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

Authors: M. De la Sen
Journal: Journal of Mathematics
 (2018)

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## 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.


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## Body

## 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 T driven by one parameterizing sequence of scalars, say αnn=0∞⊂0,1 (Mann iterative scheme), or two ones, say αnn=0∞,βnn=0∞⊂0,1 (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 T, 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 T, the usual assumptions being invoked are basically that T 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 0,1 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 0,1, 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]).
LetS be a nonempty subset of a normed linear space E and let T:S⟶S be a mapping. Then, T is said to be k-strict asymptotically pseudocontractive if there exist a sequence knn=1∞⊂1,∞, such that knn=1∞⟶1, and a constant k∈0,1 such that for all x,y∈S and n≥1(1)Tnx-Tny2≤kn2x-y2+kx-Tnx-y-Tny2.Definition 2 (see [1]).
LetT:S⟶S be a self-mapping with S being a nonempty subset of a normed linear space E and a fixed point set FT≠⌀. Then, T is said to beasymptotically demicontractiveif there exist a sequence knn=1∞⊂1,∞ such that knn=1∞⟶1 and a constant k∈0,1 such that for all x,y∈S, p∈FT, and n≥0, one has(2)Tnx-p2≤kn2x-p2+kx-Tnx2.Note that if T is k-strict asymptotically pseudocontractive with FT≠⌀, then T is asymptotically demicontractive since, by taking y=p∈FT in (1), one gets (2). Note also that the extended sequence knn=0∞=k0∪knn=1∞⊂1,∞ can be considered in (1)–(2), with k0∈1,∞, instead of knn=1∞, since x-T0x=0, for any x∈S so that (1)–(2) still hold trivially for n=0 since x-y2=T0x-y2≤k02x-y2≤x-y2.
LetS be a nonempty bounded closed subset of a normed linear space E of dimension r and let T:S⟶S be an asymptotically demicontractive mapping. Consider the sequence xnn=0∞, with xn∈Cr, defined by(3)x0∈Syn=Ir-Bnxn+BnTnxnxn+1=Ir-Anxn+AnTnynfor n≥0, where An∈Cr×r and Bn∈Cr×r for n≥0 and Ir is the rth identity matrix. Note that (3) is a generalized modified Ishikawa iterative process in the sense that it is applicable to real or complex scalar r=1 and vector r≥2 sequences and An and Bn are, in general, nondiagonal nonsparse matrices with, in general, distinct diagonal entries if r≥2. Equaion (3) may be equivalently rewritten as follows:(4)x0∈Syn=1-βnxn+βnTnxn+vnxn+1=1-αnxn+αnTnyn+unvn=vnxn=βnIr-Bnxn-Tnxnun=unxn,ynxn=αnIr-Anxn-Tnynfor n≥0, where αnn=0∞ and βnn=0∞ are real parameterizing sequences in 0,1 which drive the iterative scheme together with the auxiliary self-mapping T. Note that the sequences unn=0∞ and vnn=0∞ 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 xnn=1∞ and ynn=1∞ and the errors of the diagonal parts of the matrix sequences Ann=0∞ and Bnn=0∞ related to the case of being diagonal with identical diagonal entries defined by the sequences αnIrn=0∞ and βnIrn=0∞, respectively.The following result is concerned with the convergence of the solution of the iterated scheme when the sequencesxn-Tnxnn=0∞ and xn-Tnynn=0∞ converge to the limit kernels of the parameterizing matrix sequences errors related to a diagonal matrix with identical entries.Theorem 3.
LetS be a nonempty subset of a normed linear space E of dimension r and let the auxiliary self-mapping T:S⟶S be an asymptotically demicontractive mapping. Then, the following properties hold:
( i )   lim  sup n ⟶ ∞ ⁡ T n x n - p + v n 2 - 1 + θ n ν n k n 2 x n - p 2 - 1 + θ n ν n k + β n I r - B n 2 θ n ν n x n - T n x n 2 ≤ 0 (5) l i m  s u p n ⟶ ∞  T n x n - p + u n 2 - 1 + ϑ n u n k n 2 x n - p 2 - 1 + ϑ n u n k + α n I r - A n 2 ϑ n u n x n - T n x n 2 ≤ 0for any sequence xnn=0∞⊂S and some real sequence knn=0∞⊂1,∞, such that knn=0∞⟶1, some real constant k∈0,1, and any p∈FT, where(6)θnνn=0ifνn=01ifνn≠0;⁡ϑnun=0ifun=01ifun≠0forn≥0.
( i i ) Assume that xn-Tnxnn=0∞⟶pB with pB∈KB where KB exists and it is defined as a theoretic limit set by(7)KB=liminfn⟶∞⋃n≥1⋂j≥nKerβjIr-Bj=limsupn⟶∞⋂n≥1⋃j≥nKerβjIr-Bj≠⌀.Then, vnn=0∞⟶0. This holds, in particular, if k∈0,1/2 leading to KB=0, so that pB=0, and for some p∈FT,xnn=0∞⟶p which leads to Tnxnn=0∞⟶p.
( i i i ) Assume that xnn=0∞⟶p for some p∈FT and Ann=0∞ is pointwise convergent to the nonempty limit nonsingular r-matrix
A ∞ = lim n ⟶ ∞ ⁡ ⋃ n ≥ 1 ⋂ j ≥ n A j = lim  inf n ⟶ ∞ ⁡ ⋃ n ≥ 1 ⋂ j ≥ n A j = lim  sup n ⟶ ∞ ⁡ ⋂ n ≥ 1 ⋃ j ≥ n A j. Then,(8)Tnynn=0∞⟶p,Tnyn-xnn=0∞⟶0,Tnyn-xn+1n=0∞⟶0,unn=0∞⟶0.If, in addition, vnn=0∞⟶0, then ynn=0∞⟶p.
( i v ) Assume that xn-Tnynn=0∞⟶pA∈KA such that the set KA exists and it is defined by(9)KA=liminfn⟶∞⋃n≥1⋂j≥nKerαjIr-Aj=limsupn⟶∞⋂n≥1⋃j≥nKerαjIr-Aj≠⌀.Then, unn=0∞⟶0 and xn-Tnynn=0∞⟶0. This holds, in particular, if for some p∈FT, xnn=0∞⟶p, k∈0,1/2; then KA=0, so that pA=0, which leads to Tnynn=0∞⟶p and ynn=0∞⟶p.Proof.
SinceT:S⟶S is asymptotically demicontractive, FT≠⌀. Then, there exists a sequence knn=0∞⊂1,∞ such that knn=0∞⟶1 and a real constant k∈0,1 such that, for any x∈S, p∈FT, and one has:(10)Tnx-p2≤kn2x-p2+kx-Tnx2;n≥0Then, one gets from (4) and (10) that, for any xnn=0∞⊂S,(11)Tnxn-p+vn2≤1+θnνnTnxn-p2+2θnνnvn2≤1+θnνnkn2xn-p2+kxn-Tnxn2+2θnνnvn2≤1+θnνnkn2xn-p2+1+θnνnk+θnνnβnIr-Bn2xn-Tnxn2≤2kn2xn-p2+k+βnIr-Bn2xn-Tnxn2;n≥0(12)Tnxn-p+un2≤1+ϑnunTnxn-p2+2ϑnunun2≤1+ϑnunkn2xn-p2+kxn-Tnxn2+2ϑnunun2≤1+ϑnunkn2xn-p2+1+ϑnunk+ϑnunαnIr-An2xn-Tnxn2≤2kn2xn-p2+k+αIr-An2xn-Tnxn2;n≥0.Property (i) follows directly from (11)-(12). On the other hand, under the existence condition of the set-theoretic limit KB, it follows that the theoretic limit set exists as the nonempty limit KB=limn⟶∞⁡⋃n≥1⋂j≥nKerβjIr-Bj⊃pB and, since vn=βnIr-Bnxn-Tnxn for n≥0, vnn=0∞⟶0 if xn-Tnxnn=0∞⟶pB since (a) KBn=KerβnIr-Bn for n≥0 is closed since each linear operator βnIr-Bn is continuous and of finite dimension, so its Kernel is closed for any n≥0 and, (b) pB∈KerKB⊆⋂j>n0nKerKBj for all n≥0 some nonnegative integer n0=n0n and each integer n≥0. Thus, pB is in all except finitely often (and also in infinitely often) sets KerKBj since it belongs to KB because of the identity of limit supremum and limit infimum which defines KB. Then, xn-Tnxnn=0∞⟶pB, limn⟶∞vn=βnIr-Bnxn-Tnxn=KBpB=0. Also, for any p∈FT, one has(13)Tnxn-xn2=Tnxn-p+p-xn2≤2Tnxn-p2+p-xn2≤2kn2xn-p2+kxn-Tnxn2+p-xn2;n≥0and since knn=0∞⟶1, limsupn⟶∞1-2kxn-Tnxn2-4p-xn2≤0. Thus, xn-Tnxnn=0∞⟶0 if k∈0,1/2 and xnn=0∞⟶p. Then, Tnxnn=0∞⟶p. This holds, in particular, if KB=0 (then pB=0), and one concludes that vnn=0∞⟶0 from the third identity of (4). Property (ii) has been proved.
Note that ifxnn=0∞⟶p, then the replacements of xn+1-Tnyn and unn=0∞ in (4) yield (14)0=limn⟶∞xn+1-Tnyn-1-αnxn-Tnyn-un=limn⟶∞p-Tnyn-1-αnp-Tnyn-un=limn⟶∞αnp-Tnyn-un=limn⟶∞αnp-Tnyn+An-αnIrp-Tnyn=limn⟶∞Anp-Tnyn=limn⟶∞A∞p-Tnyn=A∞limn⟶∞p-Tnyn=0and limn⟶∞⁡p-Tnyn∈KerA∞=0 since A∞ exists and it is nonsingular. Therefore, Tnynn=0∞⟶p, Tnyn-xnn=0∞⟶0, Tnyn-xn+1n=0∞⟶0, and unn=0∞⟶0. If, in addition, vnn=0∞⟶0, then ynn=0∞⟶p from (4). Property (iii) has been proved. On the other hand, if xn-Tnynn=0∞⟶pA∈KA, then unn=0∞⟶0 and xn-Tnynn=0∞⟶0 from (4). If, for some p∈FT, xnn=0∞⟶p and k∈0,1/2, then pA=0, KA=0, Tnynn=0∞⟶p, and then ynn=0∞⟶p from (4); thus Property (iv) is proved.Note that no Property of Theorem3 assumes any of the constraints liminfn⟶∞⁡αn>0 or liminfn⟶∞⁡βn>0. Note also that, if xnn=0∞⟶p∈FT in Theorem 3, then ynn=0∞⟶p is achievable if Ann=0∞⟶A∞ (nonsingular) pointwise or if KA=0 and k<1/2, each case under certain supplementary conditions. Note that(15)xn-Tnyn=xn-Tnxn+Tnxn-p+p-Tnyn≤Tnxn-p+p-Tnyn+xn-Tnxn≤kn2xn-p2+kxn-Tnxn+kn2yn-p2+kyn-Tnyn+xn-Tnxn≤knxn-p+yn-p+kxn-Tnxn+yn-Tnyn+xn-Tnxn;n≥0(16)limsupn⟶∞xn-Tnyn-xn-p-yn-p-1+kxn-Tnxn-kyn-Tnyn≤0(17)limsupn⟶∞yn-Tnxn-yn-p-xn-p-1+kyn-Tnyn-kxn-Tnxn≤0.If xnn=0∞⟶p and xn-Tnxnn=0∞⟶0, then(18)limsupn⟶∞xn-Tnyn-yn-p-kyn-Tnyn≤0(19)limsupn⟶∞yn-Tnxn-yn-p-1+kyn-Tnyn≤0.The subsequent result establishes that, under a similar condition as (2) of asymptotic demicontractiveness of an uniformly L-Lipschitzian operator [1] on a Hilbert space for given alternative specific constraints on kn and k, the boundedness of xnn=0∞ implies that of xn-Txnn=0∞ and xn-Tnxnn=0∞. Also, the strong convergence of xnn=0∞ to a fixed point of T implies the convergence to zero of xn-Txnn=0∞ and xn-Tnxnn=0∞.Theorem 4.
LetH be a Hilbert space and let T:H⟶H be an operator on a Hilbert space H with a fixed point p∈FT which is uniformly L-Lipschitzian and satisfies (2) with knn=0∞⊂1,∞ and k∈0,1/2. Then, the sequences xn-Tnxnn=0∞ and xn-Txnn=0∞ are bounded if xnn=0∞ is bounded. If, furthermore, xnn=0∞⟶p, then xn-Tnxnn=0∞⟶0 and xn-Txnn=0∞⟶0. If, in addition, T is continuous at p, then Txnn=0∞⟶p.Proof.
Note that(20)Tnxn-p≤2knxn-p+kxn-Tnxn;n≥0,xn-Tnxn≤xn-p+p-Tnxn≤1+2knxn-p+2kxn-Tnxn;n≥0.Thus,(21)xn-Tnxn≤1+2kn1-2kxn-p;n≥0(22)xn-Txn≤Tnxn-Txn+xn-Tnxn≤1+2kn1-2kxn-p+L1+2kn-11-2kxn-1-p;n≥0so that if xnn=0∞ is bounded, then xn-Tnxnn=0∞ and xn-Txnn=0∞ are bounded since knn=0∞⊂1,∞ and k∈0,1/2. Furthermore, if xnn=0∞⟶p, then xn-Tnxnn=0∞⟶0 and xn-Txnn=0∞⟶0. If, in addition, T is continuous at p, then Txnn=0∞⟶p since xnn=0∞⟶p and xn-Txnn=0∞⟶0.
Now, an illustrative example with two “ad hoc” specific results (Lemmas 6 and7) is given and discussed for the modified Ishikawa iterative scheme under structured errors for the case of a linear bounded operator T on Rr.
Example 5. Let the iterative scheme(23)xn+1=Ir-An+AnTnIr-Bn+BnTnxn;x0∈Rrfor n≥0. Consider the simplest case that T defined on Rr is linear and bounded with norm upper-bound K. Note that the assumption that T is a self-mapping on a bounded set to which, furthermore, the initial condition of the iteration belongs is removed. Now, assume that(24)An=αnIr+A~n;Bn=βnIr+B~nfor n≥0 with αn, βn∈0,1 for n≥0. Note that A~n=Adn-αnIr+Aodn and B~n=Bdn-βnIr+Bodn, where Adn and Bdn are the diagonal parts of An and Bn, which are zero if all their nonzero entries equalize αn and, respectively, βn, and Aodn and Bodn are their off-diagonal parts. Calculations via (4) and (24) yield(25)xn+1=1-αn+αn1-βn+αnβnTnTnxn+Δnxn;n≥0where(26)Δn=Δnxn,ynxn=αnTnB~nTnxn-xn+A~nTnynxn-xn=Δnxn=αnTnB~n+A~nTnB~n+βnA~nTn+A~nTnxn-xn;n≥0which is zero if A~n=B~n=0 for any n≥0. Assume that εn=maxεan,εbn where εan≥A~n and εbn≥B~n for n≥0. Then, since T≤K, note that Δn≤hnxn; n≥0 with(27)hn=1+Knεn1+αn+βn+εnKn;n≥0.Let thebig Landau’s “O” andsmall Landau’s“o” such that for, in general, complex vector sequences fn and gn, fn=Ogn if fn≤K1gn+K2; n≥0 for some nonnegative real constants K1 and K2, and fn=ogn if fn=Ogn and fn/gn⟶0. Note the following:
( 1 )   h n ⟶ 0 if 1+Knεn1+αn+βn+εnKn=o1. In particular, this constraint holds if(a)
K<1 and εn≤ε-<∞ for n≥0,(b)
K=1 and εnn=0∞⟶0,(c)
K>1 and εnn=0∞⟶0 according to εn=oK-2n,(d)
εnn=0∞⟶∞ and K<1 satisfying Kn=oεn-2; that is, εnn=0∞, equivalently, is unbounded according to εn=oK-n/2.
( 2 )   h n x n n = 0 ∞ ⟶ 0 if 1+Knεn1+αn+βn+εnKn=oxn-1. In particular, this holds if K<1 and εn=oxn-1, if K>1 and εn=oK-2n1+αn+βnxn-1, and if K=1 and εn=o1+αn+βn-1xn-1.
( 3 )   h n n = 0 ∞ is bounded if there exists a nonnegative finite real constant C such that hn=anbn≤C for n≥0, where an=1+Knεn and bn=1+αn+βn+εnKn for n≥0. This holds, in particular, if an=Obn-1 and bn is bounded or if bn=Oan-1 and an is bounded. Note from (25) that(28)xn+1≤1-αn+αn1-βn+αnβnKnKn+hnxn;n≥0.The following two results hold directly from (28) and the given conditions in the case when hnn=0∞⟶0, hnxnn=0∞⟶0 and boundedness of hnn=0∞ in Example 5.
Lemma 6.The solution sequencexnn=0∞is bounded for any finite initial conditionx0if the following constraints hold: (29)liminfn⟶∞hn+αn1-βn+αnβnKnKn-αn>-1(30)limsupn⟶∞hn+αn1-βn+αnβnKnKn-αn<0.The second constraint holds, in particular, if(31)1+Knεn1+αn+βn+εnKn=maxo1,o1+αn+βn-1xn-1,(32)limsupn⟶∞αn1-βn+αnβnKnKn-αn<0since then eitherhnn=0∞⟶0orhnxnn=0∞⟶0, orhn≤Cforn≥0, and(33)liminfn⟶∞αn1-βn+αnβnKnKn-αn>-1(34)limsupn⟶∞αn1-βn+αnβnKnKn-αn<-C.Lemma 7.
Assume thatT:Rr⟶Rr and that (29)-(30) hold for the sequence hnn=0∞, such that Δn≤hnxn, and for hn≡0 for n≥0 and x0∈S for some given bounded set S⊂Rr.
Then, the following properties hold:(35)i∑i=0n∏j=i+1n1-αjΔixix0<∞(a)
if the constraints (29)-(30) also hold for the perturbation-free case, i.e., if hnn=0∞≡0, or(b)
if∑i=0∞∏j=i+1∞1-αj<∞ (it suffices that ∑i=0∞1-αi<∞) and 1+Knεn1+αn+βn+εnKn=o1.(ii)Δnxnn=0∞⟶0 if the set-theoretic limit KerΔ∞0=limn⟶∞⁡KerΔn0 exists, KerΔ∞0∩S≠⌀, and Tnxn-xnn=0∞⟶pΔ0∈KerΔ∞0∩S, a particular case arises if 0∈S, KerΔ∞0=0, and Tnxn-xnn=0∞⟶0.(iii)Δnxnn=0∞⟶0 if Δn0⟶Δ∞0 as n⟶∞ pointwise and Tnxn-xnn=0∞⟶pΔ0∈KerΔ∞0∩S.(iv) Assume that the theoretic limit setTS∞=limn⟶∞⁡TnS exists; that is,(36)TS∞=limn⟶∞⋃n≥1⋂j≥nTjS=liminfn⟶∞⋃n≥1⋂j≥nTjS=limsupn⟶∞⋂n≥1⋃j≥nTjSwith TS∞⊆S, either KerΔ∞0=limn⟶∞⁡KerΔn0 exists or Δn0⟶Δ∞0 pointwise as n⟶∞, 0∈KerΔ∞0∩S, and xn⟶p∈FT, Then, Δnxn⟶0.Proof.
Assume thatxn0 is then sequence generated from the iteration scheme with Δn≡0 for x00=x0. One gets from (25) that the perturbed and disturbance-free solutions are(37)xn+1=xn+10+∑i=0n∏j=i+1n1-αjΔixi;n≥0(38)xn+10=∏j=0n1-αj+∑i=0n∏j=i+1n1-αjαi1-βi+αiβiTiTix0;n≥0.Since (29)-(30) hold for the sequence hnn=0∞ and for hn≡0, one has from Lemma 6 that, for any x0∈S: xnn=0∞, xn0n=0∞ are bounded and then xn+1-xn+10=∑i=0n∏j=i+1n1-αjΔixix0<∞ for n≥0. Thus, Property (i) holds under conditions (a). On the other hand, if 1+Knεn1+αn+βn+εnKn=o1, then hnxnn=0∞⟶0 since xnn=0∞ is bounded from Lemma 6 and hn⟶0, and then Δn⟶0. Thus, supn≥0⁡Δn<∞ since convergent sequences are bounded. Since ∑i=0∞∏j=i+1∞1-αj<∞, it follows that (35) holds, and then Property (i) holds under conditions (b). Property (ii) follows directly since Δxn=Δn0xn⟶0 as n⟶∞ from (26) since the set- theoretic limit KerΔ∞0=limn⟶∞⁡KerΔn0 exists, and Tnxn-xnn=0∞⟶pΔ0∈KerΔ∞0∩S≠⌀. To prove Property (iii), note that if Δn0n=0∞⟶Δ∞0, then Δn0χn-Δ∞0χnn=0∞⟶0 for χn∈DomΔn0. Since Tnxn-xnn=0∞⟶pΔ0∈KerΔ∞0∩S, then(39)Δn0·pΔ0+Tnxn-xn-pΔ0-Δ∞0·pΔ0+Tnxn-xn-pΔ0⟶Δn0·pΔ0-Δ∞0·pΔ0=0.Since pΔ0∈KerΔ∞0∩S, then(40)Δ∞0·pΔ0=0,Δn0·pΔ0n=0∞⟶0,Δn0·Tnxn-xnn=0∞=ΔnTnxn-xnn=0∞⟶0.Property (iv) follows if either Property (ii) or Property (iii) holds and, furthermore, there exists the set-theoretic limit TS∞⊆S, 0∈KerΔ∞0∩S, and xnn=0∞⟶p∈FT⊂S since Tnxn-xnn=0∞⟶0.Inspired by Example 5, we get the following result, to be then used, on sufficient conditions for asymptotic vanishing of the perturbation sequenceΔnxnn=0∞ by removing the assumption of linearity of T.Theorem 8.
Assume thatE is a normed linear space and that Tnn=0∞ is a sequence of bounded operators on S⊂E. Define(41a)Δn=Δnxn=xn+1-1-αnxn+αnTn1-βnxn+αnβnTnxn(41b)hn=1+L1+βnA~n+αnLB~n+LA~nB~n;εn=maxA~n,B~nfor n≥0.Then, the following Items hold:
( i )   Δ n n = 0 ∞ ⟶ 0 if εn=oxn-1, and hn⟶0 if either (a)A~n⟶0 and B~n⟶0 as n⟶∞, or (b)A~n⟶0 and αn⟶0 as n⟶∞.
( i i ) Define the sequences x~n=Tnxn-xn and G~n=αnTnB~n+A~n for n≥0 and assume that x~nn=0∞⟶x~, G~nn=0∞⟶G~ pointwise, such that G~ is closed, and G~nx~nn=0∞⟶vG~. Then, G~x~=vG~ and(ii. a)
Δn-un=G~nxn for n≥0(ii. b)
IfG~nxnn=0∞⟶0, equivalently if Δn-unn=0∞⟶0, then x~=0 if G~ is nonsingular.(ii. c)
Ifx~=0 and G~.0=0, in particular if G~=0, then Δn-unn=0∞⟶0.
( i i i ) If B~nn=0∞⟶B~ pointwise, such that B~ is closed, and B~nx~nn=0∞⟶vB~, then B~x~=vB~, and if x~=0 and B~.0=0, in particular if B~=0, then vnn=0∞⟶0.
( i v ) If G~nn=0∞⟶G~ pointwise, B~nn=0∞⟶B~ pointwise, G~ and B~ are closed, x~nn=0∞⟶x~, G~nx~nn=0∞⟶vG~, and B~nx~nn=0∞⟶vB~, then x∈DomG~∩DomB~ and rankG~B~=rankG~vG~B~vB~. If vG~=vB~=0, then x~∈KerG~∩KerB~, and x~=0 if G~ or B~ is nonsingular.Proof.
From the last two equations of (4), we have(42)vn=B~nTnxn-xn;n≥0un=A~nTnyn-xn=A~nTn1-βnxn+βnTnxn-B~nxn+B~nTnxn-xn;n≥0(43)Δn=Δnxn=A~nIr+TnβnIr+B~n+αnTnB~nTnxn-xn=αnTnB~n+A~nTnxn-xn+un;n≥0without requiring the linearity of T
forn≥0. Since Tn=supx∈S,x≤1⁡(Tnx/x)≤L for n≥0, one gets that Δn≤hnxn for n≥0, hence Property (i) of the lemma. Property (ii. a) follows by direct calculations, Property (ii. b) follows since if x~nn=0∞⟶x~ and G~nxnn=0∞⟶0, then x~∈DomG~ and G~x~=vG~ since G~ is closed. As a result, if vG~=0, then x~∈KerG~ and x~=0 if G~ is nonsingular since then KerG~=0. Property (ii. c) follows since G~n⟶G~ pointwise as n⟶∞, x~=0, and G~.0=0, in particular if G~=0, then(44)G~nx~n=G~x~+G~n-G~x~+G~nx~n-x~=G~n-G~x~+G~nx~n-x~,n≥0and then G~nx~n=Δn-un⟶0 as n⟶∞. Property (iii) follows under a similar reasoning to that used to prove Property (ii. c). Property (iv) follows from Properties [(ii)-(iii)] and the Rouché-Frobenius theorem from Linear Algebra.Assume thatΔnn=0∞⟶0. Then unn=0∞⟶0 via (43) if Tnn≥1 and T are operators on E, Tnn=0∞⟶T, such that T is closed, pointwise and either(a)
αn⟶0 as n⟶∞, or(b)
yn-xn⟶0 and yn-Tnxn⟶0 as n⟶∞ and T.0=0 (then Tnyn-xn⟶Tyn-xn⟶T.0=0, Tnyn-Tnxn⟶Tyn-Tnxn⟶T.0=0 and 0∈DomT), or(c)
yn-xn⟶0, βn⟶0 as n⟶∞.Note thatyn-xn⟶0 and yn-Tnxn⟶0 as n⟶∞ if T:S⟶S, p∈FT∩S, and xn,yn⟶p as n⟶∞.Definition 9 (see [1]).
LetS be a nonempty subset of a normed linear space E. T:S⟶S is said to beuniformlyL- Lipschitzian with constantL>0 if Tnx-Tny≤Lx-y for n≥1.
Note that ifT is asymptotically nonexpansive, then it is 0-strict pseudocontractive and uniformly supn≥1⁡kn-Lipschitzian, [1]. Note also that if T is uniformly L- Lipschitzian and 0∈S, then the sequence of operators Tnn=0∞ is uniformly bounded since by taking y=0, one has Tnx≤Lx for all x∈S and n≥1.Lemma 10.
LetT:S⟶S be asymptotically demicontractive. The following properties hold:
( i ) If T Lipschitzian with constant LT (i.e., T is LT-Lipschitzian), then for any p∈FT(45)limsupn⟶∞Tnx-Tny-kx-p+y-p1+LTn≤0.
( i i ) If T is k-strict asymptotically pseudocontractive and uniformly L-Lipschitzian, then for any p∈FT(46)Tnx-Tny≤k1+Lx-p+y-pforallx,y∈S,n≥1.Let T:S⟶S be k-strict asymptotically pseudocontractive. The following properties hold:
( i i i ) If Tis k-strict asymptotically pseudocontractive and LT-Lipschitzian with constant, then (47)limsupn⟶∞Tnx-Tny-1+k1+LTnx-y≤0forallx,y∈S.
( i v ) If T is k-strict asymptotically pseudocontractive and uniformly L-Lipschitzian then (48)Tnx-Tny≤1+k1+Lx-yforallx,y∈S,n≥0.Proof.
One gets from (2) that(49)Tnx-Tny≤Tnx-p+Tny-p≤knx-p+y-p+kx-Tnx+y-Tny≤kn+kx-p+y-p+kTnp-Tnx+Tnp-Tny≤kn+k1+LTnx-p+y-p,for all x,y∈S and n≥0. Property (i) follows directly since knn=0∞⟶1. Property (ii) follows by replacing LTn⟶L in (49). On the other hand, if T is k-strict asymptotically pseudocontractive and LT-Lipschitzian, then for a constant k∈0,1 and some real sequence knn=0∞⊂1,∞⟶1, one gets Property (iii) since for any x,y∈S(50)Tnx-Tny≤knx-y+kx-y+Tny-Tnx≤kn+k1+LTnx-y;n≥0.Property (iv) follows by replacing LTn⟶L in (50).Lemma 11 (see [1]).
Letx, y, and z be points in Hilbert space and λ∈0,1. Then(51)λx+1-λy-z2=λx-z2+1-λy-z2-λ1-λx-y2.


## 3. Main Results

The main result follows.Theorem 12.
LetS be a nonempty bounded closed subset of a Hilbert space H and let T:S⟶S be a completely continuous uniformly L-Lipschitzian asymptotically demicontractive mapping. Suppose that the following conditions hold:(a)
kn-1n=0∞ is summable,(b)
0<a≤αn≤1-k1-βn, n≥0,(c)
βnIr-Bnn=0∞ and αnIr-Ann=0∞ are summable,and either(d1)
0<b≤βn<minβ0n,1, n≥0, where(52)β0n=12kL2kn2+kL2λn1+λn+k2+4k+kn2-kkn21/2-kn2+kL2λn1+λn+kfor anyn≥0, where λn=1 if vn≠0 and λn=0 if vn=0, or(d2)
0<b≤βn<1-k, n≥0 (implying that k<1-b), and βnβn+λn1+λn-(1-βn)/L2n=0∞ is summable with positive sum, or(d3)
0<b≤βn<(L2λn1+λn+12+4L2-L2λn-1)/2L2, n≥0 (implying that k<1-b), and βn+k-1n=0∞ is summable with positive sum. Then, xnn=0∞ converges strongly to some fixed point of T in S.Proof.
One gets from the third equation of (4) and  ((41a) and (41b)) that(53)xn+1=1-αnxn+αnTnyn+Δn-αnTnvn;n≥0,(54)xn+1-p^n=1-αnxn-p+αnTnyn-p;n≥0where p^n=p+un and un=Δn-αnTnvn, n≥0. Equations (53)-(54) agree with  ((41a) and (41b)) and (4). Since T is asymptotically demicontractive, FT≠⌀. Let p∈FT. One gets from Lemma 11, with z=0, that(55)xn+1-p^n2=xn+1-p-un2=1-αnxn-p2+αnTnyn-p2-αn1-αnxn-Tnyn2;n≥0.The following relations will be used:(56)xn+1-p^n2=xn+1-p+αnTnvn-Δn2≥xn+1-p-αnTnvn-Δn2=xn+1-p2+αnTnvn-Δn-σnxn+1-pαnTnvn-Δn;n≥0where σn=2 if αnTnvn-Δn≠0 and σn=0 if αnTnvn-Δn=0 for any n≥0. Then, one has from (55) and (56), again since T is asymptotically demicontractive, that(57)xn+1-p2≤1-αnxn-p2+αnkn2yn-p2+kTnyn-yn2-1-αnxn-Tnyn2+σnxn+1-p-αnTnvn-ΔnαnTnvn-Δn;n≥0.On the other hand, one gets from the second equation of (4) that(58)yn-p^yn=1-βnxn-p+βnTnxn-p;n≥0where(59)p^yn=p+vn=p^n+vn+αnTnvn-Δn;n≥.0The following relations are afterward used:(60)yn-p^yn2=yn-p-vn2≥yn-p-vn2=yn-p2+vn-1+λnyn-pvn;n≥0where λn=1 if vn≠0 and λn=0 if vn=0 for any n≥0. Again, using Lemma 11, (58), (59), and (60), one gets that(61)yn-p2+vn-λnyn-pvn=yn-p^yn2=1-βnxn-p2+βnTnxn-p2-βn1-βnxn-Tnxn2≤1-βnxn-p2+βnkn2xn-p2+kTnxn-xn2-βn1-βnxn-Tnxn2;n≥0.Then, (62)yn-p2≤1+βnkn2-1xn-p2+βnβn+k-1xn-Tnxn2+λnyn-p-vnvn;n≥0.Since T is asymptotically demicontractive and L-uniformly Lipschitzian, one gets from (4), Lemma 11, and the second relation of (4) that(63)yn-Tnyn2+λnvn2-λn1+λnyn-Tnynvn=yn-Tnyn-vn2=1-βnxn-Tnyn+βnTnxn-Tnyn2=1-βnxn-Tnyn2+βnTnxn-Tnyn2-βn1-βnxn-Tnxn2≤1-βnxn-Tnyn2+L2βnxn-yn2-βn1-βnxn-Tnxn2≤1-βnxn-Tnyn2+L2βnβnxn-Tnxn-vn2-βn1-βnxn-Tnxn2≤1-βnxn-Tnyn2+βnL2βn21+λn+βn-1xn-Tnxn2+L2βnλn1+λnvn2;n≥0which leads to(64)yn-Tnyn2≤1-βnxn-Tnyn2+βnL2βn21+λn+βn-1xn-Tnxn2+λnL2βn1+λn-1vn+1+λnyn-Tnynvn;n≥0.Now, one gets from (62) and (64) into (57) that(65)xn+1-p2≤xn-p2+αnkn21+βnkn2-1-1xn-p2+αnβnkn2βn+k-1+kL2βnβn+λn+βn-1xn-Tnxn2+αnk1-βn+αn-1xn-Tnyn2+αnkλnL2βn1+λn-1-kn2vn+kn2λn1+λnyn-pvn+σnxn+1-p-αnTnvn-ΔnαnTnvn-Δn;n≥0.Thus, the constraints on the sequences kn, αnn=0∞ and βnn=0∞, Bnn=0∞ and Ann=0∞ being implicit in the conditions (a) to (c) of the theorem statement lead to(66)xn+1-p2-xn-p2≤αnkn21+βnkn2-1-1xn-p2+αnβnkn2βn+k-1+kL2βnβn+λn+βn-1xn-Tnxn2-αnk1-βn+αn-1xn-Tnyn2+αnkλnL2βn1+λn-1-kn2vn+kn2λn1+λnyn-pvn+σnxn+1-p-αnTnvn-ΔnαnTnvn-Δn;n≥0so that (67)∑n=0∞xn+1-p2-xn-p2≤M1M2+supn≥0xn-Tnxn2∑n=0∞αnβnkn2βn+k-1+kL2βn1+λnβn+λn+βn-1+M3for some finite real constant M3; since unn=0∞, vnn=0∞, and Δnn=0∞ are summable as a result following from condition (c), the definitions of unn=0∞, vnn=0∞, and Δnn=0∞ from (4) and (43) with M1=supn≥0⁡αnkn+11+kn2βnxn-p2<+∞, T:S⟶S, and S are bounded so xn-p and xn-Tnxnn=0∞ are bounded and knn=0∞ is bounded since T:S⟶S is asymptotically demicontractive and M2=∑n=0∞kn-1<∞. If condition (d1) holds then the convex parabola pn=pnβn=kn2βn+k-1+kL2βn1+λnβn+λn+βn-1≤0 for any n≥0 such that βn∈max0,β1n,β0n where β1n≤0 and β0n>0 are the roots of pnβn=0. Therefore, one gets from (67), since minn≥0⁡αnβn≥ab>0, that(68)∑n=0∞xn+1-p2-xn-p2+supn≥0αnβnpn∑n=0∞xn-Tnxn2≤M4=M1M2+M3.As a result, xn-pn=0∞⟶x∗-p, xnn=0∞⟶x∗∈S, since S is closed, and xn-Tnxnn=0∞⟶0. Conditions (d2) allow modifying (68) as (69)∑n=0∞xn+1-p2-xn-p2+supn≥0αnβnkn21-k-βn∑n=0∞xn-Tnxn2≤M6=M4+M5where M5=k∑n=0∞αnβnL2βnβn+λn+βn-1supn≥0⁡xn-Tnxn2 and the same conclusions on the convergence of the involved sequences follow. On the other hand, conditions (d3) allow modifying (68) as(70)∑n=0∞xn+1-p2-xn-p2+ksupn≥0αnβnL2βn1+λnβn+λn+βn-1∑n=0∞xn-Tnxn2≤M8=M4+M7where M7=∑n=0∞αnβnkn2k+βn-1supn≥0⁡xn-Tnxn2 and the same conclusions follow again on the convergence of the involved sequences.
Now, it is proven thatxn-Txnn=0∞⟶0 proceeding by contradiction. Assume that xn-Txnn=0∞ does not converge to 0 as n⟶∞. Then, there is a subsequence xni-Txnii=0∞⊆xn-Tnxnn=0∞ which does not converge to zero. Since T is uniformly L-Lipschitzian, one has(71)Lxn-Tnxn≥Tn+1xn-Txn≥xn-Txn-xn-Tn+1xn≥xn-Txn-xn-Tnxn+Tnxn-Tn+1xn≥xn-Txn-xn-Tnxn-Lxn-Txn=1-Lxn-Txn-xn-Tnxn≥1-Lxn-Txn-xn-Tnxn;n≥0and then, since xn-Tnxnn=0∞⟶0, the subsequence xni-Tnxnii=0∞⊆xn-Tnxnn=0∞⟶0 with ni+1>ni for any i≥0 such that xni-Txni≤(1+L/1-L)xni-Tnxni. Thus, xni-Txnii=0∞⟶0, hence a contradiction. Then, xn-Tnxnn=0∞⟶0 implies that xn-Txnn=0∞⟶0. Since the set S is bounded and closed and T is completely continuous, Txnn=0∞ has a subsequence Txnii=0∞ such that Txnii=0∞⟶q for some q∈S. Since xni-Txnii=0∞⟶0, xnii=0∞⟶q and q∈FT since T is continuous. Now, the constraints (68) to (70) imply in any case that xn+1-p2≤xn-p2+sn for n≥0 such that ∑n=0∞sn<∞ with xn-pn=0∞⟶x∗-p and xni-qi=0∞⊆xn-qn=0∞⟶0. One concludes from Lemma 1.7 of [1] that xn-qn=0∞⟶x∗-q=0, hence xn-pn=0∞⟶x∗=q.The following result is a parallel one to Theorem12 under the assumption that T satisfies anasymptotically demicontractive-like condition (2), in the sense that (2) holds with kn∈0,1 and kn⟶1 and it has a unique fixed point.Theorem 13.
LetS be a nonempty bounded closed subset of a of a Hilbert space H and let T:S⟶S be a uniformly L-Lipschitzian satisfying (2) and being continuous at p∈FT such that S∩FT=p and knn=0∞⊆0,1, knn=0∞⟶1 and k∈0,1. Suppose also that the following conditions hold:(a)
αnkn21-βn1-kn2-1n=0∞⟶0 and limn⟶∞⁡∑j=0nαjkj21-βj1-kn2-1=+∞,(b)
0<a≤αn≤min1-k1-βn,1/(1-kn2), n≥0,(c)
βnIr-Bnn=0∞ and αnIr-Ann=0∞ are summable,and either (d1) or (d2) below holds:(d1)
βn∈min0,(a-1-k)/k,min(1-αn1-kn2)/αnkn21-kn2,β1n, n≥0, where(72)β1n=12kL2kL2λn+k+kn2+4kL2k1-kn2+kn21/2(d2)
βn∈min(a-1-k)/k,β1n,1n≥0 and αnβnkn2βn+k-1+kL2βnβn+λn+βn-1n=0∞ is summable.
Then,xnn=0∞⟶p.Proof.
It is possible to obtain from (66), under the given conditions, that(73)xn+1-p2≤1-αnkn21-βn1-kn2-1xn-p2+αnβnkn2βn+k-1+kL2βnβn+λn+βn-1xn-Tnxn2-αnk1-βn+αn-1xn-Tnyn2+αnkλnL2βn-1-kn2vn+kn2λnyn-pvn+σnxn+1-p-αnTnvn-ΔnαnTnvn-Δn≤1-αnkn21-βn1-kn2-1xn-p2+γn;n≥0where(74)γn=γ1n+αnkλnL2βn-1-kn2vn+kn2λnyn-pvn+σnxn+1-p-αnTnvn-ΔnαnTnvn-Δn;n≥0and γ1n=0 under condition (d1) since the second right-hand-side term of (73) is nonpositive and can be omitted in the analysis and (75)γ1n=αnβnkn2βn+k-1+kL2βnβn+λn+βn-1xn-Tnxn2under condition (d2) is nonnegative and γ1nn=0∞ is summable since xn-Tnxnn=0∞ is bounded since S 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, γnn=0∞ is summable and has nonnegative elements while αnkn21-βn1-kn2-1n=0∞⟶0 and limn⟶∞⁡∑k=0nαkkj21-βj1-kn2-1=+∞ from conditions (a). The result xn-pn=0∞⟶0 follows from Venter’s theorem, [14].Corollary 14.
Theorem13 holds if the second constraint of the condition (a) is modified as follows:(76a)limsupn⟶∞n-∑j=0nαjkj21-βj1-kn2-1-gn≤0(76b)liminfn⟶∞n-∑j=0nαjkj21-βj1-kn2-1-g0n≥0for some real sequences gnn=0∞⊆0,Kg⊂0,∞ and g0nn=0∞⊆0,Kg fulfilling g0n≤gn for n≥0.Proof.
Note that ((76a) and (76b)) are guaranteed if(77)n-g0n≥∑j=0nαjkj21-βj1-kn2-1≥n-gn≥n-Kg;n≥0and there exists some finite integer n0≥max1,Kg such that (78)∑j=0nαjkj21-βj1-kn2-1≥0forn≥n0and, since αnn=0∞⊂0,1 and knn=0∞⟶1, one has αnkn21-βn1-kn2-1n=0∞⟶0 since (79)limn⟶∞∑j=0nαjkj21-βj1-kn2-1=+∞.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 mappingT 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 Rn which generates sequences xnn=0∞ as xn=Tnx with x=x0 for x0∈R0+n, such that FT⊃0, where R0+=z∈R:z0≥0, subject to(80)xn2=kn2x2+kxn-x2-νnx2≤kn2x2+kxn-x2;n≥0for some sequences knn=0∞⊂1,∞ such that knn=0∞⟶1and νnn=0∞⊂0,kn2x2+kxn-x2, and some real constant k∈0,1. From the above constraint, it follows that (81)xn2=kn2+2k1-2kx2-νn1-2kx2;n≥0provided that k∈0,1/2. Assume also that kn2=1-γn+μn for n≥0 with μn≥γn≥0 for n≥0, ∑n=0∞γn=+∞, μnn=0∞, γnn=0∞⟶0, and(82)νnn=0∞⊆0,minμn+2k2-γn,1+μn-γn+2ksince kn2=1-γn+μn, with νnn=0∞⟶4k. One has from (81) that such an equation can be rewritten as (83)xn+12=kn+12+2k-νn+1kn2+2k-νnxn2;n≥0and also as (84)xn2=1+μn-γn+2k1-2kx2-νn1-2k=1-γnx2+μn-νn+2k2-γn1-2kx2;n≥0.Thus, xnn=0∞ converges to some fixed point of the self-mapping T on R0+n, such that xnn=0∞⟶x; 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 T on R0+n, 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-mappingT on R0+n with FT⊃0 such that any sequence xnn=0∞ generated as xn+1=Txn=Tn+1x0, n≥0, for a given x0∈R0+n, such that for some norm · (for instance, the Euclidean norm, zn=xn  for n≥0) satisfies the following norm constraint:(85)zn+1=1+γn-δnzn+ωn;n≥0where γnn=0∞, δnn=0∞, and ωnn=0∞ are parameterizing real sequences subject to the constraints:(a)
δn≥γn≥0, n≥0;(b)
δnzn+kzn-z0≥ωn≥0, n≥0, for some k∈0,1;(c)
δnn=0∞⟶0; γnn=0∞⟶0;(d)
∑n=0∞δn=+∞;(e)
∑n=0∞γn=+∞;(f)
∑n=0∞δn-γnzn<+∞;(g)
∑n=0∞ωn+γnzn<+∞.
Note that (81) is identical to(86)zn+1=1-δnzn+γnzn+ωn;n≥0.Note from (a), (b), (c), (d), (g), and Venter’s theorem, [14], that znn=0∞⟶0 which implies that xnn=0∞⟶0. Note also that, since zn≥0 for n≥0, (85), or (86), and the constraints (b) and (f) imply that (87)zn+1=zn+1≤1+γnzn+kzn-z0;n≥0with knn=0∞⊂1,∞, where kn=1+γn, n≥0, so that knn=0∞⟶1 from the constraint (c). Since (87) implies that the squared norm of the elements of xnn=0∞ satisfies the asymptotic demi-contractiveness condition (2), it follows that the self-mapping T on S≡R0+n is asymptotically demicontractive and xnn=0∞⟶0 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 numbersnkk=0cnS with cnS=cardnkk=0cnS≤χ0 (χ0 denoting the infinity cardinal of denumerable sets) and an associated set or sequence of nonnegative real numbers Mnn=0cnS such that(88)∑k=0cnS∑n=nknk+1ωn-γnzn-kzn-z0≤∑k=0cnS∑n=nknk+1δn-γnzn=∑n=0∞δn-γnzn=M=∑k=0cnSMk<+∞which holds if the parameterizing sequences fulfil the constraints:(89)δn-γn≤Mkxn;n=nk,nk+1,…,nk+1-1,k=0,1,…,cnSwith the convention that ncnS+1-1=+∞. Note that the condition (g) with (b) is guaranteed, for instance, if (90)∑n=0∞δn+γn+kzn=Ma=∑k=0cnSaMak<+∞for a finite set or for sequence of nonnegative real numbers Mann=0cnSa, which is guaranteed if (91)δn-γn+k≤Mkxn;n=nk,nk+1,…,nk+1-1,k=0,1,…,cnSa.If a generalized modified Ishikawa’s iterative process with matrix parameterizing sequences is implemented with such an auxiliary mapping T on R0+n, then any such obtained generated sequence converges asymptotically to zero under the given constraints of the above results given in the paper body.



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*Source: 3840784-2018-07-19.xml*