A Note on Implicit and Explicit Mann Iterative Processes for Asymptotically 𝜙-Strongly Pseudocontractive Mappings

Abstract

In this paper we prove the equivalence and the strong convergence of an explicit Mann iterative process and a modified implicit iterative process for 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 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.

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 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.

Their scope was to extend the well-known Browder’s fixed point theorem [2] to this class of mappings.

This class is really more general than the class of nonexpansive mappings (see [1]).

In 1991, Schu [3] introduced the class of asymptotically pseudocontractive mappings.

Definition 1.2 (see [3]). 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,(1.2) where 𝐽 is the normalized duality mapping.

Obviously every asymptotically nonexpansive mapping is asymptotically pseudocontractive but the converse is not valid: it is well known that 𝑇∶[0,1]→[0,1] defined by 𝑇𝑥=(1−𝑥^2/3)^3/2 is not Lipschitz but asymptotically pseudocontractive [4].

In [3], Schu proved the following.

Theorem 1.3 (see [3]). 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.(1.3) Then, {𝑥_𝑛}_𝑛 converges strongly to some fixed point of 𝑇.

From 1991 to 2009, no fixed point theorem for asymptotically pseudocontractive mappings had been proved. First Zhou, in [5], completed this lack in the setting of Hilbert spaces proving (1) a fixed-point theorem for an asymptotically pseudocontractive mapping that is also uniformly L-Lipschitzian and uniformly asymptotically regular; (2) that the set of fixed points of 𝑇 is closed and convex; (3) the strong convergence of a CQ-iterative method. The literature on asymptotical-type mappings is wide (see e.g., [6–11]).

In 1974, Deimling [12], studying the zeros of accretive operators, introduced the class of 𝜑-strongly accretive operators.

Definition 1.4. An operator 𝐴 defined on a subset 𝐾 of a Banach space 𝑋 is called 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 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.

Since an operator 𝐴 is a strongly accretive operators if and only if (𝐼−𝐴) is a strongly pseudocontractive mappings (i.e., ⟨(𝐼−𝐴)𝑥−(𝐼−𝐴)𝑦,𝑗(𝑥−𝑦)⟩≤𝑘‖𝑥−𝑦‖², 𝑘<1), taking in account Definition 1.4, it is natural to study the class of 𝜑-pseudocontractive mappings, that is, the maps such that⟨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 this mappings contains, at most, only one point.

Here our attention is on the class of the asymptotically 𝜙-strongly pseudocontractions define as follows.

Definition 1.5. If 𝑋 is a Banach space and 𝐾 is a subset of 𝑋, a mapping 𝑇∶𝐾→𝐾 is said asymptotically 𝜙-strongly pseudocontraction if ⟨𝑇^𝑛𝑥−𝑇^𝑛𝑦,𝑗(𝑥−𝑦)⟩≤𝑘_𝑛‖𝑥−𝑦‖²−𝜙(‖𝑥−𝑦‖),(1.6) where 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦), {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 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−𝑦‖²−𝜙(‖𝑥−𝑦‖),(1.7) so, passing 𝑛 to +∞, it results‖𝑥−𝑧‖²≤‖𝑥−𝑧‖²−𝜙(‖𝑥−𝑦‖)⟹−𝜙(‖𝑥−𝑦‖)≥0.(1.8) Since 𝜙∶[0,∞)→[0,∞) is strictly increasing and 𝜙(0)=0, then 𝑥=𝑧.

We now give two examples.

Example 1.6. The mapping 𝑇𝑥=𝑥/(𝑥+1), where 𝑥∈[0,1], is asymptotically 𝜙-strongly pseudocontraction with 𝑘_𝑛=1, for all 𝑛∈ℕ and 𝜙(𝑡)=𝑡³/(1+𝑡). However, 𝑇 is not strongly pseudocontractive, see [13].

Example 1.7. The mapping 𝑇𝑥=𝑥/(1+𝛼𝑥), where 𝑥∈[0,1] and 𝛼 is closing to zero, is asymptotically 𝜙-strongly pseudocontraction with 𝑘_𝑛=1+1/𝛼𝑛, for all 𝑛∈ℕ and 𝜙(𝑥)=𝑥³/(1+𝑥). However, 𝑇 is not strongly pseudocontractive and nor is the 𝜙-strongly pseudocontraction.

Proof. First we prove that 𝑇 is not strongly pseudocontractive. For arbitrary 𝑘<1, there exists 𝑥,𝑦∈[0,1], such that 1(1+𝛼𝑥)(1+𝛼𝑦)>𝑘.(1.9) So we have ⟨𝑇𝑥−𝑇𝑦,𝑗(𝑥−𝑦)⟩=1(1+𝛼𝑥)(1+𝛼𝑦)(𝑥−𝑦)²>𝑘‖𝑥−𝑦‖².(1.10) Next we prove that 𝑇 is not 𝜙-strongly pseudocontractive. Taking 𝑦=0, we have, for all 𝑥∈[0,1], ⟨𝑇𝑥−𝑇𝑦,𝑗(𝑥−𝑦)⟩=𝑥²‖𝑥−𝑦‖₂(1+𝛼𝑥)−𝜙(‖𝑥−𝑦‖)=𝑥²1+𝑥,(1.11) Therefore, 𝑇 is not 𝜙-strongly pseudocontractive. Finally, we prove that 𝑇 is asymptotically 𝜙-strongly pseudocontraction.
For arbitrary 𝑥,𝑦∈[0,1], without loss of generality, let 𝑥>𝑦. Then, ⟨𝑇^𝑛𝑥−𝑇^𝑛𝑦,𝑗(𝑥−𝑦)⟩=(𝑥−𝑦)²𝑘(1+𝛼𝑛𝑥)(1+𝛼𝑛𝑦)_𝑛,‖𝑥−𝑦‖²−𝜙(‖𝑥−𝑦‖)=1+1𝛼𝑛(𝑥−𝑦)²−(𝑥−𝑦)³=1+11+(𝑥−𝑦)𝛼𝑛−1+(𝑥(−𝑥−𝑦)𝑦)(𝑥−𝑦)²=1+1/𝛼𝑛+1/𝛼𝑛(𝑥−𝑦)1+(𝑥−𝑦)(𝑥−𝑦)².(1.12) We only need to prove that 1+𝑥−𝑦≤(1+𝛼𝑛𝑥)(1+𝛼𝑛𝑦)1+1𝛼𝑛+𝛼1𝑛(𝑥−𝑦).(1.13) Using 𝑥>𝑦, this is easy.

Let us consider the well-known Mann iterative process defined as follows: for any given 𝑧₀∈𝑋, the sequence {𝑧_𝑛}_𝑛 defined by𝑧_𝑛+1=1−𝛼_𝑛𝑧_𝑛+𝛼_𝑛𝑇^𝑛𝑧_𝑛,𝑛≥0.(1.14)

We can also introduce a modified implicit iterative process as follows: suppose 𝑇 is continuous, (𝑤^𝑛)_𝑛 is bounded, and let 𝑧^0 be an initial point. Thus, we define𝑧^𝑛=1−𝛼_𝑛−𝛾_𝑛𝑧^𝑛−1+𝛼_𝑛𝑇^𝑛𝑧^𝑛+𝛾_𝑛𝑤^𝑛,𝑛≥1,(1.15) where {𝛼_𝑛}_𝑛 is a real sequence in [0,1] satisfying 𝛼_𝑛𝑘_𝑛<1 for all 𝑛≥1.

The algorithm is well defined. In fact, if 𝑇 is a continuous asymptotically strongly 𝜙-pseudocontraction, for every fixed 𝑛, the mapping 𝑆_𝑛 defined by 𝑆_𝑛𝑥∶=(1−𝛼_𝑛−𝛾_𝑛)𝑧_𝑛−1+𝛼_𝑛𝑇^𝑛𝑥+𝛾_𝑛𝑤_𝑛 is such that⟨𝑆_𝑛𝑥−𝑆_𝑛𝑦,𝑗(𝑥−𝑦)⟩=⟨𝑇^𝑛𝑥−𝑇^𝑛𝑦,𝑗(𝑥−𝑦)⟩≤𝛼_𝑛𝑘_𝑛‖𝑥−𝑦‖²,(1.16) that is, 𝑆_𝑛 is a continuous 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.

This modified implicit method is inspired to a wide literature.

In 1995, Liu [15] introduced the following modified Ishikawa method:𝑥_𝑛+1𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇^𝑛𝑥_𝑛+𝑣_𝑛,=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇^𝑛𝑦_𝑛+𝑢_𝑛,𝑛≥0,(1.17) and he called it Ischikawa iteration process with errors (obviously posing 𝛽_𝑛=0, he obtains the Mann iteration process with errors). This new class of methods with errors was studied, among the others, also by Chang in [16] in 2001, Chang et. al [17] in 2006, Gu in [18], Huang in [19], and Huang and Bu in 2007 [20].

In 2001, Chidume and Osilike [21] proved the strong convergence of the iterative method𝑦𝑥_𝑛=_𝑛+1𝑎_𝑛𝑥_𝑛+𝑏_𝑛𝑆𝑥_𝑛+𝑐_𝑛𝑢_𝑛,=𝑎^𝑛𝑥_𝑛+𝑏^𝑛𝑆𝑦_𝑛𝑐^𝑛𝑣_𝑛,(1.18) 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. Note that Chidume and Osilike did not use term with errors to indicate their methods.

In 2003, Chidume and Zegeye [22] studied the following iterative method:𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇𝑥_𝑛−𝛼_𝑛𝜃_𝑛𝑥_𝑛−𝑥₁,(1.19) where 𝑇 is a Lipschitzian pseudocontractive map with fixed points. The authors proved the strong convergence of the method to a fixed point of 𝑇 under opportune hypotheses on the control sequences (𝜃_𝑛)_𝑛,(𝜆_𝑛)_𝑛.

If we pose 𝑤_𝑛=𝑧₀ in (1.14) and 𝛾_𝑛=𝛼_𝑛𝜃_𝑛, the modified Mann iterative process coincides with the Chideme and Zegeye’s iterative method.

In this paper, we prove the equivalence between the implicit and the explicit modified Mann iterative method which involves an asymptotically 𝜙-strongly pseudocontractive mapping.

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 shall denote the single-valued duality mapping by 𝑗.

For the sake of completeness, we recall some definitions and conclusions.

Definition 2.1. An 𝑋 is said to be a uniformly smooth Banach space if the smooth module of 𝑋, 𝜌_𝑋(𝑡)=sup12(‖𝑥−𝑦‖+‖𝑥+𝑦‖)−1∶‖𝑥‖≤1,‖𝑦‖≤𝑡,(2.1) satisfies lim_𝑡→0(𝜌_𝑋(𝑡)/𝑡)=0.

Lemma 2.2 (see [23]). Let 𝑋 be a Banach space, and let 𝐽∶𝑋→2^𝑋∗ be the normalized duality mapping. Then, for any 𝑥,𝑦∈𝑋, we have ‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝑗(𝑥+𝑦)⟩,∀𝑗(𝑥+𝑦)∈𝐽(𝑥+𝑦).(2.2)

Next lemma is a key for our proofs.

Lemma 2.3 (see [19]). 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 [19] but changing, in (2.4), (𝑎_𝑛+1−𝑒_𝑛) with |𝑎_𝑛+1−𝑒_𝑛|.

Lemma 2.4 (see [24]). 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 ∑_𝑛𝑐_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 we have ∑_𝑛𝑎_𝑛=∞ and limsup_𝑛(𝑏_𝑛/𝑎_𝑛)≤0, then 𝑠_𝑛→0 as 𝑛→∞.

Remark 2.5. If, in Lemma 2.3, we choose 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^𝑛1+2𝑏_𝑛𝑘≤1−2𝑏^𝑛𝑘1+2𝑏_𝑛𝑘𝑎^2𝑛+𝑐^𝑛1+2𝑏_𝑛𝑘,(2.6) 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 in account the papers of Chang and Chidume et al. [16, 21, 25].

Theorem 3.1. Let 𝑋 be a uniformly smooth Banach space, and let 𝑇∶𝑋→𝑋 be asymptotically 𝜙-strongly pseudocontractive mapping with fixed point 𝑥^∗ and bounded range.
Let {𝑣_𝑛}_𝑛 be the sequence defined by 𝑣𝑣₀∈𝑋,_𝑛+1=1−𝛼_𝑛𝑣_𝑛+𝛼_𝑛𝑇^𝑛𝑣_𝑛,𝑛≥0,(3.1) where {𝛼_𝑛}_𝑛⊂[0,1] satisfies(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≤≤11−−𝛼𝛼_𝑛𝑛²‖‖𝑣_𝑛−𝑥^∗‖‖²+2𝛼_𝑛𝑇^𝑛𝑣_𝑛−𝑥^∗,𝑗𝑣_𝑛+1−𝑥^∗+2𝛼_𝑛²‖‖𝑣_𝑛−𝑥^∗‖‖²+2𝛼_𝑛𝑇_𝑛𝑣_𝑛−𝑥^∗,𝑗𝑣_𝑛+1−𝑥^∗−𝑗𝑣_𝑛−𝑥^∗≤1−𝛼_𝑛𝑇_𝑛𝑣_𝑛−𝑥^∗,𝑗𝑣_𝑛−𝑥^∗+2𝛼_𝑛²‖‖𝑣_𝑛−𝑥^∗‖‖²+2𝛼_𝑛𝑇_𝑛𝑣_𝑛−𝑥^∗,𝑗𝑣_𝑛+1−𝑥^∗−𝑗𝑣_𝑛−𝑥^∗=1+𝛼𝑘_𝑛2𝑛‖‖𝑣_𝑛−𝑥^∗‖‖²−2𝛼_𝑛𝜙‖‖𝑣_𝑛−𝑥^∗‖‖+2𝛼_𝑛−2𝛼_𝑛‖‖𝑣_𝑛−𝑥^∗‖‖²+2𝛼_𝑛𝑇_𝑛𝑣_𝑛−𝑥^∗,𝑗𝑣_𝑛+1−𝑥^∗−𝑗𝑣_𝑛−𝑥^∗=‖‖𝑣_𝑛𝑘_𝑛‖‖𝑣_𝑛−𝑥^∗‖‖²−2𝛼_𝑛𝜙‖‖𝑣_𝑛−𝑥^∗‖‖−𝑥^∗‖‖²+𝛼^2𝑛−2𝛼_𝑛+2𝛼_𝑛𝑘_𝑛‖‖𝑣_𝑛−𝑥^∗‖‖²−2𝛼_𝑛𝜙‖‖𝑣_𝑛−𝑥^∗‖‖+2𝛼_𝑛𝜇_𝑛,(3.2) where 𝜇_𝑛∶=⟨𝑇_𝑛𝑣_𝑛−𝑥^∗,𝑗(𝑣_𝑛+1−𝑥^∗)−𝑗(𝑣_𝑛−𝑥^∗)⟩. Let 𝑀∶=maxsup_𝑛‖‖𝑣_𝑛−𝑥^∗‖‖,sup_𝑛‖‖𝑇^𝑛𝑣_𝑛−𝑥^∗‖‖.(3.3) We have ‖‖𝑣_𝑛+1−𝑥^∗‖‖²≤‖‖𝑣_𝑛−𝑥^∗‖‖²+𝛼^2𝑛+2𝛼_𝑛𝑘_𝑛−1𝑀−2𝛼_𝑛𝜙‖‖𝑣_𝑛−𝑥^∗‖‖+2𝛼_𝑛𝜇_𝑛=‖‖𝑣_𝑛−𝑥^∗‖‖²−𝛼_𝑛𝜙‖‖𝑣_𝑛−𝑥^∗‖‖−𝛼_𝑛𝜙‖‖𝑣_𝑛−𝑥^∗‖‖−2𝜇_𝑛−𝛼_𝑛+2𝑘_𝑛−1𝑀,(3.4) so we can observe the following.(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⟶0,as𝑛⟶∞,(3.5) and, since 𝑗 is norm to norm uniformly continuous, then 𝑗‖‖𝑣_𝑛+1−𝑥^∗‖‖−𝑗‖‖𝑣_𝑛−𝑥^∗‖‖⟶0,as𝑛⟶∞.(3.6)(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, 𝜙‖‖𝑣_𝑛−𝑥^∗‖‖≥𝜙(𝜎)>0.(3.7) 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>𝑁, ‖‖𝑣_𝑛+1−𝑥^∗‖‖²≤‖‖𝑣_𝑛−𝑥^∗‖‖²−𝛼_𝑛𝜙‖‖𝑣_𝑛−𝑥^∗‖‖≤‖‖𝑣_𝑛−𝑥^∗‖‖²−𝛼_𝑛𝜙(𝜎)(3.8) and so 𝛼_𝑛𝜙(𝜎)≤‖‖𝑣_𝑛−𝑥^∗‖‖²−‖‖𝑣_𝑛+1−𝑥^∗‖‖²∀𝑛>𝑁.(3.9) In the same manner, we obtain that 𝜙(𝜎)^𝑚_𝑖_=𝑁𝛼_𝑖≤^𝑚_𝑖_=𝑁‖‖𝑣^𝑖−𝑥^∗‖‖²−‖‖𝑣_𝑖+1−𝑥^∗‖‖²=‖‖𝑣^𝑁−𝑥^∗‖‖²−‖‖𝑣_𝑚−𝑥^∗‖‖².(3.10) 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 (3.4) 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 results that ||𝛼_𝑛||<12𝑀min𝜖,𝜙(2𝜖/2),||𝑘^𝑛−1||<𝜙(𝜖/2)8𝑀,||𝜇^𝑛||<𝜙(𝜖/2)8.(3.11) 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 results that ‖𝑣_𝑛∗+1−𝑥^∗‖≥𝜖. Thus, 𝜖≤‖‖𝑣_𝑛∗+1−𝑥^∗‖‖≤‖‖𝑣_𝑛∗−𝑥^∗‖‖+𝛼_𝑛∗<‖‖𝑣_𝑛∗−𝑥^∗‖‖+𝜖2𝑀𝑀=‖‖𝑣^𝑛∗−𝑥^∗‖‖+𝜖2,(3.12) that is, ‖𝑣_𝑛∗−𝑥^∗‖>𝜖/2. By the strict increasing of 𝜙, then 𝜙(‖𝑣_𝑛∗−𝑥^∗‖)>𝜙(𝜖/2). By (3.4), it results ‖‖𝑣_𝑛∗+1−𝑥^∗‖‖²<𝜖²−𝛼_𝑛∗𝜙‖‖𝑣_𝑛∗−𝑥^∗‖‖−𝛼_𝑛∗𝜙‖‖𝑣_𝑛^∗−𝑥^∗‖‖−2𝜇_𝑛∗−𝛼_𝑛∗+2𝑘_𝑛∗−1𝑀.(3.13) We can note that 2𝜇_𝑛∗+𝛼_𝑛∗+2𝑘_𝑛∗−1𝑀≤𝜙(𝜖/2)4+𝜙4(𝑀𝜖/2)+𝜙4(𝑀𝜖/2)𝑀,(3.14) so 𝜙‖‖𝑣_𝑛∗−𝑥^∗‖‖−2𝜇_𝑛∗−𝛼_𝑛∗+2𝑘_𝑛∗−1𝑀≥𝜙𝜖2−3𝜙(4𝜖/2)>0.(3.15) Moreover, 𝜙(‖𝑣_𝑛∗−x^∗‖)>𝜙(𝜖/2)/2>0, it results that ‖‖𝑣_𝑛∗+1−𝑥^∗‖‖²≤𝜖².(3.16) 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.

Remark 3.2. Our result is similar to Schu’s theorem. However, our results hold in a more general setting of uniformly smooth Banach spaces, while the Schu’s result holds for completely continuous, uniformly Lipschitzian mappings which are asymptotically pseudocontractive.

Theorem 3.3. Let 𝑋 be a uniformly smooth Banach space, and let 𝑇∶𝑋→𝑋 be a continuous and asymptotically 𝜙-strongly pseudocontractive mapping with fixed point 𝑥^∗ and bounded range.
Let {𝑧_𝑛} and {𝑧^𝑛} be the sequences defined by (1.14) and (1.15), 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=∞, and such that 𝛼_𝑛𝑘_𝑛<1, for every 𝑛∈ℕ.
Let us 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 Mann iteration sequence (1.14) converges to the fixed point 𝑥^∗.(ii)The modified implicit iteration sequence (1.15) converges to the fixed point 𝑥^∗.

Proof. By the boundedness of the range of 𝑇 and by Lemma 2.4, one obtains that our schemes are bounded. Let us define 𝑀=sup_𝑛⎧⎪⎨⎪⎩‖‖𝑇^𝑛𝑧_𝑛−𝑇^𝑛𝑧^𝑛‖‖‖‖𝑧,‖‖_𝑛+1𝑇^𝑛𝑧_𝑛−𝑧_𝑛‖‖,‖‖𝑇^𝑛𝑧^𝑛−𝑧^𝑛‖‖,‖‖𝑇^𝑛𝑧^𝑛−𝑧^𝑛−1‖‖−𝑧^𝑛‖‖,‖‖𝑤^𝑛−𝑧^𝑛−1‖‖,‖‖𝑤^𝑛‖‖⎫⎪⎬⎪⎭.(3.17) By the iteration schemes (1.14) and (1.15), we have ‖‖𝑧_𝑛+1−𝑧^𝑛‖‖²≤‖‖1−𝛼_𝑛𝑧_𝑛−𝑧^𝑛−1+𝛼_𝑛𝑇^𝑛𝑧_𝑛−𝑇^𝑛𝑧^𝑛+𝛾_𝑛𝑧^𝑛−1−𝑤^𝑛‖‖²≤≤11−−𝛼𝛼_𝑛𝑛²‖‖𝑧_𝑛−𝑧^𝑛−1‖‖²+2𝛼_𝑛T^𝑛𝑧_𝑛−𝑇^𝑛𝑧^𝑛+𝛾_𝑛𝑧^𝑛−1−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𝑇^𝑛𝑧_𝑛−𝑇^𝑛𝑧^𝑛,𝑗𝑧_𝑛+1−𝑧^𝑛−𝑗𝑧_𝑛−𝑧^𝑛+2𝛾_𝑛𝑀²≤1−𝛼_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,(3.18) where 𝜎_𝑛=‖𝑗(𝑧_𝑛+1−𝑧^𝑛)−𝑗(𝑧_𝑛−𝑧^𝑛)‖. By (1.14), we have ‖‖𝑧_𝑛+1−𝑧^𝑛−𝑧_𝑛−𝑧^𝑛‖‖=‖‖𝑧_𝑛+1−𝑧_𝑛‖‖=‖‖𝛼_𝑛𝑇^𝑛𝑧_𝑛−𝑧_𝑛‖‖≤𝛼_𝑛𝑀.(3.19) 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.(3.20) 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,(3.21) and so ‖‖𝑧_𝑛+1−𝑧^𝑛‖‖≤=‖‖‖‖𝑧𝑧_𝑛𝑛−𝑧^𝑛+𝑧^𝑛−𝑧^𝑛−1−𝛼_𝑛𝑧_𝑛−𝑧^𝑛−1+𝛼_𝑛𝑇^𝑛𝑧_𝑛−𝑇^𝑛𝑧^𝑛+𝛾_𝑛𝑧^𝑛−1−𝑤^𝑛‖‖≤‖‖𝑧_𝑛−𝑧^𝑛‖‖+‖‖𝑧^𝑛−𝑧^𝑛−1‖‖+𝛼_𝑛‖‖𝑧_𝑛−𝑧^𝑛−1‖‖+𝛼_𝑛‖‖𝑇^𝑛𝑧_𝑛−𝑇^𝑛𝑧^𝑛‖‖+𝛾_𝑛‖‖𝑧^𝑛−1−𝑤^𝑛‖‖−𝑧^𝑛‖‖+3𝛼_𝑛+𝛾_𝑛𝑀.(3.22) 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 index for which ‖𝑧_𝑛+1−𝑧^𝑛‖−𝑒_𝑛≤0 contains infinite terms,(ii)the set of index 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 (3.20) in (3.18), 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,(3.23) where 𝑘=sup_𝑛𝑘_𝑛. Again by (3.20), 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.(3.24) 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.(3.25) 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 it results ‖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,(3.26) so ‖𝑧_𝑛∗−𝑧^𝑛_∗‖>𝜖/2. In consequence of this, 𝜙(‖𝑧_𝑛∗−𝑧^𝑛_∗‖)>𝜙(𝜖/2).
In (3.23), we note that7𝑘𝛼_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),(3.27) so 𝜙‖‖𝑧_𝑛∗−𝑧^𝑛_∗‖‖−7𝑘𝛼_𝑛∗𝑀²−2𝑀²𝑘_𝑛^∗−1−6𝛾_𝑛∗𝑀²𝑘−2𝑀𝜎_𝑛∗≥𝜙(𝜖/2)2,(3.28) Hence, in (3.23), remains ‖‖𝑧_𝑛∗+1−𝑧^𝑛_∗‖‖²≤𝜖²−𝛼_𝑛∗𝜙‖‖𝑧_𝑛∗−𝑧^𝑛_∗‖‖+2𝛾_𝑛∗𝑀².(3.29) In the same manner, 𝜙‖‖𝑧_𝑛∗−𝑧^𝑛_∗‖‖−2𝛾^𝑛∗𝛼_𝑛∗𝑀²>𝜙𝜖2−𝜙(2𝜖/2)>0,(3.30) so ‖‖𝑧_𝑛∗+1−𝑧^𝑛_∗‖‖²≤𝜖².(3.31) This is absurd. By the same idea and by using the induction method, we obtain that ‖𝑧_𝑛∗+𝑖−𝑧^𝑛_∗+𝑖−1‖<𝜖, for every 𝑖≥0. This assure that ‖𝑧_𝑛+1−𝑧^𝑛‖→0. In the second case (ii), definitively, ‖𝑧_𝑛+1−𝑧^𝑛‖−𝑒_𝑛≥0, then, from the strict increasing function 𝜙, we have 𝜙‖‖𝑧_𝑛−𝑧^𝑛‖‖≥𝜙‖‖𝑧_𝑛+1−𝑧^𝑛‖‖−𝑒_𝑛.(3.32) Substituting (3.32) and (3.20) into (3.18) 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.(3.33) By virtue of Lemma 2.3, we obtain that lim_𝑛→∞‖𝑧_𝑛−𝑧^𝑛−1‖=0.