Volume 2011, Article ID 274820, 19 pagesdoi:10.1155/2011/274820 Research Article Hybrid Algorithm for Finding Common Elements of the Set of Generalized Equilibrium Problems and the Set o
Trang 1Volume 2011, Article ID 274820, 19 pages
doi:10.1155/2011/274820
Research Article
Hybrid Algorithm for Finding Common
Elements of the Set of Generalized Equilibrium
Problems and the Set of Fixed Point Problems of Strictly Pseudocontractive Mapping
Atid Kangtunyakarn
Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand
Correspondence should be addressed to Atid Kangtunyakarn,beawrock@hotmail.com
Received 8 November 2010; Accepted 14 December 2010
Academic Editor: Qamrul Hasan Ansari
Copyrightq 2011 Atid Kangtunyakarn 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 prove the strong convergence theorem for finding a common element of the set of fixed point problems of strictly pseudocontractive mapping in Hilbert spaces and two sets of generalized equilibrium problems by using the hybrid method
1 Introduction
Let C be a closed convex subset of a real Hilbert space H, and let F : C × C → R be
a bifunction Recall that the equilibrium problem for a bifunction F is to find x ∈ C such that
F
x, y
≥ 0, ∀y ∈ C. 1.1
The set of solutions of1.1 is denoted by EPF Given a mapping T : C → H, let Fx, y
Tx, y − x for all x, y ∈ C Then, z ∈ EPF if and only if Tz, y − z ≥ 0 for all y ∈ C; that is, z is a solution of the variational inequality Let A : C → H be a nonlinear mapping The variational inequality problem is to find a u ∈ C such that
v − u, Au ≥ 0 1.2
Trang 2for all v ∈ C The set of solutions of the variational inequality is denoted by VIC, A Now,
we consider the following generalized equilibrium problem:
Find z ∈ C such that F
z, y
Az, y − z
≥ 0, ∀y ∈ C. 1.3
The set of z ∈ C is denoted by EPF, A, that is,
EPF, A z ∈ C : F
z, y
Az, y − z
≥ 0, ∀y ∈ C. 1.4
In the case of A ≡ 0, EPF, A is denoted by EPF In the case of F ≡ 0, EPF, A is also
denoted by VIC, A Numerous problems in physics, optimization, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and economics are reduced to find a solution of1.3; see, for instance, 1 3
A mapping A of C into H is called inverse strongly monotone mapping, see 4, if there
exists a positive real number α such that
x − y, Ax − Ay ≥ αAx − Ay2 1.5
for all x, y ∈ C The following definition is well known.
Definition 1.1 A mapping T : C → C is said to be a κ-strict pseudocontraction if there exists
κ ∈ 0, 1 such that
Tx − Ty2≤x − y2 κI − Tx − I − Ty2
, ∀x, y ∈ C. 1.6
A mapping T is called nonexpansive if
for all x, y ∈ C.
We know that κ-strict pseudocontraction includes a class of nonexpansive mappings.
If κ 1, T is said to be a pseudocontractive mapping T is strong pseudocontraction if there exists a positive constant λ ∈ 0, 1 such that T λI is pseudocontraction In a real Hilbert space H, 1.6 is equivalent to
Tx − Ty, x − y
≤x − y2−1− κ
2 I − Tx − I − Ty2
, ∀x, y ∈ DT. 1.8
T is pseudocontraction if and only if
Tx − Ty, x − y
≤x − y2
, ∀x, y ∈ DT. 1.9
Then, T is strong pseudocontraction if there exists positive constant λ ∈ 0, 1
Tx − Ty, x − y
≤ 1 − λx − y2
, ∀x, y ∈ DT. 1.10
Trang 3The class of κ-strict pseudocontractions falls into the one between classes of
nonex-pansive mappings, and the pseudocontraction mappings, and the class of strong
pseudocon-traction mappings is independent of the class of κ-strict pseudoconpseudocon-traction.
We denote by FT the set of fixed points of T If C ⊂ H is bounded, closed, and convex, and T is a nonexpansive mapping of C into itself, then FT is nonempty; for instance, see 5 Browder and Petryshyn6 show that if a κ-strict pseudocontraction T has a fixed point in C, then starting with an initial x0∈ C, the sequence {x n} generated by the recursive formula:
x n1 αx n 1 − αTx n , 1.11
where α is a constant such that 0 < α < 1, converges weakly to a fixed point of T Marino and
Xu7 have extended Browder and Petryshyns above-mentioned result by proving that the sequence{x n} generated by the following Manns algorithm 8:
x n1 α n x n 1 − α n Tx n 1.12
converges weakly to a fixed point of T provided the control sequence {α n}∞
n0 satisfies the
conditions that κ < α n < 1 for all n and ∞
n0 α n − κ1 − α n ∞ In 1974, S Ishikawa proved the following strong convergence theorem of pseudocontractive mapping
Theorem 1.2 see 9 Let C be a convex compact subset of a Hilbert space H, and let T : C → C
be a Lipschitzian pseudocontractive mapping For any x1 ∈ C, suppose that the sequence {x n } is
defined by
y n 1− β n
x n β n Tx n ,
x n1 1 − α n x n α n Ty n , ∀n ∈ N, 1.13
where {α n }, {β n } are two real sequences in 0, 1 satisfying
i α n ≤ β n , for all n ∈ N,
ii limn → ∞ β n 0 ,
iii ∞
n1 α n β n ∞.
Then {x n } converges strongly to a fixed point of T.
In order to prove a strong convergence theorem of Mann algorithm1.12 associated with strictly pseudocontractive mapping, in 2006, Marino and Xu7 proved the following
theorem for strict pseudocontractive mapping in Hilbert space by using CQ method.
Trang 4Theorem 1.3 see 7 Let C be a closed convex subset of a Hilbert space H Let T : C → C be
a κ-strict pseudocontraction for some 0 ≤ κ < 1, and assume that the fixed point set FT of T is nonempty Let {x n}∞n1 be the sequence generated by the following CQ algorithm:
x1∈ C,
y n α n x n 1 − α n Tx n ,
C n z ∈ C :y n − z2≤ 1 − α n κ − α n x n − Tx n 2
,
Q n {z ∈ C : x n − z, x1− x n },
x n1 P C n ∩Q n x1.
1.14
Assume that the control sequence {α n}∞
n1 is chosen so that α n < 1 for all n ∈ N Then {x n}
converges strongly to P FT x1 Very recently, in 2010, [10] established the hybrid algorithm for Lipschitz pseudocontractive mapping as follows:
For C1 C, x1 P C1x1,
y n 1 − α n x n α n Tz n ,
z n1− β n
x n β n Tx n ,
C n1 z ∈ C n :α n I − Ty n2≤ 2α n
x n − z, I − Ty n
2α n β n L x n − Tx n y n − x n α n I − Ty n ,
x n1 P C n1 x1, ∀n ∈ N.
1.15
Under suitable conditions of {α n } and {β n }, they proved that the sequence {x n } defined by 1.15
converges strongly to P FT x1.
Many authors study the problem for finding a common element of the set of fixed point problem and the set of equilibrium problem in Hilbert spaces, for instance, [2, 3, 11–15] The motivation of
1.14, 1.15, and the research in this direction, we prove the strong convergence theorem for finding
solution of the set of fixed points of strictly pseudocontractive mapping and two sets of generalized equilibrium problems by using the hybrid method.
2 Preliminaries
In order to prove our main results, we need the following lemmas Let C be closed convex subset of a real Hilbert space H, and let P C be the metric projection of H onto C; that is, for
x ∈ H, P C x satisfies the property
x − P C x min
y∈C
The following characterizes the projection P C
Trang 5Lemma 2.1 see 5 Given that x ∈ H and y ∈ C, then P C x y if and only if the following inequality holds:
x − y, y − z
≥ 0, ∀z ∈ C. 2.2 The following lemma is well known
Lemma 2.2 Let H be Hilbert space, and let C be a nonempty closed convex subset of H Let T :
C → C be κ-strictly pseudocontractive, then the fixed point set FT of T is closed and convex so that the projection P FT is well defined.
Lemma 2.3 demiclosedness principle see 16 If T is a κ-strict pseudocontraction on closed
convex subset C of a real Hilbert space H, then I − T is demiclosed at any point y ∈ H.
To solve the equilibrium problem for a bifunction F : C × C → R, 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 all x, y, z ∈ C,
lim
t → 0F
tz 1 − tx, y≤ Fx, y
A4 for all x ∈ C, y → Fx, y is convex and lower semicontinuous.
The following lemma appears implicitly in1
Lemma 2.4 see 1 Let C be a nonempty closed convex subset of H, and let F be a bifunction of
C × C into R satisfying A1–A4 Let r > 0, and x ∈ H Then, there exists z ∈ C such that
F
z, y
1
r
y − z, z − x
for all x ∈ C.
Lemma 2.5 see 11 Assume that F : C × C → R satisfies A1–A4 For r > 0 and x ∈ H,
define a mapping T r : H → C as follows:
T r x
z ∈ C : F
z, y
1
r
y − z, z − x
≥ 0, ∀y ∈ C , 2.5
for all z ∈ H Then, the following hold:
1 T r is single-valued;
Trang 62 T r is firmly nonexpansive, that is,
T r x − T r y2≤T r x − T r
y
, x − y
, ∀x, y ∈ H, 2.6
3 FT r EPF;
4 EPF is closed and convex.
Lemma 2.6 see 17 Let C be a closed convex subset of H Let {x n } be a sequence in H and u ∈ H.
Let q P C u; if {x n } is such that ωx n ⊂ C and satisfy the condition
x n − u ≤u − q, ∀n ∈ N, 2.7
then x n → q, as n → ∞.
Lemma 2.7 see 7 For a real Hilbert space H, the following identities hold: if {x n } is a sequence
in H weak convergence to z, then
lim sup
n → ∞
x n − y2 lim sup
n → ∞
x n − z 2z − y2
for all y ∈ H.
3 Main Result
Theorem 3.1 Let C be a nonempty closed convex subset of a Hilbert space H Let F and G be
bifunctions from C × C into R satisfying A1–A4, respectively Let A : C → H be an α-inverse
strongly monotone mapping, and let B : C → H be a β-inverse strongly monotone mapping Let
T : C → C be a κ-strict pseudocontraction mapping with F FT ∩ EP F, A ∩ EP G, B / ∅ Let {x n } be a sequence generated by x1∈ C C1and
F u n , u Ax n , u − u n 1
r n u − u n , u n − x n ≥ 0, ∀u ∈ C,
G v n , v Bx n , v − v n 1
s n v − v n , v n − x n ≥ 0, ∀v ∈ C,
z n δ n u n 1 − δ n v n ,
y n α n z n 1 − α n Tz n ,
C n1z ∈ C n:y n − z ≤ x n − z ,
x n1 P C n1 x1, ∀n ≥ 1,
3.1
where {α n}∞n0 is sequence in 0, 1, r n ∈ a, b ⊂ 0, 2α, and s n ⊂ c, d ⊂ 0, 2β satisfy the
following conditions:
Trang 7i limn → ∞ δ n δ ∈ 0, 1,
ii 0 ≤ κ ≤ α n < 1, for all n ≥ 1.
Then x n converges strongly to PFx1.
Proof First, we show that I −r n A is nonexpansive Let x, y ∈ C Since A is α-inverse strongly
monotone mapping and r n < 2α, we have
I − r n A x − I − r n A y2x − y − r n
Ax − Ay2
x − y2− 2r n
x − y, Ax − Ay
r2
nAx − Ay2
≤x − y2− 2αr nAx − Ay2 r2
nAx − Ay2
x − y2 r n r n − 2αAx − Ay2
≤x − y2
.
3.2
ThusI − r n A is nonexpansive, so are I − s n B, T r n I − r n A, and T s n I − s n B Since
F u n , u Ax n , u − u n 1
r n u − u n , u n − x n ≥ 0, ∀u ∈ C, 3.3 then we have
F u n , u 1
r n u − u n , u n − I − r n A x n ≥ 0. 3.4
By Lemma2.5, we have u n T r n I − r n Ax n By the same argument as above, we conclude
that v n T s n I − s n Bx n
Let z ∈ F Then Fz, y y − z, Az ≥ 0 and Gz, y y − z, Bz ≥ 0 Hence
F
z, y
1
r n
y − z, z − z r n Az
≥ 0,
G
z, y
1
s n
y − z, z − z s n Bz
≥ 0.
3.5
Again by Lemma2.5, we have z T r n z − r n Az T s n z − s n Bz By nonexpansiveness of
T r n I − r n A and T s n I − s n B, we have
u n − z T r n I − r n A x n − T r n I − r n A z
≤ x n − z ,
v n − z T s n I − s n A x n − T s n I − s n A z
≤ x n − z
3.6
Trang 8By3.6, we have
z n − z ≤ x n − z 3.7
Next, we show that C n is closed and convex for every n ∈ N It is obvious that C nis closed
In fact, we know that, for z ∈ C n,
y n − z ≤ x n − z is equivalent toy n − x n2 2y n − x n , x n − z≤ 0. 3.8
So, we have that for all z1, z2∈ C n and t ∈ 0, 1, it follows that
y n − x n2 2y n − x n , x n − tz1 1 − tz2
t2
y n − x n , x n − z1
y n − x n2
1 − t2
y n − x n , x n − z2
y n − x n2
≤ 0.
3.9
Then, we have that C n is convex By Lemmas2.5and2.2, we conclude thatF is closed and
convex This implies that PFis well defined Next, we show thatF ⊂ C n for every n ∈ N Taking p ∈ F, we have
y n − p2α n
z n − p 1 − α nTz n − p2
α nz n − p2
1 − α nTz n − p2− α n 1 − α n z n − Tz n 2
≤ α nz n − p2
1 − α nz
n − p2 κI − Tz n − I − Tp2
− α n 1 − α n z n − Tz n 2
α nz n − p2 1 − α nz n − p2 κ1 − α n z n − Tz n 2
− α n 1 − α n z n − Tz n 2
z n − p2
κ − α n 1 − α n z n − Tz n 2
≤z n − p2
≤x n − p2
.
3.10
It follows that p ∈ C n Then, we haveF ⊂ C n , for all n ∈ N Since x n P C n x1, for every w ∈ C n,
we have
x n − x1 ≤ w − x1 , ∀n ∈ N. 3.11
Trang 9In particular, we have
x n − x1 ≤ PFx1− x1 3.12
By3.11, we have that {x n } is bounded, so are {u n }, {v n }, {z n }, {y n } Since x n1 P C n1 x1 ∈
C n1 ⊂ C n and x n P C n x1, we have
0≤ x1− x n , x n − x n1
x1− x n , x n − x1 x1− x n1
≤ − x n − x1 2 x n − x1 x1− x n1
3.13
It is implied that
x n − x1 ≤ x n1 − x1 3.14 Hence, we have that limn → ∞ x n − x1 exists Since
x n − x n1 2 x n − x1 x1− x n1 2
x n − x1 2 2x n − x1, x1− x n1 x1− x n1 2
x n − x1 2 2x n − x1, x1− x n x n − x n1 x1− x n1 2
x n − x1 2− 2 x n − x1 2 2x n − x1, x n − x n1 x1− x n1 2
≤ x1− x n1 2− x n − x1 2
,
3.15
it is implied that
lim
Since x n1 P C n1 x1∈ C n1, we have
y n − x n1 ≤ x n − x n1 , 3.17 And by3.16, we have
lim
n → ∞y n − x n1 0. 3.18 Since
y n − x n ≤ y n − x n1 x n1 − x n , 3.19
Trang 10by3.16 and 3.18, we have
lim
Next, we show that
lim
n → ∞ u n − x n 0, lim
n → ∞ v n − x n 0. 3.21
Let p ∈ F, by 3.10 and 3.7, we have
y n − p2α n
z n − p 1 − α nTz n − p2
α nz n − p2 1 − α nTz n − p2− α n 1 − α n z n − Tz n 2
≤ α nz n − p2
1 − α nz
n − p2 κI − Tz n − I − Tp2
− α n 1 − α n z n − Tz n 2
α nz n − p2
1 − α nz n − p2
κ1 − α n z n − Tz n 2
− α n 1 − α n z n − Tz n 2
α nz n − p2
1 − α nz n − p2
κ − α n 1 − α n z n − Tz n 2
≤ α nx n − p2
1 − α nz n − p2
≤ α nx n − p2
1 − α nδ nu n − p2
1 − δ nv n − p2
.
3.22
Since u n T r n I − r n Ax n , p T r n I − r n Ap, we have
u n − p2T r
n I − r n A x n − T r n I − r n A p2
≤ I − r n A x n − I − r n A p 2
x n − r n Ax n − p r n Ap2
x n − p − r n
Ax n − Ap2
x n − p2 r2
nAx n − Ap2− 2r n
x n − p, Ax n − Ap
≤x n − p2 r2
nAx n − Ap2− 2r n αAx n − Ap2
x n − p2 r n r n − 2αAx n − Ap2
.
3.23
Trang 11Since v n T s n I − s n Bx n , p T s n I − s n Bp, we have
v n − p2T s
n I − s n B x n − T s n I − s n B p2
≤I − s n B x n − I − s n B p2
x n − s n Bx n − p s n Bp2
x n − p − s n
Bx n − Bp2
x n − p2 s2
nBx n − Bp2− 2s n
x n − p, Bx n − Bp
≤x n − p2 s2
nBx n − Bp2− 2s n βBx n − Bp2
x n − p2 s n
s n − 2βBx n − Bp2
.
3.24
Substituting3.23 and 3.24 into 3.22,
y n − p2≤ α nx n − p2
1 − α nδ nu n − p2
1 − δ nv n − p2
≤ α nx n − p2
1 − α nδ n
x
n − p2 r n r n − 2αAx n − Ap2
1 − δ nx
n − p2 s n
s n − 2βBx n − Bp2
α nx n − p2
1 − α nδ nx n − p2 δ n r n r n − 2αAx n − Ap2
1 − δ nx n − p2 s n 1 − δ ns n − 2βBx n − Bp2
α nx n − p2
1 − α n
×x
n − p2 δ n r n r n − 2αAx n − Ap2 s n 1 − δ ns n − 2βBx n − Bp2
α nx n − p2 1 − α nx n − p2 1 − α n δ n r n r n − 2αAx n − Ap2
s n 1 − α n 1 − δ ns n − 2βBx n − Bp2
x n − p2
1 − α n δ n r n r n − 2αAx n − Ap2
s n 1 − α n 1 − δ ns n − 2βBx n − Bp2
.
3.25
... element of the set of fixed point problem and the set of equilibrium problem in Hilbert spaces, for instance, [2, 3, 11–15] The motivation of1.14, 1.15, and the research... finding
solution of the set of fixed points of strictly pseudocontractive mapping and two sets of generalized equilibrium problems by using the hybrid method.
2 Preliminaries... Hilbert space, and let C be a nonempty closed convex subset of H Let T :
C → C be κ-strictly pseudocontractive, then the fixed point set FT of T is closed and convex so that