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# Convergence Theorems on Asymptotically Pseudocontractive Mappings in the Intermediate Sense

*Fixed Point Theory and Applications*
**volume 2010**, Article number: 186874 (2010)

## Abstract

A new nonlinear mapping is introduced. The convergence of Ishikawa iterative processes for the class of asymptotically pseudocontractive mappings in the intermediate sense is studied. Weak convergence theorems are established. A strong convergence theorem is also established without any compact assumption by considering the so-called hybrid projection methods.

## 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 . The symbols and are denoted by strong convergence and weak convergence, respectively. denotes the weak -limit set of . Let be a nonempty closed and convex subset of and a mapping. In this paper, we denote the fixed point set of by .

Recall that is said to be*nonexpansive* if

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

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] as a generalization of the class of nonexpansive mappings. They proved that if is a nonempty closed convex and 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:

Observe that if we define

then as . It follows that (1.3) is reduced to

The class of mappings which are asymptotically nonexpansive in the intermediate sense was introduced by Bruck et al. [2]. It is known [3] that if is a nonempty close convex 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 contains properly the class of asymptotically nonexpansive mappings.

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

The class of strict pseudocontractions was introduced by Browder and Petryshyn [4] in a real Hilbert space. Marino and Xu [5] proved that the fixed point set of strict pseudocontractions is closed convex, and they also obtained a weak convergence theorem for strictly pseudocontractive mappings by Mann iterative process; see [5] for more details.

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

The class of asymptotically strict pseudocontractions was introduced by Qihou [6] in 1996 (see also [7]). Kim and Xu [8] proved that the fixed point set of asymptotically strict pseudocontractions is closed convex. They also obtained that the class of asymptotically strict pseudocontractions is demiclosed at the origin; see [8, 9] for more details.

Recently, Sahu et al. [10] introduced a class of new mappings: asymptotically strict pseudocontractive mappings in the intermediate sense. Recall that is said to be an *asymptotically strict pseudocontraction in the intermediate sense* if

where and such that as Put

It follows that as Then, (1.8) is reduced to the following:

They obtained a weak convergence theorem of modified Mann iterative processes for the class of mappings. Moreover, a strong convergence theorem was also established in a real Hilbert space by considering the so-called hybrid projection methods; see [10] for more details.

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

The class of asymptotically pseudocontractive mapping was introduced by Schu [11] (see also [12]). In [13], Rhoades gave an example to show that the class of asymptotically pseudocontractive mappings contains properly the class of asymptotically nonexpansive mappings; see [13] for more details. In 1991, Schu [11] established the following classical results.

Theorem JS.

Let be a Hilbert space: closed bounded and covnex; ; completely continuous, uniformly -Lipschitzian and asymptotically pseudocontractive with sequence ; for all ; ; , are sequences in ; for all , some and some ; ; for all , define

then converges strongly to some fixed point of .

Recently, Zhou [14] showed that every uniformly Lipschitz and asymptotically pseudocontractive mapping which is also uniformly asymptotically regular has a fixed point. Moreover, the fixed point set is closed and convex.

In this paper, we introduce and consider the following mapping.

Definition 1.1.

A mapping is said to be a*asymptotically pseudocontractive mapping in the intermediate sense* if

where is a sequence in such that as Put

It follows that as Then, (1.13) is reduced to the following:

In real Hilbert spaces, we see that (1.15) is equivalent to

We remark that if for each , then the class of asymptotically pseudocontractive mappings in the intermediate sense is reduced to the class of asymptotically pseudocontractive mappings.

In this paper, we consider the problem of convergence of Ishikawa iterative processes for the class of mappings which are asymptotically pseudocontractive in the intermediate sense.

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

Lemma 1.2 (see [15]).

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

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

Lemma 1.3.

In a real Hilbert space, the following inequality holds:

From now on, we always use to denotes .

Lemma 1.4.

Let be a nonempty close convex subset of a real Hilbert space and a uniformly -Lipschitz and asymptotically pseudocontractive mapping in the intermediate sense with sequences and as defined in (1.15). Then is a closed convex subset of .

Proof.

To show that is convex, let and . Put , where . Next, we show that Choose and define for each . From the assumption that is uniformly -Lipschitz, we see that

For any , it follows that

This implies that

Letting and in (1.21), respectively, we see that

It follows that

Letting in (1.23), we obtain that . Since is uniformly -Lipschitz, we see that This completes the proof of the convexity of . From the continuity of , we can also obtain the closedness of . The proof is completed.

Lemma 1.5.

Let be a nonempty close convex subset of a real Hilbert space and a uniformly -Lipschitz and asymptotically pseudocontractive mapping in the intermediate sense such that is nonempty. Then is demiclosed at zero.

Proof.

Let be a sequence in such that and as Next, we show that and . Since is closed and convex, we see that It is sufficient to show that Choose and define for arbitrary but fixed From the assumption that is uniformly -Lipschitz, we see that

It follows from the assumption that

Note that

Since and (1.25), we arrive at

On the other hand, we have

Note that

Substituting (1.27) and (1.28) into (1.29), we arrive at

This implies that

Letting in (1.31), we see that . Since is uniformly -Lipschitz, we can obtain that This completes the proof.

## 2. Main Results

Theorem 2.1.

Let be a nonempty closed convex bounded subset of a real Hilbert space and a uniformly -Lipschitz and asymptotically pseudocontractive mapping in the intermediate sense with sequences and defined as in (1.15). Assume that is nonempty. Let be a sequence generated in the following manner:

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

(a), , where for each

(b) for some and some

then the sequence generated by (*) converges weakly to fixed point of .

Proof.

Fix . From (1.16) and Lemma 1.3, we see that

From (2.1) and (2.2), we arrive at

It follows that

From condition (b), we see that there exists such that

Note that

In view of Lemma 1.2, we see that exists. For any , we see that

from which it follows that

Note that

Thanks to (2.8), we obtain that

Note that

From (2.8) and (2.10), we obtain that

Since is bounded, we see that there exists a subsequence such that . From Lemma 1.5, we see that .

Next we prove that converges weakly to . Suppose the contrary. Then we see that there exists some subsequence such that converges weakly to and . From Lemma 1.5, we can also prove that . Put Since satisfies Opial property, we see that

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

Next, we modify Ishikawa iterative processes to obtain a strong convergence theorem without any compact assumption.

Theorem 2.2.

Let be a nonempty closed convex bounded subset of a real Hilbert space , the metric projection from onto and a uniformly -Lipschitz and asymptotically pseudocontractive mapping in the intermediate sense with sequences and as defined in (1.15). Let for each Assume that is nonempty. Let and be sequences in . Let be a sequence generated in the following manner:

where for each . Assume that the control sequences and are chosen such that for some and some Then the sequence generated in (**) converges strongly to a fixed point of .

Proof.

The proof is divided into seven steps.

Step 1.

Show that is closed and convex for each

It is obvious that is closed and convex and is closed for each . We, therefore, only need to prove that is convex for each . Note that

is equivalent to

It is easy to see that is convex for each . Hence, we obtain that is closed and convex for each This completes Step 1.

Step 2.

Show that for each .

Let . From Lemma 1.3 and the algorithm (**), we see that

Substituting (2.17) and (2.18) into (2.16), we arrive at

where for each . This implies that for each . That is, for each

Next, we show that for each We prove this by inductions. It is obvious that . Suppose that for some . Since is the projection of onto , we see that

By the induction assumption, we know that . In particular, for any , we have

which implies that . That is, . This proves that for each . Hence, for each . This completes Step 2.

Step 3.

Show that exists.

In view of the algorithm (**), we see that and which give that

This shows that the sequence is nondecreasing. Note that is bounded. It follows that exists. This completes Step 3.

Step 4.

Show that as

Note that and . This implies that

from which it follows that

Hence, we have as This completes Step 4.

Step 5.

Show that as

In view of , we see that

On the other hand, we have

Combining (2.25) and (2.26) and noting , we get that

From the assumption, we see that there exists such that

For any , it follows from (2.27) that

Note that as Thanks to Step 4, we obtain that

This completes Step 5.

Step 6.

Show that as

Note that

From Step 5, we can conclude the desired conclusion. This completes Step 6.

Step 7.

Show that , where as

Note that Lemma 1.5 ensures that . From and , we see that

From Lemma of Yanes and Xu [16], we can obtain Step 7. This completes the proof.

Remark 2.3.

The results of Theorem 2.2 are more general which includes the corresponding results of Kim and Xu [17], Marino and Xu [5], Qin et al. [18], Sahu et al. [10], Zhou [14, 19] as special cases.

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## Acknowledgment

This work was supported by the Kyungnam University Research Fund 2009.

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Qin, X., Cho, S. & Kim, J. Convergence Theorems on Asymptotically Pseudocontractive Mappings in the Intermediate Sense.
*Fixed Point Theory Appl* **2010, **186874 (2010). https://doi.org/10.1155/2010/186874

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### Keywords

- Hilbert Space
- Iterative Process
- Convergence Theorem
- Nonexpansive Mapping
- Opial Property