APPROXIMATING COMMON ELEMENTS OF THE SET OF AMENABLE SEMIGROUP AND ZERO POINT SETS AND THE SOLUTIONS SETS OF SYSTEMS OF EQUILIBRIUM PROBLEMS

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235

APPROXIMATING COMMON ELEMENTS OF THE SET OF AMENABLE SEMIGROUP AND ZERO POINT SETS AND THE SOLUTIONS SETS OF SYSTEMS OF EQUILIBRIUM PROBLEMS

Qiang Li* & Dapeng Gao

Department of Mathematics and Information, China West Normal University, Sichuan 637009, P. R. China

ABSTRACT

In this paper, we introduce a new iterative scheme for finding a common element of the set of fixed point of left amenable semigroup, the zero point set of the operator which is the sum of inverse strongly monotone operators and maximal monotone operators, and the set of solutions for systems of equilibrium problems in Hilbert spaces by using a hybrid steepest descent methods. Then strong convergence of the scheme to a common element of the three sets is proved. Our results improve and generalize some well-known results in the literature.

Keywords: Common fixed point; non-expansive mapping; amenable semigroup; system of equilibrium problem;

strictly pseudo-contractive mapping; hybrid steepest descent method.

1. INTRODUCTION

Let

H

be a real Hilbert space whose inner product and norm are denoted by

 , 

^and

 

, respectively. When

}

{ x

_n is a sequence in

H

, we denote strong convergence of

{ x

_n

}

to

x  H

by

x

_n

 x

and weak convergence by

x

_n

 x

. Let

C

be a nonempty closed convex subset of

H

and let

F : C  C  R

be a bi-function, where

R

is the set of real numbers. The equilibrium problem for

F : C  C  R

is to find

x

^

 C

such that

. 0,

= ) ,

( x y y C

F

^

 

(1.1)

The set of solutions of (1.1) is denoted by

EP (F )

^{. Let}

{ F

_i

, i = 1,2,  , N }

be a finite family of bi-functions from

C  C

to

R

^{, where}

R

is the set of real numbers. The system of equilibrium problems for

{ F

₁

, F

₂

,  , F

_N

}

is to find a common element

x

^

 C

such that

 



 



















. 0,

= ) , (

, 0,

= ) , (

, 0,

= ) , (

C y y

x F

C y y

x F

C y y

x F

N 2 1



(1.2) We denote the set of solutions of (1.2) by



^M_j₌₁

SEP ( F

_j

)

, where

SEP ( F

_j

)

is the set of solutions to the equilibrium problems, that is,

. 0,

= ) ,

( x y y C

F

_i ^

 

^(1.3)

If

N = 1

, then the problem (1.2) is reduced to the equilibrium problems.

If

N = 1

^and

F ( x

^

, y ) =  Bx

^

, y  x

^



, then the problem (1.2) is reduced to the variational inequality problem of finding

x

^

 C

such that

. 0,

, y x y C

Bx     



^ ^ (1.4)

The set of solutions of (1.4) is denoted by

VI ( C , B )

. Many problems in applied sciences, such as monotone inclusion problems, saddle point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and equilibrium problems as special cases. Some methods have been proposed to solve

) , ( C B

VI

,

EP (F )

and

SEP ( F

_i

)

; see, for example [1,2] and references therein. The above formulations (1.2) extends this formulism to such problems, covering in particular various forms of feasibility problems.

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236

Definition 1.1. Let

A : C  H

be nonlinear mappings. Then

A

is called (1)monotone if

 Ax  Ay , x  y   0,  x , y  C

;

(2)

 

inverse strongly monotone if there exists a constant

 > 0

such that

; , ,

, x y Ax Ay

²

x y C

Ay

Ax       

   

(3)



-strictly pseudo-contractive if there exists a constant

  (0,1)

such that

; , , ) ( )

(

²

2

x y I A x I A y x y C

Ay

Ax              

 

(4)For the variational inequality, the following is true:

0. >

), (

= )

,

( C A u P u  A 

VI

u  

_C



(5)Let

M : H  2

^H be a set-valued mapping. The set

D (M )

defined by

D ( M ) = { x  H : Mx   }

is said to be the domain of

M

. The set

R (M )

defined by

R ( M ) = 

_x_H

Mx

is said to be the range of

M

. The set

) (M

G

defined by

G ( M ) = {( x , y )  H  H : x  D ( M ), y  R ( M )}

is said to be the graph of

M

. Recall that

M

is said to be monotone if

).

( ) , ( ), , ( 0,

, f g x f y g G M

y

x      



M

is said to be maximal monotone if it is not properly contained in any other mono-tone operator. Equivalently,

M

is maximal monotone if

R ( I  rM ) = H

for all

r > 0

. The class of monotone mappings is one of the most important classes of mappings. With in the past several decades, many authors have been devoting to the studies on the existence and convergence of zero points for maximal monotone mappings, and the references therein. For a maximal monotone operator

M

^on

H

^and

r > 0

, we may define the single-valued resolvent

) ( :

) (

= I rM

¹

H D M

J

_r



^



. It is known that

J

_r is firmly non-expansive and

( M )

^¹

(0) = F ( J

_r

)

, where

) ( J

_r

F

denotes the fixed point set of

J

_r^.

In 2003, Takahashi and Toyoda [3] proved the following weak convergence theorem.

Theorem 1.1. Let

C

be a nonempty closed convex subset of a real Hilbert space space

H

^{. Let}

A

^{be an}

 

inverse strongly monotone mapping from

C

into

H

and

S

be a non-expansive mapping from

C

into itself such that

F ( S )  VI ( C , A )  

^{. Let}

{ x

_n

}

be a sequence generated by

0, ),

( ) (1

= ,

₁

0

 C x

_

x   SP x  Ax  n 

x

_n



_n _n



_n _C _n



_n _n

where



_n

 [ b a , ]

for some

a , b  (0,2  )

^and



_n

 [ d c , ]

for some

c , d  (0,1)

^{. Then}

{ x

_n

}

converges weakly to

z  F ( S )  VI ( C , A )

^{, where}

z = lim

n__

P

_F₍_S₎__VI₍_C_,_A₎

x

_n.

In 2007, Tada and Takahashi [4] obtained the following weak convergence theorem.

Theorem 1.2. Let

C

H

. Let

F

be a bi- function from

C  C

to

R

satisfying

A (1)  A (4)

and

S

be a non-expansive mapping from

C

into

H

such that

F ( S )  EP ( F )  

. Let

{ x

_n

}

and

{ u

_n

}

be sequences generated by

x

₁

= x  H

^{and let}



 























 n n n n

n

n n n n

n n

Su x

x

C u x

u u r u u u F that such C

u

) (1

=

, 0,

1 , ) , (

1

 

(1.5) for each

n  1

, where



_n

 [ b a , ]

for some

a , b  (0,1)

and

r

_n

 (0,  )

satisfies

liminf

n__

r

n

> 0

. Then

}

{ x

_n converges weakly to

w  F ( S )  EP ( F )

, where

z = lim

_n__

P

_F₍_S₎__EP₍_F₎₎

x

_n.

Very recently,Jitpeera et al.[5],introduced the iterative scheme based on viscosity and Cesro mean

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237

 



 







 































, 0,

1 ) 1 )

((1 )

(

=

), (

) (1

=

, 0,

1 , ) ( ) ( ) , (

0

=

1

T y n

A n I

x y

rf x

Bu u

P u

y

C y x

u u r y

u y y

u

n i n

i n

n n

n n n n

n n n C n n

n n

n n n n

n n

















(1.6) where

B : C  H

a

 

inverse strongly monotone,

 : C  R  {  }

is a proper lower semi-continuous and convex function,

T

ⁱ

: C  C

is a non-expansive mapping for all

i = 1,2,3,  , n

^,

{ 

_n

}

^,

{ 

_n

}

^,

{ 

_n

}  (0,1)

^,

) (0,2 }

{ 

_n

 

^{, and}

{ r

_n

}  (0,  )

satisfy the following conditions (i)

lim

n__



n

= 0

and



^_n₌₁



_n

= 

;

(ii)

lim

n__



n

= 0

;

(iii)

0 < liminf

n__



_n

 limsup

_n__



_n

< 1

; (iv)



_n

 [ e , g ]  (0,2  )

,

lim

n__

| 

n_1

 

n

|= 0

; (v)

liminf

n__

r

n

> 0

and

lim

n__

| r

n_1

 r

n

|= 0

.

They show that if

 := 

ⁿ_i₌₁

Fix ( T

ⁱ

)  VI ( C , B )  MEP (  ,  )

is nonempty, then the sequence

x

_n converges strongly to the

z = P

_

( I  A  rf )( z )

which is the unique solution of the variational inequality

. 0,

, )

(       

 rf A z x z y

In this paper, motivated and inspired by Takahashi and Toyoda

[3]

, Tada and Takahashi

[4]

, and Jitpeera et. al [5], we introduce a new iterative scheme for finding a common element of the set of fixed point of left amenable semigroup, the zero point set of the operator which is the sum of inverse strongly monotone operators and maximal monotone operators, and the set of solutions for systems of equilibrium problems in Hilbert spaces by using a hybrid steepest descent methods. Then strong convergence of the scheme to a common element of the three sets is proved.

Our results improve and generalize some well-known results in the literature.

2. PRELIMINARIES

Lemma 2.1 ([6]) Let

S

be a semigroup and

C

be a nonempty closed convex subset of a reflexive Banach space

E

. Let

 = { T

_t

x : t  S }

be a non-expansive semigroup on

H

such that

{ T

_t

x : t  S }

is bounded for some

C

x 

, let

X

be a subspace of

B (S )

such that

1  X

and the mapping

t   T

_t

x , y

^



is an element of

X

for each

x  C

and

y

^

 E

^, and



is a mean on

X

. If we write

T

_

x

instead of

 T

_t

xd  (t )

, then the followings hold .

(i)

T

_ is non-expansive mapping from

C

into

C

; (ii)

T

_

x = x

for each

x  Fix (  )

^;

(iii)

T

_

x  c o { T

_t

x : t  S }

for each

x  C

.

Lemma 2.2 ([7]) Let

H

be a real Hilbert spaces, there hold the following identities:

(i)for each

x  H

and

x

^

 C

^,

x

^

= P

_C

x   x  x

^

, y  x

^

  0

for all

y  C

^;

(ii)

P

_C

: H  C

is non-expansive,that is,

 P

_C

x  P

_C

y    x  y 

for all

x , y  H

^;

(iii)

P

_C is firmly non-expansive,that is,

 P

_C

x  P

_C

y 

²

  P

_C

x  P

_C

y , x  y 

for all

x , y  H

^;

(iv)

 tx  (1  t ) y 

²

= t  x 

²

 (1  t )  y 

²

 t (1  t )  x  y 

²

,  t  [0,1]

, for all

x , y  H

; (v)

 x  y 

²

  x 

²

 2  y , x  y 

.

Lemma 2.3 ([8]) Let

{ x

_n

}

and

{ y

_n

}

be bounded sequences in a Banach space

E

and let

{ 

_n

}

be a sequence in

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238

[0,1]

with

0 < liminf

n__



_n

 limsup

_n__



_n

< 1

. Suppose

x

_n_₁

= 

_n

x

_n

 (1  

_n

) y

_n for all integers

n  0

and

0. ) (

limsup

_₁

 

_₁

 













_n _n _n _n

n

x x y y

Then,

lim

n__

 y

_n

 x

_n

 = 0

.

Lemma 2.4 ([9]) Let

{ a

_n

}

be a sequence of nonnegative real numbers such that

0. , )

1

 (1   



b a b c n

a

_n _n _n _n _n

where

{ b

_n

}

and

{ c

_n

}

are sequences of real numbers satisfying the following conditions

(i)

(0,1), = ,

0

=



 

^n ⁿ

n

b

(ii)either

limsup

_n__

c

_n

 0

or



^n=0

^| b

n

c

n

^|< 

. Then,

lim

_n__

a

_n

= 0

.

Lemma 2.5 ([10]) Assume

A

be a strongly positive linear bounded operator on

H

with coefficient

 > 0

and

<

1

0   A 

^ . Then

 I   A   1   

.

Throughout this article, we assume that a bi-function

F : C  C  R

satisfies the following conditions:

(A1)

F ( x , x ) = 0

for all

x  C

;

(A2)

F

is monotone, i.e.,

F ( x , y )  F ( y , x )  0

^{for all}

x , y  C

^;

(A3)for each

x , y , z  C

,

lim

t⁰

F ( tz  (1  t ) x , y )  F ( x , y )

; (A4)for each

x  C

,

y  F ( x , y )

is convex and lower semi-continuous.

Lemma 2.6 ([11]) Let

C

H

and let

F

be a bi-function from

C  C

to

R

satisfying

A (1)  A (4)

^{. Let}

r > 0

^and

x  H

. Then, there exists

z  C

such that

, 0,

1 , ) ,

( y z z x y C

y r z

F        

Lemma 2.7 ([12]) Let

C

H

,

A : C  H

^be

a mapping and

M : H  2

^H be a maximal monotone mapping. Then

0. >

(0), ) (

= )) (

( J I rA A M

¹

r

F

_r

 

^



Lemma 2.8 ([13]) Let

H

be a real Hilbert space space. For

q

which solves the variational inequality

0 ,

)

(    

 rf F q x q

nk



,

f ^ 

H,

p  F (T )

, the following statement is true :

,

= ) (

0 ,

)

( rf F q x q P I F rf q q

nk

     



 

_



where

 := F ( S )  ( A  M )

^¹

(0)  ( B  W )

^¹

(0)  

^M_j₌₁

SEP ( F

_j

)  

. Let

I

_C be the indicator function of

C

,i.e.,

 









 . ,

,

= 0, )

( x C

C x x

I

_C

Lemma 2.9 ([14]) Let

C

H

. Let

P

_C be the metric projection from

H

onto

C

,

 I

_C be the sub-differential of

I

_C, where

I

_C is as defined in (2.1) and

)

1

(

=  

_C ^

r

I r I

J

. Then

. , ,

=

= J x y P x x H y C

y

_r



_C

 

3. MAIN RESULTS

Theorem 3.1 Let

C

H

,

A : C  H

^{be an}

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239

 

inverse strongly monotone,

B : C  H

be an

 

inverse strongly monotone,

f

be a contraction of

C

into itself with coefficient

  (0,1)

^,

F

is a strongly positive linear bounded operator on

H

with coefficient

 > 0

and



 < 

<

0

. Let

{ F

_k

, k = 1,2,  , M }

be a finite family of bi-functions from

C  C

to

R

which satisfy

(4) (1) A

A 

^,

S

a semigroup and

 = { T

_t

x : t  S }

be a non-expansive semigroup from

C

^into

C

such that







_

( )

= )

(

_t _S

Fix T

_t

Fix 

^{. Let}

X

be a left invariant subspace of

B (S )

such that

1  X

, and the function



t  T

_t

x , y 

is an element of

X

^{for each}

x  C

and

y  H

^,

{ 

_n

}

a left regular sequence of means on

X

such that

lim

n

 

_n_₁

 

_n

 = 0

^.

M : H  2

^H^and

N : H  2

^H be maximal monotone operators such that

C M

D ( ) 

^and

D ( W )  C

^. ^Assume ^that















 := F ( S ) ( A M )

^¹

(0) ( B W )

^¹

(0)

^M_j₌₁

SEP ( F

_j

)

. If the sequences

{ x

_n

}

,

{ y

_n

}

and

{ u

_n

}

are generated iteratively by

x

₁

 C

and

 

 













 n n n n n n n n

n

n n n n

s n n

n n n

n n

n F r n F r n FM

n rM n

y T F I

x y

rf x

Bu s u J Au

u J y

x T T T

u



















) )

((1 )

(

=

) (

) (1 ) (

=

1

1 1, 2 2,

,



where

J = ( I 

_n

M )

^¹

n



 ,

J = ( I  s

_n

W )

^¹

sn ,

{ 

_n

}

is a sequence in

(0,2  )

,

{ s

_n

}

is a sequence in

(0,2  )

,

M k n

r

k_,

}

₌₁

{

are a real sequence in

(0,  )

^and

{ 

_n

}

^,

{ 

_n

}

^,

{ 

_n

}

are three sequences in

(0,1)

. Assume that the following restrictions are satisfied

(C1)

0 < liminf

n__



_n

 limsup

_n__



_n

< 1

; (C2)

lim

n__



n

= 0

and



^_n₌₁



_n

= 

;

(C3)

0 < a  

_n

 b < 2 

,

0 < c  s

_n

 d < 2 

and

0 < k  

_n

 e < 1

; (C4)

lim

n__

| 

n_1

 

n

|= lim

n__

| s

n_1

 s

n

|= lim

n__

| 

n_1

 

n

|= 0

;

(C5)

liminf

n__

r

k,n

> 0

and

lim

n__

| r

k,n_1

 r

k,n

|= 0

for each

k  {1,2,3,  , M }

^{, where}

a

,

b

,

c

,

d

,

k

,

e

are real numbers. Then the sequence

{ x

_n

}

,

{ y

_n

}

and

{ u

_n

}

converges strongly to a point

q  

, which is the unique solution of the variational inequality

. 0,

, )

(       

  F rf q p q p

(3.2)

Equivalently, we have

q = P

_

( I   F  rf )( q )

. Proof We shall divide the proof into several steps.

By taking

= (

_n _n _n

)

n

J

n

u Au

v

_

 

^,

= (

_n _n _n

)

sn

n

J u s Bu

w 

^{and for}

k  {1,2,3,  , M }

and for all

n  N

, we shall equivalently write scheme

(3.1)

as follows:

 

 











 n n n n n n n n

n

n n n

n n

n M n n

y T F I

x y

rf x

w v

y

x u











) )

((1 )

(

=

) (1

=

1

T

Step 1. We show that the mapping

P

_

( I   F  rf )

has a unique fixed point.

Since

f

be a contraction of

C

into itself with coefficient

  (0,1)

. Then, we have

(6)

240

. , , ))

( (1

=

) (1

) ( ) (

) )(

( ( ) ))(

( (

C y x y x r

y x y

x r

y x F I y f x f r

y F I rf x F I rf y

F I rf P x F I rf P







































_









































Since

0 < 1  (    r  ) < 1

, it follows that

P

_

( I   F  rf )

is a contraction of

C

into itself. Therefore, by the Banach Contraction Mapping Principle, has a unique fixed point, say