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Algorithm of a new variational inclusion problem and strictly pseudononspreading mapping with application

Abstract

The purpose of this research is to modify the variational inclusion problems and prove a strong convergence theorem for finding a common element of the set of fixed points of a κ-strictly pseudononspreading mapping and the set of solutions of a finite family of variational inclusion problems and the set of solutions of a finite family of equilibrium problems in Hilbert space. By using our main result, we prove a strong convergence theorem involving a κ-quasi-strictly pseudo-contractive mapping in Hilbert space. We give a numerical example to support some of our results.

1 Introduction

Throughout this article, let H be a real Hilbert space with inner product , and norm . Let C be a nonempty closed convex subset of H. Let S:CC be a nonlinear mapping. A point uC is called a fixed point of Su=u if Su=u. The set of fixed points of Su=u is denoted by Fix(S):={uC:Su=u}. A mapping Su=u is called nonexpansive if

SuSvuv,u,vC.

In 2008, Kohsaka and Takahashi [1] introduced the nonspreading mapping in Hilbert space H as follows:

2 S u S v 2 S u v 2 + u S v 2 ,u,vC.
(1.1)

It is shown in [2] that (1.1) is equivalent to

S u S v 2 u v 2 +2uSu,vSv,u,vC.
(1.2)

The mapping S:CC is called a κ-strictly pseudononspreading mapping if there exists κ[0,1) such that

S u S v 2 u v 2 +κ ( I S ) u ( I S ) v 2 +2uSu,vSv,u,vC.

This mapping was introduced by Osilike and Isiogugu [3] in 2011. Clearly every nonspreading mapping is a κ-strictly pseudononspreading mapping.

Remark 1.1 Let C be a nonempty closed convex subset of H. Then a mapping S:CC is a κ-strictly pseudononspreading if and only if

1 κ 2 ( I S ) u ( I S ) v 2 ( I S ) u ( I S ) v , u v + ( I S ) u , ( I S ) v ,

for all u,vC.

Proof Let u,vC and Su=u be a κ-strictly pseudononspreading mapping, then there exists κ[0,1) such that

S u S v 2 u v 2 +κ ( I S ) u ( I S ) v 2 +2uSu,vSv.
(1.3)

Since

( I S ) u ( I S ) v 2 = S u S v 2 2SuSv,uv+ u v 2 ,
(1.4)

then we have

S u S v 2 = ( I S ) u ( I S ) v 2 +2SuSv,uv u v 2 .
(1.5)

From (1.3) and (1.5), we have

( I S ) u ( I S ) v 2 + 2 S u S v , u v u v 2 u v 2 + κ ( I S ) u ( I S ) v 2 + 2 u S u , v S v .

It follows that

( 1 κ ) ( I S ) u ( I S ) v 2 2 u v 2 2 S u S v , u v + 2 u S u , v S v = 2 ( I S ) u ( I S ) v , u v + 2 u S u , v S v .

Then

1 κ 2 ( I S ) u ( I S ) v 2 ( I S ) u ( I S ) v , u v + ( I S ) u , ( I S ) v .

On the other hand, let u,vC and

1 κ 2 ( I S ) u ( I S ) v 2 ( I S ) u ( I S ) v , u v + u S u , v S v = u v 2 S u S v , u v + u S u , v S v .

Then

(1κ) ( I S ) u ( I S ) v 2 2 u v 2 2SuSv,uv+2uSu,vSv.

It follows that

2 S u S u , u v 2 u v 2 ( 1 κ ) ( I S ) u ( I S ) v 2 + 2 u S u , v S v .
(1.6)

From (1.4), we have

2SuSv,uv= S u S v 2 + u v 2 ( I S ) u ( I S ) v 2 .
(1.7)

From (1.6) and (1.7), we have

S u S v 2 + u v 2 ( I S ) u ( I S ) v 2 2 u v 2 ( 1 κ ) ( I S ) u ( I S ) v 2 + 2 u S u , v S v .

Then

S u S v 2 u v 2 +κ ( I S ) u ( I S ) v 2 +2 ( I S ) u , ( I S ) v .

 □

Example 1.2 Let S:[1,)[1,) be defined by

Su=sinu,u[1,).

Then Su=u is a κ-strictly pseudononspreading mapping where κ[0,1).

Example 1.3 Let S:[0,)[0,) be defined by

Su= 4 u 2 5 + 4 u ,uC.

Then Su=u is a 23 25 -strictly pseudononspreading mapping.

The mapping A:CH is called α-inverse strongly monotone if there exists a positive real number α such that

AuAv,uvα A u A v 2 ,

for all u,vC.

Let B:HH be a mapping and M:H 2 H be a multi-valued mapping. The variational inclusion problem is to find xH such that

θBx+Mx,
(1.8)

where θ is a zero vector in H. The set of the solution of (1.8) is denoted by VI(H,B,M). It is well known that the variational inclusion problems are widely studied in mathematical programming, complementarity problems, variational inequalities, optimal control, mathematical economics, and game theory, etc. Many authors have increasingly investigated such a problem (1.8); see for instance [47] and references therein.

Let M:H 2 H be a multi-valued maximal monotone mapping, then the single-valued mapping J M , λ :HH defined by

J M , λ (x)= ( I + λ M ) 1 (x),xH,

is called the resolvent operator associated with M, where λ is any positive number and I is an identity mapping; see [7].

Let Ψ:C×CR be a bifunction. The equilibrium problem for Ψ is to determine its equilibrium point. The set of solution of equilibrium problem is denoted by

EP(Ψ)= { u C : Ψ ( u , v ) 0 , v C } .
(1.9)

Finding a solution of an equilibrium problem can be applied to many problems in physics, optimization, and economics. Many researchers have proposed some methods to solve the equilibrium problem; see, for example, [8, 9] and the references therein.

In 2008, Zhang et al. [7] introduced an iterative scheme for finding a common element of the set of solutions of the variational inclusion problem with multi-valued maximal monotone mapping and inverse strongly monotone mappings and the set of fixed points of nonexpansive mappings in Hilbert space as follows:

{ v n = J M , λ ( w n λ A w n ) , w n + 1 = α n w + ( 1 α n ) S v n , n 0 ,

and they proved a strong convergence theorem of the sequence { w n } under suitable conditions of the parameters { α n } and λ.

In 2013, Kangtunyakarn [10] introduced an iterative algorithm for finding a common element of the set of fixed points of a κ-strictly pseudononspreading mapping and the set of solutions of a finite family of variational inequality problems as follows:

w n + 1 = α n u+ β n P C ( I λ n ( I S ) ) w n + γ n S w n ,nN,

and proved a strong convergence theorem of the sequence { w n } under suitable conditions of the parameters { α n }, { β n }, { γ n }, and { λ n }.

Very recently, Suwannaut and Kangtunyakarn [11] have modified (1.9) as follows:

EP ( i = 1 N a i Ψ i ) = { u C : ( i = 1 N a i Ψ i ) ( u , v ) 0 , v C } ,
(1.10)

where Ψ i :C×CR is for bifunctions and a i >0 with i = 1 N a i =1 for every i=1,2,,N. It is obvious that (1.10) reduces to (1.9), if Ψ i =Ψ, for all i=1,2,,N. They also introduced an iterative method for finding a common element of the set of fixed points of an infinite family of κ i -strictly pseudo-contractive mappings and the set of solutions of a finite family of an equilibrium problem and a variational inequalities problem as follows:

{ i = 1 N a i Ψ i ( z n , y ) + 1 r n y z n , z n w n 0 , y C , w n + 1 = β n ( α n μ + ( 1 α n ) S n w n ) + ( 1 β n ) P C ( I ρ n i = 1 N b i A i ) z n , n 1 .

Under some appropriate conditions, they proved a strong convergence theorem of the sequence { w n } converging to an element of a set i = 1 N EP( Ψ i ) i = 1 N VP( A i ,C) i = 1 Fix( S i ) where A i is a strongly positive linear bounded operator for every i=1,2,,N.

For i=1,2,,N, let A i :HH be a single-valued mapping and let M:H 2 H be a multi-valued mapping. From the concept of (1.8), we introduce the problem of finding uH such that

θ i = 1 N a i A i u+Mu,
(1.11)

for all a i (0,1) with i = 1 N a i =1 and θ is a zero vector. This problem is called the modified variational inclusion. The set of solutions of (1.11) is denoted by VI(H, i = 1 N a i A i ,M). If A i A for all i=1,2,,N, then (1.11) reduces to (1.8).

In this paper, motivated by the research described above, we prove fixed point theory involving the modified variational inclusion and introduce iterative scheme for finding a common element of the set of fixed points of a κ-strictly pseudononspreading mapping and the set of solutions of a finite family of variational inclusion problems and the set of solutions of a finite family of equilibrium problem. By using the same method as our main theorem, we prove a strong convergence theorem for finding a common element of the set of fixed points of a κ-quasi-strictly pseudo-contractive mapping and the set of solutions of a finite family of variational inclusion problems and the set of solutions of a finite family of equilibrium problem in Hilbert space. Applying such a problem, we have a convergence theorem associated with a nonspreading mapping. In the last section, we also give numerical examples to support some of our results.

2 Preliminaries

In this paper, we denote weak and strong convergence by the notations ‘’ and ‘→’, respectively. In a real Hilbert space H, recall that the (nearest point) projection P C from H onto C assigns to each uH the unique point P C uC satisfying the property

u P C v= min v C uv.

For a proof of the main theorem, we will use the following lemmas.

Lemma 2.1 ([12])

Given uH and vC, then P C u=v if and only if we have the inequality

uv,vz0,zC.

Lemma 2.2 ([13])

Let { p n } be a sequence of nonnegative real numbers satisfying

p n + 1 (1 a n ) p n + b n ,n0,

where { a n } is a sequence in (0,1) and { b n } is a sequence such that

  1. (1)

    n = 1 a n =,

  2. (2)

    lim sup n b n a n 0 or n = 1 | b n |<.

Then lim n p n =0.

Lemma 2.3 Let H be a real Hilbert space. Then

u + v 2 u 2 +2v,u+v,

for all u,vH.

Lemma 2.4 ([14])

Let H be a Hilbert space. Then for all u,vH and α i [0,1] for i=1,2,,n such that α 0 + α 1 ++ α n =1 the following equality holds:

α 0 w 0 + α 1 w 1 + + α n w n 2 = i = 0 n α i w i 2 0 i , j n α i α j w i w j 2 .

For finding solutions of the equilibrium problem, assume a bifunction Ψ:C×CR to satisfy the following conditions:

  1. (A1)

    Ψ(u,v)=0 for all uC;

  2. (A2)

    Ψ is monotone, i.e., Ψ(u,v)+Ψ(v,u)0 for all u,vC;

  3. (A3)

    for each u,v,zC,

    lim t 0 Ψ ( t z + ( 1 t ) u , v ) Ψ(u,v);
  4. (A4)

    for each uC, vΨ(u,v) is convex and lower semicontinuous.

Lemma 2.5 ([15])

Let C be a nonempty closed convex subset of H and let Ψ be a bifunction of C×C into satisfying (A1)-(A4). Let r>0 and uH. Then there exists zC such that

Ψ(z,v)+ 1 r vz,zu0,vC.

Lemma 2.6 ([8])

Assume that Ψ:C×CR satisfies (A1)-(A4). For r>0, define a mapping Θ r :HC as follows:

Θ r (u)= { z C : Ψ ( z , v ) + 1 r v z , z u 0 , v C } ,

for all uH. Then the following hold:

  1. (i)

    Θ r is single-valued;

  2. (ii)

    Θ r is firmly nonexpansive, i.e., for any u,vH,

    Θ r ( u ) Θ r ( v ) 2 Θ r ( u ) Θ r ( v ) , u v ;
  3. (iii)

    Fix( Θ r )=EP(Ψ);

  4. (iv)

    EP(Ψ) is closed and convex.

Lemma 2.7 ([11])

Let C be a nonempty closed convex subset of a real Hilbert space H. For i=1,2,,N, let Ψ i :C×CR be bifunctions satisfying (A1)-(A4) with i = 1 N EP( Ψ i ). Then

EP ( i = 1 N a i Ψ i ) = i = 1 N EP( Ψ i ),

where a i (0,1) for every i=1,2,,N and i = 1 N a i =1.

Remark 2.8 ([11])

From Lemma 2.7,

Fix( Θ r )=EP ( i = 1 N a i Ψ i ) = i = 1 N EP( Ψ i ),

where a i (0,1), for each i=1,2,,N, and i = 1 N a i =1.

Lemma 2.9 Let C be a nonempty closed convex subset of a real Hilbert space H. Let i = 1 N a i Ψ i :C×CR be bifunctions satisfying (A1)-(A4) where a i (0,1), for each i=1,2,,N and i = 1 N a i =1. For every nN, let 0<c r n d with r n r as n. Then Θ r n u Θ r u0 as n for all uH.

Proof For every nN, let 0<c r n d with r n r as n, from which it follows that 0<crd. For every uH, by Lemma 2.6, we have

i = 1 N a i Ψ i ( Θ r n u,v)+ 1 r n v Θ r n u, Θ r n uu0,vC

and

i = 1 N a i Ψ i ( Θ r u,v)+ 1 r v Θ r u, Θ r n uu0,vC.

In particular, we have

i = 1 N a i Ψ i ( Θ r n u, Θ r u)+ 1 r n Θ r u Θ r n u, Θ r n uu0
(2.1)

and

i = 1 N a i Ψ i ( Θ r u, Θ r n u)+ 1 r Θ r n u Θ r u, Θ r uu0.
(2.2)

Summing up (2.1) and (2.2) and using (A2), we have

1 r Θ r n u Θ r u, Θ r uu+ 1 r n Θ r u Θ r n u, Θ r n uu0.

It follows that

Θ r n u Θ r u , Θ r u u r Θ r n u u r n 0.

This implies that

0 Θ r u Θ r n u , Θ r n u u r n r ( Θ r u u ) = Θ r u Θ r n u , Θ r n u Θ r u + ( 1 r n r ) ( Θ r u u ) .

It follows that

Θ r u Θ r n u 2 |1 r n r | Θ r u Θ r n u ( Θ r u + u ) .

Then we have

Θ r u Θ r n u 1 r |r r n |L,

where L=sup{ Θ r u+u}. Since r n r as n, we have

Θ r n u Θ r u0as n.

 □

Remark 2.10 Let S:HH be a κ-strictly pseudononspreading mapping with Fix(S). Define T:HH by Tu:=((1λ)I+λS)u, where λ(0,1κ). Then the following hold:

  1. (i)

    Fix(S)=Fix(T)=Fix(Iλ(IS));

  2. (ii)

    for every uH and vFix(S),

    Tuvuv.

Proof (i) It is easy to see that Fix(S)=Fix(T)=Fix(Iλ(IS)).

(ii) Next, we show that Tuvuv. For every uH and vFix(S), we have

T u v 2 = ( I λ ( I S ) ) u v 2 = ( 1 λ ) ( u v ) + λ ( S u v ) 2 = ( 1 λ ) u v 2 + λ S u v 2 λ ( 1 λ ) S u u 2 ( 1 λ ) u v 2 + λ ( u v 2 + κ ( I S ) u 2 ) λ ( 1 λ ) S u u 2 = u v 2 + κ λ S u u 2 λ ( 1 λ ) S u u 2 = u v 2 + λ ( λ ( 1 κ ) ) S u u 2 u v 2 .

 □

Lemma 2.11 ([7])

uH is a solution of variational inclusion (1.8) if and only if u= J M , λ (uλBu), λ>0, i.e.,

VI(H,B,M)=Fix ( J M , λ ( I λ B ) ) ,λ>0.

Further, if λ(0,2α], then VI(H,B,M) is a closed convex subset in H.

Lemma 2.12 ([7])

The resolvent operator J M , λ associated with M is single-valued, nonexpansive for all λ>0 and 1-inverse strongly monotone.

Lemma 2.13 Let H be a real Hilbert space and let M:H 2 H be a multi-valued maximal monotone mapping. For every i=1,2,,N, let A i :HH be α i -inverse strongly monotone mapping with η= min i = 1 , 2 , , N { α i } and i = 1 N VI(H, A i ,M). Then

VI ( H , i = 1 N a i A i , M ) = i = 1 N VI(H, A i ,M),

where i = 1 N a i =1, and 0< a i <1 for every i=1,2,,N. Moreover, J M , λ (Iλ i = 1 N a i A i ) is a nonexpansive mapping, for all 0<λ<2η.

Proof Clearly i = 1 N VI(H, A i ,M)VI(H, i = 1 N a i A i ,M).

Let u 0 VI(H, i = 1 N a i A i ,M) and let u i = 1 N VI(H, A i ,M). From Lemma 2.11, we have

u 0 Fix ( J M , λ ( I λ i = 1 N a i A i ) ) .

Since i = 1 N VI(H, A i ,M)VI(H, i = 1 N a i A i ,M), we have u VI(H, i = 1 N a i A i ,M). From Lemma 2.11, we have

u Fix ( J M , λ ( I λ i = 1 N a i A i ) ) .

From the nonexpansiveness of J M , λ , we have

u u 0 2 = J M , λ ( I λ i = 1 N a i A i ) u J M , λ ( I λ i = 1 N a i A i ) u 0 2 ( I λ i = 1 N a i A i ) u ( I λ i = 1 N a i A i ) u 0 2 = ( u u 0 ) λ ( i = 1 N a i A i u i = 1 N a i A i u 0 ) 2 u u 0 2 2 λ i = 1 N a i u u 0 , A i u A i u 0 + λ 2 i = 1 N a i A i u A i u 0 2 u u 0 2 2 λ i = 1 N a i α i A i u A i u 0 2 + λ 2 i = 1 N a i A i u A i u 0 2 u u 0 2 2 λ η i = 1 N a i A i u A i u 0 2 + λ 2 i = 1 N a i A i u A i u 0 2 = u u 0 2 + λ i = 1 N a i ( λ 2 η ) A i u A i u 0 2 .
(2.3)

This implies that

λ i = 1 N a i (2ηλ) A i u A i u 0 2 0.

Then

A i u = A i u 0 ,i=1,2,,N.
(2.4)

Since u 0 VI(H, i = 1 N a i A i ,M), we have

θM u 0 + i = 1 N a i A i u 0 .
(2.5)

From u VI(H, i = 1 N a i A i ,M), we have

θM u + i = 1 N a i A i u .
(2.6)

From (2.5) and (2.6), we have

θM u 0 + i = 1 N a i A i u 0 M u i = 1 N a i A i u .
(2.7)

From (2.4) and (2.7), we have

θM u 0 M u .
(2.8)

Since u i = 1 N VI(H, A i ,M) and we have (2.4) and (2.8),

θM u 0 M u +M u + A i u =M u 0 + A i u 0 ,

for all i=1,2,,N. It implies that u 0 i = 1 N VI(H, A i ,M).

Hence

VI ( H , i = 1 N a i A i , M ) i = 1 N VI(H, A i ,M).

Applying (2.3), we can conclude that J M , λ (Iλ i = 1 N a i A i ) is a nonexpansive mapping for all i=1,2,,N. □

3 Main result

Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let M:H 2 H be a multi-valued maximal monotone mapping. For every i=1,2,,N, let Ψ i :C×CR be a bifunction satisfying (A1)-(A4) and A i :HH be α i -inverse strongly monotone mapping with η= min i = 1 , 2 , , N { α i }. Let S:HH be a κ-strictly pseudononspreading mapping. Assume Φ:=Fix(S) i = 1 N EP( Ψ i ) i = 1 N VI(H, A i ,M). Let the sequences { w n } and { z n } be generated by w 1 ,μH and

{ i = 1 N a i Ψ i ( z n , y ) + 1 r n y z n , z n w n 0 , y C , w n + 1 = α n μ + β n w n + γ n J M , λ ( I λ i = 1 N b i A i ) w n w n + 1 = + η n ( I ρ n ( I S ) ) w n + δ n z n , n 1 ,
(3.1)

where { α n },{ β n },{ γ n },{ η n },{ δ n }(0,1) and λ>0 with α n + β n + γ n + η n + δ n =1, 0<α<1, and 0 a i , b i 1, for every i=1,2,,N, r n [c,d](0,1), 0<p β n , γ n , η n , δ n q<1, ρ n (0,1κ) for all n1. Suppose the following conditions hold:

  1. (i)

    lim n α n =0 and n = 1 α n =,

  2. (ii)

    n = 1 ρ n <,

  3. (iii)

    0<λ<2η, where η= min i = 1 , 2 , , N { α i },

  4. (iv)

    i = 1 N a i = i = 1 N b i =1,

  5. (v)

    n = 1 | α n + 1 α n |<, n = 1 | β n + 1 β n |<, n = 1 | γ n + 1 γ n |<, n = 1 | ρ n + 1 ρ n |<, n = 1 | δ n + 1 δ n |<, n = 1 | r n + 1 r n |<.

Then the sequences { w n } and { z n } converge strongly to ω= P Φ μ.

Proof The proof of Theorem 3.1 will be divided into five steps:

Step 1. We show that the sequence { w n } is bounded.

Since i = 1 N a i Ψ i satisfies (A1)-(A4), and

i = 1 N a i Ψ i ( z n ,y)+ 1 r n y z n , z n w n 0,yC,

by Lemma 2.6 and Remark 2.8, we have z n = Θ r n w n and Fix( Θ r n )= i = 1 N EP( Ψ i ).

Let ωΦ. From Lemma 2.11 and Lemma 2.13, we have

ω= J M , λ ( I λ i = 1 N b i A i ) ω.

From the nonexpansiveness of J M , λ (Iλ i = 1 N b i A i ), we have

J M , λ ( I λ i = 1 N b i A i ) w n ω w n ω.
(3.2)

From Remark 2.10, we have

( I ρ n ( I S ) ) w n ω 2 = ( 1 ρ n ) w n + ρ n S w n ω 2 w n ω 2 .
(3.3)

From the definition of w n , (3.2), and (3.3), we have

w n + 1 ω = α n μ + β n w n + γ n J M , λ ( I λ i = 1 N b i A i ) w n + η n ( I ρ n ( I S ) ) w n + δ n z n ω α n μ ω + β n w n ω + γ n J M , λ ( I λ i = 1 N b i A i ) w n ω + η n ( I ρ n ( I S ) ) w n ω + δ n z n ω α n μ ω + ( 1 α n ) w n ω max { μ ω , w 1 ω } = K .

By mathematical induction, we have w n zK, nN. It implies that { w n } is bounded and so is { z n }.

By continuing in the same direction as in Step 1 of Theorem 3.1 in [10], we have

S w n ω 1 + κ 1 κ w n ω.
(3.4)

From (3.4), we can conclude that {S w n } is bounded.

Step 2. Put G= i = 1 N b i A i and P=IS. We will show that lim n w n + 1 w n =0. From the definition of w n , we have

w n + 1 w n = α n μ + β n w n + γ n J M , λ ( I λ G ) w n + η n ( I ρ n P ) w n + δ n z n α n 1 μ β n 1 w n 1 γ n 1 J M , λ ( I λ G ) w n 1 η n 1 ( I ρ n 1 P ) w n 1 δ n 1 z n 1 | α n α n 1 | μ + β n w n w n 1 + | β n β n 1 | w n 1 + γ n J M , λ ( I λ G ) w n J M , λ ( I λ G ) w n 1 + | γ n γ n 1 | J M , λ ( I λ G ) w n 1 + η n ( I ρ n P ) w n ( I ρ n 1 P ) w n 1 + | η n η n 1 | ( 1 ρ n 1 P ) w n 1 + δ n z n z n 1 + | δ n δ n 1 | z n 1 | α n α n 1 | μ + β n w n w n 1 + | β n β n 1 | w n 1 + γ n w n w n 1 + | γ n γ n 1 | J M , λ ( I λ G ) w n 1 + | η n η n 1 | ( I ρ n 1 P ) w n 1 + η n ( w n w n 1 + ρ n P w n P w n 1 + | ρ n ρ n 1 | P w n 1 ) + δ n z n z n 1 + | δ n δ n 1 | z n 1 | α n α n 1 | μ + β n w n w n 1 + | β n β n 1 | w n 1 + γ n w n w n 1 + | γ n γ n 1 | J M , λ ( I λ G ) w n 1 + | η n η n 1 | ( I ρ n 1 P ) w n 1 + η n w n w n 1 + ρ n P w n P w n 1 + | ρ n ρ n 1 | P w n 1 + δ n z n z n 1 + | δ n δ n 1 | z n 1 .
(3.5)

By continuing in the same direction as in Step 2 of Theorem 3.1 in [11], we have

z n z n 1 w n w n 1 + 1 d | r n r n 1 | z n w n .
(3.6)

By substituting (3.6) into (3.5), we obtain

w n + 1 w n | α n α n 1 | μ + β n w n w n 1 + | β n β n 1 | w n 1 + γ n w n w n 1 + | γ n γ n 1 | J M , λ ( I λ G ) w n 1 + | η n η n 1 | ( I ρ n 1 P ) w n 1 + η n w n w n 1 + ρ n P w n P w n 1 + | ρ n ρ n 1 | P w n 1 + δ n z n z n 1 + | δ n δ n 1 | z n 1 | α n α n 1 | μ + β n w n w n 1 + | β n β n 1 | w n 1 + γ n w n w n 1 + | γ n γ n 1 | J M , λ ( I λ G ) w n 1 + | η n η n 1 | ( I ρ n 1 P ) w n 1 + η n w n w n 1 + ρ n P w n P w n 1 + | ρ n ρ n 1 | P w n 1 + δ n ( w n w n 1 + 1 d | r n r n 1 | z n w n ) + | δ n δ n 1 | z n 1 | α n α n 1 | μ + ( 1 α n ) w n w n 1 + | β n β n 1 | w n 1 + | γ n γ n 1 | J M , λ ( I λ G ) w n 1 + | η n η n 1 | ( I ρ n 1 P ) w n 1 + ρ n P w n P w n 1 + | ρ n ρ n 1 | P w n 1 + 1 d | r n r n 1 | z n w n + | δ n δ n 1 | z n 1 .
(3.7)

Applying Lemma 2.2, (3.7), and the conditions (i), (ii), (v), we have

lim n w n + 1 w n =0.
(3.8)

Step 3. We show that lim n z n w n = lim n (I ρ n P) w n w n = lim n J M , λ (IλG) w n w n =0. By the definition of w n , (3.2), and (3.3), we have

w n + 1 ω 2 = α n μ + β n w n + γ n J M , λ ( I λ G ) w n + η n ( I ρ n P ) w n + δ n z n ω 2 α n μ ω 2 + β n w n ω 2 + γ n J M , λ ( I λ G ) w n ω 2 + η n ( I ρ n P ) w n ω 2 + δ n z n ω 2 β n δ n w n z n 2 β n γ n J M , λ ( I λ G ) w n w n 2 = α n μ ω 2 + ( 1 α n ) w n ω 2 β n δ n w n z n 2 β n γ n J M , λ ( I λ G ) w n w n 2 α n μ ω 2 + w n ω 2 β n δ n w n ω n 2 β n γ n J M , λ ( I λ G ) w n w n 2 .

It implies that

β n δ n z n w n 2 α n μ ω 2 + w n ω 2 w n + 1 ω 2 β n γ n J M , λ ( I λ G ) w n w n 2 α n μ ω 2 + ( w n ω + w n + 1 ω ) w n + 1 w n .

From the condition (i) and (3.8), we have

lim n z n w n =0.
(3.9)

By continuing in the same direction as (3.9), we have

lim n J M , λ ( I λ G ) w n w n =0.
(3.10)

From the definition of w n , we have

w n + 1 w n = α n ( μ w n ) + γ n ( J M , λ ( I λ G ) w n w n ) + η n ( ( I ρ n P ) w n w n ) + δ n ( z n w n ) .

From the condition (i), (3.8), (3.9), and (3.10), we have

lim n ( I ρ n ( I S ) ) w n w n =0.
(3.11)

Step 4. We will show that lim sup n μω, w n ω0, where ω= P Φ μ.

To show this, choose a subsequence { w n k } of { w n } such that

lim sup n μω, w n ω= lim k μω, w n k ω.
(3.12)

Without loss of generality, we can assume that w n k ξ as k. From (3.9), we obtain z n k ξ as k.

First, we will show that ξ i = 1 N VI(H, A i ,M). Assume that ξ i = 1 N VI(H, A i ,M). By Lemmas 2.11 and 2.13, i = 1 N VI(H, A i ,M)=Fix( J M , λ ((IλG))). Then ξ J M , λ (IλG)ξ, where G= i = 1 N b i A i . By the nonexpansiveness of J M , λ ((IλG)), (3.10), and Opial’s condition, we obtain

lim k inf w n k ξ < lim k inf w n k J M , λ ( ( I λ G ) ) ξ lim k inf ( w n k J M , λ ( ( I λ G ) ) w n k + J M , λ ( ( I λ G ) ) w n k J M , λ ( ( I λ G ) ) ξ ) lim k inf w n k ξ .

This is a contradiction. Then we have

ξ i = 1 N VI(H, A i ,M).
(3.13)

Next, we will show that ξFix(S). Assume that ξFix(S). From Remark 2.10(i), we get Fix(S)=Fix(I ρ n k (IS)). Then ξ(I ρ n k (IS))ξ. From the condition (ii), (3.11), and Opial’s condition, we obtain

lim k inf w n k ξ < lim k inf w n k ( I ρ n k ( I S ) ) ξ lim k inf ( w n k ( I ρ n k ( I S ) ) w n k + ( I ρ n k ( I S ) ) w n k ( I ρ n k ( I S ) ) ξ ) lim k inf ( w n k ( I ρ n k ( I S ) ) w n k + w n k ξ + ρ n k ( I S ) w n k ( I S ) ξ ) = lim k inf w n k ξ .

This is a contradiction. Then we have

ξFix(S).
(3.14)

Since 0<c r n d, nN, then we have r n k r as k with 0<crd. Applying Lemma 2.9, we have Θ r n k w n k Θ r w n k 0 as k. Next, we will show that ξ i = 1 N EP( Ψ i ). Assume that ξ i = 1 N EP( Ψ i ). From Remark 2.8, we have ξFix( Θ r ). By Opial’s condition and (3.9), we have

lim k inf w n k ξ < lim k inf w n k Θ r ξ lim k inf ( w n k Θ r n k w n k + Θ r n k w n k Θ r w n k + Θ r w n k Θ r ξ ) lim k inf w n k ξ .

This is a contradiction. Then we have

ξ i = 1 N EP( Ψ i ).
(3.15)

From (3.13), (3.14), and (3.15), we can conclude that ξΦ.

Since w n k ξ as k and ξΦ. By (3.12) and Lemma 2.1, we have

lim sup n μ ω , w n ω = lim k μ ω , w n k ω = μ ω , ξ ω 0 .
(3.16)

Step 5. Finally, we will show that lim n w n =ω, where ω= P Φ μ. From the definition of w n , we have

w n + 1 ω 2 = α n μ + β n w n + γ n J M , λ ( I λ G ) w n + η n ( I ρ n P ) w n + δ n z n ω 2 α n ( μ ω ) + β n ( w n ω ) + γ n ( J M , λ ( I λ G ) w n ω ) + η n ( ( I ρ n P ) w n ω ) + δ n ( z n ω ) 2 ( β n w n ω + γ n J M , λ ( I λ G ) w n ω + η n ( I ρ n P ) w n ω + δ n z n ω ) 2 + 2 α n μ ω , w n + 1 ω ( 1 α n ) w n ω 2 + 2 α n μ ω , w n + 1 ω .

From the condition (i), (3.16), and Lemma 2.2, we can conclude that the sequence { w n } converges strongly to ω= P Φ μ. By (3.9), we find that { z n } converges strongly to ω= P Φ μ. This completes the proof. □

As a direct proof of Theorem 3.1, we obtain the following results.

Corollary 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H and let M:H 2 H be a multi-valued maximal monotone mapping. For every i=1,2,,N, let Ψ i :C×CR be a bifunction satisfying (A1)-(A4) and let A:HH be an α-inverse strongly monotone mapping. Let S:HH be a κ-strictly pseudononspreading mapping. Assume Φ:=Fix(S) i = 1 N EP( Ψ i )VI(H,A,M). Let the sequences { w n } and { z n } be generated by w 1 ,μH and

{ i = 1 N a i Ψ i ( z n , y ) + 1 r n y z n , z n w n 0 , y C , w n + 1 = α n μ + β n w n + γ n J M , λ ( I λ A ) w n w n + 1 = + η n ( I ρ n ( I S ) ) w n + δ n z n , n 1 ,
(3.17)

where { α n },{ β n },{ γ n },{ η n },{ δ n }(0,1) and λ>0 with α n + β n + γ n + η n + δ n =1, 0<α<1, and 0 a i 1, for every i=1,2,,N, r n [c,d](0,1), 0<p β n , γ n , η n , δ n q<1, ρ n (0,1κ) for all n1. Suppose the following conditions hold:

  1. (i)

    lim n α n =0 and n = 1 α n =,

  2. (ii)

    n = 1 ρ n <,

  3. (iii)

    0<λ<2α,

  4. (iv)

    i = 1 N a i =1,

  5. (v)

    n = 1 | α n + 1 α n |<, n = 1 | β n + 1 β n |<, n = 1 | γ n + 1 γ n |<, n = 1 | ρ n + 1 ρ n |<, n = 1 | δ n + 1 δ n |<, n = 1 | r n + 1 r n |<.

Then the sequences { w n } and { z n } converge strongly to ω= P Φ μ.

Proof Put A i A for all i=1,2,,N in Theorem 3.1. So, from Theorem 3.1, we obtain the desired result. □

Corollary 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H and let M:H 2 H be a multi-valued maximal monotone mapping. Let Ψ:C×CR be a bifunction satisfying (A1)-(A4). For every i=1,2,,N, A i :HH be α i -inverse strongly monotone mapping with η= min i = 1 , 2 , , N { α i }. Let S:HH be a κ-strictly pseudononspreading mapping. Assume Φ:=Fix(S)EP(F) i = 1 N VI(H, A i ,M). Let the sequences { w n } and { z n } be generated by w 1 ,μH and

{ Ψ ( z n , y ) + 1 r n y z n , z n w n 0 , y C , w n + 1 = α n μ + β n w n + γ n J M , λ ( I λ i = 1 N b i A i ) w n w n + 1 = + η n ( I ρ n ( I S ) ) w n + δ n z n , n 1 ,
(3.18)

where { α n },{ β n },{ γ n },{ η n },{ δ n }(0,1) and λ>0 with α n + β n + γ n + η n + δ n =1, 0<α<1, and 0 b i 1, for every i=1,2,,N, r n [c,d](0,1), 0<p β n , γ n , η n , δ n q<1, ρ n (0,1κ) for all n1. Suppose the following conditions hold:

  1. (i)

    lim n α n =0 and n = 1 α n =,

  2. (ii)

    n = 1 ρ n <,

  3. (iii)

    0<λ<2η, where η= min i = 1 , 2 , 3 , , N { α i },

  4. (iv)

    i = 1 N b i =1,

  5. (v)

    n = 1 | α n + 1 α n |<, n = 1 | β n + 1 β n |<, n = 1 | γ n + 1 γ n |<, n = 1 | ρ n + 1 ρ n |<, n = 1 | δ n + 1 δ n |<, n = 1 | r n + 1 r n |<.

Then the sequences { w n } and { z n } converge strongly to ω= P Φ μ.

Proof Take Ψ= Ψ i , i=1,2,,N. By Theorem 3.1, we obtain the desired conclusion. □

4 Applications

In this section, we utilize our main theorem to prove a strong convergence theorem for finding a common element of the set of fixed points of a κ-quasi-strictly pseudo-contractive mapping and the set of solutions of a finite family of variational inclusion problems and the set of solutions of a finite family of equilibrium problem in Hilbert space. To obtain this result, we recall some definitions, lemmas, and remarks as follows.

Definition 4.1 Let C be a subset of a real Hilbert space H and let S:CC be a mapping. Then Su=u is said to be κ-quasi-strictly pseudo-contractive if there exists a constant κ[0,1) such that

S u p 2 u p 2 +κ u S u 2 ,uC and pFix(S).

Su=u is said to be quasi-nonexpansive if

Supup,uC and pFix(S).

The class of κ-quasi-strictly pseudo-contractions includes the class of quasi-nonexpansive mappings.

Remark 4.1 If S:CC be a κ-strictly pseudononspreading mapping with Fix(S), then Su=u is a κ-quasi-strictly pseudo-contractive mapping.

Example 4.2 Let S:[0,1][0,1] be defined by

Su= 2 u + 1 3 ,for all u[0,1].

Then Su=u is a κ-strictly pseudononspreading mapping where κ[0,1). Since 1Fix(S), Su=u is also κ-quasi-strictly pseudo-contractive mapping.

Next, we give the example to show that the converse of Remark 4.1 is not true.

Example 4.3 Let S:[2,2][2,2] be defined by

Su= 5 3 u,u[2,2].

First, show that Su=u is a κ-quasi-strictly pseudo-contractive mapping for all u[2,2].

Observe that Fix(S)={0}. Let u[2,2], we have

| S u S 0 | 2 =| 5 3 u0 | 2 = 25 9 | u | 2

and

| u 0 | 2 + 1 4 | ( I S ) u | 2 = | u | 2 + 1 4 | u + 5 3 u | 2 = | u | 2 + 1 4 | 8 3 u | 2 = | u | 2 + 64 9 ( 1 4 ) | u | 2 = ( 25 9 ) | u | 2 .

Then Su=u is a 1 4 -quasi-strictly pseudo-contractive mapping. Next, we show that Su=u is not a 1 4 -strictly pseudononspreading mapping.

Choose u= 3 2 and v= 3 2 , we have

| S ( 3 2 ) S ( 3 2 ) | 2 = | 5 3 ( 3 2 ) + 5 3 ( 3 2 ) | 2 | S ( 3 2 ) S ( 3 2 ) | 2 = | 10 2 | 2 | S ( 3 2 ) S ( 3 2 ) | 2 = 25 , | u v | 2 = | 3 2 + 3 2 | 2 = 9 , 1 4 | ( I S ) ( 3 2 ) ( I S ) ( 3 2 ) | 2 = 1 4 | ( 3 2 ) + 5 3 ( 3 2 ) ( ( 3 2 ) + 5 3 ( 3 2 ) ) | 2 1 4 | ( I S ) ( 3 2 ) ( I S ) ( 3 2 ) | 2 = 1 4 | 8 | 2 1 4 | ( I S ) ( 3 2 ) ( I S ) ( 3 2 ) | 2 = 16

and

2 ( I S ) ( 3 2 ) , ( I S ) ( 3 2 ) = 2 ( 3 2 ) + 5 3 ( 3 2 ) , ( ( 3 2 ) + 5 3 ( 3 2 ) ) = 2 ( 4 ) ( 4 ) = 32 .

Then we have

| S u S v | 2 > | u v | 2 + 1 4 |(IS)u(IS)v | 2 +2uSu,vSv.

By changing Su=u from being a κ-strictly pseudononspreading mapping with Fix(S) into a κ-quasi-strictly pseudo-contractive mapping, we obtain the same result as in Remark 2.10.

Remark 4.4 Let S:HH be a κ-quasi-strictly pseudo-contractive mapping with Fix(S). Define T:HH by Tu:=((1λ)I+λS)u, where λ(0,1κ). Then the following hold:

  1. (i)

    Fix(S)=Fix(T)=Fix(Iλ(IS));

  2. (ii)

    for every uH and vFix(S),

    Tuvuv.

In 2009, Kangtunyakarn and Suantai [16] introduced the S-mapping generated by S 1 , S 2 ,, S N and α 1 , α 2 ,, α N as follows.

Definition 4.2 ([16])

Let C be a nonempty convex subset of a real Banach space. Let { S i } i = 1 N be a finite family of (nonexpansive) mappings of C into itself. For each j=1,2, , let α j =( α 1 j , α 2 j , α 3 j )I×I×I where I=[0,1] and α 1 j + α 2 j + α 3 j =1. Define the mapping S:CC as follows:

U 0 = I , U 1 = α 1 1 S 1 U 0 + α 2 1 U 0 + α 3 1 I , U 2 = α 1 2 S 2 U 1 + α 2 2 U 1 + α 3 2 I , U 3 = α 1 3 S 3 U 2 + α 2 3 U 2 + α 3 3 I , U n 1 = α 1 N 1 S N 1 U N 2 + α 2 N 1 U N 2 + α 3 N 1 I , S = U n = α 1 N S N U n 1 + α 2 N U n 1 + α 3 N I .

This mapping is called the S-mapping generated by S 1 , S 2 ,, S N and α 1 , α 2 ,, α N .

Lemma 4.5 ([17])

Let C be a nonempty closed convex subset of a real Hilbert space H. Let { S i } i = 1 N be a finite family of nonspreading mappings of C into itself with i = 1 N Fix( S i ) and let α j =( α 1 j , α 2 j , α 3 j )I×I×I where I=[0,1], α 1 j + α 2 j + α 3 j =1, α 1 j , α 3 j (0,1) for all j=1,2,,N1 and α 1 N (0,1], α 3 N [0,1), α 2 j (0,1) for all j=1,2,,N. Let S be the S-mapping generated by S 1 , S 2 ,, S N and α 1 , α 2 ,, α N . Then Fix(S)= i = 1 N Fix( S i ) and S is a quasi-nonexpansive mapping.

Remark 4.6 From Lemma 4.5 it still holds if CH.

Theorem 4.7 Let C be a nonempty closed convex subset of a real Hilbert space H. Let M:H 2 H be a multi-valued maximal monotone mapping. For every i=1,2,,N, let Ψ i :C×CR be a bifunction satisfying (A1)-(A4) and A i :HH be α i -inverse strongly monotone mapping with η= min i = 1 , 2 , , N { α i }. Let S:HH be a κ-quasi-strictly pseudo-contractive mapping. Assume Φ:=Fix(S) i = 1 N EP( Ψ i ) i = 1 N VI(H, A i ,M). Let the sequences { w n } and { z n } be generated by w 1 ,μH and

{ i = 1 N a i Ψ i ( z n , y ) + 1 r n y z n , z n w n 0 , y C , w n + 1 = α n μ + β n w n + γ n J M , λ ( I λ i = 1 N b i A i ) w n w n + 1 = + η n ( I ρ n ( I S ) ) w n + δ n z n , n 1 ,
(4.1)

where { α n },{ β n },{ γ n },{ η n },{ δ n }(0,1), and λ>0 with α n + β n + γ n + η n + δ n =1, 0<α<1 and 0 a i , b i 1, for every i=1,2,,N, r n [c,d](0,1), 0<p β n , γ n , η n , δ n q<1, ρ n (0,1κ) for all n1. Suppose the following conditions hold:

  1. (i)

    lim n α n =0 and n = 1 α n =,

  2. (ii)

    n = 1 ρ n <,

  3. (iii)

    0<λ<2η, where η= min i = 1 , 2 , , N { α i },

  4. (iv)

    i = 1 N a i = i = 1 N b i =1,

  5. (v)

    n = 1 | α n + 1 α n |<, n = 1 | β n + 1 β n |<, n = 1 | γ n + 1 γ n |<, n = 1 | ρ n + 1 ρ n |<, n = 1 | δ n + 1 δ n |<, n = 1 | r n + 1 r n |<.

Then the sequences { w n } and { z n } converge strongly to ω= P Φ μ.

Proof Using Remark 4.4 and the same method of proof in Theorem 3.1, we have the desired conclusion. □

Theorem 4.8 Let C be a nonempty closed convex subset of a real Hilbert space H. Let M:H 2 H be a multi-valued maximal monotone mapping. For every i=1,2,,N, let Ψ i :C×CR be a bifunction satisfying (A1)-(A4), and let A i :HH be α i -inverse strongly monotone mapping with η= min i = 1 , 2 , , N { α i }. Let S i :HH, for i=1,2,,N be a finite family of nonspreading mappings with Ψ:= i = 1 N Fix( S i ) i = 1 N EP( Ψ i ) i = 1 N VI(H, A i ,M). Let θ j =( α 1 j , α 2 j , α 3 j )I×I×I, j=1,2,,N, where I=[0,1], α 1 j + α 2 j + α 3 j =1, α 1 j , α 3 j (0,1) for all j=1,2,,N1, and α 1 N (0,1], α 3 N [0,1), α 2 j (0,1) for all j=1,2,,N, and let S be the S-mapping generated by S 1 , S 2 ,, S N and θ 1 , θ 2 ,, θ N . Let the sequences { w n } and { z n } be generated by w 1 ,μH and

{ i = 1 N a i Ψ i ( z n , y ) + 1 r n y z n , z n w n 0 , y C , w n + 1 = α n μ + β n w n + γ n J M , λ ( I λ i = 1 N b i A i ) w n w n + 1 = + η n ( I ρ n ( I S ) ) w n + δ n z n , n 1 ,
(4.2)

where { α n },{ β n },{ γ n },{ η n },{ δ n }(0,1) and λ>0 with α n + β n + γ n + η n + δ n =1, 0<α<1 and 0 a i , b i 1, for every i=1,2,,N, r n [c,d](0,1), 0<p β n , γ n , η n , δ n q<1, ρ n (0,1) for all n1. Suppose the following conditions hold:

  1. (i)

    lim n α n =0 and n = 1 α n =,

  2. (ii)

    n = 1 ρ n <,

  3. (iii)

    0<λ<2η, where η= min i = 1 , 2 , , N { α i },

  4. (iv)

    i = 1 N a i = i = 1 N b i =1,

  5. (v)

    n = 1 | α n + 1 α n |<, n = 1 | β n + 1 β n |<, n = 1 | γ n + 1 γ n |<, n = 1 | ρ n + 1 ρ n |<, n = 1 | δ n + 1 δ n |<, n = 1 | r n + 1 r n |<.

Then the sequences { w n } and { z n } converge strongly to ω= P Φ μ.

Proof From Theorem 4.7 and Remark 4.6, we obtain the desired conclusion. □

5 Numerical results

The purpose of this section we give a numerical example to support our some result. The following example is given for supporting Theorem 3.1.

Example 5.1 Let be the set of real numbers. For every i=1,2,,N, let Ψ i :R×RR, A i :RR be defined by

Ψ i ( u , v ) = i ( v u ) ( 3 u + v ) , A i u = i u 10 ,

for all u,vR and let S:RR be defined by

Su={ 3 u 5 if  u [ 0 , ) , u if  u ( , 0 ) .

For every i=1,2,,N, suppose that J M , λ =I, λ= 1 N , a i = 3 4 i + 1 N 4 N , b i = 8 9 i + 1 N 9 N . Let { w n } and { z n } be generated by (3.1), where α n = 1 20 n , β n = 3 ( 20 n 1 ) 220 n , γ n = 2 ( 20 n 1 ) 220 n , η n = 5 ( 20 n 1 ) 220 n , δ n = 20 n 1 220 n , r n = 3 n 5 n + 6 , and ρ n = 1 2 n 2 for every nN. Then the sequences { w n } and { z n } converge strongly to 0.

Solution. It is easy to see that Su=u is a κ-strictly pseudononspreading mapping. Since a i = 3 4 i + 1 N 4 N , we obtain

i = 1 N a i Ψ i (u,v)= i = 1 N ( 3 4 i + 1 N 4 N ) i(vu)(v+3u).

It is easy to check that Ψ i satisfies all the conditions of Theorem 3.1 and EP( i = 1 N a i Ψ i )= i = 1 N EP( Ψ i )={0}. Then we have

Fix(S) i = 1 N EP( Ψ i )={0}.
(5.1)

Put S 1 = i = 1 N ( 3 4 i + 1 N 4 N )i, then we have

0 i = 1 N a i Ψ i ( z n , y ) + 1 r n y z n , z n w n 0 = S 1 ( y z n ) ( y + 3 z n ) + 1 r n ( y z n ) ( z n w n ) 0 S 1 r n ( y z n ) ( y + 3 z n ) + ( y z n ) ( z n w n ) 0 = S 1 r n y 2 + ( z n + 2 r n S 1 z n w n ) y + z n w n 3 r n S 1 z n 2 z n 2 .

Let G(y)= S 1 r n y 2 +( z n +2 r n S 1 z n w n )y+ z n w n 3 r n S 1 z n 2 z n 2 . G(y) is a quadratic function of y with coefficient a= S 1 r n , b= z n +2 r n S 1 z n w n , and c= z n w n 3 r n S 1 z n 2 z n 2 . Determine the discriminant Δ of G as follows:

Δ = b 2 4 a c = ( z n + 2 r n S 1 z n w n ) 2 4 ( S 1 r n ) ( z n w n 3 r n S 1 z n 2 z n 2 ) = z n 2 + 8 r n S 1 z n 2 + 16 r n 2 S 1 2 z n 2 2 z n w n 8 r n S 1 z n w n + w n 2 = ( z n + 4 S 1 r n z n w n ) 2 .

We know that G(y)0, yR. If it has at most one solution in , then Δ0, so we obtain

z n = w n 1 + 4 S 1 r n ,
(5.2)

where S 1 = i = 1 N ( 3 4 i + 1 N 4 N )i.

Since A i u= i u 10 and b i = 8 9 i + 1 N 9 N ,

i = 1 N b i A i u= i = 1 N ( 8 9 i + 1 N 9 N ) i u 10 .

From (5.1) and the definition of A i , we have

Fix(S) i = 1 N EP( Ψ i ) i = 1 N VI(H, A i ,M)={0}.
(5.3)

For every nN, α n = 1 20 n , β n = 3 ( 20 n 1 ) 220 n , γ n = 2 ( 20 n 1 ) 220 n , η n = 20 n 1 220 n , δ n = 5 ( 20 n 1 ) 220 n , r n = 3 n 5 n + 6 , and ρ n = 1 2 n 2 . Then the sequences { α n }, { β n }, { γ n },