Volume 2012, Article ID 235474,30pages doi:10.1155/2012/235474
Research Article
Viscosity Approximations by the Shrinking Projection Method of Quasi-Nonexpansive
Mappings for Generalized Equilibrium Problems
Rabian Wangkeeree and Nimit Nimana
Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
Correspondence should be addressed to Rabian Wangkeeree,[email protected] Received 16 July 2012; Accepted 27 August 2012
Academic Editor: Juan Torregrosa
Copyrightq2012 R. Wangkeeree and N. Nimana. 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 viscosity approximations by using the shrinking projection method established by Takahashi, Takeuchi, and Kubota, for finding a common element of the set of solutions of the generalized equilibrium problem and the set of fixed points of a quasi-nonexpansive mapping.
Furthermore, we also consider the viscosity shrinking projection method for finding a common element of the set of solutions of the generalized equilibrium problem and the set of fixed points of the super hybrid mappings in Hilbert spaces.
1. Introduction
LetHbe a real Hilbert space with inner product·,·and norm·andCa nonempty closed convex subset of H and letT be a mapping ofCinto H. Then, T : C → His said to be nonexpansive ifTx−Ty ≤ x−yfor allx, y ∈ C. A mappingT : C → His said to be quasi-nonexpansive ifTx−y ≤ x−yfor allx∈ Candy ∈ FT : {x∈ C: Tx x}.
Recall that a mappingΨ:C → His said to beδ-inverse strongly monotone if there exists a positive real numberδsuch that
Ψx−Ψy, x−y
≥δΨx−Ψy2, ∀x, y∈C. 1.1 IfΨ is an δ-inverse strongly monotone mapping of Cinto H, then it is obvious that Ψis 1/δ-Lipschitz continuous.
LetF :C×C → Rbe a bifunction andΨ:C → Hbeδ-inverse strongly monotone mapping. The generalized equilibrium problemfor short, GEPforFandΨis to findz∈C such that
F z, y
Ψz, y−z
≥0, ∀y∈C. 1.2
The problem1.2was studied by Moudafi1. The set of solutions for the problem1.2is denoted by GEPF,Ψ, that is,
GEPF,Ψ
z∈C:F z, y
Ψz, y−z
≥0,∀y∈C
. 1.3
If Ψ ≡ 0 in 1.2, then GEP reduces to the classical equilibrium problem and GEPF,0is denoted by EPF, that is,
EPF
z∈C:F z, y
≥0,∀y∈C
. 1.4
IfF ≡ 0 in1.2, then GEP reduces to the classical variational inequality and GEP0,Ψis denoted by VIΨ, C, that is,
VIΨ, C
z∈C:
Ψz, y−z
≥0,∀y∈C
. 1.5
The problem1.2is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, min-max problems, and the Nash equilibrium problems in noncooperative games, see, for example, Blum and Oettli2and Moudafi3.
In 2005, Combettes and Hirstoaga4introduced an iterative algorithm of finding the best approximation to the initial data and proved a strong convergence theorem. In 2007, by using the viscosity approximation method, S. Takahashi and W. Takahashi 5 introduced another iterative scheme for finding a common element of the set of solutions of the equilibrium problem and the set of fixed points of a nonexpansive mapping. Subsequently, algorithms constructed for solving the equilibrium problems and fixed point problems have further developed by some authors. In particular, Ceng and Yao6introduced an iterative scheme for finding a common element of the set of solutions of the mixed equilibrium problem and the set of common fixed points of finitely many nonexpansive mappings.
Maing´e and Moudafi 7 introduced an iterative algorithm for equilibrium problems and fixed point problems. Wangkeeree 8 introduced a new iterative scheme for finding the common element of the set of common fixed points of nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of the variational inequality.
Wangkeeree and Kamraksa 9 introduced an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions of a general system of variational inequalities for a cocoercive mapping in a real Hilbert space. Their results extend and improve many results in the literature.
In 1953, Mann10introduced the following iterative procedure to approximate a fixed point of a nonexpansive mappingT in a Hilbert spaceHas follows:
xn1αnxn 1−αnTxn, ∀n∈N, 1.6
where the initial pointx1is taken inCarbitrarily and{αn}is a sequence in0,1. Wittmann 11obtained the strong convergence results of the sequence{xn}defined by1.6toPFx1
under the following assumptions:
C1limn→ ∞αn0;
C2 ∞n1αn∞;
C3 ∞n1|αn1−αn|<∞,
where PFT is the metric projection of H onto FT. In 2000, Moudafi 12 introduced the viscosity approximation method for nonexpansive mappings see 13 for further developments in both Hilbert and Banach spaces. Letfbe a contraction onH. Starting with an arbitrary initialx1∈H, define a sequence{xn}recursively by
xn1αnfxn 1−αnTxn, n≥1, 1.7 where{αn}is a sequence in0,1. It is proved12,13that under conditionsC1,C2, and C3imposed on{αn}, the sequence{xn}generated by1.7strongly converges to the unique fixed pointx∗ofPFTfwhich is a unique solution of the variational inequality
I−f
x∗, x−x∗
≥0, x∈C. 1.8
Suzuki14 considered the Meir-Keeler contractions, which is extended notion of contrac- tions and studied equivalency of convergence of these approximation methods.
Using the viscosity approximation method, in 2007, S. Takahashi and W. Takahashi 5introduced an iterative scheme for finding a common element of the solution set of the classical equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. LetT : C → H be a nonexpansive mapping. Starting with arbitrary initial x1∈H, define sequences{xn}and{un}recursively by
F un, y
1 rn
y−un, un−xn
≥0, ∀y∈C,
xn1αnfxn 1−αnTun, ∀n∈N.
1.9
They proved that under certain appropriate conditions imposed on {αn} and {rn}, the sequences{xn}and{un}converge strongly toz∈FT∩EPF, wherezPFT∩EPFfz.
On the other hand, in 2008, Takahashi et al.15has adapted Nakajo and Takahashi’s 16idea to modify the process1.6so that strong convergence has been guaranteed. They proposed the following modification for a family of nonexpansive mappings in a Hilbert space:x0∈H,C1C,u1PC1x0and
ynαnun 1−αnTnun, Cn1
z∈Cn:yn−z≤ un−z , un1PCn1x0, n∈N,
1.10
where 0 ≤ αn ≤ a < 1 for all n ∈ N. They proved that if {Tn} satisfies the appropriate conditions, then {un} generated by 1.10 converges strongly to a common fixed point of Tn.
Very recently, Kimura and Nakajo17considered viscosity approximations by using the shrinking projection method established by Takahashi et al. 15 and the modified shrinking projection method proposed by Qin et al.18, for finding a common fixed point of countably many nonlinear mappings, and they obtained some strong convergence theorems.
Motivated by these results, we introduce the viscosity shrinking projection method for finding a common element of the set of solutions of the generalized equilibrium problem and the set of fixed points of a quasi-nonexpansive mapping. Furthermore, we also consider the viscosity shrinking projection method for finding a common element of the set of solutions of the generalized equilibrium problem and the set of fixed points of the super hybrid mappings in Hilbert spaces.
2. Preliminaries
Throughout this paper, we denote byNthe set of positive integers and byRthe set of real numbers. LetH be a real Hilbert space with inner product·,·and norm · . We denote the strong convergence and the weak convergence of{xn}tox∈Hbyxn → xandxn x, respectively. From19, we know the following basic properties. Forx, y∈Handλ ∈Rwe have
λx 1−λy2λx2 1−λy2−λ1−λx−y2. 2.1
We also know that foru, v, x, y∈H, we have 2
u−v, x−y
u−y2v−x2− u−x2−v−y2. 2.2 For every pointx∈H, there exists a unique nearest point ofC, denoted byPCx, such thatx−PCx ≤ x−yfor ally∈C.PCis called the metric projection fromHontoC. It is well known thatzPCx⇔ x−z, z−y ≥0, for allx∈Handz, y∈C. We also know that PCis firmly nonexpansive mapping fromHontoC, that is,
PCx−PCy2≤
PCx−PCy, x−y
, ∀x, y∈H, 2.3 and so is nonexpansive mapping.
For solving the generalized equilibrium problem, let us assume that F 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,lim
t↓0 Ftz 1−tx, y≤Fx, y;
A4for eachx∈C, y→Fx, yis convex and lower semicontinuous.
In order to prove our main results, we also need the following lemmas.
Lemma 2.1see2. LetCbe a nonempty closed convex subset ofHand letFbe a bifunction from C×CintoRsatisfying (A1), (A2), (A3), and (A4). Then, for anyr > 0 andx ∈H, there exists a uniquez∈Csuch that
F z, y
1 r
y−z, z−x
≥0, ∀y∈C. 2.4
Lemma 2.2see4. LetCbe a nonempty closed convex subset ofHand letFbe a bifunction from C×CintoRsatisfying (A1), (A2), (A3), and (A4). Then, for anyr >0 andx∈H, define a mapping Trx:H → Cas follows:
Trx
z∈C:F z, y
1 r
y−z, z−x
≥0,∀y∈C
∀x∈H, r ∈R. 2.5
Then the following hold:
iTr is single-valued;
iiTr is firmly nonexpansive, that is, Trx−Try2≤
Trx−Try, x−y
, ∀x, y∈H; 2.6 iiiFTr EPF;
ivEPFis closed and convex.
Remark 2.3see20. UsingiiinLemma 2.2and2.2, we have 2Trx−Try2≤2
Trx−Try, x−y
Trx−y2Try−x2− Trx−x2−Try−y2.
2.7
So, fory∈FTrandx∈H, we have
Trx−u2Trx−x2≤ x−u2. 2.8 Remark 2.4. For anyx∈Handr >0, byLemma 2.1, there existsz∈Csuch that
F z, y
1 r
y−z, z−x
≥0, ∀y∈C. 2.9
Replacingxwithx−rΨx∈Hin2.9, we have F
z, y
Ψx, y−z 1
r
y−z, z−x
≥0, ∀y∈C, 2.10
whereΨ:C → His an inverse-strongly monotone mapping.
For a sequence{Cn}of nonempty closed convex subsets of a Hilbert spaceH, define s−LinCnandw−LsnCnas follows.
x∈s−LinCnif and only if there exists{xn} ⊂Hsuch thatxn → xand thatxn ∈Cn
for alln∈N.
x∈w−LsnCnif and only if there exists a subsequence{Cni}of{Cn}and a subsequence {yi} ⊂Hsuch thatyi yand thatyi∈Cni for alli∈N.
IfC0satisfies
C0s−LinCnw−LsnCn, 2.11
it is said that {Cn} converges to C0 in the sense of Mosco 21 and we write C0 M− limn→ ∞Cn. It is easy to show that if{Cn} is nonincreasing with respect to inclusion, then {Cn}converges to ∞
n1Cn in the sense of Mosco. For more details, see 21. Tsukada22 proved the following theorem for the metric projection.
Theorem 2.5see Tsukada22. LetHbe a Hilbert space. Let{Cn}be a sequence of nonempty closed convex subsets ofH. IfC0 M−limn→ ∞Cnexists and is nonempty, then for eachx ∈H, {PCnx}converges strongly toPC0x, wherePCn andPC0 are the metric projections ofHontoCnand C0, respectively.
On the other hand, a mappingfof a complete metric spaceX, dinto itself is said to be a contraction with coefficientr ∈0,1ifdfx, fy≤rdx, yfor allx, y∈C. It is well known thatfhas a unique fixed point23. Meir-Keeler24defined the following mapping called Meir-Keeler contraction. LetX, dbe a complete metric space. A mappingf:X → X is called a Meir-Keeler contraction if for allε >0, there existsδ >0 such thatε≤dx, y< εδ impliesdfx, fy< ε for allx, y ∈X. It is well known that Meir-Keeler contraction is a generalization of contraction and the following result is proved in24.
Theorem 2.6see Meir-Keeler24. A Meir-Keeler contraction defined on a complete metric space has a unique fixed point.
We have the following results for Meir-Keeler contractions defined on a Banach space by Suzuki14.
Theorem 2.7see Suzuki14. Letfbe a Meir-Keeler contraction on a convex subsetCof a Banach spaceE. Then, for everyε >0, there existsr∈0,1such thatx−y ≥εimpliesfx−fy ≤ rx−yfor allx, y∈C.
Lemma 2.8see Suzuki14. LetCbe a convex subset of a Banach spaceE. LetTbe a nonexpansive mapping onC, and letfbe a Meir-Keeler contraction onC. Then the following hold.
iT◦fis a Meir-Keeler contraction onC.
iiFor eachα∈0,1, a mappingx→1−αTxαfxis a Meir-Keeler contraction onC.
3. Main Results
In this section, using the shrinking projection method by Takahashi et al. 15, we prove a strong convergence theorem for a quasi-nonexpansive mapping with a generalized equilibrium problem in a Hilbert space. Before proving it, we need the following lemmas.
Lemma 3.1. LetC be a nonempty closed convex subset of a Hilbert space H and δ > 0 and let Ψ:C → Hbeδ-inverse strongly monotone. If 0< λ≤2δ, thenI−λΨis a nonexpansive mapping.
Proof. Forx, y∈C, we can calculate
I−λΨx−I−λΨy2x−y−λΨx−Ψy2 x−y2−2λ
x−y,Ψx−Ψy
λ2Ψx−Ψy2
≤x−y2−2λδΨx−Ψy2λ2Ψx−Ψy2 x−y2λλ−2δΨx−Ψy2
≤x−y2.
3.1
ThereforeI−λΨis nonexpansive. This completes the proof.
Lemma 3.2. Let C be a nonempty closed convex subset ofH, and let T be a quasi-nonexpansive mapping ofCintoH. Then,FTis closed and convex.
Proof. We first show thatFTis closed. Let{zn}be any sequence inFTwithzn → z. We claim thatz∈FT. SinceCis closed, we havez∈C. We observe that
z−Tz ≤ z−znzn−Tz
≤ z−znz−zn 2z−zn.
3.2
Sincezn → z, we obtain thatz−Tz ≤0 and hencezTz. This show thatz∈FT. Next, we show that FTis convex. Let x, y ∈ FT and α ∈ 0,1. We claim that αx 1−αy∈FT. Puttingzαx 1−αy, we have
z−Tz2αx 1−αy−Tz2 αx−Tz 1−α
y−Tz2
αx−Tz2 1−αy−Tz2−α1−αx−y2
≤αx−z2 1−αy−z2−α1−αx−y2
α1−α
x−y2 1−αα
y−x2−α1−αx−y2
α1−α2α21−α−α1−αx−y2 α1−α1−αα−1x−y20.
3.3 HenceFTis convex. This completes the proof.
Theorem 3.3. Let H be a Hilbert space and let Cbe a nonempty closed convex subset of H. Let F : C×C → R be a bifunction satisfying (A1), (A2), (A3), and (A4) and letΨbe a δ-inverse strongly monotone mapping fromCintoH. LetT :C → Cbe a quasi-nonexpansive mapping which is demiclosed onC, that is, if{wk} ⊂ C, wk wand I−Twk → 0, thenw ∈FT. Assume thatΩ:GEPF,Ψ∩FT/∅andfis a Meir-Keeler contraction ofCinto itself. Let the sequence {xn} ⊂Cbe defined by
C1C, x1x∈C, F
zn, y
Ψxn, y−zn
1 λn
zn−xn, y−zn
≥0, ∀y∈C,
ynαnxn 1−αnTzn, Cn1
z∈Cn:yn−z≤ xn−z , xn1PCn1fxn, ∀n∈N,
3.4
wherePCn1 is the metric projection ofH ontoCn1and {αn} ⊂ 0,1and {λn} ⊂ 0,2δare real sequences satisfying
lim inf
n→ ∞ αn<1, 0< a≤λn≤b <2δ, 3.5 for somea, b∈R. Then,{xn}converges strongly toz0∈Ω, which satisfiesz0PΩfz0.
Proof. Since Ω is a closed convex subset of C, we have that PΩ is well defined and nonexpansive. Furthermore, we know thatfis Meir-Keeler contraction and we know from Lemma 2.8 i that PΩf of C onto Ω is a Meir-Keeler contraction on C. By Theorem 2.6, there exists a unique fixed point z0 ∈ C such that z0 PΩfz0. Next, we observe that zn Tλnxn−λnΨxnfor eachn∈Nand takez∈FT∩GEPF,Ψ. FromzTλnz−λnΨz andLemma 2.2, we have that for anyn∈N,
zn−z2Tλnxn−λnΨxn−Tλnz−λnΨz2
≤ xn−λnΨxn−z−λnΨz2
≤ xn−z2.
3.6
Next, we divide the proof into several steps.
Step 1.Cnis closed convex and{xn}is well defined for everyn∈N.
It is obvious from the assumption thatC1 : Cis closed convex andΩ⊂ C1. For any k∈N, suppose thatCkis closed and convex, andΩ⊂Ck. Note that for allz∈Ck,
yk−z2≤ xk−z2⇐⇒yk−z2− xk−z2≤0
⇐⇒yk2z2−2yk, z − xk2− z22xk, z ≤0
⇐⇒yk2−2yk−xk, z − xk2 ≤0.
3.7
It is easy to see thatCk1is closed. Next, we prove thatCk1is convex. For anyu, v∈Ck1and α∈0,1, we claim thatz:αu 1−αv∈Ck1. Sinceu∈Ck1, we haveyk−u ≤ xk−u and soyk−u2≤ xk−u2, that is,yk2−2yk−xk, u − xk2≤0. Similarly,v∈Ck1, we getyk2−2yk−xk, v − xk2≤0.
Thus,
αyk2−2
yk−xk, αu
−αxk2≤0, 1−αyk2−2
yk−xk,1−αv
−1−αxk2≤0.
3.8
Combining the above inequalities, we obtain yk2−2
yk−xk, αu 1−αv
− xk2 ≤0. 3.9 Thereforeyk−z ≤ xk−z. This shows thatz∈Ck1 and henceCk1 is convex. Therefore Cnis closed and convex for alln∈N.
Next, we show thatΩ ⊂ Cn, for alln∈ N. For anyk ∈N, suppose thatv ∈ Ω ⊂Ck. SinceT is quasi-nonexpansive and from3.6, we have
yk−v2αkxk 1−αkTzk−v2 αkxk−v 1−αkTzk−v2
≤αkxk−v2 1−αkTzk−v2
≤αkxk−v2 1−αkzk−v2
≤αkxk−v2 1−αkxk−v2 xk−v2.
3.10
So, we havev ∈ Ck1. By principle of mathematical induction, we can conclude thatCn is closed and convex, andΩ⊂Cn, for alln∈N. Hence, we have
∅ / Ω⊂Cn1⊂Cn, 3.11 for alln∈N. Therefore{xn}is well defined.
Step 2. limn→ ∞xn−u0 for someu∈∞
n1Cnandfu−u, u−y ≥0 for ally∈Ω.
Since ∞
n1Cn is closed convex, we also have that P∞n1Cn is well defined and so P∞n1Cnfis a Meir-Keeler contraction onC. ByTheorem 2.6, there exists a unique fixed point u ∈ ∞
n1Cn ofP∞n1Cnf. SinceCn is a nonincreasing sequence of nonempty closed convex subsets ofHwith respect to inclusion, it follows that
∅/ Ω⊂∞
n1
CnM− lim
n→ ∞Cn. 3.12
Settingun:PCnfuand applyingTheorem 2.5, we can conclude that
nlim→ ∞unP∞n1Cnfu u. 3.13
Next, we will prove that limn→ ∞xn−u0. Assume to contrary that lim supn→ ∞xn−u/0, there existsε >0 and a subsequence{xnj−u}of{xn−u}such that
xnj−u≥ε, ∀j∈N, 3.14
which gives that
lim sup
j→ ∞
xnj−u≥ε >0. 3.15
We choose a positive numberε>0 such that lim sup
j→ ∞
xnj−u> ε>0. 3.16
For suchε, by the definition of Meir-Keeler contraction, there existsδε>0 with εδε <lim sup
j→ ∞
xnj−u, 3.17
such that
x−y< εδε implies fx−f
y< ε, 3.18 for allx, y∈C. Again for suchε, byTheorem 2.7, there existsrε ∈0,1such that
x−y≥εδε implies fx−f
y< rεx−y. 3.19 Sinceun → u, there existsn0∈Nsuch that
un−u< δε, ∀n≥n0. 3.20 By the idea of Suzuki14and Kimura and Nakajo17, we consider the following two cases.
CaseI. Assume that there existsn1≥n0such that
xn1−u< εδε. 3.21
Thus, we get
xn11−u ≤ xn11−un11un11−u
PCn11fxn1−PCn11fuun11−u
≤fxn1−fuun11−u
< εδε.
3.22
By induction on{n}, we can obtain that
xn−u< εδε, 3.23
for alln≥n0. In particular, for allj≥n0, we havenj≥j ≥n0and
xnj−u< εδε. 3.24
This implies that
lim sup
j→ ∞
xnj−u≤εδε <lim sup
j→ ∞
xnj−u, 3.25
which is a contradiction. Therefore, we conclude thatxn−u → 0 asn → ∞.
CaseII. Assume that
xn−u ≥εδε, ∀n≥n0. 3.26 By3.19, we have
fxn−fu< rεxn−u, ∀n≥n0. 3.27
Thus, we have
xn1−un1PCn1fxn−PCn1fu
≤fxn−fu
≤rεxn−u
≤rεxn−unun−u,
3.28
for everyn≥n0. In particular, we have
xnj1−unj1≤rεxnj−unj
unj−u
, 3.29
for everyj ≥n0 nj≥j≥n0. Let us consider lim sup
n→ ∞
xnj−unj
lim sup
j→ ∞
xnj1−unj1
≤rεlim sup
j→ ∞
xnj−unj
unj−u
≤rεlim sup
j→ ∞
xnj−unj
rεlim sup
j→ ∞
unj−u rεlim sup
j→ ∞
xnj−unj
<lim sup
j→ ∞
xnj−unj
,
3.30
which gives a contradiction. Hence, we obtain that
nlim→ ∞xn−u0, 3.31
and therefore{xn}is bounded. Moreover, {fxn},{zn}, and {yn}are also bounded. Since xn1PCn1fxn, we have
fxn−xn1, xn1−y
≥0, ∀y∈Cn1. 3.32 SinceΩ⊂Cn1, we get
fxn−xn1, xn1−y
≥0, ∀n∈N, y∈Ω. 3.33 We have fromxn → uthat
fu−u, u−y
≥0, ∀y∈Ω. 3.34
Step 3. There exists a subsequence{xni −zni}of{xn−zn} such thatxni−zni → 0 as i → ∞.
We have from3.13and3.31that
xn−xn1 ≤ xn−uu−un1un1−xn1
xn−uu−un1PCn1fxn−PCn1fu
≤ xn−uu−un1fxn−fu−→0.
3.35
Fromxn1∈Cn1, we have that
yn−xn1≤ xn−xn1, 3.36
and soyn−xn1 → 0. We also have
yn−xn≤yn−xn1xn1−xn −→0. 3.37 From lim infn→ ∞αn<1, there exists a subsequence{αni}of{αn}andα0with 0≤α0 <1 such thatαni → α0. Sincexn−ynxn−αnxn−1−αTzn 1−αnxn−Tzn, we have
Tzni−xni −→0 asi−→ ∞. 3.38
UsingLemma 2.2iiand3.6, we have zn−z2Tλnxn−λnΨxn−Tλnz−λnΨz2
≤ xn−λnΨxn−z−λnΨz, zn−z − xn−λnΨxn−z−λnΨz, z−zn 1
2
xn−λnΨxn−z−λnΨz2zn−z2
−xn−λnΨxn−z−λnΨz z−zn2
≤ 1 2
xn−z2zn−z2− xn−zn−λnΨxn−Ψz2
1 2
xn−z2zn−z2− xn−zn22λnxn−zn,Ψxn−Ψz −λ2nΨxn−Ψz2 . 3.39
So, we have
zn−z2 ≤ xn−z2− xn−zn22λnxn−zn,Ψxn−Ψz −λ2nΨxn−Ψz2. 3.40
Let us consider
yn−z2αnxn−z 1−αnTzn−z2
≤αnxn−z2 1−αnTzn−z2
≤αnxn−z2 1−αnzn−z2
αnxn−z2 1−αnTλnI−λnΨxn−TλnI−λnΨz2
≤αnxn−z2 1−αnI−λnΨxn−I−λnΨz2 αnxn−z2 1−αnxn−z−λnΨxn−Ψz2 αnxn−z2 1−αnxn−z2 1−αnλ2nΨxn−Ψz2
−21−αnλnxn−z,Ψxn−Ψz
≤ xn−z2 1−αnλ2nΨxn−Ψz2−21−αnλnδΨxn−Ψz2 xn−z2 1−αnλn−2δλnΨxn−Ψz2
≤ xn−z2 1−αnb−2δbΨxn−Ψz2.
3.41
In particular, we have
1−αni2δ−bbΨxni−Ψz2≤ xni−z2−yni −z2
≤xni−yni22xni−yniyni−z.
3.42
Sinceαni → α0withα0<1 andxni−yni → 0, we obtain that
Ψxni−Ψz −→0. 3.43
Using3.40, we have
yn−z2≤ αnxn−z2 1−αnzn−z2
≤ αnxn−z2 1−αn
×
xn−z2− xn−zn22λnxn−zn,Ψxn−Ψz −λ2nΨxn−Ψz2
≤ αnxn−z2 1−αn
xn−z2− xn−zn22λnxn−znΨxn−Ψz
≤ xn−z2−1−αnxn−zn221−αnλnxnznΨxn−Ψz
≤ xn−z2−1−αnxn−zn221−αnbMΨxn−Ψz,
3.44 whereM:sup{xnyn:n∈N}.
So, we have
1−αnixni−zni2≤ xni−z2−yni−z221−αnibMΨxni−Ψz
≤xni−yni22xni−yniyni−z21−αnibMΨxni−Ψz.
3.45
We have fromαni → α0,3.37, and3.43that
xni−zni −→0. 3.46
Step 4. Finally, we prove thatu∈Ω:FT∩GEPF,Ψ.
Sinceyn αnxn 1−αnTzn, we haveyn−Tzn αnxn−Tzn. So, from3.38we have
yni−Tzniαnixni−Tzni −→0. 3.47 Sincezni −Tzni ≤ zni−xnixni −yniyni −Tzni, from3.37,3.46, and3.47we have
zni−Tzni −→0. 3.48
Sincexni → u, we havezni → u. So, from3.48and the demiclosed property ofT, we have
u∈FT. 3.49
We next show thatu∈GEPF,Ψ. SinceznTλnxn−λnΨxn, for anyy∈Cwe have
F zn, y
Ψxn, y−zn
1 λn
y−zn, zn−xn
≥0. 3.50
FromA2, we have
−F y, zn
Ψxn, y−zn
1 λn
y−zn, zn−xn
≥0 3.51
and so
Ψxn, y−zn
1 λn
y−zn, zn−xn
≥F y, zn
. 3.52
Replacingnbyni, we have
Ψxni, y−zni
y−zni,zni−xni
λni
≥F y, zni
. 3.53
Note thatΨis 1/δ-Lipschitz continuous, and from3.46, we have
Ψzni−Ψxni −→0. 3.54
Fort ∈0,1andy ∈C, letz∗t ty 1−tu. SinceCis convex, we havez∗t ∈C. So, from 3.53we have
z∗t−zni,Ψz∗t ≥ z∗t−zni,Ψz∗t − z∗t−zni,Ψxni −
z∗t−zni,zni−xni
λni
Fz∗t, zni z∗t−zni,Ψz∗t−Ψxni −
z∗t−zni,zni−xni
λni
Fz∗t, zni z∗t−zni,Ψz∗t−Ψzniz∗t −zni,Ψzni −Ψxni
−
z∗t−zni,zni−xni
λni
Fz∗t, zni.
3.55
Fromz∗t−zni,Ψz∗t−Ψzni ≥0, we have
z∗t−zni,Ψz∗t ≥ z∗t−zni,Ψzni−Ψxni −
z∗t−zni,zni−xni
λni
Fz∗t, zni. 3.56
Thus,
z∗t−zni,Ψz∗t − z∗t−zniΨzni−Ψxni ≥ −z∗t−zni
zni−xni
λni
Fz∗t, zni. 3.57
From Step 3 and3.54, we obtain
z∗t−u,Ψz∗t ≥Fz∗t, u. 3.58
FromA1,A4, and3.58, we have
0Fz∗t, z∗t F
z∗t, ty 1−tu
≤tF z∗t, y
1−tFz∗t, u
≤tF z∗t, y
1−tz∗t−u,Ψz∗t
≤tF z∗t, y
1−tty−u,Ψz∗t,
3.59
and hence
0≤F z∗t, y
1−t
y−u,Ψz∗t
. 3.60
Lettingt↓0 and fromA3, we have that for eachy∈C,
0≤lim
t↓0
F z∗t, y
1−t
y−u,Ψz∗t lim
t↓0
F
ty 1−tu, y
1−t
y−u, tΨy 1−tΨu
≤F u, y
y−u,Ψu .
3.61
This implies thatu ∈ GEPF,Ψ. So, we haveu ∈ FT∩GEPF,Ψ. We obtain from3.34 thatuz0and hence,{xn}converges strongly toz0. This completes the proof.
By Theorem 3.3, we can obtain some new and interesting strong convergence theorems. Now we give some examples as follows.
Settingfxn x,∀n∈NinTheorem 3.3, we obtain the following result.
Corollary 3.4. Let H be a Hilbert space and letC be a nonempty closed convex subset ofH. Let F : C×C → R be a bifunction satisfying (A1), (A2), (A3), and (A4) and letΨbe a δ-inverse strongly monotone mapping fromCintoH. LetT :C → Cbe a quasi-nonexpansive mapping which is demiclosed onC. Assume thatΩ/∅ and letC1 Cand{xn} ⊂ Cbe a sequence generated by x1x∈Cand
F zn, y
Ψxn, y−zn
1 λn
zn−xn, y−zn
≥0, ∀y∈C,
ynαnxn 1−αnTzn, Cn1
z∈Cn:yn−z≤ xn−z , xn1 PCn1x, ∀n∈N,
3.62
wherePCn1is the metric projection ofHontoCn1and{αn} ⊂0,1and{λn} ⊂0,2δare sequences such that
lim inf
n→ ∞ αn<1, 0< a≤λn≤b <2δ, 3.63
for somea∈R. Then{xn}converges strongly toz0PΩz0.
SettingΨ≡0 inTheorem 3.3, we obtain the following result.
Corollary 3.5. Let H be a Hilbert space and letC be a nonempty closed convex subset ofH. Let F :C×C → Rbe a bifunction satisfying (A1), (A2), (A3), and (A4). LetT :C → Cbe a quasi- nonexpansive mapping which is demiclosed onC. Assume that EPF∩FT/∅ andf is a Meir- Keeler contraction ofCinto itself. LetC1 Cand{xn} ⊂Cbe a sequence generated byx1 x∈C and
F zn, y
1 λn
zn−xn, y−zn
≥0, ∀y∈C,
ynαnxn 1−αnTzn, Cn1
z∈Cn:yn−z≤ xn−z , xn1PCn1fxn, ∀n∈N,
3.64
wherePCn1is the metric projection ofHontoCn1and{αn} ⊂0,1and{λn} ⊂0,∞are sequences such that
lim inf
n→ ∞ αn<1, 0< a≤λn, 3.65 for somea∈R. Then{xn}converges strongly toz0∈FT∩EPF.
SettingΨ ≡ 0 andfxn xfor all n ∈ NinTheorem 3.3, we obtain the following result.
Corollary 3.6. Let H be a Hilbert space and letC be a nonempty closed convex subset ofH. Let F :C×C → Rbe a bifunction satisfying (A1), (A2), (A3), and (A4). LetT :C → Cbe an quasi- nonexpansive mapping which is demiclosed onCand assume that EPF∩FT/∅. LetC1Cand {xn} ⊂Cbe a sequence generated byx1x∈Cand
F zn, y
1 λn
zn−xn, y−zn
≥0, ∀y∈C,
ynαnxn 1−αnTzn, Cn1
z∈Cn:yn−z≤ xn−z , xn1 PCn1x, ∀n∈N,
3.66
wherePCn1is the metric projection ofHontoCn1and{αn} ⊂0,1and{λn} ⊂0,∞are sequences such that
lim inf
n→ ∞ αn<1, 0< a≤λn, 3.67 for somea∈R. Then{xn}converges strongly toz0∈FT∩EPF.
Next, using the CQ hybrid method introduced by Nakajo and Takahashi 16, we prove a strong convergence theorem of a quasi-nonexpansive mapping for solving the generalized equilibrium problem.
