Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 312602, 18 pages
doi:10.1155/2010/312602
Research Article
Hybrid Projection Algorithms for
Generalized Equilibrium Problems and
Strictly Pseudocontractive Mappings
Jong Kyu Kim,1 Sun Young Cho,2 and Xiaolong Qin3
1
Department of Mathematics Education, Kyungnam University, Masan 631-701, Republic of Korea
Department of Mathematics, Gyeongsang National University, Chinju 660-701, Republic of Korea
3
Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China
2
Correspondence should be addressed to Jong Kyu Kim, jongkyuk@kyungnam.ac.kr
Received 12 October 2009; Accepted 19 July 2010
Academic Editor: András Rontó
Copyright q 2010 Jong Kyu Kim 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 problem of finding a common element in the solution
set of equilibrium problems and in the fixed point set of a strictly pseudocontractive mapping.
Strong convergence of the purposed hybrid projection algorithm is obtained in Hilbert spaces.
1. Introduction and Preliminaries
Let H be a real Hilbert space with inner product ·, · and norm · . Let C be a nonempty
closed convex subset of H and S : C → C 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
S is said to be k-strictly pseudocontractive if there exists a constant k ∈ 0, 1 such that
Sx − Sy2 ≤ x − y2 kx − Sx − y − Sy 2 ,
∀x, y ∈ C.
1.2
∀x, y ∈ C.
1.3
S is said to be pseudocontractive if
Sx − Sy2 ≤ x − y2 x − Sx − y − Sy 2 ,
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Journal of Inequalities and Applications
The class of strictly pseudocontractive mappings was introduced by Browder and Petryshyn
1 in 1967. It is easy to see that the class of strictly pseudocontractive mappings falls into the
class of nonexpansive mappings and the class of pseudocontractions.
Let A : C → H be a mapping. Recall that A is said to be monotone if
Ax − Ay, x − y ≥ 0,
∀x, y ∈ C.
1.4
A is said to be inverse-strongly monotone if there exists a constant α > 0 such that
2
Ax − Ay, x − y ≥ αAx − Ay ,
∀x, y ∈ C.
1.5
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.
1.6
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 .
1.7
To study the generalized equilibrium problems 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 ;
1.8
t↓0
A4 for each x ∈ C, y → Fx, y is convex and weakly lower semicontinuous.
Next, we give two special cases of the problem 1.6.
I If A ≡ 0, then the generalized equilibrium problem 1.6 is reduced to the following
equilibrium problem:
Find x ∈ C such that F x, y ≥ 0,
∀y ∈ C.
1.9
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 .
1.10
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3
II If F ≡ 0, then the problem 1.6 is reduced to the following classical variational
inequality. Find x ∈ C such that
Ax, y − x ≥ 0,
∀y ∈ C.
1.11
It is known that x ∈ C is a solution to 1.11 if and only if x is a fixed point of the
mapping PC I − ρA, where ρ > 0 is a constant and I is the identity mapping.
Recently, many authors studied the problems 1.6 and 1.9 based on iterative
methods; see, for example, 2–18.
In 2007, Tada and Takahashi 17 considered the problem 1.9 and proved the
following result.
Theorem TT. 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
1
F un , y y − un , un − xn ≥ 0,
rn
∀y ∈ C,
wn 1 − αn xn αn Sun ,
Cn {z ∈ H : wn − z ≤ xn − z},
1.12
Dn {z ∈ H : xn − z, x − xn ≥ 0},
xn1 PCn ∩Dn x,
for every n ≥ 1, where {αn } ⊂ a, 1 for some a ∈ 0, 1 and {rn } ⊂ 0, ∞ satisfies lim infn → ∞ rn >
0. Then, {xn } converges strongly to PFS∩EPF x.
In this paper, we consider the generalized equilibrium problem 1.6 and a strictly
pseudocontractive mapping based on the shrinking projection algorithm which was first
introduced by Takahashi et al. 18. A strong convergence of common elements of the
fixed point sets of the strictly pseudocontractive mapping and of the solution sets of the
generalized equilibrium problem is established in the framework of Hilbert spaces. The
results presented in this paper improve and extend the corresponding results announced
by Tada and Takahashi 17.
In order to prove our main results, we also need the following definitions and lemmas.
Lemma 1.1 see 19. Let C be a nonempty closed convex subset of a Hilbert space H and T : C →
C a k-strict pseudocontraction. Then T is 1 k/1 − k-Lipschitz and I − T is demiclosed, this is, if
{xn } is a sequence in C with xn x and xn − T xn → 0, then x ∈ FT .
The following lemma can be found in 2, 3.
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Lemma 1.2. 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.
1.13
Further, define
Tr x 1
z ∈ C : F z, y y − z, z − x ≥ 0, ∀y ∈ C
r
1.14
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 ;
1.15
c FTr EPF;
d EPF is closed and convex.
Lemma 1.3 see 1. Let C be a nonempty closed convex subset of a real Hilbert space H and S :
C → C a k-strict pseudocontraction with a fixed point. Define S : C → C by Sa x ax 1 − aSx
for each x ∈ C. If a ∈ k, 1, then Sa is nonexpansive with FSa 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 satisfies A1–A4. Let A : C → H be an α-inversestrongly monotone mapping, B : C → H a β-inverse-strongly monotone mapping, and S : C → C
Journal of Inequalities and Applications
5
a k-strict pseudocontraction. Let {rn } and {sn } be two positive real sequences. Assume that F :
EPF1 , A∩FP F2 , B∩FS is not empty. Let {xn } be a sequence generated in the following manner:
x1 ∈ C,
C1 C,
F1 un , u Axn , u − un 1
u − un , un − xn ≥ 0,
rn
∀u ∈ C,
1
v − vn , vn − xn ≥ 0, ∀v ∈ C,
sn
zn γn un 1 − γn vn ,
yn αn xn 1 − αn βn zn 1 − βn Szn ,
Cn1 w ∈ Cn : yn − w ≤ xn − w ,
F2 vn , v Bxn , v − vn xn1 PCn1 x1 ,
Υ
n ≥ 1,
where {αn }, {βn }, and {γn } are sequences in 0, 1. Assume that {αn }, {βn }, {γn }, {rn }, and {sn }
satisfy the following restrictions:
a 0 ≤ αn ≤ a < 1;
b 0 ≤ k ≤ βn < b < 1;
c 0 ≤ c ≤ γn ≤ d < 1;
d 0 < e ≤ rn ≤ f < 2α and 0 < e ≤ sn ≤ f < 2β.
Then the sequence {xn } generated in Υ converges strongly to some point x, where x PF x1 .
Proof. Note that un can be rewritten as
un Trn xn − rn Axn ,
∀n ≥ 1
2.1
vn Tsn xn − sn Bxn ,
∀n ≥ 1.
2.2
and vn can be rewritten as
Fix p ∈ F. It follows that
p Sp Trn p − rn Ap Tsn p − sn Bp ,
∀n ≥ 1.
2.3
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Journal of Inequalities and Applications
Note that I − rn A is nonexpansive for each n ≥ 1. Indeed, for any x, y ∈ C, we see from the
restriction d that
I − rn Ax − I − rn Ay2 x − y − rn Ax − Ay 2
2
2
x − y − 2rn x − y, Ax − Ay rn2 Ax − Ay
2
2
≤ x − y − rn 2α − rn Ax − Ay
2
≤ x − y .
2.4
This shows that I − rn A is nonexpansive for each n ≥ 1. In a similar way, we can obtain that
I − sn B is nonexpansive for each n ≥ 1. It follows that
un − p ≤ xn − p,
un − p ≤ xn − p.
2.5
This implies that
zn − p ≤ γn un − p 1 − γn vn − p ≤ xn − p.
2.6
Now, we are in a position to show that Cn is closed and convex for each n ≥ 1. From the
assumption, we see that C1 C is closed and convex. Suppose that Cm is closed and convex
for some m ≥ 1. We show that Cm1 is closed and convex for the same m. Indeed, for any
w ∈ Cm , we see that
ym − w ≤ xm − w
2.7
2
ym − xm 2 − 2 w, ym − xm ≥ 0.
2.8
is equivalent to
Thus Cm1 is closed and convex. This shows that Cn is closed and convex for each n ≥ 1.
Next, we show that F ⊂ Cn for each n ≥ 1. From the assumption, we see that F ⊂ C C1 . Suppose that F ⊂ Cm for some m ≥ 1. Putting
Sn βn I 1 − βn S,
∀n ≥ 1,
2.9
we see from Lemma 1.3 that Sn is a nonexpansive mapping for each n ≥ 1. For any w ∈ F ⊂
Cm , we see from 2.6 that
ym − w αm xm 1 − αm Sm zm − w
≤ αm xm − w 1 − αm zm − w
≤ xm − w.
2.10
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7
This shows that w ∈ Cm1 . This proves that F ⊂ Cn for each n ≥ 1. Note xn PCn x1 . For each
w ∈ F ⊂ Cn , we have
x1 − xn ≤ x1 − w.
2.11
x1 − xn ≤ x1 − PF x1 .
2.12
In particular, we have
This implies that {xn } is bounded. Since xn PCn x1 and xn1 PCn1 x1 ∈ Cn1 ⊂ Cn , we have
0 ≤ x1 − xn , xn − xn1 x1 − xn , xn − x1 x1 − xn1 2.13
≤ −x1 − xn 2 x1 − xn x1 − xn1 .
It follows that
xn − x1 ≤ xn1 − x1 .
2.14
This proves that limn → ∞ xn − x1 exists. Notice that
xn − xn1 2 xn − x1 x1 − xn1 2
xn − x1 2 2xn − x1 , x1 − xn1 x1 − xn1 2
xn − x1 2 2xn − x1 , x1 − xn xn − xn1 x1 − xn1 2
2.15
xn − x1 2 − 2xn − x1 2 2xn − x1 , xn − xn1 x1 − xn1 2
≤ x1 − xn1 2 − xn − x1 2 .
It follows that
lim xn − xn1 0.
n→∞
2.16
Since xn1 PCn1 x1 ∈ Cn1 , we see that
yn − xn1 ≤ xn − xn1 .
2.17
yn − xn ≤ yn − xn1 xn − xn1 ≤ 2xn − xn1 .
2.18
This implies that
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Journal of Inequalities and Applications
From 2.16, we obtain that
lim xn − yn 0.
2.19
xn − yn xn − αn xn − 1 − αn Sn zn 1 − αn xn − Sn zn .
2.20
n→∞
On the other hand, we have
From the assumption 0 ≤ αn ≤ a < 1 and 2.19, we have
lim xn − Sn zn 0.
n→∞
2.21
For any p ∈ F, we have
un − p2 Tr I − rn Axn − Tr I − rn Ap2
n
n
2
xn − p − rn Axn − Ap 2
2
xn − p − 2rn xn − p, Axn − Ap rn2 Axn − Ap
2
2
≤ xn − p − rn 2α − rn Axn − Ap .
2.22
In a similar way, we also have
vn − p2 ≤ xn − p2 − sn 2β − sn Bxn − Bp2 .
2.23
Note that
yn − p2 αn xn 1 − αn Sn zn − p2
2
2
≤ αn xn − p 1 − αn Sn zn − p
2
2
≤ αn xn − p 1 − αn zn − p
2
2
2
≤ αn xn − p 1 − αn γn un − p 1 − αn 1 − γn vn − p .
2.24
Substituting 2.22 and 2.23 into 2.24, we arrive at
yn − p2 ≤ xn − p2 − 1 − αn γn rn 2α − rn Axn − Ap2
2
− 1 − αn 1 − γn sn 2β − sn Bxn − Bp .
2.25
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9
It follows that
2 2 2
1 − αn γn rn 2α − rn Axn − Ap ≤ xn − p − yn − p
≤ xn − p yn − p xn − yn .
2.26
In view of the restrictions a–d and 2.19, we obtain that
lim Axn − Ap 0.
n→∞
2.27
It also follows from 2.25 that
2 2 2
1 − αn 1 − γn sn 2β − sn Bxn − Bp ≤ xn − p − yn − p
≤ xn − p yn − p xn − yn .
2.28
By virtue of the restrictions a–d and 2.19, we get that
lim Bxn − Bp 0.
n→∞
2.29
On the other hand, we have from Lemma 1.1 that
un − p2 Tr I − rn Axn − Tr I − rn Ap2
n
n
≤ I − rn Axn − I − rn Ap, un − p
1 I −rn Axn − I −rn Ap2 un −p2 − I −rn Axn −I −rn Ap− un −p 2
2
1 xn − p2 un − p2 − xn − un − rn Axn − Ap2
≤
2
1 xn −p2 un −p2 − xn −un 2 −2rn xn −un , Axn −Ap rn2 Axn −Ap2 .
2
2.30
This implies that
un − p2 ≤ xn − p2 − xn − un 2 2rn xn − un Axn − Ap.
2.31
In a similar way, we can also obtain that
vn − p2 ≤ xn − p2 − xn − vn 2 2sn xn − vn Bxn − Bp.
2.32
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Journal of Inequalities and Applications
Substituting 2.31 and 2.32 into 2.24, we obtain that
yn − p2 ≤ xn − p2 − 1 − αn γn xn − un 2 2rn 1 − αn γn xn − un Axn − Ap
− 1 − αn 1 − γn xn − vn 2 2sn 1 − αn 1 − γn xn − vn Bxn − Bp
2
≤ xn − p − 1 − αn γn xn − un 2 2rn xn − un Axn − Ap
− 1 − αn 1 − γn xn − vn 2 2sn xn − vn Bxn − Bp.
2.33
It follows that
2 2
1 − αn γn xn − un 2 ≤ xn − p − yn − p 2rn xn − un Axn − Ap
2sn xn − vn Bxn − Bp
≤ xn − p yn − p xn − yn 2rn xn − un Axn − Ap
2sn xn − vn Bxn − Bp.
2.34
In view of the restrictions a and c, we obtain from 2.27 and 2.29 that
lim xn − un 0.
n→∞
2.35
It also follows from 2.33 that
2 2
1 − αn 1 − γn xn − vn 2 ≤ xn − p − yn − p 2rn xn − un Axn − Ap
2sn xn − vn Bxn − Bp
≤ xn − p yn − p xn − yn 2rn xn − un Axn − Ap
2sn xn − vn Bxn − Bp.
2.36
Thanks to the restrictions a and c, we obtain from 2.27 and 2.29 that
lim xn − vn 0.
2.37
zn − xn ≤ γn un − xn 1 − γn vn − xn .
2.38
n→∞
Note that
From 2.35 and 2.37, we see that
lim xn − zn 0.
n→∞
2.39
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11
On the other hand, we see from 2.21 that
βn zn − xn 1 − βn Szn − xn −→ 0
2.40
as n → ∞. In view of 2.39 and the restriction b, we obtain that
lim xn − Szn 0.
n→∞
2.41
Note that
Sxn − xn ≤ Sxn − Szn Szn − xn ≤
1k
xn − zn Szn − xn .
1−k
2.42
It follows from 2.39 and 2.41 that
lim xn − Sxn 0.
n→∞
2.43
Since {xn } is bounded, we assume that a subsequence {xni } of {xn } converges weakly to ξ.
Next, we show that ξ ∈ FS ∩ EPF1 , A ∩ EPF2 , B. First, we prove that ξ ∈ EPF1 , A.
Since un Trn xn − rn Axn for any u ∈ C, we have
F1 un , u Axn , u − un 1
u − un , un − xn ≥ 0.
rn
2.44
From the condition A2, we see that
Axn , u − un 1
u − un , un − xn ≥ F1 u, un .
rn
2.45
Replacing n by ni , we arrive at
uni − xni
,
u
−
u
,
≥ F1 u, uni .
u
−
u
Axni
ni ni
rni
2.46
For any t with 0 < t ≤ 1 and u ∈ C, let ut tu 1 − tξ. Since u ∈ C and ξ ∈ C, we have ut ∈ C.
It follows from 2.46 that
un − xni
F1 ut , uni ut − uni , Aut ≥ ut − uni , Aut − Axni , ut − uni − ut − uni , i
rni
un −xni
F1 ut , uni .
ut −uni , Aut −Auni ut −uni , Auni −Axni − ut −uni , i
rni
2.47
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Journal of Inequalities and Applications
Since A is Lipschitz continuous, we obtain from 2.35 that Auni − Axni → 0 as i → ∞. On
the other hand, we get from the monotonicity of A that
ut − uni , Aut − Auni ≥ 0.
2.48
It follows from A4 and 2.47 that
ut − ξ, Aut ≥ F1 ut , ξ.
2.49
From A1, A4, and 2.49, we see that
0 F1 ut , ut ≤ tF1 ut , u 1 − tF1 ut , ξ
≤ tF1 ut , u 1 − tut − ξ, Aut 2.50
tF1 ut , u 1 − ttu − ξ, Aut ,
which yields that
F1 ut , u 1 − tu − ξ, Aut ≥ 0.
2.51
Letting t → 0 in the above inequality, we arrive at
F1 ξ, u u − ξ, Aξ ≥ 0.
2.52
This shows that ξ ∈ EPF1 , A. In a similar way, we can obtain that ξ ∈ EPF2 , B.
Next, we show that ξ ∈ FS. We can conclude from Lemma 1.1 the desired conclusion
easily. This proves that ξ ∈ F. Put x PF x1 . Since x PF x1 ⊂ Cn1 and xn1 PCn1 x1 , we
have
x1 − xn1 ≤ x1 − x.
2.53
On the other hand, we have
x1 − x ≤ x1 − ξ
≤ lim infx1 − xni i→∞
≤ lim supx1 − xni i→∞
≤ x1 − x.
2.54
Journal of Inequalities and Applications
13
We, therefore, obtain that
x1 − ξ lim x1 − xni x1 − x.
2.55
i→∞
This implies xni → ξ x. Since {xni } is an arbitrary subsequence of {xn }, we obtain that
xn → x as n → ∞. This completes the proof.
If S is nonexpansive, then we have from Theorem 2.1 the following result immediately.
Corollary 2.2. 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 satisfies A1–A4. Let A : C → H be an αinverse-strongly monotone mapping, B : C → H a β-inverse-strongly monotone mapping, and
S : C → C a nonexpansive mapping. Let {rn } and {sn } be two positive real sequences. Assume that
F : EPF1 , A ∩ FP F2 , B ∩ FS is not empty. Let {xn } be a sequence generated in the following
manner:
x1 ∈ C,
C1 C,
F1 un , u Axn , u − un 1
u − un , un − xn ≥ 0,
rn
∀u ∈ C,
F2 vn , v Bxn , v − vn 1
v − vn , vn − xn ≥ 0,
sn
∀v ∈ C,
2.56
yn αn xn 1 − αn S γn un 1 − γn vn ,
Cn1 w ∈ Cn : yn − w ≤ xn − w ,
xn1 PCn1 x1 ,
n ≥ 1,
where {αn } and {γn } are sequences in 0, 1. Assume that {αn }, {γn }, {rn }, and {sn } satisfy the
following restrictions:
a 0 ≤ αn ≤ a < 1;
b 0 ≤ c ≤ γn ≤ d < 1;
c 0 < e ≤ rn ≤ f < 2α and 0 < e ≤ sn ≤ f < 2β.
Then the sequence {xn } converges strongly to some point x, where x PF x1 .
As applications of Theorem 2.1, we consider the problems 1.9 and 1.11.
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Journal of Inequalities and Applications
Theorem 2.3. 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, and
S : C → C a k-strict pseudocontraction. Let {rn } and {sn } be two positive real sequences. Assume
that F : V IC, A ∩ V IC, B ∩ FS is not empty. Let {xn } be a sequence generated in the following
manner:
x1 ∈ C,
C1 C,
zn γn PC I − rn Axn 1 − γn PC I − sn Bxn ,
yn αn xn 1 − αn βn zn 1 − βn Szn ,
Cn1 w ∈ Cn : yn − w ≤ xn − w ,
xn1 PCn1 x1 ,
2.57
n ≥ 1,
where {αn }, {βn }, and {γn } are sequences in 0, 1. Assume that {αn }, {βn }, {γn }, {rn }, and {sn }
satisfy the following restrictions:
a 0 ≤ αn ≤ a < 1;
b 0 ≤ k ≤ βn < b < 1;
c 0 ≤ c ≤ γn ≤ d < 1;
d 0 < e ≤ rn ≤ f < 2α and 0 < e ≤ sn ≤ f < 2β.
Then the sequence {xn } converges strongly to some point x, where x PF x1 .
Proof. Putting F1 F2 ≡ 0, we see that
Axn , u − un 1
u − un , un − xn ≥ 0,
rn
∀u ∈ C,
2.58
is equivalent to
xn − rn Axn − un , un − u ≥ 0,
∀u ∈ C.
2.59
This implies that un PC xn − rn Axn . We also have vn PC xn − sn Bxn . We can obtain from
Theorem 2.1 the desired results immediately.
Corollary 2.4. 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,
and S : C → C a nonexpansive mapping. Let {rn } and {sn } be two positive real sequences. Assume
Journal of Inequalities and Applications
15
that F : V IC, A ∩ V IC, B ∩ FS is not empty. Let {xn } be a sequence generated in the following
manner:
x1 ∈ C,
C1 C,
zn γn PC I − rn Axn 1 − γn PC I − sn Bxn ,
Cn1
yn αn xn 1 − αn Szn ,
w ∈ Cn : yn − w ≤ xn − w ,
xn1 PCn1 x1 ,
2.60
n ≥ 1,
where {αn } and {γn } are sequences in 0, 1. Assume that {αn }, {γn }, {rn }, and {sn } satisfy the
following restrictions:
a 0 ≤ αn ≤ a < 1;
b 0 ≤ c ≤ γn ≤ d < 1;
c 0 < e ≤ rn ≤ f < 2α and 0 < e ≤ sn ≤ f < 2β.
Then the sequence {xn } converges strongly to some point x, where x PF x1 .
Theorem 2.5. 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 satisfies A1–A4. Let S : C → C be a kstrict pseudocontraction. Let {rn } and {sn } be two positive real sequences. Assume that F :
EPF1 ∩ FP F2 ∩ FS is not empty. Let {xn } be a sequence generated in the following manner:
x1 ∈ C,
C1 C,
F1 un , u 1
u − un , un − xn ≥ 0,
rn
∀u ∈ C,
1
v − vn , vn − xn ≥ 0, ∀v ∈ C,
sn
zn γn un 1 − γn vn ,
yn αn xn 1 − αn βn zn 1 − βn Szn ,
Cn1 w ∈ Cn : yn − w ≤ xn − w ,
F2 vn , v xn1 PCn1 x1 ,
2.61
n ≥ 1,
where {αn }, {βn }, and {γn } are sequences in 0, 1. Assume that {αn }, {βn }, {γn }, {rn }, and {sn }
satisfy the following restrictions:
a 0 ≤ αn ≤ a < 1;
b 0 ≤ k ≤ βn < b < 1;
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Journal of Inequalities and Applications
c 0 ≤ c ≤ γn ≤ d < 1;
d 0 < e ≤ rn ≤ f < ∞ and 0 < e ≤ sn ≤ f < ∞.
Then the sequence {xn } converges strongly to some point x, where x PF x1 .
Proof. Putting A B 0, we can obtain from Theorem 2.1 the desired conclusion
immediately.
Remark 2.6. Theorem 2.5 is generalization of Theorem TT. To be more precise, we consider a
pair of bifunctions and a strictly pseudocontractive mapping.
Let T : C → C be a k-strict pseudocontraction. It is known that I − T is a 1 − k/2inverse-strongly monotone mapping. The following results are not hard to derive.
Theorem 2.7. 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 satisfies A1–A4. Let TA : C → C be a
kα -strict pseudocontraction, B : C → C a kβ -strict pseudocontraction, and S : C → C a
k-strict pseudocontraction. Let {rn } and {sn } be two positive real sequences. Assume that F :
EPF1 , I − TA ∩ FP F2 , I − TB ∩ FS is not empty. Let {xn } be a sequence generated in the following
manner:
x1 ∈ C,
C1 C,
1
u − un , un − xn ≥ 0,
rn
∀u ∈ C,
1
v − vn , vn − xn ≥ 0,
sn
zn γn un 1 − γn vn ,
yn αn xn 1 − αn βn zn 1 − βn Szn ,
Cn1 w ∈ Cn : yn − w ≤ xn − w ,
∀v ∈ C,
F1 un , u I − TA xn , u − un F2 vn , v I − TB xn , v − vn xn1 PCn1 x1 ,
2.62
n ≥ 1,
where {αn }, {βn }, and {γn } are sequences in 0, 1. Assume that {αn }, {βn }, {γn }, {rn }, and {sn }
satisfy the following restrictions:
a 0 ≤ αn ≤ a < 1;
b 0 ≤ k ≤ βn < b < 1;
c 0 ≤ c ≤ γn ≤ d < 1;
d 0 < e ≤ rn ≤ f < 1 − kα and 0 < e ≤ sn ≤ f < 1 − kβ .
Then the sequence {xn } converges strongly to some point x, where x PF x1 .
Journal of Inequalities and Applications
17
Acknowledgment
This work was supported by a National Research Foundation of Korea Grant funded by the
Korean Government 2009-0076898.
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