Volume 2010, Article ID 264052,10pages doi:10.1155/2010/264052
Research Article
General Iterative Algorithm for Nonexpansive
Semigroups and Variational Inequalities in
Hilbert Spaces
Jinlong Kang,
1, 2Yongfu Su,
1and Xin Zhang
11Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China 2Department of Foundation, Xian Communication Institute, Xian 710106, China
Correspondence should be addressed to Yongfu Su,[email protected]
Received 24 October 2009; Revised 26 December 2009; Accepted 12 February 2010
Academic Editor: Jong Kim
Copyrightq2010 Jinlong Kang 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 a general iterative method for finding the solution of the variational inequality problem over the fixed point set of a nonexpansive semigroup in a Hilbert space. We prove that the sequence converges strongly to a common element of the above two sets under some parameters controlling conditions. Our results improve and generalize many known corresponding results.
1. Introduction
LetHbe a real Hilbert space with inner product·,·and norm · , respectively. LetCbe a nonempty closed convex subset ofHand letPC be the metric projection ofH ontoC. A
mappingTofCinto itself is said to be nonexpansive ifTx−Ty ≤ x−yfor eachx, y∈C. We denote byFTthe set of fixed points ofT. A familyS{Ts: 0≤s <∞}of mappings of
Cinto itself is called a nonexpansive semigroup onCif it satisfies the following conditions:
iT0xxfor allx∈C;
iiTst TsTtfor allx, y∈Cands, t≥0;
iiiTsx−Tsy ≤ x−yfor allx, y∈Cands≥0;
ivfor allx∈C,s →Tsxis continuous.
LetAbe a strongly positive bounded linear operator onH: that is, there is a constant
γwith property
Ax, x ≥γx2 ∀x∈H. 1.1
A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert spaceH:
min
x∈K
1
2Ax, x − x, b, 1.2
whereK is the fixed point set of a nonexpansive mappingT onHandbis a given point in
H. In 2001, Yamada1presented the hybrid steepest descent method for problem1.2. In 2003, Xu2proved that the sequence{xn}defined by the iterative method below, with the
initial guessx0∈H, chosen arbitrarily:
xn1αnb I−αnATxn, n≥0 1.3
converges strongly to the unique solution of the minimization problem1.2 provided the sequence{αn}satisfies certain conditions.
On the other hand, Moudafi 3 introduced the viscosity approximation method for nonexpansive mappingssee4 for further developments in both Hilbert and Banach spaces. Letf be a contraction onHsuch thatfx−fy ≤ αx−y, whereα ∈ 0,1is a constant. Starting with an arbitrary initialx0∈H, define a sequence{xn}recursively by
xn1βnfxn
1−βn
Txn, n≥0, 1.4
where{βn}is a sequence in0,1. It is proved3,4that under certain appropriate conditions
imposed on {βn}, the sequence {xn} generated by 1.4 converges strongly to the unique
solutionxinKof the variational inequality
I−fx, x −x≥0, x∈K. 1.5
Recently, Marino and Xu 5 combine the iterative method 1.3 with the viscosity approximation1.4and consider the following general iterative method:
They proved that if the sequence{αn}of parameters satisfies appropriate conditions,
then the sequence{xn}generated by1.6converges strongly to the unique solution of the
variational inequality
A−γfx, x −x ≥0, x∈FT, 1.7
which is the optimality condition for the minimization problem
min
x∈K
1
2Ax, x −hx, 1.8
wherehis a potential function forγfi.e.,hx γfforx∈H.
Note thatI −f and A−γf in problems1.5and 1.7are strongly monotone and Lipschitz continuous. Therefore, problems1.5and1.7can be solved by1,7,8. In7,8, algorithms to accelerate the hybrid steepest descent method have been proposed.
Quite recently, for the nonexpansive semigroupsS {Ts : 0 ≤ s < ∞}, Plubtieng and Punpaeng9study the iteration process{xn}defined by
xn1αnfxn βnxn
1−αn−βn 1
sn sn
0
Tsxnds, n≥0, 1.9
wherex0∈C,{αn},{βn}are two sequences in0,1, and{sn}is a positive real divergent real
sequence and prove a strong convergence theorem.
In this paper, motivated and inspired by the above results, we prove a strong convergence of the iterative scheme in a real Hilbert space by
xn1αnγfxn βnxn
1−βn
I−αnA 1
sn sn
0
Tsxnds, n≥0. 1.10
Furthermore, we show that if the sequences{αn}and{βn}of parameters satisfy appropriate
conditions, then the sequence {xn} converges strongly to the unique solution of the
variational inequality
A−γfx, x −x≥0, x∈FT, 1.11
which is the optimality condition for the minimization problem
min
x∈FT
1
2Ax, x −hx, 1.12
2. Preliminaries
Recall that given a closed convex subset K of a real Hilbert space H, the nearest point projection PK fromH onto K assigns to eachx ∈ H its nearest point denoted by PKx in
KfromxtoK; that is,PKxis the unique point inKwith the property
x−PKx ≤x−y ∀y∈K. 2.1
The following Lemmas2.1and2.2are well known.
Lemma 2.1. LetKbe a closed convex subset of a real Hilbert spaceH.Givenx∈Handz∈K.Then
zPKxif and only if there holds the following relation:
x−z, y−z ≤0 ∀y∈K. 2.2
Lemma 2.2. LetHbe a real Hilbert space. There hold the following identities.
ixy2≤ x22y, xy,∀x, y∈H.
iitx 1−ty2tx2 1−ty2−t1−tx−y2 ∀x, y∈H,t∈0,1.
Definition 2.3Opial’s condition10. A space X is said to satisfy Opial’s condition if for each sequence{xn}∞n1inXwhich converges weakly to pointx∈X, we have
lim inf
n→ ∞ xn−x<lim infn→ ∞ xn−y, ∀y∈X, y /x. 2.3
It is well known that Hilbert spaces satisfy Opial’s condition.
Lemma 2.4Browder6. LetEbe a uniformly convex Banach space,Ka nonempty closed convex subset ofE, andT:K → Ea nonexpansive mapping. ThenI−T is demiclosed at zero.
Theorem 2.5Shimizu and Takahashi11. LetCbe a nonempty closed convex bounded subset of a real Hilbert spaceHand letS {Ts: 0 ≤ s < ∞}be a nonexpansive semigroup onC. For
x∈Candt >0. Then, for any0≤h <∞,
lim
t→ ∞supx∈C
1
t
t
0
Tsx ds−Th 1 t
t
0
Tsx ds0. 2.4
Theorem 2.6Marino and Xu5. Assume thatAis a strong positive linear bounded operator on a Hilbert spaceHwith coefficientγ >0and0< ρ≤ A−1. ThenI−ρA ≤1−ργ.
Theorem 2.7Xu12. Assume that{αn}is a sequence of nonnegative real numbers such that
αn1≤
1−γn
where{γn}is a sequence in (0,1) and{δn}is a sequence inRsuch that
i∞n1γn∞;
iilim supn→ ∞δn/γn≤0or∞n1|δn|<∞.
Then limn→ ∞αn0.
3. Main Results
Theorem 3.1. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. LetS {Ts: 0 ≤ s < ∞}be a semigroup of nonexpansive mapping on Csuch thatFSis nonempty. Let{αn}
and{βn}be the sequences of real numbers in0,1satisfyinglimn→ ∞αn 0,limn→ ∞βn 0and
∞
n1αn ∞. Letf be a contraction ofCinto itself with a coefficientα∈ 0,1,{sn}be a positive
real divergent sequence, andAa strong positive bounded linear operator onCwith coefficientγ >0, and0< γ < γ/α. Let the sequence{xn}be defined byx0∈Cand
xn1αnγfxn βnxn
1−βn
I−αnA 1
sn sn
0
Tsxnds, n≥0. 3.1
Then{xn}converges strongly tox, wherexis the unique solution inFSof the variational inequality
A−γfx, x −x ≥0, x∈FS ∗
or equivalent toxPFSI−Aγfx, wherePis a metric projection mapping fromHontoFS.
Proof. Since limn→ ∞αn0 by the assumption, we may assume, without loss of generality, that
αn<A−1for alln. FromTheorem 2.6, we know that if 0< ρ≤ A−1, thenI−ρA ≤1−ργ.
Note thatFSis a nonempty closed convex set. We first show that{xn}is bounded.
Letq∈FS. Thus, we compute that
xn1−qαnγfxn βnxn
1−βn
I−αnA 1
sn sn
0
Tsxnds−q
≤αnγfxn−Aqβnxn−q1−βn
I−αnA s1nsn
0
Tsxnds−q ≤αnγfxn−γf
qγfq−Aqβnxn−q
1−βn
−αnγ 1
sn sn
0
Tsxn−qds
≤αnγαxn−qαnγf
q−Aq1−αnγxn−q
1−γ−γααnxn−q
γ−γααn
1
γ−γαγf
q−Aq
≤maxxn−q, 1
γ−γαγf
q−Aq
.
By induction, we get
xn−q≤maxx0−q, 1
γ−γαγf
q−Aq
, n≥0. 3.3
Therefore, {xn} is bounded.{1/sn sn
0 Tsxnds} and {fxn} are also bounded. Put z0
PFSx0andD {z∈C:z−z0 ≤ x0−z0 1/γ−γαγfz0−Az0}. ThenDis a
nonempty closed bounded convex subset ofC. SinceTsis nonexpansive for anys∈0,∞,
DisTs-invariant for eachs∈0,∞and contains{xn}. Without loss of generality, we may
assume thatS{Ts: 0≤s <∞}is a nonexpansive semigroup onD. ByTheorem 2.5, we get
lim
n→ ∞
s1nsn
0
Tsxnds−Th
1
sn sn
0
Tsxnds
0, 3.4
for everyh∈0,∞. Next we showxn−Thxn → 0 asn → ∞.Notice that
xn1−Thxn1 ≤
xn1−
1
sn sn
0
Tsxnds
1
sn sn
0
Tsxnds−Th
1
sn sn
0
Tsxnds
Th
1
sn sn
0
Tsxnds
−Thxn1
≤2xn1− 1
sn sn
0
Tsxnds
1
sn sn
0
Tsxnds−Th
1
sn sn
0
Tsxnds
≤2αn
γfxn−A1
sn sn
0
Tsxnds 2βn
xn− 1
sn sn
0
Tsxnds 1 sn sn 0
Tsxnds−Th
1
sn sn
0
Tsxnds
.
3.5
Fromαn → 0,βn → 0, and3.4, we getxn1−Thxn1 → 0, and hence
xn−Thxn −→0. 3.6
Letxbe the unique solution of the variational inequality∗; we show that
lim sup
n→ ∞
A−rfx, x n−x
≥0, x∈FS. 3.7
Since{xn} ∈Dis bounded, there is a subsequence{xnj}of{xn}such that
lim
j→ ∞
A−rfx, x nj−xlim sup
n→ ∞
andxnj q. By Opial’s condition, we haveq∈FS. In fact, ifq /Thqfor someh∈0,∞,
we have
lim inf
j→ ∞
xnj−q
<lim inf
j→ ∞
xnj−Thq
≤lim inf
j→ ∞
x
nj−Thxnj
Thxnj−Thq
≤lim inf
j→ ∞ xnj−q
.
3.9
This is a contradiction. Therefore, we haveqThqfor someh≥0, that isq∈FS. Hence, by∗, we obtain
lim sup
n→ ∞
A−rfx, x n−x
A−rfx, q−x ≥0 3.10
as required. Finally we shall show thatxn → x. For eachn≥0, we have
xn1−x2
αn
γfxn−Ax
βnxn−x
1−βn
I−αnA
1
sn sn
0
Tsxnds−x
2
≤1−βn
I−αnA
1
sn sn
0
Tsxnds−x
βnxn−x 2
2αnγfxn−Ax, x n1−x
1−βnI−αnA2 s1nsn
0
Tsxnds−x
2β2nxn−x2
2βn1−βn
I−αnA
1
sn sn
0
Tsxnds−x
, xn−x
2αnγfxn−Ax, xn1−x
≤1−βn
−αnγ 2 1
sn sn
0
Tsxn−x2dsβ2nxn−x2
2βn1−βn
I−αnAxn−x22αnγfxn−fx, xn1−x 2αnγfx−Ax, x n1−x
≤1−βn
−αnγ 2
xn−x2β2nxn−x22βn
1−βn
−αnγ
xn−x2
2αnγαxn−xxn1−x2αnγfx−Ax, x n1−x
≤1−αnγ 2
xn−x2αnγα
xn−x2xn1−x2
2αnγfx−Ax, x n1−x
1−αnγ
2
αnγα
which implies that
xn1−x2≤
1−2αnγ
αnγ
2
αnγα
1−αnγα xn−x
2 2αn
1−αnγαγfx−Ax, x n1−x
1−2
γ−γααn
1−αnγα
xn−x2
αnγ 2
1−αnγαxn−x
2
2αn
1−αnγαγfx−Ax, x n1−x
≤
1−2
γ−γααn
1−αnγα
xn−x2
2γ−γααn
1−αnγα
×
αnγ2M
2γ−γα
1
γ−γα
γfx−Ax, x n1−x
1−δnxn−x2δnBn,
3.12
whereM sup{xn−x2 :n∈N},δn 2γ−γααn/1−αnγα, andBn : αnγ2M/2γ−
γα 1/γ−γαγfx−Ax, x n1−x. It is easily to see thatδn → 0,
∞
n1δn ∞and
lim supn→ ∞Bn≤0 by3.10. Finally by usingTheorem 2.7, we can obtain that{xn}converges
strongly to a fixed pointxofT. This completes the proof.
4. Applications
Theorem 4.1. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. LetS {Ts: 0 ≤ s < ∞}be a strongly continuous semigroup of nonexpansive mapping onCsuch thatFSis nonempty. Let{αn}be a sequence of real numbers in0,1satisfyinglimn→ ∞αn0and
∞
n1αn
∞. Letf be a contraction ofCinto itself with a coefficientα∈ 0,1,{sn}a positive real divergent
sequence, andAa strong positive bounded linear operator onCwith coefficientγ >0and0< γ < γ/α. Let the sequences{xn}defined byx0∈Cand
xn1αnγfxn I−αnA
1
sn sn
0
Tsxnds, n≥0. 4.1
Then{xn}converges strongly tox, wherexis the unique solution inFSof the variational inequality
A−γfx, x −x ≥0, x∈FS 4.2
Proof. Takingβn 0 inTheorem 3.1, we get the desired conclusion easily.
Corollary 4.2Marino and Xu5. Let Cbe a nonempty closed convex subset of a real Hilbert spaceH. LetTbe a nonexpansive mapping onCsuch thatFTis nonempty. Let{αn}be a sequence
of real numbers satisfying0 < αn < 1,limn→ ∞αn 0and ∞n1αn ∞. Letf be a contraction
of C into itself with a coefficient α ∈ 0,1 and A be a strong positive bounded linear operator on C with coefficient γ > 0 and 0 < γ < γ/α. Then the sequence {xn} defined by x0 ∈ C and
xn1αnγfxn I−αnATxn, n≥0. 4.3
Then{xn}converges strongly tox, wherexis the unique solution inFSof the variational inequality
A−γfx, x −x ≥0, x∈FS 4.4
or equivalent x PFTI −A γfx, where P is a metric projection mapping from H into
FT.
Proof. TakingS {Ts : 0 ≤ s < ∞} {T} andβn 0 in the inTheorem 3.1, we get the
desired conclusion easily.
Corollary 4.3Plubtieng and Punpaeng9. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. LetS {Ts: 0≤s < ∞}be a strongly continuous semigroup of nonexpansive mapping onCsuch thatFSis nonempty. Let{αn}and{βn}be the sequences of real numbers in
0,1satisfyinglimn→ ∞αn0,limn→ ∞βn 0, and∞n1αn ∞. Letfbe a contraction ofCinto
itself with a coefficientα∈0,1and let{sn}be a positive real divergent sequence. Let the sequence
{xn}be defined byx0∈Cand
xn1αnfxn βnxn
1−αn−βn 1
sn sn
0
Tsxnds, n≥0. 4.5
Then {xn} converges strongly to x, where x is the unique solution in FS of the variational
inequality
I−fx, x −x≥0, x∈FS 4.6
or equivalent toxPFSfx, wherePis a metric projection mapping fromHintoFS.
References
1 I. Yamada, “The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings,” in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (Haifa, 2000), D. Butnariu, Y. Censor, and S. Reich, Eds., vol. 8 ofStud. Comput. Math., pp. 473–504, North-Holland, Amsterdam, The Netherlands, 2001.
2 H. K. Xu, “An iterative approach to quadratic optimization,” Journal of Optimization Theory and Applications, vol. 116, no. 3, pp. 659–678, 2003.
3 A. Moudafi, “Viscosity approximation methods for fixed-points problems,”Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46–55, 2000.
4 H. K. Xu, “Viscosity approximation methods for nonexpansive mappings,”Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004.
5 G. Marino and H. K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,”
Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006.
6 F. E. Browder, “Semicontractive and semiaccretive nonlinear mappings in Banach spaces,”Bulletin of the American Mathematical Society, vol. 74, pp. 660–665, 1968.
7 P. L. Combettes, “A block-iterative surrogate constraint splitting method for quadratic signal recovery,”IEEE Transactions on Signal Processing, vol. 51, no. 7, pp. 1771–1782, 2003.
8 H. Iiduka and I. Yamada, “A use of conjugate gradient direction for the convex optimization problem over the fixed point set of a nonexpansive mapping,”SIAM Journal on Optimization, vol. 19, no. 4, pp. 1881–1893, 2009.
9 S. Plubtieng and R. Punpaeng, “Fixed-point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces,”Mathematical and Computer Modelling, vol. 48, no. 1-2, pp. 279–286, 2008.
10 Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,”Bulletin of the American Mathematical Society, vol. 73, pp. 595–597, 1967.
11 T. Shimizu and W. Takahashi, “Strong convergence to common fixed points of families of nonexpansive mappings,”Journal of Mathematical Analysis and Applications, vol. 211, no. 1, pp. 71–83, 1997.
