Strong Convergence Theorems of the Ishikawa Process with Errors for Strictly Pseudocontractive Mapping of Browder-Petryshyn Type in Banach Spaces

Abstract

We prove several strong convergence theorems for the Ishikawa iterative sequence with errors to a fixed point of strictly pseudocontractive mapping of Browder-Petryshyn type in Banach spaces and give sufficient and necessary conditions for the convergence of the scheme to a fixed point of the mapping. The results presented in this work give an affirmative answer to the open question raised by Zeng et al. 2006, and generalize the corresponding result of Zeng et al. 2006, Osilike and Udomene 2001, and others.

1. Introduction and Preliminaries

Let be a real Banach space and its dual. denotes the generalized duality pairing between and . Let be the normalized duality mapping defined by the following:

(1.1)

It is well known that if is smooth, then is single-valued. In this paper, we denote a single-valued selection of the normalized duality mapping by . denotes the identity operator. is the fixed point set of , that is, .

Definition 1.1 (see [1]).

A mapping is said to be strictly pseudocontractive if there exists and , such that

(1.2)

Remark 1.2.

• (i) Without loss of generality, we may assume . Inequality (1.2) can be written in the form

  

  (1.3)

• (ii) If is a Hilbert space, then inequality (1.2) is equivalent to the following inequality:

  

  (1.4)

• (iii) is a Lipschitz continuous mapping, that is, , s.t, . In fact, by (1.3), we have

  

  (1.5)

so that,

(1.6)

where .

Definition 1.3.

A mapping is said to be

(i)compact, if for any bounded sequence in , there exists a strongly convergent subsequence of , or

(ii)demicompact, if for any bounded sequence in , whenever is strongly convergent, there exists a strongly convergent subsequence of .

Let us recall some important iterative processes.

Definition 1.4 (Ishikawa iterative process with errors in the sense of Liu [2]).

Let be a nonempty convex subset of with . For any , the sequence is defined as follows:

(1.7)

where and are appropriate sequences in , and , , are appropriate sequences in .

If for all , then (1.7) reduces to Mann iterative process with errors as follows:

(1.8)

Definition 1.5 (Ishikawa iterative process with errors in the sense of Xu [3]).

Let be a nonempty convex subset of . For any , the sequence is defined as follows:

(1.9)

where and are bounded sequences in , and , , , are real sequences in satisfying , , for all .

If for all , then (1.9) reduces to Mann iterative process with errors as follows:

(1.10)

Remark 1.6.

• (i) If in (1.7) or in (1.9), then (1.7) and (1.9) reduce to Ishikawa iterative process [4],

  

  (1.11)

• (ii) If in (1.8) or in (1.10), then (1.8) and (1.10) reduce to Mann iterative process [5],

  

  (1.12)

In 1974, Rhoades [6] proved strong convergence theorem by the Mann iterative process to a fixed point of strictly pseudocontractive mapping defined on a nonempty compact convex subset of a Hilbert space. In 2001, Osilike and Udomene [7] proved weak and strong convergence theorems for strictly pseudocontractive mapping in a real -uniformly smooth Banach space which is also uniform convex.

In 2006, Zeng et al. [8] established the sufficient and necessary conditions on the strong convergence to a fixed point of strictly pseudocontractive mapping in a real -uniformly smooth Banach space. They got the following main results.

Theorem 1.7.

Let and be a real -uniformly smooth Banach space, let be a nonempty closed convex subset of with , and let be a strictly pseudocontractive mapping with . Let be a bounded sequence in . Let and be real sequences in satisfying the following conditions:

(i);

(ii), and , where and is a constant depending on .

From an arbitrary , let be defined by the following:

(1.13)

Then converges strongly to a fixed point of if and only if is bounded and , where .

In the end of Zeng et al. [8], they raised an open question.

Open Question 1.

Can the Ishikawa iterative process with errors (1.7) be extended to Theorem 1.7?

At the same year, Zeng et al. [9] proved the following strong convergence theorem for strictly pseudocontractive mappings.

Theorem 1.8.

Let and be a real -uniformly smooth Banach space. Let be a nonempty closed convex subset of , and let be compact or demicompact, and strictly pseudocontractive with . Let be a bounded sequence in . Let , , and be real sequences in satisfying the following conditions:

(i), for all ;

(ii), and ;

(iii) and , where .

From an arbitrary , let be defined by the following:

(1.14)

If is the bounded sequence, then converges strongly to a fixed point of .

They raised another open question.

Open Question 2.

Can the Ishikawa iterative process with errors (1.9) be extended to Theorem 1.8?

We have answered the Open Question 1 in [10]. The purpose of this paper is to answer the Open Question 2, and we prove some strong convergence theorems for strictly pseudocontractive mapping in Banach spaces, which improve Theorem 1.8 in the following:

(i)-uniformly smooth Banach spaces can be replaced by general Banach spaces.

(ii)Remove the boundedness assumption of .

• (iii) Iterative process (1.14) can be replaced by Ishikawa iterative process with errors (1.9).

Respectively, our results improve and generalize the corresponding results of Zeng el al. [8], Osilike and Udomene [7], and others.

In the sequel, we will need the following lemmas.

Lemma 1.9 (see [11]).

Let , , be sequences of nonnegative real numbers satisfying the inequality

(1.15)

If , , we have (i) exists. (ii) In particular, if , then .

Lemma 1.10 (see [12]).

Let be a Banach space and be the normalized duality mapping, then for any , the following conclusions hold

(i), for all ;

(ii), for all .

2. Main Results

In the rest of paper, we denote by the Lipschitz constant.

Lemma 2.1.

Let be a nonempty closed convex subset of a real Banach space . Let be a strictly pseudocontractive mapping with . Let ; is defined by (1.9) and satisfying the following conditions:

(i);

(ii).

Then

(1)there exist two sequences , in , such that , , and

(2.1)

Furthermore, exists.

(2)For any integer , there exists a constant , such that

(2.2)

Proof.

• (1) Let . Since and are bounded sequences in , we have

  

  (2.3)

Since is a strictly pseudocontractive mapping, by Remark 1.2(i),

(2.4)

By Kato [13], the above inequality is equivalent to

(2.5)

Let and from (1.9), we have

(2.6)

It follows that

(2.7)

Observe that

(2.8)

From (2.7) and (2.8), we have

(2.9)

By inequality (2.5), we get

(2.10)

So

(2.11)

Furthermore, set , then

(2.12)

We make the following estimations.

(2.13)

(2.14)

(2.15)

(2.16)

Substituting (2.14), (2.15), and (2.16) in (2.11), we obtain

(2.17)

where

(2.18)

By conditions (i) and (ii), we have , . It follows from Lemma 1.9 that exists. This completes the proof of part (1).

• (2) If , then . For any integer and from part (1), we have

  

  (2.19)

where . This completes the proof of part (2).

Lemma 2.2.

Let be a nonempty closed convex subset of a real Banach space . Let be a strictly pseudocontractive mapping with . Let be defined as in Lemma 2.1. Then there exists a subsequence of , such that

(2.20)

Proof.

Let . It follows from (1.3), (1.9), and Lemma 1.10(i) that

(2.21)

Let

(2.22)

Then (2.21) becomes

(2.23)

From Lemma 2.1(1), exists. So is bounded. By inequalities (2.14), (2.15), and (2.16), the sequences , , are all bounded. Notice the conditions of and , then . It follows from (2.23) that

(2.24)

so

(2.25)

Hence, . Since , so we have .

By virtue of Lemma 1.10(ii), we obtain

(2.26)

therefore,

(2.27)

Observe the right side of the above inequality, since

(2.28)

and , , are all bounded. Together with , then , that is, there exists a subsequence of , such that

(2.29)

Theorem 2.3.

Let be a nonempty closed convex subset of a real Banach space . Let be a strictly pseudocontractive mapping with . Let be defined as in Lemma 2.1, then converges strongly to a fixed point of if and only if , where .

Proof.

The necessity is obvious. So, we will prove the sufficiency. From Lemma 2.1(1), we have

(2.30)

Therefore,

(2.31)

Note that , . By Lemma 1.9 and , we get .

Next, we prove is a cauchy sequence. For each , there exists a natural number , such that

(2.32)

where is the constant in Lemma 2.1 (2). Hence, there exists and a natural number , such that

(2.33)

From Lemma 2.1(2) and (2.33), for all , we have

(2.34)

Hence, is a cauchy sequence. Since is a closed subset of , so converges strongly to a .

Finally, we prove . In fact, since . So, for any , there exists , such that . Then we have

(2.35)

By the arbitrary of , we know that . Therefore, .

A mapping is said to satisfy Condition(A) [14], if there exists a nondecreasing function with , , for all such that for all .

Theorem 2.4.

Let be a nonempty closed convex subset of a real Banach space . Let be a strictly pseudocontractive mapping with , and satisfy Condition(A). Let be defined as in Lemma 2.1. Then converges strongly to a fixed point of .

Proof.

By Lemma 2.2, there exists a subsequence of , such that

(2.36)

By Condition(A), . Since is a nondecreasing function and , therefore . The rest of the proof is the same to Theorem 2.3.

Theorem 2.5.

Let be a nonempty closed convex subset of a real Banach space . Let be a compact and strictly pseudocontractive mapping with . Let be defined as in Lemma 2.1. Then converges strongly to a fixed point of .

Proof.

From Lemma 2.1 (1), it follows that exists, for any . By Lemma 2.2, there exists a subsequence of such that . Since is bounded and is compact, has a strongly convergent subsequence. Without loss of generality, we may assume that converges strongly to . Next, we prove .

(2.37)

that is, . By the Lipschitz continuity of , it follows that

(2.38)

This means that . By Lemmas 1.9(ii) and 2.1(i), the sequence converges strongly to .

Theorem 2.6.

Let be a nonempty closed convex subset of a real Banach space . Let be a demicompact and strictly pseudocontractive mapping with . Let be defined as in Lemma 2.1. Then converges strongly to a fixed point of .

Proof.

From Lemma 2.1(1), it follows that exists, for any . By Lemma 2.2, there exists a subsequence of such that . Since is bounded together with being demicompact, there exists a subsequence of which converges strongly to some . Taking into account that and the Lipschitz continuity of , we have . By Lemma 1.9, converges strongly to .