Open Access

Strong convergence theorems for equilibrium problems involving a family of nonexpansive mappings

Fixed Point Theory and Applications20142014:200

https://doi.org/10.1186/1687-1812-2014-200

Received: 7 May 2014

Accepted: 10 September 2014

Published: 25 September 2014

Abstract

We give new hybrid variants of extragradient methods for finding a common solution of an equilibrium problem and a family of nonexpansive mappings. We present a scheme that combines the idea of an extragradient method and a successive iteration method as a hybrid variant. Then, this scheme is modified by projecting on a suitable convex set to get a better convergence property under certain assumptions in a real Hilbert space.

MSC:65K10, 65K15, 90C25, 90C33.

Keywords

equilibrium problems fixed point pseudomonotone Lipschitz-type continuous extragradient method nonexpansive mappings

1 Introduction

In this paper, we always assume that is a real Hilbert space with the inner product , and the induced norm . Let C be a nonempty closed convex subset of and the bifunction f : C × C R . Then f is called strongly monotone on C with β > 0 iff
f ( x , y ) + f ( y , x ) β x y 2 x , y C ;
monotone on C iff
f ( x , y ) + f ( y , x ) 0 x , y C ;
pseudomonotone on C iff
f ( x , y ) 0 implies f ( y , x ) 0 x , y C ;
Lipschitz-type continuous on C in the sense of Mastroeni [1] iff there exist positive constants c 1 > 0 , c 2 > 0 such that
f ( x , y ) + f ( y , z ) f ( x , z ) c 1 x y 2 c 2 y z 2 x , y , z C .
An equilibrium problem, shortly E P ( f , C ) , is to find a point in
Sol ( f , C ) = { x C : f ( x , y ) 0 y C } .
Let a mapping T of C into itself. Then T is called contractive with constant δ ( 0 , 1 ) iff
T ( x ) T ( y ) δ x y x , y C .
The mapping T is called strictly pseudocontractive iff there exists a constant k [ 0 , 1 ) such that
T ( x ) T ( y ) 2 x y 2 + k ( I T ) ( x ) ( I T ) ( y ) 2 .

In the case k = 0 , the mapping T is called nonexpansive on C. We denote by Fix ( T ) the set of fixed points of T.

Let T i : C C , i Γ , be a family of nonexpansive mappings where Γ stands for an index set. In this paper, we are interested in the problem of finding a common element of the solution set of problem E P ( f , C ) and the set of fixed points F = i Γ Fix ( T i ) , namely:
Find  x F Sol ( f , C ) ,
(1.1)

where the function f and the mappings T i , i Γ , satisfy the following conditions:

(A1) f ( x , x ) = 0 for all x C and f is pseudomonotone on C,

(A2) f is Lipschitz-type continuous on C with constants c 1 > 0 and c 2 > 0 ,

(A3) f is upper semicontinuous on C,

(A4) For each x C , f ( x , ) is convex and subdifferentiable on C,

(A5) F Sol ( f , C ) .

Under these assumptions, for each r > 0 and x C , there exists a unique element z C such that
f ( z , y ) + 1 r y z , z x 0 y C .
(1.2)
Problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, equilibrium equilibriums, fixed point problems (see, e.g., [27]). Recently, it has become an attractive field for many researchers in both theory and its solution methods (see, e.g., [3, 4, 812] and the references therein). Most of these algorithms are based on inequality (1.2) for solving the underlying equilibrium problem when F Sol ( f , C ) . Motivated by this idea for finding a common point of Sol ( f , C ) and the fixed point set Fix ( T ) of a nonexpansive mapping T, Takahashi and Takahashi [13] first introduced an iterative scheme by the viscosity approximation method. The sequence { x n } is defined by
{ x 0 C , f ( u n , y ) + 1 r n y u n , u n x n 0 y C , x n + 1 = α n g ( x n ) + ( 1 α n ) T ( u n ) n 0 ,
where g : C C is contractive. Under certain conditions over the parameters { α n } and { r n } , they showed that the sequences { x n } and { u n } strongly converge to z = Pr Fix ( T ) Sol ( f , C ) g ( z ) , where Pr C denotes the projection on C. At each iteration n in all of these algorithms, it requires to solve approximation auxiliary equilibrium problems for finding a common solution of an equilibrium problem and a fixed point problem. In order to avoid this requirement, Anh [14] recently proposed a hybrid extragradient algorithm for finding a common point of the set Fix ( T ) Sol ( f , C ) . Starting with an arbitrary initial point x 0 C , iteration sequences are defined by
{ y k = argmin { λ k f ( x k , y ) + 1 2 y x k 2 : y C } , t k = argmin { λ k f ( y k , t ) + 1 2 t x k 2 : t C } , x k + 1 = α k x 0 + ( 1 α k ) T ( x k ) .
(1.3)

Under certain conditions onto parameters { λ k } and { α k } , he showed that the sequences { x k } , { y k } and { t k } weakly converge to the point x Fix ( T ) Sol ( f , C ) in a real Hilbert space. At each main iteration n of the scheme, he only solved strongly convex problems on C, but the proof of convergence was still done under the assumptions that x n + 1 x n 0 .

For finding a common point of a family of nonexpansive mappings T i ( i Γ ), as a corollary of Theorem 2.1 in [15], Zhou proposed the following iteration scheme:
{ x 0 H  chosen arbitrarily, C 1 , i = C , C 1 = i Γ C 1 , i , x 1 = Pr C 1 ( x 0 ) , y n , i = ( 1 α n , i ) x n + α n , i T i ( x n ) , C n + 1 , i = { z C n , i : α n , i ( 1 2 α n , i ) x n T i ( x n ) 2 x n z , y n , i T i ( y n , i ) } , C n + 1 = i Γ C n + 1 , i , x n + 1 = Pr C n + 1 ( x 0 ) .
(1.4)

Under the restrictions of the control sequences 0 < lim inf n α n , i lim sup n α n , i a i < 1 2 , he showed that the sequence { x n } defined by (1.4) strongly converges to x = Pr F ( x 0 ) in a real Hilbert space , where F = i Γ Fix ( T i ) .

In this paper, motivated by Ceng et al. [16, 17], Wang and Guo [18], Zhou [15], Nadezhkina and Takahashi [10], Cho et al. [19], Takahashi and Takahashi [13], Anh [6, 12] and Anh et al. [20, 21], we introduce several modified hybrid extragradient schemes to modify the iteration schemes (1.3) and (1.4) to obtain new strong convergence theorems for a family of nonexpansive mappings and the equilibrium problem E P ( f , C ) in the framework of a real Hilbert space .

To investigate the convergence of this scheme, we recall the following technical lemmas which will be used in the sequel.

Lemma 1.1 ([14], Lemma 3.1)

Let C be a nonempty closed convex subset of a real Hilbert space . Let f : C × C R be a pseudomonotone and Lipschitz-type continuous bifunction. For each x C , let f ( x , ) be convex and subdifferentiable on C. Suppose that the sequences { x n } , { y n } , { t n } are generated by scheme (1.3) and x Sol ( f , C ) . Then
t n x 2 x n x 2 ( 1 2 λ n c 1 ) x n y n 2 ( 1 2 λ n c 2 ) y n t n 2 n 0 .

Lemma 1.2 Let C be a closed convex subset of a real Hilbert space , and let Pr C be the metric projection from on to C (i.e., for x H , Pr C is the only point in C such that x Pr C x = inf { x z : z C } ). Given x H and z C . Then z = Pr C x if only if there holds the relation x z , y z 0 for all y C .

Lemma 1.3 Let be a real Hilbert space. Then the following equations hold:
  1. (i)

    x y 2 = x 2 y 2 2 x y , y for all x , y H .

     
  2. (ii)

    t x + ( 1 t ) y 2 = t x 2 + ( 1 t ) y 2 t ( 1 t ) x y 2 for all t [ 0 , 1 ] and x , y H .

     

2 Convergence theorems

Now, we prove the main convergence theorem.

Theorem 2.1 Let C be a nonempty closed convex subset of a real Hilbert space . Suppose that assumptions (A1)-(A5) are satisfied and { T i } i Γ is a family of nonexpansive mappings from C into itself and a nonempty common fixed points set F. Let { x n } be a sequence generated by the following scheme:
{ x 0 H  chosen arbitrarily , C 1 , i = D 1 , i = C , C 1 = i Γ C 1 , i , D 1 = i Γ D 1 , i , x 1 = Pr C 1 D 1 x 0 , y n = argmin { λ n f ( x n , y ) + 1 2 y x n 2 : y C } , z n = argmin { λ n f ( y n , y ) + 1 2 z x n 2 : z C } , y n , i = ( 1 α n , i ) z n + α n , i T i z n , C n + 1 , i = { z C n , i : α n , i ( 1 2 α n , i ) z n T i z n 2 z n z , y n , i T i y n , i } , C n + 1 = i Γ C n + 1 , i , D n + 1 , i = { z D n , i : y n , i z x n z } , D n + 1 = i Γ D n + 1 , i , x n + 1 = Pr C n + 1 D n + 1 x 0 , 0 < lim inf α n , i lim sup α n , i < 1 , { λ n } [ a , b ]  for some  a , b ( 0 , 1 L ) ,  where  L = max { 2 c 1 , 2 c 2 } .

Then the sequences { x n } , { y n } and { z n } strongly converge to the same point Pr F Sol ( f , C ) x 0 .

Proof The proof of this theorem is divided into several steps.

Step 1. Claim that C n and D n are closed and convex for all n 0 .

We have to show that for any fixed point but arbitrary i Γ , C n , i is closed and convex for every n 0 . This can be proved by induction on n. It is obvious that C 1 , i = C is closed and convex. Assume that C n , i is closed and convex for some n N = { 1 , 2 , } . We have that the set
A = { z C : α n , i ( 1 2 α n , i ) z n T i z n 2 z n z , y n , i T i y n , i }
is closed and convex, and C n + 1 , i = C n , i A , hence C n + 1 , i is closed and convex. Then C n is closed and convex for all n 0 . We can write D n + 1 , i under the form
D n + 1 , i = { z D n , i : y n , i x n 2 + 2 y n , i x n , x n z 0 } .

Then D n + 1 , i is closed and convex. Thus, D n is closed and convex.

Step 2. Claim that F Sol ( f , C ) C n D n for all n N .

First, we show that F C n by induction on n. It suffices to show that F C n , i .

We have F C = C 1 , i is obvious. Suppose F C n , i for some n N . We have to show that F C n + 1 , i . Indeed, let w F , by inductive hypothesis, we have w C n , i and
z n T i z n 2 = z n T i z n , z n T i z n = 1 α n , i z n y n , i , z n T i z n = 1 α n , i z n y n , i , z n T i z n ( y n , i T i y n , i ) + 1 α n , i z n y n , i , y n , i T i y n , i = 1 α n , i z n y n , i , z n T i z n ( y n , i T i y n , i ) + 1 α n , i z n w + w y n , i , y n , i T i y n , i = 1 α n , i z n y n , i , z n y n , i + 1 α n , i z n y n , i , T i y n , i T i z n + 1 α n , i z n w , y n , i T i y n , i + 1 α n , i w y n , i , y n , i T i y n , i 2 α n , i z n y n , i 2 + 1 α n , i z n w , y n , i T i y n , i + 1 α n , i w y n , i , y n , i T i y n , i .
(2.1)
On the other hand, for all w F and y n , i C , we have
w y n , i 2 T i w T i y n , i , w y n , i = w T i y n , i , w y n , i = w y n , i + y n , i T i y n , i , w y n , i = w y n , i 2 + y n , i T i y n , i , w y n , i ,
and hence
w y n , i , y n , i T i y n , i 0 .
Combining this with (2.1), we obtain
z n T i z n 2 2 α n , i z n y n , i 2 + 1 α n , i z n w , y n , i T i y n , i 2 α n , i z n T i z n 2 + 1 α n , i z n w , y n , i T i y n , i .
This follows that
α n , i ( 1 2 α n , i ) z n T i z n 2 z n w , y n , i T i y n , i .

By the definition of C n + 1 , i , we have w C n + 1 , i , and so F C n + 1 , i for all i Γ , which deduces that F C n . This shows that F Sol ( f , C ) C n for all n N .

Next, we will prove F Sol ( f , C ) D n by induction on n N . It suffices to show that F Sol ( f , C ) D n , i . Indeed, F C = D 1 , i so F Sol ( f , C ) D 1 , i . Suppose that F Sol ( f , C ) D n , i . Let x F Sol ( f , C ) , then x D n , i . Using Lemma 1.1, we get
y n , i x 2 = ( 1 α n , i ) z n + α n , i T i z n x 2 ( 1 α n , i ) z n x 2 + α n , i T i z n T i x 2 z n x 2 x n x 2 ( 1 2 λ n c 1 ) x n y n 2 ( 1 2 λ n c 2 ) y n z n 2 x n x 2 .
(2.2)

Then we have x D n + 1 , i and hence F Sol ( f , C ) D n + 1 , i . This shows that F Sol ( f , C ) D n , which yields that F Sol ( f , C ) C n D n for all n N .

Step 3. Claim that the sequence { x n } is bounded and there exists the limit lim n x n x 0 = c .

From x n = Pr C n D n x 0 , it follows that
x 0 x n , x n y 0 y C n D n .
(2.3)
Then, using Step 2, we have F Sol ( f , C ) C n D n and
x 0 x n , x n w 0 w F Sol ( f , C ) .
(2.4)
Combining this and assumption (A5), the projection Pr F Sol ( f , C ) x 0 is well defined and there exits a unique point p such that p = Pr F Sol ( f , C ) x 0 . So, we have
0 x 0 x n , x n p = x 0 x n , x n x 0 + x 0 p x 0 x n 2 + x 0 x n x 0 p ,
and hence
x 0 x n x 0 p .
Then the sequence { x n } is bounded. So, the sequences { y n } , { z n } , { y n , i } , { T i y n , i } also are bounded. Since x n + 1 C n + 1 D n + 1 C n D n and (2.3), we have
0 x 0 x n , x n x n + 1 = x 0 x n , x n x 0 + x 0 x n + 1 x 0 x n 2 + x 0 x n x 0 x n + 1 ,

and hence x 0 x n x 0 x n + 1 . This together with the boundedness of { x n } implies that the limit lim n x n x 0 = c exists.

Step 4. We claim that lim n x n = q C .

Since C m D m C n D n , x m = Pr C m D m x 0 C n D n for any positive integer m n and (2.3), we have
x 0 x n , x n x n + m 0 .
Then
x n x n + m 2 = x n x 0 + x 0 x n + m 2 = x n x 0 2 + x 0 x n + m 2 2 x 0 x n , x 0 x n + m x 0 x n + m 2 x n x 0 2 2 x 0 x n , x n x n + m x 0 x n + m 2 x n x 0 2 .
(2.5)

Passing the limit in (2.5) as n , we get lim n x n x n + m = 0 m N . Hence, { x n } is a Cauchy sequence in a real Hilbert space and so lim n x n = q C .

Step 5. We claim that q = Pr F Sol ( f , C ) x 0 , where q = lim n x n .

First we show that q F Sol ( f , C ) . Since x n + 1 = Pr C n + 1 D n + 1 x 0 , we have x n + 1 D n + 1 . Then x n + 1 D n + 1 , i and
y n , i x n + 1 x n x n + 1 ,
which yields that
x n y n , i x n x n + 1 + x n + 1 y n , i 2 x n x n + 1 .
Combining this and lim n x n x m = 0 for all m N , we get
lim n x n y n , i = 0 .
(2.6)
For each x Sol ( f , C ) F , by (2.2) we have
( 1 2 b c 1 ) x n y n 2 ( 1 2 λ n c 1 ) x n y n 2 x n x 2 y n , i x 2 = ( x n x + y n , i x ) ( x n x y n , i x ) ( x n x + y n , i x ) ( x n y n , i ) .
Using this, the boundedness of sequences { x n } , { y n , i } and (2.6), we obtain
lim n x n y n = 0 .
(2.7)
By a similar way, we also have lim n z n y n = 0 . Then it follows from the inequality
x n z n x n y n + y n z n
that
lim n x n z n = 0 .
(2.8)
On the other hand, we have
y n , i z n y n , i x n + x n z n .
Combining this, (2.6) and (2.8), we obtain lim n y n , i z n = 0 . By the definition of the sequence { y n , i } , we have
y n , i z n = α n , i T i z n z n ,
and hence
lim n T i z n z n = 0 ,
which yields that
T i x n x n T i x n T i z n + T i z n z n + x n z n 2 x n z n + T i z n z n 0 as  n
and
lim n T i x n x n = 0 .

It follows from Step 4 that lim n T i x n = q . Hence q F .

Now we show that q Sol ( f , C ) . By Step 5, we have y n q as n .

Since y n is the unique solution of the strongly convex problem
min { 1 2 y x n 2 + λ n f ( x n , y ) : y C } ,
we get
0 2 ( λ n f ( x n , y ) + 1 2 y x n 2 ) ( y n ) + N C ( y n ) .
From this it follows that
0 = λ n w + y n x n + w ¯ ,
where w 2 f ( x n , ) ( y n ) and w ¯ N C ( y n ) . By the definition of the normal cone N C , we have
y n x n , y y n λ n w , y n y y C .
(2.9)
On the other hand, since f ( x n , ) is subdifferentiable on C, by the well-known Moreau-Rockafellar theorem, there exists w 2 f ( x n , ) ( y n ) such that
f ( x n , y ) f ( x n , y n ) w , y y n y C .
Combining this with (2.9), we have
λ n ( f ( x n , y ) f ( x n , y n ) ) y n x n , y n y y C .
Then, using { λ n } [ a , b ] ( 0 , 1 L ) , (2.7), x n q , y n q as n and the upper semicontinuity of f, we have
f ( q , y ) 0 q C .
This means that q Sol ( f , C ) . By taking the limit in (2.4), we have
x 0 q , q w 0 w F Sol ( f , C ) ,

which implies that q = Pr F Sol ( f , C ) x 0 . Thus, the subsequences { x n } , { y n } , { z n } strongly converge to the same point q = Pr F Sol ( f , C ) x 0 . This completes the proof. □

Now, notice that w F
z n T i z n 2 = z n w + w T i z n 2 = z n w 2 + w T i z n 2 + 2 z n w , w T i z n 2 z n w 2 + 2 z n w , w z n + z n T i z n = 2 z n w 2 2 z n w 2 + 2 z n w , z n T i z n = 2 z n w , z n T i z n .
Hence
y n , i w 2 = ( 1 α n , i ) ( z n w ) + α n , i ( T i z n w ) 2 = ( 1 α n , i ) z n w 2 + α n , i T i z n w 2 α n , i ( 1 α n , i ) T i z n z n 2 = ( 1 α n , i ) z n w 2 + α n , i T i z n z n + z n w 2 α n , i ( 1 α n , i ) T i z n z n 2 = ( 1 α n , i ) z n w 2 + α n , i T i z n z n 2 + α n , i z n w 2 + 2 α n , i T i z n z n , z n w α n , i ( 1 α n , i ) T i z n z n 2 z n w 2 + 2 α n , i z n w , z n T i z n + 2 α n , i T i z n z n , z n w α n , i ( 1 α n , i ) T i z n z n 2 = z n w 2 α n , i ( 1 α n , i ) T i z n z n 2 .
(2.10)

From (2.10) and using the methods in Theorem 2.1, we can get the following convergence result.

Theorem 2.2 Let C be a nonempty closed convex subset of a real Hilbert space . Suppose that assumptions (A1)-(A5) are satisfied and { T i } i Γ is a family of nonexpansive mappings from C into itself and a nonempty common fixed points set F. Let { x n } be a sequence generated by the following scheme:
{ x 0 H  chosen arbitrarily , C 1 , i = D 1 , i = C , C 1 = i Γ C 1 , i , D 1 = i Γ D 1 , i , x 1 = Pr C 1 D 1 x 0 , y n = argmin { λ n f ( x n , y ) + 1 2 y x n 2 : y C } , z n = argmin { λ n f ( y n , y ) + 1 2 z x n 2 : z C } , y n , i = ( 1 α n , i ) z n + α n , i T i z n , C n + 1 , i = { z C n , i : y n , i z 2 z n z 2 α n , i ( 1 α n , i ) z n T i z n 2 } , C n + 1 = i Γ C n + 1 , i , D n + 1 , i = { z D n , i : y n , i z x n z } , D n + 1 = i Γ D n + 1 , i , x n + 1 = Pr C n + 1 D n + 1 x 0 , 0 < lim inf α n , i lim sup α n , i < 1 , { λ n } [ a , b ]  for some  a , b ( 0 , 1 L ) ,  where  L = max { 2 c 1 , 2 c 2 } .

Then the sequences { x n } , { y n } and { z n } converge strongly to the same point Pr F Sol ( f , C ) x 0 .

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Haiphong University

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Copyright

© Thanh; licensee Springer. 2014

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