Strong Convergence Theorems for Two Total Quasi-ϕ-Asymptotically Nonexpansive Mappings in Banach Spaces

Abstract

The purpose of this paper is to establish some strong convergence theorems for a common fixed point of two total quasi--asymptotically nonexpansive mappings in Banach space by means of the hybrid method in mathematical programming. The results presented in this paper extend and improve on the corresponding ones announced by Martinez-Yanes and Xu (2006), Plubtieng and Ungchittrakool (2007), Qin et al. (2009), and many others.

1. Introduction

Let be a smooth Banach space and let be the topological dual of . Let the function be defined by where is the normalized duality mapping from to . Let be a nonempty subset of and a nonlinear mapping. Recall that a mapping is called nonexpansive if . A point is said to be a fixed point of [1] if . Let be the set of fixed points of . A point is said to be an asymptotic fixed point of [1] if contains a sequence which converges weakly to such that the strong . The set of asymptotic fixed points of will be denoted by .

Recall that a mapping is called relatively nonexpansive [2] if and ; closed if for any sequence with and , ; quasi--nonexpansive [3] if and ; quasi--asymptotically nonexpansive [3] if and there exists a sequence with such that ; total quasi--asymptotically nonexpansive [4–6] if and there exist nonnegative real sequences with and a strictly increasing continuous function such that ; uniformly L-Lipschitz continuous [4] if there exists a constant such that .

It is well known that total quasi--asymptotically nonexpansive mappings contain relatively nonexpansive mappings, quasi--nonexpansive mappings, and quasi--asymptotically nonexpansive mappings as its special cases; see [4, 6, 7] for more details.

Iterative approximation of fixed points for nonexpansive mappings has been considered by many papers for either the Mann iteration [8] or the Ishkawa iteration [9]; see, for example, [9–12] and the references therein.

In 1967, Halpern [11] considered the following explicit iteration which was referred to as Halpern iteration: where is nonexpansive. He proves the strong convergence of to a point of provided that , where .

Recently, many authors improved and generalized the results of Halpern [11] by means of different methods; see, for example, [2, 4, 10, 12–17] and the references therein. In general, there are two ways as follows.(i)One of the methods is to combine Halpern iteration with Mann iteration; see, for example, C.E. Chidume and C.O. Chidume [10] and Hu [12].(ii)Another method is to use the hybrid projection algorithm method; see, for example, Martinez-Yanes and Xu [15], Plubtieng and Ungchittrakool [16], and Qin et al. [17].

Motivated and inspired by [2, 4, 10, 12–17], the purpose of this paper is to modify Halpern iteration (2) by means of both methods above for two total quasi--asymptotically nonexpansive mappings and then to prove the strong convergence in the framework of Banach spaces. The results presented in this paper extend and improve the corresponding results of Martinez-Yanes and Xu [15], Plubtieng and Ungchittrakool [16], Qin et al. [17], and others.

Throughout the paper, we denote for weak convergence and for strong convergence.

2. Preliminaries

Let be a real Banach space endowed with the norm and let be the topological dual of . For all and , the value of at is denoted by and is called the duality pairing. Then, the normalized duality mapping is defined by It is well known that the operator is well defined and is the identity mapping if and only if is a Hilbert space. But in general, is nonlinear and multiple-valued.

The following basic properties for a Banach space can be found in [18].(i)If is uniformly smooth, then is unformly continuous on each bounded subset of .(ii)If is reflexive and strictly convex, then is norm-weak-continuous.(iii)If is a smooth, strictly convex, and reflexive Banach space, then is single-valued, one-to-one, and onto.(iv)A Banach space is uniformly smooth if and only if is uniformly convex.(v)Each uniformly convex Banach space has the Kadec-Klee property; that is, for any sequence , if and , then .(vi)Every uniformly smooth Banach space is reflexive.

Moreover, we have the Lyapunov functional defined by (1). It is obvious from the definition of the function that we have the following property.

Property 1 (see [19]). If the function is defined by (1), then we have (i); (ii); (iii); (iv).

It follows Albert [20] that the generalized projection is defined by . So we have the following lemmas.

Lemma 1 (see [20]). Let be a smooth, strictly convex, and reflexive Banach space and let be a nonempty closed convex subset of . Then the following conclusions hold: (i); (ii) if and , then ;(iii) for , if and only if .

Remark 2. If is a real Hilbert space, then and (the metric projection of onto a closed convex subset ).

Lemma 3 (see [20]). Let be a uniformly convex and smooth Banach space and let , be sequences of . If (as ) and either or is bounded, then (as ).

Lemma 4 (see [4]). Let be a smooth, strictly convex, and reflexive Banach space with Kadec-Klee property, and let be a nonempty closed convex subset of . Let be a closed and totally quasi--asymptotically nonexpansive mapping with nonnegative real sequences and a strictly increasing continuous function such that and . If , then the fixed point set of is a closed and convex subset of .

Lemma 5 (see [21]). Let be a uniformly convex Banach space and let be a closed ball of . Then there exists a continuous strictly increasing convex function with such that for all and with .

3. Main Results

To prove the main results, we need the following hypotheses for the sequences and :(i) for all and ,(ii), for all , and .

The main results of this paper are stated as follows.

Theorem 6. Let be a real strictly convex and uniformly smooth Banach space with Kadec-Klee property, and let be a nonempty closed convex subset of . Let be two closed, uniformly L-Lipschitz continuous and totally quasi--asymptotically nonexpansive mappings with nonnegative real sequences and a strictly increasing continuous function such that and . Let be a sequence defined by where . Assume that the sequences and satisfy hypotheses (i) and (ii). If is bounded and , then the sequence converges strongly to .

Proof. We divide the proof of Theorem 6 into six steps.(I)We first show that is a closed and convex subset in .
By Lemma 4, it is trivial to show that and are two closed and convex subsets of . Therefore is closed and convex in .(II)Next we prove that is a closed and convex subset in for all .
As a matter of fact, by hypothesis, is closed and convex. Suppose that is closed and convex for some . From the definition of , we may know that and thus is closed and convex. Therefore, by induction principle, are closed and convex for all . This also shows that is well defined.(III)Now we show that .
Indeed, it is obvious that . Suppose that for some . Hence for any , by Lyapunov functional (1) and Lemma 5, we have Moreover, it follows from Property 1(ii) that we have Combining (7) with (8), we have This shows that . Therefore, by induction principle, we have for all .(IV)Next we prove that converges strongly to some point .In fact, since , it follows from Lemma 1(ii) that we have Again since for all , we have It follows from Lemma 1(i) that for each and for each we have Therefore is bounded, and hence is bounded.
On the other hand, since and , we have for all . This implies that is nondecreasing, and so exists. By the construction of , we have and for any positive integer . Therefore, using Lemma 1(i), we have for all . Since exists, we obtain that Thus, by Lemma 3, we have This implies that the sequence is a Cauchy sequence in . Since is a nonempty closed subset of Banach space , this implies that it is complete. Hence there exists an such that By the way, it is easy from (16) to see that (V)Now we prove that .
Since and by the structure of , we have Hence, by means of limits (14), (17) and and using Lemma 3, we get that . But Thus . This implies that
Meanwhile, it follows from (7) and (8) that we have where is a continuous strictly increasing convex function with . Thus, by virtue of (16), (17), (20), (21), , and , we have , and hence . Since is also uniformly norm-to-norm continuous on bounded sets, we get Next, using the convexity of , Property 1(iv), and hypothesis (ii), we obtain that Hence, from Property 1(ii), hypothesis (i), and (23), we have Thus, by Lemma 3, we have and hence Moreover, we observe that Since and is uniformly smooth, it follows from (25) and (26) that we have Next, by the assumption that is uniformly -Lipschitz continuous, we have Combining (15) and (25) with (29), we have Hence, it follows from (27) that we have that is, Furthermore, in view of (27) and the closeness of , it yields that . Similarly, we have . Therefore, this implies that .(VI)Finally we prove that and so .
Let . Since and , we have . This implies that Since , this implies that . Therefore, . This completes the proof.

If or for all in Theorem 6, then we have the following corollary.

Corollary 7. Let be the same as Theorem 6. Let be closed, uniformly L-Lipschitz continuous, and totally quasi--asymptotically nonexpansive mappings with nonnegative real sequences and a strictly increasing continuous function such that and . Let be a sequence defined by where and is the generalized projection from onto . Assume that the sequence satisfies hypothesis (i). If is bounded and , then the sequence converges strongly to .

It is well known that the quasi--asymptotically nonexpansive mappings are a special case of the total quasi--asymptotically nonexpansive mappings. So we obtain the following theorem.

Theorem 8. Let be the same as Theorem 6. Let be two closed, uniformly L-Lipschitz continuous, and quasi--asymptotically nonexpansive mappings with nonnegative real sequences such that . Let be a sequence defined by where . Assume that the sequences and satisfy hypotheses (i) and (ii). If is bounded and , then the sequence converges strongly to .

Proof. The proof is similar to that of Theorem 6 and hence we omit it. This completes the proof.

If or for all in Theorem 8, then we have the following corollary.

Corollary 9. Let be the same as Theorem 6. Let be closed, uniformly L-Lipschitz continuous, and quasi--asymptotically nonexpansive mappings with nonnegative real sequences such that . Let be a sequence defined by where and is the generalized projection from onto . Assume that the sequence satisfies hypothesis . If is bounded, then the sequence converges strongly to .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper.