Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2010, Article ID 189814, 20 pages
doi:10.1155/2010/189814
Research Article
Convergence Theorems for a Maximal Monotone
Operator and a V -Strongly Nonexpansive Mapping
in a Banach Space
Hiroko Manaka
Department of Mathematics, Graduate School of Environment and Information Sciences,
Yokohama National University, Tokiwadai, Hodogayaku, Yokohama 240-8501, Japan
Correspondence should be addressed to Hiroko Manaka, h-manaka@ynu.ac.jp
Received 25 November 2009; Revised 31 March 2010; Accepted 16 July 2010
Academic Editor: S. Reich
Copyright q 2010 Hiroko Manaka. 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.
Let E be a smooth Banach space with a norm ·. Let V x, y x2 y2 −2x, Jy for any x, y ∈ E,
where ·, · stands for the duality pair and J is the normalized duality mapping. With respect to this
bifunction V ·, ·, a generalized nonexpansive mapping and a V -strongly nonexpansive mapping
are defined in E. In this paper, using the properties of generalized nonexpansive mappings, we
prove convergence theorems for common zero points of a maximal monotone operator and a V strongly nonexpansive mapping.
1. Introduction
Let E be a smooth Banach space with a norm · and let C be a nonempty, closed and convex
subset of E. We use the following bifunction V ·, · studied by Alber 1, as well as Kamimura
and Takahashi 2. Let V ·, · : E×E → 0, ∞ be defined by V x, y x2 −2x, Jyy2 for
any x, y ∈ E, where ·, · stands for the duality pair and J is the normalized duality mapping.
Note that the duality mapping is single valued in a smooth Banach space see 3. From the
definition of V ·, · the following properties are trivial.
Lemma 1.1. a For all a, b, c ∈ E,
V a, b ≤ V a, b V b, c V a, c − 2a − b, Jb − Jc.
1.1
b If a sequence {xn } ⊂ E satisfies limn → ∞ V xn , p < ∞ for some p ∈ E, then {xn } is
bounded.
2
Abstract and Applied Analysis
Let FT be the fixed points set of T . Ibaraki and Takahashi defined a generalized
nonexpansive mapping in a Banach space see 4.
Definition 1.2. A mapping T : C → C is said to be generalized nonexpansive if FT /
∅ and
V T x, p ≤ V x, p for all x ∈ C and p ∈ FT .
In this paper, we prove strong convergence theorem for finding common fixed points
of a family of generalized nonexpansive mappings. In addition, we prove strong convergence
theorem for finding zeroes of a generalized nonexpansive mapping and a maximal monotone
operator. Now, we define a V -strongly nonexpansive mapping as follows.
Definition 1.3. A mapping T : C → E is called V -strongly nonexpansive if there exists a
constant λ > 0 such that
V T x, T y ≤ V x, y − λV I − T x, I − T y ,
1.2
for all x, y ∈ C, where I is the identity mapping on E. More explicitly, if 1.2 holds, then T is
said to be V -strongly nonexpansive with λ.
If T is V -strongly nonexpansive with λ, then T is V -strongly nonexpansive with any
γ ∈ 0, λ. It is trivial that a V -strongly nonexpansive mapping is generalized nonexpansive
if FT /
∅. In the following section, we show that in a Hilbert space H a firmly nonexpansive
mapping is V -strongly nonexpansive with λ 1 and a V -strongly nonexpansive mapping is
strongly nonexpansive if FT /
∅. Motivated by the results of Manaka and Takahashi 5, we
prove weak convergence theorem for common zero points of a maximal monotone operator
and a V -strongly nonexpansive mapping in a Banach space.
2. Preliminaries
Let D be a nonempty subset of a Banach space E. A mapping R : E → D is said to be sunny,
if for all x ∈ E and t ≥ 0,
RRx tx − Rx Rx.
2.1
A mapping R : E → D is called a retraction if Rx x for all x ∈ D see 6. It is known
that a generalized nonexpansive and sunny retraction of E onto D is uniquely determined if
E is a smooth and strictly convex Banach space cf., 7. Ibaraki and Takahashi proved the
following results in 4.
Lemma 2.1 cf., 4. Let E be a reflexive, strictly convex, and smooth Banach space, and let T be
a generalized nonexpansive mapping from E into itself. Then there exists a sunny and generalized
nonexpansive retraction on FT .
Abstract and Applied Analysis
3
Lemma 2.2 cf., 4. Let D be a nonempty subset of a reflexive, strictly convex, and smooth Banach
space E. Let R be a retraction from E onto D. Then R is sunny and generalized nonexpansive if and
only if
x − Rx, JRx − Jy ≥ 0,
2.2
for all x ∈ E and y ∈ D.
A generalized resolvent Jr of a maximal monotone operator B ⊂ E∗ × E is defined by
Jr I rBJ−1 for any real number r > 0. It is well known that Jr : E → E is single valued
if E is reflexive, smooth, and strictly convex see 8. It is also known that Jr satisfies
x − Jr x − y − Jr y , JJr x − JJr y ≥ 0,
∀x, y ∈ E.
2.3
This implies that
x − Jr x, JJr x − Jp ≥ 0,
∀x ∈ E, p ∈ FJr .
2.4
Therefore, from Lemma 1.1a, we obtain the following proposition.
Proposition 2.3. a If a sunny retraction R is generalized nonexpansive, then R satisfies
V x, Rx V Rx, y V x, y − 2 x − Rx, JRx − Jy
≤ V x, y , ∀x, y ∈ D.
2.5
b For each r > 0, a generalized resolvent Jr satisfies
V x, Jr x V Jr x, p ≤ V x, p ,
∀x ∈ E, p ∈ FJr .
2.6
Remark 2.4. The property in Proposition 2.3b means that Jr is generalized nonexpansive for
any r > 0.
We recall some nonlinear mappings in Banach spaces see, e.g., 9–12.
Definition 2.5. Let D be a nonempty, closed, and convex subset of E. A mapping T : D → E
is said to be firmly nonexpansive if
T x − T y2 ≤ x − y, j T x − T y ,
for all x, y ∈ D and some jT x − T y ∈ JT x − T y.
2.7
4
Abstract and Applied Analysis
In 12, Reich introduced a class of strongly nonexpansive mappings which is defined
with respect to the Bregman distance D·, · corresponding to a convex continuous function
f in a reflexive Banach space E. Let S be a convex subset of E, and let T : S → S be a selfmapping of S. A point p in the closure of S is said to be an asymptotically fixed point of T if S
contains a sequence {xn } which converges weakly to p and the sequence {xn −T xn } converges
denotes the asymptotically fixed points set of T .
strongly to 0. FT
Definition 2.6. The Bregman distance corresponding to a function f : E → R is defined by
D x, y fx − f y − f y x − y ,
2.8
where f is the Gâteaux differentiable and f x stands for the derivative of f at the point x.
/
We say that the mapping T is strongly nonexpansive if FT
∅ and
D p, T x ≤ D p, x ,
, x ∈ S,
∀p ∈ FT
2.9
and if it holds that limn → ∞ DT xn , xn 0 for a bounded sequence {xn } such that
.
limn → ∞ Dp, xn − Dp, T xn 0 for any p ∈ FT
We remark that the symbols xn → u and xn u mean that {xn } converges
strongly and weakly to u, respectively. Taking the function · 2 as the convex, continuous,
and Gâteaux differentiable function f, we obtain the fact that the Bregman distance D·, ·
coincides with V ·, ·. Especially in a Hilbert space, Dx, y V x, y x − y2 . Bruck and
Reich defined strongly nonexpansive mappings in a Hilbert space H as follows cf., 10.
Definition 2.7. A mapping T : D → H is said to be strongly nonexpansive if T is
nonexpansive with FT /
∅ and if it holds that
xn − yn − T xn − T yn −→ 0
2.10
when {xn } and {yn } are sequences in D such that {xn − yn } is bounded and limn → ∞ xn −
yn − T xn − T yn 0.
The relation among firmly nonexpansive mappings, strongly nonexpansive mappings
and V -strongly nonexpansive mappings is shown in the following proposition.
Proposition 2.8. In a Hilbert space H, the following hold.
a A firmly nonexpansive mapping is V -strongly nonexpansive with λ 1.
/
b A V -strongly nonexpansive mapping T with FT
∅ is strongly nonexpansive.
Abstract and Applied Analysis
5
Proof. a Suppose that T is firmly nonexpansive. Since J I in a Hilbert space, it holds that
2 2 2
2 x − y, T x − T y − T x − T y x − y − I − T x − I − T y ,
2.11
for all x, y ∈ D. Therefore, it is obvious that T is firmly nonexpansive if and only if T satisfies
x − y2 − T − Ix − T − Iy2 ≥ T x − T y2 ,
2.12
for all x, y ∈ D. Hence we obtain a.
b Suppose that T is V -strongly nonexpansive with λ. Then, it is trivial that T
is nonexpansive and 2.9 holds. Suppose that the sequences {xn } and {yn } satisfy the
conditions in Definition 2.7. Then {T xn − T yn } is also bounded. Since T is V -strongly
nonexpansive with λ, we have that
2
0 ≤ λxn − yn − T xn − T yn λV I − T xn , I − T yn
≤ V xn , yn − V T xn , T yn
2 2
xn − yn − T xn − T yn xn − yn T xn − T yn xn − yn − T xn − T yn 2.13
−→ 0.
Hence, xn − yn − T xn − T yn → 0 for λ > 0. This means that T is strongly nonexpansive.
In a Banach space, V -strongly nonexpansive mappings have the following properties.
Proposition 2.9. In a smooth Banach space E, the following hold.
a For c ∈ −1, 1, T cI is V -strongly nonexpansive. For c 1, T I is V -strongly
nonexpansive for any λ > 0. For c ∈ −1, 1, T cI is V -strongly nonexpansive for any
λ ∈ 0, 1 c/1 − c.
b If T is V -strongly nonexpansive with λ, then, for any α ∈ −1, 1 with α /
0, αT is also
V -strongly nonexpansive with α2 λ.
c If T is V -strongly nonexpansive with λ ≥ 1, then A I − T is V -strongly nonexpansive
with λ−1 .
d Suppose that T is V -strongly nonexpansive with λ and that α ∈ −1, 1 satisfies α2 λ ≥ 1.
−1
Then I − αT is V -strongly nonexpansive with α2 λ . Moreover, if Tα I − αT , then
V Tα x, Tα y ≤ V x, y − λ−1 V T x, T y .
2.14
6
Abstract and Applied Analysis
Proof. a Let T cI for any c ∈ −1, 1, and denote Il V T x, T y and Ir V x, y − λV I −
T x, I − T y. Since Jcx cJx, we have
2
Il V T x, T y c2 x2 y − 2 x, Jy c2 V x, y ,
Ir V x, y − λV I − T x, I − T y
2
2
x2 − λ1 − cx2 y − λ1 − cy
− 2 x, Jy 2λ 1 − cx, J 1 − cy
2 1 − λ1 − c2 x2 y − 2 1 − λ1 − c2 x, Jy
2
1 − λ1 − c2 x2 y − 2 x, Jy
1 − λ1 − c2 V x, y .
2.15
For c 1, it holds that Il ≤ Ir for all λ > 0. For c ∈ −1, 1, we obtain
Il ≤ Ir ⇐⇒ c2 ≤ 1 − λ1 − c2 ⇐⇒ 0 < λ1 − c2 ≤ 1 − c2
⇐⇒ 0 < λ ≤
1 − c1 c
2
1 − c
1 c
.
1 − c
2.16
Therefore, T cI is V -strongly nonexpansive for any λ ∈ 0, 1 c/1 − c.
b If T is V -strongly nonexpansive with λ > 0, then, for α ∈ −1, 1 with α /
0,
2
V αT x, αT y αT x2 αT y − 2 αT x, J αT y
2
α2 T x2 T y − 2 T x, J T y
α2 V T x, T y
≤ α2 V x, y − λV I − T x, I − T y
V x, y − 1 − α2 V x, y − α2 λV I − T x, I − T y
≤ V x, y − α2 λV I − T x, I − T y .
This means that αT is V -strongly nonexpansive with α2 λ.
2.17
Abstract and Applied Analysis
7
c Suppose that T is V -strongly nonexpansive with λ ≥ 1 and let A I − T . Then we
have that
V I − Ax, I − Ay V T x, T y
≤ V x, y − λV I − T x, I − T y
V x, y − λV Ax, Ay .
2.18
This inequality implies that
V Ax, Ay ≤ λ−1 V x, y − V I − Ax, I − Ay
λ−1 V x, y − λ−1 V I − Ax, I − Ay
≤ V x, y − λ−1 V I − Ax, I − Ay .
2.19
Thus A is V -strongly nonexpansive with λ−1 .
d From b and the assumption, αT is V -strongly nonexpansive with α2 λ ≥ 1, and
−1
from c we have that I − αT is V -strongly nonexpansive with α2 λ . Furthermore we
obtain that
−1 V Tα x, Tα y ≤ V x, y − α2 λ V I − Tα x, I − Tα y
−1 V x, y − α2 λ V αT x, αT y
−1
V x, y − α2 λ
α2 V T x, T y
2.20
V x, y − λ−1 V T x, T y .
This completes the proof.
In Banach spaces, we have the following example of V -strongly nonexpansive
mappings.
Example 2.10. Let E R × R be a Banach space with a norm · defined by
x |x1 | |x2 |,
∀x x1 , x2 ∈ E.
2.21
The normalized duality mapping J is given by
1
1
Jx x
x1 ,
x2 ,
|x1 |
|x2 |
∀x ∈ E.
2.22
8
Abstract and Applied Analysis
Hence, we have for x, y ∈ E that
2
V x, y x2 y − 2 x, Jy
2
x1 y1 x2 y2
2
x y − 2y .
y1 y2 2.23
We define a mapping T : E → E as follows:
Tx ⎧
⎪
⎨x
if x ≤ 1,
1
⎪
⎩
x
x
if x ≥ 1.
2.24
We will show that this mapping is V -strongly nonexpansive for any λ ≤ 1.
a Suppose that x, y ∈ E with x ≤ 1 and y ≥ 1. Then, we have
V T x, T y V x, T y
2
x1 T y 1 x2 T y 2
x T y − 2 T y
Ty Ty 1
2
x1 y1 x2 y2
x2 1 − 2 .
y1 y2 2
2.25
Since
y − 1
y − 1
y1 , y2 ,
y − Ty y
y
2.26
we have that
2
V x − T x, y − T y V 0, y − T y y − T y
y − 1 2
y1 y2 y
2
y − 1 .
2.27
Abstract and Applied Analysis
9
Hence, we obtain that
V x, y − V T x, T y − λV x − T x, y − T y
2
x1 y1 x2 y2
2
y
y
x
x
1
1
2
2
2
2
x y − 2y − x − 1 2 − λ y − 1
y1 y2 y1 y2 2
2
x1 y1 x2 y2
y − 1 − 2 y − 1 − λ y − 1
y1 y2 x1 y1 x2 y2 2
y 1 −2 y −1
− λ y − 1
≥ y −1
y1 y2 y − 1 y 1 − 2x − λy λ
≥ y − 1 1 − λy 1 − 2 λ
y − 1 1 − λ y − 1
2
1 − λ y − 1 ≥ 0,
2.28
for any λ ∈ 0, 1. This means that T is V -strongly nonexpansive for any λ ∈ 0, 1.
b Suppose that x, y ∈ E with x ≥ 1 and y ≤ 1. Then, we have
y x1 y1 x2 y2
2
,
V T x, T y V T x, y 1 y − 2
y2 x y1 x − 1
V x − T x, y − T y V
x, 0 x − 12 .
x
2.29
Hence, we have that
V x, y − V T x, T y − λV x − T x, y − T y
y x1 y1 x2 y2
2
2
x1 y1 x2 y2
2
−1− y 2
− λx − 12
x y − 2 y
y1 y2 y2 x y1 x1 y1 x2 y2
1
2
x − 1 − 2 y 1 −
y1 y2 x
x − 1
≥ x − 1x 1 − λx λ − 2 y
x
x
10
Abstract and Applied Analysis
x − 1 1 − λx 1 λ − 2y
≥ x − 1{1 − λx 1 λ − 2}
1 − λx − 12 ≥ 0,
2.30
for any λ ∈ 0, 1. This means that T is V -strongly nonexpansive for any λ ∈ 0, 1.
c Suppose that x, y ∈ E with x, y ≥ 1. Then, we have
x1 y1 x2 y2
2
V T x, T y 1 1 −
y2 x y1 x1 y1 x2 y2
2
,
2−
y2 x y1 2.31
2
V x − T x, y − T y x − 12 y − 1
x − 1 x1 y1 x2 y2
.
− 2 y − 1
y1 y2 x
Hence, we have that
V x, y − V T x, T y − λV x − T x, y − T y
2
x1 y1 x2 y2
x1 y1 x2 y2
2
2
x y − 2 y
−2
y1 y2 y2 x y1 x − 1 x1 y1
2 2
2λ y − 1
− λ x − 1 y − 1
y1 x
x2 y2
y2 2
x − 1x 1 y − 1 y 1 − λx − 12 − λ y − 1
x1 y1 x2 y2 2
−
xy − 1 − λ y − 1 x − 1
x
y1
y2
x − 1{x 1 − λx − 1} y − 1 y 1 − λ y − 1
x1 y1 x2 y2 2
−
xy − 1 − λx − 1 y − 1 .
y2 x y1 2.32
Now, we note that
xy − 1 − λx − 1 y − 1 ≥ 0,
2.33
Abstract and Applied Analysis
11
for any x, y ≥ 1 and for any λ ∈ 0, 1. Therefore, we obtain that
V x, y − V T x, T y − λV x − T x, y − T y
≥ x − 1{x 1 − λx − 1} y − 1 y 1 − λ y − 1
2
x xy − 1 − λx − 1 y − 1
x
2
1 − λ x − y ≥ 0,
−
2.34
for any λ ∈ 0, 1. This means that T is V -strongly nonexpansive for any λ ∈ 0, 1.
It is clear that if x, y ≤ 1 then T is V -strongly nonexpansive; therefore, from a,
b, and c, we obtain the conclusion that T is V -strongly nonexpansive with λ ≤ 1.
Next, we present some lemmas which are used in the proofs of our theorems. Let N be
the set of natural numbers.
Lemma 2.11. Let {an } and {tn } be sequences of nonnegative real numbers and satisfy the inequality
an1 ≤ 1 − tn an tn M for any n ∈ N and a constant M > 0. If n∈N tn < ∞, then limn → ∞ an
exists.
Kamimura and Takahashi showed the following useful lemmas see 2.
Lemma 2.12. Let E be a smooth and uniformly convex Banach space. Then, there exists a continuous,
strictly increasing, and convex function g : 0, ∞ → 0, ∞ such that g0 0, and for each real
number r > 0,
0 ≤ g x − y ≤ V x, y ,
2.35
for all x, y ∈ Br {z ∈ E : z ≤ r}.
From this lemma, it is obvious that the following lemma holds.
Lemma 2.13. Let E be a smooth and uniformly convex Banach space and let {xn } and {yn }
be sequences in E such that either {xn } or {yn } is bounded. If limn → ∞ V xn , yn 0, then
limn → ∞ xn − yn 0.
We present the following lemma which plays an important role in our theorems cf.
Butnariu and Resmerita 13 .
Lemma 2.14. Let E be a smooth and uniformly convex Banach space and C a nonempty, convex, and
closed subset of E. Suppose that T : C → E satisfies
V T x, T y ≤ V x, y ,
∀x, y ∈ C.
2.36
If a weakly convergent sequence {zn }n∈N ⊂ C satisfies that limn → ∞ V T zn , zn 0, then zn z ∈
FT .
12
Abstract and Applied Analysis
3. Main Results
In this section, we prove three strong convergence theorems. In the first result, we prove
strong convergence theorem for finding common fixed points of a family of generalized
nonexpansive mappings. In the next result, we prove strong convergence theorem for finding
zeroes of a generalized nonexpansive mapping and a maximal monotone operator. In the
last result, we prove weak convergence theorem for finding zeroes of a maximal monotone
operator and a V -strongly nonexpansive mapping. As consequence, we prove convergence
theorem for common zeroes of a maximal monotone operator and a firmly nonexpansive
mapping in a Hilbert space.
Theorem 3.1. Let E be a reflexive, smooth, and strictly convex Banach space, and let {Tn }n∈N be a
∅ and that R is a sunny
family of generalized nonexpansive mappings. Suppose that n∈N FTn F /
and generalized nonexpansive retraction from E to F. Let a sequence {xn } be defined as follows. For
any x1 x ∈ E,
xn1 RTn xn ,
for any n ∈ N.
3.1
Then, {xn } converges strongly to a point x∗ in F.
Proof. Since Rxn is a point in F for all n ∈ N, from Proposition 2.3a, we have for all n ∈ N
that
0 ≤ V xn1 , Rxn1 ≤ V xn1 , Rxn1 V Rxn1 , Rxn ≤ V xn1 , Rxn V RTn xn , Rxn .
3.2
Since R and Tn are generalized nonexpansive, we get that
V RTn xn , Rxn ≤ V Tn xn , Rxn ≤ V xn , Rxn .
3.3
Hence, we have that
0 ≤ V xn1 , Rxn1 ≤ V xn , Rxn ,
∀n ∈ N,
3.4
and therefore, limn → ∞ V xn , Rxn < ∞. Furthermore, Proposition 2.3a implies that
V xnk , Rxnk V Rxnk , Rxn ≤ V xnk , Rxn .
3.5
V Rxnk , Rxn ≤ V xnk , Rxn − V xnk , Rxnk .
3.6
This is equivalent to
Abstract and Applied Analysis
13
Setting m n k for all n, k ∈ N, then we have that
V Rxm , Rxn ≤ V xm , Rxn − V xm , Rxm ≤ V RTm−1 xm−1 , Rxn − V xm , Rxm ≤ V Tm−1 xm−1 , Rxn − V xm , Rxm 3.7
≤ V xm−1 , Rxn − V xm , Rxm ≤ ···
≤ V xn , Rxn − V xm , Rxm −→ 0,
as n, m −→ ∞.
Since V xn1 , p V RTn xn , p ≤ V x, p for any p ∈ F, Lemma 1.1b implies that {xn } is
bounded. Thus, from Lemma 2.12, we can take the continuous and strictly increasing function
g : 0, ∞ → 0, ∞ with g0 0 such that
gRxm − Rxn ≤ V Rxm , Rxn ≤ V xn , Rxn − V xm , Rxm −→ 0,
as n, m −→ ∞.
3.8
Since Rxn xn for all n ≥ 1, we have gxm − xn gRxm − Rxn → 0. Therefore, {xn } is a
Cauchy sequence. Since E is complete and F is closed, this sequence {xn } converges strongly
to point x∗ ∈ F.
Noting that the generalized resolvent Jr I rBJ−1 of a maximal monotone operator
B for r > 0 is a generalized nonexpansive mapping see Remark 2.4, we obtain the following
result.
Theorem 3.2. Let E be a reflexive, smooth, and strictly convex Banach space. Let T : E → E
be generalized nonexpansive and let B ⊂ E∗ × E be a maximal monotone operator. Suppose that
∅ and that R is a sunny and generalized nonexpansive retraction from E to F FT ∩ BJ−1 0 /
FT ∩ BJ−1 0. Let an iterative sequence {xn } be defined as follows: for any x x1 ∈ E,
xn1 RT Jrn xn ,
∀n ∈ N,
3.9
where {rn } is a sequence of nonnegative real numbers. Then, the sequence {xn } converges strongly to
a point x∗ in FT ∩ BJ−1 0.
Proof. From Propositions 2.3a and 2.3b, we have for all n ∈ N that
V xn1 , Rxn1 ≤ V xn1 , Rxn1 V Rxn1 , Rxn ≤ V xn1 , Rxn V RT Jrn xn , Rxn ≤ V T Jrn xn , Rxn ≤ V Jrn xn , Rxn ≤ V xn , Rxn .
3.10
14
Abstract and Applied Analysis
Thus limn → ∞ V xn , Rxn < ∞. Similarly, as in the proof of the previous theorem, we show
that {xn } is a Cauchy sequence, and we obtain that {xn } converges strongly to point x∗ in
F FT ∩ BJ−1 0.
The duality mapping J of a Banach space E with the Gâteaux differentiable norm is
said to be weakly sequentially continuous if xn x in E implies that {Jxn } converges weak
star to Jx in E∗ cf., 14. This happens, for example, if E is a Hilbert space, finite dimensional
and smooth, or lp if 1 < p < ∞ cf., 15. Next, we prove the main theorem.
Theorem 3.3. Let E be a reflexive, smooth and strictly convex Banach space. Suppose that the duality
mapping J of E is weakly sequentially continuous. Let C be a nonempty, closed, and convex subset
of E. Let B : E∗ → 2E be a maximal monotone operator and let Jrn I rn BJ−1 be a generalized
resolvent of B for a sequence {rn } ⊂ 0, ∞. Suppose that A : C → E is a V -strongly nonexpansive
∅ and that RC : E → C is a sunny and
mapping with λ ≥ 1 such that C0 A−1 0 ∩ BJ−1 0 /
generalized nonexpansive retraction. For an α ∈ −1, 1 such that α2 λ ≥ 1, let an iterative sequence
{xn } ⊂ C be defined as follows: for any x x1 ∈ C and n ∈ N,
xn1
yn RC I − αAxn ,
RC βn x 1 − βn Jrn yn ,
3.11
where {βn } ⊂ 0, 1 and {rn } ⊂ 0, ∞ satisfy that
βn < ∞,
lim inf rn > 0.
n→∞
n≥1
3.12
Then, there exists an element u ∈ C0 such that
xn u,
RC0 xn −→ u.
3.13
Proof. For simplicity, we denote RC and RC0 by R and R0 , respectively. Let zn βn x 1 −
βn Jrn yn for all n ∈ N. Since R is generalized nonexpansive, we have for any p ∈ C0 and all
n ∈ N that
V xn1 , p V Rzn , p ≤ V zn , p .
3.14
The convexity of · 2 implies that
V zn , p V βn x 1 − βn Jrn yn , p
2 2
βn x 1 − βn Jrn yn p − 2 βn x 1 − βn Jrn yn , Jp
2 2
≤ βn x2 1 − βn Jrn yn p − 2βn x, Jp − 2 1 − βn Jrn yn , Jp
2
2 2 βn x2 − 2 x, Jp p 1 − βn Jrn yn − 2 Jrn yn , Jp p
βn V x, p 1 − βn V Jrn yn , p .
3.15
Abstract and Applied Analysis
15
Thus, we obtain that
V zn , p ≤ βn V x, p 1 − βn V Jrn yn , p ,
3.16
and furthermore, since Jrn is generalized nonexpansive, we have that
V zn , p ≤ βn V x, p 1 − βn V yn , p .
3.17
Let Aα I − αA. Then, from Proposition 2.9d, Aα is V -strongly nonexpansive with
α2 λ and Aα is also generalized nonexpansive. Hence, we have that
V yn , p V RAα xn , p ≤ V Aα xn , p ≤ V xn , p .
3.18
Thus, we have from 3.14, 3.16, 3.17, and 3.18 that
V xn1 , p ≤ V zn , p ≤ βn V x, p 1 − βn V Jrn yn , p
≤ βn V x, p 1 − βn V yn , p
≤ βn V x, p 1 − βn V Aα xn , p
≤ βn V x, p 1 − βn V xn , p .
3.19
From Lemma 2.11, there exists α limn → ∞ V xn , p < ∞. Since limn → ∞ βn 0, we have that
α lim V xn , p lim V zn , p ,
n→∞
3.20
n→∞
lim V Jrn yn , p lim V yn , p lim V Aα xn , p .
n→∞
n→∞
n→∞
3.21
Hence, {xn }, {zn }, {Jrn yn }, {yn }, and {Aα xn } are bounded from Lemma 1.1b. Since E is
uniformly convex, the boundedness of {xn } implies that there exists a subsequence {xnj } ⊂
{xn } such that xnj u ∈ C. Moreover, we can take the index sequence {nj }j≥1 satisfies
limj → ∞ rnj −1 > 0. We will show that u ∈ BJ−1 0. From Proposition 2.3a,
V xn1 , p V Rzn , p ≤ V zn , Rzn V Rzn , p ≤ V zn , p ,
3.22
and furthermore, from 3.16 and Proposition 2.3b, we obtain that
V zn , p ≤ βn V x, p 1 − βn V Jrn yn , p
≤ βn V x, p 1 − βn V yn , Jrn yn V Jrn yn , p
≤ βn V x, p 1 − βn V yn , p .
3.23
16
Abstract and Applied Analysis
These inequalities and 3.22 imply with limn → ∞ βn 0 and 3.21 that
α ≤ α lim V zn , Rzn n→∞
≤ α ≤ lim V yn , Jrn yn α ≤ α,
3.24
n→∞
that is,
lim V zn , xn1 lim V zn , Rzn lim V yn , Jrn yn 0.
n→∞
n→∞
n→∞
3.25
Lemma 2.13 implies that
lim Jrn yn − yn 0.
lim xn1 − zn 0,
n→∞
n→∞
3.26
Furthermore, since
lim zn − Jrn yn lim βn x 1 − βn Jrn yn − Jrn yn n→∞
n→∞
lim βn x − Jrn yn 0,
3.27
n→∞
we have from 3.26 and 3.27 that
xn1 − Jr yn ≤ xn1 − zn zn − Jr yn n
n
−→ 0
n −→ ∞.
3.28
Hence, for an index sequence {nj }j≥1 of {xnj } such that xnj u ∈ C and limj → ∞ rnj −1 > 0, we
obtain that
Jrnj −1 ynj −1 u,
ynj −1 u,
as j ∞.
3.29
Since 1/rJr−1 − I 1/rI rBJ − I BJ, there exists wnj ∈ BJJrnj −1 ynj −1 such that
wnj 1 rnj −1
ynj −1 − Jrnj −1 ynj −1 ,
for any j ≥ 1.
3.30
1 Jrnj −1 ynj −1 − ynj −1 0.
3.31
Since limj → ∞ rnj −1 > 0, 3.26 implies that
lim wnj lim
j →∞
j → ∞ rnj −1
Abstract and Applied Analysis
17
For p, q ∈ BJ ⊂ E × E, the monotonicity of B implies that
lim q − wnj , Jp − JJrnj −1 ynj −1 ≥ 0,
j →∞
3.32
and we have, since J is weakly sequentially continuous, that
q, Jp − Ju ≥ 0.
3.33
The maximality of B implies that u ∈ BJ−1 0.
Now, we will show that u ∈ A−1 0. From Proposition 2.9d and p ∈ FAα , we get
that
V Aα xn , p V Aα xn , Aα p
≤ V xn , p − λ−1 V Axn , Ap .
3.34
Thus, we have from 3.17 that
V xn1 , p ≤ V zn , p ≤ βn V x, p 1 − βn V yn , p
βn V x, p 1 − βn V RAα xn , p
≤ βn V x, p 1 − βn V Aα xn , p
≤ βn V x, p 1 − βn V xn , p − λ−1 V Axn , Ap .
3.35
This implies that
0 ≤ 1 − βn λ−1 V Axn , Ap
≤ βn V x, p 1 − βn V xn , p − V xn1 , p
βn V x, p − V xn , p V xn , p − V xn1 , p
−→ 0,
3.36
as n −→ ∞.
Therefore we have that
lim V Axn , Ap 0.
n→∞
3.37
18
Abstract and Applied Analysis
From Lemma 2.13, we get that limn → ∞ Axn − Ap limn → ∞ Axn 0 that is, Axn → 0 as
n → ∞. From Lemma 1.1a and the boundedness of {Aα xn }, we have that
0 ≤ V xn , Aα xn V xn , p − V Aα xn , p 2 xn − Aα xn , JAα xn − Jp
V xn , p − V Aα xn , p 2α Axn , JAα xn − Jp
≤ V xn , p − V Aα xn , p 2αAxn JAα xn − Jp
3.38
≤ V xn , p − V Aα xn , p 2αAxn M,
for some M > 0. From 3.21, we have that limn → ∞ {V xn , p−V Aα xn , p} 0, and we obtain
that
lim V xn , Aα xn 0,
3.39
lim xn − Aα xn 0.
3.40
n→∞
and this means that
n→∞
From Lemma 2.14, we obtain xn u0 ∈ FAα . Since xnj u, this means that xnj u0 u; hence, we have u ∈ FAα ; that is, u ∈ A−1 0. Therefore, we obtain that u ∈ A−1 0 ∩
BJ−1 0 C0 .
Let un R0 xn for any n ∈ N. Since R0 is a sunny generalized nonexpansive retraction,
xn − un , Jun − Jy ≥ 0,
∀y ∈ C0 .
3.41
Similarly as in the proof of Theorem 3.2, we can show that {un } is a Cauchy sequence, and
therefore there exists u∗ ∈ C0 such that un → u∗ . Set y u in 3.41. Since xn u, we get that
u − u∗ , Ju∗ − Ju ≥ 0.
3.42
This means that u u∗ by the strict convexity of J; that is, R0 xn → u. This completes the
proof.
In a Hilbert space, we obtain the following theorem as a corollary of the main
Theorem 3.3 by applying Proposition 2.8a.
Corollary 3.4. Let H be a Hilbert space, and let C be a nonempty, closed, and convex subset of H.
Let B : H → 2H be a maximal monotone operator, and let Jrn I rn B−1 be a resolvent of B
for a sequence {rn } ⊂ 0, ∞. Suppose that A : C → H is a firmly nonexpansive mapping with
Abstract and Applied Analysis
19
∅. Suppose that RC is a sunny and generalized nonexpansive retraction to C.
C0 A−1 0 ∩ B−1 0 /
Let an iterative sequence {xn } ⊂ C be defined as follows: for any x x1 ∈ C and n ∈ N,
xn1
yn RC I − αAxn ,
RC βn x 1 − βn Jrn yn ,
3.43
where {βn } ⊂ 0, 1 and {rn } ⊂ 0, ∞ satisfy that
βn < ∞,
n≥1
lim infrn > 0.
n→∞
3.44
Then, there exists an element u ∈ C0 such that
xn u,
RC0 xn −→ u.
3.45
Acknowledgments
The author expresses her hearty thanks to Professor Wataru Takahashi and the referees for
giving valuable suggestions during the preparation of this paper.
References
1 Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and
applications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol.
178 of Lecture Notes in Pure and Appl. Math., pp. 15–50, Marcel Dekker, New York, NY, USA, 1996.
2 S. Kamimura and W. Takahashi, “Strong convergence of a proximal-type algorithm in a Banach
space,” SIAM Journal on Optimization, vol. 13, no. 3, pp. 938–945, 2002.
3 E. Zeidler, Nonlinear Functional Analysis and Applications II B: Nonlinear Monotone Operator, Springer,
New York, NY, USA, 1990.
4 T. Ibaraki and W. Takahashi, “A new projection and convergence theorems for the projections in
Banach spaces,” Journal of Approximation Theory, vol. 149, no. 1, pp. 1–14, 2007.
5 H. Manaka and W. Takahashi, “Weak convergence theorems for maximal monotone operators with
nonspreading mappings in a Hilbert space,” to appear in CUBO.
6 R. E. Bruck Jr., “Nonexpansive retracts of Banach spaces,” Bulletin of the American Mathematical Society,
vol. 76, pp. 384–386, 1970.
7 W. Takahashi, “Convergence theorems for nonlinear projections in Banach spaces,” RIMS Kokyuroku,
vol. 1396, pp. 49–59, 2004 Japanese.
8 T. Ibaraki, Y. Kimura, and W. Takahashi, “Convergence theorems for generalized projections and
maximal monotone operators in Banach spaces,” Abstract and Applied Analysis, no. 10, pp. 621–629,
2003.
9 H. H. Bauschke, “Fenchel duality, Fitzpatrick functions and the extension of firmly nonexpansive
mappings,” Proceedings of the American Mathematical Society, vol. 135, no. 1, pp. 135–139, 2007.
10 R. E. Bruck and S. Reich, “Nonexpansive projections and resolvents of accretive operators in Banach
spaces,” Houston Journal of Mathematics, vol. 3, no. 4, pp. 459–470, 1977.
11 K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced
Mathematics, Cambridge University Press, Cambridge, Mass, USA, 1990.
12 S. Reich, “A weak convergence theorem for the alternating method with Bregman distances,” in
Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178 of Lecture Notes
in Pure and Appl. Math., pp. 313–318, Marcel Dekker, New York, NY, USA, 1996.
20
Abstract and Applied Analysis
13 D. Butnariu and E. Resmerita, “Bregman distances, totally convex functions, and a method for solving
operator equations in Banach spaces,” Abstract and Applied Analysis, vol. 2006, Article ID 84919, 39
pages, 2006.
14 S. Reich, “Approximating zeros of accretive operators,” Proceedings of the American Mathematical
Society, vol. 51, no. 2, pp. 381–384, 1975.
15 F. E. Browder, “Fixed point theorems for nonlinear semicontractive mappings in Banach spaces,”
Archive for Rational Mechanics and Analysis, vol. 21, pp. 259–269, 1966.