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Strong Convergence of an Iterative Method for Inverse Strongly Accretive Operators
Journal of Inequalities and Applications volume 2008, Article number: 420989 (2008)
Abstract
We study the strong convergence of an iterative method for inverse strongly accretive operators in the framework of Banach spaces. Our results improve and extend the corresponding results announced by many others.
1. Introduction and Preliminaries
Let be a real Hilbert space with norm
and inner product
,
a nonempty closed convex subset of
and
a monotone operator of
into
. The classical variational inequality problem is formulated as finding a point
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ1_HTML.gif)
for all . Such a point
is called a solution of the variational inequality (1.1). Next, the set of solutions of the variational inequality (1.1) is denoted by
. In the case when
,
holds, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ2_HTML.gif)
Recall that an operator of
into
is said to be inverse strongly monotone if there exists a positive real number
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ3_HTML.gif)
for all (see [1–4]). For such a case,
is said to be
-inverse strongly monotone.
Recall that is nonexpansive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ4_HTML.gif)
for all It is known that if
is a nonexpansive mapping of
into itself, then
is
-inverse strongly monotone and
, where
denotes the set of fixed points of
.
Let be the projection of
onto the convex subset
. It is known that projection operator
is nonexpansive. It is also known that
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ5_HTML.gif)
for Moreover,
is characterized by the properties
and
for all
One can see that the variational inequality problem (1.1) is equivalent to some fixed-point problem. The element is a solution of the variational inequality (1.1) if and only if
satisfies the relation
where
is a constant.
To find a solution of the variational inequality for an inverse strongly monotone operator, Iiduka et al. [2] proved the following weak convergence theorem.
Theorem 1.
Let be a nonempty closed convex subset of a real Hilbert space
and let
be an
-inverse strongly monotone operator of
into
with
. Let
be a sequence defined as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ6_HTML.gif)
for all , where
is the metric projection from
onto
,
is a sequence in
and
is a sequence in
. If
and
are chosen so that
for some
with
and
for some
with
, then the sequence
defined by (1.6) converges weakly to some element of
.
Next, we assume that is a nonempty closed and convex subset of a Banach space
. Let
be the dual space of
and let
denote the pairing between
and
. For
, the generalized duality mapping
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ7_HTML.gif)
for all . In particular,
is called the normalized duality mapping. It is known that
for all
. If
is a Hilbert space, then
. Further, we have the following properties of the generalized duality mapping
:
-
(1)
for all
with
;
-
(2)
for all
and
;
-
(3)
for all
.
Let . A Banach space
is said to be uniformly convex if, for any
, there exists
such that, for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ8_HTML.gif)
It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space is said to be smooth if the limit
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ9_HTML.gif)
exists for all . It is also said to be uniformly smooth if the limit (1.9) is attained uniformly for
. The norm of
is said to be Fréchet differentiable if, for any
, the limit (1.9) is attained uniformly for all
. The modulus of smoothness of
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ10_HTML.gif)
where is a function. It is known that
is uniformly smooth if and only if
. Let
be a fixed real number with
. A Banach space
is said to be
-uniformly smooth if there exists a constant
such that
for all
.
Note that
-
(1)
is a uniformly smooth Banach space if and only if
is single-valued and uniformly continuous on any bounded subset of
;
-
(2)
all Hilbert spaces,
(or
) spaces (
), and the Sobolev spaces,
(
), are
-uniformly smooth, while
(or
) and
spaces (
) are
-uniformly smooth.
Recall that an operator of
into
is said to be accretive if there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ11_HTML.gif)
for all .
For recall that an operator
of
into
is said to be
-inverse strongly accretive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ12_HTML.gif)
for all . Evidently, the definition of the inverse strongly accretive operator is based on that of the inverse strongly monotone operator.
Let be a subset of
and let
be a mapping of
into
. Then
is said to be sunny if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ13_HTML.gif)
whenever for
and
. A mapping
of
into itself is called a retraction if
. If a mapping
of
into itself is a retraction, then
for all
, where
is the range of
. A subset
of
is called a sunny nonexpansive retract of
if there exists a sunny nonexpansive retraction from
onto
. We know the following lemma concerning sunny nonexpansive retraction.
Lemma 1.1 (see [5]).
Let be a closed convex subset of a smooth Banach space
, let
be a nonempty subset of
, and let
be a retraction from
onto
. Then
is sunny and nonexpansive if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ14_HTML.gif)
for all and
.
Recently, Aoyama et al. [6] first considered the following generalized variational inequality problem in a smooth Banach space. Let be an accretive operator of
into
. Find a point
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ15_HTML.gif)
for all . In order to find a solution of the variational inequality (1.15), the authors proved the following theorem in the framework of Banach spaces.
Theorem 1.
Let be a uniformly convex and
-uniformly smooth Banach space and
a nonempty closed convex subset of
. Let
be a sunny nonexpansive retraction from
onto
,
and
an
-inverse strongly accretive operator of
into
with
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ16_HTML.gif)
If and
are chosen such that
for some
and
for some
with
, then the sequence
defined by the following manners:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ17_HTML.gif)
converges weakly to some element of
, where
is the
-uniformly smoothness constant of
.
In this paper, motivated by Aoyama et al. [6], Iiduka et al. [2], Takahahsi and Toyoda [4], we introduce an iterative method to approximate a solution of variational inequality (1.15) for an -inverse strongly accretive operators. Strong convergence theorems are obtained in the framework of Banach spaces under appropriate conditions on parameters.
We also need the following lemmas for proof of our main results.
Lemma 1.2 (see [7]).
Let be a given real number with
and let
be a
-uniformly smooth Banach space. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ18_HTML.gif)
for all , where
is the
-uniformly smoothness constant of
.
The following lemma is characterized by the set of solutions of variational inequality (1.15) by using sunny nonexpansive retractions.
Lemma 1.3 (see [6]).
Let be a nonempty closed convex subset of a smooth Banach space
. Let
be a sunny nonexpansive retraction from
onto
and let
be an accretive operator of
into
. Then, for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ19_HTML.gif)
Lemma 1.4 (see [8]).
Let be a nonempty bounded closed convex subset of a uniformly convex Banach space
and let
be nonexpansive mapping of
into itself. If
is a sequence of
such that
weakly and
, then
is a fixed point of
.
Lemma 1.5 (see [9]).
Let ,
be bounded sequences in a Banach space
and let
be a sequence in
which satisfies the following condition:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ20_HTML.gif)
Suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ21_HTML.gif)
for all and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ22_HTML.gif)
Then .
Lemma 1.6 (see [10]).
Assume that is a sequence of nonnegative real numbers such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ23_HTML.gif)
for all , where
is a sequence in
and
is a sequence in
such that
(i);
(ii) or
.
Then .
2. Main Results
Theorem 2.1.
Let be a uniformly convex and
-uniformly smooth Banach space and
a nonempty closed convex subset of
. Let
be a sunny nonexpansive retraction from
onto
,
an arbitrarily fixed point, and
an
-inverse strongly accretive operator of
into
such that
. Let
and
be two sequences in
and let
a real number sequence in
for some
satisfying the following conditions:
(i) and
;
(ii);
(iii).
Then the sequence defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ24_HTML.gif)
converges strongly to , where
is a sunny nonexpansive retraction of
onto
.
Proof.
First, we show that is nonexpansive for all
. Indeed, for all
and
, from Lemma 1.2, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ25_HTML.gif)
Therefore, one obtains that is a nonexpansive mapping for all
. For all
, it follows from Lemma 1.3 that
. Put
. Noticing that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ26_HTML.gif)
one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ27_HTML.gif)
from which it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ28_HTML.gif)
Now, an induction yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ29_HTML.gif)
Hence, is bounded, and so is
. On the other hand, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ30_HTML.gif)
Put , that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ31_HTML.gif)
Next, we compute Observing that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ32_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ33_HTML.gif)
Combining (2.7) with (2.10), one obtains
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ34_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ35_HTML.gif)
Hence, from Lemma 1.5, we obtain . From (2.7) and the condition (ii), one arrives at
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ36_HTML.gif)
On the other hand, from (2.1), one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ37_HTML.gif)
which combines with (2.13), and from the conditions (i), (ii), one sees that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ38_HTML.gif)
Next, we show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ39_HTML.gif)
To show (2.16), we choose a sequence of
that converges weakly to
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ40_HTML.gif)
Next, we prove that . Since
for some
, it follows that
is bounded and so there exists a subsequence
of
which converges to
. We may assume, without loss of generality, that
. Since
is nonexpansive, it follows from
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ41_HTML.gif)
It follows from (2.15) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ42_HTML.gif)
From Lemma 1.4, we have . It follows from Lemma 1.3 that
. Now, from (2.17) and Lemma 1.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ43_HTML.gif)
From (2.1), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ44_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ45_HTML.gif)
Applying Lemma 1.6 to (2.22), we can conclude the desired conclusion. This completes the proof.
As an application of Theorem 2.1, we have the following results in the framework of Hilbert spaces.
Corollary 2.2.
Let be a Hilbert space and
a nonempty closed convex subset of
. Let
be a metric projection from
onto
,
an arbitrarily fixed point, and
an
-inverse strongly monotone operator of
into
such that
. Let
and
be two sequences in
and let
be a real number sequence in
for some
satisfying the following conditions:
(i) and
;
(ii);
(iii).
Then the sequence defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F420989/MediaObjects/13660_2008_Article_1814_Equ46_HTML.gif)
converges strongly to .
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Hao, Y. Strong Convergence of an Iterative Method for Inverse Strongly Accretive Operators. J Inequal Appl 2008, 420989 (2008). https://doi.org/10.1155/2008/420989
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DOI: https://doi.org/10.1155/2008/420989