Strong and Weak Convergence of Modified Mann Iteration for New Resolvents of Maximal Monotone Operators in Banach Spaces

Abstract

We prove strong and weak convergence theorems for a new resolvent of maximal monotone operators in a Banach space and give an estimate of the convergence rate of the algorithm. Finally, we apply our convergence theorem to the convex minimization problem. The result present in this paper extend and improve the corresponding result of Ibaraki and Takahashi (2007), and Kim and Xu (2005).

1. Introduction

Let 𝐸 be a Banach space with norm ‖⋅‖, let 𝐸^∗ denote the dual of 𝐸, and let ⟨𝑥,𝑓⟩ denote the value of 𝑓∈𝐸^∗ at 𝑥∈𝐸. Let 𝑇∶𝐸→𝐸^∗ be an operator. The problem of finding 𝑣∈𝐸 satisfying 0∈𝑇𝑣 is connected with the convex minimization problems. When 𝑇 is maximal monotone, a well-known method for solving the equation 0∈𝑇𝑣 in Hilbert space 𝐻 is the proximal point algorithm (see [1]): 𝑥₁=𝑥∈𝐻 and 𝑥_𝑛+1=𝐽_𝑟𝑛𝑥_𝑛,𝑛=1,2,…, (1.1) where 𝑟_𝑛⊂(0,∞) and 𝐽_𝑟=(𝐼+𝑟𝑇)⁻¹ for all 𝑟>0 is the resolvent operator for T. Rockafellar [1] proved the weak convergence of the algorithm (1.1).

The modifications of the proximal point algorithm for different operators have been investigated by many authors. Recently, Kohsaka and Takahashi [2] considered the algorithm (1.2) in a smooth and uniformly convex Banach space and Kamimura et al. [3] considered the algorithm (1.3) in a uniformly smooth and uniformly convex Banach space 𝐸; 𝑥₁=𝑥∈𝐸,𝑢∈𝐸 and 𝑥_𝑛+1=𝐽⁻¹(𝛼_𝑛𝐽𝑢+(1−𝛼_𝑛)𝐽𝐽_𝑟𝑛𝑥_𝑛𝑥),𝑛=1,2,…,_𝑛+1=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝐽_𝑟𝑛𝑥_𝑛),𝑛=1,2,…, (1.2)(1.3) where 𝐽_𝑟=(𝐽+𝑟𝑇)⁻¹𝐽,𝐽 is the duality mapping of 𝐸. They showed that the algorithm (1.2) converges strongly to some element of 𝑇⁻¹0 and the algorithm (1.3) converges weakly to some element of 𝑇⁻¹0 provided that the sequences {𝛼_𝑛} and {𝑟_𝑛} of real numbers are chosen appropriately. These results extend the Kamimura and Takahashi [4] results in Hilbert spaces to those in Banach spaces.

In 2008, motivated by Kim and Xu [5], Li and Song [6] studied a combination of the schemes of (1.2) and (1.3); 𝑥₁=𝑥∈𝐸 and 𝑦_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝐽_𝑟𝑛𝑥_𝑛𝑥),_𝑛+1=𝐽⁻¹(𝛽_𝑛𝐽𝑥+(1−𝛽_𝑛)𝐽𝑦_𝑛), (1.4) for every 𝑛=1,2,…, where 𝐽_𝑟=(𝐽+𝑟𝑇)⁻¹𝐽,𝐽 is the duality mapping of 𝐸. They also proved strong and weak convergence theorems and give an estimate for the rate of convergence of the algorithm (1.4).

Very recently, Ibaraki and Takahashi [7] introduced the Mann iteration and Harpern iteration for new resovents of maximal monotone operator in a uniformly smooth and uniformly convex Banach space 𝐸; 𝑥₁=𝑥∈𝐸𝑢∈𝐸 and 𝑥_𝑛+1=𝛼_𝑛𝑢+(1−𝛼_𝑛)𝐽_𝑟𝑛𝑥_𝑛𝑥,𝑛=1,2,…,_𝑛+1=𝛼_𝑛𝑥_𝑛+(1−𝛼_𝑛)𝐽_𝑟𝑛𝑥_𝑛,𝑛=1,2,…, (1.5)(1.6) where 𝐽_𝑟=(𝐼+𝑟𝐵𝐽)⁻¹,𝐽 is the duality mapping of 𝐸, and 𝐵⊂𝐸^∗×𝐸 is maximal monotone. They proved that Algorithm (1.5) converges strongly to some element of (𝐵𝐽)⁻¹0 and Algorithm (1.6) converges weakly to some element of (𝐵𝐽)⁻¹0 provided that the sequences {𝛼_𝑛} and {𝑟_𝑛} of real numbers are chosen appropriately.

Inspired and motivated by Li and Song [6] and Ibaraki and Takahashi [7], we study a combination of the schemes of (1.5) and (1.6); 𝑥₁=𝑥∈𝐸 and 𝑦_𝑛=𝛼_𝑛𝑥_𝑛+(1−𝛼_𝑛)𝐽_𝑟𝑛𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥+(1−𝛽_𝑛)𝑦_𝑛, (1.7) for every 𝑛=1,2,…, where 𝐽_𝑟=(𝐼+𝑟𝐵𝐽)⁻¹,𝐽 is the duality mapping of 𝐸, and 𝐵⊂𝐸^∗×𝐸 is maximal monotone. When 𝛼_𝑛≡0, Algorithm (1.7) reduces to (1.5) and, when 𝛽_𝑛≡0, Algorithm (1.7) reduces to (1.6). Then, we prove strong and weak convergence theorems of the sequence and we also estimate the rate of the convergence of algorithm (1.7). Finally, by using our main result, we consider the problem of finding minimizes of convex functions defined on Banach spaces.

2. Preliminaries

Let 𝐸 be a real Banach space with dual space 𝐸^∗. When {𝑥_𝑛} is a sequence in 𝐸, we denote strong convergence of {𝑥_𝑛} to 𝑥∈𝐸 by 𝑥_𝑛→𝑥 and weak convergence by 𝑥_𝑛⇀𝑥, respectively. As usual, we denote the duality pairing of 𝐸^∗ by ⟨𝑥,𝑥^∗⟩, when 𝑥^∗∈𝐸^∗ and 𝑥∈𝐸, and the closed unit ball by 𝑈_𝐸, and denote by ℝ and ℕ the set of all real numbers and the set of all positive integers, respectively. The set ℝ₊ stands for [0,+∞) and ℝ₊=ℝ₊∪{+∞}. An operator 𝑇⊂𝐸×𝐸^∗ is said to be monotone if ⟨𝑥−𝑦,𝑥^∗−𝑦^∗⟩≥0 whenever (𝑥,𝑥^∗),(𝑦,𝑦^∗)∈𝑇. We denote the set {𝑥∈𝐸∶0∈𝑇𝑥} by 𝑇⁻¹0. A monotone 𝑇 is said to be maximal if its graph 𝐺(𝑇)={(𝑥,𝑦)∶𝑦∈𝑇𝑥} is not properly contained in the graph of any other monotone operator. If 𝑇 is maximal monotone, then the solution set 𝑇⁻¹0 is closed and convex. If 𝐸 is reflexive and strictly convex, then a monotone operator 𝑇 is maximal if and only if 𝑅(𝐽+𝜆𝑇)=𝐸^∗ for each 𝜆>0 (see [8, 9] for more details).

The normalized duality mapping 𝐽 from 𝐸 into 𝐸^∗ is defined by 𝐽(𝑥)={𝑥^∗∈𝐸^∗∶⟨𝑥,𝑥^∗⟩=‖𝑥‖²=‖𝑥^∗‖²},∀𝑥∈𝐸. (2.1) We recall [10] that 𝐸 is reflexive if and only if 𝐽 is surjective; 𝐸 is smooth if and only if 𝐽 is single-valued.

Let 𝐸 be a smooth Banach space. Consider the following function: (see [11]) 𝜙(𝑥,𝑦)=‖𝑥‖²−2⟨𝑥,𝐽𝑦⟩+‖𝑦‖²,∀𝑥,𝑦∈𝐸. (2.2) It is obvious from the definition of 𝜙 that (‖𝑥‖−‖𝑦‖)²≤𝜙(𝑥,𝑦)≤(‖𝑥‖+‖𝑦‖)², for all 𝑥,𝑦∈𝐸. We also know that 𝜙(𝑥,𝑦)+𝜙(𝑦,𝑥)=2⟨𝑥−𝑦,𝐽𝑥−𝐽𝑦⟩,foreach𝑥,𝑦∈𝐸. (2.3) We recall [12] that the functional ‖⋅‖² is called totally convex at 𝑥 if the function 𝜈(𝑥,𝑡)∶[0,∞)→[0,∞] defined by 𝜈(𝑥,𝑡)=inf{𝜙(𝑦,𝑥)∶𝑦∈𝐸,‖𝑦−𝑥‖=𝑡}, (2.4) is positive whenever 𝑡>0. The functional ‖⋅‖² is called totally convex on bounded sets if for each bounded nonempty subset 𝐴 of 𝐸, the function 𝜈(𝐴,𝑡)∶[0,∞)→[0,∞] defined by 𝜈(𝐴,𝑡)=inf{𝜈(𝑥,𝑡)∶𝑥∈𝐴} is positive on (0,∞).

It is well known that if a Banach space 𝐸 is uniformly convex, then ‖⋅‖² is totally convex on any bounded nonempty set. It is known that (see [12]) if ‖⋅‖² is totally convex on a bounded set 𝐴, then 𝜈(𝐴,𝑐𝑡)≥𝑐𝜈(𝐴,𝑡) for 𝑐≥1 and 𝑡≥0, and 𝜈(𝐴,⋅) is strictly increasing on [0,∞).

Lemma 2.1 (see [13]). Let 𝐸 be a uniformly convex, smooth Banach space, and let {𝑥_𝑛} and {𝑦_𝑛} be sequences in 𝐸. If {𝑥_𝑛} or {𝑦_𝑛} is bounded and lim_𝑛→∞𝜙(𝑥_𝑛,𝑦_𝑛)=0, then lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0.

Let 𝐸 be a reflexive, strictly convex, smooth Banach space, and 𝐽 the duality mapping from 𝐸 into 𝐸^∗. Then 𝐽⁻¹ is also single-valued, one-to-one, surjective, and it is the duality mapping from 𝐸^∗ into 𝐸. We make use of the following mapping 𝑉 studied in Alber [11]: 𝑉(𝑥,𝑥^∗)=‖𝑥‖²−2⟨𝑥,𝑥^∗⟩+‖𝑥^∗‖² (2.5) for all 𝑥∈𝐸 and 𝑥^∗∈𝐸^∗. In other words, 𝑉(𝑥,𝑥^∗)=𝜙(𝑥,𝐽⁻¹(𝑥)) for all 𝑥∈𝐸 and 𝑥^∗∈𝐸^∗.

Lemma 2.2 (see [7]). Let 𝐸 be a reflexive, strictly convex, smooth Banach space, and let 𝑉 be as in (2.5). Then 𝑉(𝑥,𝑥^∗)+2⟨𝑦,𝐽𝑥−𝑥^∗⟩≤𝑉(𝑥+𝑦,𝑥^∗) (2.6) for all 𝑥,𝑦∈𝐸 and 𝑥^∗∈𝐸^∗.

Let 𝐸 be a smooth Banach space and let 𝐷 be a nonempty closed convex subset of 𝐸. A mapping 𝑅∶𝐷→𝐷 is called generalized nonexpansive if 𝐹(𝑅)≠∅ and 𝜙(𝑅𝑥,𝑦)≤𝜙(𝑥,𝑦) for each 𝑥∈𝐷 and 𝑦∈𝐹(𝑅), where 𝐹(𝑅) is the set of fixed points of 𝑅. Let 𝐶 be a nonempty closed subset of 𝐸. A mapping 𝑅∶𝐸→𝐶 is said to be sunny if 𝑅(𝑅𝑥+𝑡(𝑥−𝑅𝑥))=𝑅𝑥,∀𝑥∈𝐸,∀𝑡≥0. (2.7) A mapping 𝑅∶𝐸→𝐶 is said to be a retraction if 𝑅𝑥=𝑥, for all 𝑥∈𝐶. If 𝐸 is smooth and strictly convex, then a sunny generalized nonexpansive retraction of 𝐸 onto 𝐶 is uniquely decided if it exists (see [14]). We also know that if 𝐸 is reflexive, smooth, and strictly convex and 𝐶 is a nonempty closed subset of 𝐸, then there exists a sunny generalized nonexpansive retraction 𝑅_𝐶 of 𝐸 onto 𝐶 if and only if 𝐽(𝐶) is closed and convex. In this case, 𝑅_𝐶 is given by 𝑅_𝐶=𝐽⁻¹Π_𝐽(𝐶)𝐽 see [15]. Let 𝐶 be a nonempty closed subset of a Banach space 𝐸. Then 𝐶 is said to be a sunny generalized nonexpansive retract (resp., a generalized nonexpansive retract) of 𝐸 if there exists a sunny generalized nonexpansive retraction (resp, a generalized nonexpansive retraction) of 𝐸 onto 𝐶 (see [14] for more detials). The set of fixed points of such a generalized nonexpansive retraction is 𝐶. The following lemma was obtained in [14].

Lemma 2.3 (see [14]). Let 𝐶 be a nonempty closed subset of a smooth and strictly convex Banach space 𝐸. Let 𝑅_𝐶 be a retraction of 𝐸 onto 𝐶. Then 𝑅_𝐶 is sunny and generalized nonexpansive if and only if ⟨𝑥−𝑅_𝐶𝑥,𝐽𝑅_𝐶𝑥−𝐽𝑦⟩≥0, (2.8) for each 𝑥∈𝐸 and 𝑦∈𝐶, where 𝐽 is the duality mapping of 𝐸. Let 𝐸 be a reflexive, strictly convex, and smooth Banach space with its dual 𝐸^∗. If a monotone operator 𝐵⊂𝐸^∗×𝐸 is maximal, then (𝐵𝐽)⁻¹0 is closed and 𝐸=𝑅(𝐼+𝑟𝐵𝐽) for all 𝑟>0 (see [14]). So, for each 𝑟>0 and 𝑥∈𝐸, we can consider the set 𝐽_𝑟(𝑥)={𝑧∈𝐸∶𝑥∈𝑧+𝑟𝐵𝐽𝑧}. From [14], 𝐽_𝑟𝑥 consists of one point. We denote such a 𝐽_𝑟 by (𝐼+𝑟𝐵𝐽)⁻¹. However 𝐽_𝑟 is called a generalized resolvent of 𝐵. We also know that (𝐵𝐽)⁻¹0=𝐹(𝐽_𝑟) for each 𝑟>0, where 𝐹(𝐽_𝑟) is the set of fixed points of 𝐽_𝑟 and 𝐽_𝑟 is generalized nonexpansive for each 𝑟>0 (see [14]). The Yosida approximtion of 𝐵 is defined by 𝐴_𝑟=(𝐼−𝐽_𝑟)/𝑟. We know that (𝐽𝐽_𝑟𝑥,𝐴_𝑟𝑥)∈𝐵; (see [14] for more detials). The following result was obtained in [14].

Theorem 2.4 (see [14]). Let 𝐸 be a uniformly convex Banach space with a Fréchet differentiable norm and let 𝐵⊂𝐸^∗×𝐸 be a maximal monotone operator with 𝐵⁻¹0≠∅. Then the following hold: (1)for each 𝑥∈𝐸, lim_𝑟→∞𝐽_𝑟𝑥 exists and belongs to (𝐵𝐽)⁻¹0,(2)if 𝑅𝑥∶=lim_𝑟→∞𝐽_𝑟𝑥 for each 𝑥∈𝐸, then 𝑅 is a sunny generalized nonexpansive retraction of 𝐸 onto (𝐵𝐽)⁻¹0.

Lemma 2.5 (see [7]). Let 𝐸 be a reflexive, strictly convex, and smooth Banach space, let 𝐵⊂𝐸^∗×𝐸 be a maximal monotone operator with 𝐵⁻¹0≠∅, and 𝐽_𝑟=(𝐼+𝑟𝐵𝐽)⁻¹ for all 𝑟>0. Then 𝜙(𝑥,𝐽_𝑟𝑥)+𝜙(𝐽_𝑟𝑥,𝑢)≤𝜙(𝑥,𝑢), (2.9) for all 𝑟>0, 𝑢∈(𝐵𝐽)⁻¹0, and 𝑥∈𝐸.

Lemma 2.6 (see [16]). Let {𝑠_𝑛} be a sequence of nonnegative real numbers satisfying 𝑠_𝑛+1≤(1−𝛼_𝑛)𝑠_𝑛+𝛼_𝑛𝑡_𝑛+𝑟_𝑛,𝑛≥1, (2.10) where {𝛼_𝑛}, {𝑡_𝑛}, and {𝑟_𝑛} satisfy the conditions: {𝛼_𝑛}⊂[0,1], ∑^∞_𝑛=1𝛼_𝑛=∞, limsup_𝑛→∞𝑡_𝑛≤0, and 𝑟_𝑛≥0, ∑^∞_𝑛=1𝑟_𝑛<∞. Then, lim_𝑛→∞𝑠_𝑛=0.

Lemma 2.7 (see [17]). Let {𝛼_𝑛} and {𝛽_𝑛} be sequence of nonnegative real numbers satisfying 𝛼_𝑛+1≤𝛼_𝑛+𝛽_𝑛, (2.11) for all 𝑛∈ℕ. If ∑^∞_𝑛=1𝛽_𝑛<+∞. Then {𝛼_𝑛} has a limit in [0,+∞).

3. Convergence Theorems

In this section, we first prove a strong convergence theorem for the algorithm (1.7) which extends the previous result of Ibaraki and Takahashi [7] and we next prove a weak convergence theorem for algorithm (1.7) under different conditions on data, respectively.

Theorem 3.1. Let 𝐸 be a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable. Let 𝐵⊂𝐸^∗×𝐸 be a maximal monotone operator with 𝐵⁻¹0≠∅ and let 𝐽_𝑟=(𝐼+𝑟𝐵𝐽)⁻¹ for all 𝑟>0. Let {𝑥_𝑛} be a sequence generated by 𝑥₁=𝑥∈𝐸 and 𝑦_𝑛=𝛼_𝑛𝑥_𝑛+(1−𝛼_𝑛)𝐽_𝑟𝑛𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥+(1−𝛽_𝑛)𝑦_𝑛, (3.1) for every 𝑛=1,2,…, where {𝛼_𝑛}, {𝛽_𝑛}⊂[0,1], {𝑟_𝑛}⊂(0,∞) satisfy lim_𝑛→∞𝛼_𝑛=0, lim_𝑛→∞𝛽_𝑛=0, ∑^∞_𝑛=1𝛽_𝑛=∞ and lim_𝑛→∞𝑟_𝑛=∞. Then the sequence {𝑥_𝑛} converges strongly to 𝑅_(𝐵𝐽)⁻¹₀(𝑥), where 𝑅_(𝐵𝐽)⁻¹₀ is a sunny generalized nonexpansive retraction of 𝐸 onto (𝐵𝐽)⁻¹0.

Proof. Note that 𝐵⁻¹0≠∅ implies (𝐵𝐽)⁻¹0≠∅. In fact, if 𝑢^∗∈𝐵⁻¹0, we obtain 0∈𝐵𝑢^∗ and hence 0∈𝐵𝐽𝐽⁻¹𝑢^∗. So, we have 𝐽⁻¹𝑢^∗∈(𝐵𝐽)⁻¹0. We denote a sunny generalized nonexpansive retraction 𝑅_(𝐵𝐽)⁻¹₀ of 𝐸 onto (𝐵𝐽)⁻¹0 by 𝑅. Let 𝑧∈(𝐵𝐽)⁻¹0. We first prove that {𝑥_𝑛} is bounded. From Lemma 2.5 and the convexity of ‖⋅‖², we have 𝜙(𝑦_𝑛,𝑧)=𝜙(𝛼_𝑛𝑥_𝑛+(1−𝛼_𝑛)𝐽_𝑟𝑛𝑥_𝑛,𝑧)≤𝛼_𝑛𝜙(𝑥_𝑛,𝑧)+(1−𝛼_𝑛)𝜙(𝐽_𝑟𝑛𝑥_𝑛,𝑧)≤𝛼_𝑛𝜙(𝑥_𝑛,𝑧)+(1−𝛼_𝑛){𝜙(𝑥_𝑛,𝑧)−𝜙(𝑥_𝑛,𝐽_𝑟𝑛𝑥_𝑛)}≤𝛼_𝑛𝜙(𝑥_𝑛,𝑧)+(1−𝛼_𝑛)𝜙(𝑥_𝑛,𝑧)=𝜙(𝑥_𝑛,𝑧), (3.2) for all 𝑛∈ℕ. By (3.2), we have 𝜙(𝑥_𝑛+1,𝑧)=𝜙(𝛽_𝑛𝑥+(1−𝛽_𝑛)𝑦_𝑛,𝑧)≤𝛽_𝑛𝜙(𝑥,𝑧)+(1−𝛽_𝑛)𝜙(𝑦_𝑛,𝑧)≤𝛽_𝑛𝜙(𝑥,𝑧)+(1−𝛽_𝑛)𝜙(𝑥_𝑛,𝑧), (3.3) for all 𝑛∈ℕ. Hence, by induction, we have 𝜙(𝑥_𝑛,𝑧)≤𝜙(𝑥,𝑧) for all 𝑛∈ℕ and, therefore, {𝜙(𝑥_𝑛,𝑧)} is bounded. This implies that {𝑥_𝑛} is bounded. Since 𝜙(𝑦_𝑛,𝑧)≤𝜙(𝑥_𝑛,𝑧) and 𝜙(𝐽_𝑟𝑛𝑥_𝑛,𝑧)≤𝜙(𝑥_𝑛,𝑧) for all 𝑛∈ℕ, it follows that {𝑦_𝑛} and {𝐽_𝑟𝑛𝑥_𝑛} are also bounded. We next prove that lim_𝑛→∞sup⟨𝑥−𝑅𝑥,𝐽𝑥_𝑛−𝐽𝑅𝑥⟩≤0. (3.4) Put 𝑢_𝑛=𝑥_𝑛+1 for all 𝑛∈ℕ. Since {𝐽𝑢_𝑛} is bounded, without loss of generality, we have a subsequence {𝐽𝑢_𝑛𝑖} of {𝐽𝑢_𝑛} such that lim_𝑖→∞⟨𝑥−𝑅𝑥,𝐽𝑢_𝑛𝑖−𝐽𝑅𝑥⟩=lim_𝑛→∞sup⟨𝑥−𝑅𝑥,𝐽𝑢_𝑛−𝐽𝑅𝑥⟩, (3.5) and {𝐽𝑢_𝑛𝑖} converges weakly to some 𝑣^∗∈𝐸^∗. From the definition of {𝑥_𝑛}, we have 𝑢_𝑛−𝑦_𝑛=𝛽_𝑛(𝑥−𝑦_𝑛),𝑦_𝑛−𝐽_𝑟𝑛𝑥_𝑛=𝛼_𝑛(𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛) (3.6) for all 𝑛∈ℕ. Since {𝑦_𝑛} is bounded and 𝛽_𝑛→0 as 𝑛→∞, it follows that lim_𝑛→∞‖𝑢_𝑛−𝑦_𝑛‖=lim_𝑛→∞𝛽_𝑛‖𝑥−𝑦_𝑛‖=0. (3.7) Moreover, we note that lim_𝑛→∞‖𝑦_𝑛−𝐽_𝑟𝑛𝑥_𝑛‖=lim_𝑛→∞𝛼_𝑛‖𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛‖=0. (3.8) By (3.7) and (3.8), we have lim_𝑛→∞‖𝑢_𝑛−𝐽_𝑟𝑛𝑥_𝑛‖=0. (3.9) Since 𝐸 has a uniformly Gâteaux differentiable norm, the duality mapping 𝐽 is norm to 𝑤𝑒𝑎𝑘^∗ uniformly continuous on each bounded subset of 𝐸. Therefore, we obtain from (3.9) that 𝐽𝑢_𝑛𝑖−𝐽𝐽_𝑟𝑛𝑖𝑥_𝑛𝑖⇀0,as𝑖→∞. (3.10) This implies that 𝐽𝐽_𝑟𝑛𝑖𝑥_𝑛𝑖⇀𝑣^∗ as 𝑖→∞. On the other hand, from 𝑟_𝑛→∞ as 𝑛→∞, we have lim_𝑛→∞‖𝐴_𝑟𝑛𝑥_𝑛‖=lim_𝑛→∞1𝑟_𝑛‖𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛‖=0. (3.11) If (𝑦^∗,𝑦)∈𝐵, then it holds from the monotonicity of 𝐵 that ⟨𝑦−𝐴_𝑟𝑛𝑖𝑥_𝑛𝑖,𝑦^∗−𝐽𝐽_𝑟𝑛𝑖𝑥_𝑛𝑖⟩≥0, (3.12) for all 𝑖∈ℕ. Letting 𝑖→∞, we get ⟨𝑦,𝑦^∗−𝑣^∗⟩≥0. Then, the maximal of 𝐵 implies 𝑣^∗∈𝐵⁻¹0. Put 𝑣=𝐽⁻¹𝑣^∗. Applying Lemma 2.3, we obtain lim_𝑛→∞sup⟨𝑥−𝑅𝑥,𝐽𝑢_𝑛−𝐽𝑅𝑥⟩=lim_𝑖→∞⟨𝑥−𝑅𝑥,𝐽𝑢_𝑛𝑖−𝐽𝑅𝑥⟩=⟨𝑥−𝑅𝑥,𝑣^∗−𝐽𝑅𝑥⟩=⟨𝑥−𝑅𝑥,𝐽𝑣−𝐽𝑅𝑥⟩≤0. (3.13) Finally, we prove that 𝑥_𝑛→𝑅𝑥 as 𝑛→∞. From Lemma 2.2, the convexity of ‖⋅‖² and (3.2), we have 𝜙(𝑥_𝑛+1,𝑅𝑥)=𝑉(𝛽_𝑛𝑥+(1−𝛽_𝑛)𝑦_𝑛,𝐽𝑅𝑥)≤𝑉(𝛽_𝑛𝑥+(1−𝛽_𝑛)𝑦_𝑛−𝛽_𝑛(𝑥−𝑅𝑥),𝐽𝑅𝑥)−2⟨−𝛽_𝑛(𝑥−𝑅𝑥),𝐽𝑥_𝑛+1−𝐽𝑅𝑥⟩=𝑉(𝛽_𝑛𝑅𝑥+(1−𝛽_𝑛)𝑦_𝑛,𝐽𝑅𝑥)+2𝛽_𝑛⟨𝑥−𝑅𝑥,𝐽𝑥_𝑛+1−𝐽𝑅𝑥⟩=𝜙(𝛽_𝑛𝑅𝑥+(1−𝛽_𝑛)𝑦_𝑛,𝑅𝑥)+2𝛽_𝑛⟨𝑥−𝑅𝑥,𝐽𝑥_𝑛+1−𝐽𝑅𝑥⟩≤𝛽_𝑛𝜙(𝑅𝑥,𝑅𝑥)+(1−𝛽_𝑛)𝜙(𝑦_𝑛,𝑅𝑥)+2𝛽_𝑛⟨𝑥−𝑅𝑥,𝐽𝑥_𝑛+1−𝐽𝑅𝑥⟩≤(1−𝛽_𝑛)𝜙(𝑥_𝑛,𝑅𝑥)+𝛽_𝑛𝜎_𝑛, (3.14) for all 𝑛∈ℕ, where 𝜎_𝑛=2⟨𝑥−𝑅𝑥,𝐽𝑥_𝑛+1−𝐽𝑅𝑥⟩. It easily verified from the assumption and (3.4) that ∑^∞_𝑛=1𝛽_𝑛=∞ and limsup_𝑛→∞𝜎_𝑛≤0. Hence, by Lemma 2.6, lim_𝑛→∞𝜙(𝑥_𝑛,𝑅𝑥)=0. Applying Lemma 2.1, we obtain lim_𝑛→∞‖𝑥_𝑛−𝑅𝑥‖=0. Therefore, {𝑥_𝑛} converges strongly to 𝑅_(𝐵𝐽)⁻¹₀(𝑥). Put 𝛼_𝑛≡0 in Theorem 3.1, then we obtain the following result.

Corollary 3.2 (see Ibaraki and Takahashi [7]). Let 𝐸 be a uniformly convex and uniformly smooth Banach space and let 𝐵⊂𝐸^∗×𝐸 be a maximal monotone operator with 𝐵⁻¹0≠∅, let 𝐽_𝑟=(𝐼+𝑟𝐵𝐽)⁻¹ for all 𝑟>0, and let {𝑥_𝑛} be a sequence generated by 𝑥₁=𝑥∈𝐸 and 𝑥_𝑛+1=𝛽_𝑛𝑥+(1−𝛽_𝑛)𝐽_𝑟𝑛𝑥_𝑛, (3.15) for every 𝑛=1,2,…, where {𝛽_𝑛}⊂[0,1], {𝑟_𝑛}⊂(0,∞) satisfy lim_𝑛→∞𝛽_𝑛=0, ∑^∞_𝑛=1𝛽_𝑛=∞ and lim_𝑛→∞𝑟_𝑛=∞.Then the sequence {𝑥_𝑛} converges strongly to 𝑅_(𝐵𝐽)⁻¹₀(𝑥), where 𝑅_(𝐵𝐽)⁻¹₀ is the generalized projection of 𝐸 onto (𝐵𝐽)⁻¹0.

Theorem 3.3. Let 𝐸 be a uniformly convex and smooth Banach space whose duality mapping 𝐽 is weakly sequentially continuous. Let 𝐵⊂𝐸^∗×𝐸 be a maximal monotone operator with 𝐵⁻¹0≠∅ and let 𝐽_𝑟=(𝐼+𝑟𝐵𝐽)⁻¹ for all 𝑟>0. Let {𝑥_𝑛} be a sequence generated by 𝑥₁=𝑥∈𝐸 and 𝑦_𝑛=𝛼_𝑛𝑥_𝑛+(1−𝛼_𝑛)𝐽_𝑟𝑛𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥+(1−𝛽_𝑛)𝑦_𝑛, (3.16) for every 𝑛=1,2,…, where {𝛼_𝑛}, {𝛽_𝑛}⊂[0,1], {𝑟_𝑛}⊂(0,∞) satisfy ∑^∞_𝑛=1𝛽_𝑛<∞, limsup_𝑛→∞𝛼_𝑛<1 and liminf_𝑛→∞𝑟_𝑛>0. Then the sequence {𝑥_𝑛} converges weakly to an element of (𝐵𝐽)⁻¹0.

Proof. Let 𝑣∈(𝐵𝐽)⁻¹0. Then, from (3.3), we have 𝜙(𝑥_𝑛+1,𝑣)≤(1−𝛽_𝑛)𝜙(𝑥_𝑛,𝑣)+𝛽_𝑛𝜙(𝑥,𝑣)≤𝜙(𝑥_𝑛,𝑣)+𝛽_𝑛𝜙(𝑥,𝑣), (3.17) for all 𝑛∈ℕ. By Lemma 2.7, lim_𝑛→∞𝜙(𝑥_𝑛,𝑣) exists. From (‖𝑥_𝑛‖−‖𝑣‖)²≤𝜙(𝑥_𝑛,𝑣) and 𝜙(𝐽_𝑟𝑛𝑥_𝑛,𝑣)≤𝜙(𝑥_𝑛,𝑣), we note that {𝑥_𝑛} and {𝐽_𝑟𝑛𝑥_𝑛} are bounded. From (3.3) and (3.2), we have 𝜙(𝑥_𝑛+1,𝑣)≤𝛽_𝑛𝜙(𝑥,𝑣)+(1−𝛽_𝑛)𝜙(𝑦_𝑛,𝑣)≤𝛽_𝑛𝜙(𝑥,𝑣)+(1−𝛽_𝑛){𝛼_𝑛𝜙(𝑥_𝑛,𝑣)+(1−𝛼_𝑛)(𝜙(𝑥_𝑛,𝑣)−𝜙(𝑥_𝑛,𝐽_𝑟𝑛𝑥_𝑛))}=𝛽_𝑛𝜙(𝑥,𝑣)+(1−𝛽_𝑛){𝜙(𝑥_𝑛,𝑣)−(1−𝛼_𝑛)𝜙(𝑥_𝑛,𝐽_𝑟𝑛𝑥_𝑛)}=𝛽_𝑛𝜙(𝑥,𝑣)+(1−𝛽_𝑛)𝜙(𝑥_𝑛,𝑣)−(1−𝛽_𝑛)(1−𝛼_𝑛)𝜙(𝑥_𝑛,𝐽_𝑟𝑛𝑥_𝑛), (3.18) for all 𝑛∈ℕ and hence, (1−𝛽_𝑛)(1−𝛼_𝑛)𝜙(𝑥_𝑛,𝐽_𝑟𝑛𝑥_𝑛)≤𝛽_𝑛𝜙(𝑥,𝑣)+(1−𝛽_𝑛)𝜙(𝑥_𝑛,𝑣)−𝜙(𝑥_𝑛+1,𝑣)=𝛽_𝑛(𝜙(𝑥,𝑣)−𝜙(𝑥_𝑛,𝑣))+𝜙(𝑥_𝑛,𝑣)−𝜙(𝑥_𝑛+1,𝑣), (3.19) for all 𝑛∈ℕ. Since lim_𝑛→∞𝛽_𝑛=0 and limsup_𝑛→∞𝛼_𝑛<1, lim_𝑛→∞𝜙(𝑥_𝑛,𝐽_𝑟𝑛𝑥_𝑛)=0. Applying Lemma 2.1, we obtain lim_𝑛→∞‖𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛‖=0. (3.20) Since {𝑥_𝑛} is bounded, we have a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that 𝑥_𝑛𝑖⇀𝑤∈𝐸 as 𝑖→∞. Then it follows from (3.20) that 𝐽_𝑟𝑛𝑖𝑥_𝑛𝑖⇀𝑤 as 𝑖→∞. On the other hand, from (3.20) and liminf_𝑛→∞𝑟_𝑛>0, we have lim_𝑛→∞‖𝐴_𝑟𝑛𝑥_𝑛‖=lim_𝑛→∞1𝑟_𝑛‖𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛‖=0. (3.21) Let (𝑧^∗,𝑧)∈𝐵. Then, it holds from monotonicity of 𝐵 that ⟨𝑧−𝐴_𝑟𝑛𝑖𝑥_𝑛𝑖,𝑧^∗−𝐽𝐽_𝑟𝑛𝑖𝑥_𝑛𝑖⟩≥0, (3.22) for all 𝑖∈ℕ. Since 𝐽 is weakly sequentially continuous, letting 𝑖→∞, we get ⟨𝑧,𝑧^∗−𝐽𝑤⟩≥0. Then, the maximality of 𝐵 implies 𝐽𝑤∈𝐵⁻¹0. Thus, 𝑤∈(𝐵𝐽)⁻¹0.
Let {𝑥_𝑛𝑖} and {𝑥_𝑛𝑗} be two subsequences of {𝑥_𝑛} such that 𝑥_𝑛𝑖⇀𝑤₁ and 𝑥_𝑛𝑗⇀𝑤₂. By similar argument as above, we obtain 𝑤₁,𝑤₂∈(𝐵𝐽)⁻¹0. Put 𝑎∶=lim_𝑛→∞(𝜙(𝑥_𝑛,𝑤₁)−𝜙(𝑥_𝑛,𝑤₂)).
Note that 𝜙(𝑥_𝑛,𝑤₁)−𝜙(𝑥_𝑛,𝑤₂)=2⟨𝑥_𝑛,𝐽𝑤₂−𝐽𝑤₁⟩+‖𝑤₁‖²−‖𝑤₂‖²,𝑛=1,2,…. From 𝑥_𝑛𝑖⇀𝑤₁ and 𝑥_𝑛𝑗⇀𝑤₂, we have 𝑎=2⟨𝑤₁,𝐽𝑤₂−𝐽𝑤₁⟩+‖𝑤₁‖²−‖𝑤₂‖²,𝑎=2⟨𝑤₂,𝐽𝑤₂−𝐽𝑤₁⟩+‖𝑤₁‖²−‖𝑤₂‖², (3.23)(3.24) respectively. Combining (3.23) and (3.24), we have ⟨𝑤₁−𝑤₂,𝐽𝑤₁−𝐽𝑤₂⟩=0. (3.25) Since 𝐽 is strictly monotone, it follows that 𝑤₁=𝑤₂. Therefore, {𝑥_𝑛} converges weakly to an element of (𝐵𝐽)⁻¹0.

Put 𝛽_𝑛≡0 in Theorem 3.3, then we obtain the following result.

Corollary 3.4 (see Ibaraki and Takahashi [7]). Let 𝐸 be a uniformly convex and smooth Banach space whose duality mapping 𝐽 is weakly sequentially continuous. Let 𝐵⊂𝐸^∗×𝐸 be a maximal monotone operator with 𝐵⁻¹0≠∅, let 𝐽_𝑟=(𝐼+𝑟𝐵𝐽)⁻¹ for all 𝑟>0 and let {𝑥_𝑛} be a sequence generated by 𝑥₁=𝑥∈𝐸 and 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝐽_𝑟𝑛𝑥_𝑛, (3.26) for every 𝑛=1,2,…, where {𝛼_𝑛},{𝛽_𝑛}⊂[0,1], {𝑟_𝑛}⊂(0,∞) satisfy ∑^∞_𝑛=1𝛽_𝑛<∞, limsup_𝑛→∞𝛼_𝑛<1 and liminf_𝑛→∞𝑟_𝑛>0. Then the sequence {𝑥_𝑛} converges weakly to an element of (𝐵𝐽)⁻¹0.

4. Rate of Convergence for the Algorithm

In this section, we study the rate of the convergence of the algorithm (1.7). We use the following notations in [6, 18]:

𝑁₀∶={𝜑∶ℝ₊→ℝ₊Ω∣𝑡↦𝜑(𝑡)isnondecreasingfor𝑡≥0,𝜑(0)=0},₀∶={𝜑∶ℝ₊→ℝ₊∣𝜑(0)=0,lim_𝑡→0+Γ𝜑(𝑡)=0},₀∶={𝜑∶ℝ₊→ℝ₊Σ∣𝜑islscandconvexand𝜑(𝑡)=0⟺𝑡=0},₁∶={𝜑∶ℝ₊→ℝ₊∣𝜑islscandconvex,𝜑(0)=0,lim_𝑡→0+𝑡⁻¹𝜑(𝑡)=0}. (4.1) We recall [18] that, for a function 𝜑∶ℝ₊→ℝ₊ satisfying 𝜑(0)=0, its pseudoconjugate 𝜑^#∶ℝ₊→ℝ₊, defined by 𝜑^#(𝑠)∶=sup{𝑠𝑡−𝜑(𝑡)∣𝑡≥0}∈ℝ, (4.2) is lower semicontinuous, convex and satisfies 𝜑^#(0)=0,𝜑^#(𝑠)≥0 for all 𝑠≥0.

For a function 𝜑∈𝑁₀, its greatest quasi-inverse 𝜑^ℎ∶ℝ₊→ℝ₊, defined by 𝜑^ℎ(𝑠)∶=sup{𝑡≥0∣𝜑(𝑡)≤𝑠}, (4.3) is nondecreasing. It is known [18] that 𝜑^ℎ∈𝑁₀∩Ω₀ if 𝜑(𝑡)=0⇔𝑡=0.

For a function 𝜑∶ℝ→ℝ, its lower semicontinuous convex hull, denoted by co𝜑, is defined by epi(co𝜑)= cl(co(epi𝜑)). It is obvious that co𝜑 is lower semicontinuous convex and co𝜑≤𝜑.

Proposition 4.1. Let 𝐸 be uniformly convex and uniformly smooth. Then, for every 𝑟>0, there exists 𝜎_𝑟∈Σ₁ such that, for all 𝑥,𝑦∈𝑟𝑈_𝐸, ⟨𝑦−𝑥,𝐽𝑦−𝐽𝑥⟩≤𝜎_𝑟(‖𝐽𝑦−𝐽𝑥‖). (4.4)

Proof. Since 𝐸 is uniformly convex, 𝑓(𝑥)=(1/2)‖𝑥‖² is uniform convex on 𝑟𝑈_𝐸 for all 𝑟>0. Since the norm of 𝐸 is Fréchet differentiable, its Fréchet derivative ∇𝑓(𝑥)=𝐽𝑥. In [18, Proposition ?3.6.5] for 𝑓 and 𝐵=𝑟𝑈_𝐸, where 𝑟 is an arbitrary positive real number, we get the function 𝜗_𝑟(⋅)∶[0,+∞)→[0,+∞), defined by 𝜗_𝑟1(𝑡)∶=inf{2‖𝑦‖²−12‖𝑥‖²−⟨𝑦−𝑥,𝐽𝑥⟩∶𝑥∈𝑟𝑈_𝐸,𝑦∈𝐸,‖𝑦−𝑥‖=𝑡}, (4.5) satisfies that 𝜗_𝑟(𝑡)=0 if and only if 𝑡=0, and 𝑡⁻¹𝜗_𝑟(𝑡) is nondecreasing. Thus, 12‖𝑦‖²−12‖𝑥‖²−⟨𝑦−𝑥,𝐽𝑥⟩≥𝜗_𝑟(‖𝑦−𝑥‖) (4.6) for all 𝑥∈𝑟𝑈_𝐸, 𝑦∈𝐸 and hence 12‖𝑦+𝑥‖²≥12‖𝑥‖²+⟨𝑦,𝐽𝑥⟩+co𝜗_𝑟(‖𝑦‖) (4.7) for all 𝑥∈𝑟𝑈_𝐸, 𝑦∈𝐸. It follows that 1⟨𝑥,𝐽𝑥⟩−2‖𝑥‖²+⟨𝑥,𝐽𝑦−𝐽𝑥⟩+⟨𝑦,𝐽𝑦−𝐽𝑥⟩−co𝜑𝜗_𝑟1(‖𝑦‖)≥⟨𝑦+𝑥,𝐽𝑦⟩−2‖𝑦+𝑥‖² (4.8) for all 𝑥∈𝑟𝑈_𝐸, 𝑦∈𝐸. Since ⟨𝑥,𝐽𝑥⟩=(1/2)‖𝑥‖²+(1/2)‖𝑥‖², we have 12‖𝑥‖²+⟨𝑦+𝑥,𝐽𝑦−𝐽𝑥⟩−co𝜗_𝑟1(‖𝑦‖)≥⟨𝑦+𝑥,𝐽𝑦⟩−2‖𝑦+𝑥‖² (4.9) for all 𝑥∈𝑟𝑈_𝐸 and 𝑦∈𝐸.
Taking the supremum on both sides of (4.9) over 𝑦∈𝐸, by [18, Lemma ?3.3.1(v)] (if 𝑓(𝑥)∶=𝜑(‖𝑥‖), where 𝜑∈𝑁₀, then 𝑓^∗(𝑥^∗)=𝜑^#(‖𝑥^∗‖)), we get that 12‖𝑥‖²+⟨𝑥,𝐽𝑦−𝐽𝑥⟩+(co𝜗_𝑟)^#1(‖𝐽𝑦−𝐽𝑥‖)≥2‖𝑦‖² (4.10) for all 𝑥∈𝑟𝑈_𝐸 and 𝑦∈𝐸. Since 𝜗_𝑟(𝑡) is nondecreasing and lim_𝑡→∞𝑡⁻¹𝜗_𝑟(𝑡)≥𝜗_𝑟(1)>0, we have co𝜗_𝑟∈Γ₀. It follows from [20, Lemma ?3.3.1(iii)] that (co𝜗_𝑟)^#∈Σ₁.
Interchanging 𝑥 and 𝑦 in (4.10) for 𝑥,𝑦∈𝑟𝑈_𝐸, it also holds that 12‖𝑦‖²+⟨𝑦,𝐽𝑥−𝐽𝑦⟩+(co𝜗_𝑟)^#1(‖𝐽𝑦−𝐽𝑥‖)≥2‖𝑥‖². (4.11) Thus, by taking 𝜎_𝑟∶=2(co𝜗_𝑟)^#∈Σ₁, and adding side by side (4.10) and (4.11), we obtained ⟨𝑦−𝑥,𝐽𝑦−𝐽𝑥⟩≤𝜎_𝑟(‖𝐽𝑦−𝐽𝑥‖),∀𝑥,𝑦∈𝑟𝑈_𝐸. (4.12)

Theorem 4.2. Let 𝐸 be a uniformly convex and uniformly smooth Banach space. Suppose that 𝐵⊂𝐸^∗×𝐸 is maximal monotone with 𝐵⁻¹0={𝑣^∗} and 𝐵⁻¹ is Lipschitz continuous at 0 with modulus 𝑙≥0. Let {𝑥_𝑛} be a sequence generated by 𝑥₁=𝑥∈𝐸 and 𝑦_𝑛=𝛼_𝑛𝑥_𝑛+(1−𝛼_𝑛)𝐽_𝑟𝑛𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥+(1−𝛽_𝑛)𝑦_𝑛, (4.13) for every 𝑛=1,2,…, where {𝛼_𝑛}, {𝛽_𝑛}⊂[0,1], {𝑟_𝑛}⊂(0,∞) satisfying ?lim_𝑛→∞𝑟_𝑛=∞. If either ∑^∞_𝑛=1𝛽_𝑛<∞ and limsup_𝑛→∞𝛼_𝑛<1 or lim_𝑛→∞𝛽_𝑛=0, ∑^∞_𝑛=1𝛽_𝑛=∞ and lim_𝑛→∞𝛼_𝑛=0, then {𝑥_𝑛} converges strongly to 𝑣∶=𝐽⁻¹𝑣^∗and 𝜙(𝑥_𝑛,𝑣) converges to 0.
Moreover, there exists an integer 𝑁>0 such that 𝜙(𝑥_𝑛+1,𝑣)≤𝜏_𝑛𝜙(𝑥,𝑣)+𝜃_𝑛+𝛿_𝑛,∀𝑛≥𝑁, (4.14) where 𝜏_𝑛=𝛽_𝑛+∑^𝑛−1_𝑖=𝑁𝛽_𝑖∏^𝑛_𝑗=𝑖+1(1−𝛽_𝑗)𝛼_𝑗, 𝜃_𝑛=∏^𝑛_𝑖=𝑁(1−𝛽_𝑖)𝛼_𝑖, 𝛿_𝑛=𝑙((1−𝛼_𝑛)/𝑟_𝑛+∑^𝑛−1_𝑖=𝑁((1−𝛼_𝑖)/𝑟_𝑖)∏^𝑛_𝑗=𝑖+1(1−𝛽_𝑗−1)𝛼_𝑗) and lim_𝑛→∞𝜏_𝑛=lim_𝑛→∞𝜃_𝑛=lim_𝑛→∞𝛿_𝑛=0. Also, one obtains 𝜙(𝑥_𝑛+1,𝑣)≤𝛽_𝑛𝜙(𝑥,𝑣)+(𝛼_𝑛𝑙+∘(𝑟_𝑛))𝑘(𝜙(𝑥_𝑛,𝑣)), (4.15) for all 𝑛≥𝑁, where 𝑘(𝑡)=max{𝑡,𝜈^ℎ_𝑟(𝑡)}∈𝑁₀∩Ω₀, and 𝜈_𝑟(𝑡)∶=𝜈(𝑟𝑈_𝐸,𝑡), 𝜈^ℎ_𝑟 is the greatest quasi-inverse of 𝜈_𝑟(𝑡), and 𝑟 is a positive number such that {𝑣}∪{𝑥_𝑛}∪{𝐽_𝑟𝑛𝑥_𝑛}⊂𝑟𝑈_𝐸.

Proof. Put 𝑣=𝐽⁻¹𝑣^∗. Since 𝐵⁻¹0={𝑣^∗}, we have (𝐵𝐽)⁻¹0={𝑣}. We separate the proof into two cases.Case 1. ∑^∞_𝑛=1𝛽_𝑛<∞, and limsup_𝑛→∞𝛼_𝑛<1.
According to Theorem 3.3, we have lim_𝑛→∞𝜙(𝑥_𝑛,𝑣) exists, {𝑥_𝑛} and {𝐽_𝑟𝑛} are bounded, and hence {𝑣}∪{𝑥_𝑛}∪{𝐽_𝑟𝑛𝑥_𝑛}⊂𝑟𝑈_𝐸 for some 𝑟>0. Since 𝐵⁻¹ is Lipschitz continuous at 0 with modulus 𝑙≥0, for some 𝜏>0, we have ‖𝑧^∗−𝑣^∗‖≤𝑙‖𝑤‖ whenever 𝑧^∗∈𝐵⁻¹(𝑤) and ‖𝑤‖≤𝜏. Since 𝑟_𝑛→∞, we may assume 𝑙/𝑟_𝑛<1 for all 𝑛≥1. From Theorem 3.3, we have ‖𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛‖→0 and ‖𝐴_𝑟𝑛𝑥_𝑛‖→0 as 𝑛→∞. Hence, there exists an integer 𝑁>0 such that ‖𝐴_𝑟𝑛𝑥_𝑛‖≤𝜏 for all 𝑛≥𝑁. Since 𝐽𝐽_𝑟𝑛𝑥_𝑛∈𝐵⁻¹𝐴_𝑟𝑛𝑥_𝑛, we have ‖𝐽𝐽_𝑟𝑛𝑥_𝑛−𝑣^∗‖≤𝑙‖𝐴_𝑟𝑛𝑥_𝑛‖,∀𝑛≥𝑁. (4.16) By ‖𝐽𝑥_𝑛−𝑣^∗‖≤‖𝐽𝑥_𝑛−𝐽𝐽_𝑟𝑛𝑥_𝑛‖+‖𝐽𝐽_𝑟𝑛𝑥_𝑛−𝑣^∗‖ for all 𝑛∈ℕ. Since 𝐽 is uniformly continuous on each bounded set, (3.20), (3.21) and (4.16), we obtain lim_𝑛→∞‖𝐽𝑥_𝑛−𝑣^∗‖=0. By the uniform smoothness of 𝐸^∗, we have lim_𝑛→∞‖𝑥_𝑛−𝑣‖=lim_𝑛→∞‖𝐽⁻¹𝐽𝑥_𝑛−𝐽⁻¹𝑣^∗‖=0. Since 𝜙(𝑥_𝑛,𝑣)+𝜙(𝑣,𝑥_𝑛)=2⟨𝑥_𝑛−𝑣,𝐽𝑥_𝑛−𝐽𝑣⟩=2⟨𝑥_𝑛−𝑣,𝐽𝑥_𝑛−𝑣^∗⟩ for all 𝑛∈ℕ, we get 𝜙(𝑥_𝑛,𝑣)≤2⟨𝑥_𝑛−𝑣,𝐽𝑥_𝑛−𝑣^∗⟩≤2‖𝑥_𝑛−𝑣‖‖𝐽𝑥_𝑛−𝑣^∗‖→0,as𝑛→∞. (4.17) Hence, 𝜙(𝑥_𝑛,𝑣)→0 as 𝑛→∞. It follows from Proposition 4.1 and (4.16) that there exists 𝜎(𝑡)∈Σ₁, which implies 𝜎(𝑙𝑡)≤𝑙𝜎(𝑡) for all 𝑡≥0 and 𝑙∈[0,1], such that 𝜙(𝐽_𝑟𝑛𝑥_𝑛,𝑣)≤𝜙(𝐽_𝑟𝑛𝑥_𝑛,𝑣)+𝜙(𝑣,𝐽_𝑟𝑛𝑥_𝑛)=2⟨𝐽_𝑟𝑛𝑥_𝑛−𝑣,𝐽𝐽_𝑟𝑛𝑥_𝑛−𝐽𝑣⟩≤𝜎(‖𝐽𝐽_𝑟𝑛𝑥_𝑛−𝐽𝑣‖)=2𝜎(‖𝐽𝐽_𝑟𝑛𝑥_𝑛−𝑣^∗‖)≤2𝜎(𝑙‖𝐴_𝑟𝑛𝑥_𝑛𝑥‖)=2𝜎(𝑙‖^𝑛−𝐽_𝑟𝑛𝑥_𝑛𝑟_𝑛‖)≤2𝑙𝑟_𝑛𝜎(‖𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛‖) (4.18) for all 𝑛≥𝑁. It follows from 𝜎(𝑡)∈Σ₁ and (3.20) that lim_𝑛→∞𝜎(‖𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛‖)=0. (4.19) From (3.3), (3.2), and (4.18), we have 𝜙(𝑥_𝑛+1,𝑣)≤𝛽_𝑛𝜙(𝑥,𝑣)+(1−𝛽_𝑛){𝛼_𝑛𝜙(𝑥_𝑛,𝑣)+(1−𝛼_𝑛)𝜙(𝐽_𝑟𝑛𝑥_𝑛,𝑣)}≤𝛽_𝑛𝜙(𝑥,𝑣)+(1−𝛽_𝑛)𝛼_𝑛𝜙(𝑥_𝑛,𝑣)+(1−𝛽_𝑛)(1−𝛼_𝑛)2𝑙𝑟_𝑛𝜎(‖𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛‖) (4.20) for all 𝑛≥𝑁. Since 𝜙(𝑥_𝑛,𝑣)→0 and (4.19), we may assume 𝜙(𝑥_𝑛,𝑣)≤1 and 2𝜎(‖𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛‖)≤1 for all 𝑛≥𝑁. By (4.20) and induction, we obtain 𝜙(𝑥_𝑛+1,𝑣)≤(𝛽_𝑛+^𝑛−1_𝑖=𝑁𝛽_𝑖^𝑛_𝑗=𝑖+1(1−𝛽_𝑗)𝛼_𝑗)𝜙(𝑥,𝑣)+(^𝑛_𝑖=𝑁(1−𝛽_𝑖)𝛼_𝑖)𝜙(𝑥_𝑁,𝑣)+(1−𝛽_𝑛)𝑙((1−𝛼_𝑛)𝑟_𝑛+^𝑛−1_𝑖=𝑁(1−𝛼_𝑖)𝑟_𝑖2𝜎(‖𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛‖)^𝑛_𝑗=𝑖+1(1−𝛽_𝑗−1)𝛼_𝑗)≤𝜏_𝑛𝜙(𝑥,𝑣)+𝜃_𝑛+𝛿_𝑛, (4.21) for all 𝑛≥𝑁, where 𝜏_𝑛=𝛽_𝑛+∑^𝑛−1_𝑖=𝑁𝛽_𝑖∏^𝑛_𝑗=𝑖+1(1−𝛽_𝑗)𝛼_𝑗, 𝜃_𝑛=∏^𝑛_𝑖=𝑁(1−𝛽_𝑖)𝛼_𝑖, 𝛿_𝑛=𝑙((1−𝛼_𝑛)/𝑟_𝑛+∑^𝑛−1_𝑖=𝑁((1−𝛼_𝑖)/𝑟_𝑖)∏^𝑛_𝑗=𝑖+1(1−𝛽_𝑗−1)𝛼_𝑗).
Next, we prove 𝜏_𝑛, 𝜃_𝑛 and 𝛿_𝑛 tend to 0. By 𝛼_𝑛,𝛽_𝑛∈[0,1] and ∑^∞_𝑖=1𝛽_𝑖<∞, we get 0≤𝜏_𝑛=𝛽_𝑛+^𝑛−1_𝑖=𝑁𝛽_𝑖^𝑛_𝑗=𝑖+1(1−𝛽_𝑗)𝛼_𝑗≤𝛽_𝑛+^𝑛−1_𝑖=𝑁𝛽_𝑖=^𝑛_𝑖=𝑁𝛽_𝑖≤^∞_𝑖=1𝛽_𝑖. (4.22) Thus {𝜏_𝑛} is bounded. Since 𝛼_𝑛∈[0,1] and limsup_𝑛→∞𝛼_𝑛<1, there exists some 𝛼(0<𝛼<1) such that 0≤𝛼_𝑛≤𝛼 whenever 𝑛≥𝑁. Then, we get 0≤limsup_𝑛→∞𝜏_𝑛≤limsup_𝑛→∞𝛽_𝑛+limsup_𝑛→∞^𝑛−1_𝑖=𝑁𝛽_𝑖^𝑛_𝑗=𝑖+1(1−𝛽_𝑗)𝛼_𝑗=lim_𝑛→∞𝛽_𝑛+limsup_𝑛→∞(1−𝛽_𝑛)𝛼_𝑛(𝛽_𝑛−1+^𝑛−2_𝑖=𝑁𝛽_𝑖^𝑛−1_𝑗=𝑖+1(1−𝛽_𝑗)𝛼_𝑗)=limsup_𝑛→∞(1−𝛽_𝑛)𝛼_𝑛𝜏_𝑛−1≤limsup_𝑛→∞𝛼_𝑛𝜏_𝑛−1≤𝛼limsup_𝑛→∞𝜏_𝑛−1=𝛼limsup_𝑛→∞𝜏_𝑛, (4.23) which implies lim_𝑛→∞𝜏_𝑛=limsup_𝑛→∞𝜏_𝑛=0. (4.24) Meanwhile, we also have 0≤𝜃_𝑛=^𝑛_𝑖=𝑁(1−𝛽_𝑖)𝛼_𝑖≤^𝑛_𝑖=𝑁𝛼_𝑖≤𝛼^{𝑛−𝑁+1}→0. (4.25) On the other hand, 𝛿_𝑛=𝑙((1−𝛼_𝑛)𝑟_𝑛+^𝑛−1_𝑖=𝑁(1−𝛼_𝑖)𝑟_𝑖^𝑛_𝑗=𝑖+1(1−𝛽_𝑗−1)𝛼_𝑗)≤(1−𝛼_𝑛)𝑙𝑟_𝑛+𝛼_𝑛𝑙((1−𝛼_𝑛−1)𝑟_𝑛−1+^𝑛−2_𝑖=𝑁(1−𝛼_𝑖)𝑟_𝑖^𝑛−1_𝑗=𝑖+1(1−𝛽_𝑗−1)𝛼_𝑗)=(1−𝛼_𝑛)𝑙𝑟_𝑛+𝛼_𝑛𝛿_𝑛−1. (4.26) Since limsup_𝑛→∞𝛼_𝑛<1, ∑^∞_𝑛=1(1−𝛼_𝑛)=+∞, it follows from Lemma 2.6 that 𝛿_𝑛→0.
From 𝜎(𝑡)∈Σ₁, which implies that lim_𝑡→0⁺(𝜎(𝑡)/𝑡)=0, and (3.20). It follows that lim_𝑛→∞𝜎(‖𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛‖)‖𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛‖=0. (4.27) By (4.18), we have 𝜙(𝐽_𝑟𝑛𝑥_𝑛𝑙,𝑣)≤𝑟_𝑛2𝜎(‖𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛=𝑙‖)𝑟_𝑛2𝜎(‖𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛‖)‖𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛‖‖𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛‖𝑙=∘(𝑟_𝑛)‖𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛‖, (4.28) for all 𝑛≥𝑁, where 𝑙∘(𝑟_𝑛𝑙)∶=𝑟_𝑛2𝜎(‖𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛‖)‖𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛‖.
Since 𝐸 is uniformly convex, ‖⋅‖² is uniformly totally convex on each bounded set of 𝐸. Denote by 𝜈_𝑟(𝑡)∶=𝜈(𝑟𝑈_𝐸,𝑡) the modulus of uniformly total convexity on the bounded set 𝑟𝑈_𝐸. Then 𝜈_𝑟(𝑡)∈𝑁₀ and satisfies 𝜈(‖𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛‖)≤𝜙(𝑥_𝑛,𝐽_𝑟𝑛𝑥_𝑛). From the definition of the greatest quasi-inverse of 𝜈 , we deduce that ‖𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛‖≤𝜈^ℎ_𝑟(𝜙(𝑥_𝑛,𝐽_𝑟𝑛𝑥_𝑛)). (4.29) From Lemma 2.5, we have 𝜙(𝑥_𝑛,𝐽_𝑟𝑛𝑥_𝑛)≤𝜙(𝑥_𝑛,𝑣)−𝜙(𝐽_𝑟𝑛𝑥_𝑛,𝑣)≤𝜙(𝑥_𝑛,𝑣). (4.30) Since 𝜈^ℎ_𝑟∈𝑁₀, it holds by (4.28), (4.29), and (4.30) that 𝜙(𝐽_𝑟𝑛𝑥_𝑛𝑙,𝑣)≤∘(𝑟_𝑛)‖𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛𝑙‖≤∘(𝑟_𝑛)𝜈^ℎ_𝑟(𝜙(𝑥_𝑛,𝑣)), (4.31) for all 𝑛∈ℕ. Let 𝑘(𝑡)=max{𝑡,𝜈^ℎ_𝑟(𝑡)}. Since 𝜈^ℎ_𝑟∈𝑁₀ and 𝜈_𝑟(𝑡)≥0 for 𝑡>0, it follows in [18, Lemma ?3.3.1(i)] that 𝜈^ℎ_𝑟(𝑡)∈Ω₀ and this implies that 𝑘(𝑡)∈𝑁₀∩Ω₀. By the first inequality of (4.20), (4.31) and the definition of 𝑘(𝑡), we have 𝜙(𝑥_𝑛+1,𝑣)≤𝛽_𝑛𝜙(𝑥,𝑣)+(𝛼_𝑛𝑙+∘(𝑟_𝑛))𝑘(𝜙(𝑥_𝑛,𝑣)),∀𝑛≥𝑁. (4.32)Case 2. lim_𝑛→∞𝛽_𝑛=0,∑^∞_𝑛=1𝛽_𝑛=∞, and lim_𝑛→∞𝛼_𝑛=0.
From the proof of Theorem 3.1, we note that if (𝐵𝐽)⁻¹0={𝑣}, then {𝜙(𝑥_𝑛,𝑣)} converges to 0, {𝑥_𝑛} converges strongly to 𝑣, and lim_𝑛→∞‖𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛‖=0. By the same argument as in the proof of Case 1, we obtain 𝜙(𝑥_𝑛+1,𝑣)≤𝜏_𝑛𝜙(𝑥,𝑣)+𝜃_𝑛+𝛿_𝑛 for all 𝑛≥𝑁, where 𝜏_𝑛, 𝜃_𝑛 and 𝛿_𝑛 are those of Case 1.
It remains to show that {𝜏_𝑛}, {𝜃_𝑛}, and {𝛿_𝑛} converge to 0. Since 𝛼_𝑛,𝛽_𝑛∈[0,1] and lim_𝑛→∞𝛼_𝑛=0, lim_𝑛→∞𝛽_𝑛=0, it follows that 0≤𝜏_𝑛=𝛽_𝑛+^𝑛−1_𝑖=𝑁𝛽_𝑖^𝑛_𝑗=𝑖+1(1−𝛽_𝑗)𝛼_𝑗=𝛽_𝑛(1−𝛼_𝑛)+𝛽_𝑛𝛼_𝑛+^𝑛−1_𝑖=𝑁𝛽_𝑖^𝑛_𝑗=𝑖+1(1−𝛽_𝑗)𝛼_𝑗≤𝛽_𝑛+𝛼_𝑛(𝛽_𝑛+^𝑛−1_𝑖=𝑁𝛽_𝑖^𝑛_𝑗=𝑖+1(1−𝛽_𝑗))=𝛽_𝑛+𝛼_𝑛(1−^𝑛_𝑖=𝑁(1−𝛽_𝑖))≤𝛽_𝑛+𝛼_𝑛→0,0≤𝜃_𝑛=^𝑛_𝑖=𝑁(1−𝛽_𝑖)𝛼_𝑖≤𝛼_𝑛^𝑛_𝑖=𝑁(1−𝛽_𝑖)→0 (4.33) whenever 𝑛≥𝑁 large enough.
On the other hand, 𝛿_𝑛=𝑙((1−𝛼_𝑛)𝑟_𝑛+^𝑛−1_𝑖=𝑁(1−𝛼_𝑖)𝑟_𝑖^𝑛_𝑗=𝑖+1(1−𝛽_𝑗−1)𝛼_𝑗)≤(1−𝛼_𝑛)𝑙𝑟_𝑛+𝛼_𝑛𝛿_𝑛−1. (4.34) Since lim_𝑛→∞𝛼_𝑛=0, ∑^∞_𝑛=1(1−𝛼_𝑛)=+∞. It follows from Lemma 2.6 that 𝛿_𝑛→0. Moreover, according to the proof of Case 1, we also have that, for all 𝑛≥𝑁, 𝜙(𝑥_𝑛+1,𝑣)≤𝛽_𝑛𝜙(𝑥,𝑣)+(𝛼_𝑛𝑙+∘(𝑟_𝑛))𝑘(𝜙(𝑥_𝑛,𝑣)) (4.35) where 𝑘(𝑡) is the same as that of Case 1. Hence, the conclusion follows.

If 𝛼_𝑛≡0 for all 𝑛∈ℕ, then the algorithm (1.7) reduces to (1.5). Also, letting 𝛼_𝑛=0 in (4.15) we obtain 𝜙(𝑥_𝑛+1,𝑣)≤𝛽_𝑛𝑙𝜙(𝑥,𝑣)+∘(𝑟_𝑛)𝑘(𝜙(𝑥_𝑛,𝑣)),∀𝑛≥𝑁. (4.36)

Corollary 4.3 (see Li and Song [6]). Let 𝐻 be a Hilbert space and let 𝐵⊂𝐻×𝐻 be a maximal monotone with 𝐵⁻¹0={𝑣^∗} and 𝐵⁻¹ is Lipschitz continuous at 0 with modulus 𝑙≥0. Let {𝑥_𝑛} be a sequence generated by 𝑥₁=𝑥∈𝐻 and 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+(1−𝛼_𝑛)𝐽_𝑟𝑛𝑥_𝑛, (4.37) for every 𝑛=1,2,…, where {𝛼_𝑛}⊂[0,1], {𝑟_𝑛}⊂(0,∞) satisfy limsup_𝑛→∞𝛼_𝑛<1, and lim_𝑛→∞𝑟_𝑛=∞. Then the sequence {𝑥_𝑛} converges strongly to 𝑣. Moreover, there exists an integer 𝑁>0 such that ‖𝑥_𝑛+1−𝑣‖≤𝜃_𝑛+𝛿_𝑛∀𝑛≥𝑁, (4.38) where 𝜃_𝑛=∏^𝑛_𝑖=𝑁𝛼_𝑖, 𝛿_𝑛=𝑙((1−𝛼_𝑛)/𝑟_𝑛+∑^𝑛−1_𝑖=𝑁((1−𝛼_𝑖)/𝑟_𝑖)∏^𝑛_𝑗=𝑖+1𝛼_𝑗), and lim_𝑛→∞𝜃_𝑛=lim_𝑛→∞𝛿_𝑛=0. Meanwhile, one obtains the estimate, for all 𝑛≥𝑁, ‖𝑥_𝑛+1−𝑣‖≤𝛼_𝑛𝑙+∘(𝑟_𝑛)²‖𝑥_𝑛−𝑣‖. (4.39)

Proof. Note that, for 𝑥,𝑦∈𝐻, we have 𝐽=𝐼 and 𝜙(𝑥,𝑦)=‖𝑥−𝑦‖². Under our assumptions, the iterative sequence (4.37) reduces to a special case of the algorithm (1.7) where 𝛽_𝑛=0. From the proof of Case 1 in Theorem 4.2, we know that {𝑥_𝑛} converges strongly to 𝑣 and there exists some 𝑁>0 such that (4.20) and (4.21) hold for 𝑛≥𝑁.
Let 𝛽_𝑛=0 in the inequality (4.21). Then, 𝜏_𝑛=0. It follows from (4.21) that
‖𝑥_𝑛+1−𝑣‖≤𝜃_𝑛+𝛿_𝑛,∀𝑛≥𝑁. (4.40) Since 𝛼_𝑛∈[0,1] and limsup_𝑛→∞𝛼_𝑛<1, there exists some 𝛼(0<𝛼<1) such that 0≤𝛼_𝑛≤𝛼 whenever 𝑛≥𝑁. Then, we get 𝜃_𝑛=∏^𝑛_𝑖=𝑁𝛼_𝑖≤𝛼^{𝑛−𝑁+1}→0. Also, 𝛿_𝑛=𝑙((1−𝛼_𝑛)𝑟_𝑛+^𝑛−1_𝑖=𝑁(1−𝛼_𝑖)𝑟_𝑖^𝑛_𝑗=𝑖+1𝛼_𝑗)=(1−𝛼_𝑛)𝑙𝑟_𝑛+𝛼_𝑛𝑙((1−𝛼_𝑛−1)𝑟_𝑛−1+^𝑛−2_𝑖=𝑁(1−𝛼_𝑖)𝑟_𝑖^𝑛−1_𝑗=𝑖+1𝛼_𝑗)=(1−𝛼_𝑛)𝑙𝑟_𝑛+𝛼_𝑛𝛿_𝑛−1. (4.41) Since limsup_𝑛→∞𝛼_𝑛<1, ∑^∞_𝑛=1(1−𝛼_𝑛)=+∞. It follows from Lemma 2.6 that 𝛿_𝑛→0. Additionally, from (4.20), (4.16), and (4.30), we have ‖𝑥_𝑛+1−𝑣‖²≤𝛼_𝑛‖𝑥_𝑛−𝑣‖²+(1−𝛼_𝑛)‖𝐽_𝑟𝑛𝑥_𝑛−𝑣‖²≤𝛼_𝑛‖𝑥_𝑛−𝑣‖²+(1−𝛼_𝑛)(𝑙‖𝐴_𝑟𝑛𝑥_𝑛‖)²=𝛼_𝑛‖𝑥_𝑛−𝑣‖²+(1−𝛼_𝑛)𝑙²‖𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛𝑟_𝑛‖²≤𝛼_𝑛‖𝑥_𝑛−𝑣‖²+(1−𝛼_𝑛𝑙)(𝑟_𝑛)²‖𝑥_𝑛−𝑣‖²≤(𝛼_𝑛𝑙+(𝑟_𝑛)²)‖𝑥_𝑛−𝑣‖², (4.42) for all 𝑛≥𝑁, which implies the equality (4.39).

5. Applications

In this section, we study the problem of finding a minimizer of a proper lower semicontinuous convex function in a Banach space.

Theorem 5.1. Let 𝐸 be a uniformly convex and uniformly smooth Banach space and let 𝑓^∗∶𝐸^∗→(−∞,∞] be a proper lower semicontinuous convex function such that (𝜕𝑓^∗)⁻¹0≠∅. Let {𝑥_𝑛} be a sequence defined as follows: 𝑥₁=𝑥∈𝐸 and 𝑧^∗_𝑛=argmin_{𝑦∗∈𝐸∗}{𝑓^∗(𝑦^∗1)+2𝑟_𝑛‖𝑦^∗‖²−1𝑟_𝑛⟨𝑥_𝑛,𝑦^∗𝑦⟩};_∗𝑛=𝛼_𝑛𝑥_𝑛+(1−𝛼_𝑛)𝐽⁻¹𝑧^∗_𝑛;𝑥_𝑛+1=𝛽_𝑛𝑥+(1−𝛽_𝑛)𝑦^∗_𝑛, (5.1) for every 𝑛=1,2,…, where {𝛼_𝑛}, {𝛽_𝑛}⊂[0,1], and {𝑟_𝑛}⊂(0,∞) satisfy lim_𝑛→∞𝛼_𝑛=0, lim_𝑛→∞𝛽_𝑛=0, ∑^∞_𝑛=1𝛽_𝑛=∞, and lim_𝑛→∞𝑟_𝑛=∞. Then the sequence {𝑥_𝑛} converges strongly to 𝑅_{(𝜕𝑓∗𝐽)}⁻¹₀(𝑥), where 𝑅_{(𝜕𝑓∗𝐽)}⁻¹₀ is a sunny generalized nonexpansive retraction of 𝐸 onto (𝜕𝑓^∗𝐽)⁻¹0.

Proof. By Rockafellar's theorem [19, 20], the subdifferential mapping 𝜕𝑓^∗⊂𝐸^∗×𝐸 is maximal monotone. Let 𝐽_𝑟=(𝐼+𝑟𝜕𝑓𝐽)⁻¹ for all 𝑟>0. As in the proof of [7, Corollary? 5.1], we have 𝐽⁻¹𝑧^∗_𝑛=𝐽_𝑟𝑛𝑥_𝑛 for all 𝑛∈ℕ. Hence, by Theorem 3.1, {𝑥_𝑛} converges strongly to 𝑅_{(𝜕𝑓∗𝐽)}⁻¹₀(𝑥).

When 𝛼_𝑛=0 in Theorem 5.1 we obtain the following corollary.

Corollary 5.2 (see Ibaraki and Takahashi [7]). Let 𝐸 be a uniformly convex and uniformly smooth Banach space and let 𝑓^∗∶𝐸^∗→(−∞,∞] be a proper lower semicontinuous convex function such that (𝜕𝑓^∗)⁻¹0≠∅. Let {𝑥_𝑛} be a sequence defined as follows: 𝑥₁=𝑥∈𝐸 and 𝑦^∗_𝑛=argmin_{𝑦∗∈𝐸∗}{𝑓^∗(𝑦^∗1)+2𝑟_𝑛‖𝑦^∗‖²−1𝑟_𝑛⟨𝑥_𝑛,𝑦^∗𝑥⟩};_𝑛+1=𝛽_𝑛𝑥+(1−𝛽_𝑛)𝐽⁻¹𝑦^∗_𝑛, (5.2) for every 𝑛=1,2,…, where {𝛽_𝑛}⊂[0,1] and {𝑟_𝑛}⊂(0,∞) satisfy lim_𝑛→∞𝛽_𝑛=0,∑^∞_𝑛=1𝛽_𝑛=∞, and lim_𝑛→∞𝑟_𝑛=∞. Then the sequence {𝑥_𝑛} converges strongly to 𝑅_{(𝜕𝑓∗𝐽)}⁻¹₀(𝑥), where 𝑅_{(𝜕𝑓∗𝐽)}⁻¹₀ is a sunny generalized nonexpansive retraction of 𝐸 onto (𝜕𝑓^∗𝐽)⁻¹0.

Acknowledgments

The first author thank the National Research Council of Thailand to Naresuan University, 2009 for the financial support. Moreover, the second author would like to thank the “National Centre of Excellence in Mathematics,” PERDO, under the Commission on Higher Education, Ministry of Education, Thailand.