Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 893635, 17 pages
doi:10.1155/2012/893635
Research Article
Some Results on an Infinite Family of
Nonexpansive Mappings and an Inverse-Strongly
Monotone Mapping in Hilbert Spaces
Peng Cheng1 and Anshen Zhang2
1
School of Mathematics and Information Science, North China University of Water Resources and Electric
Power, Zhengzhou 450011, China
2
No. 13 Zhongxue, Feng-Feng Kuangqu, Hangdan 056000, China
Correspondence should be addressed to Peng Cheng, hschengp@163.com
Received 8 September 2012; Accepted 10 October 2012
Academic Editor: Xiaolong Qin
Copyright q 2012 P. Cheng and A. Zhang. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We study the problem of approximating a common element in the common fixed point set of
an infinite family of nonexpansive mappings and in the solution set of a variational inequality
involving an inverse-strongly monotone mapping based on a viscosity approximation iterative
method. Strong convergence theorems of common elements are established in the framework of
Hilbert spaces.
1. Introduction and Preliminaries
Let H be a real Hilbert space, whose inner product and norm are denoted by ·, · and · ,
respectively. Let C be a nonempty, closed, and convex subset of H. Let A : C → H be a
mapping. Let PC be the metric projection from H onto the subset C. The classical variational
inequality is to find u ∈ C such that
Au, v − u ≥ 0,
∀v ∈ C.
1.1
In this paper, we use VIC, A to denote the solution set of the variational inequality. For a
given point z ∈ H, u ∈ C satisfies the inequality
u − z, v − u ≥ 0,
∀v ∈ C,
1.2
2
Abstract and Applied Analysis
if and only if u PC z. It is known that projection operator PC is nonexpansive. It is also know
that PC satisfies
2
x − y, PC x − PC y ≥ PC x − PC y ,
∀x, y ∈ H.
1.3
One can see that the variational inequality 1.1 is equivalent to a fixed point problem. The
point u ∈ C is a solution of the variational inequality 1.1 if and only if u ∈ C satisfies the
relation u PC u − λAu, where λ > 0 is a constant.
Recall the following definitions.
a A is said to be monotone if and only if
Ax − Ay, x − y ≥ 0,
x, y ∈ C.
1.4
b A is said to be α-strongly monotone if and only if there exists a positive real number
α such that
2
Ax − Ay, x − y ≥ αx − y ,
x, y ∈ C.
1.5
c A is said to be α-inverse-strongly monotone if and only if there exists a positive
real number α such that
2
Ax − Ay, x − y ≥ αAx − Ay ,
∀x, y ∈ C.
1.6
d A mapping S : C → C is said to be nonexpansive if and only if
Sx − Sy ≤ x − y,
∀x, y ∈ C.
1.7
In this paper, we use FS to denote the fixed point set of S.
e A mapping f : C → C is said to be a κ-contraction if and only if there exists a
positive real number κ ∈ 0, 1 such that
fx − f y ≤ κx − y,
∀x, y ∈ C.
1.8
f A linear bounded operator B on H is strongly positive if and only if there exists a
positive real number γ such that
Bx, x ≥ γx2 ,
∀x ∈ H.
1.9
g A set-valued mapping T : H → 2H is called monotone if and only if for all x, y ∈
H, f ∈ T x, and g ∈ T y imply x − y, f − g ≥ 0. A monotone mapping T : H → 2H
is maximal if the graph of GT 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, x − y, f − g ≥ 0 for every y, g ∈ GT implies f ∈ T x. Let A be a monotone
Abstract and Applied Analysis
3
map of C into H and let NC v be the normal cone to C at v ∈ C, that is, NC v {w ∈ H :
v − u, w ≥ 0, for all u ∈ C} and define
Av NC v, v ∈ C,
Tv ∅,
v∈
/ C.
1.10
Then T is maximal monotone and 0 ∈ T v if and only if v ∈ VIC, A; see 1 and the
reference therein.
For finding a common element in the fixed point set of nonexpansive mappings and in
the solution set of the variational inequality involving inverse-strongly mappings, Takahashi
and Toyoda 2 introduced the following iterative process:
x0 ∈ C,
xn1 αn xn 1 − αn SPC xn − λn Axn ,
∀n ≥ 0,
1.11
where A is an α-inverse-strongly monotone mapping, {αn } is a real number sequence in 0, 1,
and {λn } is a real number sequence in 0, 2α. They showed that the sequence {xn } generated
in 1.11 weakly converges to some point z ∈ FS ∩ VIC, A provided that FS ∩ VIC, A
is nonempty.
In order to obtain a strong convergence theorem of common elements, Iiduka and
Takahashi 3 considered the problem by the following iterative process:
x0 ∈ C,
xn1 αn x 1 − αn SPC xn − λn Axn ,
∀n ≥ 0,
1.12
where x is a fixed element in C, A is an α-inverse-strongly monotone mapping, {αn } is a real
number sequence in 0, 1, and {λn } is a real number sequence in 0, 2α. They showed that
the sequence {xn } generated in 1.12 strongly converges to some point z ∈ FS ∩ VIC, A
provided that FS ∩ VIC, A is nonempty.
Iterative methods for nonexpansive mappings have been applied to solve convex
minimization problems; see, for example, 4–8 and the references therein. A typical problem
is to minimize a quadratic function over the set of the fixed points a nonexpansive mapping
S on a real Hilbert space H:
min
1
x∈FS 2
Bx, x − x, b,
1.13
where B is a linear bounded self-adjoint operator, and b is a given point in H. In 4, it is
proved that the sequence {xn } defined by the iterative method below, with the initial guess
x0 ∈ H chosen arbitrarily,
xn1 I − αn BSxn αn b,
n ≥ 0,
1.14
strongly converges to the unique solution of the minimization problem 1.13 provided that
the sequence {αn } satisfies certain conditions.
4
Abstract and Applied Analysis
Recently, Marino and Xu 5 considered the problem by viscosity approximation
method. They study the following iterative process:
x0 ∈ C,
xn1 I − αn BSxn αn γfxn ,
n ≥ 0,
1.15
where f is a contraction. They proved that the sequence {xn } generated by the above iterative
scheme strongly converges to the unique solution of the variational inequality
B − γf x∗ , x − x∗ ≥ 0,
x ∈ C,
1.16
which is the optimality condition for the minimization problem minx∈FS 1/2Bx, x − hx,
where h is a potential function for δf i.e., h x δfx for x ∈ H.
Concerning a family of nonlinear mappings has been considered by many authors; see,
for example, 9–21 and the references therein. The well-known convex feasibility problem
reduces to finding a point in the intersection of the fixed point sets of a family of nonexpansive
mappings. The problem of finding an optimal point that minimizes a given cost function over
common set of fixed points of a family of nonexpansive mappings is of wide interdisciplinary
interest and practical importance; see, for example, 16, 17.
Recently, Qin et al. 18 considered a general iterative algorithm for an infinite family
of nonexpansive mapping in the framework of Hilbert spaces. To be more precise, they
introduced the following general iterative algorithm:
x0 ∈ C,
xn1 λn γfxn βn xn 1 − βn I − λn A Wn xn ,
n ≥ 0,
1.17
where f is a contraction on H, A is a strongly positive bounded linear operator, Wn are
nonexpansive mappings which are generated by a finite family of nonexpansive mapping
T1 , T2 , . . . as follows:
Un,n1 I,
Un,n γn Tn Un,n1 1 − γn I,
Un,n−1 γn−1 Tn−1 Un,n 1 − γn−1 I,
..
.
Un,k γk Tk Un,k1 1 − γk I,
un,k−1 γk−1 Tk−1 Un,k 1 − γk−1 I,
1.18
..
.
Un,2 γ2 T2 Uu,3 1 − γ2 I,
Wn Un,1 γ1 T1 Un,2 1 − γ1 I,
where {γ1 }, {γ2 }, . . . are real numbers such that 0 ≤ γ ≤ 1, T1 , T2 , . . . become an infinite family
of mappings of C into itself. Nonexpansivity of each Ti ensures the nonexpansivity of Wn .
Abstract and Applied Analysis
5
Concerning Wn we have the following lemmas which are important to prove our main
results.
Lemma 1.1 see 19. Let C be a nonempty closed convex subset of a strictly convex Banach space
E. Let T1 , T2 , . . . be nonexpansive mappings of C into itself such that ∞
n1 FTn is nonempty, and let
γ1 , γ2 , . . . be real numbers such that 0 < γn ≤ η < 1 for any n ≥ 1. Then, for every x ∈ C and k ∈ N,
the limit limn → ∞ Un,k x exists.
Using Lemma 1.1, one can define the mapping W of C into itself as follows. Wx limn → ∞ Wn x limn → ∞ Un,1 x, for every x ∈ C. Such a W is called the W-mapping generated
by T1 , T2 , . . . and γ1 , γ2 , . . .. Throughout this paper, we will assume that 0 < γn ≤ η < 1 for all
n ≥ 1.
Lemma 1.2 see 19. Let C be a nonempty closed convex subset of a strictly convex Banach space
nonempty, and let
E. Let T1 , T2 , . . . be nonexpansive mappings of C into itself such that ∞
n1 FTn is
γ1 , γ2 , . . . be real numbers such that 0 < γn ≤ η < 1 for any n ≥ 1. Then, FW ∞
n1 FTn .
Motivated by the above results, in this paper, we study the problem of approximating
a common element in the common fixed point set of an infinite family of nonexpansive
mappings, and in the solution set of a variational inequality involving an inversestrongly monotone mapping based on a viscosity approximation iterative method. Strong
convergence theorems of common elements are established in the framework of Hilbert
spaces.
In order to prove our main results, we need the following lemmas.
Lemma 1.3 see 5. Assume B is a strongly positive linear bounded operator on a Hilbert space H
with coefficient γ > 0 and 0 < ρ ≤ B−1 . Then I − ρB ≤ 1 − ργ.
Lemma 1.4 see 22. Assume that {αn } is a sequence of nonnegative real numbers such that
αn1 ≤ 1 − γn αn δn ,
1.19
where γn is a sequence in 0, 1 and {δn } is a sequence such that
i
∞
n1 γn
∞;
ii lim supn → ∞ δn /γn ≤ 0 or
∞
n1
|δn | < ∞.
Then limn → ∞ αn 0.
Lemma 1.5 see 23. Let {xn } and {yn } be bounded sequences in a Banach space X and let {βn } be
a sequence in 0, 1 with 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1. Suppose xn1 1 − βn yn βn xn
for all integers n ≥ 0 and
lim sup yn1 − yn − xn1 − xn ≤ 0.
n→∞
Then limn → ∞ yn − xn 0.
1.20
6
Abstract and Applied Analysis
Lemma 1.6 see 14, 15. Let K be a nonempty closed convex subset of a Hilbert space H, {Ti : C →
C} be a family of infinitely nonexpansive mappings with ∞
i1 FTi , {γn } be a real sequence such that
0 < γn ≤ b < 1 for each n ≥ 1. If C is any bounded subset of K, then limn → ∞ supx∈C Wx−Wn x 0.
Lemma 1.7 see 5. Let H be a Hilbert space. Let B be a strongly positive linear bounded selfadjoint operator with the constant γ > 0 and f a contraction with the constant κ. Assume that 0 <
γ < γ/κ. Let T be a nonexpansive mapping with a fixed point xt ∈ H of the contraction x →
tγfx I − tBT x. Then {xt } converges strongly as t → 0 to a fixed point x of T , which solves the
variational inequality
A − γf x, z − x ≤ 0,
∀z ∈ FT .
1.21
Equivalently, we have PFT I − A γfx x.
2. Main Results
Theorem 2.1. Let H be a real Hilbert space and C a nonempty closed convex subset of H. Let
A : C → H be an α-inverse-strongly monotone mapping and f : C → C a κ-contraction.
Let {T }∞ be an infinite family of nonexpansive mappings from C into itself such that F :
∞ i i1
∅. Let B be a strongly positive linear bounded self-adjoint operator of C into
i1 FTi ∩ VIC, A /
itself with the constant γ > 0. Let {xn } be a sequence generated in
x1 ∈ C,
yn βn γfxn I − βn B Wn PC I − rn Axn ,
xn1 αn xn 1 − αn PC yn ,
2.1
n ≥ 1,
where Wn is generated in 1.18, {αn } and {βn } are real number sequences in 0, 1. Assume that the
control sequence {αn }, {βn }, and {rn } satisfy the following restrictions:
i limn → ∞ βn 0,
∞
n1
βn ∞;
ii 0 < lim infn → ∞ αn ≤ lim supn → ∞ αn < 1;
iii limn → ∞ |rn1 − rn | 0;
iv {rn } ⊂ a, b for some a, b with 0 < a < b < 2α.
Assume that 0 < γ < γ/κ. Then {xn } strongly converges to some point q, where q ∈ F, where
q PF γf I − Bq, which solves the variation inequality
γf q − Bq, p − q ≤ 0,
∀p ∈ F.
2.2
Abstract and Applied Analysis
7
Proof. First, we show that the mapping I − rn A is nonexpansive. Notice that
I − rn Ax − I − rn Ay2 x − y − rn Ax − Ay 2
2
2
x − y − 2rn x − y, Ax − Ay rn2 Ax − Ay
2
2
≤ x − y rn rn − 2αAx − Ay
2
≤ x − y , ∀x, y ∈ C,
2.3
which implies that the mapping I − rn A is nonexpansive. Since the condition i, we may
assume, with no loss of generality, that βn < B−1 for all n. From Lemma 1.3, we know that
if 0 < ρ ≤ B−1 , then I − ρB ≤ 1 − ργ. Letting p ∈ F, we have
yn − p βn γfxn − Bp I − βn B Wn PC I − rn Axn − p ≤ βn γfxn − Bp 1 − βn γ Wn PC I − rn Axn − p
≤ βn γ fxn − f p βn γf p − Bp 1 − βn γ xn − p
1 − βn γ − κγ xn − p βn γf p − Bp.
2.4
On the other hand, we have
xn1 − p αn xn − p 1 − αn PC yn − p ≤ αn xn − p 1 − αn yn − p
≤ αn xn − p 1 − αn 1 − βn γ − γκ xn − p βn γf p − Bp .
2.5
By simple induction, we have
Bp − γf p xn − p ≤ max x0 − p,
,
γ − γκ
2.6
which gives that the sequence {xn } is bounded, so is {yn }.
Next, we prove limn → ∞ |xn1 − xn 0. Put ρn PC I − rn Axn . Next, we compute
ρn − ρn1 PC I − rn Axn − PC I − rn1 Axn1 ≤ I − rn Axn − I − rn1 Axn1 xn − rn Axn − xn1 − rn Axn1 rn1 − rn Axn1 ≤ xn − xn1 |rn1 − rn |M1 ,
2.7
8
Abstract and Applied Analysis
where M1 is an appropriate constant such that M1 ≥ supn≥1 {Axn }. It follows that
yn − yn1 I − βn1 B Wn1 ρn1 − Wn ρn − βn1 − βn BWn ρn
γ βn1 fxn1 − fxn fxn βn1 − βn ≤ 1 − βn1 γ ρn1 − ρn Wn1 ρn − Wn ρn βn1 − βn M2 γβn1 κxn1 − xn ,
2.8
where M2 is an appropriate constant such that
M2 ≥ max sup BWn ρn , γ sup fxn .
n≥1
n≥1
2.9
Since Ti and Un,i are nonexpansive, we have from 1.18 that
Wn1 ρn − Wn ρn γ1 T1 Un1,2 ρn − γ1 T1 Un,2 ρn ≤ γ1 Un1,2 ρ − Un,2 ρn γ1 γ2 T2 Uu1,3 ρn − γ2 T2 Un,3 ρn ≤ γ1 γ2 Uu1,3 ρn − Un,3 ρn ≤ ···
2.10
≤ γ1 γ2 · · · γn Un1,n1 ρn − Un,n1 ρn ≤ M3
n
γi ,
i1
where M3 ≥ 0 is an appropriate constant such that Un1,n1 ρn − Un,n1 ρn ≤ M3 , for all n ≥ 0.
Substitute 2.7 and 2.10 into 2.8 yields that
yn − yn1 ≤ 1 − βn1 γ − κγ xn1 − xn n
γi ,
M4 |rn1 − rn | βn1 − βn 2.11
i1
where M4 is an appropriate appropriate constant such that M4 ≥ max{M1 , M2 , M3 }. From
the conditions i and iii, we have
lim sup yn1 − yn − xn1 − xn ≤ 0.
n→∞
2.12
By virtue of Lemma 1.5, we obtain that
lim yn − xn 0.
n→∞
2.13
Abstract and Applied Analysis
9
On the other hand, we have
xn1 − xn 1 − αn xn − PC yn ≤ xn − yn .
2.14
This implies from 2.13 that
lim xn1 − xn 0.
n→∞
2.15
Next, we show limn → ∞ Wρn − ρn 0. Observing that
yn − Wn ρn βn γfxn − BWn ρn
2.16
and the condition i, we can easily get
lim Wn ρn − yn 0.
n→∞
2.17
Notice that
ρn − p2 PC I − rn Axn − PC I − rn Ap2
2
≤ xn − p − rn Axn − Ap 2
2
xn − p − 2rn xn − p, Axn − Ap rn2 Axn − Ap
2
2
2
≤ xn − p − 2rn αAxn − Ap rn2 Axn − Ap
2
2
xn − p − rn 2α − rn Axn − Ap .
2.18
On the other hand, we have
yn − p2 βn γfxn − Bp I − βn B Wn ρn − p 2
2
≤ βn γfxn − Bp 1 − βn γ ρn − p
2
2 ≤ βn γfxn − Bp ρn − p 2βn γfxn − Bpρn − p,
2.19
from which it follows that
xn1 − p2 αn xn − p 1 − αn PC yn − p 2
2
2
≤ αn xn − p 1 − αn yn − p
2
≤ αn xn − p 1 − αn 2
2 × βn γfxn − Bp ρn − p 2βn γfxn − Bpρn − p .
2.20
10
Abstract and Applied Analysis
Substituting 2.18 into 2.20, we arrive at
xn1 − p2 ≤ xn − p2 βn γfxn − Bp2
2
− 1 − αn rn 2α − rn Axn − Ap
2βn γfxn − Bpρn − p.
2.21
It follows that
2
1 − αn rn 2α − rn Axn − Ap
2 2
2 ≤ βn γfxn − Bp xn − p − xn1 − p 2βn γfxn − Bpρn − p
2 ≤ βn γfxn − p xn − p xn1 − p xn − xn1 2βn γfxn − Bpρn − p.
2.22
In view of the restrictions i, and iv, we find from 2.15 that
lim Axn − Ap 0.
n→∞
2.23
Observe that
ρn − p2 PC I − rn Axn − PC I − rn Ap2
≤ I − rn Axn − I − rn Ap, ρn − p
1 I − rn Axn − I − rn Ap2 ρn − p2
2
2 −I − rn Axn − I − rn Ap − ρn − p 1 xn − p2 ρn − p2 − xn − ρn − rn Axn − Ap 2
≤
2
1 xn − p2 ρn − p2 − xn − ρn 2 − rn2 Axn − Ap2
2
2rn xn − ρn , Axn − Ap ,
2.24
which yields that
ρn − p2 ≤ xn − p2 − ρn − xn 2 2rn ρn − xn Axn − Ap.
2.25
Substituting 2.25 into 2.20, we have
xn1 − p2 ≤ xn − p2 βn γfxn − Bp2 2rn ρn − xn Axn − Ap
2
− 1 − αn ρn − xn 2βn γfxn − Bpρn − p.
2.26
Abstract and Applied Analysis
11
This implies that
2
1 − αn ρn − xn 2 2
2
≤ xn − p − xn1 − p βn γfxn − Bp 2rn ρn − xn Axn − Ap
2βn γfxn − Bpρn − p
2
≤ xn − p xn1 − p xn − xn1 βn γfxn − Bp
2rn ρn − xn Axn − Ap 2βn γfxn − Bpρn − p.
2.27
In view of the restrictions i and ii, we find from 2.15 and 2.23 that
lim ρn − xn 0.
2.28
ρn − Wn ρn ≤ xn − ρn xn − yn yn − Wn ρn .
2.29
n→∞
On the other hand, we have
It follows from 2.13, 2.17 and 2.28 that limn → ∞ Wn ρn − ρn 0. From Lemma 1.6, we
find that Wρn − Wn ρn → 0 as n → ∞. Notice that
Wρn − ρn ≤ Wn ρn − ρn Wn ρn − Wρn ,
2.30
from which it follows that
lim Wρn − ρn 0.
n→∞
2.31
Next, we show lim supn → ∞ γfq − Bq, xn − q ≤ 0, where q PF γf I − Bq. To
show it, we choose a subsequence {xni } of {xn } such that
lim sup γf q − Bq, xn − q lim γf q − Bq, xni − q .
i→∞
n→∞
2.32
As {xni } is bounded, we have that there is a subsequence {xnij } of {xni } converges weakly to
p. We may assume, without loss of generality, that xni p. Hence we have p ∈ F. Indeed, let
us first show that p ∈ VIC, A. Put
T w1 Aw1 NC w1 ,
∅,
w1 ∈ C,
w1 ∈
/ C.
2.33
Since A is inverse-strongly monotone, we see that T is maximal monotone. Let w1 , w2 ∈
GT . Since w2 − Aw1 ∈ NC w1 and ρn ∈ C, we have
w1 − ρn , w2 − Aw1 ≥ 0.
2.34
12
Abstract and Applied Analysis
On the other hand, from ρn PC I − rn Axn , we have
w1 − ρn , ρn − I − rn Axn ≥ 0
2.35
ρn − xn
w 1 − ρn ,
Axn ≥ 0.
rn
2.36
w1 − ρni , w2 ≥ w1 − ρni , Aw1
≥ w1 − ρni , Aw1
ρn − xni
− w1 − ρni , i
Axni
rni
ρn − xni
≥ w1 − ρni , Aw1 − i
− Axni
rni
w1 − ρni , Aw1 − Aρni w1 − ρni , Aρni − Axni
ρn − xni
− w1 − ρni , i
rni
ρn − xni
≥ w1 − ρni , Aρni − Axni − w1 − ρni , i
,
rni
2.37
and hence
It follows that
which implies from 2.28 that w1 − p, w2 ≥ 0. We have p ∈ T −1 0 and hence p ∈ VIC, A.
Next, let us show p ∈ ∞
i1 FTi . Since Hilbert spaces are Opial’s spaces, from 2.31, we have
lim infρni − p < lim infρni − Wp
i→∞
i→∞
lim infρni − Wρni Wn ρni − Wp
i→∞
2.38
≤ lim infWρni − Wp
i→∞
≤ lim infρni − p,
i→∞
which derives a contradiction. Thus, we have from Lemma 1.2 that p ∈ FW On the other hand, we have
∞
i1
FTi .
lim sup γf q − Bq, xn − q lim γf q − Bq, xni − q
n→∞
n→∞
γf q − Bq, p − q
≤ 0.
2.39
Abstract and Applied Analysis
13
Finally, we show xn → q strongly as n → ∞. Notice that
yn − q2 βn γfxn − Bq I − βn B Wn ρn − q 2
2
2 ≤ 1 − βn γ Wn ρn − q 2βn γfxn − Bq, yn − q
2
2 2 2 ≤ 1 − βn γ xn − q κγβn xn − q yn − q
2.40
2βn γf q − Bq, yn − q .
Therefore, we have
2
1 − βn γ βn γκ yn − q2 ≤
xn − q2 2βn
γf q − Bq, yn − q
1 − βn γκ
1 − αn γκ
2 2 1 − 2βn γ βn κγ xn − q2 βn γ xn − q2
1 − βn γκ
1 − βn γκ
2βn γf q − Bq, yn − q
1 − βn γκ
2βn γ − κγ xn − q2
≤ 1−
1 − βn γκ
2βn γ − κγ
αn γ 2
1 γf q − Bq, yn − q M5 ,
1 − βn γκ
γ − κγ
2 γ − κγ
2.41
where M5 is an appropriate constant. On the other hand, we have
xn1 − p2 αn xn − p 1 − αn PC yn − p2
2
2
≤ αn xn − p 1 − αn PC yn − p
2.42
2
2
≤ αn xn − p 1 − αn yn − p .
Substitute 2.41 into 2.42 yields that
2
2βn γ − αγ xn1 − p ≤ 1 − 1 − αn xn − q2
1 − βn γα
2βn γ − αγ
βn γ 2
1 γf q − Bq, yn − q 1 − αn M5 .
1 − βn γα
γ − αγ
2 γ − αγ
2.43
14
Abstract and Applied Analysis
Put ln 1 − αn 2βn γ − αn γ/1 − βn αγ and
tn βn γ 2
1 γf q − Bq, yn − q M5 .
γ − αγ
2 γ − αγ
2.44
xn1 − q2 ≤ 1 − ln xn − q2 ln tn .
2.45
That is,
Notice that
γf q − Bq, yn − q γf q − Bq, yn − xn γf q − Bq, xn − q
≤ γf q − Bqyn − xn γf q − Bq, xn − q .
2.46
From 2.13 and 2.39 that
lim sup γf q − Aq, yn − q ≤ 0.
n→∞
2.47
It follows from the condition i and 2.47 that
lim ln 0,
n→∞
∞
ln ∞,
lim sup tn ≤ 0.
n1
n→∞
2.48
Apply Lemma 1.4 to 2.45 to conclude xn → q as n → ∞. This completes the proof.
For a single nonexpansive mapping, we have from Theorem 2.1 the following.
Corollary 2.2. Let H be a real Hilbert space and C a nonempty closed convex subset of H. Let
A : C → H be an α-inverse-strongly monotone mapping and f : C → C a κ-contraction. Let T be
a nonexpansive mapping from C into itself such that F : FT ∩ VIC, A / ∅. Let B be a strongly
positive linear bounded self-adjoint operator of C into itself with the constant γ > 0. Let {xn } be a
sequence generated in
x1 ∈ C,
yn βn γfxn I − βn B T PC I − rn Axn ,
xn1 αn xn 1 − αn PC yn ,
2.49
n ≥ 1,
where {αn } and {βn } are real number sequences in 0, 1. Assume that the control sequence {αn },
{βn } and {rn } satisfy the following restrictions:
i limn → ∞ βn 0, ∞
n1 βn ∞;
ii 0 < lim infn → ∞ αn ≤ lim supn → ∞ αn < 1;
Abstract and Applied Analysis
15
iii limn → ∞ |rn1 − rn | 0;
iv {rn } ⊂ a, b for some a, b with 0 < a < b < 2α.
Assume that 0 < γ < γ/κ. Then {xn } strongly converges to some point q, where q ∈ F, where
q PF γf I − Bq, which solves the variation inequality
γf q − Bq, p − q ≤ 0,
∀p ∈ F.
2.50
Corollary 2.3. Let H be a real Hilbert space and C a nonempty closed convex subset of H. Let
f : C → C be a κ-contraction. Let T be a nonexpansive mapping from C into itself such that FT /
∅.
Let B be a strongly positive linear bounded self-adjoint operator of C into itself with the constant γ > 0.
Let {xn } be a sequence generated in
x1 ∈ C,
yn βn γfxn I − βn B T xn ,
xn1 αn xn 1 − αn PC yn ,
2.51
n ≥ 1,
where {αn } and {βn } are real number sequences in 0, 1. Assume that the control sequence {αn }, and
{βn } satisfy the following restrictions:
i limn → ∞ βn 0, ∞
n1 βn ∞;
ii 0 < lim infn → ∞ αn ≤ lim supn → ∞ αn < 1.
Assume that 0 < γ < γ/κ. Then {xn } strongly converges to some point q, where q ∈ FT , where
q PF γf I − Bq, which solves the variation inequality
γf q − Bq, p − q ≤ 0,
∀p ∈ FT .
2.52
If B is the identity mapping, then Theorem 2.1 is reduced to the following.
Corollary 2.4. Let H be a real Hilbert space and C a nonempty closed convex subset of H. Let
A : C → H be an α-inverse-strongly monotone mapping and f : C → C a κ-contraction. Let
∞
{Ti }∞
i1 be an infinite family of nonexpansive mappings from C into itself such that F :
i1 FTi ∩
VIC, A /
∅. Let {xn } be a sequence generated in
x1 ∈ C,
yn βn fxn 1 − βn Wn PC I − rn Axn ,
xn1 αn xn 1 − αn yn ,
2.53
n ≥ 1,
where Wn is generated in 1.18, {αn } and {βn } are real number sequences in 0, 1. Assume that the
control sequence {αn }, {βn }, and {rn } satisfy the following restrictions:
i limn → ∞ βn 0, ∞
n1 βn ∞;
ii 0 < lim infn → ∞ αn ≤ lim supn → ∞ αn < 1;
16
Abstract and Applied Analysis
iii limn → ∞ |rn1 − rn | 0;
iv {rn } ⊂ a, b for some a, b with 0 < a < b < 2α.
Then {xn } strongly converges to some point q, where q ∈ F, where q PF fq, which solves the
variation inequality
f q − q, p − q ≤ 0,
∀p ∈ F.
2.54
References
1 R. T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,” Transactions of the
American Mathematical Society, vol. 149, pp. 75–88, 1970.
2 W. Takahashi and M. Toyoda, “Weak convergence theorems for nonexpansive mappings and
monotone mappings,” Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 417–428,
2003.
3 H. Iiduka and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and inversestrongly monotone mappings,” Nonlinear Analysis. Theory, Methods & Applications, vol. 61, no. 3, pp.
341–350, 2005.
4 H. K. Xu, “An iterative approach to quadratic optimization,” Journal of Optimization Theory and
Applications, vol. 116, no. 3, pp. 659–678, 2003.
5 G. Marino and H.-K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,”
Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006.
6 X. Qin, S. Y. Cho, and S. M. Kang, “On hybrid projection methods for asymptotically quasi-ϕnonexpansive mappings,” Applied Mathematics and Computation, vol. 215, no. 11, pp. 3874–3883, 2010.
7 F. Deutsch and I. Yamada, “Minimizing certain convex functions over the intersection of the fixed
point sets of nonexpansive mappings,” Numerical Functional Analysis and Optimization, vol. 19, no.
1-2, pp. 33–56, 1998.
8 S. Y. Cho and S. M. Kang, “Approximation of fixed points of pseudocontraction semigroups based on
a viscosity iterative process,” Applied Mathematics Letters, vol. 24, no. 2, pp. 224–228, 2011.
9 S. Y. Cho and S. M. Kang, “Approximation of common solutions of variational inequalities via strict
pseudocontractions,” Acta Mathematica Scientia Series B, vol. 32, no. 4, pp. 1607–1618, 2012.
10 S. Y. Cho, X. Qin, and S. M. Kang, “Hybrid projection algorithms for treating common fixed points
of a family of demicontinuous pseudocontractions,” Applied Mathematics Letters, vol. 25, no. 5, pp.
854–857, 2012.
11 X. Qin, S.-s. Chang, and Y. J. Cho, “Iterative methods for generalized equilibrium problems and fixed
point problems with applications,” Nonlinear Analysis. Real World Applications, vol. 11, no. 4, pp. 2963–
2972, 2010.
12 A. P. Chouhan and A. S. Ranadive, “Absorbing maps and common fixed point theorem in Menger
space,” Advances in Fixed Point Theory, vol. 2, pp. 108–119, 2012.
13 J. Ye and J. Huang, “Strong convergence theorems for fixed point problems and generalized
equilibrium problems of three relatively quasi-nonexpansive mappings in Banach spaces,” Journal
of Mathematical and Computational Science, vol. 1, no. 1, pp. 1–18, 2011.
14 Z. Kadelburg and S. Radenovic, “Coupled fixed point results under tvs-cone metric and w-conedistance,” Advances in Fixed Point Theory, vol. 2, pp. 29–46, 2012.
15 Y. J. Cho, X. Qin, and S. M. Kang, “Some results for equilibrium problems and fixed point problems
in Hilbert spaces,” Journal of Computational Analysis and Applications, vol. 11, no. 2, pp. 287–294, 2009.
16 F. Deutsch and H. Hundal, “The rate of convergence of Dykstra’s cyclic projections algorithm: the
polyhedral case,” Numerical Functional Analysis and Optimization, vol. 15, no. 5-6, pp. 537–565, 1994.
17 H. H. Bauschke, “The approximation of fixed points of compositions of nonexpansive mappings in
Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 202, no. 1, pp. 150–159, 1996.
18 X. Qin, M. Shang, and Y. Su, “Strong convergence of a general iterative algorithm for equilibrium
problems and variational inequality problems,” Mathematical and Computer Modelling, vol. 48, no. 7-8,
pp. 1033–1046, 2008.
Abstract and Applied Analysis
17
19 K. Shimoji and W. Takahashi, “Strong convergence to common fixed points of infinite nonexpansive
mappings and applications,” Taiwanese Journal of Mathematics, vol. 5, no. 2, pp. 387–404, 2001.
20 R. Bhardwaj, “Some fixed point theorems in polish space using new type of contractive conditions,”
Advances in Fixed Point Theory, vol. 2, pp. 313–325, 2012.
21 X. Qin, Y. J. Cho, J. I. Kang, and S. M. Kang, “Strong convergence theorems for an infinite family of
nonexpansive mappings in Banach spaces,” Journal of Computational and Applied Mathematics, vol. 230,
no. 1, pp. 121–127, 2009.
22 H. K. Xu, “An iterative approach to quadratic optimization,” Journal of Optimization Theory and
Applications, vol. 116, no. 3, pp. 659–678, 2003.
23 T. Suzuki, “Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Applications,
vol. 305, no. 1, pp. 227–239, 2005.
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