Strong Convergence Theorem by Monotone Hybrid Algorithm for Equilibrium Problems, Hemirelatively Nonexpansive Mappings, and Maximal Monotone Operators

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Volume 2008, Article ID 617248,12pages doi:10.1155/2008/617248

Research Article

Strong Convergence Theorem by Monotone

Hybrid Algorithm for Equilibrium Problems,

Hemirelatively Nonexpansive Mappings, and

Maximal Monotone Operators

Yun Cheng and Ming Tian

College of Science, Civil Aviation University of China, Tianjin 300300, China

Correspondence should be addressed to Yun Cheng,[email protected]

Received 17 June 2008; Accepted 11 November 2008

Recommended by Wataru Takahashi

We introduce a new hybrid iterative algorithm for finding a common element of the set of fixed points of hemirelatively nonexpansive mappings and the set of solutions of an equilibrium problem and for finding a common element of the set of zero points of maximal monotone operators and the set of solutions of an equilibrium problem in a Banach space. Using this theorem, we obtain three new results for finding a solution of an equilibrium problem, a fixed point of a hemirelatively nonexpnasive mapping, and a zero point of maximal monotone operators in a Banach space.

Copyrightq2008 Y. Cheng and M. Tian. 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.

1. Introduction

LetEbe a Banach space, letCbe a closed convex subset ofE, and letf be a bifunction from

C×CtoR, whereRis the set of real numbers. The equilibrium problem is to find

x∗_∈_C _{such that}_f_x∗_{, y}_≥₀ _∀_y_∈_C. _1.1

The set of such solutionsx∗is denoted by EPf.

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Banach space. Very recently, Su et al.3proved the following theorem by a monotone hybrid method.

Theorem 1.1see Su et al.3. LetEbe a uniformly convex and uniformly smooth real Banach space, letCbe a nonempty closed convex subset ofE, and letT :C → Cbe a closed hemirelatively nonexpansive mapping such that FT/∅. Assume that αn is a sequence in 0,1 such that lim sup_n_{→ ∞}αn <1. Define a sequencexninCby the following:

x0∈C, chosen arbitrarily,

ynJ−1αnJxn 1−αnJTxn,

Cn{z∈Cn−1∩Qn−1:φz, yn≤φz, xn}, C0 {z∈C:φz, y0≤φz, x0}, Qn{z∈Cn−1∩Qn−1:xn−z, Jx0−Jxn ≥0},

Q0C, xn1 ΠCn∩Qnx0,

1.2

where J is the duality mapping on E. Then,xn converges strongly to ΠFTx0, where ΠFT is the generalized projection fromContoFT.

In this paper, motivated by Su et al. 3, we prove 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 hemirelatively nonexpansive mapping and for finding a common element of the set of zero points of maximal monotone operators and the set of solutions of an equilibrium problem in a Banach space by using the monotone hybrid method. Using this theorem, we obtain three new strong convergence results for finding a solution of an equilibrium problem, a fixed point of a hemirelatively nonexpnasive mapping, and a zero point of maximal monotone operators in a Banach space.

2. Preliminaries

LetEbe a real Banach space with dualE∗. We denote byJthe normalized duality mapping fromEto 2E∗defined by

Jxf∈E∗_:_{x, f}_x2 _f2_, _2.1

where·,·denotes the generalized duality pairing. It is well known that ifE∗ is uniformly convex, thenJ is uniformly continuous on bounded subsets ofE. In this case, J is single valued and also one to one.

LetEbe a smooth, strictly convex, and reflexive Banach space and letCbe a nonempty closed convex subset ofE. Throughout this paper, we denote byφthe function defined by

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Following Alber4, the generalized projectionΠC :E → CfromEontoCis defined by

ΠCx arg min

y∈Cφy, x ∀x∈E. 2.3

The generalized projectionΠCfromEontoCis well defined and single valued, and it satisfies

x − y2_≤_φ_{y, x}_≤_x_y2 _∀_{x, y}_∈_E. _2.4

IfEis a Hilbert space, thenφy, x y−x2_and_ΠC_{is the metric projection of}_E_onto_C_.

IfEis a reflexive strict convex and smooth Banach space, then forx, y∈E, φx, y 0 if and only ifx y. It is suﬃcient to show that ifφx, y 0, thenx y. From2.4,

we havex y. This impliesx, Jy x2 _Jy2_{. From the definition of}_J_{, we have} JxJy, that is,xy.

Let C be a closed convex subset of E and let T be a mapping from C into itself. We denote by FT the set of fixed points ofT. T is called hemirelatively nonexpansive if

φp, Tx≤φp, xfor allx∈Candp∈FT.

A pointpinCis said to be an asymptotic fixed point ofT5ifCcontains a sequencexn which converges weakly topsuch that the strong limn→ ∞Txn−xn 0. The set of asymptotic fixed points ofT will be denoted byFT. A hemirelatively nonexpansive mappingTfromC into itself is called relatively nonexpansive1,5,6ifFT FT.

We need the following lemmas for the proof of our main results.

Lemma 2.1see Alber4. LetCbe a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach spaceE. Then,

φx,ΠCyφΠCy, y≤φx, y ∀x∈C, y∈E. 2.5

Lemma 2.2see Alber4. LetCbe a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space, letx∈E, and letz∈C. Then,

z ΠCx⇐⇒ y−z, Jx−Jz ≤0 ∀y∈C. 2.6

Lemma 2.3see Kamimura and Takahashi7. LetEbe a smooth and uniformly convex Banach space and let {xn} and {yn} be sequences in E such that either {xn} or {yn} is bounded. If limn→ ∞φxn, yn 0. Thenlimn→ ∞xn−yn0.

Lemma 2.4see Xu8. LetEbe a uniformly convex Banach space and letr >0. Then, there exists a strictly increasing, continuous, and convex functiong:0,2r → Rsuch thatg0 0and

tx 1−ty2_≤_t_x2₁₋_t_y2₋_t₁₋_t_g_x₋_y _∀_{x, y}_∈_B

r, t∈0,1, 2.7

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Lemma 2.5see Kamimura and Takahashi7. LetEbe a smooth and uniformly convex Banach space and let r > 0. Then, there exists a strictly increasing, continuous, and convex functiong : 0,2r → Rsuch thatg0 0and

gx−y≤φx, y ∀x, y∈Br. 2.8

For solving the equilibrium problem, let us assume that a bifunctionf satisfies the following conditions:

A1fx, x 0 for all x∈C;

A2f is monotone, that is,fx, y fy, x≤0 for all x, y∈C; A3for all x, y, z∈C, lim sup_t_→₀ftz 1−tx, y≤fx, y; A4for all x∈C, fx,·is convex.

Lemma 2.6see Blum and Oettli9. LetCbe a closed convex subset of a smooth, strictly convex, and reflexive Banach spaceE, letf be a bifunction fromC×CtoRsatisfying (A1)–(A4), letr >0, and letx∈E. Then, there existsz∈Csuch that

fz, y 1

ry−z, Jz−Jx ≥0 ∀y∈C. 2.9

Lemma 2.7see Takahashi and Zembayashi10. LetCbe a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach spaceE, letfbe a bifunction fromC×CtoRsatisfying (A1)–(A4), and letx∈E, forr >0. Define a mappingTr :E → 2C as follows:

Trx

z∈C:fz, y 1

ry−z, Jz−Jx ≥0 ∀y∈C

∀x∈E. 2.10

Then, the following holds:

1Tr is single valued;

2Tr is a firmly nonexpansive-type mapping11, that is, for allx, y∈E,

Trx−Try, JTrx−JTry ≤ Trx−Try, Jx−Jy; 2.11

3FTr FTr Epf; 4Epfis closed and convex.

Lemma 2.8see Takahashi and Zembayashi10. LetCbe a closed convex subset of a smooth, strictly convex, and reflexive Banach spaceEand letf be a bifunction fromC×CtoR satisfying (A1)–(A4). Then, forr >0andx∈E, andq∈FTr,

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Lemma 2.9see Su et al.3. LetEbe a strictly convex and smooth real Banach space, letCbe a closed convex subset ofE, and letT be a hemirelatively nonexpansive mapping fromCinto itself. Then,FTis closed and convex.

Recall that an operatorTin a Banach space is called closed, ifxn → x, Txn → y, then

Txy.

3. Strong convergence theorem

Theorem 3.1. Let Ebe a uniformly convex and uniformly smooth real Banach space, let Cbe a nonempty closed convex subset ofE, letfbe a bifunction fromC×CtoRsatisfying (A1)–(A4), and letT:C → Cbe a closed hemirelatively nonexpansive mapping such thatFT∩EPf/∅. Define a sequence{xn}inCby the following:

ynJ−1αnJxn 1−αnJTzn,

znJ−1βnJxn 1−βnJTxn,

un∈Csuch thatfun, y _r1

ny−un, Jun−Jyn ≥0, ∀y∈C,

Cn{z∈Cn−1∩Qn−1:φz, un≤φz, xn}, C0 {z∈C:φz, u0≤φz, x0},

Qn{z∈Cn−1∩Qn−1:xn−z, Jx0−Jxn ≥0},

3.1

for everyn∈N∪ {0}, whereJis the duality mapping onE, {αn}, {βn}are sequences in0,1such thatlim infn→ ∞1−αnβn1−βn > 0 and{rn} ⊂ a,∞for somea > 0. Then,{xn}converges strongly toΠFT∩EPfx0, whereΠFT∩EPfis the generalized projection ofEontoFT∩EPf.

Proof. First, we can easily show thatCnandQnare closed and convex for eachn≥0.

Next, we show thatFT∩EPf ⊂ Cn for alln ≥ 0. Letu ∈ FT∩EPf. Putting

unTrnynfor alln∈N, fromLemma 2.8, we haveTrn relatively nonexpansive. SinceTrnare relatively nonexpansive andT is hemirelatively nonexpansive, we have

φu, zn φu, J−1_β

nJxn 1−βnJTxn u2₋₂_{u, β}

nJxn 1−βnJTxnβnJxn 1−βnJTxn2

≤ u2₋₂_β

nu, Jxn −21−βnu, JTxnβnxn2 1−βnTxn2 βnφu, xn 1−βnφu, Txn

≤φu, xn,

φu, un φu, Trnyn≤φu, yn≤αnφu, xn 1−αnφu, zn≤φu, xn.

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Hence, we have

FT∩EPf⊂Cn ∀n≥0. 3.3

Next, we show thatFT∩EPf ⊂Qnfor alln ≥0. We prove this by induction. Forn0, we have

FT∩EPf⊂Q0C. 3.4

Suppose thatFT∩EPf⊂Qn, byLemma 2.2, we have

xn1−z, Jx0−Jxn1 ≥0 ∀z∈Cn∩Qn. 3.5

As FT∩EPf ⊂ Cn ∩Qn, by the induction assumptions, the last inequality holds, in particular, for allz ∈ FT∩EPf. This, together with the definition ofQn1, implies that FT∩EPf⊂Qn1. So,{xn}is well defined.

Sincexn1 ΠCn∩Qnx0andCn∩Qn⊂Cn−1∩Qn−1for alln≥1, we have

φxn, x0≤φxn1, x0 ∀n≥0. 3.6

Therefore,{φxn, x0}is nondecreasing. In addition, from the definition ofQnandLemma 2.2,

xn ΠQnx0. Therefore, for eachu∈FT∩EPf, we have

φxn, x0 φ

ΠQnx0, x0

≤φu, x0−φu, xn≤φu, x0. 3.7

Therefore,φxn, x0and{xn}are bounded. This, together with3.6, implies that the limit of

{φxn, x0}exists. FromLemma 2.1, we have, for any positive integerm,

φxnm, xnφxnm,ΠQnx0

≤φxnm, x0−φ

ΠQnx0, x0

φxnm, x0−φxn, x0 ∀n≥0.

3.8

Therefore,

lim

n→ ∞φxnm, xn 0. 3.9

From3.9, we can prove that{xn}is a Cauchy sequence. Therefore, there exists a pointx∈C such that{xn}converges strongly tox.

Sincexn1 ∈Cn, we have

φxn1, un≤φxn1, xn. 3.10

Therefore, we have

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FromLemma 2.3, we have

lim

n→ ∞xn1−unnlim→ ∞xn1−xn0. 3.12

So, we have

lim

n→ ∞xn−un0. 3.13

SinceJis uniformly norm-to-norm continuous on bounded sets, we have

lim

n→ ∞Jxn−Jun0. 3.14

Letr sup_n∈N{xn,Txn}.SinceEis a uniformly smooth Banach space, we know thatE∗ is a uniformly convex Banach space. Therefore, fromLemma 2.4, there exists a continuous, strictly increasing, and convex functiongwithg0 0, such that

αx∗₁₋_α_y∗2 _≤_α_x∗2₁₋_α_y∗2₋_α₁₋_α_g_x∗₋_y∗ _3.15

forx∗, y∗∈Br, andα∈0,1. So, we have that foru∈FT∩EPf,

φu, zn φu, J−1_β

nJxn 1−βnJTxn u2₋₂_{u, β}

nJxn 1−βnJTxnβnJxn 1−βnJTxn2

≤φu, xn−βn1−βngJxn−JTxn,

φu, un≤αnφu, xn 1−αnφu, zn

≤φu, xn−1−αnβn1−βngJxn−JTxn.

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Therefore, we have

1−αnβn1−βngJxn−JTxn≤φu, xn−φu, un. 3.17

Since

φu, xn−φu, unxn2−un2−2u, Jxn−Jun≤xn−unxnun2uJxn−Jun, 3.18

we have

lim

n→ ∞φu, xn−φu, un 0. 3.19

From lim infn→ ∞1−αnβn1−βn>0, we have

lim

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Therefore, from the property ofg, we have

lim

n→ ∞Jxn−JTxn0. 3.21

SinceJ−1_{is uniformly norm-to-norm continuous on bounded sets, we have}

lim

n→ ∞xn−Txn0. 3.22

SinceT is a closed operator andxn → x, thenxis a fixed point ofT. On the other hand,

φun, yn φTrnyn, yn≤φu, yn−φu, Trnyn≤φu, xn−φu, Trnyn φu, xn−φu, un. 3.23

So, we have from3.19that

lim

n→ ∞φun, yn 0. 3.24

FromLemma 2.3, we have that

lim

n→ ∞un−yn0. 3.25

Fromxn → xandxn−un → 0, we haveyn → x. From3.25, we have

lim

n→ ∞Jun−Jyn0. 3.26

Fromrn≥a, we have

lim n→ ∞

Jun−Jyn

rn 0. 3.27

ByunTrnyn, we have

fun, y _r1

ny−un, Jun−Jyn ≥0 ∀y∈C. 3.28

FromA2, we have that

1

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From3.27andA4, we have

fy,x≤0 ∀y∈C. 3.30

Fortwith 0< t≤1 andy∈C, letytty 1−tx. We havefyt,x≤0. So, fromA1, we have

0fyt, yt≤tfyt, y 1−tfyt,x≤tfyt, y. 3.31

Dividing byt, we have

fyt, y≥0 ∀y∈C. 3.32

Lettingt → 0, fromA3, we have

fx, y ≥0 ∀y∈C. 3.33

Therefore,x∈EPf. Finally, we prove thatx ΠFT∩EPfx0. FromLemma 2.1, we have

φx, ΠFT∩EPfx0

φΠFT∩EPfx0, x0

≤φx, x 0. 3.34

Sincexn1 ΠCn∩Qnx0andx∈FT∩EPf⊂Cn∩Qn, for alln≥0, we get fromLemma 2.1 that

φΠFT∩EPfx0, xn1

φxn1, x0≤φ

ΠFT∩EPfx0, x0

. 3.35

By the definition of φx, y, it follows that φx, x 0 ≤ φΠFT∩EPfx0, x0 and φx, x 0 ≥ φΠFT∩EPfx0, x0, whence φx, x 0 φΠFT∩EPfx0, x0. Therefore, it follows from the

uniqueness ofΠFT∩EPfx0thatx ΠFT∩EPfx0. This completes the proof.

Corollary 3.2. LetE be a uniformly convex and uniformly smooth real Banach space, let C be a nonempty closed convex subset ofE, and letfbe a bifunction fromC×CtoRsatisfying (A1)–(A4). Define a sequence{xn}inCby the following:

ny−un, Jun−Jxn ≥0, ∀y∈C,

Cn{z∈Cn−1∩Qn−1:φz, un≤φz, xn}, C0 {z∈C:φz, u0≤φz, x0}, Qn{z∈Cn−1∩Qn−1:xn−z, Jx0−Jxn ≥0},

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for everyn∈N∪ {0}, whereJis the duality mapping onEand{rn} ⊂a,∞for somea >0. Then,

{xn}converges strongly toΠEPfx0.

Proof. PuttingT IinTheorem 3.1, we obtainCorollary 3.2.

Corollary 3.3. LetE be a uniformly convex and uniformly smooth real Banach space, let C be a nonempty closed convex subset ofE, and letT : C → Cbe a closed hemirelatively nonexpansive mapping. Define a sequence{xn}inCby the following:

x0∈C, chosen arbitrarily, ynJ−1αnJxn 1−αnJTzn,

un ΠCyn,

Cn{z∈Cn−1∩Qn−1:Φz, un≤φz, xn}, C0 {z∈C:φz, u0≤φz, x0}, Qn{z∈Cn−1∩Qn−1:xn−z, Jx0−Jxn ≥0},

3.37

for everyn∈N∪ {0}, whereJis the duality mapping onE, {αn}, {βn}are sequences in0,1such thatlim infn→ ∞1−αnβn1−βn>0. Then,{xn}converges strongly toΠFTx0.

Proof. Putting fx, y 0 for all x, y ∈ C and rn 1 for all nin Theorem 3.1, we obtain

Corollary 3.3.

Corollary 3.4. LetE be a uniformly convex and uniformly smooth real Banach space, let C be a nonempty closed convex subset ofE, letfbe a bifunction fromC×CtoRsatisfying (A1)–(A4), and letT :C → Cbe a closed relatively nonexpansive mapping such thatFT∩EPf/∅. Define a sequence{xn}inCby the following:

ynJ−1αnJxn 1−αnJTzn,

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for everyn∈N∪ {0}, whereJis the duality mapping onE, {αn}, {βn}are sequences in0,1such thatlim infn→ ∞1−αnβn1−βn > 0 and{rn} ⊂ a,∞for somea > 0. Then,{xn}converges strongly toΠFT∩EPfx0.

Proof. Since every relatively nonexpansive mapping is a hemirelatively one,Corollary 3.4is implied byTheorem 3.1.

Remark 3.5see Rockafellar 12. LetEbe a reflexive, strictly convex, and smooth Banach space and letA be a monotone operator fromE to E∗. Then,A is maximal if and only if

RJrA E∗_{for all}_{r >}_0.

LetEbe a reflexive, strictly convex, and smooth Banach space and letAbe a maximal monotone operator fromEtoE∗. UsingRemark 3.5and strict convexity ofE, we obtain that for everyr > 0 andx ∈ E, there exists a uniquexr ∈ DAsuch thatJx ∈ Jxr rAxr.If

Jrx xr, then we can define a single-valued mappingJr :E → DAbyJr JrA−1J, and such a Jr is called the resolvent ofA. We know thatA−10 FJrfor allr > 0 andJr is relatively nonexpansive mappingsee2for more details. UsingTheorem 3.1, we can consider the problem of strong convergence concerning maximal monotone operators in a Banach space.

Theorem 3.6. Let Ebe a uniformly convex and uniformly smooth real Banach space, let Cbe a nonempty closed convex subset ofE, letfbe a bifunction fromC×CtoRsatisfying (A1)–(A4), and letJr be a resolvent ofAand a closed mapping such thatA−10∩EPf/∅, wherer >0. Define a sequence{xn}inCby the following:

ynJ−1αnJxn 1−αnJJrxn,

znJ−1βnJxn 1−βnJJrxn,

3.39

for everyn∈N∪ {0}, whereJis the duality mapping onE,{αn}is a sequences in0,1such that lim inf_{n→ ∞}1−αnβn1−βn>0and{rn} ⊂a,∞for somea >0, Then,{xn}converges strongly toΠA−1₀_∩_EP_fx₀.

Proof. Since Jr is a closed relatively nonexpansive mapping and A−10 FJr, from

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Acknowledgment

This work is supported by Tianjin Natural Science Foundation in China Grant no. 06YFJMJC12500.

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