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Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2010, Article ID 836714, 29 pages
doi:10.1155/2010/836714
Research Article
Iterative Methods for Finding Common
Solution of Generalized Equilibrium Problems
and Variational Inequality Problems
and Fixed Point Problems of a Finite Family
of Nonexpansive Mappings
Atid Kangtunyakarn
Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang,
Bangkok 10520, Thailand
Correspondence should be addressed to Atid Kangtunyakarn,
Received 7 October 2010; Accepted 2 November 2010
Academic Editor: T. D. Benavides
Copyright q 2010 Atid Kangtunyakarn. 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 method for a system of generalized equilibrium problems, system of
variational inequality problems, and fixed point problems by using S-mapping generated by a
finite family of nonexpansive mappings and real numbers. Then, we prove a strong convergence
theorem of the proposed iteration under some control condition. By using our main result, we
obtain strong convergence theorem for finding a common element of the set of solution of a system
of generalized equilibrium problems, system of variational inequality problems, and the set of
common fixed points of a finite family of strictly pseudocontractive mappings.
1. Introduction
Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H.LetA :
C → H be a nonlinear mapping, and let F : C × C → R be a bifunction. A mapping T of H
into itself is called nonexpansive if Tx− Ty≤x − y for all x, y ∈ H. We denote by FT the
set of fixed points of T i.e., FT{x ∈ H : Tx x}. Goebel and Kirk 1 showed that FT
is always closed convex, and also nonempty provided T has a bounded trajectory.
A bounded linear operator A on H is called strongly positive with coefficient
γ if there
is a constant
γ>0 with the property
Ax, x
≥
γ
x
2
. 1.1
2 Fixed Point Theory and Applications
The equilibrium problem for F is to find x ∈ C such that
F
x, y
≥ 0, ∀y ∈ C. 1.2
The set of solutions of 1.2 is denoted by EPF. Many problems in physics, optimization,
and economics are seeking some elements of EPF,see2, 3. Several iterative methods have
been proposed to solve the equilibrium problem, see, for instance, 2–4. In 2005, Combettes
and Hirstoaga 3 introduced an iterative scheme of finding the best approximation to the
initial data when EPF is nonempty and proved a strong convergence theorem.
The variational inequality problem is to find a point u ∈ C such that
v − u, Au
≥ 0 ∀ v ∈ C. 1.3
The set of solutions of the variational inequality is denoted by VIC, A, and we consider the
following generalized equilibrium problem.
Find z ∈ C such that F
z, y
Az, y − z≥
0, ∀y ∈ C. 1.4
The set of such z ∈ C is denoted by EPF, A,thatis,
EP
F, A
z ∈ C : F
z, y
Az, y − z
≥ 0, ∀y ∈ C
. 1.5
In the case of A ≡ 0, EPF, AEPF. Numerous problems in physics, optimization,
variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative
games reduce to find element of 1.5
A mapping A of C into H is called inverse-strongly monotone,see5, if there exists a
positive real number α such that
x − y, Ax − Ay
≥ α
Ax − Ay
2
1.6
for all x, y ∈ C.
The problem of finding a common fixed point of a family of nonexpansive mappings
has been studied by many authors. The well-known convex feasibility problem reduces to
finding a point in the intersection of the fixed point sets of a family of nonexpansive mapping
see 6, 7.
The ploblem of finding a common element of EPF, A and the set of all common
fixed points of a family of nonexpansive mappings is of wide interdisciplinary interest and
importance. Many iterative methods are purposed for finding a common element of the
solutions of the equilibrium problem and fixed point problem of nonexpansive mappings,
see 8–10.
Fixed Point Theory and Applications 3
In 2008, S.Takahashi and W.Takahashi 11 introduced a general iterative method for
finding a common element of EPF, A and FT. They defined {x
n
} in the following way:
u, x
1
∈ C, arbitrarily;
F
z
n
,y
Ax
n
,y− z
n
1
λ
n
y − z
n
,z
n
− x
n
≥ 0, ∀y ∈ C,
x
n1
β
n
x
n
1 − β
n
T
a
n
u
1 − a
n
z
n
, ∀n ∈ N,
1.7
where A is an α-inverse strongly monotone mapping of C into H with positive real number α,
and {a
n
}∈0, 1, {β
n
}⊂0, 1, {λ
n
}⊂0, 2α, and proved strong convergence of the scheme
1.7 to z ∈
N
i1
FT
i
∩ EPF, A, where z P
N
i1
FT
i
∩EPF, A
u in the framework of a Hilbert
space, under some suitable conditions on {a
n
}, {β
n
}, {λ
n
} and bifunction F.
Very recently, in 2010, Qin, et al. 12 introduced a iterative scheme method for finding
a common element of EPF
1
,A,EPF
2
,B and common fixed point of infinite family of
nonexpansive mappings. They defined {x
n
} in the following way:
x
1
∈ C, arbitrarily;
F
1
u
n
,u
Ax
n
,u− u
n
1
r
u − u
n
,u
n
− x
n
≥ 0, ∀u ∈ C,
F
2
v
n
,v
Bx
n
,v− v
n
1
s
v − v
n
,v
n
− x
n
≥ 0, ∀v ∈ C,
y
n
δ
n
u
n
1 − δ
n
v
n
,
x
n1
α
n
f
x
n
β
n
x
n
γ
n
W
n
x
n
, ∀n ∈ N,
1.8
where f : C → C is a contraction mapping and W
n
is W-mapping generated by infinite
family of nonexpansive mappings and infinite real number. Under suitable conditions of
these parameters they proved strong convergence of the scheme 1.8 to z P
F
fz, where
F
∞
i1
FT
i
∩ EPF
1
,A ∩ EPF
2
,B.
In this paper, motivated by 11, 12, we introduced a general iterative scheme {x
n
}
defined by
F
u
n
,u
Ax
n
,u− u
n
1
r
n
u − u
n
,u
n
− x
n
≥ 0,
G
v
n
,v
Bx
n
,v− v
n
1
s
n
v − v
n
,v
n
− x
n
≥ 0,
y
n
δ
n
P
C
u
n
− λ
n
Au
n
1 − δ
n
P
C
v
n
− η
n
Bv
n
,
x
n1
α
n
f
x
n
β
n
x
n
γ
n
S
n
y
n
, ∀n ≥ 0,
1.9
4 Fixed Point Theory and Applications
where f : C → C and S
n
is S-mapping generated by T
0
, ,T
n
and α
n
,α
n−1
, ,α
0
. Under
suitable conditions, we proved strong convergence of {x
n
} to z P
F
fz,andz is solution of
Ax
∗
,x− x
∗
≥ 0,
Bx
∗
,x− x
∗
≥ 0.
1.10
2. Preliminaries
In this section, we collect and give some useful lemmas that will be used for our main result
in the next section.
Let C be closed convex subset of a real Hilbert space H,andletP
C
be the metric
projection of H onto C,thatis,forx ∈ H, P
C
x satisfies the property
x − P
C
x
min
y∈C
x − y
. 2.1
The following characterizes the projection P
C
.
Lemma 2.1 see 13. Given x ∈ H and y ∈ C.ThenP
C
x y if and only if there holds the
inequality
x − y, y − z
≥ 0 ∀z ∈ C. 2.2
Lemma 2.2 see 14. Let {s
n
} be a sequence of nonnegative real numbers satisfying
s
n1
1 − α
n
s
n
β
n
, ∀n ≥ 0 2.3
where {α
n
}, {β
n
} satisfy the conditions
1 {α
n
}⊂0, 1,
∞
n1
α
n
∞,
2 lim sup
n →∞
β
n
/α
n
≤ 0.
Then lim
n →∞
s
n
0.
Lemma 2.3 see 15. Let C be a closed convex subset of a strictly convex Banach space E.Let
{T
n
: n ∈ N} be a sequence of nonexpansive mappings on C. Suppose that
∞
n1
FT
n
is nonempty.
Let {λ
n
} be a sequence of positive numbers with Σ
∞
n1
λ
n
1. Then a mapping S on C defined by
S
x
Σ
∞
n1
λ
n
T
n
x 2.4
for x ∈ C is well defined, nonexpansive, and FS
∞
n1
FT
n
hold.
Lemma 2.4 see 16. Let E be a uniformly convex Banach space, C a nonempty closed convex
subset of E, and S : C → C a nonexpansive mapping. Then I − S is demiclosed at zero.
Fixed Point Theory and Applications 5
Lemma 2.5 see 17. Let {x
n
} and {z
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
β
n
x
n
1 − β
n
z
n
2.5
for all integer n ≥ 0 and
lim sup
n →∞
z
n1
− z
n
−
x
n1
− x
n
≤ 0. 2.6
Then lim
n →∞
x
n
− z
n
0.
For solving the equilibrium problem for a bifunction F : C × C → R, let us assume
that F satisfies the following conditions:
A1 Fx, x0 for all x ∈ C;
A2 F is monotone, that is, Fx, yFy,x ≤ 0, ∀x, y ∈ C,
A3 for all x, y, z ∈ C,
lim
t → 0
F
tz
1 − t
x, y
≤ F
x, y
, 2.7
A4 for all x ∈ C, y → Fx, y is convex and lower semicontinuous.
The following lemma appears implicitly in 2.
Lemma 2.6 see 2. Let C be a nonempty closed convex subset of H, and let F be a bifunction of
C × C into R satisfying (A1)–(A4). Let r>0 and x ∈ H. Then, there exists z ∈ C such that
F
z, y
1
r
y − z, z − x
2.8
for all x ∈ C.
Lemma 2.7 see 3. Assume that F : C× C → R satisfies (A1)–(A4). For r>0 and x ∈ H, define
a mapping T
r
: H → C as follows:
T
r
x
z ∈ C : F
z, y
1
r
y − z, z − x
≥ 0, ∀y ∈ C
. 2.9
for all z ∈ H. Then, the following hold:
1 T
r
is single-valued;
2 T
r
is firmly nonexpansive, that is,
T
r
x
− T
r
y
2
≤
T
r
x
− T
r
y
,x− y
∀x, y ∈ H; 2.10
3 FT
r
EPF;
4 EPF is closed and convex.
6 Fixed Point Theory and Applications
In 2009, Kangtunyakarn and Suantai 18 defined a new mapping and proved their
lemma as follows.
Definition 2.8. Let C be a nonempty convex subset of real Banach space. Let {T
i
}
N
i1
be a finite
family of nonexpansive mappings of C into itself. For each j 1, 2, ,N,letα
j
α
j
1
,α
j
2
,α
j
3
∈
I × I × I, where I ∈ 0, 1 and α
j
1
α
j
2
α
j
3
1. We define the mapping S : C → C as follows:
U
0
I,
U
1
α
1
1
T
1
U
0
α
1
2
U
0
α
1
3
I,
U
2
α
2
1
T
2
U
1
α
2
2
U
1
α
2
3
I,
U
3
α
3
1
T
3
U
2
α
3
2
U
2
α
3
3
I,
.
.
.
U
N−1
α
N−1
1
T
N−1
U
N−2
α
N−1
2
U
N−2
α
N−1
3
I,
S U
N
α
N
1
T
N
U
N−1
α
N
2
U
N−1
α
N
3
I.
2.11
This mapping is called S-mapping generated by T
1
, ,T
N
and α
1
,α
2
, ,α
N
.
Lemma 2.9. Let C be a nonempty closed convex subset of strictly convex. Let {T
i
}
N
i1
be a finite family
of nonexpanxive mappings of C into itself with
N
i1
FT
i
/
∅, and let α
j
α
j
1
,α
j
2
,α
j
3
∈ I × I × I,
j 1, 2, 3, ,N,whereI 0, 1, α
j
1
α
j
2
α
j
3
1, α
j
1
∈ 0, 1 for all j 1, 2, , N − 1,α
N
1
∈
0, 1 α
j
2
,α
j
3
∈ 0, 1 for all j 1, 2, ,N.LetS be the mapping generated by T
1
, ,T
N
and
α
1
,α
2
, ,α
N
.ThenFS
N
i1
FT
i
.
Lemma 2.10. Let C be a nonempty closed convex subset of Banach space. Let {T
i
}
N
i1
be a finite family
of nonexpansive mappings of C into itself and α
n
j
α
n,j
1
,α
n,j
2
,α
n,j
3
, α
j
α
j
1
,α
j
2
,α
j
3
∈ I × I × I,
where I 0, 1, α
n,j
1
α
n,j
2
α
n,j
3
1 and α
j
1
α
j
2
α
j
3
1 such that α
n,j
i
→ α
j
i
∈ 0, 1 as n →
∞ for i 1, 3 and j 1, 2, 3, ,N. Moreover, for every n ∈ N,letS and S
n
be the S-mappings
generated by T
1
,T
2
, ,T
N
and α
1
,α
2
, ,α
N
and T
1
,T
2
, ,T
N
and α
n
1
,α
n
2
, , α
n
N
, respectively.
Then lim
n →∞
S
n
x − Sx 0 for every x ∈ C.
Lemma 2.11 see 19. Let C be a nonempty closed convex subset of a Hilbert space H, and let
G : C → C be defined by
G
x
P
C
x − λAx
, ∀x ∈ C, 2.12
with ∀λ>0.Thenx
∗
∈ VIC, A if and only i f x
∗
∈ FG.
3. Main Result
Theorem 3.1. Let C be a nonempty closed convex subset of a Hilbert space H.LetF and G be two
bifunctions from C×C into R satisfying (A1)–(A4), respectively. Let A : C → H a α-inverse strongly
monotone mapping and B : C → H be a β-inverse strongly monotone mapping. Let {T
i
}
N
i1
be finite
Fixed Point Theory and Applications 7
family of nonexpansive mappings with F
N
i1
FT
i
∩ EP F, A ∩ EPG, B ∩ FG
1
∩ FG
2
/
∅,
where G
1
,G
2
: C → C are defined by G
1
xP
C
x − λ
n
Ax, G
2
xP
C
x − η
n
Bx, ∀x ∈ C.
Let f : C → C be a contraction with the coefficient θ ∈ 0, 1.LetS
n
be the S-mappings generated
by T
1
,T
2
, ,T
N
and α
n
1
,α
n
2
, ,α
n
N
,whereα
n
j
α
n,j
1
,α
n,j
2
,α
n,j
3
∈ I × I × I, I 0, 1, α
n,j
1
α
n,j
2
α
n,j
3
1 and 0 <η
1
≤ α
n,j
1
≤ θ
1
< 1 ∀n ∈ N, ∀j 1, 2, ,N− 1, 0 <η
N
≤ α
n,N
1
≤ 1 and
0 ≤ α
n,j
2
,α
n,j
3
≤ θ
3
< 1 ∀n ∈ N, ∀j 1, 2, ,N.Let{x
n
}, {u
n
}, {v
n
}, {y
n
} be sequences generated
by x
1
,u,v ∈ C
F
u
n
,u
Ax
n
,u− u
n
1
r
n
u − u
n
,u
n
− x
n
≥ 0,
G
v
n
,v
Bx
n
,v− v
n
1
s
n
v − v
n
,v
n
− x
n
≥ 0,
y
n
δ
n
P
C
u
n
− λ
n
Au
n
1 − δ
n
P
C
v
n
− η
n
Bv
n
,
x
n1
α
n
f
x
n
β
n
x
n
γ
n
S
n
y
n
, ∀n ≥ 1,
3.1
where {α
n
}, {β
n
}, {γ
n
}∈0, 1 such that α
n
β
n
γ
n
1, r
n
∈ a, b ⊂ 0, 2α, s
n
∈ c, d ⊂
0, 2β, λ
n
∈ e, f ⊂ 0, 2α, η
n
∈ g,h ⊂ 0, 2β. Assume that
i lim
n →∞
n 0 and Σ
∞
n0
α
n
∞,
ii 0 < lim inf
n →∞
β
n
≤ lim sup
n →∞
β
n
< 1,
iii lim
n →∞
δ
n
δ ∈ 0, 1,
ivΣ
∞
n0
|s
n1
−s
n
|, Σ
∞
n0
|r
n1
−r
n
|, Σ
∞
n0
|λ
n1
−λ
n
|,Σ
∞
n0
|η
n1
−η
n
|, Σ
∞
n0
|α
n1
−α
n
|, Σ
∞
n0
|β
n1
−
β
n
| < ∞,
v |α
n1,j
1
− α
n,j
1
|→0, and |α
n1,j
3
− α
n,j
3
|→0 as n →∞, for all j ∈{1, 2, 3, ,N}.
Then the sequence {x
n
}, {y
n
}, {u
n
}, {v
n
} converge strongly to z P
F
fz, and z is solution of
Ax
∗
,x− x
∗
≥ 0,
Bx
∗
,x− x
∗
≥ 0.
3.2
Proof. First, we show that I − λ
n
A, I − η
n
BI − r
n
A and I − s
n
B are nonexpansive. Let
x, y ∈ C. Since A is α-strongly monotone and λ
n
< 2α for all n ∈ N, we have
I − λ
n
A
x −
I − λ
n
A
y
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
− 2αλ
n
Ax − Ay
2
λ
2
n
Ax − Ay
2
x − y
2
λ
n
λ
n
− 2α
Ax − Ay
2
≤
x − y
2
.
3.3
8 Fixed Point Theory and Applications
Thus I − λ
n
A is nonexpansive. By using the same proof, we obtain that I − η
n
BI − r
n
A
and I − s
n
B are nonexpansive.
We will divide our proof into 6 steps.
Step 1. We will show that the sequence {x
n
} is bounded. Since
F
u
n
,u
Ax
n
,u− u
n
1
r
n
u − u
n
,u
n
− x
n
≥ 0, ∀u ∈ C,
3.4
then we have
F
u
n
,u
1
r
n
u − u
n
,u
n
−
I − r
n
A
x
n
≥ 0.
3.5
By Lemma 2.7, we have u
n
T
r
n
I − r
n
Ax
n
. By the same argument as above, we obtaine that
v
n
T
s
n
I − s
n
Bx
n
Let z ∈ F. Then Fz, yy − z, Az≥0andGz, yy − z, Bz≥0. Hence
F
z, y
1
r
n
y − z, z − z r
n
Az
≥ 0,
G
z, y
1
s
n
y − z, z − z s
n
Bz
≥ 0.
3.6
Again by Lemma 2.7, we have z T
r
n
z − r
n
AzT
s
n
z − s
n
Bz. Since z ∈ F, we have
z P
C
I − λ
n
Az P
C
I − η
n
Bz. By nonexpansiveness of T
r
n
, T
s
n
, I − r
n
A, I − s
n
B, we have
x
n1
− z
≤ α
n
f
x
n
− z
β
n
x
n
− z
γ
n
S
n
y
n
− z
≤ α
n
f
x
n
− f
z
α
n
f
z
− z
β
n
x
n
− z
γ
n
y
n
− z
≤ α
n
θ
x
n
− z
α
n
f
z
− z
β
n
x
n
− z
γ
n
δ
n
P
C
u
n
− λ
n
Au
n
− z
1 − δ
n
P
C
v
n
− η
n
Bv
n
− z
≤ α
n
θ
x
n
− z
α
n
f
z
− z
β
n
x
n
− z
γ
n
δ
n
u
n
− z
1 − δ
n
v
n
− z
α
n
θ
x
n
− z
α
n
f
z
− z
β
n
x
n
− z
γ
n
δ
n
T
r
n
I − r
n
A
x
n
− T
r
n
I − r
n
A
z
1 − δ
n
T
s
n
I − s
n
B
x
n
− T
s
n
I − s
n
B
z
≤ α
n
θ
x
n
− z
α
n
f
z
− z
β
n
x
n
− z
γ
n
x
n
− z
α
n
θ
x
n
− z
α
n
f
z
− z
1 − α
n
x
n
− z
1 − α
n
1 − θ
x
n
− z
α
n
f
z
− z
≤ max
x
n
− z
,
f
z
− z
1 − θ
.
3.7
Fixed Point Theory and Applications 9
By induction we can prove that {x
n
} is bounded and so are {u
n
}, {v
n
}, {y
n
}, {S
n
y
n
}. Without
of generality, assume that there exists a bounded set K ⊂ C such that
{
u
n
}
,
{
v
n
}
,
y
n
,
S
n
y
n
∈ K. 3.8
Step 2. We will show that lim
n →∞
x
n1
− x
n
0.
Putting k
n
x
n1
− β
n
x
n
/1 − β
n
, we have
x
n1
1 − β
n
k
n
β
n
x
n
, ∀n ≥ 0. 3.9
From definition of k
n
, we have
k
n1
− k
n
x
n2
− β
n1
x
n1
1 − β
n1
−
x
n1
− β
n
x
n
1 − β
n
α
n1
f
x
n1
γ
n1
S
n1
y
n1
1 − β
n1
−
α
n
f
x
n
γ
n
S
n
y
n
1 − β
n
α
n1
f
x
n1
1 − β
n1
− α
n1
S
n1
y
n1
1 − β
n1
−
α
n
f
x
n
1 − β
n
− α
n
S
n
y
n
1 − β
n
α
n1
1 − β
n1
f
x
n1
− S
n1
y
n1
−
α
n
1 − β
n
f
x
n
− S
n
y
n
S
n1
y
n1
− S
n
y
n
≤
α
n1
1 − β
n1
f
x
n1
− S
n1
y
n1
α
n
1 − β
n
f
x
n
− S
n
y
n
S
n1
y
n1
− S
n
y
n
≤
α
n1
1 − β
n1
f
x
n1
− S
n1
y
n1
α
n
1 − β
n
f
x
n
− S
n
y
n
S
n1
y
n1
− S
n
y
n
x
n1
− x
n
.
3.10
10 Fixed Point Theory and Applications
By definition of S
n
,fork ∈{2, 3, ,N}, we have
U
n1,k
y
n
− U
n,k
y
n
α
n1,k
1
T
k
U
n1,k−1
y
n
α
n1,k
2
U
n1,k−1
y
n
α
n1,k
3
y
n
−α
n,k
1
T
k
U
n,k−1
y
n
− α
n,k
2
U
n,k−1
y
n
− α
n,k
3
y
n
α
n1,k
1
T
k
U
n1,k−1
y
n
− T
k
U
n,k−1
y
n
α
n1,k
1
− α
n,k
1
T
k
U
n,k−1
y
n
α
n1,k
3
− α
n,k
3
y
n
α
n1,k
2
U
n1,k−1
y
n
− U
n,k−1
y
n
α
n1,k
2
− α
n,k
2
U
n,k−1
y
n
≤ α
n1,k
1
U
n1,k−1
y
n
− U
n,k−1
y
n
α
n1,k
1
− α
n,k
1
T
k
U
n,k−1
y
n
α
n1,k
3
− α
n,k
3
y
n
α
n1,k
2
U
n1,k−1
y
n
− U
n,k−1
y
n
α
n1,k
2
− α
n,k
2
U
n,k−1
y
n
α
n1,k
1
α
n1,k
2
U
n1,k−1
y
n
− U
n,k−1
y
n
α
n1,k
1
− α
n,k
1
T
k
U
n,k−1
y
n
α
n1,k
3
− α
n,k
3
y
n
α
n1,k
2
− α
n,k
2
U
n,k−1
y
n
≤
U
n1,k−1
y
n
− U
n,k−1
y
n
α
n1,k
1
− α
n,k
1
T
k
U
n,k−1
y
n
α
n1,k
3
− α
n,k
3
y
n
1 −
α
n1,k
1
α
n1,k
3
−
1 −
α
n,k
1
α
n,k
3
U
n,k−1
y
n
U
n1,k−1
y
n
− U
n,k−1
y
n
α
n1,k
1
− α
n,k
1
T
k
U
n,k−1
y
n
α
n1,k
3
− α
n,k
3
y
n
α
n,k
1
− α
n1,k
1
α
n,k
3
− α
n1,k
3
U
n,k−1
y
n
≤
U
n1,k−1
y
n
− U
n,k−1
y
n
α
n1,k
1
− α
n,k
1
T
k
U
n,k−1
y
n
α
n1,k
3
− α
n,k
3
y
n
α
n,k
1
− α
n1,k
1
U
n,k−1
y
n
α
n,k
3
− α
n1,k
3
U
n,k−1
y
n
U
n1,k−1
y
n
− U
n,k−1
y
n
α
n1,k
1
− α
n,k
1
T
k
U
n,k−1
y
n
U
n,k−1
y
n
α
n1,k
3
− α
n,k
3
y
n
U
n,k−1
y
n
.
3.11
Fixed Point Theory and Applications 11
By 3.11, we obtain that for each n ∈ N,
S
n1
y
n
− S
n
y
n
U
n1,N
y
n
− U
n,N
y
n
≤
U
n1,N−1
y
n
− U
n,N−1
y
n
α
n1,N
1
− α
n,N
1
T
N
U
n,N−1
y
n
U
n,N−1
y
n
α
n1,N
3
− α
n,N
3
y
n
U
n,N−1
y
n
≤
U
n1,N−2
y
n
− U
n,N−2
y
n
α
n1,N−1
1
− α
n,N−1
1
×
T
N−1
U
n,N−2
y
n
U
n,N−2
y
n
α
n1,N−1
3
− α
n,N−1
3
y
n
U
n,N−2
y
n
α
n1,N
1
− α
n,N
1
T
N
U
n,N−1
y
n
U
n,N−1
y
n
α
n1,N
3
− α
n,N
3
y
n
U
n,N−1
y
n
U
n1,N−2
y
n
− U
n,N−2
y
n
N
jN−1
α
n1,j
1
− α
n,j
1
T
j
U
n,j−1
y
n
U
n,j−1
y
n
N
jN−1
α
n1,j
3
− α
n,j
3
y
n
U
n,j−1
y
n
≤
.
.
.
≤
U
n1,1
y
n
− U
n,1
y
n
N
j2
α
n1,j
1
− α
n,j
1
T
j
U
n,j−1
y
n
U
n,j−1
y
n
N
j2
α
n1,j
3
− α
n,j
3
y
n
U
n,j−1
y
n
1 − α
n1,1
1
y
n
α
n1,1
1
T
1
y
n
−
1 − α
n,1
1
y
n
− α
n,1
1
T
1
y
n
N
j2
α
n1,j
1
− α
n,j
1
T
j
U
n,j−1
y
n
U
n,j−1
y
n
N
j2
α
n1,j
3
− α
n,j
3
y
n
U
n,j−1
y
n
α
n1,1
1
− α
n,1
1
T
1
y
n
− y
n
N
j2
α
n1,j
1
− α
n,j
1
T
j
U
n,j−1
y
n
U
n,j−1
y
n
N
j2
α
n1,j
3
− α
n,j
3
y
n
U
n,j−1
y
n
.
3.12
12 Fixed Point Theory and Applications
This together with the condition iv,weobtain
lim
n →∞
S
n1
y
n
− S
n
y
n
0.
3.13
By 3.10, 3.13 and conditions i, ii, iii, iv, it implies that
lim sup
n →∞
k
n1
− k
n
−
x
n1
− x
n
≤ 0.
3.14
From Lemma 2.5, 3.9, 3.14 and condition ii, we have
lim
n →∞
x
n
− k
n
0.
3.15
From 3.9, we can rewrite
x
n1
− x
n
1 − β
n
k
n
− x
n
. 3.16
By 3.15, we have
lim
n →∞
x
n1
− x
n
0.
3.17
On the other hand, we have
x
n
− S
n
y
n
≤
x
n
− x
n1
x
n1
− S
n
y
n
x
n
− x
n1
α
n
f
x
n
β
n
x
n
γ
n
S
n
y
n
− S
n
y
n
x
n
− x
n1
α
n
f
x
n
− S
n
y
n
β
n
x
n
− S
n
y
n
x
n
− x
n1
α
n
f
x
n
− S
n
y
n
β
n
x
n
− S
n
y
n
.
3.18
This implies that
1 − β
n
x
n
− S
n
y
n
≤
x
n
− x
n1
α
n
f
x
n
− S
n
y
n
. 3.19
By 3.17 and condition ii, we have
lim
n →∞
x
n
− S
n
y
n
0.
3.20
Step 3. Let z ∈ F; we show that
lim
n →∞
Au
n
− Az
lim
n →∞
Bv
n
− Bz
lim
n →∞
Ax
n
− Az
lim
n →∞
Bx
n
− Bz
0.
3.21
Fixed Point Theory and Applications 13
From definition of y
n
, we have
y
n
− z
2
δ
n
P
C
u
n
− λ
n
Au
n
− P
C
I − λ
n
A
z
1 − δ
n
P
C
v
n
− η
n
Bv
n
−P
C
I − η
n
B
z
2
≤ δ
n
P
C
u
n
− λ
n
Au
n
− P
C
I − λ
n
A
z
2
1 − δ
n
P
C
v
n
− η
n
Bv
n
− P
C
I − η
n
B
z
2
≤ δ
n
u
n
− λ
n
Au
n
− z λ
n
Az
2
1 − δ
n
v
n
− η
n
Bv
n
− z η
n
Bz
2
δ
n
u
n
− z
− λ
n
Au
n
− Az
2
1 − δ
n
v
n
− z
− η
n
Bv
n
− Bz
2
δ
n
u
n
− z
2
λ
2
n
Au
n
− Az
2
− 2λ
n
u
n
− z, Au
n
− Az
1 − δ
n
v
n
− z
2
η
2
n
Bv
n
− Bz
2
− 2η
n
v
n
− z, Bv
n
− Bz
≤ δ
n
u
n
− z
2
λ
2
n
Au
n
− Az
2
− 2λ
n
α
Au
n
− Az
2
1 − δ
n
v
n
− z
2
η
2
n
Bv
n
− Bz
2
− 2η
n
β
Bv
n
− Bz
2
δ
n
u
n
− z
2
− λ
n
2α − λ
n
Au
n
− Az
2
1 − δ
n
v
n
− z
2
− η
n
2β − η
n
Bv
n
− Bz
2
3.22
δ
n
T
r
n
I − r
n
A
x
n
− T
r
n
z − r
n
Az
2
− λ
n
2α − λ
n
Au
n
− Az
2
1 − δ
n
T
s
n
I − s
n
B
x
n
− T
s
n
z − s
n
Bz
2
− η
n
2β − η
n
Bv
n
− Bz
2
≤ δ
n
x
n
− z
2
− λ
n
2α − λ
n
Au
n
− Az
2
1 − δ
n
x
n
− z
2
− η
n
2β − η
n
Bv
n
− Bz
2
x
n
− z
2
− λ
n
δ
n
2α − λ
n
Au
n
− Az
2
− η
n
1 − δ
n
2β − η
n
Bv
n
− Bz
2
.
3.23
14 Fixed Point Theory and Applications
By 3.23, we have
x
n1
− z
2
α
n
f
x
n
− z
β
n
x
n
− z
γ
n
S
n
y
n
− z
2
≤ α
n
f
x
n
− z
2
β
n
x
n
− z
2
γ
n
S
n
y
n
− z
2
≤ α
n
f
x
n
− z
2
β
n
x
n
− z
2
γ
n
y
n
− z
2
≤ α
n
f
x
n
− z
2
β
n
x
n
− z
2
γ
n
x
n
− z
2
− λ
n
δ
n
2α − λ
n
Au
n
− Az
2
−η
n
1 − δ
n
2β − η
n
Bv
n
− Bz
2
α
n
f
x
n
− z
2
β
n
x
n
− z
2
γ
n
x
n
− z
2
− λ
n
γ
n
δ
n
2α − λ
n
Au
n
− Az
2
− η
n
γ
n
1 − δ
n
2β − η
n
Bv
n
− Bz
2
≤ α
n
f
x
n
− z
2
x
n
− z
2
− λ
n
γ
n
δ
n
2α − λ
n
Au
n
− Az
2
− η
n
γ
n
1 − δ
n
2β − η
n
Bv
n
− Bz
2
.
3.24
By 3.24, we have
λ
n
γ
n
δ
n
2α − λ
n
Au
n
− Az
2
≤ α
n
f
x
n
− z
2
x
n
− z
2
−
x
n1
− z
2
− η
n
γ
n
1 − δ
n
2β − η
n
Bv
n
− Bz
2
≤ α
n
f
x
n
− z
2
x
n
− z
x
n1
− z
x
n1
− x
n
.
3.25
From conditions i–iii and 3.17, we have
lim
n →∞
Au
n
− Az
2
0.
3.26
By using the same method as 3.26, we have
lim
n →∞
Bv
n
− Bz
2
0.
3.27
Fixed Point Theory and Applications 15
By nonexpansiveness of T
r
n
,T
s
n
,I− λ
n
A, I − η
n
B and 3.23, we have
y
n
− z
2
≤ δ
n
P
C
u
n
− λ
n
Au
n
− P
C
I − λ
n
A
z
2
1 − δ
n
P
C
v
n
− η
n
Bv
n
− P
C
I − η
n
B
z
2
≤ δ
n
I − λ
n
A
u
n
−
I − λ
n
A
z
2
1 − δ
n
I − η
n
B
v
n
−
I − η
n
B
z
2
≤ δ
n
u
n
− z
2
1 − δ
n
v
n
− z
2
δ
n
T
r
n
I − r
n
A
x
n
− T
r
n
I − r
n
A
z
2
1 − δ
n
T
s
n
I − s
n
B
x
n
−T
s
n
I − s
n
B
z
2
≤ δ
n
I − r
n
A
x
n
−
I − r
n
A
z
2
1 − δ
n
I − s
n
B
x
n
−
I − s
n
B
z
2
δ
n
x
n
− r
n
Ax
n
− z r
n
Az
2
1 − δ
n
x
n
− s
n
Bx
n
− z s
n
Bz
2
δ
n
x
n
− z
− r
n
Ax
n
− Az
2
1 − δ
n
x
n
− z
− s
n
Bx
n
− Bz
2
δ
n
x
n
− z
2
r
2
n
Ax
n
− Az
2
− 2r
n
x
n
− z, Ax
n
− Az
1 − δ
n
x
n
− z
2
s
2
n
Bx
n
− Bz
2
− 2s
n
x
n
− zBx
n
− Bz
δ
n
x
n
− z
2
r
2
n
δ
n
Ax
n
− Az
2
− 2δ
n
r
n
x
n
− z, Ax
n
− Az
1 − δ
n
x
n
− z
2
s
2
n
1 − δ
n
Bx
n
− Bz
2
− 2s
n
1 − δ
n
x
n
− zBx
n
− Bz
≤
x
n
− z
2
r
2
n
δ
n
Ax
n
− Az
2
− 2δ
n
r
n
α
Ax
n
− Az
2
s
2
n
1 − δ
n
Bx
n
− Bz
2
− 2s
n
1 − δ
n
β
Bx
n
− Bz
2
x
n
− z
2
− δ
n
r
n
2α − r
n
Ax
n
− Az
2
− s
n
1 − δ
n
2β − s
n
Bx
n
− Bz
2
.
3.28
By 3.28, we have
x
n1
− z
2
α
n
f
x
n
− z
β
n
x
n
− z
γ
n
S
n
y
n
− z
2
≤ α
n
f
x
n
− z
2
β
n
x
n
− z
2
γ
n
S
n
y
n
− z
2
≤ α
n
f
x
n
− z
2
β
n
x
n
− z
2
γ
n
y
n
− z
2
≤ α
n
f
x
n
− z
2
β
n
x
n
− z
2
γ
n
x
n
− z
2
− δ
n
r
n
2α − r
n
Ax
n
− Az
2
−s
n
1 − δ
n
2β − s
n
Bx
n
− Bz
2
α
n
f
x
n
− z
2
β
n
x
n
− z
2
γ
n
x
n
− z
2
− δ
n
γ
n
r
n
2α − r
n
Ax
n
− Az
2
16 Fixed Point Theory and Applications
− s
n
γ
n
1 − δ
n
2β − s
n
Bx
n
− Bz
2
≤ α
n
f
x
n
− z
2
x
n
− z
2
− δ
n
γ
n
r
n
2α − r
n
Ax
n
− Az
2
− s
n
γ
n
1 − δ
n
2β − s
n
Bx
n
− Bz
2
.
3.29
By 3.29, we have
δ
n
γ
n
r
n
2α − r
n
Ax
n
− Az
2
≤ α
n
f
x
n
− z
2
x
n
− z
2
−
x
n1
− z
2
− s
n
γ
n
1 − δ
n
2β − s
n
Bx
n
− Bz
2
≤ α
n
f
x
n
− z
2
x
n
− z
x
n1
− z
x
n1
− x
n
.
3.30
From 3.17 and conditions i–iii, we have
lim
n →∞
Ax
n
− Az
0.
3.31
By using the same method as 3.31, we have
lim
n →∞
Bx
n
− Bz
0.
3.32
Step 4. We will show that
lim
n →∞
y
n
− x
n
0.
3.33
Putting M
n
P
C
u
n
− λ
n
Au
n
and N
n
P
C
v
n
− η
n
Bv
n
, we will show that
lim
n →∞
u
n
− x
n
lim
n →∞
v
n
− x
n
lim
n →∞
M
n
− u
n
lim
n →∞
N
n
− v
n
0.
3.34
Let z ∈ F;by3.28, we have
y
n
− z
2
≤ δ
n
M
n
− z
2
1 − δ
n
N
n
− z
2
≤ δ
n
u
n
− z
2
1 − δ
n
v
n
− z
2
.
3.35
Fixed Point Theory and Applications 17
By nonexpansiveness of I − r
n
A, we have
u
n
− z
2
T
r
n
x
n
− r
n
Ax
n
− T
r
n
z − r
n
Az
2
≤
x
n
− r
n
Ax
n
−
z − r
n
Az
,u
n
− z
1
2
x
n
− r
n
Ax
n
−
z − r
n
Az
2
u
n
− z
2
−
x
n
− r
n
Ax
n
−
z − r
n
Az
−
u
n
− z
2
≤
1
2
x
n
− z
2
u
n
− z
2
−
x
n
− u
n
− r
n
Ax
n
− Az
2
1
2
x
n
− z
2
u
n
− z
2
−
x
n
− u
n
2
2r
n
x
n
− u
n
,Ax
n
− Az
− r
2
n
Ax
n
− Az
2
.
3.36
This implies
u
n
− z
2
≤
x
n
− z
2
−
x
n
− u
n
2
2r
n
x
n
− u
n
,Ax
n
− Az
− r
2
n
Ax
n
− Az
2
.
3.37
By using the same method as 3.37, we have
v
n
− z
2
≤
x
n
− z
2
−
x
n
− v
n
2
2s
n
x
n
− v
n
,Bx
n
− Bz
− s
2
n
Bx
n
− Bz
2
.
3.38
Substituting 3.37 and 3.38 into 3.35, we have
y
n
− z
2
≤ δ
n
u
n
− z
2
1 − δ
n
v
n
− z
2
≤ δ
n
x
n
− z
2
−
x
n
− u
n
2
2r
n
x
n
− u
n
,Ax
n
− Az
− r
2
n
Ax
n
− Az
2
1 − δ
n
x
n
− z
2
−
x
n
− v
n
2
2s
n
x
n
− v
n
,Bx
n
− Bz
− s
2
n
Bx
n
− Bz
2
≤ δ
n
x
n
− z
2
− δ
n
x
n
− u
n
2
2δ
n
r
n
x
n
− u
n
Ax
n
− Az
1 − δ
n
x
n
− z
2
−
1 − δ
n
x
n
− v
n
2
2s
n
1 − δ
n
x
n
− v
n
Bx
n
− Bz
x
n
− z
2
− δ
n
x
n
− u
n
2
2δ
n
r
n
x
n
− u
n
Ax
n
− Az
−
1 − δ
n
x
n
− v
n
2
2s
n
1 − δ
n
x
n
− v
n
Bx
n
− Bz
.
3.39
18 Fixed Point Theory and Applications
By 3.39, we have
x
n1
− z
2
≤ α
n
f
x
n
− z
2
β
n
x
n
− z
2
γ
n
y
n
− z
2
≤ α
n
f
x
n
− z
2
β
n
x
n
− z
2
γ
n
x
n
− z
2
− δ
n
x
n
− u
n
2
2δ
n
r
n
x
n
− u
n
Ax
n
− Az
−
1 − δ
n
x
n
− v
n
2
2s
n
1 − δ
n
x
n
− v
n
Bx
n
− Bz
α
n
f
x
n
− z
2
β
n
x
n
− z
2
γ
n
x
n
− z
2
− γ
n
δ
n
x
n
− u
n
2
2γ
n
δ
n
r
n
x
n
− u
n
Ax
n
− Az
−
1 − δ
n
γ
n
x
n
− v
n
2
2s
n
γ
n
1 − δ
n
x
n
− v
n
Bx
n
− Bz
≤ α
n
f
x
n
− z
2
x
n
− z
2
− γ
n
δ
n
x
n
− u
n
2
2γ
n
δ
n
r
n
x
n
− u
n
Ax
n
− Az
−
1 − δ
n
γ
n
x
n
− v
n
2
2s
n
γ
n
1 − δ
n
x
n
− v
n
Bx
n
− Bz
.
3.40
It follows that
γ
n
δ
n
x
n
− u
n
2
≤ α
n
f
x
n
− z
2
x
n
− z
2
−
x
n1
− z
2
2γ
n
δ
n
r
n
x
n
− u
n
Ax
n
− Az
−
1 − δ
n
γ
n
x
n
− v
n
2
2s
n
γ
n
1 − δ
n
x
n
− v
n
Bx
n
− Bz
≤ α
n
f
x
n
− z
2
x
n
− z
x
n1
− z
x
n1
− x
n
2γ
n
δ
n
r
n
x
n
− u
n
Ax
n
− Az
2s
n
γ
n
1 − δ
n
x
n
− v
n
Bx
n
− Bz
.
3.41
By conditions i–iii, 3.41, 3.31, 3.32,and3.17, we have
lim
n →∞
x
n
− u
n
0.
3.42
By using the same method as 3.42, we have
lim
n →∞
x
n
− v
n
0.
3.43
Fixed Point Theory and Applications 19
By nonexpansiveness of T
r
n
I − r
n
A, we have
M
n
− z
2
P
C
u
n
− λ
n
Au
n
− P
C
z − λ
n
Az
2
≤
u
n
− α
n
Au
n
−
z − α
n
Az
,M
n
− z
1
2
u
n
− α
n
Au
n
−
z − α
n
Az
2
M
n
− z
2
−
u
n
− α
n
Au
n
−
z − α
n
Az
−
M
n
− z
2
≤
1
2
u
n
− z
2
M
n
− z
2
−
u
n
− M
n
− α
n
Au
n
− Az
2
1
2
T
r
n
I − r
n
A
x
n
− T
r
n
I − r
n
A
z
2
M
n
− z
2
−
u
n
− M
n
2
2α
n
u
n
− M
n
,Au
n
− Az
− α
2
n
Au
n
− Az
2
≤
1
2
x
n
− z
2
M
n
− z
2
−
u
n
− M
n
2
2α
n
u
n
− M
n
,Au
n
− Az
−α
2
n
Au
n
− Az
2
.
3.44
Hence, we have
M
n
− z
2
≤
x
n
− z
2
−
u
n
− M
n
2
2α
n
u
n
− M
n
,Au
n
− Az
− α
2
n
Au
n
− Az
2
.
3.45
By using the same method as 3.45, we have
N
n
− z
2
≤
x
n
− z
2
−
v
n
− N
n
2
2η
n
v
n
− N
n
,Bv
n
− Bz
− η
2
n
Bv
n
− Bz
2
.
3.46
Substituting 3.45 and 3.46 into 3.35, we have
y
n
− z
2
≤ δ
n
M
n
− z
2
1 − δ
n
N
n
− z
2
≤ δ
n
x
n
− z
2
−
u
n
− M
n
2
2α
n
u
n
− M
n
,Au
n
− Az
− α
2
n
Au
n
− Az
2
1 − δ
n
x
n
− z
2
−
v
n
− N
n
2
2η
n
v
n
− N
n
,Bv
n
− Bz
− η
2
n
Bv
n
− Bz
2
≤ δ
n
x
n
− z
2
− δ
n
u
n
− M
n
2
2δ
n
α
n
u
n
− M
n
Au
n
− Az
1 − δ
n
x
n
− z
2
−
1 − δ
n
v
n
− N
n
2
2
1 − δ
n
η
n
v
n
− N
n
Bv
n
− Bz
x
n
− z
2
− δ
n
u
n
− M
n
2
2δ
n
α
n
u
n
− M
n
Au
n
− Az
−
1 − δ
n
v
n
− N
n
2
2
1 − δ
n
η
n
v
n
− N
n
Bv
n
− Bz
.
3.47
20 Fixed Point Theory and Applications
By 3.47, we have
x
n1
− z
2
≤ α
n
f
x
n
− z
2
β
n
x
n
− z
2
γ
n
y
n
− z
2
≤ α
n
f
x
n
− z
2
β
n
x
n
− z
2
γ
n
x
n
− z
2
− δ
n
u
n
− M
n
2
2δ
n
α
n
u
n
− M
n
Au
n
− Az
−
1 − δ
n
v
n
− N
n
2
2
1 − δ
n
η
n
v
n
− N
n
Bv
n
− Bz
α
n
f
x
n
− z
2
β
n
x
n
− z
2
γ
n
x
n
− z
2
− δ
n
γ
n
u
n
− M
n
2
2δ
n
γ
n
α
n
u
n
− M
n
Au
n
− Az
−
1 − δ
n
γ
n
v
n
− N
n
2
2
1 − δ
n
γ
n
η
n
v
n
− N
n
Bv
n
− Bz
≤ α
n
f
x
n
− z
2
x
n
− z
2
− δ
n
γ
n
u
n
− M
n
2
2δ
n
γ
n
α
n
u
n
− M
n
Au
n
− Az
−
1 − δ
n
γ
n
v
n
− N
n
2
2
1 − δ
n
γ
n
η
n
v
n
− N
n
Bv
n
− Bz
.
3.48
It follows that
δ
n
γ
n
u
n
− M
n
2
≤ α
n
f
x
n
− z
2
x
n
− z
2
−
x
n1
− z
2
2δ
n
γ
n
α
n
u
n
− M
n
Au
n
− Az
−
1 − δ
n
γ
n
v
n
− N
n
2
2
1 − δ
n
γ
n
η
n
v
n
− N
n
Bv
n
− Bz
≤ α
n
f
x
n
− z
2
x
n
− z
x
n1
− z
x
n1
− x
n
2δ
n
γ
n
α
n
u
n
− M
n
Au
n
− Az
2
1 − δ
n
γ
n
η
n
v
n
− N
n
Bv
n
− Bz
.
3.49
From 3.17, 3.26, 3.27, and conditions i–iii, we have
lim
n →∞
u
n
− M
n
0.
3.50
By using the same method as 3.50, we have
lim
n →∞
v
n
− N
n
0.
3.51
Fixed Point Theory and Applications 21
By 3.42 and 3.50, we have
lim
n →∞
M
n
− x
n
0.
3.52
By 3.43 and 3.51, we have
lim
n →∞
N
n
− x
n
0.
3.53
Since M
n
P
C
u
n
− λ
n
Au
n
and N
n
P
C
v
n
− η
n
Bv
n
, we have
y
n
− x
n
δ
n
M
n
− x
n
1 − δ
n
N
n
− x
n
. 3.54
By 3.52 and 3.53,weobtain
lim
n →∞
y
n
− x
n
0.
3.55
Note that
x
n
− S
n
x
n
≤
x
n
− S
n
y
n
S
n
y
n
− S
n
x
n
≤
x
n
− S
n
y
n
y
n
− x
n
.
3.56
From 3.20 and 3.55, we have
lim
n →∞
x
n
− S
n
x
n
0.
3.57
Step 5. We will show that
lim sup
n →∞
f
z
− z, x
n
− z
≤ 0,
3.58
where z P
F
fz. To show this inequality, take subsequence {x
n
i
} of {x
n
} such that
lim sup
n →∞
f
z
− z, x
n
− z
lim sup
i →∞
f
z
− z, x
n
i
− z
.
3.59
Since {x
n
i
} is bounded, there exists a subsequence {x
n
i
j
} of {x
n
i
} which converges weakly
to q. Without loss of generality, we can assume that x
n
i
q. Since C is closed convex, C is
weakly closed. So, we have q ∈ C. Let us show that q ∈ F
N
i1
FT
i
∩ EPF, A ∩ EPG, B ∩
FG
1
∩ FG
2
. We first show that q ∈ EPF, A ∩ EPG, B ∩ FG
1
∩ FG
2
.From3.42,we
have u
n
i
q. Since u
n
T
r
n
I − r
n
Ax
n
, for any y ∈ C, we have
F
u
n
,y
Ax
n
,y− u
n
1
r
n
y − u
n
,u
n
− x
n
≥ 0.
3.60
22 Fixed Point Theory and Applications
From A2, we have
Ax
n
,y− u
n
1
r
n
y − u
n
,u
n
− x
n
≥ F
y, u
n
.
3.61
This implies that
Ax
n
i
,y− u
n
i
1
r
n
i
y − u
n
i
,u
n
i
− x
n
i
≥ F
y, u
n
i
.
3.62
Put z
t
ty 1 − tq for all t ∈ 0, 1 and y ∈ C. Then, we have z
t
∈ C. So, from 3.62,we
have
z
t
− u
n
i
,Az
t
≥
z
t
− u
n
i
,Az
t
−z
t
− u
n
i
,Ax
n
i
−
z
t
− u
n
i
,
u
n
i
− x
n
i
r
n
i
F
z
t
,u
n
i
z
t
− u
n
i
,Az
t
− Au
n
i
z
t
− u
n
i
,Au
n
i
− Ax
n
i
−
z
t
− u
n
i
,
u
n
i
− x
n
i
r
n
i
F
z
t
,u
n
i
.
3.63
Since u
n
i
− x
n
i
→0, we have Au
n
i
− Ax
n
i
→0. Further, from monotonicity of A, we have
z
t
− u
n
i
,Az
t
− Au
n
i
≥0. So, from A4, we have
z
t
− q, Az
t
≥ F
z
t
,q
as i −→ ∞ . 3.64
From A1, A4,and3.64 , we also have
0 F
z
t
,z
t
≤ tF
z
t
,y
1 − t
F
z
t
,q
≤ tF
z
t
,y
1 − t
z
t
− q, Az
t
tF
z
t
,y
1 − t
t
y − q, Az
t
.
3.65
Thus
0 ≤ F
z
t
,y
1 − t
y − q, Az
t
. 3.66
Letting t → 0, we have, for each y ∈ C,
0 ≤ F
q, y
y − q, Aq
. 3.67
This implies that
q ∈ EP
F, A
. 3.68
Fixed Point Theory and Applications 23
From 3.43, we have v
ni
q. Since v
n
T
s
n
I − s
n
Bx
n
, for any y ∈ C, we have
G
v
n
,y
Bx
n
,y− v
n
1
s
n
y − v
n
,v
n
− x
n
≥ 0.
3.69
From A2, we have
Bx
n
,y− v
n
1
s
n
y − v
n
,v
n
− x
n
≥ G
y, v
n
.
3.70
This implies that
Bx
n
i
,y− v
n
i
1
s
n
i
y − v
n
i
,v
n
i
− x
n
i
≥ G
y, v
n
i
.
3.71
Put z
t
ty 1 − tq for all t ∈ 0, 1 and y ∈ C. Then, we have z
t
∈ C. So, from 3.71 we have
z
t
− v
n
i
,Bz
t
≥
z
t
− v
n
i
,Bz
t
−
z
t
− v
n
i
,Bx
n
i
−
z
t
− v
n
i
,
v
n
i
− x
n
i
s
n
i
G
z
t
,v
n
i
z
t
− v
n
i
,Bz
t
− Bv
n
i
z
t
− v
n
i
,Bv
n
i
− Bx
n
i
−
z
t
− v
n
i
,
v
n
i
− x
n
i
s
n
i
G
z
t
,v
n
i
.
3.72
Since v
n
i
− x
n
i
→0, we have Bv
n
i
− Bx
n
i
→0. Further, from monotonicity of B, we have
z
t
− v
n
i
,Bz
t
− Bv
n
i
≥0. So, from A4, we have
z
t
− q, Bz
t
≥ G
z
t
,q
. 3.73
From A1, A4,and3.64 , we also have
0 G
z
t
,z
t
≤ tG
z
t
,y
1 − t
G
z
t
,q
≤ tG
z
t
,y
1 − t
z
t
− q, Bz
t
tG
z
t
,y
1 − t
t
y − q, Bz
t
,
3.74
hence
0 ≤ G
z
t
,y
1 − t
y − q, Bz
t
. 3.75
Letting t → 0, we have, for each y ∈ C,
0 ≤ G
q, y
y − q, Bq
. 3.76
24 Fixed Point Theory and Applications
This implies that
q ∈ EP
G, B
. 3.77
Define a mapping Q : C → C by
Qx δP
C
I − λ
n
A
x
1 − δ
P
C
I − η
n
B
x, ∀x ∈ C, 3.78
where lim
n →∞
δ
n
δ ∈ 0, 1.FromLemma 2.3, we have that Q is nonexpansive with
F
Q
F
P
C
I − λ
n
A
F
P
C
I − η
n
B
. 3.79
Next, we show that
lim
n → n
x
n
− Qx
n
0.
3.80
By nonexpansiveness of I − η
n
B and I − λ
n
A, we have
x
n
− Qx
n
≤
x
n
− y
n
y
n
− Qx
n
x
n
− y
n
δ
n
P
C
u
n
− λ
n
Au
n
1 − δ
n
P
C
v
n
− η
n
Bv
n
− δP
C
I − λ
n
A
x
n
−
1 − δ
P
C
I − η
n
B
x
n
x
n
− y
n
δ
n
P
C
I − λ
n
A
u
n
− δ
n
P
C
I − λ
n
A
x
n
δ
n
P
C
I − λ
n
A
x
n
1 − δ
n
P
C
I − η
n
B
v
n
−
1 − δ
n
P
C
I − η
n
B
x
n
1 − δ
n
P
C
I − η
n
B
x
n
− δP
C
I − λ
n
A
x
n
−
1 − δ
P
C
I − η
n
B
x
n
x
n
− y
n
δ
n
P
C
I − λ
n
A
u
n
− P
C
I − λ
n
A
x
n
δ
n
− δ
P
C
I − λ
n
A
x
n
1 − δ
n
P
C
I − η
n
B
v
n
− P
C
I − η
n
B
x
n
δ − δ
n
P
C
I − η
n
B
x
n
≤
x
n
− y
n
δ
n
P
C
I − λ
n
A
u
n
− P
C
I − λ
n
A
x
n
|
δ
n
− δ
|
P
C
I − λ
n
A
x
n
1 − δ
n
P
C
I − η
n
B
v
n
− P
C
I − η
n
B
x
n
|
δ
n
− δ
|
P
C
I − η
n
B
x
n
≤
x
n
− y
n
δ
n
u
n
− x
n
|
δ
n
− δ
|
P
C
I − λ
n
A
x
n
1 − δ
n
v
n
− x
n
|
δ
n
− δ
|
P
C
I − η
n
B
x
n
≤
x
n
− y
n
δ
n
u
n
− x
n
2
|
δ
n
− δ
|
M
1
1 − δ
n
v
n
− x
n
,
3.81
Fixed Point Theory and Applications 25
where M
1
sup
n≥0
{P
C
I − λ
n
Ax
n
P
C
I − η
n
Bx
n
}.From3.17, 3.42, 3.43, 3.55,
and condition iii, we have lim
n → n
x
n
− Qx
n
0. Since x
n
i
q, it follows from 3.80 that,
lim
i →∞
x
n
i
− Qx
n
i
0. By Lemma 2.4,weobtainthat
q ∈ F
Q
F
P
C
I − λ
n
A
∩ F
P
C
I − η
n
B
F
G
1
∩ F
G
2
. 3.82
Assume that q
/
Sq. Using Opial s
,
property, 3.57 and Lemma 2.10 we have
lim inf
i →∞
x
n
i
− q
< lim inf
i →∞
x
n
i
− Sq
≤ lim inf
i →∞
x
n
i
− S
n
i
x
n
i
S
n
i
x
n
i
− S
n
i
q
S
n
i
q − Sq
≤ lim inf
i →∞
x
n
i
− q
.
3.83
This is a contradiction, so we have
q ∈
N
i1
F
T
i
F
S
.
3.84
From 3.68, 3.773.82,and3.84, we have q ∈ F. Since P
F
f is contraction with the
coefficient θ ∈ 0, 1, P
F
has a unique fixed point. Let z be a fixed point of P
F
f,thatis
z P
F
fz. Since x
n
i
qand q ∈ F, we have
lim sup
n →∞
f
z
− z, x
n
− z
lim sup
i →∞
f
z
− z, x
n
i
− z
f
z
− z, q − z
≤ 0.
3.85
Step 6. Finally, we will show that x
n
→ z as n →∞. By nonexpansiveness of T
r
n
,T
s
n
,I −
λ
n
A, I − η
n
B, I − r
n
A, I − s
n
B, we can show that y
n
− z≤x
n
− z. Then
x
n1
− z
2
α
n
f
x
n
− z
β
n
x
n
− z
γ
n
S
n
y
n
− z
,x
n1
− z
α
n
f
x
n
− z, x
n1
− z
β
n
x
n
− z, x
n1
− z
γ
n
S
n
y
n
− z, x
n1
− z
≤ α
n
f
x
n
− f
z
,x
n1
− z
α
n
f
z
− z, x
n1
− z
β
n
x
n
− z
x
n1
− z
γ
n
S
n
y
n
− z
x
n1
− z
≤ α
n
f
x
n
− f
z
x
n1
− z
α
n
f
z
− z, x
n1
− z
β
n
x
n
− z
x
n1
− z
γ
n
y
n
− z
x
n1
− z
