A Strong Convergence Theorem for a Family of Quasi--Nonexpansive Mappings in a Banach Space

Abstract

The purpose of this paper is to propose a modified hybrid projection algorithm and prove a strong convergence theorem for a family of quasi--nonexpansive mappings. The strong convergence theorem is proven in the more general reflexive, strictly convex, and smooth Banach spaces with the property (K). The results of this paper improve and extend the results of S. Matsushita and W. Takahashi (2005), X. L. Qin and Y. F. Su (2007), and others.

1. Introduction

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 [1] in 1968. For 40 years, HPIA has received rapid developments. For details, the readers are referred to papers [2–7] and the references therein.

In 2005, Matsushita and Takahashi [5] proposed the following hybrid iteration method with generalized projection for relatively nonexpansive mapping in a Banach space :

(1.1)

They proved the following convergence theorem.

Theorem 1 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 (1.1), where is the normalized duality mapping on . If is nonempty, then converges strongly to , where is the generalized projection from onto .

In 2007, Qin and Su [2] proposed the following hybrid iteration method with generalized projection for relatively nonexpansive mapping in a Banach space :

(1.2)

They proved the following convergence theorem.

Theorem 1 QS.

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 such that . Assume that and are sequences in such that and . Suppose that is given by (1.2). If is uniformly continuous, then converges strongly to .

Question 1.

Can both Theorems MT and QS be extended to more general reflexive, strictly convex, and smooth Banach spaces with the property (K)?

Question 2.

Can both Theorems MT and QS be extended to more general class of quasi--nonexpansive mappings?

The purpose of this paper is to give some affirmative answers to the questions mentioned previously, by introducing a modified hybrid projection iteration algorithm and by proving a strong convergence theorem for a family of closed and quasi--nonexpansive mappings by using new analysis techniques in the setting of reflexive, strictly convex, and smooth Banach spaces with the property (K). The results of this paper improve and extend the results of Matsushita and Takahashi [5], Qin and Su [2], and others.

2. Preliminaries

In this paper, we denote by and a Banach space and the dual space of , respectively. Let be a nonempty closed convex subset of . We denote by the normalized duality mapping from to defined by

(2.1)

where denotes the generalized duality pairing between and . It is well known that if is reflexive, strictly convex, and smooth, then is single-valued, demi-continuous and strictly monotone (see, e.g., [8, 9]).

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 [10] 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 reflexive, strictly convex, and smooth Banach space. Let us consider the functional defined as in [4, 5] by

(2.2)

Observe that, in a Hilbert space , (2.2) reduces to

The generalized projection is a map that assigns to an arbitrary point the unique minimum point of the functional ; that is, where is the unique solution to the minimization problem

(2.3)

Remark 2.1.

The existence and uniqueness of the element follow from the reflexivity of , the properties of the functional and strict monotonicity of the mapping (see, e.g., [8–12]). In Hilbert spaces, It is obvious from the definition of function that

(2.4)

Remark 2.2.

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 in turn implies that From the smoothness of , we know that is single valued, and hence we have . Since is strictly convex, is strictly monotone, in particular, is one to one, which implies that one may consult [8, 9] for the details.

Let be a closed convex subset of , and a mapping from into itself. A point in is said to be asymptotic fixed point of [13] if contains a sequence which converges weakly to such that . The set of asymptotic fixed point of will be denoted by . A mapping from into itself is said to be relatively nonexpansive [5, 14–16] if and for all and . The asymptotic behavior of a relatively nonexpansive mapping was studied in [14–16].

is said to be quasi--nonexpansive if and for all and .

Remark 2.3.

The class of quasi--nonexpansive mappings is more general than the class of relatively nonexpansive mappings [5, 14–16] which requires the strong restriction: .

We present two examples which are closed and quasi--nonexpansive.

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--nonexpansive mapping from onto with .

Example 2.5.

Let be a reflexive, strictly convex, and smooth Banach space, and is a maximal monotone mapping such that its zero set is nonempty. Then, is a closed and quasi--nonexpansive mapping from onto and .

Recall that a Banach space has the property (K) if for any sequence and , if weakly and , then . For more information concerning property (K) the reader is referred to [17] and references cited therein.

In order to prove our main result of this paper, we need to the following facts.

Lemma 2.6 (see, e.g., [10–12]).

Let be a convex subset of a real smooth Banach space , and . Then,

(2.5)

if and only if

(2.6)

Lemma 2.7 (see, e.g., [10–12]).

Let be a convex subset of a real reflexive, strictly convex, and smooth Banach space . Then the following inequality holds:

(2.7)

for all and .

Now we are in a proposition to prove the main results of this paper.

3. Main Results

Theorem 3.1.

Let be a reflexive, strictly convex, smooth Banach space such that and have the property (K). Assume that is a nonempty closed convex subset of . Let be an infinitely countable family of closed and quasi--nonexpansive mappings such that . Assume that are real sequences in such that . Define a sequence in by the following algorithm:

(3.1)

Then converges strongly to , where is the generalized projection from onto .

Proof.

We split the proof into six steps.

Step 1.

Show that is well defined for every .

To this end, we prove first that is closed and convex for any . Let be a sequence in with as , we prove that . From the definition of quasi--nonexpansive mappings, one has , which implies that as . Noticing that

(3.2)

By taking limit in (3.2), we have

(3.3)

Hence . It implies that for all . We next show that is convex. To this end, for arbitrary , putting , we prove that . Indeed, by using the definition of , we have

(3.4)

This implies that . Hence is closed and convex for all and consequently is closed and convex. By our assumption that , we have is well defined for every .

Step 2.

Show that is closed and convex for each .

It suffices to show that for any , is closed and convex for every . This can be proved by induction on . In fact, for , is closed and convex. Assume that is closed and convex for some . For , one obtains that

(3.5)

is equivalent to

(3.6)

It is easy to see that is closed and convex. Then, for all , is closed and convex. Consequently, is closed and convex for all .

Step 3.

Show that .

It suffices to show that for any , for every . For any , from the definition of quasi--nonexpansive mappings, we have , for all and . Noting that for any and , we have

(3.7)

which implies that and consequently . So . Hence is well defined for each . Therefore, the iterative algorithm (3.1) is well defined.

Step 4.

Show that , where .

From Steps 2 and 3, we obtain that is a nonempty, closed, and convex subset of . Hence is well defined for every . From the construction of , we know that

(3.8)

Let , where is the unique element that satisfies . Since , by Lemma 2.7, we have

(3.9)

By the reflexivity of , we can assume that weakly. Since , for , we have for . Since is closed and convex, by the Marzur theorem, for any . Hence . Moreover, by using the weakly lower semicontinuity of the norm on and (3.9), we obtain

(3.10)

which implies that . By using Lemma 2.6, we have

(3.11)

and hence , since is strictly monotone.

Further, by the definition of , we have

(3.12)

which shows that . By the property (K) of , we have , where .

Step 5.

Show that , for any .

Since for all and , we have

(3.13)

Since and consequently

(3.14)

Note that . Hence and consequently . This implies that is bounded. Since is reflexive, is also reflexive. So we can assume that

(3.15)

weakly. On the other hand, in view of the reflexivity of , one has , which means that for , there exists , such that . It follows that

(3.16)

where we used the weakly lower semicontinuity of the norm on . From (3.14), we have and consequently , which implies that . Hence

(3.17)

weakly. Since and has the property (K), we have

(3.18)

Since , noting that is demi-continuous, we have

(3.19)

weakly. Noticing that

(3.20)

which implies that . By using the property (K) of , we have

(3.21)

From (3.1), (3.18), (3.21), and , we have

(3.22)

Since is demi-continuous, we have

(3.23)

weakly in . Moreover,

(3.24)

which implies that . By the property (K) of , we have

(3.25)

From and the closeness property of , we have

(3.26)

which implies that .

Step 6.

Show that .

It follows from Steps 3, 4, and 5 that

(3.27)

which implies that . Hence, . Then converges strongly to . This completes the proof.

From Theorem 3.1, we can obtain the following corollary.

Corollary 3.2.

Let be a reflexive, strictly convex and smooth Banach space such that both and have the property (K). Assume that is a nonempty closed convex subset of . Let be a closed and quasi--nonexpansive mapping. Assume that is a sequence in such that . Define a sequence in by the following algorithm:

(3.28)

Then converges strongly to , where is the generalized projection from onto .

Remark 3.3.

Theorem 3.1 and its corollary improve and extend Theorems MT and QS at several aspects.

(i)From uniformly convex and uniformly smooth Banach spaces extend to reflexive, strictly convex and smooth Banach spaces with the property (K). In Theorem 3.1 and its corollary the hypotheses on are weaker than the usual assumptions of uniform convexity and uniform smoothness. For example, any strictly convex, reflexive and smooth Musielak-Orlicz space satisfies our assumptions [17] while, in general, these spaces need not to be uniformly convex or uniformly smooth.

(ii)From relatively nonexpansive mappings extend to closed and quasi--non-expansive mappings.

(iii)The continuity assumption on mapping in Theorem QS is removed.

(iv)Relax the restriction on from to .

Remark 3.4.

Corollary 3.2 presents some affirmative answers to Questions 1 and 2.

4. Applications

In this section, we present some applications of the main results in Section 3.

Theorem 4.1.

Let be a reflexive, strict, and smooth Banach space that both and have the property (K), and let be a nonempty closed convex subset of . Let be a family of proper, lower semicontinuous, and convex functionals. Assume that the common fixed point set is nonempty, where for and . Let be a sequence generated by the following manner:

(4.1)

where satisfies the restriction: and . Then defined by (4.1) converges strongly to a minimizer of the family .

Proof.

By a result of Rockafellar [18], we see that is a maximal monotone mapping for every . It follows from Example 2.5 that is a closed and quasi--nonexpansive mapping for every . Notice that

(4.2)

is equivalent to

(4.3)

and the last inclusion relation is equivalent to

(4.4)

Now the desired conclusion follows from Theorem 3.1. This completes the proof.