Hybrid Proximal-Type Algorithms for Generalized Equilibrium Problems, Maximal Monotone

Volume 2011, Article ID 973028,23pages doi:10.1155/2011/973028

Research Article

Hybrid Proximal-Type Algorithms for Generalized Equilibrium Problems, Maximal Monotone

Operators, and Relatively Nonexpansive Mappings

Lu-Chuan Zeng,

Q. H. Ansari,

and S. Al-Homidan

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

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

3Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals (KFUPM), P.O. Box 119, Dhahran 31261, Saudi Arabia

Correspondence should be addressed to S. Al-Homidan,homidan@kfupm.edu.sa Received 24 September 2010; Accepted 18 October 2010

Academic Editor: Jen Chih Yao

Copyrightq2011 Lu-Chuan 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 consider new hybrid proximal-type algorithms for finding a common element of the set EP of solutions of a generalized equilibrium problem, the setFSof fixed points of a relatively nonexpansive mappingS, and the setT⁻¹0 of zeros of a maximal monotone operatorTin a uniformly smooth and uniformly convex Banach space. Strong convergence theorems for these hybrid proximal-type algorithms are established; that is, under appropriate conditions, the sequences generated by these various algorithms converge strongly to the same point in EP∩FS∩T⁻¹0. These new results represent the improvement, generalization, and development of the previously known ones in the literature.

1. Introduction

LetEbe a real Banach space with the dualE^∗andCbe a nonempty closed convex subset of E. We denote byNandRthe sets of positive integers and real numbers, respectively. Also, we denote byJthe normalized duality mapping fromEto 2^E∗defined by

Jx

x^∗∈E^∗:x, x^∗x²x^∗2

, ∀x∈E, 1.1

where ·,· denotes the generalized duality pairing. Recall that if E is smooth, then J is single valued and ifEis uniformly smooth, thenJ is uniformly norm-to-norm continuous on bounded subsets ofE. We will still denote byJ the single valued duality mapping. Let

f : C×C → Rbe a bifunction andA: C → E^∗be a nonlinear mapping. We consider the following generalized equilibrium problem:

findu∈Csuch thatf u, y

Au, y−u

≥0, ∀y∈C. 1.2

The set of suchu∈Cis denoted by EP, that is, EP

u∈C:f u, y

Au, y−u

≥0, ∀y∈C . 1.3 WheneverEHa Hilbert space, problem1.2was introduced and studied by S. Takahashi and W. Takahashi 1. Similar problems have been studied extensively recently. See, for example,2–11. In the case ofA ≡ 0,EP is denoted by EPf. In the case of f ≡ 0, EP is also denoted by VIC, A. The problem1.2is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games and others; see, for example, 12–14. A mappingS:C → Eis called nonexpansive ifSx−Sy ≤ x−yfor allx, y∈C. Denote by FSthe set of fixed points ofS, that is,FS {x∈C:Sxx}. A mappingA:C → E^∗is calledα-inverse-strongly monotone, if there exists anα >0 such that

Ax−Ay, x−y

≥αAx−Ay², ∀x, y∈C. 1.4 It is easy to see that ifA : C → E^∗is anα-inverse-strongly monotone mapping, then it is 1/α-Lipschitzian.

LetEbe a real Banach space with the dualE^∗. A multivalued operatorT :E → 2^E∗ with domainDT {z ∈ E : Tz /∅}is called monotone ifx1−x2, y1 −y2 ≥0 for each x_i ∈ DT and y_i ∈ Tx_i, i 1,2. A monotone operator T is called maximal if its graph GT {x, y : y ∈ Tx} is not properly contained in the graph of any other monotone operator. A method for solving the inclusion 0∈Txis the proximal point algorithm. Denote byIthe identity operator onEHa Hilbert space. The proximal point algorithm generates, for any initial pointx0x∈H, a sequence{x_n}inH, by the iterative scheme

x_n1 Ir_nT⁻¹x_n, n0,1,2, . . . , 1.5 where{r_n}is a sequence in the interval0,∞. Note that this iteration is equivalent to

0∈Tx_n1 1

r_nx_n1−x_n, n0,1,2, . . . . 1.6 This algorithm was first introduced by Martinet12and generally studied by Rockafellar 15in the framework of a Hilbert space. Later many authors studied its convergence in a Hilbert space or a Banach space. See, for instance,16–21and the references therein.

LetEbe a reflexive, strictly convex, and smooth Banach space with the dualE^∗andC be a nonempty closed convex subset ofE. LetT :E → 2^E∗ be a maximal monotone operator with domainDT CandS :C → Cbe a relatively nonexpansive mapping. LetA:C → X^∗be anα-inverse-strongly monotone mapping andf:C×C → Rbe a bifunction satisfying

A1–A4:A1 fx, x 0, ∀x ∈ C; A2 f is monotone, that is, fx, y fy, x ≤ 0,

∀x, y ∈ C;A3limsup_t↓0fxtz−x, y ≤ fx, y,∀x, y, z ∈ C;A4 the functiony → fx, yis convex and lower semicontinuous. The purpose of this paper is to introduce and investigate two new hybrid proximal-type Algorithms1.1and1.2for finding an element of EP∩ FS∩ T⁻¹0.

Algorithm 1.1.

x0∈Carbitrarily chosen, 0v_n 1

r_nJx_n−Jx_n, v_n∈Tx_n, znJ⁻¹

βnJxn 1−βn

JSxn , y_nJ⁻¹α_nJx_n 1−α_nJSz_n,

u_n∈Csuch that f

un, y

Aun, y−un 1

rn

y−un, Jun−Jyn

≥0, ∀y∈C, Hn

v∈C:φv, un≤αnφv,xn 1−αnφv, zn, v−xn, vn ≤0 , W_n{v∈C:v−x_n, Jx0−Jx_n ≤0},

x_n1 Π_Hn∩Wnx0, n0,1,2, . . . ,

1.7

where{r_n}^∞_n0is a sequence in0,∞and{α_n}^∞_n0,{β_n}^∞_n0are sequences in0,1.

Algorithm 1.2.

x0∈Carbitrarily chosen, 0vn 1

r_nJxn−Jxn, vn∈Txn, ynJ⁻¹αnJx0 1−αnJSxn,

u_n∈Csuch that f

un, y

Aun, y−un 1

r_n

y−un, Jun−Jyn

≥0, ∀y∈C, H_n

v∈C:φv, u_n≤α_nφv, x0 1−α_nφv,x_n, v−x_n, v_n ≤0 , Wn{v∈C:v−xn, Jx0−Jxn ≤0},

x_n1 Π_Hn∩Wnx0, n0,1,2, . . . ,

1.8

where{r_n}^∞_n0is a sequence in0,∞and{α_n}^∞_n0is a sequence in0,1.

In this paper, strong convergence results on these two hybrid proximal-type algorithms are established; that is, under appropriate conditions, the sequence{x_n}generated byAlgorithm 1.1and the sequence{xn}generated byAlgorithm 1.2, converge strongly to the same pointΠ_EP∩FS∩T⁻¹0x0. These new results represent the improvement, generalization and development of the previously known ones in the literature including Solodov and Svaiter 22, Kamimura and Takahashi23, Qin and Su24, and Ceng et al.25.

Throughout this paper the symbolstands for weak convergence and → stands for strong convergence.

2. Preliminaries

Let E be a real Banach space with the dual E^∗. We denote by J the normalized duality mapping fromEto 2^E∗defined by

Jx

x^∗∈E^∗:x, x^∗x²x^∗2

, ∀x∈X, 2.1

where·,·denotes the generalized duality pairing. A Banach spaceEis called strictly convex ifxy/2 < 1 for allx, y ∈ Ewithx y 1 andx /y. It is said to be uniformly convex ifx_n−y_n → 0 for any two sequences{x_n},{y_n} ⊂Esuch thatx_n y_n 1 and lim_{n→ ∞}xnyn/21. LetU{x∈E:x1}be a unit sphere ofE, then the Banach spaceEis called smooth if

limt→0

xty− x

t 2.2

exists for each x, y ∈ U. IfEis smooth, thenJ is single valued. We still denote the single valued duality mapping byJ.

It is also said to be uniformly smooth if the limit is attained uniformly forx, y ∈ U.

Recall also that ifEis uniformly smooth, then Jis uniformly norm-to-norm continuous on bounded subsets ofE. A Banach spaceEis said to have the Kadec-Klee property if for any sequence{xn} ⊂E, wheneverxn x∈Eandxn → x, we havexn → x. It is known that ifEis uniformly convex, thenEhas the Kadec-Klee property; see26,27for more details.

LetCbe a nonempty closed convex subset of a real Hilbert spaceHandP_C :H → C be the metric projection ofHonto C, thenPCis nonexpansive. This fact actually characterizes Hilbert spaces and hence, it is not available in more general Banach spaces. Nevertheless, Alber 28 recently introduced a generalized projection operator Π_C in a Banach space E which is an analogue of the metric projection in Hilbert spaces.

Next, we assume thatEis a smooth Banach space. Consider the functional defined as in28,29by

φ x, y

x²−2x, Jyy², ∀x, y∈E. 2.3

It is clear that in a Hilbert spaceH,2.3reduces toφx, y x−y², for allx, y∈H.

The generalized projectionΠC:E → Cis a mapping that assigns to an arbitrary point x∈Ethe minimum point of the functionalφy, x; that is,Π_Cxx, wherexis the solution to the minimization problem

φx, x min

y∈C φ y, x

. 2.4

The existence and uniqueness of the operatorΠCfollows from the properties of the functional φx, yand strict monotonicity of the mappingJsee, e.g.,30. In a Hilbert spaceH,Π_C P_C. From28, in uniformly smooth and uniformly convex Banach spaces, we have

x −y²≤φ x, y

≤

xy², ∀x, y∈E. 2.5 LetCbe a nonempty closed convex subset ofE, and letSbe a mapping fromCinto itself.

A pointp∈Cis called an asymptotically fixed point ofS31ifCcontains a sequence{xn} which converges weakly topsuch thatSxn−xn → 0. The set of asymptotical fixed points of Swill be denoted byFS. A mapping SfromCinto itself is called relatively nonexpansive 32–34ifFS FSandφp, Sx≤φp, xfor allx∈Candp∈FS.

We remark that ifEis a reflexive, strictly convex and smooth Banach space, then for anyx, y ∈E, φx, y 0 if and only ifx y. It is sufficient to show that ifφx, y 0 then x y. From2.5, we havex y. This implies that x, Jy x² y². From the definition ofJ, we haveJxJy. Therefore, we havexy; see26,27for more details.

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

Lemma 2.1see23. LetEbe a smooth and uniformly convex Banach space and let{x_n}and{y_n} be two sequences ofE. Ifφxn, yn → 0 and either{xn}or{yn}is bounded, thenxn−yn → 0.

Lemma 2.2see23,28. LetCbe a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach spaceE, letx∈Eand letz∈C, then

z ΠCx⇐⇒

y−z, Jx−Jz

≤0, ∀y∈C. 2.6

Lemma 2.3see23,28. LetCbe a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach spaceE, then

φ

x,Π_Cy φ

Π_Cy, y

≤φ x, y

, ∀x∈C, y∈E. 2.7

Lemma 2.4see35. LetCbe a nonempty closed convex subset of a reflexive, strictly convex and smooth Banach spaceE, and letS:C → Cbe a relatively nonexpansive mapping, thenFSis closed and convex.

The following result is according to Blum and Oettli36.

Lemma 2.5see36. LetCbe a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach spaceE, letf be a bifunction fromC×CtoRsatisfying (A1)–(A4), and letr >0 andx∈E, then, there existsz∈Csuch that

f z, y

1 r

y−z, Jz−Jx

≥0, ∀y∈C. 2.8

Motivated by Combettes and Hirstoaga 37 in a Hilbert space, Takahashi and Zembayashi38established the following lemma.

Lemma 2.6 see38. LetCbe a nonempty closed convex subset of a uniformly smooth, strictly convex and reflexive Banach spaceE, and letfbe a bifunction fromC×CtoRsatisfying (A1)–(A4).

Forr >0 andx∈E, define a mappingT_r :E → Cas follows:

Trx

z∈C:f z, y

1 r

y−z, Jz−Jx

≥0, ∀y∈C

2.9

for allx∈E, then, the following hold:

iT_r is single valued;

iiTr is a firmly nonexpansive-type mapping, that is, for allx, y∈E,

T_rx−T_ry, JT_rx−JT_ry

≤

T_rx−T_ry, Jx−Jy

; 2.10

iiiFTr FT r EPf;

ivEPfis closed and convex.

UsingLemma 2.6, one has the following result.

Lemma 2.7see38. LetCbe a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach spaceE, letfbe a bifunction fromC×CtoRsatisfying (A1)–(A4), and letr >0, then, forx∈Eandq∈FT_r,

φ q, T_rx

φT_rx, x≤φ q, x

. 2.11

Utilizing Lemmas2.5,2.6and2.7as above, Chang39derived the following result.

Proposition 2.8 see39, Lemma 2.5. LetEbe a smooth, strictly convex and reflexive Banach space and Cbe a nonempty closed convex subset ofE. LetA : C → E^∗ be anα-inverse-strongly monotone mapping, letf be a bifunction fromC×CtoRsatisfying (A1)–(A4), and letr >0, then there hold the following:

Iforx∈E, there existsu∈Csuch that

f u, y

Au, y−u 1

r

y−u, Ju−Jx

≥0, ∀y∈C; 2.12

IIifEis additionally uniformly smooth andKr :E → Cis defined as

K_rx

u∈C:f u, y

Au, y−u 1

r

y−u, Ju−Jx

≥0, ∀y∈C

, ∀x∈E, 2.13

then the mappingKrhas the following properties:

iK_ris single valued,

iiKris a firmly nonexpansive-type mapping, that is, K_rx−K_ry, JK_rx−JK_ry

≤

K_rx−K_ry, Jx−Jy

, ∀x, y∈E, 2.14

iiiFK_r FK _r EP,

ivEP is a closed convex subset ofC,

vφp, K_rx φK_rx, x≤φp, x, for allp∈FK_r. Proof. Define a bifunctionF:C×C → Ras follows:

F x, y

f x, y

Ax, y−x

, ∀x, y∈C. 2.15

Then it is easy to verify thatFsatisfies the conditionsA1–A4. Therefore, The conclusions IandIIofProposition 2.8follow immediately from Lemmas2.5,2.6and2.7.

Lemma 2.9see13,14. LetE be a reflexive, strictly convex and smooth Banach space, and let T :E → 2^E∗be a maximal monotone operator withT⁻¹0/∅, then,

φz, J_rx φJ_rx, x≤φz, x, ∀r >0, z∈T⁻¹0, x∈E. 2.16

3. Main Results

Throughout this section, unless otherwise stated, we assume thatT :E → 2^E∗is a maximal monotone operator with domainDT C,S:C → Cis a relatively nonexpansive mapping, A:C → E^∗is anα-inverse-strongly monotone mapping andf :C×C → Ris a bifunction satisfying A1–A4, where C is a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach spaceE. In this section, we study the following algorithm.

Algorithm 3.1.

x0∈Carbitrarily chosen, 0vn 1

r_nJxn−Jxn, vn∈Txn, znJ⁻¹

βnJxn 1−βn

JSxn , y_nJ⁻¹α_nJx_n 1−α_nJSz_n,

un∈Csuch that f

u_n, y

Au_n, y−u_n 1

rn

y−u_n, Ju_n−Jy_n

≥0, ∀y∈C, Hn

v∈C:φv, un≤αnφv,xn 1−αnφv, zn, v−xn, vn ≤0 , W_n{v∈C:v−x_n, Jx0−Jx_n ≤0},

xn1 ΠHn∩Wnx0, n0,1,2, . . . ,

3.1

where{rn}^∞_n0is a sequence in0,∞and{αn}^∞_n0,{βn}^∞_n0are sequences in0,1.

First we investigate the condition under which the Algorithm 3.1 is well defined.

Rockafellar40proved the following result.

Lemma 3.2Rockafellar40. LetEbe a reflexive, strictly convex, and smooth Banach space and letT:E → 2^E∗be a multivalued operator, then there hold the following:

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

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

Utilizing this result, we can show the following lemma.

Lemma 3.3. LetEbe a reflexive, strictly convex, and smooth Banach space. If EP∩FS∩T⁻¹0/∅, then the sequence{x_n}generated byAlgorithm 3.1is well defined.

Proof. For eachn≥0, define two setsCnandDnas follows:

C_n

v∈C:φv, u_n≤α_nφv,x_n 1−α_nφv, z_n ,

D_n{v∈C:v−x_n, v_n ≤0}. 3.2

It is obvious thatCnis closed andDn, Wnare closed convex sets for eachn≥0. Let us show thatC_nis convex. Forv1, v2∈C_nandt∈0,1, putvtv1 1−tv2. It is sufficient to show thatv∈Cn. Indeed, observe that

φv, un≤αnφv,xn 1−αnφv, zn 3.3

is equivalent to

2αnv, Jxn21−αnv, Jzn −2v, Jun ≤αnxn² 1−αnzn²− un². 3.4

Note that there hold the following:

φv, un v²−2v, Junun², φv,x_n v²−2v, Jx_nx_n², φv, z_n v²−2v, Jz_nz_n²,

3.5

Thus we have

2αnv, Jxn21−αnv, Jzn −2v, Jun

2α_ntv1 1−tv2, Jx_n21−α_ntv1 1−tv2, Jz_n

−2tv1 1−tv2, Jun

2tα_nv1, Jx_n21−tα_nv2, Jx_n21−α_ntv1, Jz_n 21−α_n1−tv2, Jz_n −2tv1, Ju_n −21−tv2, Ju_n

≤α_nx_n² 1−α_nz_n²− un².

3.6

This implies thatv ∈ C_n. Therefore, C_n is convex and henceH_n C_n ∩ D_n is closed and convex.

On the other hand, letw ∈EP∩ FS∩ T⁻¹0 be arbitrarily chosen, thenw ∈EP, w∈ FSandw∈T⁻¹0. FromAlgorithm 3.1, it follows that

φw, un φ

w, Krnyn

≤φ w, yn φ

w, J⁻¹α_nJx_n 1−α_nJSz_n

w²−2w, α_nJx_n 1−α_nJSz_nα_nJx_n 1−α_nJSz_n²

≤ w²−2α_nw, Jx_n −21−α_nw, JSz_nα_nx_n² 1−α_nSz_n²

≤α_nφw,x_n 1−α_nφw, Sz_n

≤αnφw,xn 1−αnφw, zn.

3.7

Sow∈Cnfor alln≥0. Now, fromLemma 3.2it follows that there existsx0, v0∈E×E^∗such that 0v01/r0Jx0−Jx0andv0∈Tx0. SinceTis monotone, it follows thatx0−w, v0 ≥0, which implies thatw∈D0and hencew∈H0. Furthermore, it is clear thatw∈W0C, then w ∈H0∩ W0, and thereforex1 ΠH0∩W0x0is well defined. Suppose thatw ∈Hn−1∩ Wn−1

andxnis well defined for somen≥1. Again byLemma 3.2, we deduce thatxn, vn∈E×E^∗ such that 0 v_n 1/r_nJx_n−Jx_nand v_n ∈ Tx_n, then from the monotonicity of T we

conclude thatxn−w, vn ≥0, which implies thatw∈Dnand hencew∈Hn. It follows from Lemma 2.4that

w−xn, Jx0−Jxnw−ΠHn−1∩Wn−1x0, Jx0−JΠHn−1∩Wn−1x0 ≤0, 3.8

which implies thatw∈Wn. Consequently,w∈Hn∩Wnand so EP∩FS∩T⁻¹0⊂Hn∩Wn. Thereforexn1 ΠHn∩Wnx0 is well defined, then, by induction, the sequence{xn}generated byAlgorithm 3.1, is well defined for each integern≥0.

Remark 3.4. From the above proof, we obtain that

EP∩ FS∩ T⁻¹0⊂H_n∩ W_n 3.9

for each integern≥0.

We are now in a position to prove the main theorem.

Theorem 3.5. LetEbe a uniformly smooth and uniformly convex Banach space. Let{rn}^∞_n0 be a sequence in0,∞and{α_n}^∞_n0,{β_n}^∞_n0be sequences in0,1such that

lim inf

n→ ∞ rn>0, lim sup

n→ ∞ αn<1, lim

n→ ∞βn1. 3.10

Let EP∩ FS∩ T⁻¹0/∅. If S is uniformly continuous, then the sequence {x_n} generated by Algorithm 3.1converges strongly toΠ_EP∩FS∩T⁻¹0x0.

Proof. First of all, if follows from the definition of Wn that xn ΠWnx0. Since xn1 Π_Hn∩Wnx0∈W_n, we have

φxn, x0≤φxn1, x0, ∀n≥0. 3.11

Thus{φxn, x0}is nondecreasing. Also fromxn ΠWnx0andLemma 2.3, we have that φx_n, x0 φΠ_Wnx0, x0≤φw, x0−φw, x_n≤φw, x0 3.12

for eachw∈EP∩FS∩T⁻¹0⊂W_nand for eachn≥0. Consequently,{φx_n, x0}is bounded.

Moreover, according to the inequality

xn − x0²≤φxn, x0≤xnx0², 3.13

we conclude that {xn} is bounded. Thus, we have that lim_{n→ ∞}φxn, x0 exists. From Lemma 2.3, we derive the following:

φxn1, xn φxn1,ΠWnx0

≤φxn1, x0−φΠWnx0, x0 φx_n1, x0−φx_n, x0,

3.14

for alln≥0. This implies thatφxn1, xn → 0. So it follows fromLemma 2.1thatxn1−xn → 0. Sincex_n1 Π_Hn∩Wnx0∈H_n, from the definition ofH_n, we also have

φxn1, un≤αnφxn1,xn 1−αnφxn1, zn, xn1−xn, vn ≤0. 3.15

Observe that φx_n1, z_n φ

x_n1, J⁻¹

β_nJx_n 1−β_n

JSx_n xn1²−2

xn1, βnJxn 1−βn

JSxn

βnJxn 1−βn

JSxn²

≤ xn1²−2β_nx_n1, Jx_n −2 1−β_n

x_n1, JSx_nβ_nx_n² 1−β_n

Sx_n² βnφxn1,xn

1−βn

φxn1, Sxn.

3.16

At the same time,

φΠ_Hnx_n, x_n−φx_n, x_n Π_Hnx_n²− x_n²2x_n−Π_Hnx_n, Jx_n

≥2ΠHnxn−xn, Jxn2xn−ΠHnxn, Jxn 2x_n−Π_Hnx_n, Jx_n−Jx_n.

3.17

SinceΠHnxn∈Hnandvn 1/rnJxn−Jxn, it follows that

xn−ΠHnxn, Jxn−Jxnrnxn−ΠHnxn, vn ≥0 3.18

and hence that φΠHnxn, xn ≥ φxn, xn. Further, from xn1 ∈ Hn, we have φxn1, xn ≥ φΠ_Hnx_n, x_n, which yields

φxn1, xn≥φΠHnxn, xn≥φxn, xn. 3.19

Then it follows fromφxn1, xn → 0 thatφxn, xn → 0. Hence it follows fromLemma 2.1 thatx_n−x_n → 0. Since from3.15we derive

φx_n1,x_n−φx_n, x_n

xn1²−2xn1, Jxnxn²−

xn²−2xn, Jxnxn² x_n1²− xn²−2x_n1, Jx_n2x_n, Jx_n

x_n1²− xn²−2x_n1−x_n, Jx_n−Jx_n

−2x_n1−x_n, Jx_n2x_n, Jx_n−Jx_n

xn1 − xnxn1xn 2rnxn1−xn, vn −2xn1−xn, Jxn 2x_n, Jx_n−Jx_n

≤ xn1−xnxn1xn 2xn1−xnxn2xnJxn−Jxn

≤ xn1−xnxn1xn 2xn1−xnxn−xnxn2xnJxn−Jxn, 3.20

we have

φxn1,xn≤φxn, xn xn1−xnxn1xn

2x_n1−x_nx_n−x_nx_n2x_nJx_n−Jx_n.

3.21

Thus, fromφx_n, x_n → 0,x_n−x_n → 0, andx_n1−x_n → 0, we know thatφx_n1,x_n → 0.

Consequently from3.16,φxn1,xn → 0, andβn → 1 it follows that

φxn1, zn−→0. 3.22

So it follows from3.15,φx_n1,x_n → 0, andφx_n1, z_n → 0 thatφx_n1, u_n → 0. Utilizing Lemma 2.1we deduce that

nlim→ ∞xn1−un lim

n→ ∞xn1−xn lim

n→ ∞xn1−zn0. 3.23

Furthermore, foru∈EP∩FS∩ T⁻¹0 arbitrarily fixed, it follows fromProposition 2.8that φ

un, yn φ

Krnyn, yn

≤φ u, yn

−φ

u, Krnyn φ

u, J⁻¹α_nJx_n 1−α_nJSz_n

−φu, u_n

u²−2u, α_nJx_n 1−α_nJSz_nα_nJx_n 1−α_nJSz_n²−φu, u_n

≤ u²−2α_nu, Jx_n −21−α_nu, JSz_nα_nx_n² 1−α_nSz_n²−φu, u_n α_nφu,x_n 1−α_nφu, Sz_n−φu, u_n

≤1−αnφu, zn αnφu,xn−φu, un 1−α_nφ

u, J⁻¹

β_nJx_n 1−β_n

JSx_n

α_nφu,x_n−φu, u_n 1−α_n

u²−2

u, β_nJx_n 1−β_n

JSx_n

β_nJx_n 1−β_n

JSx_n² α_nφu,x_n−φu, u_n

≤1−αn

u²−2βnu, Jxn −2 1−βn

u, JSxnβnxn² 1−βn

Sxn² α_nφu,x_n−φu, u_n

1−αn

βnφu,xn 1−βn

φu, Sxn

αnφu,xn−φu, un

≤1−αn

βnφu,xn 1−βn

φu,xn

αnφu,xn−φu, un 1−α_nφu,x_n α_nφu,x_n−φu, u_n

φu,xn−φu, un

φu,xn−φu, xn1 φu, xn1−φu, un

xn²− xn1²2u, Jxn1−Jxnxn1²− un²2u, Jun−Jxn1

≤ x_n−x_n1x_nx_n1 2uJx_n1−Jx_n x_n1−u_nx_n1u_n 2uJu_n−Jx_n1.

3.24 SinceJis uniformly norm-to-norm continuous on bounded subsets ofE, it follows from3.23 thatJx_n1−Jx_n → 0 andJu_n−Jx_n1 → 0, which hence yieldφu_n, y_n → 0. Utilizing Lemma 2.1, we getun−yn → 0. Observe that

xn1−yn≤ xn1−unun−yn−→0, 3.25 due to3.23. Since J is uniformly norm-to-norm continuous on bounded subsets ofE, we have that

nlim→ ∞Jxn1−Jyn lim

n→ ∞Jxn1−Jxn0. 3.26

On the other hand, we have

x_n−z_n ≤ x_n−x_n1x_n1−z_n −→0. 3.27 Noticing that

Jx_n1−Jy_nJx_n1−α_nJx_n 1−α_nJSz_n

αnJxn1−Jxn 1−αnJxn1−JSzn 1−α_nJx_n1−JSz_n−α_nJx_n−Jx_n1

≥1−α_nJx_n1−JSz_n −α_nJx_n−Jx_n1,

3.28

we have

Jxn1−JSzn ≤ 1

1−α_nJxn1−JynαnJxn−Jxn1

. 3.29

From3.26and lim sup_{n→ ∞}α_n<1, we obtain

nlim→ ∞Jx_n1−JSz_n0. 3.30

SinceJ⁻¹is also uniformly norm-to-norm continuous on bounded subsets ofE^∗, we obtain

nlim→ ∞x_n1−Sz_n0. 3.31

Observe that

x_n−Sx_n ≤ x_n−x_n1x_n1−Sz_nSz_n−Sx_n. 3.32

SinceS is uniformly continuous, it follows from3.27,3.31andx_n1−x_n → 0 thatx_n− Sx_n → 0.

Now let us show thatωw{xn}⊂EP∩FS∩ T⁻¹0, where ω_w{x_n}:

x∈C:x_nk xfor some subsequence{n_k} ⊂ {n}withn_k↑ ∞ . 3.33 Indeed, since {xn} is bounded and X is reflexive, we know that ωw{xn}/∅. Take x ∈ ω_w{x_n}arbitrarily, then there exists a subsequence{x_nk}of{x_n}such thatx_nk x. Hence

x∈FS. Let us show thatx∈T⁻¹0. Sincexn−xn → 0, we have thatxnk x. Moreover, since Jis uniformly norm-to-norm continuous on bounded subsets ofEand lim inf_{n→ ∞}rn>0, we obtain

vn 1

rnJxn−Jxn−→0. 3.34

It follows fromvn∈Txnand the monotonicity ofTthat z−xn, z−vn

≥0 3.35

for allz∈DTandz∈Tz. This implies that z−x, z

≥0 3.36

for allz∈DTandz∈Tz. Thus from the maximality ofT, we infer thatx∈T⁻¹0. Therefore,

x∈FS∩ T⁻¹0. Further, let us show thatx∈EP. Sinceun−yn → 0 andxn−un → 0, from xnk xwe obtain thatynk xandunk x.

SinceJ is uniformly norm-to-norm continuous on bounded subsets ofE, fromu_n − y_n → 0 we derive

nlim→ ∞Jun−Jyn0. 3.37

From lim inf_{n→ ∞}rn>0, it follows that

nlim→ ∞

Ju_n−Jy_n

rn 0. 3.38

By the definition ofun :Krnyn, we have F

u_n, y 1

r_n

y−u_n, Ju_n−Jy_n

≥0, ∀y∈C, 3.39

where

F un, y

f un, y

Aun, y−un

. 3.40

Replacingnbyn_k, we have fromA2that 1

rnk

y−unk, Junk −Jynk

≥ −F unk, y

≥F y, unk

, ∀y∈C. 3.41

Sincey→fx, y Ax, y−xis convex and lower semicontinuous, it is also weakly lower semicontinuous. Lettingnk → ∞in the last inequality, from3.38andA4we have

F y,x

≤0, ∀y∈C. 3.42 Fort, with 0< t≤1, andy∈C, lety_tty 1−tx. Since y∈Candx∈C, theny_t∈Cand henceFyt,x ≤0. So, fromA1we have

0F yt, yt

≤tF yt, y

1−tF yt,x

≤tF yt, y

. 3.43

Dividing byt, we have

F yt, y

≥0, ∀y∈C. 3.44

Lettingt↓0, fromA3it follows that F x, y

≥0, ∀y∈C. 3.45

So,x∈EP. Therefore, we obtain thatω_w{x_n}⊂EP∩ FS∩ T⁻¹0 by the arbitrariness ofx.

Next, let us show thatωw{xn} {Π_EP∩FS∩T⁻¹0x0}andxn → Π_EP∩FS∩T⁻¹0x0. Indeed, putx Π_EP∩FS∩T⁻¹₀x0. Fromx_n1 Π_Hn∩Wnx0andx∈EP∩ FS∩ T⁻¹0 ⊂ H_n∩W_n, we haveφx_n1, x0≤φx, x0. Now from weakly lower semicontinuity of the norm, we derive for eachx∈ωw{xn}

φx, x 0 x ²−2x, x 0x0²

≤lim inf

k→ ∞

x_nk²−2x_nk, x0x0²

lim inf

k→ ∞ φx_nk, x0

≤lim sup

k→ ∞ φxnk, x0

≤φx, x0.

3.46

It follows from the definition ofΠ_EP∩FS∩T⁻¹0x0thatxxand hence

klim→ ∞φx_nk, x0 φx, x0. 3.47

So we have lim_{k→ ∞}xnk x. Utilizing the Kadec-Klee property ofE, we conclude that {x_nk} converges strongly toΠ_EP∩FS∩T⁻¹₀x0. Since{x_nk} is an arbitrary weakly convergent subsequence of {x_n}, we know that {x_n} converges strongly to Π_EP∩FS∩T⁻¹0x0. This completes the proof.

Theorem 3.5 covers 25, Theorem 3.1 by taking C E, f ≡ 0 and A ≡ 0. Also Theorem 3.5covers24, Theorem 2.1by takingf ≡0,A≡0 andT≡0.

Theorem 3.6. LetCbe a nonempty closed convex subset of a uniformly smooth and uniformly convex Banach spaceE. LetT :E → 2^E∗be a maximal monotone operator with domainDT C,S:C → Cbe a relatively nonexpansive mapping,A :C → E^∗be anα-inverse-strongly monotone mapping andf:C×C → Rbe a bifunction satisfying (A1)–(A4). Assume that{r_n}^∞_n0is a sequence in0,∞ satisfying lim inf_{n→ ∞}rn >0 and that{αn}^∞_n0is a sequences in0,1satisfying lim_{n→ ∞}αn0.

Define a sequence{x_n}by the following algorithm.

Algorithm 3.7.

x0∈Carbitrarily chosen, 0v_n 1

rnJx_n−Jx_n, v_n∈Tx_n, ynJ⁻¹αnJx0 1−αnJSxn,

un∈Csuch that f

u_n, y

Au_n, y−u_n 1

r_n

y−u_n, Ju_n−Jy_n

≥0, ∀y∈C, H_n

v∈C:φv, u_n≤α_nφv, x0 1−α_nφv,x_n, v−x_n, v_n ≤0 , Wn{v∈C:v−xn, Jx0−Jxn ≤0},

xn1 ΠHn∩Wnx0, n0,1,2, . . . ,

3.48

whereJis the single valued duality mapping onE. Let EP∩FS∩T⁻¹0/∅. IfSis uniformly continuous, then{x_n}converges strongly toΠ_EP∩FS∩T⁻¹0x0.

Proof. For eachn≥0, define two setsCnandDnas follows:

Cn

v∈C:φv, un≤αnφv, x0 1−αnφv,xn , D_n{v∈C:v−x_n, v_n ≤0}.

3.49

It is obvious that C_n is closed and D_n, W_n are closed convex sets for each n ≥ 0. Let us show thatC_n is convex and so H_n C_n∩ D_n is closed and convex. Similarly to the proof ofLemma 3.3, since

φv, un≤αnφv, x0 1−αnφv,xn 3.50

is equivalent to

2αnv, Jx021−αnv, Jxn −2v, Jun ≤αnx0² 1−αnxn²− un², 3.51