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

Auxiliary principle technique and iterative algorithm for a perturbed system of generalized multi valued mixed quasi equilibrium like problems Rahaman et al Journal of Inequalities and Applications (2[.]

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

* Correspondence:

imctpang@saturn.yzu.edu.tw

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; see e.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

© The Author(s) 2017 This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro-vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and

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

Glowinski et 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, Qiu et 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 that I = {, } is an index set For each i ∈ I, let H ibe

a Hilbert space endowed with inner product·, · and norm  · , d be the metric induced

by the norm · , and let K i be a nonempty, closed, and convex subset of H i , CB(H i) be

the family of all nonempty, closed, and bounded subsets of H i , and for a finite subset K ,

Co(K ) denotes the convex hull of K Let D(·, ·) be the Hausdorff metric on CB(H i) defined

by

D(P i , Q i) = max



sup

x i∈Pi d (x i , Q i), sup

y i∈Qi d (P i , y i)

 , ∀P i , Q i ∈ CB(H i),

where d(x i , Q i) = infy i∈Qi d (x i , y i ) and d(P i , y i) = infx i∈Pi d (x i , y i)

For each i ∈ I, let N i : H i × H i −→ R be a real-valued mapping, M i : H i × H i −→ H i

be a single-valued mapping, A i , T i , S i : K i −→ CB(H i ) and B i : K× K−→ CB(H i) be the

multi-valued mappings, η i : K i × K i −→ H i be a nonlinear single-valued mapping, and

f i : K i −→ K ibe a single-valued mapping We introduce the following perturbed system of

generalized multi-valued mixed quasi-equilibrium-like problems: Find (x , x )∈ K × K ,

u i ∈ T i (x), v i ∈ S i (x), w i ∈ B i (x, x), and z i ∈ A i (x i) 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,

()

where α i is a real constant and φ i : K i × K i−→ R is a real-valued non-differential mapping with the following properties:

Assumption (*)

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

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

(iii) φ i(·, ·) is bounded;

(iv) for any x i , y i , z i ∈ K i,

φ i (x i , y i ) – φ i (x i , z i)≤ φ i (x i , y i – z i)

Remark . Notice that the role of the term α i f i (y i ) – f i (x i), for each i ∈ I, in problem () is analogous to a choice of perturbation in the system Since α iis a real constant, the solution set of the system () is larger than the solution set of system not involving the

term α i f i (y i ) – f i (x i) It is also remarked that, combining Assumptions (iii) and (iv), it

follows that φ(·, ·) is continuous in the second argument, which is used in many research

works; see e.g., [–].

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

(i) If N= N≡ , f= f= I, the identity mappings, and α= α= , then system ()

reduces to the problem of finding (x, x)∈ K× K, u i ∈ T i (x), vi ∈ S i (x), and

w i ∈ B i (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 Qui et al [].

(ii) If A i is a single-valued identity mapping, f i is an identity mapping, α= α= ,

N i(·, ηi (f i (y i ), f i (x i ))) = N i(·, yi), and wi = –w i ∈ CB(K i), then system () reduces to the system of generalized mixed equilibrium problems involving generalized mixed

variational-like inequalities of finding (x, x)∈ K× K, u i ∈ T i (x)and v i ∈ S i (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

()

System () introduced and studied by Ding []

(iii) If for each i ∈ I, K i = H i , B i ≡ , T i (x) = xand S i (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 each i ∈ I, K i = K , N i = N , B i = φ i ≡ , α i= , Ai = A, T i = T , η i = η, f i = f and

M i (u i , v i ), η i (f i (y i ), f i (x i)) = M i (u i , f i (y i )) = M(u, f (y)), then system () equivalent to the problem of finding x ∈ K, z ∈ A(x) and u ∈ T(x) such that

N

z , η

f (y), f (x)

+ M

u , f (y)

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 operators M i , N i , T i , S i , φ i , η i , A i , B i, and

f i , for each i ∈ 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 . Let H be a Hilbert space A mapping h : H−→ R is said to be

(i) upper semicontinuous if, the set{x ∈ H : h(x) > λ} is a closed set, for every λ ∈ R;

(ii) lower semicontinuous if, the set{x ∈ H : h(x) > λ} is an open set, for every λ ∈ R;

(iii) continuous if, it is both lower semicontinuous and upper semicontinuous

Remark . If h is lower semicontinuous, upper semicontinuous, and continuous at ev-ery point of H, respectively, then h is lower semicontinuous, upper semicontinuous, and continuous on H, respectively.

Definition . Let η : K × K −→ K and f : K −→ K be the single-valued mappings Then

ηis said to be

(i) affine in the first argument if

η

λx + ( – λ)z, y

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

(ii) κ-Lipschitz continuous with respect to f if there exists a constant κ >  such that

η

f (x), f (y) κ x – y, ∀x, y ∈ K.

Definition . Let N : H × H −→ R be a real-valued mapping and A : K −→ CB(H) be a multi-valued mapping Then N is said to be

(i) monotone if

N (x, y) + N(y, x) ≤ , ∀x, y ∈ H;

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

x , y ∈ K, z ∈ A(x), and z∈ A(y),

N

z , η

f (y), f (x)

+ N

z, η

f (x), f (y)

≤ – f (y) – f (x) 

Definition . A mapping g : K −→ H is said to be

(i) ε-η-relaxed strongly monotone with respect to f if there exists ε >  such that

g

f (x)

– g

f (y)

, η

f (x), f (y)

≥ –ε f (x) – f (y) ;

(ii) σ -Lipschitz continuous with respect to f if there exists a constant σ >  such that

g

f (x)

– g

f (y) σ x – y;

(iii) hemicontinuous with respect to f if, for λ∈ [, ], the mapping

λ → g(λf (x) + ( – λ)f (y)) is continuous as λ → +, for any x, y ∈ K.

Definition . A mapping f : H −→ H is said to be β-expansive if there exists a constant

β>  such that

f (x) – f (y) β x – y.

Definition . A multi-valued mapping P : K−→ Kis said to be KKM-mapping if, for each finite subset{x, , x n } of K, Co{x, , x n} ⊆ n

i=P (x i), where Co{x, , x n} denotes the convex hull of{x, , x n}

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, thenx ∈K P (x)= ∅

Definition . The mapping M : H × H −→ H is said to be (μ, ξ)-mixed Lipschitz con-tinuous if, there exist constants μ, ξ >  such that

M (x, y) – M(x, y) μ x– x + ξy– y

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

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

DT (x), T(y)

≤ δx – y, ∀x, y ∈ H,

whereD(·, ·) is the Hausdorff metric on CB(H).

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

multi-valued mapping ∈ X and u ∈ T(x), there exists v ∈ T(y) such

that

d (u, v) DT (x), T(y)

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 each i ∈ I and given (x, x)∈ K× K, u i ∈ T i (x), v i ∈ S i (x), w i ∈ B i (x, x) and

z i ∈ A i (x i ), find (t, t)∈ K× Ksuch that, for constants ρ, 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,

()

where g i : K i −→ H iis 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 if t i = x i is a solution of the system (), then (x i , u i , v i , w i , z i) 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 K i be a nonempty , closed, and convex subset of Hilbert

space H i , N i : H i × H i −→ R be a real-valued mapping, φ i : K i × K i −→ R is a real-valued

non-differential mapping , M i : H i ×H i −→ H i be a single-valued mapping , A i , T i , S i : K i−→

CB(H i ) and B i : K× K−→ CB(H i ) be the multi-valued mappings, η i : K i × K i −→ H i be

a nonlinear single-valued mapping , and f i : K i −→ K i be a single-valued mapping Assume

that the following conditions are satisfied:

(i) N i (z i , η i (f i (x i ), f i (x i))) = , for each xi ∈ K i and N i is convex in the second argument;

(ii) N i is  i -η i -f i -strongly monotone with respect to A i and upper semicontinuous;

(iii) η i is affine , continuous in the second argument with the condition

η i (x i , y i ) + η i (y i , x i) = , for all xi , y i ∈ K i;

(iv) g i is ε i -η i -relaxed strongly monotone with respect to f i and hemicontinuous with respect to f i;

(v) f i is β i -expansive and affine;

(vi) φ i satisfies Assumption(*);

(vii) ε i = α i ρ i and ε i < ρ i  i;

(viii) if there exists a nonempty compact subset D i of H i and t

i ∈ D i ∩ K i such that for any t i ∈ K i \ D i , we have

ρ i N i



z i , η i



f i



t i

, f i (t i) +

g i



f i



t i

– g i



f i (x i)

+ ρ i



M i (u i , v i)

+ w i

, η i

f i

t

, f i (t i)

+ ρ i

φ i

x i , t

– φ i (x i , t i ) + α i f i

t

– f i (t i) 

< ,

for given u i ∈ T i (x), v i ∈ S i (x), w i ∈ B i (x, x) and z i ∈ A i (x i ) Then the perturbed

sys-tem of auxiliary generalized multi-valued mixed quasi-equilibrium-like problems () has

a unique solution

Proof For each i ∈ I, t i ∈ K i and fixed (x, x)∈ K× K, u i ∈ T i (x), v i ∈ S i (x), w i ∈

B i (x, x) and z i ∈ A i (x i ), define the multi-valued mappings P i , Q i : K i−→ K ias follows:

P i (y i) =

t i ∈ K i : ρ i N i



z i , η i



f i (y i ), f i (t i)

+

g i



f i (t i)

– g i



f i (x i)

+ ρ i

M i (u i , v i ) + w i

, η i

f i (y i ), f i (t i)

+ ρ i

φ i (x i , y i ) – φ i (x i , t i)

+ α i f i (y i ) – f i (t i) 

≥ ,

Q i (y i) =

t i ∈ K i : ρ i N i



z i , η i



f i (y i ), f i (t i)

+

g i



f i (y i)

– g i



f i (x i)

+ ρ i



M i (u i , v i ) + w i

, η i



f i (y i ), f i (t i)

+ ρ i



φ i (x i , y i ) – φ i (x i , t i)

≥ 

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

First, we claim that Q i is a KKM-mapping On the contrary, suppose that Q iis not a KKM-mapping Then there exist{y

i , , y n

i } and λ j

i ≥ , j = , , n, withn

j=λ j i=  such that

y=

n



j=

λ j i y j i∈/

n



j=

Q i



y j i

Therefore, we have

ρ i N i



z , η i



f i



y j i

, f i (y) +

g i



f i



y j i

– g i



f i (x i)

+ ρ i



M (u i , v i ) + w i

, η i



f i



y j i

, f i (y)

+ ρ i

φ i

x i , y j i

– φ i (x i , y)

< , ∀i, j and z ∈ A i (y).

Since η i and f i are affine, and N i and φ iare convex in the second argument, we have

 = ρ i N i

z , η i

f i (y), f i (y)

+

g i

f i

y j i

– g i

f i (x i)

+ ρ i

M (u i , v i ) + w i

, η i

f i (y), f i (y)

+ ρ i



φ i (x i , y) – φ i (x i , y)

= ρ i N i



z , η i



j

λ j i f i



y j i

, f i (y)



+



g i



f i



y j i

– g i



f i (x i)

+ ρ i



M (u i , v i ) + w i

,

η i

j

λ j i f i



y j i

, f i (y)



+ ρ i



φ i



x i,

j

λ j i y j i



– φ i (x i , y)



≤ ρ i N i



z,

j

λ j i η i



f i



y j i

, f i (y)  +



g i



f i



y j i

– g i



f i (x i)

+ ρ i



M (u i , v i ) + w i

,



j

λ j i η i

f i

y j i

, f i (y i) 

+ ρ i

j

λ j i φ i

x i , y j i

– φ i (x i , y)



≤λ j i

ρ i N i



z , η i



f i



y j i

, f i (y) 

λ j i

g i



f i



y j i

– g i



f i (x i)

+ ρ i



M (u i , v i)

+ w i

, η i



f i



y j i

, f i (y)

+ ρ i



j

λ j i φ i



x i , y j i

j

λ j i φ i (x i , y)



j

λ j i

ρ i N i



z , η i



f i



y j i

, f i (y) +

g i



f i



y j i

– g i



f i (x i)

+ ρ i



M (u i , v i ) + w i

,

η i



f i



y j i

, f i (y)

+ ρ i



φ i



x i , y j i

– φ i (x i , y)

< ,

which is a contradiction Therefore, y being an arbitrary element of Co {y

i , , y n

i}, we have

y ∈ Co{y

i , , y n

j=Q i (y j i ) Hence Q iis a KKM-mapping

Now, we show that

y i∈Ki Q i (y i) =

y i∈Ki P i (y i ), for every y i ∈ K i Let t i ∈ Q i (y i), therefore

by definition, we have

ρ i N i



z i , η i



f i (y i ), f i (t i)

+

g i



f i (y i)

– g i



f i (x i)

+ ρ i



M (u i , v i ) + w i

, η i



f i (y i ), f i (t i)

+ ρ i



φ i (x i , y i ) – φ i (x i , t i)

≥ , which implies that

g i



f i (y i)

, η i



f i (y i ), f i (t i)

+ ρ i N i



z i , η i



f i (t i ), f i (y)

≥g i



f i (x i)

+ ρ i



M (u i , v i ) + w i

, η i



f i (y i ), f i (t i)

+ ρ i



φ i (x i , y i ) – φ i (x i , t i)

Since g i is ε i -η i -relaxed strongly monotone with respect to f i with the condition ε i = ρ i α i,

inequality () becomes

ρ i N i

z i , η i

f i (y i ), f i (t i)

+

g i

f i (t i)

– g i

f i (x i)

+ ρ i

M (u i , v i ) + w i

, η i

f i (y i ), f i (t i)

+ ρ i



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

≥ ,

and hence we have t i ∈ P i (y i) It follows that

y i∈Ki Q i (y i)⊆y i∈Ki P i (y i)

Conversely , suppose that t i∈y i∈Ki P i (y i), then we have

ρ i N i

z i , η i

f i (y i ), f i (t i)

+

g i

f i (t i)

– g i

f i (x i)

+ ρ i

M (u i , v i ) + w i

, η i

f i (y i ), f i (t i)

+ ρ i

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

Let y λ

i = λ i t i + ( – λ i )y i , λ i ∈ [, ] Since K i is convex, we have y λ

i ∈ K i It follows from () that

ρ i N i



z i , η i



f i (y i ), f i



y λ i +

g i



f i



y λ i

– g i



f i (x i)

+ ρ i



M (u i , v i ) + w i

, η i



f i (y i ), f i



y λ i

+ ρ i



φ i (x i , y i ) – φ i



x i , y λ i

+ α i f i (y i ) – f i



y λ i 

Since η i is affine with the condition η i (f i (y i ), f i (y i )) = , f i is affine, and N i and φ iare convex

in the second argument, inequality () reduces to

λ i ρ i N i



z i , η i



f i (y i ), f i (t i)

+ λ i

g i



f i



y λ i

– g i



f i (x i)

+ ρ i



M (u i , v i ) + w i

, η i



f i (y i ), f i (t i)

+ λ ρ

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

≥ ,

which implies that

λ i ρ i N i



z i , η i



f i (y i ), f i (t i)

+ λ i

g i



λ i f i (t i ) + ( – λ i )f i (y i)

– g i



f i (x i)

+ ρ i

M (u i , v i ) + w i

, η i

f i (y i ), f i (t i)

+ λ i ρ i

φ i (x i , y i ) – φ i (x i , t i)

+ α i λ i f i (y i ) – f i (t i) 

Dividing () by λ i, we get

ρ i N i



z i , η i



f i (y i ), f i (t i)

+

g i



λ i f i (t i ) + ( – λ i )f i (y i)

– g i



f i (x i)

+ ρ i



M (u i , v i)

+ w i

, η i



f i (y i ), f i (t i)

+ ρ i



φ i (x i , y i ) – φ i (x i , t i ) + α i λ i f i (y i ) – f i (t i) 

≥ 

Since g i is hemicontinuous with respect to f i and taking λ i→ , it implies that

ρ i N i



z i , η i



f i (y i ), f i (t i)

+

g i



f i (y i)

– g i



f i (x i)

+ ρ i



M (u i , v i ) + w i

, η i



f i (y i ), f i (t i)

+ ρ i



φ i (x i , y i ) – φ i (x i , t i)

≥ 

Therefore, we have t i ∈ Q i (y i), and we conclude that

y i∈Ki Q i (y i) =

y i∈Ki P i (y i ) and P iis

also a KKM-mapping, for each y i ∈ K i

Since η i is continuous in the second argument, f i and φ i are continuous and N iis upper

semicontinuous, it follows that P i (y i ) is closed for each y i ∈ K i

Finally, we show that, for t

i ∈ D i ∩ K i , P i (t

i) is compact For this purpose, suppose that

there exists ˜t i ∈ P i (t

i ) such that ˜t i ∈ D Therefore, for ˜z/ i ∈ A i (˜t i), we have

ρ i N i

˜z i , η i

f i

t i

, f i ( ˜t i) +

g i

f i

t i

– g i

f i (x i)

+ ρ i

M i (u i , v i ) + w i

, η i

f i

t i

, f i ( ˜t i)

+ ρ i



φ i

x i , ti

– φ i (x i , ˜t i ) + α i f i



t i

– f i ( ˜t i) 

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

ρ i N i



˜z i , η i



f i



t i

, f i ( ˜t i) +

g i



f i



t i

– g i



f i (x i)

+ ρ i



M i (u i , v i ) + w i

, η i



f i



t i

, f i ( ˜t i)

+ ρ i



φ i

x i , ti

– φ i (x i , ˜t i)

+ ρ i α i f i



ti

– f i ( ˜t i) < ,

which is a contradiction to () Therefore Q i (t i)⊂ D Due to compactness of D, and closedness of P i (t

i ), we conclude that P i (t

i) is compact

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



y i∈Ki

P i (y i)= 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 each i ∈ I,

we have

ρ i N i

˜z i , η i

f i (y i ), f i ( ˜t i)

+

g i

f i ( ˜t i)

– g i

f i (x i)

+ ρ i

M i (u i , v i ) + w i

, η i

f i (y i ), f i ( ˜t i)

+ ρ

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

ρ i N i

z i , η i

f i (y i ), f i (t i)

+

g i

f i (t i)

– g i

f i (x i)

+ ρ i

M i (u i , v i ) + w i

, η i

f i (y i ), f i (t i)

+ ρ i

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

Putting y i = t i in () and y i = ˜t iin (), summing up the resulting inequalities and using

the condition η i (f i (x i ), f i (y i )) + η i (f i (y i ), f i (x i)) = , we have

ρ i



N i



˜z i , η i



f i (t i ), f i ( ˜t i)

+ N i



z i , η i



f i ( ˜t i ), f i (t i) 

+

g i



f i ( ˜t i)

– g i



f i (t i)

, η i



f i (t i ), f i ( ˜t i)

Since N i is strongly  i -η i -f i -strongly monotone with respect to A i , g i is ε i -η i-relaxed

strongly monotone with respect to f i with the condition ε i = α i ρ i, we have from ()

–ρ i  i f i (t i ) – f i ( ˜t i) + ρ i α i f i (t i ) – f i ( ˜t i) 

≥ ρ i



N i



˜z i , η i



f i (t i ), f i ( ˜t i)

+ N i



z i , η i



f i ( ˜t i ), f i (t i) 

+ ρ i α i f i (t i ) – f i ( ˜t i) 

≥g i

f i ( ˜t i)

– g i

f i (t i)

, η i

f i (t i ), f i ( ˜t i)

≥ –ε i f i (t i ) – f i ( ˜t i) , which implies that

(–ρ i  i + ε i) f i (t i ) – f i ( ˜t i) ≥ 

Since f i is β i -expansive and ε i < ρ i  i, we obtain

≤ (–ρ i  i + ε i) f i (t i ) – f i ( ˜t i) ≤ (–ρ i  i + ε i )β it i – ˜t i< ,

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

) and z

 ∈ A(x

), z

 ∈ A(x

), compute the iterative sequences{(x n

, x n

)} ⊆ K× K,{u n

i }, {v n

i }, {w n

i } and {z n

i} by the following iter-ative schemes:

ρN



z n+, η



f(y), f



x n+

+

g



f



x n+

– g



f



x n

+ ρ



M



u n, v n

+ w n

, η

f(y), f

x n+

+ ρ

φ

x n, y

– φ

x n, x n+

+ α f(y) – f



x n+ 

ρN



z n+, η



f(y), f



x n+

+

g



f



x n+

– g



f



x n

+ ρ



M



u n, v n

+ w n

, η

f(y), f

x n+

+ ρ

φ

x n, y

– φ

x n, x n+

+ α f(y) – f



x n+ 

Iterative Algorithm .... (t, t)∈ K× K is a solution of the perturbed system of auxiliary generalized multi- valued mixed quasi- equilibrium- like. .. 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 K i