Strong Convergence for a Strongly Quasi-Nonexpansive Sequence in Hilbert Spaces

Abstract

We prove a strong convergence theorem for strongly quasi-nonexpansive sequence of mappings in Hilbert spaces. Moreover, we can improve the recent results of Tian and Jin (2011). We also give a simple proof of Marino-Xu’s result (2006).

1. Introduction

Let 𝐻 be a Hilbert space with inner product ⟨⋅,⋅⟩ and induced norm ‖⋅‖. Recall that a mapping 𝑇∶𝐻→𝐻 is said to be 𝐿-Lipschitzian where 𝐿>0 if ‖𝑇𝑥−𝑇𝑦‖≤𝐿‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐻. In this paper, we are interested in nonexpansive mappings (that is, 1-Lipschitzian ones) and contractions (that is, 𝐿-Lipschitzian ones with 𝐿<1). The problem of finding a fixed point of such mappings plays an important role in many nonlinear equations appearing in both pure and applied sciences. The celebrated Banach’s contraction principle is probably known as the major tool for the case of contraction mappings. However, for nonexpansive mappings, the situation is more difficult and different.

In 2000, Moudafi [1] introduced the viscosity approximation method, starting with an arbitrary initial 𝑥₁∈𝐻, and defined a sequence {𝑥_𝑛} by 𝑥_𝑛+1=𝜀_𝑛1+𝜀_𝑛𝑓𝑥_𝑛+11+𝜀_𝑛𝑇𝑥_𝑛(𝑛≥1), (1) where 𝑇 is a nonexpansive mapping, 𝑓∶𝐻→𝐻 is a contraction, and {𝜀_𝑛} is a sequence in (0,1) satisfying (M1)lim_𝑛→∞𝜀_𝑛=0;(M2)∑^∞_𝑛=1𝜀_𝑛=∞;(M3)lim_𝑛→∞(1/𝜀_𝑛)−(1/𝜀_𝑛+1)=0.It was proved that the sequence {𝑥_𝑛} generated by (1) converges to a fixed point 𝑧 of 𝑇 and the following inequality holds: 𝑓(𝑧)−𝑧,𝑞−𝑧≤0∀𝑞∈Fix(𝑇)∶={𝑥∈𝐻∶𝑥=𝑇𝑥}. (2) In the literature, Moudafi’s scheme has been widely studied and extended (see [2, 3]). It should be noted that the convergence of Moudafi’s scheme is equivalent to that of its special setting with a constant contraction 𝑓 (see [4]). In fact, this follows from the role of the nonexpansiveness of 𝑇.

In the earlier result, the following scheme was studied by Halpern [5]; starting with an arbitrary initial 𝑥₁∈𝐻 and a given 𝑢∈𝐻, he defined a sequence {𝑥_𝑛} by 𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇𝑥_𝑛(𝑛≥1), (3) where {𝛼_𝑛} is a certain sequence in (0,1). In fact, Halpern proved in 1967 the convergence of the iterative sequence {𝑥_𝑛} where 𝛼_𝑛=𝑛^−𝜃 and 𝜃∈(0,1). Many researchers (see, e.g., [6, 7]) have improved Halpern’s result from Hilbert spaces to certain Banach spaces with the following conditions on {𝛼_𝑛}: (C1)lim_𝑛→∞𝛼_𝑛=0;(C2)∑^∞_𝑛=1𝛼_𝑛=∞;(C3)lim_𝑛→∞(𝛼_𝑛/𝛼_𝑛+1)=1 or ∑^∞_𝑛=1|𝛼_𝑛−𝛼_𝑛+1|<∞.Halpern also showed that conditions (C1) and (C2) are necessary for the convergence of the sequence generated by (3) for any given 𝑥₁,𝑢∈𝐻.

On the other hand, Chidume-Chidume [8] and Suzuki [9] independently discovered that together just conditions (C1) and (C2) are sufficient for the convergence of the following iterative sequence: 𝑥₁,𝑢∈𝐶,𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇_𝜆𝑥_𝑛(𝑛≥1), (4) where 𝑇_𝜆=𝜆𝐼+(1−𝜆)𝑇 and 𝜆∈(0,1). Recently, Saejung [10] proved that the conclusion remains true if 𝑇 is a strongly nonexpansive mapping. It is noted that in Hilbert spaces the mapping 𝑇_𝜆 is strongly nonexpansive whenever 𝜆∈(0,1). Recall that a mapping 𝑇∶𝐻→𝐻 is strongly nonexpansive (see [11, 12]) if it is nonexpansive and lim_𝑛→∞‖(𝑥_𝑛−𝑦_𝑛)−(𝑇𝑥_𝑛−𝑇𝑦_𝑛)‖=0 whenever {𝑥_𝑛}, {𝑦_𝑛} are sequences in 𝐻 such that {𝑥_𝑛−𝑦_𝑛} is bounded and lim_𝑛→∞(‖𝑥_𝑛−𝑦_𝑛‖−‖𝑇𝑥_𝑛−𝑇𝑦_𝑛‖)=0.

In the aforementioned results, it was assumed that 𝑇 has a fixed point; that is, Fix(𝑇)≠∅. Now we consider the following more general settings. A mapping 𝑇∶𝐻→𝐻 is(i)quasi-nonexpansive if Fix(𝑇)≠∅ and ‖𝑇𝑥−𝑞‖≤‖𝑥−𝑞‖ for all 𝑥∈𝐻 and 𝑞∈Fix(𝑇);(ii)strongly quasi-nonexpansive if it is quasi-nonexpansive and lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0 whenever {𝑥_𝑛} is a bounded sequence in 𝐻 such that lim_𝑛→∞(‖𝑥_𝑛−𝑞‖−‖𝑇𝑥_𝑛−𝑞‖)=0 for some 𝑞∈Fix(𝑇).In 2010, Maingé [2] proved the convergence of the sequence {𝑥_𝑛} defined by 𝑥₁∈𝐻 and 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑇_𝜔𝑥_𝑛, (5) where 𝑇_𝜔=(1−𝜔)𝐼+𝜔𝑇, 𝜔∈(0,1/2) and 𝑇 is a quasi-nonexpansive mapping under the conditions (C1) and (C2). In 2011, Wongchan and Saejung [13] improved Maingé’s result by replacing 𝑇_𝜔 with a strongly nonexpansive mapping 𝑇. Hence, the restriction 𝜔∈(0,1/2) can be extended to 𝜔∈(0,1).

There are also some other iterative schemes closely related to the schemes above studied by many authors. For example, inspired by the scheme studied by Yamada [14], Tian and Jin [15, 16] recently proposed the following iterative scheme, starting with an arbitrary initial 𝑥₁∈𝐻 and 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝜇𝐹𝑇_𝜔𝑥_𝑛(𝑛≥1), (6) where 𝑓 and 𝑇_𝜔 are the same as Maingé’s result but 𝐹∶𝐻→𝐻 is strongly monotone and Lipschitzian.

A careful reading shows that there are some connections between them. We will discuss and consolidate them into the following scheme: Started with an arbitrary initial 𝑥₁∈𝐻 and 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+𝑔𝑇_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇_𝑛𝑥_𝑛(𝑛≥1), (7) where 𝑓, 𝑔 are Lipschitzian and {𝑇_𝑛} is a certain sequence of quasi-nonexpansive mappings.

2. Preliminaries

In this section, we collect together some known lemmas which are our main tool in proving our results. Let 𝐶 be a closed and convex subset of 𝐻. Recall that the metric projection 𝑃_𝐶∶𝐻→𝐶 is defined as follows: for 𝑥∈𝐻, 𝑃_𝐶𝑥 is the only one point in 𝐶 satisfying 𝑥−𝑃_𝐶𝑥=inf𝑥−𝑦∶𝑦∈𝐶. (8)

Lemma 1 (see [17]). Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻. Then for 𝑥∈𝐻 and 𝑦∈𝐶, 𝑦=𝑃_𝐶𝑥 if and only if ⟨𝑥−𝑦,𝑦−𝑧⟩≥0 for all 𝑧∈𝐶.

Lemma 2. Let 𝐻 be a Hilbert space. Then 𝑥+𝑦²≤‖𝑥‖²+2⟨𝑦,𝑥+𝑦⟩ (9) for all 𝑥,𝑦∈𝐻.

We also need the following lemma.

Lemma 3 (see [18, Lemma 2.5]). Let {𝑎_𝑛}⊂[0,∞), {𝛼_𝑛}⊂[0,1), and {𝑏_𝑛}⊂(−∞,∞), 𝛼∈[0,1) be such that (i){𝑎_𝑛} is a bounded sequence;(ii)𝑎_𝑛+1≤(1−𝛼_𝑛)²𝑎_𝑛+2𝛼_𝑛𝛼√𝑎_𝑛√𝑎_𝑛+1+𝛼_𝑛𝑏_𝑛 for all 𝑛∈ℕ;(iii)whenever {𝑎_𝑛𝑘} is a subsequence of {𝑎_𝑛} satisfying liminf_𝑘→∞(𝑎_𝑛𝑘+1−𝑎_𝑛𝑘)≥0, it follows that limsup_𝑘→∞𝑏_𝑛𝑘≤0;(iv)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞.Then lim_𝑛→∞𝑎_𝑛=0.

Lemma 4 (see [19, Lemma 2.3]). Let {𝑠_𝑛} be a sequence of nonnegative real numbers, {𝛼_𝑛} a sequence of real numbers in [0,1] with ∑^∞_𝑛=1𝛼_𝑛=∞, {𝑢_𝑛} a sequence of nonnegative real numbers with ∑^∞_𝑛=1𝑢_𝑛<∞, and {𝑡_𝑛} a sequence of real numbers with limsup_𝑛→∞𝑡_𝑛≤0. Suppose that 𝑠_𝑛+1≤1−𝛼_𝑛𝑠_𝑛+𝛼_𝑛𝑡_𝑛+𝑢_𝑛∀𝑛∈ℕ. (10) Then lim_𝑛→∞𝑠_𝑛=0.

3. Main Results

Recall that {𝑇_𝑛∶𝐻→𝐻} is a strongly quasi-nonexpansive sequence if it satisfies the following conditions:(1)⋂^∞_𝑛=1Fix(𝑇_𝑛)≠∅;(2)‖𝑇_𝑛𝑥−𝑝‖≤‖𝑥−𝑝‖ for all 𝑥∈𝐻 and 𝑝∈⋂^∞_𝑛=1Fix(𝑇_𝑛) and for all 𝑛∈ℕ;(3)lim_𝑛→∞‖𝑥_𝑛−𝑇_𝑛𝑥_𝑛‖=0 whenever {𝑥_𝑛} is a bounded sequence in 𝐻 such that lim_𝑛→∞(‖𝑥_𝑛−𝑞‖−‖𝑇_𝑛𝑥_𝑛−𝑞‖)=0 for some 𝑞∈⋂^∞_𝑛=1Fix(𝑇_𝑛).We also say that {𝑇_𝑛} satisfies the NST-condition if whenever {𝑧_𝑛} is a bounded sequence in 𝐻 such that lim_𝑛→∞‖𝑧_𝑛−𝑇_𝑛𝑧_𝑛‖=0 it follows that every weak cluster point of {𝑧_𝑛} belongs to ⋂^∞_𝑛=1Fix(𝑇_𝑛).

Remark 5. (1)Being strongly nonexpansive the sequence and NST-condition are apparently inherited by subsequences.(2)Suppose that 𝑇_𝑛=𝑇∶𝐻→𝐻 for all 𝑛≥1.(i)If 𝑇 is a strongly nonexpansive mapping, then {𝑇_𝑛} is a strongly nonexpansive sequence.(ii)If 𝐼−𝑇 is demiclosed at zero, then {𝑇_𝑛} satisfies NST-condition.Recall that 𝐼−𝑇∶𝐻→𝐻 is demiclosed at zero if {𝑥_𝑛} is a sequence in 𝐻 such that lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0 and 𝑤−lim_𝑛→∞𝑥_𝑛=𝑝; then 𝑝∈Fix(𝑇).

We now state our main theorem.

Theorem 6. Let {𝑇_𝑛∶𝐻→𝐻} be a strongly quasi-nonexpansive sequence satisfying the NST-condition. Let 𝑓,𝑔∶𝐻→𝐻 be 𝛼- and 𝛽-Lipschitzian, respectively. Suppose that {𝑥_𝑛} is given by 𝑥₁∈𝐻 and 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+𝑔𝑇_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇_𝑛𝑥_𝑛(𝑛≥1), (11) where {𝛼_𝑛} is a sequence in (0,1) satisfying the conditions (C1) and (C2). Suppose that 𝛼+𝛽<1. Then {𝑥_𝑛} converges strongly to 𝑝=𝑃_⋂^∞_{𝑛=1Fix(𝑇𝑛)}(𝑓+𝑔)(𝑝).

Before we give the proof, we note that 𝐹∶=⋂^∞_𝑛=1Fix(𝑇_𝑛) is closed and convex. It follows from 𝛼+𝛽<1 that 𝑓+𝑔 is an (𝛼+𝛽)-contraction. Then the mapping 𝑃_𝐹(𝑓+𝑔)∶𝐹→𝐹 is a contraction. By Banach’s contraction principle, there exists a unique element 𝑝∈𝐹 such that 𝑝=𝑃_𝐹(𝑓+𝑔)(𝑝). It follows then from Lemma 1 that ⟨(𝑓+𝑔)(𝑝)−𝑝,𝑧−𝑝⟩≤0 for all 𝑧∈𝐹.

Let us consider the following three lemmas first.

Lemma 7. The sequence {𝑥_𝑛} is bounded. Hence, so are the sequences {𝑓(𝑥_𝑛)}, {𝑇_𝑛𝑥_𝑛}, and {𝑔(𝑇_𝑛𝑥_𝑛)}.

Proof. We consider the following inequality: 𝑥_𝑛+1−𝑝≤𝛼_𝑛𝑓𝑥_𝑛+𝑔𝑇_𝑛𝑥_𝑛+−𝑝1−𝛼_𝑛𝑇_𝑛𝑥_𝑛−𝑝. (12) Since each 𝑇_𝑛 is quasi-nonexpansive and 𝑝∈⋂^∞_𝑛=1Fix(𝑇_𝑛), we have 𝑇_𝑛𝑥_𝑛−𝑝≤𝑥_𝑛−𝑝. (13) It follows from the Lipschitzian conditions of 𝑓 and 𝑔, respectively that, 𝛼_𝑛𝑓𝑥_𝑛+𝑔𝑇_𝑛𝑥_𝑛−𝑝≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑝+𝛼_𝑛𝑔𝑇_𝑛𝑥_𝑛−𝑔𝑝+𝛼_𝑛𝑓𝑝+𝑔𝑝−𝑝≤𝛼𝛼_𝑛𝑥_𝑛−𝑝+𝛽𝛼_𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑝+𝑔𝑝−𝑝. (14) Then, we have 𝑥_𝑛+1≤−𝑝1−𝛼_𝑛1−𝛼+𝛽𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑝𝑝1−𝛼+𝛽+𝑔−𝑝𝑥1−𝛼+𝛽≤max_𝑛,𝑓𝑝𝑝−𝑝+𝑔−𝑝.1−𝛼+𝛽 (15) By induction, for all 𝑛≥1, we have 𝑥_𝑛+1−𝑝≤max𝑥₁,𝑓𝑝𝑝−𝑝+𝑔−𝑝1−𝛼+𝛽. (16) In particular, the sequence {𝑥_𝑛} is bounded.

Lemma 8. The following inequality holds for all 𝑛≥1: 𝑥_𝑛+1−𝑝²≤1−𝛼_𝑛²𝑥_𝑛−𝑝²+2𝛼+𝛽𝛼_𝑛𝑥_𝑛×𝑥−𝑝_𝑛+1−𝑝+2𝛼_𝑛⟨𝑓𝑝+𝑔𝑝−𝑝,𝑥_𝑛+1−𝑝⟩. (17)

Proof. It follows from Lemma 2 that 𝑥_𝑛+1−𝑝²=𝛼_𝑛𝑓𝑥_𝑛+𝑔𝑇_𝑛𝑥_𝑛−𝑝+1−𝛼_𝑛𝑇_𝑛𝑥_𝑛−𝑝²≤1−𝛼_𝑛²𝑇_𝑛𝑥_𝑛−𝑝²+2𝛼_𝑛𝑓𝑥_𝑛+𝑔𝑇_𝑛𝑥_𝑛−𝑝,𝑥_𝑛+1−𝑝. (18) Since each 𝑇_𝑛 is quasi-nonexpansive and 𝑝∈⋂^∞_𝑛=1Fix(𝑇_𝑛), 𝑇_𝑛𝑥_𝑛−𝑝²≤𝑥_𝑛−𝑝². (19) Next, we consider 𝑓𝑥_𝑛+𝑔𝑇_𝑛𝑥_𝑛−𝑝,𝑥_𝑛+1=𝑓𝑥−𝑝_𝑛−𝑓𝑝,𝑥_𝑛+1+𝑔𝑇−𝑝_𝑛𝑥_𝑛−𝑔𝑝,𝑥_𝑛+1+𝑓𝑝𝑝−𝑝+𝑔−𝑝,𝑥_𝑛+1𝑥−𝑝≤𝛼_𝑛−𝑝𝑥_𝑛+1−𝑝+𝛽𝑥_𝑛×𝑥−𝑝_𝑛+1−𝑝+𝑓𝑝+𝑔𝑝−𝑝,𝑥_𝑛+1=𝑥−𝑝𝛼+𝛽_𝑛−𝑝𝑥_𝑛+1+𝑓𝑝𝑝−𝑝+𝑔−𝑝,𝑥_𝑛+1−𝑝. (20) Hence, the result follows.

Lemma 9. If there is a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that liminf_𝑘→∞(‖𝑥_𝑛𝑘+1−𝑝‖−‖𝑥_𝑛𝑘−𝑝‖)≥0, then limsup_𝑘→∞𝑓𝑝+𝑔𝑝−𝑝,𝑥_𝑛𝑘+1−𝑝≤0. (21)

Proof. We note that lim_𝑘→∞𝛼_𝑛𝑘=0. We consider the following inequality: 0≤liminf_𝑘→∞𝑥_𝑛𝑘+1−𝑝−𝑥_𝑛𝑘−𝑝≤liminf_𝑘→∞𝛼_𝑛𝑘𝑓𝑥_𝑛𝑘−𝑔𝑇_𝑛𝑘𝑥_𝑛𝑘+−𝑝1−𝛼_𝑛𝑘𝑇_𝑛𝑘𝑥_𝑛𝑘−𝑝−𝑥_𝑛𝑘−𝑝≤liminf_𝑘→∞𝑇_𝑛𝑘𝑥_𝑛𝑘−𝑝−𝑥_𝑛𝑘−𝑝≤limsup_𝑘→∞𝑇_𝑛𝑘𝑥_𝑛𝑘−𝑝−𝑥_𝑛𝑘−𝑝≤0. (22) Then lim_𝑘→∞(‖𝑇_𝑛𝑘𝑥_𝑛𝑘−𝑝‖−‖𝑥_𝑛𝑘−𝑝‖)=0. Since {𝑇_𝑛} is strongly quasi-nonexpansive, so is {𝑇_𝑛𝑘}. This implies that lim_𝑘→∞‖𝑥_𝑛𝑘−𝑇_𝑛𝑘𝑥_𝑛𝑘‖=0. Moreover, 𝑥_𝑛𝑘+1−𝑥_𝑛𝑘≤𝑥_𝑛𝑘+1−𝑇_𝑛𝑘𝑥_𝑛𝑘+𝑇_𝑛𝑘𝑥_𝑛𝑘−𝑥_𝑛𝑘=𝛼_𝑛𝑘𝑓𝑥_𝑛𝑘+𝑔𝑇_𝑛𝑥_𝑛𝑘−𝑇_𝑛𝑘𝑥_𝑛𝑘+𝑇_𝑛𝑘𝑥_𝑛𝑘−𝑥_𝑛𝑘. (23) Then lim_𝑘→∞‖𝑥_𝑛𝑘+1−𝑥_𝑛𝑘‖=0. Since {𝑥_𝑛𝑘} is bounded, there exists a subsequence {𝑥_𝑛𝑘𝑙} of {𝑥_𝑛𝑘} such that 𝑤−lim_𝑙→∞𝑥_𝑛𝑘𝑙=𝑞 and limsup_𝑘→∞𝑓𝑝+𝑔𝑝−𝑝,𝑥_𝑛𝑘−𝑝=lim_𝑙→∞𝑓𝑝+𝑔𝑝−𝑝,𝑥_𝑛𝑘𝑙−𝑝. (24) As lim_𝑘→∞‖𝑥_𝑛𝑘−𝑥_𝑛𝑘+1‖=0, we have limsup_𝑘→∞⟨𝑓(𝑝) +𝑔(𝑝) −𝑝,𝑥_𝑛𝑘+1 −𝑝⟩=⟨𝑓(𝑝)+𝑔(𝑝)− 𝑝,𝑞−𝑝⟩. Since {𝑇_𝑛} satisfies NST-condition, we have 𝑞∈𝐹 and hence ⟨𝑓(𝑝)+𝑔(𝑝)−𝑝,𝑞−𝑝⟩≤0. Therefore, limsup_𝑘→∞𝑓𝑝+𝑔𝑝−𝑝,𝑥_𝑛𝑘+1−𝑝≤0, (25) as desired.

Proof of Theorem 6. We are ready to apply Lemma 3. Set 𝑎_𝑛∶=𝑥_𝑛−𝑝²,𝑏_𝑛∶=𝑓𝑝+𝑔𝑝−𝑝,𝑥_𝑛+1−𝑝,𝛼∶=𝛼+𝛽. (26) It follows that (i){𝑎_𝑛} is a bounded sequence (by Lemma 7);(ii)𝑎_𝑛+1≤(1−𝛼_𝑛)²𝑎_𝑛+2𝛼_𝑛𝛼√𝑎_𝑛√𝑎_𝑛+1+𝛼_𝑛𝑏_𝑛 for all 𝑛≥1 (by Lemma 8);(iii)whenever {𝑎_𝑛𝑘} is a subsequence of {𝑎_𝑛} satisfying liminf_𝑘→∞(𝑎_𝑛𝑘+1−𝑎_𝑛𝑘)≥0, it follows that limsup_𝑘→∞𝑏_𝑛𝑘≤0 (by Lemma 9).Hence, lim_𝑛→∞‖𝑥_𝑛−𝑝‖=lim_𝑛→∞𝑎_𝑛=0. This completes the proof.

4. Deduced Results

4.1. Wongchan and Saejung’s Result

Setting 𝑔≡0 and 𝑇_𝑛≡𝑇 for all 𝑛∈ℕ in the proof of Theorem 6, we immediately have the following result of Wongchan and Saejung ([13, Theorem 6]).

Corollary 10. Let 𝐶 be a closed convex subset of a Hilbert space 𝐻 and 𝑇∶𝐶→𝐶 a strongly quasi-nonexpansive mapping such that 𝐼−𝑇 is demiclosed at zero. Suppose that 𝑓∶𝐶→𝐶 is a contraction and a sequence {𝑥_𝑛} is generated by 𝑥₁∈𝐶 and 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛, (27) where {𝛼_𝑛} is a sequence in (0,1) satisfying the conditions (C1) and (C2). Then {𝑥_𝑛} converges strongly to 𝑝=𝑃_Fix(𝑇)𝑓(𝑝).

4.2. Tian and Jin’s Result I

Recall that a mapping 𝐹∶𝐻→𝐻 is 𝜂-strongly monotone if ⟨𝑥−𝑦,𝐹𝑥−𝐹𝑦⟩≥𝜂‖𝑥−𝑦‖² for all 𝑥,𝑦∈𝐻.

Lemma 11. Let 𝐹∶𝐻→𝐻 be an 𝜂-strongly monotone and 𝜅-Lipschitzian mapping. Then ‖(𝐼−𝜇𝐹)𝑥−(𝐼−𝜇𝐹)𝑦‖≤√1−2𝜏‖𝑥−𝑦‖ where 𝜏=𝜇(𝜂−(𝜇𝜅²/2)) for all 𝑥,𝑦∈𝐻. In particular, if 0<𝜇<2𝜂/𝜅², then 𝐼−𝜇𝐹 is a contraction.

Proof. Let 𝑥,𝑦∈𝐻. Then 𝐼−𝜇𝐹𝑥−𝐼−𝜇𝐹𝑦²=𝑥−𝑦−𝜇𝐹𝑥−𝐹𝑦²=𝑥−𝑦²−2𝜇𝑥−𝑦,𝐹𝑥−𝐹𝑦+𝜇²𝐹𝑥−𝐹𝑦²≤𝑥−𝑦²−2𝜇𝜂𝑥−𝑦²+𝜇²𝜅²𝑥−𝑦²=1−2𝜇𝜂−𝜇𝜅²2𝑥−𝑦²=(1−2𝜏)𝑥−𝑦². (28)

Theorem 12. Let 𝑇∶𝐻→𝐻 be a strongly quasi-nonexpansive mapping such that 𝐼−𝑇 is demiclosed at zero. Let 𝐹∶𝐻→𝐻 be an 𝜂-strongly monotone and 𝜅-Lipschitzian mapping. Let 𝑓∶𝐻→𝐻 be an 𝐿-Lipschitzian mapping and let a sequence {𝑥_𝑛} be generated by 𝑥₁∈𝐻 and 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝜇𝐹𝑇𝑥_𝑛(𝑛≥1), (29) where the sequence {𝛼_𝑛}⊂(0,1) satisfies the conditions (C1) and (C2). Suppose that 0<𝜇<2𝜂/𝜅² and 0<𝛾𝐿<1−√1−2𝜏, where 𝜏=𝜇(𝜂−(𝜇𝜅²/2)). Then {𝑥_𝑛} converges to 𝑝=𝑃_Fix(𝑇)(𝐼−𝜇𝐹+𝛾𝑓)𝑝.

Proof. First we rewrite the iteration (29) as follows: 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+𝑔𝑇𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛, (30) where 𝑓=𝛾𝑓 and 𝑔=𝐼−𝜇𝐹. Note that 𝑓 is a 𝛾𝐿-Lipschitzian and 𝑔 is a √1−2𝜏-Lipschitzian. Using 𝛾𝐿+√1−2𝐿<1 and putting 𝑇_𝑛=𝑇 for all 𝑛∈ℕ in Theorem 6 imply that {𝑥_𝑛} converges to 𝑝∈Fix(𝑇), where 𝑝=𝑃_Fix(𝑇)𝑓+𝑔𝑝=𝑃_Fix(𝑇)𝐼−𝜇𝐹+𝛾𝑓𝑝. (31)

Lemma 13 (see [12]). If 𝑇∶𝐻→𝐻 is a quasi-nonexpansive mapping, then the mapping 𝑇_𝜔∶=(1−𝜔)𝐼+𝜔𝑇 is strongly quasi-nonexpansive wherever 𝜔∈(0,1).

Using Theorem 12 and Lemma 13, we immediately have the following result which is an improvement of Tian and Jin’s result ([15, Theorem 3.1]).

Theorem 14. Let 𝑇∶𝐻→𝐻 be a quasi-nonexpansive mapping such that 𝐼−𝑇 is demiclosed at zero. Let 𝐹∶𝐻→𝐻 be an 𝜂-strongly monotone and 𝜅-Lipschitzian mapping. Let 𝑓∶𝐻→𝐻 be an 𝐿-Lipschitzian mapping and let the sequence {𝑥_𝑛} be generated by 𝑥₁∈𝐻 and 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝜇𝐹𝑇_𝜔𝑥_𝑛(𝑛≥1), (32) where 𝑇_𝜔=(1−𝜔)𝐼+𝜔𝑇, 𝜔∈(0,1) and the sequence {𝛼_𝑛}⊂(0,1) satisfies the conditions (C1) and (C2). Suppose that 0<𝜇<2𝜂/𝜅²and 0<𝛾𝐿<1−√1−2𝜏 where 𝜏=𝜇(𝜂−(𝜇𝜅²/2)). Then {𝑥_𝑛} converges to 𝑝=𝑃_Fix(𝑇)(𝐼−𝜇𝐹+𝛾𝑓)(𝑝).

Remark 15. Theorem 14 improves the result of Tian and Jin ([15, Theorem 3.1]) in the following ways. (i)We assume that 𝛾𝐿<1−√1−2𝜏 while [15, Theorem 3.1] is proved under the assumptions 𝛾𝐿<𝜏. We note that 𝜏<1−√1−2𝜏.(ii)Our result allows us to choose 𝜔 in the wider interval (0,1) while [15, Theorem 3.1] is proved under the assumptions 𝜔∈(0,1/2).

4.3. Tian and Jin’s Result II

Recall that a mapping 𝐴∶𝐻→𝐻 is strongly positive with the coefficient 𝛾>0 if ⟨𝐴𝑥,𝑥⟩≥𝛾‖𝑥‖² (33) for all 𝑥∈𝐻.

Lemma 16 (see [20]). Let 𝐴 be a strongly positive self-adjoint linear bounded operator with coefficient 𝛾>0 on 𝐻 and 0<𝜌≤‖𝐴‖⁻¹. Then ‖𝐼−𝜌𝐴‖≤1−𝜌𝛾.

Theorem 17. Let 𝑇∶𝐻→𝐻 be a strongly quasi-nonexpansive mapping such that 𝐼−𝑇 is demiclosed at zero. Let 𝐴∶𝐻→𝐻 be a bounded linear self-adjoint operator and strongly positive with the coefficient 𝛾. Let 𝑓∶𝐻→𝐻 be an 𝛼-contraction mapping and let a sequence {𝑥_𝑛} be generated by 𝑥₁∈𝐻 and 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛(𝑛≥1), (34) where the sequence {𝛼_𝑛}⊂(0,1) satisfies the conditions (C1) and (C2). Suppose that 0<𝛾𝛼<𝛾. Then {𝑥_𝑛} converges to 𝑝=𝑃_{𝐹𝑖𝑥(𝑇)}(𝐼−𝐴+𝛾𝑓)𝑝.

Proof. By Lemma 16, we can choose 𝑡∈(0,1) such that ‖𝐼−𝑡𝐴‖≤1−𝑡𝛾. Rewrite the iteration (34) as follows: 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+𝑔𝑇𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛, (35) where 𝑓∶=𝑡𝛾𝑓, 𝑔∶=𝐼−𝑡𝐴 and 𝛼_𝑛≡𝛼_𝑛/𝑡 for all 𝑛∈ℕ. Note that 𝑓 is 𝑡𝛾𝛼-Lipschitzian and 𝑔 is (1−𝑡𝛾)-Lipschitzian. It follows from 0<𝛾𝛼<𝛾 that 𝑡𝛾𝛼+1−𝑡𝛾=1−𝑡𝛾−𝛼𝛾<1. (36) Setting 𝑇_𝑛≡𝑇 for all 𝑛∈ℕ in Theorem 6 implies that {𝑥_𝑛} converges to 𝑝∈Fix(𝑇) such that 𝑝=𝑃_Fix(𝑇)(𝑓+𝑔)𝑝=𝑃_Fix(𝑇)(𝑡𝛾𝑓+𝐼−𝑡𝐴)𝑝; that is, ⟨𝑡𝛾𝑓(𝑝)+𝑝−𝑡𝐴𝑝−𝑝,𝑝−𝑤⟩≥0 for all 𝑤∈Fix(𝑇). This implies that ⟨𝛾𝑓(𝑝)−𝐴𝑝,𝑝−𝑤⟩≥0 for all 𝑤∈Fix(𝑇); that is, 𝑝=𝑃_Fix(𝑇)(𝛾𝑓+𝐼−𝐴)𝑝. This completes the proof.

Using Lemma 13 and Theorem 17, we immediately have the following result which is an improvement of Tian and Jin's result ([16, Theorem 3.1]).

Theorem 18. Let 𝑇∶𝐻→𝐻 be a quasi-nonexpansive mapping such that 𝐼−𝑇 is demiclosed at zero. Let 𝐴∶𝐻→𝐻 be a bounded linear self-adjoint operator and strongly positive with the coefficient 𝛾. Let 𝑓∶𝐻→𝐻 be an 𝛼-contraction mapping, and let the sequence {𝑥_𝑛} be generated by 𝑥₁∈𝐻 and 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑇_𝜔𝑥_𝑛(𝑛≥1), (37) where 𝑇_𝜔=(1−𝜔)𝐼+𝜔𝑇, 𝜔∈(0,1) and the sequence {𝛼_𝑛}⊂(0,1) satisfies the conditions (C1) and (C2). Suppose that 0<𝛾𝛼<𝛾. Then {𝑥_𝑛} converges to 𝑝=𝑃_Fix(𝑇)(𝐼−𝐴+𝛾𝑓)𝑝.

Remark 19. Theorem 18 improves the result of Tian and Jin ([16, Theorem 3.1]). In fact, their result was proved under the assumption 𝜔∈(0,1/2) while our result allows us to choose 𝜔 in the wider interval (0,1).

5. A Discussion on Marino-Xu’s Result

The following theorem is studied by many authors; for example, see [3].

Theorem 20. Let 𝐶 be a closed convex subset of a Hilbert space 𝐻. Suppose that (i)𝑇∶𝐶→𝐶 is a nonexpansive mapping and Fix(𝑇)≠∅;(ii){𝛼_𝑛}⊂(0,1) is a sequence satisfying the conditions (C1), (C2), and (C3).Define the following iterative sequence: 𝑢,𝑥₁𝑥∈𝐶,_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇𝑥_𝑛. (38)(39) Then {𝑥_𝑛} converges to 𝑃_Fix(𝑇)𝑢.

Using the technique in [4], we can give a simple proof of the following result proved by Marino and Xu [20].

Theorem 21. Suppose that(i)𝐴∶𝐻→𝐻 is a bounded linear self-adjoint operator and it is strongly positive with the coefficient 𝛾;(ii)𝑇∶𝐻→𝐻 is a nonexpansive mapping and Fix(𝑇)≠∅;(iii)𝑓∶𝐻→𝐻 is an 𝛼-contraction;(iv)𝛾 is a positive number such that 0<𝛾𝛼<𝛾;(v){𝛼_𝑛}⊂(0,1) is a sequence satisfying the conditions (C1), (C2), and (C3). Define the following iterative sequence: 𝑧₁𝑧∈𝐻_𝑛+1=𝛼_𝑛𝛾𝑓𝑧_𝑛+𝐼−𝛼_𝑛𝐴𝑇𝑧_𝑛. (40)(41) Then {𝑧_𝑛} converges to 𝑧∈Fix(𝑇) and ⟨𝐴𝑧−𝛾𝑓(𝑧),𝑧−𝑤⟩≤0 for all 𝑤∈Fix(𝑇).

Proof. Choose 𝑡∈(0,1) such that ‖𝐼−𝑡𝐴‖≤1−𝑡𝛾. First we show that 𝐼−𝑡𝐴+𝑡𝛾𝑓 is a contraction. To see this, let 𝑥,𝑦∈𝐻. Then 𝑦≤)()𝑦𝑓(𝑥)𝑦≤‖𝑓(𝑥)𝑦≤𝐼−𝑡𝐴+𝑡𝛾𝑓𝑥−𝐼−𝑡𝐴+𝑡𝛾𝑓(𝐼−𝑡𝐴𝑥−𝐼−𝑡𝐴+𝑡𝛾−𝑓𝐼−𝑡𝐴‖𝑥−𝑦+𝑡𝛾−𝑓1−𝑡𝛾=𝑥−𝑦+𝑡𝛾𝛼𝑥−𝑦1−𝑡𝛾−𝛾𝛼𝑥−𝑦. (42) It follows from 𝛾𝛼<𝛾 that 𝐼−𝑡𝐴+𝑡𝛾𝑓 is a contraction. Note that 𝑃_Fix(𝑇) is nonexpansive and hence 𝑃_Fix(𝑇)(𝐼−𝑡𝐴+𝑡𝛾𝑓) is a contraction from Fix(𝑇) into itself. It follows from the closedness of Fix(𝑇) and the Banach’s contraction principle that there exists a unique element 𝑧∈Fix(𝑇) such that 𝑧=𝑃_Fix(𝑇)𝐼−𝑡𝐴+𝑡𝛾𝑓(𝑧). (43) Therefore, 𝐴𝑧−𝛾𝑓(𝑧),𝑧−𝑤≤0∀𝑤∈Fix(𝑇). (44) Now we define the following iterative sequence: 𝑥₁=𝑧₁,𝑥_𝑛+1=𝛼_𝑛𝑡()𝑇(𝑧+𝛼𝐼−𝑡𝐴𝑧+𝑡𝛾𝑓)1−^𝑛𝑡𝑇𝑥_𝑛. (45) It follows from Theorem 20 that the sequence {𝑥_𝑛} converges to 𝑧=𝑃_Fix(𝑇)(𝐼−𝑡𝐴+𝑡𝛾𝑓)(𝑧). Observe that 𝑧_𝑛+1=𝛼_𝑛𝑡(𝐼−𝑡𝐴)𝑇𝑧_𝑛+𝑡𝛾𝑓𝑧_𝑛+𝛼1−^𝑛𝑡𝑇𝑧_𝑛. (46) We next consider the following expression: 𝑧_𝑛+1−𝑥_𝑛+1=𝛼1−^𝑛𝑡𝑇𝑧_𝑛−𝑇𝑥_𝑛+𝛼_𝑛𝑡(𝐼−𝑡𝐴)𝑇𝑧_𝑛𝑧+𝛼−𝑇_𝑛𝑡𝑡𝛾𝑓𝑧_𝑛(𝑧)≤𝛼−𝑓1−^𝑛𝑡𝑧_𝑛−𝑥_𝑛+𝛼_𝑛𝑡1−𝑡𝛾𝑧_𝑛−𝑧+𝛼_𝑛𝛾𝛼𝑧_𝑛−𝑧=𝛼1−^𝑛𝑡𝑧_𝑛−𝑥_𝑛+𝛼_𝑛𝑡−𝛼_𝑛𝛾−𝛾𝛼𝑧_𝑛−𝑧≤1−𝛼_𝑛𝛾−𝛾𝛼𝑧_𝑛−𝑥_𝑛+𝛼_𝑛𝑡−𝛼_𝑛𝛾−𝛾𝛼𝑥_𝑛−𝑧=1−𝛼_𝑛𝛾−𝛾𝛼𝑧_𝑛−𝑥_𝑛+𝛼_𝑛×()−𝛾−𝛾𝛼1/𝑡𝛾−𝛾𝛼𝑥𝛾−𝛾𝛼^𝑛−𝑧. (47) It follows from Lemma 4 that lim_𝑛→∞‖𝑧_𝑛−𝑥_𝑛‖=0. Therefore, we conclude that {𝑧_𝑛} converges to 𝑧∈Fix(𝑇) and ⟨𝐴𝑧−𝛾𝑓(𝑧),𝑧−𝑤⟩≤0 for all 𝑤∈Fix(𝑇). This completes the proof.

Acknowledgments

The authors thank Professor Simeon Reich and the referee for their comments and suggestions. The first author is supported by the TRF Research Career Development Grant RSA 5680002 and the second author is supported by grant fund under the program Strategic Scholarships for Frontier Research Network for the Ph.D. Program Thai Doctoral degree from the Office of the Higher Education Commission, Thailand.