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

Abstract

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 and , then and since is closed. As a result, if , then and if is nonsingular since then . Property (ii. c) follows since pointwise as , , and , in particular if , thenand then as . Property (iii) follows under a similar reasoning to that used to prove Property (ii. c). Property (iv) follows from Properties and the Rouché-Frobenius theorem from Linear Algebra.

Assume that . Then via (43) if and are operators on , , such that is closed, pointwise and either(a) as , or(b) and as and (then , and ), or(c), as

Note that and as if , , and as .

Definition 9 (see [1]). Let be a nonempty subset of a normed linear space . is said to be uniformly - Lipschitzian with constant if for .
Note that if is asymptotically nonexpansive, then it is -strict pseudocontractive and uniformly -Lipschitzian, [1]. Note also that if is uniformly - Lipschitzian and , then the sequence of operators is uniformly bounded since by taking , one has for all and .

Lemma 10. Let be asymptotically demicontractive. The following properties hold:
If Lipschitzian with constant (i.e., is -Lipschitzian), then for any If is -strict asymptotically pseudocontractive and uniformly -Lipschitzian, then for any Let be -strict asymptotically pseudocontractive. The following properties hold:
If is -strict asymptotically pseudocontractive and -Lipschitzian with constant, then If is -strict asymptotically pseudocontractive and uniformly -Lipschitzian then

Proof. One gets from (2) thatfor all and . Property (i) follows directly since . Property (ii) follows by replacing in (49). On the other hand, if is -strict asymptotically pseudocontractive and -Lipschitzian, then for a constant and some real sequence , one gets Property (iii) since for any Property (iv) follows by replacing in (50).

Lemma 11 (see [1]). Let , , and be points in Hilbert space and . Then

3. Main Results

The main result follows.

Theorem 12. Let be a nonempty bounded closed subset of a Hilbert space and let be a completely continuous uniformly -Lipschitzian asymptotically demicontractive mapping. Suppose that the following conditions hold:(a) is summable,(b), ,(c) and are summable, and either(d1), , where for any , where if and if , or(d2), (implying that ), and is summable with positive sum, or(d3), (implying that ), and is summable with positive sum. Then, converges strongly to some fixed point of in .

Proof. One gets from the third equation of (4) and ((41a) and (41b)) thatwhere and , . Equations (53)-(54) agree with ((41a) and (41b)) and (4). Since is asymptotically demicontractive, . Let . One gets from Lemma 11, with , thatThe following relations will be used:where if and if for any . Then, one has from (55) and (56), again since is asymptotically demicontractive, thatOn the other hand, one gets from the second equation of (4) thatwhereThe following relations are afterward used:where if and if for any . Again, using Lemma 11, (58), (59), and (60), one gets thatThen, Since is asymptotically demicontractive and -uniformly Lipschitzian, one gets from (4), Lemma 11, and the second relation of (4) thatwhich leads toNow, one gets from (62) and (64) into (57) thatThus, the constraints on the sequences , and , and being implicit in the conditions (a) to (c) of the theorem statement lead toso that for some finite real constant ; since , , and are summable as a result following from condition (c), the definitions of , , and from (4) and (43) with , , and are bounded so and are bounded and is bounded since is asymptotically demicontractive and . If condition (d1) holds then the convex parabola for any such that where and are the roots of . Therefore, one gets from (67), since , thatAs a result, , , since is closed, and . Conditions (d2) allow modifying (68) as where and the same conclusions on the convergence of the involved sequences follow. On the other hand, conditions (d3) allow modifying (68) aswhere and the same conclusions follow again on the convergence of the involved sequences.
Now, it is proven that proceeding by contradiction. Assume that does not converge to as Then, there is a subsequence which does not converge to zero. Since is uniformly -Lipschitzian, one hasand then, since , the subsequence with for any such that . Thus, , hence a contradiction. Then, implies that . Since the set is bounded and closed and is completely continuous, has a subsequence such that for some . Since , and since is continuous. Now, the constraints (68) to (70) imply in any case that for such that with and . One concludes from Lemma 1.7 of [1] that , hence .

The following result is a parallel one to Theorem 12 under the assumption that satisfies an asymptotically demicontractive-like condition (2), in the sense that (2) holds with and and it has a unique fixed point.

Theorem 13. Let be a nonempty bounded closed subset of a of a Hilbert space and let be a uniformly -Lipschitzian satisfying (2) and being continuous at such that and , and . Suppose also that the following conditions hold:(a) and ,(b), ,(c) and are summable, and either (d1) or (d2) below holds:(d1), , where(d2) and is summable. Then, .

Proof. It is possible to obtain from (66), under the given conditions, thatwhereand under condition (d1) since the second right-hand-side term of (73) is nonpositive and can be omitted in the analysis and under condition (d2) is nonnegative and is summable since is bounded since is bounded. The third right-hand-side term of (73) is nonpositive under condition (b), and it can be omitted in the analysis, and the last two terms of (73) are summable from condition (c). Then, is summable and has nonnegative elements while and from conditions (a). The result follows from Venter’s theorem, [14].

Corollary 14. Theorem 13 holds if the second constraint of the condition (a) is modified as follows:for some real sequences and fulfilling for .

Proof. Note that ((76a) and (76b)) are guaranteed ifand there exists some finite integer such that and, since and , one has since Therefore, condition (a) of Theorem 13 holds.

Two examples are now given concerning the use of simple auxiliary asymptotically demicontractive self-mappings in the generalized modified Ishikawa’s iterative schemes under matrix parameterizing sequences.

Example 15. The condition that the mapping is asymptotically demicontractive prior to its use as an auxiliary self-mapping in a generalized modified Ishikawa’s scheme includes many other stronger properties for such a self-mapping as, for instance, contractive, asymptotically contractive, and asymptotically pseudocontractive self-mappings. Consider a self-mapping defined on the first closed orthant of which generates sequences as with for , such that , where , subject tofor some sequences such that and , and some real constant . From the above constraint, it follows that provided that . Assume also that for with for , , , , andsince , with . One has from (81) that such an equation can be rewritten as and also as Thus, converges to some fixed point of the self-mapping on , such that ; that is, the fixed point has the same norm as that of the initial condition of the generated sequence which is, furthermore, asymptotically demicontractive and asymptotically nonexpansive in view of (80) and (81). If a generalized modified Ishikawa’s iterative process with matrix parameterizing sequences is implemented with such an auxiliary mapping on , then any such obtained generated sequence converges asymptotically to zero under the given constraints of the results in the paper body. That is the case of Theorems 3 and 4 and Theorem 13 under the respective groups of invoked extra constraints on the matrix parameterizing sequences.

Example 16. Consider a self-mapping on with such that any sequence generated as , , for a given , such that for some norm (for instance, the Euclidean norm,   for ) satisfies the following norm constraint:where , , and are parameterizing real sequences subject to the constraints:(a), ;(b), , for some ;(c); ;(d);(e);(f);(g). Note that (81) is identical toNote from (a), (b), (c), (d), (g), and Venter’s theorem, [14], that which implies that . Note also that, since for , (85), or (86), and the constraints (b) and (f) imply that with , where , , so that from the constraint (c). Since (87) implies that the squared norm of the elements of satisfies the asymptotic demi-contractiveness condition , it follows that the self-mapping on is asymptotically demicontractive and which is a similar conclusion to that obtained from Venter’s theorem.
Note that conditions (b) and (f) can be guaranteed if there exist either a finite set or sequence of nonnegative integer numbers with ( denoting the infinity cardinal of denumerable sets) and an associated set or sequence of nonnegative real numbers such thatwhich holds if the parameterizing sequences fulfil the constraints:with the convention that . Note that the condition (g) with (b) is guaranteed, for instance, if for a finite set or for sequence of nonnegative real numbers , which is guaranteed if If a generalized modified Ishikawa’s iterative process with matrix parameterizing sequences is implemented with such an auxiliary mapping on , then any such obtained generated sequence converges asymptotically to zero under the given constraints of the above results given in the paper body.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that he does not have any conflicts of interest.

Acknowledgments

This research is supported by the Spanish Government and by the European Fund of Regional Development (FEDER) through Grant DPI2015-64766-R and by UPV/EHU through Grant PGC 17/33.