The Equivalence of Convergence Results between Ishikawa and Mann Iterations with Errors for Uniformly Continuous Generalized Φ-Pseudocontractive Mappings in Normed Linear Spaces

Abstract

We prove the equivalence of the convergence of the Mann and Ishikawa iterations with errors for uniformly continuous generalized Φ-pseudocontractive mappings in normed linear spaces. Our results extend and improve the corresponding results of Xu, 1998, Kim et al., 2009, Ofoedu, 2006, Chidume and Zegeye, 2004, Chidume, 2001, Chang et al. 2002, Liu, 1995, Hirano and huang, 2003, C. E. Chidume and C. O. Chidume, 2005, and huang, 2007.

1. Introduction

Let 𝐸 be a real normed linear space, 𝐸^∗ its dual space, and 𝐽∶𝐸→2^𝐸∗ the normalized duality mapping defined by𝐽(𝑥)=𝑓∈𝐸^∗∶⟨𝑥,𝑓⟩=‖𝑥‖⋅‖𝑓‖=‖𝑓‖²,(1.1) where ⟨⋅,⋅⟩ denotes the generalized duality pairing. The single-valued normalized duality mapping is denoted by 𝑗.

Definition 1.1. A mapping 𝑇∶𝐸→𝐸 is said to be(1)strongly accretive if, for all 𝑥,𝑦∈𝐸, there exist a constant 𝑘∈(0,1) and 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that ⟨𝑇𝑥−𝑇𝑦,𝑗(𝑥−𝑦)⟩≥𝑘‖𝑥−𝑦‖²,(1.2)(2)𝜙-strongly accretive if there exist 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) and a strictly increasing function 𝜙∶[0,+∞)→[0,+∞) with 𝜙(0)=0 such that ⟨𝑇𝑥−𝑇𝑦,𝑗(𝑥−𝑦)⟩≥𝜙(‖𝑥−𝑦‖)‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐸,(1.3)(3)generalized Φ-accretive if, for all 𝑥,𝑦∈𝐸, there exist 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) and a strictly increasing function Φ∶[0,+∞)→[0,+∞) with Φ(0)=0 such that ⟨𝑇𝑥−𝑇𝑦,𝑗(𝑥−𝑦)⟩≥Φ(‖𝑥−𝑦‖).(1.4)

Definition 1.2. Let 𝑁(𝑇)={𝑥∈𝐸∶𝑇𝑥=0}≠∅. The mapping 𝑇 is called strongly quasi-accretive if, for all 𝑥∈𝐸,𝑞∈𝑁(𝑇), there exist a constant 𝑘∈(0,1) and 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that ⟨𝑇𝑥−𝑇𝑞,𝑗(𝑥−𝑞)⟩≥𝑘‖𝑥−𝑞‖²;𝑇 is called 𝜙-strongly quasi-accretive if, for all 𝑥∈𝐸,𝑞∈𝑁(𝑇), there exists a function 𝜙 such that ⟨𝑇𝑥−𝑇𝑞,𝑗(𝑥−𝑞)⟩≥𝜙(‖𝑥−𝑞‖)‖𝑥−𝑞‖, where 𝜙 is as in Definition 1.1. Finally, 𝑇 is called generalized Φ-quasi-accretive if, for each 𝑥∈𝐸,𝑞∈𝑁(𝑇), there exist 𝑗(𝑥−𝑞)∈𝐽(𝑥−𝑞) and a strictly increasing function Φ∶[0,+∞)→[0,+∞) with Φ(0)=0 such that ⟨𝑇𝑥−𝑇𝑞,𝑗(𝑥−𝑞)⟩≥Φ(‖𝑥−𝑞‖).

Closely related to the class of accretive-type mappings are those of pseudocontractive types.

Definition 1.3. A mapping 𝑇 with domain 𝐷(𝑇) and range 𝑅(𝑇) is said to be(1)strongly pseudocontractive if there exist a constant 𝑘∈(0,1) and 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that, for each 𝑥,𝑦∈𝐷(𝑇), ⟨𝑇𝑥−𝑇𝑦,𝑗(𝑥−𝑦)⟩≤𝑘‖𝑥−𝑦‖²,(1.5)(2)𝜙-strongly pseudocontractive if there exist 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) and a strictly increasing function 𝜙∶[0,+∞)→[0,+∞) with 𝜙(0)=0 such that ⟨𝑇𝑥−𝑇𝑦,𝑗(𝑥−𝑦)⟩≤‖𝑥−𝑦‖²−𝜙(‖𝑥−𝑦‖)‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐷(𝑇),(1.6)(3)generalized Φ-pseudocontractive if, for all 𝑥,𝑦∈𝐷(𝑇), there exist 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) and a strictly increasing function Φ∶[0,+∞)→[0,+∞) with Φ(0)=0 such that ⟨𝑇𝑥−𝑇𝑦,𝑗(𝑥−𝑦)⟩≤‖𝑥−𝑦‖²−Φ(‖𝑥−𝑦‖).(1.7)

Definition 1.4. Let 𝐹(𝑇)={𝑥∈𝐸∶𝑇𝑥=𝑥}≠∅. The mapping 𝑇 is called generalized Φ-hemi-pseudocontractive if, for all 𝑥∈𝐷(𝑇),𝑞∈𝐹(𝑇), there exist 𝑗(𝑥−𝑞)∈𝐽(𝑥−𝑞) and a strictly increasing function Φ∶[0,+∞)→[0,+∞) with Φ(0)=0 such that ⟨𝑇𝑥−𝑇𝑞,𝑗(𝑥−𝑞)⟩≤‖𝑥−𝑞‖²−Φ(‖𝑥−𝑞‖).(1.8)

Obviously, a mapping 𝑇 is strongly pseudocontractive, 𝜙-strongly pseudocontractive, generalized Φ-pseudocontractive, and generalized Φ-hemicontractive if and only if (𝐼−𝑇) is strongly accretive, 𝜙-strongly accretive, generalized Φ-accretive, and generalized Φ-quasi-accretive, respectively.

It is shown in [1] that the class of strongly pseudocontractive mappings is a proper subclass of 𝜙-strongly pseudocontractive mappings. Furthermore, an example in [2] shows that the class of 𝜙-strongly pseudocontractive mappings with the nonempty fixed point set is a proper subclass of Φ-hemicontractive mappings. Hence, the class of generalized Φ-hemicontractive mappings is the most general among those defined above.

Definition 1.5. The mapping 𝑇∶𝐸→𝐸 is called Lipschitz if exists a constant 𝐿>0 such that ‖𝑇𝑥−𝑇𝑦‖≤𝐿‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐸.(1.9) It is clear that if 𝑇 is Lipschitz then it must be uniformly continuous. Otherwise, it is not true.

The following iteration schemes were introduced by Xu [3] in 1998. Let 𝐾 be a nonempty convex subset of 𝐸. For any given 𝑥₀∈𝐾, the sequence {𝑥_𝑛}^∞𝑛=0 is defined by𝑦_𝑛𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+𝛽_𝑛𝑇𝑦_𝑛+𝛾_𝑛𝛿_𝑛,=𝛼^𝑛𝑥_𝑛+𝛽^𝑛𝑇𝑥_𝑛+𝛾^𝑛𝜂_𝑛,∀𝑛≥0,(1.10) is called the Ishikawa iteration sequence with errors, where {𝛿_𝑛}^∞𝑛=0,{𝜂_𝑛}^∞𝑛=0 are arbitrary bounded sequences in 𝐾 and {𝛼_𝑛}^∞𝑛=0,{𝛽_𝑛}^∞𝑛=0,{𝛾_𝑛}^∞𝑛=0,{𝛼^𝑛}^∞𝑛=0,{𝛽^𝑛}^∞𝑛=0, and {𝛾^𝑛}^∞𝑛=0 are six real sequences in [0,1] such that 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=𝛼^𝑛+𝛽^𝑛+𝛾^𝑛=1 for all 𝑛≥0 and satisfy certain conditions.

If 𝛽^𝑛=𝛾^𝑛=0, for all 𝑛≥0, then, from (1.10), we get the Mann iteration sequence with errors {𝑢_𝑛}^∞𝑛=0 defined by∀𝑢₀∈𝐾,𝑢_𝑛+1=𝛼_𝑛𝑢_𝑛+𝛽_𝑛𝑇𝑢_𝑛+𝛾_𝑛𝜀_𝑛,∀𝑛≥0,(1.11) where {𝜀_𝑛}^∞𝑛=0 is an arbitrary bounded sequence in 𝐾.

Numerous convergence results have been proved through iterative methods of approximating fixed points of Lipschitz pseudocontractive- (accretive-) type nonlinear mappings [3–10]. Most of these results have been extended to uniformly continuous mappings by some authors. Recently, C. E. Chidume and C. O. Chidume in [11] gave the most general result for uniformly continuous generalized Φ-hemicontractive mappings in normed linear. Their results are as follows.

Theorem 1.6 (see [11, Theorem 2.3]). Let 𝐸 be a real normed linear space, 𝐾 nonempty subset of 𝐸, and 𝑇∶𝐸→𝐾 a uniformly continuous generalized Φ-hemicontractive mapping, that is, there exist 𝑥^∗∈𝐹(𝑇) and a strictly increasing function Φ∶[0,+∞)→[0,+∞),Φ(0)=0, such that, for all 𝑥∈𝐾, there exists 𝑗(𝑥−𝑥^∗)∈𝐽(𝑥−𝑥^∗) such that 𝑇𝑥−𝑥^∗,𝑗𝑥−𝑥^∗≤‖𝑥−𝑥^∗‖²−Φ‖𝑥−𝑥^∗‖.(1.12)(a)If 𝑦^∗∈𝐾 is a fixed point of 𝑇, then 𝑦^∗=𝑥^∗ and so 𝑇 has at most one fixed point in 𝐾.(b)Suppose there exists 𝑥₀∈𝐾 such that the sequence {𝑥_𝑛} defined by 𝑥_𝑛+1=𝑎_𝑛𝑥_𝑛+𝑏_𝑛𝑇𝑥_𝑛+𝑐_𝑛𝑢_𝑛,∀𝑛≥0,(1.13) is contained in 𝐾, where {𝑎_𝑛},{𝑏_𝑛}, and {𝑐_𝑛} are real sequences satisfying the following conditions:(i)𝑎_𝑛+𝑏_𝑛+𝑐_𝑛=1, (ii)∑^∞𝑛=0(𝑏_𝑛+𝑐_𝑛)=∞, (iii)∑^∞𝑛=0(𝑏_𝑛+𝑐_𝑛)²<∞, (iv)∑^∞𝑛=0𝑐_𝑛<∞. Then, {𝑥_𝑛} converges strongly to 𝑥^∗. In particular, if 𝑦^∗ is a fixed point of 𝑇 in 𝐾, then {𝑥_𝑛} converges strongly to 𝑦^∗.

Unfortunately, the control conditions (iii) ∑^∞𝑛=0(𝑏_𝑛+𝑐_𝑛)²<∞ and (iv) ∑^∞𝑛=0𝑐_𝑛<∞ cannot assure that the result in [11] holds. In the proof course of p552, let 𝜖>0 be any given, we can choose (also in view of conditions (iii) and (iv)) an integer 𝑁₁>0 such that for all 𝑛>𝑁₁ the following inequality 𝑀₁𝑐_𝑛<(Φ(𝜖)/4)𝛼_𝑛 holds, where 𝛼_𝑛=𝑏_𝑛+𝑐_𝑛. The inequality above 𝑀₁𝑐_𝑛<(Φ(𝜖)/4)𝛼_𝑛 implies that 𝑐_𝑛=𝑜(𝛼_𝑛). But conditions (iii) and (iv) of [11] can not assure that 𝑐_𝑛=𝑜(𝛼_𝑛). On the one hand, let 𝑐_𝑛=1/𝑛²,𝑛=1,2,3,…;𝛼₁=0,𝛼₂=1/2,𝛼₃=0,𝛼₄=1/4,𝛼₅=0,𝛼₆=1/6,…; then ∑^∞𝑛=0𝑐_𝑛<∞, but 𝑐_𝑛≠𝑜(𝛼_𝑛). On the other hand, set 𝑐_𝑛=1/𝑛,𝛼_𝑛=2/√𝑛; then 𝑐_𝑛=𝑜(𝛼_𝑛), but ∑^∞𝑛=0𝑐_𝑛=∞.

The purpose of this paper is that we obtain the convergence result of the Mann iteration with errors, and we also prove the equivalence of convergence between the Ishikawa iteration with errors defined by (1.10) and the Mann iteration with errors defined by (1.11). We also show that the Ishikawa iteration with errors defined by (1.10) converges to the unique fixed point of 𝑇. Our results extend and improve the corresponding results of [3–12]. For this, in the sequel, we will need the following lemmas.

Lemma 1.7 (see [13]). Let 𝐸 be a real normed space. Then, for all 𝑥,𝑦∈𝐸, the following inequality holds: ‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝑗(𝑥+𝑦)⟩,∀𝑗(𝑥+𝑦)∈𝐽(𝑥+𝑦).(1.14)

Lemma 1.8 (see [14]). Let Φ∶[0,+∞)→[0,+∞) be a strictly increasing continuous function with Φ(0)=0 and {𝜃_𝑛},{𝜎_𝑛}, and {𝜆_𝑛} nonnegative three sequences that satisfy the following inequality: 𝜃^2𝑛+1≤𝜃^2𝑛−2𝜆_𝑛Φ𝜃_𝑛+1+𝜎_𝑛,𝑛≥𝑁,(1.15) where 𝜆_𝑛∈(0,1),lim_𝑛→∞𝜆_𝑛=0, and ∑^∞𝑛=0𝜆_𝑛=∞,𝜎_𝑛=𝑜(𝜆_𝑛). Then 𝜃_𝑛→0 as 𝑛→∞.

2. Main Results

Theorem 2.1. Let 𝐾 be a nonempty closed convex subset of a real normed linear space 𝐸. Suppose that 𝑇∶𝐾→𝐾 is a uniformly continuous generalized Φ-hemicontractive mapping with 𝐹(𝑇)≠∅. Let {𝑢_𝑛} be a sequence in 𝐾 defined iteratively from some 𝑢₀∈𝐾 by (1.11), where {𝜀_𝑛} is an arbitrary bounded sequence in 𝐾 and {𝛼_𝑛},{𝛽_𝑛}, and {𝛾_𝑛} are three sequences in [0,1] satisfying the following conditions:(i)𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1, (ii)∑^∞𝑛=0𝛽_𝑛=∞, (iii)lim_𝑛→∞𝛽_𝑛=0, (iv)𝛾_𝑛=𝑜(𝛽_𝑛). Then, the iteration sequence {𝑢_𝑛} converges strongly to the unique fixed point of 𝑇.

Proof. Let 𝑞∈𝐹(𝑇). The uniqueness of the fixed point of 𝑇 comes from Definition 1.4.
First, we prove that there exists 𝑢₀∈𝐾 with 𝑢₀≠𝑇𝑢₀ such that 𝑡₀=‖𝑢₀−𝑇𝑢₀‖⋅‖𝑢₀−𝑞‖∈𝑅(Φ). In fact, if 𝑢₀=𝑇𝑢₀, then we are done. Otherwise, there exists the smallest positive integer 𝑛₀∈𝑁 such that 𝑢_𝑛0≠𝑇𝑢_𝑛0. We denote 𝑢_𝑛0=𝑢₀, and then we obtain that 𝑡₀=‖𝑢₀−𝑇𝑢₀‖⋅‖𝑢₀−𝑞‖∈𝑅(Φ). Indeed, if 𝑅(Φ)=[0,+∞), then 𝑡₀∈𝑅(Φ). If 𝑅(Φ)=[0,𝐴) with 0<𝐴<+∞, then, for 𝑞∈𝐾, there exists a sequence {𝑤_𝑛}⊂𝐾 such that 𝑤_𝑛→𝑞 as 𝑛→∞ with 𝑤_𝑛≠𝑞, and we also obtain that the sequence {𝑤_𝑛−𝑇𝑤_𝑛} is bounded. So there exists 𝑛₀∈𝑁 such that ‖𝑤_𝑛−𝑇𝑤_𝑛‖⋅‖𝑤_𝑛−𝑞‖∈𝑅(Φ) for 𝑛≥𝑛₀, and then we redefine 𝑢₀=𝑤_𝑛0. Let 𝜔₀=Φ⁻¹(‖(𝑢₀−𝑞)−(𝑇𝑢₀−𝑞)‖⋅‖𝑢₀−𝑞‖)>0.
Next for 𝑛≥0 we will prove ‖𝑢_𝑛−𝑞‖≤𝜔₀ by induction. Clearly, ‖𝑢₀−𝑞‖≤𝜔₀ holds. Suppose that ‖𝑢_𝑛−𝑞‖≤𝜔₀, for some 𝑛; then we want to prove ‖𝑢_𝑛+1−𝑞‖≤𝜔₀. If it is not the case, then ‖𝑢_𝑛+1−𝑞‖>𝜔₀. Since 𝑇 is a uniformly continuous mapping, setting 𝜖₀=Φ(𝜔₀)/12𝜔₀, there exists 𝛿>0 such that ‖𝑇𝑥−𝑇𝑦‖<Φ(𝜔₀)/12𝜔₀ whenever ‖𝑥−𝑦‖<𝛿 and 𝑇 is a bounded operator. Set 𝑀=sup{‖𝑇𝑥‖∶‖𝑥−𝑞‖≤𝜔₀}+sup_𝑛‖𝜀_𝑛‖. Since lim_𝑛→∞𝛽_𝑛=0,𝛾_𝑛=𝑜(𝛽_𝑛), without loss of generality, let 𝛽_𝑛,𝛾^𝑛𝛽_𝑛<min𝜔⁰4(𝑀+‖𝑞‖),2𝑀+2𝜔𝛿₀+2‖𝑞‖,Φ𝜔⁰6𝜔₀,Φ𝜔⁰6𝜔₀+𝑀+‖𝑞‖²,𝑛≥0.(2.1)
From (1.10), we have ‖‖𝑢_𝑛+1−𝑞‖‖≤=‖‖‖‖𝛼𝑢_𝑛𝑛𝑢_𝑛−𝑞+𝛽_𝑛𝑇𝑢_𝑛+𝛾_𝑛𝜀−𝛽_𝑛+𝛾_𝑛𝑞‖‖≤𝜔₀−𝑞‖‖+𝛽_𝑛‖‖𝑇𝑢_𝑛‖‖+𝛾_𝑛‖‖𝜀_𝑛‖‖+𝛽_𝑛+𝛾_𝑛‖𝑞‖≤𝜔₀+𝛽_𝑛‖‖𝑇𝑢_𝑛‖‖+‖‖𝜀_𝑛‖‖+2‖𝑞‖+𝛽_𝑛(2𝑀+2‖𝑞‖)≤32𝜔⁰‖𝑢_𝑛+1,−𝑢_𝑛‖=≤𝛽‖‖_𝑛𝛽_𝑛𝑇𝑢_𝑛+𝛾_𝑛𝜀_𝑛−𝛽_𝑛+𝛾_𝑛𝑢_𝑛‖‖≤𝛽_𝑛‖‖𝑇𝑢_𝑛‖‖+𝛾_𝑛‖‖𝜀_𝑛‖‖+𝛽_𝑛+𝛾_𝑛‖‖𝑢_𝑛−𝑞‖‖+𝛽_𝑛+𝛾_𝑛‖𝑞‖≤𝛽_𝑛‖‖𝑇𝑢_𝑛‖‖+‖‖𝜀_𝑛‖‖+2‖‖𝑢_𝑛−𝑞‖‖+2‖𝑞‖2𝑀+2𝜔₀+2‖𝑞‖.(2.2)
Applying Lemma 1.7, the recursion (1.11), and the inequalities above, we obtain ‖‖𝑢_𝑛+1−𝑞‖‖²=‖‖𝑢_𝑛−𝑞+𝛽_𝑛𝑇𝑢_𝑛+𝛾_𝑛𝜀_𝑛−𝛽_𝑛+𝛾_𝑛𝑢_𝑛‖‖²≤≤‖‖‖‖𝑢𝑢_𝑛−𝑞‖‖²+2𝛽_𝑛𝑇𝑢_𝑛+𝛾_𝑛𝜀_𝑛−𝛽_𝑛+𝛾_𝑛𝑢_𝑛,𝑗𝑢_𝑛+1−𝑞_𝑛+2𝛽−_𝑛𝑞‖‖²+2𝛽_𝑛𝑇𝑢_𝑛+1−𝑇𝑞,𝑗𝑢_𝑛+1−𝑞+2𝛾_𝑛𝑇𝑢_𝑛−𝑇𝑢_𝑛+1+𝑢_𝑛+1−𝑢_𝑛−𝑢_𝑛+1−𝑞,𝑗𝑢_𝑛+1−𝑞≤‖‖𝑢‖‖𝑢_𝑛−𝑞‖‖+‖𝑞‖+‖‖𝜀_𝑛‖‖‖‖𝑢_𝑛+1−𝑞_𝑛‖‖+2𝛽−_𝑛𝑞‖‖²−2𝛽_𝑛Φ‖‖𝑢_𝑛+1−𝑞‖‖+2𝛽_𝑛‖‖𝑇𝑢_𝑛−𝑇𝑢_𝑛+1‖‖⋅‖‖𝑢_𝑛+1−𝑞‖‖≤𝜔₂₀‖‖𝑢_𝑛−𝑢_𝑛+1‖‖⋅‖‖𝑢_𝑛+1−𝑞‖‖+2𝛾_𝑛𝜔₀+‖𝑞‖+𝑀‖‖𝑢_𝑛+1−𝑞‖‖−2𝛽_𝑛Φ𝜔₀+2𝛽_𝑛Φ𝜔₀12𝜔₀3𝜔₀2+2𝛽^𝑛Φ𝜔₀6𝜔₀3𝜔₀2+2𝛽^𝑛Φ𝜔₀6𝜔₀3𝜔₀<𝜔₂₀2,(2.3) which is a contraction with the assumption ‖𝑢_𝑛+1−𝑞‖>𝜔₀. Then, ‖𝑢_𝑛+1−𝑞‖≤𝜔₀, that is, the sequence {𝑢_𝑛} is bounded. It leads to lim_𝑛→∞‖‖𝑢_𝑛+1−𝑢_𝑛‖‖=0,lim_𝑛→∞‖‖𝑇𝑢_𝑛+1−𝑇𝑢_𝑛‖‖=0.(2.4)
Again using Lemma 1.7, we have ‖‖𝑢_𝑛+1−𝑞‖‖²=‖‖𝑢_𝑛−𝑞+𝛽_𝑛𝑇𝑢_𝑛+𝛾_𝑛𝜀_𝑛−𝛽_𝑛+𝛾_𝑛𝑢_𝑛‖‖²≤≤‖‖‖‖𝑢𝑢_𝑛−𝑞‖‖²+2𝛽_𝑛𝑇𝑢_𝑛+𝛾_𝑛𝜀_𝑛−𝛽_𝑛+𝛾_𝑛𝑢_𝑛,𝑗𝑢_𝑛+1−𝑞_𝑛+2𝛽−_𝑛𝑞‖‖²+2𝛽_𝑛𝑇𝑢_𝑛+1−𝑇𝑞,𝑗𝑥_𝑢+1−𝑞+2𝛾_𝑛𝑇𝑢_𝑛−𝑇𝑢_𝑛+1+𝑢_𝑛+1−𝑢_𝑛−𝑢_𝑛+1−𝑞,𝑗𝑢_𝑛+1−𝑞≤‖‖𝑢‖‖𝑢_𝑛−𝑞‖‖+‖𝑞‖+‖‖𝜀_𝑛‖‖‖‖𝑢_𝑛+1−𝑞_𝑛‖‖+2𝛽−_𝑛𝑞‖‖²−2𝛽_𝑛Φ‖‖𝑢_𝑛+1−𝑞‖‖+2𝛽_𝑛‖‖𝑇𝑢_𝑛−𝑇𝑢_𝑛+1‖‖⋅‖‖𝑢_𝑛+1−𝑞‖‖≤‖‖𝑢‖‖𝑢_𝑛−𝑢_𝑛+1‖‖⋅‖‖𝑢_𝑛+1−𝑞‖‖+2𝛾_𝑛𝜔₀+‖𝑞‖+𝑀‖‖𝑢_𝑛+1−𝑞_𝑛‖‖−𝑞‖‖²−2𝛽_𝑛Φ‖‖𝑢_𝑛+1−𝑞‖‖+𝐴_𝑛,(2.5) where 𝐴_𝑛=+2𝛽2_𝑛𝛾‖‖_𝑛𝑇𝑢_𝑛−𝑇𝑢_𝑛+1‖‖⋅‖‖𝑢_𝑛+1−𝑞‖‖+2𝛽_𝑛‖‖𝑢_𝑛−𝑢_𝑛+1‖‖⋅‖‖𝑢_𝑛+1−𝑞‖‖=𝑜𝛽_𝑛𝜔₀+‖𝑞‖+𝑀‖‖𝑢_𝑛+1−𝑞‖‖.(2.6) Therefore, (2.5) becomes ‖‖𝑢_𝑛+1−𝑞‖‖²≤‖‖𝑢_𝑛−𝑞‖‖²−2𝛽_𝑛Φ‖‖𝑢_𝑛+1−𝑞‖‖+𝑜𝛽_𝑛.(2.7) From Lemma 1.8, we obtain lim_𝑛→∞‖𝑢_𝑛−𝑞‖=0.

Theorem 2.2. Let 𝐾 be a nonempty closed convex subset of a real normed linear space 𝐸 and 𝑇∶𝐾→𝐾 a uniformly continuous generalized Φ-pseudocontractive mapping with 𝐹(𝑇)≠∅. Let {𝑢_𝑛},{𝑥_𝑛} be two sequences in 𝐾 defined iteratively from some 𝑢₀≠𝑥₀∈𝐾 by (1.11) and (1.10), where {𝛿_𝑛},{𝜂_𝑛},{𝜀_𝑛} are three arbitrary bounded sequences in 𝐾 and {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛},{𝛼^𝑛},{𝛽^𝑛}, and {𝛾^𝑛} are six sequences in [0,1] satisfying the following conditions:(i)𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=𝛼^𝑛+𝛽^𝑛+𝛾^𝑛=1, (ii)∑^∞𝑛=0𝛽_𝑛=∞, (iii)lim_𝑛→∞𝛽_𝑛=lim_𝑛→∞𝛽^𝑛=lim_𝑛→∞𝛾^𝑛=0, (iv)𝛾_𝑛=𝑜(𝛽_𝑛). Then, the following two assertions are equivalent:(1)Iteration (1.11) converges strongly to the unique fixed point of 𝑇,(2)Iteration (1.10) converges strongly to the unique fixed point of 𝑇.

Proof. Let 𝑞∈𝐹(𝑇). The uniqueness of 𝑞 comes from Definition 1.4. If the Ishikawa iteration with errors converges to 𝑞, then setting 𝛽^𝑛=𝛾^𝑛=0, for all 𝑛≥0 in (1.10), we can get the convergence of the Mann iteration with errors. Conversely, we only prove that (1)⇒(2), that is, if the Mann iteration with errors converges to 𝑞, we want to prove the convergence of the Ishikawa iteration with errors.
First, we will prove that there exists 𝑥₀∈𝐾 with 𝑥₀≠𝑇𝑥₀ such that 𝑡₀=‖𝑥₀−𝑇𝑥₀‖⋅‖𝑥₀−𝑞‖∈𝑅(Φ).
In fact, if 𝑥₀=𝑇𝑥₀, then we are done. Otherwise, there exists the smallest positive integer 𝑁₀∈𝑁 such that 𝑥_𝑁0≠𝑇𝑥_𝑁0. We denote 𝑥_𝑁0=𝑥₀, and then we obtain that 𝑡₀=‖𝑥₀−𝑇𝑥₀‖⋅‖𝑥₀−𝑞‖∈𝑅(Φ). Indeed if 𝑅(Φ)=[0,+∞), then 𝑡₀∈𝑅(Φ). If 𝑅(Φ)=[0,𝐴) with 0<𝐴<+∞, then for 𝑞∈𝐾 there exists a sequence {𝑤_𝑛}⊂𝐾 such that 𝑤_𝑛→𝑞 as 𝑛→∞ with 𝑤_𝑛≠𝑞, and we also obtain that the sequence {𝑤_𝑛−𝑇𝑤_𝑛} is bounded. So there exists 𝑛₀∈𝑁 such that ‖𝑤_𝑛−𝑇𝑤_𝑛‖⋅‖𝑤_𝑛−𝑞‖∈𝑅(Φ),𝑛≥𝑛₀, and then we redefine 𝑥₀=𝑤_𝑛0. Let 𝜇₀=Φ⁻¹(‖(𝑥₀−𝑞)−(𝑇𝑥₀−𝑞)‖⋅‖𝑥₀−𝑞‖)>0.
Second, we will prove that the sequence {𝑥_𝑛−𝑞} is a bounded sequence.
Set 𝑀𝐵₁=‖𝑥−𝑞‖≤𝜇₀∶𝑥∈𝐾,𝐵₂=‖𝑥−𝑞‖≤2𝜇₀∶𝑥∈𝐾₁,=maxsup_𝑥∈𝐵2‖𝑇𝑥‖;sup_𝑛∈𝑁‖‖𝜀_𝑛‖‖;sup_𝑛∈𝑁‖‖𝛿_𝑛‖‖;sup_𝑛∈𝑁‖‖𝜂_𝑛‖‖;sup_𝑛∈𝑁‖‖𝑢_𝑛‖‖;sup_𝑛∈𝑁‖‖𝑇𝑢_𝑛‖‖.(2.8) Since 𝑇 is uniformly continuous, for 𝜖=Φ(𝜇₀)/5𝜇₀, there exists 𝛿>0 such that ‖𝑇𝑥−𝑇𝑦‖<𝜖 whenever ‖𝑥−𝑦‖<𝛿. Now let 𝑘=min{1,𝜇₀/4(𝑀₁+𝜇₀+‖𝑞‖),𝛿/2(4𝑀₁+𝜇₀+‖𝑞‖),Φ(𝜇₀)/20𝜇₀(𝑀₁+𝜇₀+‖𝑞‖)}. By the control conditions (iii) and (iv), without loss of generality, set |𝛼_𝑛−𝛼^𝑛|,𝛽_𝑛,𝛾_𝑛,𝛽^𝑛,𝛾^𝑛<𝑘, for all𝑛≥0.
Observe that if 𝑥_𝑛∈𝐵₁, we obtain 𝑦_𝑛∈𝐵₂. Indeed ‖‖𝑦_𝑛−𝑞‖‖=≤‖‖‖‖𝑥1−𝛽^𝑛−𝛾^𝑛𝑥_𝑛+𝛽^𝑛𝑇𝑥_𝑛+𝛾^𝑛𝜂_𝑛−𝑞_𝑛‖‖≤𝜇₀−𝑞‖‖+𝛽^𝑛‖‖𝑇𝑥_𝑛‖‖+𝛽^𝑛+𝛾^𝑛‖‖𝑥_𝑛−𝑞‖‖+𝛾^𝑛‖‖𝜂_𝑛‖‖+𝛽^𝑛+𝛾^𝑛‖𝑞‖≤2𝜇₀+𝑘2𝑀₁+2𝜇₀+2‖𝑞‖.(2.9)
Next, by induction, we prove 𝑥_𝑛∈𝐵₁. Clearly, from (2.1), we obtain ‖𝑥₀−𝑞‖≤𝜇₀, that is, 𝑥₀∈𝐵₁. Set ‖𝑥_𝑛−𝑞‖≤𝜇₀, for some 𝑛; then we will prove that ‖𝑥_𝑛+1−𝑞‖≤𝜇₀. If it is not the case, we assume that ‖𝑥_𝑛+1−𝑞‖>𝜇₀. From (1.10), we obtain the following inequalities: ‖‖𝑥_𝑛+1−𝑞‖‖=≤‖‖‖‖𝛼_𝑛1𝑥_𝑛−+𝛽_𝑛𝛽_𝑛𝑇𝑦_𝑛+𝛾_𝑛𝛿_𝑛−𝑞‖‖≤‖‖𝑥_𝑛−𝛾_𝑛𝑥_𝑛−𝑞+𝛽_𝑛𝑇𝑦_𝑛+𝛾_𝑛𝛿_𝑛+𝛽_𝑛+𝛾_𝑛𝑞‖‖≤𝜇₀−𝑞‖‖+𝛽_𝑛+𝛾_𝑛‖‖𝑥_𝑛−𝑞‖‖+𝛽_𝑛‖‖𝑇𝑦_𝑛‖‖+𝛾_𝑛‖‖𝛿_𝑛‖‖+𝛽_𝑛+𝛾_𝑛‖𝑞‖+𝑘2𝜇₀+2𝑀₁+2‖𝑞‖≤54𝜇⁰‖‖𝑥_𝑛+1,−𝑥_𝑛‖‖=≤‖‖𝛽𝛽_𝑛𝑛𝑇𝑦_𝑛+𝛾_𝑛𝛿_𝑛−𝛽_𝑛+𝛾_𝑛𝑥_𝑛‖‖≤𝑘2𝜇+₀𝛾_𝑛‖‖𝑥_𝑛−𝑞‖‖+𝛽_𝑛‖‖𝑇𝑦_𝑛‖‖+𝛾_𝑛‖‖𝛿_𝑛‖‖+𝛽_𝑛+𝛾_𝑛‖𝑞‖+2𝑀₁+2‖𝑞‖≤Φ𝜇⁰10𝜇₀‖‖𝑥_𝑛+1,−𝑦_𝑛‖‖≤=‖‖||𝛼_𝑛𝛼_𝑛−𝛼^𝑛𝑥_𝑛+𝛽_𝑛𝑇𝑦_𝑛−𝛽^𝑛𝑇𝑥_𝑛+𝛾_𝑛𝛿_𝑛−𝛾^𝑛𝜂_𝑛‖‖≤𝑘𝜇−₀𝛼^𝑛||⋅‖‖𝑥_𝑛−𝑞‖‖+𝛽_𝑛‖‖𝑇𝑦_𝑛‖‖+𝛽^𝑛‖‖𝑇𝑥_𝑛‖‖+𝛾_𝑛‖‖𝛿_𝑛‖‖+𝛾^𝑛‖‖𝜂_𝑛‖‖+||𝛼_𝑛−𝛼^𝑛||⋅‖𝑞‖+4𝑀₁+‖𝑞‖≤𝛿.(2.10)
Since 𝑇 is a uniformly continuous mapping, then ‖‖𝑇𝑥_𝑛+1−𝑇𝑦_𝑛‖‖≤𝜖=Φ𝜇⁰5𝜇₀.(2.11)
Since 𝛾_𝑛=𝑜(𝛽_𝑛), we let 𝛾_𝑛<𝛽_𝑛⋅18⋅Φ𝜇⁰⋅54𝜇₀².(2.12)
From (2.9)–(2.12), we have ‖‖𝑥_𝑛+1−𝑞‖‖²=‖‖1−𝛽_𝑛−𝛾_𝑛𝑥_𝑛+𝛽_𝑛𝑇𝑦_𝑛+𝛾_𝑛𝛿_𝑛−𝑞‖‖²=‖‖𝑥_𝑛−𝑞+𝛽_𝑛𝑇𝑦_𝑛+𝛾_𝑛𝛿_𝑛−𝛽_𝑛+𝛾_𝑛𝑥_𝑛‖‖²≤≤‖‖‖‖𝑥𝑥_𝑛−𝑞‖‖²+2𝛽_𝑛𝑇𝑦_𝑛+𝛾_𝑛𝛿_𝑛−𝛽_𝑛+𝛾_𝑛𝑥_𝑛,𝑗𝑥_𝑛+1−𝑞_𝑛+2−𝛽_𝑛𝑞‖‖²+2𝛽_𝑛𝑇𝑥_𝑛+1−𝑇𝑞,𝑗𝑥_𝑛+1−𝑞+2𝛽_𝑛𝑇𝑦_𝑛−𝑇𝑥_𝑛+1,𝑗𝑥_𝑛+1−𝑞−2𝛽_𝑛+𝛾_𝑛𝑥_𝑛+1−𝑥_𝑛,𝑗𝑥_𝑛+1−𝑞+2𝛾_𝑛‖‖𝛿_𝑛‖‖⋅‖‖𝑥_𝑛+1−𝑞‖‖≤‖‖𝑥+𝛾_𝑛𝑥_𝑛+1−𝑞,𝑗𝑥_𝑛+1−𝑞_𝑛+2−𝛽_𝑛𝑞‖‖²−2𝛽_𝑛Φ‖‖𝑥_𝑛+1−𝑞‖‖+2𝛽_𝑛‖‖𝑇𝑥_𝑛+1−𝑇𝑦_𝑛‖‖⋅‖‖𝑥_𝑛+1−𝑞‖‖+𝛾_𝑛‖‖𝑥_𝑛+1−𝑥_𝑛‖‖⋅‖‖𝑥_𝑛+1−𝑞‖‖+2𝛾_𝑛‖‖𝛿_𝑛‖‖‖‖𝑥_𝑛+1−𝑞‖‖+2𝛾_𝑛‖‖𝑥_𝑛+1−𝑞‖‖²<𝜇²⁰−2𝛽_𝑛Φ𝜇₀+2𝛽_𝑛Φ𝜇₀5𝜇₀⋅5𝜇⁰4+4𝛽^𝑛Φ𝜇₀10𝜇₀⋅5𝜇⁰4+2𝛽^𝑛Φ𝜇₀5𝜇₀⋅5𝜇⁰+2𝛽_𝑛4⋅18⋅Φ𝜇⁰⋅54𝜇₀²⋅5𝜇⁰4²<𝜇²⁰,(2.13) which is a contradiction with the assumption ‖𝑥_𝑛+1−𝑞‖>𝜇₀. Hence, 𝑥_𝑛∈𝐵₁, that is, the sequence {𝑥_𝑛} is bounded. From (2.9), the sequence {𝑦_𝑛} is also bounded. So we obtain lim_𝑛→∞‖‖𝑥_𝑛+1−𝑥_𝑛‖‖=lim_𝑛→∞‖‖𝑢_𝑛+1−𝑢_𝑛‖‖=lim_𝑛→∞‖‖𝑥_𝑛+1−𝑦_𝑛‖‖=0.(2.14) Since 𝑇 is a uniformly continuous mapping, then lim_𝑛→∞‖‖𝑇𝑢_𝑛+1−𝑇𝑢_𝑛‖‖=lim_𝑛→∞‖‖𝑇𝑥_𝑛+1−𝑇𝑦_𝑛‖‖=0.(2.15)
Using the recursion formula (1.10), we compute as follows: ‖‖𝑥_𝑛+1−𝑢_𝑛+1‖‖²≤+‖‖𝑥2_𝑛𝛽_𝑛−𝑢_𝑛‖‖²−2𝛽_𝑛Φ‖‖𝑥_𝑛+1−𝑢_𝑛+1‖‖+2𝛽_𝑛‖‖𝑇𝑥_𝑛+1−𝑇𝑦_𝑛‖‖⋅‖‖𝑥_𝑛+1−𝑢_𝑛+1‖‖+2𝛽‖‖_𝑛𝑇𝑢_𝑛−𝑇𝑢_𝑛+1‖‖⋅‖‖𝑥_𝑛+1−𝑢_𝑛+1‖‖+2𝛽_𝑛+𝛾_𝑛‖‖𝑢_𝑛−𝑢_𝑛+1‖‖⋅‖‖𝑥_𝑛+1−𝑢_𝑛+1‖‖=‖‖𝑥_𝑛+𝛾_𝑛‖‖𝑥_𝑛+1−𝑥_𝑛‖‖⋅‖‖𝑥_𝑛+1−𝑢_𝑛+1‖‖+2𝛾_𝑛‖‖𝛿_𝑛‖‖+‖‖𝜀_𝑛‖‖‖‖𝑥_𝑛+1−𝑢_𝑛+1‖‖−𝑢_𝑛‖‖²−2𝛽_𝑛Φ‖‖𝑥_𝑛+1−𝑢_𝑛+1‖‖+𝐴_𝑛,(2.16) where 𝐴_𝑛=+2𝛽2_𝑛‖‖𝛽𝑇_𝑛𝑥_𝑛+1−𝑇𝑦_𝑛‖‖⋅‖‖𝑥_𝑛+1−𝑢_𝑛+1‖‖+2𝛽_𝑛‖‖𝑇𝑢_𝑛−𝑇𝑢_𝑛+1‖‖⋅‖‖𝑥_𝑛+1−𝑢_𝑛+1‖‖+2𝛾_𝑛+𝛾_𝑛‖‖𝑢_𝑛−𝑢_𝑛+1‖‖⋅‖‖𝑥_𝑛+1−𝑢_𝑛+1‖‖+2𝛽_𝑛+𝛾_𝑛‖‖𝑥_𝑛+1−𝑥_𝑛‖‖⋅‖‖𝑥_𝑛+1−𝑢_𝑛+1‖‖‖‖𝛿_𝑛‖‖+‖‖𝜀_𝑛‖‖‖‖𝑥_𝑛+1−𝑢_𝑛+1‖‖.(2.17)
Since the sequence {𝑥_𝑛−𝑢_𝑛} is bounded, from (2.15), (2.16), and the control condition (iv), we get 𝐴_𝑛=𝑜(𝛽_𝑛). Then, in (2.18), set 𝜃_𝑛=‖𝑥_𝑛−𝑢_𝑛‖,𝜆_𝑛=𝛽_𝑛, and 𝜎_𝑛=𝐴_𝑛. From Lemma 1.8, we obtain lim_𝑛→∞‖𝑥_𝑛+1−𝑢_𝑛+1‖=0. Since lim_𝑛→∞‖𝑢_𝑛−𝑞‖=0, using inequality 0≤‖𝑥_𝑛−𝑞‖≤‖𝑥_𝑛−𝑢_𝑛‖+‖𝑢_𝑛−𝑞‖, then lim_𝑛→∞‖𝑥_𝑛−𝑞‖=0, and we complete the proof of Theorem 2.1.

From Theorems 2.1 and 2.2, we obtain the following corollary.

Corollary 2.3. Let 𝐾 be a nonempty convex subset of a real normed linear space 𝐸. 𝑇∶𝐾→𝐾 is a uniformly continuous generalized Φ-hemicontractive mapping with 𝐹(𝑇)≠∅. Let {𝑥_𝑛} be a sequence in 𝐾 defined iteratively from some 𝑥₀∈𝐾 by (1.10), where {𝛿_𝑛},{𝜂_𝑛} are arbitrary bounded sequences in 𝐾 and {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛},{𝛼^𝑛},{𝛽^𝑛}, and {𝛾^𝑛} are sequences in [0,1] satisfying the following conditions:(i)𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=𝛼^𝑛+𝛽^𝑛+𝛾^𝑛=1, (ii)∑^∞𝑛=0𝛽_𝑛=∞, (iii)lim_𝑛→∞𝛽_𝑛=lim_𝑛→∞𝛽^𝑛=lim_𝑛→∞𝛾^𝑛=0, (iv)𝛾_𝑛=𝑜(𝛽_𝑛). Then, the iteration sequence {𝑥_𝑛} defined by (1.10) converges strongly to the unique fixed point of 𝑇.

Corollary 2.4. Let 𝑇∶𝐸→𝐸 be a uniformly continuous generalized Φ-accretive mapping with 𝑁(𝑇)≠∅. For some 𝑢₀,𝑥₀∈𝐸, the iteration sequences {𝑢_𝑛},{𝑥_𝑛} in 𝐸 are defined as follows: 𝑢_𝑛+1𝑥=𝛼_𝑛𝑢_𝑛+𝛽_𝑛𝑆𝑢_𝑛+𝛾_𝑛𝜀_𝑛,∀𝑛≥0,(2.18)_𝑛+1𝑦_𝑛=𝛼_𝑛𝑥_𝑛+𝛽_𝑛𝑆𝑦_𝑛+𝛾_𝑛𝛿_𝑛,=𝛼^𝑛𝑥_𝑛+𝛽^𝑛𝑆𝑥_𝑛+𝛾^𝑛𝜂_𝑛,∀𝑛≥0,(2.19) where 𝑆𝑥∶=𝑥−𝑇𝑥 for all 𝑥∈𝐸. {𝛿_𝑛},{𝜂_𝑛}, and {𝜀_𝑛} are arbitrary bounded sequences in 𝐾 and {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛},{𝛼^𝑛},{𝛽^𝑛}, and {𝛾^𝑛} are sequences in [0,1] satisfying the following conditions:(i)𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=𝛼^𝑛+𝛽^𝑛+𝛾^𝑛=1, (ii)∑^∞𝑛=0𝛽_𝑛=∞, (iii)lim_𝑛→∞𝛽_𝑛=lim_𝑛→∞𝛽^𝑛=lim_𝑛→∞𝛾^𝑛=0, (iv)𝛾_𝑛=𝑜(𝛽_𝑛). Then, the following two assertions are equivalent:(1){𝑥_𝑛} converges strongly to the unique fixed point of 𝑆,(2){𝑢_𝑛} converges strongly to the unique fixed point of 𝑆.

Proof. Let 𝑆=𝐼−𝑇, and observe that 𝑇 is a uniformly continuous generalized Φ-accretive mapping if and only if 𝑆 is a uniformly continuous generalized Φ-pseudocontractive mapping. The result follows from Theorem 2.2.

Remark 2.5. Our results improve and extend the corresponding results of [11] in the following sense.(i)We point out the gaps of C. E. Chidume and C. O. Chidume [11] in their proof and provide a counterexample.(ii)In the proof method, Theorem 2.1 differs from Theorem  2.3 in [11].(iii)The control conditions ∑^∞𝑛=0(𝑏_𝑛+𝑐_𝑛)²<∞ and ∑^∞𝑛=0𝑐_𝑛<∞ in Theorem  2.3 in [11] are replaced by the condition 𝑏_𝑛=𝑜(𝑐_𝑛). Under the new condition, we obtain the convergence theorem of the Mann iteration sequence with errors.(iv)We also obtain the equivalence of convergence results between the Ishikawa and Mann iterations with errors.(v)The Mann iteration sequence with errors is extended to the Ishikawa iteration sequence with errors.

Acknowledgments

This paper is supported by Hebei Province Natural Science Foundation (A2011210033) and Shijiazhuang Tiedao University Foundation (Q64).