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Equivalence of semistability of Picard, Mann, Krasnoselskij and Ishikawa iterations
Fixed Point Theory and Applications volume 2014, Article number: 5 (2014)
Abstract
In this paper, we show that convergence of Picard, Mann, Krasnoselskij and Ishikawa iterations is equivalent in cone normed spaces. Also, we prove that semistability of these iterations is equivalent.
1 Introduction
Let be a real Banach space. A subset is called a cone in E if it satisfies the following conditions:
-
(i)
P is closed, nonempty and ,
-
(ii)
, and imply that ,
-
(iii)
and imply that .
The space E can be partially ordered by the cone P, by defining if and only if . Also, we write if , where intP denotes the interior of P. A cone P is called normal if there exists a constant such that implies . The least positive number satisfying above is called the normal constant of P.
From now on, we suppose that E is a real Banach space, P is a cone in E and ≤ is a partial ordering with respect to P.
Lemma 1.1 ([1])
Let P be a normal cone and let and be sequences in E satisfying the following inequality:
where and as . Then .
Definition 1.2 ([2])
Let X be a vector space over the field F. Assume that the function having the properties:
-
(i)
for all x in X,
-
(ii)
for all x, y in X,
-
(iii)
for all and .
Then p is called a cone seminorm on X. A cone norm is a cone seminorm p such that
-
(iv)
if .
We will denote a cone norm by and is called a cone normed space. Also, defines a cone metric on X.
Definition 1.3 ([3])
Let be a cone normed space. Then is called bounded above if there exists , such that for all .
Definition 1.4 Let be a cone normed space. Let be a sequence in X and . If for any with , there exists an integer such that for all , , then we will say converges to x and we write .
Definition 1.5 Let be a cone normed space. Let be a sequence in X and . If for any with , there exists an integer such that for all , , then is said to be a Cauchy sequence. If every Cauchy sequence is convergent in X, then X is called a cone Banach space.
Lemma 1.6 ([4])
Let be a cone metric space, P be a normal cone. Let be a sequence in X and . Then converges to x if and only if .
Lemma 1.7 Let be a cone normed space over the real Banach space E with the cone P which is normal with the normal constant k. The mapping defined by satisfies the following properties:
-
(i)
implies ,
-
(ii)
for all ,
-
(iii)
for all and ,
-
(iv)
for all ,
-
(v)
if and only if .
Moreover, let A be a bounded above subset of X, then
-
(vi)
is a bounded set.
Proof The proof is obvious. □
Definition 1.8 Let be a cone normed space over the real Banach space E with the normal cone P. The mapping N, defined in Lemma 1.7, is called a norm type with respect to .
Lemma 1.9 Let be a cone normed space over the real Banach space E with the normal cone P. Also, let be a sequence in X and . Then converges to x if and only if .
Proof Note that is a sequence in E and by Lemma 1.6, the proof is obvious. □
Definition 1.10 Let X be a cone normed space and be a map for which there exist real numbers a, b, c satisfying , and . Then T is called a Zamfirescu operator with respect to if and only if for each pair , T satisfies at least one of the following conditions:
(Z1) ,
(Z2) ,
(Z3) .
Usually, for simplicity, T is called a Zamfirescu operator if T is Zamfirescu with respect to some triple of scalers a, b and c with above restrictions. Also, T is called f-Zamfirescu operator if at least one of the relations (Z1), (Z2) and (Z3) hold for all and for all .
Remark 1.11 Let T be a Zamfirescu operator and be arbitrary. Since T is Zamfirescu, at least one of the conditions (Z1), (Z2) and (Z3) is satisfied. If (Z2) holds, then
Thus we get
Since , we have
Similarly, if (Z3) holds, then we obtain
Hence
where and .
Definition 1.12 Let X be a cone normed space. A self-map T of X is called a quasi-contraction if for some constant and for every , there exists
such that . If this inequality holds for all and , we say that T is a f-quasi-contraction.
Definition 1.13 Let X be a cone normed space, T be a self-map of X and . The Picard iteration is given by
For a sequence of self-maps , the iteration is called the Picard’s S-iteration.
Another two well-known iteration procedures for obtaining fixed points of T are Mann iteration defined by
and Ishikawa iteration defined by
where and . Also, the Krasnoselskij iteration is defined by
where .
If T is a self-map of X, then by we mean the set of fixed points of T. Also, denotes the set of nonnegative integers, i.e., .
Lemma 1.14 ([5])
Let be a complete cone metric space and P be a normal cone. Suppose that the mapping satisfies the contractive condition
for all , where is a constant. Then T has a unique fixed point in X and for each , the iterative sequence converges to the fixed point.
Lemma 1.15 ([5])
Let be a complete cone metric space and P be a normal cone. Suppose that the mapping satisfies the contractive condition
for all , where is a constant. Then T has a unique fixed point in X and for each , the iterative sequence converges to the fixed point.
Lemma 1.16 ([5])
Let be a complete cone metric space and P be a normal cone. Suppose that the mapping satisfies the contractive condition
for all , where is a constant. Then T has a unique fixed point in X and for each , the iterative sequence converges to the fixed point.
Lemma 1.17 ([2])
Let T be a quasi-contraction with . Then T is a Zamfirescu operator.
Definition 1.18 Let be a cone normed space and be a sequence of self-maps of X with . Let be a point of X and assume that is an iteration procedure involving , which yields a sequence of points from X. The iteration is said to be -semistable (or semistable with respect to ) if whenever converges to a fixed point q in and is a sequence in X with and for some sequence , then .
The iteration is said to be -stable (or stable with respect to ) if converges to a fixed point q in and whenever is a sequence in X with , we have .
Note that if for all n, then Definition 1.18 gives the definitions of T-semistability and T-stability respectively.
Lemma 1.19 ([2])
Let be a cone metric space, P be a normal cone and be a sequence of self-maps of X with . Suppose that there exist nonnegative bounded sequences , with such that
for each , and . Then the Picard’s S-iteration is semistable with respect to .
Lemma 1.20 ([2])
Let be a cone metric space, P be a normal cone and be a sequence of self-maps of X with . If for all , is a f-Zamfirescu operator with respect to with . Then the Picard’s S-iteration is semistable with respect to .
Lemma 1.21 ([2])
Under the conditions of Lemma 1.22 if is a Zamfirescu operator for all n, then the Picard’s S-iteration is semistable with respect to .
Lemma 1.22 ([2])
Let be a cone metric space, P be a normal cone and be a sequence of self-maps of X with . If for all , is a f-quasi-contraction with such that , then the Picard’s S-iteration is semistable with respect to .
For some other sources on these topics, we refer to [6–23].
2 Main results
Theorem 2.1 Let X be a cone normed space and P be a normal cone. Suppose that T is a Zamfirescu self-map of X and . Then the following are equivalent:
-
(i)
the Picard iteration converges to q,
-
(ii)
the Mann iteration converges to q.
Proof Let be given. We prove the implication . Suppose that . Now, by using (3) and (4), we have
Using (2) with , , we get
Using (2) with , , we obtain
Relations (7), (8) and (9) lead to
Set
Since , by using Lemma 1.1, we get
Thus
as . This completes the proof.
Now we prove . Suppose that . Applying (3) and (4), we have
Using (2) with , , we obtain
Therefore, from (10) and (11), we get
Put
Since , by Lemma 1.1 and relation (12), we get . Thus
as and so the proof is complete. □
Theorem 2.2 Let X be a cone normed space and P be a normal cone. Suppose that T is a Zamfirescu self-map of X and . Then the following are equivalent:
-
(i)
the Picard iteration converges to q,
-
(ii)
the Krasnoselskij iteration converges to q.
Proof For , the Mann iteration reduces to the Krasnoselskij iteration. Now apply the proof of Theorem 2.1. □
Theorem 2.3 Let X be a cone normed space and P be a normal cone. Suppose that T is a Zamfirescu operator of X and . Then the following are equivalent:
-
(i)
the Mann iteration converges to q,
-
(ii)
the Ishikawa iteration converges to q.
Proof Let and be given. We prove the implication . Suppose that . Using
and
we get . The proof is complete if we prove relation (13).
Using (2), (4) and (5) with , , we have
Using (2) with , , we have
Relations (14) and (15) lead to
Put
Note that , T is Zamfirescu and . By (2) we obtain
Hence ; that is, . Lemma 1.1 leads to
Now we will prove that . Using (2) with , , we obtain
Also, the following relation holds:
Substituting (17) in (16), we obtain
Put
From , T is Zamfirescu, and by (2) we obtain
and
Hence and ; that is, .
Lemma 1.1 and (18) lead to . Thus, we get
and the proof is complete. □
Corollary 2.4 Let X be a cone Banach space, P be a normal cone and T be a Zamfirescu self-map of X. Then T has a unique fixed point in X and the Picard, Mann, Krasnoselskij and Ishikawa iterative sequences converge to the fixed point of T.
Corollary 2.5 Let X be a cone Banach space, P be a normal cone and T be a quasi-contraction mapping of X with . Then T has a unique fixed point in X and the Picard, Mann, Krasnoselskij and Ishikawa iterative sequences converge to the fixed point of T.
Theorem 2.6 Let X be a cone Banach space and P be a normal cone. Suppose that T is a self-map of X and that every Picard and Mann iteration converges to a fixed point of T. Then the following are equivalent:
-
(i)
the Picard iteration is semistable with respect to T,
-
(ii)
the Mann iteration is semistable with respect to T.
Proof Suppose that q is a fixed point of T such that every Picard and Mann iteration converges to q. Let be an arbitrary sequence in X. For , let
and for some . We have
as . By assumption (i), we get .
Conversely, we prove . Let and for some . We have
as . Thus and so the Picard iteration is semistable with respect to T. □
Theorem 2.7 Let X be a cone Banach space and P be a normal cone. Suppose that T is a self-map of X and that every Picard and Krasnoselskij iteration converges to a fixed point of T. Then the following are equivalent:
-
(i)
the Picard iteration is semistable with respect to T,
-
(ii)
the Krasnoselskij iteration is semistable with respect to T.
Proof In Theorem 2.7, put . Then by the same method used in the proof of Theorem 2.7, we can complete the proof. □
Theorem 2.8 Let X be a cone Banach space and P be a normal cone. Suppose that in Ishikawa iteration procedure satisfies , T is a self-map of X with bounded above range and also every Picard and Ishikawa iterative sequence converges to a fixed point of T. Then the following are equivalent:
-
(i)
the Picard iteration is semistable with respect to T,
-
(ii)
the Ishikawa iteration is semistable with respect to T.
Proof Suppose that q is a fixed point of T such that every Picard and Ishikawa iterative sequence converges to q. Let and be given and set
where N is the norm type with respect to . It is assumed that T has bounded above range and so, by Lemma 1.7, .
Now we prove that . Let and for some . Observe that
By Lemma 1.7 we have
as (here k is the normal constant of P). So, by Lemma 1.9, and the condition (i) assures that . Thus the Ishikawa iteration is semistable with respect to T.
Conversely, we prove . Let and for some . We have
By Lemmas 1.7 and 1.9, we get
as , where k is the normal constant of P. So and by assumption (ii), we have . Thus the Picard iteration is semistable with respect to T. □
Theorem 2.9 Let X be a cone Banach space and P be a normal cone. Suppose that in Mann and Ishikawa procedures satisfies , T is a self-map of X with bounded above range and also every Mann and Ishikawa iterative sequence converges to a fixed point of T. Then the following are equivalent:
-
(i)
the Mann iteration is T-stable,
-
(ii)
the Ishikawa iteration is T-stable.
Proof Let q be a fixed point of T and every Mann and Ishikawa iterative sequence converge to q. Suppose that k is the normal constant of P and put
where N is the norm type with respect to . Since T has bounded above range, then . Now let be an arbitrary sequence in X. We prove . For this suppose that
where and . We show that . Note that
By Lemma 1.7 and Lemma 1.9, we obtain
as and so
Condition (i) assures that . Thus the Ishikawa iteration is T-stable.
Conversely, we prove . Suppose that
We show that . Put
and observe that
By Lemma 1.7 and Lemma 1.9, we obtain
as and hence . By assumption (ii), we get and the proof is complete. □
Corollary 2.10 Let be a cone normed space, P be a normal cone and T be a self-map of X and . Suppose that there exist nonnegative real numbers a and b with such that
for each . Assume that for , , and let every Picard, Mann, Krasnoselskij and Ishikawa iterative sequence converge to q. Then the Picard, Mann and Krasnoselskij iterations are T-semistable. Moreover, if T has bounded above range, then the Ishikawa iteration is T-semistable.
Corollary 2.11 Let be a cone normed space, P be a normal cone and T be a f-Zamfirescu or quasi-contraction self-map of X and . Assume that in Mann and Ishikawa iteration procedures satisfies and . Also, let every Picard, Mann, Krasnoselskij and Ishikawa iterative sequence converge to q. Then the Picard, Mann and Krasnoselskij iterations are T-semistable. Moreover, if T has bounded above range, then the Ishikawa iteration is T-semistable.
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An erratum to this article is available at http://dx.doi.org/10.1186/1687-1812-2014-84.
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Yousefi, B., Yadegarnejad, A., Azadi Kenary, H. et al. Equivalence of semistability of Picard, Mann, Krasnoselskij and Ishikawa iterations. Fixed Point Theory Appl 2014, 5 (2014). https://doi.org/10.1186/1687-1812-2014-5
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DOI: https://doi.org/10.1186/1687-1812-2014-5