Volume 2010, Article ID 157278,18pages doi:10.1155/2010/157278
Research Article
On a Hybrid Method for Generalized Mixed
Equilibrium Problem and Fixed Point Problem of
a Family of Quasi-
φ
-Asymptotically Nonexpansive
Mappings in Banach Spaces
Min Liu, Shih-sen Chang, and Ping Zuo
Department of Mathematics, Yibin University, Yibin, Sichuan 644007, China
Correspondence should be addressed to Shih-sen Chang,[email protected]
Received 30 November 2009; Accepted 19 January 2010
Academic Editor: Nanjing Huang
Copyrightq2010 Min Liu 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 prove a strong convergence theorem by using a hybrid method for finding a common element of the set of solutions for generalized mixed equilibrium problems, the set of fixed points of a family of quasi-φ-asymptotically nonexpansive mappings in strictly convex reflexive Banach space with the Kadec-Klee property and, aFr´echetdifferentiable norm under weaker conditions. The method of the proof is different from, S. Takahashi and W. Takahashi that by2008and that by Takahashi and Zembayashi2008and see references. It also shows that the type of projection used in the iterative method is independent of the properties of the mappings. The results presented in the paper improve and extend some recent results.
1. Introduction
LetEbe a Banach space and letCbe a closed convex subsets ofE. LetF be an equilibrium bifunction fromC×CintoR, letψ :C → Rbe a real-valued function, and letA:C → E∗ be a nonlinear mapping. The “so-called” generalized mixed equilibrium problem is to find
z∈Csuch that
Fz, yAz, y−zψy−ψz≥0, ∀y∈C,s 1.1
The set of solutions of1.1is denoted by GMEP, that is,
Sepecial Examples
iIfA0, then the problem1.1is equivalent to findz∈Csuch that
Fz, yψy−ψz≥0, ∀y∈C. 1.3
which is called the mixed equilibrium problem; see1. The set of solutions of1.3
is denoted by MEP.
iiIfF0, then the problem1.1is equivalent to findz∈Csuch that
Az, y−zψy−ψz≥0, ∀y∈C, 1.4
which is called the mixed variational inequality of Browder type. The set of solutions of1.4is denoted byV IC, A, ψ.
iiiIfψ0, then the problem1.1is equivalent to findz∈Csuch that
Fz, yAz, y−z≥0, ∀y∈C. 1.5
which is called the generalized equilibrium problem; see2. The set of solutions of 1.5is denoted by EP.
ivIfA0, ψ0, then the problem1.1is equivalent to findz∈Csuch that
Fz, y≥0, ∀y∈C, 1.6
which is called the equilibrium problem. The set of solutions of1.6is denoted by EPF.
Recently, Tada and Takahashi3and S. Takahashi and W. Takahashi4considered iterative methods for finding an element of EPF∩FS in Hilbert space. Very recently, S.Takahashi and W.Takahashi 2 introduced an iterative method for finding an element of EP ∩FS, where A : C → H is an inverse-strongly monotone mapping and S is nonexpansive mapping and then proved a strong convergence theorem in Hilbert space. On the other hand, Takahashi and Zembayashi5prove a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a relatively nonexpansive mapping in a Banach space by using the shrinking Projection method. Very recently, Kimura and Takahashi 6 prove a strong convergence theorem for a family of relatively nonexpansive mapping in a Banach space by using a hybrid method.
2. Preliminaries
Throughout this paper, we assume that all the Banach spaces are real. We denote byNandR the sets of positive integers and real numbers, respectively. LetEbe a Banach space and let
E∗be the topological dual ofE. For allx∈Eandx∗ ∈E∗, we denote the value ofx∗atxby
x, x∗ . The duality mappingJ :E → 2E∗
is defined by
Jx x∗∈E∗:x, x∗ x2x∗2, x∈E. 2.1
By Hahn-Banach theorem,Jxis nonempty; see7for more details. We denote the weak convergence and the strong convergence of a sequence{xn}toxinEbyxn xandxn → x, respectively. A Banach spaceEis said to be strictly convex ifxy/2<1 forx, y∈SE
{z ∈E:z 1}andx /y. It is also said to be uniformly convex if for eachε ∈0,2there existsδ >0 such thatxy/2<1−δforx, y∈SEandx−y ≥ε.Eis said to have the Kadec-Klee property, that is, for any sequence{xn} ⊂E, ifxn x∈Eandxn → x, then xn → x.
Definef :SE×SE×R\ {0} → Rby
fx, y, t xty− x
t 2.2
for x, y ∈ SE and t ∈ R \ {0}. A norm of E is said to be Gateaux differentiable if limt→0fx, y, thas a limit for eachx, y∈SE. In this case,Eis said to be smooth. A norm of Eis said to beFr´echetdifferentiable if limt→0fx, y, tis attained uniformly fory∈SEfor
eachx∈SE. It is known thatE∗has aFr´echetdifferentiable norm if and only ifEis strictly convex and reflexive, and has the Kadec-Klee property. We know that ifEis smooth, strictly convex, and reflexive, then the duality mappingJ is single valued, one to one, and onto. In this case, the inverse mappingJ−1 coincides with the duality mappingJ∗onE∗. See8for
more details.
Remark 2.1. IfEis a reflexive and strictly convex Banach space, thenJ−1 is hemicontinuous,
that is,J−1isnorm-weak continuous.
LetEbe a smooth, strictly convex and reflexive Banach space and letCbe a closed convex subset ofE. Throughout this paper, we denote byφthe function defined by
φy, xy2−
2y, Jxx2, ∀x, y∈E. 2.3
Let {Cn} be a sequence of nonempty closed convex subset of a reflexive Banach spaceE. We define two subsets s−LinCn and w−LsnCn as follows: x ∈ s−LinCn if and only if there exists{xn} ⊂Esuch that{xn}converges strongly toxand thatxn ∈Cnfor alln∈N. Similarly,y∈w−LsnCnif and only if there exists a subsequence{Cni}of{Cn}and a sequence
{yi} ⊂Esuch that{yi}converges weakly toyandyi∈Cnifor alli∈N. We define the Mosco
convergence9of{Cn}as follows: ifC0satisfies thatC0 s−LinCnw−LsnCn, then it is
LetCbe a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach spaceE. Then, for arbitrarily fixedx ∈ E, a functionC y → x−y2 ∈ Rhas a
unique minimizeryx ∈C. Using such a point, we define the metric projectionPC byPCx yx arg miny∈Cx−y2 for every x ∈ E. In a similar fashion, we can see that a function Cy→φx, y∈Rhas a unique minimizerzx∈C. The generalized projectionΠCofEonto Cis defined byΠCzxarg miny∈Cφx, yfor everyx∈E; see11.
The generalized projectionΠC fromEontoCis well defined and single valued and satisfies
x −y2≤φy, x≤xy2, ∀x, y∈E. 2.4
IfEis a Hilbert space, thenφy, x y−x2andΠCis the metric projectionPCofEontoC.
It is well-known that the following conclusions hold.
Lemma 2.2see11,12. LetCbe a nonempty closed convex subsets of a smooth, strictly convex
and reflexive Banach spaces. Then
φx,ΠCyφΠCy, y≤φx, y, ∀x∈C, y∈E. 2.5
Lemma 2.3. LetCbe a nonempty closed convex subsets of a smooth, strictly convex and reflexive
Banach spacesE, letx∈Eand letz∈C. Then the following conclusions hold:
az ΠCx⇔ y−z, Jx−Jz ≤0, for ally∈C.
bForx, y∈E, φx, y 0if and only ifxy.
The following theorem proved by Tsukada13plays an important role in our results.
Theorem 2.4. LetEbe a smooth, reflexive, and strictly convex Banach space having the Kadec-Klee
property. Let{Kn}be a sequence of nonempty closed convex subset ofE. IfK0 M−limn→ ∞Kn exists and is nonempty, then{PKnx}converges strongly toPK0xfor eachx∈C.
Theorem 2.4 is still valid if we replace the metric projections with the generalized projections as follows:
Theorem 2.5. LetEbe a smooth, reflexive, and strictly convex Banach spaces having the Kadec-Klee
property. Let{Kn}be a sequence of nonempty closed convex subsets ofE. IfK0 M−limn→ ∞Kn exists and is nonempty, then{ΠKnx}converges strongly toΠK0xfor eachx∈C.
Let C be a nonempty closed convex subsets ofE, and let T be a mapping from C into itself. We denoted byFTthe set of fixed points ofT.T is said to beφ-asymptotically nonexpansive, if there exists some real sequence{kn} with kn ≥ 1 andkn → 1 such that φTnx, Tny ≤ knφx, yfor all n ≥ 1 andx, y ∈ C.T is said to be quasi-φ-asymptotically nonexpansive14, if there exists some real sequence {kn} with kn ≥ 1 and kn → 1 and FT/∅such that φp, Tnx ≤ knφp, xfor alln ≥ 1,x ∈ C,andp ∈ FT.T is said to be uniformly Lipschitzian continuous if there exists someL >0 such thatTnx−Tny ≤Lx−y for alln≥1 andx, y ∈C. A pointp ∈Cis said to be an asymptotic fixed point ofT 15,16
a mappingT : C → Cis said to be relatively nonexpansive if the following conditions are satisfied:
1FTis nonempty,
2φu, Tx≤φu, x, for allu∈FT, x∈C, 3FT FT.
A mapping T : C → C is said to be quasi-φ-nonexpansive, if φp, Tx ≤
φp, x, for allx∈C, for allp∈FT.
We remark that a quasi-φ-nonexpansive mapping with a nonempty fixed point set
FTis a quasi-φ-asymptotically nonexpansive mapping, but the converse may be not true. A mappingT :C → Cis said to be closed, if for any sequence{xn} ⊂Cwithxn → x andTxn → y,Txy.
Lemma 2.6. LetEbe a strictly convex reflexive Banach space having the Kadec-Klee property and
a Fr´echet differentiable norm,Cbe a nonempty closed convex subset ofE, and letT be a uniformly Lipschitzian continuous and quasi-φ-asymptotically nonexpansive mapping fromCinto itself. Then
FTis closed and convex.
Proof. We first show thatFT is closed. To see this, let {pn} be a sequence in FTwith pn → pasn → ∞; we shall prove thatp ∈FT. In fact, from the definition ofT, we have φpn, Tp≤k1φpn, p → 0n → ∞. Therefore we have
lim n→ ∞φ
pn, Tpnlim
→ ∞
pn2−2pn, JTpTp2
p2−
2p, JTpTp2φp, Tp0,
2.6
that is,p Tp. We next show thatFTis convex. To end this, for arbitraryp, q ∈FT, t ∈ 0,1, by settingwtp 1−tq, it is sufficient to show thatTww. Indeed, by using2.3
we have
φw, Tnw w2−2w, JTnw Tnw2
w2−2tp, JTnw−21−tq, JTnwTnw2
w2tφp, Tnw 1−tφq, Tnw−tp2−
1−tq2
≤ w2k
ntφp, wkn1−tφq, w−tp2−1−tq2
kn−1
tp2
1−tq2− w2,
2.7
which implies that φw, Tnw → 0 as n → ∞. From 2.4 we have Tnw → w. Consequently JTnw → Jw. This implies that {JTnw} is bounded in E∗. Since E is reflexive, so isE∗, we can assume that
In view of the reflexive ofE, we see thatJE E∗. Hence there existsp∈Esuch thatJpf0. By virtue of the weak lower semicontinuity of norm · , we have
0lim inf n→ ∞ φw, T
nw lim inf n→ ∞
w2−2w, JTnw Tnw2
lim inf n→ ∞
w2−2w, JTnw JTnw2
≥ w2−2w, f0f02
w2−2w, JpJp2
w2−2w, Jpp2φw, p,
2.9
that is,w p. This implies thatf0 Jw. Thus from2.8we haveJTnw Jw ∈E∗. Since
JTnw → Jw and E∗ has the Kadec-Klee property, we have JTnw → Jw. Note that J−1 : E∗ → Eis hemicontinuous, it yields that Tnw w. Again sinceTnw → w, by using the Kadec-Klee property ofE, we haveTnw → w. HenceTTnw Tn1w → w as n → ∞. SinceT is uniformly Lipschitzian continuous, we havewTw. This completes the proof.
For solving the equilibrium problem for 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 sup t↓0
Ftz 1−tx, y≤Fx, y, 2.10
A4for eachx∈C, y→Fx, yis a convex and lower semicontinuous.
If an equilibrium bifunctionF : C×C → Rsatisfies conditionsA1–A4, then we have the following two important results.
Lemma 2.7see18. LetCbe a nonempty closed convex subset of a smooth, strictly convex and
reflexive Banach spaceE, letFbe an equilibrium bifunction fromC×CtoRsatisfying conditions A1–A4, letr >0,and letx∈E. Then, there existsz∈Csuch that
Fz, y1
r
y−z, Jz−Jx≥0, ∀y∈C. 2.11
Lemma 2.8 see5. Let C be a nonempty closed convex subset of a uniformly smooth, strictly
convex and reflexive Banach spaceE, and letF:C×C → Rbe an equilibrium bifunction satisfying conditionsA1–A4. Forr >0andx∈E, define a mappingTr :E → Cas follows:
Trx
z∈C:Fz, y1
r
y−z, Jz−Jx≥0, ∀y∈C
for allx∈E. Then, the following hold:
1Tr is single-valued,
2Tr is a firmly nonexpansive-type mapping, that is, for anyx, y∈E,
Trx−Try, JTrx−JTry≤Trx−Try, Jx−Jy, 2.13
3FTr FTr EPF,
4EPFis a closed and convex set.
Lemma 2.9see5. LetCbe a nonempty closed convex subset of a smooth, strictly convex and
reflexive Banach spaceE, and letF :C×C → Rbe an equilibrium bifunction satisfying conditions A1–A4. Forr >0, x∈Eandq∈FTr,
φq, TrxφTrx, x≤φq, x. 2.14
3. The Main Results
Lemma 3.1. LetEbe a strictly convex reflexive Banach space having a Fr´echet differentiable norm,
Ca nonempty closed convex subset ofE,and{Sn}a sequence of mappings ofCinto itself. Let{xn} be a strongly convergent sequence inCwith a limitx0 and{yn}a sequence inCdefined byyn J∗αnJxn 1−αnJSnxnfor eachn∈N, where{αn}is a convergent sequence in0,1with a limit
α0 ∈0,1. Suppose thatφx0, yn≤φx0, xn ξnfor alln∈Nand that{Jyn}converges weakly toy∗0 ∈E∗, wherelimn→ ∞ξn 0. Then{Jxn−JSnxn}converges strongly to0.Moreover, ifEhas the Kadec-Klee property, then{Snxn}converges strongly tox0.
Proof. Sinceφx0, yn≤φx0, xn ξnforn∈N, we have that
0≤ lim n→ ∞φ
x0, yn≤nlim→ ∞φx0, xn ξn0. 3.1
and hence limn→ ∞φx0, yn 0. Since
x0 −yn2 x02−2x0ynyn2≤φx0, yn 3.2
forn∈N, we have that limn→ ∞ynx0and that
lim n→ ∞
x0, Jynnlim→ ∞1 2
x02yn2−φx0, ynx02. 3.3
Using weak lower semicontinuity of the norm, we have that
x02 lim n→ ∞
x0, Jynx0, y0∗
≤ x0y∗
0≤ x0lim infn→ ∞ Jyn
x0lim
n→ ∞Jynx0
Therefore, we have thaty∗02 x0, y∗
0 x02,and hencey∗0 Jx0. Thus we have
that{Jyn}converges weakly toJx0. It also holds that
lim
n→ ∞Jynnlim→ ∞ynx0Jx0. 3.5
SinceEhas aFr´echetdifferentiable norm, it follows thatE∗has the Kadec-Klee property, and thus we have that{Jyn}converges strongly toJx0. Then, we have that
Jx0−JynJx0−αnJxn 1−αnJSnxn
≥ Jx0−αnJx0−1−αnJSnxn −αnJxn−Jx0
1−αnJx0−JSnxn −αnJxn−Jx0
3.6
forn∈N. Using norm-to-norm continuity ofJ, we get that
lim
n→ ∞1−αnJx0−JSnxn 1−α0nlim→ ∞Jx0−JSnxn0, 3.7
and sinceα0<1, we have that
lim
n→ ∞Jx0−JSnxn0. 3.8
We also have that{JSnxn}converges strongly toJx0,and hence we obtain that{Jxn−JSnxn}
converges strongly to 0. Further, let us suppose thatEhas the Kadec-Klee property. Then, the norm ofE∗isFr´echetdifferentiable and, therefore,J∗is norm-to-norm continuous. Hence we have that
lim
n→ ∞x0−Snxnnlim→ ∞J
∗Jx0−J∗JSnxn0, 3.9
which is the desired result.
Theorem 3.2. LetEbe a strictly convex reflexive Banach space having the Kadec-Klee property and
F λ∈ΛFTλGMEP/∅. Assume thatR sup{u :u∈F} <∞. Let{xn}be the sequence generated byx1x∈C, C1C,and
ynλ J∗αnJxn 1−αnJTλnxn ∀λ∈Λ,
unλ∈Csuch that
Funλ, yAunλ, y−unλψy−ψunλ r1 nλ
y−unλ, Junλ−Jynλ≥0,
∀y∈C, λ∈Λ
Cn1
z∈Cn: sup λ∈Λ
φz, unλ≤φz, xn ξn
,
xn1PCn1x, ∀n≥0,
3.10
whereJis the duality mapping onEandξn 1−αnsupλ∈Λknλ−1supu∈Fφu, xnfor allxn∈ C, wherelimn→ ∞supλ∈Λknλ 1. Let{αn}be a sequence in0,1such thatlim infn→ ∞αn <1
and {rnλ} ⊂ a,∞for somea > 0, then{xn}converge strongly to PFx, wherePF is the metric
projection ofEontoF.
Proof. We define a bifunctionG:C×C → Rby
Gz, yFz, yAz, y−zψy−ψz, ∀z, y∈C. 3.11
Next, we prove that the bifunctionGsatisfies conditionsA1–A4as follows
A1Gx, x 0 for allx∈C,
sinceGx, x Fx, x Ax,0 ψx−ψx Fx, x 0,for allx∈C. A2Gis monotone, that is,Gz, y Gy, z≤0 for ally, z∈C.
SinceAis a continuous and monotone operator, hence from the definition ofG we have
Gz, yGy, zFz, yAz, y−zψy−ψz Fy, z
Ay, z−yψz−ψy
Fz, yFy, zAz, y−z−Ay, y−z
≤0Az−Ay, y−z −Ay−Az, y−z
≤0.
A3for eachx, y, z∈C,
lim sup t↓0
Gtz 1−tx, y≤Gx, y. 3.13
SinceAis continuous andψis lower semicontinuous, we have
lim sup t↓0
Gtz 1−tx, y
lim sup t↓0
Ftz 1−tx, ylim sup t↓0
Atz 1−tx, y−tz 1−tx
lim sup t↓0
ψy−ψtz 1−tx
≤Fx, yAx, y−xψy−ψx Gx, y.
3.14
A4For eachx∈C, y→Gx, yis a convex and lower semicontinuous.
For eachx ∈C, for all t∈ 0,1and for ally, z∈ C, sinceFsatisfiesA4andψ is convex,
we have
Gx, ty 1−tzFx, ty 1−tzAx, ty 1−tz−xψty 1−tz−ψx
≤tFx, yAx, y−xψy−ψx 1−tFx, z Ax, z−x ψz−ψx tGx, y 1−tGx, z.
3.15
So,y→Gx, yis convex.
Similarly, we can prove thaty→Gx, yis lower semi-continuous.
Therefore, the generalized mixed equilibrium problem 1.1 is equivalent to the following equilibrium problem: findz∈Csuch that
Gz, y≥0, ∀y∈C, 3.16
then, GMEPEPG. We haveFGMEP∩λ∈ΛFTλ EPG∩λ∈ΛFTλ. So,3.10can be written as
ynλ J∗αnJxn 1−αnJTλnxn ∀λ∈Λ,
unλ∈Csuch thatGunλ, y r1 nλ
y−unλ, Junλ−Jynλ≥0, ∀y∈C, λ∈Λ,
Cn1
z∈Cn: sup λ∈Λ
φz, unλ≤φz, xn ξn
,
xn1PCn1x, ∀n≥0.
Since the bifunctionGsatisfies conditionsA1–A4, fromLemma 2.8, for givenr >0 andx∈E, we can define a mappingWr :E → 2Cas follows:
Wrx
z∈C:Gz, y1
r
y−z, Jz−Jx≥0, ∀y∈C
. 3.18
Moreover,Wrsatisfies the conclusions inLemma 2.8.
Puttingunλ Wrnλynλfor alln∈N, we have fromLemma 2.8andLemma 2.9that Wrnλ is relatively nonexpansive.
We divide the proof ofTheorem 3.2into five steps.
Step 1. We first show thatCnis closed and convex for everyn∈N. From the definition ofφ, we may show that
Cn1
z∈Cn: sup
λ∈Λφz, unλ≤φz, xn ξn
λ∈Λ
z∈Cn:φz, unλ≤φz, xn ξn
λ∈Λ
z∈C: 2z, Jxn−Junλ unλ2− xn2−ξn≤0
∩Cn,
3.19
and thusCnis closed and convex for everyn∈N.
Step 2. Next we show thatF⊂Cnfor eachn∈Nandλ∈Λ.
for any u ∈ F, since Wrnλ is relatively nonexpansive and {Tλ}, λ ∈ Λ is quasi-φ -asymptotically nonexpansive, we have
φu, unλ φ
u, Wrnλynλ
≤φu, ynλ
φu, J∗α
nJxn 1−αnJTλnxn
u2−2u, α
nJxn 1−αnJTλnxnαnJxn 1−αnJTλnxn2
≤ u2−2α
nu, Jxn −21−αnu, JTλnxnαnJxn2 1−αnJTλnxn2
αnφu, xn 1−αnφu, Tλnxn
≤αnφu, xn 1−αnknλφu, xn
φu, xn 1−αnknλ−1φu, xn
≤φu, xn ξn.
Hence, we have supλ∈Λφu, unλ≤φu, xn ξn, that is,u∈Cn. This implies that
F ⊂Cn, ∀n∈N. 3.21
Step 3. Now we prove that the limit limn→ ∞xnexists.SinceFis nonempty,Cnis a nonempty closed convex subset ofE, and, thus,PCn exists for everyn∈ N, hence{xn}is well defined.
Also, since{Cn}is a decreasing sequence of closed convex subsets ofEsuch thatC0∞n1Cn
is nonempty, it follows that
M− lim
n→ ∞CnC0 ∞
n1
Cn/∅. 3.22
ByTheorem 2.4,{xn}{PCnx}converges strongly tox0PC0x. Therefore, we have
xn1−xn −→0. 3.23
Step 4. Next we prove thatx0∈F.
aFirst, we prove thatx0∈λ∈ΛFTλ.
Sincex0 ∈ Cnfor everyn ∈N, it follows that supλ∈Λφx0, unλ≤ φx0, xn ξn for everyn∈N. Fixλ∈Λarbitrarily. From the assumption that lim infn→ ∞αn<1, we may take subsequences{αni}of{αn}and{yniλ}of{ynλ}such that limi→ ∞αni α0with 0≤α0 <1
and{Jyniλ}converges weakly to a pointy∗0∈E∗. Then, byLemma 3.1, we have that
lim
i→ ∞xni−T
ni
λxniilim
→ ∞x0−T
ni
λ xni0. 3.24
From3.23and3.24, we have
Tni1
λ xni−T
ni
λ xni≤T
ni1 λ xni−T
ni1
λ xni1Tλni1xni1−xni1
xni1−xnixni −T
ni
λ xni
≤Lλ1xni1−xniT
ni1
λ xni1−xni1xni−T
ni
λxni
−→0.
3.25
Observe that
Tni1
λ xni−x0≤Tλni1xni−T
ni
λ xniT
ni
λ xni−x0. 3.26
By using3.24,3.25, and3.26, we have
Tni1
SinceTλis uniformly Lipschitzian continuous, from3.24and3.27, we havex0Tλx0, that is,x0∈ ∩λ∈ΛFTλ.
bNext we prove thatx0∈EPG.
1In fact, sincexn → x0, we have
φxn1, xn−→0. 3.28
In view ofxn1∈Cn1, from the definition ofCn1, we have
sup
λ∈Λφxn1, unλ≤φxn1, xn ξn 3.29
From3.28andξn → 0, we have
sup
λ∈Λφxn1, unλ−→0, ∀λ∈Λ. 3.30
From2.4, it yields supλ∈Λxn1 − unλ2 → 0. Sincexn1 → x0, we have
unλ −→ x0n−→ ∞, ∀λ∈Λ. 3.31
Hence we have
Junλ −→ Jx0n−→ ∞, ∀λ∈Λ. 3.32
This implies that{Junλ}is bounded inE∗. SinceEis reflexive, and so isE∗, we can assume thatJunλ f0∈E∗. In view of the reflexive ofE, we see thatJE E∗. Hence there exists p∈Esuch thatJpf0. Since
φxn1, unλ xn12−2xn1, Junλ unλ2
xn12−2xn1, Junλ Junλ2,
3.33
taking lim infn→ ∞ on the both sides of equality above and in view of the weak lower
semi-continuity of norm · , it yields that
0≥ x02−2x0, f0f02x02−2x0, JpJp2
x02−2
x0, Jp
p2φx 0, p
, 3.34
that is,x0p. This implies thatf0Jx0, and soJunλ Jx0, for all λ∈Λ. It follows from
3.32and the Kadec-Klee property ofE∗thatJunλ → Jx0n → ∞. Note thatJ−1 :E∗ → Eis hemicontinuous, it yields that unλ x0. It follows from3.31and the Kadec-Klee
property ofEthat
lim
From3.35, we have
lim
n→ ∞xn−unλ0, ∀λ∈Λ. 3.36
SinceJis norm-to-norm continuous, hence we have that
Jxn−Junλ −→0, ∀λ∈Λ. 3.37
2Next we prove that
φu, xn−φu, unλ−→0, ∀u∈F. 3.38
From3.36and3.37, we have
φu, xn−φu, unλ xn2− unλ2−2u, Jxn−Junλ
≤xn2− unλ22|u, Jxn−Junλ |
≤ |xn − unλ|xnunλ 2u · Jxn−Junλ
≤ xn−unλxnunλ 2u · Jxn−Junλ.
−→0.
3.39
Since
φunλ, ynλφ
Wrnλynλ, ynλ
≤φu, ynλ−φ
u, Wrnλynλ
≤φu, xn ξn−φ
u, Wrnλynλ
φu, xn ξn−φu, unλ,
3.40
hence it follows from3.38and3.40that
lim n→ ∞φ
unλ, ynλ0. 3.41
From2.3and3.41it yieldsunλ − ynλ2 → 0. Sinceunλ → x0, we have ynλ−→ x0n−→ ∞. 3.42
Hence we have
This implies that{Jynλ}is bounded inE∗. SinceEis reflexive, and so isE∗, we can assume thatJynλ g0∈E∗. In view of the reflexive ofE, we see thatJE E∗. Hence there exists y∈Esuch thatJyg0. Since
φunλ, ynλunλ2−2unλ, Jynλynλ2
unλ2−2unλ, JynλJynλ2,
3.44
Taking lim infn→ ∞on the both sides of equality above and in view of the weak lower
semi-continuity of norm · , it yields that
0≥ x02−2x0, g0g02 x02−2x0, JyJy2
x02−2
x0, Jy
y2φx 0, y
, 3.45
that is, x0 y. This implies thatg0 Jx0, and so Jynλ Jx0. It follows from 3.43 and the Kadec-Klee property ofE∗thatJynλ → Jx0n → ∞. Note thatJ−1 : E∗ → Eis
hemicontinuous; it yields thatynλ x0. It follows from3.42and the Kadec-Klee property
ofEthat
lim
n→ ∞ynλ x0. 3.46
Sinceunλ → x0, from3.46, we have
lim
n→ ∞unλ−ynλ0. 3.47
SinceJis uniformly norm-to-norm continuous on bounded sets, from3.47, we have
lim
n→ ∞Junλ−Jynλ0. 3.48
Fromrnλ ≥a, we have
lim n→ ∞
Junλ−Jynλ rnλ
0. 3.49
Byunλ Wrnλynλ, we have
Gunλ, y r1 nλ
y−unλ, Junλ−Jynλ≥0, ∀y∈C. 3.50
FromA2, we have
1
rnλ
SinceGx,·is convex and lower semicontinuous, it is also weakly lower semicontinuous. So, lettingn → ∞, we have from3.51andA4that
Gy, x0
≤0, ∀y∈C. 3.52
For anytwith 0< t≤1 andy∈C, letytty 1−tx0. Sincey∈Cand, hence,Gyt, x0≤0,
from conditionsA1andA4, we have
0Gyt, yt≤tGyt, y 1−tGyt, x0
≤tGyt, y, 3.53
This implies thatGyt, y≥0. Hence from conditionA3, we haveGx0, y≥0 for ally∈C,
and hencex0∈EPG.
Step 5. Finally we prove thatxn → PFx.
Sincex0 PC0x ∈ F andFis a nonempty closed convex subset ofC0 ∞n1Cn, we
conclude that
x0PFx. 3.54
This completes the proof ofTheorem 3.2.
The proof ofTheorem 3.2shows that the properties of projections used in the iterative scheme do not interact with the properties of mappings{Tλ}. Therefore, we may prove similar results as follows by replacingTheorem 2.4withTheorem 2.5in the proof.
Theorem 3.3. LetEbe a strictly convex reflexive Banach space having the Kadec-Klee property and
a Fr´echet differentiable norm,Ca nonempty closed convex subset ofE,A : C → E∗ a continuous and monotone mapping,ψ : C → Ra lower semicontinuous and convex function,F a bifunction fromC×C to Rwhich satisfies the conditionsA1–A4and let{Tλ}λ∈Λ:C → Ca family of uniformly Lipschitzian continuous and quasi-φ- asymptotically nonexpansive mappings such that
F λ∈ΛFTλGMEP/∅. Assume thatR sup{u :u∈F} <∞. Let{xn}be the sequence generated byx1x∈C, C1C,and
ynλ J∗αnJxn 1−αnJTλnxn ∀λ∈Λ,
unλ∈Csuch that
Funλ, yAunλ, y−unλψy−ψunλ r1 nλ
y−unλ, Junλ−Jynλ≥0,
∀y∈C, λ∈Λ
Cn1
z∈Cn: sup λ∈Λ
φz, unλ≤φz, xn ξn
,
xn1 ΠCn1x, ∀n≥0.
whereJis the duality mapping onEandξn 1−αnsupλ∈Λknλ−1supu∈Fφu, xnfor allxn∈ C, wherelimn→ ∞supλ∈Λknλ 1. Let{αn}be a sequence in0,1such thatlim infn→ ∞αn <1 and{rnλ} ⊂a,∞for somea >0, then{xn}converge strongly toΠFx, whereΠF is the generalized
projection ofEontoF.
Acknowledgments
The authors would like to express their thanks to the referees for their helpful suggestions and comments.
References
1 L.-C. Ceng and J.-C. Yao, “A hybrid iterative scheme for mixed equilibrium problems and fixed point problems,”Journal of Computational and Applied Mathematics, vol. 214, no. 1, pp. 186–201, 2008.
2 S. Takahashi and W. Takahashi, “Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space,”Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 3, pp. 1025–1033, 2008.
3 A. Tada and W. Takahashi, “Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem,”Journal of Optimization Theory and Applications, vol. 133, no. 3, pp. 359–370, 2007.
4 S. Takahashi and W. Takahashi, “Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces,”Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 506–515, 2007.
5 W. Takahashi and K. Zembayashi, “Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings,”Fixed Point Theory and Applications, vol. 2008, Article ID 528476, 11 pages, 2008.
6 Y. Kimura and W. Takahashi, “On a hybrid method for a family of relatively nonexpansive mappings in a Banach space,”Journal of Mathematical Analysis and Applications, vol. 357, no. 2, pp. 356–363, 2009.
7 W. Takahashi, Nonlinear Functional Analysis, Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, Japan, 2000.
8 K. Goebel and S. Reich,Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, vol. 83 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1984.
9 U. Mosco, “Convergence of convex sets and of solutions of variational inequalities,”Advances in Mathematics, vol. 3, pp. 510–585, 1969.
10 G. Beer,Topologies on Closed and Closed Convex Sets, vol. 268 ofMathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993.
11 Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” inTheory and Applications of Nonlinear Operators of Accretive and Monotone Type, A. G. Kartsators, Ed., vol. 178 ofLecture Notes in Pure and Applied Mathematics, pp. 15–50, Marcel Dekker, New York, NY, USA, 1996.
12 S. Kamimura and W. Takahashi, “Strong convergence of a proximal-type algorithm in a Banach space,”SIAM Journal on Optimization, vol. 13, no. 3, pp. 938–945, 2002.
13 M. Tsukada, “Convergence of best approximations in a smooth Banach space,” Journal of Approximation Theory, vol. 40, no. 4, pp. 301–309, 1984.
14 Y. Su and X. Qin, “Strong convergence of modified Ishikawa iterations for nonlinear mappings,” Proceedings of the Indian Academy of Sciences. Mathematical Sciences, vol. 117, no. 1, pp. 97–107, 2007.
15 Y. Censor and S. Reich, “Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization,”Optimization, vol. 37, no. 4, pp. 323–339, 1996.
17 S. Matsushita and W. Takahashi, “Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces,”Fixed Point Theory and Applications, vol. 2004, no. 1, pp. 37–47, 2004.
