A parallel resolvent method for solving a system of nonlinear mixed variational inequalities

R E S E A R C H

Open Access

A parallel resolvent method for solving a

system of nonlinear mixed variational

inequalities

Ke Guo

_{, Ya Jiang and Shi-Qiang Feng}

*_{Correspondence:}

[email protected] College of Mathematics and Information, China West Normal University, Nanchong, Sichuan 637009, China

Abstract

In this paper, we introduce a system of generalized nonlinear mixed variational inequalities and obtain the approximate solvability by using the resolvent parallel technique. Our results may be viewed as an extension and improvement of the previously known results for variational inequalities.

Keywords: resolvent operator; parallel projection; relaxed cocoercive; generalized nonlinear mixed variational inequalities

1 Introduction and preliminaries

Variational inequality theory, which was introduced by Stampacchia [] in , has been witnessed as an interesting branch of mathematical and engineering sciences with a wide range of applications in industry, finance, economics and pure and applied sciences. In , Verma [] introduced a new system of strongly monotonic variational inequalities and studied the approximation solvability of the system based on the application of a pro-jection method. The main and basic idea in this technique is to establish the equivalence between variational inequalities and fixed point problems. This alternative equivalence has been used to develop several projection iterative methods for solving variational in-equalities and related optimization problems. Several extensions and generalizations of the system of strongly monotonic variational inequalities have been considered by many authors [–]. Inspired and motivated by research in this area, we introduce a system of generalized nonlinear mixed variational inequalities problem involving two different nonlinear operators. It is well known that if the nonlinear term in the mixed variational inequality is a proper, convex, and lower semicontinuous, then one can establish the equiv-alence between the mixed variational inequality and the fixed point problem. Using the parallel algorithm considered in [], we suggest and analyze a parallel iterative method for solving this system. Our result may be viewed as an extension and improvement of the recent results.

LetHbe a real Hilbert space whose inner product and norm are denoted by·,·and

· , respectively. LetKbe a nonempty closed convex subset ofH. LetT,T:K×K→H be two nonlinear operators. Letϕ,ϕ:H→R∪ {+∞}be proper convex lower semi-continuous functions onH. We consider a system of generalized nonlinear mixed

tional inequalities (abbreviated as SMNVI) as follows: Find (x∗,y∗)∈K×Ksuch that

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

ρT(y∗,x∗) +g(x∗) –g(y∗),g(x) –g(x∗)+ϕ(g(x)) –ϕ(g(x∗))

≥, ∀g(x)∈K,

ηT(x∗,y∗) +g(y∗) –g(x∗),g(x) –g(x∗)+ϕ(g(x)) –ϕ(g(y∗))

≥, ∀g(x)∈K,

(.)

whereg:K→Kis a mapping andρ,η> .

Note that ifϕ=ϕ=δK, andg=I, whereIis the identity operator,δK is the indicator

function ofKdefined by

δK(x) = ⎧ ⎨ ⎩

 ifx∈K,

+∞ otherwise,

then problem (.) reduces to the following system of nonlinear variational inequalities (SNVI) considered in [] of finding (x∗,y∗)∈K×Ksuch that

⎧ ⎨ ⎩

ρT(y∗,x∗) +x∗–y∗,x–x∗ ≥, ∀x∈K,

ηT(x∗,y∗) +y∗–x∗,x–x∗ ≥, ∀x∈K. (.)

IfT=T=Tandg=I, whereIis the identity operator, then problem (.) is equivalent to the following system of nonlinear mixed variational inequalities (SNVI) considered in [, ] of finding (x∗,y∗)∈K×Ksuch that

⎧ ⎨ ⎩

ρT(y∗,x∗) +x∗–y∗,x–x∗+ϕ(x) –ϕ(x∗)≥, ∀x∈K,

ηT(x∗,y∗) +y∗–x∗,x–x∗+ϕ(x) –ϕ(y∗)≥, ∀x∈K. (.)

Ifϕ=ϕ=δKandT,T:K→Hare univariate mappings, then problem (.) is reduced to the following system of nonlinear variational inequalities (SNVI) considered in [] of finding (x∗,y∗)∈K×Ksuch that

⎧ ⎨ ⎩

ρT(y∗) +g(x∗) –g(y∗),g(x) –g(x∗) ≥, ∀g(x)∈K,

ηT(x∗) +g(y∗) –g(x∗),g(x) –g(x∗) ≥, ∀g(x)∈K, (.)

whereg:K→Kis a mapping.

IfT=T=T,g=Iandϕ=ϕ=δK, whereT is a univariate mapping defined byT : K→H, then problem (.) reduces to the following system of variational inequalities (SVI)

considered in [] of finding (x∗,y∗)∈K×Ksuch that

⎧ ⎨ ⎩

ρT(y∗) +x∗–y∗,x–x∗ ≥, ∀x∈K,

ηT(x∗) +y∗–x∗,x–x∗ ≥, ∀x∈K.

(.)

Definition . Define the norm · onH×Hby

(u,v)=u+v, ∀(u,v)∈H×H.

Definition . For any maximal monotone operatorT, the resolvent operator associated withT, for anyλ> , is defined by

J_Tλ(u) = (I+λT)–(u), ∀u∈H.

Remark . It is well known that the subdifferential∂ϕof a proper convex lower semi-continuous functionϕ:H→R∪ {+∞}is a maximal monotone operator. We can define its resolvent operator by

Jλ

ϕ(u) = (I+λ∂ϕ)–(u), ∀u∈H, whereλ>  andJϕis defined everywhere.

Lemma .[] For a given u,z∈Hsatisfies the inequality

u–z,x–u+λϕ(x) –λϕ(u)≥, ∀x∈H,

if and only if u=Jλ

ϕ(z),where Jϕλ(u) = (I+λ∂ϕ)–(u)is the resolvent operator andλ> . Ifϕis the indicator function of a closed convex setK⊆H, then the resolvent operator Jλ

ϕ(·) reduces to the projection operatorPK(·). It is well known thatJϕλis nonexpansive,i.e.,

Jλ ϕ(u) –J

λ

ϕ(v)≤ u–v, ∀u,v∈H.

Based on Lemma ., similar to that in [] and [], the following statement gives equiv-alent characterization of problem (.).

Lemma . Problem(.)is equivalent to finding(x∗,y∗)∈K×K such that

⎧ ⎨ ⎩

g(x∗) =J ϕ[g(y

∗_{) –ρT(y}∗_,x∗_)], g(y∗) =J

ϕ[g(x

∗_{) –ηT}

(x∗,y∗)],

(.)

where J

ϕi= (I+∂ϕi)

–_{,i= , .}

Proof Suppose that (x∗,y∗)∈K×Kis a solution of the following generalized nonlinear mixed variational inequalities (abbreviated as SNMVI):

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

ρT(y∗,x∗) +g(x∗) –g(y∗),g(x) –g(x∗)+ρϕ(g(x)) –ρϕ(g(x∗))

≥, ∀g(x)∈K,

ηT(x∗,y∗) +g(y∗) –g(x∗),g(x) –g(x∗)+ηϕ(g(x)) –ηϕ(g(y∗))

≥, ∀g(x)∈K,

whereg:K→Kis a mapping andρ> ,ρ> ,η> ,η> . Using Lemma ., we can easily show that problem (.) is equivalent to

⎧ ⎨ ⎩

g(x∗) =Jρ ϕ[g(y

∗_{) –ρT}

(y∗,x∗)],

g(y∗) =Jη ϕ[g(x

∗_{) –ηT(x}∗_,y∗_)], (.)

whereJρ ϕ= (I+ρ

_∂ϕ)–_,Jη ϕ= (I+η

_∂ϕ)–_{. Letρ=η= . Then problem (.) reduces to} problem (.) andJ

ϕi= (I+∂ϕi)

–_{,i= , . This completes the proof.}

Remark . IfT=T=T andg=I, whereI is the identity operator, then Lemma . reduces to Lemma . in [].

Definition . A mappingT:K×K→His said to be

() relaxedg-(γ,r)-cocoercive if there exist constantsγ > andr> such that for all

x,y∈K,

T(x,u) –T(y,v),g(x) –g(y)≥(–γ)T(x,u) –T(y,v)+rg(x) –g(y), ∀u,v∈K;

() g-μ-Lipschitz continuous in the first variable if there exists a constantμ> such that for allx,y∈K,

T(x,u) –T(y,v)≤μg(x) –g(y), ∀u,v∈K.

Remark . IfTis a univariate mapping andg=I, whereIis the identity operator, then Definition . reduces to the standard definition of relaxed (γ,r)-cocoercive and Lipschitz continuous, respectively.

Definition . A mappingg:K→His said to beα-expansive if there exists a constant

α>  such that for allx,y∈H, g(x) –g(y)≥αx–y.

Lemma .[] Suppose that{δn}is a nonnegative sequence satisfying the following in-equality:

δn+≤( –λn)δn+σn, ∀n≥n,

where n is a nonnegative number, λn∈[, ]with ∞n=λn=∞,andσn=o(λn).Then

lim_n→∞δn= .

2 Algorithms

Algorithm . For arbitrarily chosen initial pointsx,y∈K(andg(x),g(y)∈K), com-pute the sequences{xn}and{yn}such that

⎧ ⎨ ⎩

g(xn+) = ( –αn)g(xn) +αnJϕ[g(yn) –ρT(yn,xn)],

g(yn+) = ( –βn)g(yn) +βnJϕ[g(xn) –ηT(xn,yn)],

(.)

whereJ

ϕi= (I+∂ϕi)

–_{,i= , , is the resolvent operator,ρ,η> ,α}

n∈[, ] andβn∈[, ]

for alln≥.

As reported in [], one of the attractive features of Algorithm . is that it is suitable for implementing on two different processor computers. In other words,xn+andyn+are solved in parallel, and Algorithm . is the so-called parallel resolvent method. We refer the interested reader to papers [–] and references therein for more examples and ideas of parallel iterative methods.

Ifϕ=ϕ=δK, andg=I,δK is the indicator function ofK, then Algorithm . reduces

to the following algorithm.

Algorithm . For arbitrarily chosen initial pointsx,y∈K, compute the sequences{xn} and{yn}such that

⎧ ⎨ ⎩

xn+= ( –αn)xn+αnPK[yn–ρT(yn,xn)], yn+= ( –βn)yn+βnPK[xn–ηT(xn,yn)],

(.)

whereρ,η> ,αn∈[, ] andβn∈[, ] for alln≥.

IfT=T=Tandg=I, then Algorithm . reduces to the following algorithm.

Algorithm . For arbitrarily chosen initial pointsx,y∈K, compute the sequences{xn} and{yn}such that

⎧ ⎨ ⎩

xn+= ( –αn)xn+αnJϕ[yn–ρT(yn,xn)],

yn+= ( –βn)yn+βnJϕ[xn–ηT(xn,yn)],

(.)

whereJϕi= (I+∂ϕi)

–_{,i= , , is the resolvent operator,ρ,η> ,α}

n∈[, ] andβn∈[, ]

for alln≥.

Ifϕ=ϕ=δKandT,T:K→Hare univariate mappings, then Algorithm . reduces to the following algorithm.

Algorithm . For arbitrarily chosen initial pointsx,y∈K(andg(x),g(y)∈K), com-pute the sequences{xn}and{yn}such that

⎧ ⎨ ⎩

g(xn+) = ( –αn)g(xn) +αnPK[g(yn) –ρT(yn)], g(yn+) = ( –βn)g(yn) +βnPK[g(xn) –ηT(xn)],

(.)

IfT=T=T,g=Iandϕ=ϕ=δK, whereT is a univariate mapping defined byT : K→H, then Algorithm . reduces to the following algorithm.

Algorithm . For arbitrarily chosen initial pointsx,y∈K(andg(x),g(y)∈K), com-pute the sequences{xn}and{yn}such that

⎧ ⎨ ⎩

xn+= ( –αn)xn+αnPK[yn–ρT(yn)], yn+= ( –βn)yn+βnPK[xn–ηT(xn)],

(.)

whereρ,η> ,αn∈[, ] andβn∈[, ] for alln≥.

3 Main results

In this section, based on Algorithm ., we now present the approximation solvability of problem (.) involving relaxedg-(γ,r)-cocoercive andg-μ-Lipschitz continuous in the first variable mappings in Hilbert settings.

Theorem . LetHbe a real Hilbert space.Let K be a nonempty closed convex subset ofH, and let Ti:K×K→Hbe relaxed g-(γi,ri)-cocoercive and g-μi-Lipschitz continuous in the first variable for i= , .Let g:K→K be anα-expansive mapping.Suppose that(x∗,y∗)∈

K×K is the unique solution to problem(.)and{xn},{yn}are generated by Algorithm..

If{αn}and{βn}are two sequences in[, ]satisfying the following conditions: (i) αn–θβn≥andβn–θαn≥such that ∞n=αn–θβn=∞,

∞

n=βn–θαn=∞,

(ii) θ=

 + ργμ– ρr+ρμsuch that <θ< , (iii) θ=

 + ηγμ– ηr+ημsuch that <θ< ,

then the sequences{xn}and{yn}converge to x∗and y∗,respectively.

Proof Since (x∗,y∗)∈K×K is the unique solution to problem (.), from Lemma . it follows that

⎧ ⎨ ⎩

g(x∗) =Jϕ[g(y

∗_{) –ρT(y}∗_,x∗_)], g(y∗) =J

ϕ[g(x

∗_{) –ηT(x}∗_,y∗_)]. (.)

We first evaluateg(xn+) –g(x∗)for alln≥. From (.) and the nonexpansive property of the resolvent operator, we have

g(xn+) –g

x∗

=( –αn)g(xn) +αnJϕ

g(yn) –ρT(yn,xn) – ( –αn)g

x∗–αnJϕ

gy∗–ρT

y∗,x∗

≤( –αn)g(xn) –g

x∗

+αnJϕ

g(yn) –ρT(yn,xn)–Jϕ

gy∗–ρT

y∗,x∗

≤( –αn)g(xn) –g

x∗+αng(yn) –g

y∗–ρT(yn,xn) –T

Notice thatTis relaxedg-(γ,r)-cocoercive andg-μ-Lipschitz continuous in the first variable. Then we have

g(yn) –gy∗–ρT(yn,xn) –T

y∗,x∗

=g(yn) –gy∗– ρg(yn) –gy∗,T(yn,xn) –T

y∗,x∗

+ρT(yn,xn) –T

y∗,x∗

≤g(yn) –gy∗+ ργT(yn,xn) –T

y∗,x∗

– ρrg(yn) –g

y∗+ρT(yn,xn) –T

y∗,x∗

≤θg(yn) –g

y∗, (.)

whereθ=

 + ργμ– ρr+ρμ<  in view of assumption (ii). Substituting (.) into (.), we have

g(xn+) –gx∗≤( –αn)g(xn) –g

x∗+αnθg(yn) –g

y∗. (.)

Similarly, sinceTis relaxedg-(γ,r)-cocoercive andg-μ-Lipschitz continuous in the first variable, we have

g(yn+) –gy∗≤( –βn)g(yn) –g

y∗+βnθg(xn) –g

x∗, (.)

whereθ=

 + ηγμ– ηr+ημ<  in view of assumption (iii). It follows from (.) and (.) that

g(xn+),g(yn+)

–gx∗,gy∗

≤ – (αn–θβn)g(xn) –g

x∗+ – (βn–θαn)g(yn) –g

y∗

=max{wn,wn}g(xn),g(yn)

–gx∗,gy∗, (.)

wherewn=  – (αn–θβn) andwn=  – (βn–θαn).

From assumption (i) and Lemma ., we can obtain

lim

n→∞g(xn+),g(yn+)

–gx∗,gy∗= ,

and so

lim

n→∞g(xn+) –g

x∗=  and lim

n→∞g(yn+) –g

y∗= ,

which implies that sequences{g(xn)}and{g(yn)}converge tog(x∗) andg(y∗), respectively.

Sincegisα-expansive, it follows that{xn}and{yn}converge tox∗andy∗, respectively. This

completes the proof.

The following theorems can be obtained from Theorem . immediately.

the first variable for i= , .Suppose that(x∗,y∗)∈K×K is the unique solution to problem

(.)and{xn},{yn}are generated by Algorithm..If{αn}and{βn}are two sequences in

[, ]satisfying the following conditions:

() αn–θβn≥andβn–θαn≥such that ∞n=αn–θβn=∞, n∞=βn–θαn=∞,

() θ=

 + ργμ– ρr+ρμsuch that <θ< , () θ=

 + ηγμ– ηr+ημsuch that <θ< ,

then the sequences{xn}and{yn}converge to x∗and y∗,respectively.

Theorem .[] LetHbe a real Hilbert space.Let K be a nonempty closed convex subset ofH,and let Ti:K→Hbe relaxed g-(γi,ri)-cocoercive and g-μi-Lipschitz continuous for i= , .Let g:K→K be anα-expansive mapping.Suppose that(x∗,y∗)∈K×K is the unique solution to problem(.)and{xn},{yn}are generated by Algorithm..If{αn}and {βn}are two sequences in[, ]satisfying the following conditions:

() ≤αn,βn≤,αn–θβn≥andβn–θαn≥such that ∞n=αn–θβn=∞, ∞

n=βn–θαn=∞,

() θ=

 + ργμ– ρr+ρμsuch that <θ< , () θ=

 + ηγμ– ηr+ημsuch that <θ< ,

then the sequences{xn}and{yn}converge to x∗and y∗,respectively.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed significantly in writing the paper. All authors read and approved the final manuscript.

Acknowledgements

This work was supported by the Natural Science Foundation of China (60804065, 11371015), the Key Project of Chinese Ministry of Education (211163), Sichuan Youth Science and Technology Foundation (2012JQ0032).

Received: 19 December 2012 Accepted: 18 October 2013 Published:08 Nov 2013

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