Convergence Theorems on Asymptotically Pseudocontractive Mappings in the Intermediate Sense

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 benonexpansive if

(1.1)

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

(1.2)

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 beasymptotically nonexpansive in the intermediate sense if it is continuous and the following inequality holds:

(1.3)

Observe that if we define

(1.4)

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

(1.5)

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 bestrictly pseudocontractive if there exists a constant such that

(1.6)

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 aasymptotically strict pseudocontraction if there exist a constant and a sequence with as such that

(1.7)

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

(1.8)

where and such that as Put

(1.9)

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

(1.10)

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 beasymptotically pseudocontractive if there exists a sequence with as such that

(1.11)

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

(1.12)

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 aasymptotically pseudocontractive mapping in the intermediate sense if

(1.13)

where is a sequence in such that as Put

(1.14)

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

(1.15)

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

(1.16)

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:

(1.17)

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

Lemma 1.3.

In a real Hilbert space, the following inequality holds:

(1.18)

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

(1.19)

For any , it follows that

(1.20)

This implies that

(1.21)

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

(1.22)

It follows that

(1.23)

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

(1.24)

It follows from the assumption that

(1.25)

Note that

(1.26)

Since and (1.25), we arrive at

(1.27)

On the other hand, we have

(1.28)

Note that

(1.29)

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

(1.30)

This implies that

(1.31)

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:

(x2a)

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

(2.1)

(2.2)

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

(2.3)

It follows that

(2.4)

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

(2.5)

Note that

(2.6)

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

(2.7)

from which it follows that

(2.8)

Note that

(2.9)

Thanks to (2.8), we obtain that

(2.10)

Note that

(2.11)

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

(2.12)

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

(2.13)

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:

(x2ax2a)

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

(2.14)

is equivalent to

(2.15)

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

(2.16)

(2.17)

(2.18)

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

(2.19)

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

(2.20)

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

(2.21)

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

(2.22)

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

(2.23)

from which it follows that

(2.24)

Hence, we have as This completes Step 4.

Step 5.

Show that as

In view of , we see that

(2.25)

On the other hand, we have

(2.26)

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

(2.27)

From the assumption, we see that there exists such that

(2.28)

For any , it follows from (2.27) that

(2.29)

Note that as Thanks to Step 4, we obtain that

(2.30)

This completes Step 5.

Step 6.

Show that as

Note that

(2.31)

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

(2.32)

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.