6th
International Science, Social Sciences, Engineering and Energy Conference
17-19 December, 2014, Prajaktra Design Hotel, Udon Thani, Thailand
I-SEEC 2014
http//iseec2014.udru.ac.th
Weak Convergence Theorems for Maximal
Monotone Operators and Nonspreading-Type Mappings
in Hilbert spaces
Wanna Sriprada,e1, Somnuk Srisawatb,e2, Tawan Ampawac,e3
a, b, c
e1
Department of Mathematics, Faculty of Science and Technology,
Rajamangala University of Technology Thanyaburi
Pathum Thani 12110, Thailand.
wanna_sriprad@hotmail.com , e2nuk_srisawat@hotmail.com , e3tawan08@gmail.com
Abstract
In this paper, we introduce an iterative process for finding a common element of the set of fixed points of
a k-srtictly presudo-nonspreading mapping and the solution sets of zero of maximal monotone mapping
and α-inverse strongly monotone mapping in a Hilbert space. Under suitable conditions, some weak
convergence theorems are proved. Our work improves previous results for nonspreading mapping.
Keywords: k-srtictly presudononspreading mapping; nonspreading mapping; Maximal monotone mapping; α-inverse strongly
monotone mapping.
1. Introduction
Let H be a real Hilbert space with inner product ,  and induced norm  and let C be a
nonempty closed convex subset of H . Let T be a mapping of C into itself. We denote by F (T ) the set of
fixed points of T , i.e. F (T ) : x  C | Tx  x. A mapping T of C into itself is said to be nonexpansive if
Tx  Ty  x  y , for all x, y  C. T is said to be firmly nonexpansive if
2
Tx  Ty  2 x  y, Tx Ty ,
for all x, y  C. A mapping T of C into itself is nonspreading if,
2 Tx  Ty
2
2
2
 Tx  y  Ty  x , for all x, y  C .
see [3, 4]. It is shown in [2] that (1.1) is equivalent to
(1.1)
2
2 Tx  Ty
2
2
 x  y  2 x  Tx, y  Ty , for all x, y  C .
Then, it easy to see that every firmly nonexpansive mapping is nonspreading. Following the terminology
of Browder-Petryshyn [1] , a mapping T of C into itself is 𝑘-strictly pseudo-nonspreading if there exists
k  [0,1) such that
2
2
2
2 Tx  Ty  x  y  2 x  Tx, y  Ty  k x  Tx  ( y  Ty) , for all x, y  C .
Clearly, every nonspreading mapping is k-strictly pseudo-nonspreading, but the converse is not true; see
[7]. Recall that a mapping A : C  H is called   inverse strongly monotone, if there exists a positive
number  such that
Au  Av, u  v   Au  Av , for all u, v  C . It is well known that if A : C  H
2
is  -inverse-strongly monotone, then A is
1
- Lipschitz continuous and monotone mapping. In addition,

if 0    2 , then I   A is a nonexpansive mapping. A set-valued mapping B : H  2H is called
monotone if for all x, y  H , f  Bx and g  By imply x  y, f  g  0. A monotone operator on H is

said to be maximal if its graph G ( B) : ( x, f )  H  H | f  B( x) of B is not properly contained in the
graph of any other monotone mapping. For any maximal monotone operator B on H and   0 we
1
defined a single-valued operator J   ( I  rB) : H  D( B) , which is called the resolvent of B for   0.
1
Let B be a maximal monotone operator on H and let B (0)  {x  H : 0  Bx} . It is known that the
1
resolvent J  is firmly nonexpansive and B (0)  F ( J  ).
Recently, in the case when T : C  C is nonspreading mapping, A : C  H is an  - inverse strongly
monotone mapping and B is a maximal monotone operator on H , Manaka and Takahashi [5] proved a
weak convergence theorem for finding a point of the set F (T ) ( A  B)1 (0) , where F (T ) is the set of
fixed points of nonspreading mapping T and ( A  B)1 (0) the set of zero points of A  B .
In this paper, motivated by Manaka and Takahashi [5] , we introduce an iteration scheme (3.1) for
finding a common elements of a set F (T ) of fixed points of a k - preudo-nonspreading mapping T and the
set ( A  B)1 (0) of zero points of A  B , where A is an  - inverse strongly monotone mapping and B is a
maximal monotone operator in a real Hilbert space H . Then, the weak convergence theorem are proved
under some parameters controlling conditions. The results obtained in this paper improve and extend the
corresponding result of Manaka and Takahashi [5] and many others.
2. Preliminaries
This section collects some lemmas which will be used in the proof for the main results in the next
section. Let H be a real Hilbert space with inner product ,  and norm  respectively. It is well-known
that a Hilbert space satisfies Opial’s condition, that is,
liminf xn  u  liminf xn  v ,
n
n
if xn ⇀ u and u  v ; see [6]. Let C be a nonempty closed convex subset of H . Then, for any x  H , there
exists a unique nearest point of C denoted by PC x such that x  PC x  x  y for all x  H and y  C
Such a PC is called the metric projection from H into C . We know that PC is firmly nonexpansive, that is
3
2
PC x  PC y  PC x  PC y, x  y for all x, y  H . Moreover x  PC x, y  PC x  0 holds for all x  H
and y  C.
Lemma 2.1 [5] Let H be a real Hilbert space and let C be a nonempty closed convex subset of H . Let
  0 . Let A be an   inverse strongly monotone mapping of C into H and let B be a maximal
monotone operator on H such that the domain of B is included in C . Let J   ( I   B)1 be the resolvent
of B for any   0 . Then, for any   0, u  ( A  B)1 (0) if and only if u  J  ( I   A)u.
Lemma 2.2 [2] Let H be a real Hilbert space. Then, the following well known results hold:
(i)
tx  (1  t ) y
2
2
2
2
 t x  (1  t ) y  t (1  t ) x  y for all x, y  H and for all t [0,1],
2
(ii) 2 x  y, z  w  x  w  y  z
2
2
2
 x  z  y  w for all x, y, z, w  H .
Lemma 2.3 [8] Let H be a real Hilbert space, let  n  be a sequence of real numbers such that
0  a   n  b  1 for all n 
and let vn  and wn  be sequence in H such that for some
c, limsup vn  c, limsup wn  c and limsup  n vn  (1  n )wn  c. Then lim vn  wn  0.
n 
n 
n 
n
Lemma 2.4 [7] Let C be a nonempty closed convex subset of a real Hilbert space H , and let S : C  C
be a k  strictly preudo-nonspeading mapping. Then I  S is demiclosed at 0 .
Lemma 2.5 [ 9] Let H be a real Hilbert space, let S be a nonempty closed convex subset of H . Let  xn 
be a sequence in H . If xn1  x  xn  x for all n 
and x  S , then PS ( xn ) converges strongly to
some z  S , where PS stands for the metric projection on H onto S .
3. Main Result
In this section, we prove a weak convergence theorem for for finding a common elements of a set of
fixed points of a k - preudo-nonspreading mapping and the set of zero points of  - inverse strongly
monotone mapping and maximal monotone operator in a real Hilbert space.
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H , let A : C  H be  inverse strongly monotone, let B : D( B)  C  2H be maximal monotone, let J   ( I   B)1 be the
resolvent of B for any   0 , and let S : C  C be a k - strictly preudo-nonspeading mapping. Assume
that  : F (S ) ( A  B)1 (0)   . Define a sequence  xn  as follows:
x1  C ,
yn  J n ( I  n A) xn ,
xn 1   n xn  (1   n )( n yn  (1   n )Syn ),  n  ,
(3.1)
4
where  n  be a sequence in [k ,1) such that liminf (1  n )  0 and  n  , n  be sequence in [0,1] such
n
that 0  c   n  d  1 and 0  a  n  b  2 . Then xn ⇀ z0   where z0  lim P ( xn ) .
n
Proof. Let p  . It follows from Lemma 2.1, that p  J n ( I  n A) p, together with (3.1) and A is an
 - inverse strongly monotone, we get that
yn  p
2
2
 J  n ( I  n A) xn  J  n ( I  n A) p
 ( I  n A) xn  ( I  n A) p
2
2
 xn  p  2n xn  p, Axn  Ap  n2 Axn  Ap
2
2
 xn  p  2n Axn  Ap  n2 Axn  Ap
2
2
(3.2)
2
2
 xn  p  n (2  n ) Axn  Ap
2
 xn  p .
Next, let Sn : n I  (1  n )S . By Lemma 2.2 (i) and S is k  strictly pseudo-nonspreading, we have
S  n yn  p
2
  n yn  (1   n ) Syn  p
2
2
  n ( yn  p )  (1   n )( Syn  p )
2
2
  n yn  p  (1   n ) Syn  p   n (1   n ) yn  Syn
2
2
2
2
  n yn  p  (1   n )  yn  p  k yn  Syn    n (1   n ) yn  Syn


2
 yn  p  k (1   n ) yn  Syn
2
  n (1   n ) yn  Syn
(3.3)
2
2
2
 yn  p .
Using (3.1), (3.2) and (3.3), we get that
xn 1  p   n xn  (1   n ) S n yn  p
  n xn  p  (1   n ) S n yn  p
(3.4)
  n xn  p  (1   n ) yn  p
 xn  p
for all n 

. Therefore, we obtain that lim x n  p exists and hence xn  ,  Axn  ,  yn  and S n yn
x 
are bounded. From Lemma 2.2(i) , (3.2) and (3.3) we get that
xn 1  p
2
2
  n xn  p  (1   n ) S  n yn  p
2
  n xn  p  (1   n ) yn  p
2
2
2
2
2
  n xn  p  (1   n )  xn  p  n (2  n ) Axn  Ap 


2
2
 xn  p  (1   n )n (2  n ) Axn  Ap .

5
It implies that
0  (1  d )a(2  b) Axn  Ap
This mean that
2
2
 xn  p  xn1  p
2
lim Axn  Ap
 , as n   .
2
n 
0.
(3.5)
Since J n is firmly nonexpansive, we have
yn  p
2
 J  n ( I  n A) xn  J  n ( I  n A) p
2
 y n  p, ( I  n A) xn  ( I  n A) p



1
2
2
2
yn  p  ( I  n A) xn  ( I  n A) p  yn  p  ( I  n A) xn  ( I  n A) p
2
1
2
2
2
2

yn  p  xn  p  yn  xn  2n yn  xn , Axn  Ap  n2 Axn  Ap .
2


It implies that
yn  p
2
2
 xn  p  yn  xn
2
2
 2n yn  xn , Axn  Ap  n2 Axn  Ap .
(3.6)
Together with (3.1) and (3.3), we have
xn 1  p
2
2
2
  n xn  p  (1   n ) S  n yn  p
2
  n xn  p  (1   n ) yn  p
2

2
2
  n xn  p  (1   n )  xn  p  yn  xn
2
 xn  p  (1   n ) yn  xn
Therefore, we obtain that
(1  d ) yn  xn
2
2
2
 2n yn  xn , Axn  Ap  n2 Axn  Ap
2

2
 2n (1   n ) yn  xn , Axn  Ap  n2 (1   n ) Axn  Ap .

 xn  p  xn 1  p  Axn  Ap 2b(1  c ) yn  xn  b 2 (1  c ) Axn  Ap .
2
2
Since xn  ,  yn  are bounded, lim Axn  Ap  0 and lim xn  p exists, we have lim yn  xn  0 .
n 
n 
n 
Since A is Lipschitz continuous, we also have lim Ayn  Axn  0 .
n 
 
Since  xn  is bounded, there exists a subsequence xn j of  xn  converges weakly to z . We will show
that z   . First, we prove that z  ( A  B)1 (0). Since yn  Jn ( I  n A) xn , we have that
yn  ( I   n B) 1 ( I   n A) xn
 ( I  n A) xn  ( I  n B) yn  yn  n Byn
 xn  yn  n Axn  n Byn

1
n
( xn  yn  n Axn )  Byn .
Since B is monotone, for (u, v)  B , we have
yn  u,
1
n
( xn  yn  n Axn )  v  0,
6
and so
yn  u, xn  yn  n ( Axn  v)  0.
Since xn j ⇀ z , A is an   inverse strongly monotone and Axn  Ap by (3.5),
xn j  z, Axn j  Az   Axn j  Az
2
implies that Axn j  Az as j  . Moreover, since lim yn  xn  0 , we get that yn j ⇀ z .
n 
Then, we have that
lim yn j  u , xn j  yn j  n j ( Axn j  v)  0
j 
and hence z  u,  Az  v  0 . Since B is maximal monotone, ( Az )  Bz. That is z  ( A  B)1 (0).
Next, we will show that z  F ( S ). Let u   . Since Sn yn  u  yn  u  xn  u , we have
lim sup S  n yn  u  c, where c  lim xn  u . Further, we have
n 
n 
lim  n ( xn  u)  (1   n )(Sn yn  u)  lim xn1  u  c.
n
n
By Lemma 2.3, we get that
lim Sn yn  xn  0.
n
We also get that Sn yn  yn  Sn yn  xn  xn  yn . Hence, lim Sn yn  yn  0.
n
Since liminf (1  n )  0 and (1   n ) Syn  yn  Sn yn  yn , for all n  , lim Syn  yn  0.
n
n 
Since yn j ⇀ z and lim Syn  yn  0 , it follows from Lemma 2.4 that z  F ( S ).
n
 
Thus, z  . Let x nk be another subsequence of  x n  such that xnk ⇀ z . We will show that z  z .
Assume that z  z . By the Opial condition, we get
lim xn  z  lim inf xn j  z
n 
j 
 lim inf xn j  z
j 
 lim xn  z
n 
 lim inf xnk  z
k 
 lim inf xnk  z  lim xn  z .
k 
n 
This is a contradiction. Thus, z  z . This implies that xn ⇀ z  . Moreover, since for any p ,
xn1  p  xn  p , n  , by Lemma 2.5, there exists z0   such that P ( xn )  z0 . By the
property of metric projection, we obtain
z  P ( xn ), xn  P ( xn )  0 .
Therefore, we have
z  z0 , z  z0  z  z0  0.
This means that z  z0 , i.e. xn ⇀ z  lim P xn .
n

7
If we set k  0 and n  k  0 for all n 
, then we get the following corollary.
Corollary 3.2 [5] Let C be a nonempty closed convex subset of a real Hilbert space H , let A : C  H be
 - inverse strongly monotone, let B : D( B)  C  2H be maximal monotone, let J  ( I   B)1 be the
resolvent of B for any   0 , and let S : C  C be a nonspeading mapping. Assume
that  : F (S ) ( A  B)1 (0)   . Let x1  C , define
xn1 n xn  (1  n )S ( Jn ( I  n A) xn ),  n  ,
where  n  , n  be sequence in [0,1] such that 0  c   n  d  1 and 0  a  n  b  2 .
Then xn ⇀ z0   where z0  lim P ( xn ) .
n
.
Acknowledgements
The authors would like to thank the faculty of science and technology, Rajamangala University of
Technology Thanyaburi for the financial support.
References
[1] Browder F E, Petryshyn W V. Construction of fixed points of nonlinear mappings in Hilbert space. Journal of Mathematical
Analysis and Applications 1967; 20:197-228.
[2] Iemoto S, Takahashi W. Approximating commom fixed points of nonexpansive mappings and nonspreading mappings in a
Hilbert space. Nonlinear Anal 2009;71: 2080-2089.
[3] Kosaka F, Takahashi W. Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach
Spaces. SIAM. J.Optim 2008;19:824-835.
[4] Kosaka F, Takahashi W. Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in
Banach spaces. Arch. Math. (Basel) 2008; 91:166-177.
[5] Manaka H, and Takahashi W. Weak convergence theorems for maximal monotone operators with nonspreading mappings
in a Hilbert space. Cubo 2011; 13:11-24.
[6] Opial Z. Weak covergence of the sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math.
Soc.1967;73:591-597.
[7] Osilike MO, Isiogugu FO. Weak and strong convergence theorems for nonspreading-type mappings in Hilbert spaces.
Nonlinear Analysis. Theory, Methods and Applications A 2011; 74:1814-1822.
[8] Schu J. Weak and strong convergence to fixed points of asymptotically nonexpansive map-pings. Bull. Austral. Math. Soc.
1991;43:153–159.
[9] Takahashi W, Toyoda M, Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim.
Theory Appl. 2003; 118:417–428.