1. Introduction and Preliminaries

Let be a Banach space with the dual . We denote by the normalized duality mapping from to defined by

(1.1)

where denotes the generalized duality pairing. A Banach space is said to be strictly convex if for all with and It is said to be uniformly convex if for any two sequences in such that and . Let be the unit sphere of . Then the Banach space is said to be smooth provided that

(1.2)

exists for each It is also said to be uniformly smooth if the limit (1.2) is attained uniformly for . It is well known that if is uniformly smooth, then is uniformly norm-to-norm continuous on each bounded subset of . It is also well known that is uniformly smooth if and only if is uniformly convex.

Recall that a Banach space has the Kadec-Klee property if for any sequences and with and then as ; for more details on Kadec-Klee property, the readers is referred to [1, 2] and the references therein. It is well known that if is a uniformly convex Banach space, then enjoys the Kadec-Klee property.

Let be a nonempty closed and convex subset of a Banach space and  :  →  a mapping. The mapping is said to be closed if for any sequence such that and , then . A point is a fixed point of provided . In this paper, we use to denote the fixed point set of and use and to denote the strong convergence and weak convergence, respectively.

Recall that the mapping is said to be nonexpansive if

(1.3)

It is well known that if is a nonempty bounded closed and convex subset of a uniformly convex Banach space , then every nonexpansive self-mapping on has a fixed point. Further, the fixed point set of is closed and convex.

As we all know 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 [3] 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 by

(1.4)

Observe that, in a Hilbert space , (1.4) is reduced 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

(1.5)

Existence and uniqueness of the operator follow from the properties of the functional and strict monotonicity of the mapping (see, e.g., [1–4]). In Hilbert spaces, It is obvious from the definition of function that

(1.6)

Remark 1.1.

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 (1.6), we have . This implies that From the definition of we have . Therefore, we have see [1, 2] for more details.

Let be a nonempty closed convex subset of and a mapping from into itself. A point in is said to be an asymptotic fixed point of [5] if contains a sequence which converges weakly to such that . The set of asymptotic fixed points of will be denoted by . A mapping from into itself is said to be relatively nonexpansive [3, 6, 7] if and for all and . The mapping is said to be hemirelatively nonexpansive [8–12] if and for all and . The asymptotic behavior of a relatively nonexpansive mappings was studied in [3, 6, 7].

Remark 1.2.

The class of hemirelatively nonexpansive mappings is more general than the class of relatively nonexpansive mappings which requires the restriction: . From Su et al. [11], we see that every hemirelatively nonexpansive mapping is relatively nonexpansive, but the inverse is not true. Hemirelatively nonexpansive mapping is also said to be quasi--nonexpansive; see [13–17].

Recently, fixed point iterations of relatively nonexpansive mappings and hemirelatively nonexpansive mappings have been considered by many authors; see, for example [14–25] and the references therein. In 2005, Matsushita and Takahashi [8] considered fixed point problems of a single relatively nonexpansive mapping in a Banach space. To be more precise, they proved the following 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; let be a sequence of real numbers such that and . Suppose that is given by

(1.7)

where is the duality mapping on . If is nonempty, then converges strongly to , where is the generalized projection from onto

In 2007, Plubtieng and Ungchittrakool [9] further improved Theorem MT by considering a pair of relatively nonexpansive mappings. To be more precise, they proved the following theorem.

Theorem PU

Let be a uniformly convex and uniformly smooth Banach space, and let be a nonempty closed convex subset of Let and be two relatively nonexpansive mappings from into itself with being nonempty. Let a sequence be defined by

(1.8)

with the following restrictions:

for each and

for each , and .

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

Very recently, Su et al. [11] improved Theorem PU partially by considering a pair of hemirelatively nonexpansive mappings. To be more precise, they obtained the following results.

Theorem SWX

Let be a uniformly convex and uniformly smooth Banach space, and let be a nonempty closed convex subset of Let and be two closed hemirelatively nonexpansive mappings from into itself with being nonempty. Let a sequence be defined by

(1.9)

with the following restrictions:

(1);

(2);

(3) for some .

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

In this paper, motivated by Theorems MT, PU, and SWX, we consider the problem of finding a common fixed point of a pair of hemirelatively nonexpansive mappings by shrinking projection methods which were introduced by Takahashi et al. [26] in Hilbert spaces. Strong convergence theorems of common fixed points are established in a Banach space. The results presented in this paper mainly improve the corresponding results announced in Matsushita and Takahashi [8], Nakajo and Takahahsi [27], and Su et al. [11].

In order to prove our main results, we need the following lemmas.

Lemma 1.3 (see [3]).

Let be a nonempty closed convex subset of a smooth Banach space and . Then, if and only if

(1.10)

Lemma 1.4 (see [3]).

Let be a reflexive, strictly convex and smooth Banach space, a nonempty closed convex subset of , and Then

(1.11)

The following lemma can be deduced from Matsushita and Takahashi [8].

Lemma 1.5.

Let be a strictly convex and smooth Banach space, a nonempty closed convex subset of and a hemirelatively nonexpansive mapping. Then is a closed convex subset of .

Lemma 1.6 (see [28]).

Let be a uniformly convex Banach space and a closed ball of Then there exists a continuous strictly increasing convex function with such that

(1.12)

for all and with

2. Main Results

Theorem 2.1.

Let be a uniformly smooth and strictly convex Banach space which enjoys the Kadec-Klee property and a nonempty closed and convex subset of Let and be two closed and hemirelatively nonexpansive mappings such that is nonempty. Let be a sequence generated in the following manner:

(2.1)

where , , , and are real sequences in satisfying the following restrictions:

and .

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

Proof.

First, we show that is closed and convex for each It is obvious that is closed and convex. Suppose that is closed and convex for some . For , we see that is equivalent to

(2.2)

It is easy to see that is closed and convex. Then, for each , is closed and convex. Now, we are in a position to show that for each Indeed, is obvious. Suppose that for some . Then, for all , we have

(2.3)

It follows that

(2.4)

which shows that . This implies that for each On the other hand, we obtain from Lemma 1.4 that

(2.5)

for each and for each This shows that the sequence is bounded. From (1.6), we see that the sequence is also bounded. Since the space is reflexive, we may, without loss of generality, assume that . Note that is closed and convex for each . It is easy to see that for each Note that

(2.6)

It follows that

(2.7)

This implies that

(2.8)

Hence, we have as In view of the Kadec-Klee property of we obtain that as

Next, we show that By the construction of we have that and It follows that

(2.9)

Letting in (2.9), we obtain that . In view of , we arrive at It follows that

(2.10)

From (1.6), we can obtain that

(2.11)

It follows that

(2.12)

This implies that is bounded. Note that is reflexive and is also reflexive. We may assume that In view of the reflexivity of , we see that This shows that there exists an such that It follows that

(2.13)

Taking , the both sides of equality above yield that

(2.14)

That is, which in turn implies that It follows that From (2.12) and since enjoys the Kadec-Klee property, we obtain that

(2.15)

Note that is demicontinuous. It follows that From (2.11) and since enjoys the Kadec-Klee property, we obtain that

(2.16)

Note that

(2.17)

It follows that

(2.18)

Since is uniformly norm-to-norm continuous on any bounded sets, we have

(2.19)

On the other hand, we see from the definition of that

(2.20)

In view of the assumption on and (2.19), we see that

(2.21)

On the other hand, since is demicontinuous, we have In view of

(2.22)

we arrive at as By virtue of the Kadec-Klee property of , we obtain that as Note that

(2.23)

In view of (2.21), we arrive at Since is demicontinuous, we have Note that

(2.24)

It follows that as . Since enjoys the Kadec-Klee property, we obtain that Note that

(2.25)

It follows that

(2.26)

Let . Fixing we have from Lemma 1.6 that

(2.27)

It follows that

(2.28)

On the other hand, we have

(2.29)

It follows from (2.21) and (2.26) that

(2.30)

In view of (2.28) and the assumption , we see that

(2.31)

It follows from the property of that

(2.32)

Note that

(2.33)

On the other hand, we have

(2.34)

From (2.32) and (2.33), we arrive at

(2.35)

Note that is demicontinuous. It follows that On the other hand, we have

(2.36)

In view of (2.35), we obtain that as . Since enjoys the Kadec-Klee property, we obtain that

(2.37)

It follows from the closedness of that By repeating (2.27)–(2.37), we can obtain that . This shows that

Finally, we show that From we have

(2.38)

Taking the limit as in (2.38), we obtain that

(2.39)

and hence by Lemma 1.3. This completes the proof.

Remark 2.2.

Theorem 2.1 improves Theorem SWX in the following aspects:

from the point of view on computation, we remove the set in Theorem SWX;

from the point of view on the framework of spaces, we extend Theorem SWX from a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex Banach space which enjoys the Kadec-Klee property. Note that every uniformly convex Banach space enjoys the Kadec-Klee property.

If for each , then Theorem 2.1 is reduced to the following.

Corollary 2.3.

Let be a uniformly smooth and strictly convex Banach space which enjoys the Kadec-Klee property and a nonempty closed and convex subset of Let  :  and  :  be two closed and hemirelatively nonexpansive mappings such that is nonempty. Let be a sequence generated in the following manner:

(2.40)

where , , and are real sequences in satisfying the following restrictions:

 

  and .

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

If , then Corollary 2.3 is reduced to the following.

Corollary 2.4.

Let be a uniformly smooth and strictly convex Banach space which enjoys the Kadec-Klee property and a nonempty closed and convex subset of . Let be a closed and hemirelatively nonexpansive mapping with a nonempty fixed point set. Let be a sequence generated in the following manner:

(2.41)

where is a real sequence in satisfying . Then converges strongly to , where is the generalized projection from onto .