Volume 2009, Article ID 462489,12pages doi:10.1155/2009/462489
Research Article
Strong Convergence Theorems for Countable
Lipschitzian Mappings and Its Applications in
Equilibrium and Optimization Problems
Liping Yang
1and Yongfu Su
21Department of Fundamental, Tianjin Institute of Urban Construction, Tianjin 300384, China 2Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
Correspondence should be addressed to Yongfu Su,[email protected]
Received 21 October 2008; Revised 20 December 2008; Accepted 5 March 2009
Recommended by Naseer Shahzad
The purpose of this paper is to propose a modified hybrid method in mathematical programming and to obtain some strong convergence theorems for common fixed points of a countable family of Lipschitzian mappings. Further, we apply our results to solve the equilibrium and optimization problems. The results of this paper improved and extended the results of W. Nilsrakoo and S. Saejung2008and some others in some respects.
Copyrightq2009 L. Yang and Y. Su. 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.
1. Introduction and Preliminaries
LetHbe a real Hilbert space with inner product·,·and norm · and letCbe a nonempty subset ofH. A mappingT :C → Cis said to be Lipschitzian if there exists a positive constant
Lsuch that
Tx−Ty≤Lx−y, ∀x, y∈C. 1.1
In this case, T is also said to be L-Lipschitzian. Throughout the paper, we assume that every Lipschitzian mapping is L-Lipschitzian with L ≥ 1. If L 1, then T is known as a nonexpansive mapping. We denote byFTthe set of fixed points ofT. There are many methods for approximating the fixed points of a nonexpansive mapping. In 1953, Mann1
introduced the following iteration method:
where the initial guess element x0 ∈ C is arbitrary, and {αn} is a real sequence in0,1. Mann iteration has been extensively investigated for nonexpansive mappings. One of the fundamental convergence results is proved by Qin and Su 2. In an infinite-dimensional Hilbert space, Mann iteration could conclude only weak convergence3. Attempts to modify the Mann iteration method 1.2 so that strong convergence is guaranteed have recently been made. Nakajo and Takahashi4proposed the following modification of Mann iteration method1.2:
x0∈Cchosen arbitrarily, ynαnxn1−αnTxn,
Cn z∈C:yn−z≤xn−z,
Qnz∈C:xn−z, x0−xn≥0,
xn1 PCn∩Qn
x0
,
1.3
wherePK denotes the metric projection from H onto a closed convex subsetK of H. The iterative method1.3is said to be hybrid method orCQmethod. In recent years, the hybrid method1.3has been modified by many authors for other nonlinear operators2,5–10.
In 2008, Nilsrakoo and Saejung 11 used the hybrid method to obtain a strong convergence theorem for countable Lipschitzian mappings{Tn}∞n0as follows.
Theorem NS. Let Cbe a nonempty bounded closed convex subset of a real Hilbert space H. Let
{Tn}∞n0be a sequence ofLn-Lipschitzian mappings fromCinto itself withLn≥1and let ∞n0FTn
be nonempty. Assume that{αn} is a sequence in 0,1 with lim supn→ ∞αn < 1. Let{xn} be a sequence inCdefined as follows:
x0∈Cchosen arbitrarily, ynαnxn1−αnTnxn,
Cnz∈C:yn−z2≤xn−z2θn,
Qnz∈C:xn−z, x0−xn≥0,
xn1 PCn∩Qn
x0
,
1.4
where
θn1−αnL2n−1
diamC2−→0, 1.5
as n → ∞. Let ∞n1sup{Tn1z−Tnz : z ∈ B} < ∞for any bounded subset B of C, and letT be a mapping of Cinto itself defined byTz limn→ ∞Tnz, for allz ∈ Cand suppose that
FT ∞
n0FTn, then{xn}converges strongly toPFTx0.
The purpose of this paper is to propose a modified hybrid method in mathematical programming and to obtain some strong convergence theorems for common fixed points of a countable family of Lipschitzian mappings. Finally, we apply our results to solve the equilibrium and optimization problems. The results of this paper improved and extended the Nilsrakoo and Saejung results in some respects.
Recall that given a closed convex subsetKof a real Hilbert spaceH, the nearest point projectionPKfromHontoK assigns to eachx∈ Hits nearest point denoted byPKxinK fromxtoK; that is,PKxis the unique point inKwith the property
x−PKx≤ x−y ∀y∈K. 1.6
The following lemma is well known.
Lemma 1.1. LetK be a closed convex subset of a real Hilbert spaceH. Givenx ∈ H andz ∈ K. ThenzPKxif and only if there holds the relation:
x−z, y−z ≤0 ∀y∈K. 1.7
Definition 1.2. LetCbe a nonempty closed convex subset of a real Hilbert spaceH and let
{Tn}∞n0be a sequence ofLn-Lipschitzian mappings fromCinto itself withLn ≥1.{Tn}∞n0is
said to satisfy theSUcondition, if the following conditions hold:
1for any strong convergence sequence{xn} ⊂ C, the sequence{Tnxn}is also strong convergent;
2the common fixed points set ∞n0FTnis nonempty;
3FT ∞n0FTn, whereT :C → Cis defined byTxlimn→ ∞Tnx, for allx∈C,
FTdenotes the fixed points set ofT.
In Section 3 of this paper, we will give an important example of sequence of Ln -Lipschitzian mappings which satisfies theSUcondition.
Lemma 1.3. LetCbe a nonempty closed convex subset of a real Hilbert spaceHand let{Tn}∞n0be a sequence ofLn-Lipschitzian mappings fromCinto itself withLn ≥1. If{Tn}∞n0 satisfies the (SU) condition. Then
1{Ln}is bounded impliesTis uniformly continuous on theC; 2limn→ ∞LnLimpliesT isL-Lipschitzian;
3limn→ ∞Ln1 impliesT is nonexpansive.
Proof. Observe that, for allx, y∈C, we have
Tx−Ty lim
n→ ∞Tnx−Tny≤nlim→ ∞Lnx−y. 1.8
2. Main Results
Theorem 2.1. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Let{Tn}∞n0be a sequence ofLn-Lipschitzian mappings fromCinto itself withLn → 1, asn → ∞. Assume{Tn}∞n0 satisfies the (SU) condition and{αn}is a sequence in0,1withlim supn→ ∞αn <1. Let{xn}∞n0be
a sequence inCdefined as follows:
x0∈Cchosen arbitrarily, ynαnxn1−αnTnxn,
Cnz∈Cn−1
Qn−1:yn−z≤xn−zθn, n≥1,
C0
z∈C:y0−z≤x0−zθ0
,
Qnz∈Cn−1
Qn−1:
xn−z, x0−xn≥0, n≥1,
Q0C, xn1 PCn∩Qn
x0
,
2.1
where
θn1−αnL2n−1sup z∈A
xn−z2 ,
Ay∈FT:y−PFTx0≤1
.
2.2
Then{xn}converges strongly toPFTx0, wherePKis the metric projection fromHonto closed convex subsetK.
Proof. We first prove thatCnandQnare closed and convex for eachn≥0. From the definition ofCnandQn, it is obvious thatCnis closed andQn is closed and convex for eachn≥ 0. We prove thatCnis convex. Sinceyn−z ≤ xn−zθnis equivalent to
2xn−yn, z≤xn2−yn2θn, 2.3
it follows thatCnis convex. Next, we show thatFT⊂Cn, for alln≥0. For anyp∈FT, we have
yn−p2α
nxn1−αnTnxn−p2
≤αnxn−p21−αnTnxn−p2
αnxn−p21−αnL2nxn−p2
≤xn−p21−αnL2n−1xn−p2
≤xn−p2θn,
so thatp ∈Cn, therefore, we haveFT ⊂Cn, for alln≥ 0. Next, we show thatFT⊂ Qn, for alln ≥ 0. We prove this by induction. Forn 0, we haveFT ⊂ C Q0. Suppose that FT⊂Qn, thenFT⊂Cn∩Qnand there exists a unique elementxn1 ∈Cn∩Qnsuch that xn1PCn∩Qnx0. By usingLemma 1.1, we have
xn1−z, x0−xn1
≥0, 2.5
for allz∈Cn∩Qn. In particular,
xn1−p, x0−xn1
≥0, 2.6
for allp∈FT. It follows from the definition ofQn1thatFT⊂Qn1. By using the principle
of induction, we claim thatFT⊂Qn, for alln≥0. Therefore, we haveFT⊂Cn∩Qn, for alln≥0. Now the sequence{xn}is well defined.
It follows from the definition ofQnthatxnPQnx0. Therefore, we have
xn−x0≤z−x0, 2.7
for allz∈FT⊂Qn. This implies that the{xn−x0}is bounded. On the other hand, from xn1PCn∩Qnx0 ∈Qn, we have
xn−x0 ≤ xn1−x0, 2.8
for alln≥0. Therefore,{xn−x0}is nondecreasing and bounded. So that limn→ ∞xn−x0
exists.
Note again thatxnPQnx0, hence for any positive integerm, we havexnm∈Qnm−1⊂ Qnwhich implies thatxnm−xn, xn−x0 ≥0. Therefore, we have
xnm−xn2x
nm−x0−xn−x02
xnm−x02−xn−x02−2
xnm−xn, xn−x0
≤xnm−x02−xn−x02.
2.9
From this inequality, we know that{xn}is a Cauchy sequence inC, so that there exists a point
p∈Csuch that limn→ ∞xnp.
Sincexn1 ∈ Cn, thenyn−xn12 ≤ xn−xn12θnthis together withθn → 0, as → ∞, implies that limn→ ∞ynp. From lim supn→ ∞αn<1, we get
xn−Tnxn≤ 1 1−αn
asn → ∞. SinceTis nonexpansive, therefore,
xn−Txn≤xn−TnxnTnxn−Txn
≤xn−TnxnTnxn−TnpTnp−TpTp−Txn
≤xn−TnxnLnxn−pTnp−Tpp−xn−→0,
2.11
asn → ∞. Then limn→ ∞xnp∈FT.
Finally, we claim thatpp0PFTx0. If not, we havex0−p>x0−p0. There must
exists a positive integerN, ifn > N, thenx0−xn>x0−p0, which leads to
x0−p02x
0−xnxn−p02
x0−xn2xn−p022
xn−p0, x0−xn.
2.12
It follows thatxn−p0, x0−xn<0 implies thatp0/∈QnThis is a contradiction, hencepp0.
This completes the proof.
The following theorem directly follows fromTheorem 2.1.
Theorem 2.2. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Let{Tn}∞n0be a
sequence of nonexpansive mappings fromCinto itself. Assume{Tn}∞n0 satisfies the (SU) condition
and{αn}is a sequence in0,1withlim supn→ ∞αn <1. Let{xn}∞n0 be a sequence inCdefined as follows:
x0∈Cchosen arbitrarily, ynαnxn1−αnTnxn,
Cnz∈Cn−1
Qn−1:yn−z≤xn−z, n≥1,
C0
z∈C:y0−z≤x0−z,
Qnz∈Cn−1
Qn−1:
xn−z, x0−xn≥0, n≥1,
Q0C, xn1 PCn∩Qn
x0
.
2.13
3. Application for Equilibrium and Optimization
LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Letfbe a bifunction of
C×CintoR, whereRis the set of real numbers. The equilibrium problem forF :C×C → R is to findx∈Csuch that
fx, y≥0, ∀y∈C. 3.1
The set of solutions of3.1is denoted by EPF. Given a mappingT :C → H, letfx, y
Tx, y−x, for allx, y∈C. Then,z∈EPfif and only ifTz, y−z ≥0, for ally∈C, that is,zis a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of3.1. Some methods have been proposed to solve the equilibrium problem; see, for instance,12–16.
For solving the equilibrium problem for a bifunctionf:C×C → R, let us assume that
fsatisfies the following conditions:
A1fx, x 0, for allx∈C;
A2fis monotone, that is,fx, y fy, x≤0, for allx, y∈C; A3for eachx, y, z∈C,
lim
t↓0ftz 1−tx, y≤fx, y; 3.2
A4for eachx∈C,y→fx, yis convex and lower semicontinuous.
We need the following lemmas for the proof of our main results.
Lemma 3.1see11–13. LetCbe a nonempty closed convex subset ofHand letfbe a bifunction ofC×CintoRsatisfying (A1)–(A4). Letr >0andx∈H. Then, there existsz∈Csuch that
fz, y 1ry−z, z−x ≥0, ∀y∈C. 3.3
Lemma 3.2see11–13. Assume thatf :C×C → Rsatisfies (A1)–(A4). Forr >0andx∈H, define a mappingTr :H → Cas follows:
Trx
z∈C: fz, y 1
ry−z, z−x ≥0,∀y∈C
, 3.4
for allz∈H. Then, the following hold:
1Tr is single-valued;
2Tr is firmly nonexpansive, that is, for anyx, y∈H,
Trx−Try2≤T
3FTr EPf;
4EPfis closed and convex.
Remark 3.3. Tris also nonexpansive, for allr >0.
Now, we prove the following lemma which is very important for the main results of this section.
Lemma 3.4. LetCbe a nonempty closed convex subset ofHand letf be a bifunction ofC×Cinto Rsatisfying (A1)–(A4). Let{rn}be a positive real sequence such thatlimn→ ∞rn r >0. Then the sequence{Trn}satisfies the (SU) condition.
Proof. 1Let{xn}be a convergent sequence inC. LetznTrnxn, for alln, then
fzn, y1ry−zn, zn−xn≥0, ∀y∈C, 3.6
fznm, y1ry−znm, znm−xnm≥0, ∀y∈C. 3.7
Puttingyznmin3.6andyznin3.7, we have
fzn, znm 1rznm−zn, zn−xn≥0, ∀y∈C,
fznm, zn1rzn−znm, znm−xnm≥0, ∀y∈C.
3.8
So, fromA2we have
znm−zn,znr−xn n −
znm−xnm
rnm
≥0, 3.9
and hence
znm−zn, zn−xn−rrn nm
znm−xnm≥0. 3.10
Thus, we have
znm−zn, zn−znmznm−xn−rrn nm
znm−xnm≥0, 3.11
which implies that
znm−zn2≤z
nm−zn, znm−xn−rrn nm
znm−xnm
znm−zn,
1− rn
rnm
znmrrn
nmxnm−xn
.
Therefore, we get
znm−zn≤1− rn
rnm
znmrrn
nmxnm−xn
. 3.13
On the other hand, for anyp∈EPf, fromznTrnxn, we have
zn−pTr
nxn−p≤xn−p, 3.14
so that{zn}is bounded. Since limn→ ∞rn r > 0, this together with3.13implies that the
{zn}is a Cauchy sequence. HenceTrnxnznis convergent. 2By usingLemma 3.2, we know that
∞
n0 FTrn
EPf/∅. 3.15
3 From 1 we know that, limn→ ∞Trnx exists, for allx ∈ C. So, we can define a
mappingTfromCinto itself by
Tx lim
n→ ∞Trnx, ∀x∈C. 3.16
It is obvious that theTis nonexpansive. It is easy to see that
EPf ∞
n1 FTrn
⊂FT. 3.17
On the other hand, letw∈FT, wnTrnw, we have
fwn, y1ry−wn, wn−w≥0, ∀y∈C. 3.18
ByA2we know
1
r
y−wn, wn−w≥fy, wn, ∀y∈C. 3.19
Sincewn → Tw wand fromA4, we havefy, w≤ 0, for ally∈C. Then, fort ∈0,1 andy∈C,
0fty 1−tw, ty 1−tw
≤tfty 1−tw, y 1−tfty 1−tw, w
≤tfty 1−tw, y.
Therefore, we have
fty 1−tw, y≥0. 3.21
Lettingt↓0 and usingA3, we get
fw, y≥0, ∀y∈C, 3.22
and hence w ∈ EPf. From the aforementioned two respects, we know that FT ∞
n0FTrn. This completes the proof.
Theorem 3.5. Let C be a nonempty closed convex subset of a real Hilbert space H. Let f be a bifunction fromC×CintoR −∞,∞satisfying (A1)–(A4) and EPf/∅. Let{xn}and{un} be sequences generated byx0∈Cand
x0∈Cchosen arbitrarily,
fun, xn r1 n
y−un, un−xn≤0, ∀y∈C,
ynαnxn1−αnun,
Cnz∈Cn−1
Qn−1:yn−z≤xn−z, n≥1,
C0
z∈C:y0−z≤x0−z,
Qnz∈Cn−1
Qn−1:
xn−z, x0−xn≥0, n≥1,
Q0C, xn1 PCn∩Qn
x0
.
3.23
Assumelimn→ ∞rn r > 0 andlim supn→ ∞αn < 1. Then{xn}converges strongly to PEPfx0,
wherePEPfis the metric projection fromHontoEPf.
Proof. By usingLemma 3.4, we know that {Trn} satisfies the SUcondition. Then FT
EPf. By usingTheorem 2.2, we obtain the result ofTheorem 3.5
Now, we study a kind of optimization problem by using the aforementioned results of this paper. That is, we will give an iterative algorithm of solution for the following optimization problem with nonempty set of solutions:
minhx,
x∈C, 3.24
bifunction from C×C to R defined by fx, y hy−hx. We consider the following equilibrium problem, that is to findx∈Csuch that
fx, y≥0, ∀y∈C. 3.25
It is obvious that EPF A, where EPFdenotes the set of solutions of equilibrium Problem 3.25. In addition, it is easy to see thatfx, ysatisfies the conditionsA1–A4inSection 2. Therefore, fromTheorem 3.5, we can obtain the following theorem.
Theorem 3.6. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Lethxbe a convex and lower semicontinuous functional defined onC. Let{xn}and{un}be sequences generated byx0∈Cand
x0∈Cchosen arbitrarily,
hxn−hunr1 n
y−un, un−xn≤0, ∀y∈C,
ynαnxn1−αnun,
Cnz∈Cn−1
Qn−1:yn−z≤xn−z, n≥1,
C0
z∈C:y0−z≤x0−z,
Qnz∈Cn−1
Qn−1 :xn−z, x0−xn ≥0, n≥1,
Q0C, xn1 PCn∩Qn
x0
.
Assumelimn→ ∞rnr >0andlim supn→ ∞αn<1. Then{xn}converges strongly toPAx0.
Remark 3.7. It is easy to see that this paper has some new methods and results than the results of Nilsrakoo and Saejung11:
1proposed a modified hybrid iterative scheme, so that the new simple method of proof has been used;
2removed the bounded restriction for closed convex setC; 3relax the conditions of sequence{Tn};
4give an application for optimization problem;
5the sequence of setsCn, Qnsatisfy the following relation:
FT⊂ · · ·Cn1∩Qn1⊂Cn∩Qn, n≥0, 3.26
Acknowledgment
This project is supported by the National Natural Science Foundation of China under grant10771050.
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