Open Access

On variational inequality, fixed point and generalized mixed equilibrium problems

Journal of Inequalities and Applications20142014:203

https://doi.org/10.1186/1029-242X-2014-203

Received: 14 January 2014

Accepted: 8 May 2014

Published: 22 May 2014

Abstract

In this article, variational inequality, fixed point, and generalized mixed equilibrium problems are investigated based on an extragradient iterative algorithm. Weak convergence of the extragradient iterative algorithm is obtained in Hilbert spaces.

Keywords

fixed pointequilibrium problemmonotone mappingnonexpansive mappingprojection

1 Introduction

In this paper, we always assume that H is a real Hilbert space with the inner product , and the norm , and C is a nonempty, closed, and convex subset of H. is denoted by the set of real numbers. Let F be a bifunction of C × C into . Consider the problem: find a p such that
F ( p , y ) 0 , y C .
(1.1)
In this paper, the solution set of the problem is denoted by EP ( F ) , i.e.,
EP ( F ) = { p C : F ( p , y ) 0 , y C } .

The above problem is first introduced by Ky Fan [1]. In the sense of Blum and Oettli [2], the Ky Fan problem is also called an equilibrium problem.

Recently, the ‘so-called’ generalized mixed equilibrium problem has been investigated by many authors: The generalized mixed equilibrium problem is to find p C such that
F ( p , y ) + A p , y p + φ ( y ) φ ( p ) 0 , y C ,
(1.2)
where φ : C R is a real valued function and A : C H is mapping. We use GMEP ( F , A , φ ) to denote the solution set of the equilibrium problem. That is,
GMEP ( F , A , φ ) : = { p C : F ( p , y ) + A p , y p + φ ( y ) φ ( z ) 0 , y C } .

Next, we give some special cases.

If A = 0 , then the problem (1.2) is equivalent to find p C such that
F ( p , y ) + φ ( y ) φ ( z ) 0 , y C ,
(1.3)

which is called the mixed equilibrium problem.

If F = 0 , then the problem (1.2) is equivalent to find p C such that
A p , y p + φ ( y ) φ ( z ) 0 , y C ,
(1.4)

which is called the mixed variational inequality of Browder type.

If φ = 0 , then the problem (1.2) is equivalent to find p C such that
F ( p , y ) + A p , y p 0 , y C ,
(1.5)

which is called the generalized equilibrium problem.

If A = 0 and φ = 0 , then the problem (1.2) is equivalent to (1.1).

For solving the above equilibrium problems, let us assume that the bifunction F : C × C R satisfies the following conditions:

(A1) F ( x , x ) = 0 , x C ;

(A2) F is monotone, i.e., F ( x , y ) + F ( y , x ) 0 , x , y C ;

(A3)
lim sup t 0 F ( t z + ( 1 t ) x , y ) F ( x , y ) , x , y , z C ;

(A4) for each x C , y F ( x , y ) is convex and weakly lower semicontinuous.

Equilibrium problems have intensively been studied. It has been shown that equilibrium problems cover fixed point problems, variational inequality problems, inclusion problems, saddle problems, complementarity problem, minimization problem, and Nash equilibrium problem; see [120] and the references therein.

Let S : C C be a mapping. In this paper, we use F ( S ) to stand for the set of fixed points. Recall that the mapping S is said to be nonexpansive if
S x S y x y , x , y C .
S is said to be κ-strictly pseudocontractive if there exists a constant κ [ 0 , 1 ) such that
S x S y 2 x y 2 + κ x y S x + S y 2 , x , y C .

It is clear that the class of κ-strictly pseudocontractive includes the class of nonexpansive mappings as a special case. The class of κ-strictly pseudocontractive mappings was introduced by Browder and Petryshyn [21]; for existence and approximation of fixed points of the class of mappings, see [2229] and the references therein.

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 κ-inverse strongly monotone if there exists a constant α > 0 such that
A x A y , x y κ A x A y 2 , x , y C .

It is clear that the κ-inverse being strongly monotone is monotone and Lipschitz continuous.

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 well 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 . The class of monotone operators is one of the most important classes of operators. Within the past several decades, many authors have been devoting their efforts to the studies of the existence and convergence of zero points for maximal monotone operators.

Let F ( x , y ) = A x , y x , x , y C . We see that the problem (1.1) is reduced to the following classical variational inequality. Find x C such that
A x , y x 0 , y C .
(1.6)

It is well known that x C is a solution to (1.6) if and only if x is a fixed point of the mapping P C ( I ρ A ) , where ρ > 0 is a constant, and I is the identity mapping. If C is bounded, closed, and convex, then the solution set of the variational inequality (1.6) is nonempty.

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

Lemma 1.1 [21]

Let S : C C be a κ-strictly pseudocontractive mapping. Define S t : C C by S t x = t x + ( 1 t ) S x for each x C . Then, as t [ κ , 1 ) , S t is nonexpansive such that F ( S t ) = F ( S ) .

Lemma 1.2 [2]

Let C be a nonempty, closed, and convex subset of H, and F : C × C R 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 1.3 [30]

Let A be a monotone mapping of C into H and N C v 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 .

Lemma 1.4 [31]

Let { a n } n = 1 be real numbers in [ 0 , 1 ] such that n = 1 a n = 1 . Then we have the following:
i = 1 a i x i 2 i = 1 a i x i 2

for any given bounded sequence { x n } n = 1 in H.

Lemma 1.5 [32]

Let 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 r 0 . Then lim n x n y n = 0 .

Lemma 1.6 [21]

Let C be a nonempty, closed, and convex subset of H, and S : C C a strictly pseudocontractive mapping. If { x n } is a sequence in C such that x n x and lim n x n S x n = 0 , then x = S x .

Lemma 1.7 [33]

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.

2 Main results

Theorem 2.1 Let C be a nonempty, closed, and convex subset of H, S : C C a κ-strictly pseudocontractive mapping with a nonempty fixed point set, and A : C H an L-Lipschitz continuous and monotone mapping. Let F m be a bifunction from C × C to which satisfies (A1)-(A4), B m : C H a continuous and monotone mapping, φ m : C R a lower semicontinuous and convex function for each m 1 . Assume that F : = m = 1 GMEP ( F m , B m , φ m ) VI ( C , A ) F ( S ) is not empty. Let { α n } , { β n } , and { δ n , m } be real number sequences in ( 0 , 1 ) . Let { λ n } , { r n , m } be positive real number sequences. Let { x n } be a sequence generated in the following manner:
{ x 1 H , x n + 1 = α n x n + ( 1 α n ) ( β n I + ( 1 β n ) S ) Proj C ( m = 1 δ n , m z n , m λ n A y n ) , n 1 , y n = Proj C ( m = 1 δ n , m z n , m λ n A m = 1 δ n , m z n , m ) ,
where z n , m is such that
F m ( z n , m , z ) + B m z n , m , z z n , m + φ m ( z ) φ m ( z n , m ) + 1 r n , m z z n , m , z n , m x n 0 , z C .
Assume that { α n } , { β n } , { δ n , m } , { λ n } , { r n , m } satisfy the following restrictions:
  1. (a)

    0 < a α n b < 1 ;

     
  2. (b)

    κ β n c < 1 ;

     
  3. (c)

    m = 1 δ n , m = 1 , and 0 < d δ n , m 1 ;

     
  4. (d)

    lim inf n r n , m > 0 and e λ n f , where e , f ( 0 , 1 / L ) .

     

Then the sequence { x n } weakly converges to some point x ¯ F .

Proof The proof is split into five steps.

Step 1. Show that the sequence { x n } is bounded.

Define G m ( p , y ) = F m ( p , y ) + B m p , y p + φ m ( y ) φ m ( p ) , p , y C . Next, we prove that the bifunction G m satisfies the conditions (A1)-(A4). Therefore, generalized mixed equilibrium problem is equivalent to the following equilibrium problem: find p C such that G m ( p , y ) 0 , y C . It is clear that G m satisfies (A1). Next, we prove G m is monotone. Since B m is a continuous and monotone operator, we find from the definition of G that
G m ( y , z ) + G m ( z , y ) = F m ( y , z ) + B m y , z y + φ m ( z ) φ m ( y ) + F m ( z , y ) + B m z , y z + φ m ( y ) φ m ( z ) = F m ( z , y ) + F m ( y , z ) + B m z , y z + B m y , z y B m z B m y , y z 0 .
Next, we show G m satisfies (A3), that is,
lim sup t 0 G m ( t z + ( 1 t ) x , y ) G m ( x , y ) , x , y , z C .
Since B m is continuous and φ m is lower semicontinuous, we have
lim sup t 0 G m ( t z + ( 1 t ) x , y ) = lim sup t 0 F m ( t z + ( 1 t ) x , y ) + lim sup t 0 B m ( t z + ( 1 t ) x ) , y ( t z + ( 1 t ) x ) + lim sup t 0 ( φ m ( y ) φ m ( t z + ( 1 t ) x ) ) F m ( x , y ) + B m x , y x + φ m ( y ) φ m ( x ) = G m ( x , y ) .
Next, we show that, for each x C , y G m ( x , y ) is a convex and lower semicontinuous. For each x C , for all t ( 0 , 1 ) and for all y , z C , since F m satisfies (A4) and φ m is convex, we have
G m ( x , t y + ( 1 t ) z ) = F m ( x , t y + ( 1 t ) z ) + B m x , t y + ( 1 t ) z x + φ m ( t y + ( 1 t ) z ) φ m ( x ) t ( F m ( x , y ) + B m x , y x + φ m ( y ) φ m ( x ) ) + ( 1 t ) ( F m ( x , z ) + B m x , z x + φ m ( z ) φ m ( x ) ) = t G m ( x , y ) + ( 1 t ) G m ( x , z ) .
Thus, y G m ( x , y ) is convex. Similarly, we find that y G m ( x , y ) is also lower semicontinuous. Put u n = Proj C ( m = 1 N δ n , m z n , m λ n A y n ) and v n = m = 1 N δ n , m z n , m . Letting p F , we see that
u n p 2 v n λ n A y n p 2 v n λ n A y n u n 2 = v n p 2 v n u n 2 + 2 λ n ( A y n A p , p y n + A p , p y n + A y n , y n u n ) v n p 2 v n y n 2 y n u n 2 + 2 v n λ n A y n y n , u n y n .
Notice that A is L-Lipschitz continuous and y n = Proj C ( v n λ n A v n ) . It follows that
v n λ n A y n y n , u n y n λ n L v n y n u n y n .
It follows that
u n p 2 v n p 2 + ( λ n 2 L 2 1 ) v n y n 2 .
(2.1)
On the other hand, we have
v n p 2 m = 1 δ n , m z n , m p 2 m = 1 δ n , m T r n , m x n p 2 x n p 2 ,
(2.2)
where T r n , m = { z C : G m ( z , y ) + 1 r y z , z x 0 , y C } . Substituting (2.2) into (2.1), we obtain
u n p 2 x n p 2 + ( λ n 2 L 2 1 ) v n y n 2 .
Putting S n = β n I + ( 1 β n ) S , we find from Lemma 1.1 that S n is nonexpansive and F ( S n ) = F ( S ) . It follows that
x n + 1 p 2 α n x n p 2 + ( 1 α n ) S n u n p 2 α n x n p 2 + ( 1 α n ) u n p 2 α n x n p 2 + ( 1 α n ) ( x n p 2 + ( λ n 2 L 2 1 ) v n y n 2 ) x n p 2 + ( 1 α n ) ( λ n 2 L 2 1 ) v n y n 2 x n p 2 .
(2.3)

It follows from Lemma 1.7 that the lim n x n p exists. This shows that { x n } is bounded. Since { x n } is bounded, we may assume that a subsequence { x n i } of { x n } converges weakly to ξ.

Step 2. Show that ξ VI ( C , A )

From (2.3), we find that β n ( 1 λ n 2 L 2 ) v n y n 2 x n p 2 x n + 1 p 2 . In view of the restrictions (b) and (d), we see that lim n v n y n = 0 . Since y n u n λ L v n y n , we have that lim n y n u n = 0 . It follows that
lim n v n u n = 0 .
(2.4)
Notice that
z n , m p 2 = T r n , m x n T r n , m p 2 T r n , m x n T r n , m p , x n p = 1 2 ( z n , m p 2 + x n p 2 z n , m x n 2 ) .
This implies that z n , m p 2 x n p 2 z n , m x n 2 . Since v n = m = 1 δ n , m z n , m , where m = 1 δ n , m = 1 , we find that
v n p 2 m = 1 δ n , m z n , m p 2 x n p 2 m = 1 δ n , m z n , m x n 2 .
It follows that
x n + 1 p 2 α n x n p 2 + ( 1 α n ) S n u n p 2 α n x n p 2 + ( 1 α n ) u n p 2 α n x n p 2 + ( 1 α n ) v n p 2 x n p 2 ( 1 α n ) m = 1 δ n , m z n , m x n 2 .
This implies that ( 1 α n ) δ n , m z n , m x n 2 x n p 2 x n + 1 p 2 . In view of the restrictions (a) and (c), we find that
lim n z n , m x n = 0 .
(2.5)
Let T be the maximal monotone mapping defined by
T x = { A x + N C x , x C , , x C .
For any given ( x , y ) G ( T ) , we have y A x N C x . So, we have x m , y A x 0 , for all m C . On the other hand, we have u n = Proj C ( v n λ n A y n ) . We obtain
x u n , u n v n λ n + A y n 0 .
In view of the monotonicity of A, we see that
x u n i , y x u n i , A x x u n i , A x x u n i , u n i v n i λ n i + A y n i = x u n i , A x A u n i + x u n i , A u n i A y n i x u n i , u n i v n i λ n i x u n i , A u n i A y n i x u n i , u n i v n i λ n i
in view of v n x n m = 1 δ n , m z n , m x n . It follows from (2.5) that lim n v n x n = 0 . Notice that u n x n u n v n + v n x n . It follows that
lim n u n x n = 0 .
(2.6)

This in turn implies that u n i ξ . It follows that x ξ , y 0 . Notice that T is maximal monotone and hence 0 T ξ . This shows from Lemma 1.3 that ξ VI ( C , A ) .

Step 3. Show that ξ GMEP ( F m , B m , φ m ) .

It follows from (2.5) that { z n i , m } converges weakly to ξ for each m 1 . Since z n , m = T r n , m x n , we have
G m ( z n , m , z ) + 1 r n , m z z n , m , z n , m x n 0 , z C .
From the assumption (A2), we see that
z z n i , m , z n i , m x n i r n i , m G m ( z , z n i , m ) , z C .
In view of the assumption (A4), we find from (2.5) that G m ( z , ξ ) 0 , z C . For t m with 0 < t m 1 and z C , let z t m = t m z + ( 1 t m ) ξ , for each 1 m N . Since z C and ξ C , we have z t m C . It follows that G m ( z t m , ξ ) 0 . Notice that
0 = G m ( z t m , z t m ) t m G m ( z t m , z ) + ( 1 t m ) G m ( z t m , ξ ) t m G m ( z t m , z ) ,

which yields G m ( z t m , z ) 0 , z C . Letting t m 0 , one sees that G m ( ξ , z ) 0 , z C . This implies that ξ GMEP ( F m , B m , φ m ) for each m 1 . This proves that ξ m = 1 GMEP ( F m , B m , φ m ) .

Step 4. Show that ξ F ( S ) .

Since lim n x n p exists, we put lim n x n p = d > 0 . It follows that
lim n x n + 1 p = lim n α n ( x n p ) + ( 1 α n ) ( S n u n p ) = d .
Notice that lim sup n S n u n p d . From Lemma 1.5, we see that
lim n x n S n u n = 0 .
(2.7)
Since
S n x n x n S n x n S n u n + S n u n x n x n u n + S n u n x n ,
we find from (2.6) and (2.7) that
lim n x n S n x n = 0 .
(2.8)

In view of S x n x n S x n S n x n + S n x n x n , we find from (2.8) that lim n x n S x n = 0 . This implies from Lemma 1.6 that ξ F ( S ) . This completes the proof that ξ F .

Step 5. Show that the whole sequence { x n } weakly converges to ξ.

Let { x n j } be another subsequence of { x n } converging weakly to ξ , where ξ ξ . In the same way, we can show that ξ F . Since the space H enjoys Opial’s condition, we, therefore, obtain
d = lim inf i x n i ξ < lim inf i x n i ξ = lim inf j x j ξ < lim inf j x j ξ = d .

This is a contradiction. Hence ξ = ξ . This completes the proof. □

3 Applications

In this section, we consider solutions of the mixed equilibrium problem (1.3), which includes the Ky Fan inequality as a special case.

The so-called mixed equilibrium problem is to find p C such that
F ( p , y ) + φ ( y ) φ ( z ) 0 , y C .

The mixed equilibrium problem includes the Ky Fan inequality, fixed point problems, saddle problems, and complementary problems as special cases.

Theorem 3.1 Let C be a nonempty, closed, and convex subset of H, S : C C a κ-strictly pseudocontractive mapping with a nonempty fixed point set, and A : C H a L-Lipschitz continuous and monotone mapping. Let F m be a bifunction from C × C to which satisfies (A1)-(A4), and φ m : C R a lower semicontinuous and convex function for each m 1 . Assume that F : = m = 1 MEP ( F m , φ m ) VI ( C , A ) F ( S ) is not empty. Let { α n } , { β n } and { δ n , m } be real number sequences in ( 0 , 1 ) . Let { λ n } , { r n , m } be positive real number sequences. Let { x n } be a sequence generated in the following manner:
{ x 1 H , x n + 1 = α n x n + ( 1 α n ) ( β n I + ( 1 β n ) S ) Proj C ( m = 1 δ n , m z n , m λ n A y n ) , n 1 , y n = Proj C ( m = 1 δ n , m z n , m λ n A m = 1 δ n , m z n , m ) ,
where z n , m is such that
F m ( z n , m , z ) + φ m ( z ) φ m ( z n , m ) + 1 r n , m z z n , m , z n , m x n 0 , z C .
Assume that { α n } , { β n } , { δ n , m } , { λ n } , { r n , m } satisfy the following restrictions:
  1. (a)

    0 < a α n b < 1 ;

     
  2. (b)

    κ β n c < 1 ;

     
  3. (c)

    m = 1 δ n , m = 1 , and 0 < d δ n , m 1 ;

     
  4. (d)

    lim inf n r n , m > 0 and e λ n f , where e , f ( 0 , 1 / L ) .

     

Then the sequence { x n } weakly converges to some point x ¯ F .

Proof If B m = 0 , we draw the desired conclusion immediately from Theorem 2.1. □

Further, if S is nonexpansive, we find from Theorem 3.1 the following result.

Corollary 3.2 Let C be a nonempty, closed, and convex subset of H, S : C C a nonexpansive mapping with a nonempty fixed point set, and A : C H an L-Lipschitz continuous and monotone mapping. Let F m be a bifunction from C × C to which satisfies (A1)-(A4), and φ m : C R a lower semicontinuous and convex function for each m 1 . Assume that F : = m = 1 MEP ( F m , φ m ) VI ( C , A ) F ( S ) is not empty. Let { α n } , { β n } , and { δ n , m } be real number sequences in ( 0 , 1 ) . Let { λ n } , { r n , m } be positive real number sequences. Let { x n } be a sequence generated in the following manner:
{ x 1 H , x n + 1 = α n x n + ( 1 α n ) S Proj C ( m = 1 δ n , m z n , m λ n A y n ) , n 1 , y n = Proj C ( m = 1 δ n , m z n , m λ n A m = 1 δ n , m z n , m ) ,
where z n , m is such that
F m ( z n , m , z ) + φ m ( z ) φ m ( z n , m ) + 1 r n , m z z n , m , z n , m x n 0 , z C .
Assume that { α n } , { β n } , { δ n , m } , { λ n } , { r n , m } satisfy the following restrictions:
  1. (a)

    0 < a α n b < 1 ;

     
  2. (b)

    m = 1 δ n , m = 1 , and 0 < d δ n , m 1 ;

     
  3. (c)

    lim inf n r n , m > 0 and e λ n f , where e , f ( 0 , 1 / L ) .

     

Then the sequence { x n } weakly converges to some point x ¯ F .

If A = 0 , we find from Theorem 2.1 the following result.

Theorem 3.3 Let C be a nonempty, closed, and convex subset of H, S : C C a κ-strictly pseudocontractive mapping with a nonempty fixed point set. Let F m be a bifunction from C × C to which satisfies (A1)-(A4), B m : C H a continuous and monotone mapping, φ m : C R a lower semicontinuous and convex function for each m 1 . Assume that F : = m = 1 GMEP ( F m , B m , φ m ) F ( S ) is not empty. Let { α n } , { β n } , and { δ n , m } be real number sequences in ( 0 , 1 ) . Let { r n , m } be a positive real number sequence. Let { x n } be a sequence generated in the following manner:
x 1 H , x n + 1 = α n x n + ( 1 α n ) ( β n I + ( 1 β n ) S ) m = 1 δ n , m z n , m , n 1 ,
where z n , m is such that
F m ( z n , m , z ) + B m z n , m , z z n , m + φ m ( z ) φ m ( z n , m ) + 1 r n , m z z n , m , z n , m x n 0 , z C .
Assume that { α n } , { β n } , { δ n , m } , and { r n , m } satisfy the following restrictions:
  1. (a)

    0 < a α n b < 1 ;

     
  2. (b)

    κ β n c < 1 ;

     
  3. (c)

    m = 1 δ n , m = 1 , and 0 < d δ n , m 1 ;

     
  4. (d)

    lim inf n r n , m > 0 .

     

Then the sequence { x n } weakly converges to some point x ¯ F .

Declarations

Acknowledgements

The authors are grateful to the reviewers for useful suggestions which improved the contents of the article.

Authors’ Affiliations

(1)
School of Information Engineering, North China University of Water Resources and Electric Power
(2)
School of Mathematics and Information Science, North China University of Water Resources and Electric Power

References

  1. Fan K: A minimax inequality and its application. 3. In Inequalities. 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. 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
  4. 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
  5. Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 2005, 6: 117–136.MathSciNetMATHGoogle Scholar
  6. Qin X, Agarwal RP: Shrinking projection methods for a pair of asymptotically quasi- ϕ -nonexpansive mappings. Numer. Funct. Anal. Optim. 2010, 31: 1072–1089. 10.1080/01630563.2010.501643MathSciNetView ArticleMATHGoogle Scholar
  7. 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
  8. Cho SY, Li W, Kang SM: Convergence analysis of an iterative algorithm for monotone operators. J. Inequal. Appl. 2013., 2013: Article ID 199Google Scholar
  9. 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
  10. Qin X, Chang SS, Cho YJ: Iterative methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal., Real World Appl. 2010, 11: 2963–2972. 10.1016/j.nonrwa.2009.10.017MathSciNetView ArticleMATHGoogle Scholar
  11. 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
  12. Chen JH: Iterations for equilibrium and fixed point problems. J. Nonlinear Funct. Anal. 2013., 2013: Article ID 4Google Scholar
  13. Yuan Q, Cho SY: Proximal point algorithms for zero points of nonlinear operators. Fixed Point Theory Appl. 2014., 2014: Article ID 42Google Scholar
  14. 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
  15. Cho SY, Kang SM, 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
  16. Qin X, Cho SY, Wang L: A regularization method for treating zero points of the sum of two monotone operators. Fixed Point Theory Appl. 2014., 2014: Article ID 75Google Scholar
  17. 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
  18. Qin X, Cho YJ, Kang SM: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J. Comput. Appl. Math. 2009, 225: 20–30. 10.1016/j.cam.2008.06.011MathSciNetView 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., 2011: Article ID 10Google Scholar
  20. Qin X, Cho SY, Kang SM: Strong convergence of shrinking projection methods for quasi- ϕ -nonexpansive mappings and equilibrium problems. J. Comput. Appl. Math. 2010, 234: 750–760. 10.1016/j.cam.2010.01.015MathSciNetView ArticleMATHGoogle Scholar
  21. Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl. 1967, 20: 197–228. 10.1016/0022-247X(67)90085-6MathSciNetView ArticleMATHGoogle Scholar
  22. Qin X, Cho SY, Kang SM: An extragradient-type method for generalized equilibrium problems involving strictly pseudocontractive mappings. J. Glob. Optim. 2011, 49: 679–693. 10.1007/s10898-010-9556-2MathSciNetView ArticleMATHGoogle Scholar
  23. Cheng P, Wu H: On asymptotically strict pseudocontractions and equilibrium problems. J. Inequal. Appl. 2013., 2013: Article ID 251Google Scholar
  24. Jung JS: Iterative methods for mixed equilibrium problems and strictly pseudocontractive mappings. Fixed Point Theory Appl. 2012., 2012: Article ID 184Google Scholar
  25. 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
  26. Qin X, Shang M, Kang SM: Strong convergence theorems of modified Mann iterative process for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 2009, 70: 1257–1264. 10.1016/j.na.2008.02.009MathSciNetView ArticleMATHGoogle Scholar
  27. Kim JK, Buong N: A new iterative method for equilibrium problems and fixed point problems for infinite family of nonself strictly pseudocontractive mappings. Fixed Point Theory Appl. 2013., 2013: Article ID 286Google Scholar
  28. Cho SY, Qin X, Wang L: Strong convergence of a splitting algorithm for treating monotone operators. Fixed Point Theory Appl. 2014., 2014: Article ID 94Google Scholar
  29. Wang G, Sun S: Hybrid projection algorithms for fixed point and equilibrium problems in a Banach space. Adv. Fixed Point Theory 2013, 3: 578–594.Google Scholar
  30. 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
  31. Yang L, Zhao F, Kim JK: Hybrid projection method for generalized mixed equilibrium problem and fixed point problem of infinite family of asymptotically quasi- ϕ -nonexpansive mappings in Banach spaces. Appl. Math. Comput. 2012, 218: 6072–6082. 10.1016/j.amc.2011.11.091MathSciNetView ArticleMATHGoogle Scholar
  32. Schu J: Weak and strong convergence of fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 1991, 43: 153–159. 10.1017/S0004972700028884MathSciNetView ArticleMATHGoogle Scholar
  33. 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

Copyright

© Li and Zhao; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.