Open Access

On strong convergence of an iterative algorithm for common fixed point and generalized equilibrium problems

Journal of Inequalities and Applications20142014:263

https://doi.org/10.1186/1029-242X-2014-263

Received: 19 April 2014

Accepted: 28 June 2014

Published: 22 July 2014

Abstract

In this article, an iterative algorithm for finding a common element in the solution set of generalized equilibrium problems and in the common fixed point set of a family of nonexpansive mappings. Strong convergence of the algorithm is established in the framework of Hilbert spaces.

MSC:47H05, 47H09, 47J25.

Keywords

equilibrium problemfixed pointnonexpansive mappingvariational inequality

1 Introduction and preliminaries

Equilibrium problems which were introduced by Ky Fan [1] and further studied by Blum and Oettli [2] have intensively been investigated based on iterative methods. The equilibrium problems have emerged as an effective and powerful tool for studying a wide class of problems which arise in economics, ecology, transportation, network, elasticity, and optimization; see [36] and the references therein. It is well known that the equilibrium problems cover fixed point problems, variational inequality problems, saddle problems, inclusion problems, complementarity problems, and minimization problems; see [715] and the references therein.

In this paper, an iterative algorithm is proposed for treating common fixed point and generalized equilibrium problems. It is proved that the sequence generated in the algorithm converges strongly to a common element in the solution set of generalized equilibrium problems and in the common fixed point set of a family of nonexpansive mappings.

From now on, we always assume that H is a real Hilbert space, whose inner product and norm are denoted by , and , respectively. Let C be a nonempty closed convex subset of H and let P C be the projection of H onto C.

Let S : C C be a mapping. Throughout this paper, we use F ( S ) to denote the fixed point set of the mapping S. Recall that S : C C is said to be nonexpansive iff
S x S y x y , x , y C .
S : C C is said to be firmly nonexpansive iff
S x S y 2 S x S y , x y , x , y C .

It is easy to see that every firmly nonexpansive mapping is nonexpansive.

Let A : C H be a mapping. Recall that A is said to be monotone iff
A x A y , x y 0 , x , y C .
A is said to be inverse-strongly monotone iff there exists a constant α > 0 such that
A x A y , x y δ A x A y 2 , x , y C .
For such a case, A is also said to be α-inverse-strongly monotone. It is known that if S : C C is nonexpansive, then A = I S is 1 2 -inverse-strongly monotone. Recall that a set-valued mapping T : H 2 H is called 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 of G ( T ) of T is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mapping T is maximal if and only if for ( x , f ) H × H , x y , f g 0 for every ( y , g ) G ( T ) implies f T x . Let B be a monotone map of C into H and let 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
T v = { B 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 u C ; see [16] and the references therein

Recall that the classical variational inequality is to find u C such that
A u , v u 0 , v C .
(1.1)
In this paper, we use V I ( C , A ) to denote the solution set of the variational inequality (1.1). One can see that the variational inequality (1.1) is equivalent to a fixed point problem. The element u C is a solution of the variational inequality (1.1) if and only if u C is a fixed point of the mapping P C ( I λ A ) , where λ > 0 is a constant and I denotes the identity mapping. If A is an α-inverse strongly monotone, we remark here that the mapping P C ( I λ A ) is nonexpansive iff 0 < λ < 2 α . Indeed,
P C ( I λ A ) x P C ( I λ A ) y 2 ( I λ A ) x ( I λ A ) y 2 = x y 2 2 λ x y , A x A y + λ 2 A x A y 2 x y 2 λ ( 2 α λ ) A x A y 2 .

This alternative equivalent formulation has played a significant role in the studies of the variational inequalities and related optimization problems.

Let A be an inverse-strongly monotone mapping, F a bifunction of C × C into , where is the set of real numbers. We consider the following equilibrium problem:
Find  z C  such that  F ( z , y ) + A z , y z 0 , y C .
(1.2)
In this paper, the set of such z C is denoted by E P ( F , A ) , i.e.,
E P ( F , A ) = { z C : F ( z , y ) + A z , y z 0 , y C } .
If the case of A 0 , the zero mapping, the problem (1.2) is reduced to
Find  z C  such that  F ( z , y ) 0 , y C .
(1.3)

In this paper, we use E P ( F ) to denote the solution set of the problem (1.3). The problem of (1.2) and (1.3) have been considered by many authors; see, for example, [1729] and the references therein. In the case of F 0 , the problem (1.2) is reduced to the classical variational inequality (1.1).

To study the equilibrium problems, we assume that the bifunction F : C × C R 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.

     

The well-known convex feasibility problem which captures applications in various disciplines such as image restoration, and radiation therapy treatment planning is to find a point in the intersection of common fixed point sets of a family of nonlinear mappings. In this paper, we propose an iterative algorithm for finding a common element in the solution set of the generalized equilibrium problem (1.2) and in the common fixed point set of a family of nonexpansive mappings. Strong convergence of the algorithm is established in the framework of Hilbert spaces.

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

A space X is said to satisfy Opial’s condition [30] if for each sequence { x n } n = 1 in X which converges weakly to point x X , we have
lim inf n x n x < lim inf n x n y , y X , y x .
It is well known that the above inequality is equivalent to
lim sup n x n x < lim sup n x n y , y X , y x .

The following lemma can be found in [2].

Lemma 1.1 Let C be a nonempty closed convex subset of H ad 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, if T r x = { z C : F ( z , y ) + 1 r y z , z x 0 , y C } , then the following hold:
T r ( x ) = { z C : F ( z , y ) + 1 r y z , z x 0 , y C }
for all z H . Then the following hold:
  1. (1)

    T r is single-valued;

     
  2. (2)
    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. (3)

    F ( T r ) = E P ( F ) ;

     
  4. (4)

    E P ( F ) is closed and convex.

     

Lemma 1.2 [20]

Let C, H, F and T r be as in Lemma 1.1. Then the following holds:
T s x T t x 2 s t s T s x T t x , T s x x

for all s , t > 0 and x H .

Lemma 1.3 [31]

Assume that { α n } is a sequence of nonnegative real numbers such that
α n + 1 ( 1 γ n ) α n + δ n ,
where { γ n } is a sequence in ( 0 , 1 ) and { δ n } is a sequence such that
  1. (a)

    n = 1 γ n = ;

     
  2. (b)

    lim sup n δ n / γ n 0 or n = 1 | δ n | < .

     

Then lim n α n = 0 .

Definition 1.4 [32]

Let { S i : C C } be a family of infinitely nonexpansive mappings and { γ i } be a nonnegative real sequence with 0 γ i < 1 , i 1 . For n 1 define a mapping W n : C C as follows:
U n , n + 1 = I , U n , n = γ n S n U n , n + 1 + ( 1 γ n ) I , U n , n 1 = γ n 1 S n 1 U n , n + ( 1 γ n 1 ) I , U n , k = γ k S k U n , k + 1 + ( 1 γ k ) I , u n , k 1 = γ k 1 S k 1 U n , k + ( 1 γ k 1 ) I , U n , 2 = γ 2 S 2 U n , 3 + ( 1 γ 2 ) I , W n = U n , 1 = γ 1 S 1 U n , 2 + ( 1 γ 1 ) I .
(1.4)

Such a mapping W n is nonexpansive from C to C and it is called a W-mapping generated by S n , S n 1 , , S 1 and γ n , γ n 1 , , γ 1 .

Lemma 1.5 [32]

Let C be a nonempty closed convex subset of a Hilbert space H, { S i : C C } be a family of infinitely nonexpansive mappings with i = 1 F ( S i ) , { γ i } be a real sequence such that 0 < γ i l < 1 , i 1 . Then
  1. (1)

    W n is nonexpansive and F ( W n ) = i = 1 F ( S i ) , for each n 1 ;

     
  2. (2)

    for each x C and for each positive integer k, the limit lim n U n , k exists;

     
  3. (3)
    the mapping W : C C defined by
    W x : = lim n W n x = lim n U n , 1 x , x C ,
    (1.5)
     

is a nonexpansive mapping satisfying F ( W ) = i = 1 F ( S i ) and it is called the W-mapping generated by S 1 , S 2 , and γ 1 , γ 2 ,  .

Lemma 1.6 [27]

Let C be a nonempty closed convex subset of a Hilbert space H, { S i : C C } be a family of infinitely nonexpansive mappings with i = 1 F ( S i ) , { γ i } be a real sequence such that 0 < γ i l < 1 , i 1 . If K is any bounded subset of C, then
lim n sup x K W x W n x = 0 .

Throughout this paper, we always assume that 0 < γ i l < 1 , i 1 .

Lemma 1.7 [33]

Let { x n } and { y n } be bounded sequences in a Hilbert space H and let { β n } be a sequence in [ 0 , 1 ] with 0 < lim inf n β n lim sup n β n < 1 . Suppose that x n + 1 = ( 1 β n ) y n + β n x n for all n 0 and
lim sup n ( y n + 1 y n x n + 1 x n ) 0 .

Then lim n y n x n = 0 .

2 Main results

Theorem 2.1 Let C be a nonempty closed convex subset of a Hilbert space H and 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 { S i : C C } be a family of infinitely nonexpansive mappings. Assume that Ω : = i = 1 F ( S i ) E P ( F , A ) . Let f : C C be a κ-contractive mapping. Let { x n } be a sequence generated in the following process: let it be a sequence generated in
{ x 1 C , chosen arbitrarily , F ( y n , y ) + A x n , y y n + 1 r n y y n , y n x n 0 , y C , x n + 1 = β n x n + ( 1 β n ) W n ( α n f ( W n x n ) + ( 1 α n ) y n ) , n 1 ,
where { W n } is the mapping sequence defined by (1.4), { α n } , and { β n } are sequences in [ 0 , 1 ] and { r n } is a positive number sequence. Assume that the above control sequences satisfy the following restrictions:
  1. (a)

    0 < a β n b < 1 , 0 < c r n d < 2 α ;

     
  2. (b)

    lim n α n = 0 , and n = 1 α n = ;

     
  3. (c)

    lim n ( r n r n + 1 ) = 0 .

     

Then { x n } converges strongly to a point x Ω , where x = P Ω f ( x ) .

Proof First, we show that the sequence { x n } and { y n } are bounded. Fixing x Ω , we find that
y n x 2 = T r n ( x n r n A x n ) T r n ( x r n A x ) 2 ( x n r n A x n ) ( x r n A x ) 2 = ( x n x ) r n ( A x n A x ) 2 = x n x 2 2 r n x n x , A x n A x + r n 2 A x n A x 2 x n x 2 2 r n α A x n A x 2 + r n 2 A x n A x 2 = x n x 2 + r n ( r n 2 α ) A x n A x 2 .
(2.1)
Using the restriction (a), we find that
y n x x n x .
(2.2)
From the above, we also find that the mappings I r n A is nonexpansive. Putting z n = α n f ( W n x n ) + ( 1 α n ) y n , we find from (2.2) that
z n x = α n f ( W n x n ) + ( 1 α n ) y n x α n f ( W n x n ) x + ( 1 α n ) y n x α n κ x n x + α n f ( x ) x + ( 1 α n ) y n x ( 1 α n ( 1 κ ) ) x n x + α n f ( x ) x .
(2.3)
It follows from (2.3) that
x n + 1 x β n x n x + ( 1 β n ) W n z n x β n x n x + ( 1 β n ) z n x ( 1 α n ( 1 β n ) ( 1 κ ) ) x n x + α n ( 1 β n ) f ( x ) x max { x 1 x , f ( x ) x 1 κ } .
This shows that the sequence { x n } is bounded, and so are { y n } and { z n } . Without loss of generality, we can assume that there exists a bounded set K C such that x n , y n , z n K ;
y n + 1 y n = T r n + 1 ( x n + 1 r n + 1 A x n + 1 ) T r n + 1 ( x n r n A x n ) + T r n + 1 ( x n r n A x n ) T r n ( x n r n A x n ) ( x n + 1 r n + 1 A x n + 1 ) ( x n r n A x n ) + T r n + 1 ( x n r n A x n ) T r n ( x n r n A x n ) x n + 1 x n + | r n + 1 r n | A x n + T r n + 1 ( x n r n A x n ) T r n ( x n r n A x n ) .
(2.4)
It follows that
z n + 1 z n α n + 1 f ( W n + 1 x n + 1 ) f ( W n x n ) + | α n + 1 α n | ( f ( W n + 1 x n + 1 ) + y n ) + ( 1 α n + 1 ) y n + 1 y n α n + 1 κ W n + 1 x n + 1 W n x n + | α n + 1 α n | ( f ( W n + 1 x n + 1 ) + y n ) + x n + 1 x n + | r n + 1 r n | A x n + T r n + 1 ( x n r n A x n ) T r n ( x n r n A x n ) .
(2.5)
Note that
W n + 1 z n + 1 W n z n = W n + 1 z n + 1 W z n + 1 + W z n + 1 W z n + W z n W n z n W n + 1 z n + 1 W z n + 1 + W z n + 1 W z n + W z n W n z n sup x K { W n + 1 x W x + W x W n x } + z n + 1 z n .
(2.6)
Combing (2.5) with (2.6) yields
W n + 1 z n + 1 W n z n x n + 1 x n sup x K { W n + 1 x W x + W x W n x } + α n + 1 κ W n + 1 x n + 1 W n x n + | α n + 1 α n | ( f ( W n + 1 x n + 1 ) + y n ) + | r n + 1 r n | A x n + T r n + 1 ( x n r n A x n ) T r n ( x n r n A x n ) .
From the restrictions (a), (b), and (c), we find from Lemma 1.6 that
lim sup n { W n + 1 z n + 1 W n z n x n + 1 x n } 0 .
Using Lemma 1.7, we obtain
lim n W n z n x n = 0 .
(2.7)
It follows that
lim n x n + 1 x n = 0 .
(2.8)
Using (2.1), we find that
x n + 1 x 2 = β n x n + ( 1 β n ) W n z n x 2 β n x n x 2 + ( 1 β n ) z n x 2 β n x n x 2 + ( 1 β n ) ( α n f ( W n x n ) x 2 + ( 1 α n ) y n x 2 ) x n x 2 + α n f ( W n x n ) x 2 + r n ( r n 2 α ) ( 1 α n ) ( 1 β n ) A x n A x 2 ,
which in turn yields
r n ( 2 α r n ) ( 1 α n ) ( 1 β n ) A x n A x 2 x n x 2 x n + 1 x 2 + α n f ( W n x n ) x 2 ( x n x + x n + 1 x ) x n x n + 1 + α n f ( W n x n ) x 2 .
Using (2.8), we find from the restrictions (a), (b), and (c) that
lim n A x n A x = 0 .
(2.9)
On the other hand, we see that
y n x 2 = T r n ( I r n A ) x n T r n ( I r n A ) x 2 ( I r n A ) x n ( I r n A ) x , y n x 1 2 ( x n x 2 + y n x 2 ( x n y n ) r n ( A x n A x ) 2 ) = 1 2 ( x n x 2 + y n x 2 x n y n 2 + 2 r n x n y n , A x n A x r n 2 A x n A x 2 ) .
Hence, we have
y n x 2 x n x 2 x n y n 2 + 2 r n x n y n A x n A x .
It follows that
x n + 1 x 2 x n x 2 + α n f ( W n x n ) x 2 ( 1 α n ) ( 1 β n ) x n y n 2 + 2 r n ( 1 α n ) ( 1 β n ) x n y n A x n A x x n x 2 + α n f ( W n x n ) x 2 ( 1 α n ) ( 1 β n ) x n y n 2 + 2 r n x n y n A x n A x .
This implies that
( 1 α n ) ( 1 β n ) x n y n 2 ( x n x + x n + 1 x ) x n x n + 1 + α n f ( W n x n ) x 2 + 2 r n x n y n A x n A x .
Using (2.8) and (2.9), we find from the restrictions (a), (b), and (c) that
lim n x n y n = 0 .
(2.10)
Since z n = α n f ( W n x n ) + ( 1 α n ) y n , we find that
lim n z n y n = 0 .
(2.11)
Notice that
x n + 1 x n = ( 1 β n ) W n z n x n .
This implies from (2.8)
lim n W n z n x n = 0 .
(2.12)
Note that
W n z n z n z n y n + y n x n + x n W n z n .
From (2.10), (2.11), and (2.12), we obtain
lim n W n z n z n = 0 .
(2.13)
Since the mapping P Ω f is contractive, we denote the unique fixed point by x. Next, we prove that lim sup n f ( x ) x , z n x 0 . To see this, we choose a subsequence { z n i } of { z n } such that
lim sup n f ( x ) x , z n x = lim i f ( x ) x , z n i x .

Since { z n i } is bounded, there exists a subsequence { z n i j } of { z n i } which converges weakly to z. Without loss of generality, we may assume that z n i z . Indeed, we also have y n i f .

First, we show that z i = 1 F ( S i ) . Suppose the contrary, W z z . Note that
z n W z n W z n W n z n + W n z n z n sup x K { W x W n x } + W n z n z n .
Using Lemma 1.6, we obtain from (2.13) that lim n z n W z n = 0 . By Opial’s condition, we see that
lim inf i z n i z < lim inf i z n i W z lim inf i { z n i W z n i + W z n i W z } lim inf i { z n i W z n i + z n i z } .

This implies that lim inf i z n i z < lim inf i z n i z , which leads to a contradiction. Thus, we have z i = 1 F ( S i ) .

Next, we show that f E P ( F , A ) . Note that y n z . Since y n = T r n ( x n r n A x n ) , we have
F ( y n , y ) + A x n , y y n + 1 r n y y n , y n x n 0 , y C .
From the condition (A2), we see that
A x n , y y n + 1 r n y y n , y n x n F ( y , y n ) , y C .
Replacing n by n i , we arrive at
A x n i , y y n i + y y n i , y n i x n i r n i F ( y , y n i ) , y C .
(2.14)
For t with 0 < t 1 and ρ C , let ρ t = t ρ + ( 1 t ) z . Since ρ C and z C , we have ρ t C . It follows from (2.14) that
ρ t y n i , A ρ t ρ t y n i , A ρ t A x n i , ρ t y n i ρ t y n i , y n i x n i r n i + F ( ρ t , y n i ) = ρ t y n i , A ρ t A y n , i + ρ t y n i , A y n , i A x n i ρ t y n i , y n i x n i r n i + F ( ρ t , y n i ) .
(2.15)
Using (2.10), we have A y n , i A x n i 0 as i . On the other hand, we get from the monotonicity of A that ρ t y n i , A ρ t A y n , i 0 . It follows from (A4) and (2.15) that
ρ t z , A ρ t F ( ρ t , z ) .
(2.16)
From (A1) and (A4), we see from (2.16) that
0 = F ( ρ t , ρ t ) t F ( ρ t , ρ ) + ( 1 t ) F ( ρ t , z ) t F ( ρ t , ρ ) + ( 1 t ) ρ t z , A ρ t = t F ( ρ t , ρ ) + ( 1 t ) t ρ z , A ρ t ,
which yields F ( ρ t , ρ ) + ( 1 t ) ρ f , A 3 ρ t 0 . Letting t 0 in the above inequality, we arrive at F ( z , ρ ) + ρ z , A z 0 . This shows that f E P ( F , A ) . It follows that
lim sup n f ( x ) x , z n x 0 .
(2.17)
Finally, we show that x n x , as n . Note that
z n x 2 = α n f ( W n x n ) x , z n x + ( 1 α n ) y n x , z n x ( 1 α n ( 1 κ ) ) x n x z n x + α n f ( x ) x , z n x 1 α n ( 1 κ ) 2 ( x n x 2 + z n x 2 ) + α n f ( x ) x , z n x .
Hence, we have
z n x 2 ( 1 α n ( 1 κ ) ) x n x 2 + 2 α n f ( x ) x , z n x .
This implies that
x n + 1 x 2 = β n x n + ( 1 β n ) W n z n x 2 β n x n x 2 + ( 1 β n ) z n x 2 ( 1 α n ( 1 β n ) ( 1 κ ) ) x n x 2 + 2 α n ( 1 β n ) f ( x ) x , z n x .

Using Lemma 1.3 and (2.17), we find from the restrictions (a), (b), and (c) that lim n x n x = 0 . This completes the proof. □

3 Applications

For a single mapping, we find from Theorem 2.1 the following result.

Theorem 3.1 Let C be a nonempty closed convex subset of a Hilbert space H and 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 S be a nonexpansive mapping. Assume that Ω : = F ( S ) E P ( F , A ) . Let f : C C be a κ-contractive mapping. Let { x n } be a sequence generated in the following process: let it be a sequence generated in
{ x 1 C , chosen arbitrarily, F ( y n , y ) + A x n , y y n + 1 r n y y n , y n x n 0 , y C , x n + 1 = β n x n + ( 1 β n ) S ( α n f ( S x n ) + ( 1 α n ) y n ) , n 1 ,
where { α n } and { β n } are sequences in [ 0 , 1 ] and { r n } is a positive number sequence. Assume that the above control sequences satisfy the following restrictions:
  1. (a)

    0 < a β n b < 1 , 0 < c r n d < 2 α ;

     
  2. (b)

    lim n α n = 0 , and n = 1 α n = ;

     
  3. (c)

    lim n ( r n r n + 1 ) = 0 .

     

Then { x n } converge strongly to a point x Ω , where x = P Ω f ( x ) .

If S is the identity, we find the following result on the generalized equilibrium problem.

Corollary 3.2 Let C be a nonempty closed convex subset of a Hilbert space H and let F be a bifunction from C × C to which satisfies (A1)-(A4). Let A : C H be an α-inverse-strongly monotone mapping. Assume that E P ( F , A ) . Let f : C C be a κ-contractive mapping. Let { x n } be a sequence generated in the following process: let it be a sequence generated in
{ x 1 C , chosen arbitrarily, F ( y n , y ) + A x n , y y n + 1 r n y y n , y n x n 0 , y C , x n + 1 = β n x n + ( 1 β n ) ( α n f ( x n ) + ( 1 α n ) y n ) , n 1 ,
where { α n } and { β n } are sequences in [ 0 , 1 ] and { r n } is a positive number sequence. Assume that the above control sequences satisfy the following restrictions:
  1. (a)

    0 < a β n b < 1 , 0 < c r n d < 2 α ;

     
  2. (b)

    lim n α n = 0 , and n = 1 α n = ;

     
  3. (c)

    lim n ( r n r n + 1 ) = 0 .

     

Then { x n } converge strongly to a point x E P ( F , A ) , where x = P E P ( F , A ) f ( x ) .

Next, we give a result on the equilibrium problem (1.3).

Theorem 3.3 Let C be a nonempty closed convex subset of a Hilbert space H and let F be a bifunction from C × C to which satisfies (A1)-(A4). Let { S i : C C } be a family of infinitely nonexpansive mappings. Assume that Ω : = i = 1 F ( S i ) E P ( F ) . Let f : C C be a κ-contractive mapping. Let { x n } be a sequence generated in the following process: let it be a sequence generated in
{ x 1 C , chosen arbitrarily, F ( y n , y ) + 1 r n y y n , y n x n 0 , y C , x n + 1 = β n x n + ( 1 β n ) W n ( α n f ( W n x n ) + ( 1 α n ) y n ) , n 1 ,
where { W n } is the mapping sequence defined by (1.4), { α n } and { β n } are sequences in [ 0 , 1 ] and { r n } is a positive number sequence. Assume that the above control sequences satisfy the following restrictions:
  1. (a)

    0 < a β n b < 1 , 0 < c r n d < + ;

     
  2. (b)

    lim n α n = 0 , and n = 1 α n = ;

     
  3. (c)

    lim n ( r n r n + 1 ) = 0 .

     

Then { x n } converge strongly to a point x Ω , where x = P Ω f ( x ) .

Proof By putting A 3 0 , the zero operator, we can easily get the desired conclusion. This completes the proof. □

Next, we give a result on the classical variational inequality.

Theorem 3.4 Let C be a nonempty closed convex subset of a Hilbert space H. Let A : C H be an α-inverse-strongly monotone mapping and let { S i : C C } be a family of infinitely nonexpansive mappings. Assume that Ω : = i = 1 F ( S i ) V I ( C , A ) . Let f : C C be a κ-contractive mapping. Let { x n } be a sequence generated in the following process: let it be a sequence generated in
{ x 1 C , chosen arbitrarily, y n = P C ( x n r n A x n ) , x n + 1 = β n x n + ( 1 β n ) W n ( α n f ( W n x n ) + ( 1 α n ) y n ) , n 1 ,
where { W n } is the mapping sequence defined by (1.4), { α n } and { β n } are sequences in [ 0 , 1 ] and { r n } is a positive number sequence. Assume that the above control sequences satisfy the following restrictions:
  1. (a)

    0 < a β n b < 1 , 0 < c r n d < 2 α ;

     
  2. (b)

    lim n α n = 0 , and n = 1 α n = ;

     
  3. (c)

    lim n ( r n r n + 1 ) = 0 .

     

Then { x n } converge strongly to a point x Ω , where x = P Ω f ( x ) .

Proof Putting F 0 , we see from Theorem 2.1 that
A x n , y y n + 1 r n y y n , y n x n 0 , y C , y C , n 1 .
This implies that
y y n , x n r n A x n y n 0 , y C .
It follows that
y n = P C ( x n r n A x n ) .

This completes the proof. □

Finally, we utilize the results presented in the paper to study the following optimization problem:
min x C h ( x ) ,
(3.1)
where C is a nonempty closed convex subset of a Hilbert space, and h : C R is a convex and lower semi-continuous functional. We use Ω to denote the solution set of the problem (3.1). Let F : C × C R be a bifunction defined by F ( x , y ) = h ( y ) h ( x ) . We consider the following equilibrium problem: to find x C such that
F ( x , y ) 0 , y C .

It is easy to see that the bifunction F satisfies conditions (A1)-(A4) and E P ( F ) = Ω .

Theorem 3.5 Let C be a nonempty closed convex subset of a Hilbert space H and let h : C R be defined as above. Assume that Ω . Let f : C C be a κ-contractive mapping. Let { x n } be a sequence generated in the following process: let it be a sequence generated in
{ x 1 C , chosen arbitrarily, h ( y ) h ( u n ) + 1 r n y y n , y n x n 0 , y C , x n + 1 = β n x n + ( 1 β n ) ( α n f ( x n ) + ( 1 α n ) y n ) , n 1 ,
where { α n } and { β n } are sequences in [ 0 , 1 ] and { r n } is a positive number sequence. Assume that the above control sequences satisfy the following restrictions:
  1. (a)

    0 < a β n b < 1 , 0 < c r n d < + ;

     
  2. (b)

    lim n α n = 0 , and n = 1 α n = ;

     
  3. (c)

    lim n ( r n r n + 1 ) = 0 .

     

Then { x n } converges strongly to a point x Ω , where x = P Ω f ( x ) .

Declarations

Acknowledgements

The author is grateful to the anonymous referees for useful suggestions which improved the contents of the article.

Authors’ Affiliations

(1)
Department of Mathematics and Sciences, Shijiazhuang University of Economics

References

  1. Fan K: A minimax inequality and applications. In Inequality III. Edited by: Shisha O. Academic Press, New York; 1972:103-113.Google Scholar
  2. Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123-145.MathSciNetMATHGoogle Scholar
  3. 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
  4. Iiduka H: Strong convergence for an iterative method for the triple-hierarchical constrained optimization problem. Nonlinear Anal. 2009, 71: e1292-e1297. 10.1016/j.na.2009.01.133MathSciNetView ArticleMATHGoogle Scholar
  5. Park S: On generalizations of the Ekeland-type variational principles. Nonlinear Anal. 2000, 39: 881-889. 10.1016/S0362-546X(98)00253-3MathSciNetView ArticleMATHGoogle Scholar
  6. Khanh PQ, Luu LM: On the existence of solutions to vector quasivariational inequalities and quasicomplementarity problems with applications to traffic network equilibria. J. Optim. Theory Appl. 2004, 123: 533-548. 10.1007/s10957-004-5722-3MathSciNetView ArticleMATHGoogle Scholar
  7. 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
  8. Qin X, Agarwal RP: Shrinking projection methods for a pair of asymptotically quasi- ϕ -nonexpansive mappings. Numer. Funct. Anal. Optim. 2001, 31: 1072-1089.MathSciNetView ArticleMATHGoogle Scholar
  9. Park S: Some equilibrium problems in generalized convex spaces. Acta Math. Vietnam. 2001, 26: 349-364.MathSciNetMATHGoogle Scholar
  10. Park S:A review of the KKM theory on ϕ A -space or GFC-spaces. Adv. Fixed Point Theory 2013, 3: 355-382.Google Scholar
  11. Cho SY, Qin X, Wang L: Strong convergence of a splitting algorithm for treating monotone operators. Fixed Point Theory Appl. 2014. Article ID 94, 2014: Article ID 94Google Scholar
  12. Sun L: Hybrid methods for common solutions in Hilbert spaces with applications. J. Inequal. Appl. 2014. Article ID 183, 2014: Article ID 183Google Scholar
  13. Qin X, Su Y: Approximation of a zero point of accretive operator in Banach spaces. J. Math. Anal. Appl. 2007, 329: 415-424. 10.1016/j.jmaa.2006.06.067MathSciNetView ArticleMATHGoogle Scholar
  14. 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
  15. Cho SY, Li W, Kang SM: Convergence analysis of an iterative algorithm for monotone operators. J. Inequal. Appl. 2013. Article ID 199, 2013: Article ID 199Google Scholar
  16. Rockafellar RT: Characterization of the subdifferentials of convex functions. Pac. J. Math. 1966, 17: 497-510. 10.2140/pjm.1966.17.497MathSciNetView ArticleMATHGoogle Scholar
  17. Kim JK, Cho SY, Qin X: Hybrid projection algorithms for generalized equilibrium problems and strictly pseudocontractive mappings. J. Inequal. Appl. 2010. Article ID 312602, 2010: Article ID 312602Google Scholar
  18. Kim JK, Anh PN, Nam YM: Strong convergence of an extended extragradient method for equilibrium problems and fixed point problems. J. Korean Math. Soc. 2012, 49: 187-200. 10.4134/JKMS.2012.49.1.187MathSciNetView ArticleMATHGoogle Scholar
  19. Kim JK: Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi- ϕ -nonexpansive mappings. Fixed Point Theory Appl. 2011. Article ID 10, 2011: Article ID 10Google Scholar
  20. 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
  21. Takahashi S, Takahashi W: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal. 2008, 69: 1025-1033. 10.1016/j.na.2008.02.042MathSciNetView ArticleMATHGoogle Scholar
  22. Zhang M: Strong convergence of a viscosity iterative algorithm in Hilbert spaces. J. Nonlinear Funct. Anal. 2014. Article ID 1, 2014: Article ID 1Google Scholar
  23. Zhang M: Iterative algorithms for common elements in fixed point sets and zero point sets with applications. Fixed Point Theory Appl. 2012. Article ID 21, 2012: Article ID 21Google Scholar
  24. Zhang QN: Common solutions of equilibrium and fixed point problems. J. Inequal. Appl. 2013. Article ID 425, 2013: Article ID 425Google Scholar
  25. Chen JH: Iterations for equilibrium and fixed point problems. J. Nonlinear Funct. Anal. 2013. Article ID 4, 2013: Article ID 4Google Scholar
  26. 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
  27. Chang SS, Lee HWJ, Chan CK: A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal. 2009, 70: 3307-3319. 10.1016/j.na.2008.04.035MathSciNetView ArticleMATHGoogle Scholar
  28. Chang SS, Chan CK, Lee HWJ, Yang L: A system of mixed equilibrium problems, fixed point problems of strictly pseudo-contractive mappings and nonexpansive semi-groups. Appl. Math. Comput. 2010, 216: 51-60. 10.1016/j.amc.2009.12.060MathSciNetView ArticleMATHGoogle Scholar
  29. Qin X, Cho SY, Wang L: Algorithms for treating equilibrium and fixed point problems. Fixed Point Theory Appl. 2013. Article ID 308, 2013: Article ID 308Google Scholar
  30. Opial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 1967, 73: 595-597.MathSciNetView ArticleMATHGoogle Scholar
  31. Liu LS: Ishikawa and Mann iteration process with errors for nonlinear strongly accretive mappings in Banach spaces. J. Math. Anal. Appl. 1995, 194: 114-125. 10.1006/jmaa.1995.1289MathSciNetView ArticleMATHGoogle Scholar
  32. Shimoji K, Takahashi W: Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwan. J. Math. 2001, 5: 387-404.MathSciNetMATHGoogle Scholar
  33. Suzuki T: Moudafi’s viscosity approximations with Meir-Keeler contractions. J. Math. Anal. Appl. 2007, 325: 342-352. 10.1016/j.jmaa.2006.01.080MathSciNetView ArticleMATHGoogle Scholar

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© Song; licensee Springer. 2014

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