A Modified Regularization Method for the Proximal Point Algorithm

Abstract

Under some weaker conditions, we prove the strong convergence of the sequence generated by a modified regularization method of finding a zero for a maximal monotone operator in a Hilbert space. In addition, an example is also given in order to illustrate the effectiveness of our generalizations. The results presented in this paper can be viewed as the improvement, supplement, and extension of the corresponding results.

1. Introduction

Let 𝐻 be a real Hilbert space and 𝐶 a nonempty closed convex subset of 𝐻, and let 𝐹∶𝐻→𝐻 be a nonlinear operator. The variational inequality problem is formulated as finding a point 𝑥^∗∈𝐶 such that ⟨𝐹𝑥^∗,𝑣−𝑥^∗⟩≥0,∀𝑣∈𝐶.(1.1)

In 1964, Stampacchia [1] introduced and studied variational inequality initially. It is now well known that variational inequalities cover as diverse disciplines as partial differential equations, optimal control, optimization, mathematical programming, mechanics, and finance, see [1–5].

Let 𝑇 be an operator with domain 𝐷(𝑇) and range 𝑅(𝑇) in 𝐻. A multivalued operator 𝑇∶𝐻→2^𝐻 is called monotone if ⟨𝑢−𝑣,𝑥−𝑦⟩≥0,(1.2) for any 𝑢∈𝑇𝑥,𝑣∈𝑇𝑦, and maximal monotone if it is monotone and its graph 𝐺(𝑇)={(𝑥,𝑦)∶𝑥∈𝐷(𝑇),𝑦∈𝑇𝑥}(1.3) is not properly contained in the graph of any other monotone operator.

One of the major problems in the theory of monotone operators is to find a point in the zero set, which can be formulated as finding a point 𝑥 so that 𝑥∈𝑇⁻¹(0), where 𝑇⁻¹(0) denotes the zero set of the operator 𝑇. A variety of problems, including convex programming and variational inequalities, can be formulated as finding a zero of maximal monotone operators. A classical way to solve such problem is Rockafellar's proximal point algorithm [6], which generates an iterative sequence as 𝑥_𝑛+1=𝐽^𝑇𝑐𝑥_𝑛+𝑒_𝑛,(1.4)

where, for 𝑐>0, 𝐽^𝑇𝑐 denotes the resolvent of 𝑇 given by 𝐽^𝑇𝑐=(𝐼+𝑐𝑇)⁻¹, with 𝐼 being the identity map on the space 𝐻. If 𝑇⁻¹(0)≠∅, it is known that the sequence generated by (1.4) converges weakly to some point in 𝑇⁻¹(0).

Motivated by Lehdili and Moudafi's prox-Tikhonov method [7], Xu [8] considered the following regularization iterative form: for a fixed point 𝑢∈𝐻, 𝑥_𝑛+1=𝐽^𝑇𝑐_𝑛1−𝑡_𝑛𝑥_𝑛+𝑡_𝑛𝑢+𝑒_𝑛,𝑛≥0,(1.5)

where 𝑡_𝑛∈(0,1) and {𝑒_𝑛} is a sequence of errors. Then, the iterative sequence converges strongly to 𝑃_𝑇⁻¹₍₀₎𝑢, provided that(C1)lim_𝑛→∞𝑡_𝑛=0,(C2)∑^∞𝑛=0|𝑡_𝑛+1−𝑡_𝑛|<∞,(C3)0<𝑐≤𝑐_𝑛≤𝑐,(C4)∑^∞𝑛=0|𝑐_𝑛+1−𝑐_𝑛|<∞,(C5)∑^∞𝑛=0𝑡_𝑛=∞, ∑^∞𝑛=0‖𝑒_𝑛‖<∞.

Recently, Song and Yang [9] removed some strict restrictions in Xu [8]. Under conditions (C1), (C2), (C4) (or ∑^∞𝑛=0|1−(𝑐_𝑛/𝑐_𝑛+1)|<+∞), (C5), and (C3′)(C3′)0<liminf_𝑛→∞𝑐_𝑛, they proved that the sequence generated by (1.5) converges strongly to 𝑃_𝑇⁻¹₍₀₎𝑢.

Very recently, under conditions (C1), (C3) (or C3′), (C5), and (C4′) (C4′)lim_𝑛→∞|1−(𝑐_𝑛/𝑐_𝑛+1)|=0. Wang [10] proved the strong convergence of the sequence generated by (1.5). It is easy to see that conditions (C3′) and (C4′) are strictly weaker than conditions (C3) and (C4), respectively.

We remind the reader of the following fact: in order to guarantee the strong convergence of the iterative sequence {𝑥_𝑛}, there is at least one parameter sequence converging to zero (i.e., 𝑡_𝑛→0) in the result of Xu [8], Song and Yang [9], and Wang [10]. So the above results bring us to the following natural questions.

Question 1. Can we obtain the strong convergence theorem without the parameter sequence {𝑡_𝑛} converging to zero?

Question 2. Can we get that the sequence {𝑥_𝑛} converges strongly to 𝑥^∗∈𝑇⁻¹(0), which solves uniquely some variational inequalities?

In this work, motivated by the above results, we consider the following modified regularization method for the proximal point algorithm: for an arbitrary 𝑥₀∈𝐻, 𝑧_𝑛=𝑥𝐼_𝑛+1−𝑡_𝑛𝐹𝑥_𝑛+𝑡_𝑛𝑢+𝑒_𝑛,=𝐽^𝑇𝑐_𝑛𝑧_𝑛,𝑛≥0,(1.6)

where 𝐹 is a 𝑘-Lipschitzian and 𝜂-strongly monotone operator on 𝐻 and 𝑢 is a fixed point in 𝐻. Without the parameter sequence {𝑡_𝑛} converging to zero, we prove that the sequence {𝑥_𝑛} generated by the iterative algorithm (1.6) converges strongly to 𝑥^∗∈𝑇⁻¹(0), which solves uniquely the variational inequality ⟨𝐹𝑥^∗−𝑢,𝑥^∗−𝑝⟩≤0,  for all 𝑝∈𝑇⁻¹(0). In addition, an example is also given in order to illustrate the effectiveness of our generalizations. The results presented in this paper can be viewed as the improvement, supplement, and extension of the results obtained in [6–10].

2. Preliminaries

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖. For the sequence {𝑥_𝑛} in 𝐻, we write 𝑥_𝑛⇀𝑥 to indicate that the sequence {𝑥_𝑛} converges weakly to 𝑥. 𝑥_𝑛→𝑥 means that {𝑥_𝑛} converges strongly to 𝑥.

A mapping 𝐹∶𝐻→𝐻 is called 𝑘-Lipschitzian if there exists a positive constant 𝑘 such that ‖𝐹𝑥−𝐹𝑦‖≤𝑘‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐻.(2.1)

𝐹 is said to be 𝜂-strongly monotone if there exists a positive constant 𝜂 such that ⟨𝐹𝑥−𝐹𝑦,𝑥−𝑦⟩≥𝜂‖𝑥−𝑦‖²,∀𝑥,𝑦∈𝐻.(2.2)

Let 𝐴 be a strongly positive bounded linear operator on 𝐻, that is, there exists a constant 𝛾̃>0 such that ⟨𝐴𝑥,𝑥⟩≥𝛾̃‖𝑥‖²,∀𝑥∈𝐻.(2.3)

A typical problem is that of minimizing a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space 𝐻: min_{𝑥∈Fix(𝑊)}12⟨𝐴𝑥,𝑥⟩−⟨𝑥,𝑏⟩,(2.4)

where 𝑏 is a given point in 𝐻 and Fix(𝑊) is the set of the fixed points of nonexpansive mapping 𝑊.

Remark 2.1 (see [11]). From the definition of 𝐴, we note that a strongly positive bounded linear operator 𝐴 is a ‖𝐴‖-Lipschitzian and 𝛾̃-strongly monotone operator.

Let 𝑇 be a maximal monotone operator on a real Hilbert space 𝐻 such that 𝑆∶=𝑇⁻¹(0)≠∅. For 𝑐>0, we use 𝐽^𝑇𝑐 to denote the resolvent of 𝑇, that is, 𝐽^𝑇𝑐=(𝐼+𝑐𝑇)⁻¹.(2.5)

It is well known that 𝐽^𝑇𝑐 is firmly nonexpansive and consequently nonexpansive; moreover, 𝑆=Fix(𝐽^𝑇𝑐)={𝑥∈𝐻∶𝑥=𝐽^𝑇𝑐𝑥}.

The following lemma is known as the resolvent identity of maximal monotone operators.

Lemma 2.2 (see [8]). Let 𝑐,𝑡>0. Then, for any 𝑥∈𝐻, 𝐽^𝑇𝑐𝑥=𝐽^𝑇𝑡𝑡𝑐𝑥+1−𝑐𝑡𝐽^𝑇𝑐𝑥.(2.6)

In order to prove our main results, we need the following lemmas.

Lemma 2.3 (see [11]). Let 𝐹 be a 𝑘-Lipschitzian and 𝜂-strongly monotone operator on a Hilbert space 𝐻 with 0<𝜂≤𝑘 and 0<𝑡<𝜂/𝑘². Then, 𝑆=(𝐼−𝑡𝐹)∶𝐻→𝐻 is a contraction with contraction coefficient 𝜏_𝑡=√1−𝑡(2𝜂−𝑡𝑘²).

Lemma 2.4 (see [12]). 𝑇 is firmly nonexpansive if and only if 2𝑇−𝐼 is nonexpansive.

Lemma 2.5 (see [13]). Let 𝐻 be a Hilbert space, 𝐶 a closed convex subset of 𝐻, and 𝑇∶𝐶→𝐶 a nonexpansive mapping with Fix(𝑇)≠∅; if {𝑥_𝑛} is a sequence in 𝐶 weakly converging to 𝑥 and if {(𝐼−𝑇)𝑥_𝑛} converges strongly to 𝑦, then (𝐼−𝑇)𝑥=𝑦.

Lemma 2.6 (see [14]). Let {𝑥_𝑛} and {𝑧_𝑛} be bounded sequences in Banach space 𝐸 and {𝛾_𝑛} a sequence in [0,1] which satisfies the following condition: 0<liminf_𝑛→∞𝛾_𝑛≤limsup_𝑛→∞𝛾_𝑛<1.(2.7) Suppose that 𝑥_𝑛+1=𝛾_𝑛𝑥_𝑛+(1−𝛾_𝑛)𝑧_𝑛, 𝑛≥0, and limsup_𝑛→∞(‖𝑧_𝑛+1−𝑧_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0. Then, lim_𝑛→∞‖𝑧_𝑛−𝑥_𝑛‖=0.

Lemma 2.7 (see [15, 16]). Let {𝑠_𝑛} be a sequence of nonnegative real numbers satisfying 𝑠_𝑛+1≤1−𝜆_𝑛𝑠_𝑛+𝜆_𝑛𝛿_𝑛+𝛾_𝑛,𝑛≥0,(2.8) where {𝜆_𝑛}, {𝛿_𝑛}, and {𝛾_𝑛} satisfy the following conditions: (i) {𝜆_𝑛}⊂[0,1] and ∑^∞𝑛=0𝜆_𝑛=∞, (ii) limsup_𝑛→∞𝛿_𝑛≤0 or ∑^∞𝑛=0𝜆_𝑛𝛿_𝑛<∞, and (iii) 𝛾_𝑛≥0(𝑛≥0),∑^∞𝑛=0𝛾_𝑛<∞. Then, lim_𝑛→∞𝑠_𝑛=0.

3. Main Results

Let 𝐹 be a 𝑘-Lipschitzian and 𝜂-strongly monotone operator on 𝐻 with 0<𝜂≤𝑘 and 𝐽^𝑇𝑐 the resolvent of 𝑇. Let 𝑡∈(0,𝜂/𝑘²) and 𝜏_𝑡=√1−𝑡(2𝜂−𝑡𝑘²)∈(0,1), and consider a mapping 𝑉_𝑡 on 𝐻 defined by 𝑉_𝑡𝑥=𝐽^𝑇𝑐[(𝐼−𝑡𝐹)𝑥+𝑡𝑢],𝑥∈𝐻,(3.1)

where 𝑐>0 is a fixed constant and 𝑢∈𝐻 is a fixed point. It is easy to see that 𝑉_𝑡 is a contraction. Indeed, from Lemma 2.3, we have ‖‖𝑉_𝑡𝑥−𝑉_𝑡𝑦‖‖≤≤=𝜏‖‖‖_𝑡𝐽(𝐼^𝑇𝑐−[(𝐼𝑡𝐹−)𝑡𝑥𝐹−)𝑥(𝐼+−𝑡𝑢𝑡𝐹]−)𝑦𝐽‖^𝑇𝑐[(𝐼−𝑡𝐹)𝑦+𝑡𝑢]‖‖‖𝑥−𝑦‖,(3.2)

for all 𝑥,𝑦∈𝐻. Hence, it has a unique fixed point, denoted by 𝑣_𝑡, which uniquely solves the fixed point equation 𝑣_𝑡=𝐽^𝑇𝑐(𝐼−𝑡𝐹)𝑣_𝑡+𝑡𝑢,𝑣_𝑡∈𝐻.(3.3)

Theorem 3.1. For any 𝑐>0 and 𝑢∈𝐻, let the net {𝑣_𝑡} be generated by (3.3). Then, as 𝑡→0, the net {𝑣_𝑡} converges strongly to 𝑣^∗ of 𝑆, which solves uniquely the variational inequality ⟨𝐹𝑣^∗−𝑢,𝑣^∗−𝑝⟩≤0,∀𝑝∈𝑆.(3.4)

Proof. We first show the uniqueness of a solution of the variational inequality (3.4), which is indeed a consequence of the strong monotonicity of 𝐹. Suppose 𝑣^∗∈𝑆 and 𝑣̃∈𝑆 both are solutions to (3.4); then, ⟨⟨𝐹𝐹𝑣𝑣^∗̃−−𝑢𝑢,,𝑣𝑣^∗̃−−𝑣𝑣_∗⟩̃≤0,(3.5)⟩≤0.(3.6)
Adding (3.5) to (3.6), we get ⟨𝐹𝑣^∗−𝐹𝑣,̃𝑣^∗−𝑣⟩̃≤0.(3.7)
The strong monotonicity of 𝐹 implies that 𝑣^∗=𝑣̃ and the uniqueness is proved. Below we use 𝑣^∗∈𝑆 to denote the unique solution of (3.4). Next, we prove that {𝑣_𝑡} is bounded. Taking 𝑝∈𝑆, from (3.3) and using Lemma 2.3, we have ‖‖𝑣_𝑡−𝑝‖‖≤=‖‖‖‖𝐽(𝐼^𝑇𝑐−(𝐼𝑡𝐹−)𝑣𝑡𝐹_𝑡)𝑣_𝑡+𝑡𝑢−𝑝‖‖≤𝜏_𝑡−(𝐼−𝑡𝐹)𝑝+𝑡(𝑢−𝐹𝑝)‖‖‖‖𝑣_𝑡−𝑝‖‖+𝑡‖𝑢−𝐹𝑝‖,(3.8)
that is, ‖‖𝑣_𝑡−𝑝‖‖≤𝑡1−𝜏_𝑡‖𝑢−𝐹𝑝‖.(3.9)
Observe that lim_𝑡→0+𝑡1−𝜏_𝑡=1𝜂.(3.10)
From 𝑡→0, we may assume, without loss of generality, that 𝑡≤𝜂/𝑘²−𝜖, where 𝜖 is an arbitrarily small positive number. Thus, we have that 𝑡/(1−𝜏_𝑡) is continuous, for all 𝑡∈[0,𝜂/𝑘²−𝜖]. Therefore, we obtain sup𝑡1−𝜏_𝑡∶𝑡∈0,𝜂𝑘²−𝜖<+∞.(3.11) From (3.9) and (3.11), we have {𝑣_𝑡} bounded and so is {𝐹𝑣_𝑡}. On the other hand, from (3.3), we obtain ‖‖𝑣_𝑡−𝐽^𝑇𝑐𝑣_𝑡‖‖≤=‖‖‖‖𝐽(𝐼^𝑇𝑐−(𝐼𝑡𝐹−)𝑣𝑡𝐹_𝑡)𝑣_𝑡+𝑡𝑢−𝐽^𝑇𝑐𝑣_𝑡‖‖=𝑡‖‖𝑢−𝐹𝑣_𝑡+𝑡𝑢−𝑣_𝑡‖‖‖‖→0(𝑡→0).(3.12) To prove that 𝑣_𝑡→𝑣^∗, for a given  𝑝∈𝑆, using Lemma 2.3, we have ‖‖𝑣_𝑡−𝑝‖‖²=‖‖𝐽^𝑇𝑐(𝐼−𝑡𝐹)𝑣_𝑡+𝑡𝑢−𝑝‖‖²≤‖‖(𝐼−𝑡𝐹)𝑣_𝑡−(𝐼−𝑡𝐹)𝑝+𝑡(𝑢−𝐹𝑝)‖‖²≤≤𝜏𝜏^2𝑡_𝑡‖‖𝑣_𝑡−𝑝‖‖²+𝑡²‖𝑢−𝐹𝑝‖²+2𝑡⟨(𝐼−𝑡𝐹)𝑣_𝑡−(𝐼−𝑡𝐹)𝑝,𝑢−𝐹𝑝⟩≤𝜏_𝑡‖‖𝑣_𝑡−𝑝‖‖²+𝑡²‖𝑢−𝐹𝑝‖²+2𝑡⟨𝑣_𝑡−𝑝,𝑢−𝐹𝑝⟩+2𝑡²⟨𝐹𝑝−𝐹𝑣_𝑡,𝑢−𝐹𝑝⟩≤𝜏_𝑡‖‖𝑣_𝑡−𝑝‖‖²+𝑡²‖𝑢−𝐹𝑝‖²+2𝑡⟨𝑣_𝑡−𝑝,𝑢−𝐹𝑝⟩+2𝑡²𝑘‖‖𝑝−𝑣_𝑡‖‖‖𝑢−𝐹𝑝‖‖‖𝑣_𝑡−𝑝‖‖²+2𝑡²𝑀+2𝑡⟨𝑣_𝑡−𝑝,𝑢−𝐹𝑝⟩,(3.13) where 𝑀=max{‖𝑢−𝐹𝑝‖²,2𝑘‖𝑝−𝑣_𝑡‖‖𝑢−𝐹𝑝‖}. Therefore, ‖‖𝑣_𝑡−𝑝‖‖²≤2𝑡²1−𝜏_𝑡𝑀+2𝑡1−𝜏_𝑡⟨𝑣_𝑡−𝑝,𝑢−𝐹𝑝⟩.(3.14) From 𝜏_𝑡=√1−𝑡(2𝜂−𝑡𝑘²), we have lim_𝑡→0(𝑡²/(1−𝜏_𝑡))=0. Moreover, if 𝑣_𝑡⇀𝑝, we have lim_𝑡→0((2𝑡/(1−𝜏_𝑡))⟨𝑣_𝑡−𝑝,𝑢−𝐹𝑝⟩)=0.
Since {𝑣_𝑡} is bounded, we see that if {𝑡_𝑛} is a sequence in (0,𝜂/𝑘²−𝜖] such that 𝑡_𝑛→0 and 𝑣_𝑡𝑛⇀𝑣̃, then, by (3.14), we see that 𝑣_𝑡𝑛→𝑣̃. Moreover, by (3.12) and using Lemma 2.5, we have 𝑣̃∈𝑆. We next prove that 𝑣̃ solves the variational inequality (3.4). From (3.3) and 𝑝∈𝑆, we have ‖‖𝑣_𝑡−𝑝‖‖²≤‖‖(𝐼−𝑡𝐹)𝑣_𝑡+𝑡𝑢−𝑝‖‖²=‖‖𝑣_𝑡−𝑝‖‖²+𝑡²‖‖𝑢−𝐹𝑣_𝑡‖‖²+2𝑡⟨𝑣_𝑡−𝑝,𝑢−𝐹𝑣_𝑡⟩,(3.15) that is, ⟨𝐹𝑣_𝑡−𝑢,𝑣_𝑡−𝑝⟩≤𝑡2‖‖𝑢−𝐹𝑣^𝑡‖‖².(3.16) Now replacing 𝑡 in (3.16) with 𝑡_𝑛 and letting 𝑛→∞, we have ⟨𝐹𝑣̃−𝑢,𝑣̃−𝑝⟩≤0.(3.17) That is, 𝑣̃∈𝑆 is a solution of (3.4), and hence 𝑣̃=𝑣^∗ by uniqueness. In a summary, we have shown that each cluster point of {𝑣_𝑡} (at 𝑡→0) equals 𝑣^∗. Therefore, 𝑣_𝑡→𝑣^∗ as 𝑡→0.

Setting 𝐹=𝐴 in Theorem 3.1, we can obtain the following result.

Corollary 3.2. For any 𝑐>0 and 𝑢∈𝐻, let 𝐴 be a strongly positive bounded linear operator with coefficient 0<𝛾̃≤‖𝐴‖. For each 𝑡∈(0,𝛾̃/‖𝐴‖²), let the net {𝑣_𝑡} be generated by 𝑣_𝑡=𝐽^𝑇𝑐[(𝐼−𝑡𝐴)𝑣_𝑡+𝑡𝑢]. Then, as 𝑡→0, the net {𝑣_𝑡} converges strongly to 𝑣^∗ of 𝑆 which solves uniquely the variational inequality ⟨𝐴𝑣^∗−𝑢,𝑣^∗−𝑝⟩≤0,∀𝑝∈𝑆.(3.18)

Setting 𝐹=𝐼 and 𝑣^∗=𝑃_𝑆𝑢 in Theorem 3.1, we can obtain the following result.

Corollary 3.3 (Xu [8, Theorem 3.1]). For any 𝑐>0 and 𝑢∈𝐻. For each 𝑡∈(0,1), let the net {𝑣_𝑡} be generated by 𝑣_𝑡=𝐽^𝑇𝑐[(1−𝑡)𝑣_𝑡+𝑡𝑢]. Then, as 𝑡→0, {𝑣_𝑡} converges strongly to the projection of 𝑢 onto 𝑆; that is, lim_𝑡→0𝑣_𝑡=𝑃_𝑆𝑢. Moreover, this limit is attained uniformly for 𝑐>0.

The next result gives a strong convergence theorem on algorithm (1.6) with a weaker restriction on the sequence {𝑡_𝑛}.

Theorem 3.4. Let 𝑇 be a maximal monotone operator on a Hilbert space 𝐻 with 𝑆≠∅. Let 𝐹 be a 𝑘-Lipschitzian and 𝜂-strongly monotone operator on 𝐻 with 0<𝜂≤𝑘. Let {𝑡_𝑛} be a sequence in (0,1), {𝑐_𝑛} a sequence in (0,+∞), and 𝜖 an arbitrarily small positive number. Assume that the control conditions (C1^), (C3^), (C4^), and (C5) hold for {𝑡_𝑛}, {𝑐_𝑛}, and {𝑒_𝑛}(C1^)0<𝑡_𝑛≤𝜂/𝑘²−𝜖, forall𝑛≥𝑛₀ for some integer 𝑛₀≥0.
For an arbitrary point 𝑥₀∈𝐻, let the sequence {𝑥_𝑛} be generated by (1.6). Then, 𝑧_𝑛→𝑥^∗⟺𝑡_𝑛𝑢−𝐹𝑥_𝑛→0(𝑛→∞),(3.19) where 𝑥^∗∈𝑆 solves the variational inequality ⟨𝐹𝑥^∗−𝑢,𝑥^∗−𝑝⟩≤0,∀𝑝∈𝑆.(3.20)

Proof. On the one hand, suppose that 𝑡_𝑛(𝑢−𝐹𝑥_𝑛)→0(𝑛→∞). We proceed with the following steps.
Step 1. We claim that {𝑥_𝑛} is bounded. In fact, taking 𝑝∈𝑆, from (1.6) and (C1^) and using Lemma 2.3, we have ‖‖𝑥_𝑛+1−𝑝‖‖≤=‖‖‖‖𝐽𝐼^𝑇𝑐_𝑛𝑧−_𝑛𝑡−_𝑛𝑝‖‖≤‖‖𝐼−𝑡_𝑛𝐹𝑥_𝑛+𝑡_𝑛𝑢+𝑒_𝑛−𝑝‖‖≤𝜏_𝑡𝑛𝐹𝑥_𝑛−𝐼−𝑡_𝑛𝐹𝑝+𝑡_𝑛(𝑢−𝐹𝑝)+𝑒_𝑛‖‖≤1‖‖−𝑥_𝑛1−−𝑝‖‖𝜏_𝑡𝑛+𝑡_𝑛‖𝑢−𝐹𝑝‖+‖‖𝑒_𝑛‖‖≤max‖‖𝑥_𝑛‖‖𝑥_𝑛−𝑝‖‖+1−𝜏_𝑡𝑛1−𝑡^𝑛𝜏_𝑡𝑛‖𝑢−𝐹𝑝‖+‖‖𝑒_𝑛‖‖−𝑝‖‖,𝑡^𝑛1−𝜏_𝑡𝑛‖𝑢−𝐹𝑝‖+‖‖𝑒_𝑛‖‖,(3.21) for all𝑛≥𝑛₀ for some integer 𝑛₀≥0, where 𝜏_𝑡𝑛=1−𝑡_𝑛(2𝜂−𝑡_𝑛𝑘²)∈(0,1). By induction, we have ‖‖𝑥_𝑛−𝑝‖‖≤max‖‖𝑥₀−𝑝‖‖,‖𝑢−𝐹𝑝‖𝑀₁+^𝑛−1_𝑗=0‖‖𝑒_𝑗‖‖,(3.22) for all𝑛≥𝑛₀ for some integer 𝑛₀≥0, where 𝑀₁=sup{𝑡_𝑛/(1−𝜏_𝑡𝑛)∶0<𝑡_𝑛≤𝜂/𝑘²−𝜖}<+∞. Therefore, {𝑥_𝑛} is bounded. We also obtain that {𝑧_𝑛} and {𝐹𝑥_𝑛} are bounded.
Step 2. We claim that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0. In fact, write 𝐽_𝑛=𝐽^𝑇𝑐_𝑛 and 𝑇_𝑛=2𝐽_𝑛−𝐼. Then, 𝐽_𝑛 is firmly nonexpansive and 𝑇_𝑛 is nonexpansive (see Lemma 2.4).
Observe that 𝑥_𝑛+1=𝐽_𝑛𝑧_𝑛=𝐼+𝑇^𝑛2𝑧^𝑛=12𝑧^𝑛+12𝑇^𝑛𝑧_𝑛=12𝑥^𝑛+12𝑡^𝑛𝑢−𝐹𝑥_𝑛+𝑒_𝑛=1+𝑇_𝑛𝑧_𝑛2𝑥^𝑛+12𝑦^𝑛,(3.23) where 𝑦_𝑛=𝑡_𝑛(𝑢−𝐹𝑥_𝑛)+𝑒_𝑛+𝑇_𝑛𝑧_𝑛. Therefore, ‖‖𝑦_𝑛+1−𝑦_𝑛‖‖≤=‖‖‖‖𝑡𝑡_𝑛+1𝑛+1𝑢−𝐹𝑥_𝑛+1+𝑒_𝑛+1+𝑇_𝑛+1𝑧_𝑛+1−𝑡_𝑛𝑢−𝐹𝑥_𝑛−𝑒_𝑛−𝑇_𝑛𝑧_𝑛‖‖𝑢−𝐹𝑥_𝑛+1‖‖+‖‖𝑡_𝑛𝑢−𝐹𝑥_𝑛‖‖+‖‖𝑒_𝑛+1‖‖+‖‖𝑒_𝑛‖‖+‖‖𝑇_𝑛+1𝑧_𝑛+1−𝑇_𝑛𝑧_𝑛‖‖.(3.24)
It follows from the resolvent identity that ‖‖𝑇_𝑛+1𝑥−𝑇_𝑛𝑥‖‖==22‖‖‖‖‖𝐽𝐽_𝑛+1𝑛𝑥−𝐽_𝑛𝑥‖‖𝑐𝑐^𝑛_𝑛+1𝑥+1−𝑐^𝑛𝑐_𝑛+1𝐽^𝑛+1≤2||||1−𝑐_𝑛𝑥−𝐽_𝑛𝑥‖‖‖𝑐_𝑛+1||||‖‖𝐽_𝑛+1≤||||1−𝑐_𝑛𝑥−𝑥‖‖𝑐_𝑛+1||||‖‖𝑇_𝑛+1𝑥−𝑥‖‖(3.25) for any 𝑥∈𝐻. From (1.6), we get ‖‖𝑧_𝑛+1−𝑧_𝑛‖‖≤=‖‖‖‖𝑥𝐼−_𝑛+1𝑡_𝑛+1𝐹𝑥_𝑛+1+𝑡_𝑛+1𝑢+𝑒_𝑛+1−𝐼−𝑡_𝑛𝐹𝑥_𝑛−𝑡_𝑛𝑢−𝑒_𝑛‖‖−𝑥_𝑛‖‖+‖‖𝑡_𝑛+1𝑢−𝐹𝑥_𝑛+1‖‖+‖‖𝑡_𝑛𝑢−𝐹𝑥_𝑛‖‖+‖‖𝑒_𝑛+1‖‖+‖‖𝑒_𝑛‖‖.(3.26) By (3.25) and (3.26), we have ‖‖𝑇_𝑛+1𝑧_𝑛+1−𝑇_𝑛𝑧_𝑛‖‖≤≤‖‖||||1𝑇−_𝑛+1𝑧𝑐_𝑛+1𝑛−𝑇_𝑛𝑧_𝑛+1‖‖+‖‖𝑇_𝑛𝑧_𝑛+1−𝑇_𝑛𝑧_𝑛‖‖𝑐_𝑛+1||||‖‖𝑇_𝑛+1≤||||1−𝑐_𝑛𝑧_𝑛+1−𝑧_𝑛+1‖‖+‖‖𝑧_𝑛+1−𝑧_𝑛‖‖𝑐_𝑛+1||||‖‖𝑇_𝑛+1+‖‖𝑡_𝑛+1𝑧_𝑛+1−𝑧_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝑥_𝑛‖‖𝑢−𝐹𝑥_𝑛+1‖‖+‖‖𝑡_𝑛𝑢−𝐹𝑥_𝑛‖‖+‖‖𝑒_𝑛+1‖‖+‖‖𝑒_𝑛‖‖.(3.27) Substituting (3.27) into (3.24) at once gives ‖‖𝑦_𝑛+1−𝑦_𝑛‖‖≤2+‖‖2𝑡‖‖𝑒_𝑛+1𝑛𝑢−𝐹𝑥_𝑛+1‖‖+2‖‖𝑡_𝑛𝑢−𝐹𝑥_𝑛‖‖+2‖‖𝑒_𝑛+1‖‖‖‖+||||1−𝑐^𝑛𝑐_𝑛+1||||𝑀₂+‖‖𝑥_𝑛+1−𝑥_𝑛‖‖,(3.28) that is, ‖‖𝑦_𝑛+1−𝑦_𝑛‖‖−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤2+‖‖2𝑡‖‖𝑒_𝑛+1𝑛𝑢−𝐹𝑥_𝑛+1‖‖+2‖‖𝑡_𝑛𝑢−𝐹𝑥_𝑛‖‖+2‖‖𝑒_𝑛+1‖‖‖‖+||||1−𝑐^𝑛𝑐_𝑛+1||||𝑀₂,(3.29) where 𝑀₂=sup{‖𝑇_𝑛+1𝑧_𝑛+1−𝑧_𝑛+1‖,𝑛≥0}. Observing 𝑡_𝑛(𝑢−𝐹𝑥_𝑛)→0, 𝑒_𝑛→0, and |1−(𝑐_𝑛/𝑐_𝑛+1)|→0(𝑛→∞), it follows that limsup_𝑛→∞‖‖𝑦_𝑛+1−𝑦_𝑛‖‖−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤0.(3.30) From (3.23) and using Lemma 2.6, we have lim_𝑛→∞‖𝑦_𝑛−𝑥_𝑛‖=0. Therefore, lim_𝑛→∞‖‖𝑥_𝑛+1−𝑥_𝑛‖‖=lim_𝑛→∞12‖‖𝑦^𝑛−𝑥_𝑛‖‖=0.(3.31)Step 3. We claim that lim_𝑛→∞‖𝑥_𝑛−𝐽^𝑇𝑐𝑥_𝑛‖=0. Since liminf_𝑛→∞𝑐_𝑛>0, there exist 𝛼>0 and a positive integer 𝑁 such that for all 𝑛≥𝑁, 𝑐_𝑛≥𝛼. From Lemma 2.2, for each 𝑐∈(0,𝛼), we have ‖‖𝐽_𝑛𝑥_𝑛−𝐽^𝑇𝑐𝑥_𝑛‖‖=‖‖‖𝐽^𝑇𝑐𝑐𝑐_𝑛𝑥_𝑛+1−𝑐𝑐_𝑛𝐽^𝑛𝑥_𝑛−𝐽^𝑇𝑐𝑥_𝑛≤‖‖‖𝑐‖‖‖𝑐_𝑛𝑥_𝑛+1−𝑐𝑐_𝑛𝐽^𝑛𝑥_𝑛−𝑥_𝑛=||||1−𝑐‖‖‖𝑐_𝑛||||‖‖𝐽_𝑛≤‖‖𝐽_𝑛𝑥_𝑛−𝑥_𝑛‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝑥_𝑛‖‖.(3.32) Observe that ‖‖𝐽_𝑛𝑥_𝑛−𝑥_𝑛+1‖‖≤‖‖𝑥_𝑛−𝑧_𝑛‖‖≤‖‖𝑡_𝑛𝑢−𝐹𝑥_𝑛‖‖+‖‖𝑒_𝑛‖‖→0.(3.33) Thus, it follows from (3.32), (3.33), and Step2 that lim_𝑛→∞‖‖𝐽_𝑛𝑥_𝑛−𝐽^𝑇𝑐𝑥_𝑛‖‖=0.(3.34) Since ‖𝑥_𝑛−𝐽^𝑇𝑐𝑥_𝑛‖≤‖𝑥_𝑛−𝑥_𝑛+1‖+‖𝑥_𝑛+1−𝐽_𝑛𝑥_𝑛‖+‖𝐽_𝑛𝑥_𝑛−𝐽^𝑇𝑐𝑥_𝑛‖, then lim_𝑛→∞‖‖𝑥_𝑛−𝐽^𝑇𝑐𝑥_𝑛‖‖=0.(3.35)Step 4. We claim that limsup_𝑛→∞⟨𝑥_𝑛−𝑥^∗,𝑢−𝐹𝑥^∗⟩≤0, where 𝑥^∗=lim_𝑡→0𝑣_𝑡 and 𝑣_𝑡 is defined by (3.3). Since 𝑥_𝑛 is bounded, there exists a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} which converges weakly to 𝜔. From Step3, we obtain 𝐽^𝑇𝑐𝑥_𝑛⇀𝜔. From Lemma 2.5, we have 𝜔∈𝑆. Hence, by Theorem 3.1, we have limsup_𝑛→∞⟨𝑥_𝑛−𝑥^∗,𝑢−𝐹𝑥^∗⟩=lim_𝑘→∞𝑥_𝑛𝑘−𝑥^∗,𝑢−𝐹𝑥^∗=⟨𝜔−𝑥^∗,𝑢−𝐹𝑥^∗⟩≤0.(3.36)Step 5. We claim that {𝑧_𝑛} converges strongly to 𝑥^∗∈𝑆. From (1.6), for an appropriate constant 𝛾>0, we have‖‖𝑥_𝑛+1−𝑥^∗‖‖²=‖‖𝐽_𝑛𝑧_𝑛−𝑥^∗‖‖²≤‖‖𝑧_𝑛−𝑥^∗‖‖²=‖‖(𝐼−𝑡_𝑛𝐹)𝑥_𝑛+𝑡_𝑛𝑢+𝑒_𝑛−𝑥^∗‖‖²≤≤‖‖‖‖(𝐼𝐼−−𝑡𝑡_𝑛𝑛𝐹)𝑥_𝑛+𝑡_𝑛𝑢−𝑥^∗‖‖²+𝛾‖‖𝑒_𝑛‖‖≤𝜏_2𝑡𝑛𝐹𝑥_𝑛−𝐼−𝑡_𝑛𝐹𝑥^∗+𝑡_𝑛𝑢−𝐹𝑥^∗‖‖²+𝛾‖‖𝑒_𝑛‖‖≤𝜏_𝑡𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝑡^2𝑛‖𝑢−𝐹𝑥^∗‖²+2𝑡_𝑛𝐼−𝑡_𝑛𝐹𝑥_𝑛−𝐼−𝑡_𝑛𝐹𝑥^∗,𝑢−𝐹𝑥^∗+𝛾‖‖𝑒_𝑛‖‖+‖‖2𝑥𝑡_𝑛2𝑛−𝑥^∗‖‖²+𝑡^2𝑛‖𝑢−𝐹𝑥^∗‖²+2𝑡_𝑛⟨𝑥_𝑛−𝑥^∗−𝑡_𝑛𝐹𝑥_𝑛+𝑡_𝑛𝑢,𝑢−𝐹𝑥^∗⟩≤𝜏_𝑡𝑛⟨𝐹𝑥^∗−𝑢,𝑢−𝐹𝑥^∗⟩+𝛾‖‖𝑒_𝑛‖‖≤1‖‖−𝑥_𝑛1−−𝑥^∗𝜏‖‖²_𝑡𝑛+2𝑡_𝑛⟨𝑥_𝑛−𝑥^∗,𝑢−𝐹𝑥^∗⟩+2𝑡_𝑛‖‖𝑡_𝑛𝑢−𝐹𝑥_𝑛‖‖‖𝑢−𝐹𝑥^∗‖+𝛾‖‖𝑒_𝑛‖‖‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝜏_𝑡𝑛2𝑀₁⟨𝑥_𝑛−𝑥^∗,𝑢−𝐹𝑥^∗⟩+2𝑀₁‖‖𝑡_𝑛𝑢−𝐹𝑥_𝑛‖‖‖𝑢−𝐹𝑥^∗‖+𝛾‖‖𝑒_𝑛‖‖,(3.37) for all 𝑛≥𝑛₀ for some integer 𝑛₀≥0. For every 𝑛≥𝑛₀, put 𝜇_𝑛=1−𝜏_𝑡𝑛 and 𝛿_𝑛=2𝑀₁⟨𝑥_𝑛−𝑥^∗,𝑢−𝐹𝑥^∗⟩+2𝑀₁‖𝑡_𝑛(𝑢−𝐹𝑥_𝑛)‖‖𝑢−𝐹𝑥^∗‖. It follows that ‖‖𝑥_𝑛+1−𝑥^∗‖‖²≤1−𝜇_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝜇_𝑛𝛿_𝑛+𝛾‖‖𝑒_𝑛‖‖,∀𝑛≥𝑛₀.(3.38) It is easy to see that ∑^∞𝑛=1𝜇_𝑛=∞ and limsup_𝑛→∞𝛿_𝑛≤0. Hence, by Lemma 2.7, the sequence {𝑥_𝑛} converges strongly to 𝑥^∗∈𝑆. Observe that ‖‖𝑧_𝑛−𝑥^∗‖‖≤=‖‖‖‖𝑥_𝑛𝐼−𝑡_𝑛𝐹𝑥_𝑛+𝑡_𝑛𝑢+𝑒_𝑛−𝑥^∗‖‖−𝑥^∗‖‖+‖‖𝑡_𝑛𝑢−𝐹𝑥_𝑛‖‖+‖‖𝑒_𝑛‖‖.(3.39) Thus, it follows that the sequence {𝑧_𝑛} converges strongly to 𝑥^∗∈𝑆.
On the other hand, suppose that 𝑧_𝑛→𝑥^∗∈𝑆 as 𝑛→∞. From (1.6), we have ‖‖𝑥_𝑛+1−𝑥^∗‖‖=‖‖𝐽_𝑛𝑧_𝑛−𝑥^∗‖‖≤‖‖𝑧_𝑛−𝑥^∗‖‖→0.(3.40) Therefore, ‖‖𝑡_𝑛𝑢−𝐹𝑥_𝑛‖‖≤=‖‖‖‖𝑧𝑧_𝑛𝑛−𝑥_𝑛−𝑒_𝑛‖‖≤‖‖𝑧_𝑛−𝑥_𝑛‖‖+‖‖𝑒_𝑛‖‖−𝑥^∗‖‖+‖‖𝑥_𝑛−𝑥^∗‖‖+‖‖𝑒_𝑛‖‖→0.(3.41)

Setting 𝐹=𝐼 and 𝑥^∗=𝑃_𝑆𝑢 in Theorem 3.4, we can obtain the following result.

Corollary 3.5. Let 𝑇 be a maximal monotone operator on a Hilbert space 𝐻 with 𝑆≠∅. Let {𝑡_𝑛} be a sequence in (0,1), {𝑐_𝑛} a sequence in (0,+∞), and 𝜖 an arbitrarily small positive number. Assume that the control conditions (C1^), (C3^), (C4^), and (C5) hold for {𝑡_𝑛}, {𝑐_𝑛}, and {𝑒_𝑛}.(C1^)0<𝑡_𝑛≤1−𝜖, for all 𝑛≥𝑛₀ for some integer 𝑛₀≥0.
For an arbitrary point 𝑥₀∈𝐻, let the sequence {𝑥_𝑛} be generated by 𝑧_𝑛𝑥=_𝑛+11−𝑡_𝑛𝑥_𝑛+𝑡_𝑛𝑢+𝑒_𝑛,=𝐽^𝑇𝑐_𝑛𝑧_𝑛,𝑛≥0.(3.42) Then, 𝑧_𝑛→𝑃_𝑆𝑢⟺𝑡_𝑛𝑢−𝑥_𝑛→0(𝑛→∞).(3.43)

Corollary 3.6 ([Wang [10], Theorem  4]). Let {𝑐_𝑛}, {𝑡_𝑛}, and {𝑒_𝑛} satisfy (C1), (C3),(or (C3^)), (C4^) and (C5). In addition, if 𝑆≠∅, then the sequence generated by (1.5) converges strongly to 𝑃_𝑆𝑢.

Proof. Since lim_𝑛→∞𝑡_𝑛=0, it is easy to see that 𝑡_𝑛≤𝜂/𝑘²−𝜖, for all 𝑛≥𝑛₀ for some integer 𝑛₀≥0. Without loss of generality, we assume that 0<𝑡_𝑛≤𝜂/𝑘²−𝜖, for all 𝑛≥𝑛₀ for some integer 𝑛₀≥0. Repeating the same argument as in the proof of Theorem  4 in Wang [10], we know that {𝑥_𝑛} is bounded. Thus, we have that 𝑡_𝑛(𝑢−𝑥_𝑛)→0. Therefore, all conditions of Corollary 3.5 are satisfied. Using Corollary 3.5, we have that {𝑧_𝑛} converges strongly to 𝑃_𝑆𝑢∈𝑆, with 𝑧_𝑛=(1−𝑡_𝑛)𝑥_𝑛+𝑡_𝑛𝑢+𝑒_𝑛. Therefore, ‖‖𝑥_𝑛+1−𝑃_𝑆𝑢‖‖≤‖‖𝑧_𝑛−𝑃_𝑆𝑢‖‖→0.(3.44)

Remark 3.7. Corollary 3.5 is more general than Theorem 4 of Wang [10]. The following example is given in order to illustrate the effectiveness of our generalizations.

Example 3.8. Let 𝐻=𝑅  be the set of real numbers, 𝑢=0 and 𝑐_𝑛=1/2 for all 𝑛≥0. Define a maximal monotone operator 𝑇 as follows: 𝑇𝑥=2𝑥, for all 𝑥∈𝑅. It is easy to see that 𝐽^𝑇𝑐_𝑛=(1/2)𝐼 and 𝑆={0}. Given sequences {𝑡_𝑛} and {𝑒_𝑛}, 𝑡_𝑛=1/2 and 𝑒_𝑛=0, for all 𝑛≥0. For an arbitrary 𝑥₀∈𝑅, let {𝑥_𝑛} be defined by (3.42), that is, 𝑧_𝑛=12𝑥^𝑛𝑥_𝑛+1,=12𝑧^𝑛=14𝑥^𝑛,𝑛≥0.(3.45) Observe that ‖‖𝑥_𝑛+1−0‖‖=‖‖‖14𝑥^𝑛−0‖‖‖=14‖‖𝑥^𝑛−0‖‖.(3.46) Hence, we have ‖𝑥_𝑛+1−0‖=(1/4)^𝑛+1‖𝑥₀−0‖ for all 𝑛≥0. This implies that {𝑥_𝑛} converges strongly to 0=𝑃_𝑆0. Thus, ‖‖𝑡_𝑛𝑢−𝑥_𝑛‖‖=12‖‖𝑥^𝑛‖‖→0(𝑛→∞).(3.47) Furthermore, it is easy to see that there hold the following:(B1)0<𝑡_𝑛=1/2≤1−𝜖, forall𝑛≥𝑛₀ for some integer 𝑛₀≥0,(B2)∑^∞𝑛=0𝑡_𝑛=∑^∞𝑛=0(1/2)=∞,(B3)liminf_𝑛→∞𝑐_𝑛=1/2>0 and lim_𝑛→∞|1−(𝑐_𝑛/𝑐_𝑛+1)|=0,(B4)∑^∞𝑛=0‖𝑒_𝑛‖=∑^∞𝑛=00=0<∞.

Hence there is no doubt that all conditions of Corollary 3.5 are satisfied. Since 𝑡_𝑛=1/2↛0, the condition 𝑡_𝑛→0 of Wang [10, Theorem   4] is not satisfied. So, by Corollary 3.5, we obtain that the sequence {𝑥_𝑛} and {𝑧_𝑛} converges strongly to zero but Theorem  4 of Wang [10] cannot be applied to {𝑥_𝑛} and {𝑧_𝑛} in this example.

Acknowledgment

This paper is supported by the Natural Science Foundation of Yancheng Teachers University under Grant 11YCKL009.