Abstract

We prove the equivalence and the strong convergence of iterative processes involving generalized strongly asymptotically 𝜙-pseudocontractive mappings in uniformly smooth Banach spaces.

1. Introduction

Throughout this paper, we assume that 𝑋 is a uniformly convex Banach space and 𝑋 is the dual space of 𝑋. Let 𝐽 denote the normalized duality mapping form 𝑋 into 2𝑋 given by 𝐽(𝑥)={𝑓𝑋𝑥,𝑓=𝑥2=𝑓2} for all 𝑥𝑋, where , denotes the generalized duality pairing. It is well known that if 𝑋 is uniformly smooth, then 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.

In 1967, Browder [1] and Kato [2], independently, introduced accretive operators (see, for details, Chidume [3]). Their interest is connected with the existence of results in the theory of nonlinear equations of evolution in Banach spaces.

In 1972, Goebel and Kirk [4] introduced the class of asymptotically nonexpansive mappings as follows.

Definition 1.1. Let 𝐾 be a subset of a Banach space 𝑋. A mapping 𝑇𝐾𝐾 is said to be asymptotically nonexpansive if for each 𝑥,𝑦𝐾𝑇𝑛𝑥𝑇𝑛𝑦𝑘𝑛,𝑥𝑦(1.1) where {𝑘𝑛}𝑛[1,) is a sequence of real numbers converging to 1.

This class is more general than the class of nonexpansive mappings as the following example clearly shows.

Example 1.2 (see [4]). If 𝐵 is the unit ball of 𝑙2 and 𝑇𝐵𝐵 is defined as 𝑇𝑥1,𝑥2=,0,𝑥21,𝑎2𝑥2,𝑎3𝑥3,,(1.2) where {𝑎𝑖}𝑖(0,1) is such that 𝑖=2𝑎𝑖=1/2, it satisfies. 𝑇𝑥𝑇𝑦2𝑥𝑦,𝑇𝑛𝑥𝑇𝑛𝑦2𝑛𝑗=2𝑎𝑖𝑥𝑦.(1.3)

In 1974, Deimling [5], studying the zeros of accretive operators, introduced the class of 𝜑-strongly accretive operators.

Definition 1.3. An operator 𝐴 defined on a subset 𝐾 of a Banach space 𝑋 is said, 𝜑-strongly accretive if ),𝐴𝑥𝐴𝑦,𝑗(𝑥𝑦)𝜑(𝑥𝑦𝑥𝑦(1.4) where 𝜑++ is a strictly increasing function such that 𝜑(0)=0.

Note that in the special case in which 𝜑(𝑡)=𝑘𝑡,  𝑘(0,1), we obtain a strongly accretive operator.

Osilike [6], among the others, proved that 𝐴𝑥=𝑥(𝑥/(𝑥+1)) in + is 𝜑-strongly accretive where 𝜑(𝑡)=(𝑡2/(1+𝑡)) but not strongly accretive.

Since an operator 𝐴 is a strongly accretive operator if and only if (𝐼𝐴) is a strongly pseudocontractive mapping (i.e., (𝐼𝐴)𝑥(𝐼𝐴)𝑦,𝑗(𝑥𝑦)𝑘𝑥𝑦2,  𝑘<1), taking in to account Definition 1.3, it is natural to study the class of 𝜑-pseudocontractive mappings, that is, the maps such that𝑇𝑥𝑇𝑦,𝑗(𝑥𝑦)𝑥𝑦2()𝜑𝑥𝑦𝑥𝑦,(1.5) where 𝜑++ is a strictly increasing function such that 𝜑(0)=0. Of course, the set of fixed points for these mappings contains, at most, only one point.

Recently, has been also studied the following class of maps.

Definition 1.4. A mapping 𝑇 is a generalized 𝜙-strongly pseudocontractive mapping if 𝑇𝑥𝑇𝑦,𝑗(𝑥𝑦)𝑥𝑦2(𝜙𝑥𝑦),(1.6) where 𝜙++ is a strictly increasing function such that 𝜙(0)=0.

Choosing 𝜙(𝑡)=𝜑(𝑡)𝑡, we obtain Definition 1.3. In [7], Xiang remarked that it is a open problem if every generalized 𝜙-strongly pseudocontractive mapping is 𝜑-pseudocontractive mapping. In the same paper, Xiang obtained a fixed-point theorem for continuous and generalized 𝜙-strongly pseudocontractive mappings in the setting of the Banach spaces.

In 1991, Schu [8] introduced the class of asymptotically pseudocontractive mappings.

Definition 1.5 (see [8]). Let 𝑋 be a normed space, 𝐾𝑋 and {𝑘𝑛}𝑛[1,). A mapping 𝑇𝐾𝐾 is said to be asymptotically pseudocontractive with the sequence {𝑘𝑛}𝑛 if and only if lim𝑛𝑘𝑛=1, and for all 𝑛 and all 𝑥,𝑦𝐾, there exists 𝑗(𝑥𝑦)𝐽(𝑥𝑦) such that 𝑇𝑛𝑥𝑇𝑛𝑦,𝑗(𝑥𝑦)𝑘𝑛𝑥𝑦2,(1.7) 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 that 𝑇[0,1][0,1] defined by 𝑇𝑥=(1𝑥2/3)3/2 is not Lipschitz but asymptotically pseudocontractive [9].

In [8], Schu proved the following.

Theorem 1.6 (see [8]). Let 𝐻 be a Hilbert space and 𝐴𝐻 closed and convex; 𝐿>0; 𝑇𝐴𝐴 completely continuous, uniformly 𝐿-Lipschitzian, and asymptotically pseudocontractive with sequence {𝑘𝑛}𝑛[1,);  𝑞𝑛=2𝑘𝑛1 for all 𝑛; 𝑛(𝑞2𝑛1)<; {𝛼𝑛}𝑛,{𝛽𝑛}𝑛[0,1]; 𝜖𝛼𝑛𝛽𝑛𝑏 for all 𝑛, some 𝜖>0 and some 𝑏(0,𝐿2[1+𝐿21]); 𝑥1𝐴; for all 𝑛, define 𝑧𝑛=𝛽𝑛𝑇𝑛𝑥𝑛+1𝛽𝑛𝑥𝑛,𝑥𝑛+1=𝛼𝑛𝑇𝑛𝑧𝑛+1𝛼𝑛𝑥𝑛,(1.8) then {𝑥𝑛}𝑛 converges strongly to some fixed point of 𝑇.

Until 2009, no results on fixed-point theorems for asymptotically pseudocontractive mappings have been proved. First, Zhou in [10] completed this lack in the setting of Hilbert spaces proving a fixed-point theorem for an asymptotically pseudocontractive mapping that is also uniformly 𝐿-Lipschitzian and uniformly asymptotically regular and that the set of fixed points of 𝑇 is closed and convex. Moreover, Zhou proved the strong convergence of a CQ-iterative method involving this kind of mappings.

In this paper, our attention is on the class of the generalized strongly asymptotically 𝜙-pseudocontraction defined as follows.

Definition 1.7. If 𝑋 is a Banach space and 𝐾 is a subset of 𝑋, a mapping 𝑇𝐾𝐾 is said to be a generalized asymptotically 𝜙-strongly pseudocontraction if 𝑇𝑛𝑥𝑇𝑛𝑦,𝑗(𝑥𝑦)𝑘𝑛𝑥𝑦2(),𝜙𝑥𝑦(1.9) where {𝑘𝑛}𝑛[1,) is converging to one and 𝜙[0,)[0,) is strictly increasing and such that 𝜙(0)=0.

One can note that (i)if 𝑇 has fixed points, then it is unique. In fact, if 𝑥,𝑧 are fixed points for 𝑇, then for every 𝑛, 𝑥𝑧2=𝑇𝑛𝑥𝑇𝑛𝑧,𝑗(𝑥𝑧)𝑘𝑛𝑥𝑧2𝜙(𝑥𝑧),(1.10) so passing 𝑛 to +, it results that 𝑥𝑧2𝑥𝑧2𝜙(𝑥𝑧)𝜙(𝑥𝑧)0.(1.11) Since 𝜙[0,)[0,) is strictly increasing and 𝜙(0)=0, then 𝑥=𝑧. (ii)the mapping 𝑇𝑥=𝑥/(𝑥+1), where 𝑥[0,1], is generalized asymptotically strongly 𝜙-pseudocontraction with 𝑘𝑛=1, for all 𝑛 and 𝜙(𝑡)=𝑠3/(1+𝑠). However, 𝑇 is not strongly pseudocontractive; see [6].

We study the equivalence between three kinds of iterative methods involving the generalized asymptotically strongly 𝜙-pseudocontractions.

Moreover, we prove that these methods are equivalent and strongly convergent to the unique fixed point of the generalized strongly asymptotically 𝜙-pseudocontraction 𝑇, under suitable hypotheses.

We will briefly introduce some of the results in the same line of ours. In 2001, [11] Chidume and Osilike proved the strong convergence of the iterative method 𝑦𝑛=𝑎𝑛𝑥𝑛+𝑏𝑛𝑆𝑥𝑛+𝑐𝑛𝑢𝑛,𝑥𝑛+1=𝑎𝑛𝑥𝑛+𝑏𝑛𝑆𝑦𝑛𝑐𝑛𝑣𝑛,(1.12) where 𝑎𝑛+𝑏𝑛+𝑐𝑛=𝑎𝑛+𝑏𝑛+𝑐𝑛=1,  𝑆𝑥=𝑥𝑇𝑥+𝑓 (𝑇 a 𝜙-strongly accretive operator), and 𝑓𝑋, to a solution of the equation 𝑇𝑥=𝑓.

In 2003, Chidume and Zegeye [12] studied the following iterative method: 𝑥𝑛+1=1𝜆𝑛𝑥𝑛+𝜆𝑛𝑇𝑥𝑛𝜆𝑛𝜃𝑛𝑥𝑛𝑥1,(1.13) 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 suitable hypotheses on the control sequences (𝜃𝑛)𝑛,(𝜆𝑛)𝑛.

Taking in to account Chidume and Zegeye [12] and Chang [13], we introduce the modified Mann and Ishikawa iterative processes as follows: for any given 𝑥0𝑋, the sequence {𝑥𝑛}𝑛 is defined by 𝑦𝑛=1𝛽𝑛𝑥𝑛+𝛽𝑛𝑇𝑛𝑥𝑛𝛿𝑛𝑥𝑛𝑣𝑛,𝑥𝑛+1=1𝛼𝑛𝑥𝑛+𝛼𝑛𝑇𝑛𝑦𝑛𝛾𝑛𝑥𝑛𝑢𝑛,𝑛0,(1.14) where {𝛼𝑛}𝑛, {𝛾𝑛}𝑛, {𝛽𝑛}𝑛, and {𝛿𝑛}𝑛 are four sequences in (0,1) satisfying the conditions 𝛼𝑛+𝛾𝑛1 and 𝛽𝑛+𝛿𝑛1 for all 𝑛0.

In particular, if 𝛽𝑛=𝛿𝑛=0 for all 𝑛0, we can define a sequence {𝑧𝑛}𝑛 by 𝑧0𝑧𝑋,𝑛+1=1𝛼𝑛𝑧𝑛+𝛼𝑛𝑇𝑛𝑧𝑛𝛾𝑛𝑧𝑛𝑤𝑛,𝑛0,(1.15) which is called the modified Mann iteration sequence.

We also introduce an implicit iterative process as follows: 𝑧𝑛=1𝛼𝑛𝑧𝑛1+𝛼𝑛𝑇𝑛𝑧𝑛𝛾𝑛𝑧𝑛1𝑤𝑛,𝑛1,(1.16) where {𝛼𝑛}𝑛,{𝛾𝑛}𝑛 are two real sequences in [0,1] satisfying 𝛼𝑛+𝛾𝑛1 and 𝛼𝑛𝑘𝑛<1 for all 𝑛1, {𝑤𝑛}𝑛 is a sequence in 𝑋, and 𝑧0 is an initial point.

The algorithm is well defined. Indeed, if 𝑇 is a asymptotically strongly 𝜙-pseudocontraction, one can observe that, for every fixed 𝑛, the mapping 𝑆𝑛 defined by 𝑆𝑛𝑥=(1𝛼𝑛𝛾𝑛)𝑧𝑛1+𝛼𝑛𝑇𝑛𝑥+𝛾𝑤𝑛 is such that 𝑆𝑛𝑥𝑆𝑛𝑦,𝑗(𝑥𝑦)=𝑇𝑛𝑥𝑇𝑛𝑦,𝑗(𝑥𝑦)𝛼𝑛𝑘𝑛𝑥𝑦2,(1.17) that is, 𝑆𝑛 is a strongly pseudocontraction, for every fixed 𝑛, then (see Theorem  13.1 in [14]) there exists a unique fixed point of 𝑆𝑛 for each 𝑛.

These kind of iterative processes (also called by Chang iterative processes with errors) have been developed in [1518], while equivalence theorem for Mann and Ishikawa methods has been studied, in [19, 20], among the others.

In [21], Huang established equivalences between convergence of the modified Mann iteration process with errors (1.15) and convergence of modified Ishikawa iteration process with errors (1.14) for strongly successively 𝜙-pseudocontractive mappings in uniformly smooth Banach space.

In the next section, we prove that, in the setting of the uniformly smooth Banach space, if 𝑇 is an asymptotically strongly 𝜙-pseudocontraction, not only (1.14) and (1.15) are equivalent but also (1.16) is equivalent to the others. Moreover, we prove also that (1.14), (1.15), and (1.16) strongly converge to the unique fixed point of 𝑇, if it exists.

2. Preliminaries

We recall some definitions and conclusions.

Definition 2.1. 𝑋 is said to be a uniformly smooth Banach space if the smooth module of 𝑋𝜌𝑋1(𝑡)=sup2(𝑥𝑦+𝑥+𝑦)1𝑥1,𝑦𝑡(2.1) satisfies lim𝑡0𝜌𝑋(𝑡)/𝑡=0.

Lemma 2.2 (see [22]). Let 𝑋 be a Banach space, and let 𝑗𝑋2𝑋 be the normalized duality mapping, then for any 𝑥,𝑦𝑋, one has 𝑥+𝑦2𝑥2+2𝑦,𝑗(𝑥+𝑦),𝑗(𝑥+𝑦)𝐽(𝑥+𝑦).(2.2)

The next lemma is one of the main tools for our proofs.

Lemma 2.3 (see [21]). Let 𝜙[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 lim𝑛𝑏𝑛=0,𝑐𝑛𝑏=𝑜𝑛,𝑛=1𝑏𝑛=,lim𝑛𝑒𝑛=0.(2.3) Suppose that there exists an integer 𝑁1>0 such that 𝑎2𝑛+1𝑎2𝑛2𝑏𝑛𝜙||𝑎𝑛+1𝑒𝑛||+𝑐𝑛,𝑛𝑁1,(2.4) then lim𝑛𝑎𝑛=0.

Proof. The proof is the same as in [21], but we substitute (𝑎𝑛+1𝑒𝑛) with |𝑎𝑛+1𝑒𝑛|, in (2.4).

Lemma 2.4 (see [23]). Let {𝑠𝑛}𝑛, {𝑐𝑛}𝑛+, {𝑎𝑛}𝑛(0,1), and {𝑏𝑛}𝑛 be sequences such that 𝑠𝑛+11𝑎𝑛𝑠𝑛+𝑏𝑛+𝑐𝑛,(2.5) for all 𝑛0. Assume that 𝑛|𝑐𝑛|<, then the following results hold:(1)if 𝑏𝑛𝛽𝑎𝑛 (where 𝛽0), then {𝑠𝑛}𝑛 is a bounded sequence;(2)if one has 𝑛𝑎𝑛= and limsup𝑛𝑏𝑛/𝑎𝑛0, then 𝑠𝑛0 as 𝑛.

Remark 2.5. If in Lemma 2.3 choosing 𝑒𝑛=0, for all 𝑛, 𝜙(𝑡)=𝑘𝑡2 (𝑘<1), then the inequality (2.4) becomes 𝑎2𝑛+1𝑎2𝑛2𝑏𝑛𝑘𝑎2𝑛+1+𝑐𝑛𝑎2𝑛+111+2𝑏𝑛𝑘𝑎2𝑛+𝑐𝑛1+2𝑏𝑛𝑘=12𝑏𝑛𝑘1+2𝑏𝑛𝑘𝑎2𝑛+𝑐𝑛1+2𝑏𝑛𝑘.(2.6) Setting 𝛼𝑛=2𝑏𝑛𝑘/(1+2𝑏𝑛𝑘) and 𝛽𝑛=𝑐𝑛/(1+2𝑏𝑛𝑘) and by the hypotheses of Lemma 2.3, we get 𝛼𝑛0 as 𝑛, 𝑛𝛼𝑛=, and limsup𝑛𝛽𝑛/𝛼𝑛=0. That is, we reobtain Lemma 2.4 in the case of 𝑐𝑛=0.

3. Main Results

The ideas of the proofs of our main Theorems take in to account the papers of Chang and Chidume et al. [11, 13, 24].

Theorem 3.1. Let 𝑋 be a uniformly smooth Banach space, and let 𝑇𝑋𝑋 be generalized strongly asymptotically 𝜙-pseudocontractive mapping with fixed point 𝑥 and bounded range.
Let {𝑥𝑛} and {𝑧𝑛} be the sequences defined by (1.14) and (1.15), respectively, where {𝛼𝑛},{𝛾𝑛},{𝛽𝑛},{𝛿𝑛}[0,1] satisfy(H1)lim𝑛𝛼𝑛=lim𝑛𝛽𝑛=lim𝑛𝛿𝑛=0 and 𝛾𝑛=𝑜(𝛼𝑛),(H2)𝑛=1𝛼𝑛=,and the sequences {𝑢𝑛},{𝑣𝑛},{𝑤𝑛} are bounded in 𝑋, then for any initial point 𝑧0,𝑥0𝑋, the following two assertions are equivalent: (i)the modified Ishikawa iteration sequence with errors (1.14) converges to 𝑥; (ii)the modified Mann iteration sequence with errors (1.15) converges to 𝑥.

Proof. First of all, we note that by boundedness of the range of 𝑇, of the sequences {𝑤𝑛},{𝑢𝑛} and by Lemma 2.4, it results that {𝑧𝑛} and {𝑥𝑛} are bounded sequences. So, we can set 𝑀=sup𝑛𝑇𝑛𝑧𝑛𝑇𝑛𝑦𝑛,𝑇𝑛𝑥𝑛𝑥𝑛,𝑇𝑛𝑦𝑛𝑥𝑛,𝑇𝑛𝑧𝑛𝑧𝑛,𝑧𝑛𝑥𝑛,𝑢𝑛𝑥𝑛,𝑣𝑛𝑥𝑛,𝑤𝑛𝑧𝑛,𝑤𝑛𝑢𝑛.(3.1) By Lemma 2.2, we have 𝑧𝑛+1𝑥𝑛+12=1𝛼𝑛𝛾𝑛𝑧𝑛𝑥𝑛+𝛼𝑛𝑇𝑛𝑧𝑛𝑇𝑛𝑦𝑛+𝛾𝑛𝑤𝑛𝑢𝑛21𝛼𝑛𝛾𝑛2𝑧𝑛𝑥𝑛2𝛼+2𝑛𝑇𝑛𝑧𝑛𝑇𝑛𝑦𝑛+𝛾𝑛𝑤𝑛𝑢𝑛𝑧,𝑗𝑛+1𝑥𝑛+11𝛼𝑛2𝑧𝑛𝑥𝑛2+2𝛼𝑛𝑇𝑛𝑧𝑛𝑇𝑛𝑦𝑛𝑧,𝑗𝑛𝑦𝑛+2𝛼𝑛𝑇𝑛𝑧𝑛𝑇𝑛𝑦𝑛𝑧,𝑗𝑛+1𝑥𝑛+1𝑧𝑗𝑛𝑦𝑛+2𝛾𝑛𝑤𝑛𝑢𝑛𝑧,𝑗𝑛+1𝑥𝑛+11𝛼𝑛2𝑧𝑛𝑥𝑛2+2𝛼𝑛𝑘𝑛𝑧𝑛𝑦𝑛22𝛼𝑛𝜙𝑧𝑛𝑦𝑛+2𝛼𝑛𝑇𝑛𝑧𝑛𝑇𝑛𝑦𝑛𝑗𝑧𝑛+1𝑥𝑛+1𝑧𝑗𝑛𝑦𝑛+2𝛾𝑛𝑤𝑛𝑢𝑛𝑧𝑛+1𝑥𝑛+11𝛼𝑛2𝑧𝑛𝑥𝑛2+2𝛼𝑛𝑘𝑛𝑧𝑛𝑦𝑛22𝛼𝑛𝜙𝑧𝑛𝑦𝑛+2𝛼𝑛𝜎𝑛𝑀+2𝛾𝑛𝑀2,(3.2) where 𝜎𝑛=𝑗(𝑧𝑛+1𝑥𝑛+1)𝑗(𝑧𝑛𝑦𝑛). Using (1.14) and (1.15), we have 𝑧𝑛+1𝑥𝑛+1𝑧𝑛𝑦𝑛𝑥𝑛+1𝑦𝑛+𝑧𝑛+1𝑧𝑛=𝛼𝑛𝑇𝑛𝑦𝑛𝑥𝑛+𝛾𝑛𝑢𝑛𝑥𝑛𝛽𝑛𝑇𝑛𝑥𝑛𝑥𝑛𝛿𝑛𝑣𝑛𝑥𝑛+𝛼𝑛𝑇𝑛𝑧𝑛𝑧𝑛+𝛾𝑛𝑤𝑛𝑧𝑛𝛼2𝑀𝑛+𝛾𝑛+𝛽𝑛+𝛿𝑛0,(𝑛).(3.3) In view of the uniformly continuity of 𝑗, we obtain that 𝜎𝑛0 as 𝑛. Furthermore, it follows from the definition of {𝑦𝑛} that for all 𝑛0𝑧𝑛𝑦𝑛2=𝑧𝑛𝑥𝑛+𝛽𝑛(𝑇𝑛𝑥𝑛+𝑥𝑛)+𝛿𝑛(𝑣𝑛+𝑥𝑛)2𝑧𝑛𝑥𝑛+𝛽𝑛𝑇𝑛𝑥𝑛𝑥𝑛+𝛿𝑛𝑣𝑛𝑥𝑛2𝑧𝑛𝑥𝑛+𝛽𝑛+𝛿𝑛𝑀2=𝑧𝑛𝑥𝑛2+𝛽𝑛+𝛿𝑛2𝑧𝑛𝑥𝑛𝛽𝑀+𝑛+𝛿𝑛𝑀2𝑧𝑛𝑥𝑛2𝛽+3𝑛+𝛿𝑛𝑀2,(3.4)𝑦𝑛𝑥𝑛𝛽𝑛𝑇𝑛𝑥𝑛𝑥𝑛+𝛿𝑛𝑣𝑛𝑥𝑛𝛽𝑛+𝛿𝑛𝑀0,(𝑛),(3.5) so 𝑧𝑛+1𝑥𝑛+1=𝑧𝑛𝑥𝑛𝛼𝑛+𝛾𝑛𝑧𝑛𝑥𝑛+𝛼𝑛𝑇𝑛𝑧𝑛𝑇𝑛𝑦𝑛+𝛾𝑛𝑤𝑛𝑢𝑛𝑧𝑛𝑥𝑛+𝛼𝑛+𝛾𝑛𝑧𝑛𝑥𝑛+𝛼𝑛𝑇𝑛𝑧𝑛𝑇𝑛𝑦𝑛+𝛾𝑛𝑤𝑛𝑢𝑛𝑧𝑛𝑦𝑛+𝑦𝑛𝑥𝑛𝛼+2𝑛+𝛾𝑛𝑧𝑀.𝑛𝑦𝑛+𝛽𝑛+𝛿𝑛𝛼𝑀+2𝑛+𝛾𝑛𝑀.(3.6) Therefore, we have 𝑧𝑛𝑦𝑛𝑧𝑛+1𝑥𝑛+1𝑒𝑛,(3.7) where 𝑒𝑛=(𝛽𝑛+𝛿𝑛)𝑀+2(𝛼𝑛+𝛾𝑛)𝑀. By (H1), we have that 𝑒𝑛0 as 𝑛. If 𝑧𝑛+1𝑥𝑛+1𝑒𝑛0 for an infinite number of indices, we can extract a subsequence such that 𝑧𝑛𝑘𝑥𝑛𝑘𝑒𝑛𝑘10. For this subsequence, 𝑧𝑛𝑘𝑥𝑛𝑘0, as 𝑘.
In this case, we can prove that 𝑧𝑛𝑥𝑛0, that is, the thesis.
Firstly, we note that substituting (3.4) into (3.2), we have 𝑧𝑛+1𝑥𝑛+12𝑧𝑛𝑥𝑛2+𝛼2𝑛𝑧𝑛𝑥𝑛2+2𝛼𝑛𝑘𝑛𝑧1𝑛𝑥𝑛2+6𝛼𝑛𝑘𝑛𝛽𝑛+𝛿𝑛𝑀22𝛼𝑛𝜙𝑧𝑛𝑦𝑛+2𝛼𝑛𝜎𝑛𝑀+2𝛾𝑛𝑀2𝑧𝑛𝑥𝑛22𝛼𝑛𝜙𝑧𝑛𝑦𝑛+𝛼2𝑛𝑀2+2𝛼𝑛𝑘𝑛𝑀12+6𝛼𝑛𝑘𝛽𝑛+𝛿𝑛𝑀2+2𝛼𝑛𝜎𝑛𝑀+2𝛾𝑛𝑀2=𝑧𝑛𝑥𝑛2𝛼𝑛𝜙𝑧𝑛𝑦𝑛+2𝛾𝑛𝑀2𝛼𝑛𝜙𝑧𝑛𝑦𝑛𝛼𝑛𝑀2𝑘2𝑛𝑀126𝑘𝛽𝑛+𝛿𝑛𝑀22𝜎𝑛𝑀,(3.8) where 𝑘=sup𝑛(𝑘𝑛).
Moreover, we observe that 𝑧𝑛𝑗𝑦𝑛𝑗𝑧𝑛𝑗𝑥𝑛𝑗+𝑦𝑛𝑗𝑥𝑛𝑗0as.𝑗(3.9) Thus, for every fixed 𝜖>0, there exists 𝑗1 such that for all 𝑗>𝑗1𝑧𝑛𝑗𝑦𝑛𝑗𝑧<2𝜖𝑛𝑗𝑥𝑛𝑗<𝜖.(3.10) Since {𝛼𝑛}𝑛, {(𝑘𝑛1)}𝑛, {(𝛽𝑛+𝛿𝑛)}𝑛, {𝜎𝑛}𝑛, and {𝛾𝑛}𝑛 are null sequences (and in particular 𝛾𝑛=𝑜(𝛼𝑛)), for the previous fixed 𝜖>0, there exists an index 𝑁 such that, for all 𝑛𝑁, ||𝛼𝑛||𝜖<min,16𝑀𝜙(𝜖/2)8𝑀2,||𝛾𝑛||<𝜖,||||𝛾16𝑀𝑛𝛼𝑛||||<𝜙(𝜖/2)4𝑀2,||𝑘𝑛||<1𝜙(𝜖/2)16𝑀2,||𝛽𝑛+𝛿𝑛||𝜖<min,4𝑀𝜙(𝜖/2)48𝑘𝑀2,||𝜎𝑛||<𝜙(𝜖/2),16𝑀(3.11) for all 𝑛>𝑁.
Take 𝑛>max{𝑁,𝑛𝑗1} such that 𝑛=𝑛𝑘 for a certain 𝑘.
We prove, by induction, that 𝑧𝑛+𝑖𝑥𝑛+𝑖<𝜖, for every 𝑖. Let 𝑖=1. Suppose that 𝑧𝑛+1𝑥𝑛+1𝜖.
By (3.6), we have 𝑧𝜖𝑛+1𝑥𝑛+1𝑧𝑛𝑦𝑛+𝛽𝑛+𝛿𝑛𝛼𝑀+2𝑛+𝛾𝑛𝑀𝑧𝑛𝑦𝑛𝜖+𝑀𝜖4𝑀+2𝑀+𝜖16𝑀=𝑧16𝑀𝑛𝑦𝑛+𝜖4+𝜖4=𝑧𝑛𝑦𝑛+𝜖2.(3.12) Thus, 𝑧𝑛𝑦𝑛𝜖/2. Since 𝜙 is strictly increasing, 𝜙(𝑧𝑛𝑦𝑛)𝜙(𝜖/2).
From (3.8), we obtain that 𝑧𝑛+1𝑥𝑛+12<𝜖2𝛼𝑛𝜙𝑧𝑛𝑦𝑛2𝑀2𝛾𝑛𝛼𝑛𝛼𝑛𝜙𝑧𝑛𝑦𝑛𝛼𝑛𝑀2𝑘2𝑛𝑀126𝑘𝛽𝑛+𝛿𝑛𝑀22𝜎𝑛𝑀.(3.13) One can note that 𝛼𝑛𝑀2𝑘+2𝑛𝑀12+6𝑘𝛽𝑛+𝛿𝑛𝑀2+2𝜎𝑛𝑀𝜙(𝜖/2)8+𝜙(𝜖/2)8+𝜙(𝜖/2)8+𝜙(𝜖/2)8.(3.14) hence 𝜙𝑧𝑛𝑦𝑛𝛼𝑛𝑀2𝑘2𝑛𝑀126𝑘𝛽𝑛+𝛿𝑛𝑀22𝜎𝑛𝜖𝑀𝜙2𝜙(𝜖/2)2.>0(3.15) In the same manner, 𝜙𝑧𝑛𝑦𝑛2𝑀2𝛾𝑛𝛼𝑛>𝜙(𝜖/2)𝜙(𝜖/2)2.>0(3.16) Thus, 𝑧𝑛+1𝑥𝑛+12<𝜖2.(3.17) So we have 𝑧𝑛+1𝑥𝑛+1<𝜖, which contradicts 𝑧𝑛+1𝑥𝑛+1𝜖. By the same idea, we can prove that 𝑧𝑛+2𝑥𝑛+2<𝜖 and then, by inductive step, 𝑧𝑛+𝑖𝑥𝑛+𝑖𝜖, for all 𝑖. This is enough to ensure that 𝑧𝑛𝑥𝑛0.
If there are only finite indices for which 𝑧𝑛+1𝑥𝑛+1𝑒𝑛0, then definitively 𝑧𝑛+1𝑥𝑛+1𝑒𝑛0. By the strict increasing function 𝜙, we have definitively 𝜙𝑧𝑛𝑦𝑛𝑧𝜙𝑛+1𝑥𝑛+1𝑒𝑛.(3.18) Again substituting (3.4) and (3.18) into (3.2) and simplifying, we have 𝑧𝑛+1𝑥𝑛+12𝑧𝑛𝑥𝑛2+𝛼2𝑛𝑧𝑛𝑥𝑛2+2𝛼𝑛𝑘𝑛𝑧1𝑛𝑥𝑛2+6𝛼𝑛𝑘𝑛𝛽𝑛+𝛿𝑛𝑀22𝛼𝑛𝜙𝑧𝑛+1𝑥𝑛+1𝑒𝑛+2𝛼𝑛𝜎𝑛𝑀+2𝛾𝑛𝑀2𝑧𝑛𝑥𝑛22𝛼𝑛𝜙𝑧𝑛+1𝑥𝑛+1𝑒𝑛+𝛼2𝑛𝑀2+2𝛼𝑛𝑘𝑛𝑀12+6𝛼𝑛𝑘𝑛𝛽𝑛+𝛿𝑛𝑀2+2𝛼𝑛𝜎𝑛𝑀+2𝛾𝑛𝑀2.(3.19) Suppose that 𝑎𝑛=𝑧𝑛𝑥𝑛,𝑏𝑛=𝛼𝑛,and𝑐𝑛=𝛼2𝑛𝑀2+2𝛼𝑛(𝑘𝑛1)𝑀2+6𝛼𝑛𝑘𝑛(𝛽𝑛+𝛿𝑛)𝑀2+2𝛼𝑛𝜎𝑛𝑀+2𝛾𝑛𝑀2. It follows from lim𝑛𝜎𝑛=0, lim𝑛𝑒𝑛=0, lim𝑛𝑘𝑛=1, and the hypothesis that we have, 𝑛=1𝑏𝑛= and 𝑐𝑛=𝑜(𝑏𝑛), 𝑒𝑛0 as 𝑛. By virtue of Lemma 2.3, we obtain that lim𝑛𝑎𝑛=0. Hence, lim𝑛𝑧𝑛𝑥𝑛=0.

Theorem 3.2. Let 𝑋 be a uniformly smooth Banach space, and let 𝑇𝑋𝑋 be generalized strongly asymptotically 𝜙-pseudocontractive mapping with fixed point 𝑥 and bounded range.
Let {𝑧𝑛} and {𝑧𝑛} be the sequences defined by (1.15) and (1.16), respectively, where {𝛼𝑛},{𝛾𝑛}[0,1] are null sequences satisfying (H1)lim𝑛𝛼𝑛=0 and 𝛾𝑛=𝑜(𝛼𝑛), (H2)𝑛=1𝛼𝑛=, of Theorem 3.1 and such that 𝛼𝑛𝑘𝑛<1, for every 𝑛.
Suppose moreover that the sequences {𝑤𝑛},{𝑤𝑛} are bounded in 𝑋, then for any initial point 𝑧0,𝑧0𝑋, the following two assertions are equivalent:
(i)the modified Mann iteration sequence with errors (1.15) converges to the fixed point 𝑥, (ii)the implicit iteration sequence with errors (1.16) converges to the fixed point 𝑥.

Proof. As in Theorem 3.1, by the boundedness of the range of 𝑇 and by Lemma 2.4, one obtains that our schemes are bounded. We define 𝑀=sup𝑛𝑇𝑛𝑧𝑛𝑇𝑛𝑧𝑛,𝑇𝑛𝑧𝑛𝑧𝑛,𝑇𝑛𝑧𝑛𝑧𝑛,𝑇𝑛𝑧𝑛𝑧𝑛1𝑧𝑛+1𝑧𝑛,𝑧𝑛𝑤𝑛,𝑤𝑛𝑧𝑛1,𝑤𝑛𝑤𝑛.(3.20) By the iteration schemes (1.15) and (1.16), we have 𝑧𝑛+1𝑧𝑛21𝛼𝑛𝛾𝑛𝑧𝑛𝑧𝑛1+𝛼𝑛𝑇𝑛𝑧𝑛𝑇𝑛𝑧𝑛+𝛾𝑛𝑤𝑛𝑤𝑛21𝛼𝑛𝛾𝑛2𝑧𝑛𝑧𝑛12𝛼+2𝑛𝑇𝑛𝑧𝑛𝑇𝑛𝑧𝑛+𝛾𝑛𝑤𝑛𝑤𝑛𝑧,𝑗𝑛+1𝑧𝑛1𝛼𝑛2𝑧𝑛𝑧𝑛12+2𝛼𝑛𝑇𝑛𝑧𝑛𝑇𝑛𝑧𝑛𝑧,𝑗𝑛𝑧𝑛+2𝛼𝑛𝑇𝑛𝑧𝑛𝑇𝑛𝑧𝑛𝑧,𝑗𝑛+1𝑧𝑛𝑧𝑗𝑛𝑧𝑛+2𝛾𝑛𝑤𝑛𝑤𝑛𝑧𝑛+1𝑧𝑛1𝛼𝑛2𝑧𝑛𝑧𝑛12+2𝛼𝑛𝑘𝑛𝑧𝑛𝑧𝑛22𝛼𝑛𝜙𝑧𝑛𝑧𝑛+2𝛼𝑛𝑀𝜎𝑛+2𝛾𝑛𝑀2,(3.21) where 𝜎𝑛=𝑗(𝑧𝑛+1𝑧𝑛)𝑗(𝑧𝑛𝑧𝑛). By (1.15), we get 𝑧𝑛+1𝑧𝑛𝑧𝑛𝑧𝑛=𝑧𝑛+1𝑧𝑛=𝛼𝑛𝑇𝑛𝑧𝑛𝑧𝑛+𝛾𝑛𝑤𝑛𝑧𝑛𝛼𝑛+𝛾𝑛𝑀.(3.22) It follows from (H1) that (𝑧𝑛+1𝑧𝑛)(𝑧𝑛𝑧𝑛)0 as 𝑛, which implies that 𝜎𝑛0 as 𝑛. Moreover, for all 𝑛0, 𝑧𝑛𝑧𝑛2𝑧𝑛𝑧𝑛1+𝑧𝑛1𝑧𝑛2𝑧𝑛𝑧𝑛1+𝛼𝑛𝑇𝑛𝑧𝑛𝑧𝑛1+𝛾𝑛𝑤𝑛𝑧𝑛12𝑧𝑛𝑧𝑛1+𝛼𝑛+𝛾𝑛𝑀2=𝑧𝑛𝑧𝑛12+𝛼𝑛+𝛾𝑛𝑧2𝑀𝑛𝑧𝑛1+𝛼𝑛+𝛾𝑛𝑀2𝑧𝑛𝑧𝑛12𝛼+3𝑛+𝛾𝑛𝑀2.(3.23) Again by the boundedness of all components, we have that 𝑧𝑛𝑧𝑛1=𝛼𝑛𝑇𝑛𝑧𝑛𝑧𝑛1+𝛾𝑛𝑤𝑛𝑧𝑛1𝛼𝑛+𝛾𝑛𝑀,(3.24) and so 𝑧𝑛+1𝑧𝑛=𝑧𝑛𝑧𝑛+𝑧𝑛𝑧𝑛1𝛼𝑛+𝛾𝑛𝑧𝑛𝑧𝑛1+𝛼𝑛𝑇𝑛𝑧𝑛𝑇𝑛𝑧𝑛+𝛾𝑛𝑤𝑛𝑤𝑛𝑧𝑛𝑧𝑛+𝑧𝑛𝑧𝑛1+𝛼𝑛+𝛾𝑛𝑧𝑛𝑧𝑛1+𝛼𝑛𝑇𝑛𝑧𝑛𝑇𝑛𝑧𝑛+𝛾𝑛𝑤𝑛𝑤𝑛𝑧𝑛𝑧𝑛𝛼+3𝑛+𝛾𝑛𝑀.(3.25) Hence, we have that 𝑧𝑛𝑧𝑛𝑧𝑛+1𝑧𝑛𝑒𝑛, where 𝑒𝑛=3(𝛼𝑛+𝛾𝑛)𝑀. Note that 𝑒𝑛0 as 𝑛. As in proof of Theorem 3.1, we distinguish two cases:(i)the set of indices for which 𝑧𝑛+1𝑧𝑛𝑒𝑛0 contains infinite terms; (ii)the set of indices for which 𝑧𝑛+1𝑧𝑛𝑒𝑛0 contains finite terms. In the first case, (i) we can extract a subsequence such that 𝑧𝑛𝑘𝑧𝑛𝑘10, as 𝑘. Substituting (3.23) in (3.21), we have that 𝑧𝑛+1𝑧𝑛21+𝛼2𝑛2𝛼𝑛𝑧𝑛𝑧𝑛12+2𝛼𝑛𝑘𝑛𝑧𝑛𝑧𝑛12𝛼+6𝑛+𝛾𝑛𝑀2𝛼𝑛𝑘𝑛2𝛼𝑛𝜙𝑧𝑛𝑧𝑛+2𝛼𝑛𝑀𝜎𝑛+2𝛾𝑛𝑀2𝑧𝑛𝑧𝑛12+𝛼2𝑛+2𝛼𝑛𝑘𝑛𝑧1𝑛𝑧𝑛12𝛼+6𝑛+𝛾𝑛𝑀2𝛼𝑛𝑘2𝛼𝑛𝜙𝑧𝑛𝑧𝑛+2𝛼𝑛𝑀𝜎𝑛+2𝛾𝑛𝑀2𝑧𝑛𝑧𝑛12𝛼𝑛𝜙𝑧𝑛𝑧𝑛+2𝛾𝑛𝑀2𝛼𝑛𝜙𝑧𝑛𝑧𝑛𝛼𝑛𝑀22𝑀2𝑘𝑛𝛼16𝑛+𝛾𝑛𝑀2𝑘2𝑀𝜎𝑛=𝑧𝑛𝑧𝑛12𝛼𝑛𝜙𝑧𝑛𝑧𝑛+2𝛾𝑛𝑀2𝛼𝑛𝜙𝑧𝑛𝑧𝑛7𝑘𝛼𝑛𝑀22𝑀2𝑘𝑛16𝛾𝑛𝑀2𝑘2𝑀𝜎𝑛,(3.26) where 𝑘=sup𝑛𝑘𝑛. Again by (3.23), for every 𝜖>0, there exists an index 𝑙 such that if 𝑗>𝑙, 𝑧𝑛𝑗𝑧𝑛𝑗1𝑧<𝜖,𝑛𝑗𝑧𝑛𝑗<2𝜖.(3.27) By hypotheses on the control sequence, with the same 𝜖>0, there exists an index 𝑁 such that definitively ||𝛼𝑛||𝜖<min,12𝑀𝜙(𝜖/2)56𝑀2𝑘,||𝛾𝑛||𝜖<min,12𝑀𝜙(𝜖/2)48𝑀2𝑘,||||𝛾𝑛𝛼𝑛||||<𝜙(𝜖/2)4𝑀2,||𝑘𝑛||<1𝜙(𝜖/2)16𝑀2,||𝜎𝑛||<𝜙(𝜖/2).16𝑀(3.28) So take 𝑛>max{𝑛𝑙,𝑁} with 𝑛=𝑛𝑗 for a certain 𝑗.
We can prove that 𝑧𝑛+1𝑧𝑛0 as 𝑛 proving that, for every 𝑖0, the result is 𝑧𝑛+𝑖𝑧𝑛+𝑖1<𝜖.
Let 𝑖=1. If we suppose that 𝑧𝑛+1𝑧𝑛𝜖, it results that 𝑧𝜖𝑛+1𝑧𝑛𝑧𝑛𝑧𝑛𝛼+3𝑛+𝛾𝑛𝑧𝑀<𝑛𝑧𝑛+𝜖2,(3.29) so 𝑧𝑛𝑧𝑛>𝜖/2. In consequence of this, 𝜙(𝑧𝑛𝑧𝑛)>𝜙(𝜖/2).
In (3.26), we note that 7𝑘𝛼𝑛𝑀2+2𝑀2𝑘𝑛1+6𝛾𝑛𝑀2𝑘+2𝑀𝜎𝑛7𝑘𝑀2𝜙(𝜖/2)56𝑀2𝑘+2𝑀2𝜙(𝜖/2)16𝑀2+6𝑀2𝑘𝜙(𝜖/2)48𝑀2𝑘+2𝑀𝜙(𝜖/2)=16𝑀𝜙(𝜖/2)84=𝜙(𝜖/2)2,(3.30) so 𝜙𝑧𝑛𝑧𝑛7𝑘𝛼𝑛𝑀22𝑀2𝑘𝑛16𝛾𝑛𝑀2𝑘2𝑀𝜎𝑛𝜙(𝜖/2)2.(3.31) hence in (3.26) remains 𝑧𝑛+1𝑧𝑛2𝜖2𝛼𝑛𝜙𝑧𝑛𝑧𝑛+2𝛾𝑛𝑀2<𝜖2,(3.32) as in Theorem 3.1. This is a contradiction. By the same idea, and using the inductive hypothesis, we obtain that 𝑧𝑛+𝑖𝑧𝑛+𝑖1<𝜖, for every 𝑖0. This ensures that 𝑧𝑛+1𝑧𝑛0. In the second case (ii), definitively, 𝑧𝑛+1𝑧𝑛𝑒𝑛0, then from the strictly increasing function 𝜙, we have 𝜙𝑧𝑛𝑧𝑛𝑧𝜙𝑛+1𝑧𝑛𝑒𝑛.(3.33) Substituting (3.33) and (3.23) into (3.21) and simplifying, we have 𝑧𝑛+1𝑧𝑛2𝑧𝑛𝑧𝑛12+𝛼2𝑛𝑧𝑛𝑧𝑛12+2𝛼𝑛𝑘𝑛𝑧1𝑛𝑧𝑛12+6𝛼𝑛𝑘𝑛𝛼𝑛+𝛾𝑛𝑀22𝛼𝑛𝜙𝑧𝑛+1𝑧𝑛𝑒𝑛+2𝛼𝑛𝜎𝑛𝑀+2𝛾𝑛𝑀2𝑧𝑛𝑧𝑛122𝛼𝑛𝜙𝑧𝑛+1𝑧𝑛𝑒𝑛+𝛼2𝑛𝑀2+2𝛼𝑛𝑘𝑛𝑀12+6𝛼𝑛𝑘𝑛𝛼𝑛+𝛾𝑛𝑀2+2𝛼𝑛𝜎𝑛𝑀+2𝛾𝑛𝑀2.(3.34) By virtue of Lemma 2.3, we obtain that lim𝑛𝑧𝑛𝑧𝑛1=0.

Theorem 3.3. Let 𝑋 be a uniformly smooth Banach space, and let 𝑇𝑋𝑋 be generalized strongly asymptotically 𝜙-pseudocontractive mapping with fixed point 𝑥 and bounded range.
Let {𝑧𝑛}𝑛 be the sequences defined by (1.15) where {𝛼𝑛}𝑛,{𝛾𝑛}𝑛[0,1] satisfy(i)lim𝑛𝛼𝑛=lim𝑛𝛾𝑛=0, (ii)𝑛=1𝛼𝑛=, 𝑛=1𝛾𝑛<.and the sequence {𝑤𝑛}𝑛 is bounded on 𝑋, then for any initial point 𝑧0𝑋, the sequence {𝑧𝑛}𝑛 strongly converges to 𝑥.

Proof. Firstly, we observe that, by the boundedness of the range of 𝑇, of the sequence {𝑤𝑛}𝑛, and by Lemma 2.4, we have that {𝑧𝑛}𝑛 is bounded.
By Lemma 2.2, we observe that 𝑧𝑛+1𝑥21𝛼𝑛𝛾𝑛2𝑧𝑛𝑥2+2𝛼𝑛𝑇𝑛𝑧𝑛𝑥𝑧,𝑗𝑛+1𝑥+2𝛾𝑛𝑤𝑛𝑥𝑧,𝑗𝑛+1𝑥1𝛼𝑛2𝑧𝑛𝑥2+2𝛼𝑛𝑇𝑛𝑧𝑛𝑥𝑧,𝑗𝑛+1𝑥𝑧𝑗𝑛𝑥+2𝛼𝑛𝑇𝑛𝑧𝑛𝑥𝑧,𝑗𝑛𝑥+2𝛾𝑛𝑤𝑛𝑥𝑧,𝑗𝑛+1𝑥1𝛼𝑛2𝑧𝑛𝑥2+2𝛼𝑛𝑇𝑛𝑛𝑧𝑛𝑥𝑧,𝑗𝑛+1𝑥𝑧𝑗𝑛𝑥+2𝛼𝑛𝑘𝑛𝑧𝑛𝑥22𝛼𝑛𝜙𝑧𝑛𝑥+2𝛾𝑛𝑤𝑛𝑥𝑧,𝑗𝑛+1𝑥=1+𝛼2𝑛2𝛼𝑛𝑧𝑛𝑥2+2𝛼𝑛𝑇𝑛𝑧𝑛𝑥𝑧,𝑗𝑛+1𝑥𝑧𝑗𝑛𝑥+2𝛼𝑛𝑘𝑛𝑧𝑛𝑥22𝛼𝑛𝜙𝑧𝑛𝑥+2𝛾𝑛𝑤𝑛𝑥𝑧,𝑗𝑛+1𝑥=𝑧𝑛𝑥2+𝛼2𝑛2𝛼𝑛+2𝛼𝑛𝑘𝑛𝑧𝑛𝑥22𝛼𝑛𝜙𝑧𝑛𝑥+2𝛼𝑛𝜇𝑛+2𝛾𝑛𝑤𝑛𝑥𝑧,𝑗𝑛+1𝑥,(3.35) where 𝜇𝑛=𝑇𝑛𝑧𝑛𝑥,𝑗(𝑧𝑛+1𝑥)𝑗(𝑧𝑛𝑥). Let 𝑀=maxsup𝑛𝑧𝑛𝑥,sup𝑛𝑇𝑛𝑧𝑛𝑥,sup𝑛𝑤𝑛𝑥,sup𝑛𝑤𝑛𝑥𝑧,𝑗𝑛+1𝑥.(3.36) We have 𝑧𝑛+1𝑥2𝑧𝑛𝑥2+𝛼2𝑛+2𝛼𝑛𝑘𝑛1𝑀2𝛼𝑛𝜙𝑧𝑛𝑥+2𝛼𝑛𝜇𝑛+2𝛾𝑛𝑀=𝑧𝑛𝑥2𝛼𝑛𝜙𝑧𝑛𝑥+2𝛾𝑛𝑀𝛼𝑛𝜙𝑧𝑛𝑥2𝜇𝑛𝛼𝑛𝑘+2𝑛𝑀,1(3.37) so we can observe that(1)𝜇𝑛0 as 𝑛. Indeed from the inequality ||𝑧𝑛+1𝑥𝑧𝑛𝑥||𝑧𝑛+1𝑧𝑛𝛼𝑛+𝛾𝑛𝑀0,as,𝑛(3.38) and since 𝑗 is norm to norm uniformly continuous, then 𝑗(𝑧𝑛+1𝑥)𝑗(𝑧𝑛𝑥)0, as 𝑛, (2)inf𝑛(𝑧𝑛𝑥)=0. Indeed, if we supposed that 𝜎=inf𝑛(𝑧𝑛𝑥)>0, by the monotonicity of 𝜙, 𝜙𝑧𝑛𝑥𝜙(𝜎)>0.(3.39) Thus, by (1) and by the hypotheses on 𝛼𝑛 and 𝑘𝑛, the value 𝛼𝑛[𝜙(𝑧𝑛𝑥)2𝜇𝑛(𝛼𝑛+2(𝑘𝑛1))𝑀] is definitively negative. In this case, we conclude that there exists 𝑁>0 such that for every 𝑛>𝑁, 𝑧𝑛+1𝑥2𝑧𝑛𝑥2𝛼𝑛𝜙𝑧𝑛𝑥+2𝛾𝑛𝑀𝑧𝑛𝑥2𝛼𝑛𝜙(𝜎)+2𝛾𝑛𝑀,(3.40) and so 𝛼𝑛𝑧𝜙(𝜎)𝑛𝑥2𝑧𝑛+1𝑥2+2𝛾𝑛𝑀𝑛>𝑁.(3.41) In the same way we obtain that 𝜙(𝜎)𝑚𝑖=𝑁𝛼𝑖𝑚𝑖=𝑁𝑧𝑖𝑥2𝑧𝑖+1𝑥2+2𝑚𝑖=𝑁𝛾𝑖𝑀=𝑧𝑁𝑥2𝑧𝑚𝑥2+2𝑀𝑚𝑖=𝑁𝛾𝑖.(3.42) By the hypotheses 𝑛𝛾𝑛< and 𝑛𝛼𝑛=, the previous is a contradiction, and it follows that inf𝑛(𝑧𝑛𝑥)=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 𝑗𝑛𝑘(𝜖), 𝑧𝑛𝑗𝑥<𝜖.
Now, we will prove that the 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 𝑛𝑛(𝜖), it results that ||𝛼𝑛||<1𝜖4𝑀min𝜖,𝜙2,||𝛾𝑛||<𝜖,||||𝛾4𝑀𝑛𝛼𝑛||||<𝜙(𝜖/2),||𝑘4𝑀𝑛||<𝜙1(𝜖/2),||𝜇8𝑀𝑛||<𝜙(𝜖/2)8.(3.43) 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+𝑖𝑥<𝜖 for every 𝑖. Let 𝑖=1. If not, it results that 𝑧𝑛+1𝑥𝜖. Thus, 𝑧𝜖𝑛+1𝑥𝑧𝑛𝑥+𝛼𝑛𝑀+𝛾𝑛<𝑧𝑀,𝑛𝑥+𝜖𝜖4𝑀𝑀+𝑧4𝑀𝑀=𝑛𝑥+𝜖2,(3.44) that is, 𝑧𝑛𝑥>𝜖/2. By the strict increasing of 𝜙, 𝜙(𝑧𝑛𝑥)>𝜙(𝜖/2).
By (3.37), it results that 𝑧𝑛+1𝑥2<𝜖2𝛼𝑛𝜙𝑧𝑛𝑥𝛾2𝑀𝑛𝛼𝑛𝛼𝑛𝜙𝑧𝑛𝑥2𝜇𝑛𝛼𝑛𝑘+2𝑛𝑀.1(3.45) We can note that 2𝜇𝑛+𝛼𝑛𝑘+2𝑛1𝑀𝜙(𝜖/2)4+𝜙(𝜖/2)+4𝑀𝜙(𝜖/2)4𝑀𝑀,(3.46) so 𝜙𝑧𝑛𝑥2𝜇𝑛𝛼𝑛𝑘+2𝑛1𝑀>𝜙(𝜖/2)3𝜙(𝜖/2)4>0.(3.47) Moreover, 𝜙(𝑧𝑛𝑥)2𝑀𝛾𝑛/𝛼𝑛>𝜙(𝜖/2)/2>0, so it results that 𝑧𝑛+1𝑥2<𝜖2.(3.48) This is a contradiction. Thus, 𝑧𝑛+1𝑥<𝜖.
In the same manner, by induction, one obtains that, for every 𝑖1, 𝑧𝑛+𝑖𝑥<𝜖. So 𝑧𝑛𝑥0.

Corollary 3.4. Let 𝑋 be a uniformly smooth Banach space, and let 𝑇𝑋𝑋 be generalized strongly asymptotically 𝜙-pseudocontractive mapping with bounded range and fixed point 𝑥. The sequences {𝑥𝑛}𝑛, {𝑧𝑛}𝑛, and {𝑧𝑛}𝑛 are defined by (1.14), (1.15), and (1.16), respectively, where the sequences {𝛼𝑛}𝑛, {𝛽𝑛}𝑛, {𝛾𝑛}𝑛, {𝛿𝑛}𝑛[0,1] satisfy(i)lim𝑛𝛼𝑛=lim𝑛𝛽𝑛=lim𝑛𝛿𝑛=0,(ii)𝑛=1𝛼𝑛=,𝑛=1𝛾𝑛<,and the sequences {𝑢𝑛}𝑛, {𝑣𝑛}𝑛, {𝑤𝑛}𝑛, and {𝑤𝑛}𝑛 are bounded in X. Then for any initial point 𝑥0,𝑧0,𝑧0𝑋, the following two assertions are equivalent and true: (i)the modified Ishikawa iteration sequence with errors (1.14) converges to the fixed point 𝑥;(ii)the modified Mann iteration sequence with errors (1.15) converges to the fixed point 𝑥;(iii)the implicit iteration sequence with errors (1.16) converges to the fixed point 𝑥.