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Strong Convergence Theorems of Common Fixed Points for a Family of Quasi-
-Nonexpansive Mappings
Fixed Point Theory and Applications volume 2010, Article number: 754320 (2009)
Abstract
We consider a modified Halpern type iterative algorithm for a family of quasi--nonexpansive mappings in the framework of Banach spaces. Strong convergence theorems of the purposed iterative algorithms are established.
1. Introduction
Let be a Banach space,
a nonempty closed and convex subset of
, and
a nonlinear mapping. Recall that
is nonexpansive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F754320/MediaObjects/13663_2009_Article_1337_Equ1_HTML.gif)
A point is a fixed point of
provided
. Denote by
the set of fixed points of
, that is,
.
One classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping; see ([1, 2]). More precisely, take and define a contraction
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F754320/MediaObjects/13663_2009_Article_1337_Equ2_HTML.gif)
where is a fixed element. Banach Contraction Mapping Principle guarantees that
has a unique fixed point
in
. It is unclear, in general, what the behavior of
is as
even if
has a fixed point. However, in the case of
having a fixed point, Browder [1] proved the following well-known strong convergence theorem.
Theorem 1 B.
Let be a bounded closed convex subset of a Hilbert space
and
a nonexpansive mapping on
. Fix
and define
as
for any
. Then
converges strongly to an element of
nearest to
.
Motivated by Theorem B, Halpern [3] considered the following explicit iteration:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F754320/MediaObjects/13663_2009_Article_1337_Equ3_HTML.gif)
and obtained the following theorem.
Theorem 1 H.
Let be a bounded closed convex subset of a Hilbert space
and
a nonexpansive mapping on
. Define a real sequence
in
by
,
. Then the sequence
defined by (1.3) converges strongly to the element of
nearest to
.
In [4], Lions improved the result of Halpern [3], still in Hilbert spaces, by proving the strong convergence of to a fixed point of
provided that the control sequence
satisfies the following conditions:
(C1)
(C2)
(C3)
It was observed that both the Halpern's and Lion's conditions on the real sequence excluded the canonical choice
. This was overcome by Wittmann [5], who proved, still in Hilbert spaces, the strong convergence of
to a fixed point of
if
satisfies the following conditions:
(C1)
(C2)
(C4)
In [6], Shioji and Takahashi extended Wittmann's results to the setting of Banach spaces under the assumptions (C1), (C2), and (C4) imposed on the control sequences . In [7], Xu remarked that the conditions (C1) and (C2) are necessary for the strong convergence of the iterative sequence defined in (1.3) for all nonexpansive self-mappings. It is well known that the iterative algorithm (1.3) is widely believed to have slow convergence because the restriction of condition (C2). Thus, to improve the rate of convergence of the iterative process (1.3), one cannot rely only on the process itself.
Recently, hybrid projection algorithms have been studied for the fixed point problems of nonlinear mappings by many authors; see, for example, [8–24]. In 2006, Martinez-Yanes and Xu [10] proposed the following modification of the Halpern iteration for a single nonexpansive mapping in a Hilbert space. To be more precise, they proved the following theorem.
Theorem 1 MYX.
Let be a real Hilbert space,
a closed convex subset of
, and
a nonexpansive mapping such that
. Assume that
is such that
Then the sequence
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F754320/MediaObjects/13663_2009_Article_1337_Equ4_HTML.gif)
converges strongly to
Very recently, Qin and Su [17] improved the result of Martinez-Yanes and Xu [10] from Hilbert spaces to Banach spaces. To be more precise, they proved the following theorem.
Theorem 1 QS.
Let be a uniformly convex and uniformly smooth Banach space,
a nonempty closed convex subset of
, and
a relatively nonexpansive mapping. Assume that
is a sequence in
such that
. Define a sequence
in
by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F754320/MediaObjects/13663_2009_Article_1337_Equ5_HTML.gif)
where is the single-valued duality mapping on
. If
is nonempty, then
converges to
In this paper, motivated by Kimura and Takahashi [8], Martinez-Yanes and Xu [10], Qin and Su [17], and Qin et al. [19], we consider a hybrid projection algorithm to modify the iterative process (1.3) to have strong convergence under condition (C1) only for a family of closed quasi--nonexpansive mappings.
2. Preliminaries
Let be a Banach space with the dual space
. We denote by
the normalized duality mapping from
to
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F754320/MediaObjects/13663_2009_Article_1337_Equ6_HTML.gif)
where denotes the generalized duality pairing. It is well known that, if
is strictly convex, then
is single-valued and, if
is uniformly convex, then
is uniformly continuous on bounded subsets of
.
We know that, if is a nonempty closed convex subset of a Hilbert space
and
is the metric projection of
onto
, then
is nonexpansive. This fact actually characterizes Hilbert spaces and, consequently, it is not available in more general Banach spaces. In this connection, Alber [25] recently introduced a generalized projection operator
in a Banach space
, which is an analogue of the metric projection in Hilbert spaces.
A Banach space is said to be strictly convex if
for all
with
and
. The space
is said to be uniformly convex if
for any two sequences
and
in
such that
and
. Let
be the unit sphere of
. Then the space
is said to be smooth if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F754320/MediaObjects/13663_2009_Article_1337_Equ7_HTML.gif)
exists for each It is also said to be uniformly smooth if the limit is attained uniformly for
. It is well known that, if
is uniformly smooth, then
is uniformly norm-to-norm continuous on each bounded subset of
.
In a smooth Banach space , we consider the functional defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F754320/MediaObjects/13663_2009_Article_1337_Equ8_HTML.gif)
Observe that, in a Hilbert space , (2.3) reduces to
for all
The generalized projection
is a mapping that assigns to an arbitrary point
the minimum point of the functional
that is,
where
is the solution to the minimization problem:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F754320/MediaObjects/13663_2009_Article_1337_Equ9_HTML.gif)
The existence and uniqueness of the operator follows from some properties of the functional
and the strict monotonicity of the mapping
(see, e.g., [25–28]). In Hilbert spaces,
It is obvious from the definition of the function
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F754320/MediaObjects/13663_2009_Article_1337_Equ10_HTML.gif)
Remark 2.1.
If is a reflexive, strictly convex, and smooth Banach space, then, for any
,
if and only if
. In fact, it is sufficient to show that, if
, then
. From (2.5), we have
. This implies
From the definition of
one has
. Therefore, we have
(see [27, 29] for more details).
Let be a nonempty closed and convex subset of
and
a mapping from
into itself. A point
is said to be an asymptotic fixed point of
([30]) if
contains a sequence
which converges weakly to
such that
. The set of asymptotic fixed points of
will be denoted by
. A mapping
from
into itself is said to be relatively nonexpansive ([27, 31, 32]) if
and
for all
and
. The asymptotic behavior of a relatively nonexpansive mapping was studied by some authors ([27, 31, 32]).
A mapping is said to be
-nonexpansive ([18, 19, 24]) if
for all
. The mapping
is said to be quasi-
-nonexpansive ([18, 19, 24]) if
and
for all
and
.
Remark 2.2.
The class of quasi--nonexpansive mappings is more general than the class of relatively nonexpansive mappings, which requires the strong restriction:
.
In order to prove our main results, we need the following lemmas.
Lemma 2.3 (see [28]).
Let be a uniformly convex and smooth Banach space and
,
two sequences of
. If
and either
or
is bounded, then
Let be a nonempty closed convex subset of a smooth Banach space
and
. Then
if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F754320/MediaObjects/13663_2009_Article_1337_Equ11_HTML.gif)
Let be a reflexive, strictly convex, and smooth Banach space,
a nonempty closed convex subset of
and
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F754320/MediaObjects/13663_2009_Article_1337_Equ12_HTML.gif)
Let be a uniformly convex and smooth Banach space,
a nonempty, closed, and convex subset of
and
a closed quasi-
-nonexpansive mapping from
into itself. Then
is a closed and convex subset of
.
3. Main Results
From now on, we use to denote an index set. Now, we are in a position to prove our main results.
Theorem 3.1.
Let be a nonempty closed and convex subset of a uniformly convex and uniformly smooth Banach space
and
a family of closed quasi-
-nonexpansive mappings such that
. Let
be a real sequence in
such that
. Define a sequence
in
in the following manner:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F754320/MediaObjects/13663_2009_Article_1337_Equ13_HTML.gif)
then the sequence defined by (3.1) converges strongly to
.
Proof.
We first show that and
are closed and convex for each
. From the definitions of
and
, it is obvious that
is closed and
is closed and convex for each
. We, therefore, only show that
is convex for each
. Indeed, note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F754320/MediaObjects/13663_2009_Article_1337_Equ14_HTML.gif)
is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F754320/MediaObjects/13663_2009_Article_1337_Equ15_HTML.gif)
This shows that is closed and convex for each
and
Therefore, we obtain that
is convex for each
.
Next, we show that for all
. For each
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F754320/MediaObjects/13663_2009_Article_1337_Equ16_HTML.gif)
which yields that for all
and
It follows that
. This proves that
for all
.
Next, we prove that for all
We prove this by induction. For
we have
Assume that
for some
. Next, we show that
for the same
. Since
is the projection of
onto
we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F754320/MediaObjects/13663_2009_Article_1337_Equ17_HTML.gif)
Since by the induction assumption, (3.5) holds, in particular, for all
. This together with the definition of
implies that
for all
Noticing that
and
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F754320/MediaObjects/13663_2009_Article_1337_Equ18_HTML.gif)
We, therefore, obtain that is nondecreasing. From Lemma 2.5, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F754320/MediaObjects/13663_2009_Article_1337_Equ19_HTML.gif)
This shows that is bounded. It follows that the limit of
exists. By the construction of
, we see that
and
for any positive integer
Notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F754320/MediaObjects/13663_2009_Article_1337_Equ20_HTML.gif)
Taking the limit as in (3.8), we get that
From Lemma 2.3, one has
as
It follows that
is a Cauchy sequence in
. Since
is a Banach space and
is closed and convex, we can assume that
as
.
Finally, we show that To end this, we first show
. By taking
in (3.8), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F754320/MediaObjects/13663_2009_Article_1337_Equ21_HTML.gif)
From Lemma 2.3, we arrive at
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F754320/MediaObjects/13663_2009_Article_1337_Equ22_HTML.gif)
Noticing that , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F754320/MediaObjects/13663_2009_Article_1337_Equ23_HTML.gif)
It follows from the assumption on and (3.9) that
for each
. From Lemma 2.3, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F754320/MediaObjects/13663_2009_Article_1337_Equ24_HTML.gif)
On the other hand, we have By the assumption on
, we see that
for each
Since
is also uniformly norm-to-norm continuous on bounded sets, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F754320/MediaObjects/13663_2009_Article_1337_Equ25_HTML.gif)
On the other hand, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F754320/MediaObjects/13663_2009_Article_1337_Equ26_HTML.gif)
From (3.10)–(3.13), we obtain From the closedness of
, we get
Finally, we show that From
, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F754320/MediaObjects/13663_2009_Article_1337_Equ27_HTML.gif)
Taking the limit as in (3.15), we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F754320/MediaObjects/13663_2009_Article_1337_Equ28_HTML.gif)
and hence by Lemma 2.4. This completes the proof.
Remark 3.2.
Comparing the hybrid projection algorithm (3.1) in Theorem 3.1 with algorithm (1.5) in Theorem QS, we remark that the set is constructed based on the set
instead of
for each
We obtain that the sequence generated by the algorithm (3.1) is a Cauchy sequence. The proof is, therefore, different from the one presented in Qin and Su [17].
As a corollary of Theorem 3.1, for a single quasi--nonexpansive mapping, we have the following result immediately.
Corollary 3.3.
Let be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth Banach space
and
a closed quasi-
-nonexpansive mappings with a fixed point. Let
be a real sequence in
such that
. Define a sequence
in
in the following manner:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F754320/MediaObjects/13663_2009_Article_1337_Equ29_HTML.gif)
then the sequence converges strongly to
.
Remark 3.4.
Corollary 3.3 mainly improves Theorem of Qin and Su [17] from the class of relatively nonexpansive mappings to the class of quasi-
-nonexpansive mappings, which relaxes the strong restriction:
In the framework of Hilbert spaces, Theorem 3.1 is reduced to the following result.
Corollary 3.5.
Let be a nonempty closed and convex subset of a Hilbert space
and
a family of closed quasi-nonexpansive mappings such that
. Let
be a real sequence in
such that
. Define a sequence
in
in the following manner:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F754320/MediaObjects/13663_2009_Article_1337_Equ30_HTML.gif)
then the sequence converges strongly to
.
Remark 3.6.
Corollary 3.5 includes the corresponding result of Martinez-Yanes and Xu [10] as a special case. To be more precise, Corollary 3.5 improves Theorem 3.1 of Martinez-Yanes and Xu [10] from a single mapping to a family of mappings and from nonexpansive mappings to quasi-nonexpansive mappings, respectively.
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Acknowledgment
This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).
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Qin, X., Cho, Y., Cho, S. et al. Strong Convergence Theorems of Common Fixed Points for a Family of Quasi--Nonexpansive Mappings.
Fixed Point Theory Appl 2010, 754320 (2009). https://doi.org/10.1155/2010/754320
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DOI: https://doi.org/10.1155/2010/754320