Volume 2010, Article ID 296759, 16 pagesdoi:10.1155/2010/296759 Research Article Strong Convergence Theorem for Equilibrium Problems and Fixed Points of a Nonspreading Mapping in Hilbert
Trang 1Volume 2010, Article ID 296759, 16 pages
doi:10.1155/2010/296759
Research Article
Strong Convergence Theorem for Equilibrium
Problems and Fixed Points of a Nonspreading
Mapping in Hilbert Spaces
Somyot Plubtieng and Sukanya Chornphrom
Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
Correspondence should be addressed to Somyot Plubtieng,somyotp@nu.ac.th
Received 30 June 2010; Revised 10 October 2010; Accepted 13 December 2010
Academic Editor: Brailey Sims
Copyrightq 2010 S Plubtieng and S Chornphrom This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
We introduce an iterative method for finding a common element of the set of solutions of equilibrium problems and the set of fixed points of a nonspreading mapping in a Hilbert space Then, we prove a strong convergence theorem which is connected with the work of S Takahashi and W Takahashi2007 and Iemoto and Takahashi 2009
1 Introduction
Let H be a real Hilbert space with inner product ·, · and norm · , respectively, and let C
be a closed convex subset of H Let F : C × C → Êbe bifunction, whereÊis the set of real
numbers The equilibrium problem for F : C × C → Êis to find x ∈ C such that
F
x, y
The set of solution of1.1 is denoted by EPF Given a mapping A : C → H, let
Fx, y Ax, y − x for all x, y ∈ C Then, z ∈ EPF if and only if Az, y − z ≥ 0 for all
y ∈ C, that is, z is a solution of the variational inequality Numerous problems in physics,
optimization, and economics reduce to find a solution of1.1; see, for example, 1 9 and the references therein
A mapping T of C into itself is said to be nonexpansive if Tx − Ty ≤ x − y for all
x, y ∈ C, and a mapping F is said to be firmly nonexpansive if Fx − Fy2 ≤ x − y, Fx − Fy for all x, y ∈ C Let E be a smooth, strictly convex and reflexive Banach space, and let J be the
Trang 2duality mapping of E and C a nonempty closed convex subset of E A mapping S : C → C is said to be nonspreading if
φ
Sx, Sy
φSy, Sx
≤ φSx, y
φSy, x
1.2
for all x, y ∈ C, where φx, y x2−2x, Jyy2
for all x, y ∈ E; see, for instance, Kohsaka
and Takahashi10 In the case when E is a Hilbert space, we know that φx, y x − y2
for all x, y ∈ E Then a nonspreading mapping S : C → C in a Hilbert space H is defined as
follows:
2Sx − Sy2≤Sx − y2x − Sy2 1.3
for all x, y ∈ C Let FQ be the set of fixed points of Q, and FQ nonempty; a mapping
Q : C → C is said to be quasi-nonexpansive if Qx − y ≤ x − y for all x ∈ C and y ∈ FQ.
Remark 1.1 In a Hilbert space, we know that every firmly nonexpansive mapping is
nonspreading and that if the set of fixed points of a nonspreading mapping is nonempty, the nonspreading mapping is quasi-nonexpansive; see10,11
In 1953, Mann12 introduced the iteration as follows: a sequence {x n} defined by
where the initial guess element x0 ∈ C is arbitrary and {α n} is a real sequence in
0, 1 Mann iteration has been extensively investigated for nonexpansive mappings In an
infinite-dimensional Hilbert space, Mann iteration can conclude only weak convergencesee
12,13 Fourteen years later, Halpern 14 introduced the following iterative scheme for
approximating a fixed point of T:
for all n ∈ Æ, where x1 x ∈ C and {α n } is a sequence of 0, 1 Strong convergence of this
type iterative sequence has been widely studied: Wittmann15 discussed such a sequence
in a Hilbert space
On the other hand, Kohsaka and Takahashi10 proved an existence theorem of fixed point for nonspreading mappings in a Banach space Recently, Lemoto and Takahashi16
studied the approximation theorem of common fixed points for a nonexpansive mapping T
of C into itself and a nonspreading mapping S of C into itself in a Hilbert space In particular, this result reduces to approximation fixed points of a nonspreading mapping S of C into itself
in a Hilbert space by using iterative scheme
Some methods have been proposed to solve the equilibrium problem and fixed point problem of nonexpansive mapping: see, for instance,1,2,6, 7,17–20 and the references
Trang 3therein In 1997, Combettes and Hirstoaga 3 introduced an iterative scheme of finding the best approximation to the initial data when EPF is nonempty and proved a strong convergence theorem Recently, S Takahashi and W Takahashi8 introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solution of equilibrium problems and the set of fixed points of a nonexpansive mapping
in a Hilbert space Let S : C → H be a nonexpansive mapping In 2008, Plubtieng and
Punpaeng7 introduced a new iterative sequence for finding a common element of the set of solution of equilibrium problems and the set of fixed points of a nonexpansive mapping in a Hilbert space which is the optimality condition for the minimization problem Very recently,
S Takahashi and W Takahashi9 introduced an iterative method for finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points
of a nonexpansive mapping in a Hilbert space and then obtain that the sequence converges strongly to a common element of two sets
In this paper, motivated by S Takahashi and W Takahashi 8 and Lemoto and Takahashi16, we introduce an iterative sequence and prove a strong convergence theorem for finding solution of equilibrium problems and the set of fixed points of a nonspreading mapping in Hilbert spaces
2 Preliminaries
Let H be a real Hilbert space When {x n } is a sequence in H, x n x implies that xnconverges
weakly to x and x n → x means the strong convergence Let C be a nonempty closed convex subset of H For every point x ∈ H, there exists a unique nearest point in C; denote by P Cx,
such that
PC is called the metric projection of H onto C We know that P Cis nonexpansive Further, for
x ∈ H and z ∈ C,
z PC x ⇐⇒
x − z, z − y
Moreover, P Cx is characterized by the following properties: PCx ∈ C and
x − PCx, y − PC y
≤ 0,
x − y2
for all x ∈ H, y ∈ C We also know that H satisfies Opial’s condition 21, that is, for any sequence{x n } ⊂ H with x n x, the inequality
lim inf
n → ∞ x n − x < lim inf
n → ∞ xn − y 2.4
holds for every y ∈ H with x / y; see 21,22 for more details
Trang 4The following lemmas will be useful for proving the convergence result of this paper.
Lemma 2.1 see 23 Let E, ·, · be an inner product space Then for all x, y, z ∈ E and α, β, γ ∈
0, 1 with α β γ 1, one has
αx βy γz2
αx2 βy2
γz2− αβx − y2
− αγx − z2− βγy − z2
. 2.5
Lemma 2.2 see 10 Let H be a Hilbert space, C a nonempty closed convex subset of H Let S be
a nonspreading mapping of C into itself Then the following are equivalent.
1 There exists x ∈ C such that {S n x} is bounded;
2 FS is nonempty.
Lemma 2.3 see 10 Let H be a Hilbert space, C a nonempty closed convex subset of H Let S be
a nonspreading mapping of C into itself Then FS is closed and convex.
Lemma 2.4 Let H be a real Hilbert space Then for all x, y ∈ H,
1 x y2≤ x2 2y, x y;
2 x y2≥ x2 2y, x.
Lemma 2.5 see 24 Let {a n }, {b n } ⊂ 0, ∞, and let {c n } ⊂ 0, 1 be sequences of real numbers
such that
an1 ≤ 1 − c n a n b n, for all n ∈Æ,
∞
n1 cn ∞ and∞
n1 bn < ∞.
Then, limn → ∞an 0
Lemma 2.6 see 16 Let H be a Hilbert space, C a closed convex subset of H, and S : C → C
a nonspreading mapping with FS / ∅ Then S is demiclosed, that is, x n u and xn − Sx n → 0
imply u ∈ FS.
Lemma 2.7 see 16 Let H be a Hilbert space, C a nonempty closed convex subset of a real Hilbert
space H, and let S be a nonspreading mapping of C into itself, and let A I − S Then
Ax − Ay2≤x − y, Ax − Ay
1 2
Ax2Ay2
Lemma 2.8 see 25 Assume {a n } is a sequence of nonnegative real numbers such that
where {α n } is a sequence in 0, 1 and {δ n } is a sequence inÊsuch that
1∞
n1 αn ∞;
2 lim supn → ∞ δ n/αn ≤ 0 or∞n1 |δ n | < ∞.
Then limn → ∞an 0.
Trang 5For solving the equilibrium problems for a bifunction F : C × C → Ê, let us 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 t↓0Ftz 1 − tx, y ≤ Fx, y;
A4 for each x ∈ C, y → Fx, y is convex and lower semicontinuous.
The following lemma appears implicitly in26
Lemma 2.9 see 26 Let C be a nonempty closed convex subset of H, and let F be a bifunction of
C × C intoÊsatisfying (A1)–(A4) Let r > 0 and x ∈ H Then, there exists z ∈ C such that
F
z, y
1
r
y − z, z − x
The following lemma was also given in4
Lemma 2.10 see 4 Assume that F : C × C → Ê satisfies (A1)–(A4) For r > 0 and x ∈ H, define a mapping Tr : H → C as follows:
Tr x z ∈ C : F
z, y
1
r
y − z, z − x
≥ 0, ∀y ∈ C
2.9
for all z ∈ H Then, the following hold:
1 T r is single-valued;
2 T r is firmly nonexpansive, that is, for any x, y ∈ H, Trx − Tr y2 ≤ T r x − Try, x − y;
3 FT r EPF;
4 EPF is closed and convex.
Lemma 2.11 see 27 Let Γ n be a sequence of real numbers that does not decrease at infinity, in
the sense that there exists a subsequenceΓn jj≥0 ofΓn which satisfies Γ n j < Γn j1for all j ≥ 0 Also consider the sequence of integers τn n≥n0defined by
Then τn n≥n0 is a nondecreasing sequence verifying limn → ∞τn ∞, and the following properties are satisfied for all n ≥ n0:
Γτn ≤ Γτn1, Γn≤ Γτn1. 2.11
3 Main Result
In this section, we prove a strong convergence theorem for finding a common element of the set of fixed points of a nonspreading mapping and the set of solutions of the equilibrium problems
Trang 6Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H Let F be a
bifunctions from C × C → Êsatisfying (A1)–(A4), and let S be a nonspreading mapping of C into itself such that FS ∩ EP F / ∅ Let u ∈ C, and let {x n } and {u n } be sequences generated by x1∈ C
and
F
un, y
1
rn
y − un, un − x n
≥ 0, ∀y ∈ C,
xn1 β nxn1− β n
S α nu 1 − α n u n ,
3.1
for all n ∈Æ, where {α n }, {β n } ∈ 0, 1 and {r n } ∈ 0, ∞ satisfy
limn → ∞αn 0,∞
n1 αn ∞, 0 < a ≤ β n ≤ b < 1,
∞
n1 |α n − α n−1 | < ∞,∞n1 |β n − β n−1 | < ∞,
lim infn → ∞rn > 0, and∞
n1 |r n1 − r n | < ∞.
Then {x n } converges strongly to z ∈ FS ∩ EPF, where z P FS∩EPFu.
Proof Let p ∈ FS ∩ EPF From un T r n xn, we have
un − p T r n xn − T r n p ≤ x n − p 3.2
for all n ∈Æ Put y n α nu 1 − αn u n We divide the proof into several steps
Step 1 We claim that the sequences {x n }, {u n }, {y n }, and {Sy n} are bounded First, we note that
Syn − p ≤ y n − p
αnu 1 − α n u n − p
≤ α nu − p 1 − α nun − p
≤ α nu − p 1 − α nxn − p,
3.3
and so
xn1 − p β nxn1− β n
Syn − p
≤ β nxn − p 1 − β nSyn − p
≤ β nxn − p 1 − β nyn − p
β nxn − p 1 − β nαnu 1 − α n u n − p
≤ β nxn − p 1 − β n
αnu − p 1 − α nun − p
≤ β nxn − p 1 − β n
αnu − p 1 − α nxn − p
1− α n
1− β nxn − p α n
1− β nu − p.
3.4
Trang 7Putting M max{x n − p, u − p}, we note that x n − p ≤ M for all n ∈ Æ In fact, it is obvious thatx1− p ≤ M Assume that x k − p ≤ M for all k ∈Æ Thus, we have
xk1 − p ≤ 1 − α k
1− β kxk − p α k
1− β ku − p
≤1− α k
1− β k
M αk
1− β k
M
M.
3.5
By induction, we obtain thatx n − p ≤ M for all n ∈Æ So,{x n } is bound Hence, {u n }, {y n}, and{Sy n} are also bounded
Step 2 Put tn β nyn 1 − β n Sy n We claim thatx n1 − t n → 0 as n → ∞ We note that
x n1 − x n βnxn
1− β n
Syn
−βn−1xn−11− β n−1
Syn−1
βnxn − β nxn−1 β nxn−1 − β n−1xn−1
1− β n
Syn−1− β n
Syn−1
1− β n
Syn−1−1− β n−1
Syn−1
≤ β n x n − x n−1 βn − β n−1 x n−1 1− β nSyn − Sy n−1
1− β n
−1− β n−1 Syn−1
≤ β n x n − x n−1 βn − β n−1 x n−1 1− β nyn − y n−1 β n−1 − β n Syn−1
β n x n − x n−1 βn − β n−1 x n−1 1− β n
× α nu 1 − α n u n − α n−1u − 1 − α n−1 u n−1 βn − β n−1 Syn−1
≤ β n x n − x n−1 βn − β n−1 x n−1 1− β n
× α nu − αn−1u 1 − α n u n − 1 − α n−1 u n−1 βn − β n−1 Syn−1
β n x n − x n−1 βn − β n−1 x n−1 1− β n
|α n − α n−1 |u
1− β n
1 − α n u n − 1 − α n u n−1 1 − α n u n−1 − 1 − α n−1 u n−1
βn − β n−1 Syn−1
≤ β n x n − x n−1 βn − β n−1 x n−1 1− β n
|α n − α n−1 |u
1− β n
1 − α n u n − u n−1 1− β n
|1 − α n − 1 − α n−1 |u n−1
βn − β n−1 Syn−1
β n x n − x n−1 βn − β n−1 x n−1 1− β n
|α n − α n−1 |u
1− β n
1 − α n u n − u n−1 1− β n
|α n − α n−1 |u n−1
βn − β n−1 Syn−1
β n x n − x n−1 βn − β n−1 K1
1− β n
|α n − α n−1 |K1
1− β n
1 − α n u n − u n−1 1− β n
|α n − α n−1 |K1 βn − β n−1 K1,
3.6
Trang 8where K1 sup{x n Sy n u u n−1 : n ∈Æ} On the other hand, from u n T r n xnand
un1 T r n1 xn1, we have
F
un, y
1
rn
y − un, un − x n
F
un1, y
rn1
y − un1, un1 − x n1
for all y ∈ C Putting y u n1in3.7 and y u nin3.8, we have
F u n, un1 1
rn u n1 − u n, un − x n ≥ 0,
F u n1, un 1
rn1 u n − u n1, un1 − x n1 ≥ 0.
3.9
So, fromA2, we note that
un1 − u n, un − x n
rn −un1 − x n1
rn1
and hence
un1 − u n, un − u n1 u n1 − x n− rn
rn1 u n1 − x n1
Without loss of generality, let us assume that there exists a real number d such that r n > d > 0
for all n ∈Æ Thus, we have
u n1 − u n2≤ un1 − u n, xn1 − x n
1− rn
rn1
u n1 − x n1
≤ u n1 − u n x n1 − x n
1 −rn1 rn u n1 − x n1
,
3.12
and hence
u n1 − u n ≤ x n1 − x n 1
rn1 |r n1 − r n |u n1 − x n1
≤ x n1 − x n 1
d |r n1 − r n |L,
3.13
Trang 9where L sup{u n − x n : n ∈Æ} So, from 3.6, we note that
x n1 − x n ≤ β n x n − x n−1 2 βn − β n−1 K1 2
1− β n
|α n − α n−1 |K1
1− β n
1 − α n
x n − x n−1 1
d |r n − r n−1 |L
βn1− β n
1 − α nx n − x n−1 2 βn − β n−1 K1 2
1− β n
|α n − α n−1 |K1
1− β n
1 − α n1
d |r n − r n−1 |L
1−1− β n
αn
x n − x n−1 2 βn − β n−1 K1 2
1− β n
|α n − α n−1 |K1
1− β n
1 − α nL
d |r n − r n−1 |.
3.14
lim
n → ∞ x n1 − x n 0 3.15
for p ∈ FS ∪ EPF We note from u n T r n xnthat
un − p2Tr
n xn − T r n p2 ≤Tr n xn − T r n p, xn − pun − p, x n − p
1 2
u
n − p2xn − p2
− x n − u n2
,
3.16
and hence
un − p2≤xn − p2
Therefore, from the convexity of · 2, we have
xn1 − p2βnxn
1− β n
Syn − p2
≤ β nxn − p21− β nSyn − p2
≤ β nxn − p21− β nyn − p2
β nxn − p21− β nαnu 1 − α n u n − p2
≤ β nxn − p2 α n
1− β nu − p21− β n
1 − α nun − p2
≤ β nxn − p2 α n
1− β nu − p21− β n
1 − α nx
n − p2
− x n − u n2
1−1− β n
αnxn − p2 α n
1− β nu − p21− β n
1 − α n x n − u n2
,
3.18
Trang 10and hence
1− β n
1 − α n x n − u n2 ≤ α n
1− β nu − p2− α n
1− β nxn − p2
xn − p2−xn1 − p2
α n
1− β nu − p2− α n
1− β nxn − p2
xn − p − x n1 − pxn − p x n1 − p
≤ α n
1− β nu − p2− α n
1− β nxn − p2
x n − x n1xn − p x n1 − p.
3.19
So, we havex n − u n → 0 Indeed, since y n α nu 1 − αn u n, it follows that
lim
n → ∞xn − y n lim
n → ∞ x n − α nu 1 − α n u n
lim
n → ∞ α n 1 − α n x n − α nu 1 − α n u n
≤ lim
n → ∞ α n x n − u 1 − α n x n − u n
lim
n → ∞ αn x n − u lim
n → ∞ 1 − α n x n − u n
0.
3.20
Then, we note that
x n1 − t n βnxn
1− β n
Syn
−βnyn1− β n
Syn
βn
xn − y n
1− β n
Syn − Sy n
Since, 0 < a ≤ β n ≤ b < 1 and x n − y n → 0, it follows that
lim
n → ∞ x n1 − t n 0. 3.22
Step 3 Put A I − S From Ap 0, it follows byLemma 2.7that
tn − p2βnyn
1− β n
Syn
− p2
yn − p
−1− β n
yn − Sy n2
yn − p
−1− β n
Ayn2
yn − p2− 21− β n
yn − p, Ay n − Ap1− β n
2Ayn2