Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 827082, 16 pages
doi:10.1155/2010/827082
Research Article
Convergence of Iterative Sequences for
Generalized Equilibrium Problems Involving
Inverse-Strongly Monotone Mappings
Shin Min Kang,1 Sun Young Cho,2 and Zeqing Liu3
1
Department of Mathematics and the RINS, Gyeongsang National University, Jinju 660-701, South Korea
Department of Mathematics, Gyeongsang National University, Jinju 660-701, South Korea
3
Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China
2
Correspondence should be addressed to Sun Young Cho, ooly61@yahoo.co.kr
Received 11 December 2009; Accepted 25 January 2010
Academic Editor: Yeol Je Cho
Copyright q 2010 Shin Min Kang 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.
The purpose of this paper is to consider the weak convergence of an iterative sequence for finding a
common element in the set of solutions of generalized equilibrium problems, in the set of solutions
of classical variational inequalities, and in the set of fixed points of nonexpansive mappings.
1. Introduction and Preliminaries
Throughout this paper, we always assume that H is a real Hilbert space with the inner
product ·, · and the norm · and C is a nonempty closed convex subset of H. Let S : C → C
be a nonlinear mapping. In this paper, we use FS to denote the fixed point set of S. Recall
that the mapping S is said to be nonexpansive if
Sx − Sy ≤ x − y,
∀x, y ∈ C.
1.1
Let A : C → H be a mapping. Recall that A is said to be monotone if
Ax − Ay, x − y ≥ 0,
∀x, y ∈ C.
1.2
2
Journal of Inequalities and Applications
A is said to be inverse-strongly monotone if there exists a constant α > 0 such that
Ax − Ay, x − y ≥ αAx − Ay2 ,
∀x, y ∈ C.
1.3
A set-valued mapping R : H → 2H is said to be monotone if for all x, y ∈ H, f ∈ Rx and
g ∈ Ry imply x − y, f − g > 0. A monotone mapping R : H → 2H is maximal if the graph
GR of R is not properly contained in the graph of any other monotone mapping. It is known
that a monotone mapping R is maximal if and only if, for any x, f ∈ H ×H, x −y, f −g ≥ 0
for all y, g ∈ GR implies f ∈ Rx.
Let F be a bifunction of C × C into R, where R denotes the set of real numbers and
A : C → H an inverse-strongly monotone mapping. In this paper, we consider the following
generalized equilibrium problem:
Find x ∈ C such that F x, y Ax, y − x ≥ 0,
∀y ∈ C.
1.4
In this paper, the set of such an x ∈ C is denoted by EPF, A, that is,
EPF, A x ∈ C : F x, y Ax, y − x ≥ 0, ∀y ∈ C .
1.5
Next, we give two special cases of problem 1.4.
I If A ≡ 0, then the generalized equilibrium problem 1.4 is reduced to the following
equilibrium problem:
Find x ∈ C such that F x, y ≥ 0,
∀y ∈ C.
1.6
In this paper, the set of such an x ∈ C is denoted by EPF, that is,
EPF x ∈ C : F x, y ≥ 0, ∀y ∈ C .
1.7
Numerous problems in physics, optimization, and economics reduce to finding a solution of
the equilibrium problem.
To study problems 1.4 and 1.6, we may assume that F satisfies the following
conditions:
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 sup F tz 1 − tx, y ≤ F x, y ;
t↓0
A4 for each x ∈ C, y → Fx, y is convex and weakly lower semicontinuous.
1.8
Journal of Inequalities and Applications
3
II If F ≡ 0, then problem 1.4 is reduced to the classical variational inequality. Find
x ∈ C such that
Ax, y − x ≥ 0,
∀y ∈ C.
1.9
It is known that x ∈ C is a solution to 1.9 if and only if x is a fixed point of the mapping
PC I − λA, where λ > 0 is a constant and I is the identity mapping.
In 2003, Takahashi and Toyoda 1 considered the variational inequality 1.9 and
proved the following theorem.
Theorem 1.1. Let C be a closed convex subset of a real Hilbert space H. Let A be an α-inversestrongly monotone mapping of C into H, and let S be a nonexpansive mapping of C into itself such
that F FS ∩ V IC, A /
∅. Let {xn } be a sequence generated by
x0 ∈ C,
xn1 αn xn 1 − αn SPC xn − λn Axn ,
∀n ≥ 0,
1.10
where λn ∈ a, b for some a, b ∈ 0, 2α and αn ∈ c, d for some c, d ∈ 0, 1. Then, {xn } converges
weakly to z ∈ FS ∩ V IC, A, where z limn → ∞ PF xn .
In 2007, Tada and Takahashi 2 considered the equilibrium problem 1.6 and proved
the following result.
Theorem 1.2. Let C be a nonempty closed convex subset of H. Let F be a bifunction from C × C to R
satisfying (A1)–(A4) and let S be a nonexpansive mapping of C into H such that FS ∩ EPF /
∅.
Let {xn } and {un } be sequences generated by x1 x ∈ H and let
un ∈ C such that Fun , u 1
u − un , un − xn ≥ 0,
rn
xn1 αn xn 1 − αn Sun ,
∀u ∈ C,
1.11
∀n ≥ 1,
where {αn } ⊂ a, b for some a, b ∈ 0, 1 and {rn } ⊂ 0, ∞ satisfies lim infn → ∞ rn > 0. Then, {xn }
converges weakly to w ∈ FS ∩ EPF, where w limn → ∞ PFS∩EPF xn .
Very recently, Moudafi 3 considered the following iterative process:
x0 ∈ C,
F1 un , u 1
u − un , un − xn ≥ 0,
rn
∀u ∈ C,
F2 vn , v 1
v − vn , vn − xn ≥ 0,
rn
∀v ∈ C,
xn1 un vn
,
2
1.12
∀n ≥ 1,
where F1 and F2 are bifunctions and {rn } is a control sequence. A weak convergence theorem
was established; see 3 for more details.
4
Journal of Inequalities and Applications
Weak convergence of iterative sequences has been studied recently for the problems
1.4, 1.6, and 1.9; see 1–14 and the references therein. In this paper, we consider the
generalized equilibrium problem 1.4 and a nonexpansive mapping based on an iterative
process. We show that the sequence generated in the purposed iterative process converges
weakly to a common element in the set of solutions of the variational inequality 1.9, in
the fixed point sets of a nonexpansive mapping and in the solution sets of the generalized
equilibrium problem 1.4. The results presented in this paper improve and extend the
corresponding results announced by Takahashi and Toyoda 1 and Tada and Takahashi 2.
In order to prove our main results, we also need the following lemmas.
Lemma 1.3 see 1. Let C be a nonempty closed convex subset of H. Let {xn } be a sequence in H.
Suppose that, for all y ∈ C,
xn1 − y ≤ xn − y,
∀n ≥ 1,
1.13
then {PC xn } converges strongly to some z ∈ C.
The following lemma can be found in 15.
Lemma 1.4. Let T be a monotone mapping of C into H and NC v the normal cone to C at v ∈ C, that
is,
NC v {w ∈ H : v − u, w ≥ 0, ∀u ∈ C}
1.14
and define a mapping R on C by
Rv ⎧
⎨T v NC v,
v ∈ C,
⎩∅,
v/
∈ C.
1.15
Then R is maximal monotone and 0 ∈ Rv if and only if T v, u − v ≥ 0 for all u ∈ C.
The following lemma can be found in 16, 17.
Lemma 1.5. Let C be a nonempty closed convex subset of H and let F : C × C → R be a bifunction
satisfying (A1)–(A4). Then, for any r > 0 and x ∈ H, there exists z ∈ C such that
1
F z, y y − z, z − x ≥ 0,
r
∀y ∈ C.
1.16
Further, define
1
Tr x z ∈ C : F z, y y − z, z − x ≥ 0, ∀y ∈ C
r
1.17
Journal of Inequalities and Applications
5
for all r > 0 and x ∈ H. Then, the following hold:
a Tr is single-valued;
b Tr is firmly nonexpansive, that is, for any x, y ∈ H,
Tr x − Tr y2 ≤ Tr x − Tr y, x − y ;
1.18
c FTr EPF;
d EPF is closed and convex.
Lemma 1.6 see 18. Let H be a Hilbert space and 0 < p ≤ tn ≤ q < 1 for all n ≥ 1. Suppose that
{xn } and {yn } are sequences in H such that
lim supxn ≤ r,
n→∞
lim supyn ≤ r,
n→∞
1.19
lim tn xn 1 − tn yn r
n→∞
hold for some r ≥ 0, then limn → ∞ xn − yn 0.
Lemma 1.7 see 19. Let H be a real Hilbert space, C a nonempty closed convex subset of H, and
S : C → C a nonexpansive mapping. Then the mapping I − S is demiclosed at zero, that is, if {xn } is
a sequence in C such that xn x and xn − Sxn → 0, then x ∈ FS.
2. Main Results
Theorem 2.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F1 and F2
be two bifunctions from C × C to R which satisfy (A1)–(A4). Let A : C → H be an α-inversestrongly monotone mapping, B : C → H a β-inverse-strongly monotone mapping, T : C → H
an λ-inverse-strongly monotone mapping, and S : C → C a nonexpansive mapping. Assume that
∅. Let {αn } and {βn } be sequences in 0, 1. Let
F : EPF1 , A ∩ EPF2 , B ∩ V IC, T ∩ FS /
{an } be a sequence in 0, 2α, {bn } a sequence in 0, 2β, and {tn } a sequence in 0, 2λ. Let {xn } be
a sequence generated in the following manner:
x1 ∈ C,
F1 un , u Axn , u − un ∀u ∈ C,
1
v − vn , vn − xn ≥ 0, ∀v ∈ C,
bn
yn βn un 1 − βn vn ,
αn xn 1 − αn SPC yn − tn T yn , ∀n ≥ 1.
F2 vn , v Bxn , v − vn xn1
1
u − un , un − xn ≥ 0,
an
Δ
6
Journal of Inequalities and Applications
Assume that the sequences {αn }, {βn }, {an }, {bn }, and {tn } satisfy the following restrictions:
a 0 < a ≤ αn ≤ a < 1, 0 < b ≤ βn ≤ c < 1;
b 0 < d ≤ an ≤ e < 2α, 0 < f ≤ bn ≤ g < 2β, 0 < h ≤ tn ≤ j < 2λ
for some a , a, b, c, d, e, f, g, h, j ∈ R, then the sequence {xn } generated in Δ converges weakly to
some point x ∈ F, where x limn → ∞ PF xn .
Proof. Fix p ∈ F. It follows that
p Sp Tan I − an Ap Tbn I − bn Bp PC I − tn T p,
∀n ≥ 1.
2.1
Note that I − tn T is nonexpansive for each n ≥ 1. Indeed, for any x, y ∈ C, we see that
I − tn T x − I − tn T y2 x − y − tn T x − T y 2
2
2
x − y − 2tn x − y, T x − T y t2n T x − T y
2
≤ x − y − tn 2λ − tn T x − T y2
2
≤ x − y .
2.2
In a similar way, we can obtain that I − an A and I − bn B are nonexpansive for each n ≥ 1. Note
that
un − p ≤ Tan I − an Axn − p ≤ xn − p ,
vn − p ≤ Tbn I − bn Bxn − p ≤ xn − p.
2.3
Put zn PC yn − tn T yn for each n ≥ 1. It follows from 2.3 that
xn1 − p ≤ αn xn − p 1 − αn Szn − p
≤ αn xn − p 1 − αn zn − p
≤ αn xn − p 1 − αn yn − p
≤ αn xn − p 1 − αn βn un − p 1 − βn vn − p
2.4
≤ xn − p.
This implies that limn → ∞ xn − p exists. This shows that {xn } is bounded, so are {yn }, {un },
and {vn }.
On the other hand, we have
un − p2 Ta I − an Axn − p2 ≤ xn − p2 − an 2α − an Axn − Ap2 ,
n
vn − p2 Tb I − bn Bxn − p2 ≤ xn − p2 − bn 2β − bn Bxn − Bp2 .
n
2.5
2.6
Journal of Inequalities and Applications
7
Combining 2.5 with 2.6 yields that
xn1 − p2 ≤ αn xn − p2 1 − αn Szn − p2
2
2
≤ αn xn − p 1 − αn zn − p
2
2
≤ αn xn − p 1 − αn yn − p
2
2 2 ≤ αn xn − p 1 − αn βn un − p 1 − βn vn − p
2
2
2 ≤ αn xn − p 1 − αn βn xn − p − an 2α − an Axn − Ap
2
2 1 − βn xn − p − bn 2β − an Bxn − Bp
2
2
≤ xn − p − 1 − αn βn an 2α − an Axn − Ap
2
− 1 − αn 1 − βn bn 2β − an Bxn − Bp .
2.7
This implies that
2 2 2
1 − αn βn an 2α − an Axn − Ap ≤ xn − p − xn1 − p .
2.8
In view of the restrictions a and b, we obtain that
lim Axn − Ap 0.
2.9
2 2 2
1 − αn 1 − βn bn 2β − an Bxn − Bp ≤ xn − p − xn1 − p .
2.10
n→∞
It also follows from 2.7 that
In view of the restrictions a and b, we obtain that
lim Bxn − Bp 0.
n→∞
2.11
8
Journal of Inequalities and Applications
On the other hand, we see from Lemma 1.5 that
un − p2 Ta I − an Axn − Ta I − an Ap2
n
n
≤ I − an Axn − I − an Ap, un − p
1 I − an Axn − I − an Ap2 un − p2
2
2 −I − an Axn − I − an Ap − un − p
1 xn − p2 un − p2 − xn − un − an Axn − Ap 2
≤
2
1 xn − p2 un − p2
2
2 .
− xn − un 2 − 2an xn − un , Axn − Ap a2n Axn − Ap
2.12
This implies that
un − p2 ≤ xn − p2 − xn − un 2 2an xn − un Axn − Ap.
2.13
In a similar way, we can obtain that
vn − p2 ≤ xn − p2 − xn − vn 2 2bn xn − vn Bxn − Bp.
2.14
It follows from 2.13 and 2.14 that
xn1 − p2 ≤ αn xn − p2 1 − αn Szn − p2
2
2
≤ αn xn − p 1 − αn zn − p
2
2
≤ αn xn − p 1 − αn yn − p
2
2 2 ≤ αn xn − p 1 − αn βn un − p 1 − βn vn − p
2
≤ αn xn − p 1 − αn 2
× βn xn − p − xn − un 2 2an xn − un Axn − Ap
2
1 − βn xn − p − xn − vn 2 2bn xn − vn Bxn − Bp
2
≤ xn − p − 1 − αn βn xn − un 2 2an xn − un Axn − Ap
− 1 − αn 1 − βn xn − vn 2 2bn xn − vn Bxn − Bp.
2.15
Journal of Inequalities and Applications
9
This shows that
2 2
1 − αn βn xn − un 2 ≤ xn − p − xn1 − p 2an xn − un Axn − Ap
2bn xn − vn Bxn − Bp.
2.16
In view of the restriction a, we obtain from 2.9 and 2.11 that
lim xn − un 0.
2.17
n→∞
From 2.15, we also have
2 2
1 − αn 1 − βn xn − vn 2 ≤ xn − p − xn1 − p 2an xn − un Axn − Ap
2bn xn − vn Bxn − Bp.
2.18
In view of the restriction a, we obtain from 2.9 and 2.11 that
lim xn − vn 0.
2.19
n→∞
Since {xn } is bounded, we see that there exits a subsequence {xni } of {xn } which converges
weakly to x. It follows from 2.17 that uni converges weakly to x. Note that
F1 un , u Axn , u − un 1
u − un , un − xn ≥ 0,
an
∀u ∈ C.
2.20
From A2, we see that
Axn , u − un 1
u − un , un − xn ≥ F1 u, un ,
an
∀u ∈ C.
2.21
Replacing n by ni , we arrive at
Axni , u − uni 1
u − uni , uni − xni ≥ F1 u, uni ,
ani
∀u ∈ C.
2.22
For t with 0 < t ≤ 1 and u ∈ C, let ut tu 1 − tx. Since u ∈ C and x ∈ C, we have ut ∈ C. It
follows from 2.22 that
un − xni
ut − uni , Aut ≥ ut − uni , Aut − Axni , ut − uni − ut − uni , i
F1 ut , uni ani
ut − uni , Aut − Auni ut − uni , Auni − Axni uni − xni
− ut − uni ,
F1 ut , uni .
ani
2.23
10
Journal of Inequalities and Applications
From 2.17, we have Auni − Axni → 0 as i → ∞. On the other hand, we obtain from the
monotonicity of A that ut − uni , Aut − Auni ≥ 0. It follows from A4 that
ut − x, Aut ≥ F1 ut , x.
2.24
From A1, A4, and 2.24, we obtain that
0 F1 ut , ut ≤ tF1 ut , u 1 − tF1 ut , x
≤ tF1 ut , u 1 − tut − x, Aut 2.25
tF1 ut , u 1 − ttu − x, Aut ,
which yields that
F1 ut , u 1 − tu − x, Aut ≥ 0.
2.26
Letting t → 0 in the above inequality, we arrive at
F1 x, u u − x, Ax ≥ 0.
2.27
This shows that x ∈ EPF1 , A. In a similar way, we can obtain that x ∈ EPF2 , B.
Next, we claim that x ∈ V IC, T xn1 − p2 ≤ αn xn − p2 1 − αn SPC yn − tn T yn − p2
2
2
≤ αn xn − p 1 − αn yn − tn T yn − p − tn T p
2
2
2 ≤ αn xn − p 1 − αn yn − p − tn 2λ − tn T yn − T p
2.28
2
2
≤ xn − p − 1 − αn tn 2λ − tn T yn − T p .
It follows that
2 2 2
1 − αn tn 2λ − tn T yn − T p ≤ xn − p − xn1 − p .
2.29
This implies from conditions a and b that
lim T yn − T p 0.
n→∞
2.30
Journal of Inequalities and Applications
11
Since PC is firmly nonexpansive, we have
zn − p2 PC I − tn T yn − PC I − tn T p2
≤ I − tn T yn − I − tn T p, zn − p
1 I − tn T yn − I − tn T p2 zn − p2
2
2 − I − tn T yn − I − tn T p − zn − p
1 yn − p2 zn − p2 − yn − zn − tn T yn − T p 2
2
1 yn −p2 zn −p2 − yn −zn 2 − 2tn yn −zn , T yn −T p t2n T yn −T p2 .
2
2.31
≤
So, we obtain that
zn − p2 ≤ xn − p2 − yn − zn 2 2tn yn − zn T yn − T p.
2.32
It follows that
2
2
xn1 − p2 ≤ αn xn − p 1 − αn Szn − p
2
2
≤ αn xn − p 1 − αn zn − p
2
2 2
≤ αn xn − p 1 − αn xn − p − yn − zn 2tn yn − zn T yn − T p .
2.33
Therefore we have
2 2
2 1 − αn yn − zn ≤ xn − p − xn1 − p 2tn yn − zn T yn − T p.
2.34
From the restriction a and 2.30, we get that
lim yn − zn 0.
2.35
xn − zn ≤ xn − yn yn − zn ≤ βn xn − un 1 − βn xn − vn yn − zn .
2.36
n→∞
Note that
From 2.17, 2.19, and 2.35, we obtain that
lim xn − zn 0.
n→∞
2.37
12
Journal of Inequalities and Applications
Define a mapping R by
Rv ⎧
⎨T v NC v,
v ∈ C,
⎩∅,
v/
∈ C.
2.38
Let v, w ∈ GR. Since w − T v ∈ NC v and zn ∈ C, we obtain that
v − zn , w − T v ≥ 0.
2.39
As zn PC yn − tn T yn and v ∈ C, we get that
zn − yn
v − zn ,
T yn ≥ 0.
tn
2.40
From 2.39 and 2.40, we obtain that
v − zni , w ≥ v − zni , T v
zni − yni
≥ v − zni , T v − v − zni ,
T yni
tni
zni − yni
v − zni , T v − T yni −
tni
2.41
zn − yni
v − zni , T v − T zni v − zni , T zni − T yni − v − zni , i
tni
zn − yni
≥ v − zni , T zni − T yni − v − zni , i
.
tni
Note that T is Lipschitz. On the other hand, we see from 2.37 that zni x. Hence, we get
that
v − x, w ≥ 0.
2.42
Since R is maximal monotone, we obtain that x ∈ R−1 0. From Lemma 1.4, we get that x ∈
V IC, T .
Finally, we show that x ∈ FS. Note that
xn1 − p ≤ αn xn − p 1 − αn Szn − p ,
Szn − p ≤ zn − p ≤ xn − p.
2.43
Journal of Inequalities and Applications
13
Since limn → ∞ xn − p exists, we may assume that limn → ∞ xn − p r for some positive
constant r. Then we see that
lim supxn1 − p ≤ r,
n→∞
lim supxn − p ≤ r,
n→∞
lim supSzn − p ≤ r.
2.44
n→∞
In view of Lemma 1.6, we get that
lim xn − Szn 0.
2.45
n→∞
Furthermore, we know that
Sxn − xn ≤ Sxn − Szn Szn − xn ≤ xn − zn Szn − xn .
2.46
From 2.37 and 2.45, we obtain that
lim Sxn − xn 0.
2.47
n→∞
Note that xni x and Sxni − xni → 0 as i → ∞. From Lemma 1.7, we arrive at x ∈ FS.
x. In
Assume that there exists another subsequence {xnj } of {xn }, converges to x , where x /
view of the Opial’s condition, we see that
lim xn − x lim infxni − x < lim infxni − x n→∞
i→∞
i→∞
lim xn − x lim infxnj − x n→∞
j →∞
2.48
< lim xnj − x lim xn − x.
j →∞
n→∞
This is a contradiction. So, we have x x.
Let hn PF xn . Since x ∈ F, we have
xn − hn , hn − x ≥ 0.
2.49
From 2.4 and Lemma 1.3, we get that {hn } converges strongly to some υ ∈ F. Since {xn }
converges weakly to x, we have
x − υ, υ − x ≥ 0.
2.50
14
Journal of Inequalities and Applications
Hence we obtain that
x υ lim PF xn .
2.51
n→∞
This completes the proof.
Corollary 2.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a
bifunction from C × C to R which satisfies (A1)–(A4). Let A : C → H be an α-inverse-strongly
monotone mapping, T : C → H an λ-inverse-strongly monotone mapping, and S : C → C a
nonexpansive mapping. Assume that F : EPF, A ∩ V IC, T ∩ FS /
∅. Let {αn } be a sequence
in 0, 1. Let {an } be a sequence in 0, 2α, and {tn } a sequence in 0, 2λ. Let {xn } be a sequence
generated in the following manner:
x1 ∈ C,
Fun , u Axn , u − un 1
u − un , un − xn ≥ 0,
an
xn1 αn xn 1 − αn SPC un − tn T un ,
∀u ∈ C,
2.52
∀n ≥ 1.
Assume that the sequences {αn }, {an }, and {tn } satisfy the following restrictions:
a 0 < a ≤ αn ≤ a < 1;
b 0 < d ≤ an ≤ e < 2α, 0 < h ≤ tn ≤ j < 2λ
for some a , a, d, e, h, j ∈ R, then the sequence {xn } converges weakly to some point x ∈ F, where
x limn → ∞ PF xn .
Proof. Putting F1 F2 F, A B, and an bn in Theorem 2.1, we see that yn un . From the
proof of Theorem 2.1, we can conclude the desired conclusion immediately.
Corollary 2.3. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a
bifunction from C × C to R which satisfies (A1)–(A4). Let A : C → H be an α-inverse-strongly
monotone mapping and S : C → C a nonexpansive mapping. Assume that F : EPF, A∩FS /
∅.
Let {αn } be a sequence in 0, 1. Let {an } be a sequence in 0, 2α. Let {xn } be a sequence generated
in the following manner:
x1 ∈ C,
Fun , u Axn , u − un 1
u − un , un − xn ≥ 0,
an
xn1 αn xn 1 − αn Sun ,
∀n ≥ 1.
Assume that the sequences {αn } and {an } satisfy the following restrictions:
a 0 < a ≤ αn ≤ a < 1;
b 0 < d ≤ an ≤ e < 2α
∀u ∈ C,
2.53
Journal of Inequalities and Applications
15
for some a , a, d, e ∈ R, then the sequence {xn } converges weakly to some point x ∈ F, where x limn → ∞ PF xn .
Proof. Putting T 0 in Corollary 2.2, we can conclude the desired conclusion immediately.
Remark 2.4. Corollary 2.3 is a generalization of Theorem 1.2 in Section 1. More precisely,
Corollary 2.3 is reduced to Theorem 1.2 if A 0.
Corollary 2.5. Let C be a nonempty closed convex subset of a real Hilbert space H. Let A : C → H
be an α-inverse-strongly monotone mapping, B : C → H a β-inverse-strongly monotone mapping,
T : C → H an λ-inverse-strongly monotone mapping, and S : C → C a nonexpansive mapping.
Assume that F : V IC, A ∩ V IC, B ∩ V IC, T ∩ FS / ∅. Let {αn } and {βn } be sequences in
0, 1. Let {an } be a sequence in 0, 2α, {bn } a sequence in 0, 2β, and {tn } a sequence in 0, 2λ.
Let {xn } be a sequence generated in the following manner:
x1 ∈ C,
yn βn PC xn − an Axn 1 − βn PC xn − bn Bxn ,
xn1 αn xn 1 − αn SPC yn − tn T yn , ∀n ≥ 1.
2.54
Assume that the sequences {αn }, {βn }, {an }, {bn }, and {tn } satisfy the following restrictions:
a 0 < a ≤ αn ≤ a < 1, 0 < b ≤ βn ≤ c < 1;
b 0 < d ≤ an ≤ e < 2α, 0 < f ≤ bn ≤ g < 2β, 0 < h ≤ tn ≤ j < 2λ
for some a , a, b, c, d, e, f, g, h, j ∈ R, then the sequence {xn } converges weakly to some point x ∈ F,
where x limn → ∞ PF xn .
Proof. Putting F1 F2 0, we see that
Axn , u − un 1
u − un , un − xn ≥ 0
an
2.55
is equivalent to
un PC xn − an Axn ,
∀n ≥ 1.
2.56
∀n ≥ 1.
2.57
In the same way, we can obtain that
vn PC xn − bn Bxn ,
From the proof of Theorem 2.1, we can conclude the desired conclusion immediately.
Remark 2.6. Corollary 2.5 is a generalization of Theorem 1.1 in Section 1. More precisely,
Corollary 2.5 is reduced to Theorem 1.1 if T 0, A B, and an bn for each n ≥ 1.
16
Journal of Inequalities and Applications
Acknowledgments
This work was supported by the Korea Research Foundation Grant funded by the Korean
Government KRF-2008-313-C00050 and the Science Research Foundation of Educational
Department of Liaoning Province 2009A419.
References
1 W. Takahashi and M. Toyoda, “Weak convergence theorems for nonexpansive mappings and
monotone mappings,” Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 417–428,
2003.
2 A. Tada and W. Takahashi, “Weak and strong convergence theorems for a nonexpansive mapping and
an equilibrium problem,” Journal of Optimization Theory and Applications, vol. 133, no. 3, pp. 359–370,
2007.
3 A. Moudafi, “On the convergence of splitting proximal methods for equilibrium problems in Hilbert
spaces,” Journal of Mathematical Analysis and Applications, vol. 359, no. 2, pp. 508–513, 2009.
4 C. Jaiboon, P. Kumam, and U. W. Humphries, “Weak convergence theorem by an extragradient
method for variational inequality, equilibrium and fixed point problems,” Bulletin of the Malaysian
Mathematical Sciences Society, vol. 32, no. 2, pp. 173–185, 2009.
5 H. Iiduka and W. Takahashi, “Weak convergence theorems by Cesáro means for nonexpansive
mappings and inverse-strongly-monotone mappings,” Journal of Nonlinear and Convex Analysis, vol.
7, no. 1, pp. 105–113, 2006.
6 H. Iiduka, W. Takahashi, and M. Toyoda, “Approximation of solutions of variational inequalities for
monotone mappings,” PanAmerican Mathematical Journal, vol. 14, no. 2, pp. 49–61, 2004.
7 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.
8 W. Nilsrakoo and S. Saejung, “Weak and strong convergence theorems for countable Lipschitzian
mappings and its applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 8, pp.
2695–2708, 2008.
9 N. Nadezhkina and W. Takahashi, “Weak convergence theorem by an extragradient method for
nonexpansive mappings and monotone mappings,” Journal of Optimization Theory and Applications,
vol. 128, no. 1, pp. 191–201, 2006.
10 S. Plubtieng and P. Kumam, “Weak convergence theorem for monotone mappings and a countable
family of nonexpansive mappings,” Journal of Computational and Applied Mathematics, vol. 224, no. 2,
pp. 614–621, 2009.
11 J.-W. Peng and J.-C. Yao, “Weak convergence of an iterative scheme for generalized equilibrium
problems,” Bulletin of the Australian Mathematical Society, vol. 79, no. 3, pp. 437–453, 2009.
12 X. Qin, Y. J. Cho, and S. M. Kang, “Convergence theorems of common elements for equilibrium
problems and fixed point problems in Banach spaces,” Journal of Computational and Applied
Mathematics, vol. 225, no. 1, pp. 20–30, 2009.
13 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.
14 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.
15 R. T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,” Transactions of the
American Mathematical Society, vol. 149, pp. 75–88, 1970.
16 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.
17 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.
18 J. Schu, “Weak and strong convergence to fixed points of asymptotically nonexpansive mappings,”
Bulletin of the Australian Mathematical Society, vol. 43, no. 1, pp. 153–159, 1991.
19 G. Marino and H.-K. Xu, “Weak and strong convergence theorems for strict pseudo-contractions in
Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 336–346, 2007.