Báo cáo hóa học: " Research Article Iterative Schemes for Generalized Equilibrium Problem and Two Maximal Monotone Operators" pdf
Bạn đang xem bản rút gọn của tài liệu. Xem và tải ngay bản đầy đủ của tài liệu tại đây (640.37 KB, 34 trang )
Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 896252, 34 pages
doi:10.1155/2009/896252
Research Article
Iterative Schemes for Generalized Equilibrium
Problem and Two Maximal Monotone Operators
L. C. Zeng,
1, 2
Y. C. Lin,
3
and J. C. Yao
4
1
Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
2
Science Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China
3
Department of Occupational Safety and Health, China Medical University, Taichung 404, Taiwan
4
Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan
Correspondence should be addressed to J. C. Yao,
Received 20 July 2009; Accepted 27 October 2009
Recommended by Yeol Je Cho
The purpose of this paper is to introduce and study two new hybrid proximal-point algorithms
for finding a common element of the set of solutions to a generalized equilibrium problem and the
sets of zeros of two maximal monotone operators in a uniformly smooth and uniformly convex
Banach space. We established strong and weak convergence theorems for these two modified
hybrid proximal-point algorithms, respectively.
Copyright q 2009 L. C. Zeng et al. 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.
1. Introduction
Let X be a real Banach space and X
∗
its dual space. The normalized duality mapping J : X →
2
X
∗
is defined as
J
x
:
x
∗
∈ X
∗
:
x
∗
,x
x
2
x
∗
2
, ∀x ∈ X, 1.1
where ·, · denotes the generalized duality pairing. Recall that if X is a smooth Banach space
then J is singlevalued. Throughout this paper, we will still denote by J the single-valued
normalized duality mapping. Let C be a nonempty closed convex subset of X, f a bifunction
from C × C to R,andA : C → X
∗
a nonlinear mapping. The generalized equilibrium problem
is to find x ∈ C such that
f
x, y
Ax, y − x
≥ 0, ∀y ∈ C. 1.2
2 Journal of Inequalities and Applications
The set of solutions of 1.2 is denoted by EP. Problem 1.2 and similar problems have been
extensively studied; see, for example, 1–11. Whenever A 0, problem 1.2 reduces to the
equilibrium problem of finding x ∈ C such that
f
x, y
≥ 0, ∀y ∈ C. 1.3
The set of solutions of 1.3 is denoted by EPf. Whenever f 0, problem 1.2 reduces to
the variational inequality problem of finding x ∈ C such that
Ax, y − x
≥ 0, ∀y ∈ C. 1.4
Whenever X H a Hilbert space, problem 1.2 was very recently introduced and
considered by Kamimura and Takahashi 12
. Problem 1.2 is very general in the sense that it
includes, as spacial cases, optimization problems, variational inequalities, minimax problems,
the Nash equilibrium problem in noncooperative games, and others; see, for example, 13,
14. A mapping S : C → X is called nonexpansive if Sx − Sy≤x − y for all x, y ∈ C.
Denote by FS the set of fixed points of S,thatis,FS{x ∈ C : Sx x}. Iterative schemes
for finding common elements of EP and fixed points set of nonexpansive mappings have
been studied recently; see, for example, 12, 15–17 and the references therein.
On the other hand, a classical method of solving 0 ∈ Tx in a Hilbert space H is the
proximal point algorithm which generates, for any starting point x
0
∈ H, a sequence {x
n
} in
H by the iterative scheme
x
n1
J
r
n
x
n
,n 0, 1, 2, , 1.5
where {r
n
} is a sequence in 0, ∞, J
r
I rT
−1
for each r>0 is the resolvent operator for
T,andI is the identity operator on H. This algorithm was first introduced by Martinet 14
and generally studied by Rockafellar 18 in the framework of a Hilbert space H. Later many
authors studied 1.5 and its variants in a Hilbert space H or in a Banach space X; see, for
example, 13, 19–23 and the references therein.
Let X be a uniformly smooth and uniformly convex Banach space and let C be a
nonempty closed convex subset of X.Letf be a bifunction from C × C to R satisfying the
following conditions A1–A4 which were imposed in 24:
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, f x, · is convex and lower semicontinuous.
Let T : X → 2
X
∗
be a maximal monotone operator such that
A5 T
−1
0 ∩ EPf
/
∅.
The purpose of this paper is to introduce and study two new iterative algorithms
for finding a common element of the set EP of solutions for the generalized equilibrium
problem 1.2 and the set T
−1
0 ∩
T
−1
0 for maximal monotone operators T,
T in a uniformly
smooth and uniformly convex Banach space X. First, motivated by Kamimura and Takahashi
Journal of Inequalities and Applications 3
12, Theorem 3.1 , Ceng et al. 16, Theorem 3.1 ,andZhang17, Theorem 3.1 ,we
introduce a sequence {x
n
} that, under some appropriate conditions, is strongly convergent
to Π
T
−1
0∩
T
−1
0∩EP
x
0
in Section 3. Second, inspired by Kamimura and Takahashi 12, Theorem
3.1 , Ceng et al. 16, Theorem 4.1 ,andZhang17, Theorem 3.1 , we define a sequence
weakly convergent to an element z ∈ T
−1
0 ∩
T
−1
0 ∩ EP, where z lim
n →∞
Π
T
−1
0∩
T
−1
0∩EP
x
n
in
Section 4. Our results represent a generalization of known results in the literature, including
Takahashi and Zembayashi 15, Kamimura and Takahashi 12, Li and Song 22, Ceng and
Yao 25, and Ceng et al. 16. In particular, compared with Theorems 3.1and4.1in16, our
results i.e., Theorems 3.2 and 4.2 in this paper extend the problem of finding an element of
T
−1
0 ∩ EPf to the one of finding an element of T
−1
0 ∩
T
−1
0 ∩ EP. Meantime, the algorithms
in this paper are very different from those in 16because of considering the complexity
involving the problem of finding an element of T
−1
0 ∩
T
−1
0 ∩ EP.
2. Preliminaries
In the sequel, we denote the strong convergence, weak convergence and weak
∗
convergence
of a sequence {x
n
} to a point x ∈ X by x
n
→ x, x
n
xand x
n
∗
x, respectively.
A Banach space X is said to be strictly convex, if x y/2 < 1 for all x, y ∈ U {z ∈
X : z 1} with x
/
y. X is said to be uniformly convex if for each ∈ 0, 2 there exists
δ>0 such that x y/2 ≤ 1 − δ for all x, y ∈ U with x − y≥. Recall that each uniformly
convex Banach space has the Kadec-Klee property, that is,
x
n
x
x
n
−→
x
⇒ x
n
−→ x 2.1
The proof of the main results of Sections 3 and 4 will be based on the following
assumption.
Assumption A. Let X be a uniformly smooth and uniformly convex Banach space and let C
be a nonempty closed convex subset of X.Letf be a bifunction from C × C to R satisfying
the same conditions A1–A4 as in Section 1.LetT,
T : X → 2
X
∗
be two maximal monotone
operators such that
A5
T
−1
0 ∩
T
−1
0 ∩ EP
/
∅.
Recall that if C is a nonempty closed convex subset of a Hilbert space H, then the
metric projection P
C
: H → C of H onto C is nonexpansive. This fact actually characterizes
Hilbert spaces and hence, it is not available in more general Banach spaces. In this connection,
Alber 26 recently introduced a generalized projection operator Π
C
in a Banach space X
which is an analogue of the metric projection in Hilbert spaces. Consider the functional
defined as in 26 by
φ
x, y
x
2
− 2
x, Jy
y
2
, ∀x, y ∈ X.
2.2
It is clear that in a Hilbert space H, 2.2 reduces to φx, yx − y
2
, ∀x, y ∈ H.
4 Journal of Inequalities and Applications
The generalized projection Π
C
: X → C is a mapping that assigns to an arbitrary point
x ∈ X the minimum point of the functional φy, x;thatis,Π
C
x x, where x is the solution
to the minimization problem
φ
x, x
min
y∈C
φ
y, x
.
2.3
The existence and uniqueness of the operator Π
C
follows from the properties of the
functional φx, y and strict monotonicity of the mapping J see, e.g., 27. In a Hilbert space,
Π
C
P
C
.From26, in a smooth strictly convex and reflexive Banach space X, we have
y
−
x
2
≤ φ
y, x
≤
y
x
2
, ∀x, y ∈ X.
2.4
Moreover, by the property of subdifferential of convex functions, we easily get the
following inequality:
φ
x, y
≤ φ
x, J
−1
Jy Jz
− 2
y − x, Jz
, ∀x, y, z ∈ X. 2.5
Let S be a mapping from C into itself. A point p in C is called an asymptotically fixed
point of S if C contains a sequence {x
n
} which converges weakly to p such that Sx
n
− x
n
→ 0
28. The set of asymptotically fixed points of S will be denoted by
FS. A mapping C from
S into itself is called relatively nonexpansive if
FSFS and φp, Sx ≤ φp, x, for all
x ∈ C and p ∈ FS15.
Observe that, if X is a reflexive strictly convex and smooth Banach space, then for any
x, y ∈ X, φx, y0 if and only if x y. To this end, it is sufficient to show that if φx, y0
then x y. Actually, from 2.4, we have x y which implies that x, Jy x
2
y
2
.
From the definition of J, we have Jx Jy and therefore, x y;see29 for more details.
We need the following lemmas for the proof of our main results.
Lemma 2.1 Kamimura and Takahashi 12. Let X be a smooth and uniformly convex Banach
space and let {x
n
} and {y
n
} be two sequences of X.Ifφx
n
,y
n
→ 0 and either {x
n
} or {y
n
} is
bounded, then x
n
− y
n
→ 0.
Lemma 2.2 Alber 26, Kamimura and Takahashi 12. Let C be a nonempty closed convex
subset of a smooth strictly convex and reflexive Banach space X.Letx ∈ X and let z ∈ C.Then
z Π
C
x ⇐⇒
y − z, Jx − Jz
≤ 0, ∀y ∈ C. 2.6
Lemma 2.3 Alber 26, Kamimura and Takahashi 12. Let C be a nonempty closed convex
subset of a smooth strictly convex and reflexive Banach space X.Then
φ
x, Π
C
y
φ
Π
C
y, y
≤ φ
x, y
, ∀x ∈ C, y ∈ X. 2.7
Journal of Inequalities and Applications 5
Lemma 2.4 Rockafellar 18. Let X be a reflexive strictly convex and smooth Banach space and let
T : X → 2
X
∗
be a multivalued operator. Then there hold the following hold:
i T
−1
0 is closed and convex if T is maximal monotone such that T
−1
0
/
∅;
ii T is maximal monotone if and only if T is monotone with RJ rTX
∗
for all r>0.
Lemma 2.5 Xu 30. Let X 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 − t
y
2
≤ t
x
2
1 − t
y
2
− t
1 − t
g
x − y
,
2.8
for all x, y ∈ B
r
and t ∈ 0, 1,whereB
r
{z ∈ X : z≤r}.
Lemma 2.6 Kamimura and Takahashi 12. Let X be a smooth and 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
g
x − y
≤ φ
x, y
, ∀x, y ∈ B
r
. 2.9
The following result is due to Blum and Oettli 24.
Lemma 2.7 Blum and Oettli 24. Let C be a nonempty closed convex subset of a smooth strictly
convex and reflexive Banach space X,letf be a bifunction from C × C to R satisfying (A1)–(A4), and
let r>0 and x ∈ X. Then, there exists z ∈ C such that
f
z, y
1
r
y − z, Jz − Jx
≥ 0, ∀y ∈ C.
2.10
Motivated by Combettes and Hirstoaga 31 in a Hilbert space, Takahashi and
Zembayashi 15 established the following lemma.
Lemma 2.8 Takahashi and Zembayashi 15. Let C be a nonempty closed convex subset of a
smooth strictly convex and reflexive Banach space X, and let f be a bifunction from C × C to R
satisfying (A1)–(A4). For r>0 and x ∈ X, define a mapping T
r
: X → C as follows:
T
r
x
z ∈ C : f
z, y
1
r
y − z, Jz − Jx
≥ 0, ∀y ∈ C
2.11
for all x ∈ X. Then, the following hold:
i T
r
is singlevalued;
ii T
r
is a firmly nonexpansive-type mapping, that is, for all x, y ∈ X,
T
r
x − T
r
y, JT
r
x − JT
r
y
≤
T
r
x − T
r
y, Jx − Jy
; 2.12
6 Journal of Inequalities and Applications
iii FT
r
FT
r
EPf;
iv EPf is closed and convex.
Using Lemma 2.8, one has the following result.
Lemma 2.9 Takahashi and Zembayashi 15. Let C be a nonempty closed convex subset of a
smooth strictly convex and reflexive Banach space X,letf be a bifunction from C × C to R satisfying
(A1)–(A4), and let r>0. Then, for x ∈ X and q ∈ FT
r
,
φ
q, T
r
x
φ
T
r
x, x
≤ φ
q, x
. 2.13
Utilizing Lemmas 2.7, 2.8 and 2.9 as previously mentioned, Zhang 17 derived the
following result.
Proposition 2.10 Zhang 21, Lemma . Let X be a smooth strictly convex and reflexive Banach
space and let C be a nonempty closed convex subset of X.LetA : C → X
∗
be an α -inverse-strongly
monotone mapping, let f be a bifunction from C × C to R satisfying (A1)–(A4), and let r>0.Then
the following hold:
for x ∈ X, there exists u ∈ C such that
f
u, y
Au, y − u
1
r
y − u, Ju − Jx
≥ 0, ∀y ∈ C,
2.14
if X is additionally uniformly smooth and K
r
: X → C is defined as
K
r
x
u ∈ C : f
u, y
Au, y − u
1
r
y − u, Ju − Jx
≥ 0, ∀y ∈ C
, ∀x ∈ X, 2.15
then the mapping K
r
has the following properties:
i K
r
is singlevalued,
ii K
r
is a firmly nonexpansive-type mapping, that is,
K
r
x − K
r
y, JK
r
x − JK
r
y
≤
K
r
x − K
r
y, Jx − Jy
, ∀x, y ∈ X, 2.16
iii FK
r
FK
r
EP,
iv EP is a closed convex subset of C,
v φp, K
r
xφK
r
x, x ≤ φp, x, for all p ∈ FK
r
.
Let T,
T : X → 2
X
∗
be two maximal monotone operators in a smooth Banach space X.We
denote the resolvent operators of T and
T by J
r
J rT
−1
J and
J
r
J r
TJ for each r>0,
respectively. Then J
r
: X → DT and
J
r
: X → D
T are two single-valued mappings. Also,
T
−1
0 FJ
r
and
T
−1
0 F
J
r
for each r>0,whereFJ
r
and F
J
r
are the sets of fixed points
of J
r
and
J
r
, respectively. For each r>0, the Yosida approximations of T and
T are defined by A
r
J − JJ
r
/r and
A
r
J − J
J
r
/r, respectively. It is known that
A
r
x ∈ T
J
r
x
,
A
r
x ∈
T
J
r
x
, ∀r>0,x∈ X. 2.17
Journal of Inequalities and Applications 7
Lemma 2.11 Kohsaka and Takahashi 13. Let X be a reflexive strictly convex and smooth
Banach space and let T : X → 2
X
∗
be a maximal monotone operator with T
−1
0
/
∅.Then
φ
z, J
r
x
φ
J
r
x, x
≤ φ
z, x
, ∀r>0,z∈ T
−1
0,x∈ X.
2.18
Lemma 2.12 Tan and Xu 32. Let {a
n
} and {b
n
} be two sequences of nonnegative real numbers
satisfying the inequality: a
n1
≤ a
n
b
n
for all n ≥ 0.If
∞
n0
b
n
< ∞,thenlim
n →∞
a
n
exists.
3. Strong Convergence Theorem
In this section, we prove a strong convergence theorem for finding a common element of the
set of solutions for a generalized equilibrium problem and the set T
−1
0∩
T
−1
0 for two maximal
monotone operators T and
T.
Lemma 3.1. Let X be a reflexive strictly convex and smooth Banach space and let T : X → 2
X
∗
be a
maximal monotone operator. Then for each r ∈ 0, ∞, the following holds:
Ju − Jv,J
r
u − J
r
v
≥
JJ
r
u − JJ
r
v, J
r
u − J
r
v
, ∀u, v ∈ X, 3.1
where J
r
J rT
−1
J and J is the duality mapping on X. In particular, whenever X H areal
Hilbert space, J
r
is a nonexpansive mapping on H.
Proof. Since for each u, v ∈ X
J
r
u
J rT
−1
Ju, J
r
v
J rT
−1
Jv,
3.2
we have that
1
r
·
Ju − JJ
r
u
∈ TJ
r
u,
1
r
·
Jv − JJ
r
v
∈ TJ
r
v.
3.3
Thus, from the monotonicity of T it follows that
1
r
·
Ju − JJ
r
u
−
1
r
·
Jv − JJ
r
v
,J
r
u − J
r
v
≥ 0, 3.4
and hence
Ju − Jv,J
r
u − J
r
v
≥
JJ
r
u − JJ
r
v, J
r
u − J
r
v
. 3.5
Theorem 3.2. Suppose that Assumption A is fulfilled and let x
0
∈ X be chosen arbitrarily. Consider
the sequence
x
n1
Π
H
n
∩W
n
x
0
,n 0, 1, 2, , 3.6
8 Journal of Inequalities and Applications
where
H
n
z ∈ C : φ
z, K
r
n
y
n
≤
α
n
β
n
− α
n
β
n
− α
n
β
n
α
n
α
n
β
n
φ
z, x
0
1 − α
n
1 − α
n
1 −
β
n
α
n
1 − α
n
φ
z, x
n
,
W
n
{
z ∈ C :
x
n
− z, Jx
0
− Jx
n
≥ 0
}
,
x
n
J
−1
α
n
Jx
0
1 − α
n
β
n
Jx
n
1 − β
n
JJ
r
n
x
n
,
y
n
J
−1
α
n
J x
n
1 − α
n
J
J
r
n
J
−1
β
n
Jx
0
1 −
β
n
J x
n
,
3.7
K
r
is defined by 2.15, {α
n
}, {β
n
}, {α
n
}, {
β
n
}⊂0, 1 satisfy
lim
n →∞
α
n
0, lim
n →∞
β
n
0, lim inf
n →∞
β
n
1 − β
n
> 0, lim inf
n →∞
α
n
1 − α
n
> 0,
3.8
and {r
n
}⊂0, ∞ satisfies lim inf
n →∞
r
n
> 0. Then, the sequence {x
n
} converges strongly to
Π
T
−1
0∩
T
−1
0∩EP
x
0
provided
J
r
n
v
n
−
J
r
n
x
n
→0 for any sequence {v
n
}⊂X with v
n
− x
n
→0,
where Π
T
−1
0∩
T
−1
0∩EP
is the generalized projection of X onto T
−1
0 ∩
T
−1
0 ∩ EP.
Remark 3.3. In Theorem 3.2,ifX H a real Hilbert space, then {
J
r
n
} is a sequence of
nonexpansive mappings on H. This implies that as n →∞,
J
r
n
v
n
−
J
r
n
x
n
≤
v
n
− x
n
−→ 0. 3.9
In this case, we can remove the requirement that
J
r
n
v
n
−
J
r
n
x
n
→0 for any sequence {v
n
}⊂
X with v
n
− x
n
→0.
Proof of Theorem 3.2. For the sake of simplicity, we define
u
n
: K
r
n
y
n
,z
n
: J
−1
β
n
Jx
n
1 − β
n
JJ
r
n
x
n
, z
n
:
J
r
n
J
−1
β
n
Jx
0
1 −
β
n
J x
n
,
3.10
so that
x
n
J
−1
α
n
Jx
0
1 − α
n
Jz
n
,y
n
J
−1
α
n
J x
n
1 − α
n
J z
n
.
3.11
We divide the proof into several steps.
Step 1. We claim that H
n
∩ W
n
is closed and convex for each n ≥ 0.
Indeed, it is obvious that H
n
is closed and W
n
is closed and convex for each n ≥ 0. Let
us show that H
n
is convex. For z
1
,z
2
∈ H
n
and t ∈ 0, 1,putz tz
1
1 − tz
2
.Itissufficient
Journal of Inequalities and Applications 9
to show that z ∈ H
n
. We first write γ
n
α
n
β
n
− α
n
β
n
− α
n
β
n
α
n
α
n
β
n
for each n ≥ 0. Next,
we prove that
φ
z, u
n
≤ γ
n
φ
z, x
0
1 − γ
n
φ
z, x
n
3.12
is equivalent to
2γ
n
z, Jx
0
2
1 − γ
n
z, Jx
n
−2z, Ju
n
≤γ
n
x
0
2
1 − γ
n
x
n
2
−u
n
2
.
3.13
Indeed, from 2.4 we deduce that the following equations hold:
φ
z, x
0
z
2
− 2
z, Jx
0
x
0
2
,
φ
z, x
n
z
2
− 2
z, Jx
n
x
n
2
,
φ
z, u
n
z
2
− 2
z, Ju
n
u
n
2
,
3.14
which combined with 3.12 yield that 3.12 is equivalent to 3.13. Thus we have
2γ
n
z, Jx
0
2
1 − γ
n
z, Jx
n
− 2
z, Ju
n
2γ
n
tz
1
1 − t
z
2
,Jx
0
2
1 − γ
n
tz
1
1 − t
z
2
,Jx
n
− 2
tz
1
1 − t
z
2
,Ju
n
2tγ
n
z
1
,Jx
0
2
1 − t
γ
n
z
2
,Jx
0
2
1 − γ
n
t
z
1
,Jx
n
2
1 − γ
n
1 − t
z
2
,Jx
n
− 2t
z
1
,Ju
n
− 2
1 − t
z
2
,Ju
n
≤ γ
n
x
0
2
1 − γ
n
x
n
2
−
u
n
2
.
3.15
This implies that z ∈ H
n
. Therefore, H
n
is closed and convex.
Step 2. We claim that T
−1
0 ∩
T
−1
0 ∩ EP ⊂ H
n
∩ W
n
for each n ≥ 0andthat{x
n
} is well defined.
Indeed, take w ∈ T
−1
0 ∩
T
−1
0 ∩ EP arbitrarily. Note that u
n
K
r
n
y
n
is equivalent to
u
n
∈ C such that f
u
n
,y
Au
n
,y− u
n
1
r
n
y − u
n
,Ju
n
− Jy
n
≥ 0, ∀y ∈ C.
3.16
10 Journal of Inequalities and Applications
Then from Lemma 2.11 we obtain
φ
w, z
n
φ
w, J
−1
β
n
Jx
n
1 − β
n
JJ
r
n
x
n
w
2
− 2
w, β
n
Jx
n
1 − β
n
JJ
r
n
x
n
β
n
Jx
n
1 − β
n
JJ
r
n
x
n
2
≤
w
2
− 2β
n
w, Jx
n
− 2
1 − β
n
w, JJ
r
n
x
n
β
n
x
n
2
1 − β
n
J
r
n
x
n
2
β
n
φ
w, x
n
1 − β
n
φ
w, J
r
n
x
n
≤ β
n
φ
w, x
n
1 − β
n
φ
w, x
n
φ
w, x
n
,
3.17
φ
w, x
n
φ
w, J
−1
α
n
Jx
0
1 − α
n
Jz
n
w
2
− 2
w, α
n
Jx
0
1 − α
n
Jz
n
α
n
Jx
0
1 − α
n
Jz
n
2
≤
w
2
− 2α
n
w, Jx
0
− 2
1 − α
n
w, Jz
n
α
n
x
0
2
1 − α
n
z
n
2
α
n
φ
w, x
0
1 − α
n
φ
w, z
n
≤ α
n
φ
w, x
0
1 − α
n
φ
w, x
n
.
3.18
Moreover, we have
φ
w, z
n
φ
w,
J
r
n
J
−1
β
n
Jx
0
1 −
β
n
J x
n
≤ φ
w, J
−1
β
n
Jx
0
1 −
β
n
x
n
w
2
− 2
w,
β
n
Jx
0
1 −
β
n
J x
n
β
n
Jx
0
1 −
β
n
J x
n
2
≤
w
2
− 2
β
n
w, Jx
0
− 2
1 −
β
n
w, J x
n
β
n
x
0
2
1 −
β
n
x
n
2
β
n
φ
w, x
0
1 −
β
n
φ
w, x
n
≤
β
n
φ
w, x
0
1 −
β
n
α
n
φ
w, x
0
1 − α
n
φ
w, x
n
β
n
1 −
β
n
α
n
φ
w, x
0
1 −
β
n
1 − α
n
φ
w, x
n
,
φ
w, y
n
φ
w, J
−1
α
n
J x
n
1 − α
n
J z
n
≤
w
2
− 2α
n
w, J x
n
− 2
1 − α
n
w, J z
n
α
n
x
n
2
1 − α
n
z
n
2
Journal of Inequalities and Applications 11
α
n
φ
w, x
n
1 − α
n
φ
w, z
n
≤ α
n
α
n
φ
w, x
0
1 − α
n
φ
w, x
n
1 − α
n
β
n
1 −
β
n
α
n
φ
w, x
0
1 −
β
n
1 − α
n
φ
w, x
n
α
n
β
n
− α
n
β
n
− α
n
β
n
α
n
α
n
β
n
φ
w, x
0
1 − α
n
1 − α
n
1 −
β
n
α
n
1 − α
n
φ
w, x
n
γ
n
φ
w, x
0
1 − γ
n
φ
w, x
n
, 3.19
where γ
n
α
n
β
n
− α
n
β
n
− α
n
β
n
α
n
α
n
β
n
.Sow ∈ H
n
for all n ≥ 0. Now, let us show that
T
−1
0 ∩
T
−1
0 ∩ EP ⊂ W
n
∀n ≥ 0.
3.20
We prove this by induction. For n 0, we have T
−1
0 ∩
T
−1
0 ∩ EP ⊂ C W
0
. Assume that
T
−1
0 ∩
T
−1
0 ∩ EP ⊂ W
n
. Since x
n1
is the projection of x
0
onto H
n
∩ W
n
, by Lemma 2.2 we have
x
n1
− z, Jx
0
− Jx
n1
≥ 0, ∀z ∈ H
n
∩ W
n
. 3.21
As T
−1
0 ∩
T
−1
0 ∩ EP ⊂ H
n
∩ W
n
by the induction assumption, the last inequality holds, in
particular, for all z ∈ T
−1
0 ∩
T
−1
0 ∩ EP. This, together with the definition of W
n1
implies that
T
−1
0 ∩
T
−1
0 ∩ EP ⊂ W
n1
. Hence 3.20 holds for all n ≥ 0. So, T
−1
0 ∩
T
−1
0 ∩ EP ⊂ H
n
∩ W
n
for
all n ≥ 0. This implies that the sequence {x
n
} is well defined.
Step 3. We claim that {x
n
} is bounded and that φx
n1
,x
n
→ 0asn →∞.
Indeed, it follows from the definition of W
n
that x
n
Π
W
n
x
0
. Since x
n
Π
W
n
x
0
and x
n1
Π
H
n
∩W
n
x
0
∈ W
n
,soφx
n
,x
0
≤ φx
n1
,x
0
for all n ≥ 0; that is, {φx
n
,x
0
} is
nondecreasing. It follows from x
n
Π
W
n
x
0
and Lemma 2.3 that
φ
x
n
,x
0
φ
Π
W
n
x
0
,x
0
≤ φ
p, x
0
− φ
p, x
n
≤ φ
p, x
0
3.22
for each p ∈ T
−1
0 ∩
T
−1
0 ∩ EP ⊂ W
n
for each n ≥ 0. Therefore, {φx
n
,x
0
} is bounded which
implies that the limit of {φx
n
,x
0
} exists. Since
x
n
−
x
0
2
≤ φ
x
n
,x
0
≤
x
n
x
0
2
, ∀n ≥ 0,
3.23
12 Journal of Inequalities and Applications
so {x
n
} is bounded. From Lemma 2.3, we have
φ
x
n1
,x
n
φ
x
n1
, Π
W
n
x
0
≤ φ
x
n1
,x
0
− φ
Π
W
n
x
0
,x
0
φ
x
n1
,x
0
− φ
x
n
,x
0
,
3.24
for each n ≥ 0. This implies that
lim
n →∞
φ
x
n1
,x
n
0.
3.25
Step 4. We claim that lim
n →∞
x
n
−u
n
0, lim
n →∞
x
n
−J
r
n
x
n
0, and lim
n →∞
x
n
−
J
r
n
x
n
0.
Indeed, from x
n1
Π
H
n
∩W
n
x
0
∈ H
n
, we have
φ
x
n1
,u
n
≤
α
n
β
n
− α
n
β
n
− α
n
β
n
α
n
α
n
β
n
φ
x
n1
,x
0
1 − α
n
1 − α
n
1 −
β
n
α
n
1 − α
n
φ
x
n1
,x
n
3.26
for all n ≥ 0. Therefore, from α
n
→ 0,
β
n
→ 0andφx
n1
,x
n
→ 0, it follows that
lim
n →∞
φx
n1
,u
n
0. Since lim
n →∞
φx
n1
,x
n
lim
n →∞
φx
n1
,u
n
0andX is uniformly
convex and smooth, we have from Lemma 2.1 that
lim
n →∞
x
n1
− x
n
lim
n →∞
x
n1
− u
n
0,
3.27
and therefore, lim
n →∞
x
n
− u
n
0. Since J is uniformly norm-to-norm continuous on
bounded subsets of X and x
n
− u
n
→ 0, then lim
n →∞
Jx
n
− Ju
n
0.
Let us set Ω : T
−1
0∩
T
−1
0∩EP. Then, according to Lemma 2.4 and Proposition 2.10,we
know that Ω is a nonempty closed convex subset of X such that Ω ⊂ C.Fixu ∈ Ω arbitrarily.
As in the proof of Step 2 we can show that φu, z
n
≤ φu, x
n
,φu, x
n
≤ α
n
φu, x
0
1 −
α
n
φu, x
n
,φu, z
n
≤
β
n
φu, x
0
1 −
β
n
φu, x
n
,φu, y
n
≤ α
n
φu, x
n
1 − α
n
φu, z
n
,
and φu, u
n
≤ α
n
φu, x
n
1 − α
n
φu, z
n
. Hence it follows from the boundedness of {x
n
}
that {z
n
}, {x
n
}, {z
n
}, {y
n
},and{u
n
} are also bounded. Let r sup{x
n
, x
n
, J
r
n
x
n
, z
n
:
n ≥ 0}. Since X is a uniformly smooth Banach space, we know that X
∗
is a uniformly convex
Banach space. Therefore, by Lemma 2.5 there exists a continuous, strictly increasing, and
convex function g with g00 such that
αx
∗
1 − α
y
∗
2
≤ α
x
∗
2
1 − α
y
∗
2
− α
1 − α
g
x
∗
− y
∗
,
3.28
Journal of Inequalities and Applications 13
for x
∗
,y
∗
∈ B
∗
r
and α ∈ 0, 1. So, we have that
φ
u, z
n
φ
u, J
−1
β
n
Jx
n
1 − β
n
JJ
r
n
x
n
u
2
− 2
u, β
n
Jx
n
1 − β
n
JJ
r
n
x
n
β
n
Jx
n
1 − β
n
JJ
r
n
x
n
2
≤
u
2
− 2β
n
u, Jx
n
− 2
1 − β
n
u, JJ
r
n
x
n
β
n
x
n
2
1 − β
n
J
r
n
x
n
2
− β
n
1 − β
n
g
Jx
n
− JJ
r
n
x
n
β
n
φ
u, x
n
1 − β
n
φ
u, J
r
n
x
n
− β
n
1 − β
n
g
Jx
n
− JJ
r
n
x
n
≤ β
n
φ
u, x
n
1 − β
n
φ
u, x
n
− β
n
1 − β
n
g
Jx
n
− JJ
r
n
x
n
φ
u, x
n
− β
n
1 − β
n
g
Jx
n
− JJ
r
n
x
n
,
3.29
φ
u, z
n
φ
u,
J
r
n
J
−1
β
n
Jx
0
1 −
β
n
J x
n
≤ φ
u, J
−1
β
n
Jx
0
1 −
β
n
J x
n
≤
u
2
− 2
u,
β
n
Jx
0
1 −
β
n
J x
n
β
n
x
0
2
1 −
β
n
x
n
2
β
n
φ
u, x
0
1 −
β
n
φ
u, x
n
≤
β
n
φ
u, x
0
1 −
β
n
φ
u, x
n
− β
n
1 − β
n
g
Jx
n
− JJ
r
n
x
n
β
n
φ
u, x
0
1 −
β
n
φ
u, x
n
−
1 −
β
n
β
n
1 − β
n
g
Jx
n
− JJ
r
n
x
n
,
3.30
and hence
φ
u, x
n
φ
u, J
−1
α
n
Jx
0
1 − α
n
Jz
n
u
2
− 2
u, α
n
Jx
0
1 − α
n
Jz
n
α
n
Jx
0
1 − α
n
Jz
n
2
≤
u
2
− 2α
n
u, Jx
0
− 2
1 − α
n
u, Jz
n
α
n
x
0
2
1 − α
n
z
n
2
α
n
φ
u, x
0
1 − α
n
φ
u, z
n
≤ α
n
φ
u, x
0
1 − α
n
φ
u, x
n
− β
n
1 − β
n
g
Jx
n
− JJ
r
n
x
n
α
n
φ
u, x
0
1 − α
n
φ
u, x
n
−
1 − α
n
β
n
1 − β
n
g
Jx
n
− JJ
r
n
x
n
,
3.31
φ
u, u
n
φ
u, K
r
n
y
n
≤ φ
u, y
n
using Proposition 2.10
φ
u, J
−1
α
n
J x
n
1 − α
n
J z
n
14 Journal of Inequalities and Applications
u
2
− 2
u, α
n
J x
n
1 − α
n
J z
n
α
n
J x
n
1 − α
n
J z
n
2
≤
u
2
− 2α
n
u, J x
n
− 2
1 − α
n
u, J z
n
α
n
x
n
2
1 − α
n
z
n
2
− α
n
1 − α
n
g
J x
n
− J z
n
α
n
φ
u, x
n
1 − α
n
φ
u, z
n
− α
n
1 − α
n
g
J x
n
− J z
n
≤ α
n
α
n
φ
u, x
0
1 − α
n
φ
u, x
n
−
1 − α
n
β
n
1 − β
n
g
Jx
n
− JJ
r
n
x
n
1 − α
n
β
n
φ
u, x
0
1 −
β
n
φ
u, x
n
−
1 −
β
n
β
n
1 − β
n
g
Jx
n
− JJ
r
n
x
n
− α
n
1 − α
n
g
J x
n
− J z
n
≤
α
n
β
n
φ
u, x
0
α
n
1 − α
n
1 − α
n
1 −
β
n
φ
u, x
n
−
α
n
1 − α
n
1 − α
n
1 −
β
n
β
n
1 − β
n
g
Jx
n
− JJ
r
n
x
n
− α
n
1 − α
n
g
J x
n
− J z
n
α
n
β
n
φ
u, x
0
1 − α
n
α
n
−
β
n
1 − α
n
φ
u, x
n
−
1 − α
n
α
n
−
β
n
1 − α
n
β
n
1 − β
n
g
Jx
n
− JJ
r
n
x
n
− α
n
1 − α
n
g
J x
n
− J z
n
≤
α
n
β
n
φ
u, x
0
φ
u, x
n
−
1 − α
n
α
n
−
β
n
1 − α
n
β
n
1 − β
n
g
Jx
n
− JJ
r
n
x
n
− α
n
1 − α
n
g
J x
n
− J z
n
, 3.32
for all n ≥ 0. Consequently we have
1 − α
n
α
n
−
β
n
1 − α
n
β
n
1 − β
n
g
Jx
n
− JJ
r
n
x
n
α
n
1 − α
n
g
J x
n
− J z
n
≤
α
n
β
n
φ
u, x
0
φ
u, x
n
− φ
u, u
n
α
n
β
n
φ
u, x
0
x
n
2
−
u
n
2
− 2
u, Jx
n
− Ju
n
≤
α
n
β
n
φ
u, x
0
x
n
2
−
u
n
2
2
|
u, Jx
n
− Ju
n
|
≤
α
n
β
n
φ
u, x
0
|
x
n
−
u
n
|
x
n
u
n
2
u
Jx
n
− Ju
n
≤
α
n
β
n
φ
u, x
0
x
n
− u
n
x
n
u
n
2
u
Jx
n
− Ju
n
.
3.33
Journal of Inequalities and Applications 15
Since x
n
− u
n
→ 0andJ is uniformly norm-to-norm continuous on bounded subsets of X,
we obtain Jx
n
− Ju
n
→ 0. From lim inf
n →∞
β
n
1 − β
n
> 0, lim inf
n →∞
α
n
1 − α
n
> 0, and
lim
n →∞
α
n
lim
n →∞
β
n
0 we have
lim
n →∞
g
Jx
n
− JJ
r
n
x
n
lim
n →∞
g
J x
n
− J z
n
0.
3.34
Therefore, from the properties of g we get
lim
n →∞
Jx
n
− JJ
r
n
x
n
lim
n →∞
x
n
− J
r
n
x
n
0,
lim
n →∞
J x
n
− J z
n
lim
n →∞
x
n
− z
n
0,
3.35
recalling that J
−1
is uniformly norm-to-norm continuous on bounded sunsets of X
∗
. Next let
us show that
lim
n →∞
J x
n
− J
J
r
n
x
n
lim
n →∞
x
n
−
J
r
n
x
n
0.
3.36
Observe first that
φ
u
n
,x
n
− φ
x
n1
,u
n
x
n
2
−
x
n1
2
− 2
u
n
,Jx
n
2
x
n1
,Ju
n
x
n
−
x
n1
x
n
x
n1
2
x
n1
− u
n
,Jx
n
2
x
n1
,Ju
n
− Jx
n
≤
x
n
− x
n1
x
n
x
n1
2
x
n1
− u
n
x
n
2
x
n1
Ju
n
− Jx
n
.
3.37
Since φx
n1
,u
n
→ 0, x
n1
− x
n
→0, x
n1
− u
n
→0, Ju
n
− Jx
n
→0, and {x
n
} is
bounded, so it follows that φu
n
,x
n
→ 0. Also, observe that
φ
u
n
,J
r
n
x
n
− φ
u
n
,x
n
J
r
n
x
n
2
−x
n
2
2
u
n
,Jx
n
− JJ
r
n
x
n
J
r
n
x
n
−
x
n
J
r
n
x
n
x
n
2
u
n
,Jx
n
− JJ
r
n
x
n
≤
J
r
n
x
n
− x
n
J
r
n
x
n
x
n
2
u
n
Jx
n
− JJ
r
n
x
n
.
3.38
16 Journal of Inequalities and Applications
Since φu
n
,x
n
→ 0, J
r
n
x
n
− x
n
→0, Jx
n
− JJ
r
n
x
n
→0 and the sequences
{x
n
}, {u
n
}, {J
r
n
x
n
} are bounded, so it follows that φu
n
,J
r
n
x
n
→ 0. Meantime, observe that
φ
u
n
,z
n
φ
u
n
,J
−1
β
n
Jx
n
1 − β
n
JJ
r
n
x
n
u
n
2
− 2
u
n
,β
n
Jx
n
1 − β
n
JJ
r
n
x
n
β
n
Jx
n
1 − β
n
JJ
r
n
x
n
2
≤
u
n
2
− 2β
n
u
n
,Jx
n
− 2
1 − β
n
u
n
,JJ
r
n
x
n
β
n
x
n
2
1 − β
n
J
r
n
x
n
2
β
n
φ
u
n
,x
n
1 − β
n
φ
u
n
,J
r
n
x
n
≤ φ
u
n
,x
n
φ
u
n
,J
r
n
x
n
,
3.39
and hence
φ
u
n
, x
n
φ
u
n
,J
−1
α
n
Jx
0
1 − α
n
Jz
n
u
n
2
− 2
u
n
,α
n
Jx
0
1 − α
n
Jz
n
α
n
Jx
0
1 − α
n
Jz
n
2
≤
u
n
2
− 2α
n
u
n
,Jx
0
− 2
1 − α
n
u
n
,Jz
n
α
n
x
0
2
1 − α
n
z
n
2
α
n
φ
u
n
,x
0
1 − α
n
φ
u
n
,z
n
≤ α
n
φ
u
n
,x
0
φ
u
n
,z
n
≤ α
n
φ
u
n
,x
0
φ
u
n
,x
n
φ
u
n
,J
r
n
x
n
.
3.40
Since α
n
→ 0,φu
n
,x
n
→ 0andφu
n
,J
r
n
x
n
→ 0, it follows from the boundedness of
{u
n
} that φu
n
, x
n
→ 0. Thus, in terms of Lemma 2.1, we have that u
n
− x
n
→0andso
x
n
− x
n
→0. Furthermore, since
β
n
Jx
0
1 −
β
n
J x
n
− J x
n
β
n
Jx
0
− J x
n
→0, from the
uniform norm-to-norm continuity of J
−1
on bounded subsets of X
∗
,weobtain
J
−1
β
n
Jx
0
1 −
β
n
J x
n
− x
n
−→ 0. 3.41
Observe that
x
n
−
J
r
n
x
n
≤
x
n
− z
n
z
n
−
J
r
n
x
n
x
n
− z
n
J
r
n
J
−1
β
n
Jx
0
1 −
β
n
J x
n
−
J
r
n
x
n
.
3.42
Thus, from 3.35 it follows that x
n
−
J
r
n
x
n
→0. Since J is uniformly norm-to-norm
continuous on bounded subsets of X, it follows that J x
n
− J
J
r
n
x
n
→0.
Step 5. We claim that ω
w
{x
n
} ⊂ T
−1
0 ∩
T
−1
0 ∩ EP, where
ω
w
{
x
n
}
:
x ∈ C : x
n
k
x for some subsequence
{
n
k
}
⊂
{
n
}
with n
k
↑∞
. 3.43
Journal of Inequalities and Applications 17
Indeed, since {x
n
} is bounded and X is reflexive, we know that ω
w
{x
n
}
/
∅. Take
x ∈ ω
w
{x
n
} arbitrarily. Then there exists a subsequence {x
n
k
} of {x
n
} such that x
n
k
x.
Hence it follows from x
n
− x
n
→ 0,x
n
− J
r
n
x
n
→ 0, and x
n
−
J
r
n
x
n
→ 0that{x
n
k
}, {J
r
n
k
x
n
k
},
and {
J
r
n
k
x
n
k
} converge weakly to the same point x. On the other hand, from 3.35, 3.36,
and lim inf
n →∞
r
n
> 0weobtainthat
lim
n →∞
A
r
n
x
n
lim
n →∞
1
r
n
Jx
n
− JJ
r
n
x
n
0,
3.44
lim
n →∞
A
r
n
x
n
lim
n →∞
1
r
n
J x
n
− J
J
r
n
x
n
0.
3.45
If z
∗
∈ Tz and z
∗
∈
T z, then it follows from 2.17 and the monotonicity of the operators T,
T
that for all k ≥ 1
z − J
r
n
k
x
n
k
,z
∗
− A
r
n
k
x
n
k
≥ 0,
z −
J
r
n
k
x
n
k
, z
∗
−
A
r
n
k
x
n
k
≥ 0. 3.46
Letting k →∞, we have that z − x, z
∗
≥0andz − x, z
∗
≥0. Then the maximality of the
operators T,
T implies that x ∈ T
−1
0and x ∈
T
−1
0. Next, let us show that x ∈ EP. Since we
have by 3.32
φ
u, y
n
≤
α
n
β
n
φ
u, x
0
φ
u, x
n
, 3.47
from u
n
K
r
n
y
n
and Proposition 2.10 it follows that
φ
u
n
,y
n
φ
K
r
n
y
n
,y
n
≤ φ
u, y
n
− φ
u, K
r
n
y
n
≤
α
n
β
n
φ
u, x
0
φ
u, x
n
− φ
u, u
n
.
3.48
Also, since
φ
u, x
n
− φ
u, u
n
x
n
2
−
u
n
2
2
u, Ju
n
− Jx
n
≤
|
x
n
−
u
n
|
x
n
u
n
2
u
Ju
n
− Jx
n
≤
x
n
− u
n
x
n
u
n
2
u
Ju
n
− Jx
n
,
3.49
so we get
lim
n →∞
φ
u, x
n
− φ
u, u
n
0.
3.50
18 Journal of Inequalities and Applications
Thus from 3.47, α
n
→ 0,
β
n
→ 0, and φu, x
n
− φu, u
n
→ 0, we have lim
n →∞
φu
n
,y
n
0. Since X is uniformly convex and smooth, we conclude from Lemma 2.1 that
lim
n →∞
u
n
− y
n
0.
3.51
From x
n
k
x, x
n
− u
n
→ 0, and 3.51, we have y
n
k
x and u
n
k
x. Since J is uniformly
norm-to-norm continuous on bounded subsets of X,from3.51 we derive
lim
n →∞
Ju
n
− Jy
n
0.
3.52
From lim inf
n →∞
r
n
> 0, it follows that
lim
n →∞
Ju
n
− Jy
n
r
n
0.
3.53
By the definition of u
n
: K
r
n
y
n
, we have
F
u
n
,y
1
r
n
y − u
n
,Ju
n
− Jy
n
≥ 0, ∀y ∈ C,
3.54
where
F
u
n
,y
f
u
n
,y
Au
n
,y− u
n
. 3.55
Replacing n by n
k
, we have from A2 that
1
r
n
k
y − u
n
k
,Ju
n
k
− Jy
n
k
≥−F
u
n
k
,y
≥ F
y, u
n
k
, ∀y ∈ C.
3.56
Since y → fx, yAx, y − x is convex and lower semicontinuous, it is also weakly lower
semicontinuous. Letting n
k
→∞in the last inequality, from 3.53 and A4 we have
F
y, x
≤ 0, ∀y ∈ C. 3.57
For t with 0 <t≤ 1andy ∈ C,lety
t
ty 1 − t x. Since y ∈ C and x ∈ C, y
t
∈ C and hence
Fy
t
, x ≤ 0. So, from A1 we have
0 F
y
t
,y
t
≤ tF
y
t
,y
1 − t
F
y
t
, x
≤ tF
y
t
,y
. 3.58
Dividing by t, we have
F
y
t
,y
≥ 0, ∀y ∈ C. 3.59
Journal of Inequalities and Applications 19
Letting t ↓ 0, from A3 it follows that
F
x, y
≥ 0, ∀y ∈ C. 3.60
Thus x ∈ EP. Therefore, we obtain that ω
w
{x
n
} ⊂ T
−1
0 ∩
T
−1
0 ∩ EP by the arbitrariness of x.
Step 6. We claim that {x
n
} converges strongly to w Π
T
−1
0∩
T
−1
0∩EP
x
0
.
Indeed, from x
n1
Π
H
n
∩W
n
x
0
and w ∈ T
−1
0 ∩
T
−1
0 ∩ EP ⊂ H
n
∩ W
n
, It follows that
φ
x
n1
,x
0
≤ φ
w, x
0
. 3.61
Since the norm is weakly lower semicontinuous, we have
φ
x, x
0
x
2
− 2
x, Jx
0
x
0
2
≤ lim inf
k →∞
x
n
k
2
− 2
x
n
k
,Jx
0
x
0
2
lim inf
k →∞
φ
x
n
k
,x
0
≤ lim sup
k →∞
φ
x
n
k
,x
0
≤ φ
w, x
0
.
3.62
From the definition of Π
T
−1
0∩
T
−1
0∩EP
, we have x w. Hence lim
k →∞
φx
n
k
,x
0
φw, x
0
and
0 lim
k →∞
φ
x
n
k
,x
0
− φ
w, x
0
lim
k →∞
x
n
k
2
−
w
2
− 2
x
n
k
− w, Jx
0
lim
k →∞
x
n
k
2
−
w
2
,
3.63
which implies that lim
k →∞
x
n
k
w. Since X has the Kadec-Klee property, then x
n
k
→
w Π
T
−1
0∩
T
−1
0∩EP
x
0
. Therefore, {x
n
} converges strongly to Π
T
−1
0∩
T
−1
0∩EP
x
0
.
Remark 3.4. In Theorem 3.2,putA ≡ 0,
T ≡ 0, and
β
n
0, ∀n ≥ 0. Then, for all α, r ∈ 0, ∞
and x, y ∈ C, we have that
Ax − Ay, x − y
≥ α
Ax − Ay
2
,
K
r
x
u ∈ C : f
u, y
Au, y − u
1
r
y − u, Ju − Jx
≥ 0, ∀y ∈ C
u ∈ C : f
u, y
1
r
y − u, Ju − Jx
≥ 0, ∀y ∈ C
T
r
x
.
3.64
20 Journal of Inequalities and Applications
Moreover, the following hold:
H
n
z ∈ C : φ
z, K
r
n
y
n
≤
α
n
β
n
− α
n
β
n
− α
n
β
n
α
n
α
n
β
n
φ
z, x
0
1 − α
n
1 − α
n
1 −
β
n
α
n
1 − α
n
φ
z, x
n
,
z ∈ C : φ
z, T
r
n
y
n
≤ α
n
φ
z, x
0
1 − α
n
φ
z, x
n
,
y
n
J
−1
α
n
J x
n
1 − α
n
J
J
r
n
J
−1
β
n
Jx
0
1 −
β
n
J x
n
J
−1
α
n
J x
n
1 − α
n
0Jx
0
1 − 0
J x
n
J
−1
α
n
J x
n
1 − α
n
J x
n
J
−1
J x
n
x
n
,
3.65
and hence
y
n
x
n
J
−1
α
n
Jx
0
1 − α
n
β
n
Jx
n
1 − β
n
JJ
r
n
x
n
.
3.66
In this case, the previous Theorem 3.2 reduces to 20, Theorem 3.1 .
4. Weak Convergence Theorem
In this section, we present the following algorithm for finding a common element of the set
of solutions for a generalized equilibrium problem and the set T
−1
0 ∩
T
−1
0 for two maximal
monotone operators T and
T.
Let x
0
∈ X be chosen arbitrarily and consider the sequence {x
n
} generated by
x
n
J
−1
α
n
Jx
0
1 − α
n
β
n
JK
r
n
x
n
1 − β
n
JJ
r
n
K
r
n
x
n
,
x
n1
J
−1
α
n
JK
r
n
x
n
1 − α
n
J
J
r
n
J
−1
β
n
Jx
0
1 −
β
n
JK
r
n
x
n
,n 0, 1, 2, ,
4.1
where {α
n
}, {β
n
}, {α
n
}, {
β
n
}⊂0, 1, {r
n
}⊂0, ∞,andK
r
,r >0, is defined by 2.15.
Before proving a weak convergence theorem, we need the following proposition.
Proposition 4.1. Suppose that Assumption A is fulfilled and let {x
n
} be a sequence defined by 4.1,
where {α
n
}, {β
n
}, {α
n
}, {
β
n
}⊂0, 1 satisfy the following conditions:
∞
n0
α
n
< ∞,
∞
n0
β
n
< ∞, lim inf
n →∞
β
n
1 − β
n
> 0, lim inf
n →∞
α
n
1 − α
n
> 0.
4.2
Journal of Inequalities and Applications 21
Then, {Π
T
−1
0∩
T
−1
0∩EP
x
n
} converges strongly to z ∈ T
−1
0 ∩
T
−1
0 ∩ EP,whereΠ
T
−1
0∩
T
−1
0∩EP
is the
generalized projection of X onto T
−1
0 ∩
T
−1
0 ∩ EP.
Proof. We set Ω : T
−1
0 ∩
T
−1
0 ∩ EP and
u
n
: K
r
n
x
n
,y
n
: J
−1
β
n
Ju
n
1 − β
n
JJ
r
n
u
n
,
u
n
: K
r
n
x
n
, y
n
:
J
r
n
J
−1
β
n
Jx
0
1 −
β
n
J u
n
,
4.3
so that
x
n
J
−1
α
n
Jx
0
1 − α
n
Jy
n
,
x
n1
J
−1
α
n
J u
n
1 − α
n
J y
n
,n 0, 1, 2,
4.4
Then, in terms of Lemma 2.4 and Proposition 2.10, Ω is a nonempty closed convex subset of
X such that Ω ⊂ C. We fi rst prove that {x
n
} is bounded. Fix u ∈ Ω. Note that by the first and
third of 4.3, u
n
, u
n
∈ C,and
F
u
n
,y
1
r
n
y − u
n
,Ju
n
− Jx
n
≥ 0, ∀y ∈ C,
F
u
n
,y
1
r
n
y − u
n
,Ju
n
− J x
n
≥ 0, ∀y ∈ C.
4.5
Here, each K
r
n
is relatively nonexpansive. Then from Proposition 2.10 we obtain
φ
u, y
n
φ
u, J
−1
β
n
Ju
n
1 − β
n
JJ
r
n
u
n
u
2
− 2
u, β
n
Ju
n
1 − β
n
JJ
r
n
u
n
β
n
Ju
n
1 − β
n
JJ
r
n
u
n
2
≤
u
2
− 2β
n
u, Ju
n
− 2
1 − β
n
u, JJ
r
n
u
n
β
n
u
n
2
1 − β
n
J
r
n
u
n
2
β
n
φ
u, u
n
1 − β
n
φ
u, J
r
n
u
n
≤ β
n
φ
u, u
n
1 − β
n
φ
u, u
n
φ
u, u
n
φ
u, K
r
n
x
n
≤ φ
u, x
n
, 4.6
22 Journal of Inequalities and Applications
φ
u, y
n
φ
u,
J
r
n
J
−1
β
n
Jx
0
1 −
β
n
J u
n
≤ φ
u, J
−1
β
n
Jx
0
1 −
β
n
J u
n
u
2
− 2
u,
β
n
Jx
0
1 −
β
n
J u
n
β
n
Jx
0
1 −
β
n
J u
n
2
≤u
2
− 2
β
n
u, Jx
0
− 2
1 −
β
n
u, J u
n
β
n
x
0
2
1 −
β
n
u
n
2
β
n
φ
u, x
0
1 −
β
n
φ
u, u
n
≤
β
n
φ
u, x
0
φ
u, u
n
β
n
φ
u, x
0
φ
u, K
r
n
x
n
≤
β
n
φ
u, x
0
φ
u, x
n
,
4.7
and hence by Proposition 2.10, we have
φ
u, x
n
φ
u, J
−1
α
n
Jx
0
1 − α
n
Jy
n
u
2
− 2
u, α
n
Jx
0
1 − α
n
Jy
n
α
n
Jx
0
1 − α
n
Jy
n
2
≤
u
2
− 2α
n
u, Jx
0
− 2
1 − α
n
u, Jy
n
α
n
x
0
2
1 − α
n
y
n
2
α
n
φ
u, x
0
1 − α
n
φ
u, y
n
≤ α
n
φ
u, x
0
φ
u, y
n
≤ φ
u, x
n
α
n
φ
u, x
0
,
4.8
φ
u, x
n1
φ
u, J
−1
α
n
J u
n
1 − α
n
J y
n
u
2
− 2
u, α
n
J u
n
1 − α
n
J y
n
α
n
J u
n
1 − α
n
J y
n
2
≤
u
2
− 2α
n
u, J u
n
− 2
1 − α
n
u, J y
n
α
n
u
n
2
1 − α
n
y
n
2
α
n
φ
u, u
n
1 − α
n
φ
u, y
n
≤ α
n
φ
u, x
n
1 − α
n
β
n
φ
u, x
0
φ
u, x
n
≤ φ
u, x
n
β
n
φ
u, x
0
.
4.9
Journal of Inequalities and Applications 23
Consequently, the last two inequalities yield that
φ
u, x
n1
≤ φ
u, x
n
β
n
φ
u, x
0
≤ φ
u, x
n
α
n
φ
u, x
0
β
n
φ
u, x
0
φ
u, x
n
α
n
β
n
φ
u, x
0
4.10
for all n ≥ 0. So, f rom
∞
n0
α
n
< ∞,
∞
n0
β
n
< ∞, and Lemma 2.12, we deduce that
lim
n →∞
φu, x
n
exists. This implies that {φu, x
n
} is bounded. Thus, {x
n
} is bounded and
so are {u
n
}, {u
n
}, {J
r
n
u
n
},and{
J
r
n
u
n
}.
Define z
n
Π
Ω
x
n
for all n ≥ 0. Let us show that {z
n
} is bounded. Indeed, observe that
z
n
−
x
n
2
≤ φ
z
n
,x
n
φ
Π
Ω
x
n
,x
n
≤ φ
p, x
n
− φ
p, Π
Ω
x
n
φ
p, x
n
− φ
p, z
n
≤ φ
p, x
n
,
4.11
for each p ∈ Ω. This, together with the boundedness of {x
n
}, implies that {z
n
} is bounded
and so is φz
n
,x
0
. Furthermore, from z
n
∈ Ω and 4.10 we have
φ
z
n
,x
n1
≤ φ
z
n
,x
n
α
n
β
n
φ
z
n
,x
0
. 4.12
Since Π
Ω
is the generalized projection, then, from Lemma 2.3 we obtain
φ
z
n1
,x
n1
φ
Π
Ω
x
n1
,x
n1
≤ φ
z
n
,x
n1
− φ
z
n
, Π
Ω
x
n1
φ
z
n
,x
n1
− φ
z
n
,z
n1
≤ φ
z
n
,x
n1
.
4.13
Hence, from 4.12, it follows that φz
n1
,x
n1
≤ φz
n
,x
n
α
n
β
n
φz
n
,x
0
.
Note that
∞
n0
α
n
< ∞,
∞
n0
β
n
< ∞,and{φz
n
,x
0
} is bounded, so that
∞
n0
α
n
β
n
φz
n
,x
0
< ∞. Therefore, {φz
n
,x
n
} is a convergent sequence. On the other hand, from
4.10 we derive, for all m ≥ 0,
φ
u, x
nm
≤ φ
u, x
n
m−1
j0
α
nj
β
nj
φ
u, x
0
.
4.14
In particular, we have
φ
z
n
,x
nm
≤ φ
z
n
,x
n
m−1
j0
α
nj
β
nj
φ
z
n
,x
0
,
4.15
24 Journal of Inequalities and Applications
Consequently, from z
nm
Π
Ω
x
nm
and Lemma 2.3, we have
φ
z
n
,z
nm
φ
z
nm
,x
nm
≤ φ
z
n
,x
nm
≤ φ
z
n
,x
n
m−1
j0
α
nj
β
nj
φ
z
n
,x
0
4.16
and hence
φ
z
n
,z
nm
≤ φ
z
n
,x
n
− φ
z
nm
,x
nm
m−1
j0
α
nj
β
nj
φ
z
n
,x
0
.
4.17
Let r sup{z
n
: n ≥ 0}. From Lemma 2.6, there exists a continuous, strictly increasing, and
convex function g with g00 such that
g
x − y
≤ φ
x, y
, ∀x, y ∈ B
r
. 4.18
So, we have
g
z
n
− z
nm
≤ φ
z
n
,z
nm
≤ φ
z
n
,x
n
− φ
z
nm
,x
nm
m−1
j0
α
nj
β
nj
φ
z
n
,x
0
.
4.19
Since {φz
n
,x
n
} is a convergent sequence, {φz
n
,x
0
} is bounded and
∞
n0
α
n
β
n
is
convergent, from the property of g we have that {z
n
} is a Cauchy sequence. Since Ω is closed,
{z
n
} converges strongly to z ∈ Ω. This completes the proof.
Now, we are in a position to prove the following theorem.
Theorem 4.2. Suppose that Assumption A is fulfilled and let {x
n
} be a sequence defined by 4.1,
where {α
n
}, {β
n
}, {α
n
}, {
β
n
}⊂0, 1 satisfy the following conditions:
∞
n0
α
n
< ∞,
∞
n0
β
n
< ∞, lim inf
n →∞
β
n
1 − β
n
> 0, lim inf
n →∞
α
n
1 − α
n
> 0,
4.20
and {r
n
}⊂0, ∞ satisfies lim inf
n →∞
r
n
> 0.IfJ is weakly sequentially continuous, then {x
n
}
converges weakly to z ∈ T
−1
0 ∩
T
−1
0 ∩ EP,wherez lim
n →∞
Π
T
−1
0∩
T
−1
0∩EP
x
n
.
Journal of Inequalities and Applications 25
Proof. We consider the notations 4.3. As in the proof of Proposition 4.1, we have that
{x
n
}, {u
n
}, {J
r
n
u
n
}, {x
n
}, {u
n
},and{
J
r
n
u
n
} are bounded sequences. Let
r sup
u
n
,
J
r
n
u
n
,
u
n
,
y
n
: n ≥ 0
. 4.21
From Lemma 2.5 and as in the proof of Theorem 3.2, there exists a continuous, strictly
increasing, and convex function g with g00 such that
αx
∗
1 − α
y
∗
2
≤ α
x
∗
2
1 − α
y
∗
2
− α
1 − α
g
x
∗
− y
∗
4.22
for x
∗
,y
∗
∈ B
∗
r
and α ∈ 0, 1. Observe that f or u ∈ Ω : T
−1
0 ∩
T
−1
0 ∩ EP,
φ
u, y
n
φ
u, J
−1
β
n
Ju
n
1 − β
n
JJ
r
n
u
n
u
2
− 2
u, β
n
Ju
n
1 − β
n
JJ
r
n
u
n
β
n
Ju
n
1 − β
n
JJ
r
n
u
n
2
≤
u
2
− 2β
n
u, Ju
n
− 2
1 − β
n
u, JJ
r
n
u
n
β
n
u
n
2
1 − β
n
J
r
n
u
n
2
− β
n
1 − β
n
g
Ju
n
− JJ
r
n
u
n
≤ β
n
φ
u, u
n
1 − β
n
φ
u, J
r
n
u
n
− β
n
1 − β
n
g
Ju
n
− JJ
r
n
u
n
≤ β
n
φ
u, u
n
1 − β
n
φ
u, u
n
− β
n
1 − β
n
g
Ju
n
− JJ
r
n
u
n
φ
u, u
n
− β
n
1 − β
n
g
Ju
n
− JJ
r
n
u
n
,
φ
u, y
n
φ
u,
J
r
n
J
−1
β
n
Jx
0
1 −
β
n
J u
n
≤ φ
u, J
−1
β
n
Jx
0
1 −
β
n
J u
n
u
2
− 2
u,
β
n
Jx
0
1 −
β
n
J u
n
β
n
Jx
0
1 −
β
n
J u
n
2
≤
u
2
− 2
β
n
u, Jx
0
−2
1 −
β
n
u, J u
n
β
n
x
0
2
1 −
β
n
u
n
2
β
n
φ
u, x
0
1 −
β
n
φ
u, u
n
β
n
φ
u, x
0
1 −
β
n
φ
u, K
r
n
x
n
≤
β
n
φ
u, x
0
φ
u, x
n
.
4.23
