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Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2010, Article ID 407651, 26 pages
doi:10.1155/2010/407651
Research Article
Approximation of Common Fixed
Points of a Countable Family of Relatively
Nonexpansive Mappings
Daruni Boonchari
1
and Satit Saejung
2
1
Department of Mathematics, Mahasarakham University, Maha Sarakham 44150, Thailand
2
Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand
Correspondence should be addressed to Satit Saejung,
Received 22 June 2009; Revised 20 October 2009; Accepted 21 November 2009
Academic Editor: Tomonari Suzuki
Copyright q 2010 D. Boonchari and S. Saejung. 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 two general iterative schemes for finding a common fixed point of a countable family
of relatively nonexpansive mappings in a Banach space. Under suitable setting, we not only obtain
several convergence theorems announced by many authors but also prove them under weaker
assumptions. Applications t o the problem of finding a common element of the fixed point set
of a relatively nonexpansive mapping and the solution set of an equilibrium problem are also
discussed.
1. Introduction and Preliminaries
Let C be a nonempty subset of a Banach space E,andletT be a mapping from C into itself.
When {x
n
} is a sequence in E, we denote strong convergence of {x
n
} to x ∈ E by x
n
→ x and
weak convergence by x
n
x. We also denote the weak
∗
convergence of a sequence {x
∗
n
} to
x
∗
in the dual E
∗
by x
∗
n
∗
x
∗
.Apointp ∈ C is an asymptotic fixed point of T if there exists
{x
n
} in C such that x
n
pand x
n
− Tx
n
→ 0. We denote FT and
FT by the set of fixed
points and of asymptotic fixed points of T, respectively. A Banach space E is said to be strictly
convex if x y/2 < 1forx, y ∈ SE{z ∈ E : z 1} and x
/
y.Itisalsosaidtobe
uniformly convex if for each ∈ 0, 2, there exists δ>0 such that x y/2 < 1 − δ for
x, y ∈ SE and x − y≥. The space E is said to be smooth if the limit
lim
t → 0
x tx
−
x
t
1.1
2 Fixed Point Theory and Applications
exists for all x, y ∈ SE. It is also said to be uniformly smooth if the limit exists uniformly in
x, y ∈ SE.
Many problems in nonlinear analysis can be formulated as a problem of finding a fixed
point of a certain mapping or a common fixed point of a family of mappings. This paper deals
with a class of nonlinear mappings, so-called relatively nonexpansive mappings introduced
by Matsushita and Takahashi 1 . This t ype of mappings is closely related to the resolvent of
maximal monotone operators see 2–4.
Let E be a smooth, strictly convex and reflexive Banach space and let C be a nonempty
closed convex subset of E. Throughout this paper, we denote by φ the function defined by
φ
x, y
x
2
− 2
x, Jy
y
2
∀x, y ∈ E,
1.2
where J is the normalized duality mapping from E to the dual space E
∗
given by the following
relation:
x, Jx
x
2
Jx
2
.
1.3
We know that if E is smooth, strictly convex, and reflexive, then the duality mapping J is
single-valued, one-to-one, and onto. The duality mapping J is said to be weakly sequentially
continuous if x
n
ximplies that Jx
n
∗
Jxsee 5 for more details.
Following Matsushita and Takahashi 6, a mapping T : C → E is said to be relatively
nonexpansive if the following conditions are satisfied:
R1 FT is nonempty;
R2 φu, Tx ≤ φu, x for all u ∈ FT, x ∈ C;
R3
FTFT.
If T satisfies R1 and R2, then T is called relatively quasi-nonexpansive 7. Obviously,
relative nonexpansiveness implies relative quasi-nonexpansiveness but the converse is
not true. Relatively quasi-nonexpansive mappings are sometimes called hemirelatively
nonexpansive mappings. But we do prefer the former name because in a Hilbert space setting,
relatively quasi-nonexpansive mappings are nothing but quasi-nonexpansive.
In 2, Alber introduced the generalized projection Π
C
from E onto C as follows:
Π
C
x
arg min
y∈C
φ
y, x
∀x ∈ E.
1.4
If E is a Hilbert space, then φy, xy − x
2
and Π
C
becomes the metric projection of E
onto C. Alber’s generalized projection is an example of relatively nonexpansive mappings.
For more example, see 1, 8.
In 2004, Masushita and Takahashi 1, 6 also proved weak and strong convergence
theorems for finding a fixed point of a single relatively nonexpansive mapping. Several
iterative methods, as a generalization of 1, 6, for finding a common fixed point of the family
of relatively nonexpansive mappings have been further studied in 7, 9–14.
Fixed Point Theory and Applications 3
Recently, a problem of finding a common element of the set of solutions of an
equilibrium problem and the set of fixed points of a relatively nonexpansive mapping is
studied by Takahashi and Zembayashi in 15, 16. The purpose of this paper is to introduce
a new iterative scheme which unifies several ones studied by many authors and to deduce the
corresponding convergence theorems under the weaker assumptions. More precisely, many
restrictions as were the case in other papers are dropped away.
First, we start with some preliminaries which will be used throughout the paper.
Lemma 1.1 see 7, Lemma 2.5. Let C be a nonempty closed convex subset of a strictly convex
and smooth Banach space E and let T be a relatively quasi-nonexpansive mapping from C into itself.
Then FT is closed and convex.
Lemma 1.2 see 17,Proposition5. Let C be a nonempty closed convex subset of a smooth, strictly
convex, and reflexive Banach space E.Then
φ
x, Π
C
y
φ
Π
C
y, y
≤ φ
x, y
1.5
for all x ∈ C and y ∈ E.
Lemma 1.3 see 17. Let E be a smooth and uniformly convex Banach space and let r>0.Then
there exists a strictly increasing, continuous, and convex function h : 0, 2r → R such that h00
and
h
x − y
≤ φ
x, y
1.6
for all x, y ∈ B
r
{z ∈ E : z≤r}.
Lemma 1.4 see 17,Proposition2. Let E be a smooth and uniformly convex Banach space and let
{x
n
} and {y
n
} be sequences of E such that either {x
n
} or {y
n
} is bounded. If lim
n →∞
φx
n
,y
n
0,
then lim
n →∞
x
n
− y
n
0.
Lemma 1.5 see 2. Let C be a nonempty closed convex subset of a smooth, strictly convex, and
reflexive Banach s pace E,letx ∈ E, and let z ∈ C.Then
z Π
C
x ⇐⇒
y − z, Jx − Jz
≤ 0, ∀y ∈ C. 1.7
Lemma 1.6 see 18. Let E be a uniformly convex Banach space and let r>0. Then there exists a
strictly increasing, continuous, and convex function g : 0, 2r → R such that g00 and
tx 1 − ty
2
≤ t
x
2
1 − t
y
2
− t
1 − t
g
x − y
1.8
for all x, y ∈ B
r
and t ∈ 0, 1.
We next prove the following three lemmas which are very useful for our main results.
4 Fixed Point Theory and Applications
Lemma 1.7. Let Let C be a closed convex subset of a smooth Banach space E.LetT be a relatively
quasi-nonexpansive mapping from E into E and let {S
i
}
N
i1
be a family of relatively quasi-nonexpansive
mappings from C into itself such that FT ∩
N
i1
FS
i
/
∅. The mapping U : C → E is defined by
Ux TJ
−1
N
i1
ω
i
α
i
Jx
1 − α
i
JS
i
x
1.9
for all x ∈ C and {ω
i
}, {α
i
}⊂0, 1, i 1, 2, ,N such that
N
i1
ω
i
1.Ifx ∈ C and z ∈
FT ∩
N
i1
FS
i
,then
φ
z, Ux
≤ φ
z, x
. 1.10
Proof. The proof of this lemma can be extracted from that of Lemma 1.8;soitisomitted.
If E has a stronger assumption, we have the following lemma.
Lemma 1.8. Let C be a closed convex subset of a uniformly smooth Banach space E.Letr>0.
Then, there exists a strictly increasing, continuous, and convex function g
∗
: 0, 6r → R such that
g
∗
00 and for each relatively quasi-nonexpansive mapping T : E → E and each finite family of
relatively quasi-nonexpansive mappings {S
i
}
N
i1
: C → C such that FT ∩
N
i1
FS
i
/
∅,
N
i1
ω
i
α
i
1 − α
i
g
∗
Jz − JS
i
z
≤ φ
u, z
− φ
u, Uz
1.11
for all z ∈ C ∩ B
r
and u ∈ FT ∩
N
i1
FS
i
∩ B
r
,where
Ux TJ
−1
N
i1
ω
i
α
i
Jx
1 − α
i
JS
i
x
1.12
x ∈ C and {ω
i
}, {α
i
}⊂0, 1, i 1, 2, ,N such that
N
i1
ω
i
1.
Proof. Let r>0. From Lemma 1.6 and E
∗
is uniformly convex, then there exists a strictly
increasing, continuous, and convex function g
∗
: 0, 6r → R such that g
∗
00and
tx
∗
1 − ty
∗
2
≤ t
x
∗
2
1 − t
y
∗
2
− t
1 − t
g
∗
x
∗
− y
∗
1.13
for all x
∗
,y
∗
∈{z
∗
∈ E
∗
: z
∗
≤3r} and t ∈ 0, 1.LetT : E → E and {S
i
}
N
i1
: C → C
be relatively quasi-nonexpansive for all i 1, 2, ,N such that FT ∩
N
i1
FS
i
/
∅. For
z ∈ C ∩ B
r
and u ∈ FT ∩
N
i1
FS
i
∩ B
r
. It follows that
u
−
S
i
z
2
≤ φ
u, S
i
z
≤ φ
u, z
≤
u
z
2
≤
2r
2
1.14
Fixed Point Theory and Applications 5
and hence S
i
z≤3r. Consequently, for i 1, 2, ,N,
α
i
Jz 1 − α
i
JS
i
z
2
≤ α
i
Jz
2
1 − α
i
JS
i
z
2
− α
i
1 − α
i
g
∗
Jz − JS
i
z
.
1.15
Then
φ
u, Uz
≤ φ
u, J
−1
N
i1
ω
i
α
i
Jz
1 − α
i
JS
i
z
u
2
− 2
u,
N
i1
ω
i
α
i
Jz
1 − α
i
JS
i
z
N
i1
ω
i
α
i
Jz 1 − α
i
JS
i
z
2
≤
N
i1
ω
i
u
2
− 2
u, α
i
Jz
1 − α
i
JS
i
z
α
i
Jz
1 − α
i
JS
i
z
2
≤
N
i1
ω
i
u
2
− 2
u, α
i
Jz
1 − α
i
JS
i
z
α
i
Jz
2
1 − α
i
JS
i
z
2
− α
i
1 − α
i
g
∗
Jz − JS
i
z
N
i1
ω
i
α
i
φ
u, z
1 − α
i
φ
u, S
i
z
− α
i
1 − α
i
g
∗
Jz − JS
i
z
≤ φ
u, z
−
N
i1
ω
i
α
i
1 − α
i
g
∗
Jz − JS
i
z
.
1.16
Thus
N
i1
ω
i
α
i
1 − α
i
g
∗
Jz − JS
i
z
≤ φ
u, z
− φ
u, Uz
.
1.17
Lemma 1.9. Let C be a closed convex subset of a uniformly smooth and strictly convex Banach
space E.LetT be a relatively quasi-nonexpansive mapping from E into E and let {S
i
}
N
i1
be a family
of relatively quasi-nonexpansive mappings from C into itself such that FT ∩
N
i1
FS
i
/
∅.The
mapping U : C → E is defined by
Ux TJ
−1
N
i1
ω
i
α
i
Jx
1 − α
i
JS
i
x
1.18
for all x ∈ C and {ω
i
}, {α
i
}⊂0, 1, i 1, 2, ,Nsuch that
N
i1
ω
i
1. Then, the following hold:
1 FUFT ∩
N
i1
FS
i
,
2 U is relatively quasi-nonexpansive.
6 Fixed Point Theory and Applications
Proof. 1 Clearly, FT ∩
N
i1
FS
i
⊂ FU. We want to show the reverse inclusion. Let z ∈
FU and u ∈ FT ∩
N
i1
FS
i
. Choose
r : max
{
u
,
z
,
S
1
z
,
S
2
z
, ,
S
m
z
}
. 1.19
From Lemma 1.8, we have
N
i1
ω
i
α
i
1 − α
i
g
∗
Jz − JS
i
z
0.
1.20
From ω
i
α
i
1 − α
i
> 0 for all i 1, 2, ,N and by the properties of g
∗
, we have
Jz JS
i
z 1.21
for all i 1, 2, ,N.FromJ is one to one, we have
z S
i
z 1.22
for all i 1, 2, ,N. Consider
z Uz TJ
−1
N
i1
ω
i
α
i
Jz
1 − α
i
JS
i
z
Tz.
1.23
Thus z ∈ FT ∩
N
i1
FS
i
.
2 It follows directly from the above discussion.
2. Weak Convergence Theorem
Theorem 2.1. Let C be a nonempty closed convex subset of a uniformly smooth and uniformly convex
Banach space E.Let{T
n
}
∞
n1
: E → C be a family of relatively quasi-nonexpansive mappings and let
{S
i
}
N
i1
: C → C be a family of relatively quasi-nonexpansive mappings such that F :
∞
n1
FT
n
∩
N
i1
FS
i
/
∅. Let the sequence {x
n
} be generated by x
1
∈ C,
x
n1
T
n
J
−1
N
i1
ω
n,i
α
n,i
Jx
n
1 − α
n,i
JS
i
x
n
2.1
for any n ∈ N, {ω
n,i
}, {α
n,i
}⊂0, 1 for all n ∈ N, i 1, 2, ,N such that
N
i1
ω
n,i
1 for all
n ∈ N.Then{Π
F
x
n
} converges strongly to z ∈ F,whereΠ
F
is the generalized projection of C onto F.
Fixed Point Theory and Applications 7
Proof. Let u ∈
∞
n1
FT
n
∩
N
i1
FS
i
.Put
U
n
T
n
J
−1
N
i1
ω
n,i
α
n,i
J
1 − α
n,i
JS
i
.
2.2
From Lemma 1.7, we have
φ
u, x
n1
φ
u, U
n
x
n
≤ φ
u, x
n
. 2.3
Therefore lim
n →∞
φu, x
n
exists. This implies that {φu, x
n
}, {x
n
} and {S
i
x
n
} are bounded
for all i 1, 2, ,N.
Let y
n
≡ Π
F
x
n
.From2.3 and m ∈ N, we have
φ
y
n
,x
nm
≤ φ
y
n
,x
n
. 2.4
Consequently,
φ
y
n
,y
nm
φ
y
nm
,x
nm
≤ φ
y
n
,x
nm
≤ φ
y
n
,x
n
. 2.5
In particular,
φ
y
n1
,x
n1
≤ φ
y
n
,x
n
. 2.6
This implies that lim
n →∞
φy
n
,x
n
exists. This together with the boundedness of {x
n
} gives
r : sup
n∈N
y
n
< ∞.UsingLemma 1.3, there exists a strictly increasing, continuous, and
convex function h : 0, 2r → R such that h00and
h
y
n
− y
nm
≤ φ
y
n
,y
nm
≤ φ
y
n
,x
n
− φ
y
nm
,x
nm
. 2.7
Since {φy
n
,x
n
} is a convergent sequence, it follows from the properties of g that {y
n
} is a
Cauchy sequence. Since F is closed, there exists z ∈ F such that y
n
→ z.
We first establish weak convergence theorem for finding a common fixed point of
a countable family of relatively quasi-nonexpansive mappings. Recall that, for a family of
mappings {T
n
}
∞
n1
: C → E with
∞
n1
FT
n
/
∅, we say that {T
n
} satisfies the NST-condition
19 if for each bounded sequence {z
n
} in C,
lim
n →∞
z
n
− T
n
z
n
0 implies ω
w
{
z
n
}
⊂
∞
n1
F
T
n
,
2.8
where ω
w
{z
n
} denotes the set of all weak subsequential limits of a sequence {z
n
}.
8 Fixed Point Theory and Applications
Theorem 2.2. Let C be a nonempty closed convex subset of a uniformly smooth and uniformly
convex Banach space E.Let{T
n
}
∞
n1
: E → C be a family of relatively quasi-nonexpansive mappings
satisfying NST-condition and let {S
i
}
N
i1
: C → C be a family of relatively nonexpansive mappings
such that F :
∞
n1
FT
n
∩
N
i1
FS
i
/
∅ and suppose that
φ
u, T
n
x
φ
T
n
x, x
≤ φ
u, x
2.9
for all u ∈
∞
n1
FT
n
, n ∈ N and x ∈ E. Let the sequence {x
n
} be generated by x
1
∈ C,
x
n1
T
n
J
−1
N
i1
ω
n,i
α
n,i
Jx
n
1 − α
n,i
JS
i
x
n
2.10
for any n ∈ N, {ω
n,i
}, {α
n,i
}⊂0, 1 for all n ∈ N, i 1, 2, ,N such that
N
i1
ω
n,i
1 for all
n ∈ N, lim inf
n →∞
ω
n,i
α
n,i
1 − α
n,i
> 0 for all i 1, 2, ,N.IfJ is weakly sequentially continuous,
then {x
n
} converges weakly to z ∈ F,wherez lim
n →∞
Π
F
x
n
.
Proof. Let u ∈ F.FromTheorem 2.1, lim
n →∞
φu, x
n
exists and hence {x
n
} and {S
i
x
n
} are
bounded for all i 1, 2, ,N.Let
r sup
n∈N
{
x
n
,
S
1
x
n
,
S
2
x
n
, ,
S
N
x
n
}
.
2.11
By Lemma 1.8, t here exists a strictly increasing, continuous, and convex function g
∗
:
0, 2r → R such that g
∗
00and
N
i1
ω
n,i
α
n,i
1 − α
n,i
g
∗
Jx
n
− JS
i
x
n
≤ φ
u, x
n
− φ
u, x
n1
.
2.12
In particular, for all i 1, 2, ,N,
ω
n,i
α
n,i
1 − α
n,i
g
∗
Jx
n
− JS
i
x
n
≤ φ
u, x
n
− φ
u, x
n1
. 2.13
Hence,
∞
n1
ω
n,i
α
n,i
1 − α
n,i
g
∗
Jx
n
− JS
i
x
n
< ∞
2.14
for all i 1, 2, ,N. Since lim inf
n →∞
ω
n,i
α
n,i
1 − α
n,i
> 0 for all i 1, 2 ,N and the
properties of g, we have
lim
n →∞
Jx
n
− JS
i
x
n
0
2.15
Fixed Point Theory and Applications 9
for all i 1, 2 ,N. Since J
−1
is uniformly norm-to-norm continuous on bounded sets, we
have
lim
n →∞
x
n
− S
i
x
n
0
2.16
for all i 1, 2 ,N. Since {x
n
} is bounded, there exists a subsequence {x
n
k
} of {x
n
} such that
x
n
k
z∈ C. Since S
i
is relatively nonexpansive, z ∈
FS
i
FS
i
for all i 1, 2 ,N.
We show that z ∈
∞
n1
FT
n
.Let
y
n
J
−1
N
i1
ω
n,i
α
n,i
Jx
n
1 − α
n,i
JS
i
x
n
.
2.17
We note from 2.15 that
N
i1
ω
n,i
α
n,i
Jx
n
1 − α
n,i
JS
i
x
n
− Jx
n
≤
N
i1
ω
n,i
1 − α
n,i
JS
i
x
n
− Jx
n
−→ 0.
2.18
Since J
−1
is uniformly norm-to-norm continuous on bounded sets, it follows that
lim
n →∞
y
n
− x
n
lim
n →∞
J
−1
N
i1
ω
n,i
α
n,i
Jx
n
1 − α
n,i
JS
i
x
n
− J
−1
Jx
n
0.
2.19
Moreover, by 2.9 and the existence of lim
n →∞
φu, x
n
, we have
φ
T
n
y
n
,y
n
≤ φ
u, y
n
− φ
u, T
n
y
n
φ
u, J
−1
N
i1
ω
n,i
α
n,i
Jx
n
1 − α
n,i
JS
i
x
n
− φ
u, x
n1
≤ φ
u, x
n
− φ
u, x
n1
−→ 0.
2.20
It follows from Lemma 1.4 that lim
n →∞
T
n
y
n
− y
n
0. From 2.19 and x
n
k
z, we have
y
n
k
z. Since {T
n
} satisfies NST-condition, we have z ∈
∞
n1
FT
n
. Hence z ∈ F.
Let z
n
Π
F
x
n
.FromLemma 1.5 and z ∈ F, we have
z
n
k
− z, Jx
n
k
− Jz
n
k
≥ 0. 2.21
From Theorem 2.1, we know that z
n
→ z
∈ F. Since J is weakly sequentially continuous, we
have
z
− z, Jz − Jz
≥ 0. 2.22
10 Fixed Point Theory and Applications
Moreover, since J is monotone,
z
− z, Jz − Jz
≤ 0. 2.23
Then
z
− z, Jz − Jz
0. 2.24
Since E is strictly convex, z
z. This implies that ω
w
{x
n
} {z
} and hence x
n
z
lim
n →∞
Π
F
x
n
.
We next apply our result for finding a common element of a fixed point set of
a relatively nonexpansive mapping and the solution set of an equilibrium problem. This
problem is extensively studied in 11, 14–16.LetC be a subset of a Banach space E and
let f : C × C → R be a bifunction. The equilibrium problem for a bifunction f is to find x ∈ C
such that fx, y ≥ 0 for all y ∈ C. The set of solutions above is denoted by EPf,thatis
x ∈ EP
f
iff f
x, y
≥ 0 ∀y ∈ C. 2.25
To solve the equilibrium problem, we usually assume that a bifunction f satisfies the
following conditions C is closed and convex:
A1 fx, x0 for all x ∈ C;
A2 f is monotone, that is, fx, yfy, x ≤ 0
, for all x, y ∈ C;
A3 for all x, y, z ∈ C, lim sup
t↓0
ftz 1 − tx, y ≤ fx, y;
A4 for all x ∈ C, fx, · is convex and lower semicontinuous.
The following lemma gives a characterization of a solution of an equilibrium problem.
Lemma 2.3. Let C be a nonempty closed convex subset of a Banach space E.Letf be a bifunction from
C × C → R satisfying (A1)–(A4). Suppose that p ∈ C.Thenp ∈ EPf if and only if fy,p ≤ 0 for
all y ∈ C.
Proof. Let p ∈ EP f, then fp, y ≥ 0 for all y ∈ C.FromA2,wegetthatfy, p ≤−fp, y ≤
0 for all y ∈ C.
Conversely, assume that fy, p ≤ 0 for all y ∈ C. For any y ∈ C,let
x
t
ty
1 − t
p, for t ∈
0, 1
. 2.26
Then fx
t
,p ≤ 0 and hence
0 f
x
t
,x
t
≤ tf
x
t
,y
1 − t
f
x
t
,p
≤ tf
x
t
,y
. 2.27
Fixed Point Theory and Applications 11
So fx
t
,y ≥ 0 for all t ∈ 0, 1.FromA3, we have
0 ≤ lim sup
t↓0
f
ty
1 − t
p, y
≤ f
p, y
∀y ∈ C.
2.28
Hence p ∈ EPf.
Takahashi and Zembayashi proved the following important result.
Lemma 2.4 see 15, Lemma 2.8. Let C be a nonempty closed convex subset of a uniformly smooth,
strictly convex and reflexive Banach space E.Letf be a bifunction from C × C → R satisfying (A1)–
(A4). For r>0 and x ∈ E, define a mapping T
r
: E → C as follows:
T
r
x
z ∈ C : f
z, y
1
r
y − z, Jz − Jx
≥ 0 ∀y ∈ C
2.29
for all x ∈ E. Then, the following hold:
1 T
r
is single-valued;
2 T
r
is a firmly nonexpansive-type mapping [20], that is, for all x, y ∈ E
T
r
x − T
r
y, JT
r
x − JT
r
y
≤
T
r
x − T
r
y, Jx − Jy
; 2.30
3 FT
r
EPf;
4 EPf is closed and convex.
We now deduce Takahashi and Zembayashi’s recent result from Theorem 2.2.
Corollary 2.5 see 15, Theorem 4.1. Let C be a nonempty closed convex subset of a uniformly
smooth and uniformly convex Banach space E.Letf be a bifunction from C × C to R satisfying (A1)–
(A4) and let S be a relatively nonexpansive mapping from C into itself such that FS ∩ EPf
/
∅.
Let the sequence {x
n
} be generated by u
1
∈ E,
x
n
∈ C such that f
x
n
,y
1
r
n
y − x
n
,Jx
n
− Ju
n
≥ 0 ∀y ∈ C,
u
n1
J
−1
α
n
Jx
n
1 − α
n
JSx
n
2.31
for every n ∈ N, {α
n
}⊂0, 1 satisfying lim inf
n →∞
α
n
1 − α
n
> 0 and {r
n
}⊂a, ∞ for some
a>0.IfJ is weakly sequentially continuous, then {x
n
} converges weakly to z ∈ Π
FS∩EPf
,where
z lim
n →∞
Π
FS∩EPf
x
n
.
12 Fixed Point Theory and Applications
Proof. Put T
n
≡ T
r
n
where T
r
n
is defined by Lemma 2.4. Then
∞
n1
FT
n
EPf.By
reindexing the sequences {x
n
} and {u
n
} of this iteration, we can apply Theorem 2.2 by
showing that t he family {T
n
} satisfies the condition 2.9 and NST-condition. It is proved
in 15, Lemma 2.9 that
φ
u, T
n
x
φ
T
n
x, x
≤ φ
u, x
∀x ∈ E, u ∈
∞
n1
F
T
n
.
2.32
To see that {T
n
} satisfies NST-condition, let {z
n
} be a bounded sequence in C such that
lim
n →∞
z
n
− T
n
z
n
0andp ∈ ω
w
{z
n
}. Suppose that there exists a subsequence {z
n
k
} of
{z
n
} such that z
n
k
p. Then T
n
k
z
n
k
p∈ C. Since J is uniformly continuous on bounded
sets and r
n
k
≥ a, we have
lim
k →∞
1
r
n
k
Jz
n
k
− JT
n
k
z
n
k
0.
2.33
From the definition of T
r
n
k
, we have
f
T
n
k
z
n
k
,y
1
r
n
k
y − T
n
k
z
n
k
,JT
n
k
z
n
k
− Jz
n
k
≥ 0 ∀y ∈ C.
2.34
Since
f
y, T
n
k
z
n
k
≤−f
T
n
k
z
n
k
,y
≤
1
r
n
k
y − T
n
k
z
n
k
,JT
n
k
z
n
k
− Jz
n
k
≤
1
r
n
k
y − T
n
k
z
n
k
JT
n
k
z
n
k
− Jz
n
k
2.35
and f is lower semicontinuous and convex in the second variable, we have
f
y, p
≤ lim inf
k →∞
f
y, T
n
k
z
n
k
≤ 0.
2.36
Thus fy, p ≤ 0 for all y ∈ C.FromLemma 2.3, we have p ∈ EPf. Then {T
n
} satisfies the
NST-condition. From Theorem 2.2 where N 1, {x
n
} converges weakly to z ∈ FT
n
∩ FS
EPf ∩ FS, where z lim
n →∞
Π
EPf∩FS
x
n
.
Using the same proof as above, we have the following result.
Fixed Point Theory and Applications 13
Corollary 2.6 see 11, Theorem 3.5. Let C be a nonempty and closed convex subset of a uniformly
convex and uniformly smooth Banach space E.Letf be a bifunction from C × C to R satisfies (A1)–
(A4) and let T,S : C → C be two relatively nonexpansive mappings such that F : FT ∩ FS ∩
EPf
/
∅. Let the sequence {x
n
} be generated by the following manner:
x
n
∈ C such that f
x
n
,y
1
r
n
y − x
n
,Jx
n
− Ju
n
≥ 0 ∀y ∈ C,
u
n1
J
−1
α
n
Jx
n
β
n
JTx
n
γ
n
JSx
n
∀n ≥ 1.
2.37
Assume that {α
n
}, {β
n
}, and {γ
n
} are three sequences in 0, 1 satisfying the following restrictions:
a α
n
β
n
γ
n
1;
b lim inf
n →∞
α
n
β
n
> 0, lim inf
n →∞
α
n
γ
n
> 0;
c {r
n
}⊂a, ∞ for some a>0.
If J is weakly sequentially continuous, then {x
n
} converges weakly to z ∈ F,wherez lim
n →∞
Π
F
x
n
.
The following result also follows from Theorem 2.2.
Corollary 2.7 see 9, Theorem 5.3. Let E be a uniformly smooth and uniformly convex Banach
space and let C be a nonempty closed convex subset of E.Let{S
i
}
N
i1
be a finite family of relatively
nonexpansive mappings from C into itself such that F
N
i1
FS
i
is a nonempty and let {α
n,i
:
n, i ∈ N, 1 ≤ i ≤ N}⊂0, 1 and {ω
n,i
: n, i ∈ N, 1 ≤ i ≤ N}⊂0, 1 be sequences such that
lim inf
n →∞
α
n,i
1 − α
n,i
> 0 and lim inf
n →∞
ω
n,i
> 0 for all i ∈{1, 2, ,N} and
N
i1
ω
n,i
1 for
all n ∈ N.LetU
n
be a sequence of mappings defined by
U
n
x Π
C
J
−1
N
i1
ω
n,i
α
n,i
Jx
1 − α
n,i
JS
i
x
2.38
for all x ∈ C and let the sequence {x
n
} be generated by x
1
x ∈ C and
x
n1
U
n
x
n
n 1, 2,
. 2.39
Then the following hold:
1 the sequence {x
n
} is bounded and each weak subsequential limit of {x
n
} belongs to
N
i1
FS
i
;
2 if the duality mapping J from E into E
∗
is weakly sequentially continuous, then {x
n
}
converges weakly to the strong limit of {Π
F
x
n
}.
Proof. Since Π
C
is relatively nonexpansive, the family {Π
C
} satisfies the NST-condition.
Moreover, FΠ
C
C and
φ
x, Π
C
y
φ
Π
C
y, y
≤ φ
x, y
∀y ∈ E, x ∈ C. 2.40
Thus the conclusions of this corollary follow.
14 Fixed Point Theory and Applications
3. Strong Convergence Theorem
In this section, we prove strong convergence of an iterative sequence generated by the hybrid
method in mathematical programming. We start with the following useful common tools.
Lemma 3.1. Let C be a nonempty closed convex subset of a uniformly smooth and uniformly convex
Banach space E.Let{T
n
}
∞
n1
: E → E and {S
i
}
N
i1
: C → C be families of relatively quasi-
nonexpansive mappings such that F :
∞
n1
FT
n
∩
N
i1
FS
i
/
∅, and
φ
u, T
n
x
φ
T
n
x, x
≤ φ
u, x
3.1
for all u ∈
∞
n1
FT
n
, n ∈ N and x ∈ E.Let{x
n
}⊂C be such that {x
n
} and {S
i
x
n
} are bounded for
all i 1, 2, ,N, and
y
n
J
−1
N
i1
ω
n,i
α
n,i
Jx
n
1 − α
n,i
JS
i
x
n
,
u
n
T
n
y
n
,
3.2
where {ω
n,i
}, {α
n,i
}⊂0, 1 for all n ∈ N and i 1, 2, ,N satisfy
N
i1
ω
n,i
1 for all n ∈ N,
lim inf
n →∞
ω
n,i
1 − α
n,i
> 0 for all i 1, 2, ,N and lim
n →∞
x
n
− u
n
0. Then the following
statements hold:
1 lim
n →∞
φu, x
n
− φu, u
n
0 for all u ∈ C,
2 lim
n →∞
u
n
− y
n
0,
3 ω
w
{x
n
} ω
w
{y
n
},
4 if lim
n →∞
x
n1
− x
n
0,thenlim
n →∞
x
n
− S
i
x
n
0 for all i 1, 2, ,N,
5 if x
n
→ z,thenu
n
→ z and y
n
→ z.
Proof. 1 Since lim
n →∞
x
n
− u
n
0andJ is uniformly norm-to-norm continuous on
bounded sets,
lim
n →∞
Jx
n
− Ju
n
0.
3.3
We note here that {u
n
} is also bounded. For any u ∈ C, we have
φ
u, x
n
− φ
u, u
n
x
n
2
−
u
n
2
− 2
u, Ju
n
− Jx
n
≤
x
n
2
−
u
2
2
|
u, Ju
n
− Jx
n
|
≤
x
n
− u
n
x
n
u
n
2
u
Ju
n
− Jx
n
−→ 0.
3.4
2 Let u ∈ F.Using3.1 and the relative quasi-nonexpansiveness of each T
n
, we have
φ
u
n
,y
n
φ
T
n
y
n
,y
n
≤ φ
u, y
n
− φ
u, T
n
y
n
≤ φ
u, x
n
− φ
u, u
n
−→ 0. 3.5
Fixed Point Theory and Applications 15
By Lemma 1.4 and the boundedness of {u
n
}, we have
lim
n →∞
u
n
− y
n
0.
3.6
3 Since
x
n
− y
n
≤
x
n
− u
n
u
n
− y
n
x
n
− u
n
T
n
y
n
− y
n
−→ 0, 3.7
we have ω
w
{x
n
} ω
w
{y
n
}.
4 Assume that lim
n →∞
x
n1
− x
n
0. From lim
n →∞
x
n
− y
n
0, we get that
lim
n →∞
x
n1
− y
n
0. Since J is uniformly norm-to-norm continuous on bounded sets, we
have
lim
n →∞
Jx
n1
− Jx
n
lim
n →∞
Jx
n1
− Jy
n
0.
3.8
So,
Jx
n1
− Jy
n
Jx
n1
−
N
i1
ω
n,i
α
n,i
Jx
n
1 − α
n,i
JS
i
x
n
≥
N
i1
ω
n,i
1 − α
n,i
Jx
n1
− JS
i
x
n
− ω
n,i
α
n,i
Jx
n1
− Jx
n
.
3.9
From 3.8, we have
N
i1
ω
n,i
1 − α
n,i
Jx
n1
− JS
i
x
n
≤
Jx
n1
− Jy
n
N
i1
ω
n,i
α
n,i
Jx
n1
− Jx
n
−→ 0.
3.10
It follows from lim inf
n →∞
ω
n,i
1 − α
n,i
> 0 for all i 1, 2, ,N that
lim
n →∞
Jx
n1
− JS
i
x
n
0
3.11
for all i 1, 2, ,N. Since J
−1
is uniformly norm-to-norm continuous on bounded sets and
lim
n →∞
x
n1
− x
n
0, we have
lim
n →∞
x
n
− S
i
x
n
0
3.12
for all i 1, 2, ,N, as desired.
5 Assume that x
n
→ z. From the assumption and 2, we have
lim
n →∞
x
n
− u
n
lim
n →∞
u
n
− y
n
0.
3.13
Hence u
n
→ z and y
n
→ z.
16 Fixed Point Theory and Applications
Lemma 3.2 see 21, Lemma 2.4. Let F be a closed convex subset of a strictly convex, smooth and
reflexive Banach space E satisfying Kadec-Klee property. Let x ∈ E and {x
n
} be a sequence in E such
that ω
w
{x
n
}⊂F and φx
n
,x ≤ φΠ
F
x, x for all n ∈ N.Thenx
n
→ z Π
F
x.
Recall that a Banach space E satisfies Kadec–Klee property if whenever {u
n
} is a
sequence in E with x
n
xand x
n
→x, it follows that x
n
→ x.
3.1. The CQ-Method
Theorem 3.3. Let C be a nonempty closed convex subset of a uniformly smooth and uniformly convex
Banach space E.Let{T
n
}
∞
i1
: E → E be a family of relatively quasi-nonexpansive mappings satisfying
NST-condition and let {S
i
}
N
i1
: C → C be a family of relatively nonexpansive mappings such that
F :
∞
n1
FT
n
∩
N
i1
FS
i
/
∅, and
φ
u, T
n
x
φ
T
n
x, x
≤ φ
u, x
3.14
for all u ∈
∞
n1
FT
n
, n ∈ N and x ∈ E. Let the sequence {x
n
} be generated by
x
1
x ∈ C,
u
n
T
n
J
−1
N
i1
ω
n,i
α
n,i
Jx
n
1 − α
n,i
JS
i
x
n
,
C
n
z ∈ C : φ
z, u
n
≤ φ
z, x
n
,
Q
n
{
z ∈ C :
x
n
− z, Jx − Jx
n
≥ 0
}
,
x
n1
Π
C
n
∩Q
n
x
3.15
for every n ∈ N, {ω
n,i
}, {α
n,i
}⊂0, 1 for all n ∈ N and i 1, 2, ,N satisfying
N
i1
ω
n,i
1 for
all n ∈ N, lim inf
n →∞
ω
n,i
1 − α
n,i
> 0 for all i 1, 2, ,N.Then{x
n
} converges strongly to Π
F
x.
Proof. The proof is broken into 4 steps.
Step 1 {x
n
} is well defined. First, we show that C
n
∩ Q
n
is closed and convex. Clearly, Q
n
is
closed and convex. Since
φ
z, u
n
≤ φ
z, x
n
⇐⇒
u
n
2
−
x
n
2
− 2
z, Ju
n
− Jx
n
≤ 0,
3.16
then C
n
is closed and convex. Thus C
n
∩ Q
n
is closed and convex.
We next show that F ⊂ C
n
∩ Q
n
.Letu ∈ F. Then, from Lemma 1.7 ,
φ
u, u
n
≤ φ
u, x
n
. 3.17
Thus u ∈ C
n
. Hence F ⊂ C
n
for all n ∈ N.
Fixed Point Theory and Applications 17
Next, we show by induction that F ⊂ C
n
∩ Q
n
for all n ∈ N. Since Q
1
C, we have
F ⊂ C
1
∩ Q
1
. 3.18
Suppose that F ⊂ C
k
∩ Q
k
for some k ∈ N.Fromx
k1
Π
C
k
∩Q
k
x ∈ C
k
∩ Q
k
and the definition
of the generalized projection, we have
x
k1
− z, Jx − Jx
k1
≥ 0 3.19
for all z ∈ C
k
∩ Q
k
.FromF ⊂ C
k
∩ Q
k
,
x
k1
− p, Jx − Jx
k1
≥ 0 3.20
for all p ∈ F. Hence F ⊂ Q
k1
, and we also have F ⊂ C
k1
∩ Q
k1
. So, we have ∅
/
F ⊂ C
n
∩ Q
n
for all n ∈ N and hence the sequence {x
n
} is well defined.
Step 2 ω
w
{x
n
}⊂
N
i1
FS
i
. From the definition of Q
n
, we have x
n
Π
Q
n
x.Using
Lemma 1.2,weget
φ
x
n
,x
φ
Π
Q
n
x, x
≤ φ
u, x
− φ
u, Π
Q
n
x
≤ φ
u, x
3.21
for all u ∈ Q
n
. In particular, since x
n1
∈ Q
n
and Π
F
x ∈ F ⊂ Q
n
,
φ
x
n
,x
≤ φ
x
n1
,x
, 3.22
φ
x
n
,x
≤ φ
Π
F
x, x
3.23
for all n ∈ N. This implies that lim
n →∞
φx
n
,x exists and {x
n
} is bounded. Moreover, from
3.21 and x
n1
∈ Q
n
,
φ
x
n
,x
≤ φ
x
n1
,x
− φ
x
n1
,x
n
. 3.24
Hence
φ
x
n1
,x
n
≤ φ
x
n1
,x
− φ
x
n
,x
−→ 0. 3.25
It follows from x
n1
Π
C
n
∩Q
n
x ∈ C
n
that
φ
x
n1
,u
n
≤ φ
x
n1
,x
n
−→ 0. 3.26
From 3.25, 3.26 ,andLemma 1.4, we have
lim
n →∞
x
n1
− x
n
0 lim
n →∞
x
n1
− u
n
.
3.27
18 Fixed Point Theory and Applications
So lim
n →∞
x
n
− u
n
0. Using Lemma 3.14,wegetthat
lim
n →∞
x
n
− S
i
x
n
0
3.28
for all i 1, 2, ,N. Since each S
i
is relatively nonexpansive,
ω
w
{
x
n
}
⊂
N
i1
F
S
i
N
i1
F
S
i
.
3.29
Step 3 ω
w
{x
n
}⊂
∞
n1
FT
n
.Lety
n
J
−1
N
i1
ω
n,i
α
n,i
Jx
n
1 − α
n,i
JS
i
x
n
.From
Lemma 3.12, we have
lim
n →∞
T
n
y
n
− y
n
0,
3.30
and ω
w
{x
n
} ω
w
{y
n
}. It follows from NST-condition that ω
w
{x
n
} ω
w
{y
n
}⊂
∞
n1
FT
n
.
Step 4 x
n
→ Π
F
x. From Steps 2 and 3, we have ω
w
{x
n
}⊂F. The conclusion follows by
Lemma 3.2 and 3.23.
We apply Theorem 3.3 and the proof of Corollary 2.5 and then obtain the following
result.
Corollary 3.4. Let C, E, f, S be as in Corollary 2.5. Let the sequence {x
n
} be generated by
x
1
x ∈ C,
y
n
J
−1
α
n
Jx
n
1 − α
n
JSx
n
,
u
n
∈ C such that f
u
n
,y
1
r
n
y − u
n
,Ju
n
− Jy
n
≥ 0 ∀y ∈ C,
C
n
z ∈ C : φ
z, u
n
≤ φ
z, x
n
,
Q
n
{
z ∈ C :
x
n
− z, Jx − Jx
n
≥ 0
}
,
x
n1
Π
C
n
∩Q
n
x
3.31
for every n ∈ N, {α
n
}⊂0, 1 satisfying lim sup
n →∞
α
n
< 1 and {r
n
}⊂a, ∞ for some a>0.
Then, {x
n
} converges strongly to Π
FS∩EPf
x,whereΠ
FS∩EPf
is the generalized projection of E
onto FS ∩ EPf.
Remark 3.5. Corollary 3.4 improves the restriction on {α
n
} of 15, Theorem 3.1.Infact,itis
assumed in 15, Theorem 3.1 that lim inf
n →∞
α
n
1 − α
n
> 0.
Fixed Point Theory and Applications 19
3.2. The Monotone CQ-Method
Let C be a closed subset of a Banach space E. Recall that a mapping T : C → C is closed if for
each {x
n
} in C,ifx
n
→ x and Tx
n
→ y, then Tx y. A family of mappings {T
n
} : C → E
with
∞
n1
FT
n
/
∅ is said to satisfy the ∗-condition if for each bounded sequence {z
n
} in C,
lim
n →∞
z
n
− T
n
z
n
0,z
n
−→ z imply z ∈
∞
n1
F
T
n
.
3.32
Remark 3.6. 1 If {T
n
} satisfies NST-condition, then {T
n
} satisfies ∗-condition.
2 If T
n
≡ T and T is closed, then {T
n
} satisfies ∗-condition.
Theorem 3.7. Let C be a nonempty closed convex subset of a uniformly smooth and uniformly
convex Banach space E.Let{T
n
}
∞
n1
: E → E be a family of relatively quasi-nonexpansive mappings
satisfying ∗-condition and let {S
i
}
N
i1
: C → C be a family of closed relatively quasi-nonexpansive
mappings such that F :
∞
n1
FT
n
∩
N
i1
FS
i
/
∅, and
φ
u, T
n
x
φ
T
n
x, x
≤ φ
u, x
3.33
for all u ∈
∞
n1
FT
n
, n ∈ N, and x ∈ E. Let the sequence {x
n
} be generated by
x
0
x ∈ C, Q
0
C,
u
n
T
n
J
−1
N
i1
ω
n,i
α
n,i
Jx
n
1 − α
n,i
JS
i
x
n
,
C
0
z ∈ C : φ
z, u
0
≤ φ
z, x
0
,
C
n
z ∈ C
n−1
∩ Q
n−1
: φ
z, u
n
≤ φ
z, x
n
,
Q
n
{
z ∈ C
n−1
∩ Q
n−1
:
x
n
− z, Jx − Jx
n
≥ 0
}
,
x
n1
Π
C
n
∩Q
n
x
3.34
for every n ∈ N, {ω
n,i
}, {α
n,i
}⊂0, 1 satisfying
N
i1
ω
n,i
1 and lim inf
n →∞
ω
n,i
1 − α
n,i
> 0 for
all i 1, 2, ,N.Then{x
n
} converges strongly to Π
F
x.
Proof.
Step 1 {x
n
} is well defined. This step is almost the same as Step 1 of the proof of
Theorem 3.3,soitisomitted.
Step 2 {x
n
} is a Cauchy sequence in C. We can follow the proof of Theorem 3.3 and conclude
that
lim
n →∞
φ
x
n
,x
3.35
20 Fixed Point Theory and Applications
exists. Moreover, as x
nm
∈ Q
n
for all n, m and x
n
Π
Q
n
x,
φ
x
nm
,x
n
φ
x
nm
, Π
Q
n
x
≤ φ
x
nm
,x
− φ
Π
Q
n
x, x
φ
x
nm
,x
− φ
x
n
,x
.
3.36
Since {x
n
} is bounded, it follows from Lemma 1.3 that there exists a strictly increasing,
continuous, and convex function h such that h00and
h
x
nm
− x
n
≤ φ
x
nm
,x
− φ
x
n
,x
. 3.37
Since lim
n →∞
φx
n
,x exists, we have that {x
n
} is a Cauchy sequence. Therefore, x
n
→ z for
some z ∈ C.
Step 3 z ∈
N
i1
FS
i
. Since x
n1
Π
C
n
∩Q
n
x ∈ C
n
, we have
φ
x
n1
,u
n
≤ φ
x
n1
,x
n
−→ φ
z, z
0. 3.38
By Lemma 1.4 and the boundedness of {x
n
}, we have
lim
n →∞
x
n1
− u
n
0.
3.39
So, we have lim
n →∞
x
n
− u
n
0. Using Lemma 3.14,wegetthat
lim
n →∞
x
n
− S
i
x
n
0
3.40
for all i 1, 2, ,N. Since each S
i
is closed, z ∈
N
i1
FS
i
.
Step 4 z ∈
∞
n1
FT
n
.Lety
n
J
−1
N
i1
ω
n,i
α
n,i
Jx
n
1 − α
n,i
JS
i
x
n
.FromLemma 3.12,
we have lim
n →∞
y
n
−T
n
y
n
0andy
n
→ z. It follows from ∗-condition that z ∈
∞
n1
FT
n
.
Step 5 x
n
→ Π
F
x. From Steps 3 and 4, we have ω
w
{x
n
}⊂F. The conclusion follows by
Lemma 3.2 and 3.23.
Letting T
n
identity and S
1
S
2
··· S
N
yield the following result.
Fixed Point Theory and Applications 21
Corollary 3.8 see 12, Theorem 3.1. Let C be a nonempty closed convex subset of a uniformly
convex and uniformly smooth real Banach space E.LetT : C → C be a closed relatively quasi-
nonexpansive mapping such that FT
/
∅. Assume that {α
n
} is a sequence in 0, 1 such that
lim sup
n →∞
α
n
< 1. Define a sequence {x
n
} in C by the following algorithm:
x
0
∈ C chosen arbitrarily,
y
n
J
−1
α
n
Jx
n
1 − α
n
JTx
n
,
C
n
z ∈ C
n−1
∩ Q
n−1
: φ
z, y
n
≤ φ
z, x
n
,
C
0
z ∈ C : φ
z, y
0
≤ φ
z, x
0
,
Q
n
{
z ∈ C
n−1
∩ Q
n−1
:
x
n
− z, Jx
0
− Jx
n
≥ 0
}
,
Q
0
C,
x
n1
Π
C
n
∩Q
n
x
0
.
3.41
Then {x
n
} converges strongly to Π
FT
x
0
.
Letting T
n
identity and N 2 yield the following result.
Corollary 3.9 see 13, Theorem 3.1. Let C be a nonempty closed convex subset of a uniformly
convex and uniformly smooth real Banach space E.LetT, S be two closed relatively quasi-nonexpansive
mappings from C into itself such that F : FT ∩ FS
/
∅. Define a sequence {x
n
} in C be the
following algorithm:
x
0
∈ C chosen arbitrarily,
z
n
J
−1
β
1
n
Jx
n
β
2
n
JTx
n
β
3
n
JSx
n
,
y
n
J
−1
α
n
Jx
n
1 − α
n
Jz
n
,
C
0
z ∈ C : φ
z, y
0
≤ φ
z, x
0
,
C
n
z ∈ C
n−1
∩ Q
n−1
: φ
z, y
n
≤ φ
z, x
n
,
Q
n
{
z ∈ C
n−1
∩ Q
n−1
:
x
n
− z, Jx
0
− Jx
n
≥ 0
}
,
Q
0
C,
x
n1
Π
C
n
∩Q
n
x
0
3.42
with the conditions: β
1
n
,β
2
n
,β
3
n
∈ 0, 1 with β
1
n
β
2
n
β
3
n
1 and
1 lim inf
n →∞
β
1
n
β
2
n
> 0;
2 lim inf
n →∞
β
1
n
β
3
n
> 0;
3 0 ≤ α
n
≤ α<1 for some α ∈ 0, 1.
Then {x
n
} converges strongly to Π
F
x
0
.
22 Fixed Point Theory and Applications
Remark 3.10. Using Theorem 3.7, we can show that the conclusion of Corollary 3.9 remains
true under the more general restrictions on {α
n
}, {β
1
n
}, {β
2
n
}, and {β
3
n
}:
1 α
n
,β
1
n
∈ 0, 1 are arbitrary;
2 lim inf
n →∞
β
2
n
> 0 and lim inf
n →∞
β
3
n
> 0.
3.3. The Shrinking Projection Method
Theorem 3.11. Let C, E, {T
n
}
∞
n1
, {S
i
}
N
i1
be as in Theorem 3.7. Let the sequence {x
n
} be generated
by
x
0
∈ E chosen arbitrarily,
C
1
C,
x
1
Π
C
1
x
0
,
u
n
T
n
J
−1
N
i1
ω
n,i
α
n,i
Jx
n
1 − α
n,i
JS
i
x
n
,
C
n1
z ∈ C
n
: φ
z, u
n
≤ φ
z, x
n
,
x
n1
Π
C
n1
x
0
3.43
for every n ∈ N, {ω
n,i
}, {α
n,i
}⊂0, 1 for all n ∈ N and i 1, 2, ,N satisfies
N
i1
ω
n,i
1 for all
n ∈ N, lim inf
n →∞
ω
n,i
1 − α
n,i
> 0 for all i 1, 2, ,N.Then{x
n
} converges strongly to Π
F
x.
Proof. The proof is almost the same as the proofs of Theorems 3.3 and 3.7; so it is omitted.
In particular, applying Theorem 3.11 gives the following result.
Corollary 3.12. Let C, E, f, S be as in Corollary 2.5. Let the sequence {x
n
} be generated by x
0
x ∈ C, C
0
C and
y
n
J
−1
α
n
Jx
n
1 − α
n
JSx
n
,
u
n
∈ C such that f
u
n
,y
1
r
n
y − u
n
,Ju
n
− Jy
n
≥ 0 ∀y ∈ C,
C
n1
z ∈ C
n
: φ
z, u
n
≤ φ
z, x
n
,
x
n1
Π
C
n1
x
0
3.44
for every n ∈ N ∪{0},whereJ is the duality mapping on E. Assume that {α
n
}⊂0, 1 satisfies
lim sup
n →∞
α
n
< 1 and {r
n
}⊂a, ∞ for some a>0.Then{x
n
} converges strongly to Π
FS∩EPf
x,
where Π
FS∩EPf
is the generalized projection of E onto FS ∩ EP f.
Remark 3.13. Corollary 3.12 improves the restriction on {α
n
} of 16, Theorem 3.1. In fact, it is
assumed in 16, Theorem 3.1 that lim inf
n →∞
α
n
1 − α
n
> 0.
Fixed Point Theory and Applications 23
Corollary 3.14 see 11, Theorem 3.1. Let C be a nonempty and closed convex subset of a
uniformly convex and uniformly smooth Banach space E.Letf be a bifunction from C × C to R
satisfying (A1)–(A4) and let T, S : C → C be two closed relatively quasi-nonexpansive mappings
such that F : FT ∩ FS ∩ EPf
/
∅. Let the sequence {x
n
} be generated by the following manner:
x
0
∈ E chosen arbitrarily,
C
1
C,
x
1
Π
C
1
x
0
,
y
n
J
−1
α
n
Jx
n
β
n
JTx
n
γ
n
JSx
n
,
u
n
∈ C such that f
u
n
,y
1
r
n
y − u
n
,Ju
n
− Jy
n
≥ 0 ∀y ∈ C,
C
n1
z ∈ C
n
: φ
z, u
n
≤ φ
z, x
n
,
x
n1
Π
C
n1
x
0
.
3.45
Assume that {α
n
}, {β
n
}, and {γ
n
} are three sequences in 0, 1 satisfying the restrictions:
a α
n
β
n
γ
n
1;
b lim inf
n →∞
α
n
β
n
> 0, lim inf
n →∞
α
n
γ
n
> 0;
c {r
n
}⊂a, ∞ for some a>0.
Then {x
n
} converges strongly to Π
F
x
0
.
Remark 3.15. The conclusion of Corollary 3.14 remains true under the more general
assumption; that is, we can replace b by the following one:
b
lim inf
n →∞
β
n
> 0 and lim inf
n →∞
γ
n
> 0.
We also deduce the following result.
Corollary 3.16 see 14, Theorem 3.1. Let C, E, f, T, S be as in Corollary 3.14.Letthe
sequences {x
n
}, {y
n
}, {z
n
}, and {u
n
} be generated by the following:
x
0
∈ E chosen arbitrarily,
C
1
C,
x
1
Π
C
1
x
0
,
y
n
J
−1
δ
n
Jx
n
1 − δ
n
Jz
n
,
z
n
J
−1
α
n
Jx
n
β
n
JTx
n
γ
n
JSx
n
,
u
n
∈ C such that f
u
n
,y
1
r
n
y − u
n
,Ju
n
− Jz
n
≥ 0 ∀y ∈ C,
C
n1
z ∈ C
n
: φ
z, u
n
≤ φ
z, x
n
,
x
n1
Π
C
n1
x
0
.
3.46
24 Fixed Point Theory and Applications
Assume that {α
n
}, {β
n
}, and {γ
n
} are three sequences in 0, 1 satisfying the following restrictions:
a α
n
β
n
γ
n
1;
b 0 ≤ α
n
< 1 for all n ∈ N ∪{0} and lim sup
n →∞
α
n
< 1;
c lim inf
n →∞
α
n
β
n
> 0, lim inf
n →∞
α
n
γ
n
> 0;
d {r
n
}⊂a, ∞ for some a>0.
Then {x
n
} and {u
n
} converge strongly to Π
F
x
0
.
Remark 3.17. The conclusion of Corollary 3.16 remains true under the more general
restrictions; that is, we replace b and c by the following one:
b
lim inf
n →∞
β
n
> 0 and lim inf
n →∞
γ
n
> 0.
Corollary 3.18 see 10, Theorem 3.1. Let C be a nonempty closed convex subset of a uniformly
convex and uniformly smooth Banach space E.Let{T
i
}
N
i1
: C → C be a family of relatively
nonexpansive mappings such that F :
N
i1
FT
i
/
∅ and let x
0
∈ E. For C
1
C and x
1
Π
C
1
x
0
,
define a sequence {x
n
} of C as follows:
y
n
J
−1
α
n
Jx
n
1 − α
n
Jz
n
,
z
n
J
−1
β
1
n
Jx
n
N
i1
β
i1
n
JT
i
x
n
,
C
n1
z ∈ C
n
: φ
z, y
n
≤ φ
z, x
n
,
x
n1
Π
C
n1
x
0
,
3.47
where {α
n
}, {β
i
n
}⊂0, 1 satisfies the following restrictions:
i 0 ≤ α
n
< 1 for all n ∈ N ∪{0} and lim sup
n →∞
α
n
< 1;
ii 0 ≤ β
i
n
≤ 1 for all i 1, 2, ,N 1,
N1
i1
β
i
n
1 for all n ∈ N ∪{0}.If
a either lim inf
n →∞
β
1
n
β
i1
n
> 0 for all i 1, 2, ,Nor
b lim
n →∞
β
1
n
0 and lim inf
n →∞
β
k1
n
β
l1
n
> 0 for all i
/
j, k, l 1, 2, ,N.
then the sequence {x
n
} converges strongly to Π
F
x
0
.
Remark 3.19. The conclusion of Corollary 3.18 remains true under the more general
restrictions on {α
n
}, {β
i
n
}:
1 α
n
,β
1
n
∈ 0, 1 are arbitrary.
2 lim inf
n →∞
β
i
n
> 0 for all i 2, ,N.
Fixed Point Theory and Applications 25
Acknowledgments
The authors would like t o thank the referee for their comments on the manuscript. The first
author is supported by grant fund under the program Strategic Scholarships for Frontier
Research Network for the Ph.D. Program Thai Doctoral degree from the Office of the Higher
Education Commission, Thailand, and the second author is supported by the Thailand
Research Fund under Grant MRG5180146.
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