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Hybrid Approximate Proximal Point Algorithms for Variational Inequalities in Banach Spaces
Journal of Inequalities and Applications volume 2009, Article number: 275208 (2009)
Abstract
Let be a nonempty closed convex subset of a Banach space
with the dual
, let
be a continuous mapping, and let
be a relatively nonexpansive mapping. In this paper, by employing the notion of generalized projection operator we study the variational inequality (for short, VI(
): find
such that
for all
, where
is a given element. By combining the approximate proximal point scheme both with the modified Ishikawa iteration and with the modified Halpern iteration for relatively nonexpansive mappings, respectively, we propose two modified versions of the approximate proximal point scheme L. C. Ceng and J. C. Yao (2008) for finding approximate solutions of the VI(
). Moreover, it is proven that these iterative algorithms converge strongly to the same solution of the VI(
), which is also a fixed point of
.
1. Introduction
Let be a real Banach space with the dual
. As usually,
denotes the duality pairing between
and
. In particular, if
is a real Hilbert space, then
denotes its inner product. Let
be a nonempty closed convex subset of
and
be a mapping. Given
, let us consider the following variational inequality problem (for short,
): find an element
such that

Suppose that the (1.1) has a (unique) solution
. For any
, define the following successive sequence in a uniformly convex and uniformly smooth Banach space
:

where is the normalized duality mapping on
and
is the generalized projection operator which assigns to an arbitrary point
the minimum point of the functional
with respect to
. In [1, Theorem?8.2], Alber proved that the above sequence converges strongly to the solution
, that is,
as
, if the following conditions hold:
(i) is uniformly monotone, that is,

where is a continuous strictly increasing function for all
with
;
(ii) has
arbitrary growth, that is,

where is a continuous nondecreasing function for all
with
. Note that solution methods for the problem (1.1) has also been studied in [2–10].
Let be a nonempty closed convex subset of a real Banach space
with the dual
. Assume that
is a continuous mapping on
and
is a relatively nonexpansive mapping such that
. The purpose of this paper is to introduce and study two new iterative algorithms (1.5) and (1.6) in a uniformly convex and uniformly smooth Banach space
.
Algorithm 1.1.

where are sequences in
,
is a bounded sequence in
, and
is assumed to exist for each
,
Algorithm 1.2.

where is a sequence in
,
is a bounded sequence in
, and
is assumed to exist for each
,
.
In this paper, strong convergence results on these two iterative algorithms are established; that is, under appropriate conditions, both the sequence generated by algorithm (1.5) and the sequence
generated by algorithm (1.6) converge strongly to the same point
, which is a solution of the
. Our results represent the improvement, generalization, and development of the previously known results in the literature including Li [8], Zeng and Yao [9], Ceng and Yao [10], and Qin and Su [11].
Notation 1.

stands for weak convergence and for strong convergence.
2. Preliminaries
Let be a Banach space with the dual
. We denote by
the normalized duality mapping from
to
defined by

where denotes the generalized duality pairing. It is well known that if
is smooth, then
is single-valued and if
is uniformly smooth, then
is uniformly norm-to-norm continuous on bounded subsets of
. We will still denote the single-valued duality mapping by
.
Recall 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 hence, it is not available in more general Banach spaces. In this connection, Alber [1] 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 smooth Banach space. Consider the functional defined as in [1, 12] by

It is clear that in a Hilbert space , (2.2) reduces to
.
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

The existence and uniqueness of the operator follow from the properties of the functional
and strict monotonicity of the mapping
(see, e.g., [13]). In a Hilbert space,
.
From [1], in uniformly convex and uniformly smooth Banach spaces, we have

Let be a closed convex subset of
, and let
be a mapping from
into itself. A point
in
is called an asymptotically fixed point of
[14] if
contains a sequence
which converges weakly to
such that
. The set of asymptotical fixed points of
will be denoted by
. A mapping
from
into itself is called relatively nonexpansive [15–17] if
and
for all
and
.
A Banach space is called strictly convex if
for all
with
and
. It is said to be uniformly convex if
for any two sequences
such that
and
. Let
be a unit sphere of
. Then the Banach space
is called smooth if

exists for each . It is also said to be uniformly smooth if the limit is attained uniformly for
. Recall also that if
is uniformly smooth, then
is uniformly norm-to-norm continuous on bounded subsets of
. A Banach space is said to have the Kadec-Klee property if for any sequence
, whenever
and
, we have
. It is known that if
is uniformly convex, then
has the Kadec-Klee property; see [18, 19] for more details.
Remark 2.1 ([11]).
If is a reflexive, strictly convex, and smooth Banach space, then for any
,
if and only if
. It is sufficient to show that if
, then
. From (2.4), we have
. This implies that
. From the definition of
, we have
. Therefore, we have
; see [18, 19] for more details.
We need the following lemmas and proposition for the proof of our main results.
Lemma 2.2 (Kamimura and Takahashi [20]).
Let be a uniformly convex and smooth Banach space and let
and
be two sequences of
. If
and either
or
is bounded, then
.
Lemma 2.3 (Alber [1]).
Let be a nonempty closed convex subset of a smooth Banach space
and
. Then,
if and only if

Lemma 2.4 (Alber [1]).
Let be a reflexive, strictly convex, and smooth Banach space, let
be a nonempty closed convex subset of
and let
. Then

Lemma 2.5 (Matsushita and Takahashi [21]).
Let be a strictly convex and smooth Banach space, let
be a closed convex subset of
, and let
be a relatively nonexpansive mapping from
into itself. Then
is closed and convex.
Lemma 2.6 (Chang [7]).
Let be a smooth Banach space. Then the following inequality holds

3. Main Results
Now we are in a position to prove the main theorems of this paper.
Theorem 3.1.
Let be a uniformly convex and uniformly smooth Banach space, let
be a nonempty closed convex subset of
, let
be a continuous mapping and, let
be a relatively nonexpansive mapping such that
. Assume that
are sequences in
and
is a sequence in
such that
and
. Define a sequence
in
by the following algorithm:

where is assumed to exist for each
,
If
is uniformly continuous and
, then
converges strongly to
, which is a solution of the
(1.1).
Proof.
First of all, let us show that and
are closed and convex for each
. Indeed, from the definition of
and
, it is obvious that
is closed and
is closed and convex for each
. We claim that
is convex. For any
and any
, put
. It is sufficient to show that
. Note that the inequality

is equivalent to the one

Observe that there hold the following:

and . Thus, we have

This implies that . So,
is convex. Next let us show that
for all
. Indeed, we have for all

So for all
. Next let us show that

We prove this by induction. For , we have
. Assume that
. Since
is the projection of
onto
, by Lemma 2.3, we have

As by the induction assumption, the last inequality holds, in particular, for all
. This together with the definition of
implies that
. Hence (3.7) holds for all
. This implies that
is well defined.
On the other hand, it follows from the definition of that
. Since
, we have

Thus is nondecreasing. Also from
and Lemma 2.4, it follows that

for each for each
. Consequently,
is bounded. Moreover, according to the inequality

we conclude that is bounded and so is
. Indeed, since
is relatively nonexpansive, we derive for each

and hence is bounded. Again from
we know that
is also bounded.
On account of the boundedness and nondecreasing property of we deduce that
exists. From Lemma 2.4, we derive

for all . This implies that
. So it follows from Lemma 2.2 that
. Since
, from the definition of
, we also have

Observe that

On the other hand, since from (3.1) we have for each

utilizing Lemma 2.3 we obtain . Thus, in terms of Lemmas 2.4 and 2.6 we conclude that

Since and
, we obtain
. Thus by Lemma 2.2 we have
. From
it follows that
is bounded. At the same time, observe that

and hence

From and the boundedness of
and
, we derive
. Note that
is uniformly continuous. Hence
by virtue of
. Since

it is known that is bounded. Consequently, from (3.15),
and
it follows that

Further, it follows from (3.14), and
that

Utilizing Lemma 2.2, we obtain

Since is uniformly norm-to-norm continuous on bounded subsets of
, we have

Furthermore, we have

It follows from and
that

Noticing that

we have

From (3.24) and , we obtain

Since is also uniformly norm-to-norm continuous on bounded subsets of
, we obtain

Observe that

Since is uniformly continuous, it follows from (3.26), (3.30) and
that
.
Finally, let us show that converges strongly to
, which is a solution of the
(1.1). Indeed, assume that
is a subsequence of
such that
. Then
. Next let us show that
and convergence is strong. Put
. From
and
, we have
. Now from weakly lower semicontinuity of the norm, we derive

It follows from the definition of that
and hence

So we have . Utilizing the Kadec-Klee property of
, we conclude that
converges strongly to
. Since
is an arbitrarily weakly convergent subsequence of
, we know that
converges strongly to
. Now observe that from (3.1) we have for each

Since is uniformly norm-to-norm continuous on bounded subsets of
, from
we infer that
. Noticing that
and
is a continuous mapping, we obtain that
and
. Therefore, from
it follows that

that is,

Letting we conclude from (3.34) that

and hence

This shows that is a solution of the
(1.1). This completes the proof.
Corollary 3.2 ([11, Theorem?2.1]).
Let be a uniformly convex and uniformly smooth Banach space, let
be a nonempty closed convex subset of
, and let
be a relatively nonexpansive mapping such that
. Assume that
and
are sequences in
such that
and
. Define a sequence
in
by the following algorithm:

where is the single-valued duality mapping on
. If
is uniformly continuous, then
converges strongly to
.
Proof.
In Theorem 3.1, we know from (3.1) and Lemma 2.3 that

is equivalent to . Now, put
for all
. Then we have

for all . Thus algorithm (3.1) reduces to algorithm (3.39). By Theorem 3.1 we obtain the desired result.
Theorem 3.3.
Let be a uniformly convex and uniformly smooth Banach space, let
be a nonempty closed convex subset of
, let
be a continuous mapping, and let
be a relatively nonexpansive mapping such that
. Assume that
satisfies
and
satisfies
. Define a sequence
in
by the following algorithm:

where is assumed to exist for each
,
If
is uniformly continuous and
, then
converges strongly to
, which is a solution of the
(1.1).
Proof.
We only derive the difference. First, let us show that is closed and convex for each
. From the definition of
, it is obvious that
is closed for each
. We prove that
is convex. Similarly to the proof of Theorem 3.1, since

is equivalent to

we know that is convex. Next, let us show that
for each
. Indeed, we have for each

So for all
and
. Similarly to the proof of Theorem 3.1, we also obtain
for all
. Consequently,
for all
. Therefore, the sequence
generated by (3.42) is well defined. As in the proof of Theorem 3.1, we can obtain
. Since
, from the definition of
, we also have

As in the proof of Theorem 3.1, we can deduce from and
that
and hence
by Lemma 2.2. Further, it follows from
and the boundedness of
and
that

Since , from the definition of
, we also have

It follows from (3.47) and that

Utilizing Lemma 2.2, we have

Since is uniformly norm-to-norm continuous on bounded subsets of
we have

Note that

Therefore, from we have

Since is also uniformly norm-to-norm continuous on bounded subsets of
, we obtain

It follows that

Since is uniformly continuous, it follows from (3.50) and (3.54) that
.
Finally, let us show that converges strongly to
, which is a solution of the
(1.1). Indeed, assume that
is a subsequence of
such that
. Then
. Next let us show that
and convergence is strong. Put
. From
and
, we have
. Now from weakly lower semicontinuity of the norm, we derive

It follows from the definition of that
and hence
. So, we have
. Utilizing the Kadec-Klee property of
, we conclude that
converges strongly to
. Since
is an arbitrarily weakly convergent subsequence of
, we know that
converges strongly to
. Now observe that from (3.1), we have for each

Since is uniformly norm-to-norm continuous on bounded subsets of
, from
we infer that
. Noticing that
and
is a continuous mapping, we obtain that
and
. Observe that

It follows from that

Letting we conclude from (3.34) that

This shows that is a solution of the
(1.1). This completes the proof.
Corollary 3.4 ([11, Theorem?2.2]).
Let be a uniformly convex and uniformly smooth Banach space, let
be a nonempty closed convex subset of
, and let
be a relatively nonexpansive mapping. Assume that
is a sequence in
such that
. Define a sequence
in
by the following algorithm:

where is the single-valued duality mapping on
. If
is nonempty, then
converges strongly to
.
Proof.
In Theorem 3.3, we know from (3.42) and Lemma 2.3 that

is equivalent to . Now, put
for all
. Then we have

for all . Thus algorithm (3.42) reduces to algorithm (3.61). Thus under the lack of the uniform continuity of
it follows from (3.55) that
. By the careful analysis of the proof of Theorem 3.3, we can obtain the desired result.
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Aknowledgments
The first author was partially supported by the National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), and Science and Technology Commission of Shanghai Municipality grant (075105118), Leading Academic Discipline Project of Shanghai Normal University (DZL707), Shanghai Leading Academic Discipline Project (S30405) and Innovation Program of Shanghai Municipal Education Commission (09ZZ133). The third author was partially supported by the Grant NSF 97-2115-M-110-001.
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Ceng, L.C., Guu, S.M. & Yao, J.C. Hybrid Approximate Proximal Point Algorithms for Variational Inequalities in Banach Spaces. J Inequal Appl 2009, 275208 (2009). https://doi.org/10.1155/2009/275208
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DOI: https://doi.org/10.1155/2009/275208