Báo cáo hóa học: "Research Article An Iterative Algorithm Combining Viscosity Method with Parallel Method for a Generalized Equilibrium Problem and Strict Pseudocontractions"

Volume 2009, Article ID 794178, 21 pagesdoi:10.1155/2009/794178 Research Article An Iterative Algorithm Combining Viscosity Method with Parallel Method for a Generalized Equilibrium Prob

Volume 2009, Article ID 794178, 21 pages

doi:10.1155/2009/794178

Research Article

An Iterative Algorithm Combining

Viscosity Method with Parallel Method for a

Generalized Equilibrium Problem and Strict

Pseudocontractions

Jian-Wen Peng,1 Yeong-Cheng Liou,2 and Jen-Chih Yao3

1 College of Mathematics and Computer Science, Chongqing Normal University, Chongqing 400047, China

2 Department of Information Management, Cheng Shiu University, Kaohsiung, Taiwan 833, Taiwan

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

Correspondence should be addressed to Yeong-Cheng Liou,simplex liou@hotmail.com

Received 5 August 2008; Accepted 4 January 2009

Recommended by Hichem Ben-El-Mechaiekh

We introduce a new approximation scheme combining the viscosity method with parallel methodfor finding a common element of the set of solutions of a generalized equilibrium problem and theset of fixed points of a family of finitely strict pseudocontractions We obtain a strong convergencetheorem for the sequences generated by these processes in Hilbert spaces Based on this result,

we also get some new and interesting results The results in this paper extend and improve somewell-known results in the literature

Copyrightq 2009 Jian-Wen Peng 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

1 Introduction

Let H be a real Hilbert space with inner product  · , ·  and induced norm  · , and let C be

a nonempty-closed convex subset of H Let ϕ : H → R ∪ {∞} be a function and let F be

a bifunction from C × C to R such that C ∩ dom ϕ / ∅, where R is the set of real numbers and dom ϕ  {x ∈ H : ϕx < ∞} Flores-Baz´an 1 introduced the following generalizedequilibrium problem:

The set of solutions of 1.1 is denoted by GEP F, ϕ Flores-Baz´an 1 provided somecharacterizations of the nonemptiness of the solution set for problem1.1 in reflexive Banachspaces in the quasiconvex case Bigi et al 2 studied a dual problem associated with theproblem1.1 with C  H  R n

Let ϕx  δ C x, ∀x ∈ H Here δ C denotes the indicator function of the set C; that is,

δ C x  0 if x ∈ C and δ C x  ∞ otherwise Then the problem 1.1 becomes the followingequilibrium problem:

The set of solutions of1.2 is denoted by EPF The problem 1.2 includes, as specialcases, the optimization problem, the variational inequality problem, the fixed point problem,the nonlinear complementarity problem, the Nash equilibrium problem in noncooperativegames, and the vector optimization problem For more detail, please see 3 5 and thereferences therein

If Fx, y  gy−gx for all x, y ∈ C, where g : C → R is a function, then the problem

1.1 becomes a problem of finding x ∈ C which is a solution of the following minimization

The set of solutions of1.3 is denoted by Argming, ϕ.

If ϕ : H → R ∪ {∞} is replaced by a real-valued function φ : C → R, the problem

1.1 reduces to the following mixed equilibrium problem introduced by Ceng and Yao 6:

Find x ∈ C such that Fx, y  φy − φx ≥ 0, ∀y ∈ C. 1.4

Recall that a mapping T : C → C is said to be a κ-strict pseudocontraction 7 if thereexists 0≤ κ < 1, such that

Tx − Ty2≤ x − y2 κI − Tx − I − Ty2

where I denotes the identity operator on C When κ  0, T is said to be nonexpansive Note

that the class of strict pseudocontraction mappings strictly includes the class of nonexpansive

mappings We denote the set of fixed points of S by FixS.

Ceng and Yao6, Yao et al 8, and Peng and Yao 9,10 introduced some iterativeschemes for finding a common element of the set of solutions of the mixed equilibriumproblem 1.4 and the set of common fixed points of a family of finitely infinitelynonexpansive mappingsstrict pseudocontractions in a Hilbert space and obtained somestrong convergence theoremsweak convergence theorems Some methods have beenproposed to solve the problem 1.2; see, for instance, 3 5, 11–18 and the referencestherein Recently, S Takahashi and W Takahashi 12 introduced an iterative scheme bythe viscosity approximation method for finding a common element of the set of solutions

of problem1.2 and the set of fixed points of a nonexpansive mapping in a Hilbert spaceand proved a strong convergence theorem Su et al.13 introduced an iterative scheme bythe viscosity approximation method for finding a common element of the set of solutions

of problem 1.2 and the set of fixed points of a nonexpansive mapping and the set of

solutions of the variational inequality problem for an α-inverse strongly monotone mapping

in a Hilbert space Tada and Takahashi14 introduced two iterative schemes for finding

a common element of the set of solutions of problem 1.2 and the set of fixed points of

a nonexpansive mapping in a Hilbert space and obtained both strong convergence theoremand weak convergence theorem Ceng et al.15 introduced an iterative algorithm for finding

a common element of the set of solutions of problem1.2 and the set of fixed points of a strictpseudocontraction mapping Chang et al.16 introduced some iterative processes based onthe extragradient method for finding the common element of the set of fixed points of a family

of infinitely nonexpansive mappings, the set of problem1.2, and the set of solutions of

a variational inequality problem for an α-inverse strongly monotone mapping Colao et al.

17 introduced an iterative method for finding a common element of the set of solutions

of problem 1.2 and the set of fixed points of a finite family of nonexpansive mappings

in a Hilbert space and proved the strong convergence of the proposed iterative algorithm

to the unique solution of a variational inequality, which is the optimality condition for aminimization problem To the best of our knowledge, there is not any algorithms for solvingproblem1.1

On the other hand, Marino and Xu19 and Zhou 20 introduced and researchedsome iterative scheme for finding a fixed point of a strict pseudocontraction mapping Acedoand Xu21 introduced some parallel and cyclic algorithms for finding a common fixed point

of a family of finite strict pseudocontraction mappings and obtained both weak and strongconvergence theorems for the sequences generated by the iterative schemes

In the present paper, we introduce a new approximation scheme combining theviscosity method with parallel method for finding a common element of the set of solutions

of the generalized equilibrium problem and the set of fixed points of a family of finitely strictpseudocontractions We obtain a strong convergence theorem for the sequences generated bythese processes Based on this result, we also get some new and interesting results The results

in this paper extend and improve some well-known results in the literature

2 Preliminaries

Let H be a real Hilbert space with inner product  · , ·  and norm  ·  Let C be a closed convex subset of H Let symbols → and  denote strong and weak convergences, respectively In a real Hilbert space H, it is well known that

nonempty-λx  1 − λy2 λx2 1 − λy2− λ1 − λx − y2 2.1

for all x, y ∈ H and λ ∈ 0, 1.

For any x ∈ H, there exists a unique nearest point in C, denoted by P C x, such that

x − P C x ≤ x − y for all y ∈ C The mapping P C is called the metric projection of H onto

and



for all x ∈ H and y ∈ C.

For each B ⊆ H, we denote by convB the convex hull of B A multivalued mapping

conv{x1, x2, , x n} ⊆∞

n1G x i .

We will use the following results in the sequel.

Lemma 2.1 see 22 Let B be a nonempty subset of a Hausdorff topological vector space X and let



x ∈B G x / ∅.

For solving the generalized equilibrium problem, let us give the following

assump-tions for the bifunction F, ϕ, and the set C:

A1 Fx, x  0 for all x ∈ C;

A2 F is monotone, that is, Fx, y  Fy, x ≤ 0 for any x, y ∈ C;

A3 for each y ∈ C, x → Fx, y is weakly upper semicontinuous;

A4 for each x ∈ C, y → Fx, y is convex;

A5 for each x ∈ C, y → Fx, y is lower semicontinuous;

B1 For each x ∈ H and r > 0, there exist a bounded subset D x ⊆ C and y x ∈ C ∩ dom ϕ such that for any z ∈ C \ D x,

Lemma 2.2 Let C be a nonempty-closed convex subset of H Let F be a bifunction from C × C to

5 GEPF, ϕ is closed and convex.

Note that for each y ∈ C∩dom ϕ, Gy is nonempty since y ∈ Gy and for each y ∈ C\dom ϕ,

G y  C We will prove that G is a KKM map on C ∩ dom ϕ Suppose that there exists a finite

subset{y1, y2, , y n } of C ∩ dom ϕ and μ i ≥ 0 for all i  1, 2, , n with n

which is a contradiction Hence, G is a KKM map on C ∩ dom ϕ Note that Gy wthe weak

closure of Gy is a weakly closed subset of C for each y ∈ C Moreover, if B2 holds, then

G y w is also weakly compact for each y ∈ C If B1 holds, then for x0 ∈ E, there exists a bounded subset D x0 ⊆ C and y x0∈ C ∩ dom ϕ such that for any z ∈ C \ D x0,

It follows fromA3 and the weak lower semicontinuity of ϕ that

We observe that S r x ⊆ dom ϕ So by similar argument with that in the proof of

Lemma 2.3 in9, we can easily show that S r is single-valued and S ris a firmly type map Next, we claim that FixSr   GEPF, ϕ Indeed, we have the following:

At last, we claim that GEPF, ϕ is a closed convex Indeed, Since Sr is firmly nonexpansive,

S r is also nonexpansive By23, Proposition 5.3, we know that GEPF, ϕ  FixSr is closedand convex

Lemma 2.4 see 24,25 Assume that {α n } is a sequence of nonnegative real numbers such that

3 Strong Convergence Theorems

In this section, we show a strong convergence of an iterative algorithm based on bothviscosity approximation method and parallel method which solves the problem of finding

a common element of the set of solutions of a generalized equilibrium problem and the set offixed points of a family of finitely strict pseudocontractions in a Hilbert space

We need the following assumptions for the parameters {γ n }, {r n }, {α n }, {ζ1n }, {ζ n2 }, , {ζ n N }, and {β n}:

C1 limn→ ∞α n 0 and ∞n1α n ∞;

C2 1 > lim sup n→ ∞β n≥ lim infn→ ∞β n > 0;

C3 {γ n } ⊂ c, d for some c, d ∈ ε, 1 and lim n→ ∞|γ n1− γ n|  0;

C4 lim infn→ ∞rn > 0 and limn→ ∞|r n1− r n|  0;

C5 limn→ ∞|ζ n1 j − ζ n j |  0 for all j  1, 2, , N.

Theorem 3.1 Let C be a nonempty-closed convex subset of a real Hilbert space H Let F be a

j1FixTj  ∩ GEPF, ϕ / ∅ Assume for each n, {ζ n j }N j1 is a finite sequence of

ε  max{ε j : 1 ≤ j ≤ N} Assume that either (B1) or (B2) holds Let f be a contraction of C

thatfx − fy ≤ ax − y for all x, y ∈ C So, we have

PΩf x − PΩf y ≤ fx − fy ≤ ax − y 3.2

for all x, y ∈ C Since H is complete, there exists a unique element u0 ∈ C such that u0 

Let u ∈ Ω and let {S r n} be a sequence of mappings defined as inLemma 2.2 From

Put M0 max{x1−u, 1/1−afu−u} It is obvious that x1−u ≤ M0 Suppose

x n − u ≤ M0 From3.3, 3.5, and x n1 α nf x n   β nxn  1 − α n − β n y n, we have

xn1− u  α nf

≤ α nf

x n − fu  α nf u − u  β nx n − u  1 − α n − β n y n − u

≤ α n ax n − u  α nf u − u  β nx n − u  1 − α n − β n u n − u

≤ α naxn − u  α nf u − u  1 − α n xn − u

for every n  1, 2, Therefore, {x n} is bounded From 3.3 and 3.5, we also obtain that

{y n } and {u n} are bounded

Following26, define B n : C → C by

As shown in26, each B n is a nonexpansive mapping on C Set M1  supn≥1{u n − W nun},

Without loss of generality, let us assume that there exists a real number b such that r n > b > 0

for all n ∈ N Then,

It follows fromC1–C5 that

xn − y n ≤ 1

1− β n

xn1− x n   α nf x n  − y n . 3.23

It follows fromC1 and C2 that limn→ ∞x n − y n  0

Since x n1 α nf x n   β nxn  1 − α n − β n y n, for u∈ Ω, it follows from 3.5 and 3.3that

It follows fromC1–C3 and x n1− x n → 0 that

It follows fromC1, C2, and x n − x n1 → 0 that limn→ ∞x n − u n  0

Next, we show that

we obtain that u n i  w From x n − y n  → 0, we also obtain that y n i  w Since {u n i } ⊂ C and C is closed and convex, we obtain w ∈ C.

We first show that w ∈N

k1FixTk  To see this, we observe that we may assume by

passing to a further subsequence if necessary ζn i

k → ζ k as i → ∞ for k  1, 2, , N It is easy to see that ζ k > 0 and N k1ζk 1 We also have

It follows fromA4, A5 and the weakly lower semicontinuity of ϕ, u n i − x n i /r n i →

0, and u n i  w that

For t with 0 < t ≤ 1 and y ∈ C ∩ dom ϕ, let y t  ty  1 − tw Since y ∈ C ∩ dom ϕ and

w ∈ C ∩ dom ϕ, we obtain y t ∈ C ∩ dom ϕ, and hence Fy t, w   ϕw ≤ ϕy t So by A4 and

the convexity of ϕ, we have

Finally, we show that x n → u0, where u0 PΩf u0

FromLemma 2.5, we have

It follows fromC1, 3.43, 3.45, andLemma 2.4that limn→ ∞x n − u0  0 From

x n − u n  → 0 and y n − x n  → 0, we have u n → u0 and y n → u0 The proof is nowcomplete

Theorem 3.2 Let C be a nonempty-closed convex subset of a real Hilbert space H Let F be a

j1FixTj  ∩ GEPF, ϕ / ∅ Assume for each n, {ζ n j }N

ε  max{ε j : 1 ≤ j ≤ N} Assume that either (B1) or (B2) holds Let v be an arbitrary point in

4 Applications

By Theorems 3.1 and 3.2, we can obtain many new and interesting strong convergence

theorems Now, give some examples as follows: for j  1, 2, , N, let T1  T2  · · ·  T N  T,

by Theorems3.1and3.2, respectively, we have the following results

Theorem 4.1 Let C be a nonempty-closed convex subset of a real Hilbert space H Let F be a

(B2) holds Let f be a contraction of C into itself and let {x n }, {u n }, and {y n } be sequences generated

Theorem 4.2 Let C be a nonempty-closed convex subset of a real Hilbert space H Let F be a

We need the following two assumptions

B3 For each x ∈ H and r > 0, there exist a bounded subset D x ⊆ C and y x ∈ C such that for any z ∈ C \ D x,

Theorem 4.3 Let C be a nonempty-closed convex subset of a real Hilbert space H Let F be a

C → C be an ε j -strict pseudocontraction for some 0 ≤ ε j < 1 such thatΓ N

j1FixTj ∩EPF / ∅.

Theorem 4.4 Let C be a nonempty-closed convex subset of a real Hilbert space H Let F be a

C → C be an ε j -strict pseudocontraction for some 0 ≤ ε j < 1 such thatΓ N

j1FixTj ∩EPF / ∅.

Let Fx, y  gy − gx for all x, y ∈ C, by Theorems3.1and 3.2, respectively, weobtain the following results

Theorem 4.5 Let C be a nonempty-closed convex subset of a real Hilbert space H Let g : C → R

1 ≤ j ≤ N, let T j : C → C be an ε j-strict pseudocontraction for some 0 ≤ ε j < 1 such thatΘ 

N

j1FixTj ∩Argming, ϕ / ∅ Assume for each n, {ζ j n}N

Theorem 4.6 Let C be a nonempty-closed convex subset of a real Hilbert space H Let g : C → R

1 ≤ j ≤ N, let T j : C → C be an ε j -strict pseudocontraction for some 0 ≤ ε j < 1 such thatΘ 

N

j1FixTj ∩Argming, ϕ / ∅ Assume for each n, {ζ j n}N j1is a finite sequence of positive numbers

Let ϕx  δ C x, ∀x ∈ H, and let Fx, y  0 for all x, y ∈ C Then u n  P Cxn  x n ByTheorems3.1and3.2, we obtain the following results

Theorem 4.7 Let C be a nonempty-closed convex subset of a real Hilbert space H Let N ≥ 1 be an