The Equivalence of Convergence Results between Mann and Multistep Iterations with Errors for Uniformly Continuous Generalized Weak Φ-Pseudocontractive Mappings in Banach Spaces

Abstract

We prove the equivalence of the convergence of the Mann and multistep iterations with errors for uniformly continuous generalized weak Φ-pseduocontractive mappings in Banach spaces. We also obtain the convergence results of Mann and multistep iterations with errors. Our results extend and improve the corresponding results.

1. Introduction

Let 𝐸 be a real Banach space, 𝐸^∗ be its dual space, and 𝐽∶𝐸→2^𝐸∗ be the normalized duality mapping defined by 𝐽(𝑥)=𝑓∈𝐸^∗∶𝑥,𝑓=‖𝑥‖⋅𝑓=𝑓², (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,𝑗𝑥−𝑦≥𝑘𝑥−𝑦²; (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,𝑗𝑥−𝑦≥𝜙𝑥−𝑦𝑥−𝑦,∀𝑥,𝑦∈𝐸; (1.3)(3) generalized Φ-accretive if, for any 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,𝑗𝑥−𝑦≥Φ𝑥−𝑦. (1.4)

Remark 1.2. Let 𝑁(𝑇)={𝑥∈𝐸∶𝑇𝑥=0}≠∅. If 𝑥,𝑦∈𝐸 in the formulas of Definition 1.1 is replaced by 𝑥∈𝐸, 𝑞∈𝑁(𝑇), then 𝑇 is called strongly quasi-accretive, 𝜙-strongly quasi-accretive, generalized Φ-quasi-accretive mapping, respectively.

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

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,𝑗𝑥−𝑦≤𝑘𝑥−𝑦²; (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,𝑗𝑥−𝑦≤𝑥−𝑦²−𝜙𝑥−𝑦𝑥−𝑦,∀𝑥,𝑦∈𝐷(𝑇); (1.6)(3) generalized Φ-pseudocontractive if, for any 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,𝑗𝑥−𝑦≤𝑥−𝑦²−Φ𝑥−𝑦. (1.7)

Definition 1.4. Let 𝐹(𝑇)={𝑥∈𝐸∶𝑇𝑥=𝑥}≠∅. The mapping 𝑇 is called Φ-strongly pseudocontractive, generalized Φ-pseudocontractive, if, for all 𝑥∈𝐷(𝑇), 𝑞∈𝐹(𝑇), the formula (2), (3) in the above Definition 1.3 hold.

Definition 1.5. A mapping 𝑇 is said to be(1) generalized weak 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 ≥Φ𝑇𝑥−𝑇𝑦,𝑗𝑥−𝑦𝑥−𝑦1+𝑥−𝑦²;+Φ𝑥−𝑦 (1.8)(2) generalized weak Φ-quasi-accretive if, for all 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 ≥Φ𝑇𝑥−𝑞,𝑗𝑥−𝑞𝑥−𝑞1+𝑥−𝑞²;+Φ𝑥−𝑞 (1.9)(3) generalized weak Φ-pseudocontractive if, for any 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,𝑗𝑥−𝑦≤𝑥−𝑦²−Φ𝑥−𝑦1+𝑥−𝑦²;+Φ𝑥−𝑦 (1.10)(4) generalized weak Φ-hemicontractive if, for any 𝑥∈𝐾, 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−𝑞,𝑗𝑥−𝑞≤𝑥−𝑞²−Φ𝑥−𝑞1+𝑥−𝑞².+Φ𝑥−𝑞 (1.11)

It is very well known that a mapping 𝑇 is strongly pseudocontractive (hemicontractive), 𝜙-strongly pseudocontractive (𝜙-strongly hemicontractive), generalized Φ-pseudocontractive (generalized Φ-hemicontractive), generalized weak Φ-pseudocontractive (generalized weak Φ-hemicontractive) if and only if (𝐼−𝑇) are strongly accretive (quasi-accretive), 𝜙-strongly accretive (𝜙-strongly quasi-accretive), (𝐼−𝑇) is generalized Φ-accretive (generalized Φ-quasi-accretive), generalized weak Φ-accretive (weak 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 hemicontractive mappings with the nonempty fixed point set is a proper subclass of generalized Φ-hemicontractive mappings. Obviously, generalized Φ-hemicontractive mapping must be generalized weak Φ-hemicontractive, but, on the contrary, it is not true. We have the following example.

Example 1.6. Let 𝐸=(−∞,+∞) be real number space with usual norm and 𝐾=[0,+∞). 𝑇∶𝐾→𝐸 defined by 𝑇𝑥=𝑥+𝑥³+𝑥²√𝑥−√𝑥1+𝑥√𝑥+𝑥²,∀𝑥∈𝐾. (1.12) Then 𝑇 has a fixed point 0∈𝐹(𝑇). Φ∶[0,+∞)→[0,+∞) defined by Φ(𝑡)=𝑡^3/2 is a strictly increasing function with Φ(0)=0. For all 𝑥∈𝐾 and 0∈𝐹(𝑇), we have 𝑇𝑥−𝑇0,𝑗(𝑥−0)=𝑥+𝑥³+𝑥²√𝑥−√𝑥1+𝑥√𝑥+𝑥²=𝑥−0,𝑥−0₂+𝑥⁴+𝑥³√𝑥−𝑥√𝑥1+𝑥√𝑥+𝑥²=𝑥²−𝑥^3/21+𝑥^3/2+𝑥²=|𝑥−0|²−Φ(𝑥)1+Φ(𝑥)+𝑥²=|𝑥−0|²(𝑥)≥||−𝜎𝑥−0²−Φ(𝑥). (1.13) Then 𝑇 is a generalized weak Φ-hemicontractive map, but it is not a generalized Φ-hemicontractive map; that is, the class of generalized weak Φ-hemicontractive maps properly contains the class of generalized Φ-hemicontractive maps. Hence the class of generalized weak Φ-hemicontractive mappings is the most general among those defined above.

Definition 1.7. The mapping 𝑇∶𝐸→𝐸 is called Lipschitz, if there exists a constant 𝐿>0 such that 𝑇𝑥−𝑇𝑦≤𝐿𝑥−𝑦,∀𝑥,𝑦∈𝐸. (1.14) It is clear that if 𝑇 is Lipschitz, then it must be uniformly continuous. Otherwise, it is not true. For example, the function 𝑓(𝑥)=√𝑥,𝑥∈[0,+∞) is uniformly continuous but it is not Lipschitz.

Now let us consider the multi-step iteration with errors. Let 𝐾 be a nonempty convex subset of 𝐸, and let {𝑇_𝑖}^𝑀_𝑖=1 be a finite family of self-maps of 𝐾. For 𝑥₀∈𝐾, the sequence {𝑥_𝑛} is generated as follows: 𝑥_𝑛+1=1−𝛼_𝑛−𝛿_𝑛𝑥_𝑛+𝛼_𝑛𝑇_𝑛𝑦¹_𝑛+𝛿_𝑛𝜈_𝑛,𝑦^𝑖_𝑛=1−𝛽^𝑖_𝑛−𝜂^𝑖_𝑛𝑥_𝑛+𝛽^𝑖_𝑛𝑇_𝑛𝑦_𝑛^𝑖+1+𝜂^𝑖_𝑛𝜔^𝑖_𝑛𝑦,𝑖=1,…,𝑝−2,_𝑛𝑝−1=1−𝛽_𝑛^𝑝−1−𝜂_𝑛^𝑝−1𝑥_𝑛+𝛽_𝑛^𝑝−1𝑇_𝑛𝑥_𝑛+𝜂_𝑛^𝑝−1𝜔_𝑛^𝑝−1,𝑝≥2, (1.15) where {𝜈_𝑛},{𝜔^𝑖_𝑛} are any bounded sequences in 𝐾 and {𝛼_𝑛},{𝛿_𝑛},{𝛽^𝑖_𝑛},{𝜂^𝑖_𝑛},(𝑖=1,2,…,𝑝−1) are sequences in [0,1] satisfying certain conditions.

If 𝑝=2, (1.15) becomes the Ishikawa iteration sequence with errors {𝑥_𝑛}^∞_𝑛=0 defined iteratively by 𝑥_𝑛+1=1−𝛼_𝑛−𝛿_𝑛𝑥_𝑛+𝛼_𝑛𝑇_𝑛𝑦¹_𝑛+𝛿_𝑛𝜈_𝑛,𝑦¹_𝑛=1−𝛽_𝑛−𝜂_𝑛𝑥_𝑛+𝛽_𝑛𝑇_𝑛𝑥_𝑛+𝜂_𝑛𝜔_𝑛,∀𝑛≥0. (1.16)

If 𝛽_𝑛=𝜂_𝑛=0, for all 𝑛≥0, then from (1.16), we get the Mann iteration sequence with errors {𝑢_𝑛}^∞_𝑛=0 defined by 𝑢_𝑛+1=1−𝛼_𝑛−𝛿_𝑛𝑢_𝑛+𝛼_𝑛𝑇_𝑛𝑢_𝑛+𝛿_𝑛𝜇_𝑛,∀𝑛≥0, (1.17) where {𝜇_𝑛}⊂𝐾 is bounded.

Recently, many authors have researched the iteration approximation of fixed points by Lipschitz pseudocontractive (accretive) type nonlinear mappings and have obtained some excellent results [3–12]. In this paper we prove the equivalence between the Mann and multi-step iterations with errors for uniformly continuous generalized weak Φ-pseduocontractive mappings in Banach spaces. Our results extend and improve the corresponding results [3–12].

Lemma 1.8 (see [13]). Let 𝐸 be a real normed space. Then, for all 𝑥,𝑦∈𝐸, the following inequality holds: 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝑗𝑥+𝑦,∀𝑗𝑥+𝑦∈𝐽𝑥+𝑦. (1.18)

Lemma 1.9 (see [14]). Let {𝜌_𝑛} be nonnegative sequence which satisfies the following inequality: 𝜌_𝑛+1≤1−𝜆_𝑛𝜌_𝑛+𝜎_𝑛,𝑛≥𝑁, (1.19) where 𝜆_𝑛∈(0,1),lim_𝑛→∞𝜆_𝑛=0 and ∑^∞_𝑛=0𝜆_𝑛=∞,𝜎_𝑛=𝑜(𝜆_𝑛). Then 𝜌_𝑛→0 as 𝑛→∞.

Lemma 1.10. Let {𝜃_𝑛},{𝑐_𝑛},{𝑒_𝑛} and {𝑡_𝑛} be four nonnegative real sequences satisfying the following conditions: (i) lim_𝑛→∞𝑡_𝑛=0; (ii) ∑^∞_𝑛=0𝑡_𝑛=∞; (iii) 𝑐_𝑛=𝑜(𝑡_𝑛),𝑒_𝑛=𝑜(𝑡_𝑛). Let Φ∶[0,+∞)→[0,+∞) be a strictly increasing and continuous function with Φ(0)=0 such that 𝜃²_𝑛+1≤1+𝑐_𝑛𝜃²_𝑛−𝑡_𝑛Φ𝜃_𝑛+11+Φ𝜃_𝑛+1+𝜃²_𝑛+1+𝑒_𝑛,𝑛≥0. (1.20) If {𝜃_𝑛} is bounded, then 𝜃_𝑛→0 as 𝑛→∞.

Proof. Since lim_𝑛→∞𝑡_𝑛=0,{𝜃_𝑛} is bounded, we set 𝑅=max{sup_𝑛≥0𝑡_𝑛,sup_𝑛≥0𝜃_𝑛}, 𝛾=liminf_𝑛→∞(Φ(𝜃_𝑛+1)/(1+𝜃²_𝑛+1)[1+Φ(𝑅)+𝑅²]), then 𝛾=0. Otherwise, we assume that 𝛾>0, then there exists a constant 𝛿>0 with 𝛿=min{1,𝛾} and a natural number 𝑁₁ such that Φ𝜃_𝑛+1>𝛿+𝛿𝜃²_𝑛+11+Φ(𝑅)+𝑅²>𝛿𝜃²_𝑛+11+Φ(𝑅)+𝑅², (1.21) for 𝑛>𝑁₁.
Then, from (1.20), we get 𝜃²_𝑛+1≤1+𝑐_𝑛1+𝛿𝑡_𝑛𝜃²_𝑛+𝑒_𝑛. (1.22) Since 𝑐_𝑛=𝑜(𝑡_𝑛), there exists a nature number 𝑁₂>𝑁₁, such that 𝑐_𝑛<(𝛿/2)𝑡_𝑛,𝑛>𝑁₂. Hence (1+𝑐_𝑛)/(1+𝛿𝑡_𝑛)<1−(𝛿/2)𝑡_𝑛 and (1.22) becomes 𝜃²_𝑛+1≤𝛿1−2𝑡_𝑛𝜃²_𝑛+𝑒_𝑛. (1.23) By Lemma 1.9, we obtain that 𝜃_𝑛→0 as 𝑛→∞. Since Φ is strictly increasing and continuous with Φ(0)=0. Hence 𝛾=0, which is contradicting with the assumption 𝛾>0. Then 𝛾=0, there exists a subsequence {𝜃_𝑛𝑗} of {𝜃_𝑛} such that 𝜃_𝑛𝑗→0 as 𝑗→∞. Let 0<𝜀<1 be any given. Since 𝑐_𝑛=𝑜(𝑡_𝑛), 𝑒_𝑛=𝑜(𝑡_𝑛), then there exists a natural number 𝑁₃>𝑁₂, such that 𝜃_𝑛𝑗<𝜀,𝑐_𝑛𝑗<Φ(𝜀)2𝑀²1+𝑅²(𝑅)𝑡+Φ^𝑛𝑗,𝑒_𝑛𝑗<Φ(𝜀)21+𝑅²(𝑅)𝑡+Φ^𝑛𝑗 (1.24) for all 𝑗>𝑁₃. Next, we will show that 𝜃_{𝑛𝑗+𝑚}<𝜀 for all 𝑚=1,2,3,…. First, we want to prove that 𝜃_𝑛𝑗+1<𝜀. Suppose that it is not the case, then 𝜃_𝑛𝑗+1≥𝜀. Since Φ is strictly increasing, Φ𝜃_𝑛𝑗+1≥Φ(𝜀). (1.25) From (1.24) and (1.25), we obtain that 𝜃²_𝑛𝑗+1≤1+𝑐_𝑛𝑗𝜃²_𝑛𝑗−2𝑡_𝑛𝑗Φ(𝜀)1+𝑅²+Φ(𝑅)+𝑒_𝑛𝑗<𝜃²_𝑛𝑗−Φ(𝜀)1+𝑅²(𝑅)𝑡+Φ^𝑛𝑗<𝜃²_𝑛𝑗<𝜀². (1.26) That is 𝜃_𝑛𝑗+1<𝜀, which is a contradiction. Hence 𝜃_𝑛𝑗+1<𝜀. Now we assume that 𝜃_{𝑛𝑗+𝑚}<𝜀 holds. Using the similar way, it follows that 𝜃_{𝑛𝑗+𝑚+1}<𝜀. Therefore, this shows that 𝜃_𝑛→0 as 𝑛→∞.

2. Main Results

Theorem 2.1. Let 𝐾 be a nonempty closed convex subset of a Banach space 𝐸. Suppose that 𝑇_𝑛=𝑇_{𝑛(mod𝑀)}, and 𝑇_𝑖∶𝐾→𝐾,𝑖∈𝐼={1,2,…,𝑀} are 𝑀 uniformly continuous generalized weak Φ-hemicontractive mappings with 𝐹=⋂^𝑀_𝑖=1𝐹(𝑇_𝑖)≠∅. Let {𝑢_𝑛} be a sequence in 𝐾 defined iteratively from some 𝑢₀∈𝐾 by (1.17), where {𝜇_𝑛} is an arbitrary bounded sequence in 𝐾 and {𝛼_𝑛},{𝛿_𝑛} are two sequences in [0,1] satisfying the following conditions: (i) 𝛼_𝑛+𝛿_𝑛≤1, (ii) ∑^∞_𝑛=0𝛼_𝑛=∞, (iii) lim_𝑛→∞𝛼_𝑛=0, (iv) 𝛿_𝑛=𝑜(𝛼_𝑛). Then the iteration sequence {𝑢_𝑛} converges strongly to the unique fixed point of 𝑇.

Proof. Since 𝐹=⋂^𝑀_𝑖=1𝐹(𝑇_𝑖)≠∅, set 𝑞∈𝐹. Since the mapping 𝑇_𝑛 are generalized weak Φ-hemicontractive mappings, there exist strictly increasing functions Φ_𝑙∶[0,+∞)→[0,+∞) with Φ_𝑙(0)=0 and 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝑇_𝑙𝑥−𝑇_𝑙𝑦,𝑗𝑥−𝑦≤𝑥−𝑦²−Φ_𝑙𝑥−𝑦1+𝑥−𝑦²+Φ_𝑙𝑥−𝑦,∀𝑥,𝑦∈𝐾,𝑙∈𝐼. (2.1)
Firstly, we claim that there exists 𝑢₀∈𝐾 with 𝑢₀≠𝑇𝑢₀ such that 𝑡₀=‖𝑢₀−𝑇𝑢₀‖⋅‖𝑢₀−𝑞‖⋅[1+‖𝑢₀−𝑞‖²+Φ₁(‖𝑢₀−𝑞‖)]∈𝑅(Φ₁). In fact, if 𝑢₀=𝑇𝑢₀, then we have done. Otherwise, there exists the smallest positive integer 𝑛₀∈𝑁 such that 𝑢_𝑛0≠𝑇𝑢_𝑛0. We denote 𝑢_𝑛0=𝑢₀, then we will obtain that 𝑡₀∈𝑅(Φ₁). Indeed, if 𝑅(Φ₁)=[0,+∞), then 𝑡₀∈𝑅(Φ). If 𝑅(Φ₁)=[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‖⋅[1+‖𝑤_𝑛−𝑞‖²+Φ₁(‖𝑤_𝑛−𝑞‖)]∈𝑅(Φ₁) for 𝑛≥𝑛₀, then we redefine 𝑢₀=𝑤_𝑛0, let 𝜔₀=Φ₁⁻¹(𝑡₀)>0.
Next we shall prove ‖𝑢_𝑛−𝑞‖≤𝜔₀ for 𝑛≥0. 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_𝑙(𝜔₀)/12𝜔₀[1+Φ_𝑙((3/2)𝜔₀)+((3/2)𝜔₀)²], there exists 𝛿>0 such that ‖𝑇_𝑛𝑥−𝑇_𝑛𝑦‖<𝜖₀, whenever ‖𝑥−𝑦‖<𝛿; and 𝑇_𝑛 are bounded operators, set 𝑀=sup{‖𝑇_𝑛𝑥‖∶‖𝑥−𝑞‖≤𝜔₀}+sup_𝑛‖𝜔_𝑛‖. Since lim_𝑛→∞𝛼_𝑛=0, 𝛿_𝑛=𝑜(𝛼_𝑛), without loss of generality, let 𝛼_𝑛,𝛿_𝑛𝛼_𝑛1<min4,𝜔₀,𝛿4𝑀4𝑀+𝜔₀,Φ_𝑙𝜔₀4𝜔₀1+Φ_𝑙(3/2)𝜔₀+(3/2)𝜔₀²,Φ_𝑙𝜔₀121+Φ_𝑙(3/2)𝜔₀+(3/2)𝜔₀²𝑀𝜔₀,𝑛≥0. (2.2)
From (1.17), we have 𝑢_𝑛+1−𝑞=1−𝛼_𝑛−𝛿_𝑛𝑢_𝑛−𝑞+𝛼_𝑛𝑇_𝑛𝑢_𝑛−𝑞+𝛿_𝑛𝜔_𝑛≤𝑢−𝑞_𝑛−𝑞+𝛼_𝑛𝑇_𝑛𝑢_𝑛−𝑞+𝛿𝜔_𝑛−𝑞≤𝜔₀+𝛼_𝑛𝑇_𝑛𝑢_𝑛−𝑞+𝛿_𝑛𝜔_𝑛−𝑞≤𝜔₀+𝑀𝛼_𝑛+𝛿_𝑛≤𝜔₀+2𝑀𝛼_𝑛≤32𝜔₀,𝑢_𝑛+1−𝑢_𝑛=𝛼_𝑛𝑇_𝑛𝑢_𝑛+𝛿_𝑛𝜔_𝑛−𝛼_𝑛+𝛿_𝑛𝑢_𝑛≤𝛼_𝑛𝑇_𝑛𝑢_𝑛−𝑞+𝛿_𝑛𝜔_𝑛−𝑞+𝛼_𝑛+𝛿_𝑛𝑢_𝑛≤𝛼−𝑞_𝑛+𝛿_𝑛𝑀+𝜔₀<𝛿. (2.3)(2.4)
Since 𝑇_𝑛 are uniformly continuous mappings, so ‖𝑇_𝑛𝑢_𝑛+1−𝑇_𝑛𝑢_𝑛‖<𝜀₀.
Applying Lemma 1.8, the recursion (1.17), and the above inequalities, we obtain 𝑢_𝑛+1−𝑞²=1−𝛼_𝑛−𝛿_𝑛𝑢_𝑛−𝑞+𝛼_𝑛𝑇_𝑛𝑢_𝑛−𝑞+𝛿_𝑛𝜔_𝑛−𝑞²≤1−𝛼_𝑛−𝛿_𝑛²𝑢_𝑛−𝑞²+2𝛼_𝑛𝑇_𝑛𝑢_𝑛−𝑞,𝑗𝑢_𝑛+1−𝑞+2𝛿_𝑛𝜔_𝑛−𝑞⋅𝑢_𝑛+1≤−𝑞1−𝛼_𝑛²𝑢_𝑛−𝑞²+2𝛼_𝑛𝑇_𝑛𝑢_𝑛+1−𝑇𝑞,𝑗𝑢_𝑛+1−𝑞+2𝛼_𝑛𝑇_𝑛𝑢_𝑛−𝑇_𝑛𝑢_𝑛+1⋅𝑢_𝑛+1−𝑞+2𝛿_𝑛𝜔_𝑛−𝑞⋅𝑢_𝑛+1≤−𝑞1−𝛼_𝑛²𝑢_𝑛−𝑞²+2𝛼_𝑛𝑢_𝑛+1−𝑞²−Φ_𝑙𝑢_𝑛+1−𝑞1+Φ_𝑙𝑢_𝑛+1−𝑞+𝑢_𝑛+1−𝑞²+2𝛼_𝑛𝑇_𝑛𝑢_𝑛−𝑇_𝑛𝑢_𝑛+1⋅𝑢_𝑛+1−𝑞+2𝛿_𝑛𝜔_𝑛−𝑞⋅𝑢_𝑛+1−𝑞. (2.5) Inequality (2.5) implies 𝑢_𝑛+1−𝑞²≤𝑢_𝑛−𝑞²−2𝛼_𝑛Φ_𝑙𝑢_𝑛+1−𝑞1+Φ_𝑙𝑢_𝑛+1−𝑞+𝑢_𝑛+1−𝑞²+𝛼²_𝑛1−2𝛼_𝑛𝑢_𝑛−𝑞²+2𝛼_𝑛1−2𝛼_𝑛𝑇_𝑛𝑢_𝑛−𝑇_𝑛𝑢_𝑛+1⋅𝑢_𝑛+1−𝑞+2𝛿_𝑛1−2𝛼_𝑛𝜔_𝑛−𝑞⋅𝑢_𝑛+1−𝑞≤𝜔²₀−2𝛼_𝑛Φ_𝑙𝜔₀1+Φ_𝑙(3/2)𝜔₀+(3/2)𝜔₀²+2𝛼_𝑛Φ_𝑙𝜔₀41+Φ_𝑙(3/2)𝜔₀+(3/2)𝜔₀²𝜔²₀⋅𝜔²₀+4𝛼_𝑛Φ_𝑙𝜔₀121+Φ_𝑙(3/2)𝜔₀+(3/2)𝜔₀²𝜔₀⋅3𝜔₀2+4𝛼_𝑛Φ_𝑙𝜔₀121+Φ_𝑙(3/2)𝜔₀+(3/2)𝜔₀²𝑀𝜔₀⋅3𝑀𝜔₀2<𝜔²₀, (2.6) which is a contradiction with the assumption ‖𝑢_𝑛+1−𝑞‖>𝜔₀. Then ‖𝑢_𝑛+1−𝑞‖≤𝜔₀; that is, the sequence {𝑢_𝑛} is bounded. Let 𝑁=sup_𝑛‖𝑢_𝑛−𝑞‖. From (2.4), we have 𝑢_𝑛+1−𝑢_𝑛≤𝛼_𝑛+𝛿_𝑛𝑀+𝜔₀→0,𝑛→∞, (2.7) that is, lim_𝑛→∞‖𝑢_𝑛+1−𝑢_𝑛‖=0. Since 𝑇 is on uniformly continuous, so lim_𝑛→∞𝑇_𝑛𝑢_𝑛+1−𝑇_𝑛𝑢_𝑛=0. (2.8) Again using (2.5), we have 𝑢_𝑛+1−𝑞²≤𝑢_𝑛−𝑞²−2𝛼_𝑛Φ_𝑙𝑢_𝑛+1−𝑞1+Φ_𝑙𝑢_𝑛+1−𝑞+𝑢_𝑛+1−𝑞²+𝐴_𝑛, (2.9) where 𝐴_𝑛=𝛼²_𝑛𝑁²+2𝛼_𝑛𝑁𝑇_𝑛𝑢_𝑛−𝑇_𝑛𝑢_𝑛+1+2𝛿_𝑛𝑀𝑁. (2.10) By (2.8), the conditions (iii) and (iv), we get 𝐴_𝑛=𝑜(𝛼_𝑛). So applying Lemma 1.10 on (2.9), we obtain lim_𝑛→∞‖𝑢_𝑛−𝑞‖=0.

Theorem 2.2. Let 𝐸 be a Banach space and 𝐾 be a nonempty closed convex subset of 𝐸, 𝑇_𝑛 are as in Theorem 2.1. For 𝑥₀,𝑢₀∈𝐾, the sequence iterations {𝑥_𝑛},{𝑢_𝑛} are defined by (1.15) and (1.17), respectively. {𝛼_𝑛},{𝛿_𝑛},{𝛽^𝑖_𝑛},{𝜂^𝑖_𝑛},𝑖=1,2,…,𝑝−1 are sequences in [0,1] satisfying the following conditions:(i)0≤𝛼_𝑛+𝛿_𝑛≤1,0≤𝛽^𝑖_𝑛+𝜂^𝑖_𝑛≤1,1≤𝑖≤𝑝−1; (ii)∑^∞_𝑛=0𝛼_𝑛=∞; (iii)lim_𝑛→∞𝛼_𝑛=0; (iv)lim_𝑛→∞𝛽^𝑖_𝑛=lim_𝑛→∞𝜂^𝑖_𝑛=0,𝑖=1,…,𝑝−1; (v)𝛿_𝑛=𝑜(𝛼_𝑛).
Then the following two assertions are equivalent:(I)the iteration sequence {𝑥_𝑛} strongly converges to the common point of 𝐹(𝑇_𝑖),𝑖∈𝐼;(II)the sequence iteration {𝑢_𝑛} strongly converges to the common point of 𝐹(𝑇_𝑖),𝑖∈𝐼.

Proof. Since 𝐹=⋂^𝑀_𝑖=1𝐹(𝑇_𝑖)≠∅, set 𝑞∈𝐹. If the iteration sequence {𝑥_𝑛} strongly converges to 𝑞, then setting 𝑝=2,𝛽_𝑛=𝛿_𝑛=0, we obtain the convergence of the iteration sequence {𝑢_𝑛}. Conversely, we only prove that (II)⇒(I). The proof is divided into two parts.
Step 1. We show that {𝑥_𝑛−𝑢_𝑛} is bounded.
By the proof method of Theorem 2.1, there exists 𝑥₀∈𝐾 with 𝑥₀≠𝑇₁𝑥₀ such that 𝑟₀=‖𝑥₀−𝑇₁𝑥₀‖⋅‖𝑥₀−𝑞‖⋅[1+‖𝑥₀−𝑞‖²+Φ₁(‖𝑥₀−𝑞‖)]∈𝑅(Φ). Setting 𝑎₀=Φ₁⁻¹(𝑟₀), we have ‖𝑥₀−𝑞‖≤𝑎₀. Set 𝐵₁={‖𝑥−𝑞‖≤𝑎₀∶𝑥∈𝐾}, 𝐵₂={‖𝑥−𝑞‖≤2𝑎₀∶𝑥∈𝐾}. Since 𝑇_𝑙 are bounded mappings and {𝜔^𝑖_𝑛}(𝑖=1,…,𝑝−1),{𝜈_𝑛} are some bounded sequences in 𝐾, we can set 𝑀=max{sup_𝑥∈𝐵2‖𝑇_𝑛𝑥−𝑞‖;sup_𝑛∈𝑁‖𝜔^𝑖_𝑛−𝑞‖;sup_𝑛∈𝑁‖𝜈_𝑛−𝑞‖}. Since 𝑇_𝑖 are uniformly continuous mappings, given 𝜖₀=Φ_𝑙(𝑎₀)/4𝑎₀[1+(5𝑎₀/4)²+Φ_𝑙(5𝑎₀/4)],∃𝛿>0, such that ‖𝑇𝑥−𝑇𝑦‖<𝜖₀ whenever ‖𝑥−𝑦‖<𝛿, for all 𝑥,𝑦∈𝐵₂. Now, we define 𝜏₀=min{1/2,𝑎₀/8𝑀,𝑎₀/8(𝑀+𝑎₀),𝛿/8(𝑀+𝑎₀),Φ_𝑙(𝑎₀)/5𝑎²₀[1+(5𝑎₀/4)²+Φ_𝑙(5𝑎₀/4)],Φ_𝑙(𝑎₀)/5𝑎₀𝑀[1+(5𝑎₀/4)²+Φ_𝑙(5𝑎₀/4)]}. Since the control conditions (iii)-(iv), without loss of generality, we let 0<𝛼_𝑛,𝛿_𝑛/𝛼_𝑛,𝛽^𝑖_𝑛,𝜂^𝑖_𝑛<𝜏₀,𝑛≥0.
Now we claim that if 𝑥_𝑛∈𝐵₁, then 𝑦^𝑖_𝑛∈𝐵₂,1≤𝑖≤𝑝−1.
From (1.15), we obtain that 𝑦_𝑛^𝑝−1−𝑞≤1−𝛽_𝑛^𝑝−1−𝜂_𝑛^𝑝−1𝑥_𝑛−𝑞+𝛽_𝑛^𝑝−1𝑇_𝑛𝑥_𝑛−𝑞+𝜂_𝑛^𝑝−1𝜔_𝑛^𝑝−1≤𝑥−𝑞_𝑛−𝑞+𝛽_𝑛^𝑝−1+𝜂_𝑛^𝑝−1𝑀≤𝑥_𝑛−𝑞+2𝜏₀𝑀≤2𝑎₀,𝑦_𝑛^𝑝−2−𝑞≤1−𝛽_𝑛^𝑝−2−𝜂_𝑛^𝑝−2𝑥_𝑛−𝑞+𝛽_𝑛^𝑝−2𝑇_𝑛𝑦_𝑛^𝑝−1−𝑞+𝜂_𝑛^𝑝−2𝜔_𝑛^𝑝−2≤𝑥−𝑞_𝑛−𝑞+𝛽_𝑛^𝑝−2+𝜂_𝑛^𝑝−2𝑀≤𝑥_𝑛−𝑞+2𝜏₀𝑀≤2𝑎₀, (2.11)we also obtain that𝑦¹_𝑛−𝑞≤2𝑎₀. (2.12)
Now we suppose that ‖𝑥_𝑛−𝑞‖≤𝑎₀ holds. We will prove that ‖𝑥_𝑛+1−𝑞‖≤𝑎₀. If it is not the case, we assume that ‖𝑥_𝑛+1−𝑞‖>𝑎₀. From (1.15), we obtain that 𝑥_𝑛+1−𝑞=1−𝛼_𝑛−𝛿_𝑛𝑥_𝑛−𝑞+𝛼_𝑛𝑇_𝑛𝑦¹_𝑛−𝑞+𝛿_𝑛𝜈_𝑛≤𝑥−𝑞_𝑛−𝑞+𝛼_𝑛𝑇_𝑛𝑦¹_𝑛−𝑞+𝛿_𝑛𝜈_𝑛≤𝑥−𝑞_𝑛−𝑞+𝛼_𝑛+𝛿_𝑛𝑀≤𝑥_𝑛−𝑞+2𝜏₀𝑀≤𝑥_𝑛+1−𝑞4𝑎₀≤54𝑎₀. (2.13) Consequently, by (2.11) and (2.12), we obtain 𝑥_𝑛+1−𝑦¹_𝑛=𝛽¹_𝑛−𝛼_𝑛+𝜂¹_𝑛−𝛿_𝑛𝑥_𝑛−𝑞+𝛼_𝑛𝑇_𝑛𝑦¹_𝑛−𝑞−𝛽¹_𝑛𝑇^𝑛_𝑛𝑦²_𝑛−𝑞+𝛿_𝑛𝜈_𝑛−𝑞−𝜂¹_𝑛𝜔¹_𝑛≤𝛽−𝑞_1𝑛+𝛼_𝑛+𝜂¹_𝑛+𝛿_𝑛𝑥_𝑛−𝑞+𝛼_𝑛𝑇_𝑛𝑦¹_𝑛−𝑞+𝛽¹_𝑛𝑇_𝑛𝑦²_𝑛−𝑞+𝛿_𝑛𝜈_𝑛−𝑞+𝜂¹_𝑛𝜔¹_𝑛−𝑞≤4𝜏₀𝜇₀+𝑀≤𝛿. (2.14) Since 𝑇_𝑛 are uniformly continuous mappings, we get 𝑇_𝑛𝑥_𝑛+1−𝑇_𝑛𝑦¹_𝑛≤𝜖₀. (2.15) Using (2.1), Lemma 1.8, and the recursion formula (1.15), we have 𝑥_𝑛+1−𝑞²=1−𝛼_𝑛−𝛿_𝑛𝑥_𝑛−𝑞+𝛼_𝑛𝑇_𝑛𝑦¹_𝑛−𝑞+𝛿_𝑛𝜈_𝑛−𝑞²≤1−𝛼_𝑛²𝑥_𝑛−𝑞²+2𝛼_𝑛𝑇_𝑛𝑦¹_𝑛−𝑞,𝑗𝑥_𝑛+1−𝑞+2𝛿_𝑛𝜈_𝑛−𝑞⋅𝑥_𝑛+1≤−𝑞1−𝛼_𝑛²𝑥_𝑛−𝑞²+2𝛼_𝑛𝑇_𝑛𝑥_𝑛+1−𝑞,𝑗𝑥_𝑛+1−𝑞+2𝛼_𝑛𝑇_𝑛𝑦¹_𝑛−𝑇_𝑛𝑥_𝑛+1,𝑗𝑥_𝑛+1−𝑞+2𝛿_𝑛𝜈_𝑛−𝑞⋅𝑥_𝑛+1≤−𝑞1−𝛼_𝑛²𝑥_𝑛−𝑞²+2𝛼_𝑛𝑥_𝑛+1−𝑞²−Φ_𝑙𝑥_𝑛+1−𝑞1+Φ_𝑙𝑥_𝑛+1−𝑞+𝑥_𝑛+1−𝑞²+2𝛼_𝑛𝑇_𝑛𝑦¹_𝑛−𝑇_𝑛𝑥_𝑛+1⋅𝑥_𝑛+1−𝑞+2𝛿_𝑛𝑀⋅𝑥_𝑛+1−𝑞. (2.16) Which implies 𝑥_𝑛+1−𝑞²≤𝑥_𝑛−𝑞²−2𝛼_𝑛1−2𝛼_𝑛Φ_𝑙𝑥_𝑛+1−𝑞1+Φ_𝑙𝑥_𝑛+1−𝑞+𝑥_𝑛+1−𝑞²+𝛼²_𝑛1−2𝛼_𝑛𝑥_𝑛−𝑞²+2𝛼_𝑛1−2𝛼_𝑛𝑇_𝑛𝑦¹_𝑛−𝑇_𝑛𝑥_𝑛+1⋅𝑥_𝑛+1−𝑞+2𝛿_𝑛1−2𝛼_𝑛𝑀⋅𝑥_𝑛+1−𝑞≤𝑎²₀−2𝛼_𝑛1−2𝛼_𝑛Φ_𝑙𝑎₀1+0𝑥0𝑒𝑒225𝑎₀/4²+Φ_𝑙5𝑎₀+/42𝛼_𝑛1−2𝛼_𝑛Φ_𝑙𝑎₀4𝑎²₀1+0𝑥0𝑒𝑒225𝑎₀/4²+Φ_𝑙5𝑎₀𝑎/4²⁰+2𝛼_𝑛1−2𝛼_𝑛Φ_𝑙𝑎₀5𝑎₀1+5𝑎₀/4²+Φ_𝑙5𝑎₀⋅/45𝑎₀4+2𝛼_𝑛1−2𝛼_𝑛Φ_𝑙𝑎₀5𝑎₀𝑀1+5𝑎₀/4²+Φ_𝑙5𝑎₀⋅/45𝑎₀𝑀4<𝑎²₀ (2.17) which is a contradiction with the assumption ‖𝑥_𝑛+1−𝑞‖>𝜇₀, then ‖𝑥_𝑛+1−𝑞‖≤𝜇₀; that is, the sequence {𝑥_𝑛−𝑞} is bounded. Since 𝑢_𝑛→𝑞, as 𝑛→∞, so the sequence {𝑥_𝑛−𝑢_𝑛} is bounded.
Step 2. We prove lim_𝑛→∞‖𝑥_𝑛−𝑞‖=0.
Since {𝑥_𝑛−𝑢_𝑛} is bounded, again applying (2.11) and (2.12), we get the boundedness of {𝑦^𝑖_𝑛−𝑢_𝑛},𝑖=1,2,…,𝑝−1. Since 𝑇_𝑛=𝑇_{𝑛(mod𝑀)} are bounded mappings, set 𝐿=max{sup_𝑛≥0‖𝑥_𝑛−𝑢_𝑛‖,sup_𝑛≥0‖𝑇_𝑛𝑥_𝑛−𝑢_𝑛‖,sup_𝑛≥0‖𝑇_𝑛𝑦^𝑖_𝑛−𝑢_𝑛‖,sup_𝑛≥0‖𝜇_𝑛−𝑢_𝑛‖,sup_𝑛≥0‖𝜈_𝑛−𝑢_𝑛‖,sup_𝑛≥0‖𝜔¹_𝑛−𝑢_𝑛‖}, (𝑖=1,2,…,𝑝−1). From (1.15) and (1.17), we obtain 𝑥_𝑛+1−𝑢_𝑛+1²=1−𝛼_𝑛−𝛿_𝑛𝑥_𝑛−𝑢_𝑛+𝛼_𝑛𝑇_𝑛𝑦¹_𝑛−𝑇_𝑛𝑢_𝑛+𝛿_𝑛𝜈_𝑛−𝜇_𝑛²≤1−𝛼_𝑛²𝑥_𝑛−𝑢_𝑛²+2𝛼_𝑛𝑇_𝑛𝑦¹_𝑛−𝑇_𝑛𝑢_𝑛,𝑗𝑥_𝑛+1−𝑢_𝑛+1+2𝛿_𝑛𝜈_𝑛−𝜇_𝑛⋅𝑥_𝑛+1−𝑢_𝑛+1≤1−𝛼_𝑛²𝑥_𝑛−𝑢_𝑛²+2𝛼_𝑛𝑇_𝑛𝑥_𝑛+1−𝑇_𝑛𝑢_𝑛+1,𝑗𝑥_𝑛+1−𝑢_𝑛+1+2𝛼_𝑛𝑇_𝑛𝑦¹_𝑛−𝑇_𝑛𝑥_𝑛+1+𝑇_𝑛𝑢_𝑛+1−𝑇_𝑛𝑢_𝑛,𝑗𝑥_𝑛+1−𝑢_𝑛+1+2𝛿_𝑛𝜈_𝑛−𝜇_𝑛⋅𝑥_𝑛+1−𝑢_𝑛+1≤1+𝛼²_𝑛𝑥_𝑛−𝑞²−2𝛼_𝑛Φ_𝑙𝑥_𝑛+1−𝑢_𝑛+11+Φ_𝑙𝑥_𝑛+1−𝑢_𝑛+1+𝑥_𝑛+1−𝑢_𝑛+1²+2𝛼_𝑛𝑇_𝑛𝑦¹_𝑛−𝑇_𝑛𝑥_𝑛+1⋅𝑥_𝑛+1−𝑢_𝑛+1+2𝛼_𝑛𝑇_𝑛𝑢_𝑛+1−𝑇_𝑛𝑢_𝑛⋅𝑥_𝑛+1−𝑢_𝑛+1+2𝛿_𝑛𝑀⋅𝑥_𝑛+1−𝑢_𝑛+1𝑥_𝑛+1−𝑦¹_𝑛≤𝛽¹_𝑛+𝛼_𝑛+𝜂¹_𝑛+𝛿_𝑛𝑥_𝑛−𝑢_𝑛+𝛼_𝑛𝑇^𝑛_𝑛𝑦¹_𝑛−𝑢_𝑛+𝛽¹_𝑛𝑇^𝑛_𝑛𝑦²_𝑛−𝑢_𝑛+𝛿_𝑛𝜈_𝑛−𝑢_𝑛+𝜂¹_𝑛𝜔¹_𝑛−𝑢_𝑛≤2𝛽¹_𝑛+𝛼_𝑛+𝜂¹_𝑛+𝛿_𝑛𝐿. (2.18)(2.19)
By the conditions (iii)–(v), we have lim_𝑛→∞𝑥_𝑛+1−𝑦¹_𝑛=0. (2.20)
Since lim_𝑛→∞‖𝑢_𝑛−𝑞‖=0, so 𝑢_𝑛+1−𝑢_𝑛≤𝑢_𝑛−𝑞+𝑢_𝑛+1−𝑞. (2.21) That is: lim_𝑛→∞𝑢_𝑛+1−𝑢_𝑛=0. (2.22)
By the uniform continuity of 𝑇, we obtain lim_𝑛→∞𝑇_𝑛𝑥_𝑛+1−𝑇_𝑛𝑦¹_𝑛=0,lim_𝑛→∞𝑇_𝑛𝑢_𝑛+1−𝑇_𝑛𝑢_𝑛=0. (2.23)
From (2.23) and the conditions (iii) and (v), (2.18) becomes 𝑥_𝑛+1−𝑢_𝑛+1²≤1+𝛼²_𝑛𝑥_𝑛−𝑢_𝑛²−2𝛼_𝑛Φ_𝑙𝑥_𝑛+1−𝑢_𝑛+11+Φ_𝑙𝑥_𝑛+1−𝑢_𝑛+1+𝑥_𝑛+1−𝑢_𝑛+1²+𝑜𝛼_𝑛. (2.24)
By Lemma 1.10, we get lim_𝑛→∞‖𝑥_𝑛−𝑢_𝑛‖=0. Since lim_𝑛→∞‖𝑢_𝑛−𝑞‖=0, and the inequality 0≤‖𝑥_𝑛−𝑞‖≤‖𝑥_𝑛−𝑢_𝑛‖+‖𝑢_𝑛−𝑞‖, so lim_𝑛→∞‖𝑥_𝑛−𝑞‖=0.

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

Corollary 2.3. Let 𝐸 be a Banach space and 𝐾 be a nonempty closed convex subset of 𝐸, 𝑇_𝑛 are as in Theorem 2.1. For 𝑥₀∈𝐾, the sequence iterations {𝑥_𝑛} is defined by (1.15). {𝛼_𝑛},{𝛿_𝑛},{𝛽^𝑖_𝑛},{𝜂^𝑖_𝑛},(𝑖=1,2,…,𝑝−1) are sequences in [0,1] satisfying the following conditions:(i)0≤𝛼_𝑛+𝛿_𝑛≤1,0≤𝛽^𝑖_𝑛+𝜂^𝑖_𝑛≤1,1≤𝑖≤𝑝−1; (ii)∑^∞_𝑛=0𝛼_𝑛=∞; (iii)lim_𝑛→∞𝛼_𝑛=0; (iv)lim_𝑛→∞𝛽^𝑖_𝑛=lim_𝑛→∞𝜂^𝑖_𝑛=0,𝑖=1,…,𝑝−1; (v)𝛿_𝑛=𝑜(𝛼_𝑛).
Then the iteration sequence {𝑥_𝑛} strongly converges to the common point of 𝐹(𝑇_𝑖), 𝑖∈𝐼.

Corollary 2.4. Let 𝑇_𝑛=𝑆_{𝑛(mod𝑀)}, 𝑇_𝑙∶𝐸→𝐸,𝑙∈𝐼={1,2,…,𝑀} are 𝑀 uniformly continuous generalized weak Φ-quasi-accretive mappings. Suppose 𝑁(𝐹)=⋂^𝑀_𝑖=1𝑁(𝐹_𝑖)≠∅, that is, there exists 𝑥^∗∈𝑁(𝐹). Let {𝛼_𝑛},{𝛿_𝑛},{𝛽^𝑖_𝑛},{𝜂^𝑖_𝑛},(𝑖=1,2,…,𝑝−1) be sequences in [0,1] satisfying the following conditions:(i)0≤𝛼_𝑛+𝛿_𝑛≤1,0≤𝛽^𝑖_𝑛+𝜂^𝑖_𝑛≤1,1≤𝑖≤𝑝−1; (ii)∑^∞_𝑛=0𝛼_𝑛=∞; (iii)lim_𝑛→∞𝛼_𝑛=0; (iv)lim_𝑛→∞𝛽^𝑖_𝑛=lim_𝑛→∞𝜂^𝑖_𝑛=0,𝑖=1,…,𝑝−1; (v)𝛿_𝑛=𝑜(𝛼_𝑛).
Let the sequence {𝑥_𝑛} in 𝐸 be generated iteratively from some 𝑥₀∈𝐸 by 𝑥_𝑛+1=1−𝛼_𝑛−𝛿_𝑛𝑥_𝑛+𝛼_𝑛𝑆_𝑛𝑦¹_𝑛+𝛿_𝑛𝜈_𝑛,𝑦^𝑖_𝑛=1−𝛽^𝑖_𝑛−𝜂^𝑖_𝑛𝑥_𝑛+𝛽^𝑖_𝑛𝑆_𝑛𝑦_𝑛^𝑖+1+𝜂^𝑖_𝑛𝜔^𝑖_𝑛𝑦,𝑖=1,…𝑝−2,_𝑛𝑝−1=1−𝛽_𝑛^𝑝−1−𝜂_𝑛^𝑝−1𝑥_𝑛+𝛽_𝑛^𝑝−1𝑆_𝑛𝑥_𝑛+𝜂_𝑛^𝑝−1𝜔_𝑛^𝑝−1,𝑝≥2, (2.25) where 𝑆_𝑙𝑥∶=𝑥−𝑇_𝑙𝑥 for all 𝑥∈𝐸 and {𝜈_𝑛},{𝜔^𝑖_𝑛} are any bounded sequences in 𝐾.
Then {𝑥_𝑛} defined by (2.25) converges strongly to 𝑥^∗.

Proof. We simply observe that 𝑆_𝑙∶=𝐼−𝑇_𝑙,𝑙∈𝐼 are 𝑀 uniformly continuous generalized weak Φ-hemicontractive mappings. The result follows from Corollary 2.3.

Acknowledgments

This project was supported by Hebei Province Natural Science Foundation (A2011210033); Shijiazhuang Tiedao University Foundation (Q64).