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Approximate Fixed Points for Nonexpansive and Quasi-Nonexpansive Mappings in Hyperspaces
Fixed Point Theory and Applications volume 2009, Article number: 520976 (2009)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ1_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ2_HTML.gif)
is called the Ishikawa iteration sequence with errors provided that .
(ii)If for all
in (1.2), the sequence
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ3_HTML.gif)
is called the Ishikawa iteration sequence provided that .
(iii)If for all
in (1.2), the sequence
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ4_HTML.gif)
is called the Mann iteration sequence with errors provided that .
(iv)If for all
in (1.2), the sequence
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ5_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ6_HTML.gif)
for all and
with
.
Proof.
Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ7_HTML.gif)
For any ,
,
, by Lemma 1.7 we infer that there exist
,
,
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ8_HTML.gif)
which imply that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ9_HTML.gif)
That is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ10_HTML.gif)
Similarly we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ11_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ12_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ13_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ14_HTML.gif)
If the Ishikawa iteration sequence with errors is bounded, then
Proof.
Since is nonexpansive,
,
, and
are bounded, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ15_HTML.gif)
Let and
be arbitrary nonnegative integers. In view of (1.2), (2.3), Lemma 1.8, and the nonexpansiveness of
, we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ16_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ17_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ18_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ19_HTML.gif)
which yields that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ20_HTML.gif)
Using (1.2), (2.3)–(2.6), Lemma 1.8, and the nonexpansiveness of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ21_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ22_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ23_HTML.gif)
Lemma 1.6, (2.2), and (2.11) yield that there exists a nonnegative constant satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ24_HTML.gif)
which implies that for any there exists a positive integer
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ25_HTML.gif)
Now we prove by induction that the following inequality holds for all :
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ26_HTML.gif)
According to (1.2), (2.8), (2.9), and (2.13), we derive that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ27_HTML.gif)
Hence (2.14) holds for . Suppose that (2.14) holds for
. That is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ28_HTML.gif)
In view of (1.2), (2.8), (2.9), and (2.16), we infer that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ29_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ30_HTML.gif)
According to (2.1), (2.2), and (2.12), we know that there exists a positive integer satisfying (2.13) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ31_HTML.gif)
It follows from (2.1), (2.2), (2.13), (2.14), and (2.19) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ32_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ33_HTML.gif)
It follows from (2.21), Theorem 2.1, and the nonexpansiveness of that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ34_HTML.gif)
as . That is,
. Put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ35_HTML.gif)
In view of (1.2), Lemma 1.8 and the nonexpansiveness of , we derive that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ36_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ37_HTML.gif)
It follows from Lemma 1.6, (2.2), and (2.25) that . In view of Theorem 2.1 and Condition A, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ38_HTML.gif)
Using the continuity of , we know that
. That is,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ39_HTML.gif)
Clearly (2.27) ensures that for any there exist
and
such that
which implies from (2.24) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ40_HTML.gif)
We require for all
. Notice that for any
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ41_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ42_HTML.gif)
Using (2.28) and (2.30) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ43_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ44_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ45_HTML.gif)
and for
. Set
and
for
and
. Then
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ46_HTML.gif)
Thus the assumptions of Corollary 2.10 are satisfied. However, Theorem 4 in [3], Theorem 2 in [7] and [8] are not applicable since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ47_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ48_HTML.gif)
Put ,
,
for
and
. It follows that
is a compact subset of
,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ49_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ50_HTML.gif)
Obviously, ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ51_HTML.gif)
for . Therefore the conditions of Theorem 2.3 are fulfilled.
Example 3.4.
Let , and
be as in Example 3.2. Define
,
and
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ52_HTML.gif)
It follows that ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ53_HTML.gif)
for . Obviously, the assumptions of Theorem 2.4 are fulfilled. On the other hand,
is not nonexpansive since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F520976/MediaObjects/13663_2009_Article_1151_Equ54_HTML.gif)
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?
References
Browder FE: Nonexpansive nonlinear operators in a Banach space. Proceedings of the National Academy of Sciences of the United States of America 1965, 54: 1041–1044. 10.1073/pnas.54.4.1041
Kirk WA: A fixed point theorem for mappings which do not increase distances. The American Mathematical Monthly 1965, 72: 1004–1006. 10.2307/2313345
Deng L: Convergence of the Ishikawa iteration process for nonexpansive mappings. Journal of Mathematical Analysis and Applications 1996,199(3):769–775. 10.1006/jmaa.1996.0174
Lei D, Shenghong L: Ishikawa iteration process with errors for nonexpansive mappings in uniformly convex Banach spaces. International Journal of Mathematics and Mathematical Sciences 2000,24(1):49–53. 10.1155/S0161171200003380
Dotson WG Jr.: On the Mann iterative process. Transactions of the American Mathematical Society 1970, 149: 65–73. 10.1090/S0002-9947-1970-0257828-6
Dotson WG Jr.: Fixed points of quasi-nonexpansive mappings. Journal of the Australian Mathematical Society 1972, 13: 167–170. 10.1017/S144678870001123X
Ishikawa S: Fixed points and iteration of a nonexpansive mapping in a Banach space. Proceedings of the American Mathematical Society 1976,59(1):65–71. 10.1090/S0002-9939-1976-0412909-X
Petryshyn WV, Williamson TE Jr.: Strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings. Journal of Mathematical Analysis and Applications 1973, 43: 459–497. 10.1016/0022-247X(73)90087-5
Rhoades BE: Some fixed point iteration procedures. International Journal of Mathematics and Mathematical Sciences 1991,14(1):1–16. 10.1155/S0161171291000017
Senter HF, Dotson WG Jr.: Approximating fixed points of nonexpansive mappings. Proceedings of the American Mathematical Society 1974, 44: 375–380. 10.1090/S0002-9939-1974-0346608-8
Tan K-K, Xu HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. Journal of Mathematical Analysis and Applications 1993,178(2):301–308. 10.1006/jmaa.1993.1309
Hu T, Huang J: Convex hull of compact sets in a hyperspace. Chinese Annals of Mathematics 1999, 20A: 253–254.
Xu Y: Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive operator equations. Journal of Mathematical Analysis and Applications 1998,224(1):91–101. 10.1006/jmaa.1998.5987
Nadler SB Jr.: Multi-valued contraction mappings. Pacific Journal of Mathematics 1969, 30: 475–488.
Acknowledgment
This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00042).
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Liu, Z., Ume, J.S. & Kang, S.M. Approximate Fixed Points for Nonexpansive and Quasi-Nonexpansive Mappings in Hyperspaces. Fixed Point Theory Appl 2009, 520976 (2009). https://doi.org/10.1155/2009/520976
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DOI: https://doi.org/10.1155/2009/520976