Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 479438, 13 pages
doi:10.1155/2012/479438
Research Article
Hybrid Algorithm of Fixed Point for Weak
Relatively Nonexpansive Multivalued Mappings
and Applications
Jingling Zhang, Yongfu Su, and Qingqing Cheng
Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China
Correspondence should be addressed to Yongfu Su, suyongfu@tjpu.edu.cn
Received 22 May 2012; Accepted 5 July 2012
Academic Editor: Yonghong Yao
Copyright q 2012 Jingling Zhang et al. 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.
The purpose of this paper is to present the notion of weak relatively nonexpansive multivalued mapping and to prove the strong convergence theorems of fixed point for weak relatively
nonexpansive multivalued mappings in Banach spaces. The weak relatively nonexpansive
multivalued mappings are more generalized than relatively nonexpansive multivalued mappings.
In this paper, an example will be given which is a weak relatively nonexpansive multivalued
mapping but not a relatively nonexpansive multivalued mapping. In order to get the strong
convergence theorems for weak relatively nonexpansive multivalued mappings, a new monotone
hybrid iteration algorithm with generalized metric projection is presented and is used to
approximate the fixed point of weak relatively nonexpansive multivalued mappings. In this paper,
the notion of multivalued resolvent of maximal monotone operator has been also presented which
is a weak relatively nonexpansive multivalued mapping and can be used to find the zero point of
maximal monotone operator.
1. Introduction and Preliminaries
Iterative methods for approximating fixed points of multivalued mappings in Banach spaces
have been studied by some authors, see for instance 1–4. Let C be a nonempty closed
convex subset of a smooth Banach space E and T : C → C a multivalued mapping such
that T x is nonempty for all x ∈ C. In 4, Homaeipour and Razani have defined the relatively
nonexpansive multivalued mapping and have proved some convergence theorems.
Theorem SA 1 see 4. Let E be a uniformly convex and uniformly smooth Banach space, and C
a nonempty closed convex subset of E. Suppose T : C → C is a relatively nonexpansive multivalued
2
Abstract and Applied Analysis
mapping. Let {αn } be a sequence of real numbers such that 0 ≤ αn ≤ 1 and lim infn → ∞ αn 1 − αn > 0.
For a given x1 ∈ C, let {xn } be the iterative sequence defined by
xn1 ΠC J −1 αn Jxn 1 − αn Jzn ,
zn ∈ T xn .
1.1
If J is weakly sequentially continuous, then {xn } converges weakly to a fixed point of T .
Theorem SA 2 see 4. Let E be a uniformly convex and uniformly smooth Banach space, and C
a nonempty closed convex subset of E. Suppose T : C → C is a relatively nonexpansive multivalued
mapping. Let {αn } be a sequence of real numbers such that 0 ≤ αn ≤ 1 and lim infn → ∞ αn 1 − αn > 0.
For a given x1 ∈ C, let {xn } be the iterative sequence defined by
xn1 ΠC J −1 αn Jxn 1 − αn Jzn ,
zn ∈ T xn .
1.2
If the interior of FT is nonempty, then {xn } converges strongly to a fixed point of T .
Let E be a Banach space with dual E∗ . We denote by J the normalized duality mapping
∗
from E to 2E defined by
2 Jx f ∈ E∗ : x, f x2 f ,
1.3
where ·, · denotes the generalized duality pairing. It is well known that if E∗ is uniformly
convex, then J is uniformly continuous on bounded subsets of E.
As we all know that if C is a nonempty closed convex subset of a Hilbert space H and
PC : H → C is the metric projection of H onto C, then PC is nonexpansive. This fact actually
characterizes Hilbert spaces, and consequently it is not available in more general Banach
spaces. In this connection, Alber 5 recently introduced a generalized projection operator
ΠC in a Banach space E which is an analogue of the metric projection in Hilbert spaces.
Next, we assume that E is a smooth Banach space. Consider the functional defined as
5, 6 by
2
φ x, y x2 − 2 x, Jy y ,
for x, y ∈ E.
1.4
Observe that, in a Hilbert space H, 1.4 reduces to φx, y x − y2 , x, y ∈ H.
The generalized projection ΠC : E → C is a map that assigns to an arbitrary point
x ∈ E the minimum point of the functional φx, y, that is, ΠC x x, where x is the solution
to the following minimization problem:
φx, x minφ y, x ,
1.5
y∈C
existence and uniqueness of the operator ΠC follow from the properties of the functional
φx, y and strict monotonicity of the mapping J see, eg., 5–7. In Hilbert space, ΠC PC .
It is obvious from the definition of function φ that
y − x 2 ≤ φ y, x ≤ y x 2 ,
∀x, y ∈ E.
1.6
Abstract and Applied Analysis
3
Remark 1.1. If E is a reflexive strictly convex and smooth Banach space, then for x, y ∈ E,
φx, y 0 if and only if x y. It is sufficient to show that if φx, y 0 then x y. From
1.6, we have x y. This implies x, Jy x2 Jy2 . From the definitions of j, we
have Jx Jy, that is, x y, see 8, 9 for more details.
Let C be a nonempty closed convex subset of a smooth Banach space E and T : C → C
a multivalued mapping such that T x is nonempty for all x ∈ C. A point p is called a fixed
point of T , if p ∈ T p. The set of fixed points of T is represented by FT . A point p ∈ C is called
an asymptotic fixed point of T , if there exists a sequence {xn } in C which converges weakly to
p and limn → ∞ dxn , T xn 0, where dxn , T xn infu∈T xn xn − u. The set of asymptotic fixed
. Moreover, a multivalued mapping T : C → C is called
points of T is represented by FT
relatively nonexpansive multivalued mapping, if the following conditions are satisfied:
1 FT /
∅,
2 φp, z ≤ φp, x, for all p ∈ FT , for all x ∈ C, for all z ∈ T x,
FT .
3 FT
In 4, authors also give an example which is a relatively nonexpansive multivalued
mapping but not a nonexpansive multivalued mapping.
The purpose of this paper is to present the notion of weak relatively nonexpansive
multivalued mapping and to prove the strong convergence theorems for the weak relatively
nonexpansive multivalued mappings in Banach spaces. The weak relatively nonexpansive
multivalued mappings are more generalized than relatively nonexpansive multivalued
mappings. In this paper, an example will be given which is a weak relatively nonexpansive
multivalued mapping but not a relatively nonexpansive multivalued mapping. In order to get
the strong convergence theorems for weak relatively nonexpansive multivalued mappings, a
new monotone hybrid iteration algorithm with generalized metric projection is presented
and is used to approximate the fixed point of weak relatively nonexpansive multivalued
mappings. We first give the definition of weak relatively nonexpansive multivalued mapping
as follows.
Let C be a nonempty closed convex subset of a smooth Banach space E and T : C → C
a multivalued mapping such that T x is nonempty for all x ∈ C. A point p ∈ C is called
an strong asymptotic fixed point of T , if there exists a sequence {xn } in C which converges
strongly to p and limn → ∞ dxn , T xn 0, where dxn , T xn infu∈T xn xn − u. The set of the
. Moreover, a multivalued mapping
strong asymptotic fixed points of T is represented by FT
T : C → C is called weak relatively nonexpansive multivalued mapping, if the following
conditions are satisfied:
I FT /
∅,
II φp, z ≤ φp, x, for all p ∈ FT , for all x ∈ C, for all z ∈ T x,
FT .
III FT
We need the following Lemmas for the proof of our main results.
Lemma 1.2 see Kamimura and Takahashi 7. Let E be a uniformly convex and smooth Banach
space and let {xn },{yn } be two sequences of E. If φxn , yn → 0 and either {xn } or {yn } is bounded,
then xn − yn → 0.
4
Abstract and Applied Analysis
Lemma 1.3 see Alber 5. Let C be a nonempty closed convex subset of a smooth Banach space E
and x ∈ E. Then, x0 ΠC x if and only if
x0 − y, Jx − Jx0 ≥ 0,
for y ∈ C.
1.7
Lemma 1.4 see Alber 5. Let E be a reflexive, strictly convex and smooth Banach space, let C be
a nonempty closed convex subset of E, and let x ∈ E. Then
φ y, Πc x φΠc x, x ≤ φ y, x ,
∀y ∈ C.
1.8
Lemma 1.5. Let E be a uniformly convex and smooth Banach space, let C be a closed convex subset of
E, and let T : C → C be a weak relatively nonexpansive multivalued mapping. Then FT is closed
and convex.
Proof. First, we show FT is closed. Let {xn } be a sequence in FT such that xn → q. Since
T is weak relatively nonexpansive multivalued mapping, we have
φxn , z ≤ φ xn , q ,
∀z ∈ T q, n 1, 2, 3, . . . .
1.9
Therefore
φ q, z lim φxn , z ≤ lim φ xn , q φ q, q 0.
n→∞
n→∞
1.10
By Lemma 1.2, we have q z, hence, T q {q}, so q ∈ FT . Therefore FT is closed. Next,
we show FT is convex. Let x, y ∈ FT , put z tx 1 − ty for any t ∈ 0, 1. For w ∈ T z,
we have
φz, w z2 − 2z, Jw w2
z2 − 2 tx 1 − ty, Jw w2
z2 − 2tx, Jw − 21 − t y, Jw w2
2
z2 tφx, w 1 − tφ y, w − tx2 − 1 − ty
2
≤ z2 tφx, z 1 − tφ y, z − tx2 − 1 − ty
z2 − 2 tx 1 − ty, Jz z2
1.11
z2 − 2z, Jz z2
φz, z 0.
By Lemma 1.2, we have z w, so z ∈ T z, that is z ∈ FT . Therefore, FT is convex. This
completes the proof.
Abstract and Applied Analysis
5
Remark 1.6. Let E be a strictly convex and smooth Banach space, and C a nonempty closed
convex subset of E. Suppose T : C → C is a weak relatively nonexpansive multivalued
mapping. If p ∈ FT , then T p {p}.
2. An Example of Weak Relatively Nonexpansive Multivalued Mapping
Next, we give an example which is a weak relatively nonexpansive multivalued mapping but
not a relatively nonexpansive multivalued mapping.
Example 2.1. Let E l2 and
x0 1, 0, 0, 0, . . .
x1 1, 1, 0, 0, . . .
x2 1, 0, 1, 0, 0, . . .
2.1
x3 1, 0, 0, 1, 0, 0, . . .
···
xn 1, 0, 1, 0, 0, . . . , 0, 1, 0, 0, 0, . . .
···
It is obvious that {xn }∞
n1 converges weakly to x0 . On the other hand, we have xn − xm for any n /
m. Define a mapping T : E → 2E as follows:
⎧
⎨ y tx : n ≤ t ≤ n
,
n
n2
n1
T x ⎩
−x,
if x xn ∃n ≥ 1,
if x / xn ∀n ≥ 1.
√
2
2.2
It is also obvious that FT {0} and
0 − y y ≤ x 0 − x,
∀x ∈ E, y ∈ T x.
2.3
Since E L2 is a Hilbert space, we have
2 2
φ 0, y 0 − y y ≤ x2 0 − x2 φ0, x,
∀x ∈ E, y ∈ T x.
2.4
FT . In fact, that for any strong convergent sequence {zn } ⊂ E such
Next, we prove FT
that zn → z0 and dzn , T zn → 0 as n → ∞, then there exist sufficiently large nature number
xm , for any n, m > N. Then T zn −zn for n > N, it follows from
N such that zn /
zn − T zn dzn , T zn −→ 0
2.5
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Abstract and Applied Analysis
FT . Then T is a
that 2zn → 0 and hence zn → z0 0, this implies z0 ∈ FT , so that FT
weak relatively multivalued nonexpansive mapping.
We second claim that T is not relatively multivalued nonexpansive mapping. In fact,
that xn x0 and
dxn , T xn ≤
n
n
−
n1 n2
−→ 0
2.6
as n → ∞ hold, but x0 ∈FT .
3. Strong Convergence of Monotone Hybrid Algorithm
Theorem 3.1. Let E be a uniformly convex and uniformly smooth Banach space, let C be a nonempty
closed convex subset of E, and let T : C → C be a weak relatively multivalued nonexpansive mapping
∞
∞
∞
such that FT / ∅. Assume that {αn }n0 , {βn }n0 , {γn }n0 are three sequences in 0, 1 such that
αn βn γn 1, limn → ∞ αn 0 and 0 < γ ≤ γn ≤ 1 for some constant γ ∈ 0, 1. Define a sequence
{xn } in C by the following algorithm:
x0 ∈ C
arbitrarily,
yn J −1 αn Jx0 βn Jxn γn Jzn , zn ∈ T xn
Cn z ∈ Cn−1 : φ z, yn ≤ 1 − αn φz, xn αn φz, x0 ,
n ≥ 1,
3.1
C0 C,
xn1 ΠCn x0 ,
then {xn } converges to q ΠFT x0 .
Proof. We first show that Cn is closed and convex for all n ≥ 0. From the definitions of Cn , it is
obvious that Cn is closed for all n ≥ 0. Next, we prove that Cn is convex for all n ≥ 0. Since
φ z, yn ≤ 1 − αn φz, xn αn φz, x0 3.2
2 z, 1 − αn Jxn αn Jx0 − Jyn ≤ 1 − αn xn 2 αn x0 2 .
3.3
is equivalent to
It is easy to get that Cn is convex for all n ≥ 0.
Abstract and Applied Analysis
7
Next, we show that FT ⊂ Cn for all n ≥ 0. Indeed, for each p ∈ FT , we have
φ p, yn φ p, J −1 αn Jx0 βn Jxn γn Jzn
2
p − 2 p, αn Jx0 βn Jxn γn Jzn
2
αn Jx0 βn Jxn γn Jzn 2
≤ p − 2αn p, Jx0 − 2βn p, Jxn − 2γn p, Jzn
3.4
αn x0 2 βn xn γn zn 2
≤ αn φ p, x0 βn φ p, xn γn φ p, zn
≤ αn φ p, x0 βn φ p, xn γn φ p, xn
≤ αn φ p, x0 1 − αn φ p, xn .
So, p ∈ Cn , which implies that FT ⊂ Cn for all n ≥ 0.
Since xn1 ΠCn x0 and Cn ⊂ Cn−1 , then we get
φxn , x0 ≤ φxn1 , x0 ,
∀n ≥ 0.
3.5
Therefore, {φxn , x0 } is nondecreasing. On the other hand, by Lemma 1.4 we have
φxn , x0 φΠCn−1 x0 , x0 ≤ φ p, x0 − φ p, xn ≤ φ p, x0 ,
3.6
for all p ∈ FT ⊂ Cn−1 and for all n ≥ 1. Therefore, φxn , x0 is also bounded. This together
with 3.5 implies that the limit of {φxn , x0 } exists. Put
lim φxn , x0 d.
n→∞
3.7
From Lemma 1.4, we have, for any positive integer m, that
φxnm , xn1 φxnm , ΠCn x0 ≤ φxnm , x0 − φΠCn x0 , x0 φxnm , x0 − φxn1 , x0 ,
3.8
for all n ≥ 0. This together with 3.7 implies that
lim φxnm , xn1 0
n→∞
3.9
uniformly for all m, holds. By using Lemma 1.2, we get that
lim xnm − xn1 0
n→∞
3.10
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Abstract and Applied Analysis
uniformly for all m, holds. Then {xn } is a Cauchy sequence, therefore there exists a point
p ∈ C such that xn → p.
Since xn1 ΠCn x0 ∈ Cn , from the definition of Cn , we have
φ xn1 , yn ≤ 1 − αn φxn1 , xn αn φxn1 , x0 .
3.11
This together with 3.9 and limn → ∞ αn 0 implies that
lim φ xn1 , yn 0.
n→∞
3.12
Therefore, by using Lemma 1.2, we obtain
lim xn1 − yn 0.
n→∞
3.13
Since j is uniformly norm-to-norm continuous on bounded sets, then we have
lim Jxn1 − Jyn lim Jxn1 − Jxn 0.
n→∞
n→∞
3.14
Noticing that
Jxn1 − Jyn Jxn1 − αn Jx0 βn Jxn γn Jzn αn Jxn1 − Jx0 βn Jxn1 − Jxn γn Jxn1 − Jzn 3.15
≥ γn Jxn1 − Jzn − αn Jxn1 − Jx0 − βn Jxn1 − Jxn ,
which leads to
Jxn1 − Jzn ≤
1 Jxn1 − Jyn αn Jx0 − Jxn1 βn Jxn1 − Jxn .
γn
3.16
From 3.14 and limn → ∞ αn 0, 0 < γ ≤ γn ≤ 1, we obtain
lim Jxn1 − Jzn 0.
n→∞
3.17
Since J −1 is also uniformly norm-to-norm continuous on bounded sets, then we obtain
lim xn1 − zn 0.
3.18
xn − zn ≤ xn1 − xn xn1 − zn .
3.19
n→∞
Observe that
Abstract and Applied Analysis
9
It follows from 3.10 and 3.18 that
zn − xn −→ 0,
as n −→ ∞,
3.20
dxn , T xn −→ 0,
as n −→ ∞.
3.21
Since we have proved that xn → p, which together with 3.21 and that, T is weak relatively
multivalued nonexpansive mapping, implies that p ∈ FT .
Finally, we prove that p ΠFT x0 . From Lemma 1.4, we have
φ p, ΠFT x0 φ ΠFT x0 , x0 ≤ φ p, x0 .
3.22
On the other hand, since xn1 ΠCn x0 and FT ⊂ Cn , for all n. Also from Lemma 1.4, we
have
φ ΠFT x0 , xn1 φxn1 , x0 ≤ φ ΠFT x0 , x0 .
3.23
By the definition of φx, y, we know that
lim φxn1 , x0 φ p, x0 .
n→∞
3.24
Combining 3.22, 3.23, and 3.24, we know that φp, x0 φΠFT x0 , x0 . Therefore, it
follows from the uniqueness of ΠFT x0 that p ΠFT x0 . This completes the proof.
When αn ≡ 0 in Theorem 3.1, we obtain the following result.
Theorem 3.2. Let E be a uniformly convex and uniformly smooth Banach space, let C be a nonempty
closed convex subset of E, and let T : C → C be a weak relatively multivalued nonexpansive mapping
such that FT /
∅. Assume that {αn }∞
n0 is a sequences in 0, 1 such that 0 ≤ αn ≤ α < 1 for some
constant α ∈ 0, 1. Define a sequence {xn } in C by the following algorithm:
x0 ∈ C
arbitrarily,
yn J −1 αn Jxn 1 − αn Jzn , zn ∈ T xn
Cn z ∈ Cn−1 : φ z, yn ≤ φz, xn , n ≥ 1,
3.25
C0 C,
xn1 ΠCn x0 ,
then {xn } converges to q ΠFT x0 .
4. Applications for Maximal Monotone Operators
In this section, we apply the above results to prove some strong convergence theorem
concerning maximal monotone operators in a Banach space E.
10
Abstract and Applied Analysis
∅}
Let A be a multivalued operator from E to E∗ with domain DA {z ∈ E : Az /
and range RA {z ∈ E : z ∈ DA}. An operator A is said to be monotone if
x1 − x2 , y1 − y2 ≥ 0
4.1
for each x1 , x2 ∈ DA and y1 ∈ Ax1 , y2 ∈ Ax2 . A monotone operator A is said to be maximal
if its graph GA {x, y : y ∈ Ax} is not properly contained in the graph of any other
monotone operator. We know that if A is a maximal monotone operator, then A−1 0 is closed
and convex. The following result is also well known.
Theorem 4.1 see Rockafellar 10. Let E be a reflexive, strictly convex and smooth Banach space
and let A be a monotone operator from E to E∗ . Then A is maximal if and only if RJ rA E∗ for
all r > 0.
Let E be a reflexive, strictly convex and smooth Banach space, and let A be a maximal
monotone operator from E to E∗ . Using Theorem 4.1 and strict convexity of E, we obtain that
for every r > 0 and x ∈ E, there exists a unique xr such that
Jx ∈ Jxr rAxr .
4.2
Then we can define a single valued mapping Jr : E → DA by Jr J rA−1 J and such a
Jr is called the resolvent of A. We know that A−1 0 FJr for all r > 0, see 9, 11 for more
details. Using Theorem 3.1, we can consider the problem of strong convergence concerning
maximal monotone operators in a Banach space. Such a problem has been also studied in
1, 7, 10–20.
Theorem 4.2. Let E be a uniformly convex and uniformly smooth Banach space, let A be a maximal
∅, and let JM : E → 2E be a multivalued mapping
monotone operator from E to E∗ with A−1 0 /
defined as follows:
JM : x −→ y ∈ E : y Jr x, r ∈ M ,
x ∈ E,
4.3
where M is a set of real numbers such that inf M > 0. JM is called the multivalued resolvent of A.
Then JM is a weak relatively nonexpansive multivalued mapping.
Proof. Since
FJM p ∈ E : Jr p p, r ∈ M
p ∈ E : p ∈ A−1 0, r ∈ M
4.4
A−1 0,
so that FJM /
∅. Next, we show
φ p, z ≤ φ p, x ,
∀p ∈ FJM , ∀x ∈ E, ∀z ∈ JM x.
4.5
Abstract and Applied Analysis
11
From the monotonicity of A, we have
2
φ p, z p − 2 p, Jz z2
2
p 2 p, Jx − Jz − Jx Jr w2
2
p 2 p, Jx − Jz − 2 p, Jx z2
2
p − 2 z − p − z, Jx − Jz − Jx − 2 p, Jx z2
2
p − 2 z − p, Jx − Jz − Jx
2z, Jx − Jz − 2 p, Jx z2
2
≤ p 2z, Jx − Jz − 2 p, Jx z2
2
p − 2 p, Jx x2 − z2 2z, Jx − w2
φ p, x − φz, x
≤ φ p, x .
4.6
M FJM . Observe that FJ
M ⊇ FJM is obvious. Next, we show
Finally, we show FJ
M ⊆ FJM . Let {zn } ⊂ E be a sequence such that zn → p and limn → ∞ dzn , JM zn 0.
FJ
There exist sequences {yn } in E and {rn } ⊆ M such that
lim zn − yn 0,
n→∞
yn Jrn zn .
4.7
Since J is uniformly norm-to-norm continuous on bounded sets, we obtain
1
Jzn − Jyn −→ 0.
rn
4.8
1
Jzn − Jyn ∈ AJrn zn
rn
4.9
It follows from
and the monotonicity of A that
1
w − Jrn zn , w − Jzn − JJrn zn rn
∗
≥0
4.10
for all w ∈ DA and w∗ ∈ Aw. Letting n → ∞, we have w − p, w∗ ≥ 0 for all w ∈ DA
and w∗ ∈ Aw. Therefore, from the maximality of A, we obtain p ∈ A−1 0 FJM . Hence JM
is a weak relatively nonexpansive multivalued mapping. This completes the proof.
By using Theorems 3.1 and 4.2, we directly obtain the following result.
12
Abstract and Applied Analysis
Theorem 4.3. Let E be a uniformly convex and uniformly smooth Banach space, let A be a maximal
∅, let JM be a multivalued resolvent of A, where inf M >
monotone operator from E to E∗ with A−1 0 /
0, and let {αn }, {βn }, {γn } be three sequences of real numbers such that αn → 0 and 0 < γ ≤ γn ≤ 1
for some constant γ ∈ 0, 1. Define a sequence {xn } of C as follows:
x0 ∈ C
arbitrarily,
yn J αn Jx0 βn Jxn γn Jzn , zn ∈ JM xn
Cn z ∈ Cn−1 : φ z, yn ≤ 1 − αn φz, xn αn φz, x0 ,
−1
n ≥ 1,
4.11
C0 C,
xn1 ΠCn x0 ,
where J is the duality mapping on E. Then {xn } converges strongly to ΠA−1 0 x0 , where ΠA−1 0 is the
generalized projection from E onto A−1 0.
Acknowledgment
This project is supported by the National Natural Science Foundation of China under Grant
11071279.
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