Báo cáo hóa học: " Research Article Existence of Solutions and Convergence of a Multistep Iterative Algorithm for a System of Variational Inclusions with (H,η)-Accretive Operators" pot

Volume 2007, Article ID 93678, 20 pagesdoi:10.1155/2007/93678 Research Article Existence of Solutions and Convergence of a Multistep Iterative Algorithm for a System of Variational Inclu

Volume 2007, Article ID 93678, 20 pages

doi:10.1155/2007/93678

Research Article

Existence of Solutions and Convergence of a Multistep

Iterative Algorithm for a System of Variational Inclusions

with ( H,η)-Accretive Operators

Jian-Wen Peng, Dao-Li Zhu, and Xiao-Ping Zheng

Received 5 April 2007; Accepted 6 July 2007

Recommended by Lech Gorniewicz

We introduce and study a new system of variational inclusions with (H,η)-accretive

op-erators, which contains variational inequalities, variational inclusions, systems of varia-tional inequalities, and systems of variavaria-tional inclusions in the literature as special cases

By using the resolvent technique for the (H,η)-accretive operators, we prove the

exis-tence and uniqueness of solution and the convergence of a new multistep iterative algo-rithm for this system of variational inclusions in realq-uniformly smooth Banach spaces.

The results in this paper unify, extend, and improve some known results in the litera-ture

Copyright © 2007 Jian-Wen Peng et al 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

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 optimiza-tion and control, economics and transportaoptimiza-tion equilibrium, and 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, please see [1–50] and the references therein

Recently, some new and interesting problems, which are called to be system of vari-ational inequality problems were introduced and studied Pang [28], Cohen and Chap-lais [29], Bianchi [30] and Ansari and Yao [16] considered a system of scalar variational inequalities and Pang showed that the traffic equilibrium problem, the spatial equilib-rium problem, the Nash equilibequilib-rium, and the general equilibequilib-rium programming problem

can be modeled as a system of variational inequalities Ansari et al [31] introduced and studied a system of vector equilibrium problems and a system of vector variational in-equalities by a fixed point theorem Allevi et al [32] considered a system of generalized vector variational inequalities and established some existence results with relative pseu-domonotonicity Kassay and Kolumb´an [17] introduced a system of variational inequal-ities and proved an existence theorem by the Ky Fan lemma Kassay et al [18] studied Minty and Stampacchia variational inequality systems with the help of the Kakutani-Fan-Glicksberg fixed point theorem Peng [19,20] introduced a system of quasivaria-tional inequality problems and proved its existence theorem by maximal element the-orems Verma [21–25] 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 K Kim and S Kim [26] introduced a new sys-tem of generalized nonlinear quasivariational inequalities and obtained some existence and uniqueness results of solution for this system of generalized nonlinear quasivaria-tional inequalities in Hilbert spaces Cho et al [27] introduced and studied a new sys-tem of nonlinear variational inequalities in Hilbert spaces They proved some existence and uniqueness theorems of solutions for the system of nonlinear variational inequali-ties

As generalizations of the above systems of variational inequalities, Agarwal et al [33] introduced a system of generalized nonlinear mixed quasivariational inclusions and in-vestigated the sensitivity analysis of solutions for this system of generalized nonlinear mixed quasivariational inclusions in Hilbert spaces Kazmi and Bhat [34] introduced a system of nonlinear variational-like inclusions and gave an iterative algorithm for finding its approximate solution Fang and Huang [35] and Fang et al [36] introduced and stud-ied a new system of variational inclusions involvingH-monotone operators and

(H,η)-monotone, respectively Peng and Huang [37] proved the existence and uniqueness of solutions and the convergence of some new three-step iterative algorithms for a new sys-tem of variational inclusions in Hilbert spaces

On the other hand, Yu [10] introduced a new concept of (H,η)-accretive operators

which provide unifying frameworks forH-monotone operators in [1],H-accretive

oper-ators in [9], (H,η)-monotone operators in [35], maximalη-monotone operators in [5], generalized m-accretive operators in [8], m-accretive operators in [12], and maximal monotone operators [13,14]

Inspired and motivated by the above results, the purpose of this paper is to introduce

a new mathematical model, which is called to be a system of variational inclusions with (H,η)-accretive operators, that is, a family of variational inclusions with (H,η)-accretive

operators defined on a product set This new mathematical model contains the system of inequalities in [16,21–30] and the system of inclusions in [35–37], the variational inclu-sions in [1,2,9,11], and some variational inequalities in the literature as special cases

By using the resolvent technique for the (H,η)-accretive operators, we prove the

exis-tence of solutions for this system of variational inclusions We also prove the convergence

of a multistep iterative algorithm approximating the solution for this system of varia-tional inclusions The result in this paper unifies, extends, and improves some results in [1,2,9,11,21–30,35–37]

2 Preliminaries

We suppose thatE is a real Banach space with dual space, norm, and the generalized

dual pair denoted byE ∗,·, and·,·, respectively, 2Eis the family of all the nonempty subsets ofE, CB(E) is the families of all nonempty closed bounded subsets of E, and the

generalized duality mappingJ q:E →2E ∗is defined by

J q(x) =f ∗ ∈ E ∗:x, f ∗

=f ∗ · x ,f ∗  =  x  q −1 

, ∀ x ∈ E, (2.1) whereq > 1 is a constant In particular, J2is the usual normalized duality mapping It is known that, in general,J q(x) =  x  q −2J2(x), for all x =0, andJ q is single valued ifE ∗is strictly convex

The modulus of smoothness ofE is the function ρ E: [0,∞)→[0,∞) defined by

ρ E(t) =sup

1

2



 x + y + x − y  −1 : x  ≤1, y  ≤ t (2.2)

A Banach spaceE is called uniformly smooth if

limt

→0

ρ E(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 thatJ qis single valued ifE is uniformly smooth Xu and Roach [51] 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 constants c q > 0, such that for all x, y ∈ E,

 x + y  q ≤  x  q+qy,J q(x)+c q  y  q (2.5)

We recall some definitions needed later, for more details, please see [3,4,9,10] and the references therein

Definition 2.2 Let E be a real uniformly smooth Banach space, and let T,H : E → E be

two single-valued operators.T is said to be

(i) accretive if



T(x) − T(y),J q(x − y)≥0, ∀ x, y ∈ E; (2.6) (ii) strictly accretive ifT is accretive and



T(x) − T(y),J q(x − y)=0 iff x = y; (2.7) (iii)r-strongly accretive if there exists a constant r > 0 such that



T(x) − T(y),J q(x − y)≥ r  x − y  q, ∀ x, y ∈ E; (2.8)

(iv)r-strongly accretive with respect to H if there exists a constant r > 0 such that

T(x) − T(y),J qH(x) − H(y) ≥ r  x − y  q, ∀ x, y ∈ E; (2.9) (v)s-Lipschitz continuous if there exists a constant s > 0 such that

T(x) − T(y)  ≤ s  x − y , ∀ x, y ∈ E. (2.10)

Definition 2.3 Let E be a real uniformly smooth Banach space, let T : E → E and

η : E × E → E be two single-valued operators T is said to be

(i)η-accretive if



T(x) − T(y),J q

η(x, y) ≥0, ∀ x, y ∈ E; (2.11) (ii) strictlyη-accretive if T is η-accretive and



T(x) − T(y),J q

η(x, y) =0 iff x = y; (2.12) (iii)r-strongly η-accretive if there exists a constant r > 0 such that



T(x) − T(y),J q

η(x, y) ≥ r  x − y  q, ∀ x, y ∈ E. (2.13)

Definition 2.4 Let η : E × E → E, let T,H : E → E be single-valued operators and M : E →

2Ebe a multivalued operator.M is said to be

(i) accretive if



u − v,J q(x − y)≥0, ∀ x, y ∈ E, u ∈ M(x), v ∈ M(y); (2.14) (ii)η-accretive if



u − v,J q

η(x, y) ≥0, ∀ x, y ∈ E, u ∈ M(x), v ∈ M(y); (2.15) (iii) strictlyη-accretive if M is η-accretive, and equality holds if and only if x = y;

(iv)r-strongly η-accretive if there exists a constant r > 0 such that if



u − v,J q

η(x, y) ≥ r  x − y  q, ∀ x, y ∈ E, u ∈ M(x), v ∈ M(y); (2.16) (v)m-accretive if M is accretive and (I + ρM)(E) = E holds for all ρ > 0, where I is the

identity map onE;

(vi) generalizedη-accretive if M is η-accretive and (I + ρM)(E) = E holds for all ρ > 0;

(vii)H-accretive if M is accretive and (H + ρM)(E) = E holds for all ρ > 0;

(viii) (H,η)-accretive if M is η-accretive and (H + ρM)(E) = E holds for all ρ > 0 Remark 2.5 (i) If η(x, y) = x − y, for all x, y ∈ E, then the definition of (H,η)-accretive

operators becomes that ofH-accretive operators in [9] IfE =Ᏼ is a Hilbert space, the definition of (H,η)-accretive operator becomes that of (H,η)-monotone operators in

[36], the definition of H-accretive operators becomes that of H-monotone operators

in [1,35] Hence, the definition of (H,η)-accretive operators provides unifying

frame-works for classes ofH-accretive operators, generalized η-accretive operators, m-accretive

operators, maximal monotone operators, maximalη-monotone operators, H-monotone

operators, and (H,η)-monotone operators.

Definition 2.6 [5] Letη : E × E → E be a single-valued operator, then η( ·,·) is said to be

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

η(u,v)  ≤ τ  u − v , ∀ u,v ∈ E. (2.17)

Definition 2.7 [10] Letη : E × E → E be a single-valued operator, let H : E → E be a strictly η-accretive single-valued operator, and let M : E →2Ebe an (H,η)-accretive operator, and

letλ > 0 be a constant The resolvent operator R H,η M,λ:E → E associated with H, η, M, λ is

defined by

R H,η M,λ(u) =(H + λM) −1(u), ∀ u ∈ E. (2.18) Lemma 2.8 [10] Let η : E × E → E be a τ-Lipschitz continuous operator, H : E → E be a γ-strongly η-accretive operator, and let M : E →2E be an (H,η)-accretive operator Then, the resolvent operator R H,η M,λ:E → E is τ q −1/γ-Lipschitz continuous, that is,



R H,η M,λ(x) − R H,η M,λ(y) ≤ τ q −1

γ  x − y , ∀ x, y ∈ E. (2.19)

We extend some definitions in [6,37,46] to more general cases as follows

Definition 2.9 Let E1,E2, ,E pbe Banach spaces, letg1:E1→ E1andN1:p

j =1E j → E1

be two single-valued mappings

(i)N1 is said to be ξ-Lipschitz continuous in the first argument if there exists a

constantξ > 0 such that

N1

x1,x2, ,x p

− N1



y1,x2, ,x p  ≤ ξx1− y1,

∀ x1,y1∈ E1,x j ∈ E j(j =2, 3, , p). (2.20)

(ii)N1is said to be accretive in the first argument if



N1



x1,x2, ,x p

− N1



y1,x2, ,x p

,J q

x1− y1



≥0,

∀ x1,y1∈ E1,x j ∈ E j(j =2, 3, , p). (2.21)

(iii)N1is said to beα-strongly accretive in the first argument if there exists a constant

α > 0 such that

N1 x1,x2, ,x p

− N1 y1,x2, ,x p

,J qx1− y1 

≥ αx1− y1q

,

∀ x1,y1∈ E1,x j ∈ E j(j =2, 3, , p). (2.22)

(iv)N1is said to be accretive with respect tog in the first argument if

N1 x1,x2, ,x p

− N1 y1,x2, ,x p

,J qgx1

− gy1 

≥0,

∀ x1,y ∈ E1,x j ∈ E j(j =2, 3, , p). (2.23)

(v)N1is said to beβ-strongly accretive with respect to g in the first argument if there

exists a constantβ > 0 such that



N1



x1,x2, ,x p

− N1



y1,x2, ,x p

,J q

gx1

− gy1

≥ βx1− y1q

,

∀ x1,y1∈ E1,x j ∈ E j(j =2, 3, , p). (2.24)

In a similar way, we can define the Lipschitz continuity and the strong accretivity (ac-cretivity) of N i :p

j =1E j → E i (with respect to g i:E i → E i) in the ith argument

(i =2, 3, , p).

3 A system of variational inclusions

In this section, we will introduce a new system of variational inclusions with (

H,η)-accretive operators In what follows, unless other specified, for eachi =1, 2, , p, we

always suppose thatE i is a realq-uniformly smooth Banach space, H i,g i:E i → E i,η i:

E i × E i → E i,F i,G i:p

j =1E j → E iare single-valued mappings, and thatM i:E i →2E i is an (H i,η i)-accretive operator We consider the following problem of finding (x1,x2, ,x p)∈

p

i =1E isuch that for eachi =1, 2, , p,

0∈ F i

x1,x2, ,x p

+G i

x1,x2, ,x p

+M i

g i

x i

The problem (3.1) is called a system of variational inclusions with (H,η)-accretive

operators

Below are some special cases of problem (3.1)

(i) For each j =1, 2, , p, if E j =Ᏼj is a Hilbert space, then problem (3.1) becomes the following system of variational inclusions with (H,η)-monotone operators, which is

to find (x1,x2, ,x p)∈i p =1E isuch that for eachi =1, 2, , p,

0∈ F i

x1,x2, ,x p

+G i

x1,x2, ,x p

+M i

g i

x i

(ii) For each j =1, 2, , p, if g j ≡ I j (the identity map onE j) andG j ≡0, then prob-lem (3.1) reduces to the system of variational inclusions with (H,η)-accretive operators,

which is to find (x1,x2, ,x p)∈p j =1E jsuch that for eachi =1, 2, , p,

0∈ F i

x1,x2, ,x p

+M i

x i

(iii) Ifp =1, then problem (3.2) becomes the following variational inclusion with an (H1,η1)-monotone operator, which is to findx1∈Ᏼ1such that

0∈ F1 x1

+G1 x1

+M1 g1 x1 . (3.4) Moreover, ifη1(x1,y1)= x1− y1for allx1,y1∈Ᏼ1andH1= I1(the identity map on Ᏼ1), then problem (3.4) becomes the variational inclusion introduced and researched by Adly [11] which contains the variational inequality in [2] as a special case

Ifp =1, then problem (3.3) becomes the following variational inclusion with an (H1,

η1)-accretive operator, which is to findx1∈ E1such that

Problem (3.5) was introduced and studied by Yu [10] and contains the variational inclusions in [1,9] as special cases

If p =2, then problem (3.3) becomes the following system of variational inclusions with (H,η)-accretive operators, which is to find (x1,x2)∈ E1× E2such that

0∈ F1



x1,x2

+M1



x1

,

0∈ F2



x1,x2

+M2



x2

Problem (3.6) contains the system of variational inclusions withH-monotone

opera-tors in [35], the system of variational inclusions with (H,η)-monotone operators in [36]

as special cases

If p =3 and for each j =1, 2, 3,E j =Ᏼj is a Hilbert space andG j =0, then prob-lem (3.1) becomes the system of variational inclusions with (H,η)-monotone operators

in [37] with f j =0 andζ j =1

(iv) For each j =1, 2, , p, if E j =Ᏼj is a Hilbert space, andM j(x j)=Δη j ϕ j for all

x j ∈Ᏼj, whereϕ j:Ᏼj → R ∪ {+∞}is a proper,η j-subdifferentiable functional and Δη j ϕ j

denotes theη j-subdifferential operator of ϕj, then problem (3.3) reduces to the following system of variational-like inequalities, which is to find (x1,x2, ,x p)∈i p =1Ᏼisuch that for eachi =1, 2, , p,



F i

x1,x2, ,x p

,η i

z i,x i 

+ϕ i

z i

− ϕ i

x i

≥0, ∀ z i ∈Ᏼi (3.7) (v) For each j =1, 2, , p, if E j =Ᏼj is a Hilbert space, andM j(x j)= ∂ϕ j(x j), for all

x j ∈Ᏼj, whereϕ j:Ᏼj → R ∪ {+∞}is a proper, convex, lower semicontinuous functional and∂ϕ j denotes the subdifferential operator of ϕj, then problem (3.3) reduces to the following system of variational inequalities, which is to find (x1,x2, ,x p)∈p i =1Ᏼisuch that for eachi =1, 2, , p,

F ix1,x2, ,x p

,z i − x i

+ϕ iz i

− ϕ ix i

≥0, ∀ z i ∈Ᏼi (3.8) (vi) For each j =1, 2, , p, if M j(x j)= ∂δ K j(x j) for allx j ∈Ᏼj, whereK j ⊂Ᏼj is a nonempty, closed, and convex subsets andδ K j denotes the indicator ofK j, then prob-lem (3.8) reduces to the following system of variational inequalities, which is to find (x1,x2, ,x p)∈i p =1Ᏼisuch that for eachi =1, 2, , p,



F i

x1,x2, ,x p

,z i − x i

≥0, ∀ z i ∈ K i (3.9) Problem (3.9) was introduced and researched in [16,28–30] Ifp =2, then problems (3.7), (3.8), and (3.9), respectively, become the problems (3.2), (3.3) and (3.4) in [36]

It is easy to see that problem (3.4) in [36] contains the models of system of variational inequalities in [21–25] as special cases

It is worthy noting that problem (3.1)–(3.8) are all new problems

4 Existence and uniqueness of the solution

In this section, we will prove existence and uniqueness for solutions of problem (3.1) For our main results, we give a characterization of the solution of problem (3.1) as follows

Lemma 4.1 For i =1, 2, , p, let η i:E i × E i → E i be a single-valued operator, let H i:E i →

E i be a strictly η i -accretive operator, and let M i:E i →2E i be an (H i,η i )-accretive operator.

Then (x1,x2, ,x p)∈i p =1E i is a solution of the problem ( 3.1 ) if and only if for each i =

1, 2, , p,

g i

x i

= R H i,η i

M i,λ i



H i

g i

x i

− λ i F i

x1,x2, ,x p

− λ i G i

x1,x2, ,x p

where R H i,η i

M i,λ i =(H i+λ i M i)−1and λ i > 0 are constants.

Proof The fact directly follows fromDefinition 2.9 

LetΓ= {1, 2, , p }

Theorem 4.2 For i =1, 2, , p, let η i:E i × E i → E i be σ i -Lipschitz continuous, let H i:

E i → E i be γ i -strongly η i -accretive and τ i -Lipschitz continuous, let g i:E i → E i be β i -strongly accretive and θ i -Lipschitz continuous, let M i:E i →2E i be an (H i,η i )-accretive operator, let

F i:p

j =1E j → E i be a single-valued mapping such that F i is r i -strongly accretive with respect

to gi and s i -Lipschitz continuous in the ith argument, where gi:E i → E i is defined by gi(x i)=

H i ◦ g i(x i)= H i(g i(x i )), for all x i ∈ E i , F i is t ij -Lipschitz continuous in the jth arguments for each j ∈ Γ, j = i, G i:p

j =1E j → E i be a single-valued mapping such that G i is l ij -Lipschitz continuous in the jth argument for each j ∈ Γ If there exist constants λ i > 0 (i =1, 2, , p) such that

q

1− qβ1+c q θ1q+σ q −1

1

γ1

q τ1q θ1q − qλ1r1+c q λ1q s q1+l11λ1σ q −1

1

γ1 +

p



k =2

λ k σ q −1

k

γ k



t k1+l k1

< 1,

q

1− qβ2+c q θ2q+σ q −1

2

γ2

q τ2q θ2q − qλ2r2+c q λ2q s q2+l22λ2σ q −1

2

γ2 + 

k ∈ Γ,k =2

λ k σ q −1

k

γ k



t k2+l k2

< 1,

···

q

1− qβ p+c q θ q p+σ q −1

p

γ p

q τ q p θ q p − qλ p r p+c q λ p s q p+l pp λ p σ q −1

p

γ p +

p−1

k =1

σ q −1

k λ k

γ k



t k,p+l k,p

< 1.

(4.2) Then, problem (3.1) admits a unique solution

Proof For i =1, 2, , p and for any given λ i > 0, define a single-valued mapping T i,λ i:

p

j =1E j → E iby

T i,λ i



x1,x2, ,x p

= x i − g i

x i

+R H i,η i

M i,λ i



H i g i

x i

− λ i F i

x1,x2, ,x p

− λ i G i

x1,x2, ,x p

, (4.3) for any (x1,x2, ,x p)∈i p =1E i

For any (x1,x2, ,x p), (y1,y2, , y p)∈i p =1E i, it follows from (4.3) that for i =1,

2, , p,

T i,λ i

x1,x2, ,x p

− T i,λ i



y1,y2, , y p i

=x i − g i

x i

+R H i,η i

M i,λ i



H i

g i

x i

− λ i F i

x1,x2, ,x p

− λ i G i

x1,x2, ,x p

−y i − g i

y i

+R H i,η i

M i,λ i



H i

g i

y i

− λ i F i

y1,y2, , y p

− λ i G i

y1,y2, , y p i

≤x i − y i −

g i

x i

− g i

y i i

+R H i,η i

M i,λ i



H i

g i

x i

− λ i F i

x1,x2, ,x p

− λ i G i

x1,x2, ,x p

− R H i,η i

M i,λ i,m i



H i

g i

y i

− λ i F i

y1,y2, , y p

− λ i G i

y1,y2, , y p i .

(4.4) Fori =1, 2, , p, since g iisβ i-strongly accretive andθ i-Lipschitz continuous, we have

x i − y i −

g i

x i

− g i

y i q i

=x i − y iq

i − qg i

x i

− g i

y i

,J q

x i − y i 

+c qg i

x i

− g i

y i q i

≤1− qβ i+c q θ q i x i − y iq

i

(4.5)

It follows fromLemma 2.1that fori =1, 2, , p,



R H i,η i

M i,λ i



H i

g i

x i

− λ i F i

x1,x2, ,x p

− λ i G i

x1,x2, ,x p

− R H i,η i

M i,λ i

H ig iy i

− λ i F iy1,y2, , y p

− λ i G iy1,y2, , y p 

i

≤ σ q −1

i

γ i

H i

g i

x i

− H i

g i

y i

− λ i

F i

x1,x2, ,x p

− F i

y1,y2, , y p i

+σ q −1

i λ i

γ i

G i

x1,x2, ,x p

− G i

y1,y2, , y p i

≤ σ q −1

i

γ i

H ig ix i

− H ig iy i

− λ iF ix1,x2, ,x i −1,x i,x i+1, ,x p

− F i

x1,x2, ,x i −1,y i,x i+1, ,x p i

+σ q −1

i λ i

γ i

 

j ∈ Γ,j = i

F ix1,x2, ,x j −1,x j,x j+1, ,x p

− F i

x1,x2, ,x j −1,y j,x j+1, ,x p i

+σ q −1

i λ i

γ i

 p

j =1

G i

x1,x2, ,x j −1,x j,x j+1, ,x p

− G i

x1,x2, ,x j −1,y j,x j+1, ,x p i.

(4.6)

Fori =1, 2, , p, since H iisτ i-Lipschitz continuous, andg iisθ i-Lipschitz continuous andF iisr i-gi-strongly accretive ands i-Lipschitz continuous in theith argument, we have

H i

g i

x i

− H i

g i

y i

− λ i

F i

x1,x2, ,x i −1,x i,x i+1, ,x p

− F i

x1,x2, ,x i −1,y i,x i+1, ,x p q

i

≤H i

g i

x i

− H i

g i

y i q

i − qλ i

F i

x1,x2, ,x i −1,x i,x i+1, ,x p

− F ix1,x2, ,x i −1,y i,x i+1, ,x p

,H ig ix i

− H ig iy i 

+c q λ i qF i

x1,x2, ,x i −1,x i,x i+1, ,x p

− F i

x1,x2, ,x i −1,y i,x i+1, ,x p q

i

≤ τ i qg i

x i

− g i

y i q

i − qλ i r ix i − y iq

i +c q λ i q s q ix i − y iq

i

≤τ i q θ q i − qλ i r i+c q λ i q s q i x i − y iq

i

(4.7)

Fori =1, 2, , p, since F iist ij-Lipschitz continuous in the jth arguments (j ∈ Γ, j =

i), we have

F i

x1,x2, ,x j −1,x j,x j+1, ,x p

− F i

x1,x2, ,x j −1,y j,x j+1, ,x p i ≤ t ijx j − y jj .

(4.8)

For i =1, 2, , p, since G i is l ij-Lipschitz continuous in the jth arguments (j =1,

2, , p), we have

G i

x1,x2, ,x j −1,x j,x j+1, ,x p

− G i

x1,x2, ,x j −1,y j,x j+1, ,x p i ≤ l ijx j − y jj .

(4.9)

It follows from (4.4)–(4.9) that for eachi =1, 2, , p

T i,λ i

x1,x2, ,x p

− T i,λ i



y1,y2, , y p i

≤



q

1− qβ i+c q θ i q+σ q −1

i

γ i

q τ i q θ q i − qλ i r i+c q λ i q s q i+l ii λ i σ q −1

i

γ i



x i − y ii

+λ i σ q −1

i

γ i

 

j ∈ Γ, j = i

t ij+l ij x j − y jj.

(4.10)

For any (x1,x2, ,x p), (y1,y2,... ix i

− H ig iy i... class="page_container" data-page ="1 0">

For< i>i =1, 2, , p, since H iisτ i-Lipschitz continuous, and< i>g iisθ i-Lipschitz