On the Convergence of Iterative Processes for Generalized Strongly Asymptotically 𝜙-Pseudocontractive Mappings in Banach Spaces

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 2^𝑋∗ given by 𝐽(𝑥)={𝑓∈𝑋^∗∶⟨𝑥,𝑓⟩=‖𝑥‖²=‖𝑓‖²} for all 𝑥∈𝑋, 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 𝑇∶𝐾→𝐾 is said to be asymptotically nonexpansive if for each 𝑥,𝑦∈𝐾‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤𝑘_𝑛‖𝑥−𝑦‖,(1.1) where {𝑘_𝑛}_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 𝑇∶𝐵→𝐵 is defined as 𝑇𝑥₁,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‖≤2‖𝑥−𝑦‖,‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤2^𝑛_𝑗=2𝑎_𝑖‖𝑥−𝑦‖.(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 ⟨𝐴𝑥−𝐴𝑦,𝑗(𝑥−𝑦)⟩≥𝜑(‖𝑥−𝑦‖)‖𝑥−𝑦‖,(1.4) where 𝜑∶ℝ⁺→ℝ⁺ 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 ℝ⁺ 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 (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 in to account Definition 1.3, it is natural to study the class of 𝜑-pseudocontractive mappings, that is, the maps such that⟨𝑇𝑥−𝑇𝑦,𝑗(𝑥−𝑦)⟩≤‖𝑥−𝑦‖²−𝜑(‖𝑥−𝑦‖)‖𝑥−𝑦‖,(1.5) where 𝜑∶ℝ⁺→ℝ⁺ 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 ⟨𝑇𝑥−𝑇𝑦,𝑗(𝑥−𝑦)⟩≤‖𝑥−𝑦‖²−𝜙(‖𝑥−𝑦‖),(1.6) where 𝜙∶ℝ⁺→ℝ⁺ 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, 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,𝑗(𝑥−𝑦)⟩≤𝑘_𝑛‖𝑥−𝑦‖²,(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=(1−𝑥^2/3)^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 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(𝑞^2𝑛−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 𝑥𝑧_𝑛𝑛+1∶=𝛽_𝑛𝑇^𝑛𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛,∶=𝛼_𝑛𝑇^𝑛𝑧_𝑛+1−𝛼_𝑛𝑥_𝑛,(1.8) then {𝑥_𝑛}_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 𝑇∶𝐾→𝐾 is said to be a generalized asymptotically 𝜙-strongly pseudocontraction if ⟨𝑇^𝑛𝑥−𝑇^𝑛𝑦,𝑗(𝑥−𝑦)⟩≤𝑘_𝑛‖𝑥−𝑦‖²−𝜙(‖𝑥−𝑦‖),(1.9) where {𝑘_𝑛}_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∈ℕ, ‖𝑥−𝑧‖²=⟨𝑇^𝑛𝑥−𝑇^𝑛𝑧,𝑗(𝑥−𝑧)⟩≤𝑘_𝑛‖𝑥−𝑧‖²−𝜙(‖𝑥−𝑧‖),(1.10) so passing 𝑛 to +∞, it results that ‖𝑥−𝑧‖²≤‖𝑥−𝑧‖²−𝜙(‖𝑥−𝑧‖)⟹−𝜙(‖𝑥−𝑧‖)≥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 𝑘_𝑛=1, for all 𝑛∈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 𝑦𝑥_𝑛=_𝑛+1𝑎_𝑛𝑥_𝑛+𝑏_𝑛𝑆𝑥_𝑛+𝑐_𝑛𝑢_𝑛,=𝑎^𝑛𝑥_𝑛+𝑏^𝑛𝑆𝑦_𝑛𝑐^𝑛𝑣_𝑛,(1.12) where 𝑎_𝑛+𝑏_𝑛+𝑐_𝑛=𝑎^𝑛+𝑏^𝑛+𝑐^𝑛=1,  𝑆𝑥=𝑥−𝑇𝑥+𝑓 (𝑇 a 𝜙-strongly accretive operator), and 𝑓∈𝑋, to a solution of the equation 𝑇𝑥=𝑓.

In 2003, Chidume and Zegeye [12] studied the following iterative method: 𝑥_𝑛+1=1−𝜆_𝑛𝑥_𝑛+𝜆_𝑛𝑇𝑥_𝑛−𝜆_𝑛𝜃_𝑛𝑥_𝑛−𝑥₁,(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 (𝜃_𝑛)_𝑛,(𝜆_𝑛)_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 𝑥₀∈𝑋, the sequence {𝑥_𝑛}_𝑛 is defined by 𝑦𝑥_𝑛=_𝑛+11−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇^𝑛𝑥_𝑛−𝛿_𝑛𝑥_𝑛−𝑣_𝑛,=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇^𝑛𝑦_𝑛−𝛾_𝑛𝑥_𝑛−𝑢_𝑛,𝑛≥0,(1.14) 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 by 𝑧𝑧₀∈𝑋,_𝑛+1=1−𝛼_𝑛𝑧_𝑛+𝛼_𝑛𝑇^𝑛𝑧_𝑛−𝛾_𝑛𝑧_𝑛−𝑤_𝑛,𝑛≥0,(1.15) which is called the modified Mann iteration sequence.

We also introduce an implicit iterative process as follows: 𝑧^𝑛=1−𝛼_𝑛𝑧^𝑛−1+𝛼_𝑛𝑇^𝑛𝑧^𝑛−𝛾_𝑛𝑧^𝑛−1−𝑤^𝑛,𝑛≥1,(1.16) where {𝛼_𝑛}_𝑛,{𝛾_𝑛}_𝑛 are two real sequences in [0,1] satisfying 𝛼_𝑛+𝛾_𝑛≤1 and 𝛼_𝑛𝑘_𝑛<1 for all 𝑛≥1, {𝑤^𝑛}_𝑛 is a sequence in 𝑋, and 𝑧^0 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 𝑆_𝑛 defined by 𝑆_𝑛𝑥∶=(1−𝛼_𝑛−𝛾_𝑛)𝑧_𝑛−1+𝛼_𝑛𝑇^𝑛𝑥+𝛾𝑤_𝑛 is such that ⟨𝑆_𝑛𝑥−𝑆_𝑛𝑦,𝑗(𝑥−𝑦)⟩=⟨𝑇^𝑛𝑥−𝑇^𝑛𝑦,𝑗(𝑥−𝑦)⟩≤𝛼_𝑛𝑘_𝑛‖𝑥−𝑦‖²,(1.17) that is, 𝑆_𝑛 is a strongly pseudocontraction, for every fixed 𝑛, then (see Theorem  13.1 in [14]) there exists a unique fixed point of 𝑆_𝑛 for each 𝑛.

These kind of iterative processes (also called by Chang iterative processes with errors) have been developed in [15–18], 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 𝑋𝜌_𝑋(𝑡)=sup12(‖𝑥−𝑦‖+‖𝑥+𝑦‖)−1∶‖𝑥‖≤1,‖𝑦‖≤𝑡(2.1) satisfies lim_𝑡→0𝜌_𝑋(𝑡)/𝑡=0.

Lemma 2.2 (see [22]). 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+𝑦‖²≤‖𝑥‖²+2⟨𝑦,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 {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=𝑜𝑏_𝑛,^∞_𝑛=1𝑏_𝑛=∞,lim_𝑛→∞𝑒_𝑛=0.(2.3) Suppose that there exists an integer 𝑁₁>0 such that 𝑎^2𝑛+1≤𝑎^2𝑛−2𝑏_𝑛𝜙||𝑎_𝑛+1−𝑒_𝑛||+𝑐_𝑛,∀𝑛≥𝑁₁,(2.4) then lim_𝑛→∞𝑎_𝑛=0.

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

Lemma 2.4 (see [23]). 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_𝑛𝑠_𝑛+𝑏_𝑛+𝑐_𝑛,(2.5) 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_𝑛𝑏_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 (2.4) becomes 𝑎𝑎^2𝑛+1_2𝑛+1≤𝑎^2𝑛−2𝑏_𝑛𝑘𝑎^2𝑛+1+𝑐_𝑛⟹≤11+2𝑏_𝑛𝑘𝑎^2𝑛+𝑐^𝑛=1−2𝑏_𝑛1+2𝑏_𝑛𝑘𝑘1+2𝑏_𝑛𝑘𝑎^2𝑛+𝑐^𝑛1+2𝑏_𝑛𝑘.(2.6) Setting 𝛼_𝑛∶=2𝑏_𝑛𝑘/(1+2𝑏_𝑛𝑘) and 𝛽_𝑛∶=𝑐_𝑛/(1+2𝑏_𝑛𝑘) and by the hypotheses of Lemma 2.3, we get 𝛼_𝑛→0 as 𝑛→∞, ∑_𝑛𝛼_𝑛=∞, and limsup_𝑛𝛽_𝑛/𝛼_𝑛=0. That is, we reobtain Lemma 2.4 in the case of 𝑐_𝑛=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 𝑇∶𝑋→𝑋 be generalized strongly asymptotically 𝜙-pseudocontractive mapping with fixed point 𝑥^∗ and bounded range.
Let {𝑥_𝑛} and {𝑧_𝑛} be the sequences defined by (1.14) and (1.15), respectively, where {𝛼_𝑛},{𝛾_𝑛},{𝛽_𝑛},{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},{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 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 {𝑤_𝑛},{𝑢_𝑛} and by Lemma 2.4, it results that {𝑧_𝑛} and {𝑥_𝑛} are bounded sequences. So, we can set 𝑀=sup_𝑛⎧⎪⎪⎪⎨⎪⎪⎪⎩‖‖𝑇‖‖^𝑛𝑇𝑧_𝑛𝑛−𝑇^𝑛𝑦_𝑛‖‖,‖‖𝑇^𝑛𝑥_𝑛−𝑥_𝑛‖‖,‖‖𝑧_𝑛𝑦_𝑛−𝑥_𝑛‖‖,‖‖𝑇^𝑛𝑧_𝑛−𝑧_𝑛‖‖,−‖‖𝑥𝑤_𝑛‖‖_𝑛,‖‖𝑢_𝑛−𝑥_𝑛‖‖,‖‖𝑣_𝑛−𝑥_𝑛‖‖,−𝑧_𝑛‖‖,‖‖𝑤_𝑛−𝑢_𝑛‖‖⎫⎪⎪⎪⎬⎪⎪⎪⎭.(3.1) By Lemma 2.2, we have ‖‖𝑧_𝑛+1−𝑥_𝑛+1‖‖²=‖‖1−𝛼_𝑛−𝛾_𝑛𝑧_𝑛−𝑥_𝑛+𝛼_𝑛𝑇^𝑛𝑧_𝑛−𝑇^𝑛𝑦_𝑛+𝛾_𝑛𝑤_𝑛−𝑢_𝑛‖‖²≤≤11−−𝛼𝛼_𝑛𝑛−𝛾_𝑛²‖‖𝑧_𝑛−𝑥_𝑛‖‖²+2𝛼_𝑛𝑇^𝑛𝑧_𝑛−𝑇^𝑛𝑦_𝑛+𝛾_𝑛𝑤_𝑛−𝑢_𝑛,𝑗𝑧_𝑛+1−𝑥_𝑛+1+2𝛼_𝑛²‖‖𝑧_𝑛−𝑥_𝑛‖‖²+2𝛼_𝑛𝑇^𝑛𝑧_𝑛−𝑇^𝑛𝑦_𝑛,𝑗𝑧_𝑛−𝑦_𝑛≤1−𝛼_𝑛𝑇^𝑛𝑧_𝑛−𝑇^𝑛𝑦_𝑛,𝑗𝑧_𝑛+1−𝑥_𝑛+1−𝑗𝑧_𝑛−𝑦_𝑛+2𝛾_𝑛𝑤_𝑛−𝑢_𝑛,𝑗𝑧_𝑛+1−𝑥_𝑛+1+2𝛼_𝑛²‖‖𝑧_𝑛−𝑥_𝑛‖‖²+2𝛼_𝑛𝑘_𝑛‖‖𝑧_𝑛−𝑦_𝑛‖‖²−2𝛼_𝑛𝜙‖‖𝑧_𝑛−𝑦_𝑛‖‖≤1−𝛼‖‖_𝑛𝑇^𝑛𝑧_𝑛−𝑇^𝑛𝑦_𝑛‖‖‖‖𝑗𝑧_𝑛+1−𝑥_𝑛+1−𝑗𝑧_𝑛−𝑦_𝑛‖‖+2𝛾_𝑛‖‖𝑤_𝑛−𝑢_𝑛‖‖‖‖𝑧_𝑛+1−𝑥_𝑛+1‖‖+2𝛼_𝑛²‖‖𝑧_𝑛−𝑥_𝑛‖‖²+2𝛼_𝑛𝑘_𝑛‖‖𝑧_𝑛−𝑦_𝑛‖‖²−2𝛼_𝑛𝜙‖‖𝑧_𝑛−𝑦_𝑛‖‖𝜎_𝑛𝑀+2𝛾_𝑛𝑀²,(3.2) where 𝜎_𝑛=‖𝑗(𝑧_𝑛+1−𝑥_𝑛+1)−𝑗(𝑧_𝑛−𝑦_𝑛)‖. Using (1.14) and (1.15), we have ‖‖𝑧_𝑛+1−𝑥_𝑛+1−𝑧_𝑛−𝑦_𝑛‖‖=≤‖‖‖‖𝛼𝑥_𝑛+1𝑛−𝑦_𝑛‖‖+‖‖𝑧_𝑛+1−𝑧_𝑛‖‖+‖‖𝛼_𝑛𝑇^𝑛𝑦_𝑛−𝑥_𝑛+𝛾_𝑛𝑢_𝑛−𝑥_𝑛−𝛽_𝑛𝑇^𝑛𝑥_𝑛−𝑥_𝑛−𝛿_𝑛𝑣_𝑛−𝑥_𝑛‖‖≤2𝑀𝛼_𝑛𝑇^𝑛𝑧_𝑛−𝑧_𝑛+𝛾_𝑛𝑤_𝑛−𝑧_𝑛‖‖+𝛾_𝑛+𝛽_𝑛+𝛿_𝑛⟶0,(𝑛→∞).(3.3) In view of the uniformly continuity of 𝑗, we obtain that 𝜎_𝑛→0 as 𝑛→∞. Furthermore, it follows from the definition of {𝑦_𝑛} that for all 𝑛≥0‖‖𝑧_𝑛−𝑦_𝑛‖‖²=‖‖𝑧_𝑛−𝑥_𝑛+𝛽_𝑛(−𝑇^𝑛𝑥_𝑛+𝑥_𝑛)+𝛿_𝑛(−𝑣_𝑛+𝑥_𝑛)‖‖²≤‖‖𝑧_𝑛−𝑥_𝑛‖‖+𝛽_𝑛‖‖𝑇^𝑛𝑥_𝑛−𝑥_𝑛‖‖+𝛿_𝑛‖‖𝑣_𝑛−𝑥_𝑛‖‖²≤‖‖𝑧_𝑛−𝑥_𝑛‖‖+𝛽_𝑛+𝛿_𝑛𝑀²≤=‖‖‖‖𝑧𝑧_𝑛𝑛−𝑥_𝑛‖‖²+𝛽_𝑛+𝛿_𝑛2‖‖𝑧_𝑛−𝑥_𝑛‖‖𝑀+𝛽_𝑛+𝛿_𝑛𝑀²−𝑥_𝑛‖‖²+3𝛽_𝑛+𝛿_𝑛𝑀²,(3.4)‖‖𝑦_𝑛−𝑥_𝑛‖‖≤𝛽_𝑛‖‖𝑇^𝑛𝑥_𝑛−𝑥_𝑛‖‖+𝛿_𝑛‖‖𝑣_𝑛−𝑥_𝑛‖‖≤𝛽_𝑛+𝛿_𝑛𝑀⟶0,(𝑛→∞),(3.5) so ‖‖𝑧_𝑛+1−𝑥_𝑛+1‖‖≤=‖‖‖‖𝑧𝑧_𝑛𝑛−𝑥_𝑛−𝛼_𝑛+𝛾_𝑛𝑧_𝑛−𝑥_𝑛+𝛼_𝑛𝑇^𝑛𝑧_𝑛−𝑇^𝑛𝑦_𝑛+𝛾_𝑛𝑤_𝑛−𝑢_𝑛‖‖≤‖‖𝑧_𝑛−𝑥_𝑛‖‖+𝛼_𝑛+𝛾_𝑛‖‖𝑧_𝑛−𝑥_𝑛‖‖+𝛼_𝑛‖‖𝑇^𝑛𝑧_𝑛−𝑇^𝑛𝑦_𝑛‖‖+𝛾_𝑛‖‖𝑤_𝑛−𝑢_𝑛‖‖≤‖‖𝑧−𝑦_𝑛‖‖+‖‖𝑦_𝑛−𝑥_𝑛‖‖+2𝛼_𝑛+𝛾_𝑛𝑀._𝑛−𝑦_𝑛‖‖+𝛽_𝑛+𝛿_𝑛𝑀+2𝛼_𝑛+𝛾_𝑛𝑀.(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. By (H1), we have that 𝑒_𝑛→0 as 𝑛→∞. If ‖𝑧_𝑛+1−𝑥_𝑛+1‖−𝑒_𝑛≤0 for an infinite number of indices, 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→∞.
In this case, we can prove that ‖𝑧_𝑛−𝑥_𝑛‖→0, that is, the thesis.
Firstly, we note that substituting (3.4) into (3.2), we have ‖‖𝑧_𝑛+1−𝑥_𝑛+1‖‖²≤‖‖𝑧_𝑛−𝑥_𝑛‖‖²+𝛼^2𝑛‖‖𝑧_𝑛−𝑥_𝑛‖‖²+2𝛼_𝑛𝑘_𝑛−1‖‖𝑧_𝑛−𝑥_𝑛‖‖²+6𝛼_𝑛𝑘_𝑛𝛽_𝑛+𝛿_𝑛𝑀²−2𝛼_𝑛𝜙‖‖𝑧_𝑛−𝑦_𝑛‖‖+2𝛼_𝑛𝜎_𝑛𝑀+2𝛾_𝑛𝑀²≤‖‖𝑧_𝑛−𝑥_𝑛‖‖²−2𝛼_𝑛𝜙‖‖𝑧_𝑛−𝑦_𝑛‖‖+𝛼^2𝑛𝑀²+2𝛼_𝑛𝑘_𝑛−1𝑀²+6𝛼_𝑛𝑘𝛽_𝑛+𝛿_𝑛𝑀²+2𝛼_𝑛𝜎_𝑛𝑀+2𝛾_𝑛𝑀²=‖‖𝑧_𝑛−𝑥_𝑛‖‖²−𝛼_𝑛𝜙‖‖𝑧_𝑛−𝑦_𝑛‖‖+2𝛾_𝑛𝑀²−𝛼_𝑛𝜙‖‖𝑧^𝑛−𝑦_𝑛‖‖−𝛼_𝑛𝑀²−2𝑘_𝑛−1𝑀²−6𝑘𝛽_𝑛+𝛿_𝑛𝑀²−2𝜎_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 null sequences (and in particular 𝛾_𝑛=𝑜(𝛼_𝑛)), for the previous fixed 𝜖>0, there exists an index 𝑁 such that, for all 𝑛∈𝑁, ||𝛼_𝑛||<min𝜖||𝛾_𝑛16𝑀,𝜙8(𝑀𝜖/2²),||<𝜖16𝑀,||||𝛼𝛾^𝑛_𝑛||||<𝜙(𝜖/2)4𝑀²||𝑘_𝑛,−1||<𝜙(𝜖/2)16𝑀²||𝛽_𝑛,+𝛿_𝑛||<min𝜖||𝜎_𝑛4𝑀,4𝜙8(𝑘𝜖𝑀/2)²,||<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. Suppose that ‖𝑧_𝑛∗+1−𝑥_𝑛∗+1‖≥𝜖.
By (3.6), we have 𝜖≤‖‖𝑧_𝑛∗+1−𝑥_𝑛∗+1‖‖≤≤‖‖‖‖𝑧𝑧_{𝑛∗𝑛∗}−𝑦_𝑛∗‖‖+𝛽_𝑛∗+𝛿_𝑛∗𝑀+2𝛼_𝑛^∗+𝛾_𝑛∗𝑀−𝑦_𝑛∗‖‖+𝑀𝜖=‖‖𝑧4𝑀+2𝑀16𝜖𝑀+16𝜖𝑀_𝑛∗−𝑦_𝑛∗‖‖+𝜖4+4𝜖=‖‖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∗+1‖‖²<𝜖²−𝛼^∗𝑛𝜙‖‖𝑧^𝑛∗−𝑦_𝑛∗‖‖−2𝑀²𝛾^∗𝑛−𝛼_∗𝑛𝛼^∗𝑛𝜙‖‖𝑧^𝑛∗−𝑦_𝑛∗‖‖−𝛼^∗𝑛𝑀²−2𝑘^∗𝑛−1𝑀²−6𝑘𝛽^∗𝑛+𝛿^∗𝑛𝑀²−2𝜎_𝑛∗𝑀.(3.13) One can note that 𝛼^∗𝑛𝑀²+2𝑘^∗𝑛−1𝑀²+6𝑘𝛽^∗𝑛+𝛿^∗𝑛𝑀²+2𝜎_𝑛∗𝑀≤𝜙(𝜖/2)8+𝜙(8𝜖/2)+𝜙(8𝜖/2)+𝜙(8𝜖/2).(3.14) hence 𝜙‖‖𝑧_𝑛∗−𝑦_𝑛∗‖‖−𝛼^∗𝑛𝑀²−2𝑘_𝑛∗−1𝑀²−6𝑘𝛽_𝑛∗+𝛿_𝑛∗𝑀²−2𝜎_𝑛∗𝑀≥𝜙𝜖2−𝜙(2𝜖/2)>0.(3.15) In the same manner, 𝜙‖‖𝑧_𝑛∗−𝑦_𝑛∗‖‖−2𝑀²𝛾_𝑛∗𝛼_𝑛∗>𝜙(𝜖/2)−𝜙(𝜖/2)2>0.(3.16) Thus, ‖‖𝑧_𝑛∗+1−𝑥_𝑛∗+1‖‖²<𝜖².(3.17) So we have ‖𝑧_𝑛∗+1−𝑥_𝑛∗+1‖<𝜖, which contradicts ‖𝑧_𝑛∗+1−𝑥_𝑛∗+1‖≥𝜖. By the same idea, we can prove 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 ensure that ‖𝑧_𝑛−𝑥_𝑛‖→0.
If there are only finite indices 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 𝜙‖‖𝑧_𝑛−𝑦_𝑛‖‖≥𝜙‖‖𝑧_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+1‖‖²≤‖‖𝑧_𝑛−𝑥_𝑛‖‖²+𝛼^2𝑛‖‖𝑧_𝑛−𝑥_𝑛‖‖²+2𝛼_𝑛𝑘_𝑛−1‖‖𝑧_𝑛−𝑥_𝑛‖‖²+6𝛼_𝑛𝑘_𝑛𝛽_𝑛+𝛿_𝑛𝑀²−2𝛼_𝑛𝜙‖‖𝑧_𝑛+1−𝑥_𝑛+1‖‖−𝑒_𝑛+2𝛼_𝑛𝜎_𝑛𝑀+2𝛾_𝑛𝑀²≤‖‖𝑧_𝑛−𝑥_𝑛‖‖²−2𝛼_𝑛𝜙‖‖𝑧_𝑛+1−𝑥_𝑛+1‖‖−𝑒_𝑛+𝛼^2𝑛𝑀²+2𝛼_𝑛𝑘_𝑛−1𝑀²+6𝛼_𝑛𝑘_𝑛𝛽_𝑛+𝛿_𝑛𝑀²+2𝛼_𝑛𝜎_𝑛𝑀+2𝛾_𝑛𝑀².(3.19) Suppose that 𝑎_𝑛=‖𝑧_𝑛−𝑥_𝑛‖,𝑏_𝑛=𝛼_𝑛,and𝑐_𝑛=𝛼^2𝑛𝑀²+2𝛼_𝑛(𝑘_𝑛−1)𝑀²+6𝛼_𝑛𝑘_𝑛(𝛽_𝑛+𝛿_𝑛)𝑀²+2𝛼_𝑛𝜎_𝑛𝑀+2𝛾_𝑛𝑀². It follows from lim_𝑛→∞𝜎_𝑛=0, lim_𝑛→∞𝑒_𝑛=0, lim_𝑛→∞𝑘_𝑛=1, and the hypothesis that we have, ∑^∞𝑛=1𝑏_𝑛=∞ and 𝑐_𝑛=𝑜(𝑏_𝑛), 𝑒_𝑛→0 as 𝑛→∞. By virtue 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 generalized strongly asymptotically 𝜙-pseudocontractive mapping with fixed point 𝑥^∗ and bounded range.
Let {𝑧_𝑛} and {𝑧^𝑛} be the sequences defined by (1.15) and (1.16), respectively, where {𝛼_𝑛},{𝛾_𝑛}⊂[0,1] are null sequences satisfying (H1)lim_𝑛→∞𝛼_𝑛=0 and 𝛾_𝑛=𝑜(𝛼_𝑛), (H2)∑^∞𝑛=1𝛼_𝑛=∞, of Theorem 3.1 and such that 𝛼_𝑛𝑘_𝑛<1, for every 𝑛∈ℕ.
Suppose moreover that the sequences {𝑤_𝑛},{𝑤^𝑛} are bounded in 𝑋, then for any initial point 𝑧^0,𝑧₀∈𝑋, 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=sup_𝑛⎧⎪⎨⎪⎩‖‖𝑇‖‖^𝑛𝑧𝑧_𝑛−_𝑛+1𝑇^𝑛𝑧^𝑛‖‖,‖‖𝑇^𝑛𝑧_𝑛−𝑧_𝑛‖‖,‖‖𝑇^𝑛𝑧^𝑛−𝑧^𝑛‖‖,‖‖𝑇^𝑛𝑧^𝑛−𝑧^𝑛−1‖‖−𝑧^𝑛‖‖,‖‖𝑧_𝑛−𝑤_𝑛‖‖,‖‖𝑤^𝑛−𝑧^𝑛−1‖‖,‖‖𝑤_𝑛−𝑤^𝑛‖‖⎫⎪⎬⎪⎭.(3.20) By the iteration schemes (1.15) and (1.16), we have ‖‖𝑧_𝑛+1−𝑧^𝑛‖‖²≤‖‖1−𝛼_𝑛−𝛾_𝑛𝑧_𝑛−𝑧^𝑛−1+𝛼_𝑛𝑇^𝑛𝑧_𝑛−𝑇^𝑛𝑧^𝑛+𝛾_𝑛𝑤_𝑛−𝑤^𝑛‖‖²≤≤11−−𝛼𝛼_𝑛𝑛−𝛾_𝑛²‖‖𝑧_𝑛−𝑧^𝑛−1‖‖²+2𝛼_𝑛𝑇^𝑛𝑧_𝑛−𝑇^𝑛𝑧^𝑛+𝛾_𝑛𝑤_𝑛−𝑤^𝑛,𝑗𝑧_𝑛+1−𝑧^𝑛+2𝛼_𝑛²‖‖𝑧_𝑛−𝑧^𝑛−1‖‖²+2𝛼_𝑛𝑇^𝑛𝑧_𝑛−𝑇^𝑛𝑧^𝑛,𝑗𝑧_𝑛−𝑧^𝑛≤1−𝛼_𝑛𝑇^𝑛𝑧_𝑛−𝑇^𝑛𝑧^𝑛,𝑗𝑧_𝑛+1−𝑧^𝑛−𝑗𝑧_𝑛−𝑧^𝑛+2𝛾_𝑛‖‖𝑤_𝑛−𝑤^𝑛‖‖‖‖𝑧_𝑛+1−𝑧^𝑛‖‖²‖‖𝑧_𝑛−𝑧^𝑛−1‖‖²+2𝛼_𝑛𝑘_𝑛‖‖𝑧_𝑛−𝑧^𝑛‖‖²−2𝛼_𝑛𝜙‖‖𝑧_𝑛−𝑧^𝑛‖‖+2𝛼_𝑛𝑀𝜎_𝑛+2𝛾_𝑛𝑀²,(3.21) where 𝜎_𝑛=‖𝑗(𝑧_𝑛+1−𝑧^𝑛)−𝑗(𝑧_𝑛−𝑧^𝑛)‖. By (1.15), we get ‖‖𝑧_𝑛+1−𝑧^𝑛−𝑧_𝑛−𝑧^𝑛‖‖=‖‖𝑧_𝑛+1−𝑧_𝑛‖‖=‖‖𝛼_𝑛𝑇^𝑛𝑧_𝑛−𝑧_𝑛+𝛾_𝑛𝑤_𝑛−𝑧_𝑛‖‖≤𝛼_𝑛+𝛾_𝑛𝑀.(3.22) It follows from (H1) that ‖(𝑧_𝑛+1−𝑧^𝑛)−(𝑧_𝑛−𝑧^𝑛)‖→0 as 𝑛→∞, which implies that 𝜎_𝑛→0 as 𝑛→∞. Moreover, for all 𝑛≥0, ‖‖𝑧_𝑛−𝑧^𝑛‖‖²≤‖‖𝑧_𝑛−𝑧^𝑛−1‖‖+‖‖𝑧^𝑛−1−𝑧^𝑛‖‖²≤‖‖𝑧_𝑛−𝑧^𝑛−1‖‖+𝛼_𝑛‖‖𝑇^𝑛𝑧^𝑛−𝑧^𝑛−1‖‖+𝛾_𝑛‖‖𝑤^𝑛−𝑧^𝑛−1‖‖²≤‖‖𝑧_𝑛−𝑧^𝑛−1‖‖+𝛼_𝑛+𝛾_𝑛𝑀²=≤‖‖‖‖𝑧𝑧_𝑛𝑛−𝑧^𝑛−1‖‖²+𝛼_𝑛+𝛾_𝑛2𝑀‖‖𝑧_𝑛−𝑧^𝑛−1‖‖+𝛼_𝑛+𝛾_𝑛𝑀²−𝑧^𝑛−1‖‖²+3𝛼_𝑛+𝛾_𝑛𝑀².(3.23) Again by the boundedness of all components, we have that ‖‖𝑧^𝑛−𝑧^𝑛−1‖‖=‖‖𝛼_𝑛𝑇^𝑛𝑧^𝑛−𝑧^𝑛−1+𝛾_𝑛𝑤^𝑛−𝑧^𝑛−1‖‖≤𝛼_𝑛+𝛾_𝑛𝑀,(3.24) and so ‖‖𝑧_𝑛+1−𝑧^𝑛‖‖≤=‖‖‖‖𝑧𝑧_𝑛𝑛−𝑧^𝑛+𝑧^𝑛−𝑧^𝑛−1−𝛼_𝑛+𝛾_𝑛𝑧_𝑛−𝑧^𝑛−1+𝛼_𝑛𝑇^𝑛𝑧_𝑛−𝑇^𝑛𝑧^𝑛+𝛾_𝑛𝑤_𝑛−𝑤^𝑛‖‖≤‖‖𝑧_𝑛−𝑧^𝑛‖‖+‖‖𝑧^𝑛−𝑧^𝑛−1‖‖+𝛼_𝑛+𝛾_𝑛‖‖𝑧_𝑛−𝑧^𝑛−1‖‖+𝛼_𝑛‖‖𝑇^𝑛𝑧_𝑛−𝑇^𝑛𝑧^𝑛‖‖+𝛾_𝑛‖‖𝑤_𝑛−𝑤^𝑛‖‖−𝑧^𝑛‖‖+3𝛼_𝑛+𝛾_𝑛𝑀.(3.25) Hence, we have that ‖𝑧_𝑛−𝑧^𝑛‖≥‖𝑧_𝑛+1−𝑧^𝑛‖−𝑒_𝑛, where 𝑒_𝑛=3(𝛼_𝑛+𝛾_𝑛)𝑀. Note that 𝑒_𝑛→0 as 𝑛→∞. As in proof of Theorem 3.1, we distinguish two cases:(i)the set of indices for which ‖𝑧_𝑛+1−𝑧^𝑛‖−𝑒_𝑛≤0 contains infinite terms; (ii)the set of indices for which ‖𝑧_𝑛+1−𝑧^𝑛‖−𝑒_𝑛≤0 contains finite terms. In the first case, (i) we can extract a subsequence such that ‖𝑧_𝑛𝑘−𝑧^𝑛_𝑘−1‖→0, as 𝑘→∞. Substituting (3.23) in (3.21), we have that ‖‖𝑧_𝑛+1−𝑧^𝑛‖‖²≤1+𝛼^2𝑛−2𝛼_𝑛‖‖𝑧_𝑛−𝑧^𝑛−1‖‖²+2𝛼_𝑛𝑘_𝑛‖‖𝑧_𝑛−𝑧^𝑛−1‖‖²+6𝛼_𝑛+𝛾_𝑛𝑀²𝛼_𝑛𝑘_𝑛−2𝛼_𝑛𝜙‖‖𝑧_𝑛−𝑧^𝑛‖‖+2𝛼_𝑛𝑀𝜎_𝑛+2𝛾_𝑛𝑀²≤−‖‖𝑧2_𝑛𝛼−_𝑛𝑧^𝑛−1‖‖²+𝛼^2𝑛+2𝛼_𝑛𝑘_𝑛−1‖‖𝑧_𝑛−𝑧^𝑛−1‖‖²+6𝛼_𝑛+𝛾_𝑛𝑀²𝛼_𝑛𝑘𝜙‖‖𝑧_𝑛−𝑧^𝑛‖‖+2𝛼_𝑛𝑀𝜎_𝑛+2𝛾_𝑛𝑀²≤‖‖𝑧_𝑛−𝑧^𝑛−1‖‖²−𝛼_𝑛𝜙‖‖𝑧_𝑛−𝑧^𝑛‖‖+2𝛾_𝑛𝑀²=−‖‖𝑧𝛼_𝑛𝑛𝜙‖‖𝑧^𝑛−𝑧^𝑛‖‖−𝛼_𝑛𝑀²−2𝑀²𝑘_𝑛−1−6𝛼_𝑛+𝛾_𝑛𝑀²𝑘−2𝑀𝜎_𝑛−𝑧^𝑛−1‖‖²−𝛼_𝑛𝜙‖‖𝑧_𝑛−𝑧^𝑛‖‖+2𝛾_𝑛𝑀²−𝛼_𝑛𝜙‖‖𝑧^𝑛−𝑧^𝑛‖‖−7𝑘𝛼_𝑛𝑀²−2𝑀²𝑘_𝑛−1−6𝛾_𝑛𝑀²𝑘−2𝑀𝜎_𝑛,(3.26) where 𝑘=sup_𝑛𝑘_𝑛. Again by (3.23), for every 𝜖>0, there exists an index 𝑙 such that if 𝑗>𝑙, ‖‖𝑧_𝑛𝑗−𝑧^𝑛_𝑗−1‖‖<𝜖,‖‖𝑧_𝑛𝑗−𝑧^𝑛_𝑗‖‖<2𝜖.(3.27) By hypotheses on the control sequence, with the same 𝜖>0, there exists an index 𝑁 such that definitively ||𝛼_𝑛||<min𝜖||𝛾_𝑛12𝑀,5𝜙6(𝑀𝜖/2²𝑘),||<min𝜖12𝑀,4𝜙8(𝑀𝜖/2²𝑘),||||𝛼𝛾^𝑛𝑛||||<𝜙(𝜖/2)4𝑀²||𝑘_𝑛,−1||<𝜙(𝜖/2)16𝑀²||𝜎_𝑛,||<𝜙(𝜖/2)16𝑀.(3.28) So take 𝑛^∗>max{𝑛_𝑙,𝑁} with 𝑛^∗=𝑛_𝑗 for a certain 𝑗.
We can prove that ‖𝑧_𝑛+1−𝑧^𝑛‖→0 as 𝑛→∞ proving that, for every 𝑖≥0, the result is ‖𝑧_𝑛∗+𝑖−𝑧^𝑛_∗+𝑖−1‖<𝜖.
Let 𝑖=1. If we suppose that ‖𝑧_𝑛∗+1−𝑧^𝑛_∗‖≥𝜖, it results that 𝜖≤‖‖𝑧_𝑛∗+1−𝑧^𝑛_∗‖‖≤‖‖𝑧_𝑛∗−𝑧^𝑛_∗‖‖+3𝛼_𝑛∗+𝛾_𝑛∗𝑀<‖‖𝑧_𝑛^∗−𝑧^𝑛_∗‖‖+𝜖2,(3.29) so ‖𝑧_𝑛∗−𝑧^𝑛_∗‖>𝜖/2. In consequence of this, 𝜙(‖𝑧_𝑛∗−𝑧_𝑛∗′‖)>𝜙(𝜖/2).
In (3.26), we note that 7𝑘𝛼_𝑛∗𝑀²+2𝑀²𝑘_𝑛^∗−1+6𝛾_𝑛∗𝑀²𝑘+2𝑀𝜎_𝑛∗≤7𝑘𝑀²𝜙(𝜖/2)56𝑀²𝑘+2𝑀²𝜙(𝜖/2)16𝑀²+6𝑀²𝑘𝜙(𝜖/2)=𝜙(𝜖/2)48𝑀²𝑘+2𝑀𝜙1(6𝜖𝑀/2)84=𝜙(2𝜖/2),(3.30) so 𝜙‖‖𝑧_𝑛∗−𝑧^𝑛_∗‖‖−7𝑘𝛼_𝑛∗𝑀²−2𝑀²𝑘_𝑛^∗−1−6𝛾_𝑛∗𝑀²𝑘−2𝑀𝜎_𝑛∗≥𝜙(𝜖/2)2.(3.31) hence in (3.26) remains ‖‖𝑧_𝑛∗+1−𝑧^𝑛_∗‖‖²≤𝜖²−𝛼_𝑛𝜙‖‖𝑧_𝑛−𝑧^𝑛‖‖+2𝛾_𝑛𝑀²<𝜖²,(3.32) as in Theorem 3.1. This is a contradiction. By the same idea, and using the inductive hypothesis, we obtain that ‖𝑧_𝑛∗+𝑖−𝑧^𝑛_∗+𝑖−1‖<𝜖, for every 𝑖≥0. This ensures that ‖𝑧_𝑛+1−𝑧^𝑛‖→0. In the second case (ii), definitively, ‖𝑧_𝑛+1−𝑧^𝑛‖−𝑒_𝑛≥0, then from the strictly increasing function 𝜙, we have 𝜙‖‖𝑧_𝑛−𝑧^𝑛‖‖≥𝜙‖‖𝑧_𝑛+1−𝑧^𝑛‖‖−𝑒_𝑛.(3.33) Substituting (3.33) and (3.23) into (3.21) and simplifying, we have ‖‖𝑧_𝑛+1−𝑧^𝑛‖‖²≤‖‖𝑧_𝑛−𝑧^𝑛−1‖‖²+𝛼^2𝑛‖‖𝑧_𝑛−𝑧^𝑛−1‖‖²+2𝛼_𝑛𝑘_𝑛−1‖‖𝑧_𝑛−𝑧^𝑛−1‖‖²+6𝛼_𝑛𝑘_𝑛𝛼_𝑛+𝛾_𝑛𝑀²−2𝛼_𝑛𝜙‖‖𝑧_𝑛+1−𝑧^𝑛‖‖−𝑒_𝑛+2𝛼_𝑛𝜎_𝑛𝑀+2𝛾_𝑛𝑀²≤‖‖𝑧_𝑛−𝑧^𝑛−1‖‖²−2𝛼_𝑛𝜙‖‖𝑧_𝑛+1−𝑧^𝑛‖‖−𝑒_𝑛+𝛼^2𝑛𝑀²+2𝛼_𝑛𝑘_𝑛−1𝑀²+6𝛼_𝑛𝑘_𝑛𝛼_𝑛+𝛾_𝑛𝑀²+2𝛼_𝑛𝜎_𝑛𝑀+2𝛾_𝑛𝑀².(3.34) By virtue of Lemma 2.3, we obtain that lim_𝑛→∞‖𝑧_𝑛−𝑧^𝑛−1‖=0.

Theorem 3.3. Let 𝑋 be a uniformly smooth Banach space, and let 𝑇∶𝑋→𝑋 be generalized strongly asymptotically 𝜙-pseudocontractive mapping with fixed point 𝑥^∗ and bounded range.
Let {𝑧_𝑛}_𝑛 be the sequences defined by (1.15) where {𝛼_𝑛}_𝑛,{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. Firstly, we observe that, by the boundedness of the range of 𝑇, 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≤+12𝛾−_𝑛𝛼_𝑛−𝛾_𝑛²‖‖𝑧_𝑛−𝑥^∗‖‖²+2𝛼_𝑛𝑇^𝑛𝑧_𝑛−𝑥^∗,𝑗𝑧_𝑛+1−𝑥^∗≤1−𝛼_𝑛𝑤_𝑛−𝑥^∗,𝑗𝑧_𝑛+1−𝑥^∗+2𝛼_𝑛²‖‖𝑧_𝑛−𝑥^∗‖‖²+2𝛼_𝑛⟨𝑇^𝑛𝑧_𝑛−𝑥^∗,𝑗𝑧_𝑛+1−𝑥^∗−𝑗𝑧_𝑛−𝑥^∗⟩≤1−𝛼_𝑛𝑇^𝑛𝑧_𝑛−𝑥^∗,𝑗𝑧_𝑛−𝑥^∗+2𝛾_𝑛𝑤_𝑛−𝑥^∗,𝑗𝑧_𝑛+1−𝑥^∗+2𝛼_𝑛²‖‖𝑧_𝑛−𝑥^∗‖‖²+2𝛼_𝑛𝑇^𝑛𝑛𝑧_𝑛−𝑥^∗,𝑗𝑧_𝑛+1−𝑥^∗−𝑗𝑧_𝑛−𝑥^∗=1+𝛼𝑘_𝑛2𝑛‖‖𝑧_𝑛−𝑥^∗‖‖²−2𝛼_𝑛𝜙‖‖𝑧_𝑛−𝑥^∗‖‖+2𝛾_𝑛𝑤_𝑛−𝑥^∗,𝑗𝑧_𝑛+1−𝑥^∗+2𝛼_𝑛−2𝛼_𝑛‖‖𝑧_𝑛−𝑥^∗‖‖²+2𝛼_𝑛𝑇^𝑛𝑧_𝑛−𝑥^∗,𝑗𝑧_𝑛+1−𝑥^∗−𝑗𝑧_𝑛−𝑥^∗=‖‖𝑧_𝑛𝑘_𝑛‖‖𝑧_𝑛−𝑥^∗‖‖²−2𝛼_𝑛𝜙‖‖𝑧_𝑛−𝑥^∗‖‖+2𝛾_𝑛⟨𝑤_𝑛−𝑥^∗,𝑗𝑧_𝑛+1−𝑥^∗⟩−𝑥^∗‖‖²+𝛼^2𝑛−2𝛼_𝑛+2𝛼_𝑛𝑘_𝑛‖‖𝑧_𝑛−𝑥^∗‖‖²−2𝛼_𝑛𝜙‖‖𝑧_𝑛−𝑥^∗‖‖+2𝛼_𝑛𝜇_𝑛+2𝛾_𝑛𝑤_𝑛−𝑥^∗,𝑗𝑧_𝑛+1−𝑥^∗,(3.35) where 𝜇_𝑛∶=⟨𝑇^𝑛𝑧_𝑛−𝑥^∗,𝑗(𝑧_𝑛+1−𝑥^∗)−𝑗(𝑧_𝑛−𝑥^∗)⟩. Let 𝑀∶=maxsup_𝑛‖‖𝑧_𝑛−𝑥^∗‖‖,sup_𝑛‖‖𝑇^𝑛𝑧_𝑛−𝑥^∗‖‖,sup_𝑛‖‖𝑤_𝑛−𝑥^∗‖‖,sup_𝑛𝑤_𝑛−𝑥^∗,𝑗𝑧_𝑛+1−𝑥^∗.(3.36) We have ‖‖𝑧_𝑛+1−𝑥^∗‖‖²=≤‖‖‖‖𝑧𝑧_𝑛𝑛−𝑥^∗‖‖²+𝛼^2𝑛+2𝛼_𝑛𝑘_𝑛−1𝑀−2𝛼_𝑛𝜙‖‖𝑧_𝑛−𝑥^∗‖‖+2𝛼_𝑛𝜇_𝑛+2𝛾_𝑛𝑀−𝛼_𝑛−𝑥^∗‖‖²−𝛼_𝑛𝜙‖‖𝑧_𝑛−𝑥^∗‖‖+2𝛾_𝑛𝑀𝜙‖‖𝑧_𝑛−𝑥^∗‖‖−2𝜇_𝑛−𝛼_𝑛+2𝑘_𝑛−1𝑀,(3.37) so we can observe that(1)𝜇_𝑛→0 as 𝑛→∞. Indeed from the inequality ||‖‖𝑧_𝑛+1−𝑥^∗‖‖−‖‖𝑧_𝑛−𝑥^∗‖‖||≤‖‖𝑧_𝑛+1−𝑧_𝑛‖‖≤𝛼_𝑛+𝛾_𝑛𝑀⟶0,as𝑛⟶∞,(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. Indeed, if we supposed that 𝜎∶=inf_𝑛(‖𝑧_𝑛−𝑥^∗‖)>0, by the monotonicity of 𝜙, 𝜙‖‖𝑧_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>𝑁, ‖‖𝑧_𝑛+1−𝑥^∗‖‖²≤≤‖‖‖‖𝑧𝑧_𝑛𝑛−𝑥^∗‖‖²−𝛼_𝑛𝜙‖‖𝑧_𝑛−𝑥^∗‖‖+2𝛾_𝑛𝑀−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 way we obtain that 𝜙(𝜎)^𝑚_𝑖_=𝑁𝛼_𝑖≤^𝑚_𝑖_=𝑁‖‖𝑧^𝑖−𝑥^∗‖‖²−‖‖𝑧_𝑖+1−𝑥^∗‖‖²+2^𝑚_𝑖_=𝑁=‖‖𝑧_𝑁𝛾_𝑖𝑀−𝑥^∗‖‖²−‖‖𝑧_𝑚−𝑥^∗‖‖²+2𝑀^𝑚_𝑖_=𝑁𝛾_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 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 results that ||𝛼_𝑛||<1||𝛾4𝑀min𝜖,𝜙2𝜖_𝑛,||<𝜖4𝑀,||||𝛼𝛾^𝑛_𝑛||||<𝜙(𝜖/2)||𝑘4𝑀_𝑛,−1||<𝜙(𝜖/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 results that ‖𝑧_𝑛∗+1−𝑥^∗‖≥𝜖. Thus, 𝜖≤<‖‖‖‖𝑧𝑧_𝑛∗+1−𝑥^∗‖‖≤‖‖𝑧_𝑛∗−𝑥^∗‖‖+𝛼_𝑛∗𝑀+𝛾_𝑛∗𝑀,_𝑛∗−𝑥^∗‖‖+𝜖4𝑀𝑀+4𝑀𝜖𝑀=‖‖𝑧^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 results that ‖‖𝑧_𝑛∗+1−𝑥^∗‖‖²<𝜖²−𝛼_𝑛∗𝜙‖‖𝑧^𝑛∗−𝑥^∗‖‖−2𝑀𝛾^𝑛∗−𝛼_𝑛∗𝛼_𝑛∗𝜙‖‖𝑧_𝑛^∗−𝑥^∗‖‖−2𝜇_𝑛∗−𝛼_𝑛∗+2𝑘_𝑛∗−1𝑀.(3.45) We can note that 2𝜇_𝑛∗+𝛼_𝑛∗+2𝑘_𝑛∗−1𝑀≤𝜙(𝜖/2)4+𝜙4(𝑀𝜖/2)+𝜙4(𝑀𝜖/2)𝑀,(3.46) so 𝜙‖‖𝑧_𝑛∗−𝑥^∗‖‖−2𝜇_𝑛∗−𝛼_𝑛∗+2𝑘_𝑛∗−1𝑀>𝜙(𝜖/2)−3𝜙(𝜖/2)4>0.(3.47) Moreover, 𝜙(‖𝑧_𝑛∗−𝑥^∗‖)−2𝑀𝛾_𝑛∗/𝛼_𝑛∗>𝜙(𝜖/2)/2>0, so it results that ‖‖𝑧_𝑛∗+1−𝑥^∗‖‖²<𝜖².(3.48) This is a contradiction. 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 generalized strongly asymptotically 𝜙-pseudocontractive mapping with bounded range and fixed point 𝑥^∗. The sequences {𝑥_𝑛}_𝑛, {𝑧_𝑛}_𝑛, and {𝑧^𝑛}_𝑛 are defined by (1.14), (1.15), and (1.16), respectively, where the sequences {𝛼_𝑛}_𝑛, {𝛽_𝑛}_𝑛, {𝛾_𝑛}_𝑛, {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, {𝑤_𝑛}_𝑛, and {𝑤_𝑛′}_𝑛 are bounded in X. Then for any initial point 𝑥₀,𝑧₀,𝑧₀′∈𝑋, 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^∗.