- Research Article
- Open access
- Published:
Modified Hybrid Algorithm for a Family of Quasi-
-Asymptotically Nonexpansive Mappings
Fixed Point Theory and Applications volume 2010, Article number: 170701 (2010)
Abstract
The purpose of this paper is to propose a modified hybrid projection algorithm and prove strong convergence theorems for a family of quasi--asymptotically nonexpansive mappings. The method of the proof is different from the original one. Our results improve and extend the corresponding results announced by Zhou et al. (2010), Kimura and Takahashi (2009), and some others.
1. Introduction
Let be a real Banach space and
a nonempty closed convex subset of
. A mapping
is said to be asymptotically nonexpansive [1] if there exists a sequence
of positive real numbers with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ1_HTML.gif)
for all and all
.
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] in 1972. They proved that if is a nonempty bounded closed convex subset of a uniformly convex Banach space
, then every asymptotically nonexpansive self-mapping
of
has a fixed point. Further, the set
of fixed points of
is closed and convex. Since 1972, a host of authors have studied the weak and strong convergence problems of the iterative algorithms for such a class of mappings (see, e.g., [1–3] and the references therein).
It is well known that in an infinite-dimensional Hilbert space, the normal Mann's iterative algorithm has only weak convergence, in general, even for nonexpansive mappings. Consequently, in order to obtain strong convergence, one has to modify the normal Mann's iteration algorithm; the so-called hybrid projection iteration method is such a modification.
The hybrid projection iteration algorithm (HPIA) was introduced initially by Haugazeau [4] in 1968. For 40 years, (HPIA) has received rapid developments. For details, the readers are referred to papers in [5–11] and the references therein.
In 2003, Nakajo and Takahashi [6] proposed the following modification of the Mann iteration method for a nonexpansive mapping in a Hilbert space
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ2_HTML.gif)
where is a closed convex subset of
,
denotes the metric projection from
onto a closed convex subset
of
. They proved that if the sequence
is bounded above from one then the sequence
generated by (1.2) converges strongly to
, where
denote the fixed points set of
.
In 2006, Kim and Xu [12] proposed the following modification of the Mann iteration method for asymptotically nonexpansive mapping in a Hilbert space
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ3_HTML.gif)
where is bounded closed convex subset and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ4_HTML.gif)
They proved that if the sequence is bounded above from one, then the sequence
generated by (1.3) converges strongly to
.
They also proposed the following modification of the Mann iteration method for asymptotically nonexpansive semigroup in a Hilbert space
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ5_HTML.gif)
where is bounded closed convex subset and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ6_HTML.gif)
and is nonincreasing in
and bounded measurable function such that,
for all
as
, and for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ7_HTML.gif)
They proved that if the sequence is bounded above from one, then the sequence
generated by (1.5) converges strongly to
, where
denote the common fixed points set of
.
In 2006, Martinez-Yanes and Xu [7] proposed the following modification of the Ishikawa iteration method for nonexpansive mapping in a Hilbert space
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ8_HTML.gif)
where is a closed convex subset of
. They proved that if the sequence
is bounded above from one and
, then the sequence
generated by (1.8) converges strongly to
.
Martinez-Yanes and Xu [7] proposed also the following modification of the Halpern iteration method for nonexpansive mapping in a Hilbert space
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ9_HTML.gif)
where is a closed convex subset of
. They proved that if the sequence
, then the sequence
generated by (1.9) converges strongly to
.
In 2005, Matsushita and Takahashi [8] proposed the following hybrid iteration method with generalized projection for relatively nonexpansive mapping in a Banach space
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ10_HTML.gif)
They proved the following convergence theorem.
Theorem MT..
Let be a uniformly convex and uniformly smooth Banach space, let
be a nonempty closed convex subset of
, let
be a relatively nonexpansive mapping from
into itself, and let
be a sequence of real numbers such that
and
. Suppose that
is given by (1.10), where
is the duality mapping on
. If
is nonempty, then
converges strongly to
, where
is the generalized projection from
onto
.
In 2009, Zhou et al. [11] proposed the following modification of the hybrid iteration method with generalized projection for a family of closed and quasi--asymptotically nonexpansive mappings
in a Banach space
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ11_HTML.gif)
They proved the following convergence theorem.
Theorem ZGT.
Let
be a nonempty bounded closed convex subset of a uniformly convex and uniformly smooth Banach space
, and let
be a family of
-asymptotically nonexpansive mappings such that
. Assume that every
,
is asymptotically regular on
. Let
be a real sequence in
such that
. Define a sequence
as given by ( 1 ), then
converges strongly to
, where
,
for all
,
, and
is the generalized projection from
onto
.
Very recently, Kimura and Takahashi [13] established strong convergence theorems by the hybrid method for a family of relatively nonexpansive mappings as follows.
Theorem KT.
Let
be a strictly convex reflexive Banach space having the Kadec-Klee property and a Fréchet differentiable norm, and let
be a nonempty and closed convex subset of
and
a family of relatively nonexpensive mappings of
into itself having a common fixed point. Let
be a sequence in
such that
. For an arbitrarily chosen point
, generate a sequence
by the following iterative scheme:
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ12_HTML.gif)
for every , then
converges strongly to
, where
is the set of common fixed points of
and
is the metric projection of
onto a nonempty closed convex subset
of
.
Motivated by these results above, the purpose of this paper is to propose a Modified hybrid projection algorithm and prove strong convergence theorems for a family of -
-asymptotically nonexpansive mappings which are asymptotically regular on
. In order to get the strong convergence theorems for such a family of mappings, the classical hybrid projection iteration algorithm is modified and then is used to approximate the common fixed points of such a family of mappings. In the meantime, the method of the proof is different from the original one. Our results improve and extend the corresponding results announced by Zhou et al. [11], and Kimura and Takahashi [13], and some others.
2. Preliminaries
Let be a Banach space with dual
. Denote by
the duality product. The normalize duality mapping
from
to
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ13_HTML.gif)
for all , where
denotes the dual space of
and
the generalized duality pairing between
and
. It is well known that if
is uniformly convex, then
is uniformly continuous on bounded subsets of
.
It is also very well known 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 [14] recently introduced a generalized projection operator
in a Banach space
which is an analogue of the metric projection in Hilbert spaces.
Next, we assume that is a real smooth Banach space. Let us consider the functional defined by [7, 8] as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ14_HTML.gif)
for all . Observe that, in a Hilbert space
, (2.2) reduces to
,
,
.
The generalized projection is a map 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%2F170701/MediaObjects/13663_2010_Article_1212_Equ15_HTML.gif)
Existence and uniqueness of the operator follow from the properties of the C functional
and strict monotonicity of the mapping
(see, e.g., [14–18]). In Hilbert spaces,
. It is obvious from the definition of function
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ16_HTML.gif)
for all .
Remark 2.1.
If is a reflexive strictly convex and smooth Banach space, then for
,
if and only if
. It is sufficient to show that if
, then
. From (2.4), we have
. This implies that
From the definitions of
, we have
. That is,
see [17, 18] for more details.
Let be a closed convex subset of
and
a mapping from
into itself.
is said to be
-asymptotically nonexpansive if there exists some real sequence
with
and
such that
for all
and
.
is said to be
-asymptotically nonexpansive [9] if there exists some real sequence
with
and
and
such that
for all
,
, and
.
is said to be asymptotically regular on
if, for any bounded subset
of
, there holds the following equality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ17_HTML.gif)
We remark that a -asymptotically nonexpansive mapping with a nonempty fixed point set
is a quasi-
-asymptotically nonexpansive mapping, but the converse may be not true.
We present some examples which are closed and quasi--asymptotically nonexpansive.
Example 2.2.
Let be a real line. We define a mapping
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ18_HTML.gif)
Then is continuous quasi-nonexpansive, and hence it is closed and
nonexpansive with the constant sequence
but not asymptotically nonexpansive.
Example 2.3.
Let be a uniformly smooth and strictly convex Banach space, and
is a maximal monotone mapping such that
is nonempty. Then,
is a closed and quasi-
-asymptotically nonexpansive mapping from
onto
, and
.
Example 2.4.
Let be the generalized projection from a smooth, strictly convex, and reflexive Banach space
onto a nonempty closed convex subset
of
. Then,
is a closed and quasi-
-asymptotically nonexpansive mapping from
onto
with
.
Let be a sequence of nonempty closed convex subsets of a reflexive Banach space
. We denote two subsets
and
as follows:
if and only if there exists
such that
converges strongly to
and that
for all
. Similarly,
if and only if there exists a subsequence
of
and a sequence
such that
converges weakly to
and that
for all
. We define the Mosco convergence [19] of
as follows. If
satisfies that
, it is said that
converges to
in the sense of Mosco, and we write
. For more details, see [20].
The following theorem plays an important role in our results.
Theorem 2.5 (see Ibaraki et al. [21]).
Let be a smooth, reflexive, and strictly convex Banach space having the Kadec-Klee property. Let
be a sequence of nonempty closed convex subsets of
. If
exists and is nonempty, then
converges strongly to
for each
.
We also need the following lemmas for the proof of our main results.
Lemma 2.6 (Kamimura and Takahashi [16]).
Let be a uniformly convex and smooth Banach space, and let
,
be two sequences of
if
and either
or
is bounded, then
.
Lemma 2.7 (Alber [14]).
Let be a reflexive, strictly convex and smooth Banach space, let
be a nonempty closed convex subset of
, and let
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ19_HTML.gif)
for all .
Lemma 2.8.
Let be a uniformly convex and smooth Banach space, let
be a closed convex subset of
, and let
be a closed and
-asympotically nonexpansive mapping from
into itself. Then
is a closed convex subset of
.
3. A Modified Algorithm and Strong Convergence Theorems
Now we are in a proposition to prove the main results of this paper. In the sequel, we use the letter to denote an index set.
Theorem 3.1.
Let be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space
, and let
be a family of closed and
-asymptotically nonexpansive mappings such that
. Assume that every
,
is asymptotically regular on
. Let
,
,
be real sequences in
such that
,
. Define a sequence
in
in the following manner:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ20_HTML.gif)
Then converges strongly to
, where
is the generalized projection from
onto
.
Proof.
Firstly, we show that is closed and convex for each
.
From the definition of , it is obvious that
is closed for each
. We show that
is convex for each
. Observe that the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ21_HTML.gif)
can be written as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ22_HTML.gif)
For and
, denote
,
, and
by noting that
is convex, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ23_HTML.gif)
So we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ24_HTML.gif)
which infers that , so we get that
is convex for each
. Thus
is closed and convex for every
.
Secondly, we prove that , for all
.
Indeed, by noting that is convex and using (2.2), we have, for any
and all
, that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ25_HTML.gif)
which infers that , for all
and
, and hence
. This proves that
, for all
and
.
Thirdly, we will show that .
Since is a decreasing sequence of closed convex subsets of
such that
is nonempty, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ26_HTML.gif)
By Theorem 2.5, converges strongly to
.
Fourthly, we prove that .
Since , from the definition of
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ27_HTML.gif)
From , one obtains
as
, and it follows from
, for every
that we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ28_HTML.gif)
and hence as
by Lemma 2.6. It follows that
as
. Since
is uniformly norm-to-norm continuous on any bounded sets of
, we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ29_HTML.gif)
for every . By the definition of
and the assumption on
, we deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ30_HTML.gif)
for every and
. So we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ31_HTML.gif)
Since , we have
as
.
Since is also uniformly norm-to-norm continuous on any bounded sets of
, we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ32_HTML.gif)
Noting that as
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ33_HTML.gif)
as . Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ34_HTML.gif)
By using (3.14), (3.15), and the asymptotic regularity of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ35_HTML.gif)
as , that is,
. Now the closedness property of
gives that
is a common fixed point of the family
, thus
.
Finally, since and
is a nonempty closed convex subset of
, we conclude that
. This completes the proof.
Remark 3.2.
The boundedness assumption on in Theorem ZGT can be dropped.
Remark 3.3.
The asymptotic regularity assumption on in Theorem 3.1 can be weakened to the assumption that
as
.
Recall that is called uniformly Lipschitzian continuous if there exists some
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ36_HTML.gif)
for all and
.
Remark 3.4.
The assumption that as
can be replaced by the uniform Lipschitz continuity of
.
With above observations, we have the following convergence result.
Corollary 3.5.
Let be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space
, and let
be a family of uniformly Lipschitzian continuous and
-
-asymptotically nonexpansive mappings such that
. Let
,
,
be real sequences in
such that
,
. Define a sequence
in
in the following manner:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ37_HTML.gif)
Then converges strongly to
, where
is the generalized projection from
onto
.
Proof.
Following the proof lines of Theorem 3.1, we can prove that is nonempty closed convex,
is closed convex,
for all
and
. At this point, it is sufficient to show that
as
. Again, from the proof lines of Theorem 3.1, we have the following conclusions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ38_HTML.gif)
Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ39_HTML.gif)
so that as
. By Theorem 3.1, we have the desired conclusion. This completes the proof.
When in Theorem 3.1, we obtain the following result.
Corollary 3.6.
Let be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space
, and let
be a family of closed and
-asymptotically nonexpansive mappings such that
. Assume that every
,
is asymptotically regular on
. 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%2F170701/MediaObjects/13663_2010_Article_1212_Equ40_HTML.gif)
Then converges strongly to
, where
is the generalized projection from
onto
.
When in Theorem 3.1, we obtain the following result.
Corollary 3.7.
Let be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space
, and let
be a family of closed and
-asymptotically nonexpansive mappings such that
. Assume that every
,
is asymptotically regular on
. 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%2F170701/MediaObjects/13663_2010_Article_1212_Equ41_HTML.gif)
Then converges strongly to
, where
is the generalized projection from
onto
.
In the spirit of Theorem 3.1, we can prove the following strong convergence theorem.
Theorem 3.8.
Let be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space
, and let
be a family of closed and
-nonexpansive mappings such that
. Let
,
,
be real sequences in
such that
,
. Define a sequence
in
in the following manner:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F170701/MediaObjects/13663_2010_Article_1212_Equ42_HTML.gif)
Then converges strongly to
, where
is the generalized projection from
onto
.
Proof.
Following the proof lines of Theorem 3.1, we have the following conclusions:
-
(1)
is a nonempty closed convex subset of
;
-
(2)
is closed covex for all
;
-
(3)
, for all
;
-
(4)
;
-
(5)
for all
.
The closedness property of together with (4) and (5) implies that
converges strongly to a common fixed point
of the family
. As shown in Theorem 3.1,
. This completes the proof.
When in Theorem 3.8, we obtain the following result.
Corollary 3.9.
Let be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space, and let
be a family of closed and
-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%2F170701/MediaObjects/13663_2010_Article_1212_Equ43_HTML.gif)
Then converges strongly to
, where
is the generalized projection from
onto
.
When in Theorem 3.8, we obtain the following result.
Corollary 3.10.
Let be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space
, and let
be a family of closed and
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%2F170701/MediaObjects/13663_2010_Article_1212_Equ44_HTML.gif)
Then converges strongly to
, where
is the generalized projection from
onto
.
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This project is supported by the Zhangjiakou city technology research and development projects foundation (0911008B-3), Hebei education department research projects foundation (2006103) and Hebei north university research projects foundation (2009008).
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Xu, Y., Zhang, X., Kang, J. et al. Modified Hybrid Algorithm for a Family of Quasi--Asymptotically Nonexpansive Mappings.
Fixed Point Theory Appl 2010, 170701 (2010). https://doi.org/10.1155/2010/170701
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DOI: https://doi.org/10.1155/2010/170701