Necessary and Sufficient Condition for Mann Iteration Converges to a Fixed Point of Lipschitzian Mappings

Abstract

Suppose that 𝐸 is a real normed linear space, 𝐶 is a nonempty convex subset of 𝐸, 𝑇∶𝐶→𝐶 is a Lipschitzian mapping, and 𝑥^∗∈𝐶 is a fixed point of 𝑇. For given 𝑥₀∈𝐶, suppose that the sequence {𝑥_𝑛}⊂𝐶 is the Mann iterative sequence defined by 𝑥_𝑛+1=(1−𝛼_𝑛)𝑥_𝑛+𝛼_𝑛𝑇𝑥_𝑛,𝑛≥0, where {𝛼_𝑛} is a sequence in [0, 1], ∑^∞_𝑛=0𝛼²_𝑛<∞, ∑^∞_𝑛=0𝛼_𝑛=∞. We prove that the sequence {𝑥_𝑛} strongly converges to 𝑥^∗ if and only if there exists a strictly increasing function Φ∶[0,∞)→[0,∞) with Φ(0)=0 such that limsup_𝑛→∞inf_{𝑗(𝑥𝑛−𝑥∗)∈𝐽(𝑥𝑛−𝑥∗)}{⟨𝑇𝑥_𝑛−𝑥^∗,𝑗(𝑥_𝑛−𝑥^∗)⟩−‖𝑥_𝑛−𝑥^∗‖²+Φ(‖𝑥_𝑛−𝑥^∗‖)}≤0.

1. Introduction

Let 𝐸 be an arbitrary real normed linear space with dual space 𝐸^∗, and let 𝐶 be a nonempty subset of 𝐸. We denote by 𝐽 the normalized duality mapping from 𝐸 to 2^𝐸∗ defined by 𝐽(𝑥)=𝑓∈𝐸^∗∶𝑥,𝑓=‖𝑥‖²=𝑓²,∀𝑥∈𝐸, (1.1) where ⟨⋅,⋅⟩ denotes the generalized duality pairing.

A mapping 𝑇∶𝐶→𝐸 is called strongly pseudocontractive if there exists a constant 𝑘∈(0,1) such that, for all 𝑥,𝑦∈𝐶, there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) satisfying 𝑇𝑥−𝑇𝑦,𝑗𝑥−𝑦≤(1−𝑘)𝑥−𝑦². (1.2)𝑇 is called 𝜙-strongly pseudocontractive if there exists a strictly increasing function 𝜙∶[0,∞)→[0,∞) with 𝜙(0)=0 such that, for all 𝑥,𝑦∈𝐶, there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) satisfying 𝑇𝑥−𝑇𝑦,𝑗𝑥−𝑦≤𝑥−𝑦²−𝜙𝑥−𝑦𝑥−𝑦. (1.3)𝑇 is called generalized Φ-pseudocontractive (see, e.g., [1]) if there exists a strictly increasing function Φ∶[0,∞)→[0,∞) with Φ(0)=0 such that 𝑇𝑥−𝑇𝑦,𝑗𝑥−𝑦≤𝑥−𝑦²−Φ𝑥−𝑦 (1.4) holds for all 𝑥,𝑦∈𝐶 and for some 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦).

Let 𝐹(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥} denote the fixed point set of 𝑇. If 𝐹(𝑇)≠∅, and (1.3) and (1.4) hold for all 𝑥∈𝐶 and 𝑦∈𝐹(𝑇), then the corresponding mapping 𝑇 is called 𝜙-hemicontractive and generalized Φ-hemicontractive, respectively. It is well known that these kinds of mappings play important roles in nonlinear analysis.

𝜙-hemicontractive (resp., generalized Φ-hemicontractive) mapping is also called uniformly pseudocontractive (resp., uniformly hemicontractive) mapping in [2, 3]. It is easy to see that if 𝑇 is generalized Φ-hemicontractive mapping, then 𝐹(𝑇) is singleton.

It is known (see, e.g., [4, 5]) that the class of strongly pseudocontractive mappings is a proper subset of the class of 𝜙-strongly pseudocontractive mappings. By taking Φ(𝑠)=𝑠𝜙(𝑠), where 𝜙∶[0,∞)→[0,∞) is a strictly increasing function with 𝜙(0)=0, we know that the class of 𝜙-strongly pseudocontractive mappings is a subset of the class of generalized Φ-pseudocontractive mappings. Similarly, the class of 𝜙-hemicontractive mappings is a subset of the class of generalized Φ-hemicontractive mappings. The example in [6] demonstrates that the class of Lipschitzian 𝜙-hemicontractive mappings is a proper subset of the class of Lipschitzian generalized Φ-hemicontractive mappings.

It is well known (see, e.g., [7]) that if 𝐶 is a nonempty closed convex subset of a real Banach space 𝐸 and 𝑇∶𝐶→𝐶 is a continuous strongly pseudocontractive mapping, then 𝑇 has a unique fixed point 𝑝∈𝐶. In 2009, it has been proved in [8] that if 𝐶 is a nonempty closed convex subset of a real Banach space 𝐸 and 𝑇∶𝐶→𝐶 is a continuous generalized Φ-pseudocontractive mappings, then 𝑇 has a unique fixed point 𝑝∈𝐶.

Many results have been proved on convergence or stability of Ishikawa iterative sequences (with errors) or Mann iterative sequences (with errors) for Lipschitzian 𝜙-hemicontractive mappings or Lipschitzian generalized Φ-hemicontractive mapping (see, e.g., [4–6, 9–12] and the references therein). In 2010, Xiang et al. [6] proved the following result.

Theorem XCZ (see [6, Theorem 3.2]). Let 𝐸 be a real normed linear space, let 𝐶 be a nonempty convex subset of 𝐸, and let 𝑇∶𝐶→𝐶 be a Lipschitzian generalized Φ-hemicontractive mapping. For given 𝑥₀∈𝐶, suppose that the sequence {𝑥_𝑛}⊂𝐶 is the Mann iterative sequence defined by𝑥_𝑛+1=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇𝑥_𝑛,𝑛≥0, (1.5)where {𝛽_𝑛} is a sequence in [0,1] satisfying the following conditions: (1)∑^∞_𝑛=0𝛽_𝑛=∞, (2)∑^∞_𝑛=0𝛽²_𝑛<∞. Then {𝑥_𝑛} converges strongly to the unique fixed point of 𝑇 in 𝐶.

The main purpose of this paper is to give necessary and sufficient condition for the Mann iterative sequence which converges to a fixed point of general Lipschitzian mappings in an arbitrary real normed linear space. As an immediate consequence, we will obtain necessary and sufficient condition for the Mann iterative sequence which converges to a solution of a general Lipschitzian operator equation 𝑇𝑥=𝑓.

2. Preliminaries

The following lemmas will be used in the proof of our main results.

Lemma 2.1 (see, e.g., [12]). Let 𝐸 be a real normed linear space. Then for all 𝑥,𝑦∈𝐸, we have 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝑗𝑥+𝑦,∀𝑗𝑥+𝑦∈𝐽𝑥+𝑦. (2.1)

Lemma 2.2 (see, e.g., [13]). Let {𝑎_𝑛}, {𝑏_𝑛}, {𝑐_𝑛} be three nonnegative sequences satisfying the following condition: 𝑎_𝑛+1≤1+𝑏_𝑛𝑎_𝑛+𝑐_𝑛,∀𝑛≥𝑛₀, (2.2) where 𝑛₀ is some nonnegative integer, ∑^∞_𝑛=𝑛0𝑏_𝑛<∞, and ∑^∞_𝑛=𝑛0𝑐_𝑛<∞. Then the limit lim_𝑛→∞𝑎_𝑛 exists.

Lemma 2.3. Suppose that 𝜑∶[0,∞)→[0,∞) is a strictly increasing function with 𝜑(0)=0 and there exists a natural number 𝑛₀ such that 𝑎_𝑛, 𝑏_𝑛, 𝜀_𝑛, and 𝛼_𝑛 are nonnegative real numbers for all 𝑛≥𝑛₀ satisfying the following conditions: (i)𝑎_𝑛+1≤(1+𝑏_𝑛)𝑎_𝑛−𝛼_𝑛𝜑(𝑎_𝑛+1)+𝛼_𝑛𝜀_𝑛,forall𝑛≥𝑛₀, (ii)∑^∞_𝑛=𝑛0𝑏_𝑛<∞,lim_𝑛→∞𝜀_𝑛=0, (iii)∑^∞_𝑛=𝑛0𝛼_𝑛=∞. Then lim_𝑛→∞𝑎_𝑛=0.

Proof. Without loss of generality, let lim_𝑛→∞inf𝑎_𝑛=𝑎. Now, we will show that 𝑎=0. Consider its contrary: 𝑎>0 or 𝑎=∞. For any given 𝑟∈(0,𝑎), there exists a nonnegative integer 𝑛₁≥𝑛₀ such that 𝑎_𝑛≥𝑟>0 and 𝜀_𝑛<1/2𝜑(𝑟)≤1/2𝜑(𝑎_𝑛+1) for all 𝑛≥𝑛₁. By condition (i), we have 𝑎_𝑛+1≤1+𝑏_𝑛𝑎_𝑛−𝛼_𝑛𝜑𝑎_𝑛+1+𝛼_𝑛⋅12𝜑𝑎_𝑛+1=1+𝑏_𝑛𝑎_𝑛−12𝛼_𝑛𝜑𝑎_𝑛+1≤1+𝑏_𝑛𝑎_𝑛,∀𝑛≥𝑛₁. (2.3) Using Lemma 2.2 and condition (ii), we obtain that lim_𝑛→∞𝑎_𝑛 exists and {𝑎_𝑛} is bounded. Suppose that 𝑎_𝑛≤𝑀(forall𝑛≥𝑛₁), where 𝑀 is a nonnegative constant. It follows that 𝑎_𝑛+1≤1+𝑏_𝑛𝑎_𝑛−12𝛼_𝑛𝜑𝑎_𝑛+1≤𝑎_𝑛−12𝛼_𝑛𝜑(𝑟)+𝑀𝑏_𝑛∀𝑛≥𝑛₁. (2.4) Thus, 1∞=2𝜑(𝑟)^∞_𝑛=𝑛1𝛼_𝑛≤𝑎_𝑛1+𝑀^∞_𝑛=𝑛1𝑏_𝑛<∞, (2.5) which is a contradiction. Therefore, liminf_𝑛→∞𝑎_𝑛=0. (2.6) By condition (ii), for all 𝜀>0, there exists a nonnegative integer 𝑛₂≥𝑛₀ such that 𝜀_𝑛<𝜑(𝜀)∀𝑛≥𝑛₂,^∞_𝑛=𝑛2𝑏_𝑛<ln2. (2.7) By (2.6), there exists a natural number 𝑁≥𝑛₂ such that 𝑎_𝑁<𝜀. Now, we prove the following inequality (2.8) holds for all 𝑘≥𝑁: 𝑎_𝑘≤𝜀⋅exp^𝑘−1_𝑛=𝑁𝑏_𝑛. (2.8) It is obvious that (2.8) holds for 𝑘=𝑁. Assuming (2.8) holds for some 𝑘≥𝑁, we prove that (2.8) holds for 𝑘+1. Suppose this is not true, that is, 𝑎_𝑘+1>𝜀⋅exp(∑^𝑘_𝑛=𝑁𝑏_𝑛). Then 𝑎_𝑘+1≥𝜀 and so 𝜑(𝑎_𝑘+1)≥𝜑(𝜀). Noting that 1+𝑏_𝑘≤exp(𝑏_𝑘), it follows from condition (i), (2.7), and (2.8) that 𝑎_𝑘+1≤1+𝑏_𝑘𝑎_𝑘−𝛼_𝑘𝜑𝑎_𝑘+1+𝛼_𝑘𝜀_𝑘≤1+𝑏_𝑘𝑎_𝑘−𝛼_𝑘𝜑(𝜀)+𝛼_𝑘𝜑(𝜀)≤𝜀⋅1+𝑏_𝑘exp^𝑘−1_𝑛=𝑁𝑏_𝑛≤𝜀⋅exp^𝑘_𝑛=𝑁𝑏_𝑛, (2.9) which is a contradiction. This implies that (2.8) holds for 𝑘+1. By induction, (2.8) holds for all 𝑘≥𝑁. From (2.7), and (2.8), we have limsup_𝑘→∞𝑎_𝑘≤𝜀⋅exp^∞_𝑛=𝑁𝑏_𝑛<2𝜀. (2.10) Taking 𝜀→0, we obtain lim_𝑛→∞sup𝑎_𝑘=0. By (2.6), we have lim_𝑛→∞𝑎_𝑛=0. This completes the proof.

Remark 2.4. Lemma 2.3 is different from Lemma 3 in [14], which requires that 𝑏_𝑛=0 for all 𝑛≥0. It is also different from Lemma 2.3 in [6], which requires that ∑^∞_𝑛=𝑛0𝛼_𝑛𝜀_𝑛<∞.

3. Main Results

Theorem 3.1. Let 𝐸 be a real normed linear space, 𝐶 be a nonempty convex subset of 𝐸, let 𝑇∶𝐶→𝐶 be a Lipschitzian mapping, and let 𝑥^∗∈𝐶 be a fixed point of 𝑇. For given 𝑥₀∈𝐶, suppose that the sequence {𝑥_𝑛}⊂𝐶 is the Mann iterative sequence defined by 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇𝑥_𝑛,𝑛≥0, (3.1) where {𝛼_𝑛} is a sequence in [0,1] satisfying the following conditions: (i)∑^∞_𝑛=0𝛼²_𝑛<∞, (ii)∑^∞_𝑛=0𝛼_𝑛=∞. Then {𝑥_𝑛} converges strongly to 𝑥^∗ if and only if there exists a strictly increasing function Φ∶[0,∞)→[0,∞) with Φ(0)=0 such that limsup_𝑛→∞inf_{𝑗𝑥𝑛−𝑥∗𝑥∈𝐽𝑛−𝑥∗}𝑇𝑥_𝑛−𝑥^∗,𝑗𝑥_𝑛−𝑥^∗−𝑥_𝑛−𝑥^∗2+Φ𝑥_𝑛−𝑥^∗≤0. (3.2)

Proof. First, we prove the sufficiency of Theorem 3.1.
Suppose there exists a strictly increasing function Φ∶[0,∞)→[0,∞) with Φ(0)=0 such that (3.2) holds. Let 𝛾_𝑛=inf_{𝑗𝑥𝑛−𝑥∗𝑥∈𝐽𝑛−𝑥∗}𝑇𝑥_𝑛−𝑥^∗,𝑗𝑥_𝑛−𝑥^∗−𝑥_𝑛−𝑥^∗2+Φ𝑥_𝑛−𝑥^∗. (3.3) Then there exists 𝑗(𝑥_𝑛−𝑥^∗)∈𝐽(𝑥_𝑛−𝑥^∗) such that 𝑇𝑥_𝑛−𝑥^∗,𝑗𝑥_𝑛−𝑥^∗−𝑥_𝑛−𝑥^∗2+Φ𝑥_𝑛−𝑥^∗<𝛾_𝑛+1𝑛,∀𝑛≥1. (3.4) By (3.2), we obtain lim_𝑛→∞sup𝛾_𝑛≤0. Taking 𝜀_𝑛=1/(𝑛+1)+max{𝛾_𝑛+1,0}(forall𝑛≥0), then lim_𝑛→∞𝜀_𝑛=0. (3.5) From (3.1) and (3.4), by using Lemma 2.1, we obtain 𝑥_𝑛+1−𝑥^∗2=1−𝛼_𝑛𝑥_𝑛−𝑥^∗+𝛼_𝑛𝑇𝑥_𝑛−𝑥^∗2≤1−𝛼_𝑛²𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝑇𝑥_𝑛−𝑥^∗,𝑗𝑥_𝑛+1−𝑥^∗≤1−𝛼_𝑛²𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝑇𝑥_𝑛+1−𝑥^∗,𝑗𝑥_𝑛+1−𝑥^∗+2𝛼_𝑛⟨𝑇𝑥_𝑛−𝑇𝑥_𝑛+1,𝑗𝑥_𝑛+1−𝑥^∗⟩≤1−𝛼_𝑛²𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝑥_𝑛+1−𝑥^∗2−Φ𝑥_𝑛+1−𝑥^∗+𝛾_𝑛+1+1𝑛+1+2𝐿𝛼_𝑛𝑥_𝑛−𝑥_𝑛+1⋅𝑥_𝑛+1−𝑥^∗≤1−𝛼_𝑛²𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝑥_𝑛+1−𝑥^∗2−Φ𝑥_𝑛+1−𝑥^∗+𝜀_𝑛+2𝐿𝛼_𝑛𝑥_𝑛−𝑥_𝑛+1⋅𝑥_𝑛+1−𝑥^∗, (3.6) where 𝐿 is the Lipschitzian constant of 𝑇. It follows from (3.1) that 𝑥_𝑛−𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛−𝑇𝑥_𝑛≤𝛼_𝑛𝑥_𝑛−𝑥^∗+𝑇𝑥^∗−𝑇𝑥_𝑛≤𝛼_𝑛(1+𝐿)𝑥_𝑛−𝑥^∗. (3.7) Substituting (3.7) into (3.6), we have 𝑥_𝑛+1−𝑥^∗2≤1−𝛼_𝑛²𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝑥_𝑛+1−𝑥^∗2−2𝛼_𝑛Φ𝑥_𝑛+1−𝑥^∗+2𝛼_𝑛𝜀_𝑛+2𝐿(1+𝐿)𝛼²_𝑛𝑥_𝑛−𝑥^∗⋅𝑥_𝑛+1−𝑥^∗≤1−𝛼_𝑛²𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝑥_𝑛+1−𝑥^∗2−2𝛼_𝑛Φ𝑥_𝑛+1−𝑥^∗+2𝛼_𝑛𝜀_𝑛+𝐿(1+𝐿)𝛼²_𝑛𝑥_𝑛−𝑥^∗2+𝑥_𝑛+1−𝑥^∗2. (3.8) Setting 𝑎_𝑛=‖𝑥_𝑛−𝑥^∗‖²(forall𝑛≥0),𝜑(𝑠)=2Φ(√𝑠), it follows from (3.8) that 𝑎_𝑛+1≤1−𝛼_𝑛²𝑎_𝑛+2𝛼_𝑛𝑎_𝑛+1−𝛼_𝑛𝜑𝑎_𝑛+1+2𝛼_𝑛𝜀_𝑛+𝐿(1+𝐿)𝛼²_𝑛𝑎_𝑛+𝑎_𝑛+1=1−2𝛼_𝑛+𝛼²_𝑛+𝐿(1+𝐿)𝛼²_𝑛𝑎_𝑛+2𝛼_𝑛+𝐿(1+𝐿)𝛼²_𝑛𝑎_𝑛+1−𝛼_𝑛𝜑𝑎_𝑛+1+2𝛼_𝑛𝜀_𝑛. (3.9) It follows from condition (i) that lim_𝑛→∞[2𝛼_𝑛+𝐿(1+𝐿)𝛼²_𝑛]=0. Thus, there exists a natural number 𝑛₀ such that 2𝛼_𝑛+𝐿(1+𝐿)𝛼²_𝑛≤1/2 for all 𝑛≥𝑛₀. Let 𝑏_𝑛=1−2𝛼_𝑛+𝛼²_𝑛+𝐿(1+𝐿)𝛼²_𝑛1−2𝛼_𝑛−𝐿(1+𝐿)𝛼²_𝑛𝛼−1=^2𝑛+2𝐿(1+𝐿)𝛼²_𝑛1−2𝛼_𝑛−𝐿(1+𝐿)𝛼²_𝑛,∀𝑛≥𝑛₀. (3.10) Since 1/2≤1−2𝛼_𝑛−𝐿(1+𝐿)𝛼²_𝑛≤1 for all 𝑛≥𝑛₀, by (3.9) and (3.10), we have 𝑎_𝑛+1≤1+𝑏_𝑛𝑎_𝑛−𝛼_𝑛𝜑𝑎_𝑛+1+4𝛼_𝑛𝜀_𝑛,∀𝑛≥𝑛₀,0≤𝑏_𝑛≤2[1+2𝐿(1+𝐿)]𝛼²_𝑛,∀𝑛≥𝑛₀. (3.11) It follows from condition (i) that ∑^∞_𝑛=𝑛0𝑏_𝑛<∞. Therefore, by (3.5), condition (ii), and Lemma 2.3, we obtain that lim_𝑛→∞𝑎_𝑛=lim_𝑛→∞‖𝑥_𝑛−𝑥^∗‖²=0. That is, {𝑥_𝑛} converges strongly to 𝑥^∗.
Finally, we prove the necessity of Theorem 3.1.
Assume that {𝑥_𝑛} converges strongly to 𝑥^∗. Let 𝐿 be the Lipschitzian constant of 𝑇. For all 𝑗(𝑥_𝑛−𝑥^∗)∈𝐽(𝑥_𝑛−𝑥^∗), we have 𝑇𝑥_𝑛−𝑥^∗,𝑗𝑥_𝑛−𝑥^∗≤𝐿𝑥_𝑛−𝑥^∗2. (3.12) Taking Φ(𝑠)=√𝑠, then Φ∶[0,∞)→[0,∞) is a strictly increasing function with Φ(0)=0, and lim_𝑛→∞Φ(‖𝑥_𝑛−𝑥^∗‖)=0. From (3.12), we obtain lim_𝑛→∞inf_{𝑗𝑥𝑛−𝑥∗𝑥∈𝐽𝑛−𝑥∗}𝑇𝑥_𝑛−𝑥^∗,𝑗𝑥_𝑛−𝑥^∗−𝑥_𝑛−𝑥^∗2+Φ𝑥_𝑛−𝑥^∗=0, (3.13) which implies (3.2) holds. This completes the proof of Theorem 3.1.

Remark 3.2. If 𝑇∶𝐶→𝐶 is a generalized Φ-hemicontractive mapping, then (3.2) holds. By Theorem 3.1, we obtain Theorem XCZ.

Theorem 3.3. Let 𝐸 be a real Banach space, let 𝐶 be a nonempty closed convex subset of 𝐸, and let 𝑇∶𝐶→𝐶 be a Lipschitzian generalized Φ-pseudocontractive mapping. For given 𝑥₀∈𝐶, suppose that the sequence {𝑥_𝑛}⊂𝐶 is the Mann iterative sequence defined by 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇𝑥_𝑛,𝑛≥0, (3.14) where {𝛼_𝑛} is a sequence in [0,1] satisfying the following conditions: (1)∑^∞_𝑛=0𝛼_𝑛=∞, (2)∑^∞_𝑛=0𝛼²_𝑛<∞. Then {𝑥_𝑛} converges strongly to the unique fixed point of 𝑇 in 𝐶.

Proof. By Theorem 2.1 in [8], 𝑇 has a unique fixed point 𝑥^∗ in 𝐶. By Theorem 3.1, {𝑥_𝑛} converges strongly to 𝑥^∗. This completes the proof of Theorem 3.3.

Theorem 3.4. Let 𝐸 be a real normed linear space, let 𝑆∶𝐸→𝐸 be a Lipschitzian operator, and let 𝑓∈𝐸 and 𝑥^∗ be a solution of the equation 𝑆𝑥=𝑓. For given 𝑥₀∈𝐸, suppose that the sequence {𝑥_𝑛} is the Mann iterative sequence defined by 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑓+𝑥_𝑛−𝑆𝑥_𝑛,𝑛≥0, (3.15) where {𝛼_𝑛} is a sequences in [0,1] satisfying the following conditions: (i)∑^∞_𝑛=0𝛼²_𝑛<∞, (ii)∑^∞_𝑛=0𝛼_𝑛=∞. Then {𝑥_𝑛} converges strongly to 𝑥^∗ if and only if there exists a strictly increasing function Φ∶[0,∞)→[0,∞) with Φ(0)=0 such that liminf_𝑛→∞sup_{𝑗𝑥𝑛−𝑥∗𝑥∈𝐽𝑛−𝑥∗}𝑆𝑥_𝑛−𝑆𝑥^∗,𝑗𝑥_𝑛−𝑥^∗−Φ𝑥_𝑛−𝑥^∗≥0. (3.16)

Proof. Define 𝑇∶𝐸→𝐸 by 𝑇𝑥=𝑓+𝑥−𝑆𝑥. Since 𝑆𝑥^∗=𝑓, we have 𝑇𝑥^∗=𝑥^∗. From (3.15), we obtain 𝑥_𝑛+1=(1−𝛼_𝑛)𝑥_𝑛+𝛼_𝑛𝑇𝑥_𝑛,𝑛≥0. Since 𝑆𝑥_𝑛−𝑆𝑥^∗,𝑗𝑥_𝑛−𝑥^∗−Φ𝑥_𝑛−𝑥^∗=−𝑇𝑥_𝑛−𝑥^∗,𝑗𝑥_𝑛−𝑥^∗−𝑥_𝑛−𝑥^∗2+Φ𝑥_𝑛−𝑥^∗. (3.17) Therefore, liminf_𝑛→∞sup_{𝑗𝑥𝑛−𝑥∗𝑥∈𝐽𝑛−𝑥∗}𝑆𝑥_𝑛−𝑆𝑥^∗,𝑗𝑥_𝑛−𝑥^∗−Φ𝑥_𝑛−𝑥^∗=−limsup_𝑛→∞inf_{𝑗𝑥𝑛−𝑥∗𝑥∈𝐽𝑛−𝑥∗}𝑇𝑥_𝑛−𝑥^∗,𝑗𝑥_𝑛−𝑥^∗−𝑥_𝑛−𝑥^∗+Φ𝑥_𝑛−𝑥^∗. (3.18) The condition (3.16) is equivalent to condition (3.2). Since 𝑆 is a Lipschitzian operator, 𝑇 is a Lipschitzian mapping. By Theorem 3.1, Theorem 3.4 holds. This completes the proof of Theorem 3.4.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (NSFC) (Grant no. 71201093, no. 11001289, and no. 11171363), Humanities and Social Sciences Foundation of Ministry of Education of China (Grant no. 10YJCZH217), Promotive Research Fund for Excellent Young and Middle-Aged Scientists of Shandong Province (Grant no. BS2012SF012), and Independent Innovation Foundation of Shandong University, IIFSDU (Grant no. 2012TS194).