Strong Convergence of an Iterative Method for Generalized Mixed Equilibrium Problems and Fixed Point Problems

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doi:10.4236/am.2011.210169 Published Online October 2011 (http://www.SciRP.org/journal/am)

Strong Convergence of an Iterative Method for

Generalized Mixed Equilibrium Problems and

Fixed Point Problems

Lijun Chen, Jianhua Huang*

Institute of Mathematics and Computer, Fuzhou University, Fuzhou, China E-mail: chenlijun861010@163.com,*fjhjh57@yahoo.com.cn Received April 24, 2011; revised July 4, 2011; accepted July 11, 2011

Abstract

In this paper, we introduce a hybrid iterative method for finding a common element of the set of common solutions of generalized mixed equilibrium problems and the set of common fixed points of an finite family of nonexpansive mappings. Furthermore, we show a strong convergence theorem under some mild condi-tions.

Keywords:Generalized Mixed Equilibrium Problem, Hybrid Iterative Scheme, Fixed Point, Nonexpansive Mapping, Strong Convergence

1. Introduction

Equilibrium problems theory provides us with a natural, novel and unified framework for studying a wide class of problems arising in economics, finance, transportation, network and structural analysis, elasticity and optimiza-tion. The ideas and techniques of this theory are being used in a variety of diverse areas and proved to be pro-ductive and innovative.

Let H be a Hilbert space with inner product  , and norm . Let C be a nonempty closed convex subset of H

and T C: 2H a multivalued mapping. Let :C C R

be a real-valued function and :H C C  R



w u v, ,



 



w v u, ,



= 0

u C

be an equilibrium-like function, i.e.,

for each . The generalized mixed equilibrium problem (for short, GMEP) is to find



w u, ,v



  H C C



and wT u

 

such that



    

: , , , , 0,

GMEP  w u v  v u  u u   v C. (1.1) in particular, if T is single-valued mapping, this problem is equivalent to finding uC such that

 



T u u v, ,





   

v u,  u u, 0, v C.

      (1.2)

Denote the set of solutions of GMEP by . Now, we recall the following definitions.

A mapping f C: C is said to be contractive if there exists a constant  (0,1) such that

 

f x  f y  xy for any x y, C. A mapping

: g CC

 



is said to be firmly nonexpansive if



2

 

,

g x g y  g x g y xy . A mapping :

T CC is said to be nonexpansive if Tx Ty  xy for any x y, C. The set of fixed points of T is denoted by F T

 

.

Let

 

=1

N

i _i be a finite family of nonexpansive map-pings of C into H and . Define the map-pings

T

 

=1 Ø

N i i F T 



















,1 ,1 1 ,1

,2 ,2 2 ,1 ,2

, 1 , 1 1 , 2 , 1

, , , 1 ,

= 1

= = 1

n n n

n n n n

n N n N N n N n N

n n N n N N n N n N

U T I

U T U I

U T U

W U T U I

 

    



  



  

 

 _{ }



 _{ }





I

(1.3)

where

 

,



0,1



N n i

 

W map

=1

i for all . Such a mapping n is called

1

n

W  ping generated by T1,,TN

and

 

, ₌₁

N n i _i

 .

2. Preliminaries

Let C be a nonempty closed convex subset of a real Hil-bert space H. Then, for any xH



C

P

, there exists a unique nearest point in C, denoted by x



, such that

 

C

xP x  xy

(2)

of H into C. We know that is nonexpansive. What’s more,

C P

 

= _C , 0,

x P x x x y 

. x

y C

 

 

Let C be a convex subset of a real Hilbert space H,

:C C H

   and a Frechet differential function. Then k is said to be

:

k CR

-strongly convex if there exists a constant > 0 such that

   



 



2

, ,

2

, .

k y k x k x y x x y ,

C x y

 

 



   

If = 0, then k is said to be -convex. In particular, if 

 

y x, =yx for all y x, C, then k is said to be strongly convex.

Let C be a nonempty subset of a real Hilbert space H. A bifunction is said to be skew- symmetric if

 

, :C C R

    



u u v

 

,  0

   

, , ,

  

, , .

u v v u u v v

C

    0,

.

It is easy to see that if the skew-symmetric bifunction is linear in both arguments, then

 

,

  

 

u v , u C

 

We denote  for weak convergence and for strong convergence. A bifunction



:C C R

  



0, 0



is called weakly sequentially continuous at x y  C C if



x yn, n





x0, 0



 







n, n



y as n  for each sequence

x y in CC converging weakly to



x y0, 0



.

The function is called weakly sequentially con-tinuous on if it is weakly sequentially continuous at each point of .



,

  

C C

C



C



Let denote the set of nonempty closed bounded subset of X. For

 

CB X

 

,

A BCB X , define the Hausdorff metric  as follows:







 



, = max ,

, .

b B

b A B

,

a A

a

A B sup inf d a

sup d b a

 

 b

inf

 

In order to solve the generalized mixed equilibrium problems for an equilibrium-like bifunction

:H C C R

   

: 2

, we assume that satisfies the fol-lowing conditions with respect to the multivalued map-ping



H

T C :

 

1 for each fixed vC,



w u,



w T2, s

   

y T x, r



 

2

T x T y



  _r  _s

for all x y, C and r s, 



 

k2 k is Lipschitz continuous with constant > 0.

 

3   

 

, is weakly continuous on C C .

Recently Wei-You Zeng, Nan-Jing Huang and Chang- Wen Zhao [1] introduce and consider a new class of equilibrium problems, which is known as the generalized mixed equilibrium problems. Furthermore, they intro-duce an iterative scheme (1.4) by the viscosity approxi-mation method for finding a common element of the set of common solutions for generalized mixed equilibrium problems and the set of common fixed points of a se-quence of nonexpansive mappings in Hilbert space.

  





 



 

  



  



1 1

1

1 , ;

1

, , , , 0,

= 1

n n

n n n n n n

n n n n

T x T x

n

v v u u u k u k x v u v C

r

x W u

  

 

 



 _ _{ } 

 

 _ _



 _ _

       

 

  

 



, ,

n n

w w

w u

x f

(2.1)

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1215

Motivated and inspired by the research going on in this important field, we introduce the following hybrid iterative scheme (1.5) for finding a common element of the set of common solutions for generalized mixed equi-librium problems and the set of common fixed points of a sequence of nonexpansive mappings. We show that the approximation solution converges strongly to a unique solution of a class of variational inequalities under some mild conditions. Results obtained in this paper can be

viewed as an improvement and refinement of the recent results in this direction.

Algorithm 1.1. Let C be a nonempty closed convex subset of a real Hilbert space H, T C: CB H

 

be a multivalued mapping, f be a contraction of C into itself with coefficient 

 

0,1. Let be defined by (1.3), and . For given 1

: W C_n C > 0

r x C and w1T x

 

1 , there

exists sequences

 

xn ,

 

un in C and



wn:wnT

 

xn



in H such that for all n= 1, 2,,

  





 



 

  







1 1

1

1 , ;

1

, , , , , , 0,

=

n n n n

n n n n n n n n

n n n n n n n n n

w w T x T x

n

w u v v u u u k u k x v u v

r

x a f W x b x c W u

  

 



 _ _{ } 

 

 _ _

 

 

      

 

  

 



C

 (2.2)

where

 

an , and are three sequences in (0, 1) such that .

 

bn

a b

 

cn

= 1

n n n

It is easy to see that the iterative scheme (1.5) may be well defined.

c  

Let r be a positive number. For a given point xC

and , consider the following auxiliary prob-lem for GMEP: find such that

 

x

w T x

uC



    

 

   

, , , ,

1

, , 0,

x

w u v v u u u

k u k x v u v C

r

 



  

 

     , (2.3)

It is easy to see that if , then u is a solution of GMEP.

=

u x

We need the following important results.

Lemma 1.1. [2] Let C be a nonempty closed convex bounded subset of a real Hilbert space H and let

:C C R

   be a real-valued function satisfying

 

1 -

 

3 . Let : 2

H

T C be a multivalued mapping and

:H C C R

  

:C C



1

 

   

= : , , , , , , 0,

r x

T x u C w u v v u u u k u k x v u v C

r

  

 _ _ _ _ _ _ _ _ _{  } 

 

  (2.4)

Then there hold the following:

1) the auxiliary problem (1.6) has a unique solution; 2) r is single-valued;

3) if

T

1

  , it follows that is firmly nonex-pansive;

r T

4) F T

 

_r =;



5) is closed and convex.

Lemma 1.2. [3] Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let

 

=1

N i i

T be a finite family of nonexpansive mappings of C

into H and

 

=1 Ø

N i i F T 



, and let

 

_,

=1

N n i _i

= _i₌₁

F W



F T .

Lemma 1.3. [4] If the sequences

 

un and

 

xn are bounded and n is defined by (1.3), then the following

estimates hold:

W

1 1 1 1, ,

=1

2 ,

N

n n n n n n n i n i

i

W_u _ W u  u _ u  M



_   n 0

and

1 1 1 1, ,

=1

2 ,

N

i

W_x_ W x  x_ x  M



_   n 0

(4)

for some constant M > 0.

Lemma 1.4. [4] In a real Hilbert space H, x y z, , H

and with , there holds the following equality:

1, ,2 3 [0,1]

t t t  t₁ t₂ t₃= 1

2 2 2

1 2 3 1 2 3 .

t x t y t z t x t y t z 2

Lemma 1.5. [6] Let

 

xn and be bounded sequences in a Banach space X and let



be a

se-quence in [0,1] with .

Suppose

 

un

lim b 



n

b sup

0 <liminfn n nbn < 1

n n





1= 1

n n n

x_ b z b x

for all integers n0 and



1 1



lim sup _n _n _n _n 0.

n

z_ z x_ x

    

Then,

lim _n _n = 0.

n z x

Lemma 1.6. [5] Let is a sequence of nonnega-tive real numbers such that

 

an





1 1 , = 1

n n n n

a _   a b n , 2,

where

 

n is a sequence in (0,1), n=1n=

 _



and

lim sup_n_b_n _n0, then lim_n_a_n .



= 0

Lemma 1.7. [2] Let



xn be a sequence in a normed space



X,



such that

n

x is a Cauchy sequence.

Lemma 1.8. [7] Let A B, CB X

 

and a . Then for

A  > 1

 , there must exist a point such that .

bB

 

,b 



,

d a  A B

_{Lemma 1.9.}_{[5] In a real Hilbert space}



_H_{, there holds} the following equality:

2 2

2 , , , .

xy  x  y xy x yH

3. Main Results

Theorem 2.1. Let C be a nonempty closed convex bounded subset of a real Hilbert space H and ,



be a multivalued -Lipschitz continu-ous mapping with constant , and let

> 0 r

C C R



:

T CCB H

> 0

L :  

be a real-valued function satisfying

 

1 -

 

3 . and

:H C C

  

:C C H

R

 be an equilibrium-like function sat-isfying the conditions

 

1 -



4



. Assume that    is a Lipschitz function with lipschitz

constant > 0 which satisfies the conditions

 

1 -

 

cn are three sequences in (0,1) with satisfying the following conditions:

= 1 n c

n n

a  b

1) lim_n_a_n= 0,

;



4) n=1cn 1 cn < 

  .

=

x P f x



  . To proof Theorem 2.1, we first establish the following lemma.

Lemma 2.1. Let C be a nonempty closed convex bounded subset of a real Hilbert space H and r> 0,

 

CB H  :

T C be a multivalued -Lipschitz continu-ous mapping with constant L> 0, and let :C C R

  

i

F T

=1

N

a b

1) lim_n_a_n= 0,

=1 n n a = 





and

1 =1

n an an < 

  



;

2) 0 < liminf_n_b_n limsup_n_b_n < 1; 3) limn n i,1n,i = 0;

4)



_n₌₁c_n_₁c_n <. then

1) limn un1un = 0, limn xn1xn =0; 2) lim_n_ x_nW u_n _n = 0, lim_n _n _n = 0

Proof. 1) From the nonexpansity of , we have

x u r T

 .

(5)

1217

1 = 1 1

n n r n r n n n

u _ u T x_ T x  x_ x (3.1) and set ₌ 1

1

n n

n

n n

x b x

z

b

 

 , we obtain

























1 1 1 1 1 1

2 1 1 1 1

1 1

= =

1 1 1 1

=

1 1 1

n n n n n n n n n n n

n n n n n n

n n

n n n n

n n n

n n n n n n

n n n

n n n n

n n n

a f W x c W u a f W x c W u

x b x x b x

z z

b b b b

a a a

f W x f W x f W x

b b b

c c c

W u W u

b b b

         



 

 

  

   

 

  _  _

 

 _   _

 

 _   _  _ 

 W un n

n

By Lemma 1.3, we arrive at





















1 1 1

1 1 1 1 1

1 1 1

1 1

1 1, ,

=1

1 1

1

1 1, ,

=1 1

1 1 1 1

2

1 1 1

2 1

n n n n

n n n n n n n n n n n n n n

n n n n

N

n n n

n n n i n i n n n n

i

n n n

N n

n n n i n i

i n

a a a c

z z f W x f W x f W x W u W u W u

b b b b

a a a

x x M f W x W u

b b b

c

u u M

b

 _ _

 

  

    

  

 



 



       

   

 

 _    _  

 _ _  

 

 _    _

  



(3.2)

Hence, it follows from (2.1) that

















1 1

1 1 1, ,

=1

1 1

1

1 1, ,

=1 1

1

1 1, ,

=1 1

2

1 1 1

2 1

2

1 1

N

n n n

n n n n n i n i n n n n

i

n n n

N n

n n n i n i

i n

N

n n

i

n n

a a a

z z x x M f W x W u

b b b

c

x x M

b

a a

x x f W x W u M

b b

 _ _

 

 

  

 



 



 



 

  _    _  

 _ _  

 

 _    _

  

      

 



(3.3)

It follows from conditions (a) and (c), we have



1 1



lim sup _n _n _n _n 0.

n

z_ z x_ x

    

Hence by Lemma 1.5, we can see that

lim _n _n = 0

n z x Consequently





1

lim _n _n = lim 1 _n _n _n = 0

n x x n b z x (3.4) From (2.1), we get

1

lim _n _n = 0

n u  u (3.5) 2) In view of (1.5), we conclude that





1 1

1

,

n n n n n n n n

n n n n n n n

n n n n

x W u x x x W u

x x a f W x W u

b x W u

 



    

   

 

that is





1

, 1

n n n n n

n

n n n n n

x W u x x

b a

f W x W u

b



  



 



which implies that

lim _n _n _n = 0

n x W u (3.6) For = =1

 

N

i i

p   F T  , note that is firmly nonexpansive, we can see that

r T





2 2

= ,

1 =

2

n r n r r n r n

n n

n n n

u p T x T p T x T p x p

u p x p

u p x p u x

    

 

     2

n

and so

2 2 2

n n n n n

(6)

























2 2

1

2 2 2

n n n n n n n n n

n n n n n n n n

n n n n n n n

n n n n n n n n n

n n n n n n n

x p a f W x b x c W u p

a f W x p b x p c W u p

a f W x p b x p c u p

a f W x p b x p c x p x u

a f W x p x p c x u

     

     

       

     

2

which follows that











2

1 1

n n n n n n n n n n

c x u  x p  x_ p x_ x a f W x p

and hence

lim _n _n = 0

n x u This completes the proof.

Proof of Theorem 2.1. We divide our proof into 3 steps. Step 1. We prove that there exists xC, such that n

x x, un x 

 and wnw as n , where

 

wT x . From (1.5), (2.1) and Lemma 1.3, we compute

























1 1 1 1 1

1 1 1 1 1 1

1 1 1 1,

=1

=

2

n n n n n n n n n n n n n n n n n n

n n n n n n n n n n n n

N

n n n n n n n n i

i

x x a f W x b x c W u a f W x b x c W u

a a f W x a f W x f W x b b x

b x x c c W u c W u W u

a a f W x a  x x M 

    

    

     

   

  

 _   _{ }  

       

      

     



1 1 1 1

   





, 1

1 1 1 1 1 1, ,

=1

1 1

2

1 1

n i n n n

N

n n n n n n n n n n n i n i

i

n n n n

b b x

b x x c c W u c u u M

a x x r



 





     

 

 _ _ _ _

 

 

 

      _    _

 

  _ _  



(3.8)

where





1

= 1 1

2

n an

     , sn= 1 and





1 1 1

=1

= 2

N

n n n n n n n n n n n n n i n i

i

r a a  f W x  b b  x c c W u  M



 1, , .

By Lemma 1.7 and conditions (a)-(d), we conclude that

 

xn is a Cauchy sequence in C such that limn un =x   , there exists an element xC. On the other hand,

limn xnun = 0 implies that limn un = x   . From (1.5), we have

  







  







1 1

1

1 ,

2 , 2

n n n n

w w T x T x

n

T x T x L x x

 

 

  _ _

 

 



 

(3.9)

and for m>n1,

1 1

= =

2

m m

m n i i i i

i n i n

w w w w L x x

 

 

 



 



 (3.10)









1 1 1

1 1

= = =

1 1 1

1

= = = =

1 1

1

= =

m m m

i i i i i

i n i n i n

m m m m

i i i n m

i n i n i n i n

m m

i n i

i n i n

a x x a r

a r a a a

a a r



  

 

  

 

   



 



  

    

  



 



1

i r

Hence

1 1

=

1 1

= 1 1

m i m

i n

i i n n

i n

r

x x  x x

 

 

 

   

 



In view of (2.4) and (2.8), we obtain

,

lim m n = 0

m n w w (3.11)

which implies that

 

wn e exists an

xt we ca

is a Cauchy sequence in H and therefore ther element w in H such that

. Ne n see that limnwn=w

 





_{ }







 



 



, = inf , n

b T x

d w T x _ d w b w w

  



,

n n

d w T x

w w T x





  

Hence, we derive that

,T x

n n

w w L x x

   

(3.12)

 



,



=

(7)

1219 as

Let . Then Q is a contrac-

 

wT x

Step 2.

tion of C into i

 

T x CB H .

 

=1

= N _{F T}

i i

Q P_ _

tself. In fact, for all

f



, x yC

 

Q x Q y  f x f y  xy

qC

Therefore there exists a unique element such that q=Q q

 

. Noting that qC and

. Since

N

i i

x  F T   xn x

 

0, From and we kno

(1

n n n

.5) and

0

 

  

v x,  x

that is



. We shall show

x  x F W

 

n

_

. Assume

 

n

xF W , that is x W xn _ 

. Since

 

un is bounded,

there exists a subsequence

 

un_j of



un



which converges weakly to x. By Lemma 2.1, we conclude that

0

n n n

W u u  . From Opial’s condition, we have





lim inf < lim inf

lim inf

n_j n_j n

j j

n_j n n_j n n_j n j

n j j

u x u W x

u W u W u W x

u x

 

 

 

 



























 

2 1 2

1 1

2 ,

2 , 2 ,

2 ,

n n n n n

n n n n n n n n

n n n n n n n n n n

n n n n n

a f W x q q W u q

x q c W u q a f W x q x q

b x q c u q a f W x f q x q a f q q x q

c x q q x q a f q q x q



 

   

     

 

_    _      

      

Hence

2 1

2

=

n n n n

n

x q b x c

b

  



2

n n n n

b x q a x

 

_  _ 





_{ }

 













 





2

2 2

1 1

2

2 2

1

2 ₁

1 2

,

1 1

1 2 2

= ,

1 1 1

2 1 2 1 1

1 ,

1 1 2 1 1

= 1 ,

n n n

n n

n n n n

n n n

n n

n n n n

a a a

x q x q f q q x q

a a

a a a a

x q x q f q q x q

a a a

a a a M

x q f

a a

x q



 



  

 

   

  

 



 

     

 

   

     

 _  _ _

 

q q x q

 

 

   

  _    _  _  _   

 

   

  

where M1= sup



xnq 2:n1



,





2 1 =

1

n n

n

a a

 





 and



1



 

1

= ,

2 1 1

n

n n

a M

f q q x q



    

   . It is easy to

see that n0, n=1n=





, and lim supnn0.

Hence, by Lemma 1.6, the sequence

 

x_n converge obtain that

s

rongly o we can

st to q. C nsequently,

 

u_n =

also converges strongly to q, and so x q. This com-pletes th

closed convex e

Putting for all in Theorem 2.1, we obtain.

Corolla Let C nonempty

bounded subset of a real Hilbert space H, proof.

= i T x x

ry 2.1.

1 i

be a

 

:

T CCB H

ous mapping with c

be a multivalued -Lipschitz continu-onstant , and let



> 0

L :C C R

 

1

be a real-valued function satisfying  -

 

3 . and

:H C C R

   

isfying the conditions

be an equilibrium-like function

sat- 

1 -

 

4 and  Ø.

 

0,1 . T the sequences

et F

hen

 

xn ,

 

un , and

 

wn rated itera-tively by

(8)

  







 



 

  



 

1

;

1

, , , , , ,

=

n n n n n

n n n n n n n

w u v v u u u k u k x v u

r

x a f x b x c u

  



 

    

 

  

 

converge strongly to , and

1 1 , 1

n n n n

w w T x T x

n

 

  _ _

 _ _

 



1

  

0,

n n n   v C (3.15)

x 

1) and

n 

satisfying the following conditions:

(0,1) with an bn c_n = 1,

lim_n_a_n= 0,



n₌₁an=

1 =1 n n <

n a a



  



;

2) 0 < lim inf_n_b_nlim sup_n_b_n< 1 and

1 =1 n n <

n b b



  



;

3) n=1cn1 cn <



  



.

4. References

J Hua C.

Approxim ds for Genera ed

Equilib-rium quence of Non

expa lication

Vol.

[2] , N.-C ong and J.-C On Convergence Analysis of an I tive Algorithm for Finding Commo

eneralized Mixed Equilibriu lem oblems,” Mathematical ties and

ce Th

n Com delli

p. 1463-1471.

doi:10.1016/S0895-7177(00)00218-1

[1] W.-Y. Zeng, N.- ng and -W. Zhao, “Viscosity ation Metho lized Mix

Problems and Fixed Points of a Se -nsive Mappings,” Fixed Point Theory App s, 2008, 2008, Article ID 714939.

D. R. Sahu . W . Yao, “

tera n

s a

Solution of G m Prob

Fixed Point Pr Inequli

nd

Applications, Article in Press.

[3] W. Takahashi and K. Shimoji, “Convergen eorems for Nonexpansive Mappings and Feasibility Problems,”

Mathematical a d puter Mo ng, Vol. 32, No. 11, 2000, p

[4] L.-C. Ceng and J.-C. Yao, “A Hybrid Iterative Scheme for Mixed Equilibrium Problems and Fixed Point Prob-lems,” Journal of Computational and Applied Mathemat-ics, Vol. 214, No. 1, 2008, pp. 18

doi:10.1016/j.cam.2007.02.022

6-201.

[5] H. K. Xu, “Viscosity Approximation for Nonexpansive Mappings,” Journal of Mathematical Analysis and Ap-plications, Vol. 298, No. 1, 2004, pp. 279-291.

doi:10.1016/j.jmaa.2004.04.059

[6] T. Suzuki, “Strong Convergence of Krasnoselskii and Mann’s Type Sequences for One-Parameter Nonexpan-sive Semigroups without Bochner Integrals,” Journal of Mathematical Analysis and Applications, Vol. 305, No. 1, 2005, pp. 227-239. doi:10.1016/j.jmaa.2004.11.017

[7] S. B. Nadle Jr., “Multi-Valued Contraction Mappings,”