Volume 2010, Article ID 943275,25pages doi:10.1155/2010/943275
Research Article
Some Iterative Methods for Solving Equilibrium
Problems and Optimization Problems
Huimin He,
1Sanyang Liu,
1and Qinwei Fan
21Department of Mathematics, Xidian University, Xi’An 710071, China
2Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China
Correspondence should be addressed to Huimin He,[email protected]
Received 3 September 2010; Accepted 29 October 2010
Academic Editor: Vijay Gupta
Copyrightq2010 Huimin He 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 new iterative scheme for finding a common element of the set of solutions of the equilibrium problems, the set of solutions of variational inequality for a relaxed cocoercive mapping, and the set of fixed points of a nonexpansive mapping. The results presented in this paper extend and improve some recent results of Ceng and Yao2008, Yao2007, S. Takahashi and W. Takahashi2007, Marino and Xu2006, Iiduka and Takahashi2005, Su et al.2008, and many others.
1. Introduction
Throughout this paper, we always assume thatHis a real Hilbert space with inner product
·,·and norm · , respectively,Cis a nonempty closed and convex subset ofH, andPC is the metric projection ofHontoC. In the following, we denote by “→” strong convergence, by “” weak convergence, and by “R” the real number set. Recall that a mappingS:C → C
is called nonexpansive if
Sx−Sy ≤ x−y, ∀x, y∈C. 1.1
We denote byFSthe set of fixed points of the mappingS.
For a given nonlinear operatorA, consider the problem of findingu∈Csuch that
which is called the variational inequality. For the recent applications, sensitivity analysis, dynamical systems, numerical methods, and physical formulations of the variational inequalities, see1–24and the references therein.
For a givenz∈H,u∈Csatisfies the inequality
u−z, v−u ≥0, ∀v∈C, 1.3
if and only ifuPCz, wherePC is the projection of the Hilbert space onto the closed convex setC.
It is known that projection operator PC is nonexpansive. It is also known that PC satisfies
x−y, PCx−PCy ≥ PCx−PCy2, ∀x, y∈H. 1.4
Moreover,PCxis characterized by the propertiesPCx∈Candx−PCx, PCx−y ≥0 for all
y∈C.
Using characterization of the projection operator, one can easily show that the variational inequality1.2is equivalent to finding the fixed point problem of findingu∈C
which satisfies the relation
uPCu−λAu, 1.5
whereλ >0 is a constant.
This fixed-point formulation has been used to suggest the following iterative scheme. For a givenu0∈C,
un1PCun−λAun, n1,2, . . . , 1.6
which is known as the projection iterative method for solving the variational inequality
1.2. The convergence of this iterative method requires that the operatorAmust be strongly monotone and Lipschitz continuous. These strict conditions rule out their applications in many important problems arising in the physical and engineering sciences. To overcome these drawbacks, Noor2,3used the technique of updating the solution to suggest the two-stepor predictor-correctormethod for solving the variational inequality1.2. For a given
u0∈C,
wnPCun−λAun,
un1PCwn−λAwn, n0,1,2, . . . ,
1.7
which is also known as the modified double-projection method. For the convergence analysis and applications of this method, see the works of Noor3and Y. Yao and J.-C. Yao16.
theorem. Very recently, S. Takahashi and W. Takahashi6also introduced a new iterative scheme,
Fyn, u
1 rn
u−yn, yn−xn ≥0, ∀u∈C,
xn1αnfxn 1−αnTyn,
1.8
for approximating a common element of the set of fixed points of a nonexpansive nonself mapping and the set of solutions of the equilibrium problem and obtained a strong convergence theorem in a real Hilbert space.
Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see7–11and the references therein. 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∈C
1
2Ax, x − x, b, 1.9
whereAis a linear bounded operator,Cis the fixed point set of a nonexpansive mappingS, andbis a given point inH. In10,11, it is proved that the sequence{xn}defined by the iterative method below, with the initial guessx0∈Hchosen arbitrarily,
xn1 I−αnASxnαnb, n≥0, 1.10
converges strongly to the unique solution of the minimization problem1.9 provided the sequence{αn} satisfies certain conditions. Recently, Marino and Xu8 introduced a new iterative scheme by the viscosity approximation method12:
xn1 I−αnASxnαnγfxn, n≥0. 1.11
They proved that the sequence {xn} generated by the above iterative scheme converges strongly to the unique solution of the variational inequality
A−γfx∗, x−x∗ ≥0, x∈C, 1.12
which is the optimality condition for the minimization problem
min x∈C
1
2Ax, x −hx, 1.13
For finding a common element of the set of fixed points of nonexpansive mappings and the set of solution of variational inequalities forα-cocoercive map, Takahashi and Toyoda
13introduced the following iterative process:
xn1αnxn 1−αnSPCxn−λnAxn, 1.14
for every n 0,1,2, . . ., where Aisα-cocoercive,x0 x ∈ C,{αn}is a sequence in0,1, and{λn}is a sequence in0,2α. They showed that, ifFS∩VIC, Ais nonempty, then the sequence{xn}generated by1.14converges weakly to somez∈FS∩VIC, A. Recently, Iiduka and Takahashi14proposed another iterative scheme as follows:
xn1αnx 1−αnSPCxn−λnAxn, 1.15
for every n 0,1,2, . . ., where Aisα-cocoercive,x0 x ∈ C,{αn}is a sequence in0,1, and{λn}is a sequence in0,2α. They proved that the sequence{xn}converges strongly to
z∈FS∩VIC, A.
Recently, Chen et al.15studied the following iterative process:
xn1αnfxn 1−αnSPCxn−λnAxn 1.16
and also obtained a strong convergence theorem by viscosity approximation method. Inspired and motivated by the ideas and techniques of Noor2,3and Y. Yao and J.-C. Yao16introduce the following iterative scheme.
LetCbe a closed convex subset of real Hilbert spaceH. LetAbe anα-inverse strongly monotone mapping ofCintoH, and let S be a nonexpansive mapping ofCinto itself such thatz∈FS∩VIC, A/∅. Suppose thatx1u∈Cand{xn},{yn}are given by
ynPCxn−λnAxn,
xn1 αnuβnxnγnSPC
yn−λnAyn
, 1.17
where{αn},{βn}, and{γn}are the sequences in0,1and{λn}is a sequence in0,2α. They proved that the sequence{xn}defined by1.17converges strongly to common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for α-inverse-strongly monotone mappings under some parameters controlling conditions.
In this paper motivated by the iterative schemes considered in6,15,16, we introduce a general iterative process as follows:
Fyn, u
1 rn
u−yn, yn−xn ≥0, ∀u∈C,
xn1αnγfxn βnxn
1−βn
I−αnA
SPCI−snByn,
1.18
whereAis a linear bounded operator andBis relaxed cocoercive. We prove that the sequence
the set of fixed points of a nonexpansive mapping, the set of solutions of the variational inequalities for a relaxed cocoercive mapping, and the set of solutions of the equilibrium problems2.12, which solves another variational inequality
γfq−Aq, q−P ≤0, ∀p∈F, 1.19
whereFFS∩VIC, B∩EPFand is also the optimality condition for the minimization problem minx∈F1/2Ax, x −hx, wherehis a potential function forγfi.e.,hx γfx forx∈H. The results obtained in this paper improve and extend the recent ones announced by S. Takahashi and W. Takahashi6, Iiduka and Takahashi14, Marino and Xu8, Chen et al.15, Y. Yao and J.-C. Yao16, Ceng and Yao22, Su et al.17, and many others.
2. Preliminaries
For solving the equilibrium problem for a bifunctionF : C×C → R, let us assume thatF
satisfies 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, limt→0Ftz 1−tx, y≤Fx, y;
A4for eachx∈C,y→Fx, yis convex and lower semicontinuous.
Recall the following.
1Bis calledν-strong monotone if for allx, y∈C, we have
Bx−By, x−y ≥νx−y2, 2.1
for a constantν >0. This implies that
Bx−By ≥νx−y, 2.2
that is,Bisν-expansive, and whenν1, it is expansive.
2Bis said to beμ-cocoercive2,3if for allx, y∈C, we have
Bx−By, x−y≥μBx−By2, for a constant μ >0. 2.3
Clearly, everyμ-cocoercive mapBis 1/μ-Lipschitz continuous.
3Bis called−μ-cocoercive if there exists a constantμ >0 such that
Bx−By, x−y ≥ −μBx−By2, ∀x, y∈C. 2.4
4Bis said to be relaxedμ, ν-cocoercive if there exists two constantsμ, ν >0 such that
forμ 0, Bisν-strongly monotone. This class of maps are more general than the class of strongly monotone maps. It is easy to see that we have the following implication:ν-strongly monotonicity⇒relaxedμ, ν-cocoercivity.
We will give the practical example of the relaxed μ, ν-cocoercivity and Lipschitz continuous operator.
Example 2.1. LetTxκx, for allx∈C, for a constantκ >1; then,Tis relaxedμ, ν-cocoercive and Lipschitz continuous. Especially,Tisν-strong monotone.
Proof. 1. SinceTxκx, for allx∈C, we haveT :C → C. For for allx, y∈C, for allμ≥0, we also have the below
Tx−Ty, x−yκx−y2
≥ −μTx−Ty2 κ−1x−y2.
2.6
Takingνκ−1, it is clear thatTis relaxedμ, ν-cocoercive. 2. Obviously, for for allx, y∈C
Tx−Ty ≤κ1x−y. 2.7
Then,T isκ1 Lipschitz continuous.
Especially, Takingμ0, we observe that
Tx−Ty, x−y ≥κ−1x−y2. 2.8
Obviously,Tisν-strong monotone. The proof is completed.
5 A mapping f : H → H is said to be a contraction if there exists a coefficient
α0≤α <1such that
fx−fy ≤αx−y, ∀x, y∈H. 2.9
6An operatorAis strong positive if there exists a constantγ >0 with the property
Ax, x ≥γx2, ∀x∈H. 2.10
7A set-valued mappingT :H → 2His called monotone if for allx, y∈H,f ∈Tx, andg ∈Tyimplyx−y, f −g ≥0. A monotone mappingT :H → 2H is maximal if the graph ofGTofT is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mappingT is maximal if and only if forx, f ∈ H×H,
LetBbe a monotone map ofCintoHand letNCvbe the normal cone toCatv∈C, that is,NCv{w∈H:v−u, w ≥0,∀u∈C}and define
Tv
⎧ ⎨ ⎩
BvNCv, v∈C,
∅, v∈C. 2.11
ThenT is the maximal monotone and 0∈Tvif and only ifv∈VIC, B; see1.
Related to the variational inequality problem 1.2, we consider the equilibrium problem, which was introduced by Blum and Oettli19and Noor and Oettli20in 1994. To be more precise, letFbe a bifunction ofC×CintoR, whereRis the set of real numbers.
For given bifunctionF·,· : C×C → R, we consider the problem of findingx ∈ C
such that
Fx, y≥0, ∀y∈C 2.12
which is known as the equilibrium problem. The set of solutions of2.12is denoted by EPF. Given a mappingT : C → H, letFx, y Tx, y−xfor allx, y ∈C. Thenx ∈ EPFif and only ifTx, y−x ≥0 for ally∈ C, that is,xis a solution of the variational inequality. That is to say, the variational inequality problem is included by equilibrium problem, and the variational inequality problem is the special case of equilibrium problem.
Assume thatT is a potential function forT i.e.,∇Tx Txfor allx∈C, it is well known thatx∈Csatisfies the optimality conditionTx, y−x ≥0 for ally∈Cif and only if
find a pointx∈Csuch thatTxmin y∈CT
y. 2.13
We can rewrite the variational inequalityTx, y−x ≥0 for ally∈Cas, for anyγ >0,
x−x−γTx, y−x≥0 ∀y∈C. 2.14
If we introduce the nearest point projectionPC fromHontoC,
PCxarg min u∈C
1 2x−u
2, x∈H, 2.15
which is characterized by the inequality
CxPCx⇐⇒ x−x, y−x ≤0, ∀y∈C, 2.16
then we see from the above2.14that the minimization2.13is equivalent to the fixed point problem
PC
Therefore, they have a relation as follows:
findingx∈C, x∈EPF
Findingx∈C, Fx, y≥0, ∀y∈C, letFx, yγTx, y−x ≥0, ∀γ >0, ∀y∈C.
min y∈CT
y, where∇Tx Tx, ∀x∈C.
x∈FixPC
I−γT.
2.18
In addition to this, based on the result3ofLemma 2.7, FixTr EPF, we know if the elementx ∈ F : FixS∩EPF∩VIC, B, we havexis the solution of the nonlinear equation
x−SPC
I−γBTrx0, ∀γ >0, 2.19
where Tr is defined as in Lemma 2.7. Once we have the solutions of the equation 2.19, then it simultaneously solves the fixed points problems, equilibrium points problems, and variational inequalities problems. Therefore, the constrained setF:FixS∩EPF∩VIC, B is very important and applicable.
We now recall some well-known concepts and results. It is well-known that for all
x, y∈Handλ∈0,1there holds
λx 1−λy2λx2 1−λy2−λ1−λx−y2. 2.20
A space X is said to satisfy Opial’s condition18 if for each sequence {xn} in X which converges weakly to pointx∈X, we have
lim inf
n→ ∞ xn−x<lim infn→ ∞ xn−y, ∀y∈X, y /x. 2.21
Lemma 2.2see9,10. Assume that{αn}is a sequence of nonnegative real numbers such that
αn1≤1−γn
αnδn, 2.22
whereγnis a sequence in (0,1) and{δn}is a sequence such that
i∞n1γn∞;
iilim supn→ ∞δn/γn≤0or
∞
n1|δn|<∞.
Lemma 2.3. In a real Hilbert spaceH, the following inequality holds:
xy2 ≤ x22y, xy, ∀x, y∈H. 2.23
Lemma 2.4Marino and Xu8. Assume thatBis a strong positive linear bounded operator on a
Hilbert spaceHwith coefficientγ >0and0< ρ≤ B−1. ThenI−ρB ≤1−ργ.
Lemma 2.5see21. Let{xn}and{yn}be bounded sequences in a Banach spaceXand let{βn}be
a sequence in0,1with0<lim infn→ ∞βn≤lim supn→ ∞βn<1. Supposexn1 1−βnznβnxn
for all integersn≥0andlim supn→ ∞zn1−zn − xn1−xn≤0. Then,limn→ ∞zn−xn0.
Lemma 2.6Blum and Oettli19. LetCbe a nonempty closed convex subset ofHand letFbe a
bifunction ofC×CintoRsatisfying (A1)–(A4). Letr >0andx∈H. Then, there existsz∈Csuch that
Fz, y 1
ry−z, z−x ≥0, ∀y∈C. 2.24
Lemma 2.7Combettes and Hirstoaga4. Assume thatF:C×C → Rsatisfies (A1)–(A4). For
r >0andx∈H, define a mappingTr :H → Cas follows:
Trx
z∈C:Fz, y1
ry−z, z−x ≥0, ∀y∈C
2.25
for allz∈H. Then, the following hold:
1Tr is single-valued;
2Tr is firmly nonexpansive, that is, for anyx, y∈H,Trx−Try2 ≤ Trx−Try, x−y;
3FTr EPF;
4EPFis closed and convex.
3. Main Results
Theorem 3.1. LetCbe a nonempty closed convex subset of a Hilbert spaceH. LetFbe a bifunction
ofC×CintoRwhich satisfies (A1)–(A4), letSbe a nonexpansive mapping ofCintoH, and letBbe a
λ-Lipschitzian, relaxedμ, ν-cocoercive map ofCintoHsuch thatFFS∩EPF∩VIC, B/∅. LetAbe a strongly positive linear bounded operator with coefficientγ >0. Assume that0< γ < γ/α. Letf be a contraction ofH into itself with a coefficient α0 < α < 1and let{xn}and {yn}be
sequences generated byx1∈Hand
Fyn, η
1
rnη−yn, yn−xn ≥0, ∀η∈C,
xn1αnγfxn βnxn
1−βn
I−αnA
SPCI−snByn
for alln, where{αn},{βn} ⊂0,1and{rn},{sn} ⊂0,∞satisfy
C1limn→ ∞αn0;
C2∞n1αn∞;
C30<lim infn→ ∞βn≤lim supn→ ∞βn<1;
C4∞n1|αn1−αn|<∞,∞n1|rn1−rn|<∞and∞n1|sn1−sn|<∞;
C5lim infn→ ∞rn>0;
C6{sn} ∈a, bfor somea,bwith0≤a≤b≤2ν−μλ2/λ2.
Then, both{xn}and{yn}converge strongly toq∈F, whereqPFγf I−Aq, which
solves the following variational inequality:
γfq−Aq, p−q ≤0, ∀p∈F. 3.2
Proof. Note that from the conditionC1, we may assume, without loss of generality, that
αn≤1−βnA−1. SinceAis a strongly positive bounded linear operator onH, then
Asup{|Ax, x|:x∈H,x1}. 3.3
observe that
1−βn
I−αnA
x, x1−βn−αnAx, x
≥1−βn−αnA
≥0,
3.4
that is to say1−βnI−αnAis positive. It follows that
1−βn
I−αnAsup
1−βn
I−αnA
x, x:x∈H,x1
sup1−βn−αnAx, x:x∈H,x1
≤1−βn−αnγ.
First, we show thatI−snBis nonexpansive. Indeed, from the relaxedμ, ν-cocoercive and
λ-Lipschitzian definition onBand conditionC6, we have
I−snBx−I−snBy2
x−y−sn
Bx−By2 x−y2−2s
nx−y, Bx−Bys2nBx−By2
≤ x−y2−2sn
−μBx−By2νx−y2s2nBx−By2
≤ x−y22snλ2μx−y2−2snνx−y2λ2s2nx−y2
12snλ2μ−2snνλ2s2n
x−y2
≤ x−y2,
3.6
which implies that the mappingI−snBis nonexpansive.
Now, we observe that{xn}is bounded. Indeed, takep∈F, sinceynTrnxn, we have
yn−pTrnxn−Trnp ≤ xn−p. 3.7
PutρnPCI−snByn, sincep∈VIC, B, we havepPCI−snBp. Therefore, we have
ρn−pPCI−snByn−PCI−snBp
≤ I−snByn−I−snBp
≤ yn−p ≤ xn−p.
3.8
Due to3.5, it follows that
xn1−pαn
γfxn−Ap
βn
xn−p
1−βn
I−αnA
Sρn−p
≤1−βn−αnγ
xn−pβnxn−pαnγfxn−Ap
≤1−αnγ
xn−pαnγfxn−f
pαnγf
p−Ap
≤1−αnγ
xn−pαnγαxn−pαnγf
p−Ap 1−γ−γααn
xn−pαnγf
p−Ap.
3.9
It follows from3.9that
xn−p ≤max
x0−p,
γfp−Ap γ−γα
, n≥0. 3.10
Next, we show that
lim
n→ ∞xn1−xn0. 3.11
Observing thatynTrnxnandyn1Trn1xn1, we have
Fyn, η
1 rn
η−yn, yn−xn ≥0, ∀η∈C, 3.12
Fyn1, η r1 n1
η−yn1, yn1−xn1≥0, ∀η∈C. 3.13
Puttingηyn1in3.12andηynin3.13, we have
Fyn, yn1r1 n
yn1−yn, yn−xn ≥0, ∀η∈C,
Fyn1, yn
1 rn1
yn−yn1, yn1−xn1≥0, ∀η∈C.
3.14
It follows fromA2that
yn1−yn,
yn−xn
rn −
yn1−xn1
rn1
≥0. 3.15
That is,
yn1−yn, yn−yn1yn1−xn−
rn
rn1
yn1−xn1≥0. 3.16
Without loss of generality, let us assume that there exists a real numbermsuch thatrn> m >0 for alln. It follows that
yn1−yn2≤ yn1−yn
xn1−xn
1− rn
rn1
yn1−xn1
. 3.17
It follows that
yn1−yn ≤ xn1−xn
1− rn
rn1
yn1−xn1
≤ xn1−xn
M1
m |rn1−rn1|,
whereM1is an appropriate constant such that supn≥1yn−xn ≤M1. Note that
ρn1−ρnPCI−sn1Byn1−PCI−snByn
≤ I−sn1Byn1−I−snByn
I−sn1Byn1−I−sn1Byn sn−sn1Byn
≤ yn1−yn|sn−sn1|Byn.
3.19
Substituting3.18into3.19yields that
ρn1−ρn ≤ xn1−xnM2|rn1−rn||sn1−sn|, 3.20
whereM2is an appropriate constant such thatM2max{supn≥1Byn, M1/m}. Define
xn11−βn
znβnxn, n≥0. 3.21
Observe that from the definition ofyn, we obtain
zn1−zn
xn2−βn1xn1
1−βn1 −
xn1−βnxn 1−βn
αn1γfxn1
1−βn1I−αn1ASρn1
1−βn1
−αnγfxn
1−βn
I−αnA
Sρn 1−βn
αn1
1−βn1γfxn1−
αn
1−βnγfxn
αn 1−βn
ASρn
− αn1
1−βn1ASρn1Sρn1−Sρn
αn1
1−βn1
γfxn1−ASρn1 αn
1−βn
ASρn−γfxn
Sρn1
−Sρn.
It follows that with
zn1−zn − xn1−xn ≤
αn1 1−βn1
γfxn1ASρn1 αn
1−βn
γfxnASρn
ρn1−ρn − xn1−xn
≤ αn1
1−βn1
γfxn1ASρn1 αn
1−βn
γfxnASρn
M2|rn1−rn||sn1−sn|.
3.23
This together withC1,C3, andC4implies that
lim sup
n→ ∞ zn1−zn − xn1−xn≤0. 3.24
Hence, byLemma 2.5, we obtainzn−xn → 0 asn → ∞. Consequently,
lim
n→ ∞xn1−xnnlim→ ∞
1−βn
zn−xn0. 3.25
Note that
xn1−xnαn
γfxn−Axn
1−βn
I−αnA
Sρn−xn
≤αnγfxn−Axn
1−βn−αnγ
Sρn−xn.
3.26
This together with3.25implies that
Sρn−xn −→0. 3.27
Forp∈F, we have
yn−p2 Trnxn−Trnp
2
≤ Trnxn−Trnp, xn−p
yn−p, xn−p
1
2
yn−p2xn−p2− xn−yn2
3.28
and hence
Setλ >0 as a constant such that
λ >sup k
γfxk−ASρk,xk−p
. 3.30
By3.29and3.30, we have
xn1−p2αnγfxn βnxn
1−βn
I−αnA
Sρn−p2
1−βn
I−αnA
Sρn−p
βn
xn−p
αn
γfxn−Ap
2
1−βn
Sρn−p
−αnA
Sρn−p
βn
xn−p
αn
γfxn−Ap
2
1−βn
Sρn−p
βn
xn−p
αn
γfxn−ASρn
2
≤ 1−βn
Sρn−p
βn
xn−p
2
2αnγfxn−ASρn, xn1−p
≤ 1−βn
Sρn−p
βn
xn−p
22α
nλ2
≤1−βn
Sρn−p2βnxn−p22αnλ2
≤1−βn
ρn−p2βnxn−p22αnλ2
≤1−βn
yn−p2βnxn−p22αnλ2
≤ xn−p2−
1−βn
yn−xn22αnλ2.
3.31
It follows that
yn−xn2≤
1 1−βn
xn−p2− xn1−p22αnλ2
1
1−βn
xn−p − xn1−pxn−pxn1−p2αnλ2
≤ 1
1−βn
xn−xn1xn−pxn1−p2αnλ2
.
3.32
Byxn−xn1 → 0 andαn → 0, asn → ∞, and{xn}is bounded, we obtain that
lim
Forp∈F, we have
ρn−p2PCI−snByn−PCI−SnBp2
≤ yn−p
−sn
Byn−Bp
2
yn−p2−2snyn−p, Byn−Bps2nByn−Bp2
≤ xn−p2−2sn
−μByn−Bp2νyn−p2
s2nByn−Bp2
≤ xn−p22snμByn−Bp2−2snνyn−p2s2nByn−Bp2
≤ xn−p2
2snμs2n− 2snν
λ2
Byn−Bp2.
3.34
Observe3.31that
xn1−p2≤1−βn
ρn−p2βnxn−p22αnλ2. 3.35
Substituting3.34into3.35, we have
xn1−p2≤ x
n−p2
2snμs2n− 2snν
λ2
Byn−Bp22αnλ2. 3.36
It follows from conditionC6that
2aν
λ2 −2bμ−b 2
Byn−Bp2 ≤ xn−p2− xn1−p22αnλ2
≤ xn−xn1xn−pxn1−p2αnλ2.
3.37
From conditionC1and3.25, we have that
lim
On the other hand, we have
ρn−p2PCI−snByn−PCI−SnBp2
≤ I−snByn−I−SnBp, ρn−p
1
2
I−snByn−I−SnBp2ρn−p2
−I−snByn−I−SnBp−
ρn−p
2
≤ 1
2
yn−p2ρn−p2−
yn−ρn
−sn
Byn−Bp
2
≤ 1
2
xn−p2ρn−p2− yn−ρn2−s2nByn−Bp2
2sn
yn−ρn, Ayn−Ap
,
3.39
which yields that
ρn−p2≤ xn−p2− yn−ρn22snyn−ρnByn−Bp. 3.40
Substituting3.40into3.35yields that
xn1−p2≤ x
n−p2−
1−βn
yn−ρn2
2snyn−ρnByn−Bp2αnλ2.
3.41
It follows that
yn−ρn2 ≤ 1
1−βn
xn−p2− xn1−p2
2sn 1−βn
yn−ρnByn−Bp 2αnλ2 1−βn
≤ 1
1−βn
xn1−xn
xn−pxn1−p
2sn 1−βn
yn−ρnByn−Bp 2αnλ2 1−βn
.
3.42
From conditionC1,3.25, and3.38, we have that
lim
Observe that
yn−Syn ≤ yn−xnxn−SρnSρn−Syn
≤ yn−xnxn−Sρnρn−yn.
3.44
From3.27,3.33, and3.43, we have
lim
n→ ∞yn−Syn0. 3.45
Observe thatPFγf I−Ais a contraction. Indeed, for allx, y∈H, we have
PF
γf I−Ax−PF
γf I−Ay ≤ γf I−Ax−γf I−Ay
≤γfx−fyI−Ax−y
≤γαx−y1−γx−y
1−γ−γαx−y.
3.46
Banach’s Contraction Mapping Principle guarantees thatPFγf I−Ahas a unique fixed point, sayq∈H, that is,qPFγf I−Aq.
Next, we show that
lim sup n→ ∞ γf
q−Aq, xn−q ≤0. 3.47
To see this, we choose a subsequence{xni}of{xn}such that
lim sup n→ ∞
γfq−Aq, xn−qlim sup i→ ∞ γf
q−Aq, xni−q. 3.48
Correspondingly, there exists a subsequence{yni}of{yn}. Since{yni}is bounded, there exists a subsequence{ynij}of{yni}which converges weakly tow. Without loss of generality, we can assume thatyni w.
Next, we show thatw∈F. First, we provew∈EPF. SinceynTrnxn, we have
Fyn, η
1 rn
η−yn, yn−xn ≥0, ∀η∈C. 3.49
It follows fromA2that,
η−yn,
yn−xn
rn
≥Fη, yn
It follows that
η−yni,
yni−xni
rni
≥Fη, yni
. 3.51
Sinceyni −xni/rni → 0,yni w, andA4, we haveFη, w≤ 0 for allη ∈C. Fortwith 0< t≤1 andη∈C, letηttη 1−tw. Sinceη ∈Candw ∈C, we haveηt∈Cand hence
Fηt, w≤0. So, fromA1andA4, we have
0Fηt, ηt
≤tFηt, η
1−tFηt, w
≤tFηt, η
. 3.52
That is,Fηt, η≤0. It follows fromA3thatFw, η≥0 for allη∈Cand hencew∈EPF. Since Hilbert spaces satisfy Opial’s condition, from3.43, supposew /Sw; we have
lim inf
i→ ∞ yni−w<lim inf
i→ ∞ yni−Sw
lim inf
i→ ∞ yni−SyniSyni−Sw
≤lim inf
i→ ∞ Syni−Sw
<lim inf
i→ ∞ yni−w
3.53
which is a contradiction. Thus, we havew∈FS. Next, let us show thatw∈VIC, B. Put
Tw1
⎧ ⎨ ⎩
Bw1NCw1, w1 ∈C,
∅, w1∈C.
3.54
SinceBis relaxedμ, ν-cocoercive and from conditionC6, we have
Bx−By, x−y ≥−μBx−By2νx−y2≥ν−μλ2x−y2≥0, 3.55
which yields thatBis monotone. ThusT is maximal monotone. Letw1, w2 ∈ GT. Since
w2−Bw1 ∈NCw1andρn∈C, we have
On the other hand, fromρnPCI−snByn, we have
w1−ρn, ρn−I−snByn
≥0. 3.57
and hence
w1−ρn,
ρn−yn
sn
Byn
≥0. 3.58
It follows that
w1−ρni, w2
≥w1−ρni, Bw1
≥w1−ρni, Bw1
−
w1−ρni,
ρni−yni
sni
Byni
w1−ρni, Bw1−
ρni−yni
sni
−Byni
w1−ρni, Bw1−Bρni
w1−ρni, Bρni−Byni
−
w1−ρni,
ρni−yni
sni
≥w1−ρni, Bρni−Byni
−
w1−ρni,
ρni−yni
sni
,
3.59
which implies thatw1−w, w2 ≥ 0, We havew ∈ T−10 and hencew ∈ VIC, B. That is,
w∈F.
SinceqPFγf I−Aq, we have
lim sup n→ ∞ γf
q−Aq, xn−qlim sup i→ ∞ γf
q−Aq, xni−q
γfq−Aq, w−q ≤0.
3.60
That is,3.47holds.
Finally, we show thatxn → q, whereqPFγf I−Aq, which solves the following variational inequality:
We consider
xn1−q21−βn
I−αnA
Sρn−q
βn
xn−q
αn
γfxn−Aq
2
1−βn
I−αnA
Sρn−q
βn
xn−q
2α2
nγfxn−Aq2
2βnαnxn−q, γfxn−Aq
2αn
1−βn
I−αnA
Sρn−q
, γfxn−Aq
≤1−βnI−αnγ
Sρn−qβnxn−q
2
α2nγfxn−Aq2
2βnγαnxn−q, fxn−f
q2βnαnxn−q, γf
q−Aq
21−βn
γαnSρn−q, fxn−f
q
21−βn
αn
Sρn−q, γf
q−Aq−2α2nASρn−q
, γfq−Aq,
3.62
which implies that
xn1−q2 ≤1−α
nγ
2
2αβnγαn2α
1−βn
γαn
xn−q2
α2nγfxn−Aq22βnαnxn−q, γf
q−Aq
21−βn
αnSρn−q, γf
q−Aq −2α2nASρn−q
, γfq−Aq
≤1−2γ−αγαn
xn−q2γ2α2nxn−q2α2nγfxn−Aq2
2βnαnxn−q, γf
q−Aq21−βn
αnSρn−q, γf
q−Aq
−2α2nASρn−q
· γfq−Aq
1−2γ−αγαn
xn−q2
αn
αn
γ2xn−q2γfxn−Aq22A
Sρn−q
· γfq−Aq
2βnxn−q, γf
q−Aq21−βn
Sρn−q, γf
q−Aq.
3.63
Since{xn},{fxn}, and{Sρn}are bounded, we can take a constantM2>0 such that
M2≥γ2xn−q2γfxn−Aq22A
Sρn−q
for alln≥0. It then follows that
xn1−q2≤1−2γ−αγαn
xn−q2αnξn, 3.65
where
ξn2βnxn−q, γf
q−Aq21−βn
Sρn−q, γf
q−AqαnM2. 3.66
From3.27and3.47, we also have
lim sup n→ ∞ γf
q−Aq, Sρn−qlim sup n→ ∞ γf
q−Aq, Sρn−xnlim sup n→ ∞ γf
q−Aq, xn−q
≤lim sup n→ ∞
γfq−Aq, xn−q
≤0.
3.67
By C1,3.47, and3.67, we get lim supn→ ∞ξn ≤ 0. Now applyingLemma 2.2to 3.65 concludes thatxn → qn → ∞.
This completes the proof.
Remark 3.2. Some iterative algorithms were presented in Yamada11, Combettes24, and Iiduka-Yamada25, for example, the steepest descent method, the hybrid steepest descent method, and the conjugate gradient methods; these methods have common form
xn1xnωndn, 3.68
wherexnis thenth approximation to the solution,ωn > 0 is a step size, anddn is a search direction. In this paper, We defineT :SPCI−sBTr; the method3.1will be changed as
xn1 αnγfxn βnxn
1−βn
I−αnA
SPCI−snBTxn
xn
1−βn
−xnTxn αn
γfxn−ATxn
.
3.69
Takeωndn 1−βn−xnTxn αnγfxn−ATxn, the method3.1will be changed as
3.68.
4. Applications
Theorem 4.1. LetCbe a nonempty closed convex subset of a Hilbert spaceH. LetFbe a bifunction
ofC×CintoRwhich satisfies (A1)–(A4); letSbe a nonexpansive mapping ofCintoHsuch that
FFS∩EPF∩VIC, B/∅. LetAbe a strongly positive linear bounded operator with coefficient
γ >0. Assume that0< γ < γ/α. Letfbe a contraction ofHinto itself with a coefficientα0< α <1
and let{xn}and{Yn}be sequences generated byx1∈Hand
Fyn, η
1 rn
η−yn, yn−xn ≥0, ∀η∈C,
xn1αnγfxn βnxn
1−βn
I−αnA
Syn,
4.1
for alln, where{αn},{βn} ⊂0,1and{rn},{sn} ⊂0,∞satisfy
C1limn→ ∞αn0;
C2∞n1αn∞;
C30<lim infn→ ∞βn≤lim supn→ ∞βn<1;
C4∞n1|αn1−αn|<∞and∞n1|γn1−γn|<∞;
C5lim infn→ ∞γn>0.
Then, both{xn}and{yn}converge strongly toq∈F, whereqPFγf I−Aq, which
solves the following variational inequality:
γfq−Aq, p−q ≤0, ∀p∈F. 4.2
Proof. Taking{sn}0 inTheorem 3.1, we can get the desired conclusion easily.
Theorem 4.2. LetCbe a nonempty closed convex subset of a Hilbert spaceH, letSbe a nonexpansive mapping ofCintoH, and letBbe aλ-Lipschitzian, relaxedμ, ν-cocoercive map ofCintoHsuch thatF FS∩VIC, B/∅. LetAbe a strongly positive linear bounded operator with coefficient
γ >0. Assume that0< γ < γ/α. Letfbe a contraction ofHinto itself with a coefficientα0< α <1
and let{xn}and{yn}be sequences generated byx1∈Hand
xn1αnγfxn βnxn
1−βn
I−αnA
SPCI−snBPCxn, 4.3
for alln, where{αn},{βn} ⊂0,1and{rn},{sn} ⊂0,∞satisfy
C1limn→ ∞αn0;
C2∞n1αn∞;
C30<lim infn→ ∞βn≤lim supn→ ∞βn<1;
C4∞n1|αn1−αn|<∞and
∞
n1|sn1−sn|<∞;
C5lim infn→ ∞γn>0;
Then, both{xn}and{yn}converge strongly toq∈F, whereqPFγf I−Aq, which
solves the following variational inequality:
γfq−Aq, p−q ≤0, ∀p∈F. 4.4
Proof. PutFx, y 0 for allx, y ∈ Cand γn 1 for allnin Theorem 3.1. Then we have
ynPCxn. we can obtain the desired conclusion easily.
Acknowledgments
This work is supported by the Fundamental Research Funds for the Central Universities, no. JY10000970006 and National Science Foundation of China, no. 60974082.
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