A Weak Convergence Theorem for Total Asymptotically Pseudocontractive Mappings in Hilbert Spaces

Abstract

The modified Ishikawa iterative process is investigated for the class of total asymptotically pseudocontractive mappings. A weak convergence theorem of fixed points is established in the framework of Hilbert spaces.

1. Introduction and Preliminaries

Throughout this paper, we always assume that is a real Hilbert space, whose inner product and norm are denoted by and . and are denoted by strong convergence and weak convergence, respectively. Let be a nonempty closed convex subset of and a mapping. In this paper, we denote the fixed point set of by .

is said to be a contraction if there exists a constant such that

(1.1)

Banach contraction principle guarantees that every contractive mapping defined on complete metric spaces has a unique fixed point.

is said to be a weak contraction if

(1.2)

where is a continuous and nondecreasing function such that is positive on , , and . We remark that the class of weak contractions was introduced by Alber and Guerre-Delabriere [1]. In 2001, Rhoades [2] showed that every weak contraction defined on complete metric spaces has a unique fixed point.

is said to be nonexpansive if

(1.3)

is said to be asymptotically nonexpansive if there exists a sequence with as such that

(1.4)

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [3] as a generalization of the class of nonexpansive mappings. They proved that if is a nonempty closed convex bounded subset of a real uniformly convex Banach space and is an asymptotically nonexpansive mapping on , then has a fixed point.

is said to be asymptotically nonexpansive in the intermediate sense if it is continuous and the following inequality holds:

(1.5)

Observe that if we define

(1.6)

then as . It follows that (1.5) is reduced to

(1.7)

The class of mappings which are asymptotically nonexpansive in the intermediate sense was introduced by Bruck et al. [4] (see also [5]). It is known [6] that if is a nonempty closed convex bounded subset of a uniformly convex Banach space and is asymptotically nonexpansive in the intermediate sense, then has a fixed point. It is worth mentioning that the class of mappings which are asymptotically nonexpansive in the intermediate sense may not be Lipschitz continuous; see [5, 7].

is said to be total asymptotically nonexpansive if

(1.8)

where is a continuous and strictly increasing function with and and are nonnegative real sequences such that and as . The class of mapping was introduced by Alber et al. [8]. From the definition, we see that the class of total asymptotically nonexpansive mappings includes the class of asymptotically nonexpansive mappings and the class of asymptotically nonexpansive mappings in the intermediate sense as special cases; see [9, 10] for more details.

is said to be strictly pseudocontractive if there exists a constant such that

(1.9)

The class of strict pseudocontractions was introduced by Browder and Petryshyn [11] in a real Hilbert space. In 2007, Marino and Xu [12] obtained a weak convergence theorem for the class of strictly pseudocontractive mappings; see [12] for more details.

is said to be an asymptotically strict pseudocontraction if there exist a constant and a sequence with as such that

(1.10)

The class of asymptotically strict pseudocontractions was introduced by Qihou [13] in 1996. Kim and Xu [14] proved that the class of asymptotically strict pseudocontractions is demiclosed at the origin and also obtained a weak convergence theorem for the class of mappings; see [14] for more details.

is said to be an asymptotically strict pseudocontraction in the intermediate sense if there exist a constant and a sequence with as such that

(1.11)

Put

(1.12)

It follows that as . Then, (1.11) is reduced to the following:

(1.13)

The class of mappings was introduced by Sahu et al. [15]. They proved that the class of asymptotically strict pseudocontractions in the intermediate sense is demiclosed at the origin and also obtained a weak convergence theorem for the class of mappings; see [15] for more details.

is said to be asymptotically pseudocontractive if there exists a sequence with as such that

(1.14)

It is not hard to see that (1.14) is equivalent to

(1.15)

The class of asymptotically pseudocontractive mapping was introduced by Schu [16] (see also [17]). In [18], Rhoades gave an example to showed that the class of asymptotically pseudocontractive mappings contains properly the class of asymptotically nonexpansive mappings; see [18] for more details. Zhou [19] showed that every uniformly Lipschitz and asymptotically pseudocontractive mapping which is also uniformly asymptotically regular has a fixed point.

is said to be an asymptotically pseudocontractive mapping in the intermediate sense if there exists a sequence with as such

(1.16)

Put

(1.17)

It follows that as . Then, (1.16) is reduced to the following:

(1.18)

It is easy to see that (1.18) is equivalent to

(1.19)

The class of asymptotically pseudocontractive mappings in the intermediate sense was introduced by Qin et al. [20]. Weak convergence theorems of fixed points were established based on iterative methods; see [20] for more details.

In this paper, we introduce the following mapping.

Definition 1.1.

Recall that is said to be total asymptotically pseudocontractive if there exist sequences and with and as such that

(1.20)

where is a continuous and strictly increasing function with .

It is easy to see that (1.20) is equivalent to the following:

(1.21)

Remark 1.2.

If , then (1.20) is reduced to

(1.22)

Remark 1.3.

Put

(1.23)

If , then the class of total asymptotically pseudocontractive mappings is reduced to the class of asymptotically pseudocontractive mappings in the intermediate sense.

Recall that the modified Ishikawa iterative process which was introduced by Schu [16] generates a sequence in the following manner:

(1.24)

where is a mapping, is an initial value, and and are real sequences in .

If for each , then the modified Ishikawa iterative process (1.24) is reduced to the following modified Mann iterative process:

(1.25)

The purpose of this paper is to consider total asymptotically pseudocontractive mappings based on the modified Ishikawa iterative process. Weak convergence theorems are established in real Hilbert spaces.

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

Lemma 1.4.

In a real Hilbert space, the following inequality holds:

(1.26)

Lemma 1.5 (see [21]).

Let , , and be three nonnegative sequences satisfying the following condition:

(1.27)

where is some nonnegative integer. If and , then exists.

2. Main Results

Now, we are ready to give our main results.

Theorem 2.1.

Let be a nonempty closed convex subset of a real Hilbert space and a uniformly -Lipschitz and total asymptotically pseudocontractive mapping as defined in (1.20). Assume that is nonempty and there exist positive constants and such that for all . Let be a sequence generated in the following manner:

(2.1)

where and are sequences in . Assume that the following restrictions are satisfied:

(a) and ,

(b) for some and some .

Then, the sequence generated in (2.1) converges weakly to fixed point of .

Proof.

Fix . Since is an increasing function, it results that if and if . In either case, we can obtain that

(2.2)

In view of Lemma 1.4, we see from (2.2) that

(2.3)

where for each . Notice from Lemma 1.4 that

(2.4)

Since is an increasing function, it results that if and if . In either case, we can obtain that

(2.5)

This implies from (2.3) and (2.4) that

(2.6)

where for each . It follows that

(2.7)

From the restriction (b), we see that there exists such that

(2.8)

It follows from (2.7) that

(2.9)

Notice that and . In view of Lemma 1.5, we see that exists. For any , we see that

(2.10)

from which it follows that

(2.11)

Note that

(2.12)

In view of (2.11), we obtain that

(2.13)

Note that

(2.14)

Combining (2.11) and (2.13) yields that

(2.15)

Since is bounded, we see that there exists a subsequence such that . Next, we claim that . Choose and define for arbitrary but fixed . From the assumption that is uniformly -Lipschitz, we see that

(2.16)

It follows from (2.15) that

(2.17)

Since is an increasing function, it results that if and if . In either case, we can obtain that

(2.18)

This in turn implies that

(2.19)

Since , we see from (2.17) that

(2.20)

On the other hand, we have

(2.21)

Note that

(2.22)

Substituting (2.20) and (2.21) into (2.22), we arrive at

(2.23)

This implies that

(2.24)

Letting in (2.24), we see that . Since is uniformly -Lipschitz, we can obtain that .

Next, we prove that converges weakly to . Suppose the contrary. Then, we see that there exists some subsequence such that converges weakly to , where . It is not hard to see that that . Put . Since enjoys Opial property, we see that

(2.25)

This derives a contradiction. It follows that . This completes the proof.

Remark 2.2.

Demiclosedness principle of the class of total asymptotically pseudocontractive mappings can be deduced from Theorem 2.1.

Remark 2.3.

Since the class of total asymptotically pseudocontractive mappings includes the class of strict pseudocontractions, the class of asymptotically strict pseudocontractions, the class of pseudocontractive mappings, the class of asymptotically pseudocontractive mappings and the class of asymptotically pseudocontractive mappings in the intermediate sense as special cases, Theorem 2.1 improves the corresponding results in Marino and Xu [12], Kim and Xu [14], Sahu et al. [15], Schu [16], Zhou [19], and Qin et al. [20].

Remark 2.4.

It is of interest to improve the main results of this paper to a Banach space.