1. Introduction
Throughout this paper, we denote the strong convergence and weak convergence of the sequence
by
and
, respectively. We denote by N and R the sets of positive integers and real numbers, respectively. Let
be a nonempty closed subset of a real Banach space
. A mapping
is said to be nonexpansive if
, for all
. Let
and
denote the family of nonempty subsets and nonempty bounded closed subsets of
, respectively.
Let
be a real Banach space with dual
. We denote by
the normalized duality mapping from
to
which is defined by
, where ![](https://www.scirp.org/html/2-7401511\3fd5f51b-5a62-4c1b-8e24-f9c209111905.jpg)
and
denotes the generalized duality pairing. The Hausdorff metric on
is defined by
, for
, where
. The multi-valued mapping
is called nonexpansive if
for all
. An element
is called a fixed point of
if
. The set of fixed points of
is represented by
. In the sequel, denote
. A Banach space
is said to be strictly convex if
for all ![](https://www.scirp.org/html/2-7401511\2ff56219-b681-4933-8e4c-3e2c5bd0046a.jpg)
and
. A Banach space is said to be uniformly convex if
for any two sequences
and
. The norm of Banach space
is said to be Gâteaux differentiable if for each
, the limit
(1.1)
exists. In this case,
is said to be smooth. The norm of Banach space
is said to be Fréchet differentiable, if for each
, the limit (1.1) is attained uniformly for
and the norm is uniformly Fréchet differentiable if the limit (1.1) is attained uniformly for
. In this case,
is said to be uniformly smooth.
The following basic properties for Banach space X and for the normalized duality mapping
can be found in Cioranescu [1].
(1)
is uniformly convex if and only if
is uniformly smooth.
(2) If
is smooth, then
is single-valued and norm-to-weak* continuous.
(3) If
is reflexive, then
is onto.
(4) If
is strictly convex, then
for all
.
(5) If
has a Fréchet differentiable norm, then
is norm-to-norm continuous.
(6) If
is uniformly smooth, then
is uniformly norm-to-norm continuous on each bounded subset of
.
(7) Each uniformly convex Banach space
has the Kadec-Klee property, i.e., for any sequence
, if
and
, then
.
In 1953, Mann [2] introduced the following iterative sequence
,
![](https://www.scirp.org/html/2-7401511\0fa60da4-036d-4b70-9960-3163c95d436f.jpg)
where the initial guess
is arbitrary and
is a real sequence in
. It is known that under appropriate settings the sequence
converges weakly to a fixed point of
. However, even in a Hilbert space, Mann iteration may fail to converge strongly [3]. Some attempts to construct iteration method guaranteeing the strong convergence have been made. For example, Halpern [4] proposed the following so-called Halpern iteration,
![](https://www.scirp.org/html/2-7401511\ab03d63a-93c7-4936-a2ed-0a151eeb88c3.jpg)
where
are arbitrary given and
is a real sequence in
. Another approach was proposed by Nakajo and Takahashi [5]. They generated a sequence as follows,
(1.2)
where
is a real sequence in
and
denotes the metric projection from a Hilbert space H onto a closed convex subset K of H. It should be noted here that the iteration above works only in Hilbert space setting. To extend this iteration to a Banach space, the concept of relatively nonexpansive mappings and quasi-
-nonexpansive mappings are introduced by Aoyama et al. [6], Chang et al. [7,8], Chidume et al. [9], Matsushita et al. [10-12], Qin et al. [13], Song et al. [14], Wang et al. [15] and others.
Inspired by the work of Matsushita and Takahashi, in this paper, we introduce modifying Halpern-Mann iterations sequence for finding a fixed point of a countable family of uniformly totally quasi-
-asymptotically nonexpansive multi-valued mappings in reflexive Banach spaces
and some strong convergence theorems are proved. The results presented in the paper improve and extend the corresponding results in [7].
2. Preliminaries
In the sequel, we assume that
is a smooth, strictly convex, and reflexive Banach space and
is a nonempty closed convex subset of
. In the sequel, we always use
to denote the Lyapunov bifunction defined by
. (2.1)
It is obvious from the definition of the function
that
(2.2)
(2.3)
and
(2.4)
for all
and
.
Following Alber [16], the generalized projection
is defined by
![](https://www.scirp.org/html/2-7401511\918c60f5-5185-4d7a-96e1-656f458db624.jpg)
Many problems in nonlinear analysis can be reformulated as a problem of finding a fixed point of a nonexpansive mapping.
Remark 2.1 (see [17]) Let
be the generalized projection from a smooth, reflexive and strictly convex Banach space
onto a nonempty closed convex subset
of
, then
is a closed and quasi-
-nonexpansive from
onto
.
Lemma 2.1 (see [16]) Let
be a smooth, strictly convex and reflexive Banach space and
be a nonempty closed convex subset of
. Then the following conclusions hold(a)
if and only if
.
(b)
.
(c) If
and
, then
if and only if
.
Lemma 2.2 (see [7]) Let
be a real uniformly smooth and strictly convex Banach space with KadecKlee property, and
be a nonempty closed convex subset of
. Let
and
be two sequences in
such that
and
where ![](https://www.scirp.org/html/2-7401511\5b404ef8-d8a3-4d78-ae7a-c32d278ac5b5.jpg)
is the function defined by (1.2), then
.
Definition 2.1 A point
is said to be an asymptotic fixed point of multi-valued mapping
, if there exists a sequence
such that
and
. Denote the set of all asymptotic fixed points of
by
.
Definition 2.2
(1) A multi-valued mapping
is said to be relatively nonexpansive, if
,
, and ![](https://www.scirp.org/html/2-7401511\49ef30ac-ae2c-4150-b675-b6b698dde750.jpg)
.
(2) A multi-valued mapping
is said to be closed, if for any sequence
with
and
, then
.
Remark 2.2 If
is a real Hilbert space, then
and
is the metric projection
of
onto
.
Next, We present an example of relatively nonexpansive multi-valued mapping.
Example 2.1 (see [18]) Let
be a smooth, strictly convex and reflexive Banach space,
be a nonempty closed and convex subset of
and
be a bifunction satisfying the conditions:
(A1)
;
(A2)
;
(A3) for each
,
;
(A4) for each given
, the function
is convex and lower semicontinuous.
The “so-called” equilibrium problem for f is to find a
such that
. The set of its solutions is denoted by
.
Let
and define a multi-valued mapping
as follows,
(2.5)
then (1)
is single-valued, and so
; (2)
is a relatively nonexpansive mapping, therefore, it is a closed quasi-
-nonexpansive mapping; (3)
.
Definition 2.3
(1) A multi-valued mapping
is said to be quasi-
-nonexpansive, if
, and
.
(2) A multi-valued mapping
is said to be quasi-
-asymptotically nonexpansive, if
and there exists a real sequence
such that
(2.6)
(3) A multi-valued mapping
is said to be totally quasi-
-asymptotically nonexpansive, if
and there exist nonnegative real sequences
, with
(as
) and a strictly increasing continuous function
with
such that
(2.7)
Remark 2.3 From the definitions, it is obvious that a relatively nonexpansive multi-valued mapping is a quasi-
-nonexpansive multi-valued mapping, and a quasi-
- nonexpansive multi-valued mapping is a quasi-
-asymptotically nonexpansive multi-valued mapping, and a quasi-
-asymptotically nonexpansive multi-valued mapping is a total quasi-
-asymptotically nonexpansive multi-valued mapping, but the converse is not true.
Lemma 2.3 Let
and
be as in Lemma 2.2.
be a closed and totally quasi-
-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences
and a strictly increasing continuous function
with
,if
(as
) and
, then
is a closed and convex subset of
.
Proof. Let
be a sequence in
, such that
. Since
is totally quasi-
-asymptotically nonexpansive multi-valued mapping, we have
![](https://www.scirp.org/html/2-7401511\b02b6fd6-9d08-4907-b3a9-84960ba80e37.jpg)
for all
and for all
. Therefore,
![](https://www.scirp.org/html/2-7401511\53a0c73a-9f9f-46a3-9d42-0e0e33c5f3f9.jpg)
By Lemma 2.1(a), we obtain
. Hence,
. So, we have
. This implies
is closed.
Let
and
, and put
. Next we prove that
. Indeed, in view of the definition of
, letting
, we have
(2.8)
Since
(2.9)
Substituting (2.8) into (2.9) and simplifying it, we have
![](https://www.scirp.org/html/2-7401511\e5ecd7d3-cce6-4116-ac48-0f311fe5a048.jpg)
By Lemma 2.2, we have
. This implies that
. Since
is closed, we have
, i.e.,
. This completes the proof of Lemma 2.3. □
Definition 2.4 A mapping
is said to be uniformly
-Lipschitz continuous, if there exists a constant
such that
, where
.
Definition 2.5
(1) A countable family of mappings
is said to be uniformly quasi-
-nonexpansive, if
, and
.
(2) A countable family of mappings
is said to be uniformly quasi-
-asymptotically nonexpansive, if
, and there exists a real sequence
such that,
(2.10)
(3) A countable family of mappings
is said to be totally uniformly quasi-
-asymptotically nonexpansive multi-valued, if
and there exists nonnegative real sequences
with
(as
) and a strictly increasing and continuous function
with
such that
(2.11)
Remark 2.4 From the definitions, it is obvious that a countable family of uniformly quasi-
-nonexpansive multi-valued mappings is a countable family of uniformly quasi-
-asymptotically nonexpansive multi-valued mappings, and a countable family of uniformly quasi-
-asymptotically nonexpansive multi-valued mappings is a countable family of totally uniformly quasi-
-asymptotically multi-valued mappings, but the converse is not true.
3. Main Results
Theorem 3.1 Let
be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, D be a nonempty closed convex subset of X,
be a closed and uniformly
- Lipschitz continuous and a countable family of uniformly totally quasi-
-asymptotically nonexpansive multi-valued mappings with nonnegative real sequences![](https://www.scirp.org/html/2-7401511\1234ad36-114b-4337-9d5b-656974726f6e.jpg)
,
(as
) and a strictly increasing continuous function
with
satisfying condition (2.11). Let
be a sequence in
such that
. If
is the sequence generated by
(3.1)
where
,
is the fixed point set of
, and
is the generalized projection of
onto
.
If
and
is bounded and
, then
.
Proof. (I) First, we prove that F and
are closed and convex subsets in
. In fact, it follows from Lemma 2.3 that
is a closed and convex subsets in D. therefore F is closed and convex subsets in D. Again by the assumption,
is closed and convex. Suppose that
is closed and convex for some
. In view of the definition of
, we have
![](https://www.scirp.org/html/2-7401511\a29f2243-d14b-4b88-a375-14a9a8e61dc6.jpg)
This shows that
is closed and convex. The conclusions are proved.
(II) Next, we prove that
, for all
.
In fact, it is obvious that
. Suppose that
. Hence for any
, by (2.4), we have
(3.2)
Therefore we have
(3.3)
This shows that
and so
. The conclusions are proved.
(III) Now we prove that
converges strongly to some point
.
In fact, since
, from Lemma 2.1(c), we have
Again since
, we have
,
. It follows from Lemma 2.1(b) that for each
and for each
,
(3.4)
Therefore,
is bounded, and so is
. Since
and
, we have
.
This implies that
is nondecreasing. Hence
exists. Since X is reflexive, there exists a subsequence
such that
(some point in
). Since
is closed and convex and
. This implies that
is weakly closed and
for each
. In view of
, we have
Since the norm
is weakly lower semi-continuous, we have
![](https://www.scirp.org/html/2-7401511\828b9a2a-3862-4bd0-85c5-d0e5f41ffcd7.jpg)
and so
![](https://www.scirp.org/html/2-7401511\db041812-0154-4d54-9198-c02d08e3b998.jpg)
This shows that
and we have
. Since
, by virtue of KadecKlee property of
, we obtain that
Since
is convergent, this together with
shows that
. If there exists some subsequence
such that
, then from Lemma 2.1, we have
![](https://www.scirp.org/html/2-7401511\1eb04a12-31e7-44ce-b5ae-31e5c2054b48.jpg)
i.e.,
and hence
(3.5)
By the way, from (3.4), it is easy to see that
(3.6)
(IV) Now we prove that
.
In fact, since
, from (3.1), (3.4) and (3.5), we have
(3.7)
Since
, it follows from (3.6) and Lemma 2.2 that
(3.8)
Since
is bounded and
is a countable family of uniformly totally quasi-
-asymptotically nonexpansive multi-valued mappings,
is bounded. In view of
, from (3.1), we have
(3.9)
Since
, this implies
. From Remark 2.1, it yields that
(3.10)
Again since
(3.11)
this together with (3.9) and the Kadec-Klee-property of
shows that
(3.12)
On the other hand, by the assumptions that
is
-Lipschitz continuous for each
, we have
(3.13)
From (3.12) and
, we have that
. In view of the closeness of
, it yields that
, which implies that
.
(V) Finally we prove that
and so
.
Let
. Since
, we have
. This implies that
(3.14)
which yields that
. Therefore,
. The proof of Theorem 3.1 is completed.
By Remark 2.4, the following corollaries are obtained. □
Corollary 3.1 Let X and
be as in Theorem 3.1, and a countable family of mappings
be a closed and uniformly
-Lipschitz continuous a relatively nonexpansive multi-valued mappings. Let
in
with
. Let
be the sequence generated by
(3.15)
where
is the set of fixed points of
, and
is the generalized projection of
onto
, If
and F is bounded, then
converges strongly to
.
Corollary 3.2 Let
and
be as in Theorem 3.1, and a countable family of mappings
be a closed and uniformly
-Lipschitz continuous quasi-phi-asymptotically nonexpansive multivalued mappings with nonnegative real sequences
and
satisfying condition (2.1). Let
be a sequence in
and satisfy
. If
is the sequence generated by
(3.16)
where
is the set of fixed points of
, and
is the generalized projection of
onto
, and ![](https://www.scirp.org/html/2-7401511\661e3a72-8d40-4240-b11d-8065047cc930.jpg)
If
and F is bounded, then ![](https://www.scirp.org/html/2-7401511\0df11bc5-ee23-413c-b339-f8fb02d54f2a.jpg)
converges strongly to
.
4. Application
We utilize Corollary 3.2 to study a modified Halpern iterative algorithm for a system of equilibrium problems.
Theorem 4.1 Let
,
and
be the same as in Theorem 3.1. Let
be a bifunction satisfying conditions (A1)-(A4) as given in Example 2.6.
Let
be a mapping defined by (2.5), i.e.,
![](https://www.scirp.org/html/2-7401511\1548b495-47f9-4a08-92e7-3c53ee3c0a46.jpg)
Let
be the sequence generated by
(4.1)
If
, then
converges strongly to
which is a common solution of the system of equilibrium problems for
.
Proof. In Example 2.6, we have pointed out that
,
and
is a closed quasi-
-nonexpansive mapping. Hence (4.1) can be rewritten as follows:
![](https://www.scirp.org/html/2-7401511\5311b727-ee31-4373-a743-0f093ac7de69.jpg)
Therefore the conclusion of Theorem 4.6 can be obtained from Corollary 3.2.