Convergence analysis of an iterative algorithm for the extended regularized nonconvex variational inequalities

R E S E A R C H

Open Access

Convergence analysis of an iterative

algorithm for the extended regularized

nonconvex variational inequalities

Ying Zhao, Luoyi Shi

_{and Rudong Chen}

*_{Correspondence:}

[email protected]

Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, P.R. China

Abstract

In this paper, we suggest and analyze a new system of extended regularized

nonconvex variational inequalities and prove the equivalence between the aforesaid system and a fixed point problem. We introduce a new perturbed projection iterative algorithm with mixed errors to find the solution of the system of extended regularized nonconvex variational inequalities. Furthermore, under moderate assumptions, we research the convergence analysis of the suggested iterative algorithm.

Keywords: general nonconvex variational inclusions; convergence analysis; fixed point problem

1 Introduction

Variational inequality was introduced and studied by Stampacchia [] in . It has been recognized as a suitable mathematical model to deal with many problems arising in differ-ent fields, such as optimization theory, game theory, partial differdiffer-ential equations, and economic equilibrium mechanics; see [–] and the references therein. Because of its importance and active impact in the nonlinear analysis and optimization, variational in-equality has been explosively growing in both theory and applications; see, for example, [–]. Specially, one of the significant generalizations of variational inequality is the gen-eral variational inequality which was introduced and investigated by Noor []. Subse-quently, Balooeeet al.[, ] introduced an algorithm for solving the extended general mixed variational inequalities. However, most of the results related to the existence of so-lutions and iterative methods for variational inequality problems have been investigated and considered so far to the case where the underlying set is convex.

It is worth to mention that in many of the alluded applications, the set involved is not convex. To overcome the difficulty caused by the nonconvexity of the set, Clarkeet al.[] introduced a new class of nonconvex sets, that is, proximally smooth sets. Moreover, they were introduced by Poliquinet al.[] but called the uniformly prox-regular sets. These kinds of sets are used in many nonconvex applications, such as differential inclusions, dynamical systems, and optimization; see [, ] and the references therein. It is well known that the uniformly prox-regular sets are nonconvex and include the convex sets as special cases. In , Noor [] considered a new class of variational inequalities, called the general nonconvex variational inequalities, and introduced the convergence analysis

of the suggested iterative algorithms underlying the uniformly prox-regular sets. For more numerical methods for solving the variational inequalities and their generalizations in the context of nonconvexity, we refer the reader to [–] and the references therein.

The projection technique was introduced by Lions and Stampacchia []. It is one of the most widely used methods to study the variational inequalities, and this technique is devoted to establishing the equivalence between the variational inequalities and a fixed point problem which uses the concept of projection.

Inspired and motivated by the above works, in this paper, we introduce a new system of extended regularized nonconvex variational inequalities (SERNVI) and prove the equiva-lence between the SERNVI and a fixed point problem. Using this equivalent formulation, we consider a new perturbed projection iterative algorithm with mixed errors for finding the solution of the SERNVI. Under some moderate assumptions, we research the conver-gence analysis of the suggested iterative algorithm.

2 Preliminaries

LetHbe a real Hilbert space whose inner product and norm are denoted by·,·and · , respectively. LetKbe a nonempty closed subset ofH, the usual distance function to the subsetKis denoted byd(·,K),i.e.,d(u,K) =infv∈Ku–v. Now we recall the following basic definitions and results from nonsmooth analysis and nonlinear convex analysis.

Definition .([]) Let u∈H be a point not lying inK. Letv∈K be a point whose

distance touis minimal, i.e.,d(u,K) =u–v, thenvis called a closest point or a projection ofuontoK. The set of all such closest points is denoted byPK, that is,

PK(u) :=

v∈K:d(u,K) =u–v.

Definition .([]) The proximal normal cone ofKat a pointu∈Kis given by

N_KP(u) :=ζ∈K:∃α>  such thatu∈PK(u+αζ)

.

Clarkeet al.[] gave the following characterization of the proximal normal coneNP K(x).

Lemma . Let K be a nonempty closed subset in H.Then a vectorζ∈N_KP(u)if and only if there exists a constantα=α(ζ,u) > such that

ζ,v–u ≤αv–u, ∀v∈K. (.)

Inequality (.) is called the proximal normal inequality. Clarkeet al.[] considered the special case of the proximal normal coneNP

K(u), in whichKis closed and convex; this case is an important one.

Lemma . Let K be a nonempty,closed and convex subset in H.Thenζ ∈NP

K(u)if and only ifζ,v–u ≤for all v∈K.

Definition . The Clarke normal cone ofCat a pointu∈Kis defined as

N_KC(u) =coN_KP(u),

Clearly,NP

K(u)⊂NKC(u), but the converse is not true.NKP(u) is always closed and convex, whereasNC

K(u) is convex, but may not be closed.

To overcome the difficulty caused by the nonconvexity of the set, Clarkeet al.[] in-troduced a new class of nonconvex sets, which are said to be proximally smooth sets. Subsequently, Poliquinet al.[] considered the aforementioned set under the name of uniformly prox-regular sets. We take the following characterization verified in [] as a definition of the uniformly prox-regular sets.

Definition . For anyr∈(, +∞], a subsetKrofHis said to be normalized uniformly prox-regular (or uniformlyr-prox-regular) if every nonzero proximal normal toKrcan be realized by anr-ball. This means that, for allu¯∈Krand all =ζ ∈NKP(u) with¯ ζ= ,

ζ

ζ,u–u¯

≤ 

ru–u¯

_{, ∀u∈K}

r.

It is obvious that a closed subset of a Hilbert space is convex if and only if it is proximally smooth of radiusr> . In view of Definition ., ifr= +∞, the uniformr-prox-regularity ofKris equivalent to the convexity ofKr. So, we setKr=K, whenr= +∞. Moreover, a class of uniformly prox-regular sets is sufficiently large to include the class of convex sets, P-convex sets,C,_{submanifolds ofH, the images under aC},_{diffeomorphism of convex}

sets and many other nonconvex sets.

We now recall the following well-known proposition which summarizes some signifi-cant consequences of the uniform prox-regularity. The proof of this result can be found in [, ].

Proposition . Let r∈(,∞]and U(r) ={u∈H:d(u,Kr) <r},and let Krbe a nonempty closed and uniformly r-prox-regular subset of H.Then the following results hold:

(i) For allx∈Kr,setPKr(x)=∅;

(ii) For allr∈(,r),PKr is Lipschitz continuous with constant

r

r–r onU(r); (iii) The proximal normal cone is closed as a set-valued mapping.

3 System of extended regularized nonconvex variational inequalities

In this section, we introduce a new system of extended regularized nonconvex variational inequalities and prove the equivalence between the aforesaid system and a fixed point problem.

Let Kr be a uniformly r-prox-regular subset of H, and let gi:H→Kr, hi:H→H (i= , . . . ,N) be nonlinear single-valued mappings such thatgi(H)∈Kr. LetTi:H×H→ CB(H) (i= , . . . ,N) (CB(H) means a nonempty closed and bounded subset ofH) be non-linear set-valued mappings, letQi:H→H(i= , . . . ,N) be single-valued mappings. For any given constantsρi>  (i= , . . . ,N) and for allx∈H,gi(x)∈Kr, the following prob-lem of finding x∗_i ∈H (i= , . . . ,N) with hi(x∗i)∈Kr (i= , . . . ,N) and u∗i ∈Ti(x∗i+,x∗i) (i= , . . . ,N– ),u∗_N∈TN(x∗,x∗N) such that

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

ρiQi(u∗i) +hi(x∗i) –gi(x∗i+),gi(x) –hi(x∗i)+

λi

rgi(x) –hi(x∗i)≥ (i= , . . . ,N– ),

ρNQN(u∗N) +hN(x∗N) –gN(x∗),gN(x) –hN(x∗N)+

λN

is called the system of extended regularized nonconvex variational inequalities (SERNVI), where λi=ρiQi(u∗i) +hi(x∗i) –gi(x∗i+) (i= , . . . ,N– ),λN =ρNQN(u∗N) +hN(x∗N) – gN(x∗).

We note that ifi= , , then problem (.) is equivalent to finding (x∗_,x∗_)∈H×Hwith (h(x∗),h(x∗))∈Kr×Krandu∗∈T(x∗,x∗),u∗∈T(x∗,x∗) such that

⎧ ⎨ ⎩

ρQ(u∗) +h(x∗) –g(x∗),g(x) –h(x∗)+ λ

rg(x) –h(x∗)≥,

ρQ(u∗) +h(x∗) –g(x∗),g(x) –h(x∗)+ λ

rg(x) –h(x∗)≥,

(.)

whereλ=ρQ(u∗) +h(x∗) –g(x∗),λ=ρQ(u∗) +h(x∗) –g(x∗), which was

in-troduced by Ansariet al.[] in .

If for eachi= , ,gi=hi=g,Qi≡I, the identity operatorTi=T:H→His an univariate nonlinear operator, then problem (.) reduces to the problem of finding (x∗_,x∗_)∈H×H such that

⎧ ⎨ ⎩

ρT(x∗) +g(x∗) –g(x∗),g(x) –g(x∗)+ λ

rg(x) –g(x∗)≥,

ρT(x∗) +g(x∗) –g(x∗),g(x) –g(x∗)+ λ

rg(x) –g(x∗)≥,

(.)

whereλ=ρT(x∗) +g(x∗) –g(x∗),λ=ρT(x∗) +g(x∗) –g(x∗). Problem (.) is called

the system of general nonconvex variational inequalities and was introduced by Noor []. It is worth to mention that ifTi=T:H→His an univariate nonlinear operator,x∗i =u (i= , . . . ,N),gi=g,Qi=hi≡I, the identity operator, then problem (.) changes into the problem of findingu∈Krsuch that

ρTu+u–g(u),g(v) –u+ρTu+u–g(u)

r g(v) –u



≥, ∀v∈Kr, (.)

which is called the general nonconvex variational inequality, introduced and investigated by Noor [].

Moreover, ifgi=hi≡I, the identity operator, then problem (.) is equivalent to finding u∈Krsuch that

ρTu,v–u+ρTu r v–u

_{≥, ∀v∈K}

r, (.)

which is called the nonconvex variational inequality. For more details about the nonconvex variational inequality, we refer the reader to [, ] and the references therein.

We note that ifKr≡K, the convex set inH, then problem (.) is equivalent to finding u∈Ksuch that

Tu,v–u ≥, ∀v∈K. (.)

We now establish the equivalence between the system of extended regularized noncon-vex variational inequalities (.) and a system of nonconnoncon-vex variational inclusions.

Proposition . Let Qi,Ti,hi,gi,andρi(i= , . . . ,N)be the same as system(.)such that gi(H)∈Kr (i= , . . . ,N).Then x∗i ∈H (i= , . . . ,N)with hi(x∗i)∈Kr and u∗i ∈Ti(x∗i+,x∗i) (i= , . . . ,N– ),u∗_N∈TN(x∗,x∗N)is a solution of system(.)if and only if

⎧ ⎨ ⎩

gi(x∗i+) –hi(x∗i) –ρiQi(u∗i)∈NKPr(hi(x

∗

i)) (i= , . . . ,N– ), gN(x∗) –hN(x∗N) –ρNQN(u∗N)∈NKPr(hN(x

∗

N)),

(.)

where NP

Kr(s)denotes the P-normal cone of Krat s in the sense of nonconvex analysis.

Proof Letx∗_i ∈Hwithhi(x∗i)∈Kr(i= , . . . ,N) andu∗i ∈Ti(x∗i+,x∗i) (i= , . . . ,N– ),u∗N ∈ TN(x∗,x∗N) be a solution of system (.).

Ifgi(x∗i+) –hi(x∗i) –ρiQi(u∗i) = , since the vector zero always belongs to any normal cone, it follows that

gi

x∗_i+–hi

x∗_i–ρiQi

u∗_i∈N_KP_rhi

x∗_i.

Ifgi(x∗i+) –hi(x∗i) –ρiQi(u∗i)= , for allx∈H, we observe that

gi

x∗_i+–hi

x∗_i–ρiQi

u∗_i,gi(x) –hi

x∗_i≤λi

rgi(x) –hi

x∗_i, (.)

whereλi=ρiQi(u∗i) +hi(x∗i) –gi(x∗i+)(i= , . . . ,N– ). It follows from Lemma . and

(.) that

gi

x∗_i+–hi

x∗_i–ρiQi

u∗_i∈N_KP_rhi

x∗_i (i= , . . . ,N– ).

Similarly,

gN

x∗_–hN

x∗_N–ρNQN

u∗_N∈N_KP_rhN

x∗_N.

On the contrary, ifx∗_i ∈Hwithhi(x∗i)∈Kr(i= , . . . ,N) andu∗i ∈Ti(x∗i+,x∗i) (i= , . . . ,N– ),u∗_N∈TN(x∗,x∗N) is a solution of system (.). Using Definition ., we can get thatx∗i ∈H with hi(x∗i)∈Kr(i= , . . . ,N) andu∗i ∈Ti(x∗i+,x∗i) (i= , . . . ,N– ),u∗N ∈TN(x∗,x∗N) is a

solution of system (.). This completes the proof.

Note that problem (.) iscalled a system of general nonconvex variational inclusions (SGNVI)associated with a system of extended regularized nonconvex variational inequal-ities.

have been developed for solving the nonlinear problem; see [–] and the references therein.

Theorem . Suppose that Kr,Qi,Ti,hi,gi,andρi(i= , . . . ,N)are the same as system (.)such that gi(H)∈Kr (i= , . . . ,N).Then x∗i ∈H with hi(x∗i)∈Kr (i= , . . . ,N)and u∗_i ∈Ti(xi∗+,x∗i) (i= , . . . ,N– ),u∗N ∈TN(x∗,x∗N)is a solution of system(.)if and only if

⎧ ⎨ ⎩

hi(x∗i) =PKr[gi(x∗i+) –ρiQi(u∗i)] (i= , . . . ,N– ), hN(x∗N) =PKr[gN(x∗) –ρNQN(u∗N)],

(.)

withρi< r

+Qi(u∗i) (i= , . . . ,N– ),ρN <

r

+QN(u∗N),for some r

_{∈(,r),where P}

Kr is the

projection of H onto Kr.

Proof Supposex∗_i ∈H with hi(x∗i)∈Kr (i= , . . . ,N) and u∗i ∈Ti(x∗i+,x∗i) (i= , . . . ,N– ), u∗_N ∈TN(x∗,xN∗) is a solution of system (.). Sincegi(x∗i)∈Kr (i= , . . . ,N), and for somer∈(,r),ρi< r

+Qi(u∗i) (i= , . . . ,N– ),ρN<

r

+QN(u∗N), we can check thatgi(x

∗

i+) – ρiQi(u∗i) andgN(x∗) –ρNQN(u∗N) belong toU(r), which implies that equations (.) are well defined.

From Theorem . and the above mentioned works, we can get

⎧ ⎨ ⎩

gi(x∗i+) –hi(x∗i) –ρiQi(u∗i)∈NKPr(hi(x

∗

i)) (i= , . . . ,N– ), gN(x∗) –hN(x∗N) –ρNQN(u∗N)∈NKPr(hN(x

∗

N))

⇐⇒

⎧ ⎨ ⎩

gi(x∗i+) –ρiQi(u∗i)∈(I+NKPr)(hi(x

∗

i)) (i= , . . . ,N– ), gN(x∗) –ρNQN(uN∗)∈(I+NKPr)(hN(x

∗

N))

⇐⇒

⎧ ⎨ ⎩

hi(x∗i) =PKr[gi(x∗i+) –ρiQi(u∗i)] (i= , . . . ,N– ), hN(x∗N) =PKr[gN(x∗) –ρNQN(u∗N)],

whereIis an identity mapping andPKr= (I+NKPr)

–_{. This completes the proof.}

4 Main result

In this section, we use the foregoing equivalent alternative formulation (.) to consider new perturbed projection iterative algorithms with mixed errors for finding the solution of SERNVI (.). For convenience, we now recall some definitions, and they will be powerful tools in our analysis.

Definition . An operatorg:H→His called

(i) monotone if

g(x) –g(y),x–y≥, ∀x,y∈H;

(ii) strongly monotone if there exists a constantη> such that

(iii) Lipschitz continuous if there exists a constantσ> such that

g(x) –g(y)≤σx–y, ∀x,y∈H.

Definition . LetT:H×H→H _{be a set-valued mapping, a nonlinear single-valued}

mappingQ:H→His called

(i) monotone if for anyu∈T(x,y),v∈T(y,x)such that

Q(u) –Q(v),x–y≥, ∀x,y∈H;

(ii) relaxed(κ,λ)cocoercive if for anyu∈T(x,y),v∈T(y,x), there exist constants

κ> andλ> such that

Q(u) –Q(v),x–y≥–κQ(u) –Q(v)+λx–y, ∀x,y∈H;

(iii) μ-Lipschitz continuous if there exists a constantμ> such that

_{Q(x) –Q(x)≥}μx–x, ∀x,x∈H.

Note that the notion of the cocoercivity is applied in several directions, especially to solving the variational inequality problems using the auxiliary problem principle and the projection methods, see []; while the notion of the relaxed cocoercivity is more general than the strong monotonicity as well as cocoercivity. For more details about the relaxed cocoercive variational inequalities and variational inclusions, see, for example, [].

Definition . A two-variable set-valued mappingT :H×H→H _{is said to beπ-}

D-Lipschitz continuous in the first variable if for allx,x,y,y∈H, there exists a constant

π>  such that

DT(x,y),Tx,y≤πx–x,

whereDis the Hausdorff pseudo-metric, that is, for any two nonempty subsetsAandB ofH,

D(A,B) =maxsup x∈A

d(x,B),sup y∈B

d(y,A).

In what follows, we introduce a new perturbed projection iterative algorithm with mixed errors to find the solution of SERNVI (.). For convenience, we assume thatKris a uni-formlyr-prox-regular subset ofHwithr> , also letr∈(,r) and setδ=_r–r_r.

Algorithm . SupposeQi,Ti,hi,gi, andρi(i= , . . . ,N) are the same as system (.) such thatgi(H)∈Kr(i= , . . . ,N), the iterative sequence{(xn, . . . ,xnN)}∞n=inH × · · · × H

N-times

arbitrarily chosen initial point (x_, . . . ,x_N)∈H × · · · × H

N-times

is given as follows:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

xn_i+= ( –αn)xni +αn(xni –hi(xni) +Pkr(gi(x

n

i+) –ρiQi(uni))) +αneni +qni (i= , . . . ,N– ),

xn_N+= ( –αn)xnN+αn(xnN–hN(xnN) +Pkr(gN(x

n

) –ρNQN(unN))) +αnenN+qNn,

(.)

where {αn}∞n= is a sequence in [, ] such that

_∞

n=αn=∞, and {eni}n∞=, {qni}∞n= (i=

, . . . ,N) are sequences inHto consider a possible inexact point of the resolvent operator satisfying the following conditions:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

en_i =en_i+en_i (i= , . . . ,N),

limn→∞en, . . . ,enN∗= ,

_∞

n=en, . . . ,enN∗<∞,

∞

n=qn, . . . ,qnN∗<∞.

(.)

Moreover, by Nadler theorem [], there exist

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

un

i ∈Tin(xni+,xni) (i= , . . . ,N– ), un

i –uni+ ≤( +i)D(Ti(x n

i+,xni),Ti(xin++,xni+)), un

N∈TN(xn,xnN), un

N–unN+ ≤( + N)D(TN(x n

,xnN),TN(xn+,xnN+)).

Before establishing the convergence analysis of the aforesaid algorithm, we review the well-known property, which provides the main mathematical results of this section.

Lemma .([]) Let an,bnand cnbe three nonnegative real sequences satisfying the fol-lowing condition.There exists a natural number nsuch that

an+≤( –tn)an+bntn+cn, ∀n≥n,

where tn∈[, ],

∞

n=tn=∞,limn→∞bn= and

∞

n=cn<∞.Thenlimn→∞an= . Next, the convergence of the iterative sequence generated by Algorithm . to a solution of SERNVI (.) is demonstrated.

Theorem . Suppose Qi,Ti,hi,gi,andρi(i= , . . . ,N)are the same as system(.)such that gi(H)∈Kr(i= , . . . ,N).Furthermore,for each i= , . . . ,N,

(i) Qiis(κi,λi)-relaxed cocoercive with constantsκi> ,λi> ; (ii) Qiisμi-Lipschitz continuous with a constantμi> ;

(iii) Tiisπi–D-Lipschitz continuous in the first variable with a constantπi> ; (iv) giisηi-strongly monotone andσi-Lipschitzian with constantsηi> ,σi> ;

(v) hiisξi-strongly monotone andζi-Lipschitzian with constantsξi> ,ζi> . If the constantsρi(i= , . . . ,N)satisfy the following conditions:

ρi< r  +Qi(ui)

for some r∈(,r)and ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

|ρi–

λi–κiμi(+i)πi

μ_i(+_i)_π

i |

<

δ_(λ

i–κiμi(+i)πi)–μi(+i)πi{δ–(–θi+–δ

√

–ηi+σi)}

δμ_i(+_i)_π

i

(i= , . . . ,N– ),

|ρN–

λN–κNμN(+N)πN

μ_N(+_N)_π

N |

<

δ_(λ

N–κNμN(+N)πN)–μN(+N)πN{δ–(–θ–δ

√

–ηN+σN)}

δμ_N(+_N)_π

N

,

δ(λi–κiμi( +i)π



i)

>μi( +_i)πi

δ_{– ( –θ}

i+–δ

 – ηi+σi) (i= , . . . ,N– ),

δ(λN–κNμN( +N)πN) >μN( +_N)πN

δ_{– ( –θ} –δ

 – ηN+σN),

δ>  –θi+–δ

 – ηi+σi (i= , . . . ,N– ),

δ>  –θ–δ

 – ηN+σN,

θi=

 – ξi+ζi (i= , . . . ,N),

ξi<  +ζi (i= , . . . ,N).

(.)

Then the iterative sequence{xn_i}∞_n=(i= , . . . ,N)generated by Algorithm.strongly con-verges to(x∗_, . . . ,x∗_N).

Proof Sincegi(x∗i)∈Kr(i= , . . . ,N),ρi< r

+Qi(u∗i) (i= , . . . ,N– ) andρN <

r

+QN(u∗N), it

follows from Theorem . thatx∗_i ∈Hsatisfies equation (.). Moreover, since the solution of the system of extended regularized nonconvex variational inequalities is a singleton set, hence, for each positive integern, we obtain

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

x∗_i = ( –αn)x∗i +αn(x∗i –hi(xi∗) +Pkr(gi(x∗i+) –ρiQi(u∗i))) (i= , . . . ,N),

x∗_N= ( –αn)x∗N+αn(x∗N–hN(x∗N) +Pkr(gN(x∗) –ρNQN(u∗N))),

(.)

where the sequence{αn}is the same as in Algorithm ..

Letxn_i+,x∗_i ∈Hbe given, by Proposition ., (.) and (.), we observe that

xn_i+–x∗_i

≤( –αn)xni –xi∗+αnxni –hi

xn_i+Pkr

gi

xn_i+–ρiQi

un_i

–x∗_i –hi

x∗_i+Pkr

gi

x∗_i+–ρiQi

u∗_i+αneni+qni ≤( –αn)xni –x∗i+αnxni –x∗i –

hi

xn_i–hi

x∗_i +δgi

xn_i+–gi

x∗_i+–ρiQi

un_i–ρiQi

u∗_i+αneni+qni ≤( –αn)xni –x∗i+αnxni –x∗i –

hi

xn_i–hi

x∗_i +δxn_i+–x∗_i+–gi

xn_i+–gi

x∗_i++x_in_+–x∗_i+–ρi

Qi

un_i–Qi

u∗_i +αneni+e

n i+q

n

From the fact thathiisξi-strongly monotone andζi-Lipschitzian, we can get that

xn_i –x∗_i –hi

xn_i–hi

x∗_i =xn_i –x∗_i– hi

xn_i–hi

x∗_i,xn_i –x∗_i+hi

xn_i–hi

x∗_i ≤xn_i –x∗_i– ξixni –x∗i



+ζ_ixn_i –x∗_i = – ξi+ζixni –x∗i



. (.)

Using the fact thatgiisηi-strongly monotone andσi-Lipschitzian, we obtain

xn_i+–x∗_i+–gi

xn_i+–gi

x∗_i+ =xn_i+–x∗_i+– gi

xn_i+–gi

x∗_i+,xn_i+–x∗_i++gi

xn_i+–gi

x∗_i+ ≤ – ηi+σixni+–x∗i+



. (.)

SinceQiisμi-Lipschitz continuous andTi isπi–D-Lipschitz continuous in the first variable, it follows that

Qi

un_i–Qi

u∗_i≤μiuni –u∗i ≤μi

 + i

DTi

xn_i+,xn_i,Ti

x∗_i+,x∗_i ≤μi

 + i

πixni+–x∗i+. (.)

From (.) andQiis (κi,λi)-relaxed cocoercive with respect toQiandTi, it is easy to see that

xn_i+–x∗_i+–ρi

Qi

un_i–Qi

u∗_i =xn_i+–x∗_i+– ρi

Qi

un_i–Qi

u∗_i,xn_i+–x∗_i+ +ρ_iQi

un_i–Qi

u∗_i ≤xn_i+–x∗_i+– ρi

–κiQi

un_i–Qi

u∗_i +λixni+–x∗i+



+ρ_iμ_iun_i –u∗_i ≤xn_i+–x∗_i+– ρi

–κiμi

 + i



π_ixn_i+–x∗_i+

+λixni+–x∗i+ 

+ρ_iμ_i

 + i



π_ixn_i+–x∗_i+

=

 – ρi

λi–κiμi

 + i

 π_i

+ρ_iμ_i

 + i



π_ixn_i+–x∗_i+. (.)

Substituting (.), (.) and (.) in (.), we deduce that

xn_i+–x∗_i

≤( –αn)xin–x∗i+αn

+αneni+eni+qni

= –αn( –θi)xni –x∗i+αnφixni+–x∗i+

+αneni+eni+qni, (.)

whereφi=δ{

( – ηi+σi) +

( – ρi(λi–κiμi( +i)πi) +ρiμi( +i)πi)}, andθi=

( – ξi+ζi).

Now we define · ∗onH × · · · × H N-times

as follows:

(x, . . . ,xN)_∗=x+· · ·+xN, for all (x, . . . ,xN)∈H × · · · × H N-times

.

It is obvious that (H × · · · × H

N-times

, · ∗) is a Banach space. We derive from the definition of

· ∗and (.) that

_xn+

 ,xn+, . . . ,xnN+

–x∗_,x∗_, . . . ,x∗_N∗

≤ –αn( –θ)xn–x∗+αnφxn–x∗+αnen

+en_+qn+

 –αn( –θ)xn–x∗+αnφxn–x∗

+αnen+e

n

+q

n

+· · ·+

 –αn( –θN)xnN–x∗N +αnφNxn–x∗+αnen_N+e_Nn+qnN

≤ –αn( –ϑ)xn,xn, . . . ,xnN

–x∗_,x∗_, . . . ,x∗_N∗ +αnen, . . . ,enN∗+en, . . . ,enN∗+qn, . . . ,qnN∗

≤ –αn( –ϑ)xn,xn, . . . ,xnN

–x∗_,x∗_, . . . ,x∗_N∗ +αn( –ϑ)

(en

, . . . ,enN)∗

 –ϑ +e

n

, . . . ,e

n

N∗+qn, . . . ,qnN∗, (.)

whereϑ=+,={θ, . . . ,θN}and={φ, . . . ,φN}. From condition (.), we get that  <ϑ< . Since all conditions in (.) hold, one can easily find that all the conditions of Lemma . are satisfied. It follows from Lemma . and (.) that

xn_+,x_n+, . . . ,xn_N+→x∗_,x∗_, . . . ,x∗_N, asn→ ∞. (.) Using the definition of · ∗and (.), one can easily show that

x_, . . . ,x_N, . . . ,xn_, . . . ,xn_N–x∗_, . . . ,x∗_N, . . . ,x∗_, . . . ,x∗_N∗

= n–

j=

xj_+, . . . ,xj_N+–x∗_, . . . ,x∗_N∗+x_, . . . ,x_N–x∗_, . . . ,x∗_N∗

≤ n–

j=

 –αj( –ϑ)xj,x

j

, . . . ,x

j N

–x∗_,x∗_, . . . ,x∗_N∗

+ n–

j=

αjej_, . . . ,ej_N∗+ n–

j=

+ n–

j=

qj_, . . . ,qj_N∗+ –α( –ϑ)x,x, . . . ,xN

–x∗_,x∗_, . . . ,x_N∗_∗+αe, . . . ,e 

N∗

+e_, . . . ,e_N∗+q_, . . . ,q_N∗

= n–

j=

 –αj( –ϑ)xj,x

j

, . . . ,x

j N

–x∗_,x∗_, . . . ,x∗_N∗

+ n–

j=

αjej_, . . . ,ej_N∗+ n–

j=

eN_, . . . ,eN_N∗+ n–

j=

qj_, . . . ,qj_N∗

= n–

j=

 –αj( –ϑ)xj,x

j

, . . . ,x

j N

–x∗_,x∗_, . . . ,x∗_N∗

+αjej, . . . ,e

j

N∗+e

j

, . . . ,e

j

N∗+q

j

, . . . ,q

j N∗

. (.)

It follows from (.) and (.) that

x_, . . . ,x_N, . . . ,xn_, . . . ,xn_N–x∗_, . . . ,x∗_N, . . . ,x∗_, . . . ,x∗_N∗→, asn→ ∞.

Consequently,{xn_i}∞_n=(i= , . . . ,N) are Cauchy sequences inH. Hence, there existsx∗_i (i= , . . . ,N)∈H such thatxn

i →x∗i (i= , . . . ,N)∈H asn→ ∞. This completes the proof.

Remark Equation (.) is the key condition for the convergence analysis in our algorithm. Moreover, the problem which we are solving hasnelements. So, we consider the special case for one element. If we chooseθ=_,δ= ,λ= .,κ= ,π= ,μ= , it follows

thatρ∈(+

√

  ,

–√

 ), which implies that the algorithm introduced in our paper is feasible.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The main idea of this paper was proposed by YZ. LS and RC prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the final manuscript.

Acknowledgements

This research was supported by NSFC Grants No. 11226125; No. 11301379; No. 11671167.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 8 December 2016 Accepted: 31 March 2017 References

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