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Journal of Mathematics
Volume 2018, Article ID 3840784, 13 pages
Research Article

On Some Convergence Properties of the Modified Ishikawa Scheme for Asymptotic Demicontractive Self-Mappings with Matricial Parameterizing Sequences

Institute of Research and Development of Processes (IIDP), Facultad de Ciencia y Tecnologia, Universidad del País Vasco, Leioa (Bizkaia), P.O. Box 644, 48080 Bilbao, Spain

Correspondence should be addressed to M. De la Sen; sue.uhe@nesaled.leunam

Received 24 April 2018; Accepted 5 July 2018; Published 19 July 2018

Academic Editor: Oluwatosin T. Mewomoo

Copyright © 2018 M. De la Sen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper is focused on the modified Ishikawa iterative scheme by admitting that the parameterizing sequences might be vectors of distinct components. It is also assumed that the auxiliary self-mapping which supports the iterative scheme is asymptotically demicontractive.

1. Introduction

The study of iterative methods such as Krasnoselsky, Mann, and Ishikawa iterations as well as a large variety of extensions and their convergence properties have received special attention in the last decades. A detailed collection of existing results and new ones together with a very detailed discussion and comparative results between them is given in [1]. Ishikawa and Mann iterative schemes involve the use of an auxiliary self-mapping driven by one parameterizing sequence of scalars, say (Mann iterative scheme), or two ones, say (Ishikawa iterative scheme). It has been proved in [2] that if any of the Ishikawa and Mann iterations converges to a fixed point of the auxiliary self-mapping , then the other one converges to the same fixed point. The associated iterations are also compared to Picard’s type iterations in [2]. See also [3] and some references therein. It turns out that fixed points of certain self-mappings are very relevant in stability studies since they are equilibrium points of differential systems, difference systems, or dynamic continuous-time and discrete-time systems. Therefore, their characterization and the study of their properties as attractors are of a major importance in the modelling issues of biological systems, epidemic models, and mechanical, electrical, and control systems, in general. In the context of sequences generated from iterative schemes involving an auxiliary self-mapping , the usual assumptions being invoked are basically that is nonexpansive and defined on a nonempty closed convex subset of a normed space and that the sequences of parameters which generate the iterations are in and converge to zero and that the one being common to both schemes is a summable sequence. Further studies are developed in [4] for the case that two multivalued mappings are used as auxiliary mappings in the multivalued version of the Ishikawa iterative scheme. It is discussed in [1] that those iterations can be more robust against certain numerical errors than the Picard iterations. However, it is proved in [5] that, if the auxiliary self-mapping is a Zamfirescu operator [1], then Picard iteration through such an operator converges faster than those iterative schemes and than Mann’s iteration which converges faster than Ishikawa iteration in the framework of nonempty convex closed subsets of Banach spaces. The Mann and Ishikawa iterations under a class of errors when the auxiliary mapping is strongly accretive are studied in [6]. On the other hand, the Ishikawa iterative scheme is investigated in [7] when the auxiliary mapping is quasi-contractive and the convergence properties are linked to Kannan contractions, Chatterjea-type contractions, and Zamfirescu operators since all of them are always quasi-contractive. The convergence properties of the scheme are investigated in [8] under generalized nonexpansive mappings, while it was pointed out that although nonexpansive mappings are quasi-nonexpansive, if they have a fixed point, the converse is not true in general. Some close further formal results leading to several fixed point theorems concerning Ishikawa’s schemes are also given in [9]. On the other hand, the properties of convergence of the Ishikawa iteration are discussed in [10, 11] where the auxiliary self-maps include two auxiliary multivalued self-maps with common fixed points which are strongly pseudocontractive in the first case and satisfy a concrete general contractive condition in the second one. Also, it is found in [12] that continuous monotone and generalized quasi-nonexpansive self-mappings on nonempty compact and convex subsets of Hilbert spaces converge strongly to one of their fixed points under Ishikawa’s iterative scheme under certain standard conditions of its parameterizing sequences. On the other hand, it is proved in [13] that Mann’s iteration scheme converges strongly to the unique fixed point of the auxiliary mapping provided such a mapping is a Lipschitzian strong pseudocontraction defined on a compact convex subset of a Hilbert space under certain conditions of its single parameterizing sequence. The parameterizing sequences of those iterative schemes take real values in , even if the solution sequences of the Banach/Hilbert spaces are vector real or complex sequences, i.e., of dimension exceeding unity. In this paper, the modified Ishikawa scheme is revisited by admitting that the parameterizing sequences are real or complex sequences of matrices of, in general, distinct entries being of the same orders as the sequences generated from the iterative schemes. It is assumed that the auxiliary self-mapping is completely continuous, uniformly Lipschitzian, and asymptotically demicontractive. Such a self-map is defined, in general, on a nonempty compact subset of a Hilbert space but it is not assumed, in general, that it is convex since the uniqueness of the fixed point is not essential for the convergence purposes.

2. Some Preliminary Results

Some definitions and auxiliary results are given to be then invoked in the next section.

Definition 1 (see [1]). Let be a nonempty subset of a normed linear space and let be a mapping. Then, is said to be -strict asymptotically pseudocontractive if there exist a sequence , such that , and a constant such that for all and

Definition 2 (see [1]). Let be a self-mapping with being a nonempty subset of a normed linear space and a fixed point set . Then, is said to be asymptotically demicontractive if there exist a sequence such that and a constant such that for all , , and , one hasNote that if is -strict asymptotically pseudocontractive with , then is asymptotically demicontractive since, by taking in (1), one gets (2). Note also that the extended sequence can be considered in (1)–(2), with , instead of , since , for any so that (1)–(2) still hold trivially for since .
Let be a nonempty bounded closed subset of a normed linear space of dimension and let be an asymptotically demicontractive mapping. Consider the sequence , with , defined byfor , where and for and is the th identity matrix. Note that (3) is a generalized modified Ishikawa iterative process in the sense that it is applicable to real or complex scalar and vector sequences and and are, in general, nondiagonal nonsparse matrices with, in general, distinct diagonal entries if . Equaion (3) may be equivalently rewritten as follows:for , where and are real parameterizing sequences in which drive the iterative scheme together with the auxiliary self-mapping . Note that the sequences and take into account the contributions of the disturbances related to the standard modified Ishikawa sequence caused by the presence of couplings between the components of and and the errors of the diagonal parts of the matrix sequences and related to the case of being diagonal with identical diagonal entries defined by the sequences and , respectively.

The following result is concerned with the convergence of the solution of the iterated scheme when the sequences and converge to the limit kernels of the parameterizing matrix sequences errors related to a diagonal matrix with identical entries.

Theorem 3. Let be a nonempty subset of a normed linear space of dimension and let the auxiliary self-mapping be an asymptotically demicontractive mapping. Then, the following properties hold:
for any sequence and some real sequence , such that , some real constant , and any , where Assume that with where exists and it is defined as a theoretic limit set byThen, . This holds, in particular, if leading to , so that , and for some , which leads to .
Assume that for some and is pointwise convergent to the nonempty limit nonsingular -matrix
. Then,If, in addition, , then .
Assume that such that the set exists and it is defined byThen, and . This holds, in particular, if for some , , ; then , so that , which leads to and .

Proof. Since is asymptotically demicontractive, . Then, there exists a sequence such that and a real constant such that, for any , , and one has:Then, one gets from (4) and (10) that, for any ,Property (i) follows directly from (11)-(12). On the other hand, under the existence condition of the set-theoretic limit , it follows that the theoretic limit set exists as the nonempty limit and, since for , if since (a) for is closed since each linear operator is continuous and of finite dimension, so its Kernel is closed for any and, (b) for all some nonnegative integer and each integer . Thus, is in all except finitely often (and also in infinitely often) sets since it belongs to because of the identity of limit supremum and limit infimum which defines . Then, , . Also, for any , one hasand since , . Thus, if and . Then, . This holds, in particular, if (then ), and one concludes that from the third identity of (4). Property (ii) has been proved.
Note that if , then the replacements of and in (4) yield and since exists and it is nonsingular. Therefore, , , , and . If, in addition, , then from (4). Property (iii) has been proved. On the other hand, if , then and from (4). If, for some , and , then , , , and then from (4); thus Property (iv) is proved.

Note that no Property of Theorem 3 assumes any of the constraints or . Note also that, if in Theorem 3, then is achievable if (nonsingular) pointwise or if and , each case under certain supplementary conditions. Note thatIf and , thenThe subsequent result establishes that, under a similar condition as (2) of asymptotic demicontractiveness of an uniformly -Lipschitzian operator [1] on a Hilbert space for given alternative specific constraints on and , the boundedness of implies that of and . Also, the strong convergence of to a fixed point of implies the convergence to zero of and .

Theorem 4. Let be a Hilbert space and let be an operator on a Hilbert space with a fixed point which is uniformly -Lipschitzian and satisfies (2) with and . Then, the sequences and are bounded if is bounded. If, furthermore, , then and . If, in addition, is continuous at , then .

Proof. Note thatThus,so that if is bounded, then and are bounded since and . Furthermore, if , then and . If, in addition, is continuous at , then since and .
Now, an illustrative example with two “ad hoc” specific results (Lemmas 6 and 7) is given and discussed for the modified Ishikawa iterative scheme under structured errors for the case of a linear bounded operator on .
Example 5. Let the iterative schemefor . Consider the simplest case that defined on is linear and bounded with norm upper-bound . Note that the assumption that is a self-mapping on a bounded set to which, furthermore, the initial condition of the iteration belongs is removed. Now, assume thatfor with , for . Note that and , where and are the diagonal parts of and , which are zero if all their nonzero entries equalize and, respectively, , and and are their off-diagonal parts. Calculations via (4) and (24) yieldwherewhich is zero if for any . Assume that where and for . Then, since , note that ; withLet the big Landau’s” and small Landau’s ” such that for, in general, complex vector sequences and , if ; for some nonnegative real constants and , and if and . Note the following:
if . In particular, this constraint holds if(a) and for ,(b) and ,(c) and according to ,(d) and satisfying ; that is, , equivalently, is unbounded according to . if . In particular, this holds if and , if and , and if and .
is bounded if there exists a nonnegative finite real constant such that for , where and for . This holds, in particular, if and is bounded or if and is bounded. Note from (25) thatThe following two results hold directly from (28) and the given conditions in the case when , and boundedness of in Example 5.
Lemma 6. The solution sequence is bounded for any finite initial condition if the following constraints hold: The second constraint holds, in particular, ifsince then either or , or for , and

Lemma 7. Assume that and that (29)-(30) hold for the sequence , such that , and for for and for some given bounded set .
Then, the following properties hold:(a)if the constraints (29)-(30) also hold for the perturbation-free case, i.e., if , or(b)if (it suffices that ) and .

(ii) if the set-theoretic limit exists, , and , a particular case arises if , , and .

(iii) if as pointwise and .

(iv) Assume that the theoretic limit set exists; that is,with , either exists or pointwise as , , and , Then, .

Proof. Assume that is then sequence generated from the iteration scheme with for . One gets from (25) that the perturbed and disturbance-free solutions areSince (29)-(30) hold for the sequence and for , one has from Lemma 6 that, for any : , are bounded and then for . Thus, Property (i) holds under conditions (a). On the other hand, if , then since is bounded from Lemma 6 and , and then . Thus, since convergent sequences are bounded. Since , it follows that (35) holds, and then Property (i) holds under conditions (b). Property (ii) follows directly since as from (26) since the set- theoretic limit exists, and . To prove Property (iii), note that if , then for . Since , thenSince , thenProperty (iv) follows if either Property (ii) or Property (iii) holds and, furthermore, there exists the set-theoretic limit , , and since .

Inspired by Example 5, we get the following result, to be then used, on sufficient conditions for asymptotic vanishing of the perturbation sequence by removing the assumption of linearity of .

Theorem 8. Assume that is a normed linear space and that is a sequence of bounded operators on . Definefor .Then, the following Items hold:
if , and if either and as , or and as .
Define the sequences and for and assume that , pointwise, such that is closed, and . Then, and(ii. a) for (ii. b)If , equivalently if , then if is nonsingular.(ii. c)If and , in particular if , then . If pointwise, such that is closed, and , then , and if and , in particular if , then .
If pointwise, pointwise, and are closed, , , and , then and . If , then , and if or is nonsingular.

Proof. From the last two equations of (4), we havewithout requiring the linearity of
for . Since for , one gets that for , hence Property (i) of the lemma. Property (ii. a) follows by direct calculations, Property (ii. b) follows since if <