Existence and Convergence Theorems by an Iterative Shrinking Projection Method of a Strict Pseudocontraction in Hilbert Spaces

Abstract

The aim of this paper is to provide some existence theorems of a strict pseudocontraction by the way of a hybrid shrinking projection method, involving some necessary and sufficient conditions. The method allows us to obtain a strong convergence iteration for finding some fixed points of a strict pseudocontraction in the framework of real Hilbert spaces. In addition, we also provide certain applications of the main theorems to confirm the existence of the zeros of an inverse strongly monotone operator along with its convergent results.

1. Introduction

There are several attempts to establish an iteration method to find a fixed point of some well-known nonlinear mappings, for instant, nonexpansive mapping. We note that Mann's iterations [1] have only weak convergence even in a Hilbert space (see, e.g., [2]). Nakajo and Takahashi [3] modified the Mann iteration method so that strong convergence is guaranteed, later well known as a hybrid projection method. Since then, the hybrid method has received rapid developments. For more details, the readers are referred to papers [4–23] and the references cited therein. In 2008, Takahashi et al. [18] introduced an alternative projection method, subsequently well known as the shrinking projection method, and they showed several strong convergence theorems for a family of nonexpansive mappings; see also [24]. In 2009, Aoyama et al. [25] applied the hybrid shrinking projection method along with creating some necessary and sufficient conditions to confirm the existence of a fixed point of firmly nonexpansive mapping.

Let 𝐻 be a real Hilbert space; a mapping 𝑇 with domain 𝐷(𝑇) and range 𝑅(𝑇) in 𝐻 is called firmly nonexpansive if 𝑇𝑥−𝑇𝑦²≤𝑇𝑥−𝑇𝑦,𝑥−𝑦,∀𝑥,𝑦∈𝐷(𝑇), (1) nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐷(𝑇). (2) Throughout this paper, 𝐼 stands for an identity mapping. The mapping 𝑇 is said to be a strict pseudocontraction if there exists a constant 0≤𝑘<1 such that 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+𝑘(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦²,∀𝑥,𝑦∈𝐷(𝑇). (3) In this case, 𝑇 may be called as a 𝑘-strict pseudocontraction. It is not hard to verify that (3) is equivalent to ()()𝑦≥𝑥−𝑦,𝐼−𝑇𝑥−𝐼−𝑇1−𝑘2(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦²,∀𝑥,𝑦∈𝐷(𝑇). (4) If we set 𝐴∶=(𝐼−𝑇) that satisfies (4), then 𝐴 is said to be inverse strongly monotone. For such a case, 𝐴 may be called as (1−𝑘)/2-inverse strongly monotone (let us see Section 4).

We use 𝐹(𝑇) to denote the set of fixed points of 𝑇 (i.e., 𝐹(𝑇)={𝑥∈𝐷(𝑇)∶𝑇𝑥=𝑥}). 𝑇 is said to be a quasi-strict pseudocontraction if the set of fixed points 𝐹(𝑇) is nonempty and there exists a constant 0≤𝑘<1 such that 𝑇𝑥−𝑝²≤𝑥−𝑝²+𝑘‖𝑥−𝑇𝑥‖²,∀𝑥∈𝐷(𝑇),𝑝∈𝐹(𝑇). (5)

The class of strict pseudocontractions extends the class of nonexpansive mappings and firmly nonexpansive mappings. That is 𝑇 is nonexpansive if and only if 𝑇 is a 0-strict pseudocontraction.

By definition, it is clear that firmlynonexpansive⟹nonexpansive⟹strictpseudocontraction. (6) However, the following examples show that the converse is not true.

Example 1. Let 𝐻 be a real Hilbert space and 𝛼∈(1,∞). Define 𝑇_𝛼∶𝐻→𝐻 by 𝑇_𝛼𝑥=−𝛼𝑥,∀𝑥∈𝐻. (7) Then, 𝑇_𝛼 is a strict pseudocontraction but not a nonexpansive mapping.

Indeed, it is clear that 𝑇_𝛼 is not nonexpansive. On the other hand, let us consider 𝑇_𝛼𝑥−𝑇_𝛼𝑦²=(−𝛼𝑥)−(−𝛼𝑦)²=𝛼²𝑥−𝑦²=𝛼1+²−1(1+𝛼)²(1+𝛼)²𝑥−𝑦²=𝑥−𝑦²+𝛼²−1(1+𝛼)²(1−(−𝛼))𝑥−(1−(−𝛼))𝑦²=𝑥−𝑦²+𝛼−1𝛼+1(𝐼−𝑇_𝛼)𝑥−(𝐼−𝑇_𝛼)𝑦²≤𝑥−𝑦²+𝜅(𝐼−𝑇_𝛼)𝑥−(𝐼−𝑇_𝛼)𝑦² (8) for all 𝜅∈[(𝛼−1)/(𝛼+1),1). Thus, 𝑇_𝛼 is a strict pseudocontraction.

Example 2. Take 𝐻≠{0}, and let 𝑇=−𝐼: it is not hard to verify that 𝑇 is nonexpansive but not firmly nonexpansive.

From a practical point of view, strict pseudocontractions have more powerful applications than nonexpansive mappings do in solving inverse problems (see Scherzer [26]). Therefore, it is important to develop theory of iterative methods for strict pseudocontractions. Within the past several decades, many authors have been devoted to the studies on the existence and convergence of fixed points for strict pseudocontractions. In 1967, Browder and Petryshyn [27] introduced a convex combination method to study strict pseudocontractions in Hilbert spaces. On the other hand, Marino and Xu [11] and Zhou [28] developed some iterative scheme for finding a fixed point of a strict pseudocontraction mapping.

In 2009, Yao et al. [29] introduced the hybrid iterative algorithm for pseudo-contractive mapping in Hilbert spaces as follows.

Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 be a pseudocontraction. Let {𝛼_𝑛} be a sequence in (0,1). Let 𝑥₀∈𝐻. For 𝐶₁=𝐶 and 𝑥₁=𝑃_𝐶1(𝑥₀), define a sequence {𝑥_𝑛} of 𝐶 as follows: 𝑦_𝑛=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇𝑥_𝑛,𝐶_𝑛+1=𝑣∈𝐶_𝑛∶𝛼_𝑛(𝐼−𝑇)𝑦_𝑛²⩽2𝛼_𝑛𝑥_𝑛−𝑣,(𝐼−𝑇)𝑦_𝑛,𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥₀. (9)

Theorem 3 (Yao et al. [29]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 be an L-Lipschitz pseudocontraction such that 𝐹(𝑇)≠∅. Assume that the sequence {𝛼_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1/(𝐿+1)). Then, the sequence {𝑥_𝑛} generated by (9) converges strongly to 𝑃_𝐹(𝑇)(𝑥₀).

In 2009, Aoyama et al. [25] provided the useful and interesting lemma to confirm that the sequence generated by the shrinking projection method is well defined even if the firmly nonexpansive mapping 𝑇 has no fixed points.

Lemma 4 (Aoyama et al. [25, Lemma  4.2]). Let 𝐻 be a Hilbert space, 𝐶 a nonempty closed convex subset of 𝐻, 𝑇∶𝐶→𝐶 a firmly nonexpansive mapping, and 𝑥₀∈𝐻. Let {𝑥_𝑛} be a sequence in 𝐶 and {𝐶_𝑛} a sequence of closed convex subsets of 𝐻 generated by 𝐶₁=𝐶 and 𝑥_𝑛=𝑃_𝐶𝑛𝑥₀,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶𝑇𝑥_𝑛−𝑧,𝑥_𝑛−𝑇𝑥_𝑛≥0 (10) for all 𝑛∈ℕ. Then, 𝐶_𝑛 is nonempty for every 𝑛∈ℕ, and, consequently, {𝑥_𝑛} is well defined.

By using the lemma mentioned above, they proved the following theorem.

Theorem 5 (Aoyama et al. [25, Theorem  4.3]). Let 𝐻 be a Hilbert space, 𝐶 a nonempty closed convex subset of 𝐻, 𝑇∶𝐶→𝐶 a firmly nonexpansive mapping and 𝑥₀∈𝐻. Let {𝑥_𝑛} be a sequence in 𝐶 and {𝐶_𝑛} a sequence of closed convex subsets of 𝐻 generated by 𝐶₁=𝐶 and 𝑥_𝑛=𝑃_𝐶𝑛𝑥₀,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶𝑇𝑥_𝑛−𝑧,𝑥_𝑛−𝑇𝑥_𝑛≥0 (11) for all 𝑛∈ℕ. Then, the following are equivalent: (i)⋂^∞_𝑛=1𝐶_𝑛 is nonempty; (ii){𝑥_𝑛} is bounded; (iii)𝐹(𝑇) is nonempty.

Motivated and inspired by the results mentioned above, in this paper, we provide some existence theorems of a strict pseudocontraction by the way of the shrinking projection method, involving some necessary and sufficient conditions. Then, we prove a strong convergence theorem and present its applications to confirm the existence of the zeros of an inverse strongly monotone operator along with its convergent results.

Throughout the paper, we will using the following notations: (i)→ for strong convergence and ⇀ for weak convergence; (ii)𝜔_𝑤(𝑥_𝑛)={𝑥∶∃𝑥_𝑛𝑖⇀𝑥} denotes the weak 𝜔-limit set of {𝑥_𝑛}.

2. Preliminaries

In this section, some definitions are provided, and some relevant lemmas which are useful to prove in the next section are collected. Most of them are known and others are not hard to prove.

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖, and let 𝐶 be a closed convex subset of 𝐻. For every point 𝑥∈𝐻, there exists a unique nearest point in 𝐶, denoted by 𝑃_𝐶(𝑥), such that 𝑥−𝑃_𝐶𝑥⩽𝑥−𝑦,∀𝑦∈𝐶. (12) The mapping 𝑃_𝐶 is called the metric projection of 𝐻 onto 𝐶. It is well known that 𝑃_𝐶 is a firmly nonexpansive mapping of 𝐻 onto 𝐶, that is, 𝑃_𝐶𝑥−𝑃_𝐶𝑦²≤𝑃_𝐶𝑥−𝑃_𝐶𝑦,𝑥−𝑦,∀𝑥,𝑦∈𝐻. (13) Furthermore, for any 𝑥∈𝐻 and 𝑧∈𝐶, 𝑧=𝑃_𝐶𝑥⟺𝑥−𝑧,𝑧−𝑦≥0,∀𝑦∈𝐶. (14) Moreover, 𝑃_𝐶𝑥 is characterized by the following: 𝑥−𝑦²≥𝑥−𝑃_𝐶𝑥²+𝑦−𝑃_𝐶𝑥²,∀𝑦∈𝐶. (15) It is obvious that the following equality holds for all 𝑥,𝑦∈𝐻: 𝑥−𝑦²=‖𝑥‖²−𝑦²−2𝑥−𝑦,𝑦,∀𝑥,𝑦∈𝐻. (16)

Proposition 6 (see [11, Proposition  2.1]). Assuming that 𝐶 is a closed convex subset of a Hilbert space 𝐻, let 𝑇∶𝐶→𝐶 be a self-mapping of 𝐶. (i)If  𝑇  is a  𝑘-strict pseudocontraction, then  𝑇  satisfies the Lipschitz condition𝑇𝑥−𝑇𝑦⩽1+𝑘1−𝑘𝑥−𝑦∀𝑥,𝑦∈𝐶. (17)(ii)If  𝑇  is a  𝑘-strict pseudocontraction, then  𝐼−𝑇  is demiclosed at zero; that is, if  {𝑥_𝑛}  is a sequence in  𝐶  such that  𝑥_𝑛→𝑧  and  (𝐼−𝑇)𝑥_𝑛→0, then  (𝐼−𝑇)𝑧=0.(iii)If  𝑇  is a  𝑘-quasi-strict strict pseudocontraction, then the set of fixed points  𝐹(𝑇)  is a closed convex subset of  𝐶.

Lemma 7. Assuming that 𝐶 is a closed convex subset of a Hilbert space 𝐻, let 𝑇∶𝐶→𝐶 be a self-mapping of 𝐶. Then, the following are equivalent: (i)𝑇 is a 𝑘-strict pseudocontraction; (ii)𝐼−𝑇 is (1−𝑘)/2-inverse strongly monotone.

Lemma 8 (see [30, Theorem  7.1.8]). Let 𝐵 be a bounded closed convex subset of a Hilbert space 𝐻 and 𝐴∶𝐵→𝐻 a continuous monotone mapping. Then, there exists an element 𝑢₀∈𝐵 such that ⟨𝑣−𝑢₀,𝐴𝑢₀⟩≥0 for all 𝑣∈𝐾.

3. Main Result

In this section, motivated by Aoyama et al. [25] (see also, Matsushita and Takahashi [31]), we discuss the existence of fixed point of a strict pseudocontraction by using the shrinking projection technique acting as the tool to guarantee the existence of fixed point of a strict pseudocontraction.

Every iteration process generated by the shrinking projection method for a 𝑘-strict pseudocontraction 𝑇 is well defined even if 𝑇 is a fixed point free.

Lemma 9. Let 𝐻 be a Hilbert space, 𝐶 a nonempty closed convex subset of 𝐻, 𝑇∶𝐶→𝐶 a 𝑘-strict pseudocontraction and 𝑥₀∈𝐻. Let {𝑥_𝑛} be a sequence in 𝐶 and {𝐶_𝑛} a sequence of closed convex subsets of 𝐻 generated by 𝐶₁=𝐶 and 𝑥₁=𝑃_𝐶1𝑥₀,𝑦_𝑛=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇𝑥_𝑛,0<𝛼_𝑛<1−𝑘2,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶𝛼_𝑛(𝐼−𝑇)𝑦_𝑛²≤2𝛼_𝑛𝛼1−^𝑛𝑥1−𝑘^𝑛−𝑧,(𝐼−𝑇)𝑦_𝑛,𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥₀, (18) for all 𝑛∈ℕ. Then, 𝐶_𝑛 is nonempty for every 𝑛∈ℕ, and, consequently, {𝑥_𝑛} is well defined.

Proof. Clearly, 𝐶₁ is nonempty. Suppose that 𝐶_𝑛 is nonempty for some 𝑛∈ℕ. Since 𝐶_𝑛⊂𝐶_𝑛−1⊂⋯⊂𝐶₁, we have that 𝐶₁,𝐶₂,…,𝐶_𝑛 are nonempty and hence {𝑥₁,𝑥₂,…,𝑥_𝑛} is well defined. Put 𝑟=max{‖𝑦_𝑖‖∶𝑖=1,2,…,𝑛} and 𝐵_𝑟={𝑧∈𝐻∶‖𝑧‖≤𝑟}. Obviously, 𝐶∩𝐵_𝑟 is a nonempty bounded closed convex subset of 𝐻. Let 𝐼 denote the identity mapping on 𝐶. Since 𝐼−𝑇 is continuous and monotone, it follows from Lemma 8 that there exists 𝑢∈𝐶∩𝐵_𝑟 such that 𝑦−𝑢,(𝐼−𝑇)𝑢≥0∀𝑦∈𝐶∩𝐵_𝑟. (19) In particular, we have 𝑦_𝑖−𝑢,(𝐼−𝑇)𝑢≥0, (20) for every 𝑖=1,2…,𝑛. On the other hand, by employing the identity (16) and then adding and subtracting the terms 𝑦_𝑖 and (𝐼−𝑇)𝑢, we obtain 𝑥_𝑖−𝑢−𝛼_𝑖(𝐼−𝑇)𝑦_𝑖²=𝑥_𝑖−𝑢²−𝛼_𝑖(𝐼−𝑇)𝑦_𝑖²−2𝛼_𝑖𝑥_𝑖−𝑢−𝛼_𝑖(𝐼−𝑇)𝑦_𝑖,(𝐼−𝑇)𝑦_𝑖=𝑥_𝑖−𝑢²−𝛼_𝑖(𝐼−𝑇)𝑦_𝑖²−2𝛼_𝑖𝑥_𝑖−𝑦_𝑖−𝛼_𝑖(𝐼−𝑇)𝑦_𝑖,(𝐼−𝑇)𝑦_𝑖−2𝛼_𝑖𝑦_𝑖−𝑢,(𝐼−𝑇)𝑦_𝑖=𝑥_𝑖−𝑢²−𝛼_𝑖(𝐼−𝑇)𝑦_𝑖²+2𝑥_𝑖−𝑦_𝑖−𝛼_𝑖(𝐼−𝑇)𝑦_𝑖,𝛼_𝑖(𝐼−𝑇)𝑦_𝑖−2𝛼_𝑖𝑦_𝑖−𝑢,(𝐼−𝑇)𝑦_𝑖−(𝐼−𝑇)𝑢−2𝛼_𝑖𝑦_𝑖−𝑢,(𝐼−𝑇)𝑢. (21) By using the identity (16) again, it follows that 𝛼_𝑖(𝐼−𝑇)𝑦_𝑖²+2𝑥_𝑖−𝑦_𝑖−𝛼_𝑖(𝐼−𝑇)𝑦_𝑖,𝛼_𝑖(𝐼−𝑇)𝑦_𝑖=𝛼_𝑖(𝐼−𝑇)𝑦_𝑖²+𝑥_𝑖−𝑦_𝑖²−𝛼_𝑖(𝐼−𝑇)𝑦_𝑖²−𝑥_𝑖−𝑦_𝑖−𝛼_𝑖(𝐼−𝑇)𝑦_𝑖²=𝑥_𝑖−𝑦_𝑖²−𝑥_𝑖−𝑦_𝑖−𝛼_𝑖(𝐼−𝑇)𝑦_𝑖². (22) Substituting (22) in (21), we have 𝑥_𝑖−𝑢−𝛼_𝑖(𝐼−𝑇)𝑦_𝑖²=𝑥_𝑖−𝑢²−𝑥_𝑖−𝑦_𝑖²+𝑥_𝑖−𝑦_𝑖−𝛼_𝑖(𝐼−𝑇)𝑦_𝑖²−2𝛼_𝑖𝑦_𝑖−𝑢,(𝐼−𝑇)𝑦_𝑖−(𝐼−𝑇)𝑢−2𝛼_𝑖𝑦_𝑖()𝑢=𝑥−𝑢,𝐼−𝑇_𝑖−𝑢²−𝛼²_𝑖(𝐼−𝑇)𝑥_𝑖²+𝛼²_𝑖(𝐼−𝑇)𝑥_𝑖−(𝐼−𝑇)𝑦_𝑖²−2𝛼_𝑖𝑦_𝑖−𝑢,(𝐼−𝑇)𝑦_𝑖−(𝐼−𝑇)𝑢−2𝛼_𝑖𝑦_𝑖−𝑢,(𝐼−𝑇)𝑢. (23) By the virtue of Lemma 7 and some simple calculations, we obtain (𝐼−𝑇)𝑥_𝑖−(𝐼−𝑇)𝑦_𝑖²≤2𝑥1−𝑘^𝑖−𝑦_𝑖,(𝐼−𝑇)𝑥_𝑖−(𝐼−𝑇)𝑦_𝑖=2𝑥1−𝑘^𝑖−𝑦_𝑖,(𝐼−𝑇)𝑥_𝑖−𝑥_𝑖−𝑦_𝑖,(𝐼−𝑇)𝑦_𝑖=2𝛼1−𝑘^𝑖(𝐼−𝑇)𝑥_𝑖²−𝑥_𝑖−𝑢,(𝐼−𝑇)𝑦_𝑖−𝑢−𝑦_𝑖,(𝐼−𝑇)𝑦_𝑖=2𝛼1−𝑘^𝑖(𝐼−𝑇)𝑥_𝑖²−𝑥_𝑖−𝑢,(𝐼−𝑇)𝑦_𝑖+𝑢−𝑦_𝑖,(𝐼−𝑇)𝑢−(𝐼−𝑇)𝑦_𝑖−()𝑢=2𝐼−𝑇𝛼1−𝑘^𝑖(𝐼−𝑇)𝑥_𝑖²−𝑥_𝑖−𝑢,(𝐼−𝑇)𝑦_𝑖+𝑢−𝑦_𝑖,(𝐼−𝑇)𝑢−(𝐼−𝑇)𝑦_𝑖+𝑦_𝑖−𝑢,(𝐼−𝑇)𝑢. (24) Joining (23) for the term ‖(𝐼−𝑇)𝑥_𝑖−(𝐼−𝑇)𝑦_𝑖‖² with (24) and by 0<𝛼_𝑛<(1−𝑘)/2, the monotonicity of (𝐼−𝑇), and (20), we have 𝑥_𝑖−𝑢−𝛼_𝑖(𝐼−𝑇)𝑦_𝑖²≤𝑥_𝑖−𝑢²−𝛼²_𝑖(𝐼−𝑇)𝑥_𝑖²+𝛼²_𝑖2𝛼1−𝑘^𝑖(𝐼−𝑇)𝑥_𝑖²−𝑥_𝑖−𝑢,(𝐼−𝑇)𝑦_𝑖+𝑢−𝑦_𝑖,()𝑢−()𝑦𝐼−𝑇𝐼−𝑇_𝑖+𝑦_𝑖−𝑢,(𝐼−𝑇)𝑢−2𝛼_𝑖𝑦_𝑖−𝑢,(𝐼−𝑇)𝑦_𝑖−(𝐼−𝑇)𝑢−2𝛼_𝑖𝑦_𝑖()𝑢=𝑥−𝑢,𝐼−𝑇_𝑖−𝑢²+𝛼²_𝑖2𝛼_𝑖1−𝑘−1(𝐼−𝑇)𝑥_𝑖²−2𝛼²_𝑖𝑥1−𝑘^𝑖−𝑢,(𝐼−𝑇)𝑦_𝑖+2𝛼_𝑖𝛼_𝑖×1−𝑘−1𝑢−𝑦_𝑖,(𝐼−𝑇)𝑢−(𝐼−𝑇)𝑦_𝑖+2𝛼_𝑖𝛼_𝑖𝑦1−𝑘−1_𝑖()𝑢≤𝑥−𝑢,𝐼−𝑇_𝑖−𝑢²−2𝛼²_𝑖𝑥1−𝑘^𝑖−𝑢,(𝐼−𝑇)𝑦_𝑖. (25) Notice that 𝑥_𝑖−𝑢−𝛼_𝑖(𝐼−𝑇)𝑦_𝑖²=𝑥_𝑖−𝑢²−2𝑥_𝑖−𝑢,𝛼_𝑖(𝐼−𝑇)𝑦_𝑖+𝛼_𝑖(𝐼−𝑇)𝑦_𝑖². (26) Combining (25) and (26), we have 𝛼_𝑖(𝐼−𝑇)𝑦_𝑖²≤2𝛼_𝑖𝛼1−^𝑖𝑥1−𝑘^𝑖−𝑢,(𝐼−𝑇)𝑦_𝑖 (27) for every 𝑖=1,2,…,𝑛. This shows that 𝑢∈𝐶_𝑛+1. By induction on 𝑛, we obtain the desired result.

The following theorem provides some necessary and sufficient conditions to confirm the existence of a fixed point of a strict pseudocontraction in Hilbert spaces.

Theorem 10. Let all the assumptions be as in Lemma 9 and 0<𝑎≤𝛼_𝑛≤𝑏<(1−𝑘)/2 for all 𝑛∈ℕ. Then, the following are equivalent: (i)⋂^∞_𝑛=1𝐶_𝑛 is nonempty; (ii){𝑥_𝑛} is bounded; (iii)𝐹(𝑇) is nonempty.

Proof. [(i)⇒(ii)] By letting 𝑢∈⋂^∞_𝑛=1𝐶_𝑛, it follows from the nonexpansiveness of 𝑃_𝐶𝑛 that 𝑥_𝑛−𝑢≤𝑃_𝐶𝑛𝑥₀−𝑃_𝐶𝑛𝑢≤𝑥₀−𝑢. (28) This shows that {𝑥_𝑛} is bounded.
[(ii)⇒(iii)] Suppose that {𝑥_𝑛} is bounded; we observe that 0≤𝑥_𝑛+1−𝑥_𝑛²=𝑥_𝑛+1−𝑃_𝐶𝑛𝑥₀²≤𝑥_𝑛+1−𝑥₀²−𝑃_𝐶𝑛𝑥₀−𝑥₀²=𝑥_𝑛+1−𝑥₀²−𝑥_𝑛−𝑥₀². (29) This shows that {‖𝑥_𝑛−𝑥₀‖} is nondecreasing and then with the boundedness of {𝑥_𝑛}, we have that lim_𝑛→∞‖𝑥_𝑛−𝑥₀‖ exists. By using (29), we obtain 𝑥_𝑛+1−𝑥_𝑛⟶0as𝑛⟶∞. (30) Since 𝑥_𝑛+1∈𝐶_𝑛+1 and 0<𝑎≤𝛼_𝑛≤𝑏<(1−𝑘)/2, we have 𝛼_𝑛(𝐼−𝑇)𝑦_𝑛²≤2𝛼_𝑛𝛼1−^𝑛×𝑥1−𝑘_𝑛−𝑥_𝑛+1,(𝐼−𝑇)𝑦_𝑛⟶0as𝑛⟶∞. (31) Since {𝛼_𝑛}⊂[𝑎,𝑏], we obtain 𝑦_𝑛−𝑇𝑦_𝑛⟶0as𝑛⟶∞. (32) Furthermore, Proposition 6 (i) allows us to have 𝑥_𝑛−𝑇𝑥_𝑛≤𝑥_𝑛−𝑦_𝑛+𝑦_𝑛−𝑇𝑦_𝑛+𝑇𝑦_𝑛−𝑇𝑥_𝑛≤𝑥_𝑛−𝑦_𝑛+𝑦_𝑛−𝑇𝑦_𝑛+1+𝑘𝑦1−𝑘^𝑛−𝑥_𝑛=2𝛼_𝑛𝑥1−𝑘^𝑛−𝑇𝑥_𝑛+𝑦_𝑛−𝑇𝑦_𝑛. (33) By simple calculation, we have 𝑥_𝑛−𝑇𝑥_𝑛≤11−2𝛼_𝑛/()𝑦1−𝑘^𝑛−𝑇𝑦_𝑛≤1()𝑦1−2𝑏/1−𝑘^𝑛−𝑇𝑦_𝑛⟶0as𝑛⟶∞. (34) Since {𝑥_𝑛} is bounded, the reflexivity of 𝐻 allows a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that 𝑥_𝑛𝑖⇀𝑝∈𝐶 as 𝑖→∞. By using (34) and the demicloseness of (𝐼−𝑇), we obtain 𝑝−𝑇𝑝=0; that is, 𝑝∈𝐹(𝑇)≠∅.
[(iii)⇒(i)] Suppose that 𝐹(𝑇)≠∅. We will show that 𝐹(𝑇)⊂𝐶_𝑛 for every 𝑛∈ℕ. Let 𝑝∈𝐹(𝑇); then we have (𝐼−𝑇)𝑝=0. Let us replace 𝑢 in the proof of Lemma 9 with 𝑝; it is not difficult to see that all equalities and inequalities are satisfied until (27). This implies that 𝑝∈𝐶_𝑛 for all 𝑛∈ℕ. Therefore, 𝐹(𝑇)⊂⋂^∞_𝑛=1𝐶_𝑛≠∅.

Theorem 11. Let all the assumptions be as in Theorem 10. If ⋂^∞_𝑛=1𝐶_𝑛≠∅ (⇔{𝑥_𝑛} is bounded  ⇔𝐹(𝑇)≠∅), then the sequence {𝑥_𝑛} generated by (18) converges strongly to some points of 𝐶, and its strong limit point is a member of 𝐹(𝑇); that is, lim_𝑛→∞𝑥_𝑛=𝑃_𝐹(𝑇)𝑥₀∈𝐹(𝑇).

Proof. If ⋂^∞_𝑛=1𝐶_𝑛≠∅, then Theorem 10 guarantees that {𝑥_𝑛} is bounded and lim_𝑛→∞‖𝑥_𝑛−𝑥₀‖ exists. So, there is {𝑥_𝑛𝑖}⊂{𝑥_𝑛} such that 𝑥_𝑛𝑖⇀𝑝∈𝐶 as 𝑖→∞. By using (34) and the demicloseness of (𝐼−𝑇), we obtain 𝑝∈𝐹(𝑇). On the other hand, noticing that 𝑃_𝐹(𝑇)𝑥₀∈𝐹(𝑇)⊂𝐶_𝑛, we observe that 𝑥_𝑛−𝑥₀=𝑃_𝐶𝑛𝑥₀−𝑥₀≤𝑃_𝐹(𝑇)𝑥₀−𝑥₀ (35) for every 𝑛∈ℕ. Since ‖⋅‖² is weakly lower semicontinuous and {‖𝑥_𝑛−𝑥₀‖} is convergent, it follows from (35) that 𝑝−𝑥₀²≤liminf_𝑖→∞𝑥_𝑛𝑖−𝑥₀²=lim_𝑛→∞𝑥_𝑛−𝑥₀²≤𝑃_𝐹(𝑇)𝑥₀−𝑥₀². (36) Taking into account 𝑝∈𝐹(𝑇), we obtain 𝑝=𝑃_𝐹(𝑇)𝑥₀. This implies that 𝑥_𝑛⇀𝑃_𝐹(𝑇)𝑥₀ and ‖𝑥_𝑛−𝑥₀‖→‖𝑃_𝐹(𝑇)𝑥₀−𝑥₀‖. Hence, by using (16), we obtain 𝑥_𝑛−𝑃_𝐹(𝑇)𝑥₀²=𝑥_𝑛−𝑥₀−𝑃_𝐹(𝑇)𝑥₀−𝑥₀²=𝑥_𝑛−𝑥₀²−𝑃_𝐹(𝑇)𝑥₀−𝑥₀²−2𝑥_𝑛−𝑃_𝐹(𝑇)𝑥₀,𝑃_𝐹(𝑇)𝑥₀−𝑥₀⟶0. (37) This completes the proof.

4. Deduced Theorems and Applications

In this section, some deduced theorems and applications of the main theorem are provided in order to guarantee the existence of fixed points of a nonexpansive mapping and the existence of the zeros of an inverse strongly monotone operator. Moreover, we also have the methods that can be used to find fixed points and zero points as mentioned above.

If 𝑇∶𝐶→𝐶 is nonexpansive (⇔𝑇 is a 0-strict pseudocontraction), then we have the following corollaries.

Corollary 12. Let 𝐻 be a Hilbert space, 𝐶 a nonempty closed convex subset of 𝐻, 𝑇∶𝐶→𝐶 a nonexpansive mapping, and 𝑥₀∈𝐻. Let {𝑥_𝑛} be a sequence in 𝐶 and {𝐶_𝑛} a sequence of closed convex subsets of 𝐻 generated by 𝐶₁=𝐶 and 𝑥₁=𝑃_𝐶1𝑥₀,𝑦_𝑛=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇𝑥_𝑛,0<𝑎≤𝛼_𝑛1≤𝑏<2,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶𝛼_𝑛(𝐼−𝑇)𝑦_𝑛²≤2𝛼_𝑛1−𝛼_𝑛𝑥_𝑛−𝑧,(𝐼−𝑇)𝑦_𝑛,𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥₀ (38) for all 𝑛∈ℕ. Then, the following are equivalent: (i)⋂^∞_𝑛=1𝐶_𝑛 is nonempty; (ii){𝑥_𝑛} is bounded; (iii)𝐹(𝑇) is nonempty.

Corollary 13. Let 𝐻 be a Hilbert space, 𝐶 a nonempty closed convex subset of 𝐻, 𝑇∶𝐶→𝐶 a nonexpansive mapping, and 𝑥₀∈𝐻. Let {𝑥_𝑛} be a sequence in 𝐶 and {𝐶_𝑛} a sequence of closed convex subsets of 𝐻 generated by 𝐶₁=𝐶 and 𝑥₁=𝑃_𝐶1𝑥₀,𝑦_𝑛=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇𝑥_𝑛;0<𝑎≤𝛼_𝑛1≤𝑏<2,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶𝛼_𝑛(𝐼−𝑇)𝑦_𝑛²≤2𝛼_𝑛1−𝛼_𝑛𝑥_𝑛−𝑧,(𝐼−𝑇)𝑦_𝑛,𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥₀ (39) for all 𝑛∈ℕ. If ⋂^∞_𝑛=1𝐶_𝑛≠∅ (⇔{𝑥_𝑛} is bounded), then {𝑥_𝑛} converges strongly to some points of 𝐶, and its strong limit point is a member of 𝐹(𝑇); that is, lim_𝑛→∞𝑥_𝑛=𝑃_𝐹(𝑇)(𝑥₀)∈𝐹(𝑇) (i.e., 𝐹(𝑇)≠∅).

Recall that a mapping 𝐴 is said to be monotone if ⟨𝑥−𝑦,𝐴𝑥−𝐴𝑦⟩⩾0 for all 𝑥,𝑦∈𝐻 and inverse strongly monotone if there exists a real number 𝛾>0 such that ⟨𝑥−𝑦,𝐴𝑥−𝐴𝑦⟩⩾𝛾‖𝐴𝑥−𝐴𝑦‖² for all 𝑥,𝑦∈𝐻. For the second case, 𝐴 is said to be 𝛾-inverse strongly monotone. It follows immediately that if 𝐴 is 𝛾-inverse strongly monotone, then 𝐴 is the Lipschitz continuous; that is, ‖𝐴𝑥−𝐴𝑦‖⩽(1/𝛾)‖𝑥−𝑦‖. It is well known (see, e.g., [32]) that if 𝐴 is monotone, then the solutions of the equation 𝐴𝑥=0 correspond to the equilibrium points of some evolution systems. Therefore, it is important to focus on finding the zero point of monotone mappings. The pseudo-contractive mapping and strictly pseudo-contractive mapping are strongly related to the monotone mapping and the inverse strongly monotone mapping, respectively. It is well known that (i)𝐴 is monotone  ⇔𝑇∶=(𝐼−𝐴) is pseudo-contractive. (ii)𝐴 is inverse strongly monotone  ⇔𝑇∶=(𝐼−𝐴) is strictly pseudo-contractive.

Indeed, for (ii), we notice that the following equality always holds in a real Hilbert space: (𝐼−𝐴)𝑥−(𝐼−𝐴)𝑦²=𝑥−𝑦²+𝐴𝑥−𝐴𝑦²−2𝑥−𝑦,𝐴𝑥−𝐴𝑦∀𝑥,𝑦∈𝐻. (40) Without loss of generality, we can assume that 𝛾∈(0,1/2] and then it yields that 𝑥−𝑦,𝐴𝑥−𝐴𝑦⩾𝛾𝐴𝑥−𝐴𝑦²⟺−2𝑥−𝑦,𝐴𝑥−𝐴𝑦⩽−2𝛾‖𝐴𝑥−𝐴𝑦‖²⟺(𝐼−𝐴)𝑥−(𝐼−𝐴)𝑦²⩽𝑥−𝑦²+×1−2𝛾𝐴𝑥−𝐴𝑦²(via(40))⟺‖𝑇𝑥−𝑇𝑦‖²⩽𝑥−𝑦²+𝜅(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦²where𝑇∶=(𝐼−𝐴),𝜅∶=1−2𝛾. (41) Every iteration process generated by the shrinking projection method for a 𝛾-inverse strongly monotone 𝐴 is well defined even if 𝐴 has no zeros.

Lemma 14. Let 𝐻 be a Hilbert space, and let 𝐴∶𝐻→𝐻 be 𝛾-inverse strongly monotone (without loss of generality, one can assume that 𝛾∈(0,1/2]). Let 𝑥₀∈𝐻, 𝐶₁=𝐶 and {𝑥_𝑛} a sequence generated by 𝑥₁=𝑃_𝐶1𝑥₀,𝑦_𝑛=𝐼−𝛼_𝑛𝐴𝑥_𝑛,0<𝛼_𝑛𝐶<𝛾,_𝑛+1=𝑧∈𝐶_𝑛∶𝛼_𝑛𝐴𝑦_𝑛²≤2𝛼_𝑛𝛼1−^𝑛𝑥2𝛾^𝑛−𝑧,𝐴𝑦_𝑛,𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥₀ (42) for all 𝑛∈ℕ. Then, 𝐶_𝑛 is nonempty for every 𝑛∈ℕ, and, consequently, {𝑥_𝑛} is well defined.

Proof. Let 𝑇∶=(𝐼−𝐴). Then, 𝑇 is (1−2𝛾)-pseudocontraction, 𝑦_𝑛=𝐼−𝛼_𝑛𝐴𝑥_𝑛=𝐼−𝛼_𝑛(𝐼−(𝐼−𝐴))𝑥_𝑛=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇𝑥_𝑛, (43) and 2𝛾=1−(1−2𝛾)=1−𝑘, where 𝑘∶=1−2𝛾. Hence, by applying Lemma 9, we have the desired result.

The following theorem provides some necessary and sufficient conditions to confirm the existence of the zeros of 𝛾-inverse strongly monotone in Hilbert spaces.

Theorem 15. Let all the assumptions be as in Lemma 14 and 0<𝑎≤𝛼_𝑛≤𝑏<𝛾 for all 𝑛∈ℕ. Then, the following are equivalent: (i)⋂^∞_𝑛=1𝐶_𝑛 is nonempty; (ii){𝑥_𝑛} is bounded; (iii)𝐴⁻¹(0) is nonempty.

Proof. Let 𝑇∶=(𝐼−𝐴). Then, 𝑇 is (1−2𝛾)-pseudocontraction; it is not difficult to show that 𝐹(𝑇)=𝐴⁻¹(0) and 𝛾=(1−𝑘)/2, where 𝑘∶=1−2𝛾. Hence, by applying Theorem 10, we have the desired result.

Theorem 16. Let all the assumptions be as in Theorem 15. If ⋂^∞_𝑛=1𝐶_𝑛≠∅ (⇔{𝑥_𝑛} is bounded  ⇔𝐴⁻¹(0)≠∅), then the sequence {𝑥_𝑛} generated by (42) converges strongly to some points of 𝐻, and its strong limit point is a member of 𝐴⁻¹(0); that is, lim_𝑛→∞𝑥_𝑛=𝑃_𝐴⁻¹₍₀₎𝑥₀∈𝐴⁻¹(0).

Proof. Let 𝑇∶=(𝐼−𝐴), and by applying Theorem 11, we have the desired result.

Acknowledgments

The author would like to thank Professor Dr. Naseer Shahzad and an anonymous referee for their valuable comments and suggestions, which were helpful in improving the paper. Moreover, the author would like to thank the Centre of Excellence in Mathematics under the Commission of Higher Education, Ministry of Education, Thailand. The project was supported by the Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand.