Approximate Fixed Points for Nonexpansive and Quasi-Nonexpansive Mappings in Hyperspaces

Abstract

This paper provides a few convergence results of the Ishikawa iteration sequence with errors for nonexpansive and quasi-nonexpansive mappings in hyperspaces. The results presented in this paper improve and generalize some results in the literature.

1. Introduction and Preliminaries

Browder [1] and Kirk [2] established that a nonexpansive mapping which maps a closed bounded convex subset of a uniformly convex Banach space into itself has a fixed point in . Since then, many researchers have studied, under various conditions, the convergence of the Mann and Ishikawa iteration methods dealing with nonexpansive and quasi-nonexpansive mappings (see [3–11] and the references therein). Rhoades [9] pointed out that the Picard iteration schemes for nonexpansive mappings need not converge. Senter and Dotson [10] obtained conditions under which the Mann iteration schemes generated by nonexpansive and quasi-nonexpansiv mappings in uniformly convex Banach spaces, converge to fixed points of these mappings, respectively. Ishikawa [7] established that the Mann iteration methods can be used to approximate fixed points of nonexpansive mappings in Banach spaces. Deng [3] obtained similar results for Ishikawa iteration processes in normed linear spaces and Banach spaces.

Our aim is to prove several convergence theorems of the Ishikawa iteration sequence with errors for nonexpansive and quasi-nonexpansive mappings in hyperspaces. Our results presented in this paper extend substantially the results due to Deng [3], Ishikawa [7], and Senter and Dotson [10].

Assume that is a nonempty subset of a normed linear space and denotes the family of all nonempty convex compact subsets of , and is the Hausdorff metric induced by the norm . For , , , , , and , let

(1.1)

It is easy to see that and for all and . Hence is convex. Hu and Huang [12] proved that if is a Banach space, then is a complete metric space. Now we introduce the following concepts in hyperspaces.

Definition 1.1.

Let be a nonempty subset of and let be a mapping. Assume that , , , and are arbitrary real sequences in satisfying and for and and are any bounded sequences of the elements in .

(i)For , the sequence defined by

(1.2)

is called the Ishikawa iteration sequence with errors provided that .

(ii)If for all in (1.2), the sequence defined by

(1.3)

is called the Ishikawa iteration sequence provided that .

(iii)If for all in (1.2), the sequence defined by

(1.4)

is called the Mann iteration sequence with errors provided that .

(iv)If for all in (1.2), the sequence defined by

(1.5)

is called the Mann iteration sequence provided that

Definition 1.2.

Let be a nonempty subset of . A mapping is said to be

(i)nonexpansive if for all

(ii)quasi-nonexpansive if and for all and .

Definition 1.3.

Let be a nonempty subset of . A mapping with is said to be satisfy the following.

(i)Condition A if there is a continuous function with and for , such that for all .

(ii)Condition B if there is a nondecreasing function with and for , such that for all

Remark 1.4.

In case , where is a nonempty subset of , and is a mapping, then Definitions 1.1, 1.2, and 1.3(ii) reduce to the corresponding concepts in [1–11, 13]. It is well known that every nonexpansive mapping with nonempty fixed point set is quasi-nonexpansive, but the converse is not true; see [8]. Examples 3.1 and 3.4 in this paper reveal that the class of nonexpansive mappings with nonempty fixed point set is a proper subclass of quasi-nonexpansive mappings with both Condition A and Condition B.

The following lemmas play important roles in this paper.

Lemma 1.5 (see [12]).

Let be a Banach space and a compact subset of . Then is compact, where stands for the closure of .

Lemma 1.6 (see [4]).

Suppose that , , and are three sequences of nonnegative numbers such that for all . If and converge, then exists.

Lemma 1.7 (see [14]).

Let be a metric space. Let and be compact subsets of . Then for any , there exists such that , where is the Hausdorff metric induced by .

Lemma 1.8.

Let be a normed linear space. Then

(1.6)

for all and with .

Proof.

Set

(1.7)

For any , , , by Lemma 1.7 we infer that there exist , , such that

(1.8)

which imply that

(1.9)

That is,

(1.10)

Similarly we have

(1.11)

Thus (1.6) follows from (1.10) and (1.11). This completes the proof.

Lemma 1.9.

Let be a normed linear space and a nonempty closed subset of . If is quasi-nonexpansive, then is closed.

Proof.

Let be in with . Since is quasi-nonexpansive, it follows that

(1.12)

as . Hence . That is, is closed. This completes the proof.

2. Main Results

Our results are as follows.

Theorem 2.1.

Let be a normed linear space and let be a nonempty subset of . Assume that is nonexpansive and . Suppose that there exists a constant satisfying

(2.1)

(2.2)

If the Ishikawa iteration sequence with errors is bounded, then

Proof.

Since is nonexpansive, , , and are bounded, it follows that

(2.3)

Let and be arbitrary nonnegative integers. In view of (1.2), (2.3), Lemma 1.8, and the nonexpansiveness of , we conclude that

(2.4)

(2.5)

(2.6)

(2.7)

which yields that

(2.8)

Using (1.2), (2.3)–(2.6), Lemma 1.8, and the nonexpansiveness of , we have

(2.9)

(2.10)

which implies that

(2.11)

Lemma 1.6, (2.2), and (2.11) yield that there exists a nonnegative constant satisfying

(2.12)

which implies that for any there exists a positive integer such that

(2.13)

Now we prove by induction that the following inequality holds for all :

(2.14)

According to (1.2), (2.8), (2.9), and (2.13), we derive that

(2.15)

Hence (2.14) holds for . Suppose that (2.14) holds for . That is,

(2.16)

In view of (1.2), (2.8), (2.9), and (2.16), we infer that

(2.17)

That is, (2.14) holds for . Hence (2.14) holds for all .

We now assert that . If not, then . Let be an arbitrary positive integer and

(2.18)

According to (2.1), (2.2), and (2.12), we know that there exists a positive integer satisfying (2.13) and

(2.19)

It follows from (2.1), (2.2), (2.13), (2.14), and (2.19) that

(2.20)

as . Thus (2.3) and (2.20) yield that , which is absurd. Hence . This completes the proof.

Theorem 2.2.

Let be a Banach space and a nonempty closed subset of . Assume that is nonexpansive and there exists a compact subset of such that If (2.1) and (2.2) hold, then has a fixed point in . Moreover, given , the Ishikawa iteration sequence with errors converges to a fixed point of .

Proof.

Setting , by Lemma 1.5 and the compactness we conclude that is compact. It is evident that , which yields that is bounded. Since is closed and , we conclude that there exist a subsequence of and such that

(2.21)

It follows from (2.21), Theorem 2.1, and the nonexpansiveness of that

(2.22)

as . That is, . Put

(2.23)

In view of (1.2), Lemma 1.8 and the nonexpansiveness of , we derive that

(2.24)

for . It follows from Lemma 1.6, (2.2), (2.23), and (2.24) that exists. Using (2.21) we get that . This completes the proof.

Theorem 2.3.

Let be a Banach space and a nonempty closed subset of . Suppose that is a qusi-nonexpansive mapping and satisfies Condition A. Assume that (2.1) and (2.2) hold and is in . If is bounded, then the Ishikawa iteration sequence with errors converges to a fixed point of in .

Proof.

Let . Then . As in the proof of Theorem 2.2, we get that (2.24) holds and exists, where . Consequently, is bounded and

(2.25)

It follows from Lemma 1.6, (2.2), and (2.25) that . In view of Theorem 2.1 and Condition A, we have

(2.26)

Using the continuity of , we know that . That is, and

(2.27)

Clearly (2.27) ensures that for any there exist and such that which implies from (2.24) that

(2.28)

We require for all . Notice that for any

(2.29)

Thus (2.2) and (2.29) yield that is a Cauchy sequence in . It follows from Lemma 1.9 that there exists satisfying . For any there exists such that

(2.30)

Using (2.28) and (2.30) we have

(2.31)

for . That is, converges to . This completes the proof.

A proof similar to that of Theorem 2.3 gives the following result and is thus omitted.

Theorem 2.4.

Let be a Banach space and let be a nonempty closed subset of . Suppose that is a qusi-nonexpansive mapping and satisfies Condition B. Assume that is in and there exists a constant satisfying

(2.32)

Then the Ishikawa iteration sequence converges to a fixed point of in .

Let be a nonempty subset of . It is easy to see that is isometric to . Thus Theorems 2.1–2.4 yield the following results.

Corollary 2.5.

Let be a nonempty subset of a normed linear space . Assume that is nonexpansive and . Suppose that (2.1) and (2.2) hold. If the Ishikawa iteration sequence with errors is bounded, then .

Remark 2.6.

Corollary 2.5 extends Theorem 1 in [3] and Lemma 2 in [7] from the Ishikawa iteration scheme and Mann iteration scheme into the Ishikawa iteration scheme with errors, respectively.

Corollary 2.7.

Let be a nonempty closed subset of a Banach space . Assume that is nonexpansive and there exists a compact subset of with . Suppose that (2.1) and (2.2) hold. Then has a fixed point in . Moreover for any , the Ishikawa iteration sequence with errors converges to a fixed point of .

Remark 2.8.

Theorem 3 in [3] and Theorem 1 in [7] and [8] are special cases of Corollary 2.7.

Corollary 2.9.

Let be a nonempty closed subset of a Banach space and let be quasi-nonexpansive. Assume that (2.1) and (2.2) hold and satisfies Condition A. If is bounded, then for any , the Ishikawa iteration sequence with errors converges to a fixed point of in .

Corollary 2.10.

Let be a nonempty closed subset of a Banach space and let be quasi-nonexpansive. Assume that (2.32) holds and is in . If satisfies Condition B, then the Ishikawa iteration sequence converges to a fixed point of in .

Remark 2.11.

Corollary 2.10 extends, improves, and unifies Theorem 4 in [3], Theorem 2 in [7] and [8] in the following ways:

(i)the Mann iteration method in [7, 8], and Ishikawa iteration method in [3] are replaced by the more general Ishikawa iteration method with errors;

(ii)the nonexpansive mappings in [3, 7, 8] are replaced by the more general quasi-nonexpansive mappings.

3. Examples and Problems

Now we construct a few nontrivial examples to illustrate the results in Section 2. The following example reveals that Corollary 2.10 extends properly Theorem 4 in [3], Theorem 2 in [7] and [8].

Example 3.1.

Let with the usual norm and let . Define and by

(3.1)

and for . Set and for and . Then and

(3.2)

Thus the assumptions of Corollary 2.10 are satisfied. However, Theorem 4 in [3], Theorem 2 in [7] and [8] are not applicable since

(3.3)

that is, is not nonexpansive.

The examples below show that Theorems 2.1–2.4 extend substantially Corollaries 2.5–2.10, respectively.

Example 3.2.

Let with the usual norm and let . For any , stands for the triangle with vertices , and . Let and and be in . Define by

(3.4)

Put , , for and . It follows that is a compact subset of , and

(3.5)

for . That is, the conditions of Theorems 2.1 and 2.2 are fulfilled. Hence we can invoke our Theorems 2.1 and 2.2 show that the Ishikawa iteration sequence with errors converges to and .

Example 3.3.

Let , , , , , , , and be as in Example 3.2. Define and by

(3.6)

Obviously, ,

(3.7)

for . Therefore the conditions of Theorem 2.3 are fulfilled.

Example 3.4.

Let , and be as in Example 3.2. Define , and by

(3.8)

It follows that ,

(3.9)

for . Obviously, the assumptions of Theorem 2.4 are fulfilled. On the other hand, is not nonexpansive since

(3.10)

We conclude with the following problems.

Problem 3.5.

Can Condition A in Theorem 2.3 be replaced by Condition B?

Problem 3.6.

Can the boundedness of in Theorem 2.3 be removed?

Problem 3.7.

Can Theorem 2.4 be extended to the Ishikawa iteration method with errors?