Strong and Weak Convergence Theorems for Common Solutions of Generalized Equilibrium Problems and Zeros of Maximal

Fixed Point Theory and Applications Volume 2010, Article ID 590278,33pages doi:10.1155/2010/590278

Research Article

Strong and Weak Convergence Theorems for Common Solutions of Generalized Equilibrium Problems and Zeros of Maximal

Monotone Operators

L.-C. Zeng,

Q. H. Ansari,

David S. Shyu,

and J.-C. Yao

1Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

2Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China

3Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India

4Department of Finance, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan

5Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan

Correspondence should be addressed to J.-C. Yao,[email protected] Received 27 October 2009; Accepted 12 January 2010

Academic Editor: Tomonari Suzuki

Copyrightq2010 L.-C. Zeng 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.

The purpose of this paper is to introduce and study two modified hybrid proximal-point algorithms for finding a common element of the solution set EP of a generalized equilibrium problem and the setT⁻¹0∩T⁻¹0 for two maximal monotone operatorsTandTdefined on a Banach spaceX. Strong and weak convergence theorems for these two modified hybrid proximal-point algorithms are established.

1. Introduction

LetXbe a real Banach space with its dualX^∗. The mappingJ :X → 2^X∗defined by Jx:

x^∗∈X^∗:x^∗, xx²x^∗2

, ∀x∈X, 1.1

is called the normalized duality mapping. From the Hahn-Banach theorem, it follows that Jx/∅for eachx∈X.

A Banach spaceXis said to be strictly convex, ifxy/2<1 for allx, y∈U{z∈ X : z 1}withx /y.X is said to be uniformly convex if for each ∈ 0,2, there exists

δ >0 such thatxy/2≤1−δfor allx, y∈Uwithx−y ≥. Recall that each uniformly convex Banach space has the Kadec-Klee property, that is,

x_n x

xn −→ x ⇒x_n−→x. 1.2

It is well known that ifX^∗is strictly convex, thenJis single-valued. In the sequel, we shall still denote the single-valued normalized duality mapping byJ. LetCbe a nonempty closed convex subset ofX,f :C×C → Ra bifunction, andA:C → X^∗a nonlinear mapping.

Very recently, Zhang 1 considered and studied the generalized equilibrium problem of findingx∈Csuch that

f x, y

Ax, y −x ≥0, ∀y∈C. 1.3

The set of solutions of 1.3is denoted by EP. Problem 1.3and related problems have been studied and investigated extensively in the literature; See, for example,2–12and references therein. WheneverA ≡ 0, problem 1.3 reduces to the equilibrium problem of findingx∈Csuch that

f x, y

≥0, ∀y∈C. 1.4

The set of solutions of1.4is denoted byEPf. Wheneverf ≡0, problem1.3reduces to the variational inequality problem of findingx∈Csuch that

Ax, y −x ≥ 0, ∀y∈C. 1.5

The set of solutions of1.5is denoted byV IC, A.

WheneverX Ha Hilbert space, problem 1.3was very recently introduced and considered by S. Takahashi and W. Takahashi13. Problem1.3is very general in the sense that it includes, as spacial cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games, and others; See, for example, 1,2,4,6–9,14–17which are references therein.

A mappingS:C → Xis called nonexpansive ifSx−Sy ≤ x−yfor allx, y ∈C.

Denote byFSthe set of fixed points ofS, that is,FS {x∈ C: Sx x}. Very recently, W. Takahashi and K. Zembayashi18proposed an iterative algorithm for finding a common element of the solution set of the equilibrium problem1.4and the set of fixed points of a relatively nonexpansive mappingS in a Banach spaceX. They also studied the strong and weak convergence of the sequences generated by their algorithm. In particular, they proposed the following iterative algorithm:

x₀∈C,

y_nJ⁻¹αnJx_n 1−α_nJSxn,

u_n∈Csuch thatf u_n, y

1 r_n

y−u_n, Ju_n−Jy_n ≥0, ∀y∈C,

H_n

z∈C:φz, u_n≤φz, x_n , W_n{z∈C:xn−z, Jx−Jx_n ≥0},

x_n1 ΠHn∩Wnx, n≥0,

1.6 whereφx, y x²−2x, Jyy² for allx, y ∈ X,{αn} ⊂ 0,1, and{rn} ⊂ a,∞for somea >0. They proved that the sequence{xn}generated by the above algorithm converges strongly to ΠFS∩EPfx0, where ΠFS∩EPf is the generalized projection of X onto FS∩ EPf. They have also studied the weak convergence of the sequence{xn}generated by the following algorithm:

u₀ ∈X, xn∈Csuch thatf

xn, y 1

rn

y−xn, Jxn−Jun ≥0, ∀y∈C,

u_n1J⁻¹αnJx_n 1−α_nJSxn, n≥0,

1.7

toz∈FS∩EPf, wherezlim_{n→ ∞}Π_FS∩EPfx_n.

Let C be a nonempty closed convex subset of a uniformly smooth and uniformly convex Banach spaceX. LetA :C → X^∗ be anα-inverse-strongly monotone mapping and f:C×C → Ra bifunction satisfying the following conditions:

A1fx, x 0 for allx∈C;

A2fis monotone, that is,fx, y fy, x≤0, for allx, y∈C;

A3for allx, y, z∈C, lim sup_t↓0ftz 1−tx, y≤fx, y;

A4for allx∈C,fx,·is convex and lower semicontinuous.

LetS1, S2:C → Cbe two relatively nonexpansive mappings such thatFS1∩FS2∩ EP /∅. Let{xn}be the sequence generated by

x₀∈C, C₀C;

znJ⁻¹αnJxn 1−αnJS1xn, y_nJ⁻¹

β_nJx_n 1−β_n

JS₂z_n , un∈Csuch thatf

un, y

Aun, y−un 1 r_n

y−un, Jun−Jyn ≥0, ∀y∈C,

C_n1

v∈C_n:φv, un≤β_nφv, xn 1−β_n

φv, zn≤φv, xn

; x_n1 ΠC_n1x₀, ∀n≥0.

1.8

Zhang 1 proved the strong convergence of the sequence {xn} toΠFS1∩FS2∩EPx0 under appropriate conditions.

On the other hand, a classic method of solving 0 ∈ Tx in a Hilbert space H is the proximal point algorithm which generates, for any starting pointx₀ ∈H, a sequence{xn}in Hby the iterative scheme

x_n1Jrnxn, n0,1,2, . . . , 1.9

where{rn} is a sequence in0,∞,Jr IrT⁻¹ for eachr > 0 is the resolvent operator forT, andI is the identity operator onH. This algorithm was first introduced by Martinet 19and further studied by Rockafellar20in the framework of a Hilbert space H. Later several authors studied1.9and its variants in the setting of a Hilbert spaceHor in a Banach spaceX; See, for example,15,21–25and references therein. Very recently, Li and Song24 introduced and studied the following iterative scheme:

x₀∈X chosen arbitrarily, y_nJ⁻¹

β_nJx_n 1−β_n

JJ_rnx_n , x_n1J⁻¹

αnJx₀ 1−αnJyn

, n0,1,2, . . . ,

1.10

whereJr JrT⁻¹JandJis the duality mapping onX.

Algorithm1.10covers, as special cases, the algorithms introduced by Kohsaka and Takahashi23and Kamimura et al.22in a smooth and uniformly convex Banach spaceX.

Let X be a uniformly smooth and uniformly convex Banach space, and let C be a nonempty closed convex subset ofX. LetT :X → 2^X∗be a maximal monotone operator such that:

A5T⁻¹0∩EPf/∅.

In addition, for eachr >0, define a mappingTr :X → Cas follows:

T_rx

z∈C:f z, y

1 r

y−z, Jz−Jx ≥0, ∀y∈C

1.11

for allx∈X.

Very recently, utilizing the ideas of the above algorithms in15,16,18,21,22,24, we 17introduced two iterative methods for finding an element ofT⁻¹0∩EPfand established the following strong and weak convergence theorems.

Theorem 1.1see17. Suppose that conditions (A1)–(A5) are satisfied and letx₀ ∈X be chosen arbitrarily. Consider the sequence

x_n1 ΠHn∩Wnx0, n0,1,2, . . . , 1.12

where

Hn

z∈C:φ

z, Trnyn

≤αnφz, x0 1−αnφz, xn , Wn{z∈C:xn−z, Jx₀−Jxn ≥0},

y_nJ⁻¹

αnJx₀ 1−αn

βnJxn 1−βn

JJrnxn ,

1.13

Tr is defined by 1.11,{αn},{βn} ⊂ 0,1 satisfy lim_{n→ ∞}αn 0, lim infn→ ∞βn1−βn > 0, and {rn} ⊂ 0,∞ satisfies lim infn→ ∞rn > 0. Then, the sequence {xn} converges strongly to ΠT⁻¹0∩EPfx₀, whereΠT⁻¹0∩EPfis the generalized projection ofXontoT⁻¹0∩EPf.

Theorem 1.2see17. Suppose that conditions (A1)–(A5) are satisfied and letx₀ ∈X be chosen arbitrarily. Consider the sequence

x_n1J⁻¹

αnJx0 1−αn

βnJTrnxn 1−βn

JJrnTrnxn

, n0,1,2, . . . , 1.14 where T_r is defined by 1.11, {αn},{βn} ⊂ 0,1 satisfy the conditions _∞

n0α_n < ∞ and lim inf_{n→ ∞}βn1 − βn > 0, and {rn} ⊂ 0,∞ satisfies lim infn→ ∞rn > 0. If J is weakly sequentially continuous, then {xn} converges weakly to an element z ∈ T⁻¹0 ∩ EPf, where zlim_{n→ ∞}ΠT⁻¹0∩EPfx_n.

The purpose of this paper is to introduce and study two new iterative methods for finding a common element of the solution setEP of generalized equilibrium problem1.3 and the setT⁻¹0 ∩T⁻¹0 for maximal monotone operators T and T in a uniformly smooth and uniformly convex Banach space X. Firstly, motivated by Theorem 1.1 and a result of Zhang 1, we introduce a sequence{xn} that converges strongly toΠ_T−10∩T⁻¹0∩EPx₀ under some appropriate conditions.

Secondly, inspired byTheorem 1.2and a result of Zhang 1, we define a sequence that converges weakly to an elementz∈T⁻¹0∩T⁻¹0∩EP, wherezlimn→ ∞Π_T−10∩T⁻¹0∩EPxn

Section 4.

Our results represent a generalization of known results in the literature, including those in 16–18, 24. Our Theorems 3.1 and 4.2 are the extension and improvements of Theorems1.1and1.2in the following way:

ithe problem of finding an element ofT⁻¹0∩T⁻¹0∩EP includes the one of finding an element ofT⁻¹0∩EPfas a special case;

iithe algorithms in this paper are very different from those in 17 because of considering the complexity involving the problem of finding an element ofT⁻¹0∩ T⁻¹0∩EP.

2. Preliminaries

Throughout the paper, we denote the strong convergence, weak convergence, and weak^∗ convergence of a sequence{xn}to a pointx∈Xbyx_n → x,x_n xandx_n x, respectively.^∗ Assumption 2.1. LetX be a uniformly smooth and uniformly convex Banach space and letC be a nonempty closed convex subset ofX. LetA:C → X^∗be anα-inverse-strongly monotone

mapping and letf : C×C → Rbe a bifunction satisfying the conditions A1–A4. Let T,T:X → 2^X∗be two maximal monotone operators such that:

A5T⁻¹0∩T⁻¹0∩EP /∅.

Recall that if Cis a nonempty closed convex subset of a Hilbert space H, then the metric projectionP_C :H → CofHontoCis nonexpansive. This fact actually characterizes Hilbert spaces and hence, it is not available in more general Banach spaces. In this connection, Alber 26 recently introduced a generalized projection operatorΠC in a Banach space X which is an analogue of the metric projection in Hilbert spaces.

Consider the functional defined as in26by φ

x, y

x²−2

x, Jy y², ∀x, y∈X. 2.1 It is clear that in a Hilbert spaceH,2.1reduces toφx, y x−y², ∀x, y∈H.

The generalized projectionΠC :X → Cis a mapping that assigns to an arbitrary point x∈Xthe minimum point of the functionalφy, x, that is,ΠCxx, wherexis the solution to the minimization problem

φx, x min

y∈C φ y, x

. 2.2

The existence and uniqueness of the operatorΠC follow from the properties of the functional φx, y and strict monotonicity of the mapping J; See, for example, 27. In a Hilbert space,ΠC P_C. From26, in a smooth, strictly convex and reflexive Banach spaceX, we have

y− x2≤φ y, x

≤yx2, ∀x, y∈X. 2.3 Moreover, by the property of subdifferential of convex functions, we easily get the following inequality:

φ x, y

≤φ x, J⁻¹

JyJz

−2

y−x, Jz , ∀x, y, z∈X. 2.4

LetSbe a mapping fromCinto itself. A pointpinCis called an asymptotic fixed point ofS28ifCcontains a sequence{xn}which converges weakly topsuch thatSxn−xn → 0.

The set of asymptotic fixed points ofSis denoted byFS. A mapping SfromSinto itself is called relatively nonexpansive18,29,30ifFS FSandφp, Sx≤φp, x, for allx∈C andp∈FS.

Observe that, ifXis a reflexive, strictly convex and smooth Banach space, then for any x, y∈X, φx, y 0 if and only ifxy. To this end, it is sufficient to show that ifφx, y 0, thenxy. Actually, from2.3, we havexy, which implies thatx, Jyx²y². From the definition ofJ, we haveJxJyand therefore,xy. For further details, we refer to31.

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

Lemma 2.2see32. LetXbe a smooth and uniformly convex Banach space and let{xn}and{yn} be two sequences ofX. Ifφxn, y_n → 0 and either{xn}or{yn}is bounded, thenxn−y_n → 0.

Lemma 2.3see26,32. LetCbe a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach spaceX,x∈Xandz∈C. Then

z ΠCx⇐⇒

y−z, Jx−Jz ≤0, ∀y∈C. 2.5

Lemma 2.4see26,32. LetCbe a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach spaceX. Then

φ

x,ΠCy φ

ΠCy, y

≤φ x, y

, ∀x∈C, y∈X. 2.6

Lemma 2.5 see33. LetX be a reflexive, strictly convex and smooth Banach space and letT : X → 2^X∗be a multivalued operator. Then

iT⁻¹0 is closed and convex ifTis maximal monotone such thatT⁻¹0/∅;

iiT is maximal monotone if and only ifTis monotone withRJrT X^∗for allr >0.

Lemma 2.6see34. LetXbe a uniformly convex Banach space and letr >0. Then there exists a strictly increasing, continuous and convex functiong:0,2r → Rsuch thatg0 0 and

tx 1−ty²≤tx² 1−ty²−t1−tgx−y, 2.7

for allx, y∈Br andt∈0,1, whereBr {z∈X:z ≤r}.

Lemma 2.7see32. LetXbe a smooth and uniformly convex Banach space and letr >0. Then there exists a strictly increasing, continuous, and convex functiong :0,2r → Rsuch thatg0 0 and

gx−y≤φ x, y

, ∀x, y∈B_r. 2.8

The following result is due to Blum and Oettli14.

Lemma 2.8see14. LetCbe a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach spaceX,f :C×C → Ra bifunction satisfying conditions (A1)–(A4), andr >0 andx∈X. Then, there existsz∈Csuch that

f z, y

1 r

y−z, Jz−Jx ≥0, ∀y∈C. 2.9

Motivated by a result in35 in a Hilbert space setting, Takahashi and Zembayashi 18established the following lemma.

Lemma 2.9 see18. LetCbe a nonempty closed convex subset of a uniformly smooth, strictly convex and reflexive Banach spaceX, andf :C×C → Ra bifunction satisfying conditions (A1)–

(A4). Forr >0 andx∈X, define a mappingTr :X → Cas follows:

T_rx

z∈C:f z, y

1 r

y−z, Jz−Jx ≥0, ∀y∈C

2.10

for allx∈X. Then

iTr is single-valued;

iiTr is a firmly nonexpansive-type mapping, that is, for allx, y∈X, Trx−Try, JTrx−JTry ≤

Trx−Try, Jx−Jy ; 2.11

iiiFTr FT r EPf;

ivEPfis closed and convex.

UsingLemma 2.9, we have the following result.

Lemma 2.10see18. LetCbe a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach spaceX,f :C×C → Ra bifunction satisfying conditions (A1)–(A4), andr >0.

Then, forx∈Xandq∈FTr, φ

q, T_rx

φTrx, x≤φ q, x

. 2.12

Utilizing Lemmas2.8,2.9, and2.10, Zhang1derived the following result.

Proposition 2.11see1. LetX be a smooth, strictly convex and reflexive Banach space and let Cbe a nonempty closed convex subset ofX. LetA : C → X^∗ be anα-inverse-strongly monotone mapping,f:C×C → Ra bifunction satisfying conditions (A1)–(A4), andr >0. Then

Iforx∈X, there existsu∈Csuch that f

u, y

Au, y−u 1 r

y−u, Ju−Jx ≥0, ∀y∈C; 2.13

IIifXis additionally uniformly smooth andKr:C → Cis defined as Krx

u∈C:f u, y

Au, y−u 1 r

y−u, Ju−Jx ≥0, ∀y∈C

, ∀x∈C, 2.14

then the mappingKrhas the following properties:

iKris single-valued,

iiK_ris a firmly nonexpansive-type mapping, that is, Krx−Kry, JKrx−JKry ≤

Krx−Kry, Jx−Jy , ∀x, y∈X, 2.15

iiiFKr FK r EP,

ivEPis a closed convex subset ofC,

vφp, Krx φKrx, x≤φp, x,for allp∈FKr.

Proof. Define a bifunctionF:C×C → Rby F

x, y f

x, y

Ax, y−x , ∀x, y∈C. 2.16

It is easy to verify thatFsatisfies the conditionsA1–A4. Therefore, the conclusionsIand IIfollow immediately from Lemmas2.8,2.9, and2.10.

LetT,T:X → 2^X∗be two maximal monotone operators in a smooth Banach spaceX.

We denote the resolvent operators ofTandTbyJ_r JrT⁻¹JandJ_r JrT ⁻¹Jfor each r >0, respectively. ThenJr :X → DTandJr :X → DT are two single-valued mappings.

Also,T⁻¹0FJrandT⁻¹0FJrfor eachr >0, whereFJrandFJrare the sets of fixed points ofJr andJr, respectively. For eachr > 0, the Yosida approximations ofT andTare defined byAr J−JJr/randAr J−JJr/r, respectively. It is known that

Arx∈TJrx, Arx∈T Jrx

, for eachr >0, x∈X. 2.17

Lemma 2.12see23. LetXbe a reflexive, strictly convex and smooth Banach space, and letT : X → 2^X∗be a maximal monotone operator withT⁻¹0/∅. Then,

φz, Jrx φJrx, x≤φz, x, ∀r >0, z∈T⁻¹0, x∈X. 2.18

Lemma 2.13see36. Let{an}and{bn}be two sequences of nonnegative real numbers such that a_n1≤anbnfor alln≥0. If_∞

n0bn<∞, then limn→ ∞anexists.

3. Strong Convergence Theorem

In this section, we prove a strong convergence theorem for finding a common element of the set of solutions for a generalized equilibrium problem and the setT⁻¹0∩T⁻¹0 for two maximal monotone operatorsTandT.

Theorem 3.1. Suppose thatAssumption 2.1is satisfied. Letx₀ ∈X be chosen arbitrarily. Consider the sequence

x_n1 ΠHn∩Wnx0, n0,1,2, . . . , 3.1

where Hn

z∈C:φ

z, Krnyn

≤αnαn−αnαnφz, x0 1−αn1−αnφz, xn , Wn{z∈C:xn−z, Jx₀−Jxn ≥0},

x_nJ⁻¹

α_nJx₀ 1−α_n

β_nJx_n 1−β_n

JJ_rnx_n , ynJ⁻¹

αnJx0 1−αn

βnJxn 1−βn

JJrnxn

,

3.2

K_r is defined by2.14,{αn},{βn},{α_n},{β_n} ⊂0,1satisfy

nlim→ ∞αn0, lim

n→ ∞αn0, lim inf

n→ ∞ βn 1−βn

>0, lim inf

n→ ∞ βn

1−βn

>0, 3.3

and {rn} ⊂ 0,∞ satisfies lim infn→ ∞rn > 0. Then, the sequence {xn} converges strongly to Π_T−10∩T⁻¹0∩EPx₀, whereΠ_T−10∩T⁻¹0∩EPis the generalized projection ofXontoT⁻¹0∩T⁻¹0∩EP. Proof. For the sake of simplicity, we define

un:Krnyn, zn:J⁻¹

βnJxn 1−βn

JJrnxn

, zn:J⁻¹

βnJxn 1−βn

JJrnxn

, 3.4 so that

xnJ⁻¹αnJx0 1−αnJzn, ynJ⁻¹αnJx0 1−αnJzn. 3.5 We divide the proof into several steps.

Step 1. We claim thatH_n∩W_nis closed and convex for eachn≥0.

Indeed, it is obvious thatH_nis closed andW_nis closed and convex for eachn≥0. Let us show thatHnis convex. Forz₁, z₂∈Hnandt∈0,1, putztz₁ 1−tz2. It is sufficient to show thatz∈Hn. We first writeγnαnαn−αnαnfor eachn≥0. Next, we prove that

φz, un≤γnφz, x0

1−γn

φz, xn 3.6

is equivalent to 2γnz, Jx02

1−γn

z, Jxn −2z, Jun ≤γnx0² 1−γn

xn²− un². 3.7 Indeed, from2.1we deduce that there hold the following:

φz, x0 z²−2z, Jx0x0², φz, xn z²−2z, Jxnxn², φz, un z²−2z, Junun²,

3.8

which combined with3.6yield that3.6is equivalent to3.7. Thus we have

2γnz, Jx02 1−γn

z, Jxn −2z, Jun 2γ_ntz1 1−tz2, Jx₀2

1−γ_n

tz1 1−tz2, Jx_n

−2tz1 1−tz2, Ju_n

2tγ_nz1, Jx₀21−tγnz2, Jx₀2 1−γ_n

tz1, Jx_n 2

1−γ_n

1−tz₂, Jx_n −2tz₁, Ju_n −21−tz₂, Ju_n

≤γnx0² 1−γn

xn²− un².

3.9

This implies thatz∈H_n. Therefore,H_nis closed and convex.

Step 2. We claim thatT⁻¹0∩T⁻¹0∩EP⊂H_n∩W_nfor eachn≥0 and that{xn}is well defined.

Indeed, takew∈T⁻¹0∩T⁻¹0∩EParbitrarily. Note thatunKrnynis equivalent to

un∈Csuch thatf un, y

Aun, y−un 1 rn

y−un, Jun−Jyn ≥0, ∀y∈C. 3.10

Then fromLemma 2.12, we obtain

φw, zn φ w, J⁻¹

βnJxn 1−βn

JJrnxn w²−2

w, β_nJx_n 1−β_n

JJ_rnx_n β_nJx_n 1−β_nJJrnx_n²

≤ w²−2βnw, Jxn −2 1−βn

w, JJrnxnβnxn² 1−βn

Jrnxn² βnφw, xn

1−βn

φw, Jrnxn

≤βnφw, xn 1−βn

φw, xn φw, xn, φw,xn φ

w, J⁻¹αnJx₀ 1−αnJzn

w²−2w, αnJx₀ 1−α_nJznαnJx₀ 1−α_nJzn²

≤ w²−2α_nw, Jx0 −21−α_nw, Jz_nα_nx₀² 1−α_nz_n² αnφw, x0 1−αnφw, zn

≤αnφw, x0 1−αnφw, xn.

3.11

Moreover, we have φw,zn φ

w, J⁻¹

βnJxn 1−βn

JJrnxn

≤βnφw,xn

1−βn

φ

w,Jrnxn

≤βnφw,xn

1−βn

φw,xn φw,xn,

φ w, yn

φ

w, J⁻¹αnJx₀ 1−αnJzn

≤ w²−2αnw, Jx0 −21−αnw, Jznαnx0² 1−αnzn² α_nφw, x0 1−α_nφw,z_n

≤αnφw, x0 1−αnφw,xn

≤αnφw, x0 1−αn

αnφw, x0 1−αnφw, xn

α_n 1−α_nαnφw, x0 1−α_n1−α_nφw, xn

≤αnαn−αnαnφw, x0 1−αn1−αnφw, xn,

3.12

and hence byProposition 2.11, φw, un φ

w, K_rny_n

≤φ w, y_n

≤αnαn−αnαnφw, x0 1−αn1−αnφw, xn. 3.13 Sow∈Hnfor alln≥0. Now, let us show that

T⁻¹0∩T⁻¹0∩EP ⊂Wn ∀n≥0. 3.14 We prove this by induction. Forn 0, we haveT⁻¹0∩T⁻¹0∩EP ⊂ C W₀. Assume that T⁻¹0∩T⁻¹0∩EP ⊂Wn. Sincex_n1is the projection ofx0ontoHn∩Wn, byLemma 2.3we have xn1−z, Jx₀−Jx_n1 ≥0, ∀z∈Hn∩Wn. 3.15 AsT⁻¹0∩T⁻¹0∩EP ⊂ Hn∩Wn by the induction assumption, the last inequality holds, in particular, for allz∈T⁻¹0∩T⁻¹0∩EP. This, together with the definition ofW_n1implies that T⁻¹0∩T⁻¹0∩EP ⊂W_n1. Hence3.14holds for alln≥0. So,T⁻¹0∩T⁻¹0∩EP⊂H_n∩W_nfor alln≥0. This implies that the sequence{xn}is well defined.

Step 3. We claim that{xn}is bounded and thatφx_n1, x_n → 0 asn → ∞.

Indeed, it follows from the definition of W_n that x_n ΠWnx₀. Since x_n ΠWnx₀ and x_n1 ΠHn∩Wnx₀ ∈ Wn, soφxn, x₀ ≤ φxn1, x₀ for alln ≥ 0, that is,{φxn, x₀} is nondecreasing. It follows fromxn ΠWnx0andLemma 2.4that

φxn, x0 φΠWnx0, x0≤φ p, x0

−φ p, xn

≤φ p, x0

3.16

for eachp∈T⁻¹0∩T⁻¹0∩EP ⊂Wnfor eachn≥0. Therefore,{φxn, x0}is bounded, which implies that the limit of{φxn, x₀}exists. Since

xn − x0²≤φxn, x0≤xnx0², ∀n≥0, 3.17

so{xn}is bounded. FromLemma 2.4, we have

φxn1, xn φxn1,ΠWnx0≤φxn1, x0−φΠWnx0, x0

φx_n1, x₀−φxn, x₀, 3.18

for eachn≥0. This implies that

nlim→ ∞φx_n1, x_n 0. 3.19

Step 4. We claim that limn→ ∞xn−un0, limn→ ∞xn−Jrnxn0, and limn→ ∞xn−Jrnxn 0.

Indeed, fromx_n1 ΠHn∩Wnx₀∈H_n, we have

φxn1, un≤αnαn−αnαnφxn1, x₀ 1−αn1−αnφxn1, xn, ∀n≥0. 3.20

Therefore, fromα_n → 0α_n → 0 andφx_n1, x_n → 0, it follows that limn→ ∞φx_n1, u_n 0.

Since lim_{n→ ∞}φx_n1, x_n lim_{n→ ∞}φx_n1, u_n 0 and X is uniformly convex and smooth, we have fromLemma 2.2that

nlim→ ∞x_n1−x_n lim

n→ ∞x_n1−u_n0, 3.21

and, therefore, lim_{n→ ∞}xn −u_n 0. Since J is uniformly norm-to-norm continuous on bounded subsets ofXandxn−un → 0, then limn→ ∞Jxn−Jun0.

Let us setΩ:T⁻¹0∩T⁻¹0∩EP. Then, according toLemma 2.5andProposition 2.11, we know thatΩis a nonempty closed convex subset ofXsuch thatΩ⊂C. Fixu∈Ωarbitrarily.

As in the proof ofStep 2, we can show thatφu, zn≤φu, xn, φu,xn≤αnφu, x0 1−αnφu, xn,

φu,z_n≤φu,x_n, φ

u, y_n

≤αnα_n−α_nα_nφu, x0 1−α_n1−α_nφu, xn, φu, un≤αnαn−αnαnφu, x0 1−αn1−αnφu, xn.

3.22

Hence it follows from the boundedness of{xn}that{zn},{xn},{zn},{yn}, and{un}are also bounded. Letr sup{xn,xn,Jrnxn,Jrnxn : n ≥ 0}. SinceX is a uniformly smooth Banach space, we know thatX^∗is a uniformly convex Banach space. Therefore, byLemma 2.6 there exists a continuous, strictly increasing, and convex functiong, withg0 0, such that

αx^∗ 1−αy^∗2≤αx^∗2 1−αy^∗2−α1−αgx^∗−y^∗, 3.23

forx^∗, y^∗∈B_r^∗andα∈0,1. So, we have that

φu, z_n φ u, J⁻¹

β_nJx_n 1−β_n

JJ_rnx_n

u²−2

u, βnJxn 1−βn

JJrnxn βnJxn 1−βnJJrnxn²

≤ u²−2β_nu, Jxn −2 1−β_n

u, JJrnx_n βnxn²

1−βn

Jrnxn²−βn 1−βn

gJxn−JJrnxn βnφu, xn

1−βn

φu, Jrnxn−βn 1−βn

gJxn−JJrnxn

≤βnφu, xn 1−βn

φu, xn−βn 1−βn

gJxn−JJrnxn φu, xn−β_n

1−β_n

gJxn−JJ_rnx_n,

φu,zn φ u, J⁻¹

βnJxn 1−βn

JJrnxn

u²−2

u,β_nJx_n 1−β_n

JJ_rnx_n

β_nJx_n 1−β_nJJ_rnx_n²

≤ u²−2βnu, Jxn −2 1−βn

u, JJrnxn

β_nx_n²

1−β_nJ_rnx_n²−β_n 1−β_n

gJx_n−JJ_rnx_n βnφu,xn

1−βn

φ u,Jrnxn

−βn

1−βn

gJxn−JJrnxn

≤βnφu,xn

1−βn

φu,xn−βn

1−βn

gJxn−JJrnxn φu,x_n−β_n

1−β_n

gJx_n−JJ_rnx_n ,

3.24

and hence φu,xn φ

u, J⁻¹αnJx₀ 1−αnJzn

u²−2u, αnJx₀ 1−αnJznαnJx₀ 1−αnJzn²

≤ u²−2α_nu, Jx0 −21−α_nu, Jznα_nx₀² 1−α_nz_n² αnφu, x0 1−αnφu, zn

≤αnφu, x0 1−αn

φu, xn−βn 1−βn

gJxn−JJrnxn αnφu, x0 1−αnφu, xn−1−αnβn

1−βn

gJxn−JJrnxn, φu, un φ

u, Krnyn

≤φ

u, yn using Proposition 2.10 φ

u, J⁻¹α_nJx₀ 1−α_nJz_n

u²−2u,α_nJx₀ 1−α_nJz_nα_nJx₀ 1−α_nJz_n²

≤ u²−2αnu, Jx0 −21−αnu, Jznαnx0² 1−αnzn² α_nφu, x0 1−α_nφu,z_n

≤α_nφu, x0 1−α_n

φu,x_n−β_n 1−β_n

gJx_n−JJ_rnx_n αnφu, x0 1−αnφu,xn−1−αnβn

1−βn

gJxn−JJrnxn

≤αnφu, x0 φu,xn,

3.25

for alln≥0. Consequently, we have 1−α_nβn

1−β_n

gJxn−JJ_rnx_n

≤αnϕu, x0 1−αnϕu, xn−ϕu,xn

≤α_nϕu, x0 ϕu, xn−ϕu,x_n

αnϕu, x0 ϕu, xn−ϕu, un ϕu, un−ϕu,xn

αnϕu, x0 xn²− un²−2u, Jxn−Junϕu, un−ϕu,xn

≤αnϕu, x0 xn²− un²2|u, Jxn−Jun|ϕu, un−ϕu,xn

≤α_nϕu, x0 α_nϕu, x0 |xn − un|xnun 2uJxn−Ju_n

≤αnαnϕu, x0 xn−unxnun 2uJxn−Jun,

3.26