Auxiliary principle technique and iterative algorithm for a perturbed system of generalized multi valued mixed quasi equilibrium like problems

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R E S E A R C H

Open Access

Auxiliary principle technique and iterative

algorithm for a perturbed system of

generalized multi-valued mixed

quasi-equilibrium-like problems

Mijanur Rahaman

1

_{, Chin-Tzong Pang}

2*

_{, Mohd. Ishtyak}

1

_{and Rais Ahmad}

1

*_{Correspondence:}

[email protected]

2_{Department of Information}

Management, and Innovation Centre for Big Data and Digital Convergence, Yuan Ze University, Chung-Li, 32002, Taiwan Full list of author information is available at the end of the article

Abstract

In this article, we introduce a perturbed system of generalized mixed

quasi-equilibrium-like problems involving multi-valued mappings in Hilbert spaces. To calculate the approximate solutions of the perturbed system of generalized multi-valued mixed quasi-equilibrium-like problems, ﬁrstly we develop a perturbed system of auxiliary generalized multi-valued mixed quasi-equilibrium-like problems, and then by using the celebrated Fan-KKM technique, we establish the existence and uniqueness of solutions of the perturbed system of auxiliary generalized multi-valued mixed quasi-equilibrium-like problems. By deploying an auxiliary principle technique and an existence result, we formulate an iterative algorithm for solving the perturbed system of generalized multi-valued mixed quasi-equilibrium-like problems. Lastly, we study the strong convergence analysis of the proposed iterative sequences under monotonicity and some mild conditions. These results are new and generalize some known results in this ﬁeld.

MSC: 35M87; 47H05; 49J40; 65K15; 90C33

Keywords: quasi-equilibrium-like; perturbed system; algorithm; convergence

1 Introduction

The theory of variational inequality problem is very fruitful in connection with its appli-cations in economic problems, control and contact problems, optimizations, and many more; seee.g., [–]. In , Parida et al.[] introduced and studied the concept of variational-like inequality problem which is a salient generalization of variational inequal-ity problem, and shown its relationship with a mathematical programming problem. One of the most important topics in nonlinear analysis and several applied ﬁelds is the so-called equilibrium problem which was introduced by Blum and Oettli [] in , has extensively studied in diﬀerent generalized versions in recent past. An important and useful general-ization of equilibrium problem is a mixed equilibrium problem which is a combination of an equilibrium problem and a variational inequality problem. For more details related to variational inequalities and equilibrium problems, we refer to [–] and the references therein.

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There are many illustrious methods, such as projection techniques and their variant forms, which are recommended for solving variational inequalities but cannot be em-ployed in a similar manner to obtain the solution of mixed equilibrium problem involving non-differentiable terms. The auxiliary principle technique which was first introduced by Glowinskiet al.[] is beneficial in dodging these drawbacks related to a large number of problems like mixed equilibrium problems, optimization problems, mixed variational-like inequality problems, etc. In , Moudafi [] studied a class of bilevel monotone equilib-rium problems in Hilbert spaces and developed a proximal method with efficient iterative algorithm for solving equilibrium problems. After that, Ding [] studied a new system of generalized mixed equilibrium problems involving non-monotone multi-valued map-pings and non-differentiable mapmap-pings in Banach spaces. Very recently, Qiuet al.[] used the auxiliary principle technique to solve a system of generalized multi-valued strongly nonlinear mixed implicit quasi-variational-like inequalities in Hilbert spaces. They con-structed a new iterative algorithm and proved the convergence of the proposed iterative method.

Motivated and inspired by the research work mention above, in this article we introduce a new perturbed system of generalized mixed quasi-equilibrium-like problems involving multi-valued mappings in Hilbert spaces. We prove the existence of solutions of the per-turbed system of auxiliary generalized multi-valued mixed quasi-equilibrium-like prob-lems by using the Fan-KKM theorem. Then, by employing the auxiliary principle tech-nique and an existence result, we construct an iterative algorithm for solving the perturbed system of generalized multi-valued mixed quasi-equilibrium-like problems. Finally, the strong convergence of iterative sequences generated by the proposed algorithm is proved. The results in this article generalize, extend, and unify some recent results in the literature.

2 Preliminaries and formulation of problem

Throughout this article, we assume thatI={, }is an index set. For eachi∈I, letHibe a Hilbert space endowed with inner product·,·and norm · ,dbe the metric induced by the norm · , and letKibe a nonempty, closed, and convex subset ofHi,CB(Hi) be the family of all nonempty, closed, and bounded subsets ofHi, and for a ﬁnite subsetK,

Co(K) denotes the convex hull ofK. LetD(·,·) be the Hausdorﬀ metric onCB(Hi) deﬁned by

D(Pi,Qi) =max

sup

xi∈Pi

d(xi,Qi),sup yi∈Qi

d(Pi,yi)

, ∀Pi,Qi∈CB(Hi),

whered(xi,Qi) =infyi∈Qid(xi,yi) andd(Pi,yi) =infxi∈Pid(xi,yi).

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ui∈Ti(x),vi∈Si(x),wi∈Bi(x,x), andzi∈Ai(xi) such that ⎧

⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

N(z,η(f(y),f(x))) +M(u,v) +w,η(f(y),f(x))

+φ(x,y) –φ(x,x) +αf(y) –f(x)≥, ∀y∈K,

N(z,η(f(y),f(x))) +M(u,v) +w,η(f(y),f(x))

+φ(x,y) –φ(x,x) +αf(y) –f(x)≥, ∀y∈K,

()

whereαiis a real constant andφi:Ki×Ki−→Ris a real-valued non-diﬀerential mapping with the following properties:

Assumption (*)

(i) φi(·,·)is linear in the ﬁrst argument;

(ii) φi(·,·)is convex in the second argument;

(iii) φi(·,·)is bounded;

(iv) for anyxi,yi,zi∈Ki,

φi(xi,yi) –φi(xi,zi)≤φi(xi,yi–zi).

Remark . Notice that the role of the termαifi(yi) –fi(xi), for eachi∈I, in problem () is analogous to a choice of perturbation in the system. Sinceαiis a real constant, the solution set of the system () is larger than the solution set of system not involving the termαifi(yi) –fi(xi). It is also remarked that, combining Assumptions (iii) and (iv), it follows thatφ(·,·) is continuous in the second argument, which is used in many research works; seee.g., [–].

Some special cases of the problem () are listed below.

(i) IfN=N≡,f=f=I, the identity mappings, andα=α= , then system () reduces to the problem of ﬁnding(x,x)∈K×K,ui∈Ti(x),vi∈Si(x), and

wi∈Bi(x,x)such that

⎧ ⎨ ⎩

M(u,v) +w,η(y,x)+φ(x,y) –φ(x,x)≥, ∀y∈K,

M(u,v) +w,η(y,x)+φ(x,y) –φ(x,x)≥, ∀y∈K.

()

System () was considered and studied by Quiet al.[].

(ii) IfAiis a single-valued identity mapping,fiis an identity mapping,α=α= ,

Ni(·,ηi(fi(yi),fi(xi))) =Ni(·,yi), andwi= –wi∈CB(Ki), then system () reduces to the

system of generalized mixed equilibrium problems involving generalized mixed variational-like inequalities of ﬁnding(x,x)∈K×K,ui∈Ti(x)andvi∈Si(x) such that

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

N(x,y) +M(u,v) –w,η(y,x)+φ(x,y) –φ(x,x)≥,

∀y∈K,

N(x,y) +M(u,v) –w,η(y,x)+φ(x,y) –φ(x,x)≥,

∀y∈K.

()

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(iii) If for eachi∈I,Ki=Hi,Bi≡,Ti(x) =xandSi(x) =x, then system () reduces to the following system of mixed variational-like problems introduced and studied by Kazmi and Khan []: Find(x,x)∈H×Hsuch that

⎧ ⎨ ⎩

M(x,x),η(y,x)+φ(x,y) –φ(x,x)≥, ∀y∈H,

M(x,x),η(y,x)+φ(x,y) –φ(x,x)≥, ∀y∈H.

()

(iv) If for eachi∈I,Ki=K,Ni=N,Bi=φi≡,αi= ,Ai=A,Ti=T,ηi=η,fi=f and Mi(ui,vi),ηi(fi(yi),fi(xi))=Mi(ui,fi(yi)) =M(u,f(y)), then system () equivalent to

the problem of ﬁndingx∈K,z∈A(x)andu∈T(x)such that

Nz,ηf(y),f(x) +Mu,f(y) ≥, ∀y∈K, ()

which is called the generalized multi-valued equilibrium-like problem, introduced and studied by Dadashi and Latif [].

It should be noted that, for a suitable choice of the operatorsMi,Ni,Ti,Si,φi,ηi,Ai,Bi, and fi, for eachi∈I, in the above mentioned problems, it can easily be seen that the problem () covers many known system of generalized equilibrium problems and variational-like equilibrium problems.

Now, we give some deﬁnitions and results which will be used in the subsequent sections.

Deﬁnition . LetHbe a Hilbert space. A mappingh:H−→Ris said to be

(i) upper semicontinuous if, the set{x∈H:h(x) >λ}is a closed set, for everyλ∈R; (ii) lower semicontinuous if, the set{x∈H:h(x) >λ}is an open set, for everyλ∈R; (iii) continuous if, it is both lower semicontinuous and upper semicontinuous.

Remark . Ifhis lower semicontinuous, upper semicontinuous, and continuous at

ev-ery point ofH, respectively, thenhis lower semicontinuous, upper semicontinuous, and continuous onH, respectively.

Deﬁnition . Letη:K×K−→Kandf :K−→Kbe the single-valued mappings. Then

ηis said to be

(i) aﬃne in the ﬁrst argument if

ηλx+ ( –λ)z,y =λη(x,y) + ( –λ)η(z,y), ∀λ∈[, ],x,y,z∈K;

(ii) κ-Lipschitz continuous with respect tof if there exists a constantκ> such that

ηf(x),f(y) ≤κx–y, ∀x,y∈K.

Deﬁnition . LetN:H×H−→Rbe a real-valued mapping andA:K−→CB(H) be a

multi-valued mapping. ThenNis said to be

(i) monotone if

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(ii) -η-f-strongly monotone with respect toAif there exists> such that, for any

x,y∈K,z∈A(x), andz∈A(y),

Nz,ηf(y),f(x) +Nz,ηf(x),f(y) ≤–f(y) –f(x).

Deﬁnition . A mappingg:K−→His said to be

(i) ε-η-relaxed strongly monotone with respect tof if there existsε> such that

gf(x) –gf(y),ηf(x),f(y) ≥–εf(x) –f(y);

(ii) σ-Lipschitz continuous with respect tof if there exists a constantσ> such that

gf(x) –gf(y) ≤σx–y;

(iii) hemicontinuous with respect tof if, forλ∈[, ], the mapping

λ→g(λf(x) + ( –λ)f(y))is continuous asλ→+_{, for any}_x,_y_∈_K_.

Deﬁnition . A mappingf:H−→His said to beβ-expansive if there exists a constant β>  such that

f(x) –f(y)≥βx–y.

Deﬁnition . A multi-valued mappingP:K−→K_{is said to be KKM-mapping if, for}

each ﬁnite subset{x, . . . ,xn}ofK,Co{x, . . . ,xn} ⊆ n

i=P(xi), whereCo{x, . . . ,xn}denotes the convex hull of{x, . . . ,xn}.

Theorem .(Fan-KKM Theorem []) Let K be a subset of a topological vector space X,

and let P:K−→K_{be a KKM-mapping.}_{If for each x}_∈_K,_P(x)_{is closed and if for at least} one point x∈K,P(x)is compact,then_x_∈_KP(x)=∅.

Deﬁnition . The mappingM:H×H−→His said to be (μ,ξ)-mixed Lipschitz

con-tinuous if, there exist constantsμ,ξ>  such that M(x,y) –M(x,y)≤μx–x+ξy–y.

Deﬁnition . LetT :H−→CB(H) be a multi-valued mapping. ThenT is said to be

δ-D-Lipschitz continuous if, there exists a constantδ>  such that

DT(x),T(y) ≤δx–y, ∀x,y∈H,

whereD(·,·) is the Hausdorﬀ metric onCB(H).

Lemma .([]) Let(X,d)be a complete metric space and T:X−→CB(X)be a

multi-valued mapping.Then,for any given> ,x,y∈X and u∈T(x),there exists v∈T(y)such that

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3 Formulation of the perturbed system and existence result

In this section, ﬁrstly we consider the following perturbed system of auxiliary generalized multi-valued mixed quasi-equilibrium-like problems related to the perturbed system of generalized multi-valued mixed quasi-equilibrium-like problems (), and prove the exis-tence result.

For eachi∈I and given (x,x)∈K×K,ui∈Ti(x),vi∈Si(x),wi∈Bi(x,x) and

zi∈Ai(xi), ﬁnd (t,t)∈K×Ksuch that, for constantsρ,ρ> ,

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

ρN(z,η(f(y),f(t))) +g(f(t)) –g(f(x)) +ρ(M(u,v)

+w),η(f(y),f(t))+ρ{φ(x,y) –φ(x,t) +αf(y) –f(t)} ≥,

∀y∈K,

ρN(z,η(f(y),f(t))) +g(f(t)) –g(f(x)) +ρ(M(u,v)

+w),η(f(y),f(t))+ρ{φ(x,y) –φ(x,t)

+αf(y) –f(t)} ≥,

∀y∈K,

()

wheregi:Ki−→Hiis not necessarily the linear mapping. Problem () is called the per-turbed system of auxiliary generalized multi-valued mixed quasi-equilibrium-like prob-lems. Notice that ifti=xiis a solution of the system (), then (xi,ui,vi,wi,zi) is the solution of the system ().

Now, we establish the following existence and uniqueness of solutions of the perturbed system of auxiliary generalized multi-valued mixed quasi-equilibrium-like problems ().

Theorem . For each i∈I,let Ki be a nonempty,closed,and convex subset of Hilbert

space Hi,Ni:Hi×Hi−→Rbe a real-valued mapping,φi:Ki×Ki−→Ris a real-valued non-diﬀerential mapping,Mi:Hi×Hi−→Hibe a single-valued mapping,Ai,Ti,Si:Ki−→

CB(Hi)and Bi:K×K−→CB(Hi)be the multi-valued mappings,ηi:Ki×Ki−→Hibe a nonlinear single-valued mapping,and fi:Ki−→Kibe a single-valued mapping.Assume that the following conditions are satisﬁed:

(i) Ni(zi,ηi(fi(xi),fi(xi))) = ,for eachxi∈KiandNiis convex in the second argument;

(ii) Niisi-ηi-fi-strongly monotone with respect toAiand upper semicontinuous;

(iii) ηiis aﬃne,continuous in the second argument with the condition ηi(xi,yi) +ηi(yi,xi) = ,for allxi,yi∈Ki;

(iv) giisεi-ηi-relaxed strongly monotone with respect tofiand hemicontinuous with

respect tofi;

(v) fiisβi-expansive and aﬃne;

(vi) φisatisﬁes Assumption(*);

(vii) εi=αiρiandεi<ρii;

(viii) if there exists a nonempty compact subsetDiofHiandti ∈Di∩Kisuch that for

anyti∈Ki\Di,we have

ρiNi

zi,ηi

fi

t_i ,fi(ti)

+gi

fi

t_i –gi

fi(xi) +ρi

Mi(ui,vi)

+wi ,ηi

fi

t_i ,fi(ti) +ρi

φi

xi,ti –φi(xi,ti) +αifi

t_i –fi(ti)



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for given ui∈Ti(x),vi∈Si(x), wi∈Bi(x,x)and zi∈Ai(xi). Then the perturbed sys-tem of auxiliary generalized multi-valued mixed quasi-equilibrium-like problems()has a unique solution.

Proof For each i∈I, ti ∈Ki and ﬁxed (x,x)∈K×K, ui ∈Ti(x), vi ∈Si(x), wi ∈ Bi(x,x) andzi∈Ai(xi), deﬁne the multi-valued mappingsPi,Qi:Ki−→Kias follows:

Pi(yi) =

ti∈Ki:ρiNi

zi,ηi

fi(yi),fi(ti)

+gi

fi(ti) –gi

fi(xi)

+ρi

Mi(ui,vi) +wi ,ηi

fi(yi),fi(ti) +ρi

φi(xi,yi) –φi(xi,ti)

+αifi(yi) –fi(ti)



≥,

Qi(yi) =

ti∈Ki:ρiNi

zi,ηi

fi(yi),fi(ti)

+gi

fi(yi) –gi

fi(xi)

+ρi

Mi(ui,vi) +wi ,ηi

fi(yi),fi(ti) +ρi

≥.

In order to reach the conclusion of Theorem ., we show that all the assumptions of Fan-KKM Theorem . are satisﬁed.

First, we claim thatQiis a KKM-mapping. On the contrary, suppose thatQiis not a KKM-mapping. Then there exist{y

i, . . . ,yni}andλ j

i≥,j= , . . . ,n, with n

j=λ

j

i=  such that

y= n

j=

λj_iyj_i∈/ n

j=

Qi

yj_i .

Therefore, we have

ρiNi

z,ηi

fi

yj_i ,fi(y)

+gi

fi

yj_i –gi

fi(xi) +ρi

M(ui,vi) +wi ,ηi

fi

yj_i ,fi(y)

+ρi

φi

xi,yji –φi(xi,y)

< , ∀i,jandz∈Ai(y).

Sinceηiandfiare aﬃne, andNiandφiare convex in the second argument, we have

 = ρiNi

z,ηi

fi(y),fi(y)

+gi

fi

yj_i –gi

fi(xi) +ρi

M(ui,vi) +wi ,ηi

fi(y),fi(y)

+ρi

φi(xi,y) –φi(xi,y)

=ρiNi

z,ηi

j λj_ifi

yj_i ,fi(y)

+ gi fi yj_i –gi

fi(xi) +ρi

M(ui,vi) +wi ,

ηi

j λj_ifi

yj_i ,fi(y)

+ρi

φi

xi,

j λj_iyj_i

–φi(xi,y)

≤ρiNi

z, j

λj_iηi

fi

yj_i ,fi(y) +

gi

fi

yj_i –gi

fi(xi) +ρi

M(ui,vi) +wi ,

j λj_iηi

fi

yj_i ,fi(yi) +ρi

j λj_iφi

≤

j

λj_iρiNi

z,ηi

fi

yj_i ,fi(y)

+ j

λj_igi

fi

yj_i –gi

fi(xi) +ρi

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+wi ,ηi

fi

yj_i ,fi(y) +ρi

j λj_iφi

xi,yji –

j

λj_iφi(xi,y)

= j

λj_iρiNi

z,ηi

fi

yj_i ,fi(y)

+gi

fi

yj_i –gi

fi(xi) +ρi

M(ui,vi) +wi ,

ηi

fi

yj_i ,fi(y) +ρi

φi

< ,

which is a contradiction. Therefore,ybeing an arbitrary element ofCo{y

i, . . . ,yni}, we have y∈Co{y

i, . . . ,yni} ⊆ n

j=Qi(yji). HenceQiis a KKM-mapping. Now, we show that_y_i∈_K

iQi(yi) =

yi∈KiPi(yi), for everyyi∈Ki. Letti∈Qi(yi), therefore by deﬁnition, we have

ρiNi

zi,ηi

fi(yi),fi(ti)

+gi

fi(yi) –gi

fi(xi) +ρi

M(ui,vi) +wi ,ηi

fi(yi),fi(ti)

+ρi

≥,

which implies that

gi

fi(yi),ηi

fi(yi),fi(ti) +ρiNi

zi,ηi

fi(ti),fi(y)

≥gi

fi(xi) +ρi

M(ui,vi) +wi ,ηi

fi(yi),fi(ti) +ρi

. ()

Sincegiisεi-ηi-relaxed strongly monotone with respect tofiwith the conditionεi=ρiαi, inequality () becomes

ρiNi

zi,ηi

fi(yi),fi(ti)

+gi

fi(ti) –gi

fi(xi) +ρi

M(ui,vi) +wi ,ηi

fi(yi),fi(ti)

+ρi

φi(xi,yi) –φi(xi,ti) +αifi(yi) –fi(ti)

≥,

and hence we haveti∈Pi(yi). It follows that

yi∈KiQi(yi)⊆

yi∈KiPi(yi). Conversely, suppose thatti∈

yi∈KiPi(yi), then we have

ρiNi

zi,ηi

fi(yi),fi(ti)

+gi

fi(ti) –gi

fi(xi) +ρi

M(ui,vi) +wi

,ηi

fi(yi),fi(ti)

+ρi

φi(xi,yi) –φi(xi,ti) +αifi(yi) –fi(ti)



≥. ()

Letyλ

i =λiti+ ( –λi)yi,λi∈[, ]. SinceKiis convex, we haveyλi ∈Ki. It follows from () that

ρiNi

zi,ηi

fi(yi),fi

yλ_i +gi

fi

yλ_i –gi

fi(xi) +ρi

M(ui,vi) +wi ,ηi

fi(yi),fi

yλ_i

+ρi

φi(xi,yi) –φi

xi,yλi +αifi(yi) –fi

yλ_i ≥. ()

Sinceηiis aﬃne with the conditionηi(fi(yi),fi(yi)) = ,fiis aﬃne, andNiandφiare convex in the second argument, inequality () reduces to

λiρiNi

zi,ηi

fi(yi),fi(ti)

+λi

gi

fi

yλ_i –gi

fi(xi) +ρi

M(ui,vi) +wi ,ηi

fi(yi),fi(ti)

+λiρi

φi(xi,yi) –φi(xi,ti) +αiλifi(yi) –fi(ti)



(9)

which implies that

λiρiNi

zi,ηi

fi(yi),fi(ti)

+λi

gi

λifi(ti) + ( –λi)fi(yi) –gi

fi(xi) +ρi

M(ui,vi) +wi ,ηi

fi(yi),fi(ti) +λiρi



≥. ()

Dividing () byλi, we get

ρiNi

zi,ηi

fi(yi),fi(ti)

+gi

λifi(ti) + ( –λi)fi(yi) –gi

fi(xi) +ρi

M(ui,vi)

+wi ,ηi

fi(yi),fi(ti) +ρi



≥.

Sincegiis hemicontinuous with respect tofiand takingλi→, it implies that

ρiNi

zi,ηi

fi(yi),fi(ti)

+gi

fi(yi) –gi

fi(xi) +ρi

M(ui,vi) +wi ,ηi

fi(yi),fi(ti) +ρi

≥.

Therefore, we haveti∈Qi(yi), and we conclude that

yi∈KiQi(yi) =

yi∈KiPi(yi) andPiis also a KKM-mapping, for eachyi∈Ki.

Sinceηiis continuous in the second argument,fiandφiare continuous andNiis upper semicontinuous, it follows thatPi(yi) is closed for eachyi∈Ki.

Finally, we show that, fort

i ∈Di∩Ki,Pi(ti) is compact. For this purpose, suppose that there existst˜i∈Pi(ti) such thatt˜i∈/D. Therefore, forz˜i∈Ai(˜ti), we have

ρiNi

˜ zi,ηi

fi

t_i ,fi(t˜i)

+gi

fi

t_i –gi

fi(xi) +ρi

Mi(ui,vi) +wi ,ηi

fi

t_i ,fi(t˜i)

+ρi

φi

xi,ti –φi(xi,t˜i) +αifi

t_i –fi(t˜i)



≥. ()

But by Assumption (viii), fort˜i∈/D, we have

ρiNi

˜ zi,ηi

fi

t_i ,fi(t˜i)

+gi

fi

t_i –gi

fi(xi) +ρi

Mi(ui,vi) +wi ,ηi

fi

t_i ,fi(t˜i) +ρi

φi

xi,ti –φi(xi,t˜i)

+ρiαifi

t_i –fi(t˜i)



< ,

which is a contradiction to (). Therefore Qi(ti)⊂D. Due to compactness of D, and closedness ofPi(ti), we conclude thatPi(ti) is compact.

Thus, all the conditions of the Fan-KKM Theorem . are fulﬁlled by the mappingPi. Therefore

yi∈Ki

Pi(yi)=φ.

Hence, (t,t)∈K×K is a solution of the perturbed system of auxiliary generalized

multi-valued mixed quasi-equilibrium-like problems ().

Now, let (t,t), (t˜,t˜)∈K×Kbe any two solutions of the perturbed system of auxiliary

generalized multi-valued mixed quasi-equilibrium-like problems (). Then, for eachi∈I, we have

ρiNi

˜ zi,ηi

fi(yi),fi(t˜i)

+gi

fi(˜ti) –gi

fi(xi) +ρi

Mi(ui,vi) +wi ,ηi

fi(yi),fi(t˜i)

+ρi

φi(xi,yi) –φi(xi,t˜i) +αifi(yi) –fi(˜ti)



(10)

and

ρiNi

zi,ηi

fi(yi),fi(ti)

+gi

fi(ti) –gi

fi(xi) +ρi

Mi(ui,vi) +wi ,ηi

fi(yi),fi(ti)

+ρi

φi(xi,yi) –φi(xi,ti) +αifi(yi) –fi(ti)



≥. ()

Puttingyi=tiin () andyi=t˜iin (), summing up the resulting inequalities and using the conditionηi(fi(xi),fi(yi)) +ηi(fi(yi),fi(xi)) = , we have

ρi

Ni

˜ zi,ηi

fi(ti),fi(t˜i)

+Ni

zi,ηi

fi(t˜i),fi(ti)

+gi

fi(˜ti) –gi

fi(ti),ηi

fi(ti),fi(˜ti)

+ ρiαifi(ti) –fi(t˜i)



≥. ()

Since Ni is strongly i-ηi-fi-strongly monotone with respect to Ai, gi isεi-ηi-relaxed strongly monotone with respect tofiwith the conditionεi=αiρi, we have from ()

–ρiifi(ti) –fi(t˜i)



≥ρi

Ni

˜ zi,ηi

fi(ti),fi(t˜i)

+Ni

zi,ηi

fi(t˜i),fi(ti)



≥gi

fi(t˜i) –gi

fi(ti) ,ηi

fi(ti),fi(t˜i) ≥–εifi(ti) –fi(˜ti)



,

which implies that

(–ρii+ εi)fi(ti) –fi(t˜i)



≥.

Sincefiisβi-expansive and εi<ρii, we obtain

≤(–ρii+ εi)fi(ti) –fi(t˜i)



≤(–ρii+ εi)βiti–t˜i< ,

which shows thatt˜i=ti. This completes the proof.

4 Iterative algorithm and convergence analysis

By using Theorem . and Lemma ., we construct the following iterative algorithm for computing approximate solutions of the perturbed system of generalized multi-valued mixed quasi-equilibrium-like problems ().

Iterative Algorithm . For any given (x

,x)∈K×K,u ∈T(x),u∈T(x),v ∈

S(x),v∈S(x),w ∈B(x,x),w∈B(x,x) andz ∈A(x),z ∈A(x), compute

the iterative sequences{(xn

,xn)} ⊆K×K,{uni},{vni},{wni}and{zni}by the following iter-ative schemes:

ρN

zn_+,η

f(y),f

xn_+ +g

f

xn_+ –g

f

xn_ +ρ

M

un_,vn_ +wn_,η

f(y),f

xn_+ +ρ

φ

xn_,y –φ

xn_,xn_+

+αf(y) –f

xn_+ ≥, ∀y∈K; ()

ρN

zn_+,η

f(y),f

xn_+ +g

f

xn_+ –g

f

xn_ +ρ

M

un_,vn_ +wn_,η

f(y),f

xn_+ +ρ

φ

xn_,y –φ

xn_,xn_+

+αf(y) –f

(11)

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

un_i ∈Ti(xn); uni+–uni ≤( +n+ )D(Ti(x

n+

 ),Ti(xn));

vn

i ∈Si(xn); vni+–vni ≤( +n+)D(Si(x

n+

 ),Si(xn));

wn_i ∈Bi(xn,xn); win+–wni ≤( +n+)D(Bi(x

n+

 ,xn+),Bi(xn,xn));

zn

i ∈Ai(xn); zni+–zni ≤( +n+)D(Ai(xin+),Ai(xni)),

()

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

Now, we establish the following strong convergence result to obtain the solution of per-turbed system of generalized multi-valued mixed quasi-equilibrium-like problems ().

Theorem . For each i∈I,the mappings Ni,Mi,Ai,Ti,Si,Bi,ηi,φi,and fisatisfy the hypotheses of Theorem..Further assume that:

(i) Miis(μi,ξi)-mixed Lipschitz continuous;

(ii) giisσi-Lipschitz continuous with respect tofiandηiisκi-Lipschitz continuous with

respect tofi;

(iii) Tiisδi-D-Lipschitz continuous andSiisτi-D-Lipschitz continuous;

(iv) Biis(ζi,νi)-D-Lipschitz continuous andAiisςi-D-Lipschitz continuous. Forρ,ρ> ,if the following conditions are satisﬁed:

⎧ ⎨ ⎩

 (ρ–ε)β{

κσ+ρκ(μδ+ζ) +ργ}+₍_ρ  –ε)β{

ρκ(μδ+ζ)}< , 

(ρ–ε)β_{κσ+ρκ(ξτ+ν) +ργ}+



(ρ–ε)β_{ρκ(ξτ+ν)}< ,

()

then there exist(x,x)∈K×K,ui∈Ti(x),vi∈Si(x),wi∈Bi(x,x),and zi∈Ai(xi)such that(x,x,u,u,v,v,w,w,z,z)is the solution of the perturbed system of generalized

multi-valued mixed quasi-equilibrium-like problems()and the sequences{xn

},{xn},{uni}, {vn

i},{wni},and{zni}generated by Algorithm.converge strongly to x,x,ui,vi,wi,and zi, respectively.

Proof Firstly, from () of Algorithm ., we have, for ally∈K,

ρN

zn_,η

f(y),f

xn_ +g

f

xn_ –g

f

xn_– +ρ

M

un_–,vn_– +wn_–,η

f(y),f

xn_ +ρ

φ

xn_–,y –φ

xn_–,xn_

+αf(y) –f

xn_ ≥ ()

and

ρN

zn_+,η

f(y),f

xn_+ +g

f

xn_+ –g

f

xn_ +ρ

M

un_,vn_ +wn_,η

f(y),f

xn_+ +ρ

φ

xn_,y –φ

xn_,xn_+

+αf(y) –f

xn_+ ≥. ()

Puttingy=xn+in () andy=xnin (), and summing up the resulting inequalities, we

obtain

ρ

N

zn_,η

f

xn_+ ,f

xn_ +N

zn_+,η

f

xn_ ,f

xn_+

+g

f

xn_ –g

f

xn_– +ρ

M

un_–,vn_– +wn_–,η

f

xn_+ ,f

(12)

+g

f

xn_+ –g

f

xn_ +ρ

M

un_,vn_ +wn_,η

f

xn_ ,f

xn_+

+ρ

φ

xn_–,xn_+ –φ

xn_–,xn_ +φ

xn_,xn_ –φ

xn_,xn_+

+αf

xn_+ –f

xn_ +αf

xn_ –f

xn_+ ≥,

which implies that

g

f

xn_– –g

f

xn_ ,η

f

xn_ ,f

xn_+

+ρ

M

un_,vn_ ,η

f

xn_ ,f

xn_+ + αρf

xn_+ –f

xn_ 

+ρ

wn_–wn_–,η

f

xn_ ,f

xn_+ +ρφ

xn_ –xn_–,xn_–xn_+ ≥ρ

N

zn_,η

f

xn_+ ,f

xn_ +N

zn_+,η

f

xn_ ,f

xn_+

+g

f

xn_ –g

f

xn_+ ,η

f

xn_ ,f

xn_+ .

SinceN is-η-f-strongly monotone with respect toA,g isε-η-relaxed strongly

monotone with respect tof,φis bounded by assumption and using the Cauchy-Schwartz

inequality, we have

ρf

xn_+ –f

xn_ –εf

xn_+ –f

xn_  ≤ρ

N

zn_,η

f

xn_+ ,f

xn_ +N

zn_+,η

f

xn_ ,f

xn_+

+g

f

xn_ –g

f

xn_+ ,η

f

xn_ ,f

xn_+

≤g

f

xn_– –g

f

xn_ η

f

xn_ ,f

xn_+

+ρM

un_,vn_ –M

un_–,vn_– η

f

xn_ ,f

xn_+

+ρwn–wn–η

f

xn_ ,f

xn_+ +ργxn –xn–xn–xn+

+ αρf

xn_+ –f

xn_ . ()

By using (μ,ξ)-mixed Lipschitz continuity ofM,δi-D-Lipschitz continuity ofT and

τi-D-Lipschitz continuity ofS, it follows by Algorithm . that

M

un_,vn_ –M

un_–,vn_– ≤μun–un–+ξvn–vn–

≤μ

 +

n

DT

xn_ –T

xn_– +ξ

 +

n

DS

xn_ –S

xn_–

≤μδ

 + 

n

xn_–xn_–+ξτ

 +

n

xn_–xn_–. ()

Also by Algorithm . and (ζ,ν)-D-Lipschitz continuity ofB, we have

wn_–wn_–≤

 + n

DB

xn_,xn_ ,B

xn_–,xn_–

≤

 + n

ζxn–xn–+νxn–xn–

=ζ

 +

n

xn_–xn_–+ν

 +

n

(13)

Sinceg isσ-Lipschitz continuous with respect tof,ηisκ-Lipschitz continuous with

respect tof,fisβ-expansive with the condition ε<ρ, it follows from (), (), and

() that

(ρ– ε)βxn+–xn 

≤(ρ– ε)f

xn_+ –f

xn_ 

≤κσxn–xn–xn+–xn+ρκ

μδ

 + 

n

xn_–xn_–

+ξτ

 +

n

x_n–xn_–x_n+–xn_+κρ

ζ

 +

n

xn_–xn_–

+ν

 +

n

xn_–xn_–xn_+–xn_+ργxn–xn–xn+–xn

=κσxn–xn–xn+–xn+ρκμδ

 +

n

xn_–xn_–xn_+–xn_

+ρκξτ

 +

n

xn_–xn_–x_n+–xn_+κρζ

 +

n

xn_–xn_–xn_+

–xn_+κρν

 +

n

x_n–xn_–x_n+–xn_+ργxn–xn–xn+–xn

=

κσ+ρκμδ

 +

n

+κρζ

 +

n

+ργ

xn_–xn_–xn_+–xn_

+

ρκξτ

 +

n

+κρν

 +

n

xn_–xn_–xn_+–xn_,

which implies that

xn_+–xn_

≤ 

(ρ– ε)β

κσ+

ρκ

 +

n

(μδ+ζ) +ργ

xn_–xn_–

+

ρκ

 +

n

(ξτ+ν)

xn_–xn_–

.

Hence,

xn_+–xn_≤θ_nxn_–x_n–+ϑ_nx_n–xn_–, ()

where

θ_n=  (ρ– ε)β

κσ+

ρκ

 + 

n

(μδ+ζ) +ργ

and

ϑ_n=  (ρ– ε)β

ρκ

 +

n

(ξτ+ν)

(14)

Secondly, it follows from () of Algorithm ., for ally∈K, that

ρN

zn_,η

f(y),f

xn_ +g

f

xn_ –g

f

xn_– +ρ

M

un_–,vn_–

+wn_–,η

f(y),f

xn_ +ρ

φ

xn_–,y –φ

xn_–,xn_

+αf(y) –f

xn_ ≥

and

ρN

zn_+,η

f(y),f

xn_+ +g

f

xn_+ –g

f

xn_ +ρ

M

un_,vn_

+wn_,η

f(y),f

xn_+ +ρ

φ

xn_,y –φ

xn_,xn_+

+αf(y) –f

xn_+ ≥.

Using the same arguments as above, the imposed conditions onN,g,η,f,A,T,S,

and Algorithm ., we obtain

xn_+–xn_≤θ_nxn_–x_n–+ϑ_nx_n–xn_–, ()

where

θ_n=  (ρ– ε)β

κσ+

ρκ

 +

n

(ξτ+ν) +ργ

and

ϑ_n=  (ρ– ε)β

ρκ

 + 

n

(μδ+ζ)

.

Adding () and (), we have

xn_+–xn_+x_n+–xn_≤θ_n+ϑ_nxn_ –xn_–+θ_n+ϑ_nxn_–xn_–

≤maxθ_n,θ_nxn_ –x_n–+xn_–xn_–, ()

where

θ_n=θ_n+ϑ_n=  (ρ– ε)β

κσ+

ρκ

 +

n

(μδ+ζ) +ργ

+  (ρ– ε)β

ρκ

 +

n

(μδ+ζ)

and

θ_n=θ_n+ϑ_n=  (ρ– ε)β

κσ+

ρκ

 +

n

(ξτ+ν) +ργ

+  (ρ– ε)β

ρκ

 +

n

(ξτ+ν)

(15)

Letting

θ=

 (ρ– ε)β

κσ+ρκ(μδ+ζ) +ργ

+  (ρ– ε)β

ρκ(μδ+ζ)

and

θ=

 (ρ– ε)β

κσ+ρκ(ξτ+ν) +ργ

+  (ρ– ε)β

ρκ(ξτ+ν)

,

it can easily be seen thatθ_n→θandθn→θ, asn→ ∞. Taking into account the

con-dition (), we conclude thatmax{θ,θ}< . Hence, it follows from () that{(xn,xn)}is

a Cauchy sequence inK×K; now suppose that (xn,xn)→(x,x)∈K×K, asn→ ∞.

By Algorithm . andD-Lipschitz continuity ofTi,Si,BiandAi, for eachi∈I, we have un_i+–un_i≤

 + 

n+ 

DTi

xn_+ ,Ti

xn_

≤

 +  n+ 

δixn+–xni; vn_i+–vn_i≤

 + 

n+ 

DSi

xn_+ ,Si

xn_

≤

 +  n+ 

τixn+–xn;

wn_i+–wn_i≤

 +  n+ 

DBi

xn_+,xn_+ ,Bi

xn_,xn_

≤

 +  n+ 

ζixn+–xn+νixn+–xn ;

and

zn_i+–zn_i≤

 +  n+ 

DAi

xn_i+ ,Ai

xn_i

≤

 +  n+ 

ςixni+–xni.

Therefore, for eachi∈I,{un_i},{v_in},{wn_i}, and{zn_i}are also Cauchy sequences; now assume thatun_i →ui,vin→vi,wni →wi, andzni →zi, asn→ ∞. Asuni ∈Ti(xn), we have

dui,Ti(x) =ui–uni+d

un_i,Ti

xn_ +DTi

xn_ ,Ti(x)

≤ui–uni+δixn–x→ asn→ ∞.

Therefore, we deduce thatui∈Ti(x). Similarly, we can obtainvi∈Si(x),wi∈Bi(x,x),

andzi∈Ai(xi), for eachi∈I. By Algorithm ., we have

ρN

zn_+,η

f(y),f

xn_+ +g

f

xn_+ –g

f

xn_ +ρ

M

un_,vn_

+wn_,η

f(y),f

xn_+ +ρ

φ

xn_,y –φ

xn_,xn_+

+αf(y) –f

(16)

and

ρN

zn_+,η

f(y),f

xn_+ +g

f

xn_+ –g

f

xn_ +ρ

M

un_,vn_

+wn_,η

f(y),f

xn_+ +ρ

φ

xn_,y –φ

xn_,xn_+

+αf(y) –f

xn_+ ≥, ∀y∈K. ()

By using the continuity ofNi,Mi,gi,φi,fi, andηi, for eachi∈I, and sinceuni →ui,vni →vi, wn

i →wi,zni →zi, andxni →xiforn→ ∞, from () and (), we have, forρi> ,

N

z,η

f(y),f(x)

+M(u,v) +w,η

f(y),f(x)

+φ(x,y) –φ(x,x) +αf(y) –f(x) 

≥, ∀y∈K,

and

N

z,η

f(y),f(x)

+M(u,v) +w,η

f(y),f(x)

+φ(x,y) –φ(x,x) +αf(y) –f(x) 

≥, ∀y∈K.

Therefore (x,x,u,u,v,v,z,z,w,w) is the solution of the perturbed system of

gen-eralized multi-valued mixed quasi-equilibrium-like problems (). This completes the

proof.

5 Conclusion

In this article, a perturbed system of generalized multi-valued mixed equilibrium-like problems and a perturbed system of auxiliary generalized multi-valued mixed quasi-equilibrium-like problems are introduced in Hilbert spaces. For the corresponding aux-iliary system, we prove the existence of solutions by using relatively suitable conditions. Further, an iterative algorithm is proposed for solving our system and a strong conver-gence theorem is proved. It is noted that the solution set of our system is larger than the solution set of the system considered by Qiuet al.[], Dinget al.[], and many oth-ers. Also, our results improve and extend many well-known results for diﬀerent systems existing in the literature.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Author details

1_{Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India.}2_{Department of Information}

Management, and Innovation Centre for Big Data and Digital Convergence, Yuan Ze University, Chung-Li, 32002, Taiwan.

Received: 21 September 2016 Accepted: 15 February 2017 References

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