Weak and Strong Convergence Theorems for Asymptotically Strict Pseudocontractive Mappings in the Intermediate Sense

Abstract

We study the convergence of Ishikawa iteration process for the class of asymptotically -strict pseudocontractive mappings in the intermediate sense which is not necessarily Lipschitzian. Weak convergence theorem is established. We also obtain a strong convergence theorem by using hybrid projection for this iteration process. Our results improve and extend the corresponding results announced by many others.

1. Introduction and Preliminaries

Throughout this paper, we always assume that is a real Hilbert space with inner product and norm and denote weak and strong convergence, respectively. denotes the weak -limit set of , that is, . Let be a nonempty closed convex subset of . It is well known that for every point , there exists a unique nearest point in , denoted by , such that

(1.1)

for all . is called the metric projection of onto . is a nonexpansive mapping of onto and satisfies

(1.2)

Let be a mapping. In this paper, we denote the fixed point set of by . Recall that is said to be uniformly -Lipschitzian if there exists a constant , such that

(1.3)

is said to be nonexpansive if

(1.4)

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

(1.5)

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] as a generalization of the class of nonexpansive mappings. is said to be asymptotically nonexpansive in the intermediate sense if it is continuous and the following inequality holds:

(1.6)

Observe that if we define

(1.7)

then as . It follows that (1.6) is reduced to

(1.8)

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 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 contains properly the class of asymptotically nonexpansive mappings.

Recall that is said to be a -strict pseudocontraction if there exists a constant , such that

(1.9)

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

(1.10)

The class of asymptotically -strict pseudocontractions was introduced by Qihou [4] in 1996 (see also [5]). Kim and Xu [6] studied weak and strong convergence theorems for this class of mappings. It is important to note that every asymptotically -strict pseudocontractive mapping with sequence is a uniformly -Lipschitzian mapping with .

Recently, Sahu et al. [7] 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 with sequence if there exist a constant and a sequence with as , such that

(1.11)

Throughout this paper, we assume that

(1.12)

It follows that as and (1.11) is reduced to the relation

(1.13)

They obtained a weak convergence theorem of modified Mann iterative processes for the class of mappings which is not necessarily Lipschitzian. Moreover, a strong convergence theorem was also established in a real Hilbert space by hybrid projection methods; see [7] for more details.

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

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

Lemma 1.1 (see [8, 9]).

Let , , and be three sequences of nonnegative numbers satisfying the recursive inequality

(1.14)

If , and , then exists.

Lemma 1.2 (see [10]).

Let be a bounded sequence in a reflexive Banach space . If , then .

Lemma 1.3 (see [11]).

Let be a nonempty closed convex subset of a real Hilbert space . Given and , then if and only if , for all

Lemma 1.4 (see [11]).

For a real Hilbert space , the following identities hold:

(i),  for all ,

(ii) for all , for all ;

(iii)(Opial condition) If is a sequence in weakly convergent to , then

(1.15)

Lemma 1.5 (see [7]).

Let be a nonempty subset of a Hilbert space and an asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence . Then

(1.16)

Lemma 1.6.

Let be a nonempty subset of a Hilbert space and an asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence . Let . If , then

(1.17)

Proof.

If , for , we obtain from Lemma 1.5 that

(1.18)

Lemma 1.7 (see [7]).

Let be a nonempty subset of a Hilbert space and a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence . Let be a sequence in such that and as , then as .

Lemma 1.8 (see [7, Proposition ]).

Let be a nonempty closed convex subset of a Hilbert space and a continuous asymptotically -strict pseudocontractive mapping in the intermediate sense. Then is demiclosed at zero in the sense that if is a sequence in such that and , then .

Lemma 1.9 (see [7]).

Let be a nonempty closed convex subset of a Hilbert space and a continuous asymptotically -strict pseudocontractive mapping in the intermediate sense. Then is closed and convex.

2. Main Results

Theorem 2.1.

Let be a nonempty closed convex subset of a Hilbert space and a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence such that . Let be a sequence in generated by the following Ishikawa iterative process:

(2.1)

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

(i) and ,

(ii) for some and .

Then the sequence given by (2.1) converges weakly to an element of .

Proof.

Let . From (1.13) and Lemma 1.4, we see that

(2.2)

Without loss of generality, we may assume that for all . Since

(2.3)

it follows from Lemma 1.6 that

(2.4)

By (2.2) and (2.4), we obtain that

(2.5)

where . It follows from (2.5) and that

(2.6)

From the condition (ii) and , we see that there exists such that

(2.7)

By (2.6), we have

(2.8)

In view of Lemma 1.1 and the condition (i), we obtain that exists. For any , it is easy to see from (2.6) and (2.7) that

(2.9)

which implies that

(2.10)

Note that

(2.11)

From (2.10), we have

(2.12)

Since is uniformly continuous, we obtain from (2.10), (2.12) and Lemma 1.7 that

(2.13)

By the boundedness of , there exist a subsequence of such that . Observe that is uniformly continuous and as , for any we have as . From Lemma 1.8, we see that .

To complete the proof, it suffices to show that consists of exactly one point, namely, . Suppose there exists another subsequence of such that converges weakly to some and . As in the case of , we can also see that . It follows that and exist. Since satisfies the Opial condition, we have

(2.14)

which is a contradiction. We see and hence is a singleton. Thus, converges weakly to by Lemma 1.2.

Corollary 2.2.

Let be a nonempty closed convex subset of a Hilbert space and a uniformly continuous asymptotically -strict pseudocontractive mapping with sequence such that . Let be a sequence in generated by the following Ishikawa iterative process:

(2.15)

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

(i),

(ii) for some and .

Then the sequence given by (2.15) converges weakly to an element of .

Next, we modify Ishikawa iterative process to get a strong convergence theorem.

Theorem 2.3.

Let be a nonempty closed convex subset of a Hilbert space and a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence such that and bounded. Let and are sequences in . Let be a sequence in generated by the modified Ishikawa iterative process:

(2.16)

where , , and for each . Assume that the control sequences and are chosen such that for some and . Then the sequence given by (2.16) converges strongly to an element of .

Proof.

We break the proof into six steps.

Step 1 ( is closed and convex for each ).

It is obvious that is closed and convex and is closed for each . Note that the defining inequality in is equivalent to the inequality

(2.17)

it is easy to see that is convex for each . Hence, is closed and convex for each .

Step 2 ( for each ).

Let . Following (2.6), (2.7) and the algorithm (2.16), we have

(2.18)

where , , and for each . Hence for each .

Next, we show that for each . We prove this by induction. For , we have . Assume that for some . Since is the projection of onto , we have

(2.19)

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

(2.20)

This implies that . That is, . By the principle of mathematical induction, we get and hence for all . This means that the iteration algorithm (2.16) is well defined.

Step 3 ( exists and is bounded).

In view of (2.16), we see that and . It follows that

(2.21)

for each . We, therefore, obtain that the sequence is nondecreasing. Noticing that and , we have

(2.22)

This shows that the sequence is bounded. Therefore, the limit of exists and is bounded.

Step 4 ().

Observe that and which imply

(2.23)

Using Lemma 1.4, we obtain

(2.24)

Hence, we obtain that as .

Step 5 ( as ).

In view of , we have

(2.25)

On the other hand, we see that

(2.26)

Combing (2.25) and (2.26) and noting , we obtain that

(2.27)

From the assumption and (2.7), we see that there exists such that

(2.28)

For any , it follows from the definition of and (2.27) that

(2.29)

Noting that as and Step 4, we obtain that

(2.30)

It follows from Step 4, (2.30) and Lemma 1.7 that as .

Step 6 ( as , where ).

Since is reflexive and is bounded, we get that is nonempty. First, we show that is a singleton. Assume that is subsequence of such that . Observe that is uniformly continuous and as , for any we have as . From Lemma 1.8, we see that .

Since , we obtain that

(2.31)

for each . Observe that as . By the weak lower semicontinuity of norm, we have

(2.32)

This implies that

(2.33)

(2.34)

Hence by the uniqueness of the nearest point projection of onto . Since is an arbitrary weakly convergent subsequence, it follows that and hence . It is easy to see as (2.34) that . Since has the Kadec-Klee property, we obtain that , that is, as . This completes the proof.