A New Hybrid Algorithm for a Pair of Quasi

(1)

Volume 2011, Article ID 956852,22pages doi:10.1155/2011/956852

Research Article

A New Hybrid Algorithm for a Pair of

Quasi-

φ

-Asymptotically Nonexpansive

Mappings and Generalized Mixed

Equilibrium Problems in Banach Spaces

Jinhua Zhu and Shih-Sen Chang

Department of Mathematics, Yibin University, Yibin 644007, China

Correspondence should be addressed to Shih-Sen Chang,[email protected]

Received 22 February 2011; Accepted 20 May 2011

Academic Editor: Vittorio Colao

Copyrightq2011 J. Zhu and S.-S. Chang. 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.

The purpose of this paper is, by using a new hybrid method, to prove a strong convergence theorem for finding a common element of the set of solutions for a generalized mixed equilibrium problem, the set of solutions for a variational inequality problem, and the set of common fixed points for a pair of quasi-φ-asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in the paper improve and extend some recent results.

1. Introduction

Throughout this paper, we denote byNandRthe sets of positive integers and real numbers, respectively. We also assume thatEis a real Banach space,E∗ is the dual space ofE,Cis a nonempty closed convex subset ofE, and·,·is the pairing betweenEandE∗.

Letψ :C → Rbe a real-valued function,Θ:C×C → Ra bifunction, andA:C → E∗ a nonlinear mapping. The “so-called” generalized mixed equilibrium problem is to findu∈C such that

Θu, yAu, y−uψy−ψu≥0, ∀y∈C. 1.1

The set of solutions for1.1is denoted byΩ, that is,

(2)

Special examples are as follows.

IIfψ0, the problem1.1is equivalent to findingu∈Csuch that

Θu, yAu, y−u≥0, ∀y∈C, 1.3

which is called the generalized equilibrium problem. The set of solutions for1.3

is denoted by GEP.

IIIfA0, the problem1.1is equivalent to findingu∈Csuch that

Θu, yψy−ψu≥0, ∀y∈C, 1.4

which is called the mixed equilibrium problemMEP 1 . The set of solutions for

1.4is denoted by MEP.

IIIIfΘ 0, the problem1.1is equivalent to findingu∈Csuch that

Au, y−uψy−ψu≥0, ∀y∈C, 1.5

which is called the mixed variational inequality of Browder typeVI 2 . The set of solutions for1.5is denoted by VIC, A, ψ.

IVIfψ0 andA0, the problem1.1is equivalent to findingu∈Csuch that

Θu, y≥0, ∀y∈C, 1.6

which is called the equilibrium problem. The set of solutions for1.6is denoted by EPΘ.

VIfψ0 andΘ 0, the problem1.1is equivalent to findingu∈Csuch that

Au, y−u≥0, ∀y∈C, 1.7

which is called the variational inequality of Browder type. The set of solutions for

1.7is denoted by VIC, A.

The problem 1.1 is very general in the sense that numerous problems in physics, optimiztion and economics reduce to finding a solution for1.1. Some methods have been proposed for solving the generalized equilibrium problem and the equilibrium problem in Hilbert spacesee, e.g.,3–6 .

A mappingS:C → Eis called nonexpansive if

Sx−Sy≤x−y, ∀x, y∈C. 1.8

We denote the fixed point set ofSbyFS.

(3)

Recently, Takahashi and Zembayashi7,8 proved some weak and strong convergence theorems for finding a common element of the set of solutions for equilibrium1.6and the set of fixed points of a relatively nonexpansive mapping in a Banach space.

In 2010, Chang et al.9 proved a strong convergence theorem for finding a common element of the set of solutions for a generalized equilibrium problem 1.3 and the set of common fixed points of a pair of relatively nonexpansive mappings in a Banach space.

Motivated and inspired by 4–9 , we intend in this paper, by using a new hybrid method, to prove a strong convergence theorem for finding a common element of the set of solutions for a generalized mixed equilibrium problem1.1and the set of common fixed points of a pair of quasi-φ-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property.

2. Preliminaries

For the sake of convenience, we first recall some definitions and conclusions which will be needed in proving our main results.

The mappingJ :E → 2E∗defined by

Jx {x∗_∈_E∗_:_{x, x}∗_x_x∗_}_{, x}_∈_E, _2.1

is called the normalized duality mapping. By the Hahn-Banach theorem,Jx/∅ for each

x∈E.

In the sequel, we denote the strong convergence and weak convergence of a sequence

{xn}byxn → xandxn x, respectively.

A Banach spaceEis said to be strictly convex ifxy/2<1 for allx, y∈U{z∈E:

z1}withx /y.Eis said to be uniformly convex if, for each∈0,2 , there existsδ >0 such thatxy/2<1−δfor allx, y∈Uwith||x−y|| ≥.Eis said to be smooth if the limit

lim

t→0

_x_ty₋_x

t 2.2

exists for allx, y∈U.Eis said to be uniformly smooth if the above limit exists uniformly in

x, y∈U.

Remark 2.1. The following basic properties can be found in Cioranescu10 .

iIfEis a uniformly smooth Banach space, thenJ is uniformly continuous on each bounded subset ofE.

iiIfEis a reflexive and strictly convex Banach space, thenJ−1_{is hemicontinuous.}

iiiIfEis a smooth, strictly convex, and reflexive Banach space, thenJis singlevalued, one-to-one and onto.

ivA Banach spaceEis uniformly smooth if and only ifE∗is uniformly convex.

(4)

Next we assume thatEis a smooth, strictly convex, and reflexive Banach space andC is a nonempty closed convex subset ofE. In the sequel, we always useφ :E×E → R to denote the Lyapunov functional defined by

φx, yx2₋₂_{x, Jy}_y2_, _∀_{x, y}_∈_E. _2.3

It is obvious from the definition ofφthat

x −y2_≤_φ_{x, y}_≤_x_y2_, _∀_{x, y}_∈_E.

2.4

Following Alber11 , the generalized projectionΠC:E → Cis defined by

ΠCx arg inf_y

∈Cφ

y, x, ∀x∈E. 2.5

Lemma 2.2see11,12 . LetEbe a smooth, strictly convex, and reflexive Banach space andCa

nonempty closed convex subset ofE. Then, the following conclusions hold:

aφx,ΠCy φΠCy, y≤φx, yfor allx∈Candy∈E;

bifx∈Eandz∈C, then

z ΠCx⇐⇒z−y, Jx−Jz≥0, ∀y∈C; 2.6

cforx, y∈E,φx, y 0 if and onlyxy.

Remark 2.3. If E is a real Hilbert spaceH, then φx, y ||x−y||2 _and _Π

C is the metric

projectionPCofHontoC.

LetEbe a smooth, strictly, convex and reflexive Banach space,Ca nonempty closed convex subset ofE,T : C → Ca mapping, andFTthe set of fixed points ofT. A point

p∈Cis said to be an asymptotic fixed point ofTif there exists a sequence{xn} ⊂Csuch that

xn pand||xn−Txn|| → 0. We denoted the set of all asymptotic fixed points ofT byFT.

Definition 2.4see13 . 1A mappingT : C → Cis said to be relatively nonexpansive if

FT/∅,FT FT, and

φp, Tx≤φp, x, ∀x∈C, p∈FT. 2.7

2 A mappingT : C → Cis said to be closed if, for any sequence{xn} ⊂ Cwith

xn → xandTxn → y,Txy.

Definition 2.5see14 . 1A mappingT : C → Cis said to be quasi-φ-nonexpansive if

FT/∅and

(5)

2 A mapping T : C → C is said to be quasi-φ-asymptotically nonexpansive if

FT/∅and there exists a real sequence{kn} ⊂1,∞withkn → 1 such that

φp, Tn_x_≤_k

nφp, x, ∀n≥1, x∈C, p∈FT. 2.9

3A pair of mappingsT1, T2:C → Cis said to be uniformly quasi-φ-asymptotically

nonexpansive ifFT1

FT2/∅and there exists a real sequence{kn} ⊂1,∞withkn → 1

such that fori1,2

φp, Tn

ix

≤knφp, x, ∀n≥1, x∈C, p∈FT1∩FT2. 2.10

4A mappingT :C → Cis said to be uniformlyL-Lipschitz continuous if there exists a constantL >0 such that

_Tn_x₋_Tn_y_≤_L_x₋_y_, _∀_{x, y}_∈_C. _2.11

Remark 2.6. 1 From the definition, it is easy to know that each relatively nonexpansive

mapping is closed.

2The class of quasi-φ-asymptotically nonexpansive mappings contains properly the class of quasi-φ-nonexpansive mappings as a subclass, and the class of quasi-φ-nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse may be not true.

Lemma 2.7 see15 . LetEbe a uniformly convex Banach space, r > 0 a positive number, and

Br0a closed ball ofE. Then, for any given subset {x1, x2, . . . , xN} ⊂ Br0and for any positive

numbers{λ1, λ2, . . . , λN}with

N

i1λi1, there exists a continuous, strictly increasing, and convex

functiong:0,2r → 0,∞withg0 0 such that, for anyi, j∈ {1,2, . . . , N}withi < j,

N

n1

λnxn

2

≤N

n1

λnxn2−λiλjgxi−xj. 2.12

Lemma 2.8see15 . LetEbe a real uniformly smooth and strictly convex Banach space with the

Kadec-Klee property andCa nonempty closed convex subset ofE. LetT : C → Cbe a closed and

quasi-φ-asymptotically nonexpansive mapping with a sequence{kn} ⊂ 1,∞,kn → 1. ThenFT

is a closed convex subset ofC.

For solving the generalized mixed equilibrium problem1.1, let us assume that the functionψ :C → Ris convex and lower semicontinuous, the nonlinear mappingA:C →

E∗_{is continuous and monotone, and the bifunction}_Θ _:_C_×_C _→ _R_{satisfies the following} conditions:

A1 Θx, x 0, for allx∈C,

A2 Θis monotone, that is,Θx, y Θy, x≤0,∀x, y∈C,

A3limsupt↓0Θxtz−x, y≤Θx, y∀x, z, y∈C,

(6)

Lemma 2.9. LetEbe a smooth, strictly convex, and reflexive Banach space andCa nonempty closed

convex subset ofE. LetΘ:C×C → Ra bifunction satisfying the conditions (A1)–(A4). Letr >0

andx∈E. Then, the followings hold.

i(Blum and Oettli [3]) there existsz∈Csuch that

Θz, y 1_ry−z, Jz−Jx≥0, ∀y∈C. 2.13

ii(Takahashi and Zembayashi [8]) Define a mappingTr :E → Cby

Trx

z∈C:Θz, y1

r

y−z, Jz−Jx≥0, ∀y∈C

, x∈E. 2.14

Then, the following conclusions hold:

aTr is single-valued,

bTr is a firmly nonexpansive-type mapping, that is,∀z, y∈E,

Trz−Try, JTrz−JTry≤Trz−Try, Jz−Jy, 2.15

cFTr EPΘ FTr,

dEPΘis closed and convex,

eφq, Trx φTrx, x≤φq, x, ∀q∈FTr.

Lemma 2.10 see16 . LetE be a smooth, strictly convex, and reflexive Banach space, andCa

nonempty closed convex subset of E. Let A : C → E∗ be a continuous and monotone mapping,

ψ:C → Ra lower semicontinuous and convex function, andΘ:C×C → Ra bifunction satisfying

conditions (A1)–(A4). Letr >0 be any given number andx∈Eany given point. Then, the following

hold.

iThere existsu∈Csuch that

Θu, yAu, y−uψy−ψu 1

ry−u, Ju−Jx ≥0, ∀y∈C. 2.16

iiIf we define a mappingKr :C → Cby

Krx

u∈C:Θu, yAu, y−uψy−ψu

1_ry−u, Ju−Jx≥0, ∀y∈C

, ∀x∈C.

(7)

Then, the mappingKr has the following properties:

aKris single valued,

bKris a firmly nonexpansive-type mapping, that is,

Krz−Kry, JKrz−JKry≤Krz−Kry, Jz−Jy, ∀z, y∈E, 2.18

cFKr Ω FKr,

d Ωis closed and convex,

e

φq, KrzφKrz, z≤φq, z, ∀q∈FKr, z∈E. 2.19

Remark 2.11. It follows from Lemma 2.9that the mappingKr is a relatively nonexpansive

mapping. Thus, it is quasi-φ-nonexpansive.

3. Main Results

In this section, we will prove a strong convergence theorem for finding a common element of the set of solutions for the generalized mixed equilibrium problem1.1and the set of common fixed points for a pair of quasi-φ-asymptotically nonexpansive mappings in Banach spaces.

Theorem 3.1. LetEbe a uniformly smooth and strictly convex Banach space with the Kadec-Klee

property andCa nonempty closed convex subset ofE. LetA:C → E∗be a continuous and monotone

mapping, ψ : C → Ra lower semicontinuous and convex, function, and Θ : C×C → R a

bifunction satisfying conditions (A1)–(A4). LetS, T : C → Cbe two closed and uniformly

quasi-φ-asymptotically nonexpansive mappings with a sequence{kn} ⊂1,∞andkn → 1. Suppose that

Sand T are uniformlyL-Lipschitz continuous and thatG FTFSΩis a nonempty and

bounded subset inC. Let{xn}be the sequence generated by

x0∈C, C0C, Q0 C,

znJ−1αnJxn 1−αnJTnxn,

ynJ−1βnJxn1−βnJSnzn,

un∈Csuch that, ∀y∈C,

Θun, yAun, y−unψy−ψun _r1

n

y−un, Jun−Jyn≥0,

Cnv∈Cn−1:φv, zn≤φv, xn ξn, φv, un≤φv, xn 1kn1−βnξn,

Qn{z∈Qn−1 :xn−z, Jx0−Jxn ≥0},

xn1 ΠCn∩Qnx0, ∀n≥0,

(8)

whereJ : E → E∗ is the normalized duality mapping,{αn}and {βn}are sequences in0,1 and

{rn} ⊂ a,∞for somea > 0,ξn sup_u_∈_Gkn−1φu, xn. Suppose that the following conditions

are satisfied:

ilim infn→ ∞αn1−αn>0,

iilim infn→ ∞βn1−βn>0.

Then{xn}converges strongly toΠFS∩FT∩Ωx0, whereΠFS∩FT∩Ωis the generalized projection of

EontoFS∩FT∩Ω.

Proof. Firstly, we define two functionsH:C×C → RandKr :C → Cby

Hx, y Θx, yAx, y−xψy−ψx, ∀x, y∈C,

Krx

u∈C:Hu, y1

ry−u, Ju−Jx ≥0, ∀y∈C

, x∈C. 3.2

By Lemma 2.10, we know that the function H satisfies conditions A1–A4 and Kr has

propertiesa–e. Therefore,3.1is equivalent to

x0∈C, C0C, Q0 C,

un ∈Csuch that, ∀y∈C,

Hun, y _r1

ny−un, Jun−Jyn ≥0,

Qn{z∈Qn−1 :xn−z, Jx0−Jxn ≥0},

xn1 ΠCn∩Qnx0, ∀n≥0.

3.3

We divide the proof ofTheorem 3.1into five steps.

IFirst we prove thatCnandQnare both closed and convex subsets ofCfor alln≥0.

In fact, it is obvious thatQnis closed and convex for alln≥0. Again we have that

φv, zn≤φv, xn ξn⇐⇒2v, Jxn−Jzn ≤ xn2− zn2ξn,

φv, un≤φv, xn 1kn1−βnξn⇐⇒2v, Jxn−Jun

≤ xn2− un2 1kn1−βnξn.

3.4

(9)

IINext we prove thatFT∩FS∩Ω⊂Cn∩Qn,∀n≥0.

Putting un Krnyn, ∀n ≥ 0, by Lemma 2.10 and Remark 2.11, Krn is relatively

nonexpansive. Again sinceSandT are quasi-φ-asymptotically nonexpansive, for any given

u∈FS∩FT∩Ω, we have that

φu, zn φ

u, J−1_α

nJxn 1−αnJTnxn

u2₋₂_{u, α}

nJxn 1−αnJTnxnαnJxn 1−αnJTnxn2

≤ u2₋₂_α

nu, Jxn −21−αnu, JTnxnαnxn2

1−αnTnxn2−αn1−αngJxn−JTnxn

αnφu, xn 1−αnφu, Tnxn−αn1−αngJxn−JTnxn

≤αnφu, xn 1−αnknφu, xn−αn1−αngJxn−JTnxn

≤knφu, xn−αn1−αngJxn−JTnxn

≤φu, xn sup

p∈Gkn−1φ

p, xn−αn1−αngJxn−JTnxn

φu, xn ξn−αn1−αngJxn−JTnxn

≤φu, xn ξn.

3.5

From3.5we have that

φu, un φu, Krnyn

≤φu, yn

≤φu, J−1_β

nJxn1−βnJSnzn

u2₋₂_{u, β}

nJxn1−βnJSnznβnJxn1−βnJSnzn2

≤ u2₋₂_β

nu, Jxn −21−βnu, JSnznβnxn2

1−βSnzn2−βn1−βngJxn−JSnzn

βnφu, xn 1−βnφu, Snzn−βn1−βngJxn−JSnzn

≤βnφu, xn 1−βnknφu, zn−βn1−βngJxn−JSnzn

≤βnφu, xn 1−βnknφu, xn ξn−βn1−βngJxn−JSnzn

≤βnφu, xn 1−βnφu, xn ξn1−βnknξn−βn1−βngJxn−JSnzn

≤φu, xn 1−βnξn1−βnknξn−βn1−βngJxn−JSnzn

≤φu, xn 1kn1−βnξn−βn1−βngJxn−JSnzn

≤φu, xn 1kn1−βnξn ∀n≥0.

3.6

(10)

Now we prove thatFT∩FS∩Ω⊂Cn∩Qn,∀n≥0.

In fact, fromQ0 C, we have thatFT∩FS∩Ω ⊂ C0∩Q0. Suppose thatFT∩ FS∩Ω⊂Ck∩Qk, for somek≥0. Now we prove thatFT∩FS∩Ω⊂Ck1∩Qk1. In fact,

sincexk1 ΠCk∩Qkx0, we have that

xk1−z, Jx0−Jxk1 ≥0, ∀z∈Ck∩Qk. 3.7

SinceFT∩FS∩Ω⊂Ck∩Qk, for anyz∈FT∩FS∩Ω, we have that

xk1−z, Jx0−Jxk1 ≥0. 3.8

This shows thatz∈Qk1, and soFT∩FS∩Ω⊂Qk1. The conclusion is proved.

IIINow we prove that{xn}is bounded.

From the definition of Qn, we have that xn ΠQnx0, ∀n ≥ 0. Hence, from

Lemma 2.21,

φxn, x0 φ

ΠQnx0, x0

≤φu, x0−φ

u,ΠQnx0

≤φu, x0, ∀u∈FT∩FS∩Ω⊂Qn, ∀n≥0.

3.9

This implies that{φxn, x0}is bounded. By virtue of2.4,{xn}is bounded. Denote

Msup

n≥0

{xn}<∞. 3.10

Sincexn1 ΠCn∩Qnx0 ∈Cn∩Qn ⊂Qnandxn ΠQnx0, from the definition ofΠQn, we have

that

φxn, x0≤φxn1, x0≤Mx02, ∀n≥0. 3.11

This implies that {φxn, x0} is nondecreasing, and so the limit limn→ ∞φxn, x0 exists.

Without loss of generality, we can assume that

lim

n→ ∞φxn, x0 r ≥0. 3.12

By the way, from the definition of{ξn},2.4, and3.10, it is easy to see that

ξnsup

u∈Gkn−1φu, xn≤supu∈Gkn−1uM

2 _−→₀ _as_n_{−→ ∞}_. _3.13

IVNow, we prove that{xn}converges strongly to some pointp∈GFT∩FS∩Ω.

In fact, since{xn}is bounded inCandEis reflexive, there exists a subsequence{xni} ⊂

(11)

closed, and sop ∈Qn for eachn ≥0. Sincexn ΠQnx0, from the defintion ofΠQn, we have

that

φxni, x0≤φ

p, x0

, n≥0. 3.14

Since

lim inf

ni→ ∞ φxni, x0 lim infni→ ∞

xni2−2xni, Jx0x02

≥p2₋

2p, Jx0

x02φ

p, x0

,

3.15

we have that

φp, x0

≤lim inf

ni→ ∞ φxni, x0≤lim sup_n_i_{→ ∞} φxni, x0≤φ

p, x0

. 3.16

This implies that limni→ ∞φxni, x0 φp, x0, that is,||xni|| → ||p||. In view of the Kadec-Klee

property ofE, we obtain that limn→ ∞xni p.

Now we first prove thatxn → p n → ∞. In fact, if there exists a subsequence

{xnj} ⊂ {xn}such thatxnj → q, then we have that

φp, q lim

ni→ ∞,nj→ ∞φ

xni, xnj

≤ lim

ni→ ∞,nj→ ∞φxni, x0−φ

ΠQ_njx0, x0

lim

ni→ ∞,nj→ ∞φxni, x0−φ

xnj, x0

0 by3.12.

3.17

Therefore we have thatpq. This implies that

lim

n→ ∞xnp. 3.18

Now we first prove thatp ∈FT∩FS. In fact, by the construction ofQn, we have

thatxn ΠQnx0. Therefore, byLemma 2.2awe have that

φxn1, xn φxn1,ΠQnx0

≤φxn1, x0−φ

ΠQnx0, x0

φxn1, x0−φxn, x0−→0 asn−→ ∞.

3.19

In view ofxn1∈Cn∩Qn⊂Cnand noting the construction ofCnwe obtain

φxn1, zn≤φxn1, xn ξn,

φxn1, un≤φxn1, xn 1kn1−βnξn.

3.20

From3.13and3.19, we have that

lim

(12)

From2.4it yields that||xn1||−||un||2 → 0 and||xn1||−||zn||2 → 0. Since||xn1|| →

||p||, we have that

un −→p, zn −→p asn−→ ∞. 3.22

Hence, we have that

Jun −→Jp, Jzn −→Jp asn−→ ∞. 3.23

This implies that{Jzn}is bounded inE∗. SinceEis reflexive, and soE∗ is reflexive,

there exists a subsequence{Jzni} ⊂ {Jzn}such thatJzni p0 ∈E∗. In view of the

reflexive-ness ofE, we see thatJE E∗. Hence, there existsx∈Esuch thatJxp0. Since

φxni1, zni xni12−2xni1, Jznizni2xni12−2xni1, JzniJzni2, 3.24

taking lim infn→ ∞ on both sides of the equality above and in view of the weak lower semi-continuity of norm|| · ||, it yields that

0≥p2−2p, p0

p02 p2−2

p, JxJx2

p2₋

2p, Jxx2φp, x,

3.25

that is,p x. This implies thatp0 Jp, and soJzn Jp. It follows from3.23and the

Kadec-Klee property ofE∗thatJzni → Jpasn → ∞. Noting thatJ−1:E∗ → Eis

hemicon-tinuous, it yields thatzni p. It follows from3.22and the Kadec-Klee property ofEthat

limni→ ∞zni p.

By the same way as given in the proof of3.18, we can also prove that

lim

n→ ∞znp. 3.26

From3.18and3.26, we have that

xn−zn −→0 asn−→ ∞. 3.27

SinceJis uniformly continuous on any bounded subset ofE, we have that

Jxn−Jzn −→0 asn−→ ∞. 3.28

For anyu∈FTFSΩ, it follows from3.5that

(13)

Since

φu, xn−φu, zn xn2− zn2−2u, Jxn−Jzn

≤xn2− zn22u · Jxn−Jzn

≤ xn−znxnzn 2u · Jxn−Jzn,

3.30

From3.27and3.28, it follows that

φu, xn−φu, zn−→0 asn−→ ∞. 3.31

In view of conditioniand lim infn→ ∞αn1−αn>0, we see that

gJxn−JTnxn−→0 asn−→ ∞. 3.32

It follows from the property ofgthat

Jxn−JTnxn −→0 asn−→ ∞. 3.33

Sincexn → pandJis uniformly continuous, it yields thatJxn → Jp. Hence from3.33we

have that

JTn_x

n−→Jp asn−→ ∞. 3.34

SinceJ−1_:_E∗ _→ _E_{is hemicontinuous, it follows that}

Tn_x

n p. 3.35

On the other hand, we have that

Tn_x

n −pJTnxn −Jp≤JTnxn−Jp−→0 asn−→ ∞. 3.36

This together with3.35shows that

Tn_x

n−→p. 3.37

Furthermore, by the assumption thatTis uniformlyL-Lipschitz continuous, we have that

Tn1_x

n−Tnxn≤Tn1xn−Tn1xn1Tn1xn1−xn1xn1−xnxn−Tnxn

≤L1xn1−xnTn1xn1−xn1xn−Tnxn.

(14)

This together with3.18and3.37, yields||Tn1xn−Tnxn|| → 0asn → ∞. Hence

from3.37we have thatTn1xn → p, that is,TTnxn → p. In view of3.37and the closeness

ofT, it yields thatTpp. This implies thatp∈FT.

By the same way as given in the proof of3.23to3.31, we can also prove that

lim

n→ ∞unp, φu, xn−φu, un−→0 asn−→ ∞. 3.39

SinceunKrnyn, from2.19,3.6,3.13, and3.39, we have that

φun, ynφKrnyn, yn

≤φu, yn−φu, un

≤φu, xn−φu, un 1kn1−βnξn −→0 asn−→ ∞.

3.40

From2.4it yields that||un|| − ||yn||2 → 0. Since||un|| → ||p||, we have that

yn−→p asn−→ ∞. 3.41

Hence we have that

Jyn−→Jp asn−→ ∞. 3.42

By the same way as given in the proof of3.26, we can also prove that

lim

n→ ∞ynp. 3.43

From3.39and3.43we have that

_u_n₋_y_n_−→₀ _as_n_{−→ ∞}_. _3.44

SinceJis uniformly continuous on any bounded subset ofE, we have that

Jun−Jyn−→0 asn−→ ∞. 3.45

For anyu∈FTFSΩ, it follows from3.6,3.13, and3.39that

βn1−βngJxn−Snzn≤φu, xn−φu, un 1kn1−βnξn−→0. 3.46

In view of conditioniiand lim infn→ ∞βn1−βn>0, we see that

gJxn−JSnzn−→0 asn−→ ∞. 3.47

It follows from the property ofgthat

(15)

Sincexn → pandJis uniformly continuous, it yields,Jxn → Jp. Hence from3.48we have

that

JSn_z

n−→Jp asn−→ ∞. 3.49

SinceJ−1_:_E∗ _→ _E_{is hemicontinuous, it follows that}

Sn_z

n p. 3.50

On the other hand, we have that

_Sn_z

n −pJSnzn −Jp≤JSnzn−Jp−→0 asn−→ ∞. 3.51

This together with3.50shows that

Sn_z

n−→p. 3.52

Furthermore, by the assumption thatSis uniformlyL-Lipschitz continuous, we have that

Sn1_z_n₋_Sn_z

n≤Sn1zn−Sn1zn1Sn1zn1−zn1zn1−znzn−Snzn

≤L1zn1−znSn1zn1−zn1zn−Snzn.

3.53

This together with3.26and3.52, yields that||Sn1zn−Snzn|| → 0asn → ∞.

Hence from3.52we have thatSn1zn → p, that is,SSnzn → p. In view of3.52and the

closeness ofT, it yields thatSpp. This implies thatp∈FS.

Next we prove thatp ∈ Ω. From3.45and the assumption thatrn ≥ a,∀n ≥ 0, we

have that

lim

n→ ∞

Jun−Jyn

rn 0. 3.54

SinceunKrnyn, we have that

Hun, y_r1

n

y−un, Jun−Jyn≥0, ∀y∈C. 3.55

Replacingnbynkin3.55, from conditionA2, we have that

1

rnk

y−unk, Junk−Jynk

≥ −Hunk, y

≥Hy, unk

(16)

By the assumption thaty→ Hx, yis convex and lower semicontinuous, it is also weakly lower semicontinuous. Lettingnk → ∞in3.55, from3.54and conditionA4, we have

thatHy, p≤0,∀y∈C.

Fort∈0,1 andy∈C, lettingytty 1−tp, there areyt∈CandHyt, p≤0. By

conditionsA1andA4, we have that

0Hyt, yt≤tHyt, y 1−tHyt, p≤tHyt, y. 3.57

Dividing both sides of the above equation byt, we have thatHyt, y ≥ 0,∀y ∈ C. Letting

t↓ 0, from conditionA3, we have thatHp, y≥ 0,∀y ∈C, that is,Θp, y Ap, y−p ψy−ψp≥0,∀y∈C. Thereforep∈Ω, and sop∈FTFSΩ.

VFinally, we prove thatxn → ΠFTFSΩx0.

Letw ΠFTFSΩx0. Fromw ∈FTFSΩ⊂Cn∩Qn, andxn1 ΠCn∩Qnx0,

we have that

φxn1, x0≤φw, x0, ∀n≥0. 3.58

Since the norm is weakly lower semicontinuous, this implies that

φp, x0p2₋

2p, Jx0x02≤ lim

nk→ ∞inf

xnk

2₋₂_x

nk, Jx0x0

2

≤ lim

nk→ ∞infφxnk, x0≤nlimk→ ∞supφxnk, x0≤φw, x0.

3.59

It follows from the definition ofΠFTFSΩx0 and3.59that we havep w. Therefore, xn → ΠFTFSΩx0. This completes the proof ofTheorem 3.1.

Remark 3.2. Theorem 3.1improves and extends the corresponding results in7–9 .

aFor the framework of spaces, we extend the space from a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex Banach space with the Kadec-Klee propertynote that each uniformly convex Banach space must have the Kadec-Klee property.

bFor the mappings, we extend the mappings from nonexpansive mappings, relatively nonexpansive mappings, or weak relatively nonexpansive mappings to a pair of quasi-φ-asymptotically nonexpansive mappings.

cFor the equilibrium problem, we extend the generalized equilibrium problem to the generalized mixed equilibrium problem.

The following theorems can be obtained fromTheorem 3.1immediately.

property and C a nonempty closed convex subset of E. Let A : C → E∗ be a continuous and

monotone mapping andΘ : C×C → Ra bifunction satisfying conditions (A1)–(A4). LetS, T :

C → Cbe two closed and uniformly quasi-φ-asymptotically nonexpansive mappings with a sequence

(17)

GFTFSGEP is a nonempty and bounded subset inC. Let{xn}be the sequence generated

by

x0∈C, C0C, Q0C,

Θun, yAun, y−un_r1

n

Qn {z∈Qn−1:xn−z, Jx0−Jxn ≥0},

xn1 ΠCn∩Qnx0, ∀n≥0,

3.60

whereJ :E → E∗is the normalized duality mapping,{αn}and{βn}are sequences in0,1 , and

{rn} ⊂ a,∞for somea > 0,ξn supu∈Gkn −1φu, xn. If{αn}and {βn}satisfy conditions

(i)-(ii) inTheorem 3.1, then{xn}converges strongly toΠFS∩FT∩GEPx0, where GEP is the set for the

solutions of generalized equilibrium problem1.3.

Proof. Putting ψ 0 in Theorem 3.1, the conclusion of Theorem 3.3can be obtained from

Theorem 3.1.

property andCa nonempty closed convex subset ofE. Letψ : C → Rbe a lower semicontinuous

and convex function andΘ : C×C → Ra bifunction satisfying conditions (A1)–(A4). LetS, T :

C → Cbe two closed and uniformly quasi-φ-asymptotically nonexpansive mappings with a sequence

{kn} ⊂1,∞andkn → 1. Suppose thatSandTare uniformlyL-Lipschitz continuous and thatG

FTFSMEP is a nonempty and bounded subset inC. Let{xn}be the sequence generated by

x0∈C, C0C, Q0 C,

Θun, yψy−ψun _r1

n

Cn v∈Cn−1:φv, zn≤φv, xn ξn, φv, un≤φv, xn 1kn

1−βnξn,

Qn{z∈Qn−1 :xn−z, Jx0−Jxn ≥0},

3.61

(18)

(i)-(ii) inTheorem 3.1, then {xn}converges strongly toΠFS∩FT∩MEPx0, where MEP is the set of

solutions for mixed equilibrium problem1.4.

Proof. Putting A 0 in Theorem 3.1, the conclusion of Theorem 3.4 can be obtained from

Theorem 3.1.

property and C a nonempty closed convex subset of E. Let A : C → E∗ be a continuous and

monotone mapping andψ :C → Ra lower semicontinuous and convex function. LetS, T :C → C

be two closed and uniformly quasi-φ-asymptotically nonexpansive mappings with a sequence

{kn} ⊂ 1,∞andkn → 1. Suppose thatSandT are uniformlyL-Lipschitz continuous and that

G FTFSVIC, A, ψis a nonempty and bounded subset inC. Let{xn}be the sequence

generated by

x0∈C, C0C, Q0C,

Aun, y−unψy−ψun _r1

n

Qn {z∈Qn−1:xn−z, Jx0−Jxn ≥0},

3.62

{rn} ⊂ a,∞for somea > 0, ξn supu∈Gkn−1φu, xn. If{αn}and{βn}satisfy conditions

(i)-(ii) inTheorem 3.1, then{xn}converges strongly toΠFS∩FT∩VIC,A,ψx0, where VIC, A, ψis

the set of solutions for the mixed variational inequality1.5.

Proof. Putting Θ 0 in Theorem 3.1, the conclusion ofTheorem 3.5can be obtained from

Theorem 3.1.

property andCa nonempty closed convex subset ofE. LetΘ:C×C → Rbe a bifunction satisfying

conditions (A1)–(A4). Let S, T : C → C be two closed and uniformly quasi-φ-asymptotically

nonexpansive mappings with a sequence {kn} ⊂ 1,∞ and kn → 1. Suppose that Sand T are

uniformly L -Lipschitz continuous and thatG FTFSEPΘis a nonempty and bounded

subset inC. Let{xn}be the sequence generated by

x0∈C, C0C, Q0 C,

(19)

Θun, y_r1 n

Cn v∈Cn−1:φv, zn≤φv, xn ξn, φv, un≤φv, xn 1kn1−βnξn,

Qn{z∈Qn−1 :xn−z, Jx0−Jxn ≥0},

3.63

{rn} ⊂ a,∞for somea > 0,ξn sup_u_∈_Gkn −1φu, xn. If{αn}and {βn}satisfy conditions

(i)-(ii) inTheorem 3.1, then{xn}converges strongly toΠFS∩FT∩EPΘx0, where EPΘis the set of

solutions for the equilibrium problem1.6.

Proof. Puttingψ 0 andA0 inTheorem 3.1, the conclusion ofTheorem 3.6can be obtained

fromTheorem 3.1.

mapping and S, T : C → C two closed and uniformly quasi-φ-asymptotically nonexpansive

mappings with a sequence{kn} ⊂ 1,∞ and kn → 1. Suppose that S and T are uniformly

L-Lipschitz continuous and thatGFTFSVIC, Ais a nonempty and bounded subset inC.

Let{xn}be the sequence generated by

x0∈C, C0C, Q0 C,

Aun, y−un_r1

n

Qn{z∈Qn−1 :xn−z, Jx0−Jxn ≥0},

3.64

{rn} ⊂ a,∞for somea > 0,ξn supu∈Gkn −1φu, xn. If{αn}and {βn}satisfy conditions

(i)-(ii) inTheorem 3.1, then{xn}converges strongly toΠFS∩FT∩VIC,Ax0, where VIC, Ais the

set of solutions for the variational inequality1.7

Proof. Puttingψ 0 andΘ 0 inTheorem 3.1, the conclusion ofTheorem 3.7can be obtained

fromTheorem 3.1.

(20)

mapping,ψ :C → Ra lower semicontinuous and convex function, andΘ:C×C → Ra bifunction

satisfying conditions (A1)–(A4). LetS:C → Cbe a closed and quasi-φ-asymptotically nonexpansive

mappings with a sequence {kn} ⊂ 1,∞and kn → 1. Suppose thatS is uniformlyL-Lipschitz

continuous and thatFSΩis a nonempty and bounded subset in C. Let {xn} be the sequence

generated by

x0∈C, C0C, Q0C,

ynJ−1βnJxn1−βnJSnxn,

n

Cnv∈Cn−1:φv, un≤φv, xn ξn,

Qn {z∈Qn−1:xn−z, Jx0−Jxn ≥0},

3.65

{rn} ⊂ a,∞for somea > 0, ξn sup_u_∈_F_S_∩_Ωkn−1φu, xn. If{βn}satisfy condition (ii) in

Theorem 3.1, then{xn}converges strongly toΠFS∩Ωx0.

Proof. TakingT IinTheorem 3.1, we have thatzn xn,∀n ≥0. Hence, the conclusion of

Theorem 3.8is obtained.

mapping,ψ :C → Ra lower semicontinuous and convex function, andΘ:C×C → Ra bifunction

satisfying conditions (A1)–(A4). Suppose thatΩis a nonempty subset inC. Let{xn}be the sequence

generated by

x0∈C, C0C, Q0C,

n

y−un, Jun−Jxn≥0,

Cnv∈Cn−1:φv, un≤φv, xn,

Qn {z∈Qn−1:xn−z, Jx0−Jxn ≥0},

3.66

where{rn} ⊂a,∞for somea >0. Then{xn}converges strongly toΠΩx0.

(21)

property andCa nonempty closed convex subset ofE. LetS, T :C → Cbe two closed and uniformly

quasi-φ-asymptotically nonexpansive mappings with a sequence{kn} ⊂1,∞andkn → 1. Suppose

thatSandTare uniformlyL-Lipschitz continuous and thatFTFSis a nonempty and bounded

subset inC. Let{xn}be the sequence generated by

x0∈C, C0C, Q0C,

un ΠCyn,

Qn {z∈Qn−1:xn−z, Jx0−Jxn ≥0},

3.67

ξnsup_u_∈_F_S_∩_F_Tkn−1φu, xn. If{αn}and{βn}satisfy conditions (i)-(ii) inTheorem 3.1, then

{xn}converges strongly toΠFS∩FTx0.

Proof. TakingA Θ 0 andrn1,∀n≥0 inTheorem 3.1, the conclusion ofTheorem 3.10is

obtained.

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 F. E. Browder, “Existence and approximation of solutions of nonlinear variational inequalities,”

Proceedings of the National Academy of Sciences of the United States of America, vol. 56, pp. 1080–1086, 1966.

3 E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994.

4 P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 1, pp. 117–136, 2005.

5 A. Moudafi, “Second-order diﬀerential proximal methods for equilibrium problems,” Journal of Inequalities in Pure and Applied Mathematics, vol. 4, no. 1, article 18, 2003.

6 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.

7 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.

8 W. Takahashi and K. Zembayashi, “Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 1, pp. 45–57, 2009.

(22)

10 I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990. 11 Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and

applications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178 of Lecture Notes in Pure and Appl. Math., 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 S.-y. Matsushita and W. Takahashi, “A strong convergence theorem for relatively nonexpansive mappings in a Banach space,” Journal of Approximation Theory, vol. 134, no. 2, pp. 257–266, 2005. 14 H. Zhou, G. Gao, and B. Tan, “Convergence theorems of a modified hybrid algorithm for a family

of quasi-ϕ-asymptotically nonexpansive mappings,” Journal of Applied Mathematics and Computing, vol. 32, no. 2, pp. 453–464, 2010.

15 S.-S. Chang, J. K. Kim, and X. R. Wang, “Modified block iterative algorithm for solving convex feasibility problems in Banach spaces,” Journal of Inequalities and Applications, vol. 2010, Article ID 869684, 14 pages, 2010.

(23)

Submit your manuscripts at

http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics

Journal of

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations

International Journal of

Volume 2014

Applied MathematicsJournal of

Mathematical PhysicsAdvances in

Complex Analysis

Journal of

Optimization

Journal of

Combinatorics

http://www.hindawi.com Volume 2014 International Journal of

Journal of

Function Spaces

Abstract and Applied Analysis

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporation http://www.hindawi.com Volume 2014

The Scientific

World Journal

Discrete Dynamics in Nature and Society

http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation

Discrete Mathematics

Journal of