Báo cáo hóa học: "A NEW SYSTEM OF GENERALIZED NONLINEAR RELAXED COCOERCIVE VARIATIONAL INEQUALITIES" pptx

COCOERCIVE VARIATIONAL INEQUALITIESKE DING, WEN-YONG YAN, AND NAN-JING HUANG Received 21 November 2004; Revised 13 April 2005; Accepted 28 June 2005 We introduce and study a new system o

COCOERCIVE VARIATIONAL INEQUALITIES

KE DING, WEN-YONG YAN, AND NAN-JING HUANG

Received 21 November 2004; Revised 13 April 2005; Accepted 28 June 2005

We introduce and study a new system of generalized nonlinear relaxed cocoercive in-equality problems and construct an iterative algorithm for approximating the solutions

of the system of generalized relaxed cocoercive variational inequalities in Hilbert spaces

We prove the existence of the solutions for the system of generalized relaxed cocoercive variational inequality problems and the convergence of iterative sequences generated by the algorithm We also study the convergence and stability of a new perturbed iterative algorithm for approximating the solution

Copyright © 2006 Ke Ding et al This is an open access article distributed under the Cre-ative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

1 Introduction

Variational inequality problems have various applications in mechanics and physics, opti-mization and control, linear and nonlinear programming, economics and finance, trans-portation equilibrium and engineering science, and so forth Consequently considerable attention has been devoted to the study of the theory and efficient numerical methods for variational inequality problems (see, e.g., [2–17] and the references therein) In [15], Verma introduced a new system of nonlinear strongly monotone variational inequalities and studied the approximate of this system based on the projection method, and in [16], Verma discussed the approximate solvability of a system of nonlinear relaxed cocoercive variational inequalities in Hilbert spaces Recently, Kim and Kim [14] introduced and studied a system of nonlinear mixed variational inequalities in Hilbert spaces, and ob-tained some approximate solvability results In the recent paper [6], Cho et al 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 They also constructed an iterative algorithm for approximating the solution of the system of nonlinear variational inequalities Some related works, we refer to [2,3,5,7–10,12,13] Motivated and inspired by these works, in this paper, we introduce and study a new system of generalized nonlinear relaxed cocoercive variational

Hindawi Publishing Corporation

Journal of Inequalities and Applications

Volume 2006, Article ID 40591, Pages 1 14

DOI 10.1155/JIA/2006/40591

inequality problems and construct an iterative algorithm for approximating the solutions

of the system of generalized relaxed cocoercive variational inequalities in Hilbert spaces

We prove the existence of the solutions for the system of generalized relaxed cocoercive variational inequality problems and the convergence of iterative sequences generated by the algorithm We also study the convergence and stability of a new perturbed iterative algorithm for approximating the solution The results presented in this paper improve and extend the previously known results in this area

2 Preliminaries

LetH be a Hilbert space endowed with a norm  · and inner product (·,·), respectively Let CB(H) be the family of all nonempty subsets of H and K1,K2 be two convex and closed subsets ofH Let g1,g2,m1,m2:H → H and F, G : H × H → H be mappings We

consider the following system of generalized nonlinear variational inequality problems: findx, y ∈ H such that g i(x) ∈ K i(x) for i =1, 2, and



F(x, y), z − g1(x)

≥0, ∀ z ∈ K1(x),



G(x, y), z − g2(y)

whereK i(x) = m i(x) + K ifori =1, 2

WhenK1 andK2are both convex cones ofH, it is easy to see that problem (2.1) is equivalent to the following system of generalized nonlinear co-complementarity prob-lems: findx, y ∈ H such that g i(x) ∈ K i(x) for i =1, 2, and

F(x, y) ∈K1(x) − g1(x)∗

,

G(x, y) ∈K2(y) − g2(y)∗

,

(2.2)

whereK i(x) = m i(x) + K iand (K i(x) − g i(x)) ∗is the dual ofK i(x) − g i(x) for i =1, 2, that is,



K i(x) − g i(x)∗

=u ∈ H |(u, v) ≥0,∀ v ∈ K i(x) − g i(x)

Some examples of problems (2.1) and (2.2) are as follows

(I) IfG =0 andF(x, y) = Tx + Ax for all x, y ∈ X, where T, A : H → H are two

map-pings, then problem (2.2) reduces to findingx ∈ H such that

Tx + Ax ∈K1(x) − g1(x)∗

which is called the generalized complementarity problem The problem (2.4) was ex-tended and studied by Jou and Yao [11] in Hilbert spaces, and by Chen et al [5] in the setting of Banach spaces

(II) LetT : H × H → H be a mapping If F(x, y) = ρT(y, x) + x − y, G(x, y) = ηT(x, y)

+y − x for all x, y ∈ H, m1= m2=0,K1= K2= K, and g1= g2= I, where I is an identity

mapping andρ > 0, η > 0, then problem (2.1) reduces to findingx, y ∈ K such that



ρT(y, x) + x − y, z − x

≥0, ∀ z ∈ K,



ηT(x, y) + y − x, z − y

which is called the system of nonlinear variational inequality problems considered by Verma [16] The special case of problem (2.5) was studied by Verma [15] The problem (2.5) was extended and studied by Agarwal et al [1], Kim and Kim [14], and Cho et al [6]

(III) Ifm1= m2=0, andg1= g2= I, then problem (2.1) reduces to findingx ∈ K1and

y ∈ K2such that



F(x, y), z − x

≥0, ∀ z ∈ K1,



G(x, y), z − y

which is just the problem considered in [12] withF, G being single-valued mappings Definition 2.1 A mapping N : H × H → H is said to be

(i)α-strongly monotone with respect to first argument if there exists some α > 0 such

that



N(x, ·)− N(y, ·),x − y

≥ α  x − y 2, ∀(x, y) ∈ H × H; (2.7) (ii)ξ-Lipschitz continuous with respect to the first argument, if there exists a constant

ξ > 0 such that

N(x, ·)− N(y, ·) ≤ ξx − y, ∀(x, y) ∈ H × H. (2.8)

Similarly, we can define the strong monotonicity and Lipschitzian continuity with re-spect to the second argument ofN.

Definition 2.2 A Mapping N : H × H → H is said to be relaxed (a, b)-cocoercive with

respect to the first argument if there exists constantsa > 0 and b > 0 such that



N(x, ·)− N(y, ·),x − y

≥(− a)  x − y 2+b  x − y 2, ∀(x, y) ∈ H × H. (2.9)

Ifa =0, thenN is b-strongly monotone Similarly, we can define the relaxed (a,

b)-co-coercivity with respect to the second argument ofN.

Lemma 2.3 [4] If K ⊂ H is a closed convex subset and z ∈ H is a given point, then there exists x ∈ K such that

if and only if x = P K z, where P K is the projection of H onto K.

Lemma 2.4 [4] The projection P K is nonexpansive, that is,

P K u − P K v  ≤  u − v , ∀ u, v ∈ H. (2.11)

Lemma 2.5 [18] Let { K n } be a sequence of closed convex subsets of H such that H(K n,K) →

0 as n → ∞ , where H( ·,· ) is the Hausdor ff metric, that is, for any A,B ∈CB(H),

H(A, B) =max



sup

a ∈ A

inf

b ∈ B

 a − b , sup

b ∈ B

inf

a ∈ A

 a − b 



Then

P K

n v − P K v  −→0 (n −→ ∞),∀ v ∈ H. (2.13) Lemma 2.6 [4] If K(u) = m(u) + K for all u ∈ H, then

P K(u) v = m(u) + P K

v − m(u)

From Lemmas2.3and2.6, we have the following lemma

Lemma 2.7 If K1,K2⊂ H are two closed convex cones, and K i(·)= m( ·) +K i (i = 1, 2), then x, y ∈ H solve problem ( 2.1 ) if and only if x, y ∈ H such that

x = x − g1(x) + m1(x) + P K1

g1(x) − ρF(x, y) − m1(x)

,

y = y − g2(y) + m2(y) + P K2



g2(y) − ρG(x, y) − m2(y)

where ρ > 0 is a constant.

Lemma 2.8 [17] Let { μ n } be a real sequence of nonnegative numbers and { ν n } be a real sequence of numbers in [0, 1] with ∞

n =0ν n = ∞ If there exists a constant n1such that

μ n+1 ≤1− ν n



μ n+ν n δ n, ∀ n ≥ n1, (2.16)

where δ n ≥ 0 for all n ≥ 0, and δ n → 0 ( n → ∞ ), then lim n →∞ μ n = 0.

3 Existence and convergence

In this section, we construct an iterative algorithm to approximate the solution of prob-lem (2.1) and study the convergence of the sequence generated by the algorithm

Algorithm 3.1 For any given x0,y0∈ H, we compute

x n+1 = x n − g1 

x n

+m1 

x n

+P K1

g1 

x n

− ρF

x n,y n

− m1 

x n

,

y n+1 = y n − g2



y n

+m2



y n

+P K2

g2



y n

− ρG

x n,y n

− m2



y n

Theorem 3.2 Let g i:H → H be η i -strongly monotone and ζ i -Lipschitz continuous and

m i:H → H be γ i -Lipschitz continuous (i = 1, 2) Let F : H × H → H be l1, 2-Lipschitz con-tinuous with respect to the first, second arguments, respectively, and relaxed (a, b)-cocoercive with respect to the first argument Let G : H × H → H be n1, n2-Lipschitz continuous with respect to the first, second arguments, respectively, and relaxed (c, d)-cocoercive with respect

to the second argument If

2 1 +ζ2−2η1+ 2γ1+ 1 +ρ2l2+ 2ρal2−2ρb + ρn1< 1,

2 1 +ζ2−2η2+ 2γ2+ 1 +ρ2n2+ 2ρcn2−2ρd + ρl2< 1.

(3.2)

then there exist x ∗,y ∗ ∈ H, which solve problem ( 2.1 ) Moreover, the iterative sequences

{ x n } and { y n } generated by Algorithm 3.1 converge to x ∗ and y ∗ , respectively.

Proof From (3.1) andLemma 2.6, we have

x n+1 − x n  =  x n − g1 

x n

+m1 

x n

+P K1

g1 

x n

− ρF

x n,y n

− m1 

x n

−x n −1− g1



x n −1



+m1



x n −1



+P K1



g1



x n −1



− ρF

x n −1,y n −1



− m1



x n −1 

≤x n − x n −1−

g1



x n

− g1



x n −1 +m1

x n

− m1



x n −1 

+P K1

g1



x n

− ρF

x n,y n

− m1



x n

− P K1



g1



x n −1



− ρF

x n −1,y n −1



− m1(x n −1) .

(3.3)

Sinceg1isζ1-Lipschitz continuous andη1-strongly monotone,

x n − x n −1−

g1 

x n

− g1 

x n −1  2

≤1 +ζ2−2η1 x n − x n −1 2

From theγ1-Lipschitzian continuity ofm1, we have

m1

x n

− m1



x n −1 ≤ γ1 x n − x n −1. (3.5)

Lemma 2.4implies thatP K1is nonexpansive and it follows from the strong monotonicity

ofg1that

P K1

g1



x n

− ρF

x n,y n

− m1



x n

− P K1

g1



x n −1



− ρF

x n −1,y n −1



− m1



x n −1 

≤g1

x n

− ρF

x n,y n

− m1 

x n

−g1 

x n −1 

− ρF

x n −1,y n −1 

− m1 

x n −1 

≤x n − x n −1−

g1



x n

− g1



x n −1 +m1

x n

− m1



x n −1 

+x n − x n −1− ρ

F

x n,y n



− F

x n −1,y n+ρF

x n −1,y n



− F

x n −1,y n −1 .

(3.6)

SinceF is relaxed (a, b)-cocoercive and l1-Lipschitz continuous with respect to the first argument,

x n − x n −1− ρ

F

x n,y n

− F

x n −1,y n 2

=x n − x n −1 2

+ρ2F

x n,y n



− F

x n −1,y n 2

−2

x n − x n −1,ρ

F

x n,y n

− F

x n −1,y n

≤x n − x n −1 2

+ρ2 F

x n,y n

− F

x n −1,y n 2 + 2ρaF

x n,y n



− F

x n −1,y n 2

−2ρbx n − x n −1 2

=1 +l2ρ2+ 2ρal2−2ρbx n − x n −1 2

.

(3.7)

SinceF is l2-Lipschitz continuous with respect to the second argument,

F

x n −1,y n

− F

x n −1,y n −1 ≤ l2 y n − y n −1. (3.8)

It follows from (3.3)–(3.8) that

x n+1 − x n

≤ 2 1 +ζ2−2η1+ 2γ1+ 1 +ρ2l2+ 2ρal2−2ρb

x n − x n −1 +ρl2y n − y n −1.

(3.9) Similarly, we have

y n+1 − y n

≤ 2 1 +ζ2−2η2+ 2γ2+ 1 +ρ2n2+ 2ρcn2−2ρd

y n − y n −1 +ρn1x n − x n −1.

(3.10) Now (3.9) and (3.10) imply

x n+1 − x n+y n+1 − y n

≤ 2 1 +ζ2−2η1+ 2γ1+ 1 +ρ2l2+ 2ρal2−2ρb + ρn1



x n − x n −1 

+ 2 1 +ζ2−2η2+ 2γ2+ 1 +ρ2n2+ 2ρcn2−2ρd + ρl2



,

y n − y n −1 ≤ ωx n − x n −1+y n − y n −1,

(3.11)

where

ω =max



2 1 +ζ2−2η1+ 2γ1+ 1 +ρ2l2+ 2ρal2−2ρb + ρn1,

2 1 +ζ2−2η2+ 2γ2+ 1 +ρ2n2+ 2ρcn2−2ρd + ρl2



.

(3.12)

It follows from (3.2) thatω < 1 Thus (3.11) implies that{ x n }and{ y n }are both Cauchy sequences inH, and { x n }converges tox ∗ ∈ H, { y n }converges toy ∗ ∈ H Since m1,m2,

g1,g2,P K1,P K2,F, G are all continuous, we have

x ∗ = x ∗ − g1



x ∗

+m1



x ∗

+P K



g1(x ∗

− ρF

x ∗,y ∗

− m1



x ∗

,

y ∗ = y ∗ − g2



y ∗

+m2



y ∗

+P K

g2



y ∗

− ρG

x ∗,y ∗

− m2



y ∗, (3.13)

The result follows then fromLemma 2.7 This completes the proof 

Remark 3.3 Let ρ > 0 be a number satisfying the conditions.



ρ − b − al

2−1− e1



n1

l2− n2





<



1− e1

 2

−1 + b − al2−1− e1



n1

 2 

/

l2− n2 

l2− n2 , ρn1< 1 − e1,n1< l1,



ρ − d − cn2−



1− e2 

l2

n2− l2





<



1− e2

 2

−1 + d − cn2−1− e2



l2

 2 

/

n2− l2 

n2− l2 , ρl2< 1 − e2, 2< n2,

(3.14)

wheree1=2 1 +ζ2−2η1+ 2γ1ande2=2 1 +ζ2−2η2+ 2γ2 Then (3.2) holds

4 Perturbed algorithm and stability

In this section, we construct a new perturbed iterative algorithm for solving problem (2.1) and prove the convergence and stability of the iterative sequence generated by the algorithm

Definition 4.1 Let T be a self-map of H, x0∈ H and let x n+1 = f (T, x n) define an iteration procedure which yields a sequence of points{ x n } ∞

n =0inH Suppose that { x ∈ H : Tx =

x } = ∅and{ x n } ∞

n =0converge to a fixed pointx ∗ofT Let { u n } ⊂ H and let  n =  u n+1 −

f (T, u n) If lim n =0 implies that limu n = x ∗, then the iteration procedure defined by

x n+1 = f (T, x n) is said to beT-stable or stable with respect to T Some results for the

stability of various iterative processes, we refer to [1,10] and the references therein Let { K1

n } and { K2

n } be two sequences of closed convex subsets of H such that H(K1

n,K) →0,H(K2

n,K) →0, whenn → ∞ Now we consider the following perturbed algorithm for solving problem (2.1)

Algorithm 4.2 For any given x0,y0∈ H, we compute

x n+1 =1− t n

x n+t n

x n − g1(x n

+m1



x n

+P K1

n



g1



x n

− ρF

x n,y n

− m1



x n

+t n e n,

y n+1 =1− t n

y n+t n

y n − g2



y n

+m2



y n

+P K2

n



g2



y n

− ρG

x n,y n

− m2



y n

+t n j n, (4.1) for alln =0, 1, 2, , where { e n }and{ j n }are two sequences of the elements ofH, and the

sequence{ t n }satisfies the following conditions

0≤ t n ≤1, ∀ n ≥0,

∞



n =0

Let{ u n }and{ v n }be any sequences inH and define  n = 1

n+2

nby

1

n =u n+1 −

1− t n

u n+t n

u n − g1



u n

+m1



u n

+P K1

g1 

u n

− ρF

u n,v n

+m1 

u n

+t n e n

2

n =v n+1 −

1− t n

v n+t n

v n − g2



v n

+m2



v n

+P K2



g2



v n



− ρG

u n,v n



+m2



v n



+t n j n.

(4.3)

Theorem 4.3 Let g i:X → X be η i -strongly monotone and ζ i -Lipschitz continuous, and

m i:X → X be τ i -Lipschitz continuous for i = 1, 2 Let F : X × X → X be l1, 2-Lipschitz continuous with respect to the first and second arguments, respectively, and relaxed (a, b)-cocoercive with respect to the first argument Let G : X × X → X be n1, n2-Lipschitz continu-ous with respect to the first and second arguments, respectively, and relaxed (c, d)-cocoercive with respect to the second argument Suppose H(K n,K) →0 (n → ∞ ) and



ρ − b − al

2−1− e1



n1

l2− n2





<



1− e1

 2

−1 + b − al2−1− e1



n1

 2 

/

l2− n2 

l2− n2 , ρn1< 1 − e1,n1< l1,



ρ − d − cn

2−1− e2



l2

n2− l2





<



1− e2

 2

−1 + d − cn2−1− e2



l2

 2 

/

n2− l2 

n2− l2 , ρl2< 1 − e2, 2< n2,

(4.4)

where e i =2 1 +ζ2

i −2η i+ 2γ i for i = 1, 2 If lim n →∞  e n  = 0 and lim n →∞  j n  = 0, then

we have the following conclusions.

(I) The iterative sequences generated by Algorithm 4.2 converge to the unique solution of ( 2.1 ).

(II) Moreover, if 0 < t ≤ t n , then limu n = x ∗ , lim v n = y ∗ if and only if lim( 1

n+2

n)= 0, where 1

n and 2

n are defined by ( 4.3 ).

Proof ByTheorem 3.2, problem (2.1) admits a solution (x ∗,y ∗) It is easy to prove that (x ∗,y ∗) is the unique solution of (4.1) FromLemma 2.7, we have

x ∗ =1− t n



x ∗+t n



x ∗ − g1



x ∗

+m1



x ∗

+P K1



g1



x ∗

− ρF

x ∗,y ∗

− m1



x ∗

,

y ∗ =1− t n

y ∗+t n

y ∗ − g2



y ∗

+m2



y ∗

+P K2

g2



y ∗

− ρG

x ∗,y ∗

− m2



y ∗, (4.5) SinceP K is nonexpansive and it follows from (4.1) and (4.5) that

x n+1 − x ∗

=1− t n

x n+t n

x n − g1



x n



+m1



x n



+P K1

n



g1



x n



− ρF

x n,y n



− m1



x n



+t n e n

−1− t n

x ∗ − t n

x ∗ − g1



x ∗

+m1



x ∗

+P K1

g1



x ∗

− ρF

x ∗,y ∗

− m1



x ∗

≤1− t nx n − x ∗+t nx n − x ∗

+g1



x n



− g1



x ∗+t nm1(x) − m1

x ∗+t ne n

+t nP K1

n



g1 

x n

− ρF

x n,y n

− m1 

x n

− P K1

g1 

x ∗

− ρF

x ∗,y ∗

− m1 

x ∗

≤1− t nx n − x ∗+t nx n − x ∗

+g1



x n

− g1



x ∗+t nm1(x) − m1

x ∗+t ne n

+t nP K1

n



g1



x n



− ρF

x n,y n



− m1



x n



− P K1

n



g1



x ∗

− ρF

x ∗,y ∗

− m1



x ∗

+t nP K1

n



g1 

x ∗

− ρF

x ∗,y ∗

− m1 

x ∗

− P K1

g1 

x ∗

− ρF

x ∗,y ∗

− m1 

x ∗

≤1− t nx n − x ∗+t nx n − x ∗

+g1



x n

− g1



x ∗+t nm1(x) − m1

x ∗+t ne n

+t nx n − x ∗ −g1



x n



− g1



x ∗+t nm1

x n



− m1



x ∗

+t nx n − x ∗ − ρ

F

x n,y n

− F

x ∗,y n+ρt nF

x ∗,y n

− F

x ∗,y ∗

+t nP K1

n



g1



x ∗

− ρF

x ∗,y ∗

− m1



x ∗

− P K1

g1



x ∗

− ρF

x ∗,y ∗

− m1



x ∗.

(4.6) SinceF is l2-Lipschitz continuous with respect to the second argument,

F

x ∗,y n

− F

x ∗,y ∗  ≤ l2 y n − y ∗. (4.7)

From the strong monotonicity and Lipschitzian continuity ofg1, we obtain

x n − x ∗ −g1



x n

− g1



x ∗ 2

≤1 +ζ2−2η1 x n − x ∗ 2

The Lipschitzian continuity ofm1implies

m1

x n

− m1



x ∗  ≤ γ1 x n − x ∗. (4.9)

SinceF is relaxed (a, b)-cocoercive and l1-Lipschitz continuous with respect to the first argument,

x n − x ∗ − ρ

F

x n,y n



− F

x ∗,y n  ≤ 1 +ρ2l2+ 2ρal2−2ρbx n − x ∗. (4.10)

It follows from (4.6)–(4.10) that

x n+1 − x ∗  ≤ 2t n 1 +ζ2−2η1+ 2t n γ1+t n 1 +ρ2l2+ 2ρal2−2ρb + 1 − t n

x n − x ∗

+t n ρl2 y n − y ∗+t n b n+t ne n,

(4.11) where

b n =P K1

n



g1



x ∗

− ρF

x ∗,y ∗

− m1



x ∗

− P K1

g1



x ∗

− ρF

x ∗,y ∗

− m1



x ∗.

(4.12)

From the fact ofH(K1

n,K1)→0 andLemma 2.5, we know thatb n →0

Similarly, we have

y n+1 − y ∗  ≤ 2t n 1 +ζ2−2η2+ 2t n γ2+t n 1 +ρ2n2+ 2ρcn2−2ρd + 1 − t n



y n − y ∗

+t n ρn1 x n − x ∗+t n c n+t nj n,

(4.13) where

c n =P K2

n



g2



y ∗

− ρG

x ∗,y ∗

− m2



y ∗

− P K2

g2



y ∗

− ρF

x ∗,y ∗

− m2



y ∗,

(4.14)

andc n →0 Now (4.11) and (4.13) imply

x n+1 − x ∗+y n+1 − y ∗

≤ 2t n 1 +ζ2−2η1+ 2t n γ1+ 1− t n+t n 1 +ρ2l2+ 2ρal2−2ρb + t n ρn1



x n − x ∗

+ 2t n 1+ζ2−2η2+ 2t n γ2+ 1− t n+t n 1+ρ2n2+ 2ρcn2−2ρd + t n ρl2



y n − y ∗

+t n c n+t n b n+t ne n+t nj n.

(4.15)

SinceF is relaxed (a, b) -cocoercive and l1-Lipschitz continuous with respect to the first argument,... stability

In this section, we construct a new perturbed iterative algorithm for solving problem (2.1) and prove the convergence and stability of the iterative sequence generated by the algorithm... be a self-map of H, x0∈ H and let x n+1 = f (T, x n) define an iteration procedure which yields a sequence of points{