,wehave
lim
n →∞
x
n
− y
n
− p
1
2
x
n
− p
2
u
n
− p
2
−
u
n
− x
n
2
.
3.59
Journal of Inequalities and Applications 13
Then, from 3.7 and 3.59,wederivethat
x
n1
− p
2
−μα
n
Fp
I − μα
n
F
y
n
−
I − μα
n
2α
n
−μFp
y
n
− p
≤
u
n
− p
2
α
2
n
−μFp
n
2
α
2
n
−μFp
2
2α
n
−μFp
y
n
− p
.
3.60
Since α
n
n
− u
n
≤
Tu
n
− y
n
y
n
− u
n
≤
λ − β
n
u
μFq, q − x
n
≤0, where q P
FS∩EPφ
I − μFq is a
unique solution of the variational inequality 3.4. Indeed, take a subsequence {x
n
i
} of {x
n
}
such that
lim
i →∞
μFq, q − x
n
i
lim sup
n →∞
μFq, q − x
n
.
3.65
Since {x
n
i
μFq, q − w
≤ 0.
3.66
14 Journal of Inequalities and Applications
From x
n1
− q −α
n
μFq I − μα
n
Fy
n
− I − μα
n
Fq,wehave
x
n1
− q
2
≤
I − μα
x
n
− q
2
2α
n
−μFq, x
n1
− q
.
3.67
This implies that
x
n1
− q
2
≤
1 − 2α
n
τ
x
n
− q
2
α
n
τ
2
x
n
− q
2
2α
n
−μFq, x
n1
− q
1 − γ
n
x
n
− q
2
γ
n
δ
n
,
3.68
where M
∗
sup{x
n
−q
2
: n ∈ N}, γ
n
2α
University of China no. 2010kys02. He was also Supported in part by The Fundamental
Research Funds for the Central Universities Grant no. ZXH2009D021.
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