The Equivalence of Convergence Results of Modified Mann and Ishikawa Iterations with Errors without Bounded Range Assumption

Abstract

Let 𝐸 be an arbitrary uniformly smooth real Banach space, let 𝐷 be a nonempty closed convex subset of 𝐸, and let 𝑇∶𝐷→𝐷 be a uniformly generalized Lipschitz generalized asymptotically Φ-strongly pseudocontractive mapping with 𝑞∈𝐹(𝑇)≠∅. Let {𝑎_𝑛},{𝑏_𝑛},{𝑐_𝑛},{𝑑_𝑛} be four real sequences in [0,1] and satisfy the conditions: (i) 𝑎_𝑛+𝑐_𝑛≤1, 𝑏_𝑛+𝑑_𝑛≤1; (ii) 𝑎_𝑛,𝑏_𝑛,𝑑_𝑛→0 as 𝑛→∞ and 𝑐_𝑛=𝑜(𝑎_𝑛); (iii) Σ^∞_𝑛=0𝑎_𝑛=∞. For some 𝑥₀,𝑧₀∈𝐷, let {𝑢_𝑛},{𝑣_𝑛},{𝑤_𝑛} be any bounded sequences in 𝐷, and let {𝑥_𝑛},{𝑧_𝑛} be the modified Ishikawa and Mann iterative sequences with errors, respectively. Then the convergence of {𝑥_𝑛} is equivalent to that of {𝑧_𝑛}.

1. Introduction and Preliminary

Let 𝐸 be a real Banach space 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. It is well known that(i)if 𝐸 is a smooth Banach space, then the mapping 𝐽 is single-valued;(ii)𝐽(𝛼𝑥)=𝛼𝐽(𝑥) for all 𝑥∈𝐸 and 𝛼∈ℜ;(iii)if 𝐸 is a uniformly smooth Banach space, then the mapping 𝐽 is uniformly continuous on any bounded subset of 𝐸. Throughout this paper, we denote that 𝑗 is the single-valued normalized duality mapping, 𝐷 is a nonempty closed convex subset of 𝐸, 𝑇∶𝐷→𝐷 is a mapping, and 𝑇⁰ is the unit mapping 𝐼.In 1972, Goebel and Kirk [1] introduced the class of asymptotically nonexpansive mappings as follows.

Definition 1.1. A mapping 𝑇 is said to be asymptotically nonexpansive if for each 𝑥,𝑦∈𝐷𝑇^𝑛𝑥−𝑇^𝑛𝑦≤𝑘_𝑛𝑥−𝑦,∀𝑛≥0, (1.2) where {𝑘_𝑛}⊂[1,+∞) with lim_𝑛→∞𝑘_𝑛=1.
Schu [2], in 1991, gave the definition of asymptotically pseudocontractive mappings and proved the correlation results.

Definition 1.2. The mapping 𝑇 is called asymptotically pseudocontractive with the sequence {𝑘_𝑛}⊂[1,+∞) if and only if lim_𝑛→∞𝑘_𝑛=1, and for all 𝑛∈𝑁 and all 𝑥,𝑦∈𝐷, there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝑇^𝑛𝑥−𝑇^𝑛𝑦,𝑗𝑥−𝑦≤𝑘_𝑛𝑥−𝑦². (1.3)
It is easy to find that every asymptotically nonexpansive mapping is asymptotically pseudocontractive. However, the converse is not true in general. See example of [3].
Recently, Colao [4] combined the proof ideas of the papers of Chang [5] and C. E. Chidume and C. O. Chidume [6] and then showed the equivalent theorem results of the convergence between Mann and Ishikawa iterations with errors for generalized strongly asymptotically 𝜙-pseudocontractive mapping with bounded range. In fact, he proved the following theorem.

Theorem 1.3. Let 𝑋 be a uniformly smooth Banach space, and let 𝑇∶𝑋→𝑋 be generalized strongly asymptotically 𝜙-pseudocontractive mapping with fixed point 𝑥^∗ and bounded range. Let {𝑥_𝑛} and {𝑧_𝑛} be the sequences defined by (1.4) and (1.5), respectively, 𝑦_𝑛=1−𝛽_𝑛−𝛿_𝑛𝑥_𝑛+𝛽_𝑛𝑇^𝑛𝑥_𝑛+𝛿_𝑛𝑣_𝑛𝑥,𝑛≥0,_𝑛+1=1−𝛼_𝑛−𝛾_𝑛𝑥_𝑛+𝛼_𝑛𝑇^𝑛𝑦_𝑛+𝛾_𝑛𝑢_𝑛𝑧,𝑛≥0,_𝑛+1=1−𝛼_𝑛−𝛾_𝑛𝑧_𝑛+𝛼_𝑛𝑇^𝑛𝑧_𝑛+𝛾_𝑛𝑤_𝑛,𝑛≥0, (1.4)(1.5) where {𝛼_𝑛},{𝛾_𝑛},{𝛽_𝑛},{𝛿_𝑛}⊂[0,1] satisfy(H1) lim_𝑛→∞𝛼_𝑛=lim_𝑛→∞𝛽_𝑛=lim_𝑛→∞𝛿_𝑛=0 and 𝛾_𝑛=𝑜(𝛼_𝑛),(H2) Σ^∞_𝑛=1𝛼_𝑛=∞,and the sequences {𝑢_𝑛},{𝑣_𝑛},{𝑤_𝑛} are bounded in 𝑋, then for any initial point 𝑧₀,𝑥₀∈𝑋, the following two assertions are equivalent.(1)The modified Ishikawa iteration sequence with errors (1.4) converges to 𝑥^∗;(2)The modified Mann iteration sequence with errors (1.5) converges to 𝑥^∗.

The aim of this paper is to prove the equivalence of convergent results of above Ishikawa and Mann iterations with errors for generalized asymptotically Φ-strongly pseudocontractive mappings without bounded range assumptions in uniformly smooth real Banach spaces. For this, we need the following concepts and lemmas.

Definition 1.4 (see [4]). The mapping 𝑇 is called generalized asymptotically Φ-strongly pseudocontractive if 𝑇^𝑛𝑥−𝑇^𝑛𝑦,𝑗𝑥−𝑦≤𝑘_𝑛𝑥−𝑦²−Φ𝑥−𝑦,𝑛≥0, (1.6) where 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦), {𝑘_𝑛}⊂[1,+∞) is converging to one and Φ∶[0,+∞)→[0,+∞) is strictly increasing continuous function with Φ(0)=0.

Definition 1.5 (see [4]). For arbitrary given 𝑥₀∈𝐷, modified Ishikawa iterative process with errors {𝑥_𝑛}^∞_𝑛=0 defined by 𝑦_𝑛=1−𝑏_𝑛−𝑑_𝑛𝑥_𝑛+𝑏_𝑛𝑇^𝑛𝑥_𝑛+𝑑_𝑛𝑤_𝑛𝑥,𝑛≥0,_𝑛+1=1−𝑎_𝑛−𝑐_𝑛𝑥_𝑛+𝑎_𝑛𝑇^𝑛𝑦_𝑛+𝑐_𝑛𝑣_𝑛,𝑛≥0, (1.7) where {𝑣_𝑛},{𝑤_𝑛} are any bounded sequences in 𝐷; {𝑎_𝑛},{𝑏_𝑛},{𝑐_𝑛},{𝑑_𝑛} are four real sequences in [0,1] and satisfy 𝑎_𝑛+𝑐_𝑛≤1,𝑏_𝑛+𝑑_𝑛≤1, for all 𝑛≥0. If 𝑏_𝑛=𝑑_𝑛=0, we define modified Mann iterative process with errors {𝑧_𝑛} by 𝑧_𝑛+1=1−𝑎_𝑛−𝑐_𝑛𝑧_𝑛+𝑎_𝑛𝑇^𝑛𝑧_𝑛+𝑐_𝑛𝑢_𝑛,𝑛≥0, (1.8) where {𝑢_𝑛} is any bounded sequence in 𝐷.

Lemma 1.6 (see [7]). Let 𝐸 be a uniformly smooth real Banach space and let 𝐽∶𝐸→2^𝐸∗ be a normalized duality mapping. Then 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝐽𝑥+𝑦, (1.9) for all 𝑥,𝑦∈𝐸.

Lemma 1.7 (see [8]). Let {𝜌_𝑛}^∞_𝑛=0 be a nonnegative sequence which satisfies the following inequality: 𝜌_𝑛+1≤1−𝜆_𝑛𝜌_𝑛+𝜎_𝑛,𝑛≥0, (1.10) where 𝜆_𝑛∈[0,1] with ∑^∞_𝑛=0𝜆_𝑛=∞, 𝜎_𝑛=𝑜(𝜆_𝑛). Then 𝜌_𝑛→0 as 𝑛→∞.

2. Main Results

First of all, we give a new concept.

Definition 2.1. A mapping 𝑇∶𝐷→𝐷 is called uniformly generalized Lipschitz if there exists a constant 𝐿>0 such that 𝑇^𝑛𝑥−𝑇^𝑛𝑦≤𝐿1+𝑥−𝑦,∀𝑥,𝑦∈𝐷,∀𝑛≥0. (2.1) It is mentioned to notice that if 𝑇 has bounded range, then it is uniformly generalized Lipschitz. In fact, since 𝑅(𝑇^𝑛)⊆𝑅(𝑇), then sup_𝑥∈𝐷{‖𝑇^𝑛𝑥‖}≤sup_𝑥∈𝐷{‖𝑇𝑥‖}=𝑀₁, thus ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤2𝑀₁≤𝐿(1+‖𝑥−𝑦‖), where 𝐿=2𝑀₁. On the contrary, it is not true in general (See [6]).

In the following, we prove the main theorems of this paper.

Theorem 2.2. Let 𝐸 be an arbitrary uniformly smooth real Banach space, let 𝐷 be a nonempty closed convex subset of 𝐸, and let 𝑇∶𝐷→𝐷 be a uniformly generalized Lipschitz generalized asymptotically Φ-strongly pseudocontractive mapping with 𝑞∈𝐹(𝑇)≠∅. Let {𝑎_𝑛},{𝑏_𝑛},{𝑐_𝑛},{𝑑_𝑛} be four real sequences in [0,1] and satisfy the following conditions:(i)𝑎_𝑛+𝑐_𝑛≤1, 𝑏_𝑛+𝑑_𝑛≤1;(ii)𝑎_𝑛,𝑏_𝑛,𝑑_𝑛→0 as 𝑛→∞ and 𝑐_𝑛=𝑜(𝑎_𝑛);(iii)Σ^∞_𝑛=0𝑎_𝑛=∞. For some 𝑥₀,𝑧₀∈𝐷, let {𝑢_𝑛},{𝑣_𝑛},{𝑤_𝑛} be any bounded sequences in 𝐷, and let {𝑥_𝑛} and {𝑧_𝑛} be Ishikawa and Mann iterative sequences with errors defined by (1.7) and (1.8), respectively. Then the following conclusions are equivalent:(1){𝑥_𝑛} converges strongly to the unique fixed point 𝑞 of 𝑇;(2){𝑧_𝑛} converges strongly to the unique fixed point 𝑞 of 𝑇.

Proof. (1)⇒(2) is obvious, that is, let 𝑏_𝑛=𝑑_𝑛=0, (1.7) turns into (1.8). We only need to show that (2)⇒(1). Since 𝑇∶𝐷→𝐷 is a uniformly generalized Lipschitz generalized asymptotically Φ-strongly pseudocontractive mapping, then there exists a strictly increasing continuous function Φ∶[0,+∞)→[0,+∞) with Φ(0)=0 such that 𝑇^𝑛𝑥−𝑇^𝑛𝑦,𝐽𝑥−𝑦≤𝑘_𝑛𝑥−𝑦²−Φ𝑥−𝑦, (2.2) that is, 𝑘_𝑛𝐼−𝑇^𝑛𝑥−𝑘_𝑛𝐼−𝑇^𝑛,𝑇𝑦,𝐽𝑥−𝑦≥Φ𝑥−𝑦_𝑛𝑥−𝑇^𝑛𝑦≤𝐿1+𝑥−𝑦, (2.3)(2.4) for any 𝑥,𝑦∈𝐷. For convenience, denote 𝑘=sup_𝑛{𝑘_𝑛}.
Step 1. There exists 𝑥₀∈𝐷 and 𝑥₀≠𝑇𝑥₀ such that 𝑟₀=(𝑘+𝐿)‖𝑥₀−𝑞‖²+𝐿‖𝑥₀−𝑞‖∈𝑅(Φ)(range of Φ).
Indeed, if Φ(𝑟)→+∞ as 𝑟→+∞, then 𝑟₀∈𝑅(Φ); if sup{Φ(𝑟)∶𝑟∈[0,+∞)}=𝑟₁<+∞ with 𝑟₁<𝑟₀, then, for 𝑞∈𝐷, there exists a sequence {𝜈_𝑛} in 𝐷 such that 𝜈_𝑛→𝑞 as 𝑛→∞ with 𝜈_𝑛≠𝑞. Furthermore, there exists a natural number 𝑛₀ such that (𝑘+𝐿)‖𝜈_𝑛−𝑞‖²+𝐿‖𝜈_𝑛−𝑞‖<𝑟₁/2 for 𝑛≥𝑛₀, then we redefine 𝑥₀,𝑟₀ such that 𝑥₀=𝜈_𝑛0,𝑟₀=(𝑘+𝐿)‖𝑥₀−𝑞‖²+𝐿‖𝑥₀−𝑞‖∈𝑅(Φ). Hence, it is to ensure that Φ⁻¹(𝑟₀) is well defined.
Step 2. For any 𝑛≥0, {𝑥_𝑛} is a bounded sequence.
Set 𝑅=Φ⁻¹(𝑟₀). From (2.3), we have 𝑘_𝑛𝑥₀−𝑞−𝑇^𝑛𝑥₀−𝑞,𝐽𝑥₀−𝑞≥Φ𝑥₀−𝑞, (2.5) that is, (𝑘+𝐿)‖𝑥₀−𝑞‖²+𝐿‖𝑥₀−𝑞‖≥Φ(‖𝑥₀−𝑞‖). Thus, we obtain that ‖𝑥₀−𝑞‖≤𝑅. Denote 𝐵₁=,𝐵𝑥∈𝐷∶𝑥−𝑞≤𝑅₂=𝑥∈𝐷∶𝑥−𝑞≤2𝑅,𝑀=sup_𝑛𝑣_𝑛−𝑞+sup_𝑛𝑤_𝑛−𝑞. (2.6) Next, we want to prove that 𝑥_𝑛∈𝐵₁ for any 𝑛≥0 by induction. If 𝑛=0, then 𝑥₀∈𝐵₁. Now we 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. Since 𝐽 is uniformly continuous on bounded subset of 𝐸, then, for 𝜖₀=Φ(𝑅/4)/24𝐿(1+2𝑅), there exists 𝛿>0 such that ‖𝐽𝑥−𝐽𝑦‖<𝜖₀ when ‖𝑥−𝑦‖<𝛿,forall𝑥,𝑦∈𝐵₂. Now denote 𝜏₀𝑅=min2[𝐿()],𝑅1+2𝑅+2𝑅+𝑀4[𝐿()],𝛿1+𝑅+2𝑅+𝑀2[𝐿()],Φ()1+2𝑅+2𝑅+𝑀𝑅/424𝑅²,Φ(𝑅/4)(),Φ()24𝐿1+2𝑅𝑅/4.48𝑀𝑅 (2.7) Since 𝑎_𝑛,𝑏_𝑛,𝑐_𝑛,𝑑_𝑛→0 as 𝑛→∞, and 𝑐_𝑛=𝑜(𝑎_𝑛), without loss of generality, we assume that 0≤𝑎_𝑛,𝑏_𝑛,𝑐_𝑛,𝑑_𝑛≤𝜏₀,𝑐_𝑛<𝑎_𝑛𝜏₀ for any 𝑛≥0. Then we obtain the following estimates: 𝑇^𝑛𝑥_𝑛−𝑞≤𝐿1+𝑥_𝑛(),𝑦−𝑞≤𝐿1+𝑅_𝑛−𝑞≤1−𝑏_𝑛−𝑑_𝑛𝑥_𝑛−𝑞+𝑏_𝑛𝑇^𝑛𝑥_𝑛−𝑞+𝑑_𝑛𝑤_𝑛−𝑞≤𝑅+𝑏_𝑛𝐿1+𝑥_𝑛−𝑞+𝑑_𝑛𝑀≤𝑅+𝑏_𝑛𝐿(1+𝑅)+𝑑_𝑛𝑀≤𝑅+𝜏₀[𝐿()]𝑇1+𝑅+𝑀≤2𝑅,_𝑛𝑦_𝑛−𝑞≤𝐿1+𝑦_𝑛(),𝑥−𝑞≤𝐿1+2𝑅_𝑛−𝑇^𝑛𝑥_𝑛≤𝑥_𝑛−𝑞+𝑇^𝑛𝑥_𝑛(𝑥−𝑞≤𝐿+1+𝐿)_𝑛()𝑥−𝑞≤𝐿+1+𝐿𝑅,_𝑛−𝑞−𝑦_𝑛−𝑞≤𝑏_𝑛𝑥_𝑛−𝑇^𝑛𝑥_𝑛+𝑑_𝑛𝑤_𝑛−𝑞+𝑥_𝑛−𝑞≤𝑏_𝑛[𝐿+(1+𝐿)𝑅]+𝑑_𝑛(𝑀+𝑅)≤𝜏₀[𝐿(1+𝑅)+2𝑅+𝑀]≤𝜏₀[𝐿()]≤𝛿1+2𝑅+2𝑅+𝑀2𝑥<𝛿,_𝑛−𝑞≥𝑥_𝑛+1−𝑞−𝑎_𝑛𝑇^𝑛𝑦_𝑛−𝑥_𝑛−𝑐_𝑛𝑣_𝑛−𝑥_𝑛≥𝑥_𝑛+1−𝑞−𝑎_𝑛𝑇^𝑛𝑦_𝑛−𝑞+𝑥_𝑛−𝑞−𝑐_𝑛𝑥_𝑛−𝑞+𝑣_𝑛−𝑞>𝑅−𝑎_𝑛[𝐿(1+2𝑅)+𝑅]−𝑐_𝑛(𝑅+𝑀)≥𝑅−𝜏₀[𝐿()]𝑅1+2𝑅+𝑀+2𝑅≥𝑅−2=𝑅2,𝑦_𝑛−𝑞≥𝑥_𝑛−𝑞−𝑏_𝑛𝑇^𝑛𝑥_𝑛−𝑥_𝑛−𝑑_𝑛𝑥_𝑛−𝑤_𝑛≥𝑥_𝑛−𝑞−𝑏_𝑛[𝐿+(1+𝐿)𝑅]−𝑑_𝑛𝑥_𝑛−𝑞+𝑤_𝑛≥𝑥−𝑞_𝑛−𝑞−𝑏_𝑛[𝐿+(1+𝐿)𝑅]−𝑑_𝑛()≥𝑥𝑅+𝑀_𝑛−𝑞−𝜏₀[𝐿()]>𝑅1+𝑅+2𝑅+𝑀2−𝑅4=𝑅4,𝑥_𝑛+1−𝑞≤1−𝑎_𝑛−𝑐_𝑛𝑥_𝑛−𝑞+𝑎_𝑛𝑇^𝑛𝑦_𝑛−𝑞+𝑐_𝑛𝑣_𝑛−𝑞≤𝑅+𝜏₀[𝐿()]𝑥1+2𝑅+𝑀≤2𝑅,_𝑛+1−𝑞−𝑥_𝑛−𝑞≤𝑎_𝑛𝑇^𝑛𝑦_𝑛−𝑥_𝑛+𝑐_𝑛𝑢_𝑛−𝑥_𝑛≤𝑎_𝑛𝑇^𝑛𝑦_𝑛−𝑞+𝑥_𝑛−𝑞+𝑐_𝑛𝑣_𝑛−𝑞+𝑥_𝑛−𝑞≤𝑎_𝑛[𝐿(1+2𝑅)+𝑅]+𝑐_𝑛(𝑀+𝑅)≤𝜏₀[𝐿()]≤𝛿1+2𝑅+2𝑅+𝑀2<𝛿. (2.8) Hence, ‖𝐽(𝑥_𝑛−𝑞)−𝐽(𝑦_𝑛−𝑞)‖<𝜖₀; ‖𝐽(𝑥_𝑛+1−𝑞)−𝐽(𝑥_𝑛−𝑞)‖<𝜖₀.
Using Lemma 1.6 and formulas above, we obtain 𝑥_𝑛+1−𝑞²≤1−𝑎_𝑛−𝑐_𝑛²𝑥_𝑛−𝑞²+2𝑎_𝑛𝑇^𝑛𝑦_𝑛−𝑞,𝐽𝑥_𝑛+1−𝑞+2𝑐_𝑛𝑢_𝑛−𝑞,𝐽𝑥_𝑛+1≤−𝑞1−𝑎_𝑛²𝑥_𝑛−𝑞²+2𝑎_𝑛𝑇^𝑛𝑦_𝑛−𝑞,𝐽𝑥_𝑛+1−𝑞−𝐽𝑥_𝑛−𝑞+2𝑎_𝑛𝑇^𝑛𝑦_𝑛−𝑞,𝐽𝑥_𝑛−𝑞−𝐽𝑦_𝑛−𝑞+2𝑎_𝑛𝑇^𝑛𝑦_𝑛−𝑞,𝐽𝑦_𝑛−𝑞+2𝑐_𝑛𝑢_𝑛−𝑞,𝐽𝑥_𝑛+1≤−𝑞1−𝑎_𝑛²𝑥_𝑛−𝑞²+2𝑎_𝑛𝑇^𝑛𝑦_𝑛−𝑞⋅𝐽𝑥_𝑛+1−𝑞−𝐽𝑥_𝑛−𝑞+2𝑎_𝑛𝑇^𝑛𝑦_𝑛−𝑞⋅𝐽𝑥_𝑛−𝑞−𝐽𝑦_𝑛−𝑞+2𝑎_𝑛𝑦_𝑛−𝑞²−Φ𝑦_𝑛−𝑞+2𝑐_𝑛𝑢_𝑛−𝑞⋅𝑥_𝑛+1≤−𝑞1−𝑎_𝑛²𝑅²+4𝑎_𝑛𝐿(1+2𝑅)𝜖₀+2𝑎_𝑛𝑦_𝑛−𝑞²−Φ𝑦_𝑛−𝑞+4𝑐_𝑛𝑦𝑀𝑅,_𝑛−𝑞²≤1−𝑏_𝑛−𝑑_𝑛²𝑥_𝑛−𝑞²+2𝑏_𝑛𝑇^𝑛𝑥_𝑛−𝑞,𝐽𝑦_𝑛−𝑞+2𝑑_𝑛𝑤_𝑛−𝑞,𝐽𝑦_𝑛≤𝑥−𝑞_𝑛−𝑞²+2𝑏_𝑛𝑇^𝑛𝑥_𝑛−𝑞,𝐽𝑦_𝑛−𝑞−𝐽𝑥_𝑛−𝑞+2𝑏_𝑛𝑇^𝑛𝑥_𝑛−𝑞,𝐽𝑥_𝑛−𝑞+2𝑑_𝑛𝑤_𝑛−𝑞⋅𝑦_𝑛≤𝑥−𝑞_𝑛−𝑞²+2𝑏_𝑛𝑇^𝑥_𝑛−𝑞⋅𝐽𝑦_𝑛−𝑞−𝐽𝑥_𝑛−𝑞+2𝑏_𝑛𝑥_𝑛−𝑞²−Φ𝑥_𝑛−𝑞+2𝑑_𝑛𝑤_𝑛−𝑞⋅𝑦_𝑛−𝑞≤𝑅²+2𝑏_𝑛𝐿(1+𝑅)𝜖₀+2𝑏_𝑛𝑅²+4𝑑_𝑛𝑀𝑅. (2.9)(2.10) Substitute (2.10) into (2.9) 𝑥_𝑛+1−𝑞²≤1−𝑎_𝑛²𝑅²+4𝑎_𝑛𝐿(1+2𝑅)𝜖₀+2𝑎_𝑛𝑅²+2𝑏_𝑛𝐿(1+𝑅)𝜖₀+2𝑏_𝑛𝑅²+4𝑑_𝑛𝑀𝑅−2𝑎_𝑛Φ𝑦_𝑛−𝑞+4𝑐_𝑛𝑀𝑅≤𝑅²+𝑎²_𝑛𝑅²+4𝑎_𝑛𝐿(1+2𝑅)𝜖₀+2𝑎_𝑛2𝑏_𝑛𝐿(1+𝑅)𝜖₀+2𝑏_𝑛𝑅²+4𝑑_𝑛𝑀𝑅−2𝑎_𝑛Φ𝑅4+4𝑐_𝑛𝑀𝑅=𝑅²+2𝑎_𝑛𝑎_𝑛2𝑅²+2𝐿(1+2𝑅)𝜖₀+2𝑏_𝑛𝐿(1+𝑅)𝜖₀+2𝑏_𝑛𝑅²+4𝑑_𝑛𝑀𝑅+2𝑐_𝑛𝑀𝑅𝑎_𝑛−2𝑎_𝑛Φ𝑅4≤𝑅²+2𝑎_𝑛Φ(𝑅/4)2𝑅−Φ4≤𝑅²𝑅−Φ4𝑎_𝑛≤𝑅², (2.11) this is a contradiction. Thus 𝑥_𝑛+1∈𝐵₁, that is, {𝑥_𝑛} is a bounded sequence. So {𝑦_𝑛},{𝑇^𝑛𝑦_𝑛},{𝑇^𝑛𝑥_𝑛} are all bounded sequences. Since ‖𝑧_𝑛−𝑞‖→0 as 𝑛→∞, without loss of generality, we let ‖𝑧_𝑛−𝑞‖≤1. Therefore, ‖𝑥_𝑛−𝑧_𝑛‖ is also bounded.
Step 3. We want to prove ‖𝑥_𝑛−𝑧_𝑛‖→0 as 𝑛→∞.
Set 𝑀₀=max{sup_𝑛‖𝑇^𝑛𝑦_𝑛−𝑇^𝑛𝑧_𝑛‖,sup_𝑛‖𝑣_𝑛−𝑢_𝑛‖,sup_𝑛‖𝑥_𝑛−𝑧_𝑛‖,sup_𝑛‖𝑇^𝑛𝑥_𝑛−𝑥_𝑛‖,sup_𝑛‖𝑤_𝑛−𝑥_𝑛‖,sup_𝑛‖𝑦_𝑛−𝑧_𝑛‖,sup_𝑛‖𝑣_𝑛−𝑥_𝑛‖}.
Again using Lemma 1.6, we have 𝑥_𝑛+1−𝑧_𝑛+1²≤1−𝑎_𝑛−𝑐_𝑛²𝑥_𝑛−𝑧_𝑛²+2𝑎_𝑛𝑇^𝑛𝑦_𝑛−𝑇^𝑛𝑧_𝑛,𝐽𝑥_𝑛+1−𝑧_𝑛+1+2𝑐_𝑛𝑣_𝑛−𝑢_𝑛,𝐽𝑥_𝑛+1−𝑧_𝑛+1≤1−𝑎_𝑛²𝑥_𝑛−𝑧_𝑛²+2𝑎_𝑛𝑇^𝑛𝑦_𝑛−𝑇^𝑛𝑧_𝑛,𝐽𝑥_𝑛+1−𝑧_𝑛+1−𝐽𝑥_𝑛−𝑧_𝑛+2𝑎_𝑛𝑇^𝑛𝑦_𝑛−𝑇^𝑛𝑧_𝑛,𝐽𝑥_𝑛−𝑧_𝑛−𝐽𝑦_𝑛−𝑧_𝑛+2𝑎_𝑛𝑇^𝑛𝑦_𝑛−𝑇^𝑛𝑧_𝑛,𝐽𝑦_𝑛−𝑧_𝑛+2𝑐_𝑛𝑣_𝑛−𝑢_𝑛⋅𝑥_𝑛+1−𝑧_𝑛+1≤1−𝑎_𝑛²𝑥_𝑛−𝑧_𝑛²+2𝑎_𝑛𝑀₀𝐴_𝑛+2𝑎_𝑛𝑀₀𝐵_𝑛+2𝑎_𝑛𝑦_𝑛−𝑧_𝑛²−Φ𝑦_𝑛−𝑧_𝑛+2𝑐_𝑛𝑀²₀,𝑦_𝑛−𝑧_𝑛²≤𝑥_𝑛−𝑧_𝑛²+2𝑏_𝑛𝑇^𝑛𝑥_𝑛−𝑥_𝑛,𝐽𝑦_𝑛−𝑧_𝑛+2𝑑_𝑛𝑤_𝑛−𝑥_𝑛,𝐽𝑦_𝑛−𝑧_𝑛≤𝑥_𝑛−𝑧_𝑛²+2𝑏_𝑛𝑀²₀+2𝑑_𝑛𝑀²₀, (2.12)(2.13) where 𝐴_𝑛=‖𝐽(𝑥_𝑛+1−𝑧_𝑛+1)−𝐽(𝑥_𝑛−𝑧_𝑛)‖, 𝐵_𝑛=‖𝐽(𝑥_𝑛−𝑧_𝑛)−𝐽(𝑦_𝑛−𝑧_𝑛)‖, and 𝐴_𝑛,𝐵_𝑛→0 as 𝑛→∞.
Taking place (2.13) into (2.12), we have 𝑥_𝑛+1−𝑧_𝑛+1²≤1−𝑎_𝑛²𝑥_𝑛−𝑧_𝑛²+2𝑎_𝑛𝑀₀𝐴_𝑛+2𝑎_𝑛𝑀₀𝐵_𝑛+2𝑎_𝑛𝑥_𝑛−𝑧_𝑛²+2𝑏_𝑛𝑀²₀+2𝑑_𝑛𝑀²₀−Φ𝑦_𝑛−𝑧_𝑛+2𝑐_𝑛𝑀²₀≤𝑥_𝑛−𝑧_𝑛²+𝑎²_𝑛𝑀²₀+2𝑎_𝑛𝑀₀𝐴_𝑛+2𝑎_𝑛𝑀₀𝐵_𝑛+4𝑎_𝑛𝑏_𝑛𝑀²₀+4𝑎_𝑛𝑑_𝑛𝑀²₀−2𝑎_𝑛Φ𝑦_𝑛−𝑧_𝑛+2𝑐_𝑛𝑀²₀=𝑥_𝑛−𝑧_𝑛²+2𝑎_𝑛𝐶_𝑛−2𝑎_𝑛Φ𝑦_𝑛−𝑧_𝑛, (2.14) where 𝐶_𝑛=𝑎_𝑛𝑀²₀/2+𝑀₀𝐴_𝑛+𝑀₀𝐵_𝑛+2𝑏_𝑛𝑀²₀+2𝑑_𝑛𝑀²₀+𝑐_𝑛𝑀²₀/𝑎_𝑛→0 as 𝑛→∞.
Set inf_𝑛≥0Φ(‖𝑦_𝑛−𝑧_𝑛‖)/(1+‖𝑥_𝑛+1−𝑧_𝑛+1‖²)=𝜆, then 𝜆=0. If it is not the case, we assume that 𝜆>0. Let 0<𝛾<min{1,𝜆}, then Φ(‖𝑦_𝑛−𝑧_𝑛‖)/(1+‖𝑥_𝑛+1−𝑧_𝑛+1‖²)≥𝛾, that is, Φ(‖𝑦_𝑛−𝑧_𝑛‖)≥𝛾+𝛾‖𝑥_𝑛+1−𝑧_𝑛+1‖²≥𝛾‖𝑥_𝑛+1−𝑧_𝑛+1‖². Thus, from (2.14) that 𝑥_𝑛+1−𝑧_𝑛+1²≤𝑥_𝑛−𝑧_𝑛²+2𝑎_𝑛𝐶_𝑛−𝛾𝑥_𝑛+1−𝑧_𝑛+1², (2.15) which implies that 𝑥_𝑛+1−𝑧_𝑛+1²≤11+2𝑎_𝑛𝛾𝑥_𝑛−𝑧_𝑛²+2𝑎_𝑛𝐶_𝑛1+2𝑎_𝑛𝛾=1−2𝑎_𝑛𝛾1+2𝑎_𝑛𝛾𝑥_𝑛−𝑧_𝑛²+2𝑎_𝑛𝐶_𝑛1+2𝑎_𝑛𝛾. (2.16) Let 𝜌_𝑛=‖𝑥_𝑛−𝑧_𝑛‖²,𝜆_𝑛=2𝑎_𝑛𝛾/(1+2𝑎_𝑛𝛾),𝜎_𝑛=2𝑎_𝑛𝐶_𝑛/(1+2𝑎_𝑛𝛾). Then we get that 𝜌_𝑛+1≤1−𝜆_𝑛𝜌_𝑛+𝜎_𝑛. (2.17) Applying Lemma 1.7, we get that 𝜌_𝑛→0 as 𝑛→∞. This is a contradiction and so 𝜆=0. Therefore, there exists an infinite subsequence such that Φ(‖𝑦_𝑛𝑖−𝑧_𝑛𝑖‖)/(1+‖𝑥_𝑛𝑖+1−𝑧_𝑛𝑖+1‖²)→0 as 𝑖→∞. Since 0≤Φ(‖𝑦_𝑛𝑖−𝑧_𝑛𝑖‖)/(1+𝑀²₀)≤Φ(‖𝑦_𝑛𝑖−𝑧_𝑛𝑖‖)/(1+‖𝑥_𝑛𝑖+1−𝑧_𝑛𝑖+1‖²), then Φ(‖𝑦_𝑛𝑖−𝑧_𝑛𝑖‖)→0 as 𝑖→∞. In view of the strictly increasing and continuity of Φ, we have ‖𝑦_𝑛𝑖−𝑧_𝑛𝑖‖→0 as 𝑖→∞. From (1.7), we have 𝑥_𝑛𝑖−𝑧_𝑛𝑖≤𝑦_𝑛𝑖−𝑧_𝑛𝑖+𝑏_𝑛𝑖𝑥_𝑛𝑖−𝑇𝑥_𝑛𝑖+𝑐_𝑛𝑖𝑥_𝑛𝑖−𝑤_𝑛𝑖→0, (2.18) as 𝑖→∞. Next we want to prove ‖𝑥_𝑛−𝑧_𝑛‖→0 as 𝑛→∞. Let forall𝜀∈(0,1), there exists 𝑛_𝑖0 such that ‖𝑥_𝑛𝑖−𝑧_𝑛𝑖‖<𝜖,𝑎_𝑛,𝑎_𝑛𝑖<min{𝜖/4𝐿(1+𝑀₀),𝜖/8𝑀₀}, 𝑐_𝑛,𝑐_𝑛𝑖<𝜖/16𝑀₀, 𝑏_𝑛,𝑑_𝑛,𝑏_𝑛𝑖,𝑑_𝑛𝑖<𝜖/8𝑀₀,𝐶_𝑛,𝐶_𝑛𝑖<Φ(𝜖/4)/2, for any 𝑛_𝑖,𝑛≥𝑛_𝑖0. First, we want to prove ‖𝑥_𝑛𝑖+1−𝑧_𝑛𝑖+1‖<𝜖. Suppose it is not this case, then ‖𝑥_𝑛𝑖+1−𝑧_𝑛𝑖+1‖≥𝜖. Using (1.7), we may get the following estimates: 𝑥_𝑛𝑖−𝑧_𝑛𝑖≥𝑥_𝑛𝑖+1−𝑧_𝑛𝑖+1−𝑎_𝑛𝑖𝑇^𝑛𝑦_𝑛𝑖−𝑇^𝑛𝑧_𝑛𝑖−𝑎_𝑛𝑖𝑥_𝑛𝑖−𝑧_𝑛𝑖−𝑐_𝑛𝑖𝑣_𝑛𝑖−𝑢_𝑛𝑖−𝑐_𝑛𝑖𝑥_𝑛𝑖−𝑧_𝑛𝑖≥𝜖−𝑎_𝑛𝑖𝐿1+𝑀₀−𝑎_𝑛𝑖+2𝑐_𝑛𝑖𝑀₀>𝜖2,𝑦_𝑛𝑖−𝑧_𝑛𝑖≥𝑥_𝑛𝑖−𝑧_𝑛𝑖−𝑏_𝑛𝑖𝑇^𝑛𝑥_𝑛𝑖−𝑥_𝑛𝑖−𝑑_𝑛𝑖𝑣_𝑛𝑖−𝑥_𝑛𝑖≥𝜖2−𝑏_𝑛𝑖+𝑑_𝑛𝑖𝑀₀>𝜖4. (2.19)(2.20) Since Φ is strictly increasing, then (2.20) leads to Φ(‖𝑦_𝑛𝑖−𝑧_𝑛𝑖‖)≥Φ(𝜖/4). From (2.14), we have 𝑥_𝑛𝑖+1−𝑧_𝑛𝑖+1²≤𝑥_𝑛𝑖−𝑧_𝑛𝑖²+2𝑎_𝑛𝑖𝐶_𝑛𝑖−Φ𝑦_𝑛𝑖−𝑧_𝑛𝑖<𝜖²+2𝑎_𝑛𝑖12Φ𝜖4𝜖−Φ4≤𝜖²𝜖−Φ4𝑎_𝑛𝑖≤𝜖², (2.21) is a contradiction. Hence, ‖𝑥_𝑛𝑖+1−𝑧_𝑛𝑖+1‖<𝜖. Suppose that ‖𝑥_{𝑛𝑖+𝑚}−𝑧_{𝑛𝑖+𝑚}‖<𝜖 holds. Repeating the above course, we can easily prove that ‖𝑥_{𝑛𝑖+𝑚+1}−𝑧_{𝑛𝑖+𝑚+1}‖<𝜖 holds. Therefore, for any 𝑚 and 𝑛_𝑖≥𝑛₀, we obtain that ‖𝑥_{𝑛𝑖+𝑚}−𝑧_{𝑛𝑖+𝑚}‖<𝜖, which means ‖𝑥_𝑛−𝑧_𝑛‖→0 as 𝑛→∞. This completes the proof.

In order to make the existence of Theorem 2.2 more meaningful, we give the following theorem.

Theorem 2.3. Let 𝐸 be an arbitrary uniformly smooth real Banach space, let 𝐷 be a nonempty closed convex subset of 𝐸, and let 𝑇∶𝐷→𝐷 be a uniformly generalized Lipschitz generalized asymptotically Φ-strongly pseudocontractive mapping with 𝑞∈𝐹(𝑇)≠∅. Let {𝑎_𝑛},{𝑐_𝑛} be two real sequences in [0,1] and satisfy the conditions (i) 𝑎_𝑛+𝑐_𝑛≤1; (ii) 𝑎_𝑛→0 as 𝑛→∞ and 𝑐_𝑛=𝑜(𝑎_𝑛); (iii) Σ^∞_𝑛=0𝑎_𝑛=∞. For some 𝑧₀∈𝐷, let {𝑢_𝑛} be any bounded sequence in 𝐷 and let {𝑧_𝑛} be modified Mann iterative sequence with errors defined by (1.8). Then {𝑧_𝑛} converges strongly to the unique fixed point 𝑞 of 𝑇.

Proof. Since 𝑇∶𝐷→𝐷 is a uniformly generalized Lipschitz generalized asymptotically Φ-strongly pseudocontractive mapping, then there exists a strictly increasing continuous function Φ∶[0,+∞)→[0,+∞) with Φ(0)=0 such that 𝑘_𝑛𝐼−𝑇^𝑛𝑥−𝑘_𝑛𝐼−𝑇^𝑛,𝑇𝑦,𝐽𝑥−𝑦≥Φ𝑥−𝑦_𝑛𝑥−𝑇^𝑛𝑦≤𝐿1+𝑥−𝑦, (2.22)(2.23) for any 𝑥,𝑦∈𝐷.
Step 1. There exists 𝑧₀∈𝐷 and 𝑧₀≠𝑇𝑧₀ such that 𝑟₀=(𝑘+𝐿)‖𝑧₀−𝑞‖²+𝐿‖𝑧₀−𝑞‖∈𝑅(Φ), where 𝑘=sup_𝑛{𝑘_𝑛}. In fact, if Φ(𝑟)→+∞ as 𝑟→+∞, then 𝑟₀∈𝑅(Φ); if sup{Φ(𝑟)∶𝑟∈[0,+∞)}=𝑟₁<+∞ with 𝑟₁<𝑟₀, then, for 𝑞∈𝐷, there exists a sequence {𝜈_𝑛} in 𝐷 such that 𝜈_𝑛→𝑞 as 𝑛→∞ with 𝜈_𝑛≠𝑞. Furthermore, there exists a natural number 𝑛₀ such that (𝑘+𝐿)‖𝜈_𝑛−𝑞‖²+𝐿‖𝜈_𝑛−𝑞‖<(𝑟₁/2) for 𝑛≥𝑛₀, then we redefine 𝑧₀,𝑟₀ such that 𝑧₀=𝜈_𝑛0,𝑟₀=(𝑘+𝐿)‖𝑧₀−𝑞‖²+𝐿‖𝑧₀−𝑞‖∈𝑅(Φ).
Step 2. For any 𝑛≥0, {𝑧_𝑛} is bounded.
Set 𝑟=Φ⁻¹(𝑟₀), we have ‖𝑥₀−𝑞‖≤𝑅. Let 𝐵^′₁={𝑧∈𝐷∶‖𝑧−𝑞‖≤𝑟},𝐵^′₂={𝑧∈𝐷∶‖𝑧−𝑞‖≤2𝑟},𝑀′=sup_𝑛{‖𝑢_𝑛−𝑞‖}. Next, we prove that 𝑧_𝑛∈𝐵^′₁ for any 𝑛≥0 by induction. First 𝑧₀∈𝐵^′₁ is obvious. Suppose that 𝑧_𝑛∈𝐵^′₁ holds. We prove that 𝑧_𝑛+1∈𝐵^′₁. If it is not the case, then ‖𝑧_𝑛+1−𝑞‖>𝑟. By uniformly continuity of 𝐽 on bounded subset, we choose 𝜖₀=Φ(𝑟/2)/16𝐿(1+2𝑟), there exists 𝛿>0 such that ‖𝐽𝑥−𝐽𝑦‖<𝜖₀ when ‖𝑥−𝑦‖<𝛿,forall𝑥,𝑦∈𝐵^′₂. Now denote 𝜏₀𝑟=min2𝐿(1+𝑟)+2𝑟+𝑀^′,𝛿2𝐿(1+𝑟)+2𝑟+𝑀^′,Φ(𝑟/2)8𝑟²,Φ(𝑟/2)(),Φ()24𝐿1+2𝑟𝑟/216𝑀^′𝑟. (2.24) Since 𝑎_𝑛,𝑐_𝑛,𝑘_𝑛−1→0 as 𝑛→∞, and 𝑐_𝑛=𝑜(𝑎_𝑛), without loss of generality, let 0≤𝑎_𝑛,𝑐_𝑛,𝑘_𝑛−1≤𝜏₀,𝑐_𝑛<𝑎_𝑛𝜏₀ for any 𝑛≥0. Then we have the following estimates from (1.8): 𝑧_𝑛−𝑇^𝑛𝑧_𝑛≤𝑧_𝑛−𝑞+𝑇^𝑛𝑧_𝑛(),𝑧−𝑞≤𝑟+𝐿1+𝑟_𝑛−𝑞≥𝑧_𝑛+1−𝑞−𝑎_𝑛𝑇^𝑛𝑧_𝑛−𝑧_𝑛−𝑐_𝑛𝑢_𝑛−𝑧_𝑛>𝑟−𝑎_𝑛[𝑟+𝐿(1+𝑟)]−𝑐_𝑛𝑟+𝑀^′≥𝑟−𝜏₀𝐿(1+𝑟)+2𝑟+𝑀^′≥𝑟2,𝑧_𝑛+1−𝑞≤1−𝑎_𝑛−𝑐_𝑛𝑧_𝑛−𝑞+𝑎_𝑛𝑇^𝑛𝑧_𝑛−𝑞+𝑐_𝑛𝑢_𝑛−𝑞≤𝑟+𝜏₀𝐿(1+𝑟)+𝑀^′𝑧≤2𝑟,_𝑛+1−𝑞−𝑧_𝑛−𝑞≤𝑎_𝑛𝑇^𝑛𝑧_𝑛−𝑧_𝑛+𝑐_𝑛𝑢_𝑛−𝑧_𝑛≤𝑎_𝑛[𝑟+𝐿(1+𝑟)]+𝑐_𝑛𝑟+𝑀^′≤𝜏₀𝐿(1+𝑟)+2𝑟+𝑀^′≤𝛿2<𝛿. (2.25) Therefore, ‖𝐽(𝑧_𝑛+1−𝑞)−𝐽(𝑧_𝑛−𝑞)‖<𝜖₀.
Using Lemma 1.6 and formulas above, we obtain 𝑧_𝑛+1−𝑞²≤1−𝑎_𝑛²𝑧_𝑛−𝑞²+2𝑎_𝑛𝑇^𝑛𝑧_𝑛−𝑞,𝐽𝑧_𝑛+1−𝑞−𝐽𝑧_𝑛−𝑞+2𝑎_𝑛𝑇^𝑛𝑧_𝑛−𝑞,𝐽𝑧_𝑛−𝑞+2𝑐_𝑛𝑢_𝑛−𝑞,𝐽𝑧_𝑛+1≤−𝑞1−𝑎_𝑛²𝑧_𝑛−𝑞²+2𝑎_𝑛𝑇^𝑛𝑧_𝑛−𝑞⋅𝐽𝑧_𝑛+1−𝑞−𝐽𝑧_𝑛−𝑞+2𝑎_𝑛𝑘_𝑛𝑧_𝑛−𝑞²−Φ𝑧_𝑛−𝑞+2𝑐_𝑛𝑢_𝑛−𝑞⋅𝑧_𝑛+1≤−𝑞1−𝑎_𝑛²𝑟²+4𝑎_𝑛𝐿(1+2𝑟)𝜖₀+2𝑎_𝑛𝑘_𝑛𝑧_𝑛−𝑞²−Φ𝑧_𝑛−𝑞+4𝑐_𝑛≤𝑀′𝑟1−𝑎_𝑛²𝑟²+4𝑎_𝑛𝐿(1+2𝑟)𝜖₀+2𝑎_𝑛𝑘_𝑛𝑟²𝑟−Φ2+4𝑐_𝑛𝑀′𝑟=𝑟²+2𝑎_𝑛𝑎_𝑛2𝑟²+2𝐿(1+2𝑟)𝜖₀+𝑘_𝑛−1𝑟²+2𝑐_𝑛𝑀^′𝑟𝑎_𝑛−2𝑎_𝑛Φ𝑟2≤𝑟²+2𝑎_𝑛Φ(𝑟/2)2𝑟−Φ2≤𝑟²−𝑎_𝑛Φ𝑟2≤𝑟², (2.26) this is a contradiction. Thus 𝑧_𝑛+1∈𝐵^′₁, that is, {𝑧_𝑛} is a bounded sequence, so {𝑇^𝑛𝑧_𝑛} is also bounded. Denote 𝑀₀=sup_𝑛{‖𝑧_𝑛−𝑞‖}+sup_𝑛{‖𝑇^𝑛𝑧_𝑛−𝑞‖}+sup_𝑛{‖𝑢_𝑛−𝑞‖}.
Step 3. We prove ‖𝑧_𝑛−𝑞‖→0 as 𝑛→∞.
Again using Lemma 1.6, we have 𝑧_𝑛+1−𝑞²≤1−𝑎_𝑛−𝑐_𝑛²𝑧_𝑛−𝑞²+2𝑎_𝑛𝑇^𝑛𝑧_𝑛−𝑞,𝐽𝑧_𝑛+1−𝑞+2𝑐_𝑛𝑢_𝑛−𝑞,𝐽𝑧_𝑛+1≤−𝑞1−𝑎_𝑛²𝑧_𝑛−𝑞²+2𝑎_𝑛𝑇^𝑛𝑧_𝑛−𝑞,𝐽𝑧_𝑛+1−𝑞−𝐽𝑧_𝑛−𝑞+2𝑎_𝑛𝑇^𝑛𝑧_𝑛−𝑞,𝐽𝑧_𝑛−𝑞+2𝑐_𝑛𝑢_𝑛−𝑞⋅𝑧_𝑛+1≤−𝑞1−𝑎_𝑛²𝑧_𝑛−𝑞²+2𝑎_𝑛𝑀₀𝐷_𝑛+2𝑎_𝑛𝑘_𝑛𝑧_𝑛−𝑞²−Φ𝑧_𝑛−𝑞+2𝑐_𝑛𝑀²₀≤𝑧_𝑛−𝑞²+2𝑎_𝑛𝑘_𝑛−1𝑀²₀+𝑎_𝑛𝑀²₀2+𝑀₀𝐷_𝑛+𝑐_𝑛𝑀²₀𝑎_𝑛−Φ𝑧_𝑛≤𝑧−𝑞_𝑛−𝑞²+2𝑎_𝑛𝐸_𝑛−Φ𝑧_𝑛−𝑞, (2.27) where 𝐷_𝑛=𝐽𝑧_𝑛+1−𝑞−𝐽𝑧_𝑛−𝑞,𝐸_𝑛=𝑘_𝑛−1𝑀²₀+𝑎_𝑛𝑀²₀2+𝑀₀𝐷_𝑛+𝑐_𝑛𝑀²₀𝑎_𝑛, (2.28) and 𝐷_𝑛,𝐸_𝑛→0 as 𝑛→∞.
Set inf_𝑛≥0Φ(‖𝑧_𝑛−𝑞‖)/(1+‖𝑧_𝑛+1−𝑞‖²)=𝜆, then 𝜆=0. If it is not the case, we assume that 𝜆>0. Let 0<𝛾<min{1,𝜆}, then Φ(‖𝑧_𝑛−𝑞‖)/(1+‖𝑧_𝑛+1−𝑞‖²)≥𝛾, that is, Φ(‖𝑧_𝑛−𝑞‖)≥𝛾+𝛾‖𝑧_𝑛+1−𝑞‖²≥𝛾‖𝑧_𝑛+1−𝑞‖². Thus, from (2.14) that 𝑧_𝑛+1−𝑞²≤𝑧_𝑛−𝑞²+2𝑎_𝑛𝐸_𝑛−𝛾𝑧_𝑛+1−𝑞², (2.29) which implies that 𝑧_𝑛+1−𝑞²≤11+2𝑎_𝑛𝛾𝑧_𝑛−𝑞²+2𝑎_𝑛𝐸_𝑛1+2𝑎_𝑛𝛾=1−2𝑎_𝑛𝛾1+2𝑎_𝑛𝛾𝑧_𝑛−𝑞²+2𝑎_𝑛𝐸_𝑛1+2𝑎_𝑛𝛾. (2.30) Let 𝜌_𝑛=‖𝑧_𝑛−𝑞‖²,𝜆_𝑛=2𝑎_𝑛𝛾/(1+2𝑎_𝑛𝛾),𝜎_𝑛=2𝑎_𝑛𝐸_𝑛/(1+2𝑎_𝑛𝛾). Then we get that 𝜌_𝑛+1≤1−𝜆_𝑛𝜌_𝑛+𝜎_𝑛. (2.31) Applying Lemma 1.7, we get that 𝜌_𝑛→0 as 𝑛→∞. This is a contradiction and so 𝜆=0. Therefore, there exists an infinite subsequence such that Φ(‖𝑧_𝑛𝑖−𝑞‖)/(1+‖𝑧_𝑛𝑖+1−𝑞‖²)→0 as 𝑖→∞. Since 0≤Φ(‖𝑧_𝑛𝑖−𝑞‖)/(1+𝑀²₀)≤Φ(‖𝑧_𝑛𝑖−𝑞‖)/(1+‖𝑧_𝑛𝑖+1−𝑞‖²), then Φ(‖𝑧_𝑛𝑖−𝑞‖)→0 as 𝑖→∞. In view of the strictly increasing and continuity of Φ, we have ‖𝑧_𝑛𝑖−𝑞‖→0 as 𝑖→∞. Let 𝜀∈(0,1)  be any given, there exists 𝑛_𝑖0 such that ‖𝑧_𝑛𝑖−𝑞‖<𝜖,𝑎_𝑛𝑖,𝑎_𝑛<min{𝜖/4𝐿(1+𝑀₀),𝜖/8𝑀₀}, 𝑐_𝑛𝑖,𝑐_𝑛<𝜖/16𝑀₀, 𝐸_𝑛𝑖,𝐸_𝑛<Φ(𝜖/2)/2, for any 𝑛_𝑖,𝑛≥𝑛_𝑖0. First, we want to prove ‖𝑧_𝑛𝑖+1−𝑞‖<𝜖. Suppose it is not this case, then ‖𝑧_𝑛𝑖+1−𝑞‖≥𝜖. Using (1.8), we may get the following estimates: 𝑧_𝑛𝑖−𝑞≥𝑧_𝑛𝑖+1−𝑞−𝑎_𝑛𝑖𝑇^𝑛𝑧_𝑛𝑖−𝑞−𝑎_𝑛𝑖𝑧_𝑛𝑖−𝑞−𝑐_𝑛𝑖𝑢_𝑛𝑖−𝑞≥𝜖−𝑎_𝑛𝑖𝐿1+𝑀₀−𝑎_𝑛𝑖+2𝑐_𝑛𝑖𝑀₀>𝜖2. (2.32) Since Φ is strictly increasing, then (2.32) leads to Φ(‖𝑧_𝑛𝑖−𝑞‖)≥Φ(𝜖/2). From (2.27), we have 𝑧_𝑛𝑖+1−𝑞²≤𝑧_𝑛𝑖−𝑞²+2𝑎_𝑛𝑖𝐸_𝑛𝑖−Φ𝑧_𝑛𝑖−𝑞<𝜖²+2𝑎_𝑛𝑖12Φ𝜖2𝜖−Φ2≤𝜖²𝜖−Φ2𝑎_𝑛𝑖≤𝜖², (2.33) is a contradiction. Hence, ‖𝑧_𝑛𝑖+1−𝑞‖<𝜖. Suppose that ‖𝑧_{𝑛𝑖+𝑚}−𝑞‖<𝜖 holds. Repeating the above course, we can easily prove that ‖𝑧_{𝑛𝑖+𝑚+1}−𝑞‖<𝜖 holds. Therefore, for any 𝑚 and 𝑛_𝑖≥𝑛₀, we obtain that ‖𝑧_{𝑛𝑖+𝑚}−𝑞‖<𝜖, which means ‖𝑧_𝑛−𝑞‖→0 as 𝑛→∞. This completes the proof.

Theorem 2.4. Let 𝐸 be an arbitrary uniformly smooth real Banach space, let 𝐷 be a nonempty closed convex subset of 𝐸, and let 𝑇∶𝐷→𝐷 be a uniformly generalized Lipschitz generalized asymptotically Φ-strongly pseudocontractive mapping with 𝑞∈𝐹(𝑇)≠∅. Let {𝑎_𝑛},{𝑏_𝑛},{𝑐_𝑛},{𝑑_𝑛} be four real sequences in [0,1] and satisfy the conditions (i) 𝑎_𝑛+𝑐_𝑛≤1, 𝑏_𝑛+𝑑_𝑛≤1; (ii) 𝑎_𝑛,𝑏_𝑛,𝑑_𝑛→0 as 𝑛→∞ and 𝑐_𝑛=𝑜(𝑎_𝑛); (iii) Σ^∞_𝑛=0𝑎_𝑛=∞. For some 𝑥₀∈𝐷, let {𝑣_𝑛},{𝑤_𝑛} be two arbitrary bounded sequences in 𝐷, and let {𝑥_𝑛} be Ishikawa iterative sequence with errors defined by (1.7). Then (1.7) converges strongly to the unique fixed point 𝑞 of 𝑇.

Proof. By Theorems 2.3 and 2.2, we obtain directly the result of Theorem 2.4.

Remark 2.5. Our Theorem 2.2 extends and improves Theorem 3.1 of [4] from the bounded range of 𝑇 to uniformly generalized Lipschitz mapping, and the proof course of Theorem 2.2 is quite different from that of [4].