On the Convergence of Mann and Ishikawa Iterative Processes for Asymptotically 𝜙-Strongly Pseudocontractive Mappings

Abstract

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

1. Introduction

Let 𝑋 be a Banach space and 𝑋^∗ the dual space of 𝑋. Let 𝐽 denote the normalized duality mapping from 𝑋 into 2^𝑋∗ 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. Let 𝐾 be a subset of a Banach space 𝑋. A mapping 𝑇∶𝐾→𝐾 is said to be asymptotically nonexpansive if for each 𝑥,𝑦∈𝐾𝑇^𝑛𝑥−𝑇^𝑛𝑦≤𝑘_𝑛𝑥−𝑦, (1.1) where {𝑘_𝑛}_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]). If 𝐵 is the unit ball of 𝑙² and 𝑇∶𝐵→𝐵 is defined as 𝑇𝑥₁,𝑥₂,…=0,𝑥²₁,𝑎₂𝑥₂,𝑎₃𝑥₃,…, (1.2) where {𝑎_𝑖}_𝑖∈ℕ⊂(0,1) is such that ∏^∞_𝑖=2𝑎_𝑖=1/2, it satisfies 𝑇𝑥−𝑇𝑦≤2𝑥−𝑦,𝑇^𝑛𝑥−𝑇^𝑛𝑦≤2^𝑛_𝑖=2𝑎_𝑖𝑥−𝑦. (1.3)

In 1973, the same authors [3] proved that the Browder result remains valid for the broader class of uniformly L-Lipschitzian mappings (i.e., ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤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]). Let 𝑋 be a normed space, 𝐾⊂𝑋, and {𝑘_𝑛}_𝑛⊂[1,∞). A mapping 𝑇∶𝐾→𝐾 is said to be asymptotically pseudocontractive with the sequence {𝑘_𝑛}_𝑛 if and only if lim_𝑛→∞𝑘_𝑛=1, and for all 𝑛∈ℕ and all 𝑥,𝑦∈𝐾 there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝑇^𝑛𝑥−𝑇^𝑛𝑦,𝑗𝑥−𝑦≤𝑘_𝑛𝑥−𝑦², (1.4) 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=(1−𝑥^2/3)^3/2 is not Lipschitzian but asymptotically pseudocontractive [5].

In [4], Schu proved the following.

Theorem 1.4 (see [4]). Let 𝐻 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 {𝑘_𝑛}_𝑛∈[1,∞); 𝑞_𝑛∶=2𝑘_𝑛−1 for all 𝑛∈ℕ; ∑_𝑛(𝑞²_𝑛−1)<∞; {𝛼_𝑛}_𝑛,{𝛽_𝑛}_𝑛∈[0,1]; 𝜖≤𝛼_𝑛≤𝛽_𝑛≤𝑏 for all 𝑛∈ℕ, some 𝜖>0 and some 𝑏∈(0,𝐿⁻²[√1+𝐿²−1]); 𝑥₁∈𝐴; for all 𝑛∈ℕ define 𝑧_𝑛∶=𝛽_𝑛𝑇^𝑛𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛,𝑥_𝑛+1∶=𝛼_𝑛𝑇^𝑛𝑧_𝑛+1−𝛼_𝑛𝑥_𝑛. (1.5) Then, {𝑥_𝑛}_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 operator 𝐴 defined on a subset 𝐾 of a Banach space 𝑋 is said to be 𝜑-strongly accretive if 𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦≥𝜑𝑥−𝑦𝑥−𝑦, (1.6) 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 which 𝜑(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 operator 𝐴 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 that𝑇𝑥−𝑇𝑦,𝑗𝑥−𝑦≤𝑥−𝑦²−𝜑𝑥−𝑦𝑥−𝑦, (1.7) 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 mapping 𝑇 is a 𝜙-strongly pseudocontractive mapping if 𝑇𝑥−𝑇𝑦,𝑗𝑥−𝑦≤𝑥−𝑦²−𝜙𝑥−𝑦, (1.8) 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 generalized 𝜙-strongly pseudocontractive according to (1.7). We prefer to not use the term generalized because this class is narrower than pseudocontractive mappings.)

Choosing 𝜙(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 the asymptotically 𝜙-strongly pseudocontractive mappings defined as follows.

Definition 1.7. If 𝑋 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 𝑇^𝑛𝑥−𝑇^𝑛𝑦,𝑗𝑥−𝑦≤𝑘_𝑛𝑥−𝑦²−𝜙𝑥−𝑦, (1.9) where 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦), {𝑘_𝑛}_𝑛⊂[1,∞) is converging to one, and 𝜙∶[0,∞)→[0,∞) is strictly increasing such that 𝜙(0)=0.

One can note that if 𝑇 has fixed points, then it is unique. In fact if 𝑥,𝑧 are fixed points for 𝑇, then, for every 𝑛∈ℕ,‖𝑥−𝑧‖²=⟨𝑇^𝑛𝑥−𝑇^𝑛𝑧,𝑗(𝑥,𝑧)⟩≤𝑘_𝑛𝑥−𝑦²−𝜙𝑥−𝑦, (1.10) so, passing 𝑛 to +∞ results in‖𝑥−𝑧‖²≤‖𝑥−𝑧‖²−𝜙𝑥−𝑦⟹−𝜙𝑥−𝑦≥0. (1.11) Since 𝜙∶[0,∞)→[0,∞) is strictly increasing and 𝜙(0)=0 then, 𝑥=𝑧.

We now give two examples.

Example 1.8. The mapping 𝑇𝑥=𝑥/(𝑥+1), where 𝑥∈[0,1], is asymptotically 𝜙-strongly pseudocontractive with 𝑘_𝑛=1, for all 𝑛∈ℕ and 𝜙(𝑡)=𝑡³/(1+𝑡). However, 𝑇 is not strongly pseudocontractive, see [20].

Example 1.9. The mapping 𝑇𝑥=𝑥/(1+0.01𝑥), where 𝑥∈[0,1], is asymptotically 𝜙-strongly pseudocontractive with 𝑘_𝑛=(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 that 𝑇 is not strongly pseudocontractive. For arbitrary 𝑘<1, there exist 𝑥,𝑦∈[0,1], such that 1(1+0.01𝑥)1+0.01𝑦>𝑘. (1.12) So we have =1𝑇𝑥−𝑇𝑦,𝑗𝑥−𝑦(1+0.01𝑥)1+0.01𝑦𝑥−𝑦²>𝑘𝑥−𝑦². (1.13) Next we prove that 𝑇 is not 𝜙-strongly pseudocontractive. Taking 𝑦=0, we have, for all 𝑥∈[0,1], =𝑥𝑇𝑥−𝑇𝑦,𝑗𝑥−𝑦²(),1+0.01𝑥𝑥−𝑦²=𝑥−𝜙𝑥−𝑦²,1+𝑥 (1.14) therefore, 𝑇 is not 𝜙-strongly pseudocontractive. Finally, we prove that 𝑇 is asymptotically 𝜙-strongly Pseudocontractive.
For arbitrary 𝑥,𝑦∈[0,1], without loss of generality, let 𝑥>𝑦. Then, 𝑇^𝑛𝑥−𝑇^𝑛𝑦,𝑗𝑥−𝑦=𝑥−𝑦²(),𝑘1+0.01𝑛𝑥1+0.01𝑛𝑦_𝑛𝑥−𝑦²−𝜙𝑥−𝑦=𝑛+100𝑛𝑥−𝑦²−𝑥−𝑦³=1+𝑥−𝑦1+100𝑛−𝑥−𝑦1+𝑥−𝑦𝑥−𝑦²=1+(100/𝑛)+(100/𝑛)𝑥−𝑦1+𝑥−𝑦𝑥−𝑦². (1.15) We only need to prove that 1+𝑥−𝑦≤(1+0.01𝑛𝑥)1+0.01𝑛𝑦1+100𝑛+100𝑛𝑥−𝑦. (1.16) Using 𝑥>𝑦, this is easy.

In this paper we study the equivalence between two kinds of iterative methods involving asymptotically 𝜙-strongly Pseudocontractive mappings.

Moreover, we prove that, in opportune hypotheses, these methods are equivalent and strongly convergent to the unique fixed point of the asymptotically 𝜙-strongly Pseudocontractive 𝑇.

2. Preliminaries

Throughout this paper, we will assume that 𝑋 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 given 𝑥₀∈𝑋, the sequence {𝑥_𝑛}_𝑛 defined by𝑦_𝑛=1−𝛽_𝑛−𝛿_𝑛𝑥_𝑛+𝛽_𝑛𝑇^𝑛𝑥_𝑛+𝛿_𝑛𝑣_𝑛,𝑥_𝑛+1=1−𝛼_𝑛−𝛾_𝑛𝑥_𝑛+𝛼_𝑛𝑇^𝑛𝑦_𝑛+𝛾_𝑛𝑢_𝑛,𝑛≥0, (2.1) is called the modified Ishikawa iteration sequence, where {𝛼_𝑛}_𝑛, {𝛾_𝑛}_𝑛, {𝛽_𝑛}_𝑛, and {𝛿_𝑛}_𝑛 are four sequences in (0,1) satisfying the conditions 𝛼_𝑛+𝛾_𝑛≤1 and 𝛽_𝑛+𝛿_𝑛≤1 for all 𝑛≥0.

In particular, if 𝛽_𝑛=𝛿_𝑛=0 for all 𝑛≥0, we can define a sequence {𝑧_𝑛}_𝑛 by𝑧₀𝑧∈𝑋,_𝑛+1=1−𝛼_𝑛−𝛾_𝑛𝑧_𝑛+𝛼_𝑛𝑇^𝑛𝑧_𝑛+𝛾_𝑛𝑤_𝑛,𝑛≥0, (2.2) 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, if 𝑇 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 𝑋𝜌_𝑋(𝑡)1=sup2𝑥−𝑦+𝑥+𝑦−1∶‖𝑥‖≤1,𝑦≤𝑡 (2.3) satisfies lim_𝑡→0(𝜌_𝑋(𝑡)/𝑡)=0.

Lemma 2.2 (see [30]). Let 𝑋 be a Banach space, and let 𝐽∶𝑋→2^𝑋∗ be the normalized duality mapping. Then, for any 𝑥,𝑦∈𝑋 one has 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝑗𝑥+𝑦,∀𝑗𝑥+𝑦∈𝐽𝑥+𝑦. (2.4)

The following lemma is a key of our proofs.

Lemma 2.3 (see [29]). Let 𝜙∶[0,∞)→[0,∞) be a strictly increasing function with 𝜙(0)=0, and let {𝑎_𝑛}_𝑛,{𝑏_𝑛}_𝑛,{𝑐_𝑛}_𝑛, and {𝑒_𝑛}_𝑛 be nonnegative real sequences such that lim_𝑛→∞𝑏_𝑛=0,𝑐_𝑛=𝑜𝑏_𝑛,^∞_𝑛=1𝑏_𝑛=∞,lim_𝑛→∞𝑒_𝑛=0. (2.5) Suppose that there exists an integer 𝑁₁>0 such that 𝑎²_𝑛+1≤𝑎²_𝑛−2𝑏_𝑛𝜙𝑎_𝑛+1−𝑒_𝑛+𝑐_𝑛,∀𝑛≥𝑁₁, (2.6) then lim_𝑛→∞𝑎_𝑛=0.

Proof. The proof is the same as that in [29] but change in (2.6), (𝑎_𝑛+1−𝑒_𝑛) with |𝑎_𝑛+1−𝑒_𝑛|.

Lemma 2.4 (see [31]). Let {𝑠_𝑛}_𝑛, {𝑐_𝑛}_𝑛⊂ℝ₊, {𝑎_𝑛}_𝑛⊂(0,1), and {𝑏_𝑛}_𝑛⊂ℝ be sequences such that 𝑠_𝑛+1≤1−𝑎_𝑛𝑠_𝑛+𝑏_𝑛+𝑐_𝑛, (2.7) for all 𝑛≥0. Assume that ∑_𝑛|𝑐_𝑛|<∞. Then, the following results hold(1)If 𝑏_𝑛≤𝛽𝑎_𝑛 (where 𝛽≥0), then {𝑠_𝑛}_𝑛 is a bounded sequence.(2)If one has ∑_𝑛𝑎_𝑛=∞ and limsup_𝑛(𝑏_𝑛/𝑎_𝑛)≤0 then, 𝑠_𝑛→0 as 𝑛→∞.

Remark 2.5. If in Lemma 2.3 choosing 𝑒_𝑛=0, for all 𝑛, 𝜙(𝑡)=𝑘𝑡² (𝑘<1) then, the inequality (2.6) becomes, 𝑎²_𝑛+1≤𝑎²_𝑛−2𝑏_𝑛𝑘𝑎²_𝑛+1+𝑐_𝑛⟹𝑎²_𝑛+1≤11+2𝑏_𝑛𝑘𝑎²_𝑛+𝑐_𝑛1+2𝑏_𝑛𝑘≤1−2𝑏_𝑛𝑘1+2𝑏_𝑛𝑘𝑎²_𝑛+𝑐_𝑛1+2𝑏_𝑛𝑘, (2.8) where 𝛼_𝑛∶=2𝑏_𝑛𝑘/(1+2𝑏_𝑛𝑘) and 𝛽_𝑛∶=𝑐_𝑛/(1+2𝑏_𝑛𝑘). In the hypotheses of Lemma 2.3, 𝛼_𝑛→0 as 𝑛→∞, ∑_𝑛𝛼_𝑛=∞, and limsup_𝑛(𝛽_𝑛/𝛼_𝑛)=0. So we reobtain Lemma 2.4 in the case 𝑐_𝑛=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. Let 𝑋 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 {𝑥_𝑛} and {𝑧_𝑛} be the sequences defined by (2.1) and (2.2), respectively, where {𝛼_𝑛}, {𝛾_𝑛}, {𝛽_𝑛}, and {𝛿_𝑛}⊂[0,1] satisfy(H1) lim_𝑛→∞𝛼_𝑛=lim_𝑛→∞𝛽_𝑛=lim_𝑛→∞𝛿_𝑛=0 and 𝛾_𝑛=𝑜(𝛼_𝑛),(H2) ∑^∞_𝑛=1𝛼_𝑛=∞, and the sequences {𝑢_𝑛}, {𝑣_𝑛}, and {𝑤_𝑛} 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 of 𝑇 and of the sequences {𝑤_𝑛},{𝑢_𝑛},{𝑣_𝑛} and by Lemma 2.4, it and follows that {𝑧_𝑛} and {𝑥_𝑛} are bounded sequences. So we can set 𝑀=sup_𝑛𝑇^𝑛𝑧_𝑛−𝑇^𝑛𝑦_𝑛,𝑇^𝑛𝑥_𝑛−𝑥_𝑛,𝑇^𝑛𝑦_𝑛−𝑥_𝑛,𝑇^𝑛𝑧_𝑛−𝑧_𝑛,𝑧_𝑛−𝑥_𝑛,𝑢_𝑛−𝑥_𝑛,𝑣_𝑛−𝑥_𝑛,𝑤_𝑛−𝑧_𝑛,𝑤_𝑛−𝑢_𝑛.. (3.1) By Lemma 2.2, we have 𝑧_𝑛+1−𝑥_𝑛+1²≤1−𝛼_𝑛−𝛾_𝑛𝑧_𝑛−𝑥_𝑛+𝛼_𝑛𝑇^𝑛𝑧_𝑛−𝑇^𝑛𝑦_𝑛+𝛾_𝑛𝑤_𝑛−𝑢_𝑛²≤1−𝛼_𝑛−𝛾_𝑛²𝑧_𝑛−𝑥_𝑛²+2𝛼_𝑛𝑇^𝑛𝑧_𝑛−𝑇^𝑛𝑦_𝑛+𝛾_𝑛𝑤_𝑛−𝑢_𝑛,𝑗𝑧_𝑛+1−𝑥_𝑛+1≤1−𝛼_𝑛²𝑧_𝑛−𝑥_𝑛²+2𝛼_𝑛𝑇^𝑛𝑧_𝑛−𝑇^𝑛𝑦_𝑛,𝑗𝑧_𝑛−𝑦_𝑛+2𝛼_𝑛⟨𝑇^𝑛𝑧_𝑛−𝑇^𝑛𝑦_𝑛,𝑗𝑧_𝑛+1−𝑥_𝑛+1−𝑗𝑧_𝑛−𝑦_𝑛⟩+2𝛾_𝑛𝑤_𝑛−𝑢_𝑛,𝑗𝑧_𝑛+1−𝑥_𝑛+1≤1−𝛼_𝑛²𝑧_𝑛−𝑥_𝑛²+2𝛼_𝑛𝑘_𝑛𝑧_𝑛−𝑦_𝑛²−2𝛼_𝑛𝜙𝑧_𝑛−𝑦_𝑛+2𝛼_𝑛𝑇^𝑛𝑧_𝑛−𝑇^𝑛𝑦_𝑛𝑗𝑧_𝑛+1−𝑥_𝑛+1−𝑗𝑧_𝑛−𝑦_𝑛+2𝛾_𝑛𝑤_𝑛−𝑢_𝑛𝑧_𝑛+1−𝑥_𝑛+1≤1−𝛼_𝑛²𝑧_𝑛−𝑥_𝑛²+2𝛼_𝑛𝑘_𝑛𝑧_𝑛−𝑦_𝑛²−2𝛼_𝑛𝜙𝑧_𝑛−𝑦_𝑛+2𝛼_𝑛𝜎_𝑛𝑀+2𝛾_𝑛𝑀², (3.2) where 𝜎_𝑛=‖𝑗(𝑧_𝑛+1−𝑥_𝑛+1)−𝑗(𝑧_𝑛−𝑦_𝑛)‖. Using (2.1) and (2.2), we have 𝑧_𝑛+1−𝑥_𝑛+1−𝑧_𝑛−𝑦_𝑛≤𝑥_𝑛+1−𝑦_𝑛+𝑧_𝑛+1−𝑧_𝑛=𝛼_𝑛𝑇^𝑛𝑦_𝑛−𝑥_𝑛+𝛾_𝑛𝑢_𝑛−𝑥_𝑛−𝛽_𝑛𝑇^𝑛𝑥_𝑛−𝑥_𝑛−𝛿_𝑛𝑣_𝑛−𝑥_𝑛+𝛼_𝑛𝑇^𝑛𝑧_𝑛−𝑧_𝑛+𝛾_𝑛𝑤_𝑛−𝑧_𝑛≤2𝑀𝛼_𝑛+𝛾_𝑛+𝛽_𝑛+𝛿_𝑛⟶0(𝑛⟶∞). (3.3) In view of the uniform continuity of 𝑗 we obtain that 𝜎_𝑛→0 as 𝑛→∞. Furthermore, it follows from the definition of {𝑦_𝑛} that for all 𝑛≥0𝑧_𝑛−𝑦_𝑛²=𝑧_𝑛−𝑥_𝑛+𝛽_𝑛−𝑇^𝑛𝑥_𝑛+𝑥_𝑛+𝛿_𝑛−𝑣_𝑛+𝑥_𝑛²≤𝑧_𝑛−𝑥_𝑛+𝛽_𝑛𝑇^𝑛𝑥_𝑛−𝑥_𝑛+𝛿_𝑛𝑣_𝑛−𝑥_𝑛²≤𝑧_𝑛−𝑥_𝑛+𝛽_𝑛+𝛿_𝑛𝑀²=𝑧_𝑛−𝑥_𝑛²+𝛽_𝑛+𝛿_𝑛2𝑧_𝑛−𝑥_𝑛𝑀+𝛽_𝑛+𝛿_𝑛𝑀²≤𝑧_𝑛−𝑥_𝑛²+3𝛽_𝑛+𝛿_𝑛𝑀², (3.4)𝑦_𝑛−𝑥_𝑛≤𝛽_𝑛𝑇^𝑛𝑥_𝑛−𝑥_𝑛+𝛿_𝑛𝑣_𝑛−𝑥_𝑛≤𝛽_𝑛+𝛿_𝑛𝑀⟶0(𝑛⟶∞), (3.5) and so 𝑧_𝑛+1−𝑥_𝑛+1=𝑧_𝑛−𝑥_𝑛−𝛼_𝑛+𝛾_𝑛𝑧_𝑛−𝑥_𝑛+𝛼_𝑛𝑇^𝑛𝑧_𝑛−𝑇^𝑛𝑦_𝑛+𝛾_𝑛𝑤_𝑛−𝑢_𝑛≤𝑧_𝑛−𝑥_𝑛+𝛼_𝑛+𝛾_𝑛𝑧_𝑛−𝑥_𝑛+𝛼_𝑛𝑇^𝑛𝑧_𝑛−𝑇^𝑛𝑦_𝑛+𝛾_𝑛𝑤_𝑛−𝑢_𝑛≤𝑧_𝑛−𝑦_𝑛+𝑦_𝑛−𝑥_𝑛+2𝛼_𝑛+𝛾_𝑛≤𝑧𝑀._𝑛−𝑦_𝑛+𝛽_𝑛+𝛿_𝑛𝑀+2𝛼_𝑛+𝛾_𝑛𝑀. (3.6) Therefore, we have 𝑧_𝑛−𝑦_𝑛≥𝑧_𝑛+1−𝑥_𝑛+1−𝑒_𝑛, (3.7) where 𝑒_𝑛=(𝛽_𝑛+𝛿_𝑛)𝑀+2(𝛼_𝑛+𝛾_𝑛)𝑀. In view of (H1), we have that 𝑒_𝑛→0 as 𝑛→∞.
If ‖𝑧_𝑛+1−𝑥_𝑛+1‖−𝑒_𝑛≤0 for an infinite number of indexes we can extract a subsequence such that ‖𝑧_𝑛𝑘−𝑥_𝑛𝑘‖−𝑒_𝑛𝑘−1≤0. For this subsequence ‖z_𝑛𝑘−𝑥_𝑛𝑘‖→0 as 𝑘→∞.
We can prove that, in this case, ‖𝑧_𝑛−𝑥_𝑛‖→0, that is, the thesis.
Firstly we note that substituting (3.4) into (3.2) and simplifying, we have 𝑧_𝑛+1−𝑥_𝑛+1²≤𝑧_𝑛−𝑥_𝑛²+𝛼²_𝑛𝑧_𝑛−𝑥_𝑛²+2𝛼_𝑛𝑘_𝑛−1𝑧_𝑛−𝑥_𝑛²+6𝛼_𝑛𝑘_𝑛𝛽_𝑛+𝛿_𝑛𝑀²−2𝛼_𝑛𝜙𝑧_𝑛−𝑦_𝑛+2𝛼_𝑛𝜎_𝑛𝑀+2𝛾_𝑛𝑀²≤𝑧_𝑛−𝑥_𝑛²−2𝛼_𝑛𝜙𝑧_𝑛−𝑦_𝑛+𝛼²_𝑛𝑀²+2𝛼_𝑛𝑘_𝑛−1𝑀²+6𝛼_𝑛𝑘𝛽_𝑛+𝛿_𝑛𝑀²+2𝛼_𝑛𝜎_𝑛𝑀+2𝛾_𝑛𝑀²=𝑧_𝑛−𝑥_𝑛²−𝛼_𝑛𝜙𝑧_𝑛−𝑦_𝑛+2𝛾_𝑛𝑀²−𝛼_𝑛𝜙𝑧_𝑛−𝑦_𝑛−𝛼_𝑛𝑀²−2𝑘_𝑛−1𝑀²−6𝑘𝛽_𝑛+𝛿_𝑛𝑀²−2𝜎_𝑛𝑀, (3.8) where 𝑘∶=sup_𝑛(𝑘_𝑛).
Moreover, we observe that 𝑧_𝑛𝑗−𝑦_𝑛𝑗≤𝑧_𝑛𝑗−𝑥_𝑛𝑗+𝑦_𝑛𝑗−𝑥_𝑛𝑗⟶0as𝑗⟶∞. (3.9) Thus, for every fixed 𝜖>0, there exists 𝑗₁ such that, for all 𝑗>𝑗₁, 𝑧_𝑛𝑗−𝑦_𝑛𝑗<2𝜖,𝑧_𝑛𝑗−𝑥_𝑛𝑗<𝜖. (3.10) Since {𝛼_𝑛}_𝑛, {(𝑘_𝑛−1)}_𝑛, {(𝛽_𝑛+𝛿_𝑛)}_𝑛, {𝜎_𝑛}_𝑛, and {𝛾_𝑛}_𝑛 are real null sequences (and in particular 𝛾_𝑛=𝑜(𝛼_𝑛)), for the previous fixed 𝜖>0, there exist an index 𝑁 such that, for all 𝑛>𝑁, 𝛼_𝑛𝜖<min,𝜙()16𝑀𝜖/28𝑀²,𝛾_𝑛<𝜖,𝛾16𝑀𝛼^𝑛_𝑛<𝜙(𝜖/2)4𝑀²,𝑘_𝑛<𝜙()−1𝜖/216𝑀²,𝛽_𝑛+𝛿_𝑛𝜖<min,𝜙()4𝑀𝜖/248𝑘𝑀²,𝜎_𝑛<𝜙(𝜖/2),16𝑀 (3.11) for all 𝑛>𝑁.
Take 𝑛^∗>max{𝑁,𝑛_𝑗1} such that 𝑛^∗=𝑛_𝑘 for a certain 𝑘.
We prove, by induction, that ‖𝑧_𝑛∗+𝑖−𝑥_𝑛∗+𝑖‖<𝜖, for every 𝑖∈ℕ. Let 𝑖=1.
Let us suppose that ‖𝑧_𝑛∗+1−𝑥_𝑛∗+1‖≥𝜖.
By (3.6), we have 𝜖≤𝑧_𝑛∗+1−𝑥_𝑛∗+1≤𝑧_𝑛∗−𝑦_𝑛∗+𝛽_𝑛∗+𝛿_𝑛∗𝑀+2𝛼_𝑛∗+𝛾_𝑛∗𝑀≤𝑧_𝑛∗−𝑦_𝑛∗𝜖+𝑀𝜖4𝑀+2𝑀+𝜖8𝑀=𝑧8𝑀_𝑛∗−𝑦_𝑛∗+𝜖4+𝜖4=𝑧_𝑛∗−𝑦_𝑛∗+𝜖2. (3.12) Thus, ‖𝑧_𝑛∗−𝑦_𝑛∗‖≥𝜖/2. Since 𝜙 is strictly increasing, 𝜙(‖𝑧_𝑛∗−𝑦_𝑛∗‖)≥𝜙(𝜖/2).
From (3.8), we obtain that 𝑧_𝑛∗+1−𝑥_𝑛∗+1²<𝜖²−𝛼^∗_𝑛𝜙𝑧_𝑛∗−𝑦_𝑛∗−2𝑀²𝛾^∗_𝑛𝛼^∗_𝑛−𝛼^∗_𝑛𝜙𝑧_𝑛∗−𝑦_𝑛∗−𝛼^∗_𝑛𝑀²−2𝑘^∗_𝑛−1𝑀²−6𝑘𝛽^∗_𝑛+𝛿^∗_𝑛𝑀²−2𝜎_𝑛∗𝑀. (3.13) One can note that 𝛼^∗_𝑛𝑀²+2𝑘^∗_𝑛−1𝑀²+6𝑘𝛽^∗_𝑛+𝛿^∗_𝑛𝑀²+2𝜎_𝑛∗𝜙()𝑀≤𝜖/28+𝜙(𝜖/2)8+𝜙(𝜖/2)8+𝜙(𝜖/2)8; (3.14) hence 𝜙𝑧_𝑛∗−𝑦_𝑛∗−𝛼^∗_𝑛𝑀²−2𝑘_𝑛∗−1𝑀²−6𝑘𝛽_𝑛∗+𝛿_𝑛∗𝑀²−2𝜎_𝑛∗𝜖𝑀≥𝜙2−𝜙(𝜖/2)2>0. (3.15) In the same manner 𝜙𝑧_𝑛∗−𝑦_𝑛∗−2𝑀²𝛾_𝑛∗𝛼_𝑛∗𝜖>𝜙2−𝜙(𝜖/2)2>0. (3.16) Thus, 𝑧_𝑛∗+1−𝑥_𝑛∗+1²<𝜖². (3.17) So, we have that ‖𝑧_𝑛∗+1−𝑥_𝑛∗+1‖<𝜖 and it denies that ‖𝑧_𝑛∗+1−𝑥_𝑛∗+1‖≥𝜖. By the same idea we can prove (by contradiction) that ‖𝑧_𝑛∗+2−𝑥_𝑛∗+2‖<𝜖 and then, by inductive step, ‖𝑧_𝑛∗+𝑖−𝑥_𝑛∗+𝑖‖≤𝜖, for all 𝑖. This is enough to assure that ‖𝑧_𝑛−𝑥_𝑛‖→0.
If there are only finite indexes for which ‖𝑧_𝑛+1−𝑥_𝑛+1‖−𝑒_𝑛≤0, then definitively ‖𝑧_𝑛+1−𝑥_𝑛+1‖−𝑒_𝑛≥0. By the strict increasing function 𝜙, we have definitively 𝜙𝑧_𝑛−𝑦_𝑛≥𝜙𝑧_𝑛+1−𝑥_𝑛+1−𝑒_𝑛. (3.18) Again substituting (3.4) and (3.18) into (3.2) and simplifying, we have 𝑧_𝑛+1−𝑥_𝑛+1²≤𝑧_𝑛−𝑥_𝑛²+𝛼²_𝑛𝑧_𝑛−𝑥_𝑛²+2𝛼_𝑛𝑘_𝑛−1𝑧_𝑛−𝑥_𝑛²+6𝛼_𝑛𝑘_𝑛𝛽_𝑛+𝛿_𝑛𝑀²−2𝛼_𝑛𝜙𝑧_𝑛+1−𝑥_𝑛+1−𝑒_𝑛+2𝛼_𝑛𝜎_𝑛𝑀+2𝛾_𝑛𝑀²≤𝑧_𝑛−𝑥_𝑛²−2𝛼_𝑛𝜙𝑧_𝑛+1−𝑥_𝑛+1−𝑒_𝑛+𝛼²_𝑛𝑀²+2𝛼_𝑛𝑘_𝑛−1𝑀²+6𝛼_𝑛𝑘_𝑛𝛽_𝑛+𝛿_𝑛𝑀²+2𝛼_𝑛𝜎_𝑛𝑀+2𝛾_𝑛𝑀². (3.19) Suppose that 𝑎_𝑛=‖𝑧_𝑛−𝑥_𝑛‖, and 𝑏_𝑛=𝛼_𝑛,𝑐_𝑛=𝛼²_𝑛𝑀²+2𝛼_𝑛(𝑘_𝑛−1)𝑀²+6𝛼_𝑛𝑘_𝑛(𝛽_𝑛+𝛿_𝑛)𝑀²+2𝛼_𝑛𝜎_𝑛𝑀+2𝛾_𝑛𝑀². Since lim_𝑛→∞𝜎_𝑛=0, lim_𝑛→∞𝑒_𝑛=0, and lim_𝑛→∞𝑘_𝑛=1 and by the hypotheses, we have that ∑^∞_𝑛=1𝑏_𝑛=∞ and 𝑐_𝑛=𝑜(𝑏_𝑛), 𝑒_𝑛→0 as 𝑛→∞. In view of Lemma 2.3 we obtain that lim_𝑛→∞𝑎_𝑛=0. Hence, lim_𝑛→∞‖𝑧_𝑛−𝑥_𝑛‖=0.

Theorem 3.2. Let 𝑋 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 {𝑣_𝑛}_𝑛 be the sequences defined by 𝑣₀𝑣∈𝑋,_𝑛+1=1−𝛼_𝑛𝑣_𝑛+𝛼_𝑛𝑇^𝑛𝑣_𝑛,𝑛≥0, (3.20) where {𝛼_𝑛}_𝑛⊂[0,1] satisfy(i)lim_𝑛→∞𝛼_𝑛=0,(ii)∑^∞_𝑛=1𝛼_𝑛=∞. Then, for any initial point 𝑣₀∈𝑋, the sequence {𝑣_𝑛}_𝑛 strongly converges to 𝑥^∗.

Proof. By the boundedness of the range of 𝑇 and by Lemma 2.4, we have that {𝑣_𝑛}_𝑛 is bounded.
By Lemma 2.2 we observe that 𝑣_𝑛+1−𝑥^∗2≤1−𝛼_𝑛²𝑣_𝑛−𝑥^∗2+2𝛼_𝑛𝑇^𝑛𝑣_𝑛−𝑥^∗,𝑗𝑣_𝑛+1−𝑥^∗≤1−𝛼_𝑛²𝑣_𝑛−𝑥^∗2+2𝛼_𝑛𝑇_𝑛𝑣_𝑛−𝑥^∗,𝑗𝑣_𝑛+1−𝑥^∗−𝑗𝑣_𝑛−𝑥^∗+2𝛼_𝑛𝑇_𝑛𝑣_𝑛−𝑥^∗,𝑗𝑣_𝑛−𝑥^∗≤1−𝛼_𝑛²𝑣_𝑛−𝑥^∗2+2𝛼_𝑛𝑇_𝑛𝑣_𝑛−𝑥^∗,𝑗𝑣_𝑛+1−𝑥^∗−𝑗𝑣_𝑛−𝑥^∗+2𝛼_𝑛𝑘_𝑛𝑣_𝑛−𝑥^∗2−2𝛼_𝑛𝜙𝑣_𝑛−𝑥^∗=1+𝛼²_𝑛−2𝛼_𝑛𝑣_𝑛−𝑥^∗2+2𝛼_𝑛𝑇_𝑛𝑣_𝑛−𝑥^∗,𝑗𝑣_𝑛+1−𝑥^∗−𝑗𝑣_𝑛−𝑥^∗+2𝛼_𝑛𝑘_𝑛𝑣_𝑛−𝑥^∗2−2𝛼_𝑛𝜙𝑣_𝑛−𝑥^∗=𝑣_𝑛−𝑥^∗2+𝛼²_𝑛−2𝛼_𝑛+2𝛼_𝑛𝑘_𝑛𝑣_𝑛−𝑥^∗2−2𝛼_𝑛𝜙𝑣_𝑛−𝑥^∗+2𝛼_𝑛𝜇_𝑛, (3.21) where 𝜇_𝑛∶=⟨𝑇_𝑛𝑣_𝑛−𝑥^∗,𝑗(𝑣_𝑛+1−𝑥^∗)−𝑗(𝑣_𝑛−𝑥^∗)⟩. Let 𝑀∶=maxsup_𝑛𝑣_𝑛−𝑥^∗,sup_𝑛𝑇^𝑛𝑣_𝑛−𝑥^∗. (3.22) We have 𝑣_𝑛+1−𝑥^∗2≤𝑣_𝑛−𝑥^∗2+𝛼²_𝑛+2𝛼_𝑛𝑘_𝑛−1𝑀−2𝛼_𝑛𝜙𝑣_𝑛−𝑥^∗+2𝛼_𝑛𝜇_𝑛=𝑣_𝑛−𝑥^∗2−𝛼_𝑛𝜙𝑣_𝑛−𝑥^∗−𝛼_𝑛𝜙𝑣_𝑛−𝑥^∗−2𝜇_𝑛−𝛼_𝑛+2𝑘_𝑛−1𝑀, (3.23) and so we can observe that(1) 𝜇_𝑛→0 as 𝑛→∞, in fact from the inequality 𝑣_𝑛+1−𝑥^∗−𝑣_𝑛−𝑥^∗≤𝑣_𝑛+1−𝑣_𝑛≤𝛼_𝑛𝑀⟶0as𝑛⟶∞ (3.24) and since 𝑗 is norm-to-norm uniformly continuous, then 𝑗(‖𝑣_𝑛+1−𝑥^∗‖)−𝑗(‖𝑣_𝑛−𝑥^∗‖)→0 as 𝑛→∞,(2)inf_𝑛(‖𝑣_𝑛−𝑥^∗‖)=0, in fact, if we suppose that 𝜎∶=inf_𝑛(‖𝑣_𝑛−𝑥^∗‖)>0, by the monotonicity of 𝜙, 𝜙𝑣_𝑛−𝑥^∗≥𝜙(𝜎)>0. (3.25) Thus, by (1) and by the hypotheses on 𝛼_𝑛 and 𝑘_𝑛, the value −𝛼_𝑛[𝜙(‖𝑣_𝑛−𝑥^∗‖)−2𝜇_𝑛−(𝛼_𝑛+2(𝑘_𝑛−1))𝑀] is definitively negative. In this case we conclude that there exists 𝑁>0 such that for every 𝑛>𝑁𝑣_𝑛+1−𝑥^∗2≤𝑣_𝑛−𝑥^∗2−𝛼_𝑛𝜙𝑣_𝑛−𝑥^∗≤𝑣_𝑛−𝑥^∗2−𝛼_𝑛𝜙(𝜎), (3.26) and so 𝛼_𝑛𝜙(𝜎)≤𝑣_𝑛−𝑥^∗2−𝑣_𝑛+1−𝑥^∗2∀𝑛>𝑁. (3.27) In the same manner we obtain that 𝜙(𝜎)^𝑚_𝑖=𝑁𝛼_𝑖≤^𝑚_𝑖=𝑁𝑣_𝑖−𝑥^∗2−𝑣_𝑖+1−𝑥^∗2=𝑣_𝑁−𝑥^∗2−𝑣_𝑚−𝑥^∗2. (3.28) By the hypothesis ∑_𝑛𝛼_𝑛=∞, the previous is a contradiction and it follows that inf_𝑛(‖𝑣_𝑛−𝑥^∗‖)=0.
Then, there exists a subsequence {𝑣_𝑛𝑘}_𝑘 of {𝑣_𝑛}_𝑛 that strongly converges to 𝑥^∗. This implies that for every 𝜖>0 there exists an index 𝑛_𝑘(𝜖) such that, for all 𝑗≥𝑛_𝑘(𝜖), ‖𝑣_𝑛𝑗−𝑥^∗‖<𝜖.
Now we will prove that the entire sequence {𝑣_𝑛}_𝑛 converges to 𝑥^∗. Since the sequences in (3.37) are null sequences but ∑_𝑛𝛼_𝑛=∞, then for every 𝜖>0 there exists an index 𝑛(𝜖) such that for all 𝑛≥𝑛(𝜖) it following that: 𝛼_𝑛<1𝜙()2𝑀min𝜖,𝜖/22,𝑘_𝑛<𝜙()−1𝜖/2,𝜇8𝑀^𝑛<𝜙(𝜖/2)8. (3.29) So, fixing 𝜖>0, let 𝑛^∗>max(𝑛_𝑘(𝜖),𝑛(𝜖)) with 𝑛^∗=𝑛_𝑗 for a certain 𝑛_𝑗. We will prove, by induction, that ‖𝑣_𝑛∗+𝑖−𝑥^∗‖<𝜖 for every 𝑖∈ℕ. Let 𝑖=1. If not, it follows that ‖𝑣_𝑛∗+1−𝑥^∗‖≥𝜖. Thus 𝜖≤𝑣_𝑛∗+1−𝑥^∗≤𝑣_𝑛∗−𝑥^∗+𝛼_𝑛∗<𝑣_𝑛∗−𝑥^∗+𝜖𝑣2𝑀𝑀=_𝑛∗−𝑥^∗+𝜖2, (3.30) that is, ‖𝑣_𝑛∗−𝑥^∗‖>(𝜖/2). By the strict increasing of 𝜙, 𝜙(‖𝑣_𝑛∗−𝑥^∗‖)>𝜙(𝜖/2). By (3.37), it follows that 𝑣_𝑛∗+1−𝑥^∗2<𝜖²−𝛼_𝑛∗𝜙𝑣_𝑛∗−𝑥^∗−𝛼_𝑛∗𝜙𝑣_𝑛∗−𝑥^∗−2𝜇_𝑛∗−𝛼_𝑛∗+2𝑘_𝑛∗−1𝑀. (3.31) We can note that 2𝜇_𝑛∗+𝛼_𝑛∗+2𝑘_𝑛∗𝜙()−1𝑀≤𝜖/24+𝜙(𝜖/2)+𝜙()4𝑀𝜖/24𝑀𝑀, (3.32) and so 𝜙𝑣_𝑛∗−𝑥^∗−2𝜇_𝑛∗−𝛼_𝑛∗+2𝑘_𝑛∗𝜖−1𝑀≥𝜙2−3𝜙(𝜖/2)4>0. (3.33) Moreover, 𝜙(‖𝑣_𝑛∗−𝑥^∗‖)>𝜙(𝜖/2)/2>0, and it follows that 𝑣_𝑛∗+1−𝑥^∗2≤𝜖². (3.34) This is absurd. Thus ‖𝑣_𝑛∗+1−𝑥^∗‖<𝜖.
In the same manner, by induction, one obtains that, for every 𝑖≥1, ‖𝑣_𝑛∗+𝑖−𝑥^∗‖<𝜖. So ‖𝑣_𝑛−𝑥^∗‖→0.

Corollary 3.3. Let 𝑋 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 {𝑧_𝑛}_𝑛 be the sequences defined by (2.2), where {𝛼_𝑛}_𝑛and{𝛾_𝑛}_𝑛⊂[0,1] satisfy(i) lim_𝑛→∞𝛼_𝑛=lim_𝑛→∞𝛾_𝑛=0,(ii) ∑^∞_𝑛=1𝛼_𝑛=∞, ∑^∞_𝑛=1𝛾_𝑛<∞, and the sequence {𝑤_𝑛}_𝑛 is bounded on 𝑋. Then, for any initial point 𝑧₀∈𝑋, the sequence {𝑧_𝑛}_𝑛 strongly converges to 𝑥^∗.

Proof. Following the idea of the proof of Theorem 3.2, by the boundedness of the range of 𝑇 and by the boundedness of the sequence {𝑤_𝑛}_𝑛 and by Lemma 2.4, we have that {𝑧_𝑛}_𝑛 is bounded.
By Lemma 2.2 we observe that 𝑧_𝑛+1−𝑥^∗2≤1−𝛼_𝑛−𝛾_𝑛²𝑧_𝑛−𝑥^∗2+2𝛼_𝑛𝑇^𝑛𝑧_𝑛−𝑥^∗,𝑗𝑧_𝑛+1−𝑥^∗+2𝛾_𝑛𝑤_𝑛−𝑥^∗,𝑗𝑧_𝑛+1−𝑥^∗≤1−𝛼_𝑛²𝑧_𝑛−𝑥^∗2+2𝛼_𝑛𝑇_𝑛𝑧_𝑛−𝑥^∗,𝑗𝑧_𝑛+1−𝑥^∗−𝑗𝑧_𝑛−𝑥^∗+2𝛼_𝑛𝑇_𝑛𝑧_𝑛−𝑥^∗,𝑗𝑧_𝑛−𝑥^∗+2𝛾_𝑛𝑤_𝑛−𝑥^∗,𝑗𝑧_𝑛+1−𝑥^∗≤1−𝛼_𝑛²𝑧_𝑛−𝑥^∗2+2𝛼_𝑛𝑇_𝑛𝑧_𝑛−𝑥^∗,𝑗𝑧_𝑛+1−𝑥^∗−𝑗𝑧_𝑛−𝑥^∗+2𝛼_𝑛𝑘_𝑛𝑧_𝑛−𝑥^∗2−2𝛼_𝑛𝜙𝑧_𝑛−𝑥^∗+2𝛾_𝑛𝑤_𝑛−𝑥^∗,𝑗𝑧_𝑛+1−𝑥^∗=1+𝛼²_𝑛−2𝛼_𝑛𝑧_𝑛−𝑥^∗2+2𝛼_𝑛𝑇_𝑛𝑧_𝑛−𝑥^∗,𝑗𝑧_𝑛+1−𝑥^∗−𝑗𝑧_𝑛−𝑥^∗+2𝛼_𝑛𝑘_𝑛𝑧_𝑛−𝑥^∗2−2𝛼_𝑛𝜙𝑧_𝑛−𝑥^∗+2𝛾_𝑛𝑤_𝑛−𝑥^∗,𝑗𝑧_𝑛+1−𝑥^∗=𝑧_𝑛−𝑥^∗2+𝛼²_𝑛−2𝛼_𝑛+2𝛼_𝑛𝑘_𝑛𝑧_𝑛−𝑥^∗2−2𝛼_𝑛𝜙𝑧_𝑛−𝑥^∗+2𝛼_𝑛𝜇_𝑛+2𝛾_𝑛𝑤_𝑛−𝑥^∗,𝑗𝑧_𝑛+1−𝑥^∗, (3.35) where 𝜇_𝑛∶=⟨𝑇_𝑛𝑧_𝑛−𝑥^∗,𝑗(𝑧_𝑛+1−𝑥^∗)−𝑗(𝑧_𝑛−𝑥^∗)⟩. Let 𝑀∶=maxsup_𝑛𝑧_𝑛−𝑥^∗,sup_𝑛𝑇^𝑛𝑧_𝑛−𝑥^∗,sup_𝑛𝑤_𝑛−𝑥^∗,sup_𝑛𝑤_𝑛−𝑥^∗,𝑗𝑧_𝑛+1−𝑥^∗. (3.36) We have 𝑧_𝑛+1−𝑥^∗2≤𝑧_𝑛−𝑥^∗2+𝛼²_𝑛+2𝛼_𝑛𝑘_𝑛−1𝑀−2𝛼_𝑛𝜙𝑧_𝑛−𝑥^∗+2𝛼_𝑛𝜇_𝑛+2𝛾_𝑛𝑀=𝑧_𝑛−𝑥^∗2−𝛼_𝑛𝜙𝑧_𝑛−𝑥^∗+2𝛾_𝑛𝑀−𝛼_𝑛𝜙𝑧_𝑛−𝑥^∗−2𝜇_𝑛−𝛼_𝑛+2𝑘_𝑛−1𝑀, (3.37) and so we can observe that(1) 𝜇_𝑛→0 as 𝑛→∞, in fact from the inequality 𝑧_𝑛+1−𝑥^∗−𝑧_𝑛−𝑥^∗≤𝑧_𝑛+1−𝑧_𝑛𝛼_𝑛+𝛾_𝑛𝑀⟶0as𝑛⟶∞ (3.38) and since 𝑗 is norm-to-norm uniformly continuous, then 𝑗(‖𝑧_𝑛+1−𝑥^∗‖)−𝑗(‖𝑧_𝑛−𝑥^∗‖)→0 as 𝑛→∞,(2)inf_𝑛(‖𝑧_𝑛−𝑥^∗‖)=0, in fact, if we suppose that 𝜎∶=inf_𝑛(‖𝑧_𝑛−𝑥^∗‖)>0, by the monotonicity of 𝜙, 𝜙𝑧_𝑛−𝑥^∗≥𝜙(𝜎)>0. (3.39) Thus, by (1) and by the hypotheses on 𝛼_𝑛 and 𝑘_𝑛, the value −𝛼_𝑛[𝜙(‖𝑧_𝑛−𝑥^∗‖)−2𝜇_𝑛−(𝛼_𝑛+2(𝑘_𝑛−1))𝑀] is definitively negative. In this case, we conclude that there exists 𝑁>0 such that for every 𝑛>𝑁𝑧_𝑛+1−𝑥^∗2≤𝑧_𝑛−𝑥^∗2−𝛼_𝑛𝜙𝑧_𝑛−𝑥^∗+2𝛾_𝑛𝑀≤𝑧_𝑛−𝑥^∗2−𝛼_𝑛𝜙(𝜎)+2𝛾_𝑛𝑀, (3.40) and so 𝛼_𝑛𝜙(𝜎)≤𝑧_𝑛−𝑥^∗2−𝑧_𝑛+1−𝑥^∗2+2𝛾_𝑛𝑀∀𝑛>𝑁. (3.41) In the same manner we obtain that 𝜙(𝜎)^𝑚_𝑖=𝑁𝛼_𝑖≤^𝑚_𝑖=𝑁𝑧_𝑖−𝑥^∗2−𝑧_𝑖+1−𝑥^∗2+2^𝑚_𝑖=𝑁𝛾_𝑖𝑀=𝑧_𝑁−𝑥^∗2−𝑧_𝑚−𝑥^∗2+2𝑀^𝑚_𝑖=𝑁𝛽_𝑖. (3.42) By the hypotheses ∑_𝑛𝛾_𝑛<∞ and ∑_𝑛𝛼_𝑛=∞, the previous is a contradiction and it follows that inf_𝑛(‖𝑧_𝑛−𝑥^∗‖)=0.
Then, there exists a subsequence {𝑧_𝑛𝑘}_𝑘 of {𝑧_𝑛}_𝑛 that strongly converges to 𝑥^∗. This implies that for every 𝜖>0 there exists an index 𝑛_𝑘(𝜖) such that, for all 𝑗≥𝑛_𝑘(𝜖), ‖𝑧_𝑛𝑗−𝑥^∗‖<𝜖.
Now we will prove that the entire sequence {𝑧_𝑛}_𝑛 converges to 𝑥^∗. Since the sequences in (3.37) are null sequences and ∑_𝑛𝛾_𝑛<∞ but ∑_𝑛𝛼_𝑛=∞, then for every 𝜖>0 there exists an index 𝑛(𝜖) such that for all 𝑛≥𝑛(𝜖) it follows that: 𝛼_𝑛<1𝜖4𝑀min𝜖,𝜙2,𝛾_𝑛<𝜖,𝛾4𝑀𝛼^𝑛_𝑛<𝜙(𝜖/2),𝑘4𝑀_𝑛<𝜙()−1𝜖/2,𝜇8𝑀_𝑛<𝜙(𝜖/2)8. (3.43) So, fixing 𝜖>0, let 𝑛^∗>max(𝑛_𝑘(𝜖),𝑛(𝜖)) with 𝑛^∗=𝑛_𝑗 for a certain 𝑛_𝑗. We will prove, by induction, that ‖𝑧_𝑛∗+𝑖−𝑥^∗‖<𝜖 for every 𝑖∈ℕ. Let 𝑖=1. If not, it follows that ‖𝑧_𝑛∗+1−𝑥^∗‖≥𝜖. Thus, 𝜖≤𝑧_𝑛∗+1−𝑥^∗≤𝑧_𝑛∗−𝑥^∗+𝛼_𝑛∗𝑀+𝛾_𝑛∗𝑀<𝑧_𝑛∗−𝑥^∗+𝜖𝜖4𝑀𝑀+𝑧4𝑀𝑀=_𝑛∗−𝑥^∗+𝜖2, (3.44) that is, ‖𝑧_𝑛∗−𝑥^∗‖>𝜖/2. By the strict increasing of 𝜙, 𝜙(‖𝑧_𝑛∗−𝑥^∗‖)>𝜙(𝜖/2).
By (3.37), it follows that 𝑧_𝑛∗+1−𝑥^∗2<𝜖²−𝛼_𝑛∗𝜙𝑧_𝑛∗−𝑥^∗𝛾−2𝑀^𝑛∗𝛼_𝑛∗−𝛼_𝑛∗𝜙𝑧_𝑛∗−𝑥^∗−2𝜇_𝑛∗−𝛼_𝑛∗+2𝑘_𝑛∗−1𝑀. (3.45) We can note that 2𝜇_𝑛∗+𝛼_𝑛∗+2𝑘_𝑛∗𝜙()−1𝑀≤𝜖/24+𝜙(𝜖/2)+𝜙()4𝑀𝜖/24𝑀𝑀, (3.46) and so 𝜙𝑧_𝑛∗−𝑥^∗−2𝜇_𝑛∗−𝛼_𝑛∗+2𝑘_𝑛∗𝜖−1𝑀≥𝜙2−3𝜙(𝜖/2)4>0. (3.47) Moreover, 𝜙(‖𝑧_𝑛∗−𝑥^∗‖)−2𝑀(𝛾_𝑛∗/𝛼_𝑛∗)>𝜙(𝜖/2)/2>0 so it follows that 𝑧_𝑛∗+1−𝑥^∗2≤𝜖². (3.48) This is absurd. Thus, ‖𝑧_𝑛∗+1−𝑥^∗‖<𝜖.
In the same manner, by induction, one obtains that, for every 𝑖≥1, ‖𝑧_𝑛∗+𝑖−𝑥^∗‖<𝜖. So ‖𝑧_𝑛−𝑥^∗‖→0.

Corollary 3.4. Let 𝑋 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 {𝑥_𝑛}_𝑛, {𝑧_𝑛}_𝑛, and {𝑧^′_𝑛}_𝑛 defined by (2.1) and (2.2), respectively, where the sequences {𝛼_𝑛}_𝑛, {𝛽_𝑛}_𝑛, {𝛾_𝑛}_𝑛, and {𝛿_𝑛}_𝑛⊂[0,1] satisfy(i) lim_𝑛→∞𝛼_𝑛=lim_𝑛→∞𝛽_𝑛=lim_𝑛→∞𝛿_𝑛=0,(ii) ∑^∞_𝑛=1𝛼_𝑛=∞,∑^∞_𝑛=1𝛾_𝑛<∞,and the sequences {𝑢_𝑛}_𝑛, {𝑣_𝑛}_𝑛, and {𝑤_𝑛}_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 Example 1.9, 𝑇𝑥=𝑥/(1+0.01𝑥), where 𝑥∈[0,1], that is asymptotically 𝜙-strongly Pseudocontractive with 𝑘_𝑛=(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, 𝛼_𝑛=1/𝑛, and 𝛾_𝑛=1/𝑛². Then the sequence generated by the scheme 𝑧_𝑛+1=11−𝑛−1𝑛²𝑧_𝑛+1𝑛𝑇^𝑛𝑧_𝑛+1𝑛²𝑤_𝑛,𝑛≥0, (3.49) 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.

Acknowledgment

The authors are extremely grateful to the anonymous referees for their useful comments and suggestions.