Strong Convergence by a Hybrid Algorithm for Finding a Common Fixed Point of Lipschitz Pseudocontraction and Strict Pseudocontraction in Hilbert Spaces

Abstract

We prove a strong convergence theorem by using a hybrid algorithm in order to find a common fixed point of Lipschitz pseudocontraction and κ-strict pseudocontraction in Hilbert spaces. Our results extend the recent ones announced by Yao et al. (2009) and many others.

1. Introduction

Let 𝐻 be a real Hilbert space, and let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝑇∶𝐶→𝐶. Recall that 𝑇 is said to be a pseudocontraction if 𝑇𝑥−𝑇𝑦²⩽𝑥−𝑦²+(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦² (1.1) is equivalent to 𝑥−𝑦,(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦⩾0, (1.2) for all 𝑥,𝑦∈𝐶, and 𝑇 is said to be a strict pseudocontraction if there exists a constant 0⩽𝜅<1 such that 𝑇𝑥−𝑇𝑦²⩽𝑥−𝑦²+𝜅(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦², (1.3) for all 𝑥,𝑦∈𝐶. For the second case, we say that 𝑇 is a 𝜅-strict pseudocontraction. We use 𝐹(𝑇) to denote the set of fixed points of 𝑇.

The class of strict pseudocontractions extend the class of nonexpansive mapping. (A mapping 𝑇 is said to be nonexpansive if ‖𝑇𝑥−𝑇𝑦‖⩽‖𝑥−𝑦‖, for all 𝑥,𝑦∈𝐶) that is, 𝑇 is nonexpansive if and only if 𝑇 is a 0-strict pseudocontraction. The pseudocontractive mapping includes the strict pseudocontractive mapping.

Iterative methods for finding fixed points of nonexpansive mappings are an important topic in the theory of nonexpansive mappings and have wide applications in a number of applied areas, such as the convex feasibility problem [1–4], the split feasibility problem [5–7] and image recovery and signal processing [3, 8, 9], and so forth. However, the Picard sequence {𝑇^𝑛𝑥}^∞_𝑛=0 often fails to converge even in the weak topology. Thus, averaged iterations prevail. The Mann iteration [10] is one of the types and is defined by𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,𝑛⩾0, (1.4) where 𝑥₀∈𝐶 is chosen arbitrarily and {𝛼_𝑛}⊂[0,1]. Reich [11] proved that if 𝐸 is a uniformly convex Banach space with a Fréchet differentiable norm and if {𝛼_𝑛} is chosen such that ∑^∞_𝑛=0𝛼_𝑛(1−𝛼_𝑛)=∞, then the sequence {𝑥_𝑛} defined by (1.4) converges weakly to a fixed point of 𝑇. However, we note that Mann iterations have only weak convergence even in a Hilbert space (see e.g., [12]). From a practical point of view, strict pseudocontractions have more powerful applications than nonexpansive mappings do in solving inverse problems (see [13]). Therefore, it is important to develop theory of iterative methods for strict pseudocontractions. Indeed, Browder and Petryshyn [14] prove that if the sequence {𝑥_𝑛} is generated by the following: 𝑥_𝑛+1=𝛼𝑥_𝑛+(1−𝛼)𝑇𝑥_𝑛,𝑛⩾0, (1.5) for any starting point 𝑥₀∈𝐶, 𝛼 is a constant such that 𝜅<𝛼<1, {𝑥_𝑛} converges weakly to a fixed point of strict pseudocontraction. Marino and Xu [15] extended the result of Browder and Petryshyn [14] to Mann iteration (1.4); they proved {𝑥_𝑛} converges weakly to a fixed point of 𝑇, provided the control sequence {𝛼_𝑛} satisfies the conditions that 𝜅<𝛼_𝑛<1 for all 𝑛 and ∑^∞_𝑛=0(𝛼_𝑛−𝑘)(1−𝛼_𝑛)=∞.

The well-known strong convergence theorem for pseudocontractive mapping was proved by Ishikawa [16] in 1974. More precisely, he got the following theorem.

Theorem 1.1 (see [16]). Let 𝐶 be a convex compact subset of a Hilbert space 𝐻 and let 𝑇∶𝐶→𝐶 be a Lipschitzian pseudocontractive mapping. For any 𝑥₁∈𝐶, suppose the sequence {𝑥_𝑛} is defined by 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇𝑦_𝑛,𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇𝑥_𝑛,𝑛⩾1, (1.6) where {𝛼_𝑛}, {𝛽_𝑛} are two real sequences in [0,1] satisfying (i)𝛼_𝑛⩽𝛽_𝑛, 𝑛⩾1, (ii)lim_𝑛→∞𝛽_𝑛=0,(iii)∑^∞_𝑛=1𝛼_𝑛𝛽_𝑛=∞. Then {𝑥_𝑛} converges strongly to a fixed point of 𝑇.

Remark 1.2. (i) Since 0⩽𝛼_𝑛⩽𝛽_𝑛⩽1, 𝑛⩾1 and ∑^∞_𝑛=1𝛼_𝑛𝛽_𝑛=∞, the iterative sequence (1.6) could not be reduced to a Mann iterative sequence (1.4). Therefore, the iterative sequence (1.6) has some particular cases.
(ii) The iterative sequence (1.6) is usually called the Ishikawa iterative sequence.
(iii) Chidume and Mutangadura [17] gave an example to show that the Mann iterative sequence failed to be convergent to a fixed point of Lipschitzian pseudocontractive mapping.

In an infinite-dimensional Hilbert spaces, Mann and Ishikawa's iteration algorithms have only weak convergence, in general, even for nonexpansive mapping. In order to obtain a strong convergence theorem for the Mann iteration method (1.4) to nonexpansive mapping, Nakajo and Takahashi [18] modified (1.4) by employing two closed convex sets that are created in order to form the sequence via metric projection so that strong convergence is guaranteed. Later, it is often referred as the hybrid algorithm or the 𝐶𝑄 algorithm. After that the hybrid algorithm have been studied extensively by many authors (see e.g., [19–23]). Particularly, Martinez-Yanes and Xu [24] and Plubtieng and Ungchittrakool [20] extended the same results of Nakajo and Takahashi [18] to the Ishikawa iteration process. In 2007, Marino and Xu [15] further generalized the hybrid algorithm from nonexpansive mappings to strict pseudocontractive mappings. In 2008, Zhou [25] established the hybrid algorithm for pseudocontractive mapping in the case of the Ishikawa iteration process.

Recently, Yao et al. [26] introduced the hybrid iterative algorithm which just involved one closed convex set for pseudocontractive 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_𝑛𝑇𝑧_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. (1.7)

Theorem 1.3 (see [26]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 be a L-Lipschitz pseudocontraction such that 𝐹(𝑇)≠∅. Assume the sequence {𝛼_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1/(𝐿+1)). Then the sequence {𝑥_𝑛} generated by (1.7) converges strongly to 𝑃_𝐹(𝑇)(𝑥₀).

Very recently, Tang et al. [27] generalized the hybrid algorithm (1.7) in the case of the Ishikawa iterative precess as follows:𝑦_𝑛=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇𝑧_𝑛,𝑧_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇𝑥_𝑛,𝐶_𝑛+1=𝑣∈𝐶_𝑛∶𝛼_𝑛(𝐼−𝑇)𝑦_𝑛²⩽2𝛼_𝑛𝑥_𝑛−𝑣,(𝐼−𝑇)𝑦_𝑛+2𝛼_𝑛𝛽_𝑛𝐿𝑥_𝑛−𝑇𝑥_𝑛𝑦_𝑛−𝑥_𝑛+𝛼_𝑛(𝐼−𝑇)𝑦_𝑛,𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥₀. (1.8) Under some appropriate conditions of {𝛼_𝑛} and {𝛽_𝑛}, they proved that (1.8) converges strongly to 𝑃_𝐹(𝑇)(𝑥₀).

Motivated and inspired by the above works, in this paper, we generalize (1.7) to the Ishikawa iterative process in the case of finding the common fixed point of Lipschitz pseudocontraction and 𝜅-strict pseudocontraction. More precisely, we provide some applications of the main theorem to find the common zero point of the Lipshitz monotone mapping and 𝛾-inverse strongly monotone mapping in Hilbert spaces.

2. Preliminaries

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, (2.1) where 𝑃_𝐶 is called the metric projection of 𝐻 onto 𝐶. We know that 𝑃_𝐶 is a nonexpansive mapping. It is also known that 𝐻 satisfies Opial's condition, that is, for any sequence {𝑥_𝑛} with 𝑥_𝑛⇀𝑥, the inequality liminf_𝑛→∞𝑥_𝑛−𝑥<liminf_𝑛→∞𝑥_𝑛−𝑦 (2.2) holds for every 𝑦∈𝐻 with 𝑦≠𝑥.

For a given sequence {𝑥_𝑛}⊂𝐶, let 𝜔_𝑤(𝑥_𝑛)={𝑥∶∃𝑥_𝑛𝑗⇀𝑥} denote the weak 𝜔-limit set of {𝑥_𝑛}.

Now we collect some Lemmas which will be used in the proof of the main result in the next section. We note that Lemmas 2.1 and 2.2 are well known.

Lemma 2.1. Let 𝐻 be a real Hilbert space. There holds the following identities: (i)‖𝑥−𝑦‖²=‖𝑥‖²−‖𝑦‖²−2⟨𝑥−𝑦,𝑦⟩,forall𝑥,𝑦∈𝐻, (ii)‖𝜆𝑥+(1−𝜆)𝑦‖²=𝜆‖𝑥‖²+(1−𝜆)‖𝑦‖²−𝜆(1−𝜆)‖𝑥−𝑦‖²,forall𝑥,𝑦∈𝐻 and 𝜆∈[0,1].

Lemma 2.2. Let 𝐶 be a closed convex subset of real Hilbert space 𝐻. Given 𝑥∈𝐻 and 𝑧∈𝐶, then 𝑧=𝑃_𝐶𝑥 if and only if there holds the relation 𝑥−𝑧,𝑦−𝑧⩽0,∀𝑦∈𝐶. (2.3)

Proposition 2.3 (see [15, Proposition  2.1]). Assume 𝐶 is a closed convex subset of a Hilbert space 𝐻; let 𝑇∶𝐶→𝐶 be a self-mapping of 𝐶. If 𝑇 is a 𝜅-strict pseudocontraction, then 𝑇 satisfies the Lipschitz condition 𝑇𝑥−𝑇𝑦⩽1+𝜅1−𝜅𝑥−𝑦,∀𝑥,𝑦∈𝐶. (2.4)

Lemma 2.4 (see [28]). Let 𝐻 be a real Hilbert space, let 𝐶 be a closed convex subset of 𝐻, and let 𝑇∶𝐶→𝐶 be a continuous pseudocontractive mapping, then (i)𝐹(𝑇) is closed convex subset of 𝐶, (ii)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.

Lemma 2.5 (see [24]). Let 𝐶 be a closed convex subset of 𝐻. Let {𝑥_𝑛} be a sequence in 𝐻, and let 𝑢∈𝐻. Let 𝑞=𝑃_𝐶𝑢. If {𝑥_𝑛} is such that 𝜔_𝑤(𝑥_𝑛)⊂𝐶 and satisfies the condition 𝑥_𝑛−𝑢⩽𝑢−𝑞,∀𝑛, (2.5) then 𝑥_𝑛→𝑞.

3. Main Result

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻, let 𝑇∶𝐶→𝐶 be 𝐿_𝑇-Lipschitz pseudocontraction, and let 𝑆∶𝐶→𝐶 be 𝜅-strict pseudocontraction with 𝐹∶=𝐹(𝑆)∩𝐹(𝑇)≠∅. 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_𝑛𝑇𝑧_𝑛,𝑧_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑆𝑥_𝑛,𝐶_𝑛+1=𝑣∈𝐶_𝑛∶𝛼_𝑛(𝐼−𝑇)𝑦_𝑛²+(1−𝜅)(𝐼−𝑆)𝑥_𝑛²⩽2𝛼_𝑛𝑥_𝑛−𝑣,(𝐼−𝑇)𝑦_𝑛+2𝑥_𝑛−𝑣,(𝐼−𝑆)𝑧_𝑛+(𝐼−𝑆)𝑥_𝑛+2𝛼_𝑛𝛽_𝑛𝐿_𝑇𝑥_𝑛−𝑆𝑥_𝑛𝑦_𝑛−𝑥_𝑛+𝛼_𝑛(𝐼−𝑇)𝑦_𝑛+𝛽_𝑛2𝛽_𝑛1−𝜅²−1(𝐼−𝑆)𝑥_𝑛²,𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥₀. (3.1) Assume the sequence {𝛼_𝑛}, {𝛽_𝑛} be such that 0<𝑎⩽𝛼_𝑛⩽𝑏<1/(𝐿_𝑇+1)<1 and 0<𝛽_𝑛⩽1 for all 𝑛∈ℕ with lim_𝑛→∞𝛽_𝑛=0. Then {𝑥_𝑛} converges strongly to 𝑃_𝐹(𝑥₀).

Proof. By Lemma 2.4(i), we see that 𝐹(𝑆) and 𝐹(𝑇) are closed and convex, then 𝐹 is as well. Hence, 𝑃_𝐹 is well defined. Next, we will prove by induction that 𝐹⊂𝐶_𝑛 for all 𝑛∈ℕ. Note that 𝐹⊂𝐶=𝐶₁. Assume that 𝐹⊂𝐶_𝑘 holds for 𝑘⩾1. Let 𝑝∈𝐹, thus 𝑝∈𝐶_𝑘, and we observe that 𝑥_𝑘−𝑝−𝛼_𝑘(𝐼−𝑇)𝑦_𝑘²=𝑥_𝑘−𝑝²−𝛼_𝑘(𝐼−𝑇)𝑦_𝑘²−2𝛼_𝑘(𝐼−𝑇)𝑦_𝑘,𝑥_𝑘−𝑝−𝛼_𝑘(𝐼−𝑇)𝑦_𝑘=𝑥_𝑘−𝑝²−𝛼_𝑘(𝐼−𝑇)𝑦_𝑘²−2𝛼_𝑘(𝐼−𝑇)𝑦_𝑘−(𝐼−𝑇)𝑝,𝑦_𝑘−𝑝−2𝛼_𝑘(𝐼−𝑇)𝑦_𝑘,𝑥_𝑘−𝑦_𝑘−𝛼_𝑘(𝐼−𝑇)𝑦_𝑘⩽𝑥_𝑘−𝑝²−𝛼_𝑘(𝐼−𝑇)𝑦_𝑘²−2𝛼_𝑘(𝐼−𝑇)𝑦_𝑘,𝑥_𝑘−𝑦_𝑘−𝛼_𝑘(𝐼−𝑇)𝑦_𝑘=𝑥_𝑘−𝑝²−𝑥_𝑘−𝑦_𝑘+𝑦_𝑘−𝑥_𝑘+𝛼_𝑘(𝐼−𝑇)𝑦_𝑘²−2𝛼_𝑘(𝐼−𝑇)𝑦_𝑘,𝑥_𝑘−𝑦_𝑘−𝛼_𝑘(𝐼−𝑇)𝑦_𝑘=𝑥_𝑘−𝑝²−𝑥_𝑘−𝑦_𝑘²−𝑦_𝑘−𝑥_𝑘+𝛼_𝑘(𝐼−𝑇)𝑦_𝑘²−2𝑥_𝑘−𝑦_𝑘,𝑦_𝑘−𝑥_𝑘+𝛼_𝑘(𝐼−𝑇)𝑦_𝑘−2𝛼_𝑘(𝐼−𝑇)𝑦_𝑘,𝑥_𝑘−𝑦_𝑘−𝛼_𝑘(𝐼−𝑇)𝑦_𝑘⩽𝑥_𝑘−𝑝²−𝑥_𝑘−𝑦_𝑘²−𝑦_𝑘−𝑥_𝑘+𝛼_𝑘(𝐼−𝑇)𝑦_𝑘²+2𝑥_𝑘−𝑦_𝑘−𝛼_𝑘(𝐼−𝑇)𝑦_𝑘,𝑥_𝑘−𝑦_𝑘−𝛼_𝑘(𝐼−𝑇)𝑦_𝑘. (3.2) Consider the last term of (3.2), we obtain 𝑥_𝑘−𝑦_𝑘−𝛼_𝑘(𝐼−𝑇)𝑦_𝑘,𝑦_𝑘−𝑥_𝑘+𝛼_𝑘(𝐼−𝑇)𝑦_𝑘=𝛼_𝑘𝑥_𝑘−𝑇𝑧_𝑘−(𝐼−𝑇)𝑦_𝑘,𝑦_𝑘−𝑥_𝑘+𝛼_𝑘(𝐼−𝑇)𝑦_𝑘=𝛼_𝑘𝑥_𝑘−𝑇𝑥_𝑘+𝑇𝑥_𝑘−𝑇𝑧_𝑘−(𝐼−𝑇)𝑦_𝑘,𝑦_𝑘−𝑥_𝑘+𝛼_𝑘(𝐼−𝑇)𝑦_𝑘=𝛼_𝑘(𝐼−𝑇)𝑥_𝑘−(𝐼−𝑇)𝑦_𝑘,𝑦_𝑘−𝑥_𝑘+𝛼_𝑘(𝐼−𝑇)𝑦_𝑘+𝑇𝑥_𝑘−𝑇𝑧_𝑘,𝑦_𝑘−𝑥_𝑘+𝛼_𝑘(𝐼−𝑇)𝑦_𝑘⩽𝛼_𝑘𝐿_𝑇+1𝑥_𝑘−𝑦_𝑘𝑦_𝑘−𝑥_𝑘+𝛼_𝑘(𝐼−𝑇)𝑦_𝑘+𝛼_𝑘𝐿_𝑇𝑥_𝑘−𝑧_𝑘𝑦_𝑘−𝑥_𝑘+𝛼_𝑘(𝐼−𝑇)𝑦_𝑘=𝛼_𝑘𝐿_𝑇+1𝑥_𝑘−𝑦_𝑘𝑦_𝑘−𝑥_𝑘+𝛼_𝑘(𝐼−𝑇)𝑦_𝑘+𝛼_𝑘𝛽_𝑘𝐿_𝑇𝑥_𝑘−𝑆𝑥_𝑘𝑦_𝑘−𝑥_𝑘+𝛼_𝑘(𝐼−𝑇)𝑦_𝑘⩽𝛼_𝑘𝐿_𝑇+12𝑥_𝑘−𝑦_𝑘²+𝑦_𝑘−𝑥_𝑘+𝛼_𝑘(𝐼−𝑇)𝑦_𝑘²+𝛼_𝑘𝛽_𝑘𝐿_𝑇𝑥_𝑘−𝑆𝑥_𝑘𝑦_𝑘−𝑥_𝑘+𝛼_𝑘(𝐼−𝑇)𝑦_𝑘. (3.3) Substituting (3.3) into (3.2), we obtain 𝑥_𝑘−𝑝−𝛼_𝑘(𝐼−𝑇)𝑦_𝑘²⩽𝑥_𝑘−𝑝²−𝑥_𝑘−𝑦_𝑘²−𝑦_𝑘−𝑥_𝑘+𝛼_𝑘(𝐼−𝑇)𝑦_𝑘²+𝛼_𝑘𝐿_𝑇+1𝑥_𝑘−𝑦_𝑘²+𝑦_𝑘−𝑥_𝑘+𝛼_𝑘(𝐼−𝑇)𝑦_𝑘²+2𝛼_𝑘𝛽_𝑘𝐿_𝑇𝑥_𝑘−𝑆𝑥_𝑘𝑦_𝑘−𝑥_𝑘+𝛼_𝑘(𝐼−𝑇)𝑦_𝑘⩽𝑥_𝑘−𝑝²+2𝛼_𝑘𝛽_𝑘𝐿_𝑇𝑥_𝑘−𝑆𝑥_𝑘𝑦_𝑘−𝑥_𝑘+𝛼_𝑘(𝐼−𝑇)𝑦_𝑘. (3.4) Notice that 𝑥_𝑘−𝑝−𝛼_𝑘(𝐼−𝑇)𝑦_𝑘²=𝑥_𝑘−𝑝²−2𝛼_𝑘𝑥_𝑘−𝑝,(𝐼−𝑇)𝑦_𝑘+𝛼_𝑘(𝐼−𝑇)𝑦_𝑘². (3.5) Therefore, from (3.4) and (3.5), we get 𝛼_𝑘(𝐼−𝑇)𝑦_𝑘²⩽2𝛼_𝑘𝑥_𝑘−𝑝,(𝐼−𝑇)𝑦_𝑘+2𝛼_𝑘𝛽_𝑘𝐿_𝑇𝑥_𝑘−𝑆𝑥_𝑘𝑦_𝑘−𝑥_𝑘+𝛼_𝑘(𝐼−𝑇)𝑦_𝑘. (3.6) On the other hand, we found that 𝑥_𝑘−𝑝−𝛽_𝑘(𝐼−𝑆)𝑧_𝑘²=𝑥_𝑘−𝑝²−𝛽_𝑘(𝐼−𝑆)𝑧_𝑘²−2𝛽_𝑘(𝐼−𝑆)𝑧_𝑘,𝑥_𝑘−𝑝−𝛽_𝑘(𝐼−𝑆)𝑧_𝑘=𝑥_𝑘−𝑝²−𝛽_𝑘(𝐼−𝑆)𝑧_𝑘²−2𝛽_𝑘(𝐼−𝑆)𝑧_𝑘−(𝐼−𝑆)𝑝,𝑧_𝑘−𝑝−2𝛽_𝑘(𝐼−𝑆)𝑧_𝑘,𝑥_𝑘−𝑧_𝑘−𝛽_𝑘(𝐼−𝑆)𝑧_𝑘⩽𝑥_𝑘−𝑝²−𝛽_𝑘(𝐼−𝑆)𝑧_𝑘²−2𝛽_𝑘(𝐼−𝑆)𝑧_𝑘,𝑥_𝑘−𝑧_𝑘−𝛽_𝑘(𝐼−𝑆)𝑧_𝑘=𝑥_𝑘−𝑝²−𝛽_𝑘(𝐼−𝑆)𝑧_𝑘²+𝛽_𝑘(𝐼−𝑆)𝑧_𝑘²−𝑥_𝑘−𝑧_𝑘²+𝑥_𝑘−𝑧_𝑘−𝛽_𝑘(𝐼−𝑆)𝑧_𝑘²=𝑥_𝑘−𝑧_𝑘+𝑧_𝑘−𝑝²−𝑥_𝑘−𝑧_𝑘²+𝛽_𝑘(𝐼−𝑆)𝑥_𝑘−𝛽_𝑘(𝐼−𝑆)𝑧_𝑘²=𝑥_𝑘−𝑧_𝑘²+2𝑥_𝑘−𝑧_𝑘,𝑧_𝑘−𝑝+𝑧_𝑘−𝑝²−𝑥_𝑘−𝑧_𝑘²+𝛽_𝑘(𝐼−𝑆)𝑥_𝑘−𝛽_𝑘(𝐼−𝑆)𝑧_𝑘²=2𝑥_𝑘−𝑧_𝑘,𝑧_𝑘−𝑥_𝑘+𝑥_𝑘−𝑝+(1−𝛽_𝑘)(𝑥_𝑘−𝑝)+𝛽_𝑘(𝑆𝑥_𝑘−𝑝)²+𝛽_𝑘(𝐼−𝑆)𝑥_𝑘−𝛽_𝑘(𝐼−𝑆)𝑧_𝑘²⩽2𝑥_𝑘−𝑝,𝛽_𝑘(𝐼−𝑆)𝑥_𝑘+1−𝛽_𝑘𝑥_𝑘−𝑝²+𝛽_𝑘𝑆𝑥_𝑘−𝑝²−𝛽_𝑘1−𝛽_𝑘𝑥_𝑘−𝑆𝑥_𝑘²−2𝛽²_𝑘(𝐼−𝑆)𝑥_𝑘²+𝛽²_𝑘1+𝜅1−𝜅+1²𝑥_𝑘−𝑧_𝑘²⩽2𝑥_𝑘−𝑝,𝛽_𝑘(𝐼−𝑆)𝑥_𝑘+1−𝛽_𝑘𝑥_𝑘−𝑝²+𝛽_𝑘𝑥_𝑘−𝑝²+𝛽_𝑘𝜅(𝐼−𝑆)𝑥_𝑘²−𝛽_𝑘1−𝛽_𝑘(𝐼−𝑆)𝑥_𝑘²−2𝛽²_𝑘(𝐼−𝑆)𝑥_𝑘²+𝛽⁴_𝑘21−𝜅²(𝐼−𝑆)𝑥_𝑘²=2𝑥_𝑘−𝑝,𝛽_𝑘(𝐼−𝑆)𝑥_𝑘+𝑥_𝑘−𝑝²−𝛽_𝑘(1−𝜅)(𝐼−𝑆)𝑥_𝑘²−𝛽²_𝑘(𝐼−𝑆)𝑥_𝑘²+𝛽⁴_𝑘21−𝜅²(𝐼−𝑆)𝑥_𝑘². (3.7) Notice that 𝑥_𝑘−𝑝−𝛽_𝑘(𝐼−𝑆)𝑧_𝑘²=𝑥_𝑘−𝑝²−2𝛽_𝑘𝑥_𝑘−𝑝,(𝐼−𝑆)𝑧_𝑘+𝛽²_𝑘(𝐼−𝑆)𝑧_𝑘². (3.8) Combining (3.7) and (3.8) and then it implies that 𝛽_𝑘(1−𝜅)𝑥_𝑘−𝑆𝑥_𝑘²⩽2𝛽_𝑘𝑥_𝑘−𝑝,(𝐼−𝑆)𝑧_𝑘+(𝐼−𝑆)𝑥_𝑘+𝛽²_𝑘2𝛽_𝑘1−𝜅²−1(𝐼−𝑆)𝑥_𝑘². (3.9) Since 𝛽_𝑛>0 for all 𝑛, so we get (1−𝜅)𝑥_𝑘−𝑆𝑥_𝑘²⩽2𝑥_𝑘−𝑝,(𝐼−𝑆)𝑧_𝑘+(𝐼−𝑆)𝑥_𝑘+𝛽_𝑘2𝛽_𝑘1−𝜅²−1(𝐼−𝑆)𝑥_𝑘². (3.10) It follows from (3.6) and (3.10) that we obtain 𝛼_𝑘(𝐼−𝑇)𝑦_𝑘²+(1−𝜅)(𝐼−𝑆)𝑥_𝑘²⩽2𝛼_𝑘𝑥_𝑘−𝑣,(𝐼−𝑇)𝑦_𝑘+2𝑥_𝑘−𝑣,(𝐼−𝑆)𝑧_𝑘+(𝐼−𝑆)𝑥_𝑘+2𝛼_𝑘𝛽_𝑘𝐿_𝑇𝑥_𝑘−𝑆𝑥_𝑘𝑦_𝑘−𝑥_𝑘+𝛼_𝑘(𝐼−𝑇)𝑦_𝑘+𝛽_𝑘2𝛽_𝑘1−𝜅²−1(𝐼−𝑆)𝑥_𝑘². (3.11) Therefore, 𝑝∈𝐶_𝑘+1. By mathematical induction, we have 𝐹⊂𝐶_𝑛 for all 𝑛∈ℕ. It is easy to check that 𝐶_𝑛 is closed and convex, and then {𝑥_𝑛} is well defined. From 𝑥_𝑛=𝑃_𝐶𝑛(𝑥₀), we have ⟨𝑥₀−𝑥_𝑛,𝑥_𝑛−𝑦⟩⩾0 for all 𝑦∈𝐶_𝑛. Using 𝐹⊂𝐶_𝑛, we also have ⟨𝑥₀−𝑥_𝑛,𝑥_𝑛−𝑢⟩⩾0 for all 𝑢∈𝐹. So, for 𝑢∈𝐹, we have 0⩽𝑥₀−𝑥_𝑛,𝑥_𝑛−𝑢=𝑥₀−𝑥_𝑛,𝑥_𝑛−𝑥₀+𝑥₀𝑥−𝑢=−₀−𝑥_𝑛²+𝑥₀−𝑥_𝑛,𝑥₀𝑥−𝑢⩽−₀−𝑥_𝑛²+𝑥₀−𝑥_𝑛𝑥₀−𝑢. (3.12) Hence, ‖𝑥₀−𝑥_𝑛‖⩽‖𝑥₀−𝑢‖,forall𝑢∈𝐹. In particular, 𝑥₀−𝑥_𝑛⩽𝑥₀−𝑞,where𝑞=𝑃_𝐹𝑥₀. (3.13) This implies that {𝑥_𝑛} is bounded, and then {𝑦_𝑛}, {𝑇𝑦_𝑛}, {𝑧_𝑛}, {𝑆𝑧_𝑛}, and {𝑆𝑥_𝑛} are as well.
From 𝑥_𝑛=𝑃_𝐶𝑛(𝑥₀) and 𝑥_𝑛+1=𝑃_𝐶𝑛+1(𝑥₀)∈𝐶_𝑛+1⊂𝐶_𝑛, we have 𝑥₀−𝑥_𝑛,𝑥_𝑛−𝑥_𝑛+1⩾0. (3.14) Hence 0⩽𝑥₀−𝑥_𝑛,𝑥_𝑛−𝑥_𝑛+1=𝑥₀−𝑥_𝑛,𝑥_𝑛−𝑥₀+𝑥₀−𝑥_𝑛+1=−𝑥₀−𝑥_𝑛²+𝑥₀−𝑥_𝑛,𝑥₀−𝑥_𝑛+1⩽−𝑥₀−𝑥_𝑛²+𝑥₀−𝑥_𝑛𝑥₀−𝑥_𝑛+1, (3.15) and; therefore, 𝑥₀−𝑥_𝑛⩽𝑥₀−𝑥_𝑛+1, (3.16) which implies that lim_𝑛→∞‖𝑥_𝑛−𝑥₀‖ exists. From Lemma 2.1 and (3.14), we obtain 𝑥_𝑛+1−𝑥_𝑛²=𝑥_𝑛+1−𝑥₀−𝑥_𝑛−𝑥₀²=𝑥_𝑛+1−𝑥₀²−𝑥_𝑛−𝑥₀²−2𝑥_𝑛+1−𝑥_𝑛,𝑥_𝑛−𝑥₀⩽𝑥_𝑛+1−𝑥₀²−𝑥_𝑛−𝑥₀²⟶0. (3.17) Since 𝑥_𝑛+1∈𝐶_𝑛+1⊂𝐶_𝑛, we have 𝛼_𝑛(𝐼−𝑇)𝑦_𝑛²+(1−𝜅)(𝐼−𝑆)𝑥_𝑛²⩽2𝛼_𝑛𝑥_𝑛−𝑥_𝑛+1,(𝐼−𝑇)𝑦_𝑛+2𝑥_𝑛−𝑥_𝑛+1,(𝐼−𝑆)𝑧_𝑛+(𝐼−𝑆)𝑥_𝑛+2𝛼_𝑛𝛽_𝑛𝐿_𝑇𝑥_𝑛−𝑆𝑥_𝑛𝑦_𝑛−𝑥_𝑛+𝛼_𝑛(𝐼−𝑇)𝑦_𝑛+𝛽_𝑛2𝛽_𝑛1−𝜅²−1(𝐼−𝑆)𝑥_𝑛²⟶0as𝑛⟶∞, (3.18) therefore, we obtain 𝑦_𝑛−𝑇𝑦_𝑛⟶0,𝑥_𝑛−𝑆𝑥_𝑛⟶0. (3.19) We note that 𝑥_𝑛−𝑇𝑥_𝑛⩽𝑥_𝑛−𝑦_𝑛+𝑦_𝑛−𝑇𝑦_𝑛+𝑇𝑦_𝑛−𝑇𝑥_𝑛⩽𝐿_𝑇+1𝑥_𝑛−𝑦_𝑛+𝑦_𝑛−𝑇𝑦_𝑛⩽𝛼_𝑛𝐿_𝑇+1𝑥_𝑛−𝑇𝑧_𝑛+𝑦_𝑛−𝑇𝑦_𝑛⩽𝛼_𝑛𝐿_𝑇+1𝑥_𝑛−𝑇𝑥_𝑛+𝛼_𝑛𝐿_𝑇+1𝑇𝑥_𝑛−𝑇𝑧_𝑛+𝑦_𝑛−𝑇𝑦_𝑛⩽𝛼_𝑛𝐿_𝑇+1𝑥_𝑛−𝑇𝑥_𝑛+𝛼_𝑛𝛽_𝑛𝐿_𝑇𝐿_𝑇+1𝑥_𝑛−𝑆𝑥_𝑛+𝑦_𝑛−𝑇𝑦_𝑛, (3.20) that is, 𝑥_𝑛−𝑇𝑥_𝑛⩽𝛼_𝑛𝛽_𝑛𝐿_𝑇𝐿_𝑇+11−𝛼_𝑛𝐿_𝑇𝑥+1^𝑛−𝑆𝑥_𝑛+11−𝛼_𝑛𝐿_𝑇𝑦+1^𝑛−𝑇𝑦_𝑛⟶0,as𝑛⟶∞. (3.21)
By Lemma 2.4(ii), 𝐼−𝑇 and 𝐼−𝑆 are demiclosed at zero. Together with the fact that {𝑥_𝑛} is bounded, which guarantees that every weak limit point of {𝑥_𝑛} is a fixed point of 𝑇 and 𝑆, that is 𝜔_𝑤(𝑥_𝑛)⊂𝐹(𝑇)∩𝐹(𝑆)=𝐹, therefore, by inequality (3.13) and Lemma 2.5, we know that {𝑥_𝑛} converges strongly to 𝑞=𝑃_𝐹(𝑥₀). This completes the proof.

If 𝑆=𝐼, then we obtain the following corollary.

Corollary 3.2 (Yao et al. [26, Theorem  3.1]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 be 𝐿-Lipschitz pseudocontraction such that 𝐹(𝑇)≠∅. Assume the sequence {𝛼_𝑛} be such that 0<𝑎⩽𝛼_𝑛⩽𝑏<1/(𝐿+1)<1 for all 𝑛. Then the sequence {𝑥_𝑛} generated by (1.7) converges strongly to 𝑃_𝐹(𝑇)(𝑥₀).

If 𝑇 and 𝑆 are nonexpansive, then we also have the following corollary.

Corollary 3.3. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻, and let 𝑆,𝑇∶𝐶→𝐶 be nonexpansive mappings. Suppose that 𝐹∶=𝐹(𝑆)∩𝐹(𝑇)≠∅. Assume the sequence {𝛼_𝑛}  be such that 0<𝑎⩽𝛼_𝑛⩽𝑏<1/2 and 0<𝛽_𝑛⩽1 for all 𝑛∈ℕ with lim_𝑛→∞𝛽_𝑛=0. 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_𝑛𝑇𝑧_𝑛,𝑧_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑆𝑥_𝑛,𝐶_𝑛+1=𝑣∈𝐶_𝑛∶𝛼_𝑛(𝐼−𝑇)𝑦_𝑛²+(𝐼−𝑆)𝑥_𝑛²⩽2𝛼_𝑛𝑥_𝑛−𝑣,(𝐼−𝑇)𝑦_𝑛+2𝑥_𝑛−𝑣,(𝐼−𝑆)𝑧_𝑛+(𝐼−𝑆)𝑥_𝑛+2𝛼_𝑛𝛽_𝑛𝑥_𝑛−𝑆𝑥_𝑛𝑦_𝑛−𝑥_𝑛+𝛼_𝑛(𝐼−𝑇)𝑦_𝑛+𝛽_𝑛4𝛽²_𝑛−1(𝐼−𝑆)𝑥_𝑛²,𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥₀. (3.22) Then {𝑥_𝑛} converges strongly to 𝑃_𝐹(𝑥₀).

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 monotone and Lipschitz continuous, that is, ‖𝐴𝑥−𝐴𝑦‖⩽(1/𝛾)‖𝑥−𝑦‖. It is well known (see e.g., [29]) 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 pseudocontractive mapping and strictly pseudocontractive mapping are strongly related to the monotone mapping and inverse strongly monotone mapping, respectively. It is well known that (i)𝐴 is monotone ⇔𝑇∶=(𝐼−𝐴) is pseudocontractive, (ii)𝐴 is inverse strongly monotone ⇔𝑇∶=(𝐼−𝐴) is strictly pseudocontractive.

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. (3.23) Without loss of generality, we can assume that 𝛾∈(0,1/2], and then it yields 𝑥−𝑦,𝐴𝑥−𝐴𝑦⩾𝛾𝐴𝑥−𝐴𝑦²⟺−2𝑥−𝑦,𝐴𝑥−𝐴𝑦⩽−2𝛾𝐴𝑥−𝐴𝑦²⟺(𝐼−𝐴)𝑥−(𝐼−𝐴)𝑦²⩽𝑥−𝑦²+1−2𝛾𝐴𝑥−𝐴𝑦²((⟺via3.23))𝑇𝑥−𝑇𝑦²⩽𝑥−𝑦²+𝜅(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦²where𝑇∶=(𝐼−𝐴),𝜅∶=1−2𝛾. (3.24) Due to Theorem 3.1, we have the following corollary which generalize the corresponding results of Yao et al. [26].

Corollary 3.4. Let 𝐴∶𝐻→𝐻 be 𝐿_𝐴-Lipschitz monotone mapping and let 𝐵∶𝐻→𝐻 be an 𝛾-inverse strongly monotone which 𝐴⁻¹(0)∩𝐵⁻¹(0)≠∅. Assume the sequence {𝛼_𝑛} be such that 0<𝑎⩽𝛼_𝑛⩽𝑏<1/(𝐿_𝐴+2), 0<𝛽_𝑛⩽1 for all 𝑛∈ℕ with lim_𝑛→∞𝛽_𝑛=0 and 𝛾∈(0,1/2] such that 𝛾⩾𝛾. Let 𝑥₀∈𝐻. For 𝐶₁=𝐻 and 𝑥₁=𝑃_𝐶1(𝑥₀)=𝑥₀, define a sequence {𝑥_𝑛} as follows: 𝑦_𝑛=𝑥_𝑛−𝛼_𝑛𝑥_𝑛−𝑧_𝑛−𝛼_𝑛𝐴𝑧_𝑛,𝑧_𝑛=𝑥_𝑛−𝛽_𝑛𝐵𝑥_𝑛,𝐶_𝑛+1=𝑣∈𝐶_𝑛∶𝛼_𝑛𝐴𝑦_𝑛²+2𝛾𝐵𝑥_𝑛²⩽2𝛼_𝑛𝑥_𝑛−𝑣,𝐴𝑦_𝑛+2𝑥_𝑛−𝑣,𝐵𝑧_𝑛+𝐵𝑥_𝑛+2𝛼_𝑛𝛽_𝑛𝐿_𝐴+1𝐵𝑥_𝑛𝑦_𝑛−𝑥_𝑛+𝛼_𝑛𝐴𝑦_𝑛+𝛽_𝑛𝛽_𝑛𝛾²−1𝐵𝑥_𝑛²,𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥₀. (3.25) Then {𝑥_𝑛} converges strongly to 𝑃_𝐴⁻¹_(0)∩𝐵⁻¹₍₀₎(𝑥₀).

Proof. Let 𝑇∶=(𝐼−𝐴) and let 𝑆∶=(𝐼−𝐵). Then 𝑇 and 𝑆 are pseudocontractive and (1−2𝛾)-pseudocontractive, respectively. Moreover, 𝑇 is also (𝐿_𝐴+1)-Lipschitz, and if we set 𝜅∶=1−2𝛾, 𝑆 is also ((1−𝛾)/𝛾)-Lipschitz, and then (2/(1−𝜅))²=1/𝛾². Hence, it follows from Theorem 3.1 that we have the desired result.

If 𝐵=0 (zero mapping), then 𝑧_𝑛=𝑥_𝑛 and 𝐵⁻¹(0)=𝐻. So, we obtain the following corollary.

Corollary 3.5 (Yao et al. [26, Corollary  3.2]). Let 𝐴∶𝐻→𝐻 be a 𝐿_𝐴-Lipschitz monotone mapping for which 𝐴⁻¹(0)≠∅. Assume that the sequence {𝛼_𝑛} be as in Corollary 3.4. Then the sequence {𝑥_𝑛} generated by 𝑦_𝑛=𝑥_𝑛−𝛼_𝑛𝐴𝑧_𝑛,𝐶_𝑛+1=𝑣∈𝐶_𝑛∶𝛼_𝑛𝐴𝑦_𝑛²⩽2𝛼_𝑛𝑥_𝑛−𝑣,𝐴𝑦_𝑛,𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥₀ (3.26) strongly converges to 𝑃_𝐴⁻¹₍₀₎(𝑥₀).

Acknowledgments

The author is supported by the Naresuan University and the Thailand Research Fund under Grant no. MRG5380249. Finally, the author would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper.