Strong Convergence Theorems for Maximal Monotone Operators with Nonspreading Mappings in a Hilbert Space

Abstract

We prove the strong convergence theorems for finding a common element of the set of fixed points of a nonspreading mapping T and the solution sets of zero of a maximal monotone mapping and an α-inverse strongly monotone mapping in a Hilbert space. Manaka and Takahashi (2011) proved weak convergence theorems for maximal monotone operators with nonspreading mappings in a Hilbert space; there we introduced new iterative algorithms and got some strong convergence theorems for maximal monotone operators with nonspreading mappings in a Hilbert space.

1. Introduction

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩, and let 𝐶 be a nonempty closed convex subset of 𝐻. We denote by 𝐹(𝑇) the set of fixed point of 𝑇. Then, a mapping 𝑇∶𝐶→𝐶 is said to be nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶. The mapping 𝑇∶𝐶→𝐶 is said to be firmly nonexpansive if ‖𝑇𝑥−𝑇𝑦‖²≤⟨𝑥−𝑦,𝑇𝑥−𝑇𝑦⟩ for all 𝑥,𝑦∈𝐶; see, for instance, Browder [1] and Goebel and Kirk [2]. The mapping 𝑇∶𝐶→𝐶 is said to be firmly nonspreading [3] if 2𝑇𝑥−𝑇𝑦²≤𝑇𝑥−𝑦²+𝑥−𝑇𝑦², (1.1) for all 𝑥,𝑦∈𝐶. Iemoto and Takahashi [4] proved that 𝑇∶𝐶→𝐶 is nonspreading if and only if 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+2𝑥−𝑇𝑥,𝑦−𝑇𝑦, (1.2) for all 𝑥,𝑦∈𝐶. It is not hard to know that a nonspreading mapping is deduced from a firmly nonexpansive mapping; see [5, 6], and a firmly nonexpansive mapping is a nonexpansive mapping.

Many studies have been done for structuring the fixed point of nonexpansive mapping 𝑇. In 1953, Mann [7] introduced the iteration as follows: a sequence {𝑥_𝑛} defined by 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛, (1.3) where the initial guess 𝑥₁∈𝐶 is arbitrary and {𝑎_𝑛} is a real sequence in [0,1]. It is known that under appropriate settings, the sequence {𝑥_𝑛} converges weakly to a fixed point of 𝑇. However, even in a Hilbert space, Mann iteration may fail to converge strongly, for example see [8].

Some attempts to construct iteration method guaranteeing the strong convergence have been made. For example, Halpern [9] proposed the following so-called Halpern iteration: 𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇𝑥_𝑛, (1.4) where 𝑢,𝑥₁∈𝐶 are arbitrary and {𝑎_𝑛} is a real sequence in [0,1] which satisfies 𝛼_𝑛→0, ∑^∞_𝑛=1𝛼_𝑛=∞ and ∑^∞_𝑛=1|𝛼_𝑛−𝛼_𝑛+1|<∞. Then, {𝑥_𝑛} converges strongly to a fixed point of 𝑇; see [9, 10].

In 1975, Baillon [11] first introduced the nonlinear ergodic theorem in Hilbert space as follows: 𝑆_𝑛𝑥=^𝑛−1_𝑘=0𝑇^𝑘𝑥 (1.5) converges weakly to a fixed point of 𝑇 for some 𝑥∈𝐶.

Recently, in the case when 𝑇∶𝐶→𝐶 is a nonexpansive mapping, 𝐴∶𝐶→𝐻 is an 𝛼-inverse strongly monotone mapping, and 𝐵∈𝐻×𝐻 is a maximal monotone operator, Takahashi et al. [12] proved a strong convergence theorem for finding a point of 𝐹(𝑇)∩(𝐴+𝐵)⁻¹(0), where 𝐹(𝑇) is the set of fixed points of 𝑇 and (𝐴+𝐵)⁻¹(0) is the set of zero points of 𝐴+𝐵.

In 2011, Manaka and Takahashi [13] for finding a point of the set of fixed points of 𝑇 and the set of zero points of 𝐴+𝐵 in a Hilbert space, they introduced an iterative scheme as follows: 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑇𝐽_𝜆𝑛𝐼−𝜆_𝑛𝐴𝑥_𝑛, (1.6) where 𝑇 is a nonspreading mapping, 𝐴 is an 𝛼-inverse strongly monotone mapping, and 𝐵 is a maximal monotone operator such that 𝐽_𝜆=(𝐼−𝜆𝐵)⁻¹; {𝛽_𝑛} and {𝜆_𝑛} are sequences which satisfy 0<𝑐≤𝛽_𝑛≤𝑑<1 and 0<𝑎≤𝜆_𝑛≤𝑏<2𝛼. Then they proved that {𝑥_𝑛} converges weakly to a point 𝑝=lim_𝑛→∞𝑃_{𝐹(𝑇)∩(𝐴+𝐵)}⁻¹₍₀₎𝑥_𝑛.

Motivated by above authors, we generalize and modify the iterative algorithms (1.5) and (1.6) for finding a common element of the set of fixed points of a nonspreading mapping 𝑇 and the set of zero points of monotone operator 𝐴+𝐵 (𝐴 is an 𝛼-inverse strongly monotone mapping, and 𝐵 is a maximal monotone operator). First, we prove that the sequence generated by our iterative method is weak convergence under the property conditions. Then, we prove that the strong convergence in a Hilbert space. As expected, we get some weak and strong convergence theorems about the common element of the set of fixed points of a nonspreading mapping and the set of zero points of an 𝛼-inverse strongly monotone mapping and a maximal monotone operator in a Hilbert space.

2. Preliminaries

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩, and let 𝐶 be a nonempty closed convex subset of 𝐻. A set-valued mapping 𝐵∶𝐷(𝐵)⊆𝐻→𝐻 is said to be monotone if for any 𝑥,𝑦∈𝐷(𝐵) and 𝑥^∗∈𝐵𝑥 and 𝑦^∗∈𝐵𝑦, it holds that 𝑥−𝑦,𝑥^∗−𝑦^∗≥0. (2.1)

A monotone operator 𝐵 on 𝐻 is said to be maximal if 𝐵 has no monotone extension, that is, its graph is not properly contained in the graph of any other monotone operator on 𝐻. For a maximal monotone operator 𝐵 on 𝐻 and 𝑟>0, we may define a single-valued operator 𝐽_𝑟=(𝐼+𝑟𝐵)⁻¹∶2^𝐻→𝐷(𝐵), which is called the resolvent of 𝐵 for 𝑟>0. Let 𝐵 be a maximal monotone operator on 𝐻, and let 𝐵⁻¹(0)={𝑥∈𝐻∶0∈𝐵𝑥}. For a constant 𝛼>0, the mapping 𝐴∶𝐶→𝐻 is said to be an 𝛼-inverse strongly monotone if for any for all 𝑥,𝑦∈𝐶, 𝑥−𝑦,𝐴𝑥−𝐴𝑦≥𝛼𝐴𝑥−𝐴𝑦². (2.2)

Remark 2.1. It is not hard to know that if 𝐴 is an 𝛼-inverse strongly monotone mapping, then it is 1/𝛼-Lipschitzian and hence uniformly continuous. Clearly, the class of monotone mappings include the class of an 𝛼-inverse strongly monotone mappings.

Remark 2.2. It is well known that if 𝑇∶𝐶→𝐶 is a nonexpansive mapping, then 𝐼−𝑇 is 1/2-inverse strongly monotone, where 𝐼 is the identity mapping on 𝐻; see, for instance, [14]. It is known that the resolvent 𝐽_𝑟 is firmly nonexpansive and 𝐵⁻¹(0)=𝐹(𝐽_𝑟) for all 𝑟>0.
For a single-valued mapping 𝑇, a point 𝑝 is called a fixed point of 𝑇 if 𝑝=𝑇𝑝. For a multivalued mapping 𝑇, a point 𝑝 is called a fixed point of 𝑇 if 𝑝∈𝑇𝑝. The set of fixed points of 𝑇 is denoted by 𝐹(𝑇).
Let 𝐸 be a uniformly convex real Banach space, 𝐾 be a nonempty closed convex subset of 𝐸. A multivalued mapping 𝑇∶𝐾→𝐶𝐵(𝐾) is said to be as follows.(i)Contraction if there exists a constant 𝑘∈[0,1) such that 𝐻𝑇𝑥,𝑇𝑦≤𝑘𝑥−𝑦,∀𝑥,𝑦∈𝐾. (2.3)(ii)Nonexpansive if 𝐻𝑇𝑥,𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐾. (2.4)(iii)Quasinonexpansive if 𝐹(𝑇)≠∅ and 𝐻𝑇𝑥,𝑇𝑝≤𝑥−𝑝,∀𝑥∈𝐾,∀𝑝∈𝐹(𝑇). (2.5)
It is well known that every nonexpansive multivalued mapping 𝑇 with 𝐹(𝑇)≠∅ is multivalued quasinonexpansive. But there exist multivalued quasi-nonexpansive mappings that are not multivalued nonexpansive. It is clear that if 𝑇 is a quasi-nonexpansive multivalued mapping, then 𝐹(𝑇) is closed.
A Banach space 𝐸 is said to satisfy Opials condition if whenever {𝑥_𝑛} is a sequence in 𝐸 which converges weakly to 𝑥, then liminf_𝑛→∞𝑥_𝑛−𝑥<liminf_𝑛→∞𝑥_𝑛−𝑦,∀𝑦∈𝐸,𝑥≠𝑦. (2.6)

Lemma 2.3 (Manaka and Takahashi [13]). Let 𝐻 be a real Hilbert space, and let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝛼>0. Let 𝐴 be an 𝛼-inverse strongly monotone mapping of 𝐶 into 𝐻, and let 𝐵 be a maximal monotone operator on 𝐻 such that the domain of 𝐵 is included in 𝐶. Let 𝐽_𝜆=(𝐼+𝜆𝐵)⁻¹ be the resolvent of 𝐵 for any 𝜆>0. Then, the following hold(i)if 𝑢,𝑣∈(𝐴+𝐵)⁻¹(0), then 𝐴𝑢=𝐴𝑣;(ii)for any 𝜆>0, 𝑢∈(𝐴+𝐵)⁻¹(0) if and only if 𝑢=𝐽_𝜆(𝐼−𝜆𝐴)𝑢.

Lemma 2.4 (Schu [15]). Suppose that 𝐸 is a uniformly convex Banach space and 0<𝑝≤𝑡_𝑛≤𝑞<1 for all positive integers 𝑛. Also suppose that {𝑥_𝑛} and {𝑦_𝑛} are two sequences of 𝐸 such that limsup_𝑛→∞‖𝑥_𝑛‖≤𝑟, limsup_𝑛→∞‖𝑦_𝑛‖≤𝑟, and lim_𝑛→∞‖𝑡_𝑛𝑥_𝑛+(1−𝑡_𝑛)𝑦_𝑛‖=𝑟 hold for some 𝑟≥0. Then, lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0.

Lemma 2.5 (Liu [16] and Xu [17]). Let {𝑎_𝑛} be a sequence of nonnegative real numbers satisfying the property as follows 𝑎_𝑛+1≤1−𝑡_𝑛𝑎_𝑛+𝑏_𝑛+𝑡_𝑛𝑐_𝑛, (2.7) where {𝑡_𝑛}, {𝑏_𝑛}, and {𝑐_𝑛} satisfy the restrictions as follows(i)∑^∞_𝑛=0𝑡_𝑛=∞,(ii)∑^∞_𝑛=0𝑏_𝑛<∞,(iii)limsup_𝑛→∞𝑐_𝑛≤0.
Then, {𝑎_𝑛} converges to zero as 𝑛→∞.

3. Strong Convergence Theorem

In this section, we prove the strong convergence theorems for finding a common element in common set of the fixed sets of a nonspreading mapping and the solution sets of zero of a maximal monotone operator and an 𝛼-inverse strongly monotone operator and in a Hilbert space.

Theorem 3.1. Let 𝐶 be a nonempty convex closed subset of a real Hilbert space 𝐻, let 𝐴∶𝐶→𝐻 be an 𝛼-inverse strongly monotone, let 𝐵∶𝐷(𝐵)⊆𝐶→2^𝐻 be maximal monotone, let 𝐽_𝜆=(𝐼+𝜆𝐵)⁻¹ be the resolvent of 𝐵 for any 𝜆>0, and let 𝑇∶𝐶→𝐶 be a nonspreading mapping. Assume that 𝐹∶=𝐹(𝑇)∩(𝐴+𝐵)⁻¹(0)≠∅. We define 𝑥₁𝑧=𝑥∈𝐶,𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝐽_𝜆𝑛𝐼−𝜆_𝑛𝐴𝑥_𝑛,𝑦_𝑛=1𝑛^𝑛_𝑘=1𝑇^𝑘𝑧_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑦_𝑛, (3.1) where {𝛼_𝑛} is sequences in [0,1] such that lim_𝑛→∞𝛼_𝑛=0,∑^∞_𝑛=1𝛼_𝑛=∞. There exists 𝑎,𝑏 such that 0<𝑎≤𝜆_𝑛≤𝑏<2𝛼 for each 𝑛∈𝑁. Then, {𝑥_𝑛} converges strongly to 𝑃𝑢, and 𝑃 is the metric projection of 𝐻 onto 𝐹.

Proof. First, we prove that {𝑥_𝑛} is bounded and lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists for each 𝑝∈𝐹(𝑇). In fact, from Lemma 2.3, we have 𝑝=𝐽_𝜆𝑛(𝐼−𝜆_𝑛𝐴)𝑝, together with (3.1) and 𝐴 is an 𝛼-inverse strongly monotone, we get that 𝑧_𝑛−𝑝²=𝐽_𝜆𝑛𝐼−𝜆_𝑛𝐴𝑥_𝑛−𝐽_𝜆𝑛𝐼−𝜆_𝑛𝐴𝑝²≤𝐼−𝜆_𝑛𝐴𝑥_𝑛−𝐼−𝜆_𝑛𝐴𝑝²=𝑥_𝑛−𝑝²−2𝜆_𝑛𝑥_𝑛−𝑝,𝐴𝑥_𝑛−𝐴𝑝+𝜆²_𝑛𝐴𝑥_𝑛−𝐴𝑝²≤𝑥_𝑛−𝑝²−2𝜆_𝑛𝛼𝐴𝑥_𝑛−𝐴𝑝²+𝜆²_𝑛𝐴𝑥_𝑛−𝐴𝑝²=𝑥_𝑛−𝑝²−𝜆_𝑛2𝛼−𝜆_𝑛𝐴𝑥_𝑛−𝐴𝑝²≤𝑥_𝑛−𝑝². (3.2)
From the definition of 𝑦_𝑛 and 𝑇 is nonspreading mapping, we obtain that 𝑦_𝑛=1−𝑝𝑛^𝑛−1_𝑘=0𝑇^𝑘𝑧_𝑛≤1−𝑝𝑛^𝑛−1_𝑘=0𝑇^𝑘𝑧_𝑛≤1−𝑝𝑛^𝑛−1_𝑘=0𝑧_𝑛=𝑧−𝑝_𝑛−𝑝≤𝑥_𝑛−𝑝. (3.3)
Together with (3.1), we have that 𝑥_𝑛+1−𝑝=𝛼_𝑛𝑢+1−𝛼_𝑛𝑦_𝑛−𝑝≤𝛼_𝑛𝑢−𝑝+1−𝛼_𝑛𝑦_𝑛−𝑝≤𝛼_𝑛𝑢−𝑝+1−𝛼_𝑛𝑥_𝑛−𝑝. (3.4)
Hence, we get that 𝑥_𝑛+1−𝑝≤max𝑢−𝑝,𝑥_𝑛−𝑝, (3.5) for all 𝑛∈𝑁. This means that {𝑥_𝑛−𝑝} is bounded, so {𝑥_𝑛} is bounded. From 𝑇 is nonspreading, (3.3), and (3.2), we get that {𝑦_𝑛}, {𝑧_𝑛}, and {𝑇^𝑛𝑧_𝑛} are all bounded.
Since {𝑥_𝑛} is bounded, there exists a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that lim_𝑘→∞‖𝑥_𝑛𝑘−𝑝‖ exists. Since {𝑥_𝑛𝑘} is bounded, there exists a subsequence {𝑥_𝑛𝑘𝑖} of {𝑥_𝑛𝑘} such that 𝑥_𝑛𝑘𝑖⇀𝑤∈𝐶 as 𝑖→∞. Now, we prove that 𝑤∈𝐹. First, we prove that 𝑤∈𝐹(𝑇). Since ‖𝑥_𝑛+1−𝑦_𝑛‖=𝛼_𝑛‖𝑢−𝑦_𝑛‖, replacing 𝑛 by 𝑛_𝑘𝑖, we have ‖𝑥_{𝑛𝑘𝑖+1}−𝑦_𝑛𝑘𝑖‖=𝛼_𝑛𝑘𝑖‖𝑢−𝑦_𝑛𝑘𝑖‖. Together with 𝛼_𝑛→0 and {𝑦_𝑛} is bounded, we obtain that lim_𝑖→∞‖𝑥_{𝑛𝑘𝑖+1}−𝑦_𝑛𝑘𝑖‖=0, so we have 𝑦_𝑛𝑘𝑖⇀𝑤.
Let 𝑛∈𝑁. Since 𝑇 is nonspreading, we have that for all 𝑦∈𝐶 and 𝑘=0,1,2,…,𝑛−1, 𝑇^𝑘+1𝑧_𝑛−𝑇𝑦²≤𝑇^𝑘𝑧_𝑛−𝑦²+2𝑇^𝑘𝑧_𝑛−𝑇^𝑘+1𝑧_𝑛=𝑇,𝑦−𝑇𝑦_𝑘𝑧_𝑛−𝑇𝑦²+𝑇𝑦−𝑦²+2𝑇^𝑘𝑧_𝑛𝑇−𝑇𝑦,𝑇𝑦−𝑦+2^𝑘𝑧_𝑛−𝑇^𝑘+1𝑧_𝑛,𝑦−𝑇𝑦. (3.6)
Summing these inequalities from 𝑘=0 to 𝑛−1 and dividing by 𝑛, we have 1𝑛𝑇^𝑛𝑧_𝑛−𝑇𝑦²−𝑧_𝑛−𝑇𝑦²≤𝑇𝑦−𝑦²+2𝑦_𝑛+2−𝑇𝑦,𝑇𝑦−𝑦𝑛𝑧_𝑛−𝑇^𝑛𝑧_𝑛,𝑦−𝑇𝑦. (3.7)
Replacing 𝑛 by 𝑛_𝑘𝑖, we have 1𝑛_𝑘𝑖𝑇^𝑛𝑘𝑖𝑧_𝑛𝑘𝑖−𝑇𝑦²−𝑧_𝑛𝑘𝑖−𝑇𝑦²≤𝑇𝑦−𝑦²+2𝑦_𝑛𝑘𝑖+2−𝑇𝑦,𝑇𝑦−𝑦𝑛_𝑘𝑖𝑧_𝑛𝑘𝑖−𝑇^𝑛𝑘𝑖𝑧_𝑛𝑘𝑖,𝑦−𝑇𝑦. (3.8)
Since {𝑧_𝑛} and {𝑇^𝑛𝑧_𝑛} are bounded, we have that 0≤𝑇𝑦−𝑦²+2𝑤−𝑇𝑦,𝑇𝑦−𝑦 (3.9) as 𝑖→∞. Putting 𝑦=𝑤, we have 0≤‖𝑇𝑤−𝑤‖²+2⟨𝑤−𝑇𝑤,𝑇𝑤−𝑤⟩=−‖𝑇𝑤−𝑤‖². (3.10)
Hence, 𝑤∈𝐹(𝑇).
Next, we prove that 𝑤∈(𝐴+𝐵)⁻¹(0). From (3.2) and (3.3) we have that 𝑥_𝑛+1−𝑝²≤𝛼_𝑛𝑢−𝑝²+1−𝛼_𝑛𝑦_𝑛−𝑝²≤𝛼_𝑛𝑢−𝑝²+1−𝛼_𝑛𝑧_𝑛−𝑝²≤𝛼_𝑛𝑢−𝑝²+1−𝛼_𝑛𝑥_𝑛−𝑝²−𝜆_𝑛2𝛼−𝜆_𝑛𝐴𝑥_𝑛−𝐴𝑝²=𝛼_𝑛𝑢−𝑝²−𝑥_𝑛−𝑝²+𝑥_𝑛−𝑝²−1−𝛼_𝑛𝜆_𝑛2𝛼−𝜆_𝑛𝐴𝑥_𝑛−𝐴𝑝². (3.11)
We rewrite above inequality as follows: 1−𝛼_𝑛𝜆_𝑛2𝛼−𝜆_𝑛𝐴𝑥_𝑛−𝐴𝑝²≤𝛼_𝑛𝑢−𝑝²−𝑥_𝑛−𝑝²+𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝². (3.12)
Replacing 𝑛 by 𝑛_𝑘, we have 1−𝛼_𝑛𝑘𝜆_𝑛𝑘2𝛼−𝜆_𝑛𝑘𝐴𝑥_𝑛𝑘−𝐴𝑝²≤𝛼_𝑛𝑘𝑢−𝑝²−𝑥_𝑛𝑘−𝑝²+𝑥_𝑛𝑘−𝑝²−𝑥_𝑛𝑘+1−𝑝². (3.13)
Together with lim_𝑛→∞𝛼_𝑛=0, 0<𝑎≤𝜆_𝑛≤𝑏<2𝛼 and since lim_𝑘→∞‖𝑥_𝑛𝑘−𝑝‖ exists, we obtain that lim_𝑘→∞𝐴𝑥_𝑛𝑘−𝐴𝑝=0. (3.14)
Since 𝐽_𝜆𝑛 is firmly nonexpansive, and from (3.2), we have that 𝑧_𝑛−𝑝²=𝐽_𝜆𝑛𝐼−𝜆_𝑛𝐴𝑥_𝑛−𝐽_𝜆𝑛𝐼−𝜆_𝑛𝐴𝑝²≤𝑧_𝑛−𝑝,𝐼−𝜆_𝑛𝐴𝑥_𝑛−𝐼−𝜆_𝑛𝐴𝑝=12𝑧_𝑛−𝑝²+𝐼−𝜆_𝑛𝐴𝑥_𝑛−𝐼−𝜆_𝑛𝐴𝑝²−𝑧_𝑛−𝑝−𝐼−𝜆_𝑛𝐴𝑥_𝑛+𝐼−𝜆_𝑛𝐴𝑝²≤12𝑧_𝑛−𝑝²+𝑥_𝑛−𝑝²−𝑧_𝑛−𝑝−𝐼−𝜆_𝑛𝐴𝑥_𝑛+𝐼−𝜆_𝑛𝐴𝑝²=12𝑧_𝑛−𝑝²+𝑥_𝑛−𝑝²−𝑧_𝑛−𝑥_𝑛²−2𝜆_𝑛𝑧_𝑛−𝑥_𝑛,𝐴𝑥_𝑛−𝐴𝑝−𝜆²_𝑛𝐴𝑥_𝑛−𝐴𝑝². (3.15)
This means that 𝑧_𝑛−𝑝²≤𝑥_𝑛−𝑝²−𝑧_𝑛−𝑥_𝑛²−2𝜆_𝑛𝑧_𝑛−𝑥_𝑛,𝐴𝑥_𝑛−𝐴𝑝−𝜆²_𝑛𝐴𝑥_𝑛−𝐴𝑝². (3.16)
Together with (3.1) and (3.3), we have 𝑥_𝑛+1−𝑝²≤𝛼_𝑛𝑢−𝑝²+1−𝛼_𝑛𝑦_𝑛−𝑝²≤𝛼_𝑛𝑢−𝑝²+1−𝛼_𝑛𝑧_𝑛−𝑝²≤𝛼_𝑛𝑢−𝑝²+1−𝛼_𝑛×𝑥_𝑛−𝑝²−𝑧_𝑛−𝑥_𝑛²−2𝜆_𝑛𝑧_𝑛−𝑥_𝑛,𝐴𝑥_𝑛−𝐴𝑝−𝜆²_𝑛𝐴𝑥_𝑛−𝐴𝑝²≤𝛼_𝑛𝑢−𝑝²+𝑥_𝑛−𝑝²−𝑧_𝑛−𝑥_𝑛²−2𝜆_𝑛𝑧_𝑛−𝑥_𝑛,𝐴𝑥_𝑛−𝐴𝑝−𝜆²_𝑛𝐴𝑥_𝑛−𝐴𝑝². (3.17)
Therefore, we have 𝑧_𝑛−𝑥_𝑛²≤𝛼_𝑛𝑢−𝑝²+𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²−2𝜆_𝑛𝑧_𝑛−𝑥_𝑛,𝐴𝑥_𝑛−𝐴𝑝−𝜆²_𝑛𝐴𝑥_𝑛−𝐴𝑝². (3.18)
Replacing 𝑛 by 𝑛_𝑘, we have 𝑧_𝑛𝑘−𝑥_𝑛𝑘²≤𝛼_𝑛𝑘𝑢−𝑝²+𝑥_𝑛𝑘−𝑝²−𝑥_𝑛𝑘+1−𝑝²−2𝜆_𝑛𝑘𝑧_𝑛𝑘−𝑥_𝑛𝑘,𝐴𝑥_𝑛𝑘−𝐴𝑝−𝜆²_𝑛𝑘𝐴𝑥_𝑛𝑘−𝐴𝑝². (3.19)
Since lim_𝑘→∞‖𝑥_𝑛𝑘−𝑝‖ exists, from (3.14) and lim_𝑛→∞𝛼_𝑛=0, we obtain lim_𝑛→∞𝑧_𝑛𝑘−𝑥_𝑛𝑘=0. (3.20)
Since 𝐴 is Lipschitz continuous, we also obtain lim_𝑛→∞𝐴𝑧_𝑛𝑘−𝐴𝑥_𝑛𝑘=0. (3.21)
By the definition of 𝐽_𝜆𝑛 and (3.1), we have that 𝑧_𝑛=𝐼−𝜆_𝑛𝐵⁻¹𝐼−𝜆_𝑛𝐴𝑥_𝑛⟺𝐼−𝜆_𝑛𝐴𝑥_𝑛∈𝐼−𝜆_𝑛𝐵𝑧_𝑛=𝑧_𝑛+𝜆_𝑛𝐵𝑧_𝑛⟺𝑥_𝑛−𝑧_𝑛−𝜆_𝑛𝐴𝑥_𝑛∈𝜆_𝑛𝐵𝑧_𝑛⟺1𝜆_𝑛𝑥_𝑛−𝑧_𝑛−𝜆_𝑛𝐴𝑥_𝑛∈𝐵𝑧_𝑛. (3.22)
Since 𝐵 is monotone, so for (𝑒,𝑓)∈𝐵, we have that 𝑧_𝑛1−𝑒,𝜆_𝑛𝑥_𝑛−𝑧_𝑛−𝜆_𝑛𝐴𝑥_𝑛−𝑓≥0, (3.23) and hence 𝑧_𝑛−𝑒,𝑥_𝑛−𝑧_𝑛−𝜆_𝑛𝐴𝑥_𝑛+𝑓≥0. (3.24)
Replacing 𝑛 by 𝑛_𝑘𝑖, we have that 𝑧_𝑛𝑘𝑖−𝑒,𝑥_𝑛𝑘𝑖−𝑧_𝑛𝑘𝑖−𝜆_𝑛𝑘𝑖𝐴𝑥_𝑛𝑘𝑖+𝑓≥0. (3.25)
Since 𝐴 is an 𝛼-inverse strongly monotone, we have 𝑥_𝑛𝑘𝑖−𝑤,𝐴𝑥_𝑛𝑘𝑖−𝐴𝑤≥𝛼𝐴𝑥_𝑛𝑘𝑖−𝐴𝑤². (3.26) This means that 𝐴𝑥_𝑛𝑘𝑖→𝐴𝑤 as 𝑖→∞. From (3.20) and 𝑥_𝑛𝑘𝑖⇀𝑤, we get that 𝑧_𝑛𝑘𝑖⇀𝑤, together with (3.25), we have that 𝑤−𝑒,−𝐴𝑤−𝑓≥0. (3.27)
Since 𝐵 is maximal monotone, so (−𝐴𝑤)∈𝐵𝑤. That is, 𝑤∈(𝐴+𝐵)⁻¹(0).
Now, we prove that 𝑥_𝑛→𝑃𝑢 as 𝑛→∞. Without loss of generality, we may assume that there exists a subsequence {𝑥_{𝑛𝑘𝑖+1}} of {𝑥_𝑛+1} such that limsup_𝑛→∞𝑢−𝑃𝑢,𝑥_𝑛+1−𝑃𝑢=lim_𝑖→∞𝑢−𝑃𝑢,𝑥_{𝑛𝑘𝑖+1}−𝑃𝑢. (3.28)
Since 𝑃 is the metric projection of 𝐻 onto 𝐹 and 𝑥_{𝑛𝑘𝑖+1}⇀𝑤∈𝐹, we have lim_𝑖→∞𝑢−𝑃𝑢,𝑥_{𝑛𝑘𝑖+1}−𝑃𝑢=⟨𝑢−𝑃𝑢,𝑤−𝑃𝑢⟩≤0. (3.29)
This implies that lim_𝑛→∞𝑢−𝑃𝑢,𝑥_𝑛+1−𝑃𝑢≤0. (3.30)
From (2.1), (3.1), and (3.3), we have 𝑥_𝑛+1−𝑃𝑢²=1−𝛼_𝑛𝑦_𝑛−𝑃𝑢+𝛼_𝑛(𝑢−𝑃𝑢)²≤1−𝛼_𝑛²𝑦_𝑛−𝑃𝑢²+2𝛼_𝑛𝑢−𝑃𝑢,𝑥_𝑛+1≤−𝑃𝑢1−𝛼_𝑛𝑥_𝑛−𝑃𝑢²+2𝛼_𝑛𝑢−𝑃𝑢,𝑥_𝑛+1−𝑃𝑢. (3.31)
From Lemma 2.5 and (3.30), we have lim_𝑛→∞𝑥_𝑛−𝑃𝑢=0. (3.32)
This means that 𝑥_𝑛→𝑃𝑢 as 𝑛→∞.