Báo cáo hóa học: " Research Article Iterative Schemes for Generalized Equilibrium Problem and Two Maximal Monotone Operators" pdf

Yao,yaojc@math.nsysu.edu.tw Received 20 July 2009; Accepted 27 October 2009 Recommended by Yeol Je Cho The purpose of this paper is to introduce and study two new hybrid proximal-point a

Volume 2009, Article ID 896252, 34 pages

doi:10.1155/2009/896252

Research Article

Iterative Schemes for Generalized Equilibrium

Problem and Two Maximal Monotone Operators

L C Zeng,1, 2 Y C Lin,3 and J C Yao4

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

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

3 Department of Occupational Safety and Health, China Medical University, Taichung 404, Taiwan

4 Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan

Correspondence should be addressed to J C Yao,yaojc@math.nsysu.edu.tw

Received 20 July 2009; Accepted 27 October 2009

Recommended by Yeol Je Cho

The purpose of this paper is to introduce and study two new hybrid proximal-point algorithmsfor finding a common element of the set of solutions to a generalized equilibrium problem and thesets of zeros of two maximal monotone operators in a uniformly smooth and uniformly convexBanach space We established strong and weak convergence theorems for these two modifiedhybrid proximal-point algorithms, respectively

Copyrightq 2009 L C Zeng et al This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited

is to findx ∈ C such that

f

The set of solutions of1.2 is denoted by EP Problem 1.2 and similar problems have beenextensively studied; see, for example,1 11 Whenever A  0, problem 1.2 reduces to theequilibrium problem of findingx ∈ C such that

Whenever X  H a Hilbert space, problem 1.2 was very recently introduced and

considered by Kamimura and Takahashi12 Problem 1.2 is very general in the sense that itincludes, as spacial cases, optimization problems, variational inequalities, minimax problems,the Nash equilibrium problem in noncooperative games, and others; see, for example,13,14 A mapping S : C → X is called nonexpansive if Sx − Sy ≤ x − y for all x, y ∈ C

Denote by FS the set of fixed points of S, that is, FS  {x ∈ C : Sx  x} Iterative schemes

for finding common elements of EP and fixed points set of nonexpansive mappings havebeen studied recently; see, for example,12,15–17 and the references therein

On the other hand, a classical method of solving 0 ∈ Tx in a Hilbert space H is the proximal point algorithm which generates, for any starting point x0 ∈ H, a sequence {x n} in

H by the iterative scheme

x n1  J r n x n , n  0, 1, 2, , 1.5

where{r n } is a sequence in 0, ∞, J r  I  rT−1for each r > 0 is the resolvent operator for

T, and I is the identity operator on H This algorithm was first introduced by Martinet 14and generally studied by Rockafellar18 in the framework of a Hilbert space H Later manyauthors studied1.5 and its variants in a Hilbert space H or in a Banach space X; see, forexample,13,19–23 and the references therein

Let X be a uniformly smooth and uniformly convex Banach space and let C be a nonempty closed convex subset of X Let f be a bifunction from C × C to R satisfying the

following conditionsA1–A4 which were imposed in 24:

A1 fx, x  0 for all x ∈ C;

A2 f is monotone, that is, fx, y  fy, x ≤ 0, for all x, y ∈ C;

A3 for all x, y, z ∈ C, lim sup t↓0 ftz  1 − tx, y ≤ fx, y;

A4 for all x ∈ C, fx, · is convex and lower semicontinuous.

Let T : X → 2X∗

be a maximal monotone operator such that

A5 T−10∩ EPf / ∅.

The purpose of this paper is to introduce and study two new iterative algorithms

for finding a common element of the set EP of solutions for the generalized equilibrium

problem1.2 and the set T−10∩ T−10 for maximal monotone operators T, T in a uniformly

smooth and uniformly convex Banach space X First, motivated by Kamimura and Takahashi

12, Theorem 3.1 , Ceng et al 16, Theorem 3.1 , and Zhang 17, Theorem 3.1 , weintroduce a sequence{x n} that, under some appropriate conditions, is strongly convergent

3.1, Ceng et al 16, Theorem 4.1, and Zhang 17, Theorem 3.1 , we define a sequence

weakly convergent to an element z ∈ T−10∩ T−10∩ EP, where z  lim n → ∞ΠT−1 0∩ T −1 0∩EPx ninSection4 Our results represent a generalization of known results in the literature, includingTakahashi and Zembayashi15, Kamimura and Takahashi 12, Li and Song 22, Ceng andYao25, and Ceng et al 16 In particular, compared with Theorems 3.1 and 4.1 in 16, ourresultsi.e., Theorems3.2and4.2in this paper extend the problem of finding an element of

T−10∩ EPf to the one of finding an element of T−10∩ T−10∩ EP Meantime, the algorithms

in this paper are very different from those in 16 because of considering the complexity

involving the problem of finding an element of T−10∩ T−10∩ EP.

2 Preliminaries

In the sequel, we denote the strong convergence, weak convergence and weak∗convergence

of a sequence{x n } to a point x ∈ X by x n → x, x n  x and x n  x, respectively.∗

A Banach space X is said to be strictly convex, if x  y/2 < 1 for all x, y ∈ U  {z ∈

X : z  1} with x / y X is said to be uniformly convex if for each  ∈ 0, 2 there exists

δ > 0 such that x  y/2 ≤ 1 − δ for all x, y ∈ U with x − y ≥  Recall that each uniformly

convex Banach space has the Kadec-Klee property, that is,

x n  x

The proof of the main results of Sections 3 and 4 will be based on the followingassumption

Assumption A Let X be a uniformly smooth and uniformly convex Banach space and let C

be a nonempty closed convex subset of X Let f be a bifunction from C × C to R satisfying

the same conditionsA1–A4 as in Section1 Let T, T : X → 2 X∗

be two maximal monotoneoperators such that

The generalized projectionΠC : X → C is a mapping that assigns to an arbitrary point

x ∈ X the minimum point of the functional φy, x; that is, Π C x  x, where x is the solution

to the minimization problem

φx, x  min

y∈C φ

y, x

The existence and uniqueness of the operatorΠC follows from the properties of the

functional φx, y and strict monotonicity of the mapping J see, e.g., 27 In a Hilbert space,

ΠC  P C From26, in a smooth strictly convex and reflexive Banach space X, we have

28 The set of asymptotically fixed points of S will be denoted by FS A mapping C from

S into itself is called relatively nonexpansive if  FS  FS and φp, Sx ≤ φp, x, for all

x ∈ C and p ∈ FS 15

Observe that, if X is a reflexive strictly convex and smooth Banach space, then for any

x, y ∈ X, φx, y  0 if and only if x  y To this end, it is sufficient to show that if φx, y  0

then x  y Actually, from 2.4, we have x  y which implies that x, Jy  x2 y2

From the definition of J, we have Jx  Jy and therefore, x  y; see 29 for more details.

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

Lemma 2.1 Kamimura and Takahashi 12 Let X be a smooth and uniformly convex Banach

space and let {x n } and {y n } be two sequences of X If φx n , y n  → 0 and either {x n } or {y n } is

bounded, then x n − y n → 0.

Lemma 2.2 Alber 26, Kamimura and Takahashi 12 Let C be a nonempty closed convex

subset of a smooth strictly convex and reflexive Banach space X Let x ∈ X and let z ∈ C Then

z  Π C x ⇐⇒y − z, Jx − Jz≤ 0, ∀y ∈ C. 2.6

Lemma 2.3 Alber 26, Kamimura and Takahashi 12 Let C be a nonempty closed convex

subset of a smooth strictly convex and reflexive Banach space X Then

Lemma 2.4 Rockafellar 18 Let X be a reflexive strictly convex and smooth Banach space and let

T : X → 2 X∗

be a multivalued operator Then there hold the following hold:

i T−10 is closed and convex if T is maximal monotone such that T−10/ ∅;

ii T is maximal monotone if and only if T is monotone with RJ  rT  X∗for all r > 0.

Lemma 2.5 Xu 30 Let X be a uniformly convex Banach space and let r > 0 Then there exists a

strictly increasing, continuous, and convex function g : 0, 2r → R such that g0  0 and

tx  1 − ty 2

≤ tx2 1 − t y 2− t1 − tg x − y , 2.8

for all x, y ∈ B r and t ∈ 0, 1, where B r  {z ∈ X : z ≤ r}.

Lemma 2.6 Kamimura and Takahashi 12 Let X be a smooth and uniformly convex Banach

space and let r > 0 Then there exists a strictly increasing, continuous, and convex function g :

0, 2r → R such that g0  0 and

The following result is due to Blum and Oettli24

Lemma 2.7 Blum and Oettli 24 Let C be a nonempty closed convex subset of a smooth strictly

convex and reflexive Banach space X, let f be a bifunction from C × C to R satisfying (A1)–(A4), and let r > 0 and x ∈ X Then, there exists z ∈ C such that

Lemma 2.8 Takahashi and Zembayashi 15 Let C be a nonempty closed convex subset of a

smooth strictly convex and reflexive Banach space X, and let f be a bifunction from C × C to R satisfying (A1)–(A4) For r > 0 and x ∈ X, define a mapping T r : X → C as follows:

iii FT r   FT r   EPf;

iv EPf is closed and convex.

Using Lemma2.8, one has the following result

Lemma 2.9 Takahashi and Zembayashi 15 Let C be a nonempty closed convex subset of a

smooth strictly convex and reflexive Banach space X, let f be a bifunction from C × C to R satisfying (A1)–(A4), and let r > 0 Then, for x ∈ X and q ∈ FT r ,

Proposition 2.10 Zhang 21, Lemma  Let X be a smooth strictly convex and reflexive Banach

space and let C be a nonempty closed convex subset of X Let A : C → X∗be an α -inverse-strongly monotone mapping, let f be a bifunction from C × C to R satisfying (A1)–(A4), and let r > 0 Then the following hold:

for x ∈ X, there exists u ∈ C such that

respectively Then J r : X → DT and J r : X → D T are two single-valued mappings Also,

T−10  FJ r  and T−10  F J r  for each r > 0, where FJ r  and F J r  are the sets of fixed points

of J r and J r , respectively For each r > 0, the Yosida approximations of T and T are defined by A r 

J − JJ r /r and A r  J − J J r /r, respectively It is known that

A r x ∈ TJ r x, A r x ∈ T J r x , ∀r > 0, x ∈ X. 2.17

Lemma 2.11 Kohsaka and Takahashi 13 Let X be a reflexive strictly convex and smooth

Banach space and let T : X → 2 X∗

be a maximal monotone operator with T−10/ ∅ Then φz, J r x  φJ r x, x ≤ φz, x, ∀r > 0, z ∈ T−10, x ∈ X. 2.18

Lemma 2.12 Tan and Xu 32 Let {an } and {b n } be two sequences of nonnegative real numbers

satisfying the inequality: a n1 ≤ a n  b n for all n ≥ 0 If∞n0 b n < ∞, then lim n → ∞ a n exists.

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 set T−10∩ T−10 for two maximal

monotone operators T and T.

Lemma 3.1 Let X be a reflexive strictly convex and smooth Banach space and let T : X → 2 X∗

be a maximal monotone operator Then for each r ∈ 0, ∞, the following holds:

and {r n } ⊂ 0, ∞ satisfies lim inf n → ∞ r n > 0 Then, the sequence {x n } converges strongly to

ΠT−1 0∩ T −1 0∩EPx0 provided  J r n v n − J r n x n  → 0 for any sequence {v n } ⊂ X with v n − x n  → 0,

whereΠT−1 0∩ T −1 0∩EPis the generalized projection of X onto T−10∩ T−10∩ EP.

Remark 3.3 In Theorem 3.2, if X  H a real Hilbert space, then { J r n} is a sequence of

nonexpansive mappings on H This implies that as n → ∞,

We divide the proof into several steps

Step 1 We claim that H n ∩ W n is closed and convex for each n≥ 0

Indeed, it is obvious that H n is closed and W n is closed and convex for each n≥ 0 Let

us show that H n is convex For z1, z2∈ H n and t ∈ 0, 1, put z  tz1 1 − tz2 It is sufficient

to show that z ∈ H n We first write γ n  α n  β n − α n β n − α n β n  α n α n β n for each n≥ 0 Next,

φz, x0  z2− 2z, Jx0  x02, φz, x n   z2− 2z, Jx n   x n2, φz, u n   z2− 2z, Ju n   u n2,

This implies that z ∈ H n Therefore, H nis closed and convex

Step 2 We claim that T−10∩ T−10∩ EP ⊂ H n ∩ W n for each n ≥ 0 and that {x n} is well defined

Indeed, take w ∈ T−10∩ T−10∩ EP arbitrarily Note that u n  K r n y nis equivalent to

Then from Lemma2.11we obtain

We prove this by induction For n  0, we have T−10∩ T−10∩ EP ⊂ C  W0 Assume that

T−10∩ T−10∩ EP ⊂ W n Since x n1 is the projection of x0onto H n ∩ W n, by Lemma2.2we have

x n1 − z, Jx0− Jx n1  ≥ 0, ∀z ∈ H n ∩ W n 3.21

As T−10∩ T−10∩ EP ⊂ H n ∩ W n by the induction assumption, the last inequality holds, in

particular, for all z ∈ T−10∩ T−10∩ EP This, together with the definition of W n1implies that

T−10∩ T−10∩ EP ⊂ W n1 Hence3.20 holds for all n ≥ 0 So, T−10∩ T−10∩ EP ⊂ H n ∩ W nfor

all n ≥ 0 This implies that the sequence {x n} is well defined

Step 3 We claim that {x n } is bounded and that φx n1 , x n  → 0 as n → ∞.

Indeed, it follows from the definition of W n that x n  ΠW n x0 Since x n  ΠW n x0

and x n1  ΠH n ∩W n x0 ∈ W n , so φx n , x0 ≤ φx n1 , x0 for all n ≥ 0; that is, {φx n , x0} is

nondecreasing It follows from x n  ΠW n x0and Lemma2.3that

so{x n} is bounded From Lemma2.3, we have

As in the proof of Step2we can show that φu, z n  ≤ φu, x n , φu, x n  ≤ α n φu, x0  1 −

α n φu, x n , φu, z n  ≤ β n φu, x0  1 − β n φu, x n , φu, y n  ≤ α n φu, x n   1 − α n φu, z n,

and φu, u n  ≤ α n φu, x n   1 − α n φu, z n  Hence it follows from the boundedness of {x n}that{z n }, { x n }, { z n }, {y n }, and {u n } are also bounded Let r  sup{x n ,  x n , J r n x n ,  z n :

n ≥ 0} Since X is a uniformly smooth Banach space, we know that X∗is a uniformly convexBanach space Therefore, by Lemma 2.5 there exists a continuous, strictly increasing, and

convex function g with g0  0 such that

αx∗ 1 − αy∗ 2

≤ αx∗2 1 − α y∗ 2

− α1 − αg  x∗− y∗ , 3.28

Since x n − u n → 0 and J is uniformly norm-to-norm continuous on bounded subsets of X,

we obtain Jx n − Ju n → 0 From lim infn → ∞ β n 1 − β n  > 0, lim inf n → ∞ α n 1 − α n  > 0, and

limn → ∞ α n limn → ∞ β n 0 we have

lim

n → ∞ gJx n − JJ r n x n  lim

n → ∞ gJ x n − J z n   0. 3.34Therefore, from the properties of g we get

Since φx n1 , u n  → 0, x n1 − x n  → 0, x n1 − u n  → 0, Ju n − Jx n  → 0, and {x n} is

bounded, so it follows that φu n , x n → 0 Also, observe that

φu n , J r n x n  − φu n , x n   J r n x n2− x n2 2u n , Jx n − JJ r n x n

 J r n x n  − x n J r n x n   x n   2u n , Jx n − JJ r n x n

≤ J r x n − x n J r x n   x n   2unJx n − JJ r x n .

3.38

Since φu n , x n  → 0, J r n x n − x n  → 0, Jx n − JJ r n x n → 0 and the sequences

{x n }, {u n }, {J r n x n } are bounded, so it follows that φu n , J r n x n → 0 Meantime, observe that

Since α n → 0, φu n , x n  → 0 and φu n , J r n x n → 0, it follows from the boundedness of

{u n } that φu n , x n → 0 Thus, in terms of Lemma2.1, we have thatu n − x n → 0 and so

x n − x n  → 0 Furthermore, since  β n Jx0 1 − β n J x n − J x n   β n Jx0− J x n → 0, from the

uniform norm-to-norm continuity of J−1on bounded subsets of X∗, we obtain

Step 5 We claim that ω w {x n } ⊂ T−10∩ T−10∩ EP, where

ω w {x n} :x ∈ C : x n  x for some subsequence {n k } ⊂ {n} with n k↑ ∞. 3.43

Indeed, since{x n } is bounded and X is reflexive, we know that ω w {x n } / ∅ Take

x ∈ ω w {x n } arbitrarily Then there exists a subsequence {x n k } of {x n } such that x n k  x.

Hence it follows from x n − x n → 0, x n − J r n x n → 0, and x n − J r n x n → 0 that { x n k }, {J r nk x n k},and{ J r nk x n k } converge weakly to the same point x On the other hand, from 3.35, 3.36,

and lim infn → ∞ r n > 0 we obtain that

Letting k → ∞, we have that z − x, z∗ ≥ 0 and  z − x, z∗ ≥ 0 Then the maximality of the

operators T, T implies that x ∈ T−10 and x ∈ T−10 Next, let us show that x ∈ EP Since we

set of solutions for a generalized equilibrium problem and the set T−10∩ T−10 for two maximal

monotone. ..