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We prove the existence and uniqueness of solution and the convergence of some newn-step iterative algorithms with or without mixed errors for this system of generalized nonlinear mixed v

Volume 2010, Article ID 908490, 15 pages

doi:10.1155/2010/908490

Research Article

A New System of Generalized Nonlinear Mixed Variational Inclusions in Banach Spaces

Jian Wen Peng

College of Mathematics and Computer Science, Chongqing Normal University,

Chongqing Sichuan 400047, China

Correspondence should be addressed to Jian Wen Peng,jwpeng6@yahoo.com.cn

Received 5 July 2009; Accepted 14 September 2009

Academic Editor: Mohamed A Khamsi

Copyrightq 2010 Jian Wen Peng 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 and study a new system of generalized nonlinear mixed variational inclusions in real

q-uniformly smooth Banach spaces We prove the existence and uniqueness of solution and the

convergence of some newn-step iterative algorithms with or without mixed errors for this system

of generalized nonlinear mixed variational inclusions The results in this paper unify, extend, and improve some known results in literature

1 Introduction

Variational inclusion problems are among the most interesting and intensively studied classes

of mathematical problems and have wide applications in the fields of optimization and control, economics and transportation equilibrium, as well as engineering science For the past years, many existence results and iterative algorithms for various variational inequality and variational inclusion problems have been studied For details, see 1 25 and the references therein

Recently, some new and interesting problems, which are called to be system of variational inequality problems, were introduced and studied Pang1, Cohen and Chaplais

2, Bianchi 3, and Ansari and Yao 4 considered a system of scalar variational inequalities, and Pang showed that the traffic equilibrium problem, the spatial equilibrium problem, the Nash equilibrium, and the general equilibrium programming problem can be modeled

as a system of variational inequalities Ansari et al 5 considered a system of vector variational inequalities and obtained its existence results Allevi et al.6 considered a system

of generalized vector variational inequalities and established some existence results with relative pseudomonoyonicity Kassay and Kolumb´an7 introduced a system of variational inequalities and proved an existence theorem by the Ky Fan lemma Kassay et al.8 studied

Minty and Stampacchia variational inequality systems with the help of the Kakutani-Fan-Glicksberg fixed point theorem Peng 9, Peng and Yang 10 introduced a system of quasivariational inequality problems and proved its existence theorem by maximal element theorems Verma 11–15 introduced and studied some systems of variational inequalities and developed some iterative algorithms for approximating the solutions of system of variational inequalities in Hilbert spaces J K Kim and D S Kim16 introduced and studied

a new system of generalized nonlinear quasivariational inequalities in Hilbert spaces Cho et

al.17 introduced and studied a new system of nonlinear variational inequalities in Hilbert spaces They proved some existence and uniqueness theorems of solutions for the system of nonlinear variational inequalities

As generalizations of system of variational inequalities, Agarwal et al.18 introduced

a system of generalized nonlinear mixed quasivariational inclusions and investigated the sensitivity analysis of solutions for this system of generalized nonlinear mixed quasivariational inclusions in Hilbert spaces Peng and Zhu 19 introduce a new system

of generalized nonlinear mixed quasivariational inclusions inq-uniformly smooth Banach

spaces and prove the existence and uniqueness of solutions and the convergence of several new two-step iterative algorithms with or without errors for this system of generalized nonlinear mixed quasivariational inclusions Kazmi and Bhat 20 introduced a system

of nonlinear variational-like inclusions and proved the existence of solutions and the convergence of a new iterative algorithm for this system of nonlinear variational-like inclusions Fang and Huang21, Verma 22, and Fang et al 23 introduced and studied

a new system of variational inclusions involving H-monotone operators, A-monotone

operators andH, η-monotone operators, respectively Yan et al 24 introduced and studied

a system of set-valued variational inclusions which is more general than the model in21 Peng and Zhu25 introduced and studied a system of generalized mixed quasivariational inclusions involvingH, η-monotone operators which contains those mathematical models

in11–16,21–24 as special cases

Inspired and motivated by the results in 1 25, the purpose of this paper is to introduce and study a new system of generalized nonlinear mixed quasivariational inclusions which contains some classes of system of variational inclusions and systems of variational inequalities in the literature as special cases Using the resolvent technique for them-accretive

mappings, we prove the existence and uniqueness of solutions for this system of generalized nonlinear mixed quasivariational inclusions We also prove the convergence of some new

n-step iterative sequences with or without mixed errors to approximation the solution for this system of generalized nonlinear mixed quasivariational inclusions The results in this paper unifies, extends, and improves some results in11–16,19 in several aspects

2 Preliminaries

Throughout this paper we suppose thatE is a real Banach space with dual space, norm and

the generalized dual pair denoted byE∗, ·  and ·, ·, respectively, 2 Eis the family of all the nonempty subsets ofE, domM denotes the domain of the set-valued map M : E → 2 E ,

and the generalized duality mappingJ q :E → 2 E∗

is defined by

J q x f∗∈ E∗:

x, f∗

f∗ · x,f∗  x q−1

where q > 1 is a constant In particular, J2 is the usual normalized duality mapping It is known that, in general,J q x  x2J2x, for all x / 0, and J qis single-valued ifE∗is strictly convex

The modulus of smoothness ofE is the function ρE:0, ∞ → 0, ∞ defined by

ρE t  sup

 1

2x  y  x − y − 1 : x ≤ 1,y ≤ t 2.2

A Banach spaceE is called uniformly smooth if

lim

t → 0

ρE t

E is called q-uniformly smooth if there exists a constant c > 0, such that

ρ E t ≤ ct q , q > 1. 2.4

Note thatJqis single-valued ifE is uniformly smooth.

Xu26 and Xu and Roach 27 proved the following result

Lemma 2.1 Let E be a real uniformly smooth Banach space Then, E is q-uniformly smooth if and

only if there exists a constant cq > 0, such that for all x, y ∈ E,

x  y q ≤ x q  qy, J q x c q y q 2.5

i M is said to be accretive if, for any x, y ∈ domM, u ∈ Mx, v ∈ My, there

existsjq x − y ∈ J q x − y such that



ii M is said to be m-accretive if M is accretive and I  ρMdomM  E holds for

everyequivalently, for some ρ > 0, where I is the identity operator on E.

We recall some definitions needed later

m-accretive, for a constantρ > 0, the mapping R M

ρ u I  ρM −1u, u ∈ E, 2.7

is called the resolvent operator associated withM and ρ.

Remark 2.5 It is well known that R M

ρ is single-valued and nonexpansive mappingsee 28

single-valued operator However,T is said to be

i r-strongly accretive if there exists a constant r > 0 such that



or equivalently,



Tx − Ty, J2



ii s-Lipschitz continuous if there exists a constant s > 0 such that

Tx − Ty  ≤ sx − y, ∀x,y ∈ E. 2.10

Remark 2.7 If T is r-strongly accretive, then T is r-expanding, that is,

Tx − Ty  ≥ rx − y, ∀x,y ∈ E. 2.11

Lemma 2.8 see 30 Let {a n }, {b n }, {c n } be three real sequences, satisfying

a n1 ≤ 1 − t n a n  b n  c n , ∀n ≥ 0, 2.12

where t n ∈ 0, 1,∞

n0 t n  ∞, for all n ≥ 0, b n  ◦t n ,∞

n0 c n < ∞ Then a n → 0.

3 System of Generalized Nonlinear Mixed Variational Inequalities

In this section, we will introduce a new system of generalized nonlinear mixed variational inclusions which contains some classes of system of variational inclusions and systems of variational inequalities in literature as special cases

In what follows, unless other specified, we always suppose thatθ is a zero element in

E, and for each i  1, 2, , n, T iandS i :E → E are single-valued mappings, M i :E → 2 E

is anm-accretive operator We consider the following problem: find x∗

1, x∗

2, , x∗

n  ∈ E nsuch that

θ ∈ ρ1T1x∗

2 ρ1S1x∗

2 x∗

1− x∗

2 ρ1M1



x∗ 1

,

θ ∈ ρ2T2x∗

3 ρ2S2x∗

3 x∗

2− x∗

3 ρ2M2



x∗ 2

,

θ ∈ ρ n−1 T n−1 x∗

n  ρ n−1 S n−1 x∗

n  x∗

n−1 − x∗

n  ρ n−1 M n−1x∗

n−1

,

1 ρ nSnx∗

1 x∗

n − x∗

1 ρ nMn x∗

n ,

3.1

which is called the system of generalized nonlinear mixed variational inclusions, whereρi >

0i  1, 2, , n are constants.

In what follows, there are some special cases of the problem3.1

i If n  2, then problem 3.1 reduces to the system of nonlinear mixed quasivariational inclusions introduced and studied by Peng and Zhu19

nonlinear mixed quasivariational inclusions introduced and studied by Agarwal et al.18

ii If E  H is a Hilbert space, and for each i  1, 2, , n, M i x  ∂φ i x for all x ∈ H,

whereφ i : H → R ∪ {∞} is a proper, convex, lower semicontinuous functional, and ∂φ i

denotes the subdifferential operator of φi, then problem3.1 reduces to the following system

of generalized nonlinear mixed variational inequalities, which is to findx∗

1, x∗

2, , x∗

n ∈ Hn

such that



ρ1T1x∗

2 ρ1S1x∗

2 x∗

1− x∗

2, x − x∗ 1



≥ ρ1φ1



x∗ 1

− ρ1φ1x, ∀x ∈ H,



ρ2T2x∗

3 ρ2S2x∗

3 x∗

2− x∗

3, x − x∗ 2



≥ ρ2φ2



x∗ 2

− ρ2φ2x, ∀x ∈ H,



ρ n−1 T n−1 x∗

n  ρ n−1 S n−1 x∗

n  x∗

n−1 − x∗

n , x − x∗

n−1



≥ ρ n−1 φ n−1x∗

n−1

− ρ n−1 φ n−1 x, ∀x ∈ H,



ρ n T n x∗

1 ρ n S n x∗

1 x∗

n − x∗

1, x − x∗

n

≥ ρ n φ n x∗

n  − ρ n φ n x, ∀x ∈ H,

3.2

whereρ i > 0 i  1, 2, , n are constants.

iii If n  2, then 3.2 reduces to the problem of finding x∗

1, x∗

2 ∈ H × H such that



ρ1T1x∗

2 ρ1S1x∗

2 x∗

1− x∗

2, x − x∗ 1



≥ ρ1φ1



x∗ 1

− ρ1φ1x, ∀x ∈ H,



ρ2T2x∗

1 ρ2S2x∗

1 x∗

2− x∗

1, x − x∗ 2



≥ ρ2φ2



x∗ 2

− ρ2φ2x, ∀x ∈ H. 3.3

Moreover, if φ1  φ2  ϕ, then problem 3.3 becomes the system of generalized nonlinear mixed variational inequalities introduced and studied by J K Kim and D S Kim

in16

iv For i  1, 2, , n, if φ i  δ k i the indicator function of a nonempty closed convex subsetK i ⊂ H and T i  0, then 3.2 reduces to the problem of finding x∗

1, x∗

2, , x∗

n ∈ n

i1 Ki, such that



ρ1S1x∗

2 x∗

1− x∗

2, x − x∗ 1



≥ 0, ∀x ∈ K1,



ρ2S2x∗

3 x∗

2− x∗

3, x − x∗ 2



≥ 0, ∀x ∈ K1,



ρ n−1 S n−1 x∗

n  x∗

n−1 − x∗

n , x − x∗

n−1



≥ 0, ∀x ∈ K n−1 ,



1 x∗

n − x∗

1, x − x∗

n



≥ 0, ∀x ∈ K n.

3.4

Problem3.4 is called the system of nonlinear variational inequalities Moreover, if n 

2, then problem3.4 reduces to the following system of nonlinear variational inequalities, which is to findx∗

1, x∗

2 ∈ K1× K2such that



ρ1S1x∗

2 x∗

1− x∗

2, x − x∗ 1



≥ 0, ∀x ∈ K1,



ρ2S2x∗

1 x∗

2− x∗

1, x − x∗ 2



≥ 0, ∀x ∈ K2. 3.5

If S1  S2 and K1  K2  K, then 3.5 reduces to the problem introduced and researched by Verma11–13

Lemma 3.1 For any given x∗

i ∈ Ei  1, 2, , n, x∗

1, x∗

2, , x∗

n  is a solution of the problem 3.1

if and only if

x∗

1 R M1

ρ1 x∗

2− ρ1



T1x∗

2 S1x∗ 2



,

x∗

2 R M2

ρ2 x∗

3− ρ2



T2x∗

3 S2x∗ 3



,

.

x∗

n−1  R M n−1

ρ n−1 x∗

n − ρ n−1 T n−1 x∗

n  S n−1 x∗

n,

x∗

n  R M n

ρ n x∗

1− ρ nT n x∗

1 S n x∗ 1



,

3.6

where R M i

ρ i  I  ρ i M i−1is the resolvent operators of M i for i  1, 2, , n.

omitted

4 Existence and Uniqueness

In this section, we will show the existence and uniqueness of solution for problems3.1

Theorem 4.1 Let E be a real q-uniformly smooth Banach spaces For i  1, 2, , n, let S i:E → E

be strongly accretive and Lipschitz continuous with constants ki and μi , respectively, let Ti:E → E

be Lipschitz continuous with constant ν i , and let M i:E → E be an m-accretive mapping If for each

i  1, 2, , n,

0<q

then3.1 has a unique solution x∗

1, x∗

2, , x∗

n  ∈ E n

Proof First, we prove the existence of the solution Define a mapping F : E → E as follows:

Fx  R M1

ρ1 x2− ρ1T1x2 S1x2,

x2 R M2

ρ2 x3− ρ2T2x3 S2x3,

x n−1  R M n−1

ρ n−1 x n − ρ n−1 T n−1 x n  S n−1 x n,

xn  R M n

ρ n x − ρn T nx  Snx.

4.2

Fori  1, 2, , n, since R M i

ρ i is a nonexpansive mapping,S iis strongly accretive and Lipschitz continuous with constants k i and μ i, respectively, andT i is Lipschitz continuous with constantνi, for anyx, y ∈ E, we have

Fx − Fy 

R M1

ρ1 x2− ρ1T1x2 S1x2− R M1

ρ1 y2− ρ1



T1y2 S1y2 

≤x2− y2

− ρ1



T1x2 S1x2 −T1y2 S1y2 

≤x2− y2

− ρ1



S1x2− S1y2   ρ1T1x2− T1y2

≤q x

2− y2q − qρ1



S1x2− S1y2, Jqx2− y2



 c q ρ q1S1x2− S1y2q  ρ1ν1x2− y2





q



1− qρ1k1 c qρ q1μ q1 ρ1ν1

x

2− y2





q



1− qρ1k1 c q ρ q1μ q1 ρ1ν1



R M2

ρ2 x3− ρ2T2x3 S2x3− R M2

ρ2 y3− ρ2



T2y3 S2y3 

≤



q



1− qρ1k1 c qρ q1μ q1 ρ1ν1

x

3− y3− ρ2 S2x3− S2y3

T2x3− T2y3 

≤



q



1− qρ1k1 c qρ q1μ q1 ρ1ν1

x

3− y3− ρ2



S2x3− S2y3   ρ2T2x3− T2y3

≤



q



1− qρ1k1 c q ρ q1μ q1 ρ1ν1



×



q

x

3− y3q − qρ2



S2x3− S2y3, Jqx3− y3



 c q ρ q2S2x3− S2y3q  ρ2ν2x3− y3

≤



q



1− qρ1k1 c q ρ q1μ q1 ρ1ν1



q



1− qρ2k2 c q ρ q2μ q2 ρ2ν2



x3− y3

≤ · · · ≤n−1

i1



q



1− qρ iki  c qρ q i μ q i  ρ iνix

n − y n

i1



R M n

ρ n x − ρ n T n x  S nx− R M n

ρ n y − ρ nT n y  S n y 

≤n−1

i1



q



1− qρ iki  c qρ i q μ q i  ρ iνi x − y − ρ n S nx − Sny   ρ n T nx − Tny

≤n−1

i1



q



1− qρ i k i  c q ρ i q μ q i  ρ i ν i

×



q

x − yq − qρ nS n x − S n y, J qx − y  c q ρ q n S n x − S n yq  ρ n ν n x − y

≤n

i1



q



1− qρ i k i  c q ρ i q μ q i  ρ i ν ix − y.

4.3

It follows from4.1 that

0<n i1



q



1− qρ iki  c qρ i q μ q i  ρ iνi



Thus,4.3 implies that F is a contractive mapping and so there exists a point x∗

1 ∈ E such

that

x∗

1 Fx∗ 1

Let

x∗

i  R M i

ρ i x∗

i1 − ρ i

Tix∗

i1

 S i

x∗

i1



, i  1, 2, , n − 1

x∗

n  R M n

ρ n x∗

1− ρ n

Tnx∗ 1

 S n

x∗ 1

then by the definition ofF, we have

x∗

1 R M1

ρ1 x∗

2− ρ1



T1x∗

2 S1x∗ 2



,

x∗

2 R M2

ρ2 x∗

3− ρ2



T2x∗

3 S2x∗ 3



,

x∗

n−1  R M n−1

ρ n−1 x∗

n − ρ n−1 T n−1 x∗

n  S n−1 x∗

n,

x∗

n  R M n

ρ n x∗

1− ρ n T nx1 ∗ S nx1 ∗,

4.7

that is,x∗

1, x∗

2, , x∗

n is a solution of problem 3.1

Then, we show the uniqueness of the solution Letx1, x2, , xn be another solution

of problem3.1 It follows fromLemma 3.1that

x1 R M1

ρ1 x2− ρ1T1x2 S1x2,

x2 R M2

ρ2 x3− ρ2T2x3 S2x3,

xn−1  R M n−1

ρ n−1 xn − ρ n−1 T n−1xn  S n−1xn,

x n  R M n

ρ n x1− ρ n T n x1 S n x1.

4.8

As the proof of4.3, we have

x∗

1− x1 ≤n

i1



q



1− qρ iki  c qρ q i μ q i  ρ iνix∗

1− x1. 4.9

It follows from4.1 that

0<n i1



q



Hence,

x∗

and so fori  2, 3, , n, we have

x∗

This completes the proof

Remark 4.2 i If E is a 2-uniformly smooth space, and there exist ρ i > 0 i  1, 2, , n such

that

0< ρi < min

 2ki − ν i

c2μ2

i − ν2

i

, ν1 i



Then4.1 holds We note that the Hilbert spaces and L p or l q  spaces 2 ≤ q < ∞ are

2-uniformly smooth

ii Let n  2, byTheorem 4.1, we recover19, Theorem 3.1 SoTheorem 4.1unifies, extends, and improves19, Theorem 3.1, Corollaries 3.2 and 3.3, 16, Theorems 2.1–2.4 and the main results in13

5 Algorithms and Convergence

This section deals with an introduction of somen-step iterative sequences with or without

mixed errors for problem3.1 that can be applied to the convergence analysis of the iterative sequences generated by the algorithms

{x1,k }, {x2,k }, , {x n,k} as follows:

x1,k1  1 − α k x1,k  α k R M1

ρ1 x2,k − ρ1T1x2,k  S1x2,k α k u1,k  w k ,

x2,k  R M2

ρ2 x3,k − ρ2T2x3,k  S2x3,k u2,k,

x n−1,k  R M n−1

ρ n−1 x n,k − ρ n−1 T n−1 x n,k  S n−1 x n,k u n−1,k ,

x n,k  R M n

ρ n x1,k − ρ n T n x1,k  S n x1,k u n,k ,

5.1

wherex1,1  x0,{α k } is a sequence in 0, 1, and {u i,k } ⊂ E i  1, 2, , n, {w k } ⊂ E are the

sequences satisfying the following conditions:

∞



k1

αk ∞; ∞

k1

w k  < ∞; lim

Theorem 5.2 Let T i , S i , and M i be the same as in Theorem 4.1 , and suppose that the sequences {x1,k }, {x2,k }, , {x n,k } are generated by Algorithm 5.1 If the condition4.1 holds, then

x1,k, x2,k, , xn,k  converges strongly to the unique solution x∗

1, x∗

2, , x∗

n  of the problem 3.1.

1, x∗

2, , x∗

n,

it follows fromLemma 3.1that

x∗

1 R M1

ρ1 x∗

2− ρ1



T1x∗

2 S1x∗ 2



,

x∗

2 R M2

ρ2 x∗

3− ρ2



T2x∗

3 S2x∗ 3



,

x∗

n−1  R M n−1

ρ n−1 x∗

n − ρ n−1 T n−1 x∗

n  S n−1 x∗

n,

x∗

n  R M n

ρ n x∗

1− ρ n

Tnx∗

1 S nx∗ 1



.

5.3

3.4

Problem3.4 is called the system of nonlinear variational inequalities... n

Proof First, we prove the existence of the solution Define a mapping F... Corollaries 3.2 and 3.3, 16, Theorems 2.1–2.4 and the main results in 13