Abstract
The main purpose of this paper is to establish the convergence, almost common-stability and common-stability of the Ishikawa iteration scheme with error terms in the sense of Xu (J. Math. Anal. Appl. 224:91-101, 1998) for two Lipschitz strictly hemicontractive operators in arbitrary Banach spaces.
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1 Preliminaries
Let K be a nonempty subset of an arbitrary Banach space E and be its dual space. The symbols , and stand for the domain, the range and the set of fixed points of T respectively (for a single-valued map , is called a fixed point of T iff ). We denote by J the normalized duality mapping from E to defined by
Let T be a self-mapping of K.
Definition 1 Then 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
(1.2)
holds for each and for all . As a consequence of a result of Kato [4], it follows from the inequality (1.2) that T is pseudocontractive if and only if there exists such that
for all .
-
2.
T is said to be strongly pseudocontractive if there exists a such that
(1.4)
for all and .
-
3.
T is said to be local strongly pseudocontractive if, for each , there exists a such that
(1.5)
for all and .
-
4.
T is said to be strictly hemicontractive if and if there exists a such that
(1.6)
for all , and .
It is easy to verify that an iteration scheme which is T-stable on K is almost T-stable on K. Osilike [5] proved that an iteration scheme which is almost T-stable on X may fail to be T-stable on X.
Clearly, each strongly pseudocontractive operator is local strongly pseudocontractive.
Chidume [6] 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. Afterwards, several authors generalized this result of Chidume in various directions. Chidume [7] proved a similar result by removing the restriction . Tan and Xu [8] extended that result of Chidume to the Ishikawa iteration scheme in a p-uniformly smooth Banach space. Chidume and Osilike [2] improved the result of Chidume [6] to strictly hemicontractive mappings defined on a real uniformly smooth Banach space.
Recently, some researchers have generalized the results to real smooth Banach spaces, real uniformly smooth Banach spaces, real Banach spaces; or to the Mann iteration method, the Ishikawa iteration method; or to strongly pseudocontractive operators, local strongly pseudocontractive operators, strictly hemicontractive operators [9–19].
The main purpose of this paper is to establish the convergence, almost common-stability and common-stability of the Ishikawa iteration scheme with error terms in the sense of Xu [1] for two Lipschitz strictly hemicontractive operators in arbitrary Banach spaces. Our results extend, improve and unify the corresponding results in [2, 3, 10, 11, 15–18, 20–25].
2 Main results
We need the following results.
Lemma 3[26]
Let, , andbe nonnegative real sequences such that
with, , and. Then.
Lemma 4[27]
Let, be sequences of nonnegative real numbers and, so that
-
(i)
If , then .
-
(ii)
If , then .
Lemma 5[4]
Let. Thenfor everyif and only if there issuch that.
Lemma 6[2]
Letbe an operator with. Then T is strictly hemicontractive if and only if there existssuch that for alland, there existssatisfying
Lemma 7[24]
Let X be an arbitrary normed linear space andbe an operator.
-
(i)
If T is a local strongly pseudocontractive operator and , then is a singleton and T is strictly hemicontractive.
-
(ii)
If T is strictly hemicontractive, then is a singleton.
In the sequel, let , where t is the constant appearing in (1.6). Further L denotes the common Lipschitz constant of T and S, and I denotes the identity mapping on an arbitrary Banach space X.
Definition 8 Let K be a nonempty convex subset of X and be two operators. 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 .
-
(i)
The iteration scheme defined by is said to be common-stable on K if implies that .
-
(ii)
The iteration scheme defined by is said to be almost common-stable on K if implies that .
We now establish our main results.
Theorem 9 Let K be a nonempty closed convex subset of an arbitrary Banach space X andbe two Lipschitz strictly hemicontractive operators. Suppose that, are arbitrary bounded sequences in K, and, , , , andare any sequences insatisfying
-
(i)
,
-
(ii)
,
-
(iii)
,
-
(iv)
,
-
(v)
, ,
where s is a constant in. Suppose thatis the sequence generated from an arbitraryby
Let be any sequence in K and define by
where
Then
-
(a)
the sequence converges strongly to the common fixed point q of T and S. Also,
(b)
-
(c)
implies that , so that is almost common-stable on K,
-
(d)
implies that .
Proof From (ii), we have , where as . It follows from Lemma 7 that is a singleton; that is, for some . Set
Since T is strictly hemicontractive, it follows form Lemma 6 that
which implies that
In view of Lemma 5, we have
Also,
and
From (2.4) and (2.5), we infer that for all ,
which implies that for all ,
Substituting (2.8) in (2.7), we have
Substituting (2.9) in (2.6), we get
Put
we have
Observe that , and . It follows from Lemma 3 that .
We also have
From (2.5) and (2.10), it follows that for all ,
which implies that for all ,
Substituting (2.13) in (2.12), we have
Substituting (2.14) in (2.11), we get
for any . Thus (2.15) implies that
With
we have
Observe that , and . It follows from Lemma 3 that .
Suppose that . It follows from equation (2.15) that
as ; that is, as . □
Using the techniques in the proof of Theorem 9, we have the following results.
Theorem 10 Let X, K, T, S, s, , , , , , andbe as in Theorem 9. Suppose that, , , , andare sequences insatisfying conditions (i), (iii)-(v) of Theorem 9 with
Then the conclusions of Theorem 9 hold.
Theorem 11 Let X, K, T, S, s, , , , , , andbe as in Theorem 9. Suppose that, , , , andare sequences insatisfying condition (i), (iii) and (v) of Theorem 9 with
where m is a constant. Then
-
(a)
the sequence converges strongly to the common fixed point q of T and S. Also,
where
(b)
-
(c)
implies that .
Proof As in the proof of Theorem 9, we conclude that and
Let
Observe that and . It follows from Lemma 4 that .
Also, from (2.15), we have
Suppose that . It follows from equation (2.15) that
as ; that is, as .
Conversely, suppose that . Put
Observe that and . It follows from Lemma 4 that . □
As an immediate consequence of Theorems 9 and 11, we have the following:
Corollary 12 Let K be a nonempty closed convex subset of an arbitrary Banach space X andbe two Lipschitz strictly hemicontractive operators. Suppose that, are any sequences insatisfying
-
(vi)
,
-
(vii)
, ,
where s is a constant in. Suppose thatis the sequence generated from an arbitraryby
Let be any sequence in K and define by
where
and
Then
-
(a)
the sequence converges strongly to the common fixed point q of T and S,
-
(b)
implies that , so that is almost common-stable on K,
-
(c)
implies that .
Corollary 13 Let X, K, T, S, s, , , , andbe as in Theorem 9. Suppose that, are sequences insatisfying conditions (vi)-(vii) and (iii) of Theorem 9 with
where m is a constant. Then
-
(a)
the sequence converges strongly to the common fixed point q of T and S. Also,
(b)
-
(c)
implies that .
Example 14 Let denote the set of real numbers with the usual norm, , and define by
Set , , . Clearly, and
Clearly both T and S are Lipschitz operators on .
Also, it follows from (1.1) that
for any and . Thus T is strongly pseudocontractive and Lemma 7 ensures that T is strictly hemicontractive. Put
then it can be easily seen that
It follows from Theorem 9 that the sequence defined by (2.1) converges strongly to the common fixed point 0 of T and S in K and the iterative scheme defined by (2.1) is T-stable.
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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.
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Hussain, N., Rafiq, A. & Ciric, L.B. Stability of the Ishikawa iteration scheme with errors for two strictly hemicontractive operators in Banach spaces. Fixed Point Theory Appl 2012, 160 (2012). https://doi.org/10.1186/1687-1812-2012-160
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DOI: https://doi.org/10.1186/1687-1812-2012-160