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Strong convergence of new iterative algorithms for certain classes of asymptotically pseudocontractions
Fixed Point Theory and Applications volume 2013, Article number: 334 (2013)
Abstract
Let C be a nonempty closed convex subset of a real Hilbert space, and let be an asymptotically k-strictly pseudocontractive mapping with . Let and be real sequences in . Let be the sequence generated from an arbitrary by
where is the metric projection. Under some appropriate mild conditions on and , we prove that converges strongly to a fixed point of T. Furthermore, if is uniformly L-Lipschitzian and asymptotically pseudocontractive with , we first prove that is demiclosed at 0, and then prove that under some suitable conditions on the real sequences , and in , the sequence generated from an arbitrary by
converges strongly to a fixed point of T. No compactness assumption is imposed on T or C and no further requirement is imposed on .
MSC:47H09, 47J25, 65J15.
1 Introduction
Let H be a real Hilbert space with the inner product and the induced norm . Let C be a nonempty closed convex subset of H. A mapping is said to be L-Lipschitzian if there exists such that
T is said to be a contraction if , and T is said to be nonexpansive if . T is said to be asymptotically nonexpansive (see, for example, [1]) if there exists a sequence with such that
It is well known (see, for example, [1]) that the class of nonexpansive mappings is a proper subclass of the class of asymptotically nonexpansive mappings. T is said to be asymptotically k-strictly pseudocontractive (see, for example, [2]) if there exist and a sequence , such that
T is said to be asymptotically pseudocontractive if there exists a sequence , such that
It is well known that in real Hilbert spaces, the class of asymptotically nonexpansive maps is a proper subclass of the class of asymptotically k-strictly pseudocontractive maps. Furthermore, the class of asymptotically k-strictly pseudocontractive mappings is a proper subclass of the class of asymptotically pseudocontractive maps. T is said to be uniformly L-Lipschitzian if there exists such that
T is said to be demiclosed at p if whenever is a sequence in C which converges weakly to and converges strongly to p, then . It is well known that if is asymptotically k-strictly pseudocontractive, then T is uniformly L-Lipschitzian (see, for example, [3, 4]), and is demiclosed at 0 (see, for example, [5]). The modified Mann iteration scheme generated from an arbitrary by
where the control sequence is a real sequence in satisfying some appropriate conditions, has been used by several authors for the approximation of fixed points of asymptotically k-strictly pseudocontractive maps (see, for example, [2–10]). The iteration algorithm (1.5) is a modification of the well-known Mann iterative algorithm (see [11]) generated from an arbitrary by
where the control sequence is a real sequence in satisfying some appropriate conditions.
In real Hilbert spaces, it is known (see, for example, [3–5]) that if C is a nonempty closed convex subset of a real Hilbert space H, and is an asymptotically k-strictly pseudocontractive mapping with a sequence , , and a nonempty fixed point set , then the modified iteration sequence generated by (1.5) is an approximate fixed point sequence (i.e., as ) if . This together with the demiclosedness property of at 0 yields that converges weakly to a fixed point of T.
To obtain strong convergence of the modified Mann algorithm (1.5) to a fixed point of an asymptotically k-strictly pseudocontractive mapping, additional conditions are usually required on T and on the subset C (see, for example, [2–10]). Even for nonexpansive maps, additional conditions are required on T or C to obtain strong convergence using the Mann algorithm (1.6). In [12], Genel and Lindenstraus provided an example of a nonexpansive mapping defined on a bounded closed convex subset of a Hilbert space for which the Mann iteration does not converge to a fixed point of T. Recently Yao et al. [13] (see also [14, 15]) studied a modified Mann iteration algorithm generated from an arbitrary by
where and are real sequences in satisfying some appropriate conditions. They proved strong convergence of the modified algorithm to a fixed point of a nonexpansive mapping when . Clearly, the modified Mann iteration algorithm reduces to the normal Mann iteration algorithm when .
It is our purpose in this paper to modify algorithm (1.7) and prove that the modified algorithm converges strongly to a fixed point of an asymptotically k-strictly pseudocontractive mapping , where C is a nonempty closed convex subset of a real Hilbert space and . Furthermore, we prove that if is a uniformly L-Lipschitzian asymptotically pseudocontractive mapping, then is demiclosed at 0. We then introduce an iterative algorithm which converges strongly to a fixed point of a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with . The technique of proof of our convergence theorems follows the one proposed by Maingé [14].
2 Preliminaries
In what follows, we shall need the following results.
Lemma 2.1 [16]
Let be a sequence of nonnegative real numbers such that
where , , is a sequence of nonnegative real numbers and
-
(i)
, or equivalently, ,
-
(ii)
, and
-
(iii)
.
Then .
Let C be a closed convex subset of a real Hilbert space H. Let denote the metric projection (the proximity map) which assigns to each point the unique nearest point in C, denoted by . It is well known that
and that is nonexpansive.
It is also well known that in real Hilbert spaces H, we have the following (see, for example, [17]):
3 Main results
3.1 Strong convergence of an iterative algorithm for asymptotically k-strictly pseudocontractive maps
We now introduce the following iterative algorithm analogous to one studied in [13].
Modified averaging Mann algorithm Let C be a nonempty closed convex subset of a real Hilbert space H, and let be a given mapping. For arbitrary , our iteration sequence is given by
where and are suitable real sequences in satisfying some appropriate conditions that will be made precise in our strong convergence theorem.
We now prove the following convergence theorem.
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space, and let be an asymptotically k-strictly pseudocontractive mapping with a sequence such that . Let , and let and be sequences in satisfying the conditions:
-
(c1)
;
-
(c2)
;
-
(c3)
;
-
(c4)
, and for some ϵ.
Then the modified averaging iteration sequence generated from by (3.1) converges strongly to a fixed point of T.
Proof Observe that (1.3) is equivalent to each of the following inequalities:
Let be arbitrary. Then, using (1.3), (2.3), (3.1) and (3.3), we obtain
Hence
Since , it follows from (3.4) that is bounded. Hence is also bounded. Furthermore, it follows from (2.2) that
From (3.1) and (3.5) we obtain
Since is bounded, then
and hence using condition (c4) and (3.6) in (3.4), we obtain
Since is bounded, we have that there exists such that
From (3.7) and (3.8) we obtain
To complete the proof, we now consider the following two cases.
Case 1. Suppose that is a monotone sequence, then we may assume that is monotone decreasing. Then exists and it follows from (3.9), conditions (c1) and that
Furthermore,
Hence
and
Observe also that since T is uniformly L-Lipschitzian, we obtain
Furthermore,
Since , then the demiclosedness property of , (2.4) and the usual standard argument yield that and converge weakly to some .
Since , and since , , and for some , then using (3.4) we obtain
Thus
where , and , with . Since converges weakly to , then , and this together with condition (c1) (i.e., ) implies that as . It now follows from Lemma 2.1 that converges strongly to . Consequently, converges strongly to .
Case 2. Suppose that is not a monotone decreasing sequence, then set , and let be a mapping defined for all for some sufficiently large by
Then τ is a non-decreasing sequence such that as and for . Using (c1) and (c2) in (3.9), we obtain
Following the same argument as in Case 1, we obtain
As in Case 1, we also obtain that and converge weakly to some in . Furthermore, for all , we obtain from (3.13) that
It follows from (3.15) that
Thus
Furthermore, for , we have if (i.e., ), because for . It then follows that for all we have
This implies , and hence converges strongly to . □
Corollary 3.1 Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H with . Let be an asymptotically k-strictly pseudocontractive mapping with a sequence such that . Let , and let and be sequences in satisfying the conditions:
-
(c1)
;
-
(c2)
;
-
(c3)
;
-
(c4)
, and for some ϵ.
Then the modified averaging iteration sequence , generated from by
converges strongly to a fixed point of T.
Remark 3.1 Prototypes for our real sequences and are:
Corollary 3.2 Let C be a nonempty closed convex subset of a real Hilbert space, and let be an asymptotically nonexpansive mapping with a sequence . Let , and let and be sequences in satisfying the conditions:
-
(c1)
;
-
(c2)
;
-
(c3)
, and for some ϵ.
Then the modified averaging iteration sequence , generated from by (3.1), converges strongly to a fixed point of T.
3.2 Demiclosedness principle and strong convergence results for uniformly Lipschitzian asymptotically pseudocontractive maps
Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. For uniformly L-Lipschitzian asymptotically pseudocontractive maps , we first prove that is demiclosed at 0 and then introduce a modified averaging Ishikawa iteration algorithm and prove that it converges strongly to a fixed point of without any compactness assumption on T or C and without further requirement on . Our demiclosedness principle does not require the boundedness of C imposed in the result of [18].
Theorem 3.2 Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. Let be a uniformly L-Lipschitzian asymptotically pseudocontractive mapping. Then is demiclosed at 0.
Proof Let be a sequence in C which converges weakly to p and converges strongly to 0. We prove that . Since converges weakly, it is bounded. For each , define by
Observe that for arbitrary but fixed integer , we have
Set
where , and . Then
and
Hence
Also
From (2.4) we obtain
Thus
and hence
Observe that
Equations (3.17) and (3.18) imply that
from which it follows that
Thus
Hence
It now follows that as . Since T is continuous, we have that as , and hence . □
We now introduce the following iterative algorithm for uniformly L-Lipschitzian asymptotically pseudocontractive maps.
Modified averaging Ishikawa algorithm For arbitrary , the sequence is given by
Theorem 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H, and let be a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with a sequence , and . Let , and be real sequences in satisfying the conditions:
-
(c1)
;
-
(c2)
;
-
(c3)
, where ;
-
(c4)
.
Then the sequence generated from an arbitrary by (3.19) converges strongly to a fixed point of T.
Proof Since T is asymptotically pseudocontractive, it follows that
and
Set
Then, for arbitrary , we obtain
Thus
From (3.22) we obtain
and
Observe that
Hence
and it follows, as in the proof of Theorem 3.1, that is bounded. Observe that
Hence
Furthermore,
Thus
Observe also that
Using (3.26) and (3.28), we obtain
It now follows from (3.27) that
Using (3.29) in (3.25), we obtain
Since is bounded, we have that there exists such that
From (3.30) and (3.31) we obtain
To complete the proof, we now consider the following two cases.
Case 1. Suppose that is a monotone sequence, then we may assume that is monotone decreasing. Then exists and it follows from (3.32), conditions (c1) and that
Furthermore,
Hence
Furthermore,
Thus
and
Observe also that since T is uniformly L-Lipschitzian, we obtain
Furthermore,
Since , then the demiclosedness property of , (2.4) and the usual standard argument yield that and converge weakly to some . Since , , and for some , then using (3.25) we obtain
Thus
where as , and with . It now follows from Lemma 2.1 that converges strongly to . Consequently, converges strongly to .
Case 2. Suppose that is not a monotone decreasing sequence, then set , and let be a mapping defined for all for some sufficiently large by
Then τ is a non-decreasing sequence such that as and for . Using (c1) and (c2) in (3.32), we obtain
Following the same argument as in Case 1, we obtain
As in Case 1 we also obtain that and converge weakly to some in .
Furthermore, for all , we obtain from (3.36) that
It follows from (3.38) that
Thus
Furthermore, for , we have if (i.e., ), because for . It then follows that for all we have
This implies , and hence converges strongly to . □
Corollary 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H with , and let be a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with a sequence , and . Let , and be real sequences in satisfying the conditions:
-
(c1)
;
-
(c2)
;
-
(c3)
, where ;
-
(c4)
.
Then the sequence generated from an arbitrary by
converges strongly to a fixed point of T.
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Acknowledgements
The work was completed when the first author was visiting the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy as an Associate. He is grateful to the Centre for the invaluable facilities at the Centre and for hospitality. The authors are grateful to the referees for their valuable comments.
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MO conceived of the study. All authors carried out the research, read and approved the final manuscript.
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Osilike, M.O., Nwokoro, P.U. & Chima, E.E. Strong convergence of new iterative algorithms for certain classes of asymptotically pseudocontractions. Fixed Point Theory Appl 2013, 334 (2013). https://doi.org/10.1186/1687-1812-2013-334
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DOI: https://doi.org/10.1186/1687-1812-2013-334
Keywords
- asymptotically pseudocontractive maps
- fixed points
- strong convergence
- Hilbert spaces
- iterative algorithm