Convergence Theorems for a Maximal Monotone Operator and a 𝑉-Strongly Nonexpansive Mapping in a Banach Space

Abstract

Let E be a smooth Banach space with a norm ‖⋅‖. Let 𝑉(𝑥,𝑦)=‖𝑥‖²+‖𝑦‖²−2⟨𝑥,𝐽𝑦⟩ for any 𝑥,𝑦∈𝐸, where ⟨⋅,⋅⟩ stands for the duality pair and J is the normalized duality mapping. With respect to this bifunction 𝑉(⋅,⋅), a generalized nonexpansive mapping and a 𝑉-strongly nonexpansive mapping are defined in 𝐸. In this paper, using the properties of generalized nonexpansive mappings, we prove convergence theorems for common zero points of a maximal monotone operator and a 𝑉-strongly nonexpansive mapping.

1. Introduction

Let 𝐸 be a smooth Banach space with a norm ‖⋅‖ and let 𝐶 be a nonempty, closed and convex subset of 𝐸. We use the following bifunction 𝑉(⋅,⋅) studied by Alber [1], as well as Kamimura and Takahashi [2]. Let 𝑉(⋅,⋅)∶𝐸×𝐸→[0,∞) be defined by 𝑉(𝑥,𝑦)=‖𝑥‖²−2⟨𝑥,𝐽𝑦⟩+‖𝑦‖² for any 𝑥,𝑦∈𝐸, where ⟨⋅,⋅⟩ stands for the duality pair and 𝐽 is the normalized duality mapping. Note that the duality mapping is single valued in a smooth Banach space (see [3]). From the definition of 𝑉(⋅,⋅) the following properties are trivial.

Lemma 1.1. (a) For all 𝑎,𝑏,𝑐∈𝐸, 𝑉(𝑎,𝑏)≤𝑉(𝑎,𝑏)+𝑉(𝑏,𝑐)=𝑉(𝑎,𝑐)−2⟨𝑎−𝑏,𝐽𝑏−𝐽𝑐⟩. (1.1)
(b) If a sequence {𝑥_𝑛}⊂𝐸 satisfies lim_𝑛→∞𝑉(𝑥_𝑛,𝑝)<∞ for some 𝑝∈𝐸, then {𝑥_𝑛} is bounded.

Let 𝐹(𝑇) be the fixed points set of 𝑇. Ibaraki and Takahashi defined a generalized nonexpansive mapping in a Banach space (see [4]).

Definition 1.2. A mapping 𝑇∶𝐶→𝐶 is said to be generalized nonexpansive if 𝐹(𝑇)≠∅ and 𝑉(𝑇𝑥,𝑝)≤𝑉(𝑥,𝑝) for all 𝑥∈𝐶 and 𝑝∈𝐹(𝑇).

In this paper, we prove strong convergence theorem for finding common fixed points of a family of generalized nonexpansive mappings. In addition, we prove strong convergence theorem for finding zeroes of a generalized nonexpansive mapping and a maximal monotone operator. Now, we define a 𝑉-strongly nonexpansive mapping as follows.

Definition 1.3. A mapping  𝑇∶𝐶→𝐸  is called 𝑉-strongly nonexpansive if there exists a constant 𝜆>0 such that 𝑉(𝑇𝑥,𝑇𝑦)≤𝑉(𝑥,𝑦)−𝜆𝑉((𝐼−𝑇)𝑥,(𝐼−𝑇)𝑦), (1.2) for all 𝑥,𝑦∈𝐶, where 𝐼 is the identity mapping on 𝐸. More explicitly, if (1.2) holds, then 𝑇 is said to be 𝑉-strongly nonexpansive with 𝜆.

If 𝑇 is 𝑉-strongly nonexpansive with 𝜆, then 𝑇 is 𝑉-strongly nonexpansive with any 𝛾∈(0,𝜆]. It is trivial that a 𝑉-strongly nonexpansive mapping is generalized nonexpansive if 𝐹(𝑇)≠∅. In the following section, we show that in a Hilbert space 𝐻 a firmly nonexpansive mapping is 𝑉-strongly nonexpansive with 𝜆=1 and a 𝑉-strongly nonexpansive mapping is strongly nonexpansive if 𝐹(𝑇)≠∅. Motivated by the results of Manaka and Takahashi [5], we prove weak convergence theorem for common zero points of a maximal monotone operator and a 𝑉-strongly nonexpansive mapping in a Banach space.

2. Preliminaries

Let 𝐷 be a nonempty subset of a Banach space 𝐸. A mapping 𝑅∶𝐸→𝐷 is said to be sunny, if for all 𝑥∈𝐸 and 𝑡≥0, 𝑅(𝑅𝑥+𝑡(𝑥−𝑅𝑥))=𝑅𝑥. (2.1) A mapping 𝑅∶𝐸→𝐷 is called a retraction if 𝑅𝑥=𝑥 for all 𝑥∈𝐷 (see [6]). It is known that a generalized nonexpansive and sunny retraction of 𝐸 onto 𝐷 is uniquely determined if 𝐸 is a smooth and strictly convex Banach space (cf., [7]). Ibaraki and Takahashi proved the following results in [4].

Lemma 2.1 (cf., [4]). Let 𝐸 be a reflexive, strictly convex, and smooth Banach space, and let 𝑇 be a generalized nonexpansive mapping from 𝐸 into itself. Then there exists a sunny and generalized nonexpansive retraction on 𝐹(𝑇).

Lemma 2.2 (cf., [4]). Let 𝐷 be a nonempty subset of a reflexive, strictly convex, and smooth Banach space 𝐸. Let 𝑅 be a retraction from 𝐸 onto 𝐷. Then 𝑅 is sunny and generalized nonexpansive if and only if ⟨𝑥−𝑅𝑥,𝐽𝑅𝑥−𝐽𝑦⟩≥0, (2.2) for all 𝑥∈𝐸 and 𝑦∈𝐷.

A generalized resolvent 𝐽_𝑟 of a maximal monotone operator 𝐵⊂𝐸^∗×𝐸 is defined by 𝐽_𝑟=(𝐼+𝑟𝐵𝐽)⁻¹ for any real number 𝑟>0. It is well known that 𝐽_𝑟∶𝐸→𝐸 is single valued if 𝐸 is reflexive, smooth, and strictly convex (see [8]). It is also known that 𝐽_𝑟 satisfies ⟨𝑥−𝐽_𝑟𝑥−(𝑦−𝐽_𝑟𝑦),𝐽𝐽_𝑟𝑥−𝐽𝐽_𝑟𝑦⟩≥0,∀𝑥,𝑦∈𝐸. (2.3) This implies that ⟨𝑥−𝐽_𝑟𝑥,𝐽𝐽_𝑟𝑥−𝐽𝑝⟩≥0,∀𝑥∈𝐸,𝑝∈𝐹(𝐽_𝑟). (2.4) Therefore, from Lemma 1.1(a), we obtain the following proposition.

Proposition 2.3. (a) If a sunny retraction 𝑅 is generalized nonexpansive, then 𝑅 satisfies 𝑉(𝑥,𝑅𝑥)+𝑉(𝑅𝑥,𝑦)=𝑉(𝑥,𝑦)−2⟨𝑥−𝑅𝑥,𝐽𝑅𝑥−𝐽𝑦⟩≤𝑉(𝑥,𝑦),∀𝑥,𝑦∈𝐷. (2.5)
(b) For each 𝑟>0, a generalized resolvent 𝐽_𝑟 satisfies 𝑉(𝑥,𝐽_𝑟𝑥)+𝑉(𝐽_𝑟𝑥,𝑝)≤𝑉(𝑥,𝑝),∀𝑥∈𝐸,𝑝∈𝐹(𝐽_𝑟). (2.6)

Remark 2.4. The property in Proposition 2.3(b) means that 𝐽_𝑟 is generalized nonexpansive for any 𝑟>0.
We recall some nonlinear mappings in Banach spaces (see, e.g., [9–12]).

Definition 2.5. Let 𝐷 be a nonempty, closed, and convex subset of 𝐸. A mapping 𝑇∶𝐷→𝐸 is said to be firmly nonexpansive if ‖𝑇𝑥−𝑇𝑦‖²≤⟨𝑥−𝑦,𝑗(𝑇𝑥−𝑇𝑦)⟩, (2.7) for all 𝑥,𝑦∈𝐷 and some 𝑗(𝑇𝑥−𝑇𝑦)∈𝐽(𝑇𝑥−𝑇𝑦).

In [12], Reich introduced a class of strongly nonexpansive mappings which is defined with respect to the Bregman distance 𝐷(⋅,⋅) corresponding to a convex continuous function 𝑓 in a reflexive Banach space 𝐸. Let 𝑆 be a convex subset of 𝐸, and let 𝑇∶𝑆→𝑆 be a self-mapping of 𝑆. A point 𝑝 in the closure of 𝑆 is said to be an asymptotically fixed point of 𝑇 if 𝑆 contains a sequence {𝑥_𝑛} which converges weakly to 𝑝 and the sequence {𝑥_𝑛−𝑇𝑥_𝑛} converges strongly to 0. 𝐹(𝑇) denotes the asymptotically fixed points set of 𝑇.

Definition 2.6. The Bregman distance corresponding to a function 𝑓∶𝐸→𝑅 is defined by 𝐷(𝑥,𝑦)=𝑓(𝑥)−𝑓(𝑦)−𝑓^′(𝑦)(𝑥−𝑦), (2.8) where 𝑓 is the Gâteaux differentiable and 𝑓^′(𝑥) stands for the derivative of 𝑓 at the point 𝑥. We say that the mapping 𝑇 is strongly nonexpansive if 𝐹(𝑇)≠∅ and 𝐷(𝑝,𝑇𝑥)≤𝐷(𝑝,𝑥),∀𝑝∈𝐹(𝑇),𝑥∈𝑆, (2.9) and if it holds that lim_𝑛→∞𝐷(𝑇𝑥_𝑛,𝑥_𝑛)=0 for a bounded sequence {𝑥_𝑛} such that lim_𝑛→∞(𝐷(𝑝,𝑥_𝑛)−𝐷(𝑝,𝑇𝑥_𝑛))=0 for any 𝑝∈𝐹(𝑇).

We remark that the symbols 𝑥_𝑛→𝑢 and 𝑥_𝑛⇀𝑢 mean that {𝑥_𝑛} converges strongly and weakly to 𝑢, respectively. Taking the function ‖⋅‖² as the convex, continuous, and Gâteaux differentiable function 𝑓, we obtain the fact that the Bregman distance 𝐷(⋅,⋅) coincides with 𝑉(⋅,⋅). Especially in a Hilbert space, 𝐷(𝑥,𝑦)=𝑉(𝑥,𝑦)=‖𝑥−𝑦‖². Bruck and Reich defined strongly nonexpansive mappings in a Hilbert space 𝐻 as follows (cf., [10]).

Definition 2.7. A mapping 𝑇∶𝐷→𝐻 is said to be strongly nonexpansive if 𝑇 is nonexpansive with 𝐹(𝑇)≠∅ and if it holds that (𝑥_𝑛−𝑦_𝑛)−(𝑇𝑥_𝑛−𝑇𝑦_𝑛)→0 (2.10) when {𝑥_𝑛} and {𝑦_𝑛} are sequences in 𝐷 such that {𝑥_𝑛−𝑦_𝑛} is bounded and lim_𝑛→∞(‖𝑥_𝑛−𝑦_𝑛‖−‖𝑇𝑥_𝑛−𝑇𝑦_𝑛‖)=0.

The relation among firmly nonexpansive mappings, strongly nonexpansive mappings and 𝑉-strongly nonexpansive mappings is shown in the following proposition.

Proposition 2.8. In a Hilbert space 𝐻, the following hold.(a)A firmly nonexpansive mapping is 𝑉-strongly nonexpansive with 𝜆=1.(b)A 𝑉-strongly nonexpansive mapping 𝑇 with 𝐹(𝑇)≠∅ is strongly nonexpansive.

Proof. (a) Suppose that 𝑇 is firmly nonexpansive. Since 𝐽=𝐼 in a Hilbert space, it holds that 2⟨𝑥−𝑦,𝑇𝑥−𝑇𝑦⟩−‖𝑇𝑥−𝑇𝑦‖²=‖𝑥−𝑦‖²−‖(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦‖², (2.11) for all 𝑥,𝑦∈𝐷. Therefore, it is obvious that 𝑇 is firmly nonexpansive if and only if 𝑇 satisfies ‖𝑥−𝑦‖²−‖(𝑇−𝐼)𝑥−(𝑇−𝐼)𝑦‖²≥‖𝑇𝑥−𝑇𝑦‖², (2.12) for all 𝑥,𝑦∈𝐷. Hence we obtain (a).
(b) Suppose that 𝑇 is 𝑉-strongly nonexpansive with 𝜆. Then, it is trivial that 𝑇 is nonexpansive and (2.9) holds. Suppose that the sequences {𝑥_𝑛} and {𝑦_𝑛} satisfy the conditions in Definition 2.7. Then {𝑇𝑥_𝑛−𝑇𝑦_𝑛} is also bounded. Since 𝑇 is 𝑉-strongly nonexpansive with 𝜆, we have that 0≤𝜆‖𝑥_𝑛−𝑦_𝑛−(𝑇𝑥_𝑛−𝑇𝑦_𝑛)‖²=𝜆𝑉((𝐼−𝑇)𝑥_𝑛,(𝐼−𝑇)𝑦_𝑛)≤𝑉(𝑥_𝑛,𝑦_𝑛)−𝑉(𝑇𝑥_𝑛,𝑇𝑦_𝑛)=‖𝑥_𝑛−𝑦_𝑛‖²−‖𝑇𝑥_𝑛−𝑇𝑦_𝑛‖²=(‖𝑥_𝑛−𝑦_𝑛‖+‖𝑇𝑥_𝑛−𝑇𝑦_𝑛‖)(‖𝑥_𝑛−𝑦_𝑛‖−‖𝑇𝑥_𝑛−𝑇𝑦_𝑛‖)→0. (2.13) Hence, (𝑥_𝑛−𝑦_𝑛)−(𝑇𝑥_𝑛−𝑇𝑦_𝑛)→0 for 𝜆>0. This means that 𝑇 is strongly nonexpansive.

In a Banach space, 𝑉-strongly nonexpansive mappings have the following properties.

Proposition 2.9. In a smooth Banach space 𝐸, the following hold.(a)For 𝑐∈(−1,1], 𝑇=𝑐𝐼 is 𝑉-strongly nonexpansive. For 𝑐=1, 𝑇=𝐼 is 𝑉-strongly nonexpansive for any 𝜆>0. For 𝑐∈(−1,1), 𝑇=𝑐𝐼 is 𝑉-strongly nonexpansive for any 𝜆∈(0,(1+𝑐)/(1−𝑐)].(b)If 𝑇 is 𝑉-strongly nonexpansive with 𝜆, then, for any 𝛼∈[−1,1] with 𝛼≠0, 𝛼𝑇 is also 𝑉-strongly nonexpansive with 𝛼²𝜆.(c)If 𝑇 is 𝑉-strongly nonexpansive with 𝜆≥1, then 𝐴=𝐼−𝑇 is 𝑉-strongly nonexpansive with 𝜆⁻¹.(d)Suppose that 𝑇 is 𝑉-strongly nonexpansive with 𝜆 and that 𝛼∈[−1,1] satisfies 𝛼²𝜆≥1. Then (𝐼−𝛼𝑇) is 𝑉-strongly nonexpansive with (𝛼²𝜆)⁻¹. Moreover, if 𝑇_𝛼=𝐼−𝛼𝑇, then 𝑉(𝑇_𝛼𝑥,𝑇_𝛼𝑦)≤𝑉(𝑥,𝑦)−𝜆⁻¹𝑉(𝑇𝑥,𝑇𝑦). (2.14)

Proof. (a) Let 𝑇=𝑐𝐼 for any 𝑐∈(−1,1], and denote 𝐼_𝑙=𝑉(𝑇𝑥,𝑇𝑦) and 𝐼_𝑟=𝑉(𝑥,𝑦)−𝜆𝑉((𝐼−𝑇)𝑥,(𝐼−𝑇)𝑦). Since 𝐽(𝑐𝑥)=𝑐𝐽𝑥, we have 𝐼_𝑙=𝑉(𝑇𝑥,𝑇𝑦)=𝑐²{‖𝑥‖²+‖𝑦‖²−2⟨𝑥,𝐽𝑦⟩}=𝑐²𝐼𝑉(𝑥,𝑦),_𝑟=𝑉(𝑥,𝑦)−𝜆𝑉((𝐼−𝑇)𝑥,(𝐼−𝑇)𝑦)=‖𝑥‖²−𝜆‖(1−𝑐)𝑥‖²+‖𝑦‖²−𝜆‖(1−𝑐)𝑦‖²−2⟨𝑥,𝐽𝑦⟩+2𝜆⟨(1−𝑐)𝑥,𝐽((1−𝑐)𝑦)⟩={1−𝜆(1−𝑐)²}(‖𝑥‖²+‖𝑦‖²)−2{1−𝜆(1−𝑐)²}⟨𝑥,𝐽𝑦⟩={1−𝜆(1−𝑐)²}(‖𝑥‖²+‖𝑦‖²−2⟨𝑥,𝐽𝑦⟩)={1−𝜆(1−𝑐)²}𝑉(𝑥,𝑦). (2.15) For 𝑐=1, it holds that 𝐼_𝑙≤I_𝑟 for all 𝜆>0. For 𝑐∈(−1,1), we obtain 𝐼_𝑙≤𝐼_𝑟⇔𝑐²≤1−𝜆(1−𝑐)²⇔0<𝜆(1−𝑐)²≤1−𝑐²⇔0<𝜆≤(1−𝑐)(1+𝑐)(1−𝑐)²=(1+𝑐).(1−𝑐) (2.16) Therefore, 𝑇=𝑐𝐼 is 𝑉-strongly nonexpansive for any 𝜆∈(0,(1+𝑐)/(1−𝑐)].
(b) If 𝑇 is 𝑉-strongly nonexpansive with 𝜆>0, then, for 𝛼∈[−1,1] with 𝛼≠0, 𝑉(𝛼𝑇𝑥,𝛼𝑇𝑦)=‖𝛼𝑇𝑥‖²+‖𝛼𝑇𝑦‖²−2⟨𝛼𝑇𝑥,𝐽(𝛼𝑇𝑦)⟩=𝛼²{‖𝑇𝑥‖²+‖𝑇𝑦‖²−2⟨𝑇𝑥,𝐽(𝑇𝑦)⟩}=𝛼²𝑉(𝑇𝑥,𝑇𝑦)≤𝛼²{𝑉(𝑥,𝑦)−𝜆𝑉((𝐼−𝑇)𝑥,(𝐼−𝑇)𝑦)}=𝑉(𝑥,𝑦)−(1−𝛼²)𝑉(𝑥,𝑦)−𝛼²𝜆𝑉((𝐼−𝑇)𝑥,(𝐼−𝑇)𝑦)≤𝑉(𝑥,𝑦)−𝛼²𝜆𝑉((𝐼−𝑇)𝑥,(𝐼−𝑇)𝑦). (2.17) This means that 𝛼𝑇 is 𝑉-strongly nonexpansive with 𝛼²𝜆.
(c) Suppose that 𝑇 is 𝑉-strongly nonexpansive with 𝜆≥1 and let 𝐴=𝐼−𝑇. Then we have that 𝑉((𝐼−𝐴)𝑥,(𝐼−𝐴)𝑦)=𝑉(𝑇𝑥,𝑇𝑦)≤𝑉(𝑥,𝑦)−𝜆𝑉((𝐼−𝑇)𝑥,(𝐼−𝑇)𝑦)=𝑉(𝑥,𝑦)−𝜆𝑉(𝐴𝑥,𝐴𝑦). (2.18) This inequality implies that 𝑉(𝐴𝑥,𝐴𝑦)≤𝜆⁻¹{𝑉(𝑥,𝑦)−𝑉((𝐼−𝐴)𝑥,(𝐼−𝐴)𝑦)}=𝜆⁻¹𝑉(𝑥,𝑦)−𝜆⁻¹𝑉((𝐼−𝐴)𝑥,(𝐼−𝐴)𝑦)≤𝑉(𝑥,𝑦)−𝜆⁻¹𝑉((𝐼−𝐴)𝑥,(𝐼−𝐴)𝑦). (2.19) Thus 𝐴 is 𝑉-strongly nonexpansive with 𝜆⁻¹.
(d) From (b) and the assumption, 𝛼𝑇 is 𝑉-strongly nonexpansive with 𝛼²𝜆≥1, and from (c) we have that (𝐼−𝛼𝑇) is 𝑉-strongly nonexpansive with (𝛼²𝜆)⁻¹. Furthermore we obtain that 𝑉(𝑇_𝛼𝑥,𝑇_𝛼𝑦)≤𝑉(𝑥,𝑦)−(𝛼²𝜆)⁻¹𝑉((𝐼−𝑇_𝛼)𝑥,(𝐼−𝑇_𝛼)𝑦)=𝑉(𝑥,𝑦)−(𝛼²𝜆)⁻¹𝑉(𝛼𝑇𝑥,𝛼𝑇𝑦)=𝑉(𝑥,𝑦)−(𝛼²𝜆)⁻¹𝛼²𝑉(𝑇𝑥,𝑇𝑦)=𝑉(𝑥,𝑦)−𝜆⁻¹𝑉(𝑇𝑥,𝑇𝑦). (2.20) This completes the proof.

In Banach spaces, we have the following example of 𝑉-strongly nonexpansive mappings.

Example 2.10. Let 𝐸=𝐑×𝐑  be a Banach space with a norm ‖⋅‖ defined by ‖𝑥‖=|𝑥₁|+|𝑥₂|,∀𝑥=(𝑥₁,𝑥₂)∈𝐸. (2.21) The normalized duality mapping 𝐽 is given by 1𝐽𝑥=‖𝑥‖(|𝑥₁|𝑥₁,1|𝑥₂|𝑥₂),∀𝑥∈𝐸. (2.22) Hence, we have for 𝑥,𝑦∈𝐸 that 𝑉(𝑥,𝑦)=‖𝑥‖²+‖𝑦‖²−2⟨𝑥,𝐽𝑦⟩=‖𝑥‖²+‖𝑦‖²𝑥−2‖𝑦‖{¹𝑦₁|𝑦₁|+𝑥₂𝑦₂|𝑦₂|}. (2.23) We define a mapping 𝑇∶𝐸→𝐸 as follows: 𝑥𝑇𝑥=if1‖𝑥‖≤1,𝑥‖𝑥‖if‖𝑥‖≥1. (2.24) We will show that this mapping is 𝑉-strongly nonexpansive for any 𝜆≤1.
(a) Suppose that 𝑥,𝑦∈𝐸 with ‖𝑥‖≤1 and ‖𝑦‖≥1. Then, we have 𝑉(𝑇𝑥,𝑇𝑦)=𝑉(𝑥,𝑇𝑦)=‖𝑥‖²+‖𝑇𝑦‖²𝑥−2‖𝑇𝑦‖{¹(𝑇𝑦)₁|(𝑇𝑦)₁|+𝑥₂(𝑇𝑦)₂|(𝑇𝑦)₂|}=‖𝑥‖²𝑥+1−2{¹𝑦₁|𝑦₁|+𝑥₂𝑦₂|𝑦₂|}. (2.25) Since 𝑦−𝑇𝑦=(‖𝑦‖−1𝑦‖𝑦‖¹,‖𝑦‖−1𝑦‖𝑦‖²), (2.26) we have that 𝑉(𝑥−𝑇𝑥,𝑦−𝑇𝑦)=𝑉(0,𝑦−𝑇𝑦)=‖𝑦−𝑇𝑦‖²={(‖𝑦‖−1)‖𝑦‖(|𝑦₁|+|𝑦₂|)}²=(‖𝑦‖−1)². (2.27) Hence, we obtain that 𝑉(𝑥,𝑦)−𝑉(𝑇𝑥,𝑇𝑦)−𝜆𝑉(𝑥−𝑇𝑥,𝑦−𝑇𝑦)=‖𝑥‖²+‖𝑦‖²𝑥−2‖𝑦‖{¹𝑦₁|𝑦₁|+𝑥₂𝑦₂|𝑦₂|}−‖𝑥‖²𝑥−1+2{¹𝑦₁|𝑦₁|+𝑥₂𝑦₂|𝑦₂|}−𝜆(‖𝑦‖−1)²=‖𝑦‖²𝑥−1−2(‖𝑦‖−1){¹𝑦₁|𝑦₁|+𝑥₂𝑦₂|𝑦₂|}−𝜆(‖𝑦‖−1)²≥(‖𝑦‖−1)(‖𝑦‖+1)−2(‖𝑦‖−1){|𝑥₁𝑦₁||𝑦₁|+|𝑥₂𝑦₂||𝑦₂|}−𝜆(‖𝑦‖−1)²=(‖𝑦‖−1){‖𝑦‖+1−2‖𝑥‖−𝜆‖𝑦‖+𝜆}≥(‖𝑦‖−1){(1−𝜆)‖𝑦‖+1−2+𝜆}=(‖𝑦‖−1){(1−𝜆)(‖𝑦‖−1)}=(1−𝜆)(‖𝑦‖−1)²≥0, (2.28) for any 𝜆∈[0,1]. This means that 𝑇 is 𝑉-strongly nonexpansive for any 𝜆∈[0,1].
(b) Suppose that 𝑥,𝑦∈𝐸 with ‖𝑥‖≥1 and ‖𝑦‖≤1. Then, we have 𝑉(𝑇𝑥,𝑇𝑦)=𝑉(𝑇𝑥,𝑦)=1+‖𝑦‖²−2‖𝑦‖{𝑥‖𝑥‖¹𝑦₁|𝑦₁|+𝑥₂𝑦₂|𝑦₂|},𝑉(𝑥−𝑇𝑥,𝑦−𝑇𝑦)=𝑉((‖𝑥‖−1)‖𝑥‖𝑥,0)=(‖𝑥‖−1)². (2.29) Hence, we have that 𝑉(𝑥,𝑦)−𝑉(𝑇𝑥,𝑇𝑦)−𝜆𝑉(𝑥−𝑇𝑥,𝑦−𝑇𝑦)=‖𝑥‖²+‖𝑦‖²𝑥−2‖𝑦‖{¹𝑦₁|𝑦₁|+𝑥₂𝑦₂|𝑦₂|}−1−‖𝑦‖²+2‖𝑦‖{𝑥‖𝑥‖¹𝑦₁|𝑦₁|+𝑥₂𝑦₂|𝑦₂|}−𝜆(‖𝑥‖−1)²=‖𝑥‖²1−1−2‖𝑦‖(1−𝑥‖𝑥‖){¹𝑦₁|𝑦₁|+𝑥₂𝑦₂|𝑦₂|}≥(‖𝑥‖−1)(‖𝑥‖+1−𝜆‖𝑥‖+𝜆)−2‖𝑦‖(‖𝑥‖−1‖𝑥‖)‖𝑥‖=(‖𝑥‖−1){(1−𝜆)‖𝑥‖+1+𝜆−2‖𝑦‖}≥(‖𝑥‖−1){(1−𝜆)‖𝑥‖+1+𝜆−2}=(1−𝜆)(‖𝑥‖−1)²≥0, (2.30) for any 𝜆∈[0,1]. This means that 𝑇 is 𝑉-strongly nonexpansive for any 𝜆∈[0,1].
(c) Suppose that 𝑥,𝑦∈𝐸 with ‖𝑥‖,‖𝑦‖≥1. Then, we have 2𝑉(𝑇𝑥,𝑇𝑦)=1+1−{𝑥‖𝑥‖¹𝑦₁|𝑦₁|+𝑥₂𝑦₂|𝑦₂|}2=2−{𝑥‖𝑥‖¹𝑦₁|𝑦₁|+𝑥₂𝑦₂|𝑦₂|},𝑉(𝑥−𝑇𝑥,𝑦−𝑇𝑦)=(‖𝑥‖−1)²+(‖𝑦‖−1)²−2(‖𝑦‖−1)(‖𝑥‖−1){𝑥‖𝑥‖¹𝑦₁|𝑦₁|+𝑥₂𝑦₂|𝑦₂|}. (2.31) Hence, we have that 𝑉(𝑥,𝑦)−𝑉(𝑇𝑥,𝑇𝑦)−𝜆𝑉(𝑥−𝑇𝑥,𝑦−𝑇𝑦)=‖𝑥‖²+‖𝑦‖²𝑥−2‖𝑦‖{¹𝑦₁|𝑦₁|+𝑥₂𝑦₂|𝑦₂|2}−2+{𝑥‖𝑥‖¹𝑦₁|𝑦₁|+𝑥₂𝑦₂|𝑦₂|}−𝜆{(‖𝑥‖−1)²+(‖𝑦‖−1)²}+2𝜆(‖𝑦‖−1)(‖𝑥‖−1){𝑥‖𝑥‖¹𝑦₁|𝑦₁|+𝑥₂𝑦₂|𝑦₂|}=(‖𝑥‖−1)(‖𝑥‖+1)+(‖𝑦‖−1)(‖𝑦‖+1)−𝜆(‖𝑥‖−1)²−𝜆(‖𝑦‖−1)²−2{𝑥‖𝑥‖¹𝑦₁|𝑦₁|+𝑥₂𝑦₂|𝑦₂|−2}{‖𝑥‖‖𝑦‖−1−𝜆(‖𝑦‖−1)(‖𝑥‖−1)}=(‖𝑥‖−1){‖𝑥‖+1−𝜆(‖𝑥‖−1)}+(‖𝑦‖−1){‖𝑦‖+1−𝜆(‖𝑦‖−1)}{𝑥‖𝑥‖¹𝑦₁|𝑦₁|+𝑥₂𝑦₂|𝑦₂|}{‖𝑥‖‖𝑦‖−1−𝜆(‖𝑥‖−1)(‖𝑦‖−1)}. (2.32) Now, we note that ‖𝑥‖‖𝑦‖−1−𝜆(‖𝑥‖−1)(‖𝑦‖−1)≥0, (2.33) for any ‖𝑥‖,‖𝑦‖≥1 and for any 𝜆∈[0,1]. Therefore, we obtain that −2𝑉(𝑥,𝑦)−𝑉(𝑇𝑥,𝑇𝑦)−𝜆𝑉(𝑥−𝑇𝑥,𝑦−𝑇𝑦)≥(‖𝑥‖−1){‖𝑥‖+1−𝜆(‖𝑥‖−1)}+(‖𝑦‖−1){‖𝑦‖+1−𝜆(‖𝑦‖−1)}‖𝑥‖‖𝑥‖{‖𝑥‖‖𝑦‖−1−𝜆(‖𝑥‖−1)(‖𝑦‖−1)}=(1−𝜆)(‖𝑥‖−‖𝑦‖)²≥0, (2.34) for any 𝜆∈[0,1]. This means that 𝑇 is 𝑉-strongly nonexpansive for any 𝜆∈[0,1].
It is clear that if ‖𝑥‖,‖𝑦‖≤1 then 𝑇 is 𝑉-strongly nonexpansive; therefore, from (a), (b), and (c), we obtain the conclusion that 𝑇 is 𝑉-strongly nonexpansive with 𝜆≤1.

Next, we present some lemmas which are used in the proofs of our theorems. Let ℕ be the set of natural numbers.

Lemma 2.11. Let {𝑎_𝑛} and {𝑡_𝑛} be sequences of nonnegative real numbers and satisfy the inequality 𝑎_𝑛+1≤(1−𝑡_𝑛)𝑎_𝑛+𝑡_𝑛𝑀 for any 𝑛∈ℕ and a constant 𝑀>0. If ∑_𝑛∈ℕ𝑡_𝑛<∞, then lim_𝑛→∞𝑎_𝑛 exists.

Kamimura and Takahashi showed the following useful lemmas (see [2]).

Lemma 2.12. Let 𝐸 be a smooth and uniformly convex Banach space. Then, there exists a continuous, strictly increasing, and convex function 𝑔∶[0,∞)→[0,∞) such that 𝑔(0)=0, and for each real number 𝑟>0, 0≤𝑔(‖𝑥−𝑦‖)≤𝑉(𝑥,𝑦), (2.35) for all 𝑥,𝑦∈𝐵_𝑟={𝑧∈𝐸∶‖𝑧‖≤𝑟}.

From this lemma, it is obvious that the following lemma holds.

Lemma 2.13. Let 𝐸 be a smooth and uniformly convex Banach space and let {𝑥_𝑛} and {𝑦_𝑛} be sequences in 𝐸 such that either {𝑥_𝑛} or {𝑦_𝑛} is bounded. If lim_𝑛→∞𝑉(𝑥_𝑛,𝑦_𝑛)=0, then lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0.

We present the following lemma which plays an important role in our theorems (cf. Butnariu and Resmerita [13] ).

Lemma 2.14. Let 𝐸 be a smooth and uniformly convex Banach space and 𝐶 a nonempty, convex, and closed subset of 𝐸. Suppose that 𝑇∶𝐶→𝐸 satisfies 𝑉(𝑇𝑥,𝑇𝑦)≤𝑉(𝑥,𝑦),∀𝑥,𝑦∈𝐶. (2.36) If a weakly convergent sequence {𝑧_𝑛}_𝑛∈ℕ⊂𝐶 satisfies that lim_𝑛→∞𝑉(𝑇𝑧_𝑛,𝑧_𝑛)=0, then 𝑧_𝑛⇀𝑧∈𝐹(𝑇).

3. Main Results

In this section, we prove three strong convergence theorems. In the first result, we prove strong convergence theorem for finding common fixed points of a family of generalized nonexpansive mappings. In the next result, we prove strong convergence theorem for finding zeroes of a generalized nonexpansive mapping and a maximal monotone operator. In the last result, we prove weak convergence theorem for finding zeroes of a maximal monotone operator and a 𝑉-strongly nonexpansive mapping. As consequence, we prove convergence theorem for common zeroes of a maximal monotone operator and a firmly nonexpansive mapping in a Hilbert space.

Theorem 3.1. Let 𝐸 be a reflexive, smooth, and strictly convex Banach space, and let {𝑇_𝑛}_𝑛∈ℕ be a family of generalized nonexpansive mappings. Suppose that ⋂_𝑛∈ℕ𝐹(𝑇_𝑛)=𝐹≠∅ and that 𝑅 is a sunny and generalized nonexpansive retraction from 𝐸 to 𝐹. Let a sequence {𝑥_𝑛} be defined as follows. For any 𝑥₁=𝑥∈𝐸, 𝑥_𝑛+1=𝑅𝑇_𝑛𝑥_𝑛,forany𝑛∈ℕ. (3.1) Then, {𝑥_𝑛} converges strongly to a point 𝑥^∗ in 𝐹.

Proof. Since 𝑅𝑥_𝑛 is a point in 𝐹 for all 𝑛∈ℕ, from Proposition 2.3(a), we have for all 𝑛∈ℕ that 0≤𝑉(𝑥_𝑛+1,𝑅𝑥_𝑛+1)≤𝑉(𝑥_𝑛+1,𝑅𝑥_𝑛+1)+𝑉(𝑅𝑥_𝑛+1,𝑅𝑥_𝑛)≤𝑉(𝑥_𝑛+1,𝑅𝑥_𝑛)=𝑉(𝑅𝑇_𝑛𝑥_𝑛,𝑅𝑥_𝑛). (3.2) Since 𝑅 and 𝑇_𝑛 are generalized nonexpansive, we get that 𝑉(𝑅𝑇_𝑛𝑥_𝑛,𝑅𝑥_𝑛)≤𝑉(𝑇_𝑛𝑥_𝑛,𝑅𝑥_𝑛)≤𝑉(𝑥_𝑛,𝑅𝑥_𝑛). (3.3) Hence, we have that 0≤𝑉(𝑥_𝑛+1,𝑅𝑥_𝑛+1)≤𝑉(𝑥_𝑛,𝑅𝑥_𝑛),∀𝑛∈ℕ, (3.4) and therefore, lim_𝑛→∞𝑉(𝑥_𝑛,𝑅𝑥_𝑛)<∞. Furthermore, Proposition 2.3(a) implies that 𝑉(𝑥_𝑛+𝑘,𝑅𝑥_𝑛+𝑘)+𝑉(𝑅𝑥_𝑛+𝑘,𝑅𝑥_𝑛)≤𝑉(𝑥_𝑛+𝑘,𝑅𝑥_𝑛). (3.5) This is equivalent to 𝑉(𝑅𝑥_𝑛+𝑘,𝑅𝑥_𝑛)≤𝑉(𝑥_𝑛+𝑘,𝑅𝑥_𝑛)−𝑉(𝑥_𝑛+𝑘,𝑅𝑥_𝑛+𝑘). (3.6) Setting 𝑚=𝑛+𝑘 for all 𝑛,𝑘∈ℕ, then we have that 𝑉(𝑅𝑥_𝑚,𝑅𝑥_𝑛)≤𝑉(𝑥_𝑚,𝑅𝑥_𝑛)−𝑉(𝑥_𝑚,𝑅𝑥_𝑚)≤𝑉(𝑅𝑇_𝑚−1𝑥_𝑚−1,𝑅𝑥_𝑛)−𝑉(𝑥_𝑚,𝑅𝑥_𝑚)≤𝑉(𝑇_𝑚−1𝑥_𝑚−1,𝑅𝑥_𝑛)−𝑉(𝑥_𝑚,𝑅𝑥_𝑚)≤𝑉(𝑥_𝑚−1,𝑅𝑥_𝑛)−𝑉(𝑥_𝑚,𝑅𝑥_𝑚)≤⋯≤𝑉(𝑥_𝑛,𝑅𝑥_𝑛)−𝑉(𝑥_𝑚,𝑅𝑥_𝑚)→0,as𝑛,𝑚→∞. (3.7) Since 𝑉(𝑥_𝑛+1,𝑝)=𝑉(𝑅𝑇_𝑛𝑥_𝑛,𝑝)≤𝑉(𝑥,𝑝) for any 𝑝∈𝐹, Lemma 1.1(b) implies that {𝑥_𝑛} is bounded. Thus, from Lemma 2.12, we can take the continuous and strictly increasing function 𝑔∶[0,∞)→[0,∞) with 𝑔(0)=0 such that 𝑔(‖𝑅𝑥_𝑚−R𝑥_𝑛‖)≤𝑉(𝑅𝑥_𝑚,𝑅𝑥_𝑛)≤𝑉(𝑥_𝑛,𝑅𝑥_𝑛)−𝑉(𝑥_𝑚,𝑅𝑥_𝑚)→0,as𝑛,𝑚→∞. (3.8) Since 𝑅𝑥_𝑛=𝑥_𝑛 for all 𝑛≥1, we have 𝑔(‖𝑥_𝑚−𝑥_𝑛‖)=𝑔(‖𝑅𝑥_𝑚−𝑅𝑥_𝑛‖)→0. Therefore, {𝑥_𝑛} is a Cauchy sequence. Since 𝐸 is complete and 𝐹 is closed, this sequence {𝑥_𝑛} converges strongly to point 𝑥^∗∈𝐹.

Noting that the generalized resolvent 𝐽_𝑟=(𝐼+𝑟𝐵𝐽)⁻¹ of a maximal monotone operator 𝐵 for 𝑟>0 is a generalized nonexpansive mapping (see Remark 2.4), we obtain the following result.

Theorem 3.2. Let 𝐸 be a reflexive, smooth, and strictly convex Banach space. Let 𝑇∶𝐸→𝐸 be generalized nonexpansive and let 𝐵⊂𝐸^∗×𝐸 be a maximal monotone operator. Suppose that 𝐹(𝑇)∩(𝐵𝐽)⁻¹(0)≠∅ and that 𝑅 is a sunny and generalized nonexpansive retraction from 𝐸 to 𝐹=𝐹(𝑇)∩(𝐵𝐽)⁻¹(0). Let an iterative sequence {𝑥_𝑛} be defined as follows: for any 𝑥=𝑥₁∈𝐸, 𝑥_𝑛+1=𝑅𝑇𝐽_𝑟𝑛𝑥_𝑛,∀𝑛∈ℕ, (3.9) where {𝑟_𝑛} is a sequence of nonnegative real numbers. Then, the sequence {𝑥_𝑛} converges strongly to a point 𝑥^∗ in 𝐹(𝑇)∩(𝐵𝐽)⁻¹(0).

Proof. From Propositions 2.3(a) and 2.3(b), we have for all 𝑛∈𝑁 that 𝑉(𝑥_𝑛+1,𝑅𝑥_𝑛+1)≤𝑉(𝑥_𝑛+1,𝑅𝑥_𝑛+1)+𝑉(𝑅𝑥_𝑛+1,𝑅𝑥_𝑛)≤𝑉(𝑥_𝑛+1,𝑅𝑥_𝑛)=𝑉(𝑅𝑇𝐽_𝑟𝑛𝑥_𝑛,𝑅𝑥_𝑛)≤𝑉(𝑇𝐽_𝑟𝑛𝑥_𝑛,𝑅𝑥_𝑛)≤𝑉(𝐽_𝑟𝑛𝑥_𝑛,𝑅𝑥_𝑛)≤𝑉(𝑥_𝑛,𝑅𝑥_𝑛). (3.10) Thus lim_𝑛→∞𝑉(𝑥_𝑛,𝑅𝑥_𝑛)<∞. Similarly, as in the proof of the previous theorem, we show that {𝑥_𝑛} is a Cauchy sequence, and we obtain that {𝑥_𝑛} converges strongly to point 𝑥^∗ in 𝐹=𝐹(𝑇)∩(𝐵𝐽)⁻¹(0).

The duality mapping 𝐽 of a Banach space 𝐸 with the Gâteaux differentiable norm is said to be weakly sequentially continuous if 𝑥_𝑛⇀𝑥 in 𝐸 implies that {𝐽𝑥_𝑛} converges weak star to 𝐽𝑥 in 𝐸^∗ (cf., [14]). This happens, for example, if 𝐸 is a Hilbert space, finite dimensional and smooth, or 𝑙^𝑝 if 1<𝑝<∞ (cf., [15]). Next, we prove the main theorem.

Theorem 3.3. Let 𝐸 be a reflexive, smooth and strictly convex Banach space. Suppose that the duality mapping 𝐽 of 𝐸 is weakly sequentially continuous. Let 𝐶 be a nonempty, closed, and convex subset of 𝐸. Let 𝐵∶𝐸^∗→2^𝐸 be a maximal monotone operator and let 𝐽_𝑟𝑛=(𝐼+𝑟_𝑛𝐵𝐽)⁻¹ be a generalized resolvent of 𝐵 for a sequence {𝑟_𝑛}⊂(0,∞). Suppose that 𝐴∶𝐶→𝐸 is a 𝑉-strongly nonexpansive mapping with 𝜆≥1 such that 𝐶₀=𝐴⁻¹(0)∩(𝐵𝐽)⁻¹(0)≠∅ and that 𝑅_𝐶∶𝐸→𝐶 is a sunny and generalized nonexpansive retraction. For an 𝛼∈[−1,1] such that 𝛼²𝜆≥1, let an iterative sequence {𝑥_𝑛}⊂𝐶 be defined as follows: for any 𝑥=𝑥₁∈𝐶 and 𝑛∈ℕ, 𝑦_𝑛=𝑅_𝐶(𝐼−𝛼𝐴)𝑥_𝑛,𝑥_𝑛+1=𝑅_𝐶(𝛽_𝑛𝑥+(1−𝛽_𝑛)𝐽_𝑟𝑛𝑦_𝑛), (3.11) where {𝛽_𝑛}⊂[0,1] and {𝑟_𝑛}⊂(0,∞) satisfy that _𝑛≥1𝛽_𝑛<∞,liminf_𝑛→∞𝑟_𝑛>0. (3.12) Then, there exists an element 𝑢∈𝐶₀ such that 𝑥_𝑛⇀𝑢,𝑅_𝐶0(𝑥_𝑛)→𝑢. (3.13)

Proof. For simplicity, we denote 𝑅_𝐶 and 𝑅_𝐶0 by 𝑅 and 𝑅₀, respectively. Let 𝑧_𝑛=𝛽_𝑛𝑥+(1−𝛽_𝑛)𝐽_𝑟𝑛𝑦_𝑛 for all 𝑛∈ℕ. Since 𝑅 is generalized nonexpansive, we have for any 𝑝∈𝐶₀ and all 𝑛∈ℕ that 𝑉(𝑥_𝑛+1,𝑝)=𝑉(𝑅𝑧_𝑛,𝑝)≤𝑉(𝑧_𝑛,𝑝). (3.14) The convexity of ‖⋅‖² implies that 𝑉(𝑧_𝑛,𝑝)=𝑉(𝛽_𝑛𝑥+(1−𝛽_𝑛)𝐽_𝑟𝑛𝑦_𝑛,𝑝)=‖𝛽_𝑛𝑥+(1−𝛽_𝑛)𝐽_𝑟𝑛𝑦_𝑛‖²+‖𝑝‖²−2⟨𝛽_𝑛𝑥+(1−𝛽_𝑛)𝐽_𝑟𝑛𝑦_𝑛,𝐽𝑝⟩≤𝛽_𝑛‖𝑥‖²+(1−𝛽_𝑛)‖𝐽_𝑟𝑛𝑦_𝑛‖²+‖𝑝‖²−2𝛽_𝑛⟨𝑥,𝐽𝑝⟩−2(1−𝛽_𝑛)⟨𝐽_𝑟𝑛𝑦_𝑛,𝐽𝑝⟩=𝛽_𝑛{‖𝑥‖²−2⟨𝑥,𝐽𝑝⟩+‖𝑝‖²}+(1−𝛽_𝑛){‖𝐽_𝑟𝑛𝑦_𝑛‖²−2⟨𝐽_𝑟𝑛𝑦_𝑛,𝐽𝑝⟩+‖𝑝‖²}=𝛽_𝑛𝑉(𝑥,𝑝)+(1−𝛽_𝑛)𝑉(𝐽_𝑟𝑛𝑦_𝑛,𝑝). (3.15) Thus, we obtain that 𝑉(𝑧_𝑛,𝑝)≤𝛽_𝑛𝑉(𝑥,𝑝)+(1−𝛽_𝑛)𝑉(𝐽_𝑟𝑛𝑦_𝑛,𝑝), (3.16) and furthermore, since 𝐽_𝑟𝑛 is generalized nonexpansive, we have that 𝑉(𝑧_𝑛,𝑝)≤𝛽_𝑛𝑉(𝑥,𝑝)+(1−𝛽_𝑛)𝑉(𝑦_𝑛,𝑝). (3.17)
Let 𝐴_𝛼=(𝐼−𝛼𝐴). Then, from Proposition 2.9(d), 𝐴_𝛼 is 𝑉-strongly nonexpansive with 𝛼²𝜆 and 𝐴_𝛼 is also generalized nonexpansive. Hence, we have that 𝑉(𝑦_𝑛,𝑝)=𝑉(𝑅𝐴_𝛼𝑥_𝑛,𝑝)≤𝑉(𝐴_𝛼𝑥_𝑛,𝑝)≤𝑉(𝑥_𝑛,𝑝). (3.18) Thus, we have from (3.14), (3.16), (3.17), and (3.18) that 𝑉(𝑥_𝑛+1,𝑝)≤𝑉(𝑧_𝑛,𝑝)≤𝛽_𝑛𝑉(𝑥,𝑝)+(1−𝛽_𝑛)𝑉(𝐽_𝑟𝑛𝑦_𝑛,𝑝)≤𝛽_𝑛𝑉(𝑥,𝑝)+(1−𝛽_𝑛)𝑉(𝑦_𝑛,𝑝)≤𝛽_𝑛𝑉(𝑥,𝑝)+(1−𝛽_𝑛)𝑉(𝐴_𝛼𝑥_𝑛,𝑝)≤𝛽_𝑛𝑉(𝑥,𝑝)+(1−𝛽_𝑛)𝑉(𝑥_𝑛,𝑝). (3.19) From Lemma 2.11, there exists 𝛼=lim_𝑛→∞𝑉(𝑥_𝑛,𝑝)<∞. Since lim_𝑛→∞𝛽_𝑛=0, we have that 𝛼=lim_𝑛→∞𝑉(𝑥_𝑛,𝑝)=lim_𝑛→∞𝑉(𝑧_𝑛,𝑝),=lim_𝑛→∞𝑉(𝐽_𝑟𝑛𝑦_𝑛,𝑝)=lim_𝑛→∞𝑉(𝑦_𝑛,𝑝)=lim_𝑛→∞𝑉(𝐴_𝛼𝑥_𝑛,𝑝). (3.20)(3.21) Hence, {𝑥_𝑛}, {𝑧_𝑛}, {𝐽_𝑟𝑛𝑦_𝑛}, {𝑦_𝑛}, and {𝐴_𝛼𝑥_𝑛} are bounded from Lemma 1.1(b). Since 𝐸 is uniformly convex, the boundedness of {𝑥_𝑛} implies that there exists a subsequence {𝑥_𝑛𝑗}⊂{𝑥_𝑛} such that 𝑥_𝑛𝑗⇀𝑢∈𝐶. Moreover, we can take the index sequence {𝑛_𝑗}_𝑗≥1 satisfies lim_𝑗→∞𝑟_𝑛𝑗−1>0. We will show that 𝑢∈(𝐵𝐽)⁻¹(0). From Proposition 2.3(a), 𝑉(𝑥_𝑛+1,𝑝)=𝑉(𝑅𝑧_𝑛,𝑝)≤𝑉(𝑧_𝑛,𝑅𝑧_𝑛)+𝑉(𝑅𝑧_𝑛,𝑝)≤𝑉(𝑧_𝑛,𝑝), (3.22) and furthermore, from (3.16) and Proposition 2.3(b), we obtain that 𝑉(𝑧_𝑛,𝑝)≤𝛽_𝑛𝑉(𝑥,𝑝)+(1−𝛽_𝑛)𝑉(𝐽_𝑟𝑛𝑦_𝑛,𝑝)≤𝛽_𝑛𝑉(𝑥,𝑝)+(1−𝛽_𝑛){𝑉(𝑦_𝑛,𝐽_𝑟𝑛𝑦_𝑛)+𝑉(𝐽_𝑟𝑛𝑦_𝑛,𝑝)}≤𝛽_𝑛𝑉(𝑥,𝑝)+(1−𝛽_𝑛)𝑉(𝑦_𝑛,𝑝). (3.23) These inequalities and (3.22) imply with lim_𝑛→∞𝛽_𝑛=0 and (3.21) that 𝛼≤𝛼+lim_𝑛→∞𝑉(𝑧_𝑛,𝑅𝑧_𝑛)≤𝛼≤lim_𝑛→∞𝑉(𝑦_𝑛,𝐽_𝑟𝑛𝑦_𝑛)+𝛼≤𝛼, (3.24) that is, lim_𝑛→∞𝑉(𝑧_𝑛,𝑥_𝑛+1)=lim_𝑛→∞𝑉(𝑧_𝑛,𝑅𝑧_𝑛)=lim_𝑛→∞𝑉(𝑦_𝑛,𝐽_𝑟𝑛𝑦_𝑛)=0. (3.25) Lemma 2.13 implies that lim_𝑛→∞‖𝑥_𝑛+1−𝑧_𝑛‖=0,lim_𝑛→∞‖𝐽_𝑟𝑛𝑦_𝑛−𝑦_𝑛‖=0. (3.26) Furthermore, since lim_𝑛→∞‖𝑧_𝑛−𝐽_𝑟𝑛𝑦_𝑛‖=lim_𝑛→∞‖𝛽_𝑛𝑥+(1−𝛽_𝑛)𝐽_𝑟𝑛𝑦_𝑛−𝐽_𝑟𝑛𝑦_𝑛‖=lim_𝑛→∞𝛽_𝑛‖𝑥−𝐽_𝑟𝑛𝑦_𝑛‖=0, (3.27) we have from (3.26) and (3.27) that ‖𝑥_𝑛+1−𝐽_𝑟𝑛𝑦_𝑛‖≤‖𝑥_𝑛+1−𝑧_𝑛‖+‖𝑧_𝑛−𝐽_𝑟𝑛𝑦_𝑛‖→0(𝑛→∞). (3.28) Hence, for an index sequence {𝑛_𝑗}_𝑗≥1 of {𝑥_𝑛𝑗} such that 𝑥_𝑛𝑗⇀𝑢∈𝐶 and lim_𝑗→∞𝑟_𝑛𝑗−1>0, we obtain that 𝐽_{𝑟𝑛𝑗−1}𝑦_𝑛𝑗−1⇀𝑢,𝑦_𝑛𝑗−1⇀𝑢,as𝑗⇀∞. (3.29) Since (1/𝑟)(𝐽_𝑟⁻¹−𝐼)=(1/𝑟)(I+𝑟𝐵𝐽−𝐼)=𝐵𝐽, there exists 𝑤_𝑛𝑗∈𝐵𝐽(𝐽_{𝑟𝑛𝑗−1}𝑦_𝑛𝑗−1) such that 𝑤_𝑛𝑗=1𝑟_𝑛𝑗−1(𝑦_𝑛𝑗−1−𝐽_{𝑟𝑛𝑗−1}𝑦_𝑛𝑗−1),forany𝑗≥1. (3.30) Since lim_𝑗→∞𝑟_𝑛𝑗−1>0, (3.26) implies that lim_𝑗→∞‖𝑤_𝑛𝑗‖=lim_𝑗→∞1𝑟_𝑛𝑗−1‖𝐽_{𝑟𝑛𝑗−1}𝑦_𝑛𝑗−1−𝑦_𝑛𝑗−1‖=0. (3.31) For (𝑝,𝑞)∈𝐵𝐽⊂𝐸×𝐸, the monotonicity of 𝐵 implies that lim_𝑗→∞⟨𝑞−𝑤_𝑛𝑗,𝐽𝑝−𝐽𝐽_{𝑟𝑛𝑗−1}𝑦_𝑛𝑗−1⟩≥0, (3.32) and we have, since 𝐽 is weakly sequentially continuous, that ⟨𝑞,𝐽𝑝−𝐽𝑢⟩≥0. (3.33) The maximality of 𝐵 implies that 𝑢∈(𝐵𝐽)⁻¹(0).
Now, we will show that 𝑢∈𝐴⁻¹(0). From Proposition 2.9(d) and 𝑝∈𝐹(𝐴_𝛼), we get that 𝑉(𝐴_𝛼𝑥_𝑛,𝑝)=𝑉(𝐴_𝛼𝑥_𝑛,𝐴_𝛼𝑝)≤𝑉(𝑥_𝑛,𝑝)−𝜆⁻¹𝑉(𝐴𝑥_𝑛,𝐴𝑝). (3.34) Thus, we have from (3.17) that 𝑉(𝑥_𝑛+1,𝑝)≤𝑉(𝑧_𝑛,𝑝)≤𝛽_𝑛𝑉(𝑥,𝑝)+(1−𝛽_𝑛)𝑉(𝑦_𝑛,𝑝)=𝛽_𝑛𝑉(𝑥,𝑝)+(1−𝛽_𝑛)𝑉(𝑅𝐴_𝛼𝑥_𝑛,𝑝)≤𝛽_𝑛𝑉(𝑥,𝑝)+(1−𝛽_𝑛)𝑉(𝐴_𝛼𝑥_𝑛,𝑝)≤𝛽_𝑛𝑉(𝑥,𝑝)+(1−𝛽_𝑛){𝑉(𝑥_𝑛,𝑝)−𝜆⁻¹𝑉(𝐴𝑥_𝑛,𝐴𝑝)}. (3.35) This implies that 0≤(1−𝛽_𝑛)𝜆⁻¹𝑉(𝐴𝑥_𝑛,𝐴𝑝)≤𝛽_𝑛𝑉(𝑥,𝑝)+(1−𝛽_𝑛)𝑉(𝑥_𝑛,𝑝)−𝑉(𝑥_𝑛+1,𝑝)=𝛽_𝑛{𝑉(𝑥,𝑝)−𝑉(𝑥_𝑛,𝑝)}+𝑉(𝑥_𝑛,𝑝)−𝑉(𝑥_𝑛+1,𝑝)→0,as𝑛→∞. (3.36) Therefore we have that lim_𝑛→∞𝑉(𝐴𝑥_𝑛,𝐴𝑝)=0. (3.37) From Lemma 2.13, we get that lim_𝑛→∞‖𝐴𝑥_𝑛−𝐴𝑝‖=lim_𝑛→∞‖𝐴𝑥_𝑛‖=0 that is, 𝐴𝑥_𝑛→0 as 𝑛→∞. From Lemma 1.1(a) and the boundedness of {𝐴_𝛼𝑥_𝑛}, we have that 0≤𝑉(𝑥_𝑛,𝐴_𝛼𝑥_𝑛)=𝑉(𝑥_𝑛,𝑝)−𝑉(𝐴_𝛼𝑥_𝑛,𝑝)+2⟨𝑥_𝑛−𝐴_𝛼𝑥_𝑛,𝐽𝐴_𝛼𝑥_𝑛−𝐽𝑝⟩=𝑉(𝑥_𝑛,𝑝)−𝑉(𝐴_𝛼𝑥_𝑛,𝑝)+2𝛼⟨𝐴𝑥_𝑛,𝐽𝐴_𝛼𝑥_𝑛−𝐽𝑝⟩≤𝑉(𝑥_𝑛,𝑝)−𝑉(𝐴_𝛼𝑥_𝑛,𝑝)+2𝛼‖𝐴𝑥_𝑛‖‖𝐽𝐴_𝛼𝑥_𝑛−𝐽𝑝‖≤𝑉(𝑥_𝑛,𝑝)−𝑉(𝐴_𝛼𝑥_𝑛,𝑝)+2𝛼‖𝐴𝑥_𝑛‖𝑀, (3.38) for some 𝑀>0. From (3.21), we have that lim_𝑛→∞{𝑉(𝑥_𝑛,𝑝)−𝑉(𝐴_𝛼𝑥_𝑛,𝑝)}=0, and we obtain that lim_𝑛→∞𝑉(𝑥_𝑛,𝐴_𝛼𝑥_𝑛)=0, (3.39) and this means that lim_𝑛→∞‖𝑥_𝑛−𝐴_𝛼𝑥_𝑛‖=0. (3.40) From Lemma 2.14, we obtain 𝑥_𝑛⇀𝑢₀∈𝐹(𝐴_𝛼). Since 𝑥_𝑛𝑗⇀𝑢, this means that 𝑥_𝑛𝑗⇀𝑢₀=𝑢; hence, we have 𝑢∈𝐹(𝐴_𝛼); that is, 𝑢∈𝐴⁻¹(0). Therefore, we obtain that 𝑢∈𝐴⁻¹(0)∩(𝐵𝐽)⁻¹(0)=𝐶₀.
Let 𝑢_𝑛=𝑅₀𝑥_𝑛 for any 𝑛∈ℕ. Since 𝑅₀ is a sunny generalized nonexpansive retraction, ⟨𝑥_𝑛−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦⟩≥0,∀𝑦∈𝐶₀. (3.41) Similarly as in the proof of Theorem 3.2, we can show that {𝑢_𝑛} is a Cauchy sequence, and therefore there exists 𝑢^∗∈𝐶₀ such that 𝑢_𝑛→𝑢^∗. Set 𝑦=𝑢 in (3.41). Since 𝑥_𝑛⇀𝑢, we get that ⟨𝑢−𝑢^∗,𝐽𝑢^∗−𝐽𝑢⟩≥0. (3.42) This means that 𝑢=𝑢^∗ by the strict convexity of 𝐽; that is, 𝑅₀𝑥_𝑛→𝑢. This completes the proof.

In a Hilbert space, we obtain the following theorem as a corollary of the main Theorem 3.3 by applying Proposition 2.8(a).

Corollary 3.4. Let 𝐻 be a Hilbert space, and let 𝐶 be a nonempty, closed, and convex subset of 𝐻. Let 𝐵∶𝐻→2^𝐻 be a maximal monotone operator, and let 𝐽_𝑟𝑛=(𝐼+𝑟_𝑛𝐵)⁻¹ be a resolvent of 𝐵 for a sequence {𝑟_𝑛}⊂(0,∞). Suppose that 𝐴∶𝐶→𝐻 is a firmly nonexpansive mapping with 𝐶₀=𝐴⁻¹(0)∩𝐵⁻¹(0)≠∅. Suppose that 𝑅_𝐶 is a sunny and generalized nonexpansive retraction to 𝐶. Let an iterative sequence {𝑥_𝑛}⊂𝐶 be defined as follows: for any 𝑥=𝑥₁∈𝐶 and 𝑛∈ℕ, 𝑦_𝑛=𝑅_𝐶(𝐼−𝛼𝐴)𝑥_𝑛,𝑥_𝑛+1=𝑅_𝐶(𝛽_𝑛𝑥+(1−𝛽_𝑛)𝐽_𝑟𝑛𝑦_𝑛), (3.43) where {𝛽_𝑛}⊂[0,1] and {𝑟_𝑛}⊂(0,∞) satisfy that _𝑛≥1𝛽_𝑛<∞,liminf_𝑛→∞𝑟_𝑛>0. (3.44) Then, there exists an element 𝑢∈𝐶₀ such that 𝑥_𝑛⇀𝑢,𝑅_𝐶0(𝑥_𝑛)→𝑢. (3.45)

Acknowledgments

The author expresses her hearty thanks to Professor Wataru Takahashi and the referees for giving valuable suggestions during the preparation of this paper.