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Volume 2011, Article ID 941090, 20 pagesdoi:10.1155/2011/941090 Research Article Convergence Analysis for a System of Generalized Equilibrium Problems and a Countable Family of Strict Ps

Volume 2011, Article ID 941090, 20 pages

doi:10.1155/2011/941090

Research Article

Convergence Analysis for a System of

Generalized Equilibrium Problems and a Countable Family of Strict Pseudocontractions

Prasit Cholamjiak1 and Suthep Suantai1, 2

1 Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

2 Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Correspondence should be addressed to Suthep Suantai,scmti005@chiangmai.ac.th

Received 18 October 2010; Accepted 27 December 2010

Academic Editor: Jen Chih Yao

Copyrightq 2011 P Cholamjiak and S Suantai 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

We introduce a new iterative algorithm for a system of generalized equilibrium problems and

a countable family of strict pseudocontractions in Hilbert spaces We then prove that the sequence generated by the proposed algorithm converges strongly to a common element in the solutions set

of a system of generalized equilibrium problems and the common fixed points set of an infinitely countable family of strict pseudocontractions

1 Introduction

Let H be a real Hilbert space with the inner product ·, · and inducted norm  ·  Let C

be a nonempty, closed, and convex subset of H Let {f k}k∈Λ : C × C → Ê be a family of bifunctions, and let{A k}k∈Λ : C → H be a family of nonlinear mappings, where Λ is an

arbitrary index set The system of generalized equilibrium problems is to findx ∈ C such that

f k



x, yA k x, y − x≥ 0, ∀y ∈ C, k ∈ Λ. 1.1

IfΛ is a singleton, then 1.1 reduces to find x ∈ C such that

f

x, yA  x, y −  x

The solutions set of 1.2 is denoted by GEPf, A If f ≡ 0, then the solutions set of 1.2

is denoted by VIC, A, and if A ≡ 0, then the solutions set of 1.2 is denoted by EPf.

2 Fixed Point Theory and Applications The problem1.2 is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, and the Nash equilibrium problem

in noncooperative games; see also1,2 Some methods have been constructed to solve the system of equilibrium problemssee, e.g., 3 7 Recall that a mapping A : C → H is

said to be

1 monotone if



Ax − Ay, x − y

2 α-inverse-strongly monotone if there exists a constant α > 0 such that



Ax − Ay, x − y

≥ αAx − Ay2

It is easy to see that if A is α-inverse-strongly monotone, then A is monotone and 1/α-Lipschitz.

For solving the equilibrium problem, let us assume that f satisfies the following

conditions:

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 each x, y, z ∈ C, lim t → 0 ftz  1 − tx, y ≤ fx, y,

A4 for each x ∈ C, y → fx, y is convex and lower semicontinuous.

Throughout this paper, we denote the fixed points set of a nonlinear mapping

T : C → C by FT  {x ∈ C : Tx  x} Recall that T is said to be a κ-strict pseudocontraction if

there exists a constant 0≤ κ < 1 such that

Tx − Ty2 ≤x − y2 κI − Tx − I − Ty2

It is well known that1.5 is equivalent to



Tx − Ty, x − y

≤x − y2−1− κ

2 I − Tx − I − Ty2

It is worth mentioning that the class of strict pseudocontractions includes properly the

class of nonexpansive mappings It is also known that every κ-strict pseudocontraction is

1  κ/1 − κ-Lipschitz; see 8

In 1953, Mann9 introduced the iteration as follows: a sequence {x n} defined by

x0∈ C and

where{α n}∞n0 ⊂ 0, 1 If S is a nonexpansive mapping with a fixed point and the control

sequence {α n}∞

n0 is chosen so that∞

n0 α n 1 − α n   ∞, then the sequence {x n} defined

by 1.7 converges weakly to a fixed point of S this is also valid in a uniformly convex

Banach space with the Fr´echet differentiable norm 10

In 1967, Browder and Petryshyn11 introduced the class of strict pseudocontractions and proved existence and weak convergence theorems in a real Hilbert setting by using Mann iterative algorithm1.7 with a constant sequence α n  α for all n ≥ 0 Recently, Marino

and Xu8 and Zhou 12 extended the results of Browder and Petryshyn 11 to Mann’s iteration process1.7 Since 1967, the construction of fixed points for pseudocontractions via the iterative process has been extensively investigated by many authorssee, e.g., 13–22

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

S : C → C be a nonexpansive mapping, f : C × C → Êa bifunction, and let A : C → H be

an inverse-strongly monotone mapping

In 2008, Moudafi23 introduced an iterative method for approximating a common

element of the fixed points set of a nonexpansive mapping S and the solutions set of a

generalized equilibrium problem GEPf, A as follows: a sequence {xn } defined by x0 ∈ C

and

f

y n , y

Ax n , y − y n



 1

r n



y − y n , y n − x n



≥ 0, ∀y ∈ C,

x n1  α n x n  1 − α n Sy n , n ≥ 1,

1.8

where{α n}∞n0 ⊂ 0, 1 and {r n}∞n0 ⊂ 0, ∞ He proved that the sequence {x n} generated by

1.8 converges weakly to an element in GEPf, A ∩ FS under suitable conditions.

Due to the weak convergence, recently, S Takahashi and W Takahashi24 introduced another modification iterative method of 1.8 for finding a common element of the fixed points set of a nonexpansive mapping and the solutions set of a generalized equilibrium problem in the framework of a real Hilbert space To be more precise, they proved the following theorem

Theorem 1.1 see 24 Let C be a closed convex subset of a real Hilbert space H, and let

f : C × C → Ê be a bifunction satisfying (A1)–(A4) Let A be an α-inverse-strongly monotone mapping of C into H, and let S be a nonexpansive mapping of C into itself such that FS ∩

GEPf, A / ∅ Let u ∈ C and x1∈ C, and let {y n } ⊂ C and {x n } ⊂ C be sequences generated by

f

y n , y

Ax n , y − y n



 1

r n



y − y n , y n − x n



≥ 0, ∀y ∈ C,

x n1  β n x n1− β n



S

α n u  1 − α n y n , n ≥ 1,

1.9

where {α n}∞n1 ⊂ 0, 1, {β n}∞n1 ⊂ 0, 1 and {r n}∞n1 ⊂ 0, 2α satisfy

i limn → ∞ α n  0 and∞

n1 α n  ∞,

ii 0 < c ≤ β n ≤ d < 1,

iii 0 < a ≤ r n ≤ b < 2α,

iv limn → ∞ r n − r n1   0.

Then, {x n } converges strongly to z  P FS ∩ GEPf,A u.

4 Fixed Point Theory and Applications Recently, Yao et al.25 introduced a new modified Mann iterative algorithm which is different from those in the literature for a nonexpansive mapping in a real Hilbert space To

be more precise, they proved the following theorem

Theorem 1.2 see 25 Let C be a nonempty, closed, and convex subset of a real Hilbert space H.

Let S : C → C be a nonexpansive mapping such that FS /  ∅ Let {α n}∞

n0 , and let {β n}∞

n0 be two real sequences in 0, 1 For given x0 ∈ C arbitrarily, let the sequence {x n }, n ≥ 0, be generated

iteratively by

y n  P C 1 − α n x n ,

x n11− β n



Suppose that the following conditions are satisfied:

i limn → ∞ α n  0 and∞n0 α n  ∞,

ii 0 < lim inf n → ∞ β n≤ lim supn → ∞ β n < 1,

then, the sequence {x n } generated by 1.10 strongly converges to a fixed point of S.

We know the following crucial lemmas concerning the equilibrium problem in Hilbert spaces

Lemma 1.3 see 1 Let C be a nonempty, closed, and convex subset of a real Hilbert space H, let f

be a bifunction from C × C toÊsatisfying (A1)–(A4) Let r > 0 and x ∈ H Then, there exists z ∈ C such that

f

z, y

1

r



y − z, z − x

Lemma 1.4 see 26 Let C be a nonempty, closed, and convex subset of a real Hilbert space H Let

f be a bifunction from C × C toÊ satisfying (A1)–(A4) For x ∈ H and r > 0, define the mapping

T r f : H → 2 C as follows:

T r f x 

z ∈ C : f

z, y

 1

r



y − z, z − x

≥ 0, ∀y ∈ C

Then, the following statements hold:

1 T f

r is single-valued,

2 T f

r is firmly nonexpansive, that is, for any x, y ∈ H,



T r f x − T r f y2

≤ T r f x − T r f y, x − y

3 FT f

r   EPf,

4 EPf is closed and convex.

Let C be a nonempty, closed, and convex subset of a real Hilbert space H Let r k > 0

for each k ∈ {1, 2, , M} Let {f k}M

k1 : C × C → Êbe a family of bifunctions, let{A k}M

k1 :

C → H be a family of α k-inverse-strongly monotone mappings, and let{T n}∞n1 : C → C

be a countable family of κ-strict pseudocontractions For each k ∈ {1, 2, , M}, denote the mapping T f k ,A k

r k : C → C by T f k ,A k

r k : Tf k

r k I − r k A k , where T f k

r k : H → C is the mapping

defined as inLemma 1.4

Motivated and inspired by Marino and Xu 8, Moudafi 23, S Takahashi and W Takahashi24, and Yao et al 25, we consider the following iteration: x1∈ C and

y n  P C 1 − α n x n ,

u n  T f M ,A M

r M T f M−1 , A M−1

r M−1 · · · T f2,A2

r2 T f1,A1

r1 y n ,

x n1  β n x n1− β n



γu n1− γT n u n , n ≥ 1,

1.14

where{α n}∞n1 ⊂ 0, 1, {β n}∞n1 ⊂ 0, 1 and γ ∈ 0, 1.

In this paper, we first prove a path convergence result for a nonexpansive mapping and a system of generalized equilibrium problems Then, we prove a strong convergence theorem of the iteration process1.14 for a system of generalized equilibrium problems and

a countable family of strict pseudocontractions in a real Hilbert space Our results extend the main results obtained by Yao et al.25 in several aspects

2 Preliminaries

Let C be a nonempty, closed, and convex subset of a real Hilbert space H For each x ∈ H, there exists a unique nearest point in C, denoted by P C x, such that x −P C x  min y∈C x−y.

P C is called the metric projection of H onto C It is also known that for x ∈ H and z ∈ C,

z  P C x is equivalent to x − z, y − z ≤ 0 for all y ∈ C Furthermore,

y − P C x2

 x − P C x2≤x − y2

for all x ∈ H, y ∈ C In a real Hilbert space, we also know that

λx  1 − λy2

 λx2 1 − λy2

− λ1 − λx − y2

for all x, y ∈ H and λ ∈ 0, 1.

In the sequel, we need the following lemmas

Lemma 2.1 see 27,28 Let E be a real uniformly convex Banach space, and let C be a nonempty,

closed, and convex subset of E, and let S : C → C be a nonexpansive mapping such that FS /  ∅,

then I − S is demiclosed at zero.

Lemma 2.2 see 29 Let {x n } and {z n } be two sequences in a Banach space E such that

x n1  β n x n1− β n



z n , n ≥ 1, 2.3

6 Fixed Point Theory and Applications

where {β n}∞n1 satisfies conditions: 0 < lim inf n → ∞ β n≤ lim supn → ∞ β n < 1 If lim sup n → ∞ z n1−

z n  − x n1 − x n  ≤ 0, then x n − z n  → 0 as n → ∞.

Lemma 2.3 see 30 Assume that {a n}∞n1 is a sequence of nonnegative real numbers such that

a n1≤1− γ n



a n  γ n δ n , n ≥ 1, 2.4

where {γ n}∞n1 is a sequence in 0, 1 and {δ n}∞n1 is a sequence inÊsuch that

a∞n1 γ n  ∞; b lim sup n → ∞ δ n ≤ 0 or∞n1 |γ n δ n | < ∞.

Then, lim n → ∞ a n  0.

Lemma 2.4 see 31 Let C be a nonempty, closed, and convex subset of a real Hilbert space H Let

the mapping A : C → H be α-inverse-strongly monotone, and let r > 0 be a constant Then, we have

I − rAx −I − rAy2≤x − y2

 rr − 2αAx − Ay2

for all x, y ∈ C In particular, if 0 ≤ r ≤ 2α, then I − rA is nonexpansive.

To deal with a family of mappings, the following conditions are introduced: let C

be a subset of a real Hilbert space H, and let {T n}∞n1 be a family of mappings of C such

that∞

n1 FT n  / ∅ Then, {T n } is said to satisfy the AKTT-condition 32 if for each bounded

subset B of C,

∞



n1

Lemma 2.5 see 32 Let C be a nonempty and closed subset of a Hilbert space H, and let {T n } be

a family of mappings of C into itself which satisfies the AKTT-condition Then, for each x ∈ C, {T n x} converges strongly to a point in C Moreover, let the mapping T be defined by

Tx  lim

Then, for each bounded subset B of C,

lim sup

n → ∞

The following results can be found in33,34

Lemma 2.6 see 33,34 Let C be a closed, and convex subset of a Hilbert space H Suppose that {T n}∞n1 is a family of κ-strictly pseudocontractive mappings from C into H with∞

n1 FT n  / ∅ and {μ n}∞n1 is a real sequence in 0, 1 such that∞n1 μ n  1 Then, the following conclusions hold:

1 G :∞n1 μ n T n : C → H is a κ-strictly pseudocontractive mapping,

2 FG ∞

n1 FT n .

Lemma 2.7 see 34 Let C be a closed and convex subset of a Hilbert space H Suppose that {S i}∞i1

is a countable family of κ-strictly pseudocontractive mappings of C into itself with∞

i1 FS i  / ∅.

For each n ∈Æ, define T n : C → C by

T n x 

n



i1

μ i n S i x, x ∈ C, 2.9

where {μ i

n } is a family of nonnegative numbers satisfying

in

i1 μ i

n  1 for all n ∈Æ,

ii μ i: limn → ∞ μ i

n > 0 for all i ∈Æ,

iii∞

n1

n

i1 |μ i n1 − μ i

n | < ∞.

Then,

1 Each T n is a κ-strictly pseudocontractive mapping.

2 {T n } satisfies AKTT-condition.

3 If T : C → C is defined by

Tx 

∞



i1

μ i S i x, x ∈ C, 2.10

then Tx  lim n → ∞ T n x and FT ∞

n1 FT n ∞i1 FS i .

In the sequel, we will write{T n }, T satisfies the AKTT-condition if {T n} satisfies the

AKTT-condition and T is defined byLemma 2.5with FT ∞

n1 FT n

3 Path Convergence Results

Let C be a nonempty, closed, and convex subset of a real Hilbert space H Let S : C → C be

a nonexpansive mapping Let{f k}M

k1 : C × C → Ê be a family of bifunctions, let{A k}M

k1 :

C → H be a family of α k -inverse-strongly monotone mappings, and let r k ∈ 0, 2α k For each

k ∈ {1, 2, , M}, we denote the mapping T f k ,A k

r k : C → C by

T f k ,A k

r k : Tf k

where T f k

r k is the mapping defined as inLemma 1.4 For each t ∈ 0, 1, we define the mapping

S t : C → C as follows:

S t x  ST f M ,A M

r M T f M−1 ,A M−1

r M−1 · · · T f1,A1

By Lemmas 1.42 and 2.4, we know that T f k

r k and I − r k A k are nonexpansive for each

k ∈ {1, 2, , M} So, the mapping T f k ,A k

r is also nonexpansive for each k ∈ {1, 2, , M}.

8 Fixed Point Theory and Applications

Moreover, we can check easily that S tis a contraction Then, the Banach contraction principle

ensures that there exists a unique fixed point x t of S t in C, that is,

x t  ST f M ,A M

r M T f M−1 ,A M−1

r M−1 · · · T f1,A1

r1 P C 1 − tx t , t ∈ 0, 1. 3.3

Theorem 3.1 Let C be a nonempty, closed, and convex subset of a real Hilbert space H Let

S : C → C be a nonexpansive mapping Let {f k}M

k1 : C × C → Ê be a family of bifunctions, let {A k}M

k1 : C → H be a family of α k -inverse-strongly monotone mappings, and let r k ∈

0, 2α k  For each k ∈ {1, 2, , M}, let the mapping T f k ,A k

r k be defined by 3.1 Assume that

F : M

k1GEPfk , A k ∩ ∞

n1 FT n  / ∅ For each t ∈ 0, 1, let the net {x t } be generated by

3.3 Then, as t → 0, the net {x t } converges strongly to an element in F.

Proof First, we show that {x t } is bounded For each t ∈ 0, 1, let y t  P C 1 − tx t and

u t  T f M ,A M

r M T f M−1 ,A M−1

r M−1 · · · T f1,A1

r1 y t From3.3, we have for each p ∈ F that

x t − p  Su t − Sp ≤ u t − p ≤ y t − p ≤ 1 − tx t − p  tp. 3.4

It follows that

Hence,{x t } is bounded and so are {y t } and {u t} Observe that

as t → 0 since {x t} is bounded

Next, we show thatu t − x t  → 0 as t → 0 Denote Θ k  T f k ,A k

r k T f k−1 ,A k−1

r k−1 · · · T f1,A1

any k ∈ {1, 2, , M} and Θ0 I We note that u t ΘM y t for each t ∈ 0, 1 FromLemma 2.4,

we have for each k ∈ {1, 2, , M} and p ∈ F that



Θk y t − p2

T f k ,A k

r k Θk−1 y t − T f k ,A k

r k Θk−1 p2

T f k

r k



Θk−1 y t − r k A kΘk−1 y t



− T f k

r k



Θk−1 p − r k A kΘk−1 p2

≤

Θk−1 y t − r k A kΘk−1 y t



−Θk−1 p − r k A kΘk−1 p2

≤Θk−1 y t − p2

 r k r k − 2α kA

kΘk−1 y t − A k p2

.

3.7

It follows that

u t − p2 ΘM y t − p2

≤y t − p2M

i1

r i r i − 2α iA

iΘi−1 y t − A i p2

P C 1 − tx t  − p2M

i1

r i r i − 2α iA

iΘi−1 y t − A i p2

≤x t − p  tx t2M

i1

r i r i − 2α iA

iΘi−1 y t − A i p2

≤x t − p2 tM1M

i1

r i r i − 2α iA

iΘi−1 y t − A i p2

,

3.8

where M1 sup0<t<1 {2x t − px t   tx t2} So, we have

x t − p2 ≤u t − p2

≤x t − p2 tM1M

i1

r i r i − 2α iA

iΘi−1 y t − A i p2

,

3.9

which implies that

lim

t → 0



for each k ∈ {1, 2, , M} Since T f k

r k is firmly nonexpansive for each k ∈ {1, 2, , M}, we have for each p ∈ F and k ∈ {1, 2, , M} that



Θk y t − p2

T f k ,A k

r k Θk−1 y t − T f k ,A k

r k Θk−1 p2

T f k

r k



Θk−1 y t − r k A kΘk−1 y t



− T f k

r k



Θk−1 p − r k A kΘk−1 p2

≤ Θk−1 y t − r k A kΘk−1 y t−p − r k A k p

, Θ k y t − p

 1

2



Θk−1 y t − r k A kΘk−1 y t−p − r k A k p2

Θk y t − p2

−Θk−1 y t − r k A kΘk−1 y t−p − r k A k p

−Θk y t − p2

10 Fixed Point Theory and Applications

≤ 1

2



Θk−1 y t − p2

Θk y t − p2

−Θk−1 y t− Θk y t − r k



A kΘk−1 y t − A k p2

≤ 1

2



Θk−1 y t − p2

Θk y t − p2

−Θk−1 y t− Θk y t2

2r kΘk−1 y t− Θk y tA

kΘk−1 y t − A k p.

3.11 This implies that



Θk y t − p2

≤Θk−1 y t − p2

−Θk−1 y t− Θk y t2

 2r kΘk−1 y t− Θk y tA

kΘk−1 y t − A k p

≤Θk−1

y t − p2

−Θk−1

y t− Θk

y t2

 M2A

kΘk−1

y t − A k p,

3.12

where M2 max1≤k≤Msup0<t<1 {2r kΘk−1 y t− Θk y t} This shows that

u t − p2 ΘM y t − p2

≤y t − p2−M

i1



Θi−1 y t− Θi y t2

 M2

M



i1



A iΘi−1 y t − A i p

≤x t − p2 tM1−M

i1



Θi−1 y t− Θi y t2

 M2

M



i1



A iΘi−1 y t − A i p.

3.13

Hence,

x t − p2 ≤u t − p2

≤x t − p2 tM1−M

i1



Θi−1 y t− Θi y t2

 M2

M



i1



A iΘi−1 y t − A i p. 3.14 From3.10, we obtain

M



i1



Θi−1

y t− Θi

as t → 0 So, we can conclude that

lim

t → 0



8 Fixed Point Theory and Applications

Moreover, we can check easily that S tis a. .. generalized equilibrium problems Then, we prove a strong convergence theorem of the iteration process1.14 for a system of generalized equilibrium problems and

a countable family of strict. .. class="text_page_counter">Trang 5

Let C be a nonempty, closed, and convex subset of a real Hilbert space H Let r k > 0

for