On the Convergence of Multistep Iteration for Uniformly Continuous Φ-Hemicontractive Mappings

Abstract

It is shown that the convergence of the multistep iterative process with errors is obtained for uniformly continuous Φ-hemicontractive mappings in real Banach spaces. We also revise the problems of C. E. Chidume and C. O. Chidume (2005).

1. Introduction

Let 𝐸 be a real Banach space with norm ‖⋅‖ and let 𝐸^∗ be its dual space. The normalized duality mapping 𝐽∶𝐸→2^𝐸∗ is defined by 𝐽(𝑥)=𝑓∈𝐸^∗∶𝑥,𝑓=‖𝑥‖²=𝑓²,∀𝑥∈𝐸, (1.1) where ⟨⋅,⋅⟩ denotes the generalized duality pairing. The single-valued-normalized duality mapping is denoted by 𝑗.

A mapping 𝑇 with domain 𝐷(𝑇) and range 𝑅(𝑇) in 𝐸 is said to be strongly pseudocontractive if there is a constant 𝑘∈(0,1), and for all 𝑥,𝑦∈𝐷(𝑇),∃𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝑇𝑥−𝑇𝑦,𝑗𝑥−𝑦≤𝑘𝑥−𝑦². (1.2) The mapping 𝑇 is called Φ-pseudocontractive if there exists a strictly increasing continuous function Φ∶[0,+∞)→[0,+∞) with Φ(0)=0 such that 𝑇𝑥−𝑇𝑦,𝑗𝑥−𝑦≤𝑥−𝑦²−Φ𝑥−𝑦 (1.3) holds for all 𝑥,𝑦∈𝐷(𝑇). It is well known that the strongly pseudocontractive mapping must be the Φ-pseudocontractive mapping in the special case in which Φ(𝑡)=(1−𝑘)𝑡², but the converse is not true in general. That is, the class of strongly pseudocontractive mappings is a proper subclass of the class of Φ-pseudocontractive mappings. Let 𝐹(𝑇)={𝑥∈𝐷(𝑇)∶𝑇𝑥=𝑥}≠∅. If the inequalities (1.2) and (1.3) hold for any 𝑥∈𝐷(𝑇) and 𝑦∈𝐹(𝑇), then the corresponding mapping 𝑇 is called strongly hemicontractive and Φ-hemicontractive, respectively.

Let 𝑁(𝑇)={𝑥∈𝐸∶𝑇𝑥=0}≠∅. An operator 𝑇∶𝐷(𝑇)⊆𝐸→𝐸 is called strongly quasiaccretive, Φ-quasiaccretive if and only if 𝐼−𝑇 is strongly hemicontractive, Φ-hemicontractive, respectively, where 𝐼 denotes the identity mapping on 𝐸. That is, if 𝑇 is Φ-quasi-accretive, then 𝑁(𝑇)≠∅ and there exists a strictly increasing continuous function Φ∶[0,+∞)→[0,+∞) with Φ(0)=0 such that 𝑇𝑥−𝑇𝑦,𝑗𝑥−𝑦≥Φ𝑥−𝑦 (1.4) holds for all 𝑥∈𝐷(𝑇) and 𝑦∈𝑁(𝑇). Many authors have studied extensively the strongly convergence problems of the iterative algorithms for the class of operators.

In 2004, Rhoades and Soltuz [1] introduced the multistep iteration as follows.

Let 𝐷 be a nonempty closed convex subset of real Banach space 𝐸 and let 𝑇∶𝐷→𝐷 be a mapping. The multistep iteration {𝑥_𝑛} is defined by 𝑥₀𝑦∈𝐷,_𝑛𝑝−1=1−𝑏_𝑛^𝑝−1𝑥_𝑛+𝑏_𝑛^𝑝−1𝑇𝑥_𝑛𝑦,𝑛≥0,𝑝≥2,_𝑘𝑛=1−𝑏^𝑘_𝑛𝑥_𝑛+𝑏^𝑘_𝑛𝑇𝑦_𝑛^𝑘+1𝑥,𝑘=𝑝−2,𝑝−3,…,2,1,_𝑛+1=1−𝑎_𝑛𝑥_𝑛+𝑎_𝑛𝑇𝑦¹_𝑛,𝑛≥0, (1.5) where {𝑎_𝑛},{𝑏^𝑘_𝑛}(𝑘=1,2,…,𝑝−1) in [0,1] satisfy certain conditions. Obviously, the iteration defined above is generalization of Mann, Ishikawa, and Noor iterations.

Inspired and motivated by the work of Xu [2] and the iteration above, we discuss the following multistep iteration with errors: 𝑥₀𝑦∈𝐷,_𝑛𝑝−1=1−𝑏_𝑛^𝑝−1−𝑑_𝑛^𝑝−1𝑥_𝑛+𝑏_𝑛^𝑝−1𝑇𝑥_𝑛+𝑑_𝑛^𝑝−1𝑤_𝑛^𝑝−1𝑦,𝑛≥0,𝑝≥2,_𝑘𝑛=1−𝑏^𝑘_𝑛−𝑑^𝑘_𝑛𝑥_𝑛+𝑏^𝑘_𝑛𝑇𝑦_𝑛^𝑘+1+𝑑^𝑘_𝑛𝑤^𝑘_𝑛𝑥,𝑘=𝑝−2,𝑝−3,…,2,1,_𝑛+1=1−𝑎_𝑛−𝑐_𝑛𝑥_𝑛+𝑎_𝑛𝑇𝑦¹_𝑛+𝑐_𝑛𝑢_𝑛,𝑛≥0, (1.6) where {𝑎_𝑛},{𝑐_𝑛},{𝑏^𝑘_𝑛},{𝑑^𝑘_𝑛}(𝑘=1,2,…,𝑝−1) in [0,1] with 𝑎_𝑛+𝑐_𝑛≤1,𝑏^𝑘_𝑛+𝑑^𝑘_𝑛≤1, {𝑢_𝑛},{𝑤^𝑘_𝑛}(𝑘=1,2,…,𝑝−1) are the bounded sequences of 𝐷.

In 2005, C. E. Chidume and C. O. Chidume [3] proved the convergence theorems for fixed points of uniformly continuous generalized Φ-hemicontractive mappings and published in [3]. However, there exists a gap in the proof course of their theorems.

The aim of this paper is to show the convergence of the multistep iteration with errors for fixed points of uniformly continuous Φ-hemicontractive mappings and revise the results of C. E. Chidume and C. O. Chidume [3]. For this, we need the following Lemmas.

Lemma 1.1 (see [4]). Let 𝐸 be a real Banach space and let 𝐽∶𝐸→2^𝐸∗ be a normalized duality mapping. Then 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝑗𝑥+𝑦, (1.7) for all 𝑥,𝑦∈𝐸 and for all 𝑗(𝑥+𝑦)∈𝐽(𝑥+𝑦).

Lemma 1.2 (see [5]). Let {𝛿_𝑛}^∞_𝑛=0, {𝜆_𝑛}^∞_𝑛=0 and {𝛾_𝑛}^∞_𝑛=0 be three nonnegative real sequences and let Φ∶[0,+∞)→[0,+∞) be a strictly increasing and continuous function with Φ(0)=0 satisfying the following inequality: 𝛿²_𝑛+1≤𝛿²_𝑛−𝜆_𝑛Φ𝛿_𝑛+1+𝛾_𝑛,𝑛≥0, (1.8) where 𝜆_𝑛∈[0,1] with ∑^∞_𝑛=0𝜆_𝑛=∞, 𝛾_𝑛=𝑜(𝜆_𝑛). Then 𝛿_𝑛→0 as 𝑛→∞.

2. Main Results

Theorem 2.1. Let 𝐸 be an arbitrary real Banach space, 𝐷 a nonempty closed convex subset of 𝐸, and 𝑇∶𝐷→𝐷 a uniformly continuous Φ-hemicontractive mapping with 𝑞∈𝐹(𝑇)≠∅. Let {𝑎_𝑛},{𝑏^𝑘_𝑛},{𝑐_𝑛},{𝑑^𝑘_𝑛} be real sequences in [0,1] and satisfy the conditions:(i)𝑎_𝑛+𝑐_𝑛≤1,𝑏^𝑘_𝑛+𝑑^𝑘_𝑛≤1,𝑘=1,2,…,𝑝−1;(ii)𝑎_𝑛,𝑐_𝑛,𝑏^𝑘_𝑛,𝑑^𝑘_𝑛→0 as 𝑛→∞,𝑘=1,2,…,𝑝−1;(iii)𝑐_𝑛=𝑜(𝑎_𝑛), ∑^∞_𝑛=0𝑎_𝑛=∞.

For some 𝑥₀∈𝐷, let {𝑢_𝑛},{𝑤¹_𝑛},{𝑤²_𝑛},…,{𝑤_𝑛^𝑝−1} be any bounded sequences of 𝐷, and let {𝑥_𝑛} be the multistep iterative sequence with errors defined by (1.6). Then (1.6) converges strongly to the fixed point 𝑞 of 𝑇.

Proof. Since 𝑇∶𝐷→𝐷 is Φ-hemicontractive mapping, then there exists a strictly increasing continuous function Φ∶[0,+∞)→[0,+∞) with Φ(0)=0 such that 𝑇𝑥−𝑇𝑞,𝑗𝑥−𝑞≤𝑥−𝑞²−Φ𝑥−𝑞, (2.1) for 𝑥∈𝐷,𝑞∈𝐹(𝑇), that is 𝑇𝑥−𝑥,𝑗𝑥−𝑞≤−Φ𝑥−𝑞. (@) Choose some 𝑥₀∈𝐷 and 𝑥₀≠𝑇𝑥₀ such that ‖𝑥₀−𝑇𝑥₀‖⋅‖𝑥₀−𝑞‖∈𝑅(Φ) and denote that 𝑟₀=‖𝑥₀−𝑇𝑥₀‖⋅‖𝑥₀−𝑞‖, 𝑅(Φ) is the range of Φ. Indeed, if Φ(𝑟)→+∞ as 𝑟→+∞, then 𝑟₀∈𝑅(Φ); if sup{Φ(𝑟)∶𝑟∈[0,+∞)}=𝑟₁<+∞ with 𝑟₁<𝑟₀ (here, we only give a example. If 𝑟₀=2,Φ(𝑡)=𝑡/(1+𝑡), then sup{Φ(𝑟)∶𝑟∈[0,+∞)}=𝑟₁=1<2=𝑟₀), then for 𝑞∈𝐷, there exists a sequence {𝑤_𝑛} in 𝐷 such that 𝑤_𝑛→𝑞 as 𝑛→∞ with 𝑤_𝑛≠𝑞. Furthermore, we obtain that 𝑇𝑤_𝑛→𝑇𝑞 as 𝑛→∞. So {𝑤_𝑛−𝑇𝑤_𝑛} is the bounded sequence. Hence, there exists natural number 𝑛₀ such that ‖𝑤_𝑛−𝑇𝑤_𝑛‖⋅‖𝑤_𝑛−𝑞‖<𝑟₁/2 for 𝑛≥𝑛₀, then we redefine 𝑥₀=𝑤_𝑛0 and ‖𝑥₀−𝑇𝑥₀‖⋅‖𝑥₀−𝑞‖∈𝑅(Φ). This is to ensure that Φ⁻¹(𝑟₀) is defined well.
Step  1. We show that {𝑥_𝑛} is a bounded sequence.
Set 𝑅=Φ⁻¹(𝑟₀), then from above formula (@), we obtain that ‖𝑥₀−𝑞‖≤𝑅. Denote 𝐵₁=𝑥∈𝐷∶𝑥−𝑞≤𝑅,𝐵₂=𝑥∈𝐷∶𝑥−𝑞≤2𝑅. (2.2) Since 𝑇 is the uniformly continuous, so 𝑇 is a bounded mapping. We let 𝑀=sup_𝑥∈𝐵2𝑇𝑥−𝑞+1+maxsup_𝑛𝑤¹_𝑛−𝑞,sup_𝑛𝑤²_𝑛−𝑞,…,sup_𝑛𝑤_𝑛^𝑝−1−𝑞,sup_𝑛𝑢_𝑛−𝑞. (2.3) Next, we want to prove that 𝑥_𝑛∈𝐵₁. If 𝑛=0, then 𝑥₀∈𝐵₁. Now, assume that it holds for some 𝑛, that is, 𝑥_𝑛∈𝐵₁. We prove that 𝑥_𝑛+1∈𝐵₁. Suppose that it is not the case, then ‖𝑥_𝑛+1−𝑞‖>𝑅>𝑅/2. Since 𝑇 is uniformly continuous, then for 𝜖₀=Φ(𝑅/2)/8𝑅, there exists 𝛿>0 such that ‖𝑇𝑥−𝑇𝑦‖<𝜖₀ when ‖𝑥−𝑦‖<𝛿. Denote 𝜏₀𝑅=min1,𝑀,Φ(𝑅/2)(),𝛿8𝑅𝑀+2𝑅.2𝑀+4𝑅 (2.4) Since 𝑎_𝑛,𝑏^𝑘_𝑛,𝑐_𝑛,𝑑^𝑘_𝑛→0 as 𝑛→∞ for 𝑘=1,2,…,𝑝−1. Without loss of generality, we assume that 0≤𝑎_𝑛,𝑏^𝑘_𝑛,𝑐_𝑛,𝑑^𝑘_𝑛≤𝜏₀ for any 𝑛≥0. Since 𝑐_𝑛=𝑜(𝑎_𝑛), let 𝑐_𝑛<𝑎_𝑛𝜏₀. Now, estimate ‖𝑦^𝑘_𝑛−𝑞‖ for 𝑘=1,2,…,𝑝−1. By using (1.6), we have 𝑦_𝑛^𝑝−1−𝑞≤1−𝑏_𝑛^𝑝−1−𝑑_𝑛^𝑝−1𝑥_𝑛−𝑞+𝑏_𝑛^𝑝−1𝑇𝑥_𝑛−𝑞+𝑑_𝑛^𝑝−1𝑤_𝑛^𝑝−1−𝑞≤𝑅+𝜏₀𝑀≤2𝑅, (2.5) then 𝑦_𝑛^𝑝−1∈𝐵₂. Similarly, we have 𝑦_𝑛^𝑝−2−𝑞≤1−𝑏_𝑛^𝑝−2−𝑑_𝑛^𝑝−2𝑥_𝑛−𝑞+𝑏_𝑛^𝑝−2𝑇𝑦_𝑛^𝑝−1−𝑞+𝑑_𝑛^𝑝−2𝑤_𝑛^𝑝−2−𝑞≤𝑅+𝜏₀𝑀≤2𝑅, (2.6) then 𝑦_𝑛^𝑝−2∈𝐵₂,…. We have 𝑦¹_𝑛−𝑞≤1−𝑏¹_𝑛−𝑑¹_𝑛𝑥_𝑛−𝑞+𝑏¹_𝑛𝑇𝑦²_𝑛−𝑞+𝑑¹_𝑛𝑤¹_𝑛−𝑞≤𝑅+𝜏₀𝑀≤2𝑅, (2.7) then 𝑦¹_𝑛∈𝐵₂. Therefore, we get 𝑥_𝑛+1−𝑞≤1−𝑎_𝑛−𝑐_𝑛𝑥_𝑛−𝑞+𝑎_𝑛𝑇𝑦¹_𝑛−𝑞+𝑐_𝑛𝑢_𝑛−𝑞≤𝑅+𝜏₀𝑀≤2𝑅. (2.8) And we have 𝑥_𝑛+1−𝑥_𝑛≤𝑎_𝑛𝑇𝑦¹_𝑛−𝑥_𝑛+𝑐_𝑛𝑢_𝑛−𝑥_𝑛≤𝑎_𝑛𝑇𝑦¹_𝑛−𝑞+𝑥_𝑛−𝑞+𝑐_𝑛𝑢_𝑛−𝑞+𝑥_𝑛−𝑞≤𝜏₀𝑇𝑦¹_𝑛−𝑞+𝑢_𝑛−𝑞+2𝑥_𝑛−𝑞≤𝜏₀()≤Φ()𝑀+2𝑅𝑅/2,𝑥8𝑅_𝑛+1−𝑦¹_𝑛≤𝑎_𝑛𝑇𝑦¹_𝑛−𝑥_𝑛+𝑐_𝑛𝑢_𝑛−𝑥_𝑛+𝑏¹_𝑛𝑇𝑦²_𝑛−𝑥_𝑛+𝑑¹_𝑛𝑤¹_𝑛−𝑥_𝑛≤𝑎_𝑛𝑇𝑦¹_𝑛−𝑞+𝑥_𝑛−𝑞+𝑐_𝑛𝑢_𝑛−𝑞+𝑥_𝑛−𝑞+𝑏¹_𝑛𝑇𝑦²_𝑛−𝑞+𝑥_𝑛−𝑞+𝑑¹_𝑛𝑤¹_𝑛−𝑞+𝑥_𝑛−𝑞≤𝜏₀𝑇𝑦¹_𝑛−𝑞+𝑢_𝑛−𝑞+2𝑥_𝑛+−𝑞𝑇𝑦²_𝑛−𝑞+𝑤¹_𝑛−𝑞+2𝑥_𝑛−𝑞≤𝜏₀(2𝑀+4𝑅)≤𝛿. (2.9) Further, by using uniform continuity of 𝑇, we have 𝑇𝑥_𝑛+1−𝑇𝑦¹_𝑛<Φ(𝑅/2).8𝑅 (2.10) In view of Lemma 1.1 and the above formulas, we obtain 𝑥_𝑛+1−𝑞²=𝑥_𝑛−𝑞+𝑎_𝑛𝑇𝑦¹_𝑛−𝑥_𝑛+𝑐_𝑛𝑢_𝑛−𝑥_𝑛²≤𝑥_𝑛−𝑞²+2𝑎_𝑛𝑇𝑦¹_𝑛−𝑥_𝑛,𝑗𝑥_𝑛+1−𝑞+2𝑐_𝑛𝑢_𝑛−𝑥_𝑛,𝑗𝑥_𝑛+1≤𝑥−𝑞_𝑛−𝑞²+2𝑎_𝑛𝑇𝑥_𝑛+1−𝑥_𝑛+1+𝑥_𝑛+1−𝑥_𝑛−𝑇𝑥_𝑛+1+𝑇𝑦¹_𝑛,𝑗𝑥_𝑛+1−𝑞+2𝑐_𝑛𝑢_𝑛−𝑥_𝑛⋅𝑥_𝑛+1≤𝑥−𝑞_𝑛−𝑞²−2𝑎_𝑛Φ𝑥_𝑛+1−𝑞+2𝑎_𝑛𝑥_𝑛+1−𝑥_𝑛⋅𝑥_𝑛+1−𝑞+2𝑎_𝑛𝑇𝑥_𝑛+1−𝑇𝑦¹_𝑛⋅𝑥_𝑛+1−𝑞+2𝑐_𝑛𝑢_𝑛−𝑞+𝑥_𝑛−𝑞𝑥_𝑛+1≤𝑥−𝑞_𝑛−𝑞²−2𝑎_𝑛Φ𝑅2+2𝑎_𝑛Φ(𝑅/2)8𝑅⋅2𝑅+2𝑎_𝑛Φ(𝑅/2)8𝑅⋅2𝑅+2𝑎_𝑛𝜏₀()≤𝑥𝑅+𝑀2𝑅_𝑛−𝑞²−𝑎_𝑛Φ𝑅2+2𝑎_𝑛Φ(𝑅/2)()()≤𝑥8𝑅𝑀+2𝑅𝑅+𝑀2𝑅_𝑛−𝑞²−𝑎_𝑛2Φ𝑅2≤𝑅², (2.11) which is a contradiction. Hence, 𝑥_𝑛+1∈𝐵₁, that is, {𝑥_𝑛} is a bounded sequence; it leads to that {𝑦¹_𝑛},{𝑦²_𝑛},…,{𝑦_𝑛^𝑝−1} are all bounded sequences as well.
Step 2. We want to prove ‖𝑥_𝑛−𝑞‖→0 as 𝑛→∞.
Since 𝑎_𝑛,𝑏^𝑘_𝑛,𝑐_𝑛,𝑑^𝑘_𝑛→0 as 𝑛→∞ and {𝑥_𝑛},{𝑦¹_𝑛} are bounded. From (2.9), we obtain lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0,lim_𝑛→∞𝑥_𝑛+1−𝑦¹_𝑛=0,lim_𝑛→∞𝑇𝑥_𝑛+1−𝑇𝑦¹_𝑛=0. (2.12) By (2.11), we have 𝑥_𝑛+1−𝑞²=𝑥_𝑛−𝑞+𝑎_𝑛𝑇𝑦¹_𝑛−𝑥_𝑛+𝑐_𝑛𝑢_𝑛−𝑥_𝑛²≤𝑥_𝑛−𝑞²+2𝑎_𝑛𝑇𝑦¹_𝑛−𝑥_𝑛,𝑗𝑥_𝑛+1−𝑞+2𝑐_𝑛𝑢_𝑛−𝑥_𝑛,𝑗𝑥_𝑛+1≤𝑥−𝑞_𝑛−𝑞²+2𝑎_𝑛𝑇𝑥_𝑛+1−𝑥_𝑛+1+𝑥_𝑛+1−𝑥_𝑛−𝑇𝑥_𝑛+1+𝑇𝑦¹_𝑛,𝑗𝑥_𝑛+1−𝑞+2𝑐_𝑛𝑢_𝑛−𝑥_𝑛⋅𝑥_𝑛+1≤𝑥−𝑞_𝑛−𝑞²−2𝑎_𝑛Φ𝑥_𝑛+1−𝑞+2𝑎_𝑛𝑥_𝑛+1−𝑥_𝑛⋅𝑥_𝑛+1−𝑞+2𝑎_𝑛𝑇𝑥_𝑛+1−𝑇𝑦¹_𝑛⋅𝑥_𝑛+1−𝑞+2𝑐_𝑛𝑢_𝑛−𝑥_𝑛⋅𝑥_𝑛+1=𝑥−𝑞_𝑛−𝑞²−2𝑎_𝑛Φ𝑥_𝑛+1−𝑞+𝑜𝑎_𝑛, (2.13) where 2𝑎_𝑛‖𝑥_𝑛+1−𝑥_𝑛‖⋅‖𝑥_𝑛+1−𝑞‖+2𝑎_𝑛‖𝑇𝑥_𝑛+1−𝑇𝑦¹_𝑛‖⋅‖𝑥_𝑛+1−𝑞‖+2𝑐_𝑛‖𝑢_𝑛−𝑥_𝑛‖⋅‖𝑥_𝑛+1−𝑞‖=𝑜(𝑎_𝑛). By Lemma 1.2, we obtain that lim_𝑛→∞‖𝑥_𝑛−𝑞‖=0.

Theorem 2.2. Let 𝐸 be an arbitrary real Banach space and let 𝑇∶𝐸→𝐸 be a uniformly continuous Φ-quasi-accretive operator with 𝑞∈𝑁(𝑇)≠∅. Let {𝑎_𝑛},{𝑏^𝑘_𝑛},{𝑐_𝑛},{𝑑^𝑘_𝑛} be real sequences in [0,1] and satisfy the conditions:(i)𝑎_𝑛+𝑐_𝑛≤1,𝑏^𝑘_𝑛+𝑑^𝑘_𝑛≤1,𝑘=1,2,…,𝑝−1;(ii)𝑎_𝑛,𝑐_𝑛,𝑏^𝑘_𝑛,𝑑^𝑘_𝑛→0 as 𝑛→∞,𝑘=1,2,…,𝑝−1;(iii)𝑐_𝑛=𝑜(𝑎_𝑛), ∑^∞_𝑛=0𝑎_𝑛=∞. For some 𝑥₀∈𝐸, let {𝑢_𝑛},{𝑤¹_𝑛},{𝑤²_𝑛},…,{𝑤_𝑛^𝑝−1} be any bounded sequences of 𝐸, and let {𝑥_𝑛} be the multistep iterative sequence with errors defined by 𝑥₀𝑦∈𝐷,_𝑛𝑝−1=1−𝑏_𝑛^𝑝−1−𝑑_𝑛^𝑝−1𝑥_𝑛+𝑏_𝑛^𝑝−1𝑆𝑥_𝑛+𝑑_𝑛^𝑝−1𝑤_𝑛^𝑝−1𝑦,𝑛≥0,𝑝≥2,_𝑘𝑛=1−𝑏^𝑘_𝑛−𝑑^𝑘_𝑛𝑥_𝑛+𝑏^𝑘_𝑛𝑆𝑦_𝑛^𝑘+1+𝑑^𝑘_𝑛𝑤^𝑘_𝑛𝑥,𝑘=𝑝−2,𝑝−3,…,2,1,_𝑛+1=1−𝑎_𝑛−𝑐_𝑛𝑥_𝑛+𝑎_𝑛𝑆𝑦¹_𝑛+𝑐_𝑛𝑢_𝑛,𝑛≥0, (2.14) where 𝑆∶𝐸→𝐸 is defined by 𝑆𝑥=𝑥−𝑇𝑥 for all 𝑥∈𝐸. Then (2.14) converges strongly to the fixed point 𝑞 of 𝑆.

Proof. We find easily that 𝑆 is a uniformly continuous Φ-hemicontractive. Then the conclusion of Theorem 2.2 is obtained directly by Theorem 2.1.

Remark 2.3. In Theorems 2.1 and 2.2, if 𝑏^𝑘_𝑛=𝑑^𝑘_𝑛=0(𝑘=𝑝−1,𝑝−2,…,2,1), then, the conclusions are as follows.

Corollary 2.4. Let 𝐸 be an arbitrary real Banach space, 𝐷 a nonempty closed convex subset of 𝐸, and 𝑇∶𝐷→𝐷 a uniformly continuous Φ-hemicontractive mapping with 𝑞∈𝐹(𝑇)≠∅. Let {𝑎_𝑛},{𝑐_𝑛} be real sequences in [0,1] and satisfy the conditions (i)  𝑎_𝑛+𝑐_𝑛≤1; (ii)  𝑎_𝑛,𝑐_𝑛→0 as 𝑛→∞; (iii)  𝑐_𝑛=𝑜(𝑎_𝑛) and ∑^∞_𝑛=0𝑎_𝑛=∞. For some 𝑥₀∈𝐷, let {𝑢_𝑛} be any bounded sequence of 𝐷, and let {𝑥_𝑛} be Mann iterative sequence with errors defined by 𝑥_𝑛+1=(1−𝑎_𝑛−𝑐_𝑛)𝑥_𝑛+𝑎_𝑛𝑇𝑥_𝑛+𝑐_𝑛𝑢_𝑛,𝑛≥0. Then {𝑥_𝑛} converges strongly to the fixed point 𝑞 of 𝑇.

Corollary 2.5. Let 𝐸 be an arbitrary real Banach space and let 𝑇∶𝐸→𝐸 be a uniformly continuous Φ-quasi-accretive operator with 𝑞∈𝑁(𝑇)≠∅. Let {𝑎_𝑛},{𝑐_𝑛} be real sequences in [0,1] and satisfy the conditions (i)𝑎_𝑛+𝑐_𝑛≤1; (ii)𝑎_𝑛,𝑐_𝑛→0 as 𝑛→∞; (iii)𝑐_𝑛=𝑜(𝑎_𝑛) and ∑^∞_𝑛=0𝑎_𝑛=∞. For some 𝑥₀∈𝐸, let {𝑢_𝑛} be any bounded sequence of 𝐸, and let {𝑥_𝑛} be Mann iterative sequence with errors defined by 𝑥_𝑛+1=(1−𝑎_𝑛−𝑐_𝑛)𝑥_𝑛+𝑎_𝑛𝑆𝑥_𝑛+𝑐_𝑛𝑢_𝑛,𝑛≥0. where 𝑆∶𝐸→𝐸 is defined by 𝑆𝑥=𝑥−𝑇𝑥 for all 𝑥∈𝐸. Then {𝑥_𝑛} converges strongly to the fixed point 𝑞 of 𝑆.

Remark 2.6. It is mentioned to notice that there exists a serious shortcoming in the proof process of Theorem 2.3 of [3]. That is, 𝑀₁𝑐_𝑛≤(Φ(𝜖)/4)𝛼_𝑛 does not hold in line 15 of Claim 2 of page 552. The reason is that the conditions ∑^∞_𝑛=0𝑐_𝑛<+∞ and ∑^∞_𝑛=0𝑏_𝑛=+∞, 𝑏_𝑛→0 as 𝑛→∞ can not obtain 𝑐_𝑛=𝑜(𝑏_𝑛).
Counterexample, let the iteration parameters be 𝑎_𝑛=1−𝑏_𝑛−𝑐_𝑛,𝑏_𝑛,𝑐_𝑛 in the following: 𝑏_𝑛∶𝑏₀=𝑏₁=0,𝑏_𝑛=1𝑛𝑐,𝑛≥2,_𝑛1∶0,1,12²,13²,14,15²,16²,17²18²,19,110²,111²,112²,113²,114²,115²,1,11617²,118²1,…,23²,124²,1,12526²1,…,35²,1,13637²,… (2.15) Then, ∑^∞_𝑛=0𝑏_𝑛=+∞, ∑^∞_𝑛=0𝑐_𝑛<2∑^∞_𝑛=1(1/𝑛²)<+∞, but 𝑐_𝑛≠𝑜(𝑏_𝑛).

Application 1. Let 𝐸=𝑅 be a real number space with the usual norm and 𝐷=[0,+∞). Define 𝑇∶𝐷→𝐷 by 𝑥𝑇𝑥=³1+𝑥² (2.16) for all 𝑥∈𝐷. Then 𝑇 is uniformly continuous with 𝐹(𝑇)={0}. Define Φ∶[0,+∞)→[0,+∞) by Φ(𝑡)=𝑡²1+𝑡², (2.17) then Φ is a strictly increasing function with Φ(0)=0. For all 𝑥∈𝐷,𝑞∈𝐹(𝑇), we obtain that =𝑥𝑇𝑥−𝑇𝑞,𝑗𝑥−𝑞³1+𝑥²(=𝑥−0,𝑗𝑥−0)₃1+𝑥²=𝑥,𝑥₄1+𝑥²=𝑥−𝑞²−𝑥−𝑞²1+𝑥−𝑞²=𝑥−𝑞²−Φ𝑥−𝑞. (2.18) Therefore, 𝑇 is a Φ-hemicontractive mapping. Set 𝑎_𝑛=1𝑛+2,𝑐_𝑛=1(𝑛+2)²,𝑏^𝑘_𝑛=𝑑^𝑘_𝑛=1𝑛+2,𝑘=1,2,…,𝑝−1 (2.19) for all 𝑛≥0.

Acknowledgment

This work is supported by Hebei Natural Science Foundation under Grant no. A2011210033.