Volume 2010, Article ID 474813,11pages doi:10.1155/2010/474813
Research Article
Weak and Strong Convergence Theorems for
Equilibrium Problems and Countable Strict
Pseudocontractions Mappings in Hilbert Space
Rudong Chen,
1Xilin Shen,
2and Shujun Cui
11Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
2Department of Mathematics, Xinxiang College, Xinxiang, Henan 453002, China
Correspondence should be addressed to Rudong Chen,[email protected]
Received 27 August 2009; Accepted 10 January 2010
Academic Editor: Jong Kim
Copyrightq2010 Rudong Chen 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.
We introduce two iterative sequence for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a countable family of strict pseudocontractions in Hilbert Space. Then we study the weak and strong convergence of the sequences.
1. Introduction
LetCbe a nonempty closed convex subset of a Hilbert spaceHand letT be a self-mapping ofC. ThenT is said to be a strict pseudocontraction mappings if for allx, y∈C, there exists a constant 0≤κ <1 such that
Tx−Ty2≤x−y2κI−Tx−I−Ty2 1.1
if1.1holds, we also say thatT is aκ-strict pseudocontraction. We useFTto denote the set of fixed points of T, → to denote weakstrongconvergence, andWwxn {x : ∃xnk x}to denote theW-limit set of{xn}.
Letf:C×C → Rbe a bifunction whereRis the set of real numbers. Then, we consider the following equilibrium problem:
The set of suchz ∈ Cis denoted by EPf. Numerous problems in physics, optimization, and economics can be reduced to find a solution of1.2. Some methods have been proposed to solve the equilibrium problemsee1–3. Recently, S. Takahashi and W. Takahashi4
introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of the equilibrium problem and the set of fixed points of a nonexpansive mapping in Hilbert spaces. They also studied the strong convergence of the sequences generated by their algorithm for a solution of the EP which is also a fixed point of a nonexpansive mapping defined on a closed convex subset of a Hilbert space.
In this paper, thanks to the condition introduced by Aoyama et al.5, We introduce two iterative sequence for finding a common element of the set of solutions of an equilibrium problems and the set of fixed points of a countable family of strict pseudocontractions mappings in Hilbert Space. Then we study the weak and strong convergence of the sequences. The additional condition is inspired by Marino and Xu6and Kim and Xu7.
2. Preliminaries
For solving the equilibrium problem, let us assume that the bifunction f satisfies the following conditionssee3:
A1fx, x 0 for allx∈C;
A2f is monotone, that is, fx, y fy, x≤0 for anyx, y∈C;
A3f is upper-hemicontinuous, that is, for each x,y, z ∈ C,lim supt→0ftz 1 − tx, y≤fx, y;
A4fx,·is convex and lower semicontinuous for eachx∈C.
LetHbe a real Hilbert space. Then there hold the following well-known results:
tx 1−ty2tx2 1−ty2−t
1−tx−y2, ∀x, y∈H, ∀t∈0,1;
xy2x2−y2−
2x−y, y, ∀x, y∈H.
2.1
If{xn}is a sequence inHweakly convergent toz, then
lim sup n→ ∞
xn−y2
lim sup
n→ ∞ xn−z
2z−y2 ∀y∈H.
2.2
Recall that the nearest point projectionPC fromHontoCassigns to eachx ∈Hits nearest point denoted byPCxinC; that is,PCxis the unique point inCwith the property
x−PCx ≤x−y, ∀y∈C. 2.3
Givenx∈Handz∈C, thenzPCxif and only if there holds the following relation:
Lemma 2.1see6. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. LetT: C → Cbe aκ-strict pseudocontraction such thatFT/∅.
1(Demi-closed principle)Tis demi-closed onC, that is, ifxn x∈Candxn−Txn → 0,
thenxTx.
2T satisfies the Lipschitz condition
Tx−Ty≤Lx−y 1κ
1−κx−y ∀x, y∈C. 2.5
3The fixed point setFTofTis closed and convex so that the projectionPFTis well defined.
Lemma 2.2see5. LetCbe a nonempty closed convex subset of a Banach space and let{Tn}be a sequence of mapping ofCinto itself. Suppose∞n1supx∈CTn1x−Tnx<∞.Then, for eachy∈C, {Tny}converges strongly to some point ofC. Moreover, letT be a mapping ofCinto itself defined by
Ty lim
n→ ∞Tny ∀y∈C. 2.6
Thenlimn→ ∞supx∈CTnx−Tx0.
Lemma 2.3see8. LetCbe a closed convex subset ofH. Let{xn}be a sequence inHandu∈H. LetqPCu. If{xn}is such thatWwxn⊂Cand satisfies the condition
xn−u ≤u−q ∀n, 2.7
thenxn → q.
Lemma 2.4see9. LetCbe a nonempty closed convex subset ofH. Letf be a bifunction from C×C into Rsatisfying (A1), (A2), (A3), and(A4). Then, for anyλ >0andx∈H, there existsz∈C such that
fz, y 1
λ
y−z, z−x≥0, ∀y∈C. 2.8
Further, ifTλx{z∈C:fz, y 1/λy−z, z−x ≥0,∀y∈C}, then the following holds:
1Tλxis single-valued;
2Tλxis firmly nonexpansive, that is,
Tλx−Tλy2≤ T
λx−Tλy, x−y, ∀x, y∈H; 2.9
3FTλ EPf;
3. Weak Convergence Theorems
Theorem 3.1. Let Cbe a nonempty closed convex subset of a Hilbert space H and let {Tn} be a sequence ofκn-strict pseudocontractions mappings onCinto itself with0 ≤ κn < 1. Assume that κ max{κn : n ≥ 1}.Letf : C×C → Rbe a bifunction satisfying (A1), (A2), (A3), (A4), and EPf∩∞n1FTn/∅. Let{xn}and{zn}be sequence generated byx1∈Cand
fzn, yr1 n
y−zn, zn−xn≥0, ∀y∈C,
xn1αnzn 1−αnTnzn, ∀n≥1.
3.1
Assume that{αn} ⊂ 0,1with κδ < αn < 1−δfor all n, whereδ ∈ 0,1is a small enough constant, and{rn}is a sequence in0,∞withlim infn→ ∞rn > 0and ∞n1|rn1−rn| < ∞.Let
∞
n1supx∈BTn1x−Tnx<∞for any bounded subsetBofCand letTbe a mapping ofCinto itself
defined byTxlimn→ ∞Tnxfor allx∈Cand suppose thatFT ∞n1FTn.Then the sequences {xn}and{zn}converge weakly to an element ofFT∩EPf.
Proof. Pickp∈FT∩EPf. Then from the definition ofTrinLemma 2.4, we haveznTrnxn,
and thereforezn−pTrnxn−Trnp ≤ xn−p. It follows from3.1that
xn1−p2
1−αnTnzn−p αnzn−p2
αnzn−p2 1−αnTnzn−p2−αn1−αnzn−Tnzn2
≤αnzn−p2 1−αn zn−p2κzn−Tnzn2
−αn1−αnzn−Tnzn2
zn−p2−αn−κ1−αnzn−Tnzn2
≤xn−p2−αn−κ1−αnzn−Tnzn2.
3.2
Sinceκδ < αn<1−δfor alln, we getxn1−p ≤ xn−p; that is, the sequence{xn−p} is decreasing. Hence limn→ ∞xn−pexists. In particular,{xn}is bounded. SinceTris firmly nonexpensive,{zn}is also bounded. Also3.2implies that
zn−Tnzn2≤ δ12 xn−p2−xn1−p2
. 3.3
Taking the limit asn → ∞yields that
lim
n→ ∞zn−Tnzn0. 3.4
Since{zn}is bounded, it follows that
∞
n1 sup x∈{zn}
We applyLemma 2.2to get
zn−Tzn ≤ zn−TnznTnzn−Tzn
≤ zn−Tnznsup{Tnz−Tz:z∈ {zn}} −→0.
3.6
Next, we claim that limn→ ∞zn−xn0. Indeed, letpbe an arbitrary element ofFT∩EPf. Then as above
zn−p2T
rxn−Trp2
≤ Trxn−Trp, xn−p zn−Trp, xn−p
1
2 zn−p 2x
n−p2− xn−zn2
,
3.7
and hence
zn−p2≤x
n−p2− xn−zn2. 3.8
Therefore, from3.2, we have
xn1−p2 ≤z
n−p2−αn−κ1−αnzn−Tnzn2
≤zn−p2
≤xn−p2− xn−zn2,
3.9
and hence
xn−zn2 ≤xn−p2−xn1−p2. 3.10
So, from the existence of limn→ ∞xn−p, we have
lim
n→ ∞xn−zn0. 3.11
Next, we claim thatWwxn⊂FT∩EPf. since{xn}is bounded andHis reflexive,
Wwxnis nonempty. Letw ∈ Wwxnbe an arbitrary element. Then a subsequencexni of
{xn}converges weakly tow. Hence, from3.11we know thatzni w.Aszn−Tzn → 0,
we obtain thatTzni w. Let us showWwxn⊂EPf. SinceznTrnxn, we have
fzn, yr1 n
ByA2, we have
1
rn
y−zn, zn−xn≥fy, zn, 3.13
and hence
y−zni,
zni−xni
rni
≥fy, zni
. 3.14
FromA4, we have
0≥fy, w ∀y∈C. 3.15
Then, fort∈0,1andy∈C, fromA1, andA4, we also have
0fty 1−tw, ty 1−tw
≤tfty 1−tw, y 1−tfty 1−tw, w ≤tfty 1−tw, y,
3.16
Takingt → 0and usingA3, we get
fw, y≥0 ∀y∈C, 3.17
and hencew ∈ EPf. SinceT is a strict pseudocontraction mapping, byLemma 2.11we know that the mappingT is demiclosed at zero. Note thatzn −Tzn → 0 and zni w.
Thus,w ∈ FT. Consequently, we deduce that w ⊂ FT∩EPf. Since wis an arbitrary element, we conclude thatWwxn⊂FT∩EPf.
To see that {xn} and {zn} are actually weakly convergent, we take x,x ∈
Wwxn xni x, xmj x.Since limn→ ∞xn−p exist for everyp ∈ FT, by2.2, we
have
lim
n→ ∞xn−x 2 lim
i→ ∞xni−x 2
lim
i→ ∞xni−x
2x−x2
lim j→ ∞
xmj−x
2
x−x2
lim j→ ∞
xmj−x
2
2x−x2
lim
n→ ∞xn−x
22x−x2.
3.18
4. Strong Convergence Theorems
Theorem 4.1. LetCbe a closed convex subset of a real Hilbert spaceH. Let{Tn}be a sequence ofκn -strict pseudocontractions mappings onCinto itself with0≤κn<1. Assume thatκmax{κn:n≥ 1}.Letf :C×C → Rbe a bifunction satisfying (A1), (A2), (A3), (A4) andEPf∩∞n1FTn/∅.
ForC1Candx1PC1x0, let{xn}and{zn}be sequence generated byx0 ∈Cand
ynαnxn 1−αnTnxn,
zn∈C such that fzn, yr1
ny−zn, zn−yn ≥0, ∀y∈C,
Cn1{v∈C:zn−v ≤ xn−v},
xn1PCn1x0, n≥1.
4.1
Assume that{αn} ⊂ 0,1with κδ < αn < 1−δfor all n, whereδ ∈ 0,1is a small enough
constant, and{rn}is a sequence in0,∞with lim infn→ ∞rn > 0 and∞n1|rn1−rn| < ∞. Let
∞
n1supx∈BTn1x−Tnx<∞for any bounded subsetBofCand letTbe a mapping ofCinto itself
defined byTx limn→ ∞Tnxfor allx∈CSuppose thatFT ∞n1FTn.Then,{xn}converges strongly toPFT∩EPfx0.
Proof. First, we show thatCn is closed and convex. It is obvious thatC1 Cis closed and convex. Suppose thatCk is closed and convex for somek ≥ 1. Forz ∈ Ck, we know that zk−z ≤ xk−zis equivalent to
zk−xk22zk−xk, xk−z ≤0. 4.2
SoCk1is closed and convex. Then,Cnis closed and convex.
Next, we show by induction thatFT∩EPf⊂Cnfor alln≥1.FT∩EPf⊂C1is obvious. Suppose thatFT∩EPf⊂Ckfor somek≥1. Letp∈FT∩EPf⊂Ck. Putting
znTrnynfor alln, we know from4.1that
zn−p2T
rnyn−p
2
≤yn−p2
1−αnTnxn−p αnxn−p2
αnxn−p2 1−αnTnxn−p2−αn1−αnxn−Tnxn2
≤αnxn−p2 1−αn xn−p2κxn−Tnxn2
xn−p2−αn−κ1−αnxn−Tnxn2
≤xn−p2−δ2xn−Tnxn2
≤xn−p2,
4.3
and hencep∈Ck1. This implies thatFT∩EPf⊂Cnfor alln≥1. This implied that{xn}is well defined.
FromxnPCnx0, we have
x0−xn ≤x0−y ∀y∈Cn. 4.4
UsingFT∩EPf⊂Cn, we have
x0−xn ≤ x0−u ∀u∈FT∩EP
f, n≥1. 4.5
Then,{xn}is bounded. So are{yn}and{zn}. In particular,
x0−xn ≤x0−p wherepPFT∩EPfx0. 4.6
FromxnPCnx0andxn1PCn1x0∈Cn1⊂Cn,we have
x0−xn ≤ x0−xn1. 4.7
Since{xn−x0}is bounded, limn→ ∞xn−x0exists. FromxnPCnx0andxn1PCn1x0 ∈ Cn1⊂Cn.we also have
x0−xn, xn−xn1 ≥0. 4.8
In fact, from4.8, we have
xn−xn1xn−x0x0−xn12
x0−xn12− x0−xn2−2x0−xn, xn−xn1
≤ x0−xn12− x0−xn2.
4.9
Since limn→ ∞xn−x0exists, we have thatxn−xn1 → 0. On the other handxn1∈Cn1⊂
Cnimplies that
Further, we have
lim
n→ ∞zn−xn0. 4.11
From4.3, we have
xn−Tnxn2 ≤ δ12 xn−p2−zn−p2
. 4.12
On the other hand, we have
xn−p2−z
n−p2 xn2− zn22zn−xn, p
≤ xn−znxnzn 2pxn−zn.
4.13
Then, we have
lim
n→ ∞xn−p 2−z
n−p20. 4.14
Therefore, we have
lim
n→ ∞xn−Tnxn0. 4.15
We applyLemma 2.2to get
xn−Txn ≤ xn−TnxnTnxn−Txn
≤ xn−Tnxnsup{Tnx−Tx:x∈ {xn}} −→0.
4.16
Lastly, we show that the sequence {xn} converges to PFT∩EPfx0. Since {xn} is bounded andHis reflexive,Wwxnis nonempty. Letw ∈Wwxnbe an arbitrary element. Then a subsequence xni of {xn} converges weakly to w. From Lemma 2.1 and 4.16, we
obtain thatωwxn⊂FT. Next, we showWwxn⊂EPf. Letpbe an arbitrary element of
FT∩EPf. FromznTrnynandyn−p ≤ xn−p, we have
zn−p2≤T
ryn−Trp2
≤ Tryn−Trp, yn−p zn−Trp, yn−p
1
2 zn−p 2y
n−p2−yn−zn2
1
2 zn−p 2x
n−p2−yn−zn2
,
and hence
yn−zn2 ≤x
n−p2−zn−p2. 4.18
Therefore, we have
lim
n→ ∞yn−zn0. 4.19
As in the proof ofTheorem 3.1, we have
fzn, yr1
ny−zn, zn−yn ≥0, ∀y∈C. 4.20
ByA2, we have
1
rn
y−zn, zn−yn≥fy, zn, 4.21
and hence
y−zni,
zni−yni
rni
≥fy, zni
. 4.22
FromA4, we have
0≥fy, w ∀y∈C. 4.23
Then, fort∈0,1andy∈C, fromA1andA4, we also have
0fty 1−tw, ty 1−tw
≤tfty 1−tw, y 1−tfty 1−tw, w ≤tfty 1−tw, y.
4.24
Takingt → 0and usingA3, we get
fw, y≥0 ∀y∈C, 4.25
and hence w ∈ EPf. Lemma 2.3 and 4.6 ensure the strong convergence of {xn} to
PFT∩EPfx0. This completes the proof.
Acknowledgment
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