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Almost stability of the Mann type iteration method with error term involving strictly hemicontractive mappings in smooth Banach spaces
Journal of Inequalities and Applications volume 2012, Article number: 207 (2012)
Abstract
Let K be a nonempty closed bounded convex subset of an arbitrary smooth Banach space X and be a continuous strictly hemicontractive mapping. Under some conditions, we obtain that the Mann iteration method with error term converges strongly to a unique fixed point of T and is almost T-stable on K. As an application of our results, we establish strong convergence of a multi-step iteration process.
1 Introduction
Chidume [1] established that the Mann iteration sequence converges strongly to the unique fixed point of T in case T is a Lipschitz strongly pseudo-contractive mapping from a bounded closed convex subset of (or ) into itself. Schu [2] generalized the result in [1] to both uniformly continuous strongly pseudo-contractive mappings and real smooth Banach spaces. Park [3] extended the result in [1] to both strongly pseudocontractive mappings and certain smooth Banach spaces. Rhoades [4] proved that the Mann and Ishikawa iteration methods may exhibit different behavior for different classes of nonlinear mappings. Harder and Hicks [5, 6] revealed the importance of investigating the stability of various iteration procedures for various classes of nonlinear mappings. Harder [7] established applications of stability results to first-order differential equations. Afterwords, several generalizations have been made in various directions (see, for example, [2, 4, 8–21].
Let K be a nonempty closed bounded convex subset of an arbitrary smooth Banach space X and be a continuous strictly hemicontractive mapping. Under some conditions, we obtain that the Mann iteration method with error term converges strongly to a unique fixed point of T and is almost T-stable on K. As an application, we shall also establish strong convergence of a multi-step iteration process. The results presented here generalize the corresponding results in [2–4, 10, 11, 22].
2 Preliminaries
Let K be a nonempty subset of an arbitrary Banach space X and be its dual space. The symbols , and stand for the domain, the range and the set of fixed points of respectively (x is called a fixed point of T iff ). We denote by J the normalized duality mapping from X to defined by
Let T be a self-mapping of K.
Definition 1 The mapping T is called Lipshitzian if there exists such that
for all . If , then T is called non-expansive and if , T is called contraction.
-
1.
The mapping T is said to be pseudocontractive if the inequality
(2.1)
holds for each and for all .
-
2.
T is said to be strongly pseudocontractive if there exists such that
(2.2)
for all and .
-
3.
T is said to be local strongly pseudocontractive if for each , there exists such that
(2.3)
for all and .
-
4.
T is said to be strictly hemicontractive if and if there exists such that
(2.4)
for all , and .
Clearly, each strongly pseudocontractive operator is local strongly pseudocontractive.
Let K be a nonempty convex subset of X and be an operator. Assume that and defines an iteration scheme which produces a sequence . Suppose, furthermore, that converges strongly to . Let be any bounded sequence in K and put .
-
(1)
The iteration scheme defined by is said to be T-stable on K if implies that .
-
(2)
The iteration scheme defined by is said to be almost T-stable on K if implies that .
It is easy to verify that an iteration scheme which is T-stable on K is almost T-stable on K.
Lemma 4 [3]
Let X be a smooth Banach space. Suppose one of the following holds:
-
(1)
J is uniformly continuous on any bounded subsets of X,
-
(2)
for all x, y in X,
-
(3)
for any bounded subset D of X, there is a such that
for all , where c satisfies
Then for any and any bounded subset K, there exists such that
for all and .
Lemma 5 [10]
Let be an operator with . Then T is strictly hemicontractive if and only if there exists such that for all and , there exists satisfying
Lemma 6 [4]
Let X be an arbitrary normed linear space and be an operator.
-
(1)
If T is a local strongly pseudocontractive operator and , then is a singleton and T is strictly hemicontractive.
-
(2)
If T is strictly hemicontractive, then is a singleton.
3 Main results
We now prove our main results.
Lemma 7 Let , and be nonnegative real sequences, and let be a constant satisfying
where , for all and . Then, .
Proof By a straightforward argument, for ,
where we put . Note that . It follows from (3.1) that
For a given , there exists a positive integer k such that . Thus (3.2) ensures that
Letting yields . □
Remark 8
-
(i)
If for each , then Lemma 7 reduces to Lemma 1 of Park [3].
-
(ii)
If , then Lemma 7 reduces to Lemma 2.1 of Liu et al. [4].
Theorem 9 Let Xbe a smooth Banach space satisfying any one of the Axioms (1)-(3) of Lemma 4. Let K be a nonempty closed bounded convex subset of X and be a continuous strictly hemicontractive mapping. Suppose that is an arbitrary sequence in K and , and are any sequences in satisfying conditions (i) , (ii) , (iii) and (iv) .
For a sequence in K, suppose that is the sequence generated from an arbitrary by
and satisfying .
Let be any sequence in K and define by
where , such that .
Then
-
(a)
the sequence converges strongly to a unique fixed point q of T,
-
(b)
implies that , so that is almost T-stable on K,
-
(c)
implies that .
Proof From (ii), we have , where as .
It follows from Lemma 6 that is a singleton. That is, for some .
Set . For all , it is easy to verify that
For given any and the bounded subset K, there exists a satisfying (2.6). Note that (ii), (iii), and the continuity of T ensure that there exists an N such that
where and t satisfies (2.7). Using (3.3) and Lemma 4, we infer that
for all .
Put
we have from (3.7)
Observe that , for all . It follows from Lemma 7 that
Letting , we obtain that , which implies that as .
On the same lines, we obtain
for all .
Suppose that . In view of (3.4) and (3.8), we infer that
for all .
Now, put
and we have from (3.9)
Observe that , and for all . It follows from Lemma 7 that
Letting , we obtain that , which implies that as .
Conversely, suppose that , then (iii) and (3.8) imply that
as , that is, as . □
Using the methods of the proof of Theorem 9, we can easily prove the following.
Theorem 10 Let X, K, T and , be as in Theorem 9. Suppose that , and are sequences in satisfying conditions (i), (iii)-(iv) and
If , , , and are as in Theorem 9, then the conclusions of Theorem 9 hold.
Corollary 11 Let X be a smooth Banach space satisfying any one of the Axioms (1)-(3) of Lemma 4. Let K be a nonempty closed bounded convex subset of X and be a Lipschitz strictly hemicontractive mapping. Suppose that is an arbitrary sequence in K and , and are any sequences in satisfying conditions (i) , (ii) , (iii) and (iv) .
For a sequence in K, suppose that is the sequence generated from an arbitrary by
and satisfying .
Let be any sequence in K and define by
where , such that .
Then
-
(a)
the sequence converges strongly to a unique fixed point q of T,
-
(b)
implies that , so that is almost T-stable on K,
-
(c)
implies that .
Corollary 12 Let X, K, T and be as in Corollary 11. Suppose that , and are sequences in satisfying conditions (i), (iii)-(iv) and
If , , , and are as in Corollary 11, then the conclusions of Corollary 11 hold.
Corollary 13 Let X be a smooth Banach space satisfying any one of the Axioms (1)-(3) of Lemma 4. Let K be a nonempty closed bounded convex subset of X and be a continuous strictly hemicontractive mapping. Suppose that is a sequence in satisfying conditions (i) and (ii) .
For a sequence in K, suppose that is the sequence generated from an arbitrary by
and satisfying .
Let be any sequence in K and define by
where , such that .
Then
-
(a)
the sequence converges strongly to a unique fixed point q of T,
-
(b)
implies that , so that is almost T-stable on K,
-
(c)
implies that .
Corollary 14 Let X be a smooth Banach space satisfying any one of the Axioms (1)-(3) of Lemma 4. Let K be a nonempty closed bounded convex subset of X and be a Lipschitz strictly hemicontractive mapping. Suppose that is a sequence in satisfying conditions (i) and (ii) .
For a sequence in K, suppose that is the sequence generated from an arbitrary by
and satisfying .
Let be any sequence in K and define by
where , such that .
Then
-
(a)
the sequence converges strongly to a unique fixed point q of T,
-
(b)
implies that , so that is almost T-stable on K,
-
(c)
implies that .
4 Applications to a multi-step iteration process
Khan et al. [23] have introduced and studied a multi-step iteration process for a finite family of selfmappings. We now introduce a modified multi-step process as follows:
Let K be a nonempty closed convex subset of a real normed space E and () be a family of selfmappings.
Algorithm 1 For a given , compute the sequence by the iteration process of arbitrary fixed order ,
which is called the modified multi-step iteration process, where , .
For , we obtain the following three-step iteration process:
Algorithm 2 For a given , compute the sequence by the iteration process:
where , and are three real sequences in .
For , we obtain the Ishikawa [24] iteration process:
Algorithm 3 For a given , compute the sequence by the iteration process
where and are two real sequences in .
If , , in (4.3), we obtain the Mann iteration process [14]:
Algorithm 4 For any given , compute the sequence by the iteration process
where is a real sequence in .
Theorem 15 Let K be a nonempty closed bounded convex subset of a smooth Banach space X and () be selfmappings of K. Let be a continuous strictly hemicontractive mapping. Let , be real sequences in satisfying , and . For arbitrary , define the sequence by (4.1). Then converges strongly to a point in .
Proof By applying Corollary 13 under assumption that is continuous strictly hemicontractive mapping, we obtain Theorem 15 which proves strong convergence of the iteration process defined by (4.1). We will check only the condition by taking and ,
Now, from the condition , it can be easily seen that . □
Corollary 16 Let K be a nonempty closed bounded convex subset of a smooth Banach space X and () be selfmappings of K. Let be a Lipschitz strictly hemicontractive mapping. Let , be real sequences in satisfying , and . For arbitrary , define the sequence by (4.1). Then converges strongly to a point in .
Remark 17 Similar results can be found for the iteration processes with error terms, we omit the details.
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Acknowledgements
The first author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research. The third author gratefully acknowledges the support from the Ministry of Education and Science of Republic Serbia. The fourth author gratefully acknowledges the financial support provided by the University of Tabuk through the project of international cooperation with the University of Texas at El Paso. We are also thankful to the editor and the referees for their suggestions for the improvement of the manuscript.
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Hussain, N., Rafiq, A., Ciric, L.B. et al. Almost stability of the Mann type iteration method with error term involving strictly hemicontractive mappings in smooth Banach spaces. J Inequal Appl 2012, 207 (2012). https://doi.org/10.1186/1029-242X-2012-207
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DOI: https://doi.org/10.1186/1029-242X-2012-207