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Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2010, Article ID 756492, 22 pages
doi:10.1155/2010/756492
Research Article
Strong Convergence for Mixed Equilibrium
Problems of Infinitely Nonexpansive Mappings
Jintana Joomwong
Division of Mathematics, Faculty of Science, Maejo University, Chiang Mai 50290, Thailand
Correspondence should be addressed to J intana Joomwong,
Received 29 March 2010; Accepted 24 May 2010
Academic Editor: Tomonari Suzuki
Copyright q 2010 Jintana Joomwong. 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 introduce a new iterative scheme for finding a common element of infinitely nonexpansive
mappings, the set of solutions of a mixed equilibrium problems, and the set of solutions of the
variational inequality for an α-inverse-strongly monotone mapping in a Hilbert Space. Then, the
strong converge theorem is proved under some parameter controlling conditions. The results of
this paper extend and improve the results of Jing Zhao and Songnian He2009 and many others.
Using this theorem, we obtain some interesting corollaries.
1. Introduction
Let H be a real Hilbert space with norm · and inner product ·, ·. And let C be a nonempty
closed convex subset of H.Letϕ : C → R be a real-valued function and let Θ : C × C → R
be an equilibrium bifunction, that is, Θu, u0 for each u ∈ C. Ceng and Yao 1 considered
the following mixed equilibrium problem.
Find x
∗
∈ C such that
Θ
x
∗
,y
ϕ
y
− ϕ
x
∗
≥ 0, ∀y ∈ C. 1.1
The set of solutions of 1.1 is denoted by MEPΘ,ϕ. It is easy to see that x
∗
is the solution
of problem 1.1 and x
∗
∈ dom ϕ {x ∈ ϕx < ∞}. In particular, if ϕ ≡ 0, the mixed
equilibrium problem 1.1 reduced to the equilibrium problem.
Find x
∗
∈ C such that
Θ
x
∗
,y
≥ 0, ∀y ∈ C. 1.2
2 Fixed Point Theory and Applications
The set of solutions of 1.2 is denoted by EPΘ. If ϕ ≡ 0andΘx, yAx, y − x for
all x, y ∈ C, where A is a mapping from C to H, then the mixed equilibrium problem 1.1
becomes the following variational inequality.
Find x
∗
∈ C such that
Ax
∗
,y− x
∗
, ∀y ∈ C. 1.3
The set of solutions of 1.3 is denoted by VIA, C.
The variational inequality and the mixed equilibrium problems which include
fixed point problems, optimization problems, variational inequality problems have been
extensively studied in literature. See, for example, 2–8.
In 1997, Combettes and Hirstoaga 9 introduced an iterative method for finding
the best approximation to the initial data and proved a strong convergence theorem.
Subsequently, Takahashi and Takahashi 7 introduced another iterative scheme for finding
a common element of EPΘ and the set of fixed points of nonexpansive mappings.
Furthermore,Yao et al. 8, 10 introduced an iterative scheme for finding a common element
of EPΘ and the set of fixed points of finitely infinitely nonexpansive mappings.
Very recently, Ceng and Yao 1 considered a new iterative scheme for finding
a common element of MEPΘ,ϕ and the set of common fixed points of finitely many
nonexpansive mappings in a Hilbert space and obtained a strong convergence theorem.
Now, we recall that a mapping A : C → H is said to be
i monotone if Au − Av, u − v≥0, for all u, v ∈ C,
ii L-Lipschitz if there exists a constant L>0 such that Au − Av≤L
u −
v, for all u, v ∈ C,
iii α-inverse strongly monotone if there exists a positive real number α such that Au−
Av, u − v≥αAu − Av
2
, for all u, v ∈ C.
It is obvious that any α-inverse strongly monotone mapping A is monotone and Lipscitz. A
mapping S : C → C is called nonexpansive if Su − Sv≤u − v, for all u, v ∈ C. We denote
by FS : {x ∈ C : Sx x} the set of fixed point of S.
In 2006, Yao and Yao 11 introduced the following iterative scheme.
Let C be a closed convex subset of a real Hilbert space. Let A be an α-inverse strongly
monotone mapping of C into H and let S be a nonexpansive mapping of C into itself such
that FS ∩ VIA, C
/
∅. Suppose that x
1
u ∈ C and {x
n
} and {y
n
} are given by
y
n
P
C
x
n
− λ
n
Ax
n
,
x
n1
α
n
u β
n
x
n
γ
n
SP
C
y
n
− λ
n
Ay
n
,
1.4
where {α
n
}, {β
n
},and{γ
n
} are sequence in 0, 1 and {λ
n
} is a sequence in 0,2λ. They proved
that the sequence {x
n
} defined by 1.4 converges strongly to a common element of FS ∩
VIA, C under some parameter controlling conditions.
Moreover, Plubtieng and Punpaeng 12 introduced an iterative scheme 1.5 for
finding a common element of the set of fixed point of nonexpansive mappings, the set of
solutions of an equilibrium problems, and the set of solutions of the variational of inequality
Fixed Point Theory and Applications 3
problem for an α-inverse strongly monotone mapping in a real Hilbert space. Suppose that
x
1
u ∈ C and {x
n
}, {y
n
},and{u
n
} are given by
Θ
u
n
,y
1
r
n
y − u
n
,u
n
− x
n
≥ 0, ∀y ∈ C,
y
n
P
C
u
n
− λ
n
Au
n
,
x
n1
α
n
u β
n
x
n
γ
n
SP
C
y
n
− λ
n
Ay
n
,
1.5
where {α
n
}, {β
n
},and{γ
n
} are sequence in 0, 1, {λ
n
} is a sequence in 0,2λ,and{r
n
}⊂
0, ∞. Under some parameter controlling conditions, they proved that the sequence {x
n
}
defined by 1.5 converges strongly to P
FS∩VIA,C∩EPΘ
u.
On the other hand, Yao et al. 8 introduced an iterative scheme 1.7 for finding a
common element of the set of solutions of an equilibrium problem and the set of common
fixed point of infinitely many nonexpansive mappings in H.Let{T
n
}
∞
n1
be a sequence of
nonexpansive mappings of C into itself and let {t
n
}
∞
n1
be a sequence of real number in 0, 1.
For each n ≥ 1, define a mapping W
n
of C into itself as follows:
U
n,n1
I,
U
n,n
t
n
T
n
U
n,n1
1 − t
n
I,
U
n,n−1
t
n−1
T
n−1
U
n,n
1 − t
n−1
I,
.
.
.
U
n,k
t
k
T
k
U
n,k1
1 − t
k
I,
U
n,k−1
t
k−1
T
k−1
U
n,k
1 − t
k−1
I,
.
.
.
U
n,2
t
2
T
2
U
n,3
1 − t
2
I,
W
n
U
n,1
t
1
T
1
U
n,2
1 − t
1
I.
1.6
Such a mapping W
n
is called the W-mapping generated by T
n
,T
n−1
, ,T
1
and t
n
,t
n−1
, ,t
1
.
In 8,givenx
0
∈ H arbitrarily, the sequences {x
n
} and {u
n
} are generated by
Θ
u
n
,x
1
r
n
x − u
n
,u
n
− x
n
≥ 0, ∀x ∈ C,
x
n1
α
n
f
x
n
β
n
x
n
γ
n
W
n
u
n
.
1.7
They proved that under some parameter controlling conditions, {x
n
} generated by 1.7
converges strongly to z ∈∩
∞
n1
FT
n
∩ EPΘ, where z P
∩
∞
n1
FT
n
∩EPΘ
fz.
4 Fixed Point Theory and Applications
Subsequently, Ceng and Yao 13 introduced an iterative scheme by the viscosity
approximation method:
Θ
u
n
,x
1
r
n
x − u
n
,u
n
− x
n
≥ 0, ∀x ∈ C,
y
n
1 − γ
n
x
n
γ
n
W
n
u
n
,
x
n1
1 − α
n
− β
n
x
n
α
n
f
y
n
β
n
W
n
y
n
,
1.8
where {α
n
}, {β
n
} and {γ
n
} are sequence in 0,1 such that α
n
β
n
≤ 1. Under some parameter
controlling conditions, they proved that the sequence {x
n
} defined by 1.8 converges
strongly to z ∈∩
∞
n1
FT
n
∩ EPΘ, where z P
∩
∞
n1
FT
n
∩EPΘ
fz.
Recently, Zhao and He 14 introduced the following iterative process.
Suppose that x
1
u ∈ C,
Θ
u
n
,y
1
r
n
y − u
n
,u
n
− x
n
≥ 0, ∀y ∈ C,
y
n
s
n
P
C
u
n
− λ
n
Au
n
1 − s
n
x
n
,
x
n1
α
n
u β
n
x
n
γ
n
W
n
P
C
y
n
− λ
n
Ay
n
,
1.9
where {s
n
}, {α
n
}, {β
n
},and{γ
n
}∈0, 1 such that α
n
β
n
γ
n
1. Under some parameter
controlling conditions, they proved that the sequence {x
n
} defined by 1.9 converges
strongly to z ∈∩
∞
i1
FT
i
∩ VIA, C ∩ EPΘ, where z P
∩
∞
i1
FT
i
∩VIA,C∩EPΘ
u.
Motivated by the ongoing research in this field, in this paper we suggest and analyze
an iterative scheme for finding a common element of the set of fixed point of infinitely
nonexpansive mappings, the set of solutions of an equilibrium problem and the set of
solutions of the variational of inequality problem for an α-inverse strongly monotone
mapping in a real Hilbert space. Under some appropriate conditions imposed on the
parameters, we prove another strong convergence theorem and show that the approximate
solution converges to a unique solution of some variational inequality which is the optimality
condition for the minimization problem. The results of this paper extend and improve the
results of Zhao and He 14 and many others. For some related works, we refer the readers
to 15–22 and the references therein.
2. Preliminaries
Let H be a real Hilbert space and let C be a closed convex subset of H. Then, for any x ∈ H,
there exists a unique nearest point in C, denoted by P
C
x such that
x − P
C
x
≤x − y, ∀y ∈ C. 2.1
P
C
is called the metric projection of H onto C. It is well known that P
C
is nonexpansive
mapping and satisfies
x − y, P
C
x − P
C
y
≥
P
C
x − P
C
y
2
, ∀x, y ∈ H.
2.2
Fixed Point Theory and Applications 5
Moreover, P
C
is characterized by the following properties: P
c
x ∈ C and
x − P
C
x, y − P
C
x≤0,
x − y
2
≥
x − P
C
x
2
y − P
C
x
2
, ∀x ∈ H, y ∈ C.
2.3
It is clear that u ∈ VIA, C ⇔ u P
C
u − λAu,λ>0.
A space X is said to satisfy Opials condition if for each sequence {x
n
} in X which
converges weakly to a point x ∈ X, we have
lim inf
n →∞
x
n
− x < lim inf
n →∞
x
n
− y, ∀y ∈ X, y
/
x.
2.4
The following lemmas will be useful for proving the convergence result of this paper.
Lemma 2.1 see 23. Let {x
n
} and {y
n
} be bounded sequences in a Banach space X and let {β
n
} be
a sequence in 0, 1 with 0 < lim inf
n →∞
β
n
lim sup
n →∞
β
n
< 1. Suppose that x
n1
1−β
n
y
n
β
n
x
n
for all integer n ≥ 1 and lim sup
n →∞
y
n1
−y
n
−x
n1
−x
n
0. Then lim
n →∞
y
n
−x
n
0.
Lemma 2.2 see 24. Let H be a real Hilbert space, let C be a closed convex subset of H, and let
T : C → C be a nonexpansive mapping with FT
/
∅. If {x
n
} is a sequence in C weakly converging
to x and if I − Tx
n
converge strongly to y,thenI − Tx y.
Lemma 2.3 see 25. Assume that {a
n
} is a sequence of nonnegative real numbers such that
a
n1
≤
1 − α
n
a
n
δ
n
,n≥ 0, 2.5
where {α
n
} is a sequence in 0, 1 and {δ
n
} is a sequence in R such that
1 lim
n →∞
α
n
0 and
∞
n1
α
n
∞.
2 lim sup
n →∞
δ
n
/α
n
≤ 0 or
∞
n1
|δ
n
| < ∞.
Then lim
n →∞
a
n
0.
In this paper, for solving the mixed equilibrium problem, let us give the following
assumptions for a bifunction Θ,ϕand the set C:
A1Θx, x0 for all x ∈ C;
A2Θis monotone, that is, Θx, yΘy, x ≤ 0 for any x, y ∈ C;
A3Θis upper-hemicontinuous, that is, for each x, y, z ∈ C,
lim
t → 0
sup Θ
tz
1 − t
x, y
≤ Θ
x, y
;
2.6
A4Θx, · is convex and lower semicontinuous for each x ∈ C;
6 Fixed Point Theory and Applications
B1 for each x ∈ H and r>0, there exists a bounded subset D
x
⊂ C and y
x
∈ C such
that for any z ∈ C \ D
x
,
Θ
z, y
ϕ
y
x
1
r
n
y
x
− z, z − x
<ϕ
z
,
2.7
B2 C is a bounded set.
By a similar argument as in the proof of Lemma 2.3 in 26,wehavethefollowing
result.
Lemma 2.4. Let C be a nonempty closed convex subset of a real Hilbert space H.LetΘ be a bifunction
from C × C → R that satisfies (A1)–(A4) and let ϕ : C → R ∪{∞}be a proper lower semicontinuous
and convex function. Assume that either (B1) or (B2) holds. For r>0 and x ∈ H, define a mapping
T
r
: H → C as follows:
T
r
x
z ∈ C : Θ
z, y
ϕ
y
1
r
y − z, z − x
≥ ϕ
z
, ∀y ∈ C
2.8
for all x ∈ H. Then, the following conditions hold:
1 for each x ∈ H, T
r
x
/
∅;
2 T
r
is single-valued;
3 T
r
is firmly nonexpansive, that is, for any x, y ∈ H, T
r
x − T
r
y
2
≤T
r
x − T
r
y, x − y;
4 FT
r
MEPΘ,ϕ;
5 MEPΘ,ϕ is closed and convex.
Let {T
n
}
∞
n1
be a sequence of nonexpansive mappings of C into itself, where C is a
nonempty closed convex subset of a real Hilbert space H. Given a sequence {t
n
}
∞
n1
in 0, 1,
we define a sequence {W
n
}
∞
n1
of self-mappings on C by 1.6. Then We have the following
result.
Lemma 2.5 see 27. Let C be a nonempty closed convex subset of a real Hilbert space H.Let
{T
n
}
∞
n1
be a sequence of nonexpansive self-mappings on C such that ∩
∞
n1
FT
n
/
∅ and let {t
n
} be a
sequence in 0,b for some b ∈ 0, 1. Then, for every x ∈ C and k ≥ 1, lim
n →∞
U
n,k
x exists.
Remark 2.6 see 8. It can be shown from Lemma 2.5 that if D is a nonempty bounded subset
of C, then for >0, there exists n
0
≥ k such that for all n>n
0
,sup
x∈D
U
n,k
x − U
k
x≤,
where U
k
x lim
n →∞
U
n,k
x.
Remark 2.7 see 8. Using Lemma 2.5, we define a mapping W : C → C as follows: Wx
lim
n →∞
W
n
x lim
n →∞
U
n,1
x, for all x ∈ C. W is called the W-mapping generated by T
1
,T
2
,
and t
1
,t
2
,
Since W
n
is nonexpansive, W : C → C is also nonexpansive.
Indeed, for all x, y ∈ C, W
x
− W
y
lim
n →∞
W
n
x − W
n
y≤x − y.
If {x
n
} is a bounded sequence in C, then we put D {x
n
: n ≥ 0}. Hence it is clear
from Remark 2.6 that for any arbitrary >0, there exists n
0
≥ 1 such that for all n>n
0
,
W
n
x
n
− Wx
n
U
n,1
x
n
− U
1
x
n
≤sup
x∈D
U
n,1
x − U
1
x <.
Fixed Point Theory and Applications 7
This implies that lim
n →∞
W
n
x
n
− Wx
n
0.
Lemma 2.8 see 27. Let C be a nonempty closed convex subset of a real Hilbert space H.Let
{T
n
}
∞
n1
be a sequence of nonexpansive self-mappings on C such that ∩
∞
n1
FT
n
/
∅ and let {t
n
} be a
sequence in 0,b for some b ∈ 0, 1.ThenFW∩
∞
n1
FT
n
.
3. Main Results
Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H.Letϕ : C →
R ∪{∞} be a lower semicontinuous and convex function. Let Θ be a bifunction from C × C → R
satisfying (A1)–(A4), let A be an α-inverse-strongly monotone mapping of C into H, and let {T
n
}
∞
n1
be a sequence of nonexpansive self-mapping on C such that ∩
∞
n1
FT
n
∩ VIA, C ∩ MEPΘ,ϕ
/
∅.
Suppose that {s
n
}, {α
n
}, {β
n
}, and {γ
n
} are sequences in 0, 1,{λ
n
} is a sequence in 0, 2α such that
λ
n
∈ a, b for some a, b with 0 <a<b<2α, and {r
n
}⊂0, ∞ is a real sequence. Suppose that the
following conditions are satisfied:
i α
n
β
n
γ
n
1,
ii lim
n →∞
α
n
0 and
∞
n1
α
n
∞,
iii 0 < lim inf
n →∞
β
n
≤ lim sup
n →∞
β
n
< 1,
iv 0 < lim inf
n →∞
s
n
≤ lim sup
n →∞
s
n
< 1/2 and lim
n →∞
|s
n1
− s
n
| 0,
v lim
n →∞
|λ
n1
− λ
n
| 0,
vi lim inf
n →∞
r
n
> 0 and lim
n →∞
|r
n1
− r
n
| 0.
Let f be a contraction of C into itself with coefficient β ∈ 0, 1. Assume that either (B1) or (B2) holds.
Let the sequences {x
n
}, {u
n
}, and {y
n
} be generated by, x
1
∈ C and
Θ
u
n
,y
ϕ
y
− ϕ
u
n
1
r
n
y − u
n
,u
n
− x
n
≥ 0, ∀y ∈ C,
y
n
s
n
P
C
u
n
− λ
n
Au
n
1 − s
n
x
n
,
x
n1
α
n
f
x
n
β
n
x
n
γ
n
W
n
P
C
y
n
− λ
n
Ay
n
,
3.1
for all n ∈ N,whereW
n
is defined by 1.6 and {t
n
} is a sequence in 0,b,forsomeb ∈ 0, 1.Then
the sequence {x
n
} converges strongly to a point x
∗
∈∩
∞
n1
FT
n
∩ VIA, C ∩ MEPΘ,ϕ,where
x
∗
P
∩
∞
n1
FT
n
∩VIA,C∩MEPΘ,ϕ
fx
∗
.
Proof. For any x, y ∈ C and λ
n
∈ a, b ⊂ 0, 2α,wenotethat
I − λ
n
Ax − I − λ
n
Ay
2
x − y − λ
n
Ax − Ay
2
x − y
2
− 2λ
n
x − y, Ax − Ay
λ
2
n
Ax − Ay
2
≤
x − y
2
λ
n
λ
n
− 2α
Ax − Ay
2
≤
x − y
2
,
3.2
which implies that I − λ
n
A is nonexpansive.
8 Fixed Point Theory and Applications
Let {T
r
n
} be a sequence of mappping defined as in Lemma 2.4 and let x
∗
∈∩
∞
n1
FT
n
∩
VIA, C ∩ MEPΘ,ϕ. Then x
∗
W
n
x
∗
and x
∗
P
C
x
∗
− λ
n
Ax
∗
T
r
n
x
∗
.Putv
n
P
C
y
n
−
λ
n
Ay
n
.From3.2 we have
v
n
− x
∗
P
C
y
n
− λ
n
Ay
n
− P
C
x
∗
− λ
n
Ax
∗
≤
y
n
− λ
n
Ay
n
−
x
∗
− λ
n
Ax
∗
≤y
n
− x
∗
s
n
P
C
u
n
− λ
n
Au
n
1 − s
n
x
n
− s
n
P
C
x
∗
− λ
n
Ax
∗
−
1 − s
n
x
∗
≤ s
n
P
C
u
n
− λ
n
Au
n
− P
C
x
∗
− λ
n
Ax
∗
1 − s
n
x
n
− x
∗
≤ s
n
u
n
− x
∗
1 − s
n
x
n
− x
∗
s
n
T
r
n
x
n
− T
r
n
x
∗
1 − s
n
x
n
− x
∗
≤ s
n
x
n
− x
∗
1 − s
n
x
n
− x
∗
x
n
− x
∗
.
3.3
Hence, we obtain that
x
n1
− x
∗
α
n
f
x
n
− β
n
x
n
− γ
n
W
n
v
n
− x
∗
≤ α
n
f
x
n
− x
∗
β
n
x
n
− x
∗
γ
n
W
n
v
n
− x
∗
≤ α
n
f
x
n
− f
x
∗
α
n
f
x
∗
− x
∗
β
n
x
n
− x
∗
γ
n
v
n
− x
∗
≤ α
n
βx
n
− x
∗
α
n
f
x
∗
− x
∗
β
n
x
n
− x
∗
γ
n
x
n
− x
∗
1 − β
α
n
f
x
∗
− x
∗
1 − β
1 −
1 − β
α
n
x
n
− x
∗
≤ max
x
n
− x
∗
,
f
x
∗
− x
∗
1 − β
≤ max
x
0
− x
∗
,
f
x
∗
− x
∗
1 − β
.
3.4
Therefore {x
n
} is bounded. Consequently, {fx
n
}, {u
n
}, {y
n
}, {v
n
}, {W
n
v
n
}, {Au
n
},and
{Ay
n
} are also bounded.
Next, we claim that lim
n →∞
x
n1
− x
n
0.
Fixed Point Theory and Applications 9
Indeed, setting x
n1
β
n
x
n
1 − β
n
z
n
, for all n ≥ 1, it follows that
z
n1
− z
n
α
n1
f
x
n1
γ
n1
W
n1
v
n1
1 − β
n1
−
α
n
f
x
n
γ
n
W
n
v
n
1 − β
n
α
n1
f
x
n1
γ
n1
W
n1
v
n1
1 − β
n1
−
γ
n1
W
n1
v
n
1 − β
n1
γ
n1
W
n1
v
n
1 − β
n1
−
α
n
f
x
n
γ
n
W
n
v
n
1 − β
n
α
n1
f
x
n1
1 − β
n1
−
α
n
f
x
n
1 − β
n
γ
n1
1 − β
n1
W
n1
v
n1
− W
n1
v
n
1 − β
n1
− α
n1
1 − β
n1
W
n1
v
n
−
1 − β
n
− α
n
1 − β
n
W
n
v
n
α
n1
f
x
n1
1 − β
n1
−
α
n
f
x
n
1 − β
n
γ
n1
1 − β
n1
W
n1
v
n1
− W
n1
v
n
w
n1
v
n
− w
n
v
n
α
n
1 − β
n
W
n
v
n
−
α
n1
1 − β
n1
W
n1
v
n
.
3.5
Now, we estimate W
n1
v
n
− W
n
v
n
and W
n1
v
n1
− W
n1
v
n
.
From the definition of {W
n
}, 1.6, and since T
i
, U
n,i
are nonexpansive, we deduce that,
for each n ≥ 1,
W
n1
v
n
− W
n
v
n
t
1
T
1
U
n1,2
v
n
− t
1
T
1
U
n,2
v
n
≤ t
1
U
n1,2
v
n
− U
n,2
v
n
t
1
t
2
T
2
U
n1,3
v
n
− t
2
T
2
U
n,3
v
n
≤ t
1
t
2
U
n1,3
v
n
− U
n,3
v
n
.
.
.
≤
n
i1
t
i
U
n1,n1
v
n
− U
n,n1
v
n
≤ M
n
i1
t
i
,
3.6
for some constant M>0 such that sup{U
n1,n1
v
n
− U
n,n1
v
n
,n ≥ 1}≤M. And
10 Fixed Point Theory and Applications
we note that
W
n1
v
n1
− W
n1
v
n
≤
v
n1
− v
n
P
C
y
n1
− λ
n1
Ay
n1
− P
C
y
n
− λ
n
Ay
n
≤
y
n1
− λ
n1
Ay
n1
−
y
n
− λ
n
Ay
n
≤
I − λ
n1
A
y
n1
−
I − λ
n1
A
y
n
|
λ
n
− λ
n1
|
Ay
n
≤y
n1
− y
n
|
λ
n
− λ
n1
|
Ay
n
,
3.7
y
n1
− y
n
s
n1
P
C
u
n1
− λ
n1
Au
n1
1 − s
n1
x
n1
−s
n
P
C
u
n
− λ
n
Au
n
−
1 − s
n
x
n
s
n1
P
C
u
n1
− λ
n1
Au
n1
− s
n1
P
C
u
n
− λ
n
Au
n
s
n1
− s
n
P
C
u
n
− λ
n
Au
n
1 − s
n1
x
n1
−
1 − s
n1
s
n1
− s
n
x
n
≤ s
n1
u
n1
− λ
n1
Au
n1
−
u
n
− λ
n
Au
n
|
s
n1
− s
n
|
u
n
− λ
n
Au
n
1 − s
n1
x
n1
− x
n
|
s
n1
− s
n
|
x
n
≤ s
n1
{
u
n1
− λ
n1
Au
n1
−
u
n
− λ
n
Au
n
|
λ
n
− λ
n1
|
Au
n
}
|
s
n1
− s
n
|
u
n
λ
n
Au
n
x
n
1 − s
n1
x
n1
− x
n
≤ s
n1
u
n1
− u
n
s
n1
|
λ
n
− λ
n1
|
Au
n
|
s
n1
− s
n
|
Q
1 − s
n1
x
n1
− x
n
,
3.8
where Q sup{u
n
,λ
n
Au
n
, x
n
: n ≥ 1}.
Combining 3.7 and 3.8,weobtain
v
n1
− v
n
≤s
n1
u
n1
− u
n
s
n1
|
λ
n
− λ
n1
|
Au
n
|
s
n1
− s
n
|
Q
1 − s
n1
x
n1
− x
n
|
λ
n
− λ
n1
|
Ay
n
.
3.9
On the other hand, from u
n
T
r
n
x
n
and u
n1
T
r
n1
x
n1
,wenotethat
Θ
u
n
,y
ϕ
y
− ϕ
u
n
1
r
n
y − u
n
,u
n
− x
n
≥ 0, ∀y ∈ C,
3.10
Θ
u
n1
,y
ϕ
y
− ϕ
u
n1
1
r
n1
y − u
n1
,u
n1
− x
n1
≥ 0, ∀y ∈ C.
3.11
Fixed Point Theory and Applications 11
Putting y u
n1
in 3.10 and y u
n
in 3.11, we have
Θ
u
n
,u
n1
ϕ
u
n1
− ϕ
u
n
1
r
n
u
n1
− u
n
,u
n
− x
n
≥ 0,
Θ
u
n1
,u
n
ϕ
u
n
− ϕ
u
n1
1
r
n1
u
n
− u
n1
,u
n1
− x
n1
≥ 0.
3.12
So, from A2 we get u
n1
− u
n
, u
n
− x
n
/r
n
− u
n1
− x
n1
/r
n1
≥0.
Hence u
n1
− u
n
,u
n
− u
n1
u
n1
− x
n
− r
n
/r
n1
u
n1
− x
n1
≥0.
Without loss of generality, we may assume that there exists a real number c such that
r
n
>c>0, for all n ≥ 1. Then we get
u
n1
− u
n
2
≤
u
n1
− u
n
,x
n1
− x
n
1 −
r
n
r
n1
u
n1
− x
n1
≤u
n1
− u
n
x
n1
− x
n
1 −
r
n
r
n1
u
n1
− x
n1
,
3.13
and hence
u
n1
− u
n
≤x
n1
− x
n
1
r
n1
|
r
n1
− r
n
|
u
n1
− x
n1
≤x
n1
− x
n
1
c
|
r
n1
− r
n
|
L,
3.14
where L sup{u
n
− x
n
: n ≥ 1}. Hence from 3.9 and 3.14, we have
W
n1
v
n1
− W
n1
v
n
≤x
n1
− x
n
s
n1
L
c
|
r
n1
− r
n
|
|
λ
n
− λ
n1
|
Au
n
|
s
n1
− s
n
|
Q
|
λ
n
− λ
n1
|
Ay
n
.
3.15
Combining 3.5, 3.6,and3.15,weget
z
n1
− z
n
−x
n1
− x
n
≤
α
n1
1 − β
n1
f
x
n1
W
n1
v
n
α
n
1 − β
n
f
x
n
W
n
v
n
γ
n1
1 − β
n1
x
n1
− x
n
s
n1
L
c
|
r
n1
− r
n
|
|
λ
n
− λ
n1
|
Au
n
|
s
n1
− s
n
|
Q
|
λ
n
− λ
n1
|
Ay
n
M
n
i1
t
i
−x
n1
− x
n
12 Fixed Point Theory and Applications
≤
α
n1
1 − β
n1
f
x
n1
W
n1
v
n
α
n
1 − β
n
f
x
n
W
n
v
n
γ
n1
1 − β
n1
s
n1
L
c
|
r
n1
− r
n
|
|
λ
n
− λ
n1
|
Au
n
|
s
n1
− s
n
|
Q
|
λ
n
− λ
n1
|
Ay
n
M
n
i1
t
i
.
3.16
It follows from 3.16 and conditions i–vi and 0 <t
i
≤ b<1, for all i ≥ 1that
lim sup
n →∞
z
n1
− z
n
−x
n1
− x
n
≤ 0.
3.17
By Lemma 2.1, we have lim
n →∞
z
n
− x
n
0. Consequently,
lim
n →∞
x
n1
− x
n
lim
n →∞
1 − β
n
z
n
− x
n
0.
3.18
From conditions iv–vi, 3.7, 3.8 , 3.14,and3.18,wealsoget
lim
n →∞
u
n1
− u
n
0, lim
n →∞
y
n1
− y
n
0, lim
n →∞
v
n1
− v
n
0.
3.19
Since α
n
β
n
γ
n
1 and from the definition of {x
n
}, we have x
n1
− x
n
α
n
fx
n
−
x
n
γ
n
W
n
v
n
− x
n
. Then we have
W
n
v
n
− x
n
≤
1
γ
n
x
n1
− x
n
α
n
f
x
n
− x
n
−→ 0, as n −→ ∞ .
3.20
For x
∗
∈∩
∞
n1
FT
n
∩ VIA, C ∩ MEPΘ,ϕ, we have
u
n
− x
∗
2
T
r
n
x
n
− T
r
n
x
∗
2
≤
T
r
n
x
n
− T
r
n
x
∗
,x
n
− x
∗
u
n
− x
∗
,x
n
− x
∗
1
2
u
n
− x
∗
2
x
n
− x
∗
2
−
x
n
− u
n
2
,
3.21
and hence u
n
− x
∗
2
≤x
n
− x
∗
2
−x
n
− u
n
2
.
Fixed Point Theory and Applications 13
From 3.3, we have
x
n1
− x
∗
2
α
n
fx
n
− β
n
x
n
− γ
n
W
n
v
n
− x
∗
2
≤ α
n
fx
n
− x
∗
2
β
n
x
n
− x
∗
2
γ
n
W
n
v
n
− x
∗
2
≤ α
n
fx
n
− x
∗
2
β
n
x
n
− x
∗
2
γ
n
v
n
− x
∗
2
≤ α
n
fx
n
− x
∗
2
β
n
x
n
− x
∗
2
γ
n
s
n
x
n
− x
∗
2
− s
n
x
n
− u
n
2
1 − s
n
x
n
− x
∗
2
≤ α
n
fx
n
− x
∗
2
β
n
γ
n
x
n
− x
∗
2
− γ
n
s
n
x
n
− u
n
2
≤ α
n
fx
n
− x
∗
2
x
n
− x
∗
2
− γ
n
s
n
x
n
− u
n
2
.
3.22
That is,
x
n
− u
n
2
≤
1
γ
n
s
n
α
n
f
x
n
− x
∗
2
x
n
− x
∗
2
−
x
n1
− x
∗
2
≤
1
γ
n
s
n
α
n
f
x
n
− x
∗
2
x
n1
− x
n
x
n
− x
∗
x
n1
− x
∗
.
3.23
From ii and 3.18,weobtain
x
n
− u
n
−→0, as n −→ ∞ . 3.24
From 3.2-3.3,weget
x
n1
− x
∗
2
≤ α
n
fx
n
− x
∗
2
β
n
x
n
− x
∗
2
γ
n
W
n
v
n
− x
∗
2
≤ α
n
fx
n
− x
∗
2
β
n
x
n
− x
∗
2
γ
n
v
n
− x
∗
2
≤ α
n
fx
n
− x
∗
2
β
n
x
n
− x
∗
2
γ
n
y
n
− λ
n
Ay
n
−
x
∗
− λ
n
Ax
∗
2
≤ α
n
fx
n
− x
∗
2
β
n
x
n
− x
∗
2
γ
n
y
n
− x
∗
2
λ
n
λ
n
− 2α
Ay
n
− Ax
∗
2
≤ α
n
fx
n
− x
∗
2
β
n
x
n
− x
∗
2
γ
n
x
n
− x
∗
2
γ
n
λ
n
λ
n
− 2α
Ay
n
− Ax
∗
2
≤ α
n
fx
n
− x
∗
2
x
n
− x
∗
2
γ
n
a
b − 2α
Ay
n
− Ax
∗
2
.
3.25
14 Fixed Point Theory and Applications
Then we get,
−γ
n
a
b − 2α
Ay
n
− Ax
∗
2
≤ α
n
fx
n
− x
∗
2
x
n
− x
∗
2
−
x
n1
− x
∗
2
≤ α
n
fx
n
− x
∗
2
x
n
− x
∗
x
n1
− x
∗
x
n
− x
n1
.
3.26
Since α
n
→ 0andx
n
− x
n1
→0, we obtain
Ay
n
− Ax
∗
−→0, as n −→ ∞ . 3.27
We note that
v
n
− x
∗
2
P
C
y
n
− λ
n
Ay
n
− P
C
x
∗
− λ
n
Ax
∗
2
≤
y
n
− λ
n
Ay
n
−
x
∗
− λ
n
Ax
∗
,v
n
− x
∗
1
2
y
n
− λ
n
Ay
n
− x
∗
− λ
n
Ax
∗
2
v
n
− x
∗
2
−
y
n
− λ
n
Ay
n
− x
∗
− λ
n
Ax
∗
− v
n
− x
∗
2
≤
1
2
y
n
− x
∗
2
v
n
− x
∗
2
−
y
n
− v
n
− λ
n
Ay
n
− Ax
∗
2
1
2
y
n
− x
∗
2
v
n
− x
∗
2
−
y
n
− v
n
2
2λ
n
y
n
− v
n
,Ay
n
− Ax
∗
− λ
2
n
Ay
n
− Ax
∗
2
.
3.28
Then we derive
v
n
− x
∗
2
≤
y
n
− x
∗
2
−
y
n
− v
n
2
2λ
n
y
n
− v
n
,Ay
n
− Ax
∗
− λ
2
n
Ay
n
− Ax
∗
2
≤
x
n
− x
∗
2
−
y
n
− v
n
2
2λ
n
y
n
− v
n
,Ay
n
− Ax
∗
.
3.29
Hence
x
n1
− x
∗
2
≤ α
n
fx
n
− x
∗
2
β
n
x
n
− x
∗
2
γ
n
W
n
v
n
− x
∗
2
≤ α
n
fx
n
− x
∗
2
β
n
x
n
− x
∗
2
γ
n
v
n
− x
∗
2
≤ α
n
fx
n
− x
∗
2
β
n
x
n
− x
∗
2
γ
n
x
n
− x
∗
2
−
y
n
− v
n
2
2λ
n
y
n
− v
n
,Ay
n
− Ax
∗
≤ α
n
fx
n
− x
∗
2
x
n
− x
∗
2
− γ
n
y
n
− v
n
2
2γ
n
λ
n
y
n
− v
n
Ay
n
− Ax
∗
,
3.30
Fixed Point Theory and Applications 15
which imply that
γ
n
y
n
− v
n
2
≤ α
n
fx
n
− x
∗
2
x
n
− x
∗
2
−
x
n1
− x
∗
2
2γ
n
λ
n
y
n
− v
n
Ay
n
− Ax
∗
≤ α
n
fx
n
− x
∗
2
2γ
n
λ
n
y
n
− v
n
Ay
n
− Ax
∗
x
n
− x
∗
x
n1
− x
∗
x
n
− x
n1
.
3.31
From condition ii, 3.18,and3.27,weget
lim
n →∞
y
n
− v
n
0.
3.32
Since
u
n
− y
n
≤u
n
− x
n
x
n
− y
n
u
n
− x
n
s
n
P
C
u
n
− λ
n
Au
n
− s
n
x
n
≤u
n
− x
n
s
n
u
n
− y
n
s
n
v
n
− W
n
v
n
s
n
W
n
v
n
− x
n
,
3.33
we have
u
n
− y
n
≤
1
1 − s
n
u
n
− x
n
s
n
1 − s
n
v
n
− W
n
v
n
s
n
1 − s
n
W
n
v
n
− x
n
,
3.34
and then we obtain
W
n
v
n
− v
n
≤W
n
v
n
− x
n
x
n
− u
n
u
n
− y
n
y
n
− v
n
≤W
n
v
n
− x
n
x
n
− u
n
1
1 − s
n
u
n
− x
n
1
1 − s
n
v
n
− W
n
v
n
s
n
1 − s
n
W
n
v
n
− x
n
y
n
− v
n
.
3.35
So we get
1 − 2s
n
1 − s
n
W
n
v
n
− v
n
≤
1
1 − s
n
W
n
v
n
− x
n
2 − s
n
1 − s
n
u
n
− x
n
y
n
− v
n
.
3.36
From condition iv and 3.20, 3.24,and3.32, we have lim
n →∞
W
n
v
n
−v
n
0. Moreover,
from Remark 2.7 we get lim
n →∞
Wv
n
− v
n
0.
Next, we show that
lim sup
n →∞
f
x
∗
− x
∗
,x
n
− x
∗
≤0,
3.37
16 Fixed Point Theory and Applications
where x
∗
P
∩
∞
n1
FT
n
∩VIA,C ∩ MEPΘ,ϕ
fx
∗
. Indeed, we choose a subsequence {v
n
i
} of {v
n
}
such that
lim sup
n →∞
f
x
∗
− x
∗
,Wv
n
− x
∗
lim
i →∞
f
x
∗
− x
∗
,Wv
n
i
− x
∗
.
3.38
Since {v
n
i
} is bounded, there exists a subsequence {v
n
i
j
} of {v
n
i
} which converges weakly to
z. Without loss of generality, we can assume that v
n
i
z.
From Wv
n
− v
n
→0, we obtain Wv
n
i
z.
Next, we show that z ∈∩
∞
n1
FT
n
∩ VIA, C ∩ MEPΘ,ϕ.
First, we show that z ∈ MEPΘ,ϕ. In fact by u
n
T
r
n
x
n
∈ dom ϕ, we have
Θ
u
n
,y
ϕ
y
− ϕ
u
n
1
r
n
y − u
n
,u
n
− x
n
≥ 0, ∀y ∈ C.
3.39
From A2, we also have
ϕ
y
− ϕ
u
n
1
r
n
y − u
n
,u
n
− x
n
≥Θ
y, u
n
, ∀y ∈ C,
3.40
and hence
ϕ
y
− ϕ
u
n
y − u
n
i
,
u
n
i
− x
n
i
r
n
i
≥ Θ
y, u
n
i
, ∀y ∈ C.
3.41
From u
n
− x
n
→0, x
n
− Wv
n
→0andWv
n
− v
n
→0, we get u
n
i
→ z. It follows from
A4 that u
n
i
− x
n
i
/r
n
i
→ 0 and from the lower semicontinuity of ϕ that
Θ
y, z
ϕ
z
− ϕ
y
≤ 0, ∀y ∈ C. 3.42
For t with 0 <t≤ 1andy ∈ C,lety
t
ty 1 − tz. Since y ∈ C and z ∈ C, we have y
t
∈ C
and hence Θy
t
,zϕz − ϕy
t
≤ 0. So, from A1 and A4, we have
0 Θ
y
t
,y
t
ϕ
y
t
− ϕ
y
t
≤ tΘ
y
t
,y
1 − t
Θ
y
t
,z
tϕ
y
1 − t
ϕ
z
− ϕ
y
t
≤ t
Θ
y
t
,y
ϕ
y
− ϕ
y
t
.
3.43
Dividing by t, we have
Θ
y
t
,y
ϕ
y
− ϕ
y
t
≥ 0, ∀y ∈ C. 3.44
Fixed Point Theory and Applications 17
Letting t → 0, it follows from the weakly semicontinuity of ϕ that
Θ
z, y
ϕ
y
− ϕ
z
≥ 0, ∀y ∈ C. 3.45
Hence z ∈ MEPΘ,ϕ.
Second, we show that z ∈ FW∩
∞
n1
FT
n
. Assume z
/
∈ FW. Since u
n
i
zand
z
/
Wz, by Opial’s condition, we have
lim inf
i →∞
u
n
i
− z < lim inf
i →∞
u
n
i
− Wz
≤ lim inf
i →∞
u
n
i
− Wu
n
i
Wu
n
i
− Wz
≤ lim inf
i →∞
u
n
i
− z,
3.46
which derives a contradiction. Thus we have z ∈ FT.
Finally, by the same argument in the proof of 28, Theorem 3.1, we can show that
z ∈ VIA, C.
Hence z ∈∩
∞
n1
FT
n
∩ VIA, C ∩ MEPΘ,ϕ.
Since x
∗
P
∩
∞
n1
FT
n
∩VIA,C∩MEPΘ,ϕ
fx
∗
and x
n
− Wv
n
→0, we have
lim sup
n →∞
f
x
∗
− x
∗
,x
n
− x
∗
lim sup
n →∞
f
x
∗
− x
∗
,Wv
n
− x
∗
lim
i →∞
f
x
∗
− x
∗
,Wv
n
i
− x
∗
f
x
∗
− x
∗
,z− x
∗
≤0.
3.47
Therefore, 3.37 holds.
Finally, we show that x
n
→ x
∗
. From definition of {x
n
},weget
x
n1
− x
∗
2
α
n
fx
n
β
n
x
n
γ
n
W
n
v
n
− x
∗
2
α
n
f
x
n
β
n
x
n
γ
n
W
n
v
n
− x
∗
,x
n1
− x
∗
α
n
f
x
n
− x
∗
,x
n1
− x
∗
β
n
x
n
− x
∗
,x
n1
− x
∗
γ
n
W
n
v
n
− x
∗
,x
n1
− x
∗
≤ α
n
f
x
n
− x
∗
,x
n1
− x
∗
1
2
β
n
x
n
− x
∗
2
x
n1
− x
∗
2
1
2
γ
n
v
n
− x
∗
2
x
n1
− x
∗
2
≤ α
n
f
x
n
− x
∗
,x
n1
− x
∗
1
2
β
n
x
n
− x
∗
2
x
n1
− x
∗
2
1
2
γ
n
x
n
− x
∗
2
x
n1
− x
∗
2
18 Fixed Point Theory and Applications
α
n
f
x
n
− x
∗
,x
n1
− x
∗
1
2
1 − α
n
x
n
− x
∗
2
x
n1
− x
∗
2
≤ α
n
f
x
n
− x
∗
,x
n1
− x
∗
1
2
1 − α
n
x
n
− x
∗
2
1
2
x
n1
− x
∗
2
,
3.48
which implies that
x
n1
− x
∗
2
≤
1 − α
n
x
n
− x
∗
2
2α
n
f
x
n
− x
∗
,x
n1
− x
∗
.
3.49
By 3.47 and Lemma 2.3,wegetthat{x
n
} converges strongly to x
∗
.
This completes the proof.
Setting fx
n
≡ u and ϕ 0 in Theorem 3.1., we have the following result.
Corollary 3.2 see 14, Theorem 2.1. Let C be a nonempty closed convex subset of a real Hilbert
space H.LetΘ be a bifunction from C × C → R satisfying (A1)–(A4), let A be an α-inverse-strongly
monotone mapping of C into H, and let {T
n
}
∞
n1
be a sequence of nonexpansive self-mapping on C
such that ∩
∞
n1
FT
n
∩ VIA, C ∩ EPΘ
/
∅. Suppose that x
1
u ∈ C, {s
n
}, {α
n
}, {β
n
}, and
{γ
n
} are sequences in 0, 1,{λ
n
} is a sequence in 0, 2α such that λ
n
∈ a, b for some a, b with
0 <a<b<2α and {r
n
}⊂0, ∞ is a real sequence. Suppose that the following conditions are
satisfied:
i α
n
β
n
γ
n
1,
ii lim
n →∞
α
n
0 and
∞
n1
α
n
∞,
iii 0 < lim inf
n →∞
β
n
≤ lim sup
n →∞
β
n
< 1,
iv 0 < lim inf
n →∞
s
n
≤ lim sup
n →∞
s
n
< 1/2 and lim
n →∞
|s
n1
− s
n
| 0,
v lim
n →∞
|λ
n1
− λ
n
| 0,
vi lim inf
n →∞
r
n
> 0 and lim
n →∞
|r
n1
− r
n
| 0.
Let the sequence {x
n
} be generated by,
Θ
u
n
,y
1
r
n
y − u
n
,u
n
− x
n
≥ 0, ∀y ∈ C,
y
n
s
n
P
C
u
n
− λ
n
Au
n
1 − s
n
x
n
,
x
n1
α
n
u β
n
x
n
γ
n
W
n
P
C
y
n
− λ
n
Ay
n
,
3.50
for all n ∈ N,whereW
n
is defined by 1.6 and {t
n
} is a sequence in 0,b,forsomeb ∈ 0, 1.
Then the sequence {x
n
} converges strongly to a point x
∗
∈∩
∞
n1
FT
n
∩ VIA, C ∩ EPΘ,where
x
∗
P
∩
∞
n1
FT
n
∩VIA,C∩EPΘ
u.
Setting ϕ 0 in Theorem 3.1, we have the following result.
Fixed Point Theory and Applications 19
Corollary 3.3. Let C be a nonempty closed convex subset of a real Hilbert space H.LetΘ be a
bifunction from C × C → R satisfying (A1)–(A4), let A be an α-inverse-strongly monotone mapping
of C into H, and let {T
n
}
∞
n1
be a sequence of nonexpansive self-mapping on C such that ∩
∞
n1
FT
n
∩
VIA, C ∩ EPΘ
/
∅. Suppose that {s
n
}, {α
n
}, {β
n
}, and {γ
n
} are sequences in 0, 1,{λ
n
} is a
sequence in 0, 2α such that λ
n
∈ a, b for some a, b with 0 <a<b<2α, and {r
n
}⊂0, ∞ is a
real sequence. Suppose that the f ollowing conditions are satisfied:
i α
n
β
n
γ
n
1,
ii lim
n →∞
α
n
0 and
∞
n1
α
n
∞,
iii 0 < lim inf
n →∞
β
n
≤ lim sup
n →∞
β
n
< 1,
iv 0 < lim inf
n →∞
s
n
≤ lim sup
n →∞
s
n
< 1/2 and lim
n →∞
|s
n1
− s
n
| 0,
v lim
n →∞
|λ
n1
− λ
n
| 0,
vi lim inf
n →∞
r
n
> 0 and lim
n →∞
|r
n1
− r
n
| 0.
Let f be a contraction of C into itself with coe fficient β ∈ 0, 1 and let the sequence {x
n
} be generated
by x
1
∈ C and
Θ
u
n
,y
1
r
n
y − u
n
,u
n
− x
n
≥ 0, ∀y ∈ C,
y
n
s
n
P
C
u
n
− λ
n
Au
n
1 − s
n
x
n
,
x
n1
α
n
f
x
n
β
n
x
n
γ
n
W
n
P
C
y
n
− λ
n
Ay
n
,
3.51
for all n ∈ N,whereW
n
is defined by 1.6 and {t
n
} is a sequence in 0,b,forsomeb ∈ 0, 1.
Then the sequence {x
n
} converges strongly to a point x
∗
∈∩
∞
n1
FT
n
∩ VIA, C ∩ EPΘ,where
x
∗
P
∩
∞
n1
FT
n
∩VIA,C∩EPΘ
fx
∗
.
By Theorem 3.1, we obtain some interesting strong convergence theorems.
Setting T
n
x x then we have W
n
x x in Theorem 3.1, and we have the following
result.
Corollary 3.4. Let C be a nonempty closed convex subset of a real Hilbert space H.Letϕ : C →
R ∪{∞} be a lower semicontinuous and convex function. Let Θ be a bifunction from C × C → R
satisfying (A1)–(A4), and let A be an α-inverse-strongly monotone mapping of C into H such that
VIA, C ∩ MEPΘ,ϕ
/
∅. Suppose that {s
n
}, {α
n
}, {β
n
}, and {γ
n
} are sequences in 0, 1,{λ
n
} is
a sequence in 0, 2α such that λ
n
∈ a, b for some a, b with 0 <a<b<2α and {r
n
}⊂0, ∞ is a
real sequence. Suppose that the f ollowing conditions are satisfied:
i α
n
β
n
γ
n
1,
ii lim
n →∞
α
n
0 and
∞
n1
α
n
∞,
iii 0 < lim inf
n →∞
β
n
≤ lim sup
n →∞
β
n
< 1,
iv 0 < lim inf
n →∞
s
n
≤ lim sup
n →∞
s
n
< 1/2 and lim
n →∞
|s
n1
− s
n
| 0,
v lim
n →∞
|λ
n1
− λ
n
| 0,
vi lim inf
n →∞
r
n
> 0 and lim
n →∞
|r
n1
− r
n
| 0.
20 Fixed Point Theory and Applications
Let f be a contraction of C into itself with coefficient β ∈ 0, 1. Assume that either (B1) or (B2) holds.
Then the sequences {x
n
}, {u
n
}, and {y
n
} generated by, x
1
∈ C and
Θ
u
n
,y
ϕ
y
− ϕ
u
n
1
r
n
y − u
n
,u
n
− x
n
≥ 0, ∀y ∈ C,
y
n
s
n
P
C
u
n
− λ
n
Au
n
1 − s
n
x
n
,
x
n1
α
n
f
x
n
β
n
x
n
γ
n
P
C
y
n
− λ
n
Ay
n
3.52
converge strongly to a point x
∗
∈ VIA, C ∩ MEPΘ,ϕ,wherex
∗
P
VIA,C∩MEPΘ,ϕ
fx
∗
.
Setting Θ0,ϕ 0andr
n
1 then we have u
n
P
C
x
n
x
n
in Theorem 3.1, and we
have the following result.
Corollary 3.5. Let C be a nonempty closed convex subset of a real Hilbert space H.LetA be an
α-inverse-strongly monotone mapping of C into H and let {T
n
}
∞
n1
be a sequence of nonexpansive
self-mapping on C such that ∩
∞
n1
FT
n
∩ VIA, C
/
∅. Suppose that {s
n
}, {α
n
}, {β
n
}, and {γ
n
} are
sequences in 0, 1,{λ
n
} is a sequence in 0, 2α such that λ
n
∈ a, b for some a, b with 0 <a<b<
2α. Suppose that the following conditions are satisfied:
i α
n
β
n
γ
n
1,
ii lim
n →∞
α
n
0 and
∞
n1
α
n
∞,
iii 0 < lim inf
n →∞
β
n
≤ lim sup
n →∞
β
n
< 1,
iv 0 < lim inf
n →∞
s
n
≤ lim sup
n →∞
s
n
< 1/2 and lim
n →∞
|s
n1
− s
n
| 0,
v lim
n →∞
|λ
n1
− λ
n
| 0.
Let f be a contraction of C into itself with coe fficient β ∈ 0, 1. Let the sequences {x
n
} and {y
n
} be
generated by x
1
∈ C and
y
n
s
n
P
C
x
n
− λ
n
Ax
n
1 − s
n
x
n
,
x
n1
α
n
f
x
n
β
n
x
n
γ
n
W
n
P
C
y
n
− λ
n
Ay
n
,
3.53
for all n ∈ N,whereW
n
defined by 1.6 and {t
n
} is a sequence in 0,b,forsomeb ∈ 0, 1.
Then the sequences {x
n
} and {y
n
} converge strongly to a point x
∗
∈∩
∞
n1
FT
n
∩ VI A, C,where
x
∗
P
∩
∞
n1
FT
n
∩VIA,C
fx
∗
.
Acknowledgment
The author would like to thank the referees for their helpful comments and suggestions,
which improved the presentation of this paper.
Fixed Point Theory and Applications 21
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