An. Şt. Univ. Ovidius Constanţa
Vol. 19(1), 2011, 49–66
ON THE CONVERGENCE OF ITERATIVE
SEQUENCES FOR A FAMILY OF
NONEXPANSIVE MAPPINGS AND
INVERSE-STRONGLY MONOTONE
MAPPINGS
Sun Young Cho and Shin Min Kang
Abstract
The purpose of this paper is to introduce a general iterative process
for the problem of finding a common element in the set of common
fixed points of an infinite family of nonexpansive mappings and in the
set of solutions of variational inequalities for inverse-strongly monotone
mappings.
1. Introduction and preliminaries
Throughout this paper, we assume that H is a real Hilbert space, whose
inner product and norm are denoted by h·, ·i and k · k, respectively. Let K be
a nonempty, closed and convex subset of H and A : K → H be a nonlinear
mapping. We denote by PK be the metric projection of H onto the closed
convex subset K. The classical variational inequality problem is to find u ∈ K
such that
hAu, v − ui ≥ 0, ∀v ∈ K.
(1.1)
Key Words: Fixed point, inverse-strongly monotone mapping, nonexpansive mapping,
variational inequality.
Mathematics Subject Classification: 47H05, 47H09, 47J25
Received: January, 2010
Accepted: December, 2010
Corresponding author: Shin Min Kang
49
50
Sun Young Cho and Shin Min Kang
In this paper, we use V I(K, A) to denote the solution set of the variational
inequality (1.1). For a given z ∈ H, u ∈ K satisfies the inequality hu − z, v −
ui ≥ 0, ∀v ∈ K, if and only if u = PK z. It is known that projection operator
PK is nonexpansive. It is also known that PK satisfies
hx − y, PK x − PK yi ≥ kPK x − PK yk2 ,
∀x, y ∈ H.
One can see that the variational inequality (1.1) is equivalent to a fixed
point problem. The element u ∈ K is a solution of the variational inequality
problem (1.1) if and only if u ∈ K satisfies the relation u = PK (I − λA)u,
where λ > 0 is a constant.
Recall that the following definitions.
(1) A mapping A : K → H is said to be inverse-strongly monotone if there
exists a positive real number µ such that
hx − y, Ax − Ayi ≥ µkAx − Ayk2 ,
∀x, y ∈ K.
For such a case, A is called µ-inverse-strongly monotone.
(2) A mapping S : K → K is said to be nonexpansive if
kSx − Syk ≤ kx − yk,
∀x, y ∈ K.
In this paper, we use F (S) to denote the fixed point set of S.
(3) A mapping f : K → K is said to be a contraction if there exists a
coefficient α (0 < α < 1) such that
kf (x) − f (y)k ≤ αkx − yk,
∀x, y ∈ K.
(4) A set-valued mapping T : H → 2H is said to be monotone if for all
x, y ∈ H, f ∈ T x and g ∈ T y imply hx − y, f − gi ≥ 0. A monotone mapping
T : H → 2H is maximal if the graph of G(T ) of T is not properly contained
in the graph of any other monotone mapping. It is known that a monotone
mapping T is maximal if and only if for (x, f ) ∈ H × H, hx − y, f − gi ≥ 0
for every (y, g) ∈ G(T ) implies f ∈ T x. Let A be a monotone map of K into
H and let NK v be the normal cone to K at v ∈ K, i.e., NK v = {w ∈ H :
hv − u, wi ≥ 0, ∀u ∈ K} and define
(
Av + NK v, v ∈ K,
Tv =
∅,
v∈
/ K.
Then T is maximal monotone and 0 ∈ T v if and only if v ∈ V I(K, A); see
[24].
The classical variational inequality and fixed point problems have been
studied based on iterative methods by many authors; see [3-14,18-23,27,30,31]
ON THE CONVERGENCE OF ITERATIVE SEQUENCES FOR A FAMILY OF
NONEXPANSIVE MAPPINGS AND INVERSE-STRONGLY MONOTONE
MAPPINGS
51
For finding a common element of the set of fixed points of a nonexpansive
mapping S and the solution of the variational inequalities for a µ-inversestrongly monotone mapping, Takahashi and Toyoda [27] introduced the following iterative process
x1 ∈ K,
xn+1 = αn xn + (1 − αn )SPK (xn − λn Axn ),
n ≥ 1,
(1.2)
where A is a µ-inverse-strongly monotone mapping, {αn } is a sequence in
(0, 1), and {λn } is a sequence in (0, 2µ). They showed that, if F (S)∩V I(K, A)
is nonempty, then the sequence {xn } generated in (1.2) converges weakly to
some z ∈ F (S) ∩ V I(K, A).
Recently, Iiduka and Takahashi [8] proposed another iterative scheme as
following
xn+1 = αn x + (1 − αn )SPK (xn − λn Axn ),
n ≥ 1,
(1.3)
where x1 = x ∈ K, {αn } is a sequence in (0, 1), and {λn } is a sequence
in (0, 2µ). They proved that the sequence {xn } converges strongly to z ∈
F (S) ∩ V I(K, A).
Very recently, Chen et al. [3] studied the following iterative process
x1 ∈ K,
xn+1 = αn f (xn ) + (1 − αn )SPK (xn − λn Axn ),
n ≥ 1,
(1.4)
where A is an inverse-strongly monotone mapping and also obtained a strong
convergence theorem by so-called viscosity approximation method which first
introduced by Moudafi [13] in the framework of Hilbert spaces.
On the other hand, for solving the variational inequality problem in the
finite-dimensional Euclidean space Rn , Korpelevich [10] introduced the following so-called extra-gradient method


x0 = x ∈ K,
(1.5)
yn = PK (xn − λAxn ),


xn+1 = PK (xn − λAyn ), n ≥ 0,
where λ ∈ 0, k1 .
Recently, Nadezhkina and Takahashi [14], Yao and Yao [30] and Zeng and
Yao [31] proposed some new iterative schemes for finding common elements
in F (S) ∩ V I(K, A) by combining (1.3) and (1.5). In particular, Yao and Yao
[30] introduced the following iterative algorithm


x1 ∈ K,
(1.6)
yn = PK (I − λn A)xn ,


xn+1 = αn u + βn xn + γn SPK (yn − λn Ayn ), n ≥ 1,
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Sun Young Cho and Shin Min Kang
where S is a nonexpansive mapping and A is a inverse-strongly monotone
mapping. They proved that the sequence {xn } generated by (1.6) converges
strongly to some point in F (S) ∩ V I(K, A).
Concerning a family of nonexpansive mappings has been considered by
many authors; see [2,7,11,12,15,16,18,20,25,29] and the references therein. In
this paper, we consider the mapping Wn defined by
Un,n+1 = I,
Un,n = γn Tn Un,n+1 + (1 − γn )I,
Un,n−1 = γn−1 Tn−1 Un,n + (1 − γn−1 )I,
..
.
Un,k = γk Tk Un,k+1 + (1 − γk )I,
(1.7)
Un,k−1 = γk−1 Tk−1 Un,k + (1 − γk−1 )I,
..
.
Un,2 = γ2 T2 Un,3 + (1 − γ2 )I,
Wn = Un,1 = γ1 T1 Un,2 + (1 − γ1 )I,
where γ1 , γ2 , . . . are real numbers such that 0 ≤ γn ≤ 1 and T1 , T2 , . . . be an
infinite family of mappings of K into itself. Nonexpansivity of each Ti ensures
the nonexpansivity of Wn .
Concerning Wn we have the following lemmas which are important to prove
our main results.
Lemma 1.1. ([25]) Let K be a nonempty closed convex subset of a strictly
convex Banach space E. Let T1 , T2 , . . . be nonexpansive mappings of K into
itself such that ∩∞
n=1 F (Tn ) is nonempty and γ1 , γ2 , . . . be real numbers such
that 0 < γn ≤ b < 1 for any n ≥ 1. Then for every x ∈ K and k ∈ N , the
limit limn→∞ Un,k x exists.
Using Lemma 1.1, one can define the mapping W of K into itself as follows.
W x = lim Wn x = lim Un,1 x,
n→∞
n→∞
∀x ∈ K.
(1.8)
Such a W is called the W -mapping generated by T1 , T2 , . . . and γ1 , γ2 , . . ..
Throughout this paper, we will assume that 0 < γn ≤ b < 1 for all n ≥ 1.
Lemma 1.2. ([25]) Let K be a nonempty closed convex subset of a strictly
convex Banach space E. Let T1 , T2 , . . . be nonexpansive mappings of K into
itself such that ∩∞
n=1 F (Tn ) is nonempty and γ1 , γ2 , . . . be real numbers such
that 0 < γn ≤ b < 1 for any n ≥ 1. Then, F (W ) = ∩∞
n=1 F (Tn ).
ON THE CONVERGENCE OF ITERATIVE SEQUENCES FOR A FAMILY OF
NONEXPANSIVE MAPPINGS AND INVERSE-STRONGLY MONOTONE
MAPPINGS
53
In this paper, motivated by research work going on in this direction, we
introduce a general iterative process as following


x1 ∈ K,
(1.9)
yn = PK (I − ηn B)xn ,


xn+1 = αn f (xn ) + βn xn + γn Wn PK (I − λn A)yn , n ≥ 1,
where A and B are µi -inverse-strongly monotone mappings from K into H,
respectively for i = 1, 2, f is a contraction on K and Wn is a mapping defined
by (1.7). It is proved that the sequence {xn } generated by the above iterative
scheme converges strongly to a common element of the set of common fixed
points of an infinite nonexpansive mappings and the set of solutions of the
variational inequalities for the inverse-strongly monotone mappings, which
solves another variation inequality
hf (q) − q, p − qi ≤ 0,
∀p ∈ ∩∞
i=1 F (Ti ) ∩ V I(K, A) ∩ V I(K, B).
In order to prove our main results, we also need the following lemmas.
Lemma 1.3. ([28]) Assume that {αn } is a sequence of nonnegative real numbers such that
αn+1 ≤ (1 − γn )αn + δn ,
where {γ
} is a sequence in (0, 1) and {δn } is a sequence such that
Pn∞
(i) n=1 γn = ∞;
P∞
(ii) lim supn→∞ γδnn ≤ 0 or n=1 |δn | < ∞.
Then limn→∞ αn = 0.
Lemma 1.4. ([26]) Let {xn } and {yn } be bounded sequences in a Banach space
E and {βn } be a sequence in [0, 1] with 0 < lim inf n→∞ βn ≤ lim supn→∞ βn <
1. Suppose that xn+1 = (1 − βn )yn + βn xn for all n ≥ 0 and
lim sup(kyn+1 − yn k − kxn+1 − xn k) ≤ 0.
n→∞
Then limn→∞ kyn − xn k = 0.
Lemma 1.5. ([17]) Let E be an inner product space. Then for all x, y, z ∈ E
and α, β, γ ∈ [0, 1] with α + β + γ = 1, we have
kαx + βy + γzk2 ≤ αkxk2 + βkyk2 + γkzk2 − αγkx − zk2
− αβkx − yk2 − βγky − zk2 .
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Sun Young Cho and Shin Min Kang
2. Main results
Now, we are ready to give our main results.
Theorem 2.1. Let K be a nonempty closed convex subset of a real Hilbert
space H, A : K → H be µ1 -inverse-strongly monotone mapping and B :
K → H be µ2 -inverse-strongly monotone mappings. Let f : K → K be a
contraction with the coefficient α, where 0 < α < 1. Let {xn } be a sequence
generated by (1.9), where {αn }, {βn } and {γn } are sequences in (0, 1) and
{λn }, {ηn } are chosen such that {ηn }, {λn } ⊂ [0, 2 min{µ1 , µ2 }]. Assume that
F = ∩∞
i=1 F (Ti ) ∩ V I(K, A) ∩ V I(K, B) 6= ∅. If the control sequences {αn },
{βn }, {γn }, {λn } and {ηn } are chosen such that
(a) αn + βn + γn = 1 forP
all n ≥ 1;
∞
(b) limn→∞ αn = 0 and n=1 αn = ∞;
(c) 0 < lim inf n→∞ βn ≤ lim supn→∞ βn < 1;
(d) limn→∞ |ηn+1 − ηn | = limn→∞ |λn+1 − λn | = 0;
(e) {ηn }, {λn } ∈ [u, v] for some u, v with 0 < u < v < 2 min{µ1 , µ2 },
then {xn } converges strongly to x∗ ∈ F , where x∗ = PF f (x∗ ), which solves
the following variation inequality
hf (x∗ ) − x∗ , p − x∗ i ≤ 0,
∀p ∈ F.
Proof. First, we show that I − λn A and I − ηn B are nonexpansive for all
n ≥ 1. Indeed, we see from condition (e) that
k(I − λn A)x − (I − λn A)yk2
= kx − y − λn (Ax − Ay)k2
= kx − yk2 − 2λn hx − y, Ax − Ayi + λ2n kAx − Ayk2
≤ kx − yk2 + λn (λn − 2µ1 )kAx − Ayk2
≤ kx − yk2
from which it follows that I − λn A is nonexpansive, so is I − ηn B. Letting
p ∈ F , we have
kyn − pk = kPK (I − ηn B)xn − pk ≤ kxn − pk.
It follows that
kxn+1 − pk = kαn f (xn ) + βn xn + γn Wn PC (I − λn A)yn − pk
≤ αn kf (xn ) − pk + βn kxn − pk + γn kWn PC (I − λn A)yn − pk
≤ αn kf (xn ) − pk + βn kxn − pk + γn kyn − pk
≤ αn kf (xn ) − f (p)k + αn kf (p) − pk + (1 − αn )kxn − pk
= (1 − αn (1 − α))kxn − pk + αn kf (p) − pk.
ON THE CONVERGENCE OF ITERATIVE SEQUENCES FOR A FAMILY OF
NONEXPANSIVE MAPPINGS AND INVERSE-STRONGLY MONOTONE
MAPPINGS
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By simple inductions, we have
kp − f (p)k
,
kxn − pk ≤ max kx1 − pk,
1−α
which yields that the sequence {xn } is bounded, so is {yn }.
Next, we show the sequence limn→∞ kxn+1 − xn k = 0. Note that
kyn+1 − yn k = kPK (I − ηn+1 B)xn+1 − PK (I − ηn B)xn k
≤ k(I − ηn+1 B)xn+1 − (I − ηn B)xn k
= k(I − ηn+1 B)xn+1 − (I − ηn+1 B)xn
(2.1)
+ (I − ηn+1 B)xn − (I − ηn B)xn k
≤ kxn+1 − xn k + |ηn+1 − ηn |M1 ,
where M1 is an appropriate constant such that M1 = supn≥1 {kBxn k}. Putting
ρn = PK (I − λn A)yn , we have
kρn+1 − ρn k = kPK (I − λn+1 A)yn+1 − PK (I − λn A)yn k
≤ k(I − λn+1 A)yn+1 − (I − λn A)yn k
= k(I − λn+1 A)yn+1 − (I − λn+1 A)yn
(2.2)
+ (I − λn+1 A)yn − (I − λn A)yn k
= kyn+1 − yn k + |λn+1 − λn |kAyn k.
Substituting (2.1) into (2.2), we arrive at
kρn+1 − ρn k ≤ kxn+1 − xn k + (|ηn+1 − ηn | + |λn+1 − λn |)M2 ,
(2.3)
where M2 is an appropriate constant such that M2 ≥ max{supn≥1 kAyn k, M1 }.
Define a sequence {zn } by
zn =
xn+1 − βn xn
,
1 − βn
∀n ≥ 1.
It follows that
zn+1 − zn
xn+1 − βn xn
xn+2 − βn+1 xn+1
−
=
1 − βn+1
1 − βn
αn+1 f (xn+1 ) + γn+1 Wn+1 ρn+1
αn f (xn ) + γn Wn ρn
=
−
1 − βn+1
1 − βn
αn
αn+1
(f (xn+1 ) − Wn+1 ρn+1 ) +
(Wn ρn − f (xn ))
=
1 − βn+1
1 − βn
+ Wn+1 ρn+1 − Wn ρn .
(2.4)
56
Sun Young Cho and Shin Min Kang
This implies that
kzn+1 − zn k
αn+1
αn
≤
(kf (xn+1 )k + kWn+1 ρn+1 k) +
(kWn ρn k + kf (xn )k)
1 − βn+1
1 − βn
+ kρn+1 − ρn k + kWn+1 ρn − Wn ρn k.
(2.5)
Since Ti and Un,i are nonexpansive, we obtain from (1.7) that
kWn+1 ρn − Wn ρn k = kγ1 T1 Un+1,2 ρn − γ1 T1 Un,2 ρn k
≤ γ1 kUn+1,2 ρn − Un,2 ρn k
= γ1 kγ2 T2 Un+1,3 ρn − γ2 T2 Un,3 ρn k
≤ γ1 γ2 kUn+1,3 ρn − Un,3 ρn k
≤ ···
(2.6)
≤ γ1 γ2 · · · γn kUn+1,n+1 ρn − Un,n+1 ρn k
n
Y
≤ M3
γi ,
i=1
where M3 ≥ 0 is an appropriate constant such that kUn+1,n+1 ρn −Un,n+1 ρn k ≤
M3 for all n ≥ 1. Substituting (2.3) and (2.6) into (2.5), we arrive at
kzn+1 − zn k − kxn+1 − xn k
αn
αn+1
(kf (xn+1 )k + kWn+1 ρn+1 k) +
(kWn ρn k + kf (xn )k)
≤
1 − βn+1
1 − βn
!
n
Y
γi .
+ M4 |ηn+1 − ηn | + |λn+1 − λn | +
i=1
where M4 is an appropriate constant such that M4 = max{M2 , M3 }. From
the conditions (b), (c) and (d), we obtain that
lim sup(kzn+1 − zn k − kxn+1 − xn k) ≤ 0.
n→∞
From the condition (c) and applying Lemma 1.4, we obtain that
lim kzn − xn k = 0.
n→∞
(2.7)
Consequently, we obtain from (2.4) and the condition (c) that
lim kxn+1 − xn k = lim (1 − βn )kzn − xn k = 0.
n→∞
n→∞
(2.8)
ON THE CONVERGENCE OF ITERATIVE SEQUENCES FOR A FAMILY OF
NONEXPANSIVE MAPPINGS AND INVERSE-STRONGLY MONOTONE
MAPPINGS
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Next, we show that
lim kW xn − xn k = 0.
n→∞
(2.9)
For any p ∈ F , we have
kyn − pk2 = kPK (I − ηn B)xn − pk2
≤ k(xn − p) − ηn (Bxn − Bp)k2
= kxn − pk2 − 2ηn hxn − p, Bxn − Bpi + ηn2 kBxn − Bpk2 (2.10)
≤ kxn − pk2 − 2ηn µ2 kBxn − Bpk2 + ηn2 kBxn − Bpk2
≤ kxn − pk2 + ηn (ηn − 2µ2 )kBxn − Bpk2 .
On the other hand, we have
kρn − pk2 = kPK (I − λn A)yn − pk2
≤ k(I − λn A)yn − pk2
= kyn − p − λn (Ayn − Ap)k2
= kyn − pk2 − 2λn hyn − p, Ayn − Api + λ2n kAyn − Apk2
(2.11)
≤ kyn − pk2 − 2λn µ1 kAyn − Apk2 + λ2n kAyn − Apk2
≤ kxn − pk2 + λn (λn − 2µ1 )kAyn − Apk2 .
It follows from Lemma 1.5 that
kxn+1 − pk2 = kαn (f (xn ) − p) + βn (xn − p) + γn (Wn ρn − p)k2
≤ αn kf (xn ) − pk2 + βn kxn − pk2 + γn kρn − pk2 .
(2.12)
Substituting (2.11) into (2.12), we arrive at
kxn+1 −pk2 ≤ αn kf (xn )−pk2 +kxn −pk2 +γn λn (λn −2µ1 )kAyn −Apk2 . (2.13)
It follows from condition (e) that
γn u(2µ1 − v)kAyn − Apk2
≤ αn kf (xn ) − pk2 + kxn − pk2 − kxn+1 − pk2
≤ αn kf (xn ) − pk2 + (kxn − pk + kxn+1 − pk)kxn − xn+1 k.
From the conditions (b) and (c), we obtain from (2.8) that
lim kAyn − Apk = 0.
n→∞
(2.14)
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Sun Young Cho and Shin Min Kang
Using (2.12) again, we have
kxn+1 − pk2 ≤ αn kf (xn ) − pk2 + βn kxn − pk2 + γn kyn − pk2 ,
(2.15)
which combines with (2.10) yields that
kxn+1 − pk2 ≤ αn kf (xn ) − pk2 + kxn − pk2 + γn ηn (ηn − 2µ2 )kBxn − Bpk2 .
From condition (e), we arrive at
γn u(2µ2 − v)kBxn − Bpk2
≤ αn kf (xn ) − pk2 + kxn − pk2 − kxn+1 − pk2
≤ αn kf (xn ) − pk2 + (kxn − pk + kxn+1 − pk)kxn − xn+1 k.
It follows from the condition (b) and (2.8) that
lim kBxn − Bpk = 0.
n→∞
(2.16)
On the other hand, we have
kyn − pk2 = kPK (I − ηn B)xn − PK (I − ηn B)pk2
≤ h(I − ηn B)xn − (I − ηn B)p, yn − pi
1
= {k(I − ηn B)xn − (I − ηn B)pk2 + kyn − pk2
2
− k(I − ηn B)xn − (I − ηn B)p − (yn − p)k2 }
1
≤ {kxn − pk2 + kyn − pk2 − k(xn − yn ) − ηn (Bxn − Bp)k2 }
2
1
= {kxn − pk2 + kyn − pk2 − kxn − yn k2 − ηn2 kBxn − Bpk2
2
+ 2ηn hxn − yn , Bxn − Bpi},
which yields that
kyn − pk2 ≤kxn − pk2 − kxn − yn k2 + 2ηn kxn − yn kkBxn − Bpk.
(2.17)
In a similar way, we can prove that
kρn − pk2 ≤kxn − pk2 − kρn − yn k2 + 2λn kρn − yn kkAyn − Apk.
Substitute (2.18) into (2.12) yields that
kxn+1 − pk2 ≤ αn kf (xn ) − pk2 + kxn − pk2 − γn kρn − yn k2
+ 2γn λn kρn − yn kkAyn − Apk.
(2.18)
ON THE CONVERGENCE OF ITERATIVE SEQUENCES FOR A FAMILY OF
NONEXPANSIVE MAPPINGS AND INVERSE-STRONGLY MONOTONE
MAPPINGS
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It follows that
γn kρn − yn k2 ≤ αn kf (xn ) − pk2 + kxn − pk2 − kxn+1 − pk2
+ 2γn λn kρn − yn kkAyn − Apk
≤ αn kf (xn ) − pk2 + (kxn − pk + kxn+1 − pk)kxn − xn+1 k
+ 2γn λn kρn − yn kkAyn − Apk.
In view of the conditions (b) and (c), we see from (2.8) and (2.14) that
lim kρn − yn k = 0.
n→∞
(2.19)
Similarly, substituting (2.17) into (2.15), we can prove that
lim kxn − yn k = 0.
n→∞
(2.20)
On the other hand, we have
xn+1 − xn = αn (f (xn ) − xn ) + γn (Wn ρn − xn ).
It follows that
γn kWn ρn − xn k ≤ kxn+1 − xn k + αn kf (xn ) − xn k.
In view of conditions (b) and (c), we see from (2.8) that
lim kWn ρn − xn k = 0.
n→∞
(2.21)
Observe that
kWn xn − xn k ≤ kWn xn − Wn ρn k + kWn ρn − xn k
≤ kxn − ρn k + kWn ρn − xn k
≤ kxn − yn k + kyn − ρn k + kWn ρn − xn k.
It follows from (2.19)-(2.21) that
lim kWn xn − xn k = 0.
n→∞
(2.22)
From Remark 3.3 of [29], see also [7], we have kW xn − Wn xn k → 0 as n → ∞.
It follows that (2.9) holds. Observe that PF f is a contraction. Indeed, for all
x, y ∈ C, we have
kPF f (x) − PF f (y)k ≤ kf (x) − f (y)k ≤ αkx − yk.
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Sun Young Cho and Shin Min Kang
Banach’s contraction mapping principle guarantees that PF f has a unique
fixed point, say x∗ ∈ C. That is, x∗ = PF f (x∗ ).
Next, we show that
lim suphf (x∗ ) − x∗ , xn − x∗ i ≤ 0.
(2.23)
n→∞
To show it, we choose a subsequence {xni } of {xn } such that
lim suphf (x∗ ) − x∗ , xn − x∗ i = lim hf (x∗ ) − x∗ , xni − x∗ i.
i→∞
n→∞
As {xni } is bounded, we have that there is a subsequence {xnij } of {xni }
converges weakly to x̄. Without loss of generality, we may assume that xni ⇀
x̄. From (2.19) and (2.20), we also have yni ⇀ x̄ and ρni ⇀ x̄, respectively.
Next, we have x̄ ∈ F . Indeed, let us first show that x̄ ∈ V I(K, A). Put
(
Av + NK v, v ∈ K,
Tv =
∅,
v∈
/ K.
Then T is maximal monotone. Let (v, w) ∈ G(T ). Since w − Av ∈ NK v and
ρn ∈ K, we have
hv − ρn , w − Avi ≥ 0.
On the other hand, we see from ρn = PK (I − λn A)yn that
hv − ρn , ρn − (I − λn A)yn i ≥ 0
and hence
ρn − yn
v − ρn ,
+ Ayn ≥ 0.
λn
It follows that
hv − ρni , wi ≥ hv − ρni , Avi
ρni − yni
≥ hv − ρni , Avi − v − ρni ,
+ Ayni
λni
ρn − yni
≥ v − ρni , Av − i
− Ayni
λni
= hv − ρni , Av − Aρni i + hv − ρni , Aρni − Ayni i
ρni − yni
− v − ρni ,
λni
ρni − yni
,
≥ hv − ρni , Aρni − Ayni i − v − ρni ,
λni
ON THE CONVERGENCE OF ITERATIVE SEQUENCES FOR A FAMILY OF
NONEXPANSIVE MAPPINGS AND INVERSE-STRONGLY MONOTONE
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61
which implies that hv − x̄, wi ≥ 0. We have x̄ ∈ T −1 0 and hence x̄ ∈ V I(K, A).
In a similar way, we can show x̄ ∈ V I(K, B).
Next, let us show x̄ ∈ ∩∞
i=1 F (Ti ). Since Hilbert spaces are Opial’s spaces,
we obtain from (2.9) that
lim inf kxni − x̄k < lim inf kxni − W x̄k
i→∞
i→∞
= lim inf kxni − W xni + W xni − W x̄k
i→∞
≤ lim inf kW xni − W x̄k
i→∞
≤ lim inf kxni − x̄k,
i→∞
which derives a contradiction. Thus, we have x̄ ∈ F (W ) = ∩∞
i=1 F (Ti ). On the
other hand, we have
lim suphf (x∗ ) − x∗ , xn − x∗ i = lim hf (x∗ ) − x∗ , xni − x∗ i
n→∞
n→∞
= hf (x∗ ) − x∗ , x̄ − x∗ i ≤ 0.
That is, (2.23) holds. It follows that
lim suphf (x∗ ) − x∗ , xn+1 − x∗ i ≤ 0.
(2.24)
n→∞
Finally, we show that xn → x∗ as n → ∞. Note that
kxn+1 − x∗ k2
= hαn (f (xn ) − x∗ ) + βn (xn − x∗ ) + γn (Wn ρn − x∗ ), xn+1 − x∗ i
= αn hf (xn ) − x∗ , xn+1 − x∗ i + βn hxn − x∗ , xn+1 − x∗ i
+ γn hWn ρn − x∗ , xn+1 − x∗ i
= αn hf (xn ) − f (x∗ ), xn+1 − x∗ i + αn hf (x∗ ) − x∗ , xn+1 − x∗ i
+ βn hxn − x∗ , xn+1 − x∗ i + γn hWn ρn − x∗ , xn+1 − x∗ i
≤ αn kf (xn ) − f (x∗ )kkxn+1 − x∗ k + αn hf (x∗ ) − x∗ , xn+1 − x∗ i
+ βn kxn − x∗ kkxn+1 − x∗ k + γn kWn ρn − x∗ kkxn+1 − x∗ k
≤ αn αkxn − x∗ kkxn+1 − x∗ k + αn hf (x∗ ) − x∗ , xn+1 − x∗ i
+ βn kxn − x∗ kkxn+1 − x∗ k + γn kxn − x∗ kkxn+1 − x∗ k
= (1 − αn (1 − α))kxn − x∗ kkxn+1 − x∗ k + αn hf (x∗ ) − x∗ , xn+1 − x∗ i
≤
≤
1 − αn (1 − α)
(kxn − x∗ k2 + kxn+1 − x∗ k2 )
2
+ αn hf (x∗ ) − x∗ , xn+1 − x∗ i
1 − αn (1 − α)
1
kxn − x∗ k2 + kxn+1 − x∗ k2
2
2
+ αn hf (x∗ ) − x∗ , xn+1 − x∗ i.
62
Sun Young Cho and Shin Min Kang
It follows that
kxn+1 − x∗ k2 ≤ [1 − αn (1 − α)]kxn − x∗ k2 + 2αn hf (x∗ ) − x∗ , xn+1 − x∗ i.
From (2.24) and applying Lemma 1.3, we can obtain the desired conclusion
immediately. This completes the proof.
Let A = B and f (x) = x1 for all x ∈ K in Theorem 2.1. We can obtain
the following result easily.
Corollary 2.2. Let K be a nonempty closed convex subset of a real Hilbert
space H and A : K → H be µ-inverse-strongly monotone mappings. Let {xn }
be a sequence generated by the following iterative process


x1 ∈ K,
yn = PK (I − ηn A)xn ,


xn+1 = αn x1 + βn xn + γn Wn PK (I − λn A)yn , n ≥ 1,
where Wn is defined by (1.8), {αn }, {βn } and {γn } are sequences in (0, 1) and
{λn }, {ηn } are chosen such that {ηn }, {λn } ⊂ [0, 2 min{µ1 , µ2 }]. Assume that
F = ∩∞
i=1 F (Ti ) ∩ V I(K, A) 6= ∅. If the control sequences {αn }, {βn }, {γn },
{λn } and {ηn } are chosen such that
(a) αn + βn + γn = 1 forP
all n ≥ 1;
∞
(b) limn→∞ αn = 0 and n=1 αn = ∞;
(c) 0 < lim inf n→∞ βn ≤ lim supn→∞ βn < 1;
(d) limn→∞ |ηn+1 − ηn | = limn→∞ |λn+1 − λn | = 0;
(e) {ηn }, {λn } ∈ [u, v] for some u, v with 0 < u < v < 2 min{µ1 , µ2 },
then {xn } converges strongly to x∗ ∈ F , where x∗ = PF x1 , which solves the
following variation inequality
hx1 − x∗ , p − x∗ i ≤ 0,
∀p ∈ F.
Remark 2.3. Corollary 2.2 mainly improves the corresponding result of Yao
and Yao [30] from a single nonexpansive mapping to an infinite family nonexpansive mappings.
As some applications of our main results, we next consider another class
of important nonlinear operator: strict pseudo-contractions.
Recall that a mapping S : K → K is said to be a k-strict pseudocontraction if there exists a constant k ∈ [0, 1) such that
kSx − Syk2 ≤ kx − yk2 + kk(I − S)x − (I − S)yk2 ,
∀x, y ∈ K.
ON THE CONVERGENCE OF ITERATIVE SEQUENCES FOR A FAMILY OF
NONEXPANSIVE MAPPINGS AND INVERSE-STRONGLY MONOTONE
MAPPINGS
63
Note that the class of k-strict pseudo-contractions strictly includes the class
of nonexpansive mappings.
Put A = I − S, where S : K → K is a k-strict pseudo-contraction. Then
A is 1−k
2 -inverse-strongly monotone; see [1].
Theorem 2.4. Let K be a nonempty closed convex subset of a real Hilbert
space H, S1 : K → K be a k1 -strict pseudo-contraction and S2 : K → K
be a k2 -strict pseudo-contraction. Let f : K → K be a contraction with the
coefficient α (0 < α < 1). Let {xn } be a sequence generated by the following
iterative process


x1 ∈ K,
yn = (1 − ηn )xn + ηn S2 xn ,


xn+1 = αn f (xn ) + βn xn + γn Wn ((1 − λn )yn + λn S1 yn ), n ≥ 1,
where Wn is defined by (1.8), {αn }, {βn } and {γn } are sequences in (0, 1) and
{λn }, {ηn } are chosen such that {ηn }, {λn } ⊂ [0, 2 min{(1 − k1 ), (1 − k2 )}].
Assume that F = ∩∞
i=1 F (Ti ) ∩ F (S1 ) ∩ F (S2 ) 6= ∅. If the control sequences
{αn }, {βn }, {γn }, {λn } and {ηn } are chosen such that
(a) αn + βn + γn = 1 for all n ≥ 1;
P∞
(b) limn→∞ αn = 0 and n=1 αn = ∞;
(c) 0 < lim inf n→∞ βn ≤ lim supn→∞ βn < 1;
(d) limn→∞ |ηn+1 − ηn | = limn→∞ |λn+1 − λn | = 0;
(e) {ηn }, {λn } ∈ [u, v] for some u, v with 0 < u < v < 2 min{µ1 , µ2 },
then {xn } converges strongly to x∗ ∈ F , where x∗ = PF f (x∗ ), which solves
the following variation inequality
hf (x∗ ) − x∗ , p − x∗ i ≤ 0,
∀p ∈ F.
1
Proof. Put A = I − S1 and B = I − S2 . Then A is 1−k
2 -inverse-strongly
1−k2
monotone and B is 2 -inverse-strongly monotone, respectively. We have
F (S1 ) = V I(K, A), F (S2 ) = V I(K, B), PK (I −λn A)yn = (1−λn )yn +λn S1 yn
and PK (I −ηn B)xn = (1−ηn )xn +ηn S2 xn . It is easy to conclude from Theorem
2.1 the desired conclusion.
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Department of Mathematics,
Gyeongsang National University,
Jinju 660-701, Korea,
E-mail: ooly61@yahoo.co.kr
Department of Mathematics and RINS,
Gyeongsang National University,
Jinju 660-701, Korea,
E-mail: smkang@gnu.ac.kr