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Volume 2011, Article ID 572156, 14 pagesdoi:10.1155/2011/572156 Research Article A New Strong Convergence Theorem for Equilibrium Problems and Fixed Point Problems in Banach Spaces Weera

Volume 2011, Article ID 572156, 14 pages

doi:10.1155/2011/572156

Research Article

A New Strong Convergence Theorem for

Equilibrium Problems and Fixed Point Problems in Banach Spaces

Weerayuth Nilsrakoo

Department of Mathematics, Statistics and Computer, Faculty of Science, Ubon Ratchathani University, Ubon Ratchathani 34190, Thailand

Correspondence should be addressed to Weerayuth Nilsrakoo,nilsrakoo@hotmail.com

Received 5 June 2010; Revised 28 December 2010; Accepted 20 January 2011

Academic Editor: Fabio Zanolin

Copyrightq 2011 Weerayuth Nilsrakoo 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 sequence for finding a common element of the set of fixed points of

a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in a Banach space Then, we study the strong convergence of the sequences With an appropriate setting, we obtain the corresponding results due to Takahashi-Takahashi and Takahashi-Zembayashi Some of our results are established with weaker assumptions

1 Introduction

Throughout this paper, we denote by andthe sets of positive integers and real numbers,

respectively Let E be a Banach space, E∗the dual space of E and C a closed convex subsets

of E Let F : C × C → be a bifunction The equilibrium problem is to find x ∈ C such that

F

x, y

The set of solutions of 1.1 is denoted by EPF The equilibrium problems include

fixed point problems, optimization problems, variational inequality problems, and Nash equilibrium problems as special cases

Let E be a smooth Banach space and J the normalized duality mapping from E to E∗ Alber1 considered the following functional ϕ : E × E → 0, ∞ defined by

ϕ

x, y

 x2− 2x, Jy

y2 

x, y ∈ E. 1.2

Using this functional, Matsushita and Takahashi2,3 studied and investigated the following

mappings in Banach spaces A mapping S : C → E is relatively nonexpansive if the following

properties are satisfied:

R1 FS /,

R2 ϕp, Sx ≤ ϕp, x for all p ∈ FS and x ∈ C,

R3 FS  FS,

where FS and  FS denote the set of fixed points of S and the set of asymptotic fixed points

of S, respectively It is known that S satisfies condition R3 if and only if I − S is demiclosed

at zero, where I is the identity mapping; that is, whenever a sequence {x n } in C converges weakly to p and {x n −Sx n } converges strongly to 0, it follows that p ∈ FS In a Hilbert space

H, the duality mapping J is an identity mapping and ϕx, y  x − y2 for all x, y ∈ H Hence, if S : C → H is nonexpansive i.e., Sx − Sy ≤ x − y for all x, y ∈ C, then it is

relatively nonexpansive

Recently, many authors studied the problems of finding a common element of the set

of fixed points for a mapping and the set of solutions of equilibrium problem in the setting of Hilbert space and uniformly smooth and uniformly convex Banach space, respectivelysee, e.g.,4 21 and the references therein In a Hilbert space H, S Takahashi and W Takahashi

17 introduced the iteration as follows: sequence {x n } generated by u, x1∈ C,

F

z n , y

 1

r n



y − z n , z n − x n

≥ 0, ∀y ∈ C,

x n1  β n x n1− β n

Sα n u  1 − α n z n ,

1.3

for every n ∈ , where S is nonexpansive, {α n } and {β n } are appropriate sequences in 0, 1,

and{r n } is an appropriate positive real sequence They proved that {x n} converges strongly

to some element in FS ∩ EPF In 2009, Takahashi and Zembayashi 19 proposed the iteration in a uniformly smooth and uniformly convex Banach space as follows: a sequence

{x n } generated by u1∈ E,

x n ∈ C such that Fx n , y

 1

r n



y − x n , Jx n − Ju n

≥ 0, ∀y ∈ C,

u n1  J−1α n Jx n  1 − α n JSx n ,

1.4

for every n ∈ , S is relatively nonexpansive, {α n } is an appropriate sequence in 0, 1, and {r n } is an appropriate positive real sequence They proved that if J is weakly sequentially

continuous, then{x n } converges weakly to some element in FS ∩ EPF.

Motivated by S Takahashi and W Takahashi17 and Takahashi and Zembayashi 19,

we prove a strong convergence theorem for finding a common element of the fixed points set

of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in a uniformly smooth and uniformly convex Banach space

2 Preliminaries

We collect together some definitions and preliminaries which are needed in this paper We

say that a Banach space E is strictly convex if the following implication holds for x, y ∈ E:

x  y  1, x / y imply 

x  y2  < 1. 2.1

It is also said to be uniformly convex if for any ε > 0, there exists δ > 0 such that

x  y  1, x − y ≥ ε implyx  y

2



 ≤ 1 − δ. 2.2

It is known that if E is a uniformly convex Banach space, then E is reflexive and strictly convex We say that E is uniformly smooth if the dual space E∗ of E is uniformly convex A Banach space E is smooth if the limit lim t → 0 xty−x/t exists for all norm one elements

x and y in E It is not hard to show that if E is reflexive, then E is smooth if and only if E∗is strictly convex

Let E be a smooth Banach space The function ϕ : E × E → see 1 is defined by

ϕ

x, y

 x2− 2x, Jy

y2 

x, y ∈ E, 2.3

where the duality mapping J : E → E∗is given by

x, Jx  x2 Jx2 x ∈ E. 2.4

It is obvious from the definition of the function ϕ that



x − y2≤ ϕx, y

≤x  y2, 2.5

ϕ

x, J−1

λJy  1 − λJz ≤ λϕx, y

 1 − λϕx, z, 2.6

for all λ ∈ 0, 1 and x, y, z ∈ E The following lemma is an analogue of Xu’s inequality 22, Theorem 2 with respect to ϕ

Lemma 2.1 Let E be a uniformly smooth Banach space and r > 0 Then, there exists a continuous,

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

ϕ

x, J−1

λJy  1 − λJz ≤ λϕx, y 1 − λϕx, z − λ1 − λgJy − Jz, 2.7

for all λ ∈ 0, 1, x ∈ E, and y, z ∈ B r

It is also easy to see that if{x n } and {y n} are bounded sequences of a smooth Banach

space E, then x n − y n → 0 implies that ϕx n , y n → 0

Lemma 2.2 see 23, Proposition 2 Let E be a uniformly convex and smooth Banach space, and

let {x n } and {y n } be two sequences of E such that {x n } or {y n } is bounded If ϕx n , y n  → 0, then

x n − y n → 0.

Remark 2.3 For any bounded sequences {x n } and {y n} in a uniformly convex and uniformly

smooth Banach space E, we have

ϕ

x n , y n

−→ 0 ⇐⇒ x n − y n −→ 0 ⇐⇒ Jx n − Jy n −→ 0. 2.8

Let C be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space E It is known that1,23 for any x ∈ E, there exists a unique point x ∈ C such

that

ϕx, x  min

y∈C ϕ

y, x

Following Alber1, we denote such an element x by Π C x The mapping Π C is called the

generalized projection from E onto C It is easy to see that in a Hilbert space, the mappingΠC

coincides with the metric projection P C Concerning the generalized projection, the following are well known

Lemma 2.4 see 23, Propositions 4 and 5 Let C be a nonempty closed convex subset of a

reflexive, strictly convex and smooth Banach space E, x ∈ E, and x ∈ C Then,

a x  Π C x if and only if y − x, Jx − J x ≤ 0 for all y ∈ C,

b ϕy, Π C x  ϕΠ C x, x ≤ ϕy, x for all y ∈ C.

Remark 2.5 The generalized projection mapping ΠC above is relatively nonexpansive and

FΠ C   C.

Let E be a reflexive, strictly convex and smooth Banach space The duality mapping

J∗from E∗onto E∗∗  E coincides with the inverse of the duality mapping J from E onto E∗,

that is, J∗ J−1 We make use of the following mapping V : E × E∗ → studied in Alber1

V x, x∗  x2− 2x, x∗  x∗2, 2.10

for all x ∈ E and x∗ ∈ E∗ Obviously, V x, x∗  ϕx, J−1x∗ for all x ∈ E and x∗ ∈ E∗ We know the following lemmasee 1 and 24, Lemma 3.2

Lemma 2.6 Let E be a reflexive, strictly convex and smooth Banach space, and let V be as in 2.10.

Then,

V x, x∗  2 J−1x∗ − x, y∗

≤ Vx, x∗ y∗

for all x ∈ E and x∗, y∗∈ E∗.

Lemma 2.7 see 25,Lemma 2.1 Let {a n } be a sequence of nonnegative real numbers Suppose

that

a n1≤1− γ n

a n  γ n δ n , 2.12

for all n ∈ , where the sequences {γ n } in 0, 1 and {δ n } in satisfy conditions: lim n → ∞ γ n  0,

∞

n1 γ n  ∞, and lim sup n → ∞ δ n ≤ 0 Then, lim n → ∞ a n  0.

Lemma 2.8 see 26, Lemma 3.1 Let {an } be a sequence of real numbers such that there exists

a subsequence {n i } of {n} such that a n i < a n i1 for all i ∈ Then, there exists a nondecreasing sequence {m k } ⊂ such that m k → ∞,

a m k ≤ a m k1, a k ≤ a m k1, 2.13

for all k ∈ In fact, m k  max {j ≤ k : a j < a j1 }.

For solving the equilibrium problem, we usually assume that a bifunction F : C × C →

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 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.

The following lemma gives a characterization of a solution of an equilibrium problem

Lemma 2.9 see 19,Lemma 2.8 Let C be a nonempty closed convex subset of a reflexive, strictly

convex, and uniformly smooth Banach space E Let F : C × C →  be a bifunction satisfying conditions A1–A4 For r > 0, define a mapping T r : E → C so-called the resolvent of F as

follows:

T r x  z ∈ C : Fz, y

 1ry − z, Jz − Jx≥ 0 ∀y ∈ C



, 2.14

for all x ∈ E Then, the following hold:

i T r is single-valued,

ii T r is a firmly nonexpansive-type mapping27, that is, for all x, y ∈ E



T r x − T r y, JT r x − JT r y

≤T r x − T r y, Jx − Jy, 2.15

iii FT r   EPF,

iv EPF is closed and convex,

Lemma 2.10 see 4, Lemma 2.3 Let C be a nonempty closed convex subset of a Banach space E,

F a bifunction from C × C → satisfying conditions A1–A4 and z ∈ C Then, z ∈ EPF if and

only if Fy, z ≤ 0 for all y ∈ C.

Remark 2.11see 27 Let C be a nonempty subset of a smooth Banach space E If S : C → E

is a firmly nonexpansive-type mapping, then

ϕz, Sx ≤ ϕz, Sx  ϕSx, x ≤ ϕz, x, 2.16

for all x ∈ C and z ∈ FS In particular, S satisfies condition R2.

Lemma 2.12 see 3, Proposition 2.4 Let C be a nonempty closed convex subset of a strictly

convex and smooth Banach space E and S : C → E a relatively nonexpansive mapping Then, FS

is closed and convex.

3 Main Results

In this section, we prove a strong convergence theorem for finding a common element of the fixed points set of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in a uniformly convex and uniformly smooth Banach space

Theorem 3.1 Let C be a nonempty closed convex subset of a uniformly convex and uniformly smooth

Banach space E and F : C × C → a bifunction satisfying conditions A1–A4 and S : C → E

a relatively nonexpansive mapping such that FS ∩ EPF / Let {u n } and {x n } be sequences

generated by u ∈ C, u1∈ E and

F

x n , y

r1

n



y − x n , Jx n − Ju n

≥ 0, ∀y ∈ C,

y n ΠC J−1α n Ju  1 − α n Jx n ,

u n1  J−1

β n Jx n1− β n

JSy n

,

3.1

for all n ∈ , where {α n } ⊂ 0, 1 satisfying lim n → ∞ α n  0 and∞

n1 α n  ∞, {β n } ⊂ a, b ⊂ 0, 1,

and {r n } ⊂ c, ∞ ⊂ 0, ∞ Then, {u n } and {x n } converge strongly to Π FS∩EPF u.

Proof Note that x n can be rewritten as x n  T r n u n Since FS ∩ EPF is nonempty, closed,

and convex, we putu  Π FS∩EPF u Since Π C , T r n , and S satisfy conditionR2, by 2.6, we get

ϕ

u, y n

≤ ϕu, J−1α n Ju  1 − α n Jx n

≤ α n ϕu, u  1 − α n ϕu, x n

≤ α n ϕu, u  1 − α n ϕu, u n ,

3.2

and so

ϕu, u n1  ≤ β n ϕu, x n 1− β n

ϕ

u, Sy n

≤ β n ϕu, u n 1− β n

ϕ

u, y n

≤ α n

1− β n

ϕu, u 1− α n

1− β n

ϕu, u n

≤ maxϕu, u, ϕu, u n.

3.3

By induction, we have

ϕz, u n1 ≤ maxϕu, u, ϕu, u1, 3.4

for all n ∈ This implies that {u n } is bounded and so are {x n }, {y n }, and {Sy n} Put

z n ≡ J−1α n Ju  1 − α n Jx n . 3.5

Then, y n≡ ΠC z n UsingLemma 2.6gives

ϕ

u, y n

≤ ϕ  u,z n   V   u,Jz n

≤ V u, Jz n − α n Ju − J u − 2z n − u, −α n Ju − J u

 ϕu, J−1α n J u  1 − α n Jx n  2α n z n − u, Ju − J u

≤ α n ϕu, u  1 − α n ϕu, x n   2α n z n − u, Ju − J u

≤ 1 − α n ϕu, u n   2α n z n − u, Ju − J u

3.6

Let g : 0, 2r → 0, ∞ be a function satisfying the properties ofLemma 2.1, where r  sup{xn , Sy n  : n ∈ } Then, byRemark 2.11and3.6, we get

ϕu, u n1  ≤ β n ϕu, x n 1− β n

ϕ

u, Sy n

− β n

1− β n

g Jx n − JSy n

≤ β n

ϕu, u n  − ϕx n , u n1− β n

ϕ

u, y n

− β n

1− β n

g Jx n − JSy n

≤ β n ϕu, u n 1− β n

1 − α n ϕu, u n   2α n z n − u, Ju − J u

− β n ϕx n , u n  − β n

1− β n

g Jx n − JSy n

1− γ n

ϕu, u n   2γ n z n − u, Ju − J u

3.7

− β n ϕx n , u n  − β n

1− β n

g Jx n − JSy n

≤1− γ n

ϕu, u n   2γ n z n − u, Ju − J u, 3.8 where γ n  α n 1 − β n  for all n ∈ Notice that {γ n } ⊂ 0, 1 satisfying lim n → ∞ γ n  0 and ∞

n1 γ n ∞

The rest of the proof will be divided into two parts.

Case 1 Suppose that there exists n0 ∈ such that {ϕu, u n}∞

nn0 is nonincreasing In this situation,{ϕu, u n} is then convergent Then,

ϕu, u n  − ϕu, u n1  −→ 0. 3.9

It follows from3.7 and γ n → 0 that

β n ϕx n , u n   β n

1− β n

g Jx n − JSy n  −→ 0. 3.10 Since{β n } ⊂ a, b ⊂ 0, 1,

ϕx n , u n  −→ 0, g Jx n − JSy n  −→ 0. 3.11 Consequently, byRemark 2.3,

x n − u n −→ 0, Jx n − JSy n −→ 0, x n − Sy n −→ 0. 3.12 From2.6 and α n → 0, we obtain

ϕ

x n , y n

≤ ϕx n , z n  ≤ α n ϕx n , u  1 − α n ϕx n , x n   α n ϕx n , u −→ 0. 3.13 This implies that

x n − y n −→ 0, z n − y n −→ 0. 3.14 Therefore,

Since{y n } is bounded and E is reflexive, we choose a subsequence {y n i } of {y n} such that

y n i z and

lim sup

n → ∞



y n − u, Ju − J u lim

i → ∞



y n i − u, Ju − J u. 3.16

Then, x n i z Since x n − u n → 0 and r n ≥ c > 0, byRemark 2.3,

lim

n → ∞

1

r n Jx n − Ju n   0. 3.17 Notice that

F

x n , y

r1

n



y − x n , Jx n − Ju n

≥ 0, ∀y ∈ C. 3.18

Replacing n by n i, we have fromA2 that

1

r n i



y − x n i , Jx n i − Ju n i



≥ −Fx n i , y

≥ Fy, x n i



, ∀y ∈ C. 3.19

Letting i → ∞, we have from 3.17 and A4 that

F

y, z

FromLemma 2.10, we have z ∈ EPF Since S satisfies condition R3 and 3.15, z ∈ FS.

It follows that z ∈ FS ∩ EPF ByLemma 2.4a, we immediately obtain that

lim sup

n → ∞ y n − u, Ju − J u  z − u, Ju − J u ≤ 0. 3.21 Since z n − y n → 0,

lim sup

n → ∞ z n − u, Ju − J u ≤ 0. 3.22

It follows fromLemma 2.7and3.8 that ϕu, u n  → 0 Then, u n → u and so x n → u.

Case 2 Suppose that there exists a subsequence {n i } of {n} such that

ϕu, u n i  < ϕu, u n i1, 3.23

for all i∈ Then, byLemma 2.8, there exists a nondecreasing sequence{m k} ⊂ such that

m k → ∞,

ϕu, u m k  ≤ ϕu, u m k1, ϕu, u k  ≤ ϕu, u m k1 3.24

for all k∈ From 3.7 and γ n → 0, we have

β m k ϕx m k , u m k   β m k



1− β m k



g Jx m k − JSy m k

≤ϕu, u m k  − ϕu, u m k1− γ m k ϕu, u m k   2γ m k z m k − u, Ju − J u

≤ − γ m k ϕu, u m k   2γ m k z m k − u, Ju − J u −→ 0.

3.25

Using the same proof of Case1, we also obtain

lim sup

k → ∞ z m k − u, Ju − J u ≤ 0. 3.26

From3.8, we have

ϕu, u m1 ≤1− γ m 

ϕu, u m   2γ m z m − u, Ju − J u 3.27

Since ϕu, u m k  ≤ ϕu, u m k1, we have

γ m k ϕu, u m k  ≤ ϕu, u m k  − ϕu, u m k1  2γ m k z m k − u, Ju − J u

≤ 2γ m k



y m k − u, Ju − J u. 3.28

In particular, since γ m k > 0, we get

ϕu, u m k  ≤ 2z m k − u, Ju − J u 3.29

It follows from3.26 that ϕu, u m k → 0 This together with 3.27 gives

ϕu, u m k1 −→ 0. 3.30

But ϕu, u k  ≤ ϕu, u m k1 for all k ∈ , we conclude that u k → u, and x k → u.

From two cases, we can conclude that{u n } and {x n } converge strongly to u and the

proof is finished

ApplyingTheorem 3.1and28, Theorem 3.2, we have the following result

Theorem 3.2 Let C be a nonempty closed convex subset of a uniformly convex and uniformly smooth

Banach space E, F : C × C → a bifunction satisfying conditions (A1)–(A4), and {T i : C → E}∞

i1

a sequence of relatively nonexpansive mappings such that∞

i1 FT i  ∩ EPF / Let {u n } and {x n}

be sequences generated by3.1, where S : C → E is defined by

Sx  J−1

∞

i1

α i JT i x



for each x ∈ C. 3.31

Then, {u n } and {x n } converge strongly to Π∞

i1 FT i ∩EPF u.

Setting F ≡ 0 and r n≡ 1 inTheorem 3.1, we have the following result

Corollary 3.3 Let C be a nonempty closed convex subset of a uniformly convex and uniformly smooth

Banach space E and S : C → E a relatively nonexpansive mapping Let {u n } and {x n } be sequences

generated by u ∈ C, u1∈ E and

x n ΠC u n ,

y n ΠC J−1α n Ju  1 − α n Jx n ,

u n1  J−1

β n Jx n1− β n

JSy n

,

3.32

for all n ∈ , where {α n } ⊂ 0, 1 satisfying lim n → ∞ α n  0 and∞n1 α n  ∞, {β n } ⊂ a, b ⊂ 0, 1.

Then, {u n } and {x n } converge strongly to Π FS u.

and so

ϕu, u n1  ≤ β n ϕu, x n...

Since ϕu, u m k  ≤ ϕu, u m k1, we have

γ... y n → implies that ϕx n , y n →

Lemma 2.2