Graph convergence for the H(⋅,⋅)-mixed mappingwith an application for solving the system of generalized variationalinclusions

(1)

R E S E A R C H

Open Access

Graph convergence for the

H

(

· ,

· )

-mixed

mapping with an application for solving the

system of generalized variational inclusions

Shamshad Husain

1*

_{, Sanjeev Gupta}

1

_{and Vishnu Narayan Mishra}

2

*_{Correspondence:} [email protected] 1_{Department of Applied}

Mathematics, Faculty of Engineering & Technology, Aligarh Muslim University, Aligarh, 202002, India Full list of author information is available at the end of the article

Abstract

In this paper, we investigate a class of accretive mappings called theH(·,·)-mixed mappings in Banach spaces. We prove that the proximal-point mapping associated with theH(·,·)-mixed mapping is single-valued and Lipschitz continuous. Some examples are given to justify the deﬁnition ofH(·,·)-mixed mapping. Further, a concept of graph convergence concerned with theH(·,·)-mixed mapping is introduced in Banach spaces and some equivalence theorems between

graph-convergence and proximal-point mapping convergence for theH(·,·)-mixed mappings sequence are proved. As an application, we consider a system of

generalized variational inclusions involvingH(·,·)-mixed mappings in realq-uniformly smooth Banach spaces. Using the proximal-point mapping method, we prove the existence and uniqueness of solution and suggest an iterative algorithm for the system of generalized variational inclusions. Furthermore, we discuss the convergence criteria for the iterative algorithm under some suitable conditions.

MSC: 47J19; 49J40; 49J53

Keywords: H(·,·)-mixed mapping; graph convergence; proximal-point mapping method; system of generalized variational inclusions; iterative algorithm

1 Introduction

Variational inclusions, as the generalization of variational inequalities, have been widely studied in recent years. Some of the most interesting and important problems in the theory of variational inclusions include variational, quasi-variational, variational-like inequalities as special cases. For applications of variational inclusions, we refer to []. Various kinds of iterative methods have been studied to solve the variational inclusions. Among these methods, the proximal-point mapping technique for the study of variational inclusions has been widely used by many authors. For details, we refer to [–].

In , Huang and Fang [] were the ﬁrst to introduce the generalized m-accretive mapping and give the deﬁnition of the proximal-point mapping for the generalized

m-accretive mapping in Banach spaces. Since then a number of researchers have inves-tigated several classes of generalizedm-accretive mappings such as H-accretive,H,η -accretive, (P,η)-proximal-point, (P,η)-accretive, A-maximal relaxed accretive, (A,η )-accretive mappings. For details, we refer to [, , , , , , , ].

Recently, Zou and Huang [, ] introduced and studiedH(·,·)-accretive mappings; Kazmiet al.[–] introduced and studied generalizedH(·,·)-accretive mappings,H(·,·

(2)

η-proximal-point mappings. Very recently, Li and Huang [] studied the graph conver-gence for theH(·,·)-accretive mapping and showed the equivalence between graph con-vergence and proximal-point mapping concon-vergence for theH(·,·)-accretive mapping se-quence in a Banach space, and Verma [] studied the graph convergence for anA-maximal relaxed monotone mapping and gave the equivalence between the graph convergence and the proximal-point mapping convergence for theA-maximal relaxed monotone mapping sequence in a Hilbert space. They extended the concept of graph convergence introduced and considered by Attouch [].

Motivated by the research work going on in this direction, we consider a class of accre-tive mappings calledH(·,·)-mixed mappings, a natural generalization of accretive (mono-tone) mappings in Banach spaces. For related work, we refer to [–, , , , –]. We prove that the proximal-point mapping of theH(·,·)-mixed mapping is single-valued and Lipschitz continuous and extends the concept of proximal-point mappings associated with theH(·,·)-accretive mappings to theH(·,·)-mixed mappings. Further, we study the graph convergence for theH(·,·)-mixed mappings. We present an equivalence theorem between graph convergence and proximal-point mapping convergence for theH(·,·)-mixed map-ping sequence in Banach spaces. As an application, we consider a system of generalized variational inclusions involving theH(·,·)-mixed mappings in realq-uniformly smooth Banach spaces. Using the proximal-point mapping method, we prove the existence and uniqueness of solution and suggest an iterative algorithm for the system of generalized variational inclusions. Furthermore, we discuss the convergence criteria of the iterative algorithm under some suitable conditions. Our results can be viewed as a generalization of some known results given in [, , –].

2 Preliminaries

LetXbe a real Banach space equipped with the norm · , and letX∗be the topological dual space ofX. Let·,·be the dual pair betweenXandX∗, and let Xbe the power set ofX.

Deﬁnition .[] Forq> , a mappingJq:X→X

∗

is said to be ageneralized duality mappingif it is deﬁned by

Jq(x) =

f∗∈X∗:x,f∗=xq,f∗=xq–, ∀x∈X.

In particular,Jis theusual normalized duality mappingonX. It is known that, in

gen-eral,

Jq(x) =xq–J(x) ∀x(= )∈X.

IfX≡Ha real Hilbert space, thenJbecomes anidentity mappingonH.

Deﬁnition .[] A Banach spaceXis calledsmoothif, for everyx∈Xwithx= , there exists a uniquef ∈X∗such thatf=f(x) = .

Themodulus of smoothnessofXis a functionρX: [,∞)→[,∞) deﬁned by

ρX(t) =sup

 

x+y+x–y –  :x ≤,y ≤t

(3)

Deﬁnition .[] A Banach spaceXis called

(i) uniformly smoothif

lim t→

ρX(t)

t = ;

(ii) q-uniformly smooth, forq> , if there exists a constantc> such that ρX(t)≤ctq, t∈[,∞).

Note thatJqis single-valued ifXis uniformly smooth. Concerned with the characteristic inequalities inq-uniformly smooth Banach spaces, Xu [] proved the following result. Lemma . Let X be a real uniformly smooth Banach space.Then X is q-uniformly smooth if and only if there exists a constant cq> such that,for all x,y∈X,

x+yq≤ xq+qy,Jq(x)

+cqyq.

From Lemma  of Liu [], it is easy to have the following lemma.

Lemma . Let{an}and{bn}be two nonnegative real sequences satisfying

an+≤kan+bn

with <k< and bn→.Thenlimn→∞an= .

Deﬁnition . LetG:X→Xbe a single-valued mapping. Then

(i) Gis said to beaccretiveif

G(x) –G(y),Jq(x–y)

≥, ∀x,y∈X;

(ii) Gis said to beξ-strongly accretiveif there exists a constantξ> such that

≥ξx–yq, ∀x,y∈X;

(iii) Gis said to beμ-cocoerciveif there exists a constantμ> such that

≥μG(x) –G(y)q, ∀x,y∈X;

(iv) Gis said to beλG-Lipschitz continuousif there exists a constantλG> such that G(x) –G(y)≤λGx–y, ∀x,y∈X;

(v) Gis said to beα-expansiveif there exists a constantα> such that

G(x) –G(y)≥αx–y, ∀x,y∈X;

(4)

Deﬁnition . LetH:X×X→XandA,B:X→Xbe three single mappings. Then

(i) H(A,·)is said to beμ-cocoercive with respect toAif there exists a constantμ> 

such that

H(Ax,u) –H(Ay,u),Jq(x–y)

≥μAx–Ayq, ∀x,y,u∈X;

(ii) H(·,B)is said to beγ-relaxed accretive with respect toBif there exists a constant γ > such that

H(u,Bx) –H(u,By),Jq(x–y)

≥(–γ)x–yq, ∀x,y,u∈X;

(iii) H(A,·)is said to ber-Lipschitz continuous with respect toAif there exists a

constantr> such that

H(Ax,·) –H(Ay,·)≤rx–y, ∀x,y∈X;

(iv) H(·,B)is said to ber-Lipschitz continuous with respect toBif there exists a

constantr> such that

H(·,Bx) –H(·,By)≤rx–y, ∀x,y∈X.

Example . Let us consider the -uniformly smooth Banach spaceX=R_{with the usual}

inner product. LetA,B:R_→_R_{be deﬁned by}

Ax=

mx–mx

–mx+ mx

, By=

–my+my

–my–my

for all scalersm∈Rand for allx= (x,x),y= (y,y)∈R.

Suppose that H : R _× _R _→_R _{is deﬁned by} _H₍_Ax_,_By_{) =}_Ax₊_By_{, then} _H₍_A_,_B₎

is __m-cocoercive with respect to A and m-relaxed accretive with respect to B, and

√

m-Lipschitz continuous with respect toAand√m-Lipschitz continuous with respect toB.

Indeed, let for anyu∈X,

H(Ax,u) –H(Ay,u),x–y

=Ax–Ay,x–y

=(mx–mx, –mx+ mx) – (my–my, –my+ my),

(x–y,x–y)

=m(x–y) –m(x–y), –m(x–y) + m(x–y),

(x–y,x–y)

=m(x–y)– m(x–y)(x–y) + m(x–y),

(5)

=(mx–mx, –mx+ mx) – (my–my, –my+ my) ,

(mx–mx, –mx+ mx) – (my–my, –my+ my)

= m(x–y)– m(x–y)(x–y) + m(x–y)

≤m(x–y)– m(x–y)(x–y) + m(x–y)

= mm(x–y)– m(x–y)(x–y) + m(x–y)

= mH(Ax,u) –H(Ay,u),x–y, which implies that

H(Ax,u) –H(Ay,u),x–y≥ 

mAx–Ay

_,

that is,H(A,B) is__m-cocoercive with respect toA.

H(u,Bx) –H(u,By),x–y

=Bx–By,x–y

=(–mx+mx, –mx–mx) – (–my+my, –my–my),

(x–y,x–y)

=–m(x–y) +m(x–y), –m(x–y) –m(x–y) ,

(x–y,x–y)

= –m(x–y)–m(x–y)

= –m(x–y)+ (x–y)

≥–mx–y,

which implies that

H(u,Bx) –H(u,By),x–y≥(–m)x–y,

that is,H(A,B) ism-relaxed accretive with respect toB. H(Ax,u) –H(Ay,u) =Ax–Ay=Ax–Ay,Ax–Ay

=(mx–mx, –mx+ mx) – (my–my, –my+ my) ,

(mx–mx, –mx+ mx) – (my–my, –my+ my)

= m(x–y)– m(x–y)(x–y) + m(x–y)

≤m(x–y)+ m(x–y),

which implies that

(6)

that is,H(A,B) is√m-Lipschitz continuous with respect toA. H(u,Bx) –H(u,By)=Bx–By=Bx–By,Bx–By

=(–mx+mx, –mx–mx) – (–my+my, –my–my) ,

(–mx+mx, –mx–mx) – (–my+my, –my–my)

= m(x–y)+ m(x–y)

which implies that

H(u,Bx) –H(u,By)≤√mx–y,

that is,H(A,B) is√m-Lipschitz continuous with respect toB.

Deﬁnition . Letη:X×X→XandH,A,B:X→Xbe mappings. LetM:X→X_be a set-valued mapping. Then

(i) ηis said to beτ-Lipschitz continuousif there exists a constantτ> such that

η(x,y)≤τx–y, ∀x,y∈X;

(ii) Mis said to beaccretiveif

u–v,Jq(x–y)

≥, ∀x,y∈X,u∈Mx,v∈My;

(iii) Mis said to beμ-strongly accretiveif there exists a constantμ> such that

u–v,Jq(x–y)

≥μx–yq, ∀x,y∈X,u∈Mx,v∈My;

(iv) Mis said to bem-relaxed accretiveif there exists a constantm> such that

u–v,Jq(x–y)

≥–mx–yq, ∀x,y∈X,u∈Mx,v∈My;

(v) Mis said to beη-accretiveif

u–v,Jq

η(x,y) ≥, ∀x,y∈X,u∈Mx,v∈My;

(vi) Mis said to bestrictlyη-accretiveifMisη-accretive and equality holds if and only ifx=y;

(vii) Mis said to beγ-stronglyη-accretiveif there exists a constantγ> such that

u–v,Jq

η(x,y) ≥γx–yq, ∀x,y∈X,u∈Mx,v∈My;

(viii) Mis said to beα-relaxedη-accretiveif there exists a constantα> such that

u–v,Jq

(7)

(ix) Mis said to bem-accretiveifMis accretive and(I+ρM)(X) =Xfor allρ> , whereIdenotes the identity operator onX;

(x) Mis said to begeneralizedm-accretiveifMisη-accretive and(I+ρM)(X) =Xfor allρ> ;

(xi) Mis said to beH-accretiveifMis accretive and(H+ρM)(X) =Xfor allρ> ; (xii) Mis said to be(H,η)-accretiveifMisη-accretive and(H+ρM)(X) =Xfor all

ρ> ;

(xiii) Mis said to be(A,η)-accretiveifMism-relaxedη-accretive and(A+ρM)(X) =X

for allρ> .

Deﬁnition .[] LetA,B:X→X,H:X×X→Xbe three single-valued mappings. LetM:X→X_{be a set-valued mapping. Then}_M_{is said to be}_H₍_·_,_·_)-_accretive_{with respect} toAandBifMis accretive and (H(·,·) +ρM)(X) =Xfor allρ> .

3 H(·,·)-mixed mappings

In this section, we introduce theH(·,·)-mixed mapping and show some of its properties. Deﬁnition . LetH:X×X→X,A,B:X→Xbe three single-valued mappings. Let

H(A,B) beμ-cocoercive with respect toA,γ-relaxed accretive with respect toB. Then the set-valued mappingM:X→X_{is said be}_H₍_·_,_·_)-_mixed_{with respect to mappings}_A andBif

(i) Mism-relaxed accretive;

(ii) (H(A,B) +ρM)(X) =Xfor allρ> .

Example . LetX,H,A,Bbe the same as in Example ., andM:R_→_R_{be deﬁned}

byM(x) = (–π, –x),∀x= (x,x)∈R.

We claim that M is a -relaxed accretive mapping. Indeed, for any x= (x,x),y=

(y,y)∈R

Mx–My,x–y=(–π, –x) – (–π, –y),

(x–y), (x–y)

=, –(x–y) ,

(x–y), (x–y)

= –(x–y)

≥–(x–y)+ (x–y)

≥–x–y,

Mx–My,x–y ≥(–)x–y.

Furthermore,Mis also anH(·,·)-mixed mapping since (H(A,B) +ρM)(R) =Rfor any

ρ> .

Proposition . Let the set-valued mapping M:X→X _{be an H}₍_·_,_·₎_{-mixed mapping}

with respect to mappings A and B.If A isα-expansive andμ>γ with r=μαq–γ >m,

then the following inequality holds:

x–y,Jq(u–v)

(8)

Proof Suppose on contrary that there exists (u,x) /∈graphMsuch that

x–y,Jq(u–v)

≥, ∀(v,y)∈graph(M). (.)

SinceMis anH(·,·)-mixed mapping, we know that (H(A,B) +ρM)(X) =Xholds for every

ρ> , and so there exists (u,x)∈graph(M) such that

H(Au,Bu) +ρx=H(Au,Bu) +ρx∈X. (.)

Now

ρx–ρx=H(Au,Bu) –H(Au,Bu)∈X,

ρx–ρx,Jq(u–u)

=H(Au,Bu) –H(Au,Bu),Jq(u–u)

.

Setting (v,y) = (u,x) in (.) and then from the resultant (.) andm-relaxed accretivity

ofM, we obtain

–mu–uq≤ρ

x–x,Jq(u–u)

= –H(Au,Bu) –H(Au,Bu),Jq(u–u)

= –H(Au,Bu) –H(Au,Bu),Jq(u–u)

–H(Au,Bu) –H(Au,Bu),Jq(u–u)

. (.)

SinceH(A,B) isμ-cocoercive with respect toAandγ-relaxed accretive with respect toB, andAisα-expansive, thus (.) becomes

–mu–uq≤–μAu–Auq+γu–uq

≤–μαu–uq+γu–uq

≤–μαq–γ u–uq

= –ru–uq≤, wherer=μαq–γ

≤–(r–m)u–uq≤.

It implies thatu=usincer>m. By (.), we havex=x, a contradiction. This completes

the proof.

Theorem . Let the set-valued mapping M:X→X_{be an H}₍_·_,_·₎_{-mixed mapping with}

respect to mappings A and B.If A isα-expansive andμ>γ with r=μαq–γ >ρm,then

(H(A,B) +ρM)–_{is single-valued}_.

Proof For any givenu∈X, letx,y∈(H(A,B) +ρM)–₍_u_{). It follows that}

(9)

SinceMism-relaxed accretive, we have

–mx–yq≤ 

ρ

–H(Ax,Bx) +u––H(Ay,By) +u ,Jq(x–y)

,

–mρx–yq= –H(Ax,Bx) –H(Ay,Bx),Jq(x–y)

–H(Ay,Bx) –H(Ay,By),Jq(x–y)

,

which is like (.). Hence it follows thatx–y ≤. This implies thatx=yand so (H(A,B)+

ρM)–_{is single-valued.}

Deﬁnition . Let the set-valued mappingM:X→Xbe anH(·,·)-mixed mapping with respect to mappingsAandB. IfAisα-expansive andμ>γ withr=μαq_–_γ _>_ρ_m_{, then} theproximal-point mapping RH_ρ_,(_M·,·):X→Xis deﬁned by

RHρ,(M·,·)(u) =

H(A,B) +ρM –(u), ∀u∈X. (.)

Now we prove that the proximal-point mapping deﬁned by (.) is Lipschitz continuous.

Theorem . Let the set-valued mapping M:X→X_{be an H}₍_·_,_·₎_{-mixed mapping with}

respect to mappings A and B.If A isα-expansive andμ>γ with r=μαq–γ >ρm,then the proximal-point mapping RHρ,(M·,·):X→X isr–ρm-Lipschitz continuous,that is,

RH_ρ_,(_M·,·)(u) –R_ρH_,(_M·,·)(v)≤ 

r–ρmu–v, ∀u,v∈X.

Proof Letuandv∈Xbe any given points inX. It follows from (.) that ⎧

⎨ ⎩

RH_ρ_,(_M·,·)(u) = H((A,B) +ρM))–(u),

RHρ,(M·,·)(v) =H((A,B) +ρM))

–

(v), 

ρ(u–H

ARH_ρ_,(_M·,·)(u),BRH_ρ_,(_M·,·)(u) ∈MR_ρH_,(_M·,·)(u) , 

ρ(v–H

ARHρ,(M·,·)(v) ,B

RHρ,(M·,·)(v)

∈MRHρ,(M·,·)(v).

Letz=RHρ,(M·,·)(u) andz=RHρ,(M·,·)(v). SinceMism-relaxed accretive, we have



ρ

(u–HA(z),B(z) –

v–HA(z),B(z)

,Jq(z–z)

≥–mz–zq,

u–v–HA(z),B(z) –H

A(z),B(z)

,Jq(z–z)

≥–ρmz–zq,

which implies that

u–vz–zq–≥

u–v,Jq(z–z)

≥HA(z),B(z) –H

A(z),B(z) ,Jq(z–z)

(10)

≥HA(z),B(z) –H

A(z),B(z),Jq(z–z)

–HA(z),B(z) –H

A(z),B(z) ,Jq(z–z)

–ρmz–zq

≥μA(z) –A(z)q–γz–zq–ρmz–zq

≥μαqz–zq–γz–zq–ρmz–zq

=μαq–γ –ρm z–zq,

= (r–ρm)z–zq, wherer=μαq–γ,

and hence

u–vz–zq–≥(r–ρm)z–zq,

that is,

RHρ,(M·,·)(u) –R H(·,·)

ρ,M (v)≤ 

r–ρmu–v, ∀u,v∈X.

This completes the proof.

4 Graph convergence for anH(·,·)-mixed mapping

LetM:X→X_{be a set-valued mapping. The graph of the map}_M_{is deﬁned by} graph(M) =(x,y)∈X×X:y∈M(X).

In this section we shall introduce the graph convergence for theH(·,·)-mixed mapping. Deﬁnition . LetMn,M:X→X be the set-valued mappings such that M, Mn are

H(·,·)-mixed mappings with respect to the mappingsAandBforn= , , , . . . . The se-quence {Mn} is said to be graph convergentto M, denoted byMn

G

−→M, if for every (x,y)∈graph(M), there exists a sequence (xn,yn)∈graph(Mn) such that

xn→x, yn→y asn→ ∞.

Theorem . Let Mn,M:X→Xbe the set-valued mappings such that M,Mnare H(·,·)

-mixed mappings with respect to the mappings A and B for n= , , , . . . .Let H(A,B)be s-Lipschitz continuous with respect to A and t-Lipschitz continuous with respect to B.If A isα-expansive andμ>γ with r=μαq_–_γ _>_ρ_m_,_{then M}

n G

−→M if and only if

RHρ,(M·,·n)(u)→R

H(·,·)

ρ,M (u), ∀u∈X,ρ> ,

where

RHρ,(M·,·n)(u) =

H(A,B) +ρMn –(u), RρH,(M·,·)(u) =

H(A,B) +ρM –(u).

Proof It follows from Theorem . thatRHρ,(M·,·n)andR

H(·,·)

(11)

If part: Suppose thatMn G

−→M. For any givenx∈X, let

zn=RρH,(M·,·n)(x), z=R

H(·,·)

ρ,M (x). Then



ρ

x–H(Az,Bz)∈M(z), that is

z,

ρ

x–H(Az,Bz)∈graph(M).

In the light of Deﬁnition ., we know that there exists a sequence (z_n,y_n)∈graph(Mn) such that

z_n →z, y_n→ 

ρ

x–H(Az,Bz) asn→ ∞. (.)

Sincey_n∈Mn(zn), we have

HAz_n,Bz_n +ρy_n∈H(A,B) +ρMn

z_n

and so

z_n =HAz_n,Bz_n +ρy_n.

From the Lipschitz continuity ofMn, we get

zn–z ≤zn–zn+zn–z

=R_ρH_,(_M·,·_n)(x) –R_ρH_,_M(·,·_n)HAz_n,Bz_n +ρy_n+z_n–z

≤ 

r–ρmx–H

Az_n,Bz_n –ρy_n+z_n–z

≤ 

r–ρmx–H(Az,Bz) –ρy

n

+H(Az,Bz) –HAz_n,Bz_n +z_n–z. (.)

From the Lipschitz continuity ofH(A,B), we have H(Az,Bz) –HAz_n,Bz_n

≤H(Az,Bz) –HAz,Bz_n +HAz,Bz_n –HAz_n,Bz_n

≤(s+t)z_n–z. (.)

It follows from (.) and (.) that

zn–z ≤ 

r–ρmx–H(Az,Bz) –ρy

n+

 + 

r–ρm(s+t)

z_n–z.

By (.), we have

z_n–z→, 

ρ

(12)

and so

zn–z → asn→ ∞.

Only if part: Suppose that

R_ρH_,(_M·,·_n)→R_ρH_,(_M·,·), ∀u∈X,ρ> .

For any given (x,y)∈graph(M), we have

H(Ax,Bx) +ρy∈H(A,B) +ρM(x)

and so

x=RHρ,(M·,·)

H(Ax,Bx) +ρy.

Let

xn=RHρ,(M·,·n)

H(Ax,Bx) +ρy.

Then



ρ

H(Ax,Bx) –H(Axn,Bxn) +ρy

∈Mn(xn).

Let

yn= 

ρ

H(Ax,Bx) –H(Axn,Bxn) +ρy

.

It follows from (.) that

yn–y ≤

_ρH(Ax,Bx) –H(Axn,Bxn) +ρy

–y= 

ρH(Ax,Bx) –H(Axn,Bxn) ≤ 

ρ(s+t)xn–x. (.)

SinceR_ρH_,_M(·,·_n)→R_ρH_,(_M·,·)for anyu∈X, we know thatxn–x →. Now (.) implies that

yn→y asn→ ∞,

and soMn G

−→M. This completes the proof.

5 An application of theH(·,·)-mixed mapping for solving the system of

generalized variational inclusions

Throughout the rest of the paper, unless otherwise stated, we assume that for eachi= , ,

(13)

LetA,B:X→X,A,B:X→X,N,H:X×X→XandN,H:X×X→X

be nonlinear mappings. LetM:X→XbeH(·,·)-mixed andM:X→X beH(·,·

)-mixed mappings, respectively. We consider the following system of generalized variational inclusions (SGVI): Find (x,y)∈X×Xsuch that

⎧ ⎨ ⎩

θ∈N(x,y) +M(x);

θ∈N(x,y) +M(y),

(.)

where θ, θ are zero vectors ofX andX, respectively. The problem of type (.) was

studied by Zou and Huang [].

Deﬁnition . LetA:X→X. A mappingN:X×X→Xis said to be:

(i) κ-strongly accretive in the ﬁrst argument with respect toAif there exists a constant κ> such that

N(x,y) –N(x,y),Jq

A(x) –A(x) _≥κx–yq, ∀x,x∈X,y∈X;

(ii) LN-Lipschitz continuous in the ﬁrst argumentif there exists a constantLN> such

that

N(x,y) –N(x,y)_≤LNx–xq, ∀x,x∈X,y∈X;

(iii) lN-Lipschitz continuous in the second argumentif there exists a constantlN> such

that

N(x,y) –N(x,y)_≤lNy–yq, ∀x∈X,y,y∈X.

The following lemma, which will be used in the sequel, is an immediate consequence of the deﬁnitions ofRH(·,·)

ρ,M,R

H(·,·)

ρ,M.

Lemma . For any given(x,y)∈X×X, (x,y)is a solution of(SGVI) (.)if and only if

(x,y)satisﬁes x=RH(·,·)

ρ,M

H(A,B)(x) –ρN(x,y)

, (.)

y=RH(·,·)

ρ,M

H(A,B)(y) –ρN(x,y)

, (.)

where RH(·,·)

ρ,M= (H(A,B) +ρM)

–_{and R}H(·,·)

ρ,M= (H(A,B) +ρM)

–_,_and_ρ

,ρ> are

constants.

Proof Consider ﬁrst that an element (x,y)∈X×Xis a solution to (.). Then it follows

that

θ∈N(x,y) +M(x)

⇒ H(Ax,Bx)∈H(Ax,Bx) +ρN(x,y) +ρM(x)

⇒ H(Ax,Bx) –ρN(x,y)∈H(Ax,Bx) +ρM(x)

⇒ x=RH(·,·)

ρ,M

H(Ax,Bx) –ρN(x,y)

(14)

In a similar way, we can show that

y=RH(·,·)

ρ,M

H(Ay,By) –ρN(x,y)

.

A similar proof follows for the converse part:

x=RH(·,·)

ρ,M

H(A,B)(x) –ρN(x,y)

⇒ x=H(A,B) +ρM –

H(A,B)(x) –ρN(x,y)

⇒ H(A,B) +ρM (x)

H(Ax,Bx) –ρN(x,y)

⇒ θ∈N(x,y) +M(x).

In a similar way, we can show that

θ∈N(x,y) +M(y).

Theorem . For each i= , ,let Xibe qi-uniformly smooth Banach spaces,let Ai,Bi:

Xi→Xibe single-valued mappings.Let the set-valued mappings Mi:Xi→Xibe such that

Miare Hi(·,·)-mixed mappings with respect to mappings Aiand Bi,and Aiareαi-expansive

andμi>γiwith ri=μiαqii–γi>ρimi.Let Hi:X×X→Xibe si-Lipschitz continuous with

respect to Aiand ti-Lipschitz continuous with respect to Bi,and let Ni:X×X→Xibe

aκi-strongly accretive mapping in the ith argument,LNi-Lipschitz continuous in the ﬁrst

argument and lNi-Lipschitz continuous in the second argument.Suppose that there are two

constantsρ,ρ> satisfying the following conditions:

⎧ ⎨ ⎩

τ=a+ρLLN< ;

τ=a+ρLlN< ,

(.)

where

a=L

 – qr+cq(s+t)

q 

q_₊_{ – }_ρ

qκ+cqρ

q

 L

q



q__;

a=L

 – qr+cq(s+t)

q 

q_ ₊_{ – }_ρ

qκ+cqρ

q

 L

q





q__;

L=



r–ρm

;

L=



r–ρm

.

Then SGVI(.)has a unique solution(x,y)∈X×X.

Proof Fori= , , it follows that for (x,y)∈X×X, the proximal-point mappingsRHρ,(M·,·)

and RH(·,·)

ρ,M are



r–ρm-Lipschitz continuous and



r–ρm-Lipschitz continuous,

respec-tively.

LetR:X×X→X×Xbe deﬁned as follows:

(15)

whereP:X×X→XandQ:X×X→Xare deﬁned by

P(x,y) =RH(·,·)

ρ,M

H(A,B)(x) –ρN(x,y)

(.)

and

Q(x,y) =RH(·,·)

ρ,M

H(A,B)(x) –ρN(x,y)

(.)

forρ,ρ> , respectively.

For any (x,y), (x,y)∈X×X, it follows from (.) and (.) and the Lipschitz

conti-nuity ofRH(·,·)

ρ,MandR

H(·,·)

ρ,Mthat

P(x,y) –P(x,y)_

=RH(·,·)

ρ,M

H(A,B)(x) –ρN(x,y)

–RH(·,·)

ρ,M

H(A,B)(x) –ρN(x,y)_

≤ŁH(A,B)(x) –H(A,B)(x) –ρ

N(x,y) –N(x,y) _

+ρN(x,y) –N(x,y)_

(.)

and

Q(x,y) –Q(x,y)_

≤ŁH(A,B)(y) –H(A,B)(y) –ρ

N(x,y) –N(x,y) _

+ρN(x,y) –N(x,y)_

. (.)

Now

H(A,B)(x) –H(A,B)(x) –ρ

N(x,y) –N(x,y) _

=H(A,B)(x) –H(A,B)(x) – (x–x)_

+x–x–ρ

N(x,y) –N(x,y) _. (.)

Also,

H(A,B)(y) –H(A,B)(y) – (y–y)_

=H(A,B)(y) –H(A,B)(y) – (y–y)_

+y–y–ρ

N(x,y) –N(x,y) _. (.)

Now

H(A,B)(x) –H(A,B)(x) – (x–x)q_

≤ x–xq– q

H(A,B)(x) –H(A,B)(x) –Jq(x–x)



+cqH(A,B)(x) –H(A,B)(x)

q

(16)

SinceMis anH(·,·)-mixed mapping, thenH(A,B) isμ-cocoercive with respect toA

andγ-relaxed accretive with respect toB, and from the fact thatAisα-expansive, we

can obtain

H(A,B)(x) –H(A,B)(x) –Jq(x–x)



=H(A,B)(x) –H(A,B)(x) –Jq(x–x)



+H(A,B)(x) –H(A,B)(x) –Jq(x–x)



≤μA(x) –A(x)

q

–γx–xq

≤μαqx–xq–γx–xq

=μαq–γ x–xq. (.)

SinceH(A,B) iss-Lipschitz continuous with respect toAandt-Lipschitz continuous

with respect toB, we have

H(A,B)(x) –H(A,B)(x) – (x–x)_

≤H(A,B)(x) –H(A,B)(x)_+H(A,B)(x) –H(A,B)(x)_

≤(s+t)x–x. (.)

Using (.), (.) and (.), we have

H(A,B)(x) –H(A,B)(x) – (x–x)

q



≤ x–xq– q

H(A,B)(x) –H(A,B)(x) –Jq(x–x)



≤ – qr+cq(s+t)

q_x

–xq,

which implies that

H(A,B)(x) –H(A,B)(x) – (x–x)_

≤ – qr+cq(s+t)

q 

q__x

–x, wherer=μαq–γ. (.)

In the light of (.), we can obtain

H(A,B)(y) –H(A,B)(y) – (y–y)_

≤ – qr+cq(s+t)

qq_y

–y, wherer=μαq–γ. (.)

Again, sinceNiisκi-strongly accretive in the ﬁrst argument andLNi-Lipschitz continuous

in the ﬁrst argument andlNi-Lipschitz continuous in the second argument, then using

Lemma ., we have

_x_–x–ρ

N(x,y) –N(x,y)

q



≤ x–xq– ρq

N(x,y) –N(x,y),Jq(x–x)

(17)

+ρq

 cqN(x,y) –N(x,y)

q



≤ – ρqκ+cqρ

q

 L

q

 x–xq,

which implies that

x–x–ρ

N(x,y) –N(x,y) _≤

 – ρqκ+cqρ

q

 L

q



q__x

–x. (.)

In the light of (.), we have

y–y–ρ

N(x,y) –N(x,y) _≤

 – ρqκ+cqρ

q

 L

q





q__y

–y. (.)

Using (.), (.) and (.), we have

P(x,y) –P(x,y)_

≤L

 – qr+cq(s+t)q



q₊_{ – }_ρ

q

 L

q



q _x

–x

+LρlNy–y. (.)

Using (.), (.) and (.), we have

_Q₍_x__,_y__{) –}_Q₍_x__,_y_₎



≤L

 – qr+cq(s+t)

q 

q ₊_{ – }_ρ

q

 L

q





q _y

–y

+LρlNx–x. (.)

From (.) and (.), we have

P(x,y) –P(x,y)_+Q(x,y) –Q(x,y)_

≤τx–x+τy–y

≤max{τ,τ}

x–x+y–y , (.)

where ⎧ ⎨ ⎩

τ=a+ρLLN;

τ=a+ρLlN,

(.)

and

a=L

 – qr+cq(s+t)q



q₊_{ – }_ρ

q

 L

q



q_; a=L

 – qr+cq(s+t)q



q ₊_{ – }_ρ

q

 L

q





q_; L=



r–ρm

;

L=



r–ρm

(18)

Now deﬁne the norm · onX×Xby

(x,y)=x+y, ∀(x,y)∈X×X. (.)

We observe that (X×X, · ) is a Banach space. Hence it follows from (.), (.) and (.) that

R(x,y) –R(x,y)=max{τ,τ}(x,y) – (x,y). (.)

Sincemax{τ,τ}<  by (.), it follows from (.) thatRis a contraction mapping. Hence,

by the Banach contraction principle, there exists a unique point (x,y)∈X×Xsuch that

R(x,y) = (x,y), which implies that

x=RH(·,·)

ρ,M

H(A,B)(x) –ρN(x,y)

,

y=RH(·,·)

ρ,M

H(A,B)(y) –ρN(x,y)

.

It follows from Lemma . that (x,y) is a unique solution of SGVI (.). This completes the

proof.

6 Convergence of an iterative algorithm for SGVI (5.1)

Based on Lemma ., we suggest and analyze the following iterative algorithm for ﬁnding an approximate solution for SGVI (.).

Algorithm . For any given (x,y)∈X×X, (xn,yn)∈X×Xby an iterative scheme

xn+=RρH,(M·,·)

H(A,B)(xn) –ρN(xn,yn)

, (.)

yn+=RρH,(M·,·)

H(A,B)(yn) –ρN(xn,yn)

, (.)

wheren= , , , . . . andρ,ρ>  are constants.

Theorem . For each i= , ,let Xi be qi-uniformly smooth Banach spaces,let Ai,Bi:

Xi→Xi be single-valued mappings. Let the set-valued mappings Mni,Mi:Xi→Xi be

such that Mni,Miare Hi(·,·)-mixed mappings with respect to mappings Aiand Bisuch that

Mni

G

−→Mifor n= , , , . . . ,and Aiisαi-expansive andμi>γiwith ri=μiαiqi–γi>ρimi.

Let Hi:X×X→Xibe si-Lipschitz continuous with respect to Aiand ti-Lipschitz

con-tinuous with respect to Bi,and let Ni:X×X→Xibe aκi-strongly accretive mapping in

the ith argument,LNi-Lipschitz continuous in the ﬁrst argument and lNi-Lipschitz

contin-uous in the second argument.Suppose that there are two constantsρ,ρ> satisfying the

following conditions: ⎧

⎨ ⎩

τ=a+ρLLN< ;

τ=a+ρLlN< ,

(19)

where

a=L

 – qr+cq(s+t)

q q₊_{ – }_ρ

q

 L

q



q_; a=L

 – qr+cq(s+t)

q 

q ₊_{ – }_ρ

q

 L

q





q_; L=



r–ρm

;

L=



r–ρm

.

Then the approximate solution(xn,yn)generated by Algorithm.converges strongly to the

unique solution(x,y)of SGVI(.).

Proof By Theorem ., there exists a unique solution (x,y)∈X×X of SGVI (.). It

follows from Algorithm . and Theorem . that

xn+–x=RρH,(M·,·n)

–RH(·,·)

ρ,M

H(A,B)(x) –ρN(x,y)_

≤RH(·,·)

ρ,Mn_

–RH(·,·)

ρ,Mn_

H(A,B)(x) –ρN(x,y)_

+RH(·,·)

ρ,Mn_

H(A,B)(x) –ρN(x,y)

–RH(·,·)

ρ,M

H(A,B)(x) –ρN(x,y)_ (.)

and

yn+–y ≤RρH,M(·,·n)

–RH(·,·)

ρ,Mn

H(A,B)(y) –ρN(x,y)_

+RH(·,·)

ρ,Mn

H(A,B)(y) –ρN(x,y)

–RH(·,·)

ρ,Mn

H(A,B)(y) –ρN(x,y)_. (.)

By (.) and (.), we have

RH(·,·)

ρ,Mn_

–RH(·,·)

ρ,Mn_

H(A,B)(x) –ρN(x,y)_

≤ŁH(A,B)(xn) –H(A,B)(x) –ρ

N(xn,yn) –N(x,yn) _ +ρN(x,yn) –N(x,y)_

≤L

 – qr+cq(s+t)

q 

q₊_{ – }_ρ

q  L q   q

(20)

and

RH(·,·)

ρ,Mn

–RH(·,·)

ρ,Mn

H(A,B)(y) –ρN(x,y)_

≤ŁH(A,B)(yn) –H(A,B)(y) –ρ

N(xn,yn) –N(xn,y) _ +ρN(xn,y) –N(x,y)_

≤L

 – qr+cq(s+t)

q 

q_ ₊_{ – }_ρ

q

 L

q





q_

× yn–y+LρlNxn–x. (.)

By Theorem ., we have

RH(·,·)

ρ,Mn

H(A,B)(x) –ρN(x,y)

→RH(·,·)

ρ,M

H(A,B)(x) –ρN(x,y)

, (.)

RH(·,·)

ρ,Mn

H(A,B)(y) –ρN(x,y)

→RH(·,·)

ρ,M

H(A,B)(y) –ρN(x,y)

. (.)

Let

bn=R

H(·,·)

ρ,Mn

H(A,B)(x) –ρN(x,y)

–RH(·,·)

ρ,M

H(A,B)(x) –ρN(x,y)_, (.)

bn=R

H(·,·)

ρ,Mn_

H(A,B)(y) –ρN(x,y)

–RH(·,·)

ρ,M

H(A,B)(y) –ρN(x,y)_. (.)

From (.)-(.), we have

xn+–x≤τxn–x+bn, (.)

yn+–y≤τyn–y+bn. (.)

From (.) and (.), we have

xn+–x+yn+–y≤max{τ,τ}

xn–x+yn–y

+{bn+bn}. (.)

Since (X×X, · ) is a Banach space with the norm · deﬁned by (.), it follows from (.), (.) and (.) that

(xn+,yn+) – (x,y)=xn+–x+yn+–y

≤max{τ,τ}(xn,yn) – (x,y)+{bn+bn}. (.)

By condition (.), it follows thatmax{τ,τ}<  and Lemma . implies that(xn+,yn+) –

(x,y)→ asn→ ∞.

Thus{(xn,yn)}converges strongly to the unique solution (x,y) of SGVI (.). This

(21)

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors have equally contributed and approved the manuscript.

Author details

1_{Department of Applied Mathematics, Faculty of Engineering & Technology, Aligarh Muslim University, Aligarh, 202002,} India.2_{Department of Applied Mathematics & Humanities, Sardar Vallabhbhai National Institute of Technology, Surat, 395} 007, India.

Received: 30 July 2013 Accepted: 10 October 2013 Published: 19 November 2013

References

1. Aubin, JP, Cellina, A: Diﬀerential Inclusions. Springer, Berlin (1984)

2. Fang, Y-P, Huang, N-J:H-accretive operators and resolvent operator technique for solving variational inclusions in Banach spaces. Appl. Math. Lett.17(6), 647-653 (2004)

3. Fang, YP, Cho, Y-J, Kim, JK: (H,η)-accretive operator and approximating solutions for systems of variational inclusions in Banach spaces. Appl. Math. Lett. (to appear)

4. Huang, N-J, Fang, Y-P: Generalizedm-accretive mappings in Banach spaces. J. Sichuan Univ.38(4), 591-592 (2001) 5. Huang, N-J: A new class of generalized set-valued implicit variational inclusions in Banach spaces with an application.

Comput. Math. Appl.41(7-8), 937-943 (2001)

6. Kazmi, KR, Khan, FA: Iterative approximation of a solution of multi-valued variational-like inclusion in Banach spaces: a P-η-proximal-point mapping approach. J. Math. Anal. Appl.325(1), 665-674 (2007)

7. Kazmi, KR, Khan, FA: Sensitivity analysis for parametric generalized implicit quasi-variational-like inclusions involving P-η-accretive mappings. J. Math. Anal. Appl.337(2), 1198-1210 (2008)

8. Kazmi, KR, Ahmad, N, Shahzad, M: Convergence and stability of an iterative algorithm for a system of generalized implicit variational-like inclusions in Banach spaces. Appl. Math. Comput.218, 9208-9219 (2012)

9. Kazmi, KR, Bhat, MI, Ahmad, N: An iterative algorithm based onM-proximal mappings for a system of generalized implicit variational inclusions in Banach spaces. J. Comput. Appl. Math.233, 361-371 (2009)

10. Kazmi, KR, Khan, FA, Shahzad, M: A system of generalized variational inclusions involving generalizedH(·,·)-accretive mapping in realq-uniformly smooth Banach spaces. Appl. Math. Comput.217(2), 9679-9688 (2011)

11. Lan, H-Y, Cho, YJ, Verma, RU: Nonlinear relaxed cocoercive variational inclusions involving (A,η)-accretive mappings in Banach spaces. Comput. Math. Appl.51(9-10), 1529-1538 (2006)

12. Li, X, Huang, N-J: Graph convergence for theH(·,·)-accretive operator in Banach spaces with an application. Appl. Math. Comput.217, 9053-9061 (2011)

13. Liu, LS: Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces. J. Math. Anal. Appl.194, 114-127 (1995)

14. Peng, JW: Set-valued variational inclusions withT-accretive operator in Banach spaces. Appl. Math. Lett.16(6), 273-282 (2006)

15. Sun, JH, Zhang, LW, Xiao, XT: An algorithm based on resolvent operators for solving variational inequalities in Hilbert spaces. Nonlinear Anal. TMA69, 3344-3357 (2008)

16. Verma, RU: The generalized relaxed proximal point algorithm involvingA-maximal-relaxed accretive mappings with applications to Banach spaces. Math. Comput. Model.50(7-8), 1026-1032 (2009)

17. Verma, RU: General class of implicit variational inclusions and graph convergence onA-maximal relaxed monotonicity. J. Optim. Theory Appl.129(3), 457-467 (2012)

18. Xia, FQ, Huang, N-J: Variational inclusions with a generalH-monotone operator in Banach spaces. Comput. Math. Appl.54, 24-30 (2007)

19. Zou, Y-Z, Huang, N-J:H(·,·)-accretive operator with an application for solving variational inclusions in Banach spaces. Appl. Math. Comput.204(2), 809-816 (2008)

20. Zou, Y-Z, Huang, N-J: A new system of variational inclusions involvingH(·,·)-accretive operator in Banach spaces. Appl. Math. Comput.212, 135-144 (2009)

21. Attouch, H: Variational Convergence for Functions and Operators. Applied Mathematics Series. Pitman, London (1984)

22. Xu, HK: Inequalities in Banach spaces with applications. Nonlinear Anal.16(12), 1127-1138 (1991)

doi:10.1186/1687-1812-2013-304