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Volume 2011, Article ID 217407, 27 pagesdoi:10.1155/2011/217407 Research Article A New Hybrid Algorithm for a System of Mixed Equilibrium Problems, Fixed Point Problems for Nonexpansive

Volume 2011, Article ID 217407, 27 pages

doi:10.1155/2011/217407

Research Article

A New Hybrid Algorithm for a System of

Mixed Equilibrium Problems, Fixed Point Problems for Nonexpansive Semigroup, and Variational

Inclusion Problem

Thanyarat Jitpeera and Poom Kumam

Department of Mathematics, Faculty of Science, King Mongkut’s University of

Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand

Correspondence should be addressed to Poom Kumam,poom.kum@kmutt.ac.th

Received 14 December 2010; Accepted 15 January 2011

Academic Editor: Jen Chih Yao

Copyrightq 2011 T Jitpeera and P Kumam This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited

The purpose of this paper is to consider a shrinking projection method for finding the commonelement of the set of common fixed points for nonexpansive semigroups, the set of common fixed

points for an infinite family of a ξ-strict pseudocontraction, the set of solutions of a system of

mixed equilibrium problems, and the set of solutions of the variational inclusion problem Strongconvergence of the sequences generated by the proposed iterative scheme is obtained The resultspresented in this paper extend and improve some well-known results in the literature

1 Introduction

mixed equilibrium problems is to find x ∈ C such that

IfΓ is a singleton, the problem 1.1 reduces to find the following mixed equilibrium

problemsee also the work of Flores-Baz´an in 1 For finding x ∈ C such that,

F

x, y

 ϕy

The equilibrium problem include fixed point problems, optimization problems, tional inequalities problems, Nash equilibrium problems, noncooperative games, economics

We denote the set of fixed points of T by FT, that is FT  {x ∈ C : x  Tx}.

Definition 1.1 A family S  {Ss : 0 ≤ s ≤ ∞} of mappings of C into itself is called a

nonexpansive semigroup on C if it satisfies the following conditions:

1 S0x  x, for all x ∈ C;

2 Ss  t  SsSt, for all s, t ≥ 0;

3 Ssx − Ssy ≤ x − y, for all x, y ∈ C and s ≥ 0;

4 for all x ∈ C, s → Ssx is continuous.

We denoted by FS the set of all common fixed points of S  {Ss : s ≥ 0}, that is,

where θ is the zero vecter in H The set of solutions of problem1.8 is denoted by IB, M.

{f, x ∈ H × H : f ∈ Mx} of M is not properly contained in the graph of any other monotone mapping It is known that a monotone mapping M is maximal if and only if for

x, f ∈ H × H, x − y, f − g ≥ 0 for all y, g ∈ GM imply f ∈ Mx.

Definition 1.2 A mapping B : C → H is said to be a k-Lipschitz continous if there exists a constant k > 0 such that

Definition 1.3 A mapping B : C → H is said to be a β-inverse-strongly monotone if there exists

a constant β > 0 with the property

Remark 1.4 It is obvious that any β-inverse-strongly monotone mappings B is monotone

Definition 1.5 Let η : C × C → H is called Lipschitz continuous, if there exists a constant

2x − y2

In particular, if ηx, y  x − y for all x, y ∈ C, then K is said to be strongly convex.

Definition 1.6 Let M : H → 2Hbe a set-valued maximal monotone mapping, then the

is called the resolvent operator associated with M, where λ is any positive number and I is the

identity mapping The following characterizes the resolvent operator

R1 The resolvent operator J M,λ is single-valued and nonexpansive for all λ > 0, that is,

λ > 0; see also21, that is,

R5 IB, M is closed and convex.

for a nonexpansive mapping by using the shrinking projection method in mathematical

for finding a common element of the set of solution of generalized equilibrium problemand the set of fixed points of a nonexpansive mapping in a Hilbert space They proved thestrong convergence theorems under certain appropriate conditions imposed on parameters

by

λ and {α n } is a sequence in the interval 0, 1 Peng et al 25 introduced the iterative scheme

by the viscosity approximation method for finding a common element of the set of solutions

to the problem1.8, the set of solutions of an equilibrium problem, and the set of fixed points

of a nonexpansive mapping in a Hilbert space

common element of the set of solution for a system of equilibrium problems and the set

of common fixed points for a finite family of nonexpansive mappings and a nonexpansive

finding a common element of the set of fixed points of a family of finitely nonexpansivemappings, the set of solutions of a mixed equilibrium problem and the set of solutions

W-mapping defined by 2.8, f be a contraction mapping and A, B be inverse-strongly

theorem of the iterative sequence generated by the shrinking projection method for finding acommon element of the set of solutions of generalized mixed equilibrium problems, the set

of fixed points of a finite family of quasinonexpansive mappings, and the set of solutions ofvariational inclusion problems

of the set of solutions of mixed equilibrium problems, the set of common fixed pointsfor nonexpansive semigroup and the set of solution of quasivariational inclusions withmultivalued maximal monotone mappings and inverse-strongly monotone mappings

the common element of the set of common fixed points for a finite family of a ξ-strict

pseudocontraction, the set of solutions of a systems of equilibrium problems and the set

of solutions of variational inclusions Then, they proved strong convergence theorems ofthe iterative sequence generated by the shrinking projection method under some suitable

method for finding a common element of solution set of quasi variational inclusion problemsand of the common fixed point set of an infinite family of nonexpansive mappings

In this paper, motivated and inspired by the previously mentioned results, weintroduce an iterative scheme by the shrinking projection method for finding the commonelement of the set of common fixed points for nonexpansive semigroups, the set of common

fixed points for an infinite family of a ξ-strict pseudocontraction, the set of solutions of a

systems of mixed equilibrium problems and the set of solutions of the variational inclusionsproblem Then, we prove a strong convergence theorem of the iterative sequence generated

by the shrinking projection method under some suitable conditions The results obtained inthis paper extend and improve several recent results in this area

2 Preliminaries

Let H be a real Hilbert space and C be a nonempty closed convex subset of H Recall that

needed in the rest of this paper

Lemma 2.1 For a given z ∈ H, u ∈ C, u  PC z ⇔ u − z, v − u ≥ 0, for all v ∈ C.

It is well known that P C is a firmly nonexpansive mapping of H onto C and satisfies

Moreover, P C x is characterized by the following properties: P C x ∈ C and for all x ∈ H, y ∈ C,

Lemma 2.2 see 20 Let M : H → 2 H be a maximal monotone mapping and let B : H → H

be a Lipshitz continuous mapping Then the mapping L  M  B : H → 2 H is a maximal monotone mapping.

Lemma 2.3 see 31 Let C be a closed convex subset of H Let {x n } be a bounded sequence in H.

Assume that

Lemma 2.4 see 32 Each Hilbert space H satisfies Opial’s condition, that is, for any sequence

Lemma 2.5 see 33 Each Hilbert space H, satisfies the Kadec-Klee property, that is, for any

sequence {x n } with x n  x and x n  → x together imply x n − x → 0.

Lemma 2.6 see 34 Let C be a nonempty closed convex subset of a real Hilbert space H and let ϕ

be a lower semicontinuous and convex functional from C to R Let F be a bifunction from C × C to R

satisfying (H1)–(H3) Assume that

i η : C × C → H is k Lipschitz continuous with constant k > 0 such that;

a ηx, y  ηy, x  0, for all x, y ∈ C,

b η·, · is affine in the first variable,

c for each fixed x ∈ C, y → ηx, y is sequentially continuous from the weak topology

to the weak topology,

continuous from the weak topology to the strong topology;

For given r > 0, Let K F

4 MEPF, ϕ is closed and convex.

Lemma 2.7 see 35 Let V : C → H be a ξ-strict pseudocontraction, then

2 define a mapping T : C → H by

pseudocontractions, if there exists a constant ξ ∈ 0, 1 such that

V i x − V i y2≤x − y2 ξI − V i x − I − V i y2

Let {V i : C → C}∞i1 be a countable family of uniformly ξ-strict pseudocontractions Let

of C into itself as follows:

Lemma 2.8 see 36 Let C be a nonempty closed convex subset of a real Hilbert space H Let

T1, T2, be nonexpansive mappings of C into itself such that∞

Lemma 2.9 see 36 Let C be a nonempty closed convex subset of a Hilbert space H, {T i : C → C}

be a countable family of nonexpansive mappings with∞

i1F T i  / ∅, {μ i } be a real sequence such that

0 < μ i ≤ b < 1, for all i ≥ 1 Then FW ∞i1F T i .

Lemma 2.10 see 37 Let C be a nonempty closed convex subset of a Hilbert space H, {T i : C →

that 0 < μ i ≤ b < 1, for all i ≥ 1 If D is any bounded subset of C, then

lim

n→ ∞sup

Lemma 2.11 see 38 Let C be a nonempty bounded closed convex subset of a Hilbert space H and

Lemma 2.12 see 39 Let C be a nonempty bounded closed convex subset of H, {x n } be a sequence

in C and S  {Ss : 0 ≤ s < ∞} be a nonexpansive semigroup on C If the following conditions are

groups, the set of common fixed points for an infinite family of ξ-strict pseudocontraction,

the set of solutions of a systems of mixed equilibrium problems and the set of solutions of thevariational inclusions problem in a real Hilbert space

Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H, let {Fk : C×

(H1)–(H3) Let S  {Ss : 0 ≤ s < ∞} be a nonexpansive semigroup on C and let {t n } be a

positive real divergent sequence Let {V i : C → C}∞i1be a countable family of uniformly ξ-strict pseudocontractions, {T i : C → C}∞i1be the countable family of nonexpansive mappings defined by

and W be a mapping defined by2.9 with FW / ∅ Let A, B : C → H be γ, β-inverse-strongly

monotone mappings and M1, M2: H → 2H be maximal monotone mappings such that

Let r k > 0, k  1, 2, , N, which are constants Let {x n }, {y n }, {v n }, {z n }, and {u n } be sequences

for all n ∈ N Assume the following conditions are satisfied:

to the weak topology;

sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with a Lipschitz constant ν k > 0 such that σ k > L k ν k ;

C7 lim inf n→ ∞r k,n > 0, for each k ∈ 1, 2, 3, , N.

Then, {x n } and {u n } converge strongly to z  PΘx0.

Proof Pick any p ∈ Θ Taking Ik  K F k

Next, we will divide the proof into eight steps

Step 1 We first show by induction that Θ ⊂ C n for each n≥ 1

Applying3.5 and 3.6, we get

Step 2 Next, we show that {x n } is well defined and C n is closed and convex for any n∈ N

Hence, we get{x n} is bounded It follows by 3.5–3.7, that {v n }, {y n }, and {W n v n} are also

limn→ ∞xn − x0 exists, that is

On the other hand, from x n1 P C n1x0∈ C n1⊂ C n, which implies that

By3.5, 3.6, 3.7, and 3.27, we have for each k ∈ {1, 2, 3, , N}

Since the resolvent operator J M1,λ n is 1-inverse-strongly monotone, we obtain

From3.8 and C4, we also have

Step 7 Next, we show that q ∈ Θ : FS∩FW∩N

therefore we have q / Kn W n q, for all n ≥ m It follows from the Opial’s condition and 3.44that

We observe that A is an 1/γ-Lipschitz monotone mapping and DA  H From

Lemma 2.2, we know that M1  A is maximal monotone Let v, g ∈ GM1  A that is,

Step 8 Finally, we show that x n → z and u n → z, where z  PΘ x0.

By the weakly lower semicontinuous of the norm, we have

Theorem 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H, let {Fk : C×

(H1)–(H3) Let S  {Ss : 0 ≤ s < ∞} be a nonexpansive semigroup on C and let {t n } be a

positive real divergent sequence Let {V i : C → C}∞

i1be a countable family of uniformly ξ-strict pseudocontractions, {T i : C → C}∞i1be the countable family of nonexpansive mappings defined by

and W be a mapping defined by2.9 with FW / ∅ Let A, B : C → H be γ, β-inverse-strongly

monotone mapping Such that

Let r k > 0, k  1, 2, , N, which are constants Let {x n }, {y n }, {v n }, {z n }, and {u n } be sequences

0, 1 for all n ∈ N Assume the following conditions are satisfied:

to the weak topology;

sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with a Lipschitz constant ν k > 0 such that σ k > L k ν k ;

C7 lim inf n→ ∞r k,n > 0, for each k ∈ 1, 2, 3, , N.

Then, {x n } and {u n } converge strongly to z  PΘx0.

Proof InTheorem 3.1, take M i   iC : H → 2H , where  iC : 0 → 0, ∞ is the indicator function of C, that is,

Next, we consider another class of important mappings

Definition 3.3 A mapping S : C → C is called strictly pseudocontraction if there exists a

Now, we obtain the following result.

Theorem 3.4 Let C be a nonempty closed convex subset of a real Hilbert space H, let {Fk : C×

(H1)–(H3) Let S  {Ss : 0 ≤ s < ∞} be a nonexpansive semigroup on C and let {t n } be a

positive real divergent sequence Let {V i : C → C}∞

i1be a countable family of uniformly ξ-strict pseudocontractions, {T i : C → C}∞i1be the countable family of nonexpansive mappings defined by

and W be a mapping defined by2.9 with FW / ∅ Let A, B : C → H be γ, β-inverse-strongly

monotone mapping and S A , S B be κ γ , κ β -strictly pseudocontraction mapping of C into C for some

for all n ∈ N Assume the following conditions are satisfied:

to the weak topology;

sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with a Lipschitz constant ν k > 0 such that σ k > L k ν k ;

C3 for each k ∈ {1, 2, , N} and for all x ∈ C, there exist a bounded subset D x ⊂ C and

C7 lim inf n→ ∞r k,n > 0, for each k ∈ 1, 2, 3, , N

Then, {x n } and {u n } converge strongly to z  PΘ x0.

Proof Taking A ≡ I −S A and B ≡ I −S B , then we see that A, B is 1−κ γ /2, 1−κ β

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of common fixed points for a finite family of nonexpansive mappings and a nonexpansive

finding a common element of the set of fixed points of a family of finitely nonexpansivemappings,... points of a finite family of quasinonexpansive mappings, and the set of solutions ofvariational inclusion problems

of the set of solutions of mixed equilibrium problems, the set of common fixed. ..

On the other hand, from x n1 P C n1x0∈