28
APPROXIMATING COMMON ELEMENTS OF THE SET OF AMENABLE SEMIGROUP AND VARIATIONAL INEQUALITY AND
INFINITE FAMILY OF NON-EXPANSIVE MAPPINGS AND GENERALIZED MIXED EQUILIBRIUM PROBLEMS
Qiang LI1,* & Jun LI2
Department of Mathematics and Information, China West Normal University, Sichuan
637009
, P. R. China;Corresponding Author E-mail address: [email protected] (Qiang LI) ABSTRACT
In this paper, we introduce a new iterative scheme for finding a common element of the set of fixed point of an infinite family of non-expansive mappings and left amenable semigroup, the set of solution of generalized mixed equilibrium problem and the set of solutions of the variational inequalities for solving inverse strongly monotone mappings 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.
Key-words: common fixed point; non-expansive mapping; amenable semigroup; generalized mixed equilibrium problem; variational inequality; hybrid steepest descent method.
MR(2010) Subject Classification 47H09; 47H10; 47H20
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 inH
, we denote strong convergence of{ x
n}
tox H
byx
n x
and weak convergence byx
n x
.Let
: C R
be a real-valued function and beA : C H
a nonlinear mapping. SupposeF : C C
intoR
is an equilibrium bi-function. That is,
F ( u , u ) = 0, u C
. The generalized mixed equilibrium problem is to findx C
such that. 0,
, ) ( ) ( ) ,
( x y y x Ax y x y C
F
(1.1)We shall denote the set of solutions of this mixed equilibrium problem by
GMEP
. Thus}.
0, ,
) ( ) ( ) , ( : {
:= x C F x y y x Ax y x y C
GMEP
If
= 0
,A = 0
, then problem (1.1) reduces to equilibrium problem, which is to findx
C
such that. 0,
) ,
( x y y C
F
(1.2)If
= 0
, then problem (1.1) reduces to generalized equilibrium problem, which is to findx
C
such that. 0,
, )
,
( x y Ax y x y C
F
(1.3)If
A = 0
, then problem (1.1) reduces to mixed equilibrium problem, which is to findx
C
such that. 0,
) ( ) ( ) ,
( x y y x y C
F
(1.4)If
F ( x
, y ) = Ax
, y x
, then the problem (1.3) is reduced to the variational inequality problem of findingC
x
such that. 0,
, y x y C
Ax
(1.5)The set of solutions of (1.5) is denoted by
VI ( C , A )
. 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 A
VI
,EP (F )
andSEP ( F
i)
; see, for example [1,2] and references therein.29
Definition 1.1. Let
A : C H
be nonlinear mappings. ThenA
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
2x y C
Ay
Ax
P P
(3)
-strictly pseudo-contractive if there exists a constant (0,1)
such that; , , ) ( )
(
22
2
x y I A x I A y x y C
Ay
Ax P P P P P
P
(4)For the variational inequality, the following is true:
0;
>
), (
= )
,
( C A u P u A
VI
u
C
(5)Let
A
be an inverse strongly monotone mapping ofC
intoH
, and letN
C(v )
be the normal cone toC
atv
, that isN
C( v ) = { w H : v u , w 0}
for allu C
, and define
. ,
, ), ( )
= ( )
( v C
C v v N v v B
Q
CThen
Q
is the maximal monotone and0 Q ( v )
if and only ifx VI ( C , B ).
In 2001, Yamada [3]introduced the following iterative scheme called the hybrid steepest descent method :
0, ,
1
=
Sx BSx n
x
n n
n
n (1.6)where
x
1= x H
,{
n} (0,1)
, letB : H H
be a strongly monotone and Lipschitz continuous mapping and
is a positive real number. He proved that the sequence{ x
n}
generated by (1.6) converges strongly to the unique solution of theF ( S ) VI ( C , B ).
In 2007, Plubtieng and Punpaeng [5] considered the iteration process generated by
0, ,
) 1 (
) (1
) (
=
,
1 00
H x
f x x s T s x ds n
x
sn nn n n n
n n n
n
(1.7)where
{
n}, {
n} (0,1)
with
n
n< 1
and{ t
n}
is a positive real divergent sequence. They proved, under certain appropriate conditions on{
n}
, that{ x
n}
converges strongly to a common fixed point of one-parameter non-expansive semigroup = { T ( s ) : s 0}
.Very recently,Jitpeera et al.[5],introduced the iterative scheme based on viscosity and Ces ¨¤ ro mean
1.8 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
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 alli = 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
;30 (iv)
n [ e , g ] (0,2 )
,lim
n|
n1
n|= 0
; (v)liminf
nr
n> 0
andlim
n| r
n1 r
n|= 0
.They show that if
:=
ni=1Fix ( T
i) VI ( C , B ) MEP ( , )
is nonempty, then the sequencex
n converges strongly to thez = 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 Yamada.
[3]
, Plubtieng and Punpaeng [5], we introduce a new iterative scheme for finding a common element of the set of fixed point of an infinite family of non-expansive mappings and left amenable semigroup, the set of solution of generalized mixed equilibrium problem and the set of solutions of the variational inequalities for solving inverse strongly monotone mappings 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 andC
be a nonempty closed convex subset of a reflexive Banach spaceE
. Let} : {
= T
tx t S
be a non-expansive semigroup onH
such that{ T
tx : t S }
is bounded for somex C
, letX
be a subspace ofB (S )
such that1 X
and the mappingt T
tx , y
is an element ofX
for eachC
x
andy
E
, and
is a mean onX
. If we writeT
x
instead of Ttxd (t )
, then the followings hold .
(i)T
is non-expansive mapping from C
into C
;
(ii)
T
x = x
for eachx Fix ( )
;(iii)
T
x c o { T
tx : t S }
for eachx C
.Lemma 2.2 ([7]) Let
H
be a real Hilbert spaces, there hold the following identities:(i)for each
x H
andx
C
,x
= P
Cx x x
, y x
0
for ally C
; (ii)P
C: H C
is non-expansive,that is, P
Cx P
Cy x y
for allx , y H
;(iii)
P
C is firmly non-expansive,that is, P
Cx P
Cy
2 P
Cx P
Cy , x y
for allx , 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 allx , y H
;(v)
x y
2 x
2 2 y , x y
.Lemma 2.3 ([7]) Each Hilbert space
H
satisfies Opial's condition, that is, for any sequence{ x
n} H
withx
x
n
, the inequalityliminf ,
liminf x x < x
ny
n n n
hold for each
y H
withy x
.Lemma 2.4 ([7]) Let
H
be a Hilbert space, letC
be a nonempty closed convex subset ofH
. Let > 0
and letH
C
B :
be an
inverse strongly monotone. If0 < < 2
, thenI B
is a non-expansive mapping ofC
intoH
.Lemma 2.5 ([8]) Let
{ x
n}
and{ y
n}
be bounded sequences in a Banach spaceE
and let{
n}
be a sequence in[0,1]
with0 < liminf
n
n limsup
n
n< 1
. Supposex
n1=
nx
n (1
n) y
n for all integers 0
n
and31
0.
) (
limsup
1
1
n n
n n
n
x x y y
Then,
lim
n y
n x
n = 0
.Lemma 2.6 ([9]) Let
{ a
n}
be a sequence of nonnegative real numbers such that0.
, )
1
(1
b a b c n
a
n n n n nwhere
{ b
n}
and{ c
n}
are sequences of real numbers satisfying the following conditions(i)
(0,1), = ;
0
=
n n
n
b
b
(ii)either
limsup
nc
n 0
or
n=0| b
nc
n|<
. Then,lim
na
n= 0
.Lemma 2.7 ([10]) Assume
A
be a strongly positive linear bounded operator onH
with coefficient > 0
and<
10 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 allx C
;(A2)
F
is monotone, i.e.,F ( x , y ) F ( y , x ) 0
for allx , y C
; (A3)for eachx , y , z C
,lim
t0F ( tz (1 t ) x , y ) F ( x , y )
; (A4)for eachx C
,y F ( x , y )
is convex and lower semi-continuous.(B1)for each
x H
andr > 0
there exist a bounded subsetD
x C
andy
x C
such that for any0,
<
1 , ) ( ) ( ) ,
( y z z x
z r y
y z
F
x
x
x(B2)
C
is a bounded set.Lemma 2.8 ([11]) Assume that
F : C C R
satisfiesA (1) A (4)
and let : C R
be a proper lower semi-continuous and convex function. Assume that either(B 1)
or(B 2)
holds. Forx H
andr > 0
, define amapping
T
FC R
rn( ,)
:
as follows:}.
0, 1 ,
) ( ) ( ) , ( : {
= )
)
(
,
(
y z z x y K
z r y y
z F C z x T
Frn
for all
z C
. Then, the following hold:(1)for each
x C
, (F,)
rn
T
;(2) (F,)
rn
T
is single-valued;(3) (F,)
rn
T
is firmly non-expansive, i.e., for anyx
,y C
,;
)
,
, ( ) , ( 2 ) , ( ) ,
(
x T y T x T y x y
T
Frn F
rn F
rn F
rn
(4)
F ( T
(F, )) = GMEP ( F );
rn
(5)
GMEP (F )
is closed and convex.
Lemma 2.9 ([12]) Let
T : C H
be a k-strict pseudo-contraction. DefineS : C H
byTx x
Sx = (1 )
for eachx C
. Then, as [k ,1)
,S
is a non-expansive mapping such that) (
= ) ( S F T
F
.32
Lemma 2.10 ([13]) Let
H
be a Hilbert space, letC
be a nonempty closed convex subset ofH
. Suppose thatH C
T :
is non-expansive, then, the mappingI T
is demiclosed at zero.Lemma 2.11 ([14]) Let
H
be a real Hilbert space space.Forq
which solves the variational inequality0 ,
)
(
rf F q p q
,f H, p F (T )
, the following statement is true :
,
= ) (
0 ,
)
( rf F q p q P I F rf q q
where
:=
n=1F ( T
n) F ( S ) VI ( C , B ) GMEP
.Let
T
1, T
2,
be an infinite family of mappings ofC
into itself and let
1,
2,
be a sequence of real numbers such that0
i< 1
for everyi N
. For anyn N
, define a mappingW
n of C into C as follows:. ) (1
=
=
, ) (1
=
, ) (1
=
, ) (1
=
, ) (1
=
, ) (1
= ,
=
1 ,2
1 1 ,1
2 2,3
2 ,2
1 ,
1 1 1
,
1 , ,
1 ,
1 1 1
,
1 , ,
1 ,
I U
T U
W
I U
T U
I U
T U
I U
T U
I U
T U
I U
T U
I U
n n
n
n n
k k
n k k k
n
k k
n k k k
n
n n
n n n n
n
n n
n n n n
n n n
Such a mapping
W
n is called theW
mapping generated byT
1, T
2, , T
n and
1,
2, ,
n.
Lemma 2.12 ([15,16]) Let
C
be a nonempty closed convex subset of a real Hilbert spaceH .
LetT
1, T
2,
be an infinite family of mappings ofC
into itself such that
n=1F ( T
n)
, let
1,
2,
be real numbers such that1
<
0
n b
for everyn 1
. Then,(1)for every
x C
andk N
, the limitlim
nU
n,kx
exists;(2) the mapping
W
ofC
into itself as follows:. lim ,
lim =
= W x U
,1x x C
Wx
nn n n
is a non-expansive mapping satisfying
F ( W ) =
n=1F ( T
n)
, which is called theW
mapping generated byT
nT
T
1,
2, ,
and
1,
2, ,
n;
(3)
F ( W
n) =
n=1F ( T
n)
, for eachn 1
;(4)If
E
is any bounded subset ofC
, thenlim sup P Wx W
nx P = 0.
E n x
3. MAIN RESULTS THEOREM
3.1 Let
C
be a nonempty closed convex subset of a real Hilbert space spaceH
,A : C H
be an
inversestrongly monotone,
B : C H
be an
inverse strongly monotone,f
be a contraction ofC
into itself with coefficient (0,1)
,A
is a strongly positive linear bounded operator onH
with coefficient > 0
and
<
<
0
. Let{ F
k, k = 1,2, , M }
be a finite family of bi-functions fromC C
toR
which satisfy(4) (1) A
A
, : C R { }
be a proper lower semi-continuous and convex function with assumption33
(1)
B
orB (2)
, and{ T
i}
i=1 an infinite family of non-expansive mapping ofC
into itself.S
a semigroup and} : {
= T
tx t S
be a non-expansive semigroup fromC
intoC
such thatFix ( ) =
tSFix ( T
t)
. LetX
be a left invariant subspace ofB (S )
such that1 X
, and the functiont T
tx , y
is an element ofX
for each
x C
andy H
,{
n}
a left regular sequence of means onX
such thatlim
n
n1
n = 0
. Assume that :=
n=1F ( T
n) F ( S ) VI ( C , B ) GMEP
.If the sequences{ x
n}
,{ y
n}
and{ u
n}
are generated iteratively by
x
1 C
and
) (
) )
((1 ) (
=
) (
=
) (
) (1
=
1 ) , (
n n n n n n C n
n n n n n n
n n n F rn n
n n n C n n n n
u BW u
W P T A I
x u
rf x
y r y T u
Bx x
P x
y
where
{
n}
and{
n}
are two 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)
{
n}, {
n} [ c , d ] (0,2 )
andliminf
n|
n1
n|= liminf
n|
n1
n|= 0
; (C4){
n} [0, b ]
for someb (0,1)
andlim
n|
n1
n|= 0
;(C5)
0 < r
n< 2
andlim
n| r
n1 r
n|= 0
,where
b
,c
,d
are real numbers. Then the sequence{ x
n}
,{ y
n}
and{ u
n}
converges strongly to a pointq
, which is the unique solution of the variational inequality. 0,
, )
(
A rf q p q p
(3.2)Equivalently, we have
q = P
( I A rf )( q )
.Proof We shall divide the proof into several steps.
Seting
v
n= P
C( x
n
nBx
n)
,e
n= P
C( W
nu
n
nBW
nu
n)
.Step 1. We show that the mapping
P
( I F rf )
has a unique fixed point.Since
f
be a contraction ofC
into itself with coefficient (0,1)
. Then, we have. , , )) (
(1
=
) (1
) ( ) (
) )(
( ( ) ))(
( ( ) ))(
( ( ) ))(
( (
C y x y x r
y x y
x r
y x A I y f x f r
y A I rf x A I rf y
A I rf P x A I rf P
Since
0 < 1 ( r ) < 1
, it follows thatP
( I F rf )
is a contraction ofC
into itself. Therefore, by the Banach Contraction Mapping Principle, has a unique fixed point, sayq C
that is,).
)(
(
= P I A rf q
q
Step 2.We claim that if
{ x
n}
is a bounded sequence inC
, thenlim
1 n = 0
n n n
T
ne T
e
and
n n
n
BW
W
is non-expansive.These two assertion are proved in
[15, step 3]
and[16, step 2]
.Step 3. We show that the sequence
{ x
n}
,{ y
n}
,{ e
n}
and{ u
n}
are bounded.34 In fact, let
x
, thenx
= P
C( x
nBx
)
.Since
I
nB
is a non-expansive mapping (Lemma 2.4), we obtain.
) (
) (
=
) (
) (
) (
) (
=
x x
x B I x B I
Bx x
Bx x
Bx x
P Bx x
P x
v
n
n n
n
n n
n n
n C
n n n C n
(3.3)
.
=
) (1
) (1
x x
x x x
x
x v x
x x
y
n
n n n
n
n n n
n n
(3.4)
1).
, 2
<
(
) 2 (
) ) (
) (
) (
) (
=
) (
=
2
2 2
2
2 )
, ( )
, (
2 )
, ( 2
n r
x y
x y r
r x y
x r I y r I
x r x T y r y T
x y r y T x
u
n n
n n
n n
n n
n
n F
rn n n n F rn
n n n F rn n
(3.5)
Let
e
n= P
C( W
nu
n
nBW
nu
n)
, we can prove that.
) (
) (
=
) (
) (
) (
) (
=
x x
x u
x BW W
u BW W
x BW x
W u
BW u
W
x BW x
W P u BW u
W P x
e
n n
n n n n n n n
n n n n
n n n n
n n n C n n n n n C n
(3.6)
which yields that
) }.
( ) , (
{
) ) (
(
) ) ) (
) (
(1
=
) (1
) (
) (1
) (
) (1
) ( )
( ) (
) (1 )
(
) )(
) ((1 ) (
) )
( (
=
1
r Ax x
x rf x max
Ax x
r rf x r
x r
x x x
x Ax
x rf x
x r
x x x
x Ax
x rf x
u r
x e x
x Ax
x rf x
f u f r
x e T A I
x x Ax
u rf
x e T A I
x x Ax
u rf
x x
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 n
n n
By induction,
. ) }
( ) , (
{
1n N
r Ax x
x rf x max x
x
n
(3.7)and we obtain
{ x
n}
is bounded. So are{ y
n}
,{ u
n}
,{ e
n}
,{ v
n}
,{ BW
nu
n}
,{Bxn }
and{ f ( u
n)}
.Step 4.We show that
