# On the Convergence of Mann and Ishikawa Iterative Processes for Asymptoticallyϕ-Strongly Pseudocontractive Mappings

Authors: Xuewu Wang; Giuseppe Marino; Luigi Muglia
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We prove the equivalence and the strong convergence of (1) the modified Mann iterative process and (2) the modified Ishikawa iterative process for asymptoticallyϕ-strongly pseudocontractive mappings in a uniformly smooth Banach space.


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## Body

## 1. Introduction

LetX be a Banach space and X* the dual space of X. Let J denote the normalized duality mapping from X into 2X* given by J(x)={f∈X*:〈x,f〉=∥x∥2=∥f∥2} for all x∈X, where 〈·,·〉 denotes the generalized duality pairing.In 1972, Goebel and Kirk [1] introduced the class of asymptotically nonexpansive mappings as follows.Definition 1.1.
LetK be a subset of a Banach space X. A mapping T:K→K is said to be asymptotically nonexpansive if for each x,y∈K(1.1)‖Tnx-Tny‖≤kn‖x-y‖,
where {kn}n⊂[1,∞) is a sequence of real numbers converging to 1.Their scope was to extend the well-known Browder fixed point theorem [2] to this class of mappings.This class is really more general than the class of nonexpansive mappings.Example 1.2 (see [1]).
IfB is the unit ball of l2 and T:B→B is defined as
(1.2)T(x1,x2,…)=(0,x12,a2x2,a3x3,…),
where {ai}i∈ℕ⊂(0,1) is such that ∏i=2∞ai=1/2, it satisfies
(1.3)‖Tx-Ty‖≤2‖x-y‖,‖Tnx-Tny‖≤2∏i=2nai‖x-y‖.In 1973, the same authors [3] proved that the Browder result remains valid for the broader class of uniformly L-Lipschitzian mappings (i.e., ∥Tnx-Tny∥≤L∥x-y∥, for every n): every uniformly L-Lipschitzian self-mapping T:K→K (with L<γ and γ sufficiently close to 1) defined on a closed and convex subset of a uniformly convex Banach space has a fixed point.Taking in to account these papers one can study (1) if a suitable iterative method converges to a fixed point of the mapping and (2) the equivalence of two (given) iterative methods, that is, the first one converges if and only if the second one converges.In 1991, Schu [4] introduced the class of asymptotically pseudocontractive mappings.Definition 1.3 (see [4]).
LetX be a normed space, K⊂X, and {kn}n⊂[1,∞). A mapping T:K→K is said to be asymptotically pseudocontractive with the sequence {kn}n if and only if limn→∞kn=1, and for all n∈ℕ and all x,y∈K there exists j(x-y)∈J(x-y) such that
(1.4)〈Tnx-Tny,j(x-y)〉≤kn‖x-y‖2,
where J is the normalized duality mapping.Obviously every asymptotically nonexpansive mapping is asymptotically pseudocontractive but the converse is not valid: it is well known thatT:[0,1]→[0,1] defined by Tx=(1-x2/3)3/2 is not Lipschitzian but asymptotically pseudocontractive [5].In [4], Schu proved the following.Theorem 1.4 (see [4]).
LetH be a Hilbert space and A⊂H closed and convex; L>0; T:A→A completely continuous, uniformly L-Lipschitzian, and asymptotically pseudocontractive with sequence {kn}n∈[1,∞); qn:=2kn-1 for all n∈ℕ; ∑n(qn2-1)<∞; {αn}n,{βn}n∈[0,1]; ϵ≤αn≤βn≤b for all n∈ℕ, some ϵ>0 and some b∈(0,L-2[1+L2-1]); x1∈A; for all n∈ℕ define
(1.5)zn∶=βnTn(xn)+(1-βn)xn,xn+1∶=αnTn(zn)+(1-αn)xn.
Then, {xn}n converges strongly to some fixed point of T.From 1991 to 2009 no fixed point theorem for asymptotically pseudocontractive mappings was proved. First Zhou in [6] completed this lack in the setting of Hilbert spaces proving: (1) a fixed point theorem for an asymptotically pseudocontractive mapping that is also uniformly L-Lipschitzian and uniformly asymptotically regular, (2) that the set of fixed points of T is closed and convex, and (3) the strong convergence of a CQ-iterative method. The literature on asymptotical-type mappings is very wide (see, [7–15]).In 1967, Browder [16] and Kato [17], independently, introduced the accretive operators (see, for details, Chidume [18]). Their interest was connected with the existence of results in the theory of nonlinear equation of evolution in Banach spaces.In 1974 Deimling [19], studying the zeros of accretive operators, introduced the class of φ-strongly accretive operators.Definition 1.5.
An operatorA defined on a subset K of a Banach space X is said to be φ-strongly accretive if
(1.6)〈Ax-Ay,j(x-y)〉≥φ(‖x-y‖)‖x-y‖,
where φ:ℝ+→ℝ+ is a strictly increasing function such that φ(0)=0 and j(x-y)∈J(x-y).Note that in the special case in whichφ(t)=kt, k∈(0,1), we obtain a strongly accretive operator.However, it is not difficult to prove (see Osilike [20]) that Ax=x-x/(x+1) in ℝ+ is φ-strongly accretive with φ(s)=(s2/(1+s)) but not strongly accretive.Since an operatorA is a strongly accretive operator if and only if (I-A) is a strongly pseudocontractive mapping (i.e., 〈(I-A)x-(I-A)y,j(x-y)〉≤k∥x-y∥2,k<1), taking into account Definition 1.5, it is natural to study the class of φ-pseudocontractive mappings, that is, the mappings such that(1.7)〈Tx-Ty,j(x-y)〉≤‖x-y‖2-φ(‖x-y‖)‖x-y‖,
where φ:ℝ+→ℝ+ is a strictly increasing function such that φ(0)=0. Of course the set of fixed points for this mapping contains, at most, only one point.Recently, in same papers the following has been introduced.Definition 1.6.
A mappingT is a ϕ-strongly pseudocontractive mapping if
(1.8)〈Tx-Ty,j(x-y)〉≤‖x-y‖2-ϕ(‖x-y‖),
where j(x-y)∈J(x-y) and ϕ:ℝ+→ℝ+ is a strictly increasing function such that ϕ(0)=0. (In the literature this class is also known as generalizedϕ-strongly pseudocontractive according to (1.7). We prefer to not use the term generalized because this class is narrower than pseudocontractive mappings.)Choosingϕ(t)=φ(t)t, we obtain (1.7). In Xiang's paper [21], it was remarked that it is an open problem if every ϕ-strongly pseudocontractive mapping is φ-pseudocontractive mapping. In the same paper, Xiang obtained a fixed point theorem for continuous and ϕ-strongly pseudocontractive mappings in the setting of the Banach spaces.In this paper our attention is on the class of theasymptoticallyϕ-strongly pseudocontractive mappings defined as follows.Definition 1.7.
IfX is a Banach space and K is a subset of X, a mapping T:K→K is said to be asymptotically ϕ-strongly pseudocontractive if
(1.9)〈Tnx-Tny,j(x-y)〉≤kn‖x-y‖2-ϕ(‖x-y‖),
where j(x-y)∈J(x-y), {kn}n⊂[1,∞) is converging to one, and ϕ:[0,∞)→[0,∞) is strictly increasing such that ϕ(0)=0.One can note that ifT has fixed points, then it is unique. In fact if x,z are fixed points for T, then, for every n∈ℕ,(1.10)‖x-z‖2=〈Tnx-Tnz,j(x,z)〉≤kn‖x-y‖2-ϕ(‖x-y‖),
so, passing n to +∞ results in(1.11)‖x-z‖2≤‖x-z‖2-ϕ(‖x-y‖)⟹-ϕ(‖x-y‖)≥0.
Since ϕ:[0,∞)→[0,∞) is strictly increasing and ϕ(0)=0 then, x=z.We now give two examples.Example 1.8.
The mappingTx=x/(x+1), where x∈[0,1], is asymptotically ϕ-strongly pseudocontractive with kn=1, for all n∈ℕ and ϕ(t)=t3/(1+t). However, T is not strongly pseudocontractive, see [20].Example 1.9.
The mappingTx=x/(1+0.01x), where x∈[0,1], is asymptotically ϕ-strongly pseudocontractive with kn=(n+100)/n, for all n∈ℕ and ϕ(x)=x3/(1+x). However T is not strongly pseudocontractive, nor ϕ-strongly Pseudocontractive.Proof.
First we prove thatT is not strongly pseudocontractive. For arbitrary k<1, there exist x,y∈[0,1], such that
(1.12)1(1+0.01x)(1+0.01y)>k.
So we have
(1.13)〈Tx-Ty,j(x-y)〉=1(1+0.01x)(1+0.01y)(x-y)2>k‖x-y‖2.
Next we prove that T is not ϕ-strongly pseudocontractive. Taking y=0, we have, for all x∈[0,1],
(1.14)〈Tx-Ty,j(x-y)〉=x2(1+0.01x),‖x-y‖2-ϕ(‖x-y‖)=x21+x,
therefore, T is not ϕ-strongly pseudocontractive. Finally, we prove that T is asymptotically ϕ-strongly Pseudocontractive.
For arbitraryx,y∈[0,1], without loss of generality, let x>y. Then,
(1.15)〈Tnx-Tny,j(x-y)〉=(x-y)2(1+0.01nx)(1+0.01ny),kn‖x-y‖2-ϕ(‖x-y‖)=n+100n(x-y)2-(x-y)31+(x-y)=[1+100n-(x-y)1+(x-y)](x-y)2=1+(100/n)+(100/n)(x-y)1+(x-y)(x-y)2.
We only need to prove that
(1.16)1+x-y≤(1+0.01nx)(1+0.01ny)[1+100n+100n(x-y)].
Using x>y, this is easy.In this paper we study the equivalence between two kinds of iterative methods involving asymptoticallyϕ-strongly Pseudocontractive mappings.Moreover, we prove that, in opportune hypotheses, these methods are equivalent and strongly convergent to the unique fixed point of the asymptoticallyϕ-strongly Pseudocontractive T.


## 2. Preliminaries

Throughout this paper, we will assume thatX is a uniformly smooth Banach space. It is well known that if X is uniformly smooth, then the duality mapping J is single-valued and is norm-to-norm uniformly continuous on any bounded subset of X. In the sequel, we will denote the single valued duality mapping by j.Let us introduce the modified Mann and Ishikawa iterative processes as follows.For any givenx0∈X, the sequence {xn}n defined by(2.1)yn=(1-βn-δn)xn+βnTnxn+δnvn,xn+1=(1-αn-γn)xn+αnTnyn+γnun,n≥0,
is called the modified Ishikawa iteration sequence, where {αn}n, {γn}n, {βn}n, and {δn}n are four sequences in (0,1) satisfying the conditions αn+γn≤1 and βn+δn≤1 for all n≥0.In particular, ifβn=δn=0 for all n≥0, we can define a sequence {zn}n by(2.2)z0∈X,zn+1=(1-αn-γn)zn+αnTnzn+γnwn,n≥0,
called the modified Mann iteration sequence. In [22, 23] the methods are also called modified Mann (Ishikawa resp.) iterative processes with errors.These kind of iterative processes with errors were studied, as an example, in [23–26]. Equivalence theorems for Mann and Ishikawa methods are studied, among others, in [27, 28].Huang in [29] had established the equivalence theorems of the convergence between the modified Mann iteration process (2.2) and the modified Ishikawa iteration process (2.1) for strongly successively ϕ-pseudocontractive mappings in uniformly smooth Banach space.In next section, we prove that, in the setting of the uniformly smooth Banach space, ifT is asymptotically ϕ-strongly Pseudocontractive, then (2.1) and (2.2) are equivalent. Moreover we prove also that (2.1) and (2.2) strongly converge to the unique fixed point of T if it exists.For the sake of completeness, we recall some definitions and conclusions.Definition 2.1.
X is said to be a uniformly smooth Banach space if the smooth module of X(2.3)ρX(t)=sup⁡{12(‖x-y‖+‖x+y‖)-1:‖x‖≤1,‖y‖≤t}
satisfies limt→0(ρX(t)/t)=0.Lemma 2.2 (see [30]).
LetX be a Banach space, and let J:X→2X* be the normalized duality mapping. Then, for any x,y∈X one has
(2.4)‖x+y‖2≤‖x‖2+2〈y,j(x+y)〉,∀j(x+y)∈J(x+y).The following lemma is a key of our proofs.Lemma 2.3 (see [29]).
Letϕ:[0,∞)→[0,∞) be a strictly increasing function with ϕ(0)=0, and let {an}n,{bn}n,{cn}n, and {en}n be nonnegative real sequences such that
(2.5)lim⁡n→∞bn=0,cn=o(bn),∑n=1∞bn=∞,lim⁡n→∞en=0.
Suppose that there exists an integer N1>0 such that
(2.6)an+12≤an2-2bnϕ(|an+1-en|)+cn,∀n≥N1,
then limn→∞an=0.Proof.
The proof is the same as that in [29] but change in (2.6), (an+1-en) with |an+1-en|.Lemma 2.4 (see [31]).
Let{sn}n, {cn}n⊂ℝ+, {an}n⊂(0,1), and {bn}n⊂ℝ be sequences such that
(2.7)sn+1≤(1-an)sn+bn+cn,
for all n≥0. Assume that ∑n|cn|<∞. Then, the following results hold(1)
Ifbn≤βan (where β≥0), then {sn}n is a bounded sequence.(2)
If one has∑nan=∞ and limsupn(bn/an)≤0 then, sn→0 as n→∞.Remark 2.5.
If in Lemma2.3 choosing en=0, for all n, ϕ(t)=kt2 (k<1) then, the inequality (2.6) becomes,
(2.8)an+12≤an2-2bnkan+12+cn⟹an+12≤11+2bnkan2+cn1+2bnk≤(1-2bnk1+2bnk)an2+cn1+2bnk,
where αn∶=2bnk/(1+2bnk) and βn∶=cn/(1+2bnk). In the hypotheses of Lemma 2.3, αn→0 as n→∞, ∑nαn=∞, and limsupn(βn/αn)=0. So we reobtain Lemma 2.4 in the case cn=0.


## 3. Main Results

The ideas of the proofs of our main Theorems take into account those contained in the papers of Chang and Chidume et al. [22, 32, 33].Theorem 3.1.
LetX be a uniformly smooth Banach space, and let T:X→X be an asymptotically ϕ-strongly Pseudocontractive mapping with fixed point x* and bounded range.
Let{xn} and {zn} be the sequences defined by (2.1) and (2.2), respectively, where {αn}, {γn}, {βn}, and {δn}⊂[0,1] satisfy(H1)
limn→∞αn=limn→∞βn=limn→∞δn=0 and γn=o(αn),(H2)
∑n=1∞αn=∞, and the sequences {un}, {vn}, and {wn} are bounded in X. Then, for any initial point z0,x0∈X, the following two assertions are equivalent.(i)
The modified Ishikawa iteration sequence (2.1) converges to x*.(ii)
The modified Mann iteration sequence (2.2) converges to x*.Proof.
First of all we note that by boundedness of the range ofT and of the sequences {wn},{un},{vn} and by Lemma 2.4, it and follows that {zn} and {xn} are bounded sequences. So we can set
(3.1)M=sup⁡n{‖Tnzn-Tnyn‖,‖Tnxn-xn‖,‖Tnyn-xn‖,‖Tnzn-zn‖,‖zn-xn‖,‖un-xn‖,‖vn-xn‖,‖wn-zn‖,‖wn-un‖.}.
By Lemma 2.2, we have
(3.2)‖zn+1-xn+1‖2≤‖(1-αn-γn)(zn-xn)+αn(Tnzn-Tnyn)+γn(wn-un)‖2≤(1-αn-γn)2‖zn-xn‖2+2〈αn(Tnzn-Tnyn)+γn(wn-un),j(zn+1-xn+1)〉≤(1-αn)2‖zn-xn‖2+2αn〈Tnzn-Tnyn,j(zn-yn)〉+2αn〈Tnzn-Tnyn,j(zn+1-xn+1)-j(zn-yn)〉+2γn〈wn-un,j(zn+1-xn+1)〉≤(1-αn)2‖zn-xn‖2+2αnkn‖zn-yn‖2-2αnϕ(‖zn-yn‖)+2αn‖Tnzn-Tnyn‖‖j(zn+1-xn+1)-j(zn-yn)‖+2γn‖wn-un‖‖zn+1-xn+1‖≤(1-αn)2‖zn-xn‖2+2αnkn‖zn-yn‖2-2αnϕ(‖zn-yn‖)+2αnσnM+2γnM2,
where σn=∥j(zn+1-xn+1)-j(zn-yn)∥. Using (2.1) and (2.2), we have
(3.3)‖(zn+1-xn+1)-(zn-yn)‖≤‖xn+1-yn‖+‖zn+1-zn‖=‖αn(Tnyn-xn)+γn(un-xn)-βn(Tnxn-xn)-δn(vn-xn)‖+‖αn(Tnzn-zn)+γn(wn-zn)‖≤2M(αn+γn+βn+δn)⟶0(n⟶∞).
In view of the uniform continuity of j we obtain that σn→0 as n→∞. Furthermore, it follows from the definition of {yn} that for all n≥0(3.4)‖zn-yn‖2=‖zn-xn+βn(-Tnxn+xn)+δn(-vn+xn)‖2≤[‖zn-xn‖+βn‖Tnxn-xn‖+δn‖vn-xn‖]2≤[‖zn-xn‖+(βn+δn)M]2=‖zn-xn‖2+(βn+δn)(2‖zn-xn‖M+(βn+δn)M2)≤‖zn-xn‖2+3(βn+δn)M2,(3.5)‖yn-xn‖≤βn‖Tnxn-xn‖+δn‖vn-xn‖≤(βn+δn)M⟶0(n⟶∞),
and so
(3.6)‖zn+1-xn+1‖=‖zn-xn-(αn+γn)(zn-xn)+αn(Tnzn-Tnyn)+γn(wn-un)‖≤‖zn-xn‖+(αn+γn)‖zn-xn‖+αn‖Tnzn-Tnyn‖+γn‖wn-un‖≤‖zn-yn‖+‖yn-xn‖+2(αn+γn)M.≤‖zn-yn‖+(βn+δn)M+2(αn+γn)M.
Therefore, we have
(3.7)‖zn-yn‖≥‖zn+1-xn+1‖-en,
where en=(βn+δn)M+2(αn+γn)M. In view of (H1), we have that en→0 as n→∞.
If∥zn+1-xn+1∥-en≤0 for an infinite number of indexes we can extract a subsequence such that ∥znk-xnk∥-enk-1≤0. For this subsequence ∥znk-xnk∥→0 as k→∞.
We can prove that, in this case,∥zn-xn∥→0, that is, the thesis.
Firstly we note that substituting (3.4) into (3.2) and simplifying, we have
(3.8)‖zn+1-xn+1‖2≤‖zn-xn‖2+αn2‖zn-xn‖2+2αn(kn-1)‖zn-xn‖2+6αnkn(βn+δn)M2-2αnϕ(‖zn-yn‖)+2αnσnM+2γnM2≤‖zn-xn‖2-2αnϕ(‖zn-yn‖)+αn2M2+2αn(kn-1)M2+6αnk¯(βn+δn)M2+2αnσnM+2γnM2=‖zn-xn‖2-αnϕ(‖zn-yn‖)+2γnM2-αn[ϕ(‖zn-yn‖)-αnM2-2(kn-1)M2-6k¯(βn+δn)M2-2σnM],
where k¯:=supn(kn).
Moreover, we observe that(3.9)‖znj-ynj‖≤‖znj-xnj‖+‖ynj-xnj‖⟶0asj⟶∞.
Thus, for every fixed ϵ>0, there exists j1 such that, for all j>j1,
(3.10)‖znj-ynj‖<2ϵ,‖znj-xnj‖<ϵ.
Since {αn}n, {(kn-1)}n, {(βn+δn)}n, {σn}n, and {γn}n are real null sequences (and in particular γn=o(αn)), for the previous fixed ϵ>0, there exist an index N such that, for all n>N,
(3.11)|αn|<min⁡{ϵ16M,ϕ(ϵ/2)8M2},|γn|<ϵ16M,|γnαn|<ϕ(ϵ/2)4M2,|kn-1|<ϕ(ϵ/2)16M2,|βn+δn|<min⁡{ϵ4M,ϕ(ϵ/2)48k¯M2},|σn|<ϕ(ϵ/2)16M,
for all n>N.
Taken*>max{N,nj1} such that n*=nk for a certain k.
We prove, by induction, that∥zn*+i-xn*+i∥<ϵ, for every i∈ℕ. Let i=1.
Let us suppose that∥zn*+1-xn*+1∥≥ϵ.
By (3.6), we have
(3.12)ϵ≤‖zn*+1-xn*+1‖≤‖zn*-yn*‖+(βn*+δn*)M+2(αn*+γn*)M≤‖zn*-yn*‖+Mϵ4M+2M(ϵ8M+ϵ8M)=‖zn*-yn*‖+ϵ4+ϵ4=‖zn*-yn*‖+ϵ2.
Thus, ∥zn*-yn*∥≥ϵ/2. Since ϕ is strictly increasing, ϕ(∥zn*-yn*∥)≥ϕ(ϵ/2).
From (3.8), we obtain that
(3.13)‖zn*+1-xn*+1‖2<ϵ2-αn*(ϕ‖zn*-yn*‖-2M2γn*αn*)-αn*[ϕ(‖zn*-yn*‖)-αn*M2-2(kn*-1)M2-6k¯(βn*+δn*)M2-2σn*M].
One can note that
(3.14)αn*M2+2(kn*-1)M2+6k¯(βn*+δn*)M2+2σn*M≤ϕ(ϵ/2)8+ϕ(ϵ/2)8+ϕ(ϵ/2)8+ϕ(ϵ/2)8;
hence
(3.15)ϕ(‖zn*-yn*‖)-αn*M2-2(kn*-1)M2-6k¯(βn*+δn*)M2-2σn*M≥ϕ(ϵ2)-ϕ(ϵ/2)2>0.
In the same manner
(3.16)ϕ(‖zn*-yn*‖)-2M2γn*αn*>ϕ(ϵ2)-ϕ(ϵ/2)2>0.
Thus,
(3.17)‖zn*+1-xn*+1‖2<ϵ2.
So, we have that ∥zn*+1-xn*+1∥<ϵ and it denies that ∥zn*+1-xn*+1∥≥ϵ. By the same idea we can prove (by contradiction) that ∥zn*+2-xn*+2∥<ϵ and then, by inductive step, ∥zn*+i-xn*+i∥≤ϵ, for all i. This is enough to assure that ∥zn-xn∥→0.
If there are only finite indexes for which∥zn+1-xn+1∥-en≤0, then definitively ∥zn+1-xn+1∥-en≥0. By the strict increasing function ϕ, we have definitively
(3.18)ϕ(‖zn-yn‖)≥ϕ(‖zn+1-xn+1‖-en).
Again substituting (3.4) and (3.18) into (3.2) and simplifying, we have
(3.19)‖zn+1-xn+1‖2≤‖zn-xn‖2+αn2‖zn-xn‖2+2αn(kn-1)‖zn-xn‖2+6αnkn(βn+δn)M2-2αnϕ(‖zn+1-xn+1‖-en)+2αnσnM+2γnM2≤‖zn-xn‖2-2αnϕ(‖zn+1-xn+1‖-en)+αn2M2+2αn(kn-1)M2+6αnkn(βn+δn)M2+2αnσnM+2γnM2.
Suppose that an=∥zn-xn∥, and bn=αn,cn=αn2M2+2αn(kn-1)M2+6αnkn(βn+δn)M2+2αnσnM+2γnM2. Since limn→∞σn=0, limn→∞en=0, and limn→∞kn=1 and by the hypotheses, we have that ∑n=1∞bn=∞ and cn=o(bn), en→0 as n→∞. In view of Lemma 2.3 we obtain that limn→∞an=0. Hence, limn→∞∥zn-xn∥=0.Theorem 3.2.
LetX be a uniformly smooth Banach space, and let T:X→X be an asymptotically ϕ-strongly Pseudocontractive mapping with fixed point x* and bounded range.
Let{vn}n be the sequences defined by
(3.20)v0∈X,vn+1=(1-αn)vn+αnTnvn,n≥0,
where {αn}n⊂[0,1] satisfy(i)
limn→∞αn=0,(ii)
∑n=1∞αn=∞. Then, for any initial point v0∈X, the sequence {vn}n strongly converges to x*.Proof.
By the boundedness of the range ofT and by Lemma 2.4, we have that {vn}n is bounded.
By Lemma2.2 we observe that
(3.21)‖vn+1-x*‖2≤(1-αn)2‖vn-x*‖2+2αn〈Tnvn-x*,j(vn+1-x*)〉≤(1-αn)2‖vn-x*‖2+2αn〈Tnvn-x*,j(vn+1-x*)-j(vn-x*)〉+2αn〈Tnvn-x*,j(vn-x*)〉≤(1-αn)2‖vn-x*‖2+2αn〈Tnvn-x*,j(vn+1-x*)-j(vn-x*)〉+2αnkn‖vn-x*‖2-2αnϕ(‖vn-x*‖)=(1+αn2-2αn)‖vn-x*‖2+2αn〈Tnvn-x*,j(vn+1-x*)-j(vn-x*)〉+2αnkn‖vn-x*‖2-2αnϕ(‖vn-x*‖)=‖vn-x*‖2+(αn2-2αn+2αnkn)‖vn-x*‖2-2αnϕ(‖vn-x*‖)+2αnμn,
where μn:=〈Tnvn-x*,j(vn+1-x*)-j(vn-x*)〉. Let
(3.22)M∶=max⁡{sup⁡n‖vn-x*‖,sup⁡n‖Tnvn-x*‖}.
We have
(3.23)‖vn+1-x*‖2≤‖vn-x*‖2+(αn2+2αn(kn-1))M-2αnϕ(‖vn-x*‖)+2αnμn=‖vn-x*‖2-αnϕ(‖vn-x*‖)-αn[ϕ(‖vn-x*‖)-2μn-(αn+2(kn-1))M],
and so we can observe that(1)
μn→0 as n→∞, in fact from the inequality
(3.24)|‖vn+1-x*‖-‖vn-x*‖|≤‖vn+1-vn‖≤αnM⟶0asn⟶∞
and since j is norm-to-norm uniformly continuous, then j(∥vn+1-x*∥)-j(∥vn-x*∥)→0 as n→∞,(2)
infn(∥vn-x*∥)=0, in fact, if we suppose that σ:=infn(∥vn-x*∥)>0, by the monotonicity of ϕ,
(3.25)ϕ(‖vn-x*‖)≥ϕ(σ)>0.
Thus, by (1) and by the hypotheses on αn and kn, the value -αn[ϕ(∥vn-x*∥)-2μn-(αn+2(kn-1))M] is definitively negative. In this case we conclude that there exists N>0 such that for every n>N(3.26)‖vn+1-x*‖2≤‖vn-x*‖2-αnϕ(‖vn-x*‖)≤‖vn-x*‖2-αnϕ(σ),
and so
(3.27)αnϕ(σ)≤‖vn-x*‖2-‖vn+1-x*‖2∀n>N.
In the same manner we obtain that
(3.28)ϕ(σ)∑i=Nmαi≤∑i=Nm[‖vi-x*‖2-‖vi+1-x*‖2]=‖vN-x*‖2-‖vm-x*‖2.
By the hypothesis ∑nαn=∞, the previous is a contradiction and it follows that infn(∥vn-x*∥)=0.
Then, there exists a subsequence{vnk}k of {vn}n that strongly converges to x*. This implies that for every ϵ>0 there exists an index nk(ϵ) such that, for all j≥nk(ϵ), ∥vnj-x*∥<ϵ.
Now we will prove that the entire sequence{vn}n converges to x*. Since the sequences in (3.37) are null sequences but ∑nαn=∞, then for every ϵ>0 there exists an index n¯(ϵ) such that for all n≥n¯(ϵ) it following that:
(3.29)|αn|<12Mmin⁡{ϵ,ϕ(ϵ/2)2},|kn-1|<ϕ(ϵ/2)8M,|μn|<ϕ(ϵ/2)8.
So, fixing ϵ>0, let n*>max(nk(ϵ),n¯(ϵ)) with n*=nj for a certain nj. We will prove, by induction, that ∥vn*+i-x*∥<ϵ for every i∈ℕ. Let i=1. If not, it follows that ∥vn*+1-x*∥≥ϵ. Thus
(3.30)ϵ≤‖vn*+1-x*‖≤‖vn*-x*‖+αn*<‖vn*-x*‖+ϵ2MM=‖vn*-x*‖+ϵ2,
that is, ∥vn*-x*∥>(ϵ/2). By the strict increasing of ϕ, ϕ(∥vn*-x*∥)>ϕ(ϵ/2). By (3.37), it follows that
(3.31)‖vn*+1-x*‖2<ϵ2-αn*ϕ(‖vn*-x*‖)-αn*[ϕ(‖vn*-x*‖)-2μn*-(αn*+2(kn*-1))M].
We can note that
(3.32)2μn*+(αn*+2(kn*-1))M≤ϕ(ϵ/2)4+(ϕ(ϵ/2)4M+ϕ(ϵ/2)4M)M,
and so
(3.33)ϕ(‖vn*-x*‖)-2μn*-(αn*+2(kn*-1))M≥ϕ(ϵ2)-3ϕ(ϵ/2)4>0.
Moreover, ϕ(∥vn*-x*∥)>ϕ(ϵ/2)/2>0, and it follows that
(3.34)‖vn*+1-x*‖2≤ϵ2.
This is absurd. Thus ∥vn*+1-x*∥<ϵ.
In the same manner, by induction, one obtains that, for everyi≥1, ∥vn*+i-x*∥<ϵ. So ∥vn-x*∥→0.Corollary 3.3.
LetX be a uniformly smooth Banach space, and let T:X→X be an asymptotically ϕ-strongly Pseudocontractive mapping with fixed point x* and bounded range.
Let{zn}n be the sequences defined by (2.2), where {αn}nand{γn}n⊂[0,1] satisfy(i)
limn→∞αn=limn→∞γn=0,(ii)
∑n=1∞αn=∞, ∑n=1∞γn<∞, and the sequence {wn}n is bounded on X. Then, for any initial point z0∈X, the sequence {zn}n strongly converges to x*.Proof.
Following the idea of the proof of Theorem3.2, by the boundedness of the range of T and by the boundedness of the sequence {wn}n and by Lemma 2.4, we have that {zn}n is bounded.
By Lemma2.2 we observe that
(3.35)‖zn+1-x*‖2≤(1-αn-γn)2‖zn-x*‖2+2αn〈Tnzn-x*,j(zn+1-x*)〉+2γn〈wn-x*,j(zn+1-x*)〉≤(1-αn)2‖zn-x*‖2+2αn〈Tnzn-x*,j(zn+1-x*)-j(zn-x*)〉+2αn〈Tnzn-x*,j(zn-x*)〉+2γn〈wn-x*,j(zn+1-x*)〉≤(1-αn)2‖zn-x*‖2+2αn〈Tnzn-x*,j(zn+1-x*)-j(zn-x*)〉+2αnkn‖zn-x*‖2-2αnϕ(‖zn-x*‖)+2γn〈wn-x*,j(zn+1-x*)〉=(1+αn2-2αn)‖zn-x*‖2+2αn〈Tnzn-x*,j(zn+1-x*)-j(zn-x*)〉+2αnkn‖zn-x*‖2-2αnϕ(‖zn-x*‖)+2γn〈wn-x*,j(zn+1-x*)〉=‖zn-x*‖2+(αn2-2αn+2αnkn)‖zn-x*‖2-2αnϕ(‖zn-x*‖)+2αnμn+2γn〈wn-x*,j(zn+1-x*)〉,
where μn:=〈Tnzn-x*,j(zn+1-x*)-j(zn-x*)〉. Let
(3.36)M:=max⁡{sup⁡n‖zn-x*‖,sup⁡n‖Tnzn-x*‖,sup⁡n‖wn-x*‖,sup⁡n〈wn-x*,j(zn+1-x*)〉}.
We have
(3.37)‖zn+1-x*‖2≤‖zn-x*‖2+(αn2+2αn(kn-1))M-2αnϕ(‖zn-x*‖)+2αnμn+2γnM=‖zn-x*‖2-αnϕ(‖zn-x*‖)+2γnM-αn[ϕ(‖zn-x*‖)-2μn-(αn+2(kn-1))M],
and so we can observe that(1)
μn→0 as n→∞, in fact from the inequality
(3.38)|‖zn+1-x*‖-‖zn-x*‖|≤‖zn+1-zn‖(αn+γn)M⟶0asn⟶∞
and since j is norm-to-norm uniformly continuous, then j(∥zn+1-x*∥)-j(∥zn-x*∥)→0 as n→∞,(2)
infn(∥zn-x*∥)=0, in fact, if we suppose that σ:=infn(∥zn-x*∥)>0, by the monotonicity of ϕ,
(3.39)ϕ(‖zn-x*‖)≥ϕ(σ)>0.
Thus, by (1) and by the hypotheses on αn and kn, the value -αn[ϕ(∥zn-x*∥)-2μn-(αn+2(kn-1))M] is definitively negative. In this case, we conclude that there exists N>0 such that for every n>N(3.40)‖zn+1-x*‖2≤‖zn-x*‖2-αnϕ(‖zn-x*‖)+2γnM≤‖zn-x*‖2-αnϕ(σ)+2γnM,
and so
(3.41)αnϕ(σ)≤‖zn-x*‖2-‖zn+1-x*‖2+2γnM∀n>N.
In the same manner we obtain that
(3.42)ϕ(σ)∑i=Nmαi≤∑i=Nm[‖zi-x*‖2-‖zi+1-x*‖2]+2∑i=NmγiM=‖zN-x*‖2-‖zm-x*‖2+2M∑i=Nmβi.
By the hypotheses ∑nγn<∞ and ∑nαn=∞, the previous is a contradiction and it follows that infn(∥zn-x*∥)=0.
Then, there exists a subsequence{znk}k of {zn}n that strongly converges to x*. This implies that for every ϵ>0 there exists an index nk(ϵ) such that, for all j≥nk(ϵ), ∥znj-x*∥<ϵ.
Now we will prove that the entire sequence{zn}n converges to x*. Since the sequences in (3.37) are null sequences and ∑nγn<∞ but ∑nαn=∞, then for every ϵ>0 there exists an index n¯(ϵ) such that for all n≥n¯(ϵ) it follows that:
(3.43)|αn|<14Mmin⁡{ϵ,ϕ(ϵ2)},|γn|<ϵ4M,|γnαn|<ϕ(ϵ/2)4M,|kn-1|<ϕ(ϵ/2)8M,|μn|<ϕ(ϵ/2)8.
So, fixing ϵ>0, let n*>max(nk(ϵ),n¯(ϵ)) with n*=nj for a certain nj. We will prove, by induction, that ∥zn*+i-x*∥<ϵ for every i∈ℕ. Let i=1. If not, it follows that ∥zn*+1-x*∥≥ϵ. Thus,
(3.44)ϵ≤‖zn*+1-x*‖≤‖zn*-x*‖+αn*M+γn*M<‖zn*-x*‖+ϵ4MM+ϵ4MM=‖zn*-x*‖+ϵ2,
that is, ∥zn*-x*∥>ϵ/2. By the strict increasing of ϕ, ϕ(∥zn*-x*∥)>ϕ(ϵ/2).
By (3.37), it follows that
(3.45)‖zn*+1-x*‖2<ϵ2-αn*(ϕ(‖zn*-x*‖)-2Mγn*αn*)-αn*[ϕ(‖zn*-x*‖)-2μn*-(αn*+2(kn*-1))M].
We can note that
(3.46)2μn*+(αn*+2(kn*-1))M≤ϕ(ϵ/2)4+(ϕ(ϵ/2)4M+ϕ(ϵ/2)4M)M,
and so
(3.47)ϕ(‖zn*-x*‖)-2μn*-(αn*+2(kn*-1))M≥ϕ(ϵ2)-3ϕ(ϵ/2)4>0.
Moreover, ϕ(∥zn*-x*∥)-2M(γn*/αn*)>ϕ(ϵ/2)/2>0 so it follows that
(3.48)‖zn*+1-x*‖2≤ϵ2.
This is absurd. Thus, ∥zn*+1-x*∥<ϵ.
In the same manner, by induction, one obtains that, for everyi≥1, ∥zn*+i-x*∥<ϵ. So ∥zn-x*∥→0.Corollary 3.4.
LetX be a uniformly smooth Banach space, and let T:X→X be asymptotically ϕ-strongly Pseudocontractive mapping with bounded range and fixed point x*. The sequences {xn}n, {zn}n, and {zn′}n defined by (2.1) and (2.2), respectively, where the sequences {αn}n, {βn}n, {γn}n, and {δn}n⊂[0,1] satisfy(i)
limn→∞αn=limn→∞βn=limn→∞δn=0,(ii)
∑n=1∞αn=∞,∑n=1∞γn<∞,and the sequences {un}n, {vn}n, and {wn}n, are bounded in X. Then, for any initial point x0,z0∈X, the following two assertions are equivalents, and true.(i)
The modified Ishikawa iteration sequence (2.1) converges to the fixed point x*.(ii)
The modified Mann iteration sequence (2.2) converges to the fixed point x*.Example 3.5.
Let us consider the mapping in Example1.9, Tx=x/(1+0.01x), where x∈[0,1], that is asymptotically ϕ-strongly Pseudocontractive with kn=(n+100)/n, for all n∈ℕ and ϕ(x)=x3/(1+x). The unique fixed point of T is x=0. Let z0∈(0,1] and, for all n∈ℕ, αn=1/n, and γn=1/n2. Then the sequence generated by the scheme
(3.49)zn+1=(1-1n-1n2)zn+1nTnzn+1n2wn,n≥0,
converges to 0.Remark 3.6.
Our results are similar to those of Schu's  [4, Theorem  1.4] However, our results hold in a more general setting of uniformly smooth Banach spaces, while Schu's result holds for completely continuous, uniformly Lipschitzian mappings, which are asymptotically pseudocontractive.



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*Source: 850104-2012-03-20.xml*