About this Journal Submit a Manuscript Table of Contents
Journal of Applied Mathematics
VolumeΒ 2012Β (2012), Article IDΒ 563438, 18 pages
http://dx.doi.org/10.1155/2012/563438
Research Article

On the Convergence of Iterative Processes for Generalized Strongly Asymptotically πœ™-Pseudocontractive Mappings in Banach Spaces

Dipartimento di Matematica, UniversitΓ‘ della Calabria, 87036 Arcavacata di Rende (CS), Italy

Received 5 October 2011; Accepted 11 October 2011

Academic Editor: YonghongΒ Yao

Copyright Β© 2012 Vittorio Colao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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

1. Introduction

Throughout this paper, we assume that 𝑋 is a uniformly convex Banach space and π‘‹βˆ— is the dual space of 𝑋. Let 𝐽 denote the normalized duality mapping form 𝑋 into 2π‘‹βˆ— given by 𝐽(π‘₯)={π‘“βˆˆπ‘‹βˆ—βˆΆβŸ¨π‘₯,π‘“βŸ©=β€–π‘₯β€–2=‖𝑓‖2} for all π‘₯βˆˆπ‘‹, where βŸ¨β‹…,β‹…βŸ© 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 π‘‡βˆΆπΎβ†’πΎ is said to be asymptotically nonexpansive if for each π‘₯,π‘¦βˆˆπΎβ€–π‘‡π‘›π‘₯βˆ’π‘‡π‘›π‘¦β€–β‰€π‘˜π‘›,β€–π‘₯βˆ’π‘¦β€–(1.1) where {π‘˜π‘›}π‘›βŠ‚[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 𝐡 is the unit ball of 𝑙2 and π‘‡βˆΆπ΅β†’π΅ is defined as 𝑇π‘₯1,π‘₯2ξ€Έ=ξ€·,…0,π‘₯21,π‘Ž2π‘₯2,π‘Ž3π‘₯3ξ€Έ,,…(1.2) where {π‘Žπ‘–}π‘–βˆˆβ„•βŠ‚(0,1) is such that βˆβˆžπ‘–=2π‘Žπ‘–=1/2, it satisfies. ‖𝑇π‘₯βˆ’π‘‡π‘¦β€–β‰€2β€–π‘₯βˆ’π‘¦β€–,‖𝑇𝑛π‘₯βˆ’π‘‡π‘›π‘¦β€–β‰€2𝑛𝑗=2π‘Žπ‘–β€–π‘₯βˆ’π‘¦β€–.(1.3)

In 1974, Deimling [5], studying the zeros of accretive operators, introduced the class of πœ‘-strongly accretive operators.

Definition 1.3. An operator 𝐴 defined on a subset 𝐾 of a Banach space 𝑋 is said, πœ‘-strongly accretive if ),⟨𝐴π‘₯βˆ’π΄π‘¦,𝑗(π‘₯βˆ’π‘¦)⟩β‰₯πœ‘(β€–π‘₯βˆ’π‘¦β€–β€–π‘₯βˆ’π‘¦β€–(1.4) where πœ‘βˆΆβ„+→ℝ+ is a strictly increasing function such that πœ‘(0)=0.

Note that in the special case in which πœ‘(𝑑)=π‘˜π‘‘,β€‰β€‰π‘˜βˆˆ(0,1), we obtain a strongly accretive operator.

Osilike [6], among the others, proved that 𝐴π‘₯=π‘₯βˆ’(π‘₯/(π‘₯+1)) in ℝ+ is πœ‘-strongly accretive where πœ‘(𝑑)=(𝑑2/(1+𝑑)) but not strongly accretive.

Since an operator 𝐴 is a strongly accretive operator if and only if (πΌβˆ’π΄) is a strongly pseudocontractive mapping (i.e., ⟨(πΌβˆ’π΄)π‘₯βˆ’(πΌβˆ’π΄)𝑦,𝑗(π‘₯βˆ’π‘¦)βŸ©β‰€π‘˜β€–π‘₯βˆ’π‘¦β€–2,β€‰β€‰π‘˜<1), taking in to account Definition 1.3, it is natural to study the class of πœ‘-pseudocontractive mappings, that is, the maps such thatβŸ¨π‘‡π‘₯βˆ’π‘‡π‘¦,𝑗(π‘₯βˆ’π‘¦)βŸ©β‰€β€–π‘₯βˆ’π‘¦β€–2()βˆ’πœ‘β€–π‘₯βˆ’π‘¦β€–β€–π‘₯βˆ’π‘¦β€–,(1.5) where πœ‘βˆΆβ„+→ℝ+ is a strictly increasing function such that πœ‘(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 πœ™-strongly pseudocontractive mapping if βŸ¨π‘‡π‘₯βˆ’π‘‡π‘¦,𝑗(π‘₯βˆ’π‘¦)βŸ©β‰€β€–π‘₯βˆ’π‘¦β€–2(βˆ’πœ™β€–π‘₯βˆ’π‘¦β€–),(1.6) where πœ™βˆΆβ„+→ℝ+ is a strictly increasing function such that πœ™(0)=0.

Choosing πœ™(𝑑)=πœ‘(𝑑)𝑑, we obtain Definition 1.3. In [7], Xiang remarked that it is a open problem if every generalized πœ™-strongly pseudocontractive mapping is πœ‘-pseudocontractive mapping. In the same paper, Xiang obtained a fixed-point theorem for continuous and generalized πœ™-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, πΎβŠ‚π‘‹ and {π‘˜π‘›}π‘›βŠ‚[1,∞). A mapping π‘‡βˆΆπΎβ†’πΎ is said to be asymptotically pseudocontractive with the sequence {π‘˜π‘›}𝑛 if and only if limπ‘›β†’βˆžπ‘˜π‘›=1, and for all π‘›βˆˆβ„• and all π‘₯,π‘¦βˆˆπΎ, there exists 𝑗(π‘₯βˆ’π‘¦)∈𝐽(π‘₯βˆ’π‘¦) such that βŸ¨π‘‡π‘›π‘₯βˆ’π‘‡π‘›π‘¦,𝑗(π‘₯βˆ’π‘¦)βŸ©β‰€π‘˜π‘›β€–π‘₯βˆ’π‘¦β€–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 π‘‡βˆΆ[0,1]β†’[0,1] defined by 𝑇π‘₯=(1βˆ’π‘₯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 π΄βŠ‚π» closed and convex; 𝐿>0; π‘‡βˆΆπ΄β†’π΄ completely continuous, uniformly 𝐿-Lipschitzian, and asymptotically pseudocontractive with sequence {π‘˜π‘›}π‘›βˆˆ[1,∞);β€‰β€‰π‘žπ‘›βˆΆ=2π‘˜π‘›βˆ’1 for all π‘›βˆˆβ„•; βˆ‘π‘›(π‘ž2π‘›βˆ’1)<∞; {𝛼𝑛}𝑛,{𝛽𝑛}π‘›βˆˆ[0,1]; πœ–β‰€π›Όπ‘›β‰€π›½π‘›β‰€π‘ for all π‘›βˆˆβ„•, some πœ–>0 and some π‘βˆˆ(0,πΏβˆ’2[√1+𝐿2βˆ’1]); π‘₯1∈𝐴; for all π‘›βˆˆβ„•, define π‘§π‘›βˆΆ=𝛽𝑛𝑇𝑛π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ‘₯𝑛,π‘₯𝑛+1∢=𝛼𝑛𝑇𝑛𝑧𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘₯𝑛,(1.8) then {π‘₯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 πœ™-pseudocontraction defined as follows.

Definition 1.7. If 𝑋 is a Banach space and 𝐾 is a subset of 𝑋, a mapping π‘‡βˆΆπΎβ†’πΎ is said to be a generalized asymptotically πœ™-strongly pseudocontraction if βŸ¨π‘‡π‘›π‘₯βˆ’π‘‡π‘›π‘¦,𝑗(π‘₯βˆ’π‘¦)βŸ©β‰€π‘˜π‘›β€–π‘₯βˆ’π‘¦β€–2(),βˆ’πœ™β€–π‘₯βˆ’π‘¦β€–(1.9) where {π‘˜π‘›}π‘›βŠ‚[1,∞) is converging to one and πœ™βˆΆ[0,∞)β†’[0,∞) is strictly increasing and such that πœ™(0)=0.

One can note that (i)if 𝑇 has fixed points, then it is unique. In fact, if π‘₯,𝑧 are fixed points for 𝑇, then for every π‘›βˆˆβ„•, β€–π‘₯βˆ’π‘§β€–2=βŸ¨π‘‡π‘›π‘₯βˆ’π‘‡π‘›π‘§,𝑗(π‘₯βˆ’π‘§)βŸ©β‰€π‘˜π‘›β€–π‘₯βˆ’π‘§β€–2βˆ’πœ™(β€–π‘₯βˆ’π‘§β€–),(1.10) so passing 𝑛 to +∞, it results that β€–π‘₯βˆ’π‘§β€–2≀‖π‘₯βˆ’π‘§β€–2βˆ’πœ™(β€–π‘₯βˆ’π‘§β€–)βŸΉβˆ’πœ™(β€–π‘₯βˆ’π‘§β€–)β‰₯0.(1.11) Since πœ™βˆΆ[0,∞)β†’[0,∞) is strictly increasing and πœ™(0)=0, then π‘₯=𝑧. (ii)the mapping 𝑇π‘₯=π‘₯/(π‘₯+1), where π‘₯∈[0,1], is generalized asymptotically strongly πœ™-pseudocontraction with π‘˜π‘›=1, for all π‘›βˆˆβ„• and πœ™(𝑑)=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 πœ™-pseudocontractions.

Moreover, we prove that these methods are equivalent and strongly convergent to the unique fixed point of the generalized strongly asymptotically πœ™-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=π‘Žπ‘›π‘₯𝑛+𝑏𝑛𝑆π‘₯𝑛+𝑐𝑛𝑒𝑛,π‘₯𝑛+1=π‘Žξ…žπ‘›π‘₯𝑛+π‘ξ…žπ‘›π‘†π‘¦π‘›π‘ξ…žπ‘›π‘£π‘›,(1.12) where π‘Žπ‘›+𝑏𝑛+𝑐𝑛=π‘Žξ…žπ‘›+π‘ξ…žπ‘›+π‘ξ…žπ‘›=1,  𝑆π‘₯=π‘₯βˆ’π‘‡π‘₯+𝑓 (𝑇 a πœ™-strongly accretive operator), and π‘“βˆˆπ‘‹, to a solution of the equation 𝑇π‘₯=𝑓.

In 2003, Chidume and Zegeye [12] studied the following iterative method: π‘₯𝑛+1=ξ€·1βˆ’πœ†π‘›ξ€Έπ‘₯𝑛+πœ†π‘›π‘‡π‘₯π‘›βˆ’πœ†π‘›πœƒπ‘›ξ€·π‘₯π‘›βˆ’π‘₯1ξ€Έ,(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 (πœƒπ‘›)𝑛,(πœ†π‘›)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 π‘₯0βˆˆπ‘‹, the sequence {π‘₯𝑛}𝑛 is defined by 𝑦𝑛=ξ€·1βˆ’π›½π‘›ξ€Έπ‘₯𝑛+𝛽𝑛𝑇𝑛π‘₯π‘›βˆ’π›Ώπ‘›ξ€·π‘₯π‘›βˆ’π‘£π‘›ξ€Έ,π‘₯𝑛+1=ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘₯𝑛+π›Όπ‘›π‘‡π‘›π‘¦π‘›βˆ’π›Ύπ‘›ξ€·π‘₯π‘›βˆ’π‘’π‘›ξ€Έ,𝑛β‰₯0,(1.14) 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 𝑛β‰₯0.

In particular, if 𝛽𝑛=𝛿𝑛=0 for all 𝑛β‰₯0, we can define a sequence {𝑧𝑛}𝑛 by 𝑧0π‘§βˆˆπ‘‹,𝑛+1=ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘§π‘›+π›Όπ‘›π‘‡π‘›π‘§π‘›βˆ’π›Ύπ‘›ξ€·π‘§π‘›βˆ’π‘€π‘›ξ€Έ,𝑛β‰₯0,(1.15) which is called the modified Mann iteration sequence.

We also introduce an implicit iterative process as follows: π‘§ξ…žπ‘›=ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘§ξ…žπ‘›βˆ’1+π›Όπ‘›π‘‡π‘›π‘§ξ…žπ‘›βˆ’π›Ύπ‘›ξ€·π‘§ξ…žπ‘›βˆ’1βˆ’π‘€ξ…žπ‘›ξ€Έ,𝑛β‰₯1,(1.16) where {𝛼𝑛}𝑛,{𝛾𝑛}𝑛 are two real sequences in [0,1] satisfying 𝛼𝑛+𝛾𝑛≀1 and π›Όπ‘›π‘˜π‘›<1 for all 𝑛β‰₯1, {π‘€ξ…žπ‘›}𝑛 is a sequence in 𝑋, and π‘§ξ…ž0 is an initial point.

The algorithm is well defined. Indeed, if 𝑇 is a asymptotically strongly πœ™-pseudocontraction, one can observe that, for every fixed 𝑛, the mapping 𝑆𝑛 defined by 𝑆𝑛π‘₯∢=(1βˆ’π›Όπ‘›βˆ’π›Ύπ‘›)π‘§π‘›βˆ’1+𝛼𝑛𝑇𝑛π‘₯+𝛾𝑀𝑛 is such that βŸ¨π‘†π‘›π‘₯βˆ’π‘†π‘›π‘¦,𝑗(π‘₯βˆ’π‘¦)⟩=βŸ¨π‘‡π‘›π‘₯βˆ’π‘‡π‘›π‘¦,𝑗(π‘₯βˆ’π‘¦)βŸ©β‰€π›Όπ‘›π‘˜π‘›β€–π‘₯βˆ’π‘¦β€–2,(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 [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 πœ™-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 πœ™-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 π‘‹πœŒπ‘‹ξ‚†1(𝑑)=sup2(‖π‘₯βˆ’π‘¦β€–+β€–π‘₯+𝑦‖)βˆ’1βˆΆβ€–π‘₯‖≀1,‖𝑦‖≀𝑑(2.1) satisfies lim𝑑→0πœŒπ‘‹(𝑑)/𝑑=0.

Lemma 2.2 (see [22]). Let 𝑋 be a Banach space, and let π‘—βˆΆπ‘‹β†’2π‘‹βˆ— be the normalized duality mapping, then for any π‘₯,π‘¦βˆˆπ‘‹, one has β€–π‘₯+𝑦‖2≀‖π‘₯β€–2+2βŸ¨π‘¦,𝑗(π‘₯+𝑦)⟩,βˆ€π‘—(π‘₯+𝑦)∈𝐽(π‘₯+𝑦).(2.2)

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

Lemma 2.3 (see [21]). Let πœ™βˆΆ[0,∞)β†’[0,∞) be a strictly increasing function with πœ™(0)=0, and let {π‘Žπ‘›}𝑛,{𝑏𝑛}𝑛,{𝑐𝑛}𝑛, and {𝑒𝑛}𝑛 be nonnegative real sequences such that limπ‘›β†’βˆžπ‘π‘›=0,𝑐𝑛𝑏=π‘œπ‘›ξ€Έ,βˆžξ“π‘›=1𝑏𝑛=∞,limπ‘›β†’βˆžπ‘’π‘›=0.(2.3) Suppose that there exists an integer 𝑁1>0 such that π‘Ž2𝑛+1β‰€π‘Ž2π‘›βˆ’2π‘π‘›πœ™ξ€·||π‘Žπ‘›+1βˆ’π‘’π‘›||ξ€Έ+𝑐𝑛,βˆ€π‘›β‰₯𝑁1,(2.4) then limπ‘›β†’βˆžπ‘Žπ‘›=0.

Proof. The proof is the same as in [21], but we substitute (π‘Žπ‘›+1βˆ’π‘’π‘›) with |π‘Žπ‘›+1βˆ’π‘’π‘›|, in (2.4).

Lemma 2.4 (see [23]). Let {𝑠𝑛}𝑛, {𝑐𝑛}π‘›βŠ‚β„+, {π‘Žπ‘›}π‘›βŠ‚(0,1), and {𝑏𝑛}π‘›βŠ‚β„ be sequences such that 𝑠𝑛+1≀1βˆ’π‘Žπ‘›ξ€Έπ‘ π‘›+𝑏𝑛+𝑐𝑛,(2.5) for all 𝑛β‰₯0. Assume that βˆ‘π‘›|𝑐𝑛|<∞, then the following results hold:(1)if π‘π‘›β‰€π›½π‘Žπ‘› (where 𝛽β‰₯0), then {𝑠𝑛}𝑛 is a bounded sequence;(2)if one has βˆ‘π‘›π‘Žπ‘›=∞ and limsup𝑛𝑏𝑛/π‘Žπ‘›β‰€0, then 𝑠𝑛→0 as π‘›β†’βˆž.

Remark 2.5. If in Lemma 2.3 choosing 𝑒𝑛=0, for all 𝑛, πœ™(𝑑)=π‘˜π‘‘2 (π‘˜<1), then the inequality (2.4) becomes π‘Ž2𝑛+1β‰€π‘Ž2π‘›βˆ’2π‘π‘›π‘˜π‘Ž2𝑛+1+π‘π‘›βŸΉπ‘Ž2𝑛+1≀11+2π‘π‘›π‘˜π‘Ž2𝑛+𝑐𝑛1+2π‘π‘›π‘˜=ξ‚΅1βˆ’2π‘π‘›π‘˜1+2π‘π‘›π‘˜ξ‚Άπ‘Ž2𝑛+𝑐𝑛1+2π‘π‘›π‘˜.(2.6) Setting π›Όπ‘›βˆΆ=2π‘π‘›π‘˜/(1+2π‘π‘›π‘˜) and π›½π‘›βˆΆ=𝑐𝑛/(1+2π‘π‘›π‘˜) and by the hypotheses of Lemma 2.3, we get 𝛼𝑛→0 as π‘›β†’βˆž, βˆ‘π‘›π›Όπ‘›=∞, 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 π‘‡βˆΆπ‘‹β†’π‘‹ be generalized strongly asymptotically πœ™-pseudocontractive mapping with fixed point π‘₯βˆ— and bounded range.
Let {π‘₯𝑛} and {𝑧𝑛} be the sequences defined by (1.14) and (1.15), respectively, where {𝛼𝑛},{𝛾𝑛},{𝛽𝑛},{𝛿𝑛}βŠ‚[0,1] satisfy(H1)limπ‘›β†’βˆžπ›Όπ‘›=limπ‘›β†’βˆžπ›½π‘›=limπ‘›β†’βˆžπ›Ώπ‘›=0 and 𝛾𝑛=π‘œ(𝛼𝑛),(H2)βˆ‘βˆžπ‘›=1𝛼𝑛=∞,and the sequences {𝑒𝑛},{𝑣𝑛},{𝑀𝑛} are bounded in 𝑋, then for any initial point 𝑧0,π‘₯0βˆˆπ‘‹, the following two assertions are equivalent: (i)the modified Ishikawa iteration sequence with errors (1.14) converges to π‘₯βˆ—; (ii)the modified Mann iteration sequence with errors (1.15) converges to π‘₯βˆ—.

Proof. First of all, we note that by boundedness of the range of 𝑇, of the sequences {𝑀𝑛},{𝑒𝑛} and by Lemma 2.4, it results that {𝑧𝑛} and {π‘₯𝑛} are bounded sequences. So, we can set 𝑀=supπ‘›βŽ§βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽ©β€–β€–π‘‡π‘›π‘§π‘›βˆ’π‘‡π‘›π‘¦π‘›β€–β€–,‖‖𝑇𝑛π‘₯π‘›βˆ’π‘₯𝑛‖‖,β€–β€–π‘‡π‘›π‘¦π‘›βˆ’π‘₯𝑛‖‖,β€–β€–π‘‡π‘›π‘§π‘›βˆ’π‘§π‘›β€–β€–,β€–β€–π‘§π‘›βˆ’π‘₯𝑛‖‖,β€–β€–π‘’π‘›βˆ’π‘₯𝑛‖‖,β€–β€–π‘£π‘›βˆ’π‘₯𝑛‖‖,β€–β€–π‘€π‘›βˆ’π‘§π‘›β€–β€–,β€–β€–π‘€π‘›βˆ’π‘’π‘›β€–β€–βŽ«βŽͺβŽͺβŽͺ⎬βŽͺβŽͺβŽͺ⎭.(3.1) By Lemma 2.2, we have ‖‖𝑧𝑛+1βˆ’π‘₯𝑛+1β€–β€–2=β€–β€–ξ€·1βˆ’π›Όπ‘›βˆ’π›Ύπ‘›π‘§ξ€Έξ€·π‘›βˆ’π‘₯𝑛+π›Όπ‘›ξ€·π‘‡π‘›π‘§π‘›βˆ’π‘‡π‘›π‘¦π‘›ξ€Έ+π›Ύπ‘›ξ€·π‘€π‘›βˆ’π‘’π‘›ξ€Έβ€–β€–2≀1βˆ’π›Όπ‘›βˆ’π›Ύπ‘›ξ€Έ2β€–β€–π‘§π‘›βˆ’π‘₯𝑛‖‖2𝛼+2π‘›ξ€·π‘‡π‘›π‘§π‘›βˆ’π‘‡π‘›π‘¦π‘›ξ€Έ+π›Ύπ‘›ξ€·π‘€π‘›βˆ’π‘’π‘›ξ€Έξ€·π‘§,𝑗𝑛+1βˆ’π‘₯𝑛+1≀1βˆ’π›Όπ‘›ξ€Έ2β€–β€–π‘§π‘›βˆ’π‘₯𝑛‖‖2+2π›Όπ‘›ξ«π‘‡π‘›π‘§π‘›βˆ’π‘‡π‘›π‘¦π‘›ξ€·π‘§,π‘—π‘›βˆ’π‘¦π‘›ξ€Έξ¬+2π›Όπ‘›ξ«π‘‡π‘›π‘§π‘›βˆ’π‘‡π‘›π‘¦π‘›ξ€·π‘§,𝑗𝑛+1βˆ’π‘₯𝑛+1ξ€Έξ€·π‘§βˆ’π‘—π‘›βˆ’π‘¦π‘›ξ€Έξ¬+2π›Ύπ‘›ξ«π‘€π‘›βˆ’π‘’π‘›ξ€·π‘§,𝑗𝑛+1βˆ’π‘₯𝑛+1≀1βˆ’π›Όπ‘›ξ€Έ2β€–β€–π‘§π‘›βˆ’π‘₯𝑛‖‖2+2π›Όπ‘›π‘˜π‘›β€–β€–π‘§π‘›βˆ’π‘¦π‘›β€–β€–2βˆ’2π›Όπ‘›πœ™ξ€·β€–β€–π‘§π‘›βˆ’π‘¦π‘›β€–β€–ξ€Έ+2π›Όπ‘›β€–β€–π‘‡π‘›π‘§π‘›βˆ’π‘‡π‘›π‘¦π‘›β€–β€–β€–β€–π‘—ξ€·π‘§π‘›+1βˆ’π‘₯𝑛+1ξ€Έξ€·π‘§βˆ’π‘—π‘›βˆ’π‘¦π‘›ξ€Έβ€–β€–+2π›Ύπ‘›β€–β€–π‘€π‘›βˆ’π‘’π‘›β€–β€–β€–β€–π‘§π‘›+1βˆ’π‘₯𝑛+1‖‖≀1βˆ’π›Όπ‘›ξ€Έ2β€–β€–π‘§π‘›βˆ’π‘₯𝑛‖‖2+2π›Όπ‘›π‘˜π‘›β€–β€–π‘§π‘›βˆ’π‘¦π‘›β€–β€–2βˆ’2π›Όπ‘›πœ™ξ€·β€–β€–π‘§π‘›βˆ’π‘¦π‘›β€–β€–ξ€Έ+2π›Όπ‘›πœŽπ‘›π‘€+2𝛾𝑛𝑀2,(3.2) where πœŽπ‘›=‖𝑗(𝑧𝑛+1βˆ’π‘₯𝑛+1)βˆ’π‘—(π‘§π‘›βˆ’π‘¦π‘›)β€–. Using (1.14) and (1.15), we have ‖‖𝑧𝑛+1βˆ’π‘₯𝑛+1ξ€Έβˆ’ξ€·π‘§π‘›βˆ’π‘¦π‘›ξ€Έβ€–β€–β‰€β€–β€–π‘₯𝑛+1βˆ’π‘¦π‘›β€–β€–+‖‖𝑧𝑛+1βˆ’π‘§π‘›β€–β€–=β€–β€–π›Όπ‘›ξ€·π‘‡π‘›π‘¦π‘›βˆ’π‘₯𝑛+π›Ύπ‘›ξ€·π‘’π‘›βˆ’π‘₯π‘›ξ€Έβˆ’π›½π‘›ξ€·π‘‡π‘›π‘₯π‘›βˆ’π‘₯π‘›ξ€Έβˆ’π›Ώπ‘›ξ€·π‘£π‘›βˆ’π‘₯𝑛‖‖+β€–β€–π›Όπ‘›ξ€·π‘‡π‘›π‘§π‘›βˆ’π‘§π‘›ξ€Έ+π›Ύπ‘›ξ€·π‘€π‘›βˆ’π‘§π‘›ξ€Έβ€–β€–ξ€·π›Όβ‰€2𝑀𝑛+𝛾𝑛+𝛽𝑛+π›Ώπ‘›ξ€ΈβŸΆ0,(π‘›β†’βˆž).(3.3) In view of the uniformly continuity of 𝑗, we obtain that πœŽπ‘›β†’0 as π‘›β†’βˆž. Furthermore, it follows from the definition of {𝑦𝑛} that for all 𝑛β‰₯0β€–β€–π‘§π‘›βˆ’π‘¦π‘›β€–β€–2=β€–β€–π‘§π‘›βˆ’π‘₯𝑛+𝛽𝑛(βˆ’π‘‡π‘›π‘₯𝑛+π‘₯𝑛)+𝛿𝑛(βˆ’π‘£π‘›+π‘₯𝑛)β€–β€–2β‰€ξ€Ίβ€–β€–π‘§π‘›βˆ’π‘₯𝑛‖‖+𝛽𝑛‖‖𝑇𝑛π‘₯π‘›βˆ’π‘₯𝑛‖‖+π›Ώπ‘›β€–β€–π‘£π‘›βˆ’π‘₯𝑛‖‖2β‰€ξ€Ίβ€–β€–π‘§π‘›βˆ’π‘₯𝑛‖‖+𝛽𝑛+𝛿𝑛𝑀2=β€–β€–π‘§π‘›βˆ’π‘₯𝑛‖‖2+𝛽𝑛+𝛿𝑛2β€–β€–π‘§ξ€Έξ€·π‘›βˆ’π‘₯𝑛‖‖𝛽𝑀+𝑛+𝛿𝑛𝑀2ξ€Έβ‰€β€–β€–π‘§π‘›βˆ’π‘₯𝑛‖‖2𝛽+3𝑛+𝛿𝑛𝑀2,(3.4)β€–β€–π‘¦π‘›βˆ’π‘₯𝑛‖‖≀𝛽𝑛‖‖𝑇𝑛π‘₯π‘›βˆ’π‘₯𝑛‖‖+π›Ώπ‘›β€–β€–π‘£π‘›βˆ’π‘₯𝑛‖‖≀𝛽𝑛+π›Ώπ‘›ξ€Έπ‘€βŸΆ0,(π‘›β†’βˆž),(3.5) so ‖‖𝑧𝑛+1βˆ’π‘₯𝑛+1β€–β€–=β€–β€–π‘§π‘›βˆ’π‘₯π‘›βˆ’ξ€·π›Όπ‘›+π›Ύπ‘›π‘§ξ€Έξ€·π‘›βˆ’π‘₯𝑛+π›Όπ‘›ξ€·π‘‡π‘›π‘§π‘›βˆ’π‘‡π‘›π‘¦π‘›ξ€Έ+π›Ύπ‘›ξ€·π‘€π‘›βˆ’π‘’π‘›ξ€Έβ€–β€–β‰€β€–β€–π‘§π‘›βˆ’π‘₯𝑛‖‖+𝛼𝑛+π›Ύπ‘›ξ€Έβ€–β€–π‘§π‘›βˆ’π‘₯𝑛‖‖+π›Όπ‘›β€–β€–π‘‡π‘›π‘§π‘›βˆ’π‘‡π‘›π‘¦π‘›β€–β€–+π›Ύπ‘›β€–β€–π‘€π‘›βˆ’π‘’π‘›β€–β€–β‰€β€–β€–π‘§π‘›βˆ’π‘¦π‘›β€–β€–+β€–β€–π‘¦π‘›βˆ’π‘₯𝑛‖‖𝛼+2𝑛+𝛾𝑛≀‖‖𝑧𝑀.π‘›βˆ’π‘¦π‘›β€–β€–+𝛽𝑛+𝛿𝑛𝛼𝑀+2𝑛+𝛾𝑛𝑀.(3.6) Therefore, we have β€–β€–π‘§π‘›βˆ’π‘¦π‘›β€–β€–β‰₯‖‖𝑧𝑛+1βˆ’π‘₯𝑛+1β€–β€–βˆ’π‘’π‘›,(3.7) where 𝑒𝑛=(𝛽𝑛+𝛿𝑛)𝑀+2(𝛼𝑛+𝛾𝑛)𝑀. By (H1), we have that 𝑒𝑛→0 as π‘›β†’βˆž. If ‖𝑧𝑛+1βˆ’π‘₯𝑛+1β€–βˆ’π‘’π‘›β‰€0 for an infinite number of indices, we can extract a subsequence such that β€–π‘§π‘›π‘˜βˆ’π‘₯π‘›π‘˜β€–βˆ’π‘’π‘›π‘˜βˆ’1≀0. For this subsequence, β€–π‘§π‘›π‘˜βˆ’π‘₯π‘›π‘˜β€–β†’0, as π‘˜β†’βˆž.
In this case, we can prove that β€–π‘§π‘›βˆ’π‘₯𝑛‖→0, that is, the thesis.
Firstly, we note that substituting (3.4) into (3.2), we have ‖‖𝑧𝑛+1βˆ’π‘₯𝑛+1β€–β€–2β‰€β€–β€–π‘§π‘›βˆ’π‘₯𝑛‖‖2+𝛼2π‘›β€–β€–π‘§π‘›βˆ’π‘₯𝑛‖‖2+2π›Όπ‘›ξ€·π‘˜π‘›ξ€Έβ€–β€–π‘§βˆ’1π‘›βˆ’π‘₯𝑛‖‖2+6π›Όπ‘›π‘˜π‘›ξ€·π›½π‘›+𝛿𝑛𝑀2βˆ’2π›Όπ‘›πœ™ξ€·β€–β€–π‘§π‘›βˆ’π‘¦π‘›β€–β€–ξ€Έ+2π›Όπ‘›πœŽπ‘›π‘€+2𝛾𝑛𝑀2β‰€β€–β€–π‘§π‘›βˆ’π‘₯𝑛‖‖2βˆ’2π›Όπ‘›πœ™ξ€·β€–β€–π‘§π‘›βˆ’π‘¦π‘›β€–β€–ξ€Έ+𝛼2𝑛𝑀2+2π›Όπ‘›ξ€·π‘˜π‘›ξ€Έπ‘€βˆ’12+6π›Όπ‘›π‘˜ξ€·π›½π‘›+𝛿𝑛𝑀2+2π›Όπ‘›πœŽπ‘›π‘€+2𝛾𝑛𝑀2=β€–β€–π‘§π‘›βˆ’π‘₯𝑛‖‖2βˆ’π›Όπ‘›πœ™ξ€·β€–β€–π‘§π‘›βˆ’π‘¦π‘›β€–β€–ξ€Έ+2𝛾𝑛𝑀2βˆ’π›Όπ‘›ξ‚ƒπœ™ξ€·β€–β€–π‘§π‘›βˆ’π‘¦π‘›β€–β€–ξ€Έβˆ’π›Όπ‘›π‘€2ξ€·π‘˜βˆ’2π‘›ξ€Έπ‘€βˆ’12βˆ’6π‘˜ξ€·π›½π‘›+𝛿𝑛𝑀2βˆ’2πœŽπ‘›π‘€ξ‚„,(3.8) where π‘˜βˆΆ=sup𝑛(π‘˜π‘›).
Moreover, we observe that β€–β€–π‘§π‘›π‘—βˆ’π‘¦π‘›π‘—β€–β€–β‰€β€–β€–π‘§π‘›π‘—βˆ’π‘₯𝑛𝑗‖‖+β€–β€–π‘¦π‘›π‘—βˆ’π‘₯π‘›π‘—β€–β€–βŸΆ0as.π‘—βŸΆβˆž(3.9) Thus, for every fixed πœ–>0, there exists 𝑗1 such that for all 𝑗>𝑗1β€–β€–π‘§π‘›π‘—βˆ’π‘¦π‘›π‘—β€–β€–β€–β€–π‘§<2πœ–π‘›π‘—βˆ’π‘₯𝑛𝑗‖‖<πœ–.(3.10) Since {𝛼𝑛}𝑛, {(π‘˜π‘›βˆ’1)}𝑛, {(𝛽𝑛+𝛿𝑛)}𝑛, {πœŽπ‘›}𝑛, and {𝛾𝑛}𝑛 are null sequences (and in particular 𝛾𝑛=π‘œ(𝛼𝑛)), for the previous fixed πœ–>0, there exists an index 𝑁 such that, for all π‘›βˆˆπ‘, ||𝛼𝑛||ξ‚»πœ–<min,16π‘€πœ™(πœ–/2)8𝑀2ξ‚Ό,||𝛾𝑛||<πœ–,||||𝛾16𝑀𝑛𝛼𝑛||||<πœ™(πœ–/2)4𝑀2,||π‘˜π‘›||<βˆ’1πœ™(πœ–/2)16𝑀2,||𝛽𝑛+𝛿𝑛||ξ‚»πœ–<min,4π‘€πœ™(πœ–/2)48π‘˜π‘€2ξ‚Ό,||πœŽπ‘›||<πœ™(πœ–/2),16𝑀(3.11) for all 𝑛>𝑁.
Take π‘›βˆ—>max{𝑁,𝑛𝑗1} such that π‘›βˆ—=π‘›π‘˜ for a certain π‘˜.
We prove, by induction, that β€–π‘§π‘›βˆ—+π‘–βˆ’π‘₯π‘›βˆ—+𝑖‖<πœ–, for every π‘–βˆˆβ„•. Let 𝑖=1. Suppose that β€–π‘§π‘›βˆ—+1βˆ’π‘₯π‘›βˆ—+1β€–β‰₯πœ–.
By (3.6), we have β€–β€–π‘§πœ–β‰€π‘›βˆ—+1βˆ’π‘₯π‘›βˆ—+1β€–β€–β‰€β€–β€–π‘§π‘›βˆ—βˆ’π‘¦π‘›βˆ—β€–β€–+ξ€·π›½π‘›βˆ—+π›Ώπ‘›βˆ—ξ€Έξ€·π›Όπ‘€+2π‘›βˆ—+π›Ύπ‘›βˆ—ξ€Έπ‘€β‰€β€–β€–π‘§π‘›βˆ—βˆ’π‘¦π‘›βˆ—β€–β€–πœ–+π‘€ξ‚€πœ–4𝑀+2𝑀+πœ–16𝑀=‖‖𝑧16π‘€π‘›βˆ—βˆ’π‘¦π‘›βˆ—β€–β€–+πœ–4+πœ–4=β€–β€–π‘§π‘›βˆ—βˆ’π‘¦π‘›βˆ—β€–β€–+πœ–2.(3.12) Thus, β€–π‘§π‘›βˆ—βˆ’π‘¦π‘›βˆ—β€–β‰₯πœ–/2. Since πœ™ is strictly increasing, πœ™(β€–π‘§π‘›βˆ—βˆ’π‘¦π‘›βˆ—β€–)β‰₯πœ™(πœ–/2).
From (3.8), we obtain that β€–β€–π‘§π‘›βˆ—+1βˆ’π‘₯π‘›βˆ—+1β€–β€–2<πœ–2βˆ’π›Όβˆ—π‘›ξ‚΅πœ™β€–β€–π‘§π‘›βˆ—βˆ’π‘¦π‘›βˆ—β€–β€–ξ€Έβˆ’2𝑀2π›Ύβˆ—π‘›π›Όβˆ—π‘›ξ‚Άβˆ’π›Όβˆ—π‘›ξ‚ƒπœ™ξ€·β€–β€–π‘§π‘›βˆ—βˆ’π‘¦π‘›βˆ—β€–β€–ξ€Έβˆ’π›Όβˆ—π‘›π‘€2ξ€·π‘˜βˆ’2βˆ—π‘›ξ€Έπ‘€βˆ’12βˆ’6π‘˜ξ€·π›½βˆ—π‘›+π›Ώβˆ—π‘›ξ€Έπ‘€2βˆ’2πœŽπ‘›βˆ—π‘€ξ‚„.(3.13) One can note that π›Όβˆ—π‘›π‘€2ξ€·π‘˜+2βˆ—π‘›ξ€Έπ‘€βˆ’12+6π‘˜ξ€·π›½βˆ—π‘›+π›Ώβˆ—π‘›ξ€Έπ‘€2+2πœŽπ‘›βˆ—π‘€β‰€πœ™(πœ–/2)8+πœ™(πœ–/2)8+πœ™(πœ–/2)8+πœ™(πœ–/2)8.(3.14) hence πœ™ξ€·β€–β€–π‘§π‘›βˆ—βˆ’π‘¦π‘›βˆ—β€–β€–ξ€Έβˆ’π›Όβˆ—π‘›π‘€2ξ€·π‘˜βˆ’2π‘›βˆ—ξ€Έπ‘€βˆ’12βˆ’6π‘˜ξ€·π›½π‘›βˆ—+π›Ώπ‘›βˆ—ξ€Έπ‘€2βˆ’2πœŽπ‘›βˆ—ξ‚€πœ–π‘€β‰₯πœ™2ξ‚βˆ’πœ™(πœ–/2)2.>0(3.15) In the same manner, πœ™ξ€·β€–β€–π‘§π‘›βˆ—βˆ’π‘¦π‘›βˆ—β€–β€–ξ€Έβˆ’2𝑀2π›Ύπ‘›βˆ—π›Όπ‘›βˆ—>πœ™(πœ–/2)βˆ’πœ™(πœ–/2)2.>0(3.16) Thus, β€–β€–π‘§π‘›βˆ—+1βˆ’π‘₯π‘›βˆ—+1β€–β€–2<πœ–2.(3.17) So we have β€–π‘§π‘›βˆ—+1βˆ’π‘₯π‘›βˆ—+1β€–<πœ–, which contradicts β€–π‘§π‘›βˆ—+1βˆ’π‘₯π‘›βˆ—+1β€–β‰₯πœ–. By the same idea, we can prove that β€–π‘§π‘›βˆ—+2βˆ’π‘₯π‘›βˆ—+2β€–<πœ– and then, by inductive step, β€–π‘§π‘›βˆ—+π‘–βˆ’π‘₯π‘›βˆ—+π‘–β€–β‰€πœ–, for all 𝑖. This is enough to ensure that β€–π‘§π‘›βˆ’π‘₯𝑛‖→0.
If there are only finite indices for which ‖𝑧𝑛+1βˆ’π‘₯𝑛+1β€–βˆ’π‘’π‘›β‰€0, then definitively ‖𝑧𝑛+1βˆ’π‘₯𝑛+1β€–βˆ’π‘’π‘›β‰₯0. By the strict increasing function πœ™, we have definitively πœ™ξ€·β€–β€–π‘§π‘›βˆ’π‘¦π‘›β€–β€–ξ€Έξ€·β€–β€–π‘§β‰₯πœ™π‘›+1βˆ’π‘₯𝑛+1β€–β€–βˆ’π‘’π‘›ξ€Έ.(3.18) Again substituting (3.4) and (3.18) into (3.2) and simplifying, we have ‖‖𝑧𝑛+1βˆ’π‘₯𝑛+1β€–β€–2β‰€β€–β€–π‘§π‘›βˆ’π‘₯𝑛‖‖2+𝛼2π‘›β€–β€–π‘§π‘›βˆ’π‘₯𝑛‖‖2+2π›Όπ‘›ξ€·π‘˜π‘›ξ€Έβ€–β€–π‘§βˆ’1π‘›βˆ’π‘₯𝑛‖‖2+6π›Όπ‘›π‘˜π‘›ξ€·π›½π‘›+𝛿𝑛𝑀2βˆ’2π›Όπ‘›πœ™ξ€·β€–β€–π‘§π‘›+1βˆ’π‘₯𝑛+1β€–β€–βˆ’π‘’π‘›ξ€Έ+2π›Όπ‘›πœŽπ‘›π‘€+2𝛾𝑛𝑀2β‰€β€–β€–π‘§π‘›βˆ’π‘₯𝑛‖‖2βˆ’2π›Όπ‘›πœ™ξ€·β€–β€–π‘§π‘›+1βˆ’π‘₯𝑛+1β€–β€–βˆ’π‘’π‘›ξ€Έ+𝛼2𝑛𝑀2+2π›Όπ‘›ξ€·π‘˜π‘›ξ€Έπ‘€βˆ’12+6π›Όπ‘›π‘˜π‘›ξ€·π›½π‘›+𝛿𝑛𝑀2+2π›Όπ‘›πœŽπ‘›π‘€+2𝛾𝑛𝑀2.(3.19) Suppose that π‘Žπ‘›=β€–π‘§π‘›βˆ’π‘₯𝑛‖,𝑏𝑛=𝛼𝑛,and𝑐𝑛=𝛼2𝑛𝑀2+2𝛼𝑛(π‘˜π‘›βˆ’1)𝑀2+6π›Όπ‘›π‘˜π‘›(𝛽𝑛+𝛿𝑛)𝑀2+2π›Όπ‘›πœŽπ‘›π‘€+2𝛾𝑛𝑀2. It follows from limπ‘›β†’βˆžπœŽπ‘›=0, limπ‘›β†’βˆžπ‘’π‘›=0, limπ‘›β†’βˆžπ‘˜π‘›=1, and the hypothesis that we have, βˆ‘βˆžπ‘›=1𝑏𝑛=∞ and 𝑐𝑛=π‘œ(𝑏𝑛), 𝑒𝑛→0 as π‘›β†’βˆž. By virtue of Lemma 2.3, we obtain that limπ‘›β†’βˆžπ‘Žπ‘›=0. Hence, limπ‘›β†’βˆžβ€–π‘§π‘›βˆ’π‘₯𝑛‖=0.

Theorem 3.2. Let 𝑋 be a uniformly smooth Banach space, and let π‘‡βˆΆπ‘‹β†’π‘‹ be generalized strongly asymptotically πœ™-pseudocontractive mapping with fixed point π‘₯βˆ— and bounded range.
Let {𝑧𝑛} and {π‘§ξ…žπ‘›} be the sequences defined by (1.15) and (1.16), respectively, where {𝛼𝑛},{𝛾𝑛}βŠ‚[0,1] are null sequences satisfying (H1)limπ‘›β†’βˆžπ›Όπ‘›=0 and 𝛾𝑛=π‘œ(𝛼𝑛), (H2)βˆ‘βˆžπ‘›=1𝛼𝑛=∞, of Theorem 3.1 and such that π›Όπ‘›π‘˜π‘›<1, for every π‘›βˆˆβ„•.
Suppose moreover that the sequences {𝑀𝑛},{π‘€ξ…žπ‘›} are bounded in 𝑋, then for any initial point π‘§ξ…ž0,𝑧0βˆˆπ‘‹, the following two assertions are equivalent:
(i)the modified Mann iteration sequence with errors (1.15) converges to the fixed point π‘₯βˆ—, (ii)the implicit iteration sequence with errors (1.16) converges to the fixed point π‘₯βˆ—.

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=supπ‘›βŽ§βŽͺ⎨βŽͺβŽ©β€–β€–π‘‡π‘›π‘§π‘›βˆ’π‘‡π‘›π‘§ξ…žπ‘›β€–β€–,β€–β€–π‘‡π‘›π‘§π‘›βˆ’π‘§π‘›β€–β€–,β€–β€–π‘‡π‘›π‘§ξ…žπ‘›βˆ’π‘§ξ…žπ‘›β€–β€–,β€–β€–π‘‡π‘›π‘§ξ…žπ‘›βˆ’π‘§ξ…žπ‘›βˆ’1‖‖‖‖𝑧𝑛+1βˆ’π‘§ξ…žπ‘›β€–β€–,β€–β€–π‘§π‘›βˆ’π‘€π‘›β€–β€–,β€–β€–π‘€ξ…žπ‘›βˆ’π‘§ξ…žπ‘›βˆ’1β€–β€–,β€–β€–π‘€π‘›βˆ’π‘€ξ…žπ‘›β€–β€–βŽ«βŽͺ⎬βŽͺ⎭.(3.20) By the iteration schemes (1.15) and (1.16), we have ‖‖𝑧𝑛+1βˆ’π‘§ξ…žπ‘›β€–β€–2≀‖‖1βˆ’π›Όπ‘›βˆ’π›Ύπ‘›π‘§ξ€Έξ€·π‘›βˆ’π‘§ξ…žπ‘›βˆ’1ξ€Έ+π›Όπ‘›ξ€·π‘‡π‘›π‘§π‘›βˆ’π‘‡π‘›π‘§ξ…žπ‘›ξ€Έ+π›Ύπ‘›ξ€·π‘€π‘›βˆ’π‘€ξ…žπ‘›ξ€Έβ€–β€–2≀1βˆ’π›Όπ‘›βˆ’π›Ύπ‘›ξ€Έ2β€–β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›βˆ’1β€–β€–2𝛼+2π‘›ξ€·π‘‡π‘›π‘§π‘›βˆ’π‘‡π‘›π‘§ξ…žπ‘›ξ€Έ+π›Ύπ‘›ξ€·π‘€π‘›βˆ’π‘€ξ…žπ‘›ξ€Έξ€·π‘§,𝑗𝑛+1βˆ’π‘§ξ…žπ‘›β‰€ξ€·ξ€Έξ¬1βˆ’π›Όπ‘›ξ€Έ2β€–β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›βˆ’1β€–β€–2+2π›Όπ‘›ξ«π‘‡π‘›π‘§π‘›βˆ’π‘‡π‘›π‘§ξ…žπ‘›ξ€·π‘§,π‘—π‘›βˆ’π‘§ξ…žπ‘›ξ€Έξ¬+2π›Όπ‘›ξ«π‘‡π‘›π‘§π‘›βˆ’π‘‡π‘›π‘§ξ…žπ‘›ξ€·π‘§,𝑗𝑛+1βˆ’π‘§ξ…žπ‘›ξ€Έξ€·π‘§βˆ’π‘—π‘›βˆ’π‘§ξ…žπ‘›ξ€Έξ¬+2π›Ύπ‘›β€–β€–π‘€π‘›βˆ’π‘€ξ…žπ‘›β€–β€–β€–β€–π‘§π‘›+1βˆ’π‘§ξ…žπ‘›β€–β€–β‰€ξ€·1βˆ’π›Όπ‘›ξ€Έ2β€–β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›βˆ’1β€–β€–2+2π›Όπ‘›π‘˜π‘›β€–β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›β€–β€–2βˆ’2π›Όπ‘›πœ™ξ€·β€–β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›β€–β€–ξ€Έ+2π›Όπ‘›π‘€πœŽπ‘›+2𝛾𝑛𝑀2,(3.21) where πœŽπ‘›=‖𝑗(𝑧𝑛+1βˆ’π‘§ξ…žπ‘›)βˆ’π‘—(π‘§π‘›βˆ’π‘§ξ…žπ‘›)β€–. By (1.15), we get ‖‖𝑧𝑛+1βˆ’π‘§ξ…žπ‘›ξ€Έβˆ’ξ€·π‘§π‘›βˆ’π‘§ξ…žπ‘›ξ€Έβ€–β€–=‖‖𝑧𝑛+1βˆ’π‘§π‘›β€–β€–=β€–β€–π›Όπ‘›ξ€·π‘‡π‘›π‘§π‘›βˆ’π‘§π‘›ξ€Έ+π›Ύπ‘›ξ€·π‘€π‘›βˆ’π‘§π‘›ξ€Έβ€–β€–β‰€ξ€·π›Όπ‘›+𝛾𝑛𝑀.(3.22) It follows from (H1) that β€–(𝑧𝑛+1βˆ’π‘§ξ…žπ‘›)βˆ’(π‘§π‘›βˆ’π‘§ξ…žπ‘›)β€–β†’0 as π‘›β†’βˆž, which implies that πœŽπ‘›β†’0 as π‘›β†’βˆž. Moreover, for all 𝑛β‰₯0, β€–β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›β€–β€–2β‰€ξ€·β€–β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›βˆ’1β€–β€–+β€–β€–π‘§ξ…žπ‘›βˆ’1βˆ’π‘§ξ…žπ‘›β€–β€–ξ€Έ2β‰€ξ€·β€–β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›βˆ’1β€–β€–+π›Όπ‘›β€–β€–π‘‡π‘›π‘§ξ…žπ‘›βˆ’π‘§ξ…žπ‘›βˆ’1β€–β€–+π›Ύπ‘›β€–β€–π‘€ξ…žπ‘›βˆ’π‘§ξ…žπ‘›βˆ’1β€–β€–ξ€Έ2β‰€ξ€Ίβ€–β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›βˆ’1β€–β€–+𝛼𝑛+𝛾𝑛𝑀2=β€–β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›βˆ’1β€–β€–2+𝛼𝑛+𝛾𝑛‖‖𝑧2π‘€π‘›βˆ’π‘§ξ…žπ‘›βˆ’1β€–β€–+𝛼𝑛+𝛾𝑛𝑀2ξ€»β‰€β€–β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›βˆ’1β€–β€–2𝛼+3𝑛+𝛾𝑛𝑀2.(3.23) Again by the boundedness of all components, we have that β€–β€–π‘§ξ…žπ‘›βˆ’π‘§ξ…žπ‘›βˆ’1β€–β€–=β€–β€–π›Όπ‘›ξ€·π‘‡π‘›π‘§ξ…žπ‘›βˆ’π‘§ξ…žπ‘›βˆ’1ξ€Έ+π›Ύπ‘›ξ€·π‘€ξ…žπ‘›βˆ’π‘§ξ…žπ‘›βˆ’1‖‖≀𝛼𝑛+𝛾𝑛𝑀,(3.24) and so ‖‖𝑧𝑛+1βˆ’π‘§ξ…žπ‘›β€–β€–=β€–β€–ξ€·π‘§π‘›βˆ’π‘§ξ…žπ‘›ξ€Έ+ξ€·π‘§ξ…žπ‘›βˆ’π‘§ξ…žπ‘›βˆ’1ξ€Έβˆ’ξ€·π›Όπ‘›+π›Ύπ‘›π‘§ξ€Έξ€·π‘›βˆ’π‘§ξ…žπ‘›βˆ’1ξ€Έ+π›Όπ‘›ξ€·π‘‡π‘›π‘§π‘›βˆ’π‘‡π‘›π‘§ξ…žπ‘›ξ€Έ+π›Ύπ‘›ξ€·π‘€π‘›βˆ’π‘€ξ…žπ‘›ξ€Έβ€–β€–β‰€β€–β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›β€–β€–+β€–β€–π‘§ξ…žπ‘›βˆ’π‘§ξ…žπ‘›βˆ’1β€–β€–+𝛼𝑛+π›Ύπ‘›ξ€Έβ€–β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›βˆ’1β€–β€–+π›Όπ‘›β€–β€–π‘‡π‘›π‘§π‘›βˆ’π‘‡π‘›π‘§ξ…žπ‘›β€–β€–+π›Ύπ‘›β€–β€–π‘€π‘›βˆ’π‘€ξ…žπ‘›β€–β€–β‰€β€–β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›β€–β€–ξ€·π›Ό+3𝑛+𝛾𝑛𝑀.(3.25) Hence, we have that β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›β€–β‰₯‖𝑧𝑛+1βˆ’π‘§ξ…žπ‘›β€–βˆ’π‘’π‘›, where 𝑒𝑛=3(𝛼𝑛+𝛾𝑛)𝑀. Note that 𝑒𝑛→0 as π‘›β†’βˆž. As in proof of Theorem 3.1, we distinguish two cases:(i)the set of indices for which ‖𝑧𝑛+1βˆ’π‘§ξ…žπ‘›β€–βˆ’π‘’π‘›β‰€0 contains infinite terms; (ii)the set of indices for which ‖𝑧𝑛+1βˆ’π‘§ξ…žπ‘›β€–βˆ’π‘’π‘›β‰€0 contains finite terms. In the first case, (i) we can extract a subsequence such that β€–π‘§π‘›π‘˜βˆ’π‘§ξ…žπ‘›π‘˜βˆ’1β€–β†’0, as π‘˜β†’βˆž. Substituting (3.23) in (3.21), we have that ‖‖𝑧𝑛+1βˆ’π‘§ξ…žπ‘›β€–β€–2≀1+𝛼2π‘›βˆ’2π›Όπ‘›ξ€Έβ€–β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›βˆ’1β€–β€–2+2π›Όπ‘›π‘˜π‘›β€–β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›βˆ’1β€–β€–2𝛼+6𝑛+𝛾𝑛𝑀2π›Όπ‘›π‘˜π‘›βˆ’2π›Όπ‘›πœ™ξ€·β€–β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›β€–β€–ξ€Έ+2π›Όπ‘›π‘€πœŽπ‘›+2𝛾𝑛𝑀2β‰€β€–β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›βˆ’1β€–β€–2+𝛼2𝑛+2π›Όπ‘›ξ€·π‘˜π‘›β€–β€–π‘§βˆ’1ξ€Έξ€Έπ‘›βˆ’π‘§ξ…žπ‘›βˆ’1β€–β€–2𝛼+6𝑛+𝛾𝑛𝑀2π›Όπ‘›π‘˜βˆ’2π›Όπ‘›πœ™ξ€·β€–β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›β€–β€–ξ€Έ+2π›Όπ‘›π‘€πœŽπ‘›+2𝛾𝑛𝑀2β‰€β€–β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›βˆ’1β€–β€–2βˆ’π›Όπ‘›πœ™ξ€·β€–β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›β€–β€–ξ€Έ+2𝛾𝑛𝑀2βˆ’π›Όπ‘›ξ‚ƒπœ™ξ€·β€–β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›β€–β€–ξ€Έβˆ’π›Όπ‘›π‘€2βˆ’2𝑀2ξ€·π‘˜π‘›ξ€Έξ€·π›Όβˆ’1βˆ’6𝑛+𝛾𝑛𝑀2π‘˜βˆ’2π‘€πœŽπ‘›ξ‚„=β€–β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›βˆ’1β€–β€–2βˆ’π›Όπ‘›πœ™ξ€·β€–β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›β€–β€–ξ€Έ+2𝛾𝑛𝑀2βˆ’π›Όπ‘›ξ‚ƒπœ™ξ€·β€–β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›β€–β€–ξ€Έβˆ’7π‘˜π›Όπ‘›π‘€2βˆ’2𝑀2ξ€·π‘˜π‘›ξ€Έβˆ’1βˆ’6𝛾𝑛𝑀2π‘˜βˆ’2π‘€πœŽπ‘›ξ‚„,(3.26) where π‘˜=supπ‘›π‘˜π‘›. Again by (3.23), for every πœ–>0, there exists an index 𝑙 such that if 𝑗>𝑙, β€–β€–π‘§π‘›π‘—βˆ’π‘§ξ…žπ‘›π‘—βˆ’1‖‖‖‖𝑧<πœ–,π‘›π‘—βˆ’π‘§ξ…žπ‘›π‘—β€–β€–<2πœ–.(3.27) By hypotheses on the control sequence, with the same πœ–>0, there exists an index 𝑁 such that definitively ||𝛼𝑛||ξ‚»πœ–<min,12π‘€πœ™(πœ–/2)56𝑀2π‘˜ξ‚Ό,||𝛾𝑛||ξ‚»πœ–<min,12π‘€πœ™(πœ–/2)48𝑀2π‘˜ξ‚Ό,||||𝛾𝑛𝛼𝑛||||<πœ™(πœ–/2)4𝑀2,||π‘˜π‘›||<βˆ’1πœ™(πœ–/2)16𝑀2,||πœŽπ‘›||<πœ™(πœ–/2).16𝑀(3.28) So take π‘›βˆ—>max{𝑛𝑙,𝑁} with π‘›βˆ—=𝑛𝑗 for a certain 𝑗.
We can prove that ‖𝑧𝑛+1βˆ’π‘§ξ…žπ‘›β€–β†’0 as π‘›β†’βˆž proving that, for every 𝑖β‰₯0, the result is β€–π‘§π‘›βˆ—+π‘–βˆ’π‘§ξ…žπ‘›βˆ—+π‘–βˆ’1β€–<πœ–.
Let 𝑖=1. If we suppose that β€–π‘§π‘›βˆ—+1βˆ’π‘§ξ…žπ‘›βˆ—β€–β‰₯πœ–, it results that β€–β€–π‘§πœ–β‰€π‘›βˆ—+1βˆ’π‘§ξ…žπ‘›βˆ—β€–β€–β‰€β€–β€–π‘§π‘›βˆ—βˆ’π‘§ξ…žπ‘›βˆ—β€–β€–ξ€·π›Ό+3π‘›βˆ—+π›Ύπ‘›βˆ—ξ€Έβ€–β€–π‘§π‘€<π‘›βˆ—βˆ’π‘§ξ…žπ‘›βˆ—β€–β€–+πœ–2,(3.29) so β€–π‘§π‘›βˆ—βˆ’π‘§ξ…žπ‘›βˆ—β€–>πœ–/2. In consequence of this, πœ™(β€–π‘§π‘›βˆ—βˆ’π‘§π‘›βˆ—β€²β€–)>πœ™(πœ–/2).
In (3.26), we note that 7π‘˜π›Όπ‘›βˆ—π‘€2+2𝑀2ξ€·π‘˜π‘›βˆ—ξ€Έβˆ’1+6π›Ύπ‘›βˆ—π‘€2π‘˜+2π‘€πœŽπ‘›βˆ—β‰€7π‘˜π‘€2πœ™(πœ–/2)56𝑀2π‘˜+2𝑀2πœ™(πœ–/2)16𝑀2+6𝑀2π‘˜πœ™(πœ–/2)48𝑀2π‘˜+2π‘€πœ™(πœ–/2)=16π‘€πœ™(πœ–/2)84=πœ™(πœ–/2)2,(3.30) so πœ™ξ€·β€–β€–π‘§π‘›βˆ—βˆ’π‘§ξ…žπ‘›βˆ—β€–β€–ξ€Έβˆ’7π‘˜π›Όπ‘›βˆ—π‘€2βˆ’2𝑀2ξ€·π‘˜π‘›βˆ—ξ€Έβˆ’1βˆ’6π›Ύπ‘›βˆ—π‘€2π‘˜βˆ’2π‘€πœŽπ‘›βˆ—β‰₯πœ™(πœ–/2)2.(3.31) hence in (3.26) remains β€–β€–π‘§π‘›βˆ—+1βˆ’π‘§ξ…žπ‘›βˆ—β€–β€–2β‰€πœ–2βˆ’π›Όπ‘›πœ™ξ€·β€–β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›β€–β€–ξ€Έ+2𝛾𝑛𝑀2<πœ–2,(3.32) as in Theorem 3.1. This is a contradiction. By the same idea, and using the inductive hypothesis, we obtain that β€–π‘§π‘›βˆ—+π‘–βˆ’π‘§ξ…žπ‘›βˆ—+π‘–βˆ’1β€–<πœ–, for every 𝑖β‰₯0. This ensures that ‖𝑧𝑛+1βˆ’π‘§ξ…žπ‘›β€–β†’0. In the second case (ii), definitively, ‖𝑧𝑛+1βˆ’π‘§ξ…žπ‘›β€–βˆ’π‘’π‘›β‰₯0, then from the strictly increasing function πœ™, we have πœ™ξ€·β€–β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›β€–β€–ξ€Έξ€·β€–β€–π‘§β‰₯πœ™π‘›+1βˆ’π‘§ξ…žπ‘›β€–β€–βˆ’π‘’π‘›ξ€Έ.(3.33) Substituting (3.33) and (3.23) into (3.21) and simplifying, we have ‖‖𝑧𝑛+1βˆ’π‘§ξ…žπ‘›β€–β€–2β‰€β€–β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›βˆ’1β€–β€–2+𝛼2π‘›β€–β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›βˆ’1β€–β€–2+2π›Όπ‘›ξ€·π‘˜π‘›ξ€Έβ€–β€–π‘§βˆ’1π‘›βˆ’π‘§ξ…žπ‘›βˆ’1β€–β€–2+6π›Όπ‘›π‘˜π‘›ξ€·π›Όπ‘›+𝛾𝑛𝑀2βˆ’2π›Όπ‘›πœ™ξ€·β€–β€–π‘§π‘›+1βˆ’π‘§ξ…žπ‘›β€–β€–βˆ’π‘’π‘›ξ€Έ+2π›Όπ‘›πœŽπ‘›π‘€+2𝛾𝑛𝑀2β‰€β€–β€–π‘§π‘›βˆ’π‘§ξ…žπ‘›βˆ’1β€–β€–2βˆ’2π›Όπ‘›πœ™ξ€·β€–β€–π‘§π‘›+1βˆ’π‘§ξ…žπ‘›β€–β€–βˆ’π‘’π‘›ξ€Έ+𝛼2𝑛𝑀2+2π›Όπ‘›ξ€·π‘˜π‘›ξ€Έπ‘€βˆ’12+6π›Όπ‘›π‘˜π‘›ξ€·π›Όπ‘›+𝛾𝑛𝑀2+2π›Όπ‘›πœŽπ‘›π‘€+2𝛾𝑛𝑀2.(3.34) By virtue of Lemma 2.3, we obtain that limπ‘›β†’βˆžβ€–π‘§π‘›βˆ’π‘§ξ…žπ‘›βˆ’1β€–=0.

Theorem 3.3. Let 𝑋 be a uniformly smooth Banach space, and let π‘‡βˆΆπ‘‹β†’π‘‹ be generalized strongly asymptotically πœ™-pseudocontractive mapping with fixed point π‘₯βˆ— and bounded range.
Let {𝑧𝑛}𝑛 be the sequences defined by (1.15) where {𝛼𝑛}𝑛,{𝛾𝑛}π‘›βŠ‚[0,1] satisfy(i)limπ‘›β†’βˆžπ›Όπ‘›=limπ‘›β†’βˆžπ›Ύπ‘›=0, (ii)βˆ‘βˆžπ‘›=1𝛼𝑛=∞, βˆ‘βˆžπ‘›=1𝛾𝑛<∞.and the sequence {𝑀𝑛}𝑛 is bounded on 𝑋, then for any initial point 𝑧0βˆˆπ‘‹, the sequence {𝑧𝑛}𝑛 strongly converges to π‘₯βˆ—.

Proof. Firstly, we observe that, by the boundedness of the range of 𝑇, of the sequence {𝑀𝑛}𝑛, and by Lemma 2.4, we have that {𝑧𝑛}𝑛 is bounded.
By Lemma 2.2, we observe that ‖‖𝑧𝑛+1βˆ’π‘₯βˆ—β€–β€–2≀1βˆ’π›Όπ‘›βˆ’π›Ύπ‘›ξ€Έ2β€–β€–π‘§π‘›βˆ’π‘₯βˆ—β€–β€–2+2π›Όπ‘›ξ«π‘‡π‘›π‘§π‘›βˆ’π‘₯βˆ—ξ€·π‘§,𝑗𝑛+1βˆ’π‘₯βˆ—ξ¬+2π›Ύπ‘›ξ«π‘€π‘›βˆ’π‘₯βˆ—ξ€·π‘§,𝑗𝑛+1βˆ’π‘₯βˆ—β‰€ξ€·ξ€Έξ¬1βˆ’π›Όπ‘›ξ€Έ2β€–β€–π‘§π‘›βˆ’π‘₯βˆ—β€–β€–2+2π›Όπ‘›βŸ¨π‘‡π‘›π‘§π‘›βˆ’π‘₯βˆ—ξ€·π‘§,𝑗𝑛+1βˆ’π‘₯βˆ—ξ€Έξ€·π‘§βˆ’π‘—π‘›βˆ’π‘₯βˆ—ξ€ΈβŸ©+2π›Όπ‘›ξ«π‘‡π‘›π‘§π‘›βˆ’π‘₯βˆ—ξ€·π‘§,π‘—π‘›βˆ’π‘₯βˆ—ξ€Έξ¬+2π›Ύπ‘›ξ«π‘€π‘›βˆ’π‘₯βˆ—ξ€·π‘§,𝑗𝑛+1βˆ’π‘₯βˆ—β‰€ξ€·ξ€Έξ¬1βˆ’π›Όπ‘›ξ€Έ2β€–β€–π‘§π‘›βˆ’π‘₯βˆ—β€–β€–2+2π›Όπ‘›ξ«π‘‡π‘›π‘›π‘§π‘›βˆ’π‘₯βˆ—ξ€·π‘§,𝑗𝑛+1βˆ’π‘₯βˆ—ξ€Έξ€·π‘§βˆ’π‘—π‘›βˆ’π‘₯βˆ—ξ€Έξ¬+2π›Όπ‘›π‘˜π‘›β€–β€–π‘§π‘›βˆ’π‘₯βˆ—β€–β€–2βˆ’2π›Όπ‘›πœ™ξ€·β€–β€–π‘§π‘›βˆ’π‘₯βˆ—β€–β€–ξ€Έ+2π›Ύπ‘›ξ«π‘€π‘›βˆ’π‘₯βˆ—ξ€·π‘§,𝑗𝑛+1βˆ’π‘₯βˆ—=1+𝛼2π‘›βˆ’2π›Όπ‘›ξ€Έβ€–β€–π‘§π‘›βˆ’π‘₯βˆ—β€–β€–2+2π›Όπ‘›ξ«π‘‡π‘›π‘§π‘›βˆ’π‘₯βˆ—ξ€·π‘§,𝑗𝑛+1βˆ’π‘₯βˆ—ξ€Έξ€·π‘§βˆ’π‘—π‘›βˆ’π‘₯βˆ—ξ€Έξ¬+2π›Όπ‘›π‘˜π‘›β€–β€–π‘§π‘›βˆ’π‘₯βˆ—β€–β€–2βˆ’2π›Όπ‘›πœ™ξ€·β€–β€–π‘§π‘›βˆ’π‘₯βˆ—β€–β€–ξ€Έ+2π›Ύπ‘›βŸ¨π‘€π‘›βˆ’π‘₯βˆ—ξ€·π‘§,𝑗𝑛+1βˆ’π‘₯βˆ—ξ€ΈβŸ©=β€–β€–π‘§π‘›βˆ’π‘₯βˆ—β€–β€–2+𝛼2π‘›βˆ’2𝛼𝑛+2π›Όπ‘›π‘˜π‘›ξ€Έβ€–β€–π‘§π‘›βˆ’π‘₯βˆ—β€–β€–2βˆ’2π›Όπ‘›πœ™ξ€·β€–β€–π‘§π‘›βˆ’π‘₯βˆ—β€–β€–ξ€Έ+2π›Όπ‘›πœ‡π‘›+2π›Ύπ‘›ξ«π‘€π‘›βˆ’π‘₯βˆ—ξ€·π‘§,𝑗𝑛+1βˆ’π‘₯βˆ—,(3.35) where πœ‡π‘›βˆΆ=βŸ¨π‘‡π‘›π‘§π‘›βˆ’π‘₯βˆ—,𝑗(𝑧𝑛+1βˆ’π‘₯βˆ—)βˆ’π‘—(π‘§π‘›βˆ’π‘₯βˆ—)⟩. Let ξ‚»π‘€βˆΆ=maxsupπ‘›β€–β€–π‘§π‘›βˆ’π‘₯βˆ—β€–β€–,supπ‘›β€–β€–π‘‡π‘›π‘§π‘›βˆ’π‘₯βˆ—β€–β€–,supπ‘›β€–β€–π‘€π‘›βˆ’π‘₯βˆ—β€–β€–,supπ‘›ξ«π‘€π‘›βˆ’π‘₯βˆ—ξ€·π‘§,𝑗𝑛+1βˆ’π‘₯βˆ—ξ‚Ό.(3.36) We have ‖‖𝑧𝑛+1βˆ’π‘₯βˆ—β€–β€–2β‰€β€–β€–π‘§π‘›βˆ’π‘₯βˆ—β€–β€–2+𝛼2𝑛+2π›Όπ‘›ξ€·π‘˜π‘›βˆ’1ξ€Έξ€Έπ‘€βˆ’2π›Όπ‘›πœ™ξ€·β€–β€–π‘§π‘›βˆ’π‘₯βˆ—β€–β€–ξ€Έ+2π›Όπ‘›πœ‡π‘›+2𝛾𝑛𝑀=β€–β€–π‘§π‘›βˆ’π‘₯βˆ—β€–β€–2βˆ’π›Όπ‘›πœ™ξ€·β€–β€–π‘§π‘›βˆ’π‘₯βˆ—β€–β€–ξ€Έ+2π›Ύπ‘›π‘€βˆ’π›Όπ‘›ξ€Ίπœ™ξ€·β€–β€–π‘§π‘›βˆ’π‘₯βˆ—β€–β€–ξ€Έβˆ’2πœ‡π‘›βˆ’ξ€·π›Όπ‘›ξ€·π‘˜+2𝑛𝑀,βˆ’1ξ€Έξ€Έ(3.37) so we can observe that(1)πœ‡π‘›β†’0 as π‘›β†’βˆž. Indeed from the inequality ||‖‖𝑧𝑛+1βˆ’π‘₯βˆ—β€–β€–βˆ’β€–β€–π‘§π‘›βˆ’π‘₯βˆ—β€–β€–||≀‖‖𝑧𝑛+1βˆ’π‘§π‘›β€–β€–β‰€ξ€·π›Όπ‘›+π›Ύπ‘›ξ€Έπ‘€βŸΆ0,as,π‘›βŸΆβˆž(3.38) and since 𝑗 is norm to norm uniformly continuous, then 𝑗(‖𝑧𝑛+1βˆ’π‘₯βˆ—β€–)βˆ’π‘—(β€–π‘§π‘›βˆ’π‘₯βˆ—β€–)β†’0, as π‘›β†’βˆž, (2)inf𝑛(β€–π‘§π‘›βˆ’π‘₯βˆ—β€–)=0. Indeed, if we supposed that 𝜎∢=inf𝑛(β€–π‘§π‘›βˆ’π‘₯βˆ—β€–)>0, by the monotonicity of πœ™, πœ™ξ€·β€–β€–π‘§π‘›βˆ’π‘₯βˆ—β€–β€–ξ€Έβ‰₯πœ™(𝜎)>0.(3.39) Thus, by (1) and by the hypotheses on 𝛼𝑛 and π‘˜π‘›, the value βˆ’π›Όπ‘›[πœ™(β€–π‘§π‘›βˆ’π‘₯βˆ—β€–)βˆ’2πœ‡π‘›βˆ’(𝛼𝑛+2(π‘˜π‘›βˆ’1))𝑀] is definitively negative. In this case, we conclude that there exists 𝑁>0 such that for every 𝑛>𝑁, ‖‖𝑧𝑛+1βˆ’π‘₯βˆ—β€–β€–2β‰€β€–β€–π‘§π‘›βˆ’π‘₯βˆ—β€–β€–2βˆ’π›Όπ‘›πœ™ξ€·β€–β€–π‘§π‘›βˆ’π‘₯βˆ—β€–β€–ξ€Έ+2π›Ύπ‘›π‘€β‰€β€–β€–π‘§π‘›βˆ’π‘₯βˆ—β€–β€–2βˆ’π›Όπ‘›πœ™(𝜎)+2𝛾𝑛𝑀,(3.40) and so π›Όπ‘›β€–β€–π‘§πœ™(𝜎)β‰€π‘›βˆ’π‘₯βˆ—β€–β€–2βˆ’β€–β€–π‘§π‘›+1βˆ’π‘₯βˆ—β€–β€–2+2π›Ύπ‘›π‘€βˆ€π‘›>𝑁.(3.41) In the same way we obtain that πœ™(𝜎)π‘šξ“π‘–=π‘π›Όπ‘–β‰€π‘šξ“π‘–=π‘ξ‚ƒβ€–β€–π‘§π‘–βˆ’π‘₯βˆ—β€–β€–2βˆ’β€–β€–π‘§π‘–+1βˆ’π‘₯βˆ—β€–β€–2ξ‚„+2π‘šξ“π‘–=𝑁𝛾𝑖𝑀=β€–β€–π‘§π‘βˆ’π‘₯βˆ—β€–β€–2βˆ’β€–β€–π‘§π‘šβˆ’π‘₯βˆ—β€–β€–2+2π‘€π‘šξ“π‘–=𝑁𝛾𝑖.(3.42) By the hypotheses βˆ‘π‘›π›Ύπ‘›<∞ and βˆ‘π‘›π›Όπ‘›=∞, the previous is a contradiction, and it follows that inf𝑛(β€–π‘§π‘›βˆ’π‘₯βˆ—β€–)=0.Then, there exists a subsequence {π‘§π‘›π‘˜}π‘˜ of {𝑧𝑛}𝑛 that strongly converges to π‘₯βˆ—. This implies that for every πœ–>0, there exists an index π‘›π‘˜(πœ–) such that, for all 𝑗β‰₯π‘›π‘˜(πœ–), β€–π‘§π‘›π‘—βˆ’π‘₯βˆ—β€–<πœ–.
Now, we will prove that the sequence {𝑧𝑛}𝑛 converges to π‘₯βˆ—. Since the sequences in (3.37) are null sequences and βˆ‘π‘›π›Ύπ‘›<∞, but βˆ‘π‘›π›Όπ‘›=∞, then, for every πœ–>0, there exists an index 𝑛(πœ–) such that for all 𝑛β‰₯𝑛(πœ–), it results that ||𝛼𝑛||<1ξ‚†ξ‚€πœ–4𝑀minπœ–,πœ™2,||𝛾𝑛||<πœ–,||||𝛾4𝑀𝑛𝛼𝑛||||<πœ™(πœ–/2),||π‘˜4𝑀𝑛||<πœ™βˆ’1(πœ–/2),||πœ‡8𝑀𝑛||<πœ™(πœ–/2)8.(3.43) So, fixing πœ–>0, let π‘›βˆ—>max(π‘›π‘˜(πœ–),𝑛(πœ–)) with π‘›βˆ—=𝑛𝑗 for a certain 𝑛𝑗. We will prove, by induction, that β€–π‘§π‘›βˆ—+π‘–βˆ’π‘₯βˆ—β€–<πœ– for every π‘–βˆˆβ„•. Let 𝑖=1. If not, it results that β€–π‘§π‘›βˆ—+1βˆ’π‘₯βˆ—β€–β‰₯πœ–. Thus, β€–β€–π‘§πœ–β‰€π‘›βˆ—+1βˆ’π‘₯βˆ—β€–β€–β‰€β€–β€–π‘§π‘›βˆ—βˆ’π‘₯βˆ—β€–β€–+π›Όπ‘›βˆ—π‘€+π›Ύπ‘›βˆ—<‖‖𝑧𝑀,π‘›βˆ—βˆ’π‘₯βˆ—β€–β€–+πœ–πœ–4𝑀𝑀+‖‖𝑧4𝑀𝑀=π‘›βˆ—βˆ’π‘₯βˆ—β€–β€–+πœ–2,(3.44) that is, β€–π‘§π‘›βˆ—βˆ’π‘₯βˆ—β€–>πœ–/2. By the strict increasing of πœ™, πœ™(β€–π‘§π‘›βˆ—βˆ’π‘₯βˆ—β€–)>πœ™(πœ–/2).
By (3.37), it results that β€–β€–π‘§π‘›βˆ—+1βˆ’π‘₯βˆ—β€–β€–2<πœ–2βˆ’π›Όπ‘›βˆ—ξ‚΅πœ™ξ€·β€–β€–π‘§π‘›βˆ—βˆ’π‘₯βˆ—β€–β€–ξ€Έπ›Ύβˆ’2π‘€π‘›βˆ—π›Όπ‘›βˆ—ξ‚Άβˆ’π›Όπ‘›βˆ—ξ€Ίπœ™ξ€·β€–β€–π‘§π‘›βˆ—βˆ’π‘₯βˆ—β€–β€–ξ€Έβˆ’2πœ‡π‘›βˆ—βˆ’ξ€·π›Όπ‘›βˆ—ξ€·π‘˜+2π‘›βˆ—π‘€ξ€».βˆ’1ξ€Έξ€Έ(3.45) We can note that 2πœ‡π‘›βˆ—+ξ€·π›Όπ‘›βˆ—ξ€·π‘˜+2π‘›βˆ—βˆ’1ξ€Έξ€Έπ‘€β‰€πœ™(πœ–/2)4+ξ‚΅πœ™(πœ–/2)+4π‘€πœ™(πœ–/2)ξ‚Ά4𝑀𝑀,(3.46) so πœ™ξ€·β€–β€–π‘§π‘›βˆ—βˆ’π‘₯βˆ—β€–β€–ξ€Έβˆ’2πœ‡π‘›βˆ—βˆ’ξ€·π›Όπ‘›βˆ—ξ€·π‘˜+2π‘›βˆ—βˆ’1𝑀>πœ™(πœ–/2)βˆ’3πœ™(πœ–/2)4>0.(3.47) Moreover, πœ™(β€–π‘§π‘›βˆ—βˆ’π‘₯βˆ—β€–)βˆ’2π‘€π›Ύπ‘›βˆ—/π›Όπ‘›βˆ—>πœ™(πœ–/2)/2>0, so it results that β€–β€–π‘§π‘›βˆ—+1βˆ’π‘₯βˆ—β€–β€–2<πœ–2.(3.48) This is a contradiction. Thus, β€–π‘§π‘›βˆ—+1βˆ’π‘₯βˆ—β€–<πœ–.
In the same manner, by induction, one obtains that, for every 𝑖β‰₯1, β€–π‘§π‘›βˆ—+π‘–βˆ’π‘₯βˆ—β€–<πœ–. So β€–π‘§π‘›βˆ’π‘₯βˆ—β€–β†’0.

Corollary 3.4. Let 𝑋 be a uniformly smooth Banach space, and let π‘‡βˆΆπ‘‹β†’π‘‹ be generalized strongly asymptotically πœ™-pseudocontractive mapping with bounded range and fixed point π‘₯βˆ—. The sequences {π‘₯𝑛}𝑛, {𝑧𝑛}𝑛, and {π‘§ξ…žπ‘›}𝑛 are defined by (1.14), (1.15), and (1.16), respectively, where the sequences {𝛼𝑛}𝑛, {𝛽𝑛}𝑛, {𝛾𝑛}𝑛, {𝛿𝑛}π‘›βŠ‚[0,1] satisfy(i)limπ‘›β†’βˆžπ›Όπ‘›=limπ‘›β†’βˆžπ›½π‘›=limπ‘›β†’βˆžπ›Ώπ‘›=0,(ii)βˆ‘βˆžπ‘›=1π›Όπ‘›βˆ‘=∞,βˆžπ‘›=1𝛾𝑛<∞,and the sequences {𝑒𝑛}𝑛, {𝑣𝑛}𝑛, {𝑀𝑛}𝑛, and {𝑀𝑛′}𝑛 are bounded in X. Then for any initial point π‘₯0,𝑧0,𝑧0β€²βˆˆπ‘‹, the following two assertions are equivalent and true: (i)the modified Ishikawa iteration sequence with errors (1.14) converges to the fixed point π‘₯βˆ—;(ii)the modified Mann iteration sequence with errors (1.15) converges to the fixed point π‘₯βˆ—;(iii)the implicit iteration sequence with errors (1.16) converges to the fixed point π‘₯βˆ—.

References

  1. F. E. Browder, β€œNonlinear mappings of nonexpansive and accretive type in Banach spaces,” Bulletin of the American Mathematical Society, vol. 73, pp. 875–882, 1967. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  2. T. Kato, β€œNonlinear semigroups and evolution equations,” Journal of the Mathematical Society of Japan, vol. 19, pp. 508–520, 1967. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  3. C. Chidume, Geometric properties of Banach spaces and nonlinear iterations, vol. 1965 of Lecture Notes in Mathematics, Springer, London, UK, 2009.
  4. K. Goebel and W. A. Kirk, β€œA fixed point theorem for asymptotically nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 35, pp. 171–174, 1972. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  5. K. Deimling, β€œZeros of accretive operators,” Manuscripta Mathematica, vol. 13, pp. 365–374, 1974. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  6. M. O. Osilike, β€œIterative solution of nonlinear equations of the ϕ-strongly accretive type,” Journal of Mathematical Analysis and Applications, vol. 200, no. 2, pp. 259–271, 1996. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  7. C. H. Xiang, β€œFixed point theorem for generalized ϕ-pseudocontractive mappings,” Nonlinear Analysis, vol. 70, no. 6, pp. 2277–2279, 2009. View at Publisher Β· View at Google Scholar
  8. J. Schu, β€œIterative construction of fixed points of asymptotically nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 158, no. 2, pp. 407–413, 1991. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  9. E. U. Ofoedu, β€œStrong convergence theorem for uniformly L-Lipschitzian asymptotically pseudocontractive mapping in real Banach space,” Journal of Mathematical Analysis and Applications, vol. 321, no. 2, pp. 722–728, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  10. H. Zhou, β€œDemiclosedness principle with applications for asymptotically pseudo-contractions in Hilbert spaces,” Nonlinear Analysis, vol. 70, no. 9, pp. 3140–3145, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  11. C. E. Chidume and M. O. Osilike, β€œEquilibrium points for a system involving m-accretive operators,” Proceedings of the Edinburgh Mathematical Society. Series II, vol. 44, no. 1, pp. 187–199, 2001. View at Publisher Β· View at Google Scholar
  12. C. E. Chidume and H. Zegeye, β€œApproximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps,” Proceedings of the American Mathematical Society, vol. 132, no. 3, pp. 831–840, 2004. View at Zentralblatt MATH
  13. S. S. Chang, β€œSome results for asymptotically pseudo-contractive mappings and asymptotically nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 129, no. 3, pp. 845–853, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  14. K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.
  15. S. S. Chang, K. K. Tan, H. W. J. Lee, and C. K. Chan, β€œOn the convergence of implicit iteration process with error for a finite family of asymptotically nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 313, no. 1, pp. 273–283, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  16. F. Gu, β€œThe new composite implicit iterative process with errors for common fixed points of a finite family of strictly pseudocontractive mappings,” Journal of Mathematical Analysis and Applications, vol. 329, no. 2, pp. 766–776, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  17. Z. Huang and F. Bu, β€œThe equivalence between the convergence of Ishikawa and Mann iterations with errors for strongly successively pseudocontractive mappings without Lipschitzian assumption,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 586–594, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  18. L. S. Liu, β€œIshikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 194, no. 1, pp. 114–125, 1995. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  19. B. E. Rhoades and S. M. Soltuz, β€œThe equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive maps,” Journal of Mathematical Analysis and Applications, vol. 289, no. 1, pp. 266–278, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  20. B. E. Rhoades and S. M. Soltuz, β€œThe equivalence between Mann-Ishikawa iterations and multistep iteration,” Nonlinear Analysis, vol. 58, no. 1-2, pp. 219–228, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  21. Z. Huang, β€œEquivalence theorems of the convergence between Ishikawa and Mann iterations with errors for generalized strongly successively ϕ-pseudocontractive mappings without Lipschitzian assumptions,” Journal of Mathematical Analysis and Applications, vol. 329, no. 2, pp. 935–947, 2007. View at Publisher Β· View at Google Scholar Β· View at MathSciNet
  22. Y. Xu, β€œIshikawa and Mann iterative processes with errors for nonlinear strongly accretive operator equations,” Journal of Mathematical Analysis and Applications, vol. 224, no. 1, pp. 91–101, 1998. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  23. P.-E. Maingé, β€œApproximation methods for common fixed points of nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 469–479, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  24. C. E. Chidume and C. O. Chidume, β€œConvergence theorem for zeros of generalized lipschitz generalized phi-quasi-accretive operators,” Proceedings of the American Mathematical Society, vol. 134, no. 1, pp. 243–251, 2006. View at Publisher Β·