# A Note on Implicit and Explicit Mann Iterative Processes for Asymptoticallyϕ-Strongly Pseudocontractive Mappings

Authors: Luigi Muglia; Yonghong Yao
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

In this paper we prove the equivalence and the strong convergence of an explicit Mann iterative process and a modified implicit 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 form 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 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’s fixed point theorem [2] to this class of mappings.This class is really more general than the class of nonexpansive mappings (see [1]).In 1991, Schu [3] introduced the class of asymptotically pseudocontractive mappings.Definition 1.2 (see [3]).
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
(1.2)〈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 Lipschitz but asymptotically pseudocontractive [4].In [3], Schu proved the following.Theorem 1.3 (see [3]).
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
(1.3)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 had been proved. First Zhou, in [5], 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; (3) the strong convergence of a CQ-iterative method. The literature on asymptotical-type mappings is wide (see e.g., [6–11]).In 1974, Deimling [12], studying the zeros of accretive operators, introduced the class of φ-strongly accretive operators.Definition 1.4.
An operatorA defined on a subset K of a Banach space X is called φ-strongly accretive if
(1.4)〈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.Since an operatorA is a strongly accretive operators if and only if (I-A) is a strongly pseudocontractive mappings (i.e., 〈(I-A)x-(I-A)y,j(x-y)〉≤k∥x-y∥2, k<1), taking in account Definition 1.4, it is natural to study the class of φ-pseudocontractive mappings, that is, the maps such that(1.5)〈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 mappings contains, at most, only one point.Here our attention is on the class of theasymptoticallyϕ-strongly pseudocontractions define as follows.Definition 1.5.
IfX is a Banach space and K is a subset of X, a mapping T:K→K is said asymptotically ϕ-strongly pseudocontraction if
(1.6)〈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 and 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,(1.7)‖x-z‖2=〈Tnx-Tnz,j(x-z)〉≤kn‖x-y‖2-ϕ(‖x-y‖),
so, passing n to +∞, it results(1.8)‖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 1.6.
The mappingTx=x/(x+1), where x∈[0,1], is asymptotically ϕ-strongly pseudocontraction with kn=1, for all n∈ℕ and ϕ(t)=t3/(1+t). However, T is not strongly pseudocontractive, see [13].Example 1.7.
The mappingTx=x/(1+αx), where x∈[0,1] and α is closing to zero, is asymptotically ϕ-strongly pseudocontraction with kn=1+1/αn, for all n∈ℕ and ϕ(x)=x3/(1+x). However, T is not strongly pseudocontractive and nor is the ϕ-strongly pseudocontraction.Proof.
First we prove thatT is not strongly pseudocontractive. For arbitrary k<1, there exists x,y∈[0,1], such that
(1.9)1(1+αx)(1+αy)>k.
So we have
(1.10)〈Tx-Ty,j(x-y)〉=1(1+αx)(1+αy)(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],
(1.11)〈Tx-Ty,j(x-y)〉=x2(1+αx)‖x-y‖2-ϕ(‖x-y‖)=x21+x,
Therefore, T is not ϕ-strongly pseudocontractive. Finally, we prove that T is asymptotically ϕ-strongly pseudocontraction.
For arbitraryx,y∈[0,1], without loss of generality, let x>y. Then,
(1.12)〈Tnx-Tny,j(x-y)〉=(x-y)2(1+αnx)(1+αny),kn‖x-y‖2-ϕ(‖x-y‖)=(1+1αn)(x-y)2-(x-y)31+(x-y)=[1+1αn-(x-y)1+(x-y)](x-y)2=1+1/αn+1/αn(x-y)1+(x-y)(x-y)2.
We only need to prove that
(1.13)1+x-y≤(1+αnx)(1+αny)[1+1αn+1αn(x-y)].
Using x>y, this is easy.Let us consider the well-knownMann iterative process defined as follows: for any given z0∈X, the sequence {zn}n defined by(1.14)zn+1=(1-αn)zn+αnTnzn,n≥0.We can also introduce amodified implicit iterative process as follows: suppose T is continuous, (wn′)n is bounded, and let z0′ be an initial point. Thus, we define(1.15)zn′=(1-αn-γn)zn-1′+αnTnzn′+γnwn′,n≥1,
where {αn}n is a real sequence in [0,1] satisfying αnkn<1 for all n≥1.The algorithm is well defined. In fact, ifT is a continuous asymptotically strongly ϕ-pseudocontraction, for every fixed n, the mapping Sn defined by Snx:=(1-αn-γn)zn-1+αnTnx+γnwn is such that(1.16)〈Snx-Sny,j(x-y)〉=〈Tnx-Tny,j(x-y)〉≤αnkn‖x-y‖2,
that is, Sn is a continuous strongly pseudocontraction, for every fixed n. Then (see Theorem  13.1 in [14]), there exists a unique fixed-point of Sn for each n.This modified implicit method is inspired to a wide literature.In 1995, Liu [15] introduced the following modified Ishikawa method:(1.17)yn=(1-βn)xn+βnTnxn+vn,xn+1=(1-αn)xn+αnTnyn+un,n≥0,
and he called it Ischikawa iteration process with errors(obviously posing βn=0, he obtains the Mann iteration process with errors). This new class of methods with errors was studied, among the others, also by Chang in [16] in 2001, Chang et. al [17] in 2006, Gu in [18], Huang in [19], and Huang and Bu in 2007 [20].In 2001, Chidume and Osilike [21] proved the strong convergence of the iterative method(1.18)yn=anxn+bnSxn+cnun,xn+1=an′xn+bn′Syncn′vn,
where an+bn+cn=an′+bn′+cn′=1, Sx=x-Tx+f (T a ϕ-strongly accretive operator), and f∈X, to a solution of the equation Tx=f. Note that Chidume and Osilike did not use term with errors to indicate their methods.In 2003, Chidume and Zegeye [22] studied the following iterative method:(1.19)xn+1=(1-αn)xn+αnTxn-αnθn(xn-x1),
where T is a Lipschitzian pseudocontractive map with fixed points. The authors proved the strong convergence of the method to a fixed point of T under opportune hypotheses on the control sequences (θn)n,(λn)n.If we posewn=z0 in (1.14) and γn=αnθn, the modified Mann iterative process coincides with the Chideme and Zegeye’s iterative method.In this paper, we prove the equivalence between the implicit and the explicit modified Mann iterative method which involves an asymptoticallyϕ-strongly pseudocontractive mapping.


## 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 shall denote the single-valued duality mapping by j.For the sake of completeness, we recall some definitions and conclusions.Definition 2.1.
AnX is said to be a uniformly smooth Banach space if the smooth module of X,
(2.1)ρX(t)=sup⁡{12(‖x-y‖+‖x+y‖)-1:‖x‖≤1,‖y‖≤t},
satisfies limt→0(ρX(t)/t)=0.Lemma 2.2 (see [23]).
LetX be a Banach space, and let J:X→2X* be the normalized duality mapping. Then, for any x,y∈X, we have
(2.2)‖x+y‖2≤‖x‖2+2〈y,j(x+y)〉,∀j(x+y)∈J(x+y).Next lemma is a key for our proofs.Lemma 2.3 (see [19]).
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
(2.3)lim⁡n→∞bn=0,cn=o(bn),∑n=1∞bn=∞,lim⁡n→∞en=0.
Suppose that there exists an integer N1>0 such that
(2.4)an+12≤an2-2bnϕ(|an+1-en|)+cn,∀n≥N1,
then limn→∞an=0.Proof.
The proof is the same as in [19] but changing, in (2.4), (an+1-en) with |an+1-en|.Lemma 2.4 (see [24]).
Let{sn}n,{cn}n⊂ℝ+, {an}n⊂(0,1), and {bn}n⊂ℝ be sequences such that
(2.5)sn+1≤(1-an)sn+bn+cn,
for all n≥0. Assume that ∑ncn<∞. Then, the following results hold.(1)
Ifbn≤βan (where β≥0), then {sn}n is a bounded sequence.(2)
If we have∑nan=∞ and limsupn(bn/an)≤0, then sn→0 as n→∞.Remark 2.5.
If, in Lemma2.3, we choose en=0, for all n, ϕ(t)=kt2 (k<1), then the inequality (2.4) becomes
(2.6)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 in account the papers of Chang and Chidume et al. [16, 21, 25].Theorem 3.1.
LetX be a uniformly smooth Banach space, and let T:X→X be asymptotically ϕ-strongly pseudocontractive mapping with fixed point x* and bounded range.
Let{vn}n be the sequence defined by
(3.1)v0∈X,vn+1=(1-αn)vn+αnTnvn,n≥0,
where {αn}n⊂[0,1] satisfies(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
(3.2)‖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
(3.3)M:=max⁡{sup⁡n‖vn-x*‖,sup⁡n‖Tnvn-x*‖}.
We have
(3.4)‖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],
so we can observe the following.(1)
μn→0 as n→∞. In fact from the inequality
(3.5)|‖vn+1-x*‖-‖vn-x*‖|≤‖vn+1-vn‖≤αnM⟶0,asn⟶∞,
and, since j is norm to norm uniformly continuous, then
(3.6)j(‖vn+1-x*‖)-j(‖vn-x*‖)⟶0,asn⟶∞.(2)
infn(∥vn-x*∥)=0. In fact, if we suppose that σ:=infn(∥vn-x*∥)>0, by the monotonicity of ϕ,
(3.7)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,
(3.8)‖vn+1-x*‖2≤‖vn-x*‖2-αnϕ(‖vn-x*‖)≤‖vn-x*‖2-αnϕ(σ)
and so
(3.9)αnϕ(σ)≤‖vn-x*‖2-‖vn+1-x*‖2∀n>N.
In the same manner, we obtain that
(3.10)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 (3.4) are null sequences but ∑nαn=∞, then, for every ϵ>0, there exists an index n¯(ϵ) such that, for all n≥n¯(ϵ), it results that
(3.11)|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 results that ∥vn*+1-x*∥≥ϵ. Thus,
(3.12)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, then ϕ(∥vn*-x*∥)>ϕ(ϵ/2). By (3.4), it results
(3.13)‖vn*+1-x*‖2<ϵ2-αn*ϕ(‖vn*-x*‖)-αn*[ϕ(‖vn*-x*‖)-2μn*-(αn*+2(kn*-1))M].
We can note that
(3.14)2μn*+(αn*+2(kn*-1))M≤ϕ(ϵ/2)4+(ϕ(ϵ/2)4M+ϕ(ϵ/2)4M)M,
so
(3.15)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, it results that
(3.16)‖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.Remark 3.2.
Our result is similar to Schu’s theorem. However, our results hold in a more general setting of uniformly smooth Banach spaces, while the Schu’s result holds for completely continuous, uniformly Lipschitzian mappings which are asymptotically pseudocontractive.Theorem 3.3.
LetX be a uniformly smooth Banach space, and let T:X→X be a continuous and asymptotically ϕ-strongly pseudocontractive mapping with fixed point x* and bounded range.
Let{zn} and {zn′} be the sequences defined by (1.14) and (1.15), respectively, where {αn},{γn}⊂[0,1] are null sequences satisfying(H1)
limn→∞αn=0 and γn=o(αn),(H2)
∑n=1∞αn=∞, and such that αnkn<1, for every n∈ℕ.
Let us suppose moreover that the sequences{wn},{wn′} are bounded in X.
Then, for any initial pointz0′,z0∈X, the following two assertions are equivalent.
(i)
The Mann iteration sequence (1.14) converges to the fixed point x*.(ii)
The modified implicit iteration sequence (1.15) converges to the fixed point x*.Proof.
By the boundedness of the range ofT and by Lemma 2.4, one obtains that our schemes are bounded. Let us define
(3.17)M=sup⁡n⁡{‖Tnzn-Tnzn′‖,‖Tnzn-zn‖,‖Tnzn′-zn′‖,‖Tnzn′-zn-1′‖‖zn+1-zn′‖,‖wn′-zn-1′‖,‖wn′‖}.
By the iteration schemes (1.14) and (1.15), we have
(3.18)‖zn+1-zn′‖2≤‖(1-αn)(zn-zn-1′)+αn(Tnzn-Tnzn′)+γn(zn-1′-wn′)‖2≤(1-αn)2‖zn-zn-1′‖2+2〈αn(Tnzn-Tnzn′)+γn(zn-1′-wn′),j(zn+1-zn′)〉≤(1-αn)2‖zn-zn-1′‖2+2αn〈Tnzn-Tnzn′,j(zn-zn′)〉+2αn〈Tnzn-Tnzn′,j(zn+1-zn′)-j(zn-zn′)〉+2γnM2≤(1-αn)2‖zn-zn-1′‖2+2αnkn‖zn-zn′‖2-2αnϕ(‖zn-zn′‖)+2αnMσn+2γnM2,
where σn=∥j(zn+1-zn′)-j(zn-zn′)∥. By (1.14), we have
(3.19)‖(zn+1-zn′)-(zn-zn′)‖=‖zn+1-zn‖=‖αn(Tnzn-zn)‖≤αnM.
It follows from (H1) that ∥(zn+1-zn′)-(zn-zn′)∥→0 as n→∞, which implies that σn→0 as n→∞. Moreover, for all n≥0,
(3.20)‖zn-zn′‖2≤(‖zn-zn-1′‖+‖zn-1′-zn′‖)2≤(‖zn-zn-1′‖+αn‖Tnzn′-zn-1′‖+γn‖wn′-zn-1′‖)2≤[‖zn-zn-1′‖+(αn+γn)M]2=‖zn-zn-1′‖2+(αn+γn)[2M‖zn-zn-1′‖+(αn+γn)M2]≤‖zn-zn-1′‖2+3(αn+γn)M2.
Again by the boundedness of all components, we have that
(3.21)‖zn′-zn-1′‖=‖αn(Tnzn′-zn-1′)+γn(wn′-zn-1′)‖≤(αn+γn)M,
and so
(3.22)‖zn+1-zn′‖=‖(zn-zn′)+(zn′-zn-1′)-αn(zn-zn-1′)+αn(Tnzn-Tnzn′)+γn(zn-1′-wn′)‖≤‖zn-zn′‖+‖zn′-zn-1′‖+αn‖zn-zn-1′‖+αn‖Tnzn-Tnzn′‖+γn‖zn-1′-wn′‖≤‖zn-zn′‖+3(αn+γn)M.
Hence, we have that ∥zn-zn′∥≥∥zn+1-zn′∥-en, where en=3(αn+γn)M. Note that en→0 as n→∞. As in proof of Theorem 3.1, we distinguish two cases:(i)
the set of index for which∥zn+1-zn′∥-en≤0 contains infinite terms,(ii)
the set of index for which∥zn+1-zn′∥-en≤0 contains finite terms. In the first case (i), we can extract a subsequence such that ∥znk-znk-1′∥→0, as k→∞. Substituting (3.20) in (3.18), we have that
(3.23)‖zn+1-zn′‖2≤(1+αn2-2αn)‖zn-zn-1′‖2+2αnkn‖zn-zn-1′‖2+6(αn+γn)M2αnkn-2αnϕ(‖zn-zn′‖)+2αnMσn+2γnM2≤‖zn-zn-1′‖2+(αn2+2αn(kn-1))‖zn-zn-1′‖2+6(αn+γn)M2αnk¯-2αnϕ(‖zn-zn′‖)+2αnMσn+2γnM2≤‖zn-zn-1′‖2-αnϕ(‖zn-zn′‖)+2γnM2-αn[ϕ(‖zn-zn′‖)-αnM2-2M2(kn-1)-6(αn+γn)M2k¯-2Mσn]=‖zn-zn-1′‖2-αnϕ(‖zn-zn′‖)+2γnM2-αn[ϕ(‖zn-zn′‖)-7k¯αnM2-2M2(kn-1)-6γnM2k¯-2Mσn],
where k¯=supnkn. Again by (3.20), for every ϵ>0, there exists an index l such that, if j>l,
(3.24)‖znj-znj-1′‖<ϵ,‖znj-znj′‖<2ϵ.
By hypotheses on the control sequence, with the same ϵ>0, there exists an index N such that definitively
(3.25)|αn|<min⁡{ϵ12M,ϕ(ϵ/2)56M2k¯},|γn|<min⁡{ϵ12M,ϕ(ϵ/2)48M2k¯},|γnαn|<ϕ(ϵ/2)4M2,|kn-1|<ϕ(ϵ/2)16M2,|σn|<ϕ(ϵ/2)16M.
So take n*>max{nl,N} with n*=nj for a certain j.
We can prove that∥zn+1-zn′∥→0 as n→∞ proving that, for every i≥0 it results ∥zn*+i-zn*+i-1′∥<ϵ.
Leti= 1. If we suppose that ∥zn*+1-zn*′∥≥ϵ, it results that
(3.26)ϵ≤‖zn*+1-zn*′‖≤‖zn*-zn*′‖+3(αn*+γn*)M<‖zn*-zn*′‖+ϵ2,
so ∥zn*-zn*′∥>ϵ/2. In consequence of this, ϕ(∥zn*-zn*′∥)>ϕ(ϵ/2).
In (3.23), we note that(3.27)7k¯αn*M2+2M2(kn*-1)+6γn*M2k¯+2Mσn*≤7k¯M2ϕ(ϵ/2)56M2k¯+2M2ϕ(ϵ/2)16M2+6M2k¯ϕ(ϵ/2)48M2k¯+2Mϕ(ϵ/2)16M=ϕ(ϵ/2)84=ϕ(ϵ/2)2,
so
(3.28)ϕ(‖zn*-zn*′‖)-7k¯αn*M2-2M2(kn*-1)-6γn*M2k¯-2Mσn*≥ϕ(ϵ/2)2,
Hence, in (3.23), remains
(3.29)‖zn*+1-zn*′‖2≤ϵ2-αn*ϕ(‖zn*-zn*′‖)+2γn*M2.
In the same manner,
(3.30)ϕ(‖zn*-zn*′‖)-2γn*αn*M2>ϕ(ϵ2)-ϕ(ϵ/2)2>0,
so
(3.31)‖zn*+1-zn*′‖2≤ϵ2.
This is absurd. By the same idea and by using the induction method, we obtain that ∥zn*+i-zn*+i-1′∥<ϵ, for every i≥0. This assure that ∥zn+1-zn′∥→0. In the second case (ii), definitively, ∥zn+1-zn′∥-en≥0, then, from the strict increasing function ϕ, we have
(3.32)ϕ(‖zn-zn′‖)≥ϕ(‖zn+1-zn′‖-en).
Substituting (3.32) and (3.20) into (3.18) and simplifying, we have
(3.33)‖zn+1-zn′‖2≤‖zn-zn-1′‖2+αn2‖zn-zn-1′‖2+2αn(kn-1)‖zn-zn-1′‖2+6αnkn(αn+γn)M2-2αnϕ(‖zn+1-zn′‖-en)+2αnσnM+2γnM2≤‖zn-zn-1′‖2-2αnϕ(‖zn+1-zn′‖-en)+αn2M2+2αn(kn-1)M2+6αnkn(αn+γn)M2+2αnσnM+2γnM2.
By virtue of Lemma 2.3, we obtain that limn→∞∥zn-zn-1′∥=0.



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*Source: 912050-2012-05-08.xml*