Open Access

Strong convergence theorems for equilibrium problems and fixed point problem of multivalued nonexpansive mappings via hybrid projection method

Journal of Inequalities and Applications20122012:164

https://doi.org/10.1186/1029-242X-2012-164

Received: 18 December 2011

Accepted: 6 July 2012

Published: 23 July 2012

Abstract

In this paper, a new iterative process by the hybrid projection method is constructed. Strong convergence of the iterative process to a common element of the set of common fixed points of a finite family of generalized nonexpansive multivalued mappings and the solution set of two equilibrium problems in a Hilbert space is proved. Our results extend some important recent results.

MSC:47H10, 47H09.

Keywords

equilibrium problemhybrid projection methodstrong convergencecommon fixed pointgeneralized nonexpansive multivalued mapping

1 Introduction

Let C be a nonempty closed convex subset of a Hilbert space H. A subset C H is called proximal if for each x H there exists an element y C such that
x y = dist ( x , C ) = inf { x z : z C } .
We denote by CB ( C ) and P ( C ) the collection of all nonempty closed bounded subsets and nonempty proximal bounded subsets of C, respectively. The Hausdorff metric H on CB ( H ) is defined by
H ( A , B ) : = max { sup x A dist ( x , B ) , sup y B dist ( y , A ) } ,

for all A , B CB ( H ) .

Let T : H 2 H be a multivalued mapping. An element x H is said to be a fixed point of T, if x T x . The set of fixed points of T will be denoted by F ( T ) .

Definition 1.1 A multivalued mapping T : H CB ( H ) is called
  1. (i)
    nonexpansive if
    H ( T x , T y ) x y , x , y H .
     
  2. (ii)

    quasi-nonexpansive if F ( T ) and H ( T x , T p ) x p for all x H and all p F ( T ) .

     

Recently, J. Garcia-Falset, E. Llorens-Fuster and T. Suzuki [1] introduced a new condition on singlevalued mappings, called condition (E), which is weaker than nonexpansiveness.

Definition 1.2 A mapping T : H H is said to satisfy the condition ( E μ ) provided that
x T y μ x T x + x y , x , y H .

We say that T satisfies the condition (E) whenever T satisfies ( E μ ) for some μ 1 .

Now we modify this condition for multivalued mappings as follows (see also [2]):

Definition 1.3 A multivalued mapping T : H CB ( H ) is said to satisfy the condition ( P ) provided that
H ( T x , T y ) μ dist ( x , T x ) + η x y , x , y H ,

for some μ , η 1 .

It is obvious that every nonexpansive multivalued mapping satisfies the condition ( P ) . The theory of multivalued mappings has applications in control theory, convex optimization, differential equations and economics. Theory of nonexpansive multivalued mappings is harder than the corresponding theory of nonexpansive single valued mappings. Different iterative processes have been used to approximate fixed points of multivalued nonexpansive mappings (see [38]). Let Φ be a bifunction from C × C into R , where R is the set of real numbers. The equilibrium problem for Φ : C × C R is to find x C such that
Φ ( x , y ) 0 , y C .

The set of solutions is denoted by E P ( Φ ) . It is well known that this problem is closely related to minimax inequalities (see [9] and [10]). The equilibrium problem includes fixed point problems, optimization problems and variational inequality problems as special cases. Some methods have been proposed to solve the equilibrium problem, see, for example, [1114].

Recently, many authors have studied the problems of finding a common element of the set of fixed points of nonexpansive single valued mappings and the set of solutions of an equilibrium problem in the framework of Hilbert spaces: see, for instance, [1529] and the references therein. In this paper, a new iterative process by the hybrid projection method is constructed. Strong convergence of the iterative process to a common element of a set of common fixed points of a finite family of multivalued mappings satisfying the condition (P) and the solution set of two equilibrium problems in a Hilbert space is proved. Our results generalize some results of Tada, Takahashi [15] and many others.

2 Preliminaries

Let us recall the following definitions and results which will be used in the sequel.

Lemma 2.1 ([6])

Let H be a real Hilbert space. Then for i , j = 1 , 2 , , k we have
a 1 x 1 + a 2 x 2 + + a k x k 2 a 1 x 1 2 + a 2 x 2 2 + + a k x k 2 a i a j x i x j 2

for all x i , x j H and a i , a j [ 0 , 1 ] with i = 1 k a i = 1 .

Let C be a closed convex subset of H. For every point x H , there exists a unique nearest point in C, denoted by P C x such that
x P C x x y , y C .

P C is called the metric projection of H onto C. It is well known that P C is a nonexpansive mapping.

Lemma 2.2 ([16])

Let C be a closed convex subset of H. Given x H and a point z C . Then z = P C x if and only if
x z , z y 0 , y C .

Lemma 2.3 ([30])

Let C be a closed convex subset of H. Then for all x H and y C we have
y P C x 2 + x P C x 2 x y 2 .

For solving the equilibrium problem, we assume that the bifunction Φ satisfies the following conditions:

(A1) Φ ( x , x ) = 0 for any x C ,

(A2) Φ is monotone, i.e., Φ ( x , y ) + Φ ( y , x ) 0 for any x , y C ,

(A3) Φ is upper-hemicontinuous, i.e., for each x , y , z C ,
lim sup t 0 + Φ ( t z + ( 1 t ) x , y ) Φ ( x , y ) ,

(A4) Φ ( x , ) is convex and lower semicontinuous for each x C .

The following lemma was proved in [11].

Lemma 2.4 Let C be a nonempty closed convex subset of H and let Φ be a bifunction of C × C into R satisfying (A 1)-(A 4). Let r > 0 and x H . Then, there exists z C such that
Φ ( z , y ) + 1 r y z , z x 0 y C .

The following lemma was given in [14].

Lemma 2.5 Assume that Φ : C × C R satisfies (A 1)-(A 4). For r > 0 and x H , define a mapping T r : H C as follows:
T r x = { z C : Φ ( z , y ) + 1 r y z , z x 0 , y C } .
Then, the following hold:
  1. (i)

    T r is single valued;

     
  2. (ii)
    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. (iii)

    F ( T r ) = E P ( Φ ) ;

     
  4. (iv)

    E P ( Φ ) is closed and convex.

     

The following lemma was proved in [31] for nonexpansive multivalued mappings. The statement is true for quasi-nonexpansive multivalued mappings as well. To avoid repetition, we omit the details of the proof.

Lemma 2.6 Let C be a closed convex subset of a real Hilbert space H. Let T : C CB ( C ) be a quasi-nonexpansive multivalued mapping such that T ( p ) = { p } for all p F ( T ) . Then F ( T ) is closed and convex.

Now, following Shahzad and Zegeye [3], we remove the restriction T ( p ) = { p } for all p F ( T ) . Let T : C P ( C ) be a multivalued mapping and
P T ( x ) = { y T x : x y = dist ( x , T x ) } .

We use a similar argument as in the proof of Lemma 3.1 in [31] to obtain the following lemma.

Lemma 2.7 Let C be a closed convex subset of a real Hilbert space H. Let T : C P ( C ) be a multivalued mapping such that P T is quasi-nonexpansive. Then F ( T ) is closed and convex.

Note that for all p F ( T ) , P T ( p ) = { p } . We remark that there exist some examples of multivalued mappings for which P T is nonexpansive (see [3] for details), so that the assumption on T is not artificial.

3 The main result

Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H, Φ 1 and Φ 2 be two bifunctions of C × C into R satisfying (A 1)-(A 4). Let T i : C CB ( C ) , ( i = 1 , 2 , , m ), be a finite family of quasi-nonexpansive multivalued mappings, each satisfying the condition (P). Assume further that F = i = 1 m F ( T i ) E P ( Φ 1 ) E P ( Φ 2 ) and T i ( p ) = { p } , ( i = 1 , 2 , , m ), for each p F . For C 0 = C , let { x n } and { u n } be sequences generated by the following algorithm:
{ x 0 C , u n C such that Φ 1 ( u n , y ) + 1 r n y u n , u n x n 0 ; y C , u n C such that Φ 2 ( u n , y ) + 1 s n y u n , u n x n 0 ; y C , v n = δ n u n + ( 1 δ n ) u n , w n = a n , 0 v n + a n , 1 z n , 1 + + a n , m z n , m , y n = b n , 0 v n + b n , 1 z n , 1 + + b n , m z n , m , C n + 1 = { v C n : y n v x n v } , x n + 1 = P C n + 1 x 0 : n 0 ,
where z n , i T i v n , z n , i T i w n for i = 1 , 2 , , m . Assume that { a n , j } , { b n , j } , { δ n } and { r n } , { s n } satisfy the following conditions:
  1. (i)

    { a n , j } , { b n , j } , { δ n } [ a , b ] ( 0 , 1 ) ( j = 0 , 1 , 2 , , m ),

     
  2. (ii)

    { r n } , { s n } ( 0 , ) , and lim inf n r n > 0 and lim inf n s n > 0 .

     

Then, the sequences { x n } and { v n } converge strongly to P F x 0 .

Proof First, we show that F C n for all n 0 . Fix q F . We set
T r Φ 1 ( x ) = { z C : Φ 1 ( z , y ) + 1 r y z , z x 0 y C } .
Hence we have u n = T r n Φ 1 x n and u n = T s n Φ 2 x n . By Lemma 2.5, we have
u n q = T r n Φ 1 x n T r n Φ 1 q x n q
and
u n q = T s n Φ 2 x n T s n Φ 2 q x n q ,
which implies that
v n q δ n u n q + ( 1 δ n ) u n q x n q .
Since T i is quasi-nonexpansive, for i = 1 , 2 , , m , we have
w n q = a n , 0 v n + a n , 1 z n , 1 + + a n , m z n , m q a n , 0 v n q + a n , 1 z n , 1 q + + a n , m z n , m q a n , 0 x n q + a n , 1 dist ( z n , 1 , T 1 q ) + + a n , m dist ( z n , m , T m q ) a n , 0 x n q + a n , 1 H ( T 1 v n , T 1 q ) + + a n , m H ( T m v n , T m q ) a n , 0 x n q + a n , 1 v n q + + a n , m v n q a n , 0 x n q + a n , 1 x n q + + a n , m x n q x n q ,
and also
y n q = b n , 0 v n + b n , 1 z n , 1 + + b n , m z n , m q b n , 0 v n q + b n , 1 z n , 1 q + + b n , m z n , m q b n , 0 x n q + b n , 1 dist ( z n , 1 , T 1 q ) + + b n , m dist ( z n , m , T m q ) b n , 0 x n q + b n , 1 H ( T 1 w n , T 1 q ) + + b n , m H ( T m w n , T m q ) b n , 0 x n q + b n , 1 w n q + + b n , m w n q b n , 0 x n q + b n , 1 x n q + + b n , m x n q x n q .
Therefore q C n , which implies that
F = i = 1 m F ( T i ) E P ( Φ 1 ) E P ( Φ 2 ) C n , for all  n 0 .
We observe that C n is closed and convex (see [32]). Now we show that lim n x n x 0 exists. By Lemma 2.6 we have F is closed and convex. Put w = P F x 0 . From x n = P C n x 0 and x n + 1 C n + 1 C n we have
x n x 0 x n + 1 x 0 .
Also from w F C n and x n = P C n x 0 for all n 0 , we get that
x n x 0 w x 0 .
It follows that the sequence { x n } is bounded and nondecreasing. Hence the limit lim n x n x 0 exists. We show that lim n x n = u C . For k > n we have x k = P C k x 0 C k C n . Now by applying Lemma 2.3 we have
x k x n 2 x k x 0 2 x n x 0 2 .
Since lim n x n x 0 exists, it follows that { x n } is a Cauchy sequence, and hence there exists u C such that lim n x n = u . Putting k = n + 1 , in the above inequality we have
lim n x n + 1 x n = 0 .
From x n + 1 C n + 1 , we have
y n x n + 1 x n x n + 1 ,
so that lim n y n x n + 1 = 0 . This implies that lim n y n = u . Take q F . By Lemma 2.1, for each 1 i m , we have
w n q 2 = a n , 0 v n + a n , 1 z n , 1 + + a n , m z n , m q 2 a n , 0 v n q 2 + a n , 1 z n , 1 q 2 + + a n , m z n , m q 2 a n , i a n , o v n z n , i 2 a n , 0 v n q 2 + a n , 1 dist ( z n , 1 , T 1 q ) 2 + + a n , m dist ( z n , m , T m q ) 2 a n , i a n , o v n z n , i 2 a n , 0 v n q 2 + a n , 1 H ( T 1 v n , T 1 q ) 2 + + a n , m H ( T m v n , T m q ) 2 a n , i a n , o v n z n , i 2 a n , 0 v n q 2 + a n , 1 v n q 2 + + a n , m v n q 2 a n , i a n , o v n z n , i 2 v n q 2 a n , i a n , o v n z n , i 2 ,
and also
y n q 2 = b n , 0 v n + b n , 1 z n , 1 + + b n , m z n , m q 2 b n , 0 v n q 2 + b n , 1 z n , 1 q 2 + + b n , m z n , m q 2 b n , 0 v n q 2 + b n , 1 dist ( z n , 1 , T 1 q ) 2 + + b n , m dist ( z n , m , T m q ) 2 b n , 0 v n q 2 + b n , 1 H ( T 1 w n , T 1 q ) 2 + + b n , m H ( T m w n , T m q ) 2 b n , 0 v n q 2 + b n , 1 w n q 2 + + b n , m w n q 2 b n , 0 v n q 2 + b n , 1 v n q 2 + + b n , m v n q 2 b n , i a n , i a n , o v n z n , i 2 x n q 2 b n , i a n , i a n , o v n z n , i 2 .
(1)
So we have that
a 2 b v n z n , i 2 b n , i a n , i a n , o v n z n , i 2 v n q 2 y n q 2 x n q 2 y n q 2 ,
which implies that
lim n v n z n , i = 0 , for  i = 1 , 2 , , m .
Hence
lim n dist ( v n , T i v n ) lim n v n z n , i = 0 ( i = 1 , 2 , , m ) .
As above u n = T r n Φ 1 x n so that
u n q 2 = T r n Φ 1 x n T r n Φ 1 q 2 T r n Φ 1 x n T r n Φ 1 q , x n q = u n q , x n q = 1 2 ( u n q 2 + x n q 2 x n u n 2 )
and hence
u n q 2 x n q 2 x n u n 2 .
(2)
And also by u n = T s n Φ 2 x n we have
u n q 2 = T s n Φ 2 x n T s n Φ 2 q 2 T s n Φ 2 x n T s n Φ 2 q , x n q = u n q , x n q = 1 2 ( u n q 2 + x n q 2 x n u n 2 )
and hence
u n q 2 x n q 2 x n u n 2 .
(3)
Now we use (2) and (3) to obtain
v n q 2 δ n u n q 2 + ( 1 δ n ) u n q 2 x n q 2 δ n x n u n 2 ( 1 δ n ) x n u n 2 .
It follows from (1) and the last inequality that
y n q 2 v n q 2 x n q 2 δ n x n u n 2 ( 1 δ n ) x n u n 2 .
So we have
a x n u n 2 δ n x n u n 2 x n q 2 y n q 2 ,
and
( 1 b ) x n u n 2 ( 1 δ n ) x n u n 2 x n q 2 y n q 2 .
Since lim n x n = lim n y n = u , we obtain that
lim n u n x n = lim n u n x n = 0 ,
which implies that
lim n v n x n = 0 .
Since lim n v n x n = 0 , for i = 1 , 2 , , m , we obtain that
dist ( x n , T i x n ) x n v n + dist ( v n , T i v n ) + H ( T i v n , T i x n ) ( η + 1 ) x n v n + ( μ + 1 ) dist ( v n , T i v n ) 0 as  n .
(4)
We observe that u i = 1 m F ( T i ) . Indeed,
dist ( u , T i u ) u x n + dist ( x n , T i x n ) + H ( T i x n , T i u ) ( η + 1 ) u x n + ( μ + 1 ) dist ( x n , T i x n ) 0 as  n ,
which implies that u i = 1 m F ( T i ) . Let us show that u E P ( Φ 1 ) E P ( Φ 2 ) . From lim n x n = u and lim n x n u n = 0 , we have u n u as n . Since u n = T r n Φ 1 x n we obtain
Φ 1 ( u n , y ) + 1 r n y u n , u n x n 0 y C .
From (A2), we have
1 r n y u n , u n x n Φ 1 ( y , u n ) ,
and hence
y u n , u n x n r n Φ 1 ( y , u n ) .
Since
u n x n r n 0 ,
and u n u , from (A4) we have
0 Φ 1 ( y , u ) , y C .
For t ( 0 , 1 ] and y C , let y t = t y + ( 1 t ) u . Since y , u C , and C is convex we have y t C and hence Φ 1 ( y t , u ) 0 . So, from (A1) and (A4), we have
0 = Φ 1 ( y t , y t ) t Φ 1 ( y t , y ) + ( 1 t ) Φ 1 ( y t , u ) t Φ 1 ( y t , y ) ,
which gives Φ 1 ( y t , y ) 0 . From (A3) we have 0 Φ 1 ( u , y ) , y C and hence u E P ( Φ 1 ) . Similarly, we have u E P ( Φ 2 ) . Now we show that u = P F x 0 . Since x n = P C n x 0 , by Lemma 2.2 we have
z x n , x 0 x n 0 , z C n .
Since u F C n we get
z u , x 0 u 0 , z F .

Now by Lemma 2.2 we obtain that u = P F x 0 . □

By substituting P T i by T i and using a similar argument as in Theorem 3.1, we obtain the following result.

Theorem 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H, Φ 1 and Φ 2 be two bifunctions of C × C into R satisfying (A 1)-(A 4). Let T i : C P ( C ) , ( i = 1 , 2 , , m ), be a finite family of multivalued mappings such that each P T i is quasi-nonexpansive and satisfies the condition (P). Assume further that F = i = 1 m F ( T i ) E P ( Φ 1 ) E P ( Φ 2 ) . For C 0 = C , let { x n } and { v n } be sequences generated by the following algorithm:
{ x 0 C , u n C such that Φ 1 ( u n , y ) + 1 r n y u n , u n x n 0 ; y C , u n C such that Φ 2 ( u n , y ) + 1 s n y u n , u n x n 0 ; y C , v n = δ n u n + ( 1 δ n ) u n , w n = a n , 0 v n + a n , 1 z n , 1 + + a n , m z n , m , y n = b n , 0 v n + b n , 1 z n , 1 + + b n , m z n , m , C n + 1 = { v C n : y n v x n v } , x n + 1 = P C n + 1 x 0 : n 0 ,
where z n , i P T i ( v n ) , z n , i P T i ( w n ) for i = 1 , 2 , , m . Assume that { a n , j } , { b n , j } , { δ n } and { r n } , { s n } satisfy the following conditions:
  1. (i)

    { a n , j } , { b n , j } , { δ n } [ a , b ] ( 0 , 1 ) ( j = 0 , 1 , 2 , , m ),

     
  2. (ii)

    { r n } , { s n } ( 0 , ) , and lim inf n r n > 0 and lim inf n s n > 0 .

     

Then, the sequences { x n } and { v n } converge strongly to P F x 0 .

As a result, for single valued mappings we obtain the following theorem.

Theorem 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H and Φ 1 and Φ 2 be two bifunctions of C × C into R satisfying (A 1)-(A 4). Let T i : C C ( i = 1 , 2 , , m ), be a finite family of quasi-nonexpansive mappings, each satisfying the condition (P). Assume further that F = i = 1 m F ( T i ) E P ( Φ 1 ) E P ( Φ 2 ) . For C 0 = C , let { x n } and { v n } be sequences generated by the following algorithm:
{ x 0 C , u n C such that Φ 1 ( u n , y ) + 1 r n y u n , u n x n 0 ; y C , u n C such that Φ 2 ( u n , y ) + 1 s n y u n , u n x n 0 ; y C , v n = δ n u n + ( 1 δ n ) u n , w n = a n , 0 v n + a n , 1 T 1 v n + + a n , m T m v n , y n = b n , 0 v n + b n , 1 T 1 w n + + b n , m T m w n , C n + 1 = { v C n : y n v x n v } , x n + 1 = P C n + 1 x 0 : n 0 .
Assume that { a n , j } , { b n , j } , { δ n } and { r n } , { s n } satisfy the following conditions:
  1. (i)

    { a n , j } , { b n , j } , { δ n } [ a , b ] ( 0 , 1 ) ( j = 0 , 1 , 2 , , m ),

     
  2. (ii)

    { r n } , { s n } ( 0 , ) , and lim inf n r n > 0 and lim inf n s n > 0 .

     

Then, the sequences { x n } and { v n } converge strongly to P F x 0 .

Theorem 3.4 Let C be a nonempty closed convex subset of a real Hilbert space H. Let T i : C P ( C ) ( i = 1 , 2 , , m ), be a finite family of multivalued mappings such that P T i is quasi-nonexpansive and satisfies the condition (P). Assume further that F = i = 1 m F ( T i ) . For C 0 = C , let { x n } be the sequence generated by the following algorithm:
{ x 0 C , w n = a n , 0 x n + a n , 1 z n , 1 + + a n , m z n , m , y n = b n , 0 x n + b n , 1 z n , 1 + + b n , m z n , m , C n + 1 = { v C n : y n v x n v } , x n + 1 = P C n + 1 x 0 : n 0 ,

where z n , i P T i ( x n ) , z n , i P T i ( w n ) for i = 1 , 2 , , m . Assume that { a n , j } , { b n , j } [ a , b ] ( 0 , 1 ) ( j = 0 , 1 , 2 , , m ), Then, the sequence { x n } converges strongly to P F x 0 .

Proof Putting Φ 1 ( x , y ) = Φ 2 ( x , y ) = 0 for all x , y C and r n = s n = 1 in Theorem 3.2, we have u n = u n = x n and hence v n = x n . Now, the desired conclusion follows directly from Theorem 3.2. □

Now, we supply an example to illustrate the main result of this paper.

Example 3.5 We consider the nonempty closed convex subset C = [ 0 , 5 ] of the Hilbert space R . Define two mappings T 1 and T 2 on C as follows:
T 1 ( x ) = [ x 6 , x 2 ] , T 2 ( x ) = { , x 5 , { 1 } , x = 5 .
We note that T 1 and T 2 are quasi-nonexpansive mappings satisfying the condition (P), (for details, see [2]). Also we define two bifunctions Φ 1 and Φ 2 as follows:
{ Φ 1 : C × C R , Φ 1 ( x , y ) = y x , { Φ 2 : C × C R , Φ 2 ( x , y ) = y 2 + x y 2 x 2 .
It is easy to see that Φ 1 and Φ 2 satisfy the conditions (A1)-(A4). If we put r n = 5 and s n = 1 , then u n = T r n Φ 1 x n = 0 and u n = T s n Φ 1 x n = x n 3 s n + 1 = x n 4 (for details, see [26]). Put a n , i = b n , i = 1 3 for i = 0 , 1 , 2 and δ n = 1 2 . For any arbitrary x 0 C we have
C 1 = { v C : | y 0 v | | x 0 v | } = [ 0 , x 0 + y 0 2 ] .
Since x 0 + y 0 2 x 0 , we obtain that
x 1 = P C 1 x 0 = x 0 + y 0 2 .
By continuing this process we obtain
C n + 1 = { v C n : | y n v | | x n v | } = [ 0 , x n + y n 2 ] ,
and hence
x n + 1 = P C n + 1 x 0 = x n + y n 2 .
Now, we have the following algorithm:
{ x 0 C , v n = x n 8 , z n , i T i ( v n ) , i = 1 , 2 , w n = 1 3 v n + 1 3 z n , 1 + 1 3 z n , 2 , z n , i T i w n , i = 1 , 2 , y n = 1 3 v n + 1 3 z n , 1 + 1 3 z n , 2 , x n + 1 = x n + y n 2 : n 0 .
Putting z n , 1 = z n , 2 = v n 6 and z n , 1 = z n , 2 = w n 6 we get that
x n + 1 = 679 1 , 296 x n = ( 679 1 , 296 ) n + 1 x 0 , n 0 .

We observe that for an arbitrary x 0 C , x n is convergent to zero. We note that F = F ( T 1 ) F ( T 2 ) E P ( Φ 1 ) E P ( Φ 2 ) = { 0 } .

Remark 3.6 Since every nonexpansive mapping is quasi-nonexpansive and satisfies the condition (P), our results hold for nonexpansive mappings.

Remark 3.7 Our results generalize the results of Tada and Takahashi [15], of a nonexpansive single valued mapping to a finite family of generalized nonexpansive multivalued mappings.

Declarations

Acknowledgements

Research of the first author was supported in part by a grant from Imam Khomeini International University, under the grant number 751164-91.

Authors’ Affiliations

(1)
Department of Mathematics, Imam Khomeini International University
(2)
Young Researchers Club, Babol Branch, Islamic Azad University

References

  1. Garcia-Falset J, Llorens-Fuster E, Suzuki T: Fixed point theory for a class of generalized nonexpansive mappings. J. Math. Anal. Appl. 2011, 375: 185–195. 10.1016/j.jmaa.2010.08.069MathSciNetView ArticleGoogle Scholar
  2. Abkar A, Eslamian M: Common fixed point results in CAT(0) spaces. Nonlinear Anal. 2011, 74: 1835–1840. 10.1016/j.na.2010.10.056MathSciNetView ArticleGoogle Scholar
  3. Shahzad N, Zegeye H: Strong convergence results for nonself multimaps in Banach spaces. Proc. Am. Math. Soc. 2008, 136: 539–548.MathSciNetView ArticleGoogle Scholar
  4. Song Y, Wang H: Convergence of iterative algorithms for multivalued mappings in Banach spaces. Nonlinear Anal. 2009, 70: 1547–1556. 10.1016/j.na.2008.02.034MathSciNetView ArticleGoogle Scholar
  5. Shahzad N, Zegeye H: On Mann and Ishikawa iteration schemes for multivalued maps in Banach space. Nonlinear Anal. 2009, 71: 838–844. 10.1016/j.na.2008.10.112MathSciNetView ArticleGoogle Scholar
  6. Eslamian M, Abkar A: One-step iterative process for a finite family of multivalued mappings. Math. Comput. Model. 2011, 54: 105–111. 10.1016/j.mcm.2011.01.040MathSciNetView ArticleGoogle Scholar
  7. Nanan N, Dhompongsa S: A common fixed point theorem for a commuting family of nonexpansive mappings one of which is multivalued. Fixed Point Theory Appl. 2011, 2011: 54. 10.1186/1687-1812-2011-54MathSciNetView ArticleGoogle Scholar
  8. Abkar A, Eslamian M: Convergence theorems for a finite family of generalized nonexpansive multivalued mappings in CAT(0) spaces. Nonlinear Anal. 2012, 75: 1895–1903. 10.1016/j.na.2011.09.040MathSciNetView ArticleGoogle Scholar
  9. Fan K: A minimax inequality and applications. In Inequalities III. Edited by: Shisha O. Academic Press, New York; 1972:103–113.Google Scholar
  10. Brezis H, Nirembero L, Stampacchia G: A remark on Ky Fan’s minimax principle. Boll. Unione Mat. Ital. 1972, 6: 293–300.Google Scholar
  11. Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.MathSciNetGoogle Scholar
  12. Flam SD, Antipin AS: Equilibrium programming using proximal-link algorithms. Math. Program. 1997, 78: 29–41.MathSciNetView ArticleGoogle Scholar
  13. Moudafi A, Thera M Lecture Notes in Economics and Mathematical Systems 477. In Proximal and Dynamical Approaches to Equilibrium Problems. Springer, New York; 1999:187–201.Google Scholar
  14. Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 2005, 6: 117–136.MathSciNetGoogle Scholar
  15. Tada A, Takahashi W: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl. 2007, 133: 359–370. 10.1007/s10957-007-9187-zMathSciNetView ArticleGoogle Scholar
  16. Takahashi S, Takahashi W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 2007, 331: 506–515. 10.1016/j.jmaa.2006.08.036MathSciNetView ArticleGoogle Scholar
  17. Moudafi A: Weak convergence theorems for nonexpansive mappings and equilibrium problems. J. Nonlinear Convex Anal. 2008, 9: 37–43.MathSciNetGoogle Scholar
  18. Chang SS, Lee HWJ, Kim JK: 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 ArticleGoogle Scholar
  19. Cho YJ, Qin X, Kang JI: Convergence theorems based on hybrid methods for generalized equilibrium problems and fixed point problems. Nonlinear Anal. 2009, 71: 4203–4214. 10.1016/j.na.2009.02.106MathSciNetView ArticleGoogle Scholar
  20. Ceng LC, Al-Homidan S, Ansari QH, Yao JC: An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings. J. Comput. Appl. Math. 2009, 223: 967–974. 10.1016/j.cam.2008.03.032MathSciNetView ArticleGoogle Scholar
  21. Jaiboon C, Kumam P: Strong convergence for generalized equilibrium problems, fixed point problems and relaxed cocoercive variational inequalities. J. Inequal. Appl. 2010., 2010: Article ID 728028. doi:10.1155/2010/728028Google Scholar
  22. Jung JS: A general composite iterative method for generalized mixed equilibrium problems, variational inequality problems and optimization problems. J. Inequal. Appl. 2011., 2011: Article ID 51Google Scholar
  23. Ceng LC, Ansari QH, Yao JC: Viscosity approximation methods for generalized equilibrium problems and fixed point problems. J. Glob. Optim. 2009, 43: 487–502. 10.1007/s10898-008-9342-6MathSciNetView ArticleGoogle Scholar
  24. Zeng LC, Ansari QH, Shyu DS, Yao JC: Strong and weak convergence theorems for common solutions of generalized equilibrium problems and zeros of maximal monotone operators. Fixed Point Theory Appl. 2010., 2010: Article ID 590278Google Scholar
  25. Ceng LC, Ansari QH, Yao JC: Hybrid pseudoviscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings. Nonlinear Anal. Hybrid Syst. 2010, 4: 743–754. 10.1016/j.nahs.2010.05.001MathSciNetView ArticleGoogle Scholar
  26. Singthong U, Suantai S: Equilibrium problems and fixed point problems for nonspreading-type mappings in Hilbert space. Int. J. Nonlinear Anal. Appl. 2011, 2: 51–61.Google Scholar
  27. Ceng LC, Ansari QH, Yao JC: Hybrid proximal-type and hybrid shrinking projection algorithms for equilibrium problems, maximal monotone operators and relatively nonexpansive mappings. Numer. Funct. Anal. Optim. 2010, 31: 763–797. 10.1080/01630563.2010.496697MathSciNetView ArticleGoogle Scholar
  28. Zeng LC, Al-Homidan S, Ansari QH: Hybrid proximal-type algorithms for generalized equilibrium problems, maximal monotone operators and relatively nonexpansive mappings. Fixed Point Theory Appl. 2011., 2011: Article ID 973028Google Scholar
  29. Zeng LC, Ansari QH, Schaible S, Yao JC: Iterative methods for generalized equilibrium problems, systems of general generalized equilibrium problems and fixed point problems for nonexpansive mappings in Hilbert spaces. Fixed Point Theory 2011, 12: 293–308.MathSciNetGoogle Scholar
  30. Nakajo K, Takahashi W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 2003, 279: 372–379. 10.1016/S0022-247X(02)00458-4MathSciNetView ArticleGoogle Scholar
  31. Dhompongsa S, Kaewkhao A, Panyanak B: On Kirks strong convergence theorem for multivalued nonexpansive mappings on CAT(0) spaces. Nonlinear Anal. 2012, 75: 459–468. 10.1016/j.na.2011.08.046MathSciNetView ArticleGoogle Scholar
  32. Matinez-Yanes C, Xu HK: Strong convergence of the CQ method for fixed point processes. Nonlinear Anal. 2006, 64: 2400–2411. 10.1016/j.na.2005.08.018MathSciNetView ArticleGoogle Scholar

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