Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2010, Article ID 861031, 24 pages
doi:10.1155/2010/861031
Research Article
A New Iterative Scheme for Countable
Families of Weak Relatively Nonexpansive
Mappings and System of Generalized Mixed
Equilibrium Problems
Yekini Shehu1, 2
1
2
Department of Mathematics, University of Nigeria, Nsukka, Nigeria
Mathematics Institute, African University of Science and Technology, Abuja, Nigeria
Correspondence should be addressed to Yekini Shehu, deltanougt2006@yahoo.com
Received 21 September 2010; Revised 9 November 2010; Accepted 22 December 2010
Academic Editor: A. Zafer
Copyright q 2010 Yekini Shehu. 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 construct a new iterative scheme by hybrid methods and prove strong convergence theorem
for approximation of a common fixed point of two countable families of weak relatively
nonexpansive mappings which is also a solution to a system of generalized mixed equilibrium
problems in a uniformly convex real Banach space which is also uniformly smooth using the
properties of generalized f -projection operator. Using this result, we discuss strong convergence
theorem concerning general H-monotone mappings and system of generalized mixed equilibrium
problems in Banach spaces. Our results extend many known recent results in the literature.
1. Introduction
Let E be a real Banach space with dual E∗ , and let C be nonempty, closed and convex subset
of E. A mapping T : C → C is called nonexpansive if
Tx − Ty ≤ x − y,
∀x, y ∈ C.
1.1
A point x ∈ C is called a fixed point of T if Tx x. The set of fixed points of T is denoted by
FT : {x ∈ C : Tx x}.
2
Abstract and Applied Analysis
∗
We denote by J the normalized duality mapping from E to 2E defined by
2 Jx f ∈ E∗ : x, f x2 f .
1.2
The following properties of J are well known the reader can consult 1–3 for more details.
1 If E is uniformly smooth, then J is norm-to-norm uniformly continuous on each
bounded subset of E.
2 Jx /
∅, x ∈ E.
3 If E is reflexive, then J is a mapping from E onto E∗ .
4 If E is smooth, then J is single valued.
Throughout this paper, we denote by φ, the functional on E × E defined by
2
φ x, y x2 − 2 x, J y y ,
∀x, y ∈ E.
1.3
From 4, in uniformly convex and uniformly smooth Banach spaces, we have
2
2
x − y ≤ φ x, y ≤ x y ,
∀x, y ∈ E.
1.4
Definition 1.1. Let C be a nonempty subset of E and let {Tn }∞
n0 be a countable family of
mappings from C into E. A point p ∈ C is said to be an asymptotic fixed point of {Tn }∞
n0 if
which
converges
weakly
to
p
and
lim
x
−
T
x
C contains a sequence {xn }∞
n→∞
n
n n 0.
n0
∞
The set of asymptotic fixed points of T is denoted by F{Tn }n0 . One says that {Tn }∞
n0 is
countable family of relatively nonexpansive mappings see, e.g., 5 if the following conditions
are satisfied:
∅;
R1 F{Tn }∞
n0 /
R2 φp, Tn x ≤ φp, x, for all x ∈ C, p ∈ FTn , n ≥ 0;
n }∞
R3 ∩∞ FTn F{T
n0 .
n0
Definition 1.2. A point p ∈ C is said to be a strong asymptotic fixed point of {Tn }∞
n0 if C
contains a sequence {xn }∞
n0 which converges strongly to p and limn → ∞ xn − Tn xn 0. The
n }∞
set of strong asymptotic fixed points of T is denoted by F{T
n0 . One says that a mapping
∞
{Tn }n0 is countable family of weak relatively nonexpansive mappings see, e.g., 5 if the following
conditions are satisfied:
R1 F{Tn }∞
/ ∅;
n0 R2 φp, Tn x ≤ φp, x, for all x ∈ C, p ∈ FTn , n ≥ 0;
n }∞
R3 ∩∞ FTn F{T
n0 .
n0
Definition 1.3. Let C be a nonempty subset of E and let T be a mapping from C into E. A
point p ∈ C is said to be an asymptotic fixed point of T if C contains a sequence {xn }∞
n0 which
converges weakly to p and limn → ∞ xn − Txn 0. The set of asymptotic fixed points of T is
denoted by FT.
We say that a mapping T is relatively nonexpansive see, e.g., 6–11 if the
following conditions are satisfied:
Abstract and Applied Analysis
3
R1 FT /
∅;
R2 φp, Tx ≤ φp, x, for all x ∈ C, p ∈ FT;
R3 FT FT.
Definition 1.4. A point p ∈ C is said to be an strong asymptotic fixed point of T if C contains a
sequence {xn }∞
n0 which converges strongly to p and limn → ∞ xn − Txn 0. The set of strong
asymptotic fixed points of T is denoted by FT.
We say that a mapping T is weak relatively
nonexpansive see, e.g., 12, 13 if the following conditions are satisfied:
R1 FT / ∅;
R2 φp, Tx ≤ φp, x, for all x ∈ C, p ∈ FT;
R3 FT FT.
Definition 1.3 Definition 1.4 is a special form of Definition 1.1 Definition 1.2
as Tn ≡ T, for all n ≥ 0. Furthermore, Su et al. 5 gave an example which is a
countable family of weak relatively nonexpansive mappings but not a countable family of
relatively nonexpansive mappings. It is obvious that relatively nonexpansive mapping is
weak relatively nonexpansive mapping. In fact, for any mapping T : C → C, we have FT ⊂
FT
⊂ FT.
Therefore, if T is relatively nonexpansive mapping, then FT FT
FT.
Kang et al. 12 gave an example of a weak relatively nonexpansive mapping which is not
relatively nonexpansive.
Let F : C×C → Ê be a bifunction, A : C → E∗ a mapping and ϕ : C → Ê a real-valued
function. The generalized mixed equilibrium problem is to find x ∈ C see, e.g., 14–19 such
that
F x, y Ax, y − x ϕ y − ϕx ≥ 0,
1.5
for all y ∈ C. We will denote the solutions set of 1.5 by GMEPF, ϕ. Thus
GMEP F, A, ϕ : x∗ ∈ C : F x∗ , y Ax∗ , y − x∗ ϕ y − ϕx∗ ≥ 0, ∀y ∈ C .
1.6
If ϕ 0 and A 0, then problem 1.5 reduces to an equilibrium problem studied by many
authors see, e.g., 20–28, which is to find x∗ ∈ C such that
F x∗ , y ≥ 0
1.7
for all y ∈ C. We shall denote the solutions set of 1.7 by EPF.
If ϕ 0 and E H a real Hilbert space, then problem 1.5 reduces to a generalized
equilibrium problem studied by many authors see, e.g., 29–31, which is to find x∗ ∈ C
such that
F x∗ , y Ax∗ , y − x∗ ≥ 0
for all y ∈ C.
1.8
4
Abstract and Applied Analysis
If A 0 and E H, then problem 1.5 reduces to mixed equilibrium problem
considered by many authors see, e.g., 32–34, which is to find x∗ ∈ C such that
F x∗ , y ϕ y − ϕx∗ ≥ 0
1.9
for all y ∈ C.
The mixed equilibrium problems include fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and equilibrium problems
as special cases see, e.g., 35. Some methods have been proposed to solve the mixed
equilibrium problem see, e.g., 33, 34, 36. Numerous problems in physics, optimization
and economics reduce to find a solution of problem 1.8.
In 9, Matsushita and Takahashi introduced a hybrid iterative scheme for approximation of fixed points of relatively nonexpansive mapping in a uniformly convex real Banach
space which is also uniformly smooth: x0 ∈ C,
yn J −1 αn Jxn 1 − αn JTxn ,
Hn w ∈ C : φ w, yn ≤ φw, xn ,
1.10
Wn {w ∈ C : xn − w, Jx0 − Jxn },
xn1 ΠHn ∩Wn x0 ,
n≥0
∅.
They proved that {xn }∞
n0 converges strongly to ΠFT x0 , where FT /
In 37, Plubtieng and Ungchittrakool introduced the following hybrid projection
algorithm for a pair of relatively nonexpansive mappings: x0 ∈ C,
1
2
3
zn J −1 βn Jxn βn JTxn βn JSxn ,
yn J −1 αn Jx0 1 − αn Jzn ,
Cn z ∈ C : φ z, yn ≤ φz, xn αn x0 2 2w, Jxn − Jx0 ,
1.11
Qn {z ∈ C : xn − z, Jxn − Jx0 ≤ 0},
xn1 PCn ∩Qn x0 ,
1
2
3
1
2
3
where {αn }, {βn }, {βn }, and {βn } are sequences in 0, 1 satisfying βn βn βn 1 and
T and S are relatively nonexpansive mappings and J is the single-valued duality mapping on
E. They proved under the appropriate conditions on the parameters that the sequence {xn }
generated by 1.11 converges strongly to a common fixed point of T and S.
In 10, Takahashi and Zembayashi introduced the following hybrid iterative scheme
for approximation of fixed point of relatively nonexpansive mapping which is also a solution
to an equilibrium problem in a uniformly convex real Banach space which is also uniformly
Abstract and Applied Analysis
5
smooth: x0 ∈ C, C1 C, x1 ΠC1 x0 ,
yn J −1 αn Jxn 1 − αn JTxn ,
1
y − un , Jun − Jyn ≥ 0, ∀y ∈ C,
F un , y rn
Cn1 w ∈ Cn : φw, un ≤ φw, xn ,
xn1 ΠCn 1 x0 ,
1.12
n ≥ 1,
where J is the duality mapping on E. Then, they proved that {xn }∞
n0 converges strongly to
∅.
ΠΩ x0 , where Ω EPF ∩ FT /
Recently, Li et al. 38 introduced the following hybrid iterative scheme for
approximation of fixed points of a relatively nonexpansive mapping using the properties
of generalized f-projection operator in a uniformly smooth real Banach space which is also
uniformly convex: x0 ∈ C,
yn J −1 αn Jxn 1 − αn JTxn ,
Cn1 w ∈ Cn : G w, Jyn ≤ Gw, Jxn ,
f
xn1 ΠCn 1 x0 ,
1.13
n ≥ 1.
They proved a strong convergence theorem for finding an element in the fixed points set of
T. We remark here that the results of Li et al. 38 extended and improved on the results of
Matsushita and Takahashi 9.
Quite recently, motivated by the results of Matsushita and Takahashi 9 and Plubtieng
and Ungchittrakool 37, Su et al. 5 proved the following strong convergence theorem by
hybrid iterative scheme for approximation of common fixed point of two countable families
of weak relatively nonexpansive mappings in uniformly convex and uniformly smooth
Banach space.
Theorem 1.5. Let E be a uniformly convex real Banach space which is also uniformly smooth. Let C be
∞
a nonempty, closed and convex subset of E. Suppose {Tn }∞
n1 and {Sn }n1 are two countable families of
∞
∅.
weak relatively nonexpansive mappings of C into itself such that Ω : ∩∞
n1 FTn ∩∩n1 FSn /
∞
Suppose that {xn }n0 is iteratively generated by x0 ∈ C,
1
2
3
zn J −1 βn Jxn βn JTn xn βn JSn xn ,
yn J −1 αn Jxn 1 − αn Jzn ,
Cn w ∈ Cn−1 ∩ Qn−1 : φ w, yn ≤ φw, xn ,
C0 w ∈ C : φ w, y0 ≤ φw, x0 ,
Qn {w ∈ Cn−1 ∩ Qn−1 : xn − w, Jx0 − Jxn ≥ 0},
Q0 C,
xn1 ΠCn ∩Qn x0 ,
n ≥ 1,
1.14
6
Abstract and Applied Analysis
with the conditions
1 2
i lim infn → ∞ βn βn > 0;
1 3
ii lim infn → ∞ βn βn > 0;
iii 0 ≤ αn ≤ α < 1 for some α ∈ 0, 1.
Then, {xn }∞
n0 converges strongly to ΠΩ x0 .
Motivated by the above-mentioned results and the ongoing research, it is our purpose
in this paper to prove a strong convergence theorem for two countable families of weak
relatively nonexpansive mappings in a uniformly convex real Banach space which is also
uniformly smooth using the properties of generalized f-projection operator. Our results
extend the results of Li et al. 38, Su et al. 5 and many other recent known results in the
literature.
2. Preliminaries
Let E be a real Banach space. The modulus of smoothness of E is the function ρE : 0, ∞ →
0, ∞ defined by
1 x y x − y − 1 : x ≤ 1, y ≤ t .
ρE t : sup
2
2.1
E is uniformly smooth if and only if
lim
t→0
ρE t
0.
t
2.2
Let dim E ≥ 2. The modulus of convexity of E is the function δE : 0, 2 → 0, 1 defined by
x y
δE : inf 1 − : x y 1; x − y .
2 2.3
E is uniformly convex if for any ∈ 0, 2, there exists a δ δ > 0 such that if x, y ∈ E with
x ≤ 1, y ≤ 1, and x − y ≥ , then 1/2x y ≤ 1 − δ. Equivalently, E is uniformly
convex if and only if δE > 0 for all ∈ 0, 2. A normed space E is called strictly convex if
for all x, y ∈ E, x /
y, x y 1, we have λx 1 − λy < 1, for all λ ∈ 0, 1.
Let E be a smooth, strictly convex and reflexive real Banach space and let C be a
nonempty, closed and convex subset of E. Following Alber 39, the generalized projection
ΠC from E onto C is defined by
ΠC x : arg min φ y, x ,
y∈C
∀x ∈ E.
2.4
The existence and uniqueness of ΠC follows from the property of the functional φx, y and
strict monotonicity of the mapping J see, e.g., 3, 4, 39–41. If E is a Hilbert space, then ΠC
is the metric projection of H onto C.
Abstract and Applied Analysis
7
Next, we recall the concept of generalized f-projector operator, together with its
properties. Let G : C × E∗ → Ê ∪ {∞} be a functional defined as follows:
2
G ξ, ψ ξ2 − 2 ξ, ψ ψ 2ρfξ,
2.5
where ξ ∈ C, ψ ∈ E∗ , ρ is a positive number and f : C → Ê ∪ {∞} is proper, convex
and lower semi-continuous. From the definitions of G and f, it is easy to see the following
properties:
i Gξ, ψ is convex and continuous with respect to ψ when ξ is fixed;
ii Gξ, ψ is convex and lower semi-continuous with respect to ξ when ψ is fixed.
Definition 2.1 Wu and Huang 42. Let E be a real Banach space with its dual E∗ . Let C be
f
a nonempty, closed and convex subset of E. One says that ΠC : E∗ → 2C is a generalized
f-projection operator if
f
ΠC ψ u ∈ C : G u, ψ inf G ξ, ψ ,
ξ∈C
∀ψ ∈ E∗ .
2.6
For the generalized f-projection operator, Wu and Huang 42 proved the following
theorem basic properties.
Lemma 2.2 Wu and Huang 42. Let E be a real reflexive Banach space with its dual E∗ . Let C be
a nonempty, closed, and convex subset of E. Then the following statements hold:
f
i ΠC is a nonempty closed convex subset of C for all ψ ∈ E∗ ;
f
ii if E is smooth, then for all ψ ∈ E∗ , x ∈ ΠC if and only if
x − y, ψ − Jx ρf y − ρfx ≥ 0,
∀y ∈ C;
2.7
iii if E is strictly convex and f : C → Ê ∪ {∞} is positive homogeneous (i.e., ftx tfx
f
for all t > 0 such that tx ∈ C where x ∈ C), then ΠC is a single valued mapping.
Fan et al. 43 showed that the condition f is positive homogeneous which appeared
in Lemma 2.2 and can be removed.
Lemma 2.3 Fan et al. 43. Let E be a real reflexive Banach space with its dual E∗ and C a
f
nonempty, closed and convex subset of E. Then if E is strictly convex, then ΠC is a single-valued
mapping.
Recall that J is a single valued mapping when E is a smooth Banach space. There exists
a unique element ψ ∈ E∗ such that ψ Jx for each x ∈ E. This substitution in 4.3 gives
Gξ, Jx ξ2 − 2ξ, Jx x2 2ρfξ.
Now, we consider the second generalized f-projection operator in a Banach space.
2.8
8
Abstract and Applied Analysis
Definition 2.4. Let E be a real Banach space and C a nonempty, closed, and convex subset of
f
E. One says that ΠC : E → 2C is a generalized f-projection operator if
f
ΠC x u ∈ C : Gu, Jx inf Gξ, Jx ,
∀x ∈ E.
ξ∈C
2.9
Obviously, the definition of {Tn }∞
n0 is a countably family of weak relatively
nonexpansive mappings is equivalent to
∅;
R 1 F{Tn }∞
n0 /
R 2 Gp, JTn x ≤ Gp, Jx, for all x ∈ C, p ∈ FTn , n ≥ 0;
∞
R 3 ∩∞
n0 FTn F{Tn }n0 .
Lemma 2.5 Li et al. 38. Let E be a Banach space and f : E →
convex functional. Then there exists x∗ ∈ E∗ and α ∈ Ê such that
fx ≥ x, x∗ α,
Ê ∪{∞} a lower semi-continuous
∀x ∈ E.
2.10
f
We know that the following lemmas hold for operator ΠC .
Lemma 2.6 Li et al. 38. Let C be a nonempty, closed, and convex subset of a smooth and reflexive
Banach space E. Then the following statements hold:
f
i ΠC x is a nonempty closed and convex subset of C for all x ∈ E;
f
ii for all x ∈ E, x ∈ ΠC x if and only if
x − y, Jx − J x ρf y − ρfx ≥ 0,
∀y ∈ C;
2.11
f
iii if E is strictly convex, then ΠC x is a single-valued mapping.
Lemma 2.7 Li et al. 38. Let C be a nonempty, closed, and convex subset of a smooth and reflexive
f
Banach space E. Let x ∈ E and x ∈ ΠC . Then
φ y, x Gx,
Jx ≤ G y, Jx ,
∀y ∈ C.
2.12
Lemma 2.8 Su et al. 5. Let C be a nonempty, closed, and convex subset of a smooth, strictly
convex Banach space E. Let T be a weak relatively nonexpansive mapping of C into itself. Then FT
is closed and convex.
Also, this following lemma will be used in the sequel.
Lemma 2.9 Kamimura and Takahashi 4. Let C be a nonempty, closed, and convex subset of a
∞
smooth, uniformly convex Banach space E. Let {xn }∞
n1 and {yn }n1 be sequences in E such that either
∞
∞
{xn }n1 or {yn }n1 is bounded. If limn → ∞ φxn , yn 0, then limn → ∞ xn − yn 0.
Abstract and Applied Analysis
9
Lemma 2.10 Cho et al. 44. Let E be a uniformly convex real Banach space. For arbitrary r > 0,
let Br 0 : {x ∈ E : x ≤ r} and λ, μ, γ ∈ 0, 1 such that λ μ γ 1. Then, there exists a
continuous strictly increasing convex function
g : 0, 2r −→ Ê,
g0 0
2.13
such that for every x, y, z ∈ Br 0, the following inequality holds:
λx μy γ z2 ≤ λx2 μy2 − λμg x − y .
For solving the equilibrium problem for a bifunction F : C × C →
that F satisfies the following conditions:
2.14
Ê, let us assume
A1 Fx, x 0 for all x ∈ C;
A2 F is monotone, that is, Fx, y Fy, x ≤ 0 for all x, y ∈ C;
A3 for each x, y, z ∈ C, lim supt↓0 Ftz 1 − tx, y ≤ Fx, y;
A4 for each x ∈ C, y → Fx, y is convex and lower semicontinuous.
Lemma 2.11 Liu et al. 14 and Zhang 19. Let C be a nonempty, closed, and convex subset of
a smooth, strictly convex and reflexive Banach space E. Assume that F : C × C → Ê satisfies (A1)–
(A4), A : C → E∗ a continuous and monotone mapping, and ϕ : C → Ê a lower semicontinuous
and convex functional. For r > 0 and x ∈ E, there exists z ∈ C such that
1
Q z, y y − z, Jz − Jx ≥ 0,
r
∀y ∈ C,
2.15
where Qz, y Fz, y Az, y − z ϕy − ϕz, z, y ∈ C. Furthermore, define a mapping
Tr : E → C as follows:
Tr x 1
z ∈ C : Q z, y y − z, Jz − Jx ≥ 0, ∀y ∈ C .
r
2.16
Then, the following hold:
i Tr is single-valued;
ii for any x, y ∈ E,
Tr x − Tr y, JTr x − JTr y ≤ Tr x − Tr y, Jx − Jy ;
iii FTr GMEPF, A, ϕ;
iv GMEPF, A, ϕ is closed and convex.
2.17
10
Abstract and Applied Analysis
Lemma 2.12 Zhang 19. Let C be a nonempty, closed, and convex subset of a smooth, strictly
convex, and reflexive Banach space E. Assume that F : C × C → Ê satisfies (A1)–(A4), and let r > 0.
Then for each x ∈ E and q ∈ FTr ,
φ q, Tr x φTr x, x ≤ φ q, x .
2.18
For the rest of this paper, the sequence {xn }∞
n0 converges strongly to p shall be denoted
1 2 1
1
2
1
by xn → p as n → ∞, and we shall assume that βn , βn , βn ∈ 0, 1 such that βn βn βn 1, for all n ≥ 0.
3. Main Results
Theorem 3.1. Let E be a uniformly convex real Banach space which is also uniformly smooth. Let C be
a nonempty, closed, and convex subset of E. For each k 1, 2, . . . , m, let Fk be a bifunction from C × C
satisfying (A1)–(A4), Ak : C → E∗ a continuous and monotone mapping and ϕk : C → Ê a lower
∞
semicontinuous and convex functional. Suppose that {Tn }∞
n0 and {Sn }n0 are two countable families
of weak relatively nonexpansive mappings of C into itself such that Ω : ∩m
GMEPFk , Ak , ϕk ∩
k1
∞
FT
∩
∩
FS
∅.
Let
f
:
E
→
Ê
be
a
convex
and
lower
semicontinuous
mapping with
∩∞
/
n
n
n0
n0
is
iteratively
generated
by
x
∈
C,
C
C,
C ⊂ intDf, and suppose that {xn }∞
0
0
n0
1
2
3
zn J −1 βn Jxn βn JTn xn βn JSn xn ,
yn J −1 αn Jxn 1 − αn Jzn ,
Q
Q
Q
Q
m−1
· · · Tr2,n2 Tr1,n1 yn ,
un Trm,nm Trm−1,n
3.1
Cn1 {w ∈ Cn : Gw, Jun ≤ Gw, Jxn },
f
xn1 ΠCn1 x0 ,
n ≥ 0,
with the conditions
1 2
i lim infn → ∞ βn βn > 0;
1 3
ii lim infn → ∞ βn βn > 0;
iii 0 ≤ αn ≤ α < 1 for some α ∈ 0, 1;
iv {rk,n }∞
n1 ⊂ 0, ∞, k 1, 2, . . . , m satisfying lim infn → ∞ rk,n > 0, k 1, 2, . . . , m.
Then, {xn }∞
n0 converges strongly to ΠΩ x0 .
f
Abstract and Applied Analysis
11
∞
Proof. By Lemma 2.8, we know that ∩∞
n0 FTn ∩ ∩n0 FSn is closed and convex. We also
m
know from Lemma 2.11iv that ∩k1 GMEPFk , Ak , ϕk is closed and convex. Hence, Ω ∞
∞
∩m
k1 GMEPFk , Ak , ϕk ∩ ∩n0 FTn ∩ ∩n0 FSn is a nonempty, closed and convex subset
f
of C. Consequently, ΠΩ x0 is well defined. We first show that Cn , for all n ≥ 0 is closed and
convex. It is obvious that C0 C is closed and convex. Thus, we only need to show that Cn is
closed and convex for each n ≥ 1. Since Gz, Jun ≤ Gz, Jxn is equivalent to
3.2
2z, Jxn − z, Jun ≤ xn 2 − un 2 ,
f
this implies that Cn1 is closed and convex for all n ≥ 0. This shows that ΠCn1 x0 is well defined
Q
Q
Q
Q
k
2
1
Trk−1k−1,n · · · Tr2 ,n
Tr1 ,n
, k 1, 2, . . . , m and θn0 I for all n ≥ 1, we
for all n ≥ 0. By taking θnk Trk ,n
m
obtain un θn yn . We next show that Ω ⊂ Cn , for all n ≥ 0. For n 0, we have Ω ⊂ C C0 .
Then for each x∗ ∈ Ω, we obtain
Gx∗ , Jun ≤ G x∗ , Jθnm yn ≤ G x∗ , Jyn
Gx∗ , αn Jxn 1 − αn Jzn x∗ 2 − 2αn x∗ , Jxn − 21 − αn x∗ , Jzn αn Jxn 1 − αn Jzn 2 2ρfx∗ ≤ x∗ 2 − 2αn x∗ , Jxn − 21 − αn x∗ , Jzn αn Jxn 2 1 − αn Jzn 2 2ρfx∗ αn Gx∗ , Jxn 1 − αn Gx∗ , Jzn 1
2
3
αn Gx∗ , Jxn 1 − αn G x∗ , βn Jxn βn JTn xn βn JSn xn
1
2
≤ αn Gx∗ , Jxn 1 − αn x∗ 2 − 2βn x∗ , Jxn − 2βn x∗ , JTn xn 3
− 2βn x∗ , JSn xn 1
2
3
βn xn 2 βn Tn xn 2 βn Sn xn 2 2ρfx∗ 1
2
3
αn Gx∗ , Jxn 1 − αn βn Gx∗ , Jxn βn Gx∗ , JTn xn βn Gx∗ , JSnxn ≤ Gx∗ , Jxn .
3.3
Ω ⊂ Cn , for all n ≥ 0. It follows that {xn }∞
So, x∗ ∈ Cn . This implies that ∅ /
n0 is well defined
for all n ≥ 0.
12
Abstract and Applied Analysis
We now show that limn → ∞ Gxn , Jx0 exists. Since f : E → Ê is a convex and lower
semi-continuous, applying Lemma 2.5, we see that there exists u∗ ∈ E∗ and α ∈ Ê such that
f y ≥ y, u∗ α,
∀y ∈ E.
3.4
It follows that
Gxn , Jx0 xn 2 − 2xn , Jx0 x0 2 2ρfxn ≥ xn 2 − 2xn , Jx0 x0 2 2ρxn , u∗ 2ρα
xn 2 − 2xn , Jx0 − ρu∗ x0 2 2ρα
≥ xn 2 − 2xn Jx0 − ρu∗ x0 2 2ρα
2
2
xn − Jx0 − ρu∗ x0 2 − Jx0 − ρu∗ 2ρα.
3.5
f
Since xn ΠCn x0 , it follows from 3.5 that
2
2
Gx∗ , Jx0 ≥ Gxn , Jx0 ≥ xn − Jx0 − ρu∗ x0 2 − Jx0 − ρu∗ 2ρα
3.6
∞
for each x∗ ∈ Ω. This implies that {xn }∞
n0 is bounded and so is {Gxn , Jx0 }n0 . By the
f
construction of Cn , we have that Cm ⊂ Cn and xm ΠCm x0 ∈ Cn for any positive integer
m ≥ n. It then follows from Lemma 2.7 that
φxm , xn Gxn , Jx0 ≤ Gxm , Jx0 .
3.7
φxm , xn ≥ xm − xn 2 ≥ 0.
3.8
It is obvious that
In particular,
φxn1 , xn Gxn , Jx0 ≤ Gxn1 , Jx0 ,
φxn1 , xn ≥ xn1 − xn 2 ≥ 0,
3.9
∞
and so {Gxn , Jx0 }∞
n0 is nondecreasing. It follows that the limit of {Gxn , Jx0 }n0 exists. By
f
the fact that Cm ⊂ Cn and xm ΠCm x0 ∈ Cn for any positive integer m ≥ n, we obtain
φxm , un ≤ φxm , xn .
3.10
Abstract and Applied Analysis
13
Now, 3.7 implies that
φxm , un ≤ φxm , xn ≤ Gxm , Jx0 − Gxn , Jx0 .
3.11
Taking the limit as m, n → ∞ in 3.11, we obtain
lim φxm , xn 0.
3.12
n→∞
It then follows from Lemma 2.9 that xm − xn → 0 as m, n → ∞. Hence, {xn }∞
n0 is Cauchy.
Since E is a Banach space and C is closed and convex, then there exists p ∈ C such that xn → p
as n → ∞.
Now since φxm , xn → 0 as m, n → ∞ we have in particular that φxn1 , xn → 0 as
n → ∞ and this further implies that limn → ∞ xn1 − xn 0. Since xn1 ΠCn1 x0 ∈ Cn1 , we
have
φxn1 , un ≤ φxn1 , xn ,
∀n ≥ 0.
3.13
Then, we obtain
lim φxn1 , un 0.
n→∞
3.14
Since E is uniformly convex and smooth, we have from Lemma 2.9 that
lim xn1 − xn 0 lim xn1 − un .
3.15
xn − un ≤ xn1 − xn xn1 − un .
3.16
lim xn − un 0.
3.17
n→∞
n→∞
So,
Hence,
n→∞
Since J is uniformly norm-to-norm continuous on bounded sets and limn → ∞ xn − un 0,
we obtain
lim Jxn − un 0.
n→∞
3.18
14
Abstract and Applied Analysis
Since {xn } is bounded, so are {zn }, {JTn xn }, and {JSn xn }. Let r : supn≥0 {xn , Tn xn ,
Sn xn }. Then from Lemma 2.10, we have
Gx∗ , Jun ≤ G x∗ , Jθnm yn ≤ G x∗ , Jyn
Gx∗ , αn Jxn 1 − αn Jzn x∗ 2 − 2αn x∗ , Jxn − 21 − αn x∗ , Jzn αn Jxn 1 − αn Jzn 2 2ρfx∗ ≤ x∗ 2 − 2αn x∗ , Jxn − 21 − αn x∗ , Jzn αn Jxn 2 1 − αn Jzn 2 2ρfx∗ αn Gx∗ , Jxn 1 − αn Gx∗ , Jzn 1
2
3
αn Gx∗ , Jxn 1 − αn G x∗ , βn Jxn βn JTn xn βn JSn xn
1
2
≤ αn Gx∗ , Jxn 1 − αn x∗ 2 − 2βn x∗ , Jxn − 2βn x∗ , JTn xn 3
1
2
− 2βn x∗ , JSn xn βn xn 2 βn Tn xn 2
3
1 2
βn Sn xn 2 − βn βn gJxn − JTn xn 2ρfx∗ 1
2
αn Gx∗ , Jxn 1 − αn βn Gx∗ , Jxn βn Gx∗ , JTn xn 3
1 2
βn Gx∗ , JSnxn − βn βn gJxn − JTn xn 1
2
≤ αn Gx∗ , Jxn 1 − αn βn Gx∗ , Jxn βn Gx∗ , Jxn 3
1 2
βn Gx∗ , Jxn − βn βn gJxn − JTn xn 1 2
αn Gx∗ , Jxn 1 − αn Gx∗ , Jxn − βn βn gJxn − JTn xn 1 2
≤ Gx∗ , Jxn − 1 − αn βn βn gJxn − JTn xn .
3.19
It then follows that
1 2
1 2
1 − αβn βn gJxn − JTn xn ≤ 1 − αn βn βn gJxn − JTn xn ≤ Gx∗ , Jxn − Gx∗ , Jun .
3.20
Abstract and Applied Analysis
15
But
Gx∗ , Jxn − Gx∗ , Jun xn 2 − un 2 − 2x∗ , Jxn − Jun ≤ xn 2 − un 2 2|x∗ , Jxn − Jun |
≤ |xn − un |xn − un 2x∗ Jxn − Jun 3.21
≤ xn − un xn un 2x∗ Jxn − Jun .
From limn → ∞ xn − un 0 and limn → ∞ Jxn − Jun 0, we obtain
Gx∗ , Jxn − Gx∗ , Jun −→ 0,
n −→ ∞.
3.22
1 2
Using the condition lim infn → ∞ βn βn > 0, we have
lim gJxn − JTn xn 0.
3.23
n→∞
By property of g, we have limn → ∞ Jxn −JTn xn 0. Since J −1 is also uniformly norm-to-norm
continuous on bounded sets, we have
lim xn − Tn xn 0.
3.24
lim xn − Sn xn 0.
3.25
n→∞
Similarly, we can show that
n→∞
∞
Since xn → p and {Tn }, {Sn } are uniformly closed, we have p ∈ ∩∞
n0 FTn ∩ ∩n0 FSn .
Next, we show that p ∈ ∩m
k1 GMEPFk , Ak , ϕk . Now, by Lemma 2.12, we obtain
φ un , yn φ θnm yn , yn
≤ φ x∗ , yn − φ x∗ , θnm yn
≤ φx∗ , xn − φx∗ , un −→ 0,
3.26
n −→ ∞.
Using Lemma 2.9, we have limn → ∞ un − yn 0. Furthermore,
xn − yn ≤ xn − un un − yn −→ 0,
n −→ ∞.
3.27
16
Abstract and Applied Analysis
Since xn → p as n → ∞ and xn − yn → 0 as n → ∞, then yn → p as n → ∞. By the fact
that θnk , k 1, 2, . . . , m is relatively nonexpansive and using Lemma 2.12 again, we have that
φ θnk yn , yn ≤ φ x∗ , yn − φ x∗ , θnk yn
≤ φx∗ , xn − φ x∗ , θnk yn .
3.28
Observe that
φx∗ , un φ x∗ , θnm yn
Q
Qm−1
Q
Qk−1
Q Q
φ x∗ , Trm,nm Trm−1,n
· · · Trk,nk Trk−1,n
· · · Tr2,n2 Tr1,n1 yn
3.29
Q
Qm−1
φ x∗ , Trm,nm Trm−1,n
· · · θnk yn
≤ φ x∗ , θnk yn .
Using 3.29 in 3.28, we obtain
φ θnk yn , yn ≤ φx∗ , xn − φx∗ , un −→ 0,
n −→ ∞.
3.30
Then Lemma 2.9 implies that limn → ∞ yn − θnk yn 0, k 1, 2, . . . , m. Now
p − θnk yn ≤ yn − θnk yn yn − p −→ 0,
n −→ ∞, k 1, 2, . . . , m.
3.31
Similarly, limn → ∞ p − θnk−1 yn 0, k 1, 2, . . . , m. This further implies that
lim θnk yn − θnk−1 yn 0.
n→∞
3.32
Also, since J is uniformly norm-to-norm continuous on bounded sets and using 3.32, we
obtain
lim Jθnk yn − Jθnk−1 yn 0.
n→∞
3.33
Since lim infn → ∞ rk,n > 0, k 1, 2, . . . , m,
lim
n→∞
k
Jθn yn − Jθnk−1 yn rk,n
0.
3.34
Abstract and Applied Analysis
17
By Lemma 2.11, we have that for each k 1, 2, . . . , m
1 Qk θnk yn , y y − θnk yn , Jθnk yn − Jθnk−1 yn ≥ 0,
rk,n
∀y ∈ C.
3.35
Furthermore, using A2 we obtain
1 rk,n
y − θnk yn , Jθnk yn − Jθnk−1 yn ≥ Qk y, θnk yn .
3.36
By A4, 3.34, and θnk yn → p, we have for each k 1, 2, . . . , m
Qk y, p ≤ 0,
∀y ∈ C.
3.37
For fixed y ∈ C, let zt,y : ty 1 − tp for all t ∈ 0, 1. This implies that zt ∈ C. This yields
that Qk zt , p ≤ 0. It follows from A1 and A4 that
0 Qk zt , zt ≤ tQk zt , y 1 − tQk zt , p
≤ tQk zt , y
3.38
0 ≤ Qk zt , y .
3.39
and hence
From condition A3, we obtain
Qk p, y ≥ 0,
∀y ∈ C.
3.40
This implies that p ∈ GMEPFk , Ak , ϕk , k 1, 2, . . . , m. Thus, p ∈ ∩m
GMEPFk , Ak , ϕk .
k1
∞
∞
GMEPF
,
A
,
ϕ
∩
∩
FT
∩
∩
FS
.
Hence, we have p ∈ Ω ∩m
k
k
k
n
n
n0
n0
k1
f
GMEPFk , Ak , ϕk ∩ ∩∞
Finally, we show that p ΠΩ x0 . Since Ω ∩m
n0 FTn ∩
k1
f
∩∞
n0 FSn is a closed and convex set, from Lemma 2.6, we know that ΠΩ x0 is single valued
f
f
and denote w ΠΩ x0 . Since xn ΠCn x0 and w ∈ Ω ⊂ Cn , we have
Gxn , Jx0 ≤ Gw, Jx0 ,
∀n ≥ 0.
3.41
We know that Gξ, Jϕ is convex and lower semi-continuous with respect to ξ when ϕ is fixed.
This implies that
G p, Jx0 ≤ lim inf Gxn , Jx0 ≤ lim sup Gxn , Jx0 ≤ Gw, Jx0 .
n→∞
f
n→∞
From the definition of ΠΩ x0 and p ∈ Ω, we see that p w. This completes the proof.
3.42
18
Abstract and Applied Analysis
Corollary 3.2 Li et al. 38. Let E be a uniformly convex real Banach space which is also uniformly
smooth. Let C be a nonempty, closed, and convex subset of E. Suppose that T : C → C is a relatively
nonexpansive mapping of C into itself such that FT /
∅ and f : E → Ê is a convex and lower
semicontinuous mapping with C ⊂ intDf. Suppose that {xn }∞
n0 is iteratively generated by x0 ∈
C, C0 C
yn J −1 αn Jxn 1 − αn JTxn ,
Cn1 w ∈ Cn : G w, Jyn ≤ Gw, Jxn ,
f
xn1 ΠCn1 x0 ,
3.43
n ≥ 0,
where J is the duality mapping on E. Suppose that {αn }∞
n0 is a sequence in 0, 1 such that
f
lim supn → ∞ αn < 1. Then, {xn }∞
converges
strongly
to
Π
x .
n0
FT 0
Corollary 3.3. Let E be a uniformly convex real Banach space which is also uniformly smooth. Let C
be a nonempty, closed and convex subset of E. For each k 1, 2, . . . , m, let Fk be a bifunction from
C × C satisfying (A1)–(A4), Ak : C → E∗ a continuous and monotone mapping and ϕk : C → Ê a
∞
lower semicontinuous and convex functional. Suppose {Tn }∞
n0 and {Sn }n0 are two countable families
of weak relatively nonexpansive mappings of C into itself such that Ω : ∩m
GMEPFk , Ak , ϕk ∩
k1
∞
∞
FT
∩
∩
FS
∅.
Suppose
that
{x
}
is
iteratively
generated
by
x0 ∈ C, C0 C
∩∞
n
n /
n n0
n0
n0
1
2
3
zn J −1 βn Jxn βn JTn xn βn JSn xn ,
yn J −1 αn Jxn 1 − αn Jzn ,
Q
Q
Q
3.44
Q
m−1
un Trm,nm Trm−1,n
· · · Tr2,n2 Tr1,n1 yn ,
Cn1 w ∈ Cn : φw, un ≤ φw, xn ,
xn1 ΠCn1 x0 ,
n ≥ 0,
with the conditions
1 2
i lim infn → ∞ βn βn > 0;
1 3
ii lim infn → ∞ βn βn > 0;
iii 0 ≤ αn ≤ α < 1 for some α ∈ 0, 1;
iv {rk,n }∞
n1 ⊂ 0, ∞, k 1, 2, . . . , m satisfying lim infn → ∞ rk,n > 0, k 1, 2, . . . , m.
Then, {xn }∞
n0 converges strongly to ΠΩ x0 .
f
Proof. Take fx 0 for all x ∈ E in Theorem 3.1, Gξ, Jx φξ, x and ΠC x0 ΠC x0 . Then,
the desired conclusion follows.
Remark 3.4. Corollary 3.3 extends and improves on Theorem 1.5. In fact, the iterative
procedure 3.44 is simpler than 1.14 in the following two aspects: a the process of
computing Qn {w ∈ Cn−1 ∩ Qn−1 : xn − w, Jx0 − Jxn ≥ 0} is removed; b the process
of computing ΠCn ∩Qn is replaced by computing ΠCn .
Abstract and Applied Analysis
19
4. Applications
A mapping H from E to E∗ is said to be
i monotone if Hx − Hy, x − y ≥ 0, for all x, y ∈ E;
ii strictly monotone if H is monotone and Hx − Hy, x − y 0 if and only if x y;
iii β-Lipschitz continuous if there exists a constant β ≥ 0 such that Hx − Hy ≤
βx − y, for all x, y ∈ E.
∅} and
Let M be a set-valued mapping from E to E∗ with domain DM {z ∈ E : Mz /
range RM ∪{Mz : z ∈ DM}. A set-valued mapping M is said to be
i monotone if x1 − x2 , y1 − y2 ≥ 0 for each xi ∈ DM and yi ∈ Mxi , i 1, 2;
ii r-strongly monotone if x1 − x2 , y1 − y2 ≥ rx1 − x2 2 for each xi ∈ DM and
yi ∈ Mxi , i 1, 2;
iii maximal monotone if M is monotone and its graph GM : {x, y : y ∈ Mx} is
not properly contained in the graph of any other monotone operator;
iv a general H-monotone if M is monotone and H λME E∗ holds for every λ > 0,
where H is a mapping from E to E∗ .
We denote the set {x ∈ E : 0 ∈ Mx} by M−1 0. From Li et al. 38, we know that if H : E → E∗
∗
is strictly monotone and M : E → 2E is general H-monotone mapping, then M−1 0 is closed
and convex. Furthermore, for every λ > 0 and x∗ ∈ E∗ , there exists a unique x ∈ DM such
that x H λM−1 x∗ . Thus, we can define a single-value mapping Tλ : E → DM by
Tλ x H λM−1 Hx. It is obvious that M−1 0 FTλ for all λ > 0.
Lemma 4.1 Alber, 39. If E is a uniformly convex and uniformly smooth Banach space, δE is
the modulus of convexity of E, and ρE t is the modulus of smoothness of E, then the inequalities
2
8d δE
x − ξ
4d
hold for all x and ξ in E, where d ≤ φx, ξ ≤ 4d ρE
2
4x − ξ
d
4.1
x2 ξ2 /2.
Lemma 4.2 Xia and Huang 45. Let E be a Banach space with dual space E∗ , H : E → E∗ a
∗
strictly monotone mapping, and M : E → 2E a general H-monotone mapping. Then
i H λM−1 is a single-valued mapping;
ii if E is reflexive and M : E → 2E is r-strongly monotone, H λM−1 is Lipschitz
continuous with constant 1/λr, where r > 0.
∗
Theorem 4.3. Let E be a uniformly convex real Banach space which is also uniformly smooth with
δE ≥ k2 and ρE t ≤ ct2 for some k, c > 0. For each k 1, 2, . . . , m, let Fk be a bifunction from
E × E satisfying (A1)–(A4), Ak : E → E∗ a continuous and monotone mapping, and ϕk : E → Ê
a lower semicontinuous and convex functional. Suppose that H : E → E∗ is a strictly monotone
∗
and β-Lipschitz continuous mapping and Mi : E → 2E is a general H-monotone mapping and
ri -strongly monotone mapping with ri > 0, i 1, 2 such that Ω : ∩m
GMEPFk , Ak , ϕk ∩ M1−1 0 ∩
k1
M2−1 0 /
∅. Let TλMi H λMi −1 H, i 1, 2, let f : E → Ê a convex and lower semicontinuous
20
Abstract and Applied Analysis
mapping with Df E and suppose for each n ≥ 0 that there exists a λn > 0 such that 64cβ2 ≤
min{1/2kλ2n r12 , 1/2kλ2n r22 }. Let {xn }∞
n0 be iteratively generated by x0 ∈ E, C0 E,
1
2
3
zn J −1 βn Jxn βn JTλMn 1 xn βn JTλMn 2 xn ,
yn J −1 αn Jxn 1 − αn Jzn ,
Q
Q
Q
Q
m−1
· · · Tr2,n2 Tr1,n1 yn ,
un Trm,nm Trm−1,n
Cn1 w ∈ Cn : G w, Jyn ≤ Gw, Jxn ,
f
xn1 ΠCn1 x0 ,
4.2
n ≥ 0,
with the conditions
1 2
i lim infn → ∞ βn βn > 0;
1 3
ii lim infn → ∞ βn βn > 0;
iii 0 ≤ αn ≤ α < 1;
iv lim infn → ∞ λn > 0;
v {rk,n }∞
n1 ⊂ 0, ∞, k 1, 2, . . . , m satisfying lim infn → ∞ rk,n > 0, k 1, 2, . . . , m.
Then, {xn }∞
n0 converges strongly to ΠΩ x0 .
f
Proof. We only need to prove that {TλMn 1 } and {TλMn 2 } are countable families of weak
M1
−1
relatively nonexpansive mappings with common fixed points sets ∩∞
n0 FTλn M1 0 and
M2
M1
−1
−1
∞
∅. Secondly, we show
∩∞
n0 FTλn M2 0, respectively. Firstly, we have ∩n0 FTλn M1 0 /
that φp, TλMn 1 w ≤ φp, w, for all w ∈ E, p ∈ FTλMn 1 , n ≥ 0. Now, by Lemma 4.2 and the
Lipschitz continuity of H, we have
M1
M
Tλn p − Tλn 1 w H λn M1 −1 Hp − H λn M1 −1 Hw
≤
1 Hp − Hw
λn r1
≤
β p − w.
λn r1
4.3
Abstract and Applied Analysis
21
By 4.3 and Lemma 4.1,
φ p, TλMn 1 w φ TλMn 1 p, TλMn 1 w
⎞
⎛ 4TλMn 1 p − TλMn 1 w
2
⎠
≤ 4d ρE ⎝
d
2
≤ 64cTλMn 1 p − TλMn 1 w
≤
64cβ2 λ2n r12
φ p, w ≥ 8d2 δE
4.4
2
p − w ,
p − w 4d
≥
2
1 k p − w .
2
Since 64cβ2 ≤ 1/2kλ2n r12 , it follows from 4.4 that φp, TλMn 1 w ≤ φp, w, for all w ∈ E,
M1
M1
−1
} ∩∞
p ∈ FTλMn 1 , n ≥ 0. Thirdly, we show that F{T
n0 FTλn M1 0. We first show
λn
M1
M1
} ⊂ M−1 0. Let p ∈ F{T
}, then there exists {xn } ⊂ E such that xn → p and
that F{T
1
λn
λn
limn → ∞ xn − TλMn 1 xn 0. Since H is β-Lipschitz continuous,
M
M
Hxn − HTλn 1 xn ≤ βxn − Tλn 1 xn .
4.5
1
Hxn − HTλMn 1 xn −→ 0.
λn
4.6
Letting n → ∞, we obtain
It follows from 1/λn Hxn − HTλMn 1 xn ∈ M1 TλMn 1 and the monotonicity of M1 that
x − TλMn 1 , x∗ −
1
Hxn − HTλMn 1 xn
λn
≥0
4.7
for all ∈ DM1 and x∗ ∈ M1 x. Taking the limit as n → ∞, we obtain
x − p, x∗ ≥ 0
4.8
for all ∈ DM1 and x∗ ∈ M1 x. By the maximality of M1 , we know that p ∈ M1−1 0. On
M1 for all n ≥ 0, therefore,
the other hand, we know that FTλMn 1 M1−1 0, FTλMn 1 ⊂ FT
λn
∞ T M1 . Thus, we have proved that {T M1 } is a countable family of
M−1 0 ∩∞ FT M1 F∩
1
n0
λn
n0 λn
λn
M1
−1
weak relatively nonexpansive mappings with common fixed points sets ∩∞
n0 FTλn M1 0.
By following the same arguments, we can show that {TλMn 2 } is a countable family of weak
M2
−1
relatively nonexpansive mappings with common fixed points sets ∩∞
n0 FTλn M2 0.
22
Abstract and Applied Analysis
Let E be a uniformly convex and uniformly smooth Banach space, H J and M a
maximal monotone mapping. Then, we can define Jλ JλM−1 J for all λ > 0. We know that
Jλ is relatively nonexpansive and therefore weak relatively nonexpansive and M−1 0 FJλ for all λ > 0 see, e.g., 2, where FJλ denotes the fixed points set of Jλ . By Corollary 3.3,
we obtain the following theorem.
Theorem 4.4. Let E be a uniformly convex real Banach space which is also uniformly smooth. For each
k 1, 2, . . . , m, let Fk be a bifunction from E × E satisfying (A1)–(A4), Ak : E → E∗ a continuous
and monotone mapping, and ϕk : E → Ê a lower semicontinuous and convex functional. For each
i 1, 2, let Mi ⊂ E × E∗ a maximal monotone operator, and let JλMi J λMi −1 J for all λ > 0, and
suppose C is a nonempty closed and convex subset of E such that DMi ⊂ C ⊂ J −1 ∩λ>0 RJ λMi ,
GMEPFk , Ak , ϕk ∩ M1−1 0 ∩ M2−1 0 /
∅ and let {xn }∞
i 1, 2. Assume that Ω : ∩m
n0 be iteratively
k1
generated by x0 ∈ E, C0 E,
1
2
3
zn J −1 βn Jxn βn JJTλMn 1 xn βn JJTλMn 2 xn ,
yn J −1 αn Jxn 1 − αn Jzn ,
Q
Q
Q
Q
m−1
un Trm,nm Trm−1,n
· · · Tr2,n2 Tr1,n1 yn ,
Cn1 w ∈ Cn : φ w, yn ≤ φw, xn ,
xn1 ΠCn 1 x0 ,
4.9
n ≥ 0,
with the conditions
1 2
i lim infn → ∞ βn βn > 0;
1 3
ii lim infn → ∞ βn βn > 0;
iii 0 ≤ αn ≤ α < 1;
iv lim infn → ∞ λn > 0;
v {rk,n }∞
n1 ⊂ 0, ∞, k 1, 2, . . . , m satisfying lim infn → ∞ rk,n > 0, k 1, 2, . . . , m.
Then, {xn }∞
n0 converges strongly to ΠΩ x0 .
Acknowledgments
The author would like to express his thanks to the referees for their valuable suggestions. This
research work is dedicated to Professor Isaac U. Asuzu of the University of Nigeria, Nsukka.
References
1 C. Chidume, Geometric Properties of Banach Spaces and Nonlinear Iterations, vol. 1965 of Lecture Notes in
Mathematics, Springer, London, UK, 2009.
2 W. Takahashi, Nonlinear Functional Analysis-Fixed Point Theory and Applications, Yokohama Publishers,
Yokohama, Japan, 2000.
3 W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000.
4 S. Kamimura and W. Takahashi, “Strong convergence of a proximal-type algorithm in a Banach
space,” SIAM Journal on Optimization, vol. 13, no. 3, pp. 938–945, 2002.
Abstract and Applied Analysis
23
5 Y. Su, H.-K. Xu, and X. Zhang, “Strong convergence theorems for two countable families of
weak relatively nonexpansive mappings and applications,” Nonlinear Analysis: Theory, Methods &
Applications, vol. 73, no. 12, pp. 3890–3906, 2010.
6 D. Butnariu, S. Reich, and A. J. Zaslavski, “Asymptotic behavior of relatively nonexpansive operators
in Banach spaces,” Journal of Applied Analysis, vol. 7, no. 2, pp. 151–174, 2001.
7 D. Butnariu, S. Reich, and A. J. Zaslavski, “Weak convergence of orbits of nonlinear operators in
reflexive Banach spaces,” Numerical Functional Analysis and Optimization, vol. 24, no. 5-6, pp. 489–508,
2003.
8 Y. Censor and S. Reich, “Iterations of paracontractions and firmly nonexpansive operators with
applications to feasibility and optimization,” Optimization, vol. 37, no. 4, pp. 323–339, 1996.
9 S. Matsushita and W. Takahashi, “A strong convergence theorem for relatively nonexpansive
mappings in a Banach space,” Journal of Approximation Theory, vol. 134, no. 2, pp. 257–266, 2005.
10 W. Takahashi and K. Zembayashi, “Strong convergence theorem by a new hybrid method for
equilibrium problems and relatively nonexpansive mappings,” Fixed Point Theory and Applications,
vol. 2008, Article ID 528476, 11 pages, 2008.
11 X. Qin and Y. Su, “Strong convergence theorems for relatively nonexpansive mappings in a Banach
space,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 6, pp. 1958–1965, 2007.
12 J. Kang, Y. Su, and X. Zhang, “Hybrid algorithm for fixed points of weak relatively nonexpansive
mappings and applications,” Nonlinear Analysis: Hybrid Systems, vol. 4, no. 4, pp. 755–765, 2010.
13 H. Zegeye and N. Shahzad, “Strong convergence theorems for monotone mappings and relatively
weak nonexpansive mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 7, pp.
2707–2716, 2009.
14 M. Liu, S. Chang, and P. Zuo, “On a hybrid method for generalized mixed equilibrium problem and
fixed point problem of a family of quasi-φ-asymptotically nonexpansive mappings in Banach spaces,”
Fixed Point Theory and Applications, vol. 2010, Article ID 157278, 18 pages, 2010.
15 N. Petrot, K. Wattanawitoon, and P. Kumam, “A hybrid projection method for generalized mixed
equilibrium problems and fixed point problems in Banach spaces,” Nonlinear Analysis: Hybrid Systems,
vol. 4, no. 4, pp. 631–643, 2010.
16 S. Saewan, P. Kumam, and K. Wattanawitoon, “Convergence theorem based on a new hybrid
projection method for finding a common solution of generalized equilibrium and variational
inequality problems in Banach spaces,” Abstract and Applied Analysis, vol. 2010, Article ID 734126,
25 pages, 2010.
17 S. Saewan and P. Kumam, “Modified hybrid block iterative algorithm for convex feasibility
problems and generalized equilibrium problems for uniformly quasi-φ-asymptotically nonexpansive
mappings,” Abstract and Applied Analysis, vol. 2010, Article ID 357120, 22 pages, 2010.
18 P. Kumam and S. Saewan, “A hybrid iterative scheme for a maximal monotone operator and two
countable families of relatively quasi-nonexpansive mappings for generalized mixed equilibrium and
variational inequality problems,” Abstract and Applied Analysis, vol. 2010, Article ID 123027, 31 pages,
2010.
19 S.-S. Zhang, “Generalized mixed equilibrium problem in Banach spaces,” Applied Mathematics and
Mechanics. English Edition, vol. 30, no. 9, pp. 1105–1112, 2009.
20 P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of
Nonlinear and Convex Analysis, vol. 6, no. 1, pp. 117–136, 2005.
21 Y. Liu, “A general iterative method for equilibrium problems and strict pseudo-contractions in Hilbert
spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 10, pp. 4852–4861, 2009.
22 A. Moudafi, “Weak convergence theorems for nonexpansive mappings and equilibrium problems,”
Journal of Nonlinear and Convex Analysis, vol. 9, no. 1, pp. 37–43, 2008.
23 S. Plubtieng and R. Punpaeng, “A new iterative method for equilibrium problems and fixed
point problems of nonexpansive mappings and monotone mappings,” Applied Mathematics and
Computation, vol. 197, no. 2, pp. 548–558, 2008.
24 X. Qin, M. Shang, and Y. Su, “Strong convergence of a general iterative algorithm for equilibrium
problems and variational inequality problems,” Mathematical and Computer Modelling, vol. 48, no. 7-8,
pp. 1033–1046, 2008.
25 Y. Su, M. Shang, and X. Qin, “An iterative method of solution for equilibrium and optimization
problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 8, pp. 2709–2719, 2008.
26 W. Takahashi and K. Zembayashi, “Strong and weak convergence theorems for equilibrium problems
and relatively nonexpansive mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods &
Applications, vol. 70, no. 1, pp. 45–57, 2009.
24
Abstract and Applied Analysis
27 S. Takahashi and W. Takahashi, “Viscosity approximation methods for equilibrium problems and
fixed point problems in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 331, no.
1, pp. 506–515, 2007.
28 R. Wangkeeree, “An extragradient approximation method for equilibrium problems and fixed point
problems of a countable family of nonexpansive mappings,” Fixed Point Theory and Applications, vol.
2008, Article ID 134148, 17 pages, 2008.
29 X. Qin, Y. J. Cho, and S. M. Kang, “Viscosity approximation methods for generalized equilibrium
problems and fixed point problems with applications,” Nonlinear Analysis: Theory, Methods &
Applications, vol. 72, no. 1, pp. 99–112, 2010.
30 Y. Shehu, “Fixed point solutions of generalized equilibrium problems for nonexpansive mappings,”
Journal of Computational and Applied Mathematics, vol. 234, no. 3, pp. 892–898, 2010.
31 S. Takahashi and W. Takahashi, “Strong convergence theorem for a generalized equilibrium problem
and a nonexpansive mapping in a Hilbert space,” Nonlinear Analysis: Theory, Methods & Applications,
vol. 69, no. 3, pp. 1025–1033, 2008.
32 J.-W. Peng and J.-C. Yao, “Strong convergence theorems of iterative scheme based on the extragradient
method for mixed equilibrium problems and fixed point problems,” Mathematical and Computer
Modelling, vol. 49, no. 9-10, pp. 1816–1828, 2009.
33 S. Plubtieng and K. Sombut, “Weak convergence theorems for a system of mixed equilibrium
problems and nonspreading mappings in a Hilbert space,” Journal of Inequalities and Applications, vol.
2010, Article ID 246237, 12 pages, 2010.
34 Y. Yao, Y.-C. Liou, and J.-C. Yao, “A new hybrid iterative algorithm for fixed-point problems, variational inequality problems, and mixed equilibrium problems,” Fixed Point Theory and Applications,
vol. 2008, Article ID 417089, 15 pages, 2008.
35 E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The
Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994.
36 L.-C. Ceng and J.-C. Yao, “A hybrid iterative scheme for mixed equilibrium problems and fixed point
problems,” Journal of Computational and Applied Mathematics, vol. 214, no. 1, pp. 186–201, 2008.
37 S. Plubtieng and K. Ungchittrakool, “Strong convergence theorems for a common fixed point of two
relatively nonexpansive mappings in a Banach space,” Journal of Approximation Theory, vol. 149, no. 2,
pp. 103–115, 2007.
38 X. Li, N.-J. Huang, and D. O’Regan, “Strong convergence theorems for relatively nonexpansive
mappings in Banach spaces with applications,” Computers & Mathematics with Applications, vol. 60,
no. 5, pp. 1322–1331, 2010.
39 Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and
applications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol.
178 of Lecture Notes in Pure and Applied Mathematics, pp. 15–50, Dekker, New York, NY, USA, 1996.
40 Y. I. Alber and S. Reich, “An iterative method for solving a class of nonlinear operator equations in
Banach spaces,” Panamerican Mathematical Journal, vol. 4, no. 2, pp. 39–54, 1994.
41 I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of
Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990.
42 K. Wu and N. Huang, “The generalised f -projection operator with an application,” Bulletin of the
Australian Mathematical Society, vol. 73, no. 2, pp. 307–317, 2006.
43 J. Fan, X. Liu, and J. Li, “Iterative schemes for approximating solutions of generalized variational
inequalities in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 11, pp.
3997–4007, 2009.
44 Y. J. Cho, H. Zhou, and G. Guo, “Weak and strong convergence theorems for three-step iterations
with errors for asymptotically nonexpansive mappings,” Computers & Mathematics with Applications,
vol. 47, no. 4-5, pp. 707–717, 2004.
45 F.-Q. Xia and N.-J. Huang, “Variational inclusions with a general H-monotone operator in Banach
spaces,” Computers & Mathematics with Applications, vol. 54, no. 1, pp. 24–30, 2007.