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
≥ 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,
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
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
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
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
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
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
− 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
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
}, {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
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
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
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
∞
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
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
/
∞
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
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
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
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
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
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
I − λ
n
Ax − I − λ
n
Ay
2
x − y − λ
n
Ax − Ay
2
x − y
2
− 2λ
n
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
− x
∗
P
C
y
n
− λ
n
Ay
n
− P
C
x
∗
− λ
n
Ax
∗
≤
y
n
− λ
n
Ay
1 − s
n
x
n
− s
n
P
C
x
∗
− λ
n
Ax
∗
−
1 − s
n
x
∗
≤ s
n
P
C
u
n
− x
∗
1 − s
n
x
n
− x
∗
s
n
T
r
n
x
n
− T
r
n
x
∗
1 − s
n
n1
− x
∗
α
n
f
x
n
− β
n
x
n
− γ
n
W
n
v
n
− x
∗
≤ α
n
f
x
n
n
f
x
∗
− x
∗
β
n
x
n
− x
∗
γ
n
v
n
− x
∗
≤ α
n
βx
n
− x
∗
α
n
f
− x
∗
1 − β
1 −
1 − β
α
n
x
n
− x
∗
≤ max
x
n
− x
∗
,
f
x
∗
}, {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
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
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
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
.
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
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
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
−
y
n
− λ
n
Ay
n
≤
I − λ
n1
A
y
n1
−
I − λ
n1
A
y
n
|
λ
n
C
u
n1
− λ
n1
Au
n1
1 − s
n1
x
n1
−s
n
P
C
u
n
− λ
n
Au
n
−
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
|
u
n
− λ
n
Au
n
1 − s
n1
x
n1
− x
n
|
s
n1
− s
n
|
x
n
Au
n
}
|
s
n1
− s
n
|
u
n
λ
n
Au
n
x
n
1 − s
n1
x
n1
− x
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
− 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
− 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
,u
n
ϕ
u
n
− ϕ
u
n1
1
r
n1
u
n
− u
n1
,u
n1
− x
n1
≥ 0.
− 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
,
3.13
and hence
u
n1
− u
n
≤x
n1
− x
n
1
r
n1
: 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
− z
n
−x
n1
− x
n
≤
α
n1
1 − β
n1
f
x
n1
W
n1
v
n
α
n
1 − β
n
f
|
|
λ
n
− λ
n1
|
Au
n
|
s
n1
− s
n
|
Q
|
λ
n
− λ
n1
|
Ay
n
α
n
1 − β
n
f
x
n
W
n
v
n
γ
n1
1 − β
n1
s
n1
L
c
|
r
n1
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 →∞
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
− 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
∗
,x
n
− x
∗
1
2
u
n
− x
∗
2
x
n
− x
∗
2
−
x
n
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
v
n
− x
∗
2
≤ α
n
n
x
n
− u
n
2
1 − s
n
x
n
− x
∗
2
≤ α
n
fx
n
− x
∗
fx
n
− x
∗
2
x
n
− x
∗
2
− γ
n
s
n
x
n
− u
n
2
.
3.22
That is,
x
n
− x
∗
2
−
x
n1
− x
∗
2
≤
1
γ
n
s
n
α
n
f
x
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
γ
n
v
n
− x
∗
2
≤ α
n
fx
n
− x
∗
2
β
n
x
n
− x
∗
2
γ
2
β
n
x
n
− x
∗
2
γ
n
y
n
− x
∗
2
λ
n
λ
n
− 2α
γ
n
x
n
− x
∗
2
γ
n
λ
n
λ
n
− 2α
Ay
n
− Ax
∗
2
≤ α
n
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
∗
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
2
≤
y
n
− λ
n
Ay
n
−
x
∗
− λ
n
Ax
∗
,v
n
− x
∗
1
2
n
Ay
n
− x
∗
− λ
n
Ax
∗
− v
n
− x
∗
2
≤
1
2
y
n
− x
∗
1
2
y
n
− x
∗
2
v
n
− x
∗
2
−
y
n
− v
n
v
n
− x
∗
2
≤
y
n
− x
∗
2
−
y
n
− v
n
2
2λ
n
y
y
n
− v
n
2
2λ
n
y
n
− v
n
,Ay
n
− Ax
∗
.
3.29
Hence
x
n1
− x
∗
2
2
≤ α
n
fx
n
− x
∗
2
β
n
x
n
− x
∗
2
γ
n
v
n
− x
∗
2
−
y
n
− v
n
2
2λ
n
y
n
− v
n
,Ay
n
− Ax
∗
≤ α
n
fx
n
− x
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
fx
n
− x
∗
2
2γ
n
λ
n
y
n
− v
n
Ay
n
− Ax
∗
x
n
− x
∗
x
n1
− x
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
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
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
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
. 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
∗
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
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
→ 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
1 − t
Θ
y
t
,z
tϕ
y
1 − t
ϕ
z
− ϕ
y
t
≤ t
Θ
≥ 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
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
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
2
α
n
f
x
n
β
n
x
n
γ
n
W
n
v
n
− x
∗
,x
n1
− x
∗
α
n
f
∗
,x
n1
− x
∗
≤ α
n
f
x
n
− x
∗
,x
n1
− x
∗
1
2
β
n
x
n
− x
∗
2
≤ α
n
f
x
n
− x
∗
,x
n1
− x
∗
1
2
β
n
x
n
− 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
1 − α
n
x
n
− x
∗
2
1
2
x
n1
− x
∗
2
,
3.48
which implies that
x
n1
− x
∗
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
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
,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
} 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
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
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
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
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
|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 − s
n
x
n
,
x
n1
α
n
f
x
n
β
n
x
n
γ
n
P
C
y
n
− λ
n
Ay
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,
|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
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
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