Open Access

Weak convergence theorems of a hybrid algorithm in Hilbert spaces

Journal of Inequalities and Applications20142014:378

https://doi.org/10.1186/1029-242X-2014-378

Received: 4 August 2014

Accepted: 11 September 2014

Published: 29 September 2014

Abstract

In this paper, a hybrid algorithm is investigated for solving common solutions of a generalized equilibrium problem, a variational inequality, and fixed point problems of an asymptotically strict pseudocontraction. Weak convergence theorems are established in the framework of real Hilbert spaces.

Keywords

equilibrium problem variational inequality nonexpansive mapping common solution

1 Introduction

Monotone variational inequalities recently have been investigated as an effective and powerful tool for studying a wide class of real world problems which arise in economics, finance, image reconstruction, ecology, transportation, and network; see [19] and the references therein. Monotone variational inequalities, which include many important problems in nonlinear analysis and optimization, such as the Nash equilibrium problem, complementarity problems, fixed point problems, saddle point problems, and game theory recently have been extensively studied based on projection methods. Many well-known problems can be studied by using methods which are iterative in their nature. As an example, in computer tomography with limited data, each piece of information implies the existence of a convex set in which the required solution lies. The problem of finding a point in the intersection of these convex subsets is then of crucial interest, and it cannot be usually solved directly. Therefore, an iterative algorithm must be used to approximate such a point. Krasnoselskii-Mann iteration, which is also known as a one-step iteration, is a classic algorithm to study fixed points of nonlinear operators. However, Krasnoselskii-Mann iteration only enjoys weak convergence for nonexpansive mappings; see [10] and the references therein.

The purposes of this paper is to study common solutions of a generalized equilibrium problem, a variational inequality, and fixed point problems of an asymptotically strict pseudocontraction based on a hybrid algorithm. Weak convergence theorems are established in the framework of real Hilbert spaces. The organization of this paper is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, a hybrid algorithm is introduced and the convergence analysis is given. Weak convergence theorems are established in a real Hilbert space.

2 Preliminaries

From now on, we always assume that H is a real Hilbert space with the inner product , and the norm , C is a nonempty closed convex subset of H and P C denotes the metric projection from H onto C.

Let A : C H be a mapping. Recall that A is said to be monotone if
A x A y , x y 0 , x , y C .
A is said to be inverse-strongly monotone if there exists a constant α > 0 such that
A x A y , x y α A x A y 2 , x , y C .

For such a case, we also call it an α-inverse-strongly monotone mapping.

A set-valued mapping T : H 2 H is said to be monotone if for all x , y H , f T x and g T y imply x y , f g > 0 . A monotone mapping T : H 2 H is maximal if the graph G ( T ) of T is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping T is maximal if and only if, for any ( x , f ) H × H , x y , f g 0 for all ( y , g ) G ( T ) implies f T x . Let A be a monotone mapping of C into H and N C v be the normal cone to C at v C , i.e.,
N C v = { w H : v u , w 0 , u C }
and define a mapping T on C by
T v = { A v + N C v , v C , , v C .

Then T is maximal monotone and 0 T v if and only if A v , u v 0 for all u C ; see [6] and the references therein.

Recall that the classical variational inequality problem is to find x C such that
A x , y x 0 , y C .
(2.1)

It is known that x C is a solution to (2.1) if and only if x is a fixed point of the mapping P C ( I λ A ) , where λ > 0 is a constant and I is the identity mapping. Projection methods recently have been studied for variational inequality (2.1); see [1122] and the references therein.

Let S : C C be a nonlinear mapping. In this paper, we use F ( S ) to denote the fixed point set of S. Recall that S is said to be nonexpansive if
S x S y x y , x , y C .
S is said to be asymptotically nonexpansive if there exists a sequence { k n } [ 1 , ) with lim n k n = 1 such that
S x S y k n x y , x , y C .
S is said to be κ-strictly pseudocontractive if there exists a constant k [ 0 , 1 ) such that
S x S y 2 x y 2 + κ ( x S x ) ( y S y ) 2 , x , y C .

The class of strict pseudocontractions was introduced by Browder and Petryshyn [23]. It is clear that every nonexpansive mapping is a 0-strict pseudocontraction.

T is said to be an asymptotically κ-strict pseudocontraction if there exists a sequence { k n } [ 1 , ) with lim n k n = 1 and a constant κ [ 0 , 1 ) such that
T n x T n y 2 k n x y 2 + κ ( I T n ) x ( I T n ) y 2 , x , y C , n 1 .

The class of asymptotically strict pseudocontractions was introduced by Qihou [24]. It is clear that every asymptotically nonexpansive mapping is an asymptotically 0-strict pseudocontraction.

Let F be a bifunction of C × C into , where denotes the set of real numbers and A : C H is an inverse-strongly monotone mapping. In this paper, we consider the following generalized equilibrium problem:
Find  x C  such that  F ( x , y ) + A x , y x 0 , y C .
(2.2)
In this paper, the set of such x C is denoted by EP ( F , A ) , i.e.,
EP ( F , A ) = { x C : F ( x , y ) + A x , y x 0 , y C } .
To study the generalized equilibrium problem (2.2), we may assume that F satisfies the following conditions:
  1. (A1)

    F ( x , x ) = 0 for all x C ;

     
  2. (A2)

    F is monotone, i.e., F ( x , y ) + F ( y , x ) 0 for all x , y C ;

     
  3. (A3)
    for each x , y , z C ,
    lim sup t 0 F ( t z + ( 1 t ) x , y ) F ( x , y ) ;
     
  4. (A4)

    for each x C , y F ( x , y ) is convex and lower semi-continuous.

     
If A 0 , then the generalized equilibrium problem (2.2) is reduced to the following equilibrium problem:
Find  x C  such that  F ( x , y ) 0 , y C .
(2.3)
In this paper, the set of such x C is denoted by EP ( F ) , i.e.,
EP ( F ) = { x C : F ( x , y ) 0 , y C } .

If F 0 , then the generalized equilibrium problem (2.2) is reduced to the classical variational inequality (2.1).

Recently, equilibrium problems (2.2) and (2.3) have been investigated by many authors; see [2531] and the references therein. Motivated by the research going on in this direction, we study a hybrid algorithm for solving common solutions of variational inequality (2.1), generalized equilibrium problem (2.2), and fixed points of an asymptotically strict pseudocontraction. Possible computation errors are taken into account. Weak convergence theorems are established in the framework of real Hilbert spaces.

In order to prove our main results, we also need the following lemmas.

Lemma 2.1 [32]

Let C be a nonempty closed convex subset of H, and let F : C × C R be a bifunction satisfying (A1)-(A4). Then, for any r > 0 and x H , there exists z C such that
F ( z , y ) + 1 r y z , z x 0 , y C .
Further, define
T r x = { z C : F ( z , y ) + 1 r y z , z x 0 , y C }
for all r > 0 and x H . Then the following hold:
  1. (a)

    T r is single-valued;

     
  2. (b)
    T r is firmly nonexpansive, i.e., for any x , y H ,
    T r x T r y 2 T r x T r y , x y ;
     
  3. (c)

    F ( T r ) = EP ( F ) ;

     
  4. (d)

    EP ( F ) is closed and convex.

     

Lemma 2.2 [24]

Let C be a nonempty closed convex subset of a Hilbert space H and S : C C be an asymptotically strict pseudocontraction. Then I S is demi-closed, that is, if { x n } is a sequence in C with x n x and x n S x n 0 , then x F ( S ) .

Lemma 2.3 [33]

Let H be a Hilbert space and 0 < p t n q < 1 for all n 1 . Suppose that { x n } and { y n } are sequences in H such that
lim sup n x n d , lim sup n y n d
and
lim n t n x n + ( 1 t n ) y n = d

hold for some d 0 . Then lim n x n y n = 0 .

Lemma 2.4 [34]

Let { a n } , { b n } , and { c n } be three nonnegative sequences satisfying the following condition:
a n + 1 ( 1 + b n ) a n + c n , n n 0 ,

where n 0 is some nonnegative integer, n = 1 b n < and n = 1 c n < . Then the limit lim n a n exists.

3 Main results

Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C × C to which satisfies (A1)-(A4). Let A : C H be an α-inverse-strongly monotone mapping, and let B : C H be a β-inverse-strongly monotone mapping. Let S : C C be an asymptotically κ-strict pseudocontraction with the sequence { k n } such that n = 1 ( k n 1 ) < . Assume that Ω = F ( S ) VI ( C , B ) EP ( F , A ) is not empty. Let { α n } , { α n } , { α n } , and { β n } be real number sequences in ( 0 , 1 ) . Let { r n } and { s n } be two positive real number sequences. Let { x n } be a sequence generated in the following process:
{ x 1 C , F ( z n , z ) + A x n , z z n + 1 r n z z n , z n x n 0 , z C , y n = P C ( z n s n B z n ) , x n + 1 = α n x n + α n ( β n y n + ( 1 β n ) S n y n ) + α n e n ,
where { e n } is a bounded sequence in C. Assume that the control sequences satisfy the following restrictions:
  1. (a)

    α n + α n + α n = 1 ;

     
  2. (b)

    0 < p α n q < 1 and n = 1 α n < ;

     
  3. (c)

    0 < κ < β n b < 1 ;

     
  4. (d)

    0 < s s n s < 2 β and 0 < r r n r < 2 α ,

     

where p, q, b, s, s , r, r are real constants. Then { x n } converges weakly to some point in Ω.

Proof First, we show that the sequences { x n } , { y n } , and { z n } are bounded. Let p Ω be fixed arbitrarily. For any x , y C , we see that
( I r n A ) x ( I r n A ) y 2 = ( x y ) r n ( A x A y ) 2 = x y 2 2 r n x y , A x A y + r n 2 A x A y 2 x y 2 r n ( 2 α r n ) A x A y 2 .
(3.1)
Using the restriction (d), we see that ( I r n A ) x ( I r n A ) y x y . This implies that I r n A is nonexpansive. In the same way, we find that I s n B is also nonexpansive. Using the restriction (c), we obtain that
β n y n + ( 1 β n ) S n y n p 2 = β n y n p 2 + ( 1 β n ) S n y n S n p 2 β n ( 1 β n ) ( y n p ) ( S n y n S n p ) 2 β n y n p 2 + ( 1 β n ) ( k n y n p 2 + κ ( y n p ) ( S n y n S n p ) 2 ) β n ( 1 β n ) ( y n p ) ( S n y n S n p ) 2 = k n y n p 2 ( 1 β n ) ( β n κ ) ( y n p ) ( S n y n S n p ) 2 k n y n p 2 .
(3.2)
It follows that
x n + 1 p 2 α n x n p 2 + α n β n y n + ( 1 β n ) S n y n p 2 + α n e n p 2 α n x n p 2 + α n k n y n p 2 + α n e n p 2 = α n x n p 2 + α n k n P C ( I s n B ) z n p 2 + α n e n p 2 α n x n p 2 + α n k n T r n ( I r n A ) x n p 2 + α n e n p 2 k n x n p 2 + α n e n p 2 .
This implies from Lemma 2.4 that lim n x n p exists. This shows that { x n } is bounded, so are { y n } and { z n } . From (3.2), we have
x n + 1 p 2 α n x n p 2 + α n k n y n p 2 + α n e n p 2 α n x n p 2 + α n k n ( I s n B ) z n p 2 + α n e n p 2 α n x n p 2 + α n k n ( z n p 2 s n ( 2 β s n ) B z n B p 2 ) + α n e n p 2 k n x n p 2 s n k n α n ( 2 β s n ) B z n B p 2 + α n e n p 2 .
It follows that
s n k n α n ( 2 β s n ) B z n B p 2 k n x n p 2 x n + 1 p 2 + α n e n p 2 .
With the aid of the restrictions (b) and (d), we find that
lim n B z n B p = 0 .
(3.3)
Since P C is firmly nonexpansive, we have
y n p 2 = P C ( I s n B ) z n P C ( I s n B ) p 2 ( I s n B ) z n ( I s n B ) p , y n p = 1 2 { ( I s n B ) z n ( I s n B ) p 2 + y n p 2 ( I s n B ) z n ( I s n B ) p ( y n p ) 2 } 1 2 { z n p 2 + y n p 2 z n y n s n ( B z n B p ) 2 } = 1 2 { z n p 2 + y n p 2 z n y n 2 + 2 s n z n y n , B z n B p s n 2 B z n B p 2 } 1 2 { x n p 2 + y n p 2 z n y n 2 + 2 s n z n y n , B z n B p s n 2 B z n B p 2 } ,
which implies that
y n p 2 x n p 2 z n y n 2 + 2 s n z n y n B z n B p .
Hence, we find from (3.2) that
x n + 1 p 2 α n x n p 2 + α n k n y n p 2 + α n e n p 2 k n x n p 2 α n k n z n y n 2 + 2 α n s n k n z n y n B z n B p + α n e n p 2 .
Therefore, we obtain that
α n k n z n y n 2 k n x n p 2 x n + 1 p 2 + 2 s n k n z n y n B z n B p + α n e n p 2 .
From the restrictions (b) and (d), we find from (3.3) that
lim n z n y n = 0 .
(3.4)
It follows from (3.1) that
z n p 2 = T r n ( I r n A ) x n p 2 x n p 2 r n ( 2 α r n ) A x n A p 2 .
Hence, we have
x n + 1 p 2 α n x n p 2 + α n k n y n p 2 + α n e n p 2 α n x n p 2 + α n k n z n p 2 + α n e n p 2 k n x n p 2 α n r n ( 2 α r n ) k n A x n A p 2 + α n e n p 2 .
This implies that
α n r n ( 2 α r n ) k n A x n A p 2 k n x n p 2 x n + 1 p 2 + α n e n p 2 .
Using the restrictions (b) and (d), we obtain that
lim n A x n A p = 0 .
(3.5)
Since T r n is firmly nonexpansive, we find that
z n p 2 = T r n ( I r n A ) x n T r n ( I r n A ) p 2 ( I r n A ) x n ( I r n A ) p , z n p = 1 2 ( ( I r n A ) x n ( I r n A ) p 2 + z n p 2 ( I r n A ) x n ( I r n A ) p ( z n p ) 2 ) 1 2 ( x n p 2 + z n p 2 x n z n r n ( A x n A p ) 2 ) = 1 2 ( x n p 2 + z n p 2 ( x n z n 2 2 r n x n z n , A x n A p r n 2 A x n A p 2 ) ) ,
which implies that
z n p 2 x n p 2 x n z n 2 + 2 r n x n z n A x n A p .
It follows that
x n + 1 p 2 α n x n p 2 + α n k n y n p 2 + α n e n p 2 α n x n p 2 + α n k n z n p 2 + α n e n p 2 k n x n p 2 α n k n x n z n 2 + 2 r n α n k n x n z n A x n A p + α n e n p 2 ,
which yields that
α n k n x n z n 2 k n x n p 2 x n + 1 p 2 + 2 r n α n x n z n A x n A p + α n e n p 2 .
Using the restrictions (b) and (d), we find from (3.5) that
lim n x n z n = 0 .
(3.6)
It follows from (3.4) and (3.6) that
lim n x n y n = 0 .
(3.7)
Since { x n } is bounded, we see that there exists a subsequence { x n i } of { x n } which converges weakly to ξ. Let T be a maximal monotone mapping defined by
T x = { B x + N C x , x C , , x C .
For any given ( x , y ) Graph ( T ) , we have y B x N C x . Since y n C , by the definition of N C , we have x y n , y B x 0 . Since y n = P C ( I s n B ) z n , we see that x y n , y n ( I s n B ) z n 0 and hence
x y n , y n z n s n + B z n 0 .
It follows that
x y n i , y x y n i , B x x y n i , B x x y n i , y n i z n i s n i + B z n i = x y n i , B x B y n i + x y n i , B y n i B z n i x y n i , y n i z n i s n i x y n i , B y n i B z n i x y n i , y n i z n i s n i .
Since y n i converges weakly to ξ and B is 1 β -Lipschitz continuous, we see that x ξ , y 0 . Notice that T is maximal monotone and hence 0 T ξ . This shows that ξ VI ( C , B ) . From (3.6), we see that z n i converges weakly to ξ. It follows that
F ( z n , z ) + A x n , z z n + 1 r n z z n , z n x n 0 , z C .
From condition (A2), we see that
A x n , z z n + 1 r n z z n , z n x n F ( z , z n ) , z C .
Replacing n by n i , we arrive at
A x n i , z z n i + z z n i , z n i x n i r n i F ( z , z n i ) , z C .
(3.8)
For t with 0 < t 1 and z C , let u t = t z + ( 1 t ) ξ . Since z C and ξ C , we have u t C . In view of (3.8), we find that
u t z n i , A u t u t z n i , A u t A x n i , u t z n i u t z n i , z n i x n i r n i + F ( u t , z n i ) = u t z n i , A u t A z n i + u t z n i , A z n i A x n i u t z n i , z n i x n i r n i + F ( u t , z n i ) .
Using (3.6), we have lim i A z n i A x n i = 0 . Since A is monotone, we see that u t z n i , A u t A z n i 0 . It follows from condition (A4) that
u t ξ , A u t F ( u t , ξ ) .
(3.9)
Using conditions (A1) and (A4), we see from (3.9) that
0 = F ( u t , u t ) t F ( u t , z ) + ( 1 t ) F ( u t , ξ ) t F ( u t , z ) + ( 1 t ) u t ξ , A u t = t F ( u t , u ) + ( 1 t ) t z ξ , A u t ,
which yields that
F ( u t , z ) + ( 1 t ) z ξ , A u t 0 .
Letting t 0 , we find
F ( ξ , z ) + z ξ , A ξ 0 ,

which implies that ξ EP ( F , A ) .

Now, we are in a position to show ξ F ( S ) . Since lim n x n p exists, we may assume that lim n x n p = d > 0 . Put λ n = β n y n + ( 1 β n ) S n y n . It follows from (3.2) that lim sup n x n p + α n ( e n λ n ) d and lim sup n λ n p + α n ( e n λ n ) d . On the other hand, we have
lim n x n + 1 p = lim n α n ( ( x n p ) + α n ( e n λ n ) ) + ( 1 α n ) ( λ n p + α n ( e n λ n ) ) = d .
Using Lemma 2.3, we obtain that lim n λ n x n = 0 . Note that
S n y n x n = λ n x n 1 β n + β n ( x n y n ) 1 β n .

Hence, we have lim n S n y n x n = 0 . Note that S n x n x n S n x n S n y n + S n y n x n . Since S is Lipschitz continuous, we have lim n S n x n x n = 0 . Further, we find that lim n S x n x n = 0 . Using Lemma 2.2, we see that ξ F ( S ) . This proves that η Ω .

Finally, we show that the sequence { x n } converges weakly to ξ. Assume that there exists another subsequence { x n j } of { x n } such that { x n j } converges weakly to η. In the same way, we find η Ω . If η ξ , we see from the Opial condition [35] that
lim n x n ξ = lim inf i x n i ξ < lim inf i x n i η = lim inf n x n η = lim inf j x n j η < lim inf j x n j ξ = lim n x n ξ .

This derives a contradiction. Hence, we have η = ξ . This implies that x n ξ Ω . This completes the proof. □

Remark 3.2 The key of the weak convergence of the algorithm is due to the fact that A is inverse-strongly monotone, which yields that I r n A is nonexpansive. The nonexpansivity of the mapping I r n A plays an important role in this theorem. Therefore, it is of interest to relax the monotonicity of A such that the algorithm is still weakly convergent.

Next, we give some subresults of Theorem 3.1. If S is asymptotically nonexpansive, we find the following result.

Corollary 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C × C to which satisfies (A1)-(A4). Let A : C H be an α-inverse-strongly monotone mapping, and let B : C H be a β-inverse-strongly monotone mapping. Let S : C C be an asymptotically nonexpansive mapping with the sequence { k n } such that n = 1 ( k n 1 ) < . Assume that Ω = F ( S ) VI ( C , B ) EP ( F , A ) is not empty. Let { α n } , { α n } , and { α n } be real number sequences in ( 0 , 1 ) . Let { r n } and { s n } be two positive real number sequences. Let { x n } be a sequence generated in the following process:
{ x 1 C , F ( z n , z ) + A x n , z z n + 1 r n z z n , z n x n 0 , z C , y n = P C ( z n s n B z n ) , x n + 1 = α n x n + α n S n y n + α n e n ,
where { e n } is a bounded sequence in C. Assume that the control sequences satisfy the following restrictions:
  1. (a)

    α n + α n + α n = 1 ;

     
  2. (b)

    0 < p α n p < 1 and n = 1 α n < ;

     
  3. (c)

    0 < s s n s < 2 β and 0 < r r n r < 2 α ,

     

where p, q, s, s , r, r are real constants. Then { x n } converges weakly to some point in Ω.

Further, if S is an identity mapping, we have the following result.

Corollary 3.4 Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C × C to which satisfies (A1)-(A4). Let A : C H be an α-inverse-strongly monotone mapping, and let B : C H be a β-inverse-strongly monotone mapping. Assume that Ω = VI ( C , B ) EP ( F , A ) is not empty. Let { α n } , { α n } , and { α n } be real number sequences in ( 0 , 1 ) . Let { r n } and { s n } be two positive real number sequences. Let { x n } be a sequence generated in the following process:
{ x 1 C , F ( y n , z ) + A x n , z y n + 1 r n z y n , y n x n 0 , x n + 1 = α n x n + α n P C ( y n s n B y n ) + α n e n ,
where { e n } is a bounded sequence in C. Assume that the control sequences satisfy the following restrictions:
  1. (a)

    α n + α n + α n = 1 ;

     
  2. (b)

    0 < p α n q < 1 and n = 1 α n < ;

     
  3. (c)

    0 < s s n s < 2 β and 0 < r r n r < 2 α ,

     

where p, q, s, s , r, r are real constants. Then { x n } converges weakly to some point in Ω.

Next, we give a result on variational inequality (2.1).

Corollary 3.5 Let C be a nonempty closed convex subset of a real Hilbert space H. Let A : C H be an α-inverse-strongly monotone mapping, and let B : C H be a β-inverse-strongly monotone mapping. Assume that Ω = VI ( C , B ) VI ( C , A ) is not empty. Let { α n } , { α n } , and { α n } be real number sequences in ( 0 , 1 ) . Let { r n } and { s n } be two positive real number sequences. Let { x n } be a sequence generated in the following process:
{ x 1 C , z n = P C ( x n s n A x n ) , x n + 1 = α n x n + α n P C ( z n s n B z n ) + α n e n ,
where { e n } is a bounded sequence in C. Assume that the control sequences satisfy the following restrictions:
  1. (a)

    α n + α n + α n = 1 ;

     
  2. (b)

    0 < p α n q < 1 and n = 1 α n < ;

     
  3. (c)

    0 < s s n s < 2 β and 0 < r r n r < 2 α ,

     

where p, q, s, s , r, r are real constants. Then { x n } converges weakly to some point in Ω.

Proof Putting F 0 , we see that
A x n , z z n + 1 r n z z n , z n x n 0 , z C
is equivalent to
x n r n A x n z n , z n z 0 , z C .

This implies that z n = P C ( x n r n A x n ) . Let β n = 0 and S be the identity. Then we can obtain from Theorem 3.1 the desired results immediately. □

Finally, we consider solving common fixed points of a pair of strict pseudocontractions.

Corollary 3.6 Let C be a nonempty closed convex subset of a real Hilbert space H. Let T 1 : C C be an α-strict pseudocontraction, and let T 2 : C C be a β-strict pseudocontraction. Assume that Ω = F ( T 1 ) F ( T 2 ) is not empty. Let { α n } , { α n } , and { α n } be real number sequences in ( 0 , 1 ) . Let { r n } and { s n } be two positive real number sequences. Let { x n } be a sequence generated in the following process:
{ x 1 C , z n = ( 1 r n ) x n + r n T 2 x n , y n = ( 1 s n ) x n + s n T 1 x n , x n + 1 = α n x n + α n y n + α n e n ,
where { e n } is a bounded sequence in C. Assume that the control sequences satisfy the following restrictions:
  1. (a)

    α n + α n + α n = 1 ;

     
  2. (b)

    0 < p α n q < 1 and n = 1 α n < ;

     
  3. (c)

    0 < s s n s < 1 α and 0 < r r n r < 1 β ,

     

where p, q, s, s , r, r are real constants. Then { x n } converges weakly to some point in Ω.

Proof Put F 0 , A = I T 2 and B = I T 1 . It follows that A is 1 α 2 -inverse-strongly monotone and B is 1 β 2 -inverse-strongly monotone. We also have F ( T 1 ) = VI ( C , B ) and F ( T 2 ) = VI ( C , A ) . In view of Theorem 3.1, we find the desired result immediately. □

Declarations

Authors’ Affiliations

(1)
School of Mathematics, Physics and Information Science, Zhejiang Ocean University

References

  1. Facchinei F, Pang JS I. In Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York; 2003.Google Scholar
  2. Facchinei F, Pang JS II. In Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York; 2003.Google Scholar
  3. Fattorini HO: Infinite-Dimensional Optimization and Control Theory. Cambridge University Press, Cambridge; 1999.View ArticleMATHGoogle Scholar
  4. Rockfellar RT: Monotone operators and proximal point algorithm. SIAM J. Control Optim. 1976, 14: 877-898. 10.1137/0314056View ArticleMathSciNetGoogle Scholar
  5. Rockafellar RT: Convex Analysis. Princeton University Press, Princeton; 1970.View ArticleMATHGoogle Scholar
  6. Rockafellar RT: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 1970, 149: 75-88. 10.1090/S0002-9947-1970-0282272-5MathSciNetView ArticleMATHGoogle Scholar
  7. Tanaka Y: A constructive version of Ky Fan’s coincidence theorem. J. Math. Comput. Sci. 2012, 2: 926-936.MathSciNetGoogle Scholar
  8. He RH: Coincidence theorem and existence theorems of solutions for a system of Ky Fan type minimax inequalities in FC-spaces. Adv. Fixed Point Theory 2012, 2: 47-57.Google Scholar
  9. Byrne C: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 2008, 20: 103-120.MathSciNetView ArticleMATHGoogle Scholar
  10. Genel A, Lindenstruss J: An example concerning fixed points. Isr. J. Math. 1975, 22: 81-86. 10.1007/BF02757276View ArticleMathSciNetMATHGoogle Scholar
  11. Cho SY, Qin X, Kang SM: Iterative processes for common fixed points of two different families of mappings with applications. J. Glob. Optim. 2013, 57: 1429-1446. 10.1007/s10898-012-0017-yMathSciNetView ArticleMATHGoogle Scholar
  12. Takahashi W, Toyoda M: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 2003, 118: 417-428. 10.1023/A:1025407607560MathSciNetView ArticleMATHGoogle Scholar
  13. Cho YJ, Qin X: Systems of generalized nonlinear variational inequalities and its projection methods. Nonlinear Anal. 2008, 69: 4443-4451. 10.1016/j.na.2007.11.001MathSciNetView ArticleMATHGoogle Scholar
  14. Lv S, Wu C: Convergence of iterative algorithms for a generalized variational inequality and a nonexpansive mapping. Eng. Math. Lett. 2012, 1: 44-57.Google Scholar
  15. Chang SS, Lee HWJ, Chan CK: A new hybrid method for solving a generalized equilibrium problem, solving a variational inequality problem and obtaining common fixed points in Banach spaces, with applications. Nonlinear Anal. 2010, 73: 2260-2270. 10.1016/j.na.2010.06.006MathSciNetView ArticleMATHGoogle Scholar
  16. Cho SY, Kang SM: Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process. Appl. Math. Lett. 2011, 24: 224-228. 10.1016/j.aml.2010.09.008MathSciNetView ArticleMATHGoogle Scholar
  17. Hao Y: On variational inclusion and common fixed point problems in Hilbert spaces with applications. Appl. Math. Comput. 2010, 217: 3000-3010. 10.1016/j.amc.2010.08.033MathSciNetView ArticleMATHGoogle Scholar
  18. Yuan Q, Kim JK: Weak convergence of algorithms for asymptotically strict pseudocontractions in the intermediate sense and equilibrium problems. Fixed Point Theory Appl. 2012. Article ID 132, 2012:Google Scholar
  19. Cho SY, Kang SM: Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math. Sci. 2012, 32: 1607-1618. 10.1016/S0252-9602(12)60127-1View ArticleMathSciNetMATHGoogle Scholar
  20. Luo H, Wang Y: Iterative approximation for the common solutions of a infinite variational inequality system for inverse-strongly accretive mappings. J. Math. Comput. Sci. 2012, 2: 1660-1670.MathSciNetGoogle Scholar
  21. Cho SY, Li W, Kang SM: Convergence analysis of an iterative algorithm for monotone operators. J. Inequal. Appl. 2013. Article ID 199, 2013:Google Scholar
  22. Zegeye H, Shahzad N: Strong convergence theorem for a common point of solution of variational inequality and fixed point problem. Adv. Fixed Point Theory 2012, 2: 374-397.MathSciNetMATHGoogle Scholar
  23. Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert space. J. Math. Anal. Appl. 1967, 20: 197-228. 10.1016/0022-247X(67)90085-6MathSciNetView ArticleMATHGoogle Scholar
  24. Qihou L: Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings. Nonlinear Anal. 1996, 26: 1835-1845. 10.1016/0362-546X(94)00351-HMathSciNetView ArticleMATHGoogle Scholar
  25. Kim KS, Kim JK, Lim WH: Convergence theorems for common solutions of various problems with nonlinear mapping. J. Inequal. Appl. 2014. Article ID 2, 2014:Google Scholar
  26. Qin X, Chang SS, Cho YJ: Iterative methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal. 2010, 11: 2963-2972. 10.1016/j.nonrwa.2009.10.017MathSciNetView ArticleMATHGoogle Scholar
  27. Rodjanadid B, Sompong S: A new iterative method for solving a system of generalized equilibrium problems, generalized mixed equilibrium problems and common fixed point problems in Hilbert spaces. Adv. Fixed Point Theory 2013, 3: 675-705.Google Scholar
  28. Kang SM, Cho SY, Liu Z: Convergence of iterative sequences for generalized equilibrium problems involving inverse-strongly monotone mappings. J. Inequal. Appl. 2010. Article ID 827082, 2010:Google Scholar
  29. Chang SS, Zuo P: Shrinking projection method of common solutions for generalized equilibrium quasi- ϕ -nonexpansive mapping and relatively nonexpansive mapping. J. Inequal. Appl. 2010. Article ID 101690, 2010:Google Scholar
  30. Zhang Q, Wu H: Hybrid algorithms for equilibrium and common fixed point problems with applications. J. Inequal. Appl. 2014. Article ID 221, 2014:Google Scholar
  31. Cho SY, Qin X: On the strong convergence of an iterative process for asymptotically strict pseudocontractions and equilibrium problems. Appl. Math. Comput. 2014, 235: 430-438.MathSciNetView ArticleGoogle Scholar
  32. Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123-145.MathSciNetMATHGoogle Scholar
  33. Schu J: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 1991, 43: 153-159. 10.1017/S0004972700028884MathSciNetView ArticleMATHGoogle Scholar
  34. Tan KK, Xu HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iterative process. J. Math. Anal. Appl. 1993, 178: 301-308. 10.1006/jmaa.1993.1309MathSciNetView ArticleMATHGoogle Scholar
  35. Opial Z: Weak convergence of the sequence of successive approximation for nonexpansive mappings. Bull. Am. Math. Soc. 1967, 73: 591-597. 10.1090/S0002-9904-1967-11761-0MathSciNetView ArticleMATHGoogle Scholar

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