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We introduce a new system of general variational inequalities in Banach spaces.. The equivalence between this system of variational inequalities and fixed point problems concerning the n

Volume 2010, Article ID 246808, 13 pages

doi:10.1155/2010/246808

Research Article

Strong Convergence Theorem for a New General System of Variational Inequalities in Banach Spaces

S Imnang1, 2 and S Suantai2, 3

1 Department of Mathematics, Faculty of Science, Thaksin University, Phatthalung Campus,

Phatthalung 93110, Thailand

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

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

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

Received 26 July 2010; Revised 7 December 2010; Accepted 30 December 2010

Academic Editor: S Reich

Copyrightq 2010 S Imnang 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 system of general variational inequalities in Banach spaces The equivalence between this system of variational inequalities and fixed point problems concerning the nonexpansive mapping is established By using this equivalent formulation, we introduce an iterative scheme for finding a solution of the system of variational inequalities in Banach spaces Our main result extends a recent result acheived by Yao, Noor, Noor, Liou, and Yaqoob

1 Introduction

LetX be a real Banach space, and X∗be its dual space LetU  {x ∈ X : x  1} denote the

unit sphere ofX X is said to be uniformly convex if for each  ∈ 0, 2 there exists a constant

δ > 0 such that for any x, y ∈ U,

x − y ≥  implies x  y

2



 ≤ 1 − δ. 1.1 The norm onX is said to be Gˆateaux differentiable if the limit

lim

t → 0

x  ty − x

exists for eachx, y ∈ U and in this case X is said to have a uniformly Frechet differentiable norm

if the limit1.2 is attained uniformly for x, y ∈ U and in this case X is said to be uniformly

smooth We define a function ρ : 0, ∞ → 0, ∞, called the modulus of smoothness of X, as

follows:

ρτ  sup



1

2x  y  x − y − 1 : x,y ∈ X, x  1, y  τ. 1.3

It is known thatX is uniformly smooth if and only if lim τ → 0 ρτ/τ  0 Let q be a fixed real

number with 1< q ≤ 2 Then a Banach space X is said to be q-uniformly smooth if there exists

a constantc > 0 such that ρτ ≤ cτ qfor allτ > 0 For q > 1, the generalized duality mapping

J q:X → 2 X∗

is defined by

J q x f ∈ X∗:

x, f  x q , f  x q−1 , ∀x ∈ X. 1.4

In particular, ifq  2, the mapping J2is called the normalized duality mapping and usually, we

writeJ2  J If X is a Hilbert space, then J  I Further, we have the following properties of

the generalized duality mappingJ q:

1 J q x  x q−2 J2x for all x ∈ X with x / 0,

2 J q tx  t q−1 J q x for all x ∈ X and t ∈ 0, ∞,

3 J q −x  −J q x for all x ∈ X.

It is known that ifX is smooth, then J is single-valued, which is denoted by j Recall

that the duality mappingj is said to be weakly sequentially continuous if for each {x n } ⊂ X

withx n → x weakly, we have jx n  → jx weakly-∗ We know that if X admits a weakly

sequentially continuous duality mapping, thenX is smooth For the details, see the work of

Gossez and Lami Dozo in1

LetC be a nonempty closed convex subset of a smooth Banach space X Recall that a

mappingA : C → X is said to be accretive if



for allx, y ∈ C A mapping A : C → X is said to be α-strongly accretive if there exists a

constantα > 0 such that



Ax − Ay, jx − y ≥ αx − y2 1.6

for allx, y ∈ C A mapping A : C → X is said to be α-inverse strongly accretive if there exists a

constantα > 0 such that



Ax − Ay, jx − y ≥ αAx − Ay2 1.7 for allx, y ∈ C A mapping T : C → C is said to be nonexpansive if Tx − Ty ≤ x − y for all

x, y ∈ C The fixed point set of T is denoted by FT : {x ∈ C : Tx  x}.

LetD be a nonempty subset of C A mapping Q : C → D is said to be sunny if

wheneverQx  tx − Qx ∈ C for x ∈ C and t ≥ 0 A mapping Q : C → D is called a retraction

ifQx  x for all x ∈ D Furthermore, Q is a sunny nonexpansive retraction from C onto D if Q

is a retraction fromC onto D which is also sunny and nonexpansive.

A subsetD of C is called a sunny nonexpansive retraction of C if there exists a sunny

nonexpansive retraction fromC onto D It is well known that if X is a Hilbert space, then a

sunny nonexpansive retractionQ Cis coincident with the metric projection fromX onto C.

Conveying an idea of the classical variational inequality, denoted by VIC, A, is to find

anx∗∈ C such that



Ax∗, y − x∗

whereX  H is a Hilbert space and A is a mapping from C into H The variational inequality

has been widely studied in the literature; see, for example, the work of Chang et al in2, Zhao and He3, Plubtieng and Punpaeng 4, Yao et al 5 and the references therein LetA, B : C → H be two mappings In 2008, Ceng et al 6 considered the following problem of findingx∗, y∗ ∈ C × C such that



λAy∗ x∗− y∗, x − x∗

≥ 0, ∀x ∈ C,



μBx∗ y∗− x∗, x − y∗

≥ 0, ∀x ∈ C, 1.10

which is called a general system of variational inequalities, where λ > 0 and μ > 0 are two

constants In particular, ifA  B, then problem 1.10 reduces to finding x∗, y∗ ∈ C × C such

that



λAy∗ x∗− y∗, x − x∗

≥ 0, ∀x ∈ C,



μAx∗ y∗− x∗, x − y∗

≥ 0, ∀x ∈ C, 1.11

which is defined by Verma7 and is called the new system of variational inequalities Further,

if we add up the requirement that x∗  y∗, then problem 1.11 reduces to the classical variational inequality VIC, A

In 2006, Aoyama et al 8 first considered the following generalized variational inequality problem in Banach spaces LetA : C → X be an accretive operator Find a point

x∗∈ C such that



The problem1.12 is very interesting as it is connected with the fixed point problem for nonlinear mapping and the problem of finding a zero point of an accretive operator in Banach spaces, see9 11 and the references therein

Aoyama et al.8 introduced the following iterative algorithm in Banach spaces:

x0 x ∈ C,

y n  Q C x n − λ n Ax n ,

x n1  a n x n  1 − a n y n , n ≥ 0,

1.13

where Q C is a sunny nonexpansive retraction from X onto C Then they proved a

weak convergence theorem which is generalized simultaneously theorems of Browder and Petryshyn 12 and Gol’shte˘ın and Tret’yakov 13 In 2008, Hao 14 obtained a strong convergence theorem by using the following iterative algorithm:

x0∈ C,

y n  b n x n  1 − b n Q C I − λ n Ax n ,

x n1  a n u  1 − a n y n , n ≥ 0,

1.14

where{a n }, {b n } are two sequences in 0, 1 and u ∈ C.

Very recently, in 2009, Yao et al 5 introduced the following system of general variational inequalities in Banach spaces For given two operators A, B : C → X, they

considered the problem of findingx∗, y∗ ∈ C × C such that



Ay∗ x∗− y∗, jx − x∗ ≥ 0, ∀x ∈ C,



Bx∗ y∗− x∗, jx − y∗

≥ 0, ∀x ∈ C, 1.15

which is called the system of general variational inequalities in a real Banach space They proved a

strong convergence theorem by using the following iterative algorithm:

x0∈ C,

y n  Q C x n − Bx n ,

x n1  a n u  b n x n  c n Q C

y n − Ay n

, n ≥ 0,

1.16

where{a n }, {b n }, and {c n } are three sequences in 0, 1 and u ∈ C.

In this paper, motivated and inspired by the idea of Yao et al.5 and Cheng et al 6 First, we introduce the following system of variational inequalities in Banach spaces

LetC be a nonempty closed convex subset of a real Banach space X Let A i :C → X

for alli  1, 2, 3 be three mappings We consider the following problem of finding x∗, y∗, z∗ ∈

C × C × C such that



λ1A1y∗ x∗− y∗, jx − x∗ ≥ 0, ∀x ∈ C,



λ2A2z∗ y∗− z∗, jx − y∗

≥ 0, ∀x ∈ C,



λ3A3x∗ z∗− x∗, jx − z∗ ≥ 0, ∀x ∈ C,

1.17

which is called a new general system of variational inequalities in Banach spaces, where λ i > 0 for

alli  1, 2, 3 In particular, if A3  0, z∗  x∗, andλ i  1 for i  1, 2, 3, then problem 1.17 reduces to problem1.15 Further, if A3  0, z∗  x∗, then problem1.17 reduces to the problem1.10 in a real Hilbert space Second, we introduce iteration process for finding a solution of a new general system of variational inequalities in a real Banach space Starting with arbitrary pointsv, x1∈ C and let {x n }, {y n }, and {z n} be the sequences generated by

z n  Q C x n − λ3A3x n ,

y n  Q C z n − λ2A2z n ,

x n1  a n v  b n x n  1 − a n − b n Q C

y n − λ1A1y n

, n ≥ 1,

1.18

where λ i > 0 for all i  1, 2, 3 and {a n }, {b n } are two sequences in 0, 1 Using the

demiclosedness principle for nonexpansive mapping, we will show that the sequence{x n} converges strongly to a solution of a new general system of variational inequalities in Banach spaces under some control conditions

2 Preliminaries

In this section, we recall the well known results and give some useful lemmas that will be used in the next section

x  y q ≤ x q  qy, J q x  2Kyq 2.1

for all x, y ∈ X, where K is the q-uniformly smooth constant of X.

The following lemma concerns the sunny nonexpansive retraction

nonempty subset of C and Q : C → D be a retraction Then Q is sunny and nonexpansive if and only if



for all u ∈ C and y ∈ D.

The first result regarding the existence of sunny nonexpansive retractions on the fixed point set of a nonexpansive mapping is due to Bruck18

Remark 2.3 If X is strictly convex and uniformly smooth and if T : C → C is a nonexpansive

mapping having a nonempty fixed point setFT, then there exists a sunny nonexpansive

retraction ofC onto FT.

Lemma 2.4 see 19 Assume {a n } is a sequence of nonnegative real numbers such that

a n1≤1− γ n

where {γ n } is a sequence in 0, 1 and {δ n } is a sequence such that

i∞

n1 γ n  ∞;

ii lim supn → ∞ δ n /γ n ≤ 0 or∞n1 |δ n | < ∞.

Then lim n → ∞ a n  0.

a sequence in 0, 1 with 0 < lim inf n → ∞ b n≤ lim supn → ∞ b n < 1 Suppose x n1  1 −b n y n b n x n for all integers n ≥ 1 and lim sup n → ∞ y n1 − y n  − x n1 − x n  ≤ 0 Then, lim n → ∞ y n − x n   0.

subset of X, and T : C → C be an nonexpansive mapping Then I − T is demiclosed at 0, that is, if

x n → x weakly and x n − Tx n → 0 strongly, then x ∈ FT.

3 Main Results

In this section, we establish the equivalence between the new general system of variational inequalities1.17 and some fixed point problem involving a nonexpansive mapping Using the demiclosedness principle for nonexpansive mapping, we prove that the iterative scheme

1.18 converges strongly to a solution of a new general system of variational inequalities

1.17 in a Banach space under some control conditions In order to prove our main result, the following lemmas are needed

The next lemmas are crucial for proving the main theorem

Lemma 3.1 Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space X.

Let the mapping A : C → X be α-inverse strongly accretive Then, we have

I − λAx − I − λAy2≤x − y2 2λλK2− α Ax − Ay2, 3.1

where K is the 2-uniformly smooth constant of X In particular, if α ≥ λK2, then I − λA is a nonexpansive mapping.

Proof Indeed, for all x, y ∈ C, fromLemma 2.1, we have

I − λAx − I − λAy2x − y − λAx − Ay2

≤x − y2− 2λAx − Ay, jx − y

 2K2λ2Ax − Ay2

≤x − y2− 2λαAx − Ay2 2K2λ2Ax − Ay2

x − y2 2λλK2− α Ax − Ay2.

3.2

It is clear that, ifα ≥ λK2, thenI − λA is a nonexpansive mapping.

Lemma 3.2 Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space

X Let Q C be the sunny nonexpansive retraction from X onto C Let A i :C → X be an α i -inverse strongly accretive mapping for i  1, 2, 3 Let G : C → C be a mapping defined by

Gx  Q C Q C Q C x − λ3A3x − λ2A2Q C x − λ3A3x

−λ1A1Q C Q C x − λ3A3x − λ2A2Q C x − λ3A3x, ∀x ∈ C. 3.3

If α i ≥ λ i K2for all i  1, 2, 3, then G : C → C is nonexpansive.

Proof For all x, y ∈ C, we have

Gx − Gy  Q C Q C Q C I − λ3A3x − λ2A2Q C I − λ3A3x

−λ1A1Q C Q C I − λ3A3x − λ2A2Q C I − λ3A3x

− Q CQ CQ C I − λ3A3y − λ2A2Q C I − λ3A3y

− λ1A1Q C

Q C I − λ3A3y − λ2A2Q C I − λ3A3y

≤ Q C Q C I − λ3A3x − λ2A2Q C I − λ3A3x

− λ1A1Q C Q C I − λ3A3x − λ2A2Q C I − λ3A3x

−Q C

Q C I − λ3A3y − λ2A2Q C I − λ3A3y

− λ1A1Q C

Q C I − λ3A3y − λ2A2Q C I − λ3A3y

 I − λ1A1Q C I − λ2A2Q C I − λ3A3x

−I − λ1A1Q C I − λ2A2Q C I − λ3A3y.

3.4

FromLemma 3.1, we haveI −λ1A1Q C I −λ2A2Q C I −λ3A3 is nonexpansive which implies

by3.4 that G is nonexpansive.

the sunny nonexpansive retraction from X onto C Let A i:C → X be three nonlinear mappings For given x∗, y∗, z∗ ∈ C × C × C, x∗, y∗, z∗ is a solution of problem 1.17 if and only if x∗ ∈ FG,

y∗ Q C z∗− λ2A2z∗ and z∗ Q C x∗− λ3A3x∗, where G is the mapping defined as in Lemma 3.2 Proof Note that we can rewrite1.17 as



x∗−y∗− λ1A1y∗

, jt − x∗ ≥ 0, ∀t ∈ C,



y∗− z∗− λ2A2z∗, jt − y∗

≥ 0, ∀t ∈ C,



z∗− x∗− λ3A3x∗, jt − z∗ ≥ 0, ∀t ∈ C.

3.5

FromLemma 2.2, we can deduce that3.5 is equivalent to

x∗ Q C

y∗− λ1A1y∗

,

y∗ Q C z∗− λ2A2z∗,

z∗ Q C x∗− λ3A3x∗.

3.6

It is easy to see that3.6 is equivalent to x∗  Gx∗,y∗  Q C z∗− λ2A2z∗ and z∗  Q C x∗−

λ3A3x∗

From now on we denote byΩ∗ the set of all fixed points of the mappingG Now we

prove the strong convergence theorem of algorithm1.18 for solving problem 1.17

Theorem 3.4 Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly

smooth Banach space X which admits a weakly sequentially continuous duality mapping Let Q C be the sunny nonexpansive retraction from X onto C Let the mappings A i : C → X be α i -inverse strongly accretive with α i ≥ λ i K2, for all i  1, 2, 3 and Ω∗/ ∅ For given x1, v ∈ C, let the sequence

{x n } be generated iteratively by 1.18 Suppose the sequences {a n } and {b n } are two sequences in

0, 1 such that

C1 limn → ∞ a n  0 and∞n1 a n  ∞;

C2 0 < lim inf n → ∞ b n≤ lim supn → ∞ b n < 1.

Then {x n } converges strongly to Qv where Qis the sunny nonexpansive retraction of C onto Ω∗ Proof Let x∗∈ Ω∗andt n  Q C y n − λ1A1y n, it follows fromLemma 3.3that

x∗ Q C Q C Q C x∗− λ3A3x∗ − λ2A2Q C x∗− λ3A3x∗

−λ1A1Q C Q C x∗− λ3A3x∗ − λ2A2Q C x∗− λ3A3x∗. 3.7

Puty∗ Q C z∗− λ2A2z∗ and z∗ Q C x∗− λ3A3x∗ Then x∗ Q C y∗− λ1A1y∗ and

x n1  a n v  b n x n  1 − a n − b n t n 3.8 FromLemma 3.1, we haveI − λ i A i i  1, 2, 3 is nonexpansive Therefore

t n − x∗ Q C

y n − λ1A1y n

− Q C

y∗− λ1A1y∗

≤y n − y∗

 Q C z n − λ2A2z n  − Q C z∗− λ2A2z∗

≤ z n − z∗

 Q C x n − λ3A3x n  − Q C x∗− λ3A3x∗

≤ x n − x∗.

3.9

It follows that

x n1 − x∗  a n v  b n x n  1 − a n − b n t n − x∗

≤ a n v − x∗  b n x n − x∗  1 − a n − b n t n − x∗

≤ a n v − x∗  b n x n − x∗  1 − a n − b n x n − x∗

 a n v − x∗  1 − a n x n − x∗.

3.10

By induction, we have

x n1 − x∗ ≤ max{v − x∗, x1− x∗}. 3.11

Therefore,{x n } is bounded Hence {y n }, {z n }, {t n }, {A1y n }, {A2z n }, and {A3x n} are also bounded By nonexpansiveness ofQ CandI − λ i A i i  1, 2, 3, we have

t n1 − t n Q C

y n1 − λ1A1y n1− Q Cy n − λ1A1y n

≤y n1 − y n

 Q C z n1 − λ2A2z n1  − Q C z n − λ2A2z n

≤ z n1 − z n

 Q C x n1 − λ3A3x n1  − Q C x n − λ3A3x n

≤ x n1 − x n .

3.12

Letw n  x n1 − b n x n /1 − b n , n ∈ Thenx n1  b n x n  1 − b n w nfor alln ∈and

w n1 − w n x n2 − b n1 x n1

1− b n1 −

x n1 − b n x n

1− b n

 a n1 v  1 − a n1 − b n1 t n1

1− b n1 −

a n v  1 − a n − b n t n

1− b n

 a n1

1− b n1 v − t n1  a n

1− b n t n − v  t n1 − t n

3.13

By3.12 and 3.13, we have

w n1 − w n  − x n1 − x n ≤ a n1

1− b n1 v − t n1 

a n

1− b n t n − v

 t n1 − t n  − x n1 − x n

≤ a n1

1− b n1 v − t n1  a n

1− b n t n − v.

3.14

This together withC1 and C2, we obtain that

lim sup

n → ∞ w n1 − w n  − x n1 − x n  ≤ 0. 3.15

Hence, byLemma 2.5, we getx n − w n  → 0 as n → ∞ Consequently,

lim

n → ∞ x n1 − x n  lim

n → ∞ 1 − b n w n − x n   0. 3.16

Since

x n1 − x n  a n v − x n   1 − a n − b n t n − x n , 3.17 therefore

t n − x n  −→ 0 as n −→ ∞. 3.18 Furthermore, byLemma 3.2, we haveG : C → C is nonexpansive Thus, we have

t n − Gt n Q C

y n − λ1A1y n− Gt n

 Q C Q C z n − λ2A2z n  − λ1A1Q C z n − λ2A2z n  − Gt n

 Q C Q C Q C x n − λ3A3x n  − λ2A2Q C x n − λ3A3x n

− λ1A1Q C Q C x n − λ3A3x n  − λ2A2Q C x n − λ3A3x n  − Gt n

 Gx n  − Gt n  ≤ x n − t n ,

3.19

which impliest n − Gt n  → 0 as n → ∞.

Since

x n − Gx n  ≤ x n − t n   t n − Gt n   Gt n  − Gx n

≤ x n − t n   t n − Gt n   t n − x n , 3.20

therefore

lim

n → ∞ x n − Gx n   0. 3.21

LetQbe the sunny nonexpansive retraction ofC onto Ω∗ Now we show that

lim sup

n → ∞



v − Qv, jx n − Qv ≤ 0. 3.22

Lemma 3.2 Let C be a nonempty closed convex subset of a. ..