On the Strong Convergence of Viscosity Approximation Process for Quasinonexpansive Mappings in Hilbert Spaces

Abstract

We improve the viscosity approximation process for approximation of a fixed point of a quasi-nonexpansive mapping in a Hilbert space proposed by Maingé (2010). An example beyond the scope of the previously known result is given.

1. Introduction

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and the induced norm ‖⋅‖. In this paper, we denote the strong and weak convergence by → and ⇀, respectively. For a subset 𝐶 of 𝐻, a mapping 𝑇∶𝐶→𝐶 is said to be nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶; and it is quasinonexpansive if its fixed-point set Fix(𝑇)∶={𝑥∈𝐶∶𝑥=𝑇𝑥} is nonempty and ‖𝑇𝑥−𝑝‖≤‖𝑥−𝑝‖ for all 𝑥∈𝐶 and 𝑝∈Fix(𝑇). It is clear that every nonexpansive mapping with a nonempty fixed-point set is quasinonexpansive, but the converse is not true. The process for approximation of a fixed point of a nonexpansive or quasinonexpansive mapping is one of interesting problems in mathematics and it has been investigated by many researchers. One of the effective processes for this problem is given by Moudafi [1]. Let 𝐶 be a closed convex subset of 𝐻, and 𝑇∶𝐶→𝐶 is a nonexpansive mapping with a nonempty fixed-point set Fix(𝑇). Moudafi proposed the following scheme which is known as Moudafi's viscosity approximation process: 𝑥₁𝑥=𝑥∈𝐶arbitrarilychosen,_𝑛+1=𝜀_𝑛1+𝜀_𝑛𝑓𝑥_𝑛+11+𝜀_𝑛𝑇𝑥_𝑛, (1.1) where 𝑓∶𝐶→𝐶 is a contraction; that is, there exists an 𝛼∈[0,1) such that ‖𝑓(𝑥)−𝑓(𝑦)‖≤𝛼‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶 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 also proved that {𝑥_𝑛} converges to an element 𝑧∈Fix(𝑇) satisfying the following inequality: 𝑓(𝑧)−𝑧,𝑞−𝑧≤0 (1.2) for all 𝑞∈Fix(𝑇).

In the literature, Moudafi's scheme has been widely studied and extended (see [2–5] and references therein). For example, Xu [6] improved this result to a Banach space. The interesting improvement of this result given by Maingé [7] is our starting point. His result is given below.

Theorem 1.1. Let 𝐶 be a closed convex subset of a Hilbert space 𝐻, and 𝑇∶𝐶→𝐶 is a quasinonexpansive mapping such that 𝐼−𝑇 is demiclosed at zero, that is, 𝑧∈Fix(𝑇) whenever {𝑧_𝑛} is a sequence in 𝐶 such that 𝑧_𝑛⇀𝑧 and 𝑧_𝑛−𝑇𝑧_𝑛→0. Suppose that 𝑓∶𝐶→𝐶 is a contraction. Let {𝑥_𝑛} be a sequence in 𝐶 defined by 𝑥₁𝑥=𝑥∈𝐶arbitrarilychosen,_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛((1−𝜔)𝐼+𝜔𝑇)𝑥_𝑛, (1.3) where 𝜔∈(0,1), 𝐼 is an identity mapping, and {𝛼_𝑛} is a sequence in (0,1) satisfying (C1)lim_𝑛→∞𝛼_𝑛=0, (C2)∑^∞_𝑛=1𝛼_𝑛=∞. Then the sequence {𝑥_𝑛} converges to an element 𝑧∈Fix(𝑇) and the following inequality holds 𝑓(𝑧)−𝑧,𝑞−𝑧≤0 (1.4) for all 𝑞∈Fix(𝑇).

It should be noted that Maingé's result is more widely applicable than Moudafi's. However, after a careful reading, we find that there is a small mistake in Maingé's proof. The following fact (see [7, Remark 2.1(i3)]) is used: if 𝑇∶𝐶→𝐶 is quasinonexpansive and 𝑇_𝜔∶=(1−𝜔)𝐼+𝜔𝑇 where 𝜔∈(0,1], then 𝑥−𝑇_𝜔𝑥,𝑥−𝑞≥𝜔‖𝑥−𝑇𝑥‖² (1.5) for all 𝑥∈𝐶 and 𝑞∈Fix(𝑇). Note that the inequality above is equivalent to ⟨𝑥−𝑇𝑥,𝑥−𝑞⟩≥‖𝑥−𝑇𝑥‖². (1.6) But this fails; for example, let us consider the nonexpansive mapping 𝑇∶ℝ→ℝ defined by 𝑇𝑥=−𝑥 for all 𝑥∈ℝ. It is clear that Fix(𝑇)={0} and ⟨𝑥−𝑇𝑥,𝑥−𝑞⟩=2𝑥²≱4𝑥²=‖𝑥−𝑇𝑥‖².

Recall the following identities in a Hilbert space 𝐻: for 𝑥,𝑦∈𝐻, 𝜔∈[0,1](i)‖𝑥+𝑦‖²=‖𝑥‖²+2⟨𝑥,𝑦⟩+‖𝑦‖²; (ii)‖(1−𝜔)𝑥+𝜔𝑦‖²=(1−𝜔)‖𝑥‖²+𝜔‖𝑦‖²−(1−𝜔)𝜔‖𝑥−𝑦‖².

The correction of Maingé's result is as follows.

Proposition 1.2. Let 𝐶 be a subset of a Hilbert space and 𝑇∶𝐶→𝐶 be a mapping with a nonempty fixed-point set Fix(𝑇). Suppose that 𝑇_𝜔∶=(1−𝜔)𝐼+𝜔𝑇 where 𝜔∈(0,1]. Then 𝑇 is quasinonexpansive if and only if 𝑥−𝑇_𝜔≥𝜔𝑥,𝑥−𝑞2‖𝑥−𝑇𝑥‖² (1.7) for all 𝑥∈𝐶 and 𝑞∈Fix(𝑇).

Proof. Notice that 𝑥−𝑇_𝜔𝑥=𝜔(𝑥−𝑇𝑥) and 𝑇_𝜔𝑥−𝑞²=(𝑇_𝜔𝑥−𝑥)+(𝑥−𝑞)²=𝑇_𝜔𝑥−𝑥²+2𝑇_𝜔𝑥−𝑥,𝑥−𝑞+𝑥−𝑞²=𝜔²‖𝑥−𝑇𝑥‖²+2𝑇_𝜔𝑥−𝑥,𝑥−𝑞+𝑥−𝑞². (1.8) On the other hand, 𝑇_𝜔𝑥−𝑞²=(1−𝜔)(𝑥−𝑞)+𝜔(𝑇𝑥−𝑞)²=(1−𝜔)𝑥−𝑞²+𝜔𝑇𝑥−𝑞²−(1−𝜔)𝜔‖𝑥−𝑇𝑥‖². (1.9) Hence 2𝑥−𝑇_𝜔𝑥,𝑥−𝑞=𝜔𝑥−𝑞²−𝑇𝑥−𝑞²+𝜔‖𝑥−𝑇𝑥‖². (1.10)

Remark 1.3. Unfortunately, this effects the main result (see [7, Theorem 3.1]) in Maingé's paper. More precisely, inequality (32) of its proof (page 78, line 22) should read 12𝑥_𝑛+1−𝑧²−12𝑥_𝑛−𝑧²+12−𝜔𝜔1−𝛼_𝑛𝑥_𝑛−𝑇𝑥_𝑛²≤𝛼_𝑛𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛²−𝑥_𝑛−𝑓𝑥_𝑛,𝑥_𝑛−𝑧 (1.11) rather than 12𝑥_𝑛+1−𝑧²−12𝑥_𝑛−𝑧²+(1−𝜔)𝜔1−𝛼_𝑛𝑥_𝑛−𝑇𝑥_𝑛²≤𝛼_𝑛𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛²−𝑥_𝑛−𝑓𝑥_𝑛,𝑥_𝑛−𝑧. (1.12) Therefore, Theorem 1.1 above is valid for only 𝜔∈(0,1/2) under the same technique.

The purpose of this paper is to simultaneously present a correction of the proof of Theorem 1.1 which is valid for all 𝜔∈(0,1), and extend his scheme to a wider class of mappings including average mappings, that is, mappings of the form (1−𝜔)𝐼+𝜔𝑇. Our result is more general than Maingé's theorem. An example of a quasinonexpansive mapping which is not applied by Maingé's theorem but applied by our result is given.

2. Result

First, let us recall some lemmas which are needed for proving the main result.

Lemma 2.1 (see [8, Lemma 2.3]). Let {𝑠_𝑛} be a sequence of nonnegative real numbers, {𝛼_𝑛} a sequence of (0,1) with ∑^∞_𝑛=1𝛼_𝑛=∞, {𝛽_𝑛} a sequence of nonnegative real numbers with ∑^∞_𝑛=1𝛽_𝑛<∞, and {𝛾_𝑛} a sequence of real numbers with limsup_𝑛→∞𝛾_𝑛≤0. Suppose that 𝑠_𝑛+1≤1−𝛼_𝑛𝑠_𝑛+𝛼_𝑛𝛾_𝑛+𝛽_𝑛 (2.1) for all 𝑛∈ℕ. Then lim_𝑛→∞𝑠_𝑛=0.

The following nice result was proved by Maingé (see [7, Lemma 2.1]).

Lemma 2.2. Let {𝑠_𝑛} be a sequence of nonnegative real numbers. If there exists a subsequence {𝑠_𝑛𝑗} of {𝑠_𝑛} such that 𝑠_𝑛𝑗<𝑠_𝑛𝑗+1 for all 𝑗∈ℕ, then there exists a subsequence {𝑠_𝑚𝑘} of {𝑠_𝑛} such that 𝑠_𝑚𝑘≤𝑠_𝑚𝑘+1,𝑠_𝑘≤𝑠_𝑚𝑘+1 (2.2) for all 𝑘∈ℕ.

For a closed convex subset 𝐶 of a Hilbert space 𝐻, the metric projection 𝑃_𝐶∶𝐻→𝐶 is defined for each 𝑥∈𝐻 as the unique element 𝑃_𝐶𝑥∈𝐶 such that 𝑥−𝑃_𝐶𝑥=inf{‖𝑥−𝑧‖∶𝑧∈𝐶}. (2.3) It is well known that (see, e.g., [9]) for 𝑥∈𝐻 and 𝑦∈𝐶𝑦=𝑃_𝐶𝑥⟺𝑥−𝑦,𝑦−𝑧≥0,∀𝑧∈𝐶. (2.4) For 𝑥,𝑦∈𝐻, the following inequality is known as the subdifferential inequality: 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝑥+𝑦. (2.5)

A mapping 𝑇∶𝐶→𝐶 is said to be strongly quasinonexpansive [10] if it is quasinonexpansive and 𝑧_𝑛−𝑇𝑧_𝑛→0 whenever {𝑧_𝑛} is a bounded sequence in 𝐶 such that lim_𝑛→∞(‖𝑧_𝑛−𝑝‖−‖𝑇𝑧_𝑛−𝑝‖)=0 for some 𝑝∈Fix(𝑇). It is known that every metric projection is strongly quasinonexpansive.

We are now ready to present our main result.

Theorem 2.3. Let 𝐶 be a closed convex subset of a Hilbert space 𝐻 and 𝑇∶𝐶→𝐶 is a strongly quasinonexpansive mapping such that 𝐼−𝑇 is demiclosed at zero. Suppose that 𝑓∶𝐶→𝐶 is a contraction. Let {𝑥_𝑛} be a sequence in 𝐶 defined by 𝑥₁𝑥=𝑥∈𝐶arbitrarilychosen,_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛, (2.6) where {𝛼_𝑛} is a sequence in (0,1) satisfying (C1)lim_𝑛→∞𝛼_𝑛=0, (C2)∑^∞_𝑛=1𝛼_𝑛=∞. Then the sequence {𝑥_𝑛} converges to an element 𝑧∈Fix(𝑇) and the following inequality holds 𝑓(𝑧)−𝑧,𝑞−𝑧≤0 (2.7) for all 𝑞∈Fix(𝑇).

Before we give the proof, we note that Fix(𝑇) is closed and convex (see [11] for more general setting). Hence the mapping 𝑃_Fix(𝑇)∘𝑓∶𝐶→𝐶 is a contraction. Then it follows from the well-known Banach's contraction principle that there exists a unique element 𝑧∈𝐶 such that 𝑧=𝑃_Fix(𝑇)∘𝑓(𝑧). In particular, 𝑧∈Fix(𝑇) and ⟨𝑓(𝑧)−𝑧,𝑞−𝑧⟩≤0 for all 𝑞∈Fix(𝑇).

Let us assume that ‖𝑓(𝑥)−𝑓(𝑦)‖≤𝛼‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶 where 𝛼 is a real number in [0,1).

Lemma 2.4. The sequence {𝑥_𝑛} is bounded.

Proof. We consider the following inequality: 𝑥_𝑛+1−𝑧≤𝛼_𝑛𝑓𝑥_𝑛−𝑧+1−𝛼_𝑛𝑇𝑥_𝑛−𝑧≤𝛼_𝑛𝑓𝑥_𝑛−𝑓(𝑧)+𝛼_𝑛𝑓(𝑧)−𝑧+1−𝛼_𝑛𝑇𝑥_𝑛≤𝛼−𝑧_𝑛𝛼+1−𝛼_𝑛𝑥_𝑛−𝑧+𝛼_𝑛𝑓(𝑧)=−𝑧1−𝛼_𝑛(1−𝛼)𝑥_𝑛−𝑧+𝛼_𝑛()𝑓(𝑧)1−𝛼−𝑧𝑥1−𝛼≤max_𝑛,𝑓(𝑧)−𝑧−𝑧.1−𝛼 (2.8) By induction, we conclude that the sequence {‖𝑥_𝑛−𝑧‖} is bounded and hence so is the sequence {𝑥_𝑛}.

Lemma 2.5. The following inequality holds for all 𝑛∈ℕ: 𝑥_𝑛+1−𝑧²≤1−𝛼_𝑛²𝑥_𝑛−𝑧²+2𝛼_𝑛𝛼𝑥_𝑛−𝑧𝑥_𝑛+1−𝑧+2𝛼_𝑛𝑓(𝑧)−𝑧,𝑥_𝑛+1−𝑧. (2.9)

Proof. It follows from the subdifferential inequality that 𝑥_𝑛+1−𝑧²=𝛼_𝑛(𝑓(𝑥_𝑛)−𝑧)+(1−𝛼_𝑛)(𝑇𝑥_𝑛−𝑧)²≤1−𝛼_𝑛²𝑇𝑥_𝑛−𝑧²+2𝛼_𝑛𝑓𝑥_𝑛−𝑧,𝑥_𝑛+1≤−𝑧1−𝛼_𝑛²𝑥_𝑛−𝑧²+2𝛼_𝑛𝑓𝑥_𝑛−𝑓(𝑧),𝑥_𝑛+1−𝑧+2𝛼_𝑛𝑓(𝑧)−𝑧,𝑥_𝑛+1≤−𝑧1−𝛼_𝑛²𝑥_𝑛−𝑧²+2𝛼_𝑛𝛼𝑥_𝑛−𝑧𝑥_𝑛+1−𝑧+2𝛼_𝑛𝑓(𝑧)−𝑧,𝑥_𝑛+1−𝑧. (2.10)

Lemma 2.6. If there exists a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that liminf_𝑘→∞(‖𝑥_𝑛𝑘+1−𝑧‖−‖𝑥_𝑛𝑘−𝑧‖)≥0, then limsup_𝑘→∞⟨𝑓(𝑧)−𝑧,𝑥_𝑛𝑘+1−𝑧⟩≤0.

Proof. First, we note that 𝛼_𝑛𝑘→0 and let us consider the following inequality: 0≤liminf_𝑘→∞𝑥_𝑛𝑘+1−𝑧−𝑥_𝑛𝑘−𝑧≤liminf_𝑘→∞𝛼_𝑛𝑘𝑓𝑥_𝑛𝑘−𝑧+1−𝛼_𝑛𝑘𝑇𝑥_𝑛𝑘−𝑧−𝑥_𝑛𝑘−𝑧=liminf_𝑘→∞𝑇𝑥_𝑛𝑘−𝑧−𝑥_𝑛𝑘−𝑧≤limsup_𝑘→∞𝑇𝑥_𝑛𝑘−𝑧−𝑥_𝑛𝑘−𝑧≤0. (2.11) This implies that lim_𝑘→∞(‖𝑥_𝑛𝑘−𝑧‖−‖𝑇𝑥_𝑛𝑘−𝑧‖)=0. Since 𝑇 is a strongly quasinonexpansive mapping, 𝑥_𝑛𝑘−𝑇𝑥_𝑛𝑘→0. In particular, 𝑥_𝑛𝑘−𝑥_𝑛𝑘+1→0. Because {𝑥_𝑛𝑘} is bounded, so there exists a subsequence {𝑥_𝑛𝑘𝑙} of {𝑥_𝑛𝑘} such that 𝑥_𝑛𝑘𝑙⇀𝑞 and lim_𝑙→∞𝑓(𝑧)−𝑧,𝑥_𝑛𝑘𝑙−𝑧=limsup_𝑘→∞𝑓(𝑧)−𝑧,𝑥_𝑛𝑘−𝑧. (2.12) It follows from the demiclosedness of 𝐼−𝑇 at zero that 𝑞∈Fix(𝑇). Then lim_𝑙→∞𝑓(𝑧)−𝑧,𝑥_𝑛𝑘𝑙−𝑧=𝑓(𝑧)−𝑧,𝑞−𝑧≤0. (2.13) Hence limsup_𝑘→∞⟨𝑓(𝑧)−𝑧,𝑥_𝑛𝑘+1−𝑧⟩=limsup_𝑘→∞⟨𝑓(𝑧)−𝑧,𝑥_𝑛𝑘−𝑧⟩≤0, as desired.

Proof of Theorem 2.3. Let us consider the following two cases.Case 1. There exists an 𝑁∈ℕ such that ‖𝑥_𝑛+1−𝑧‖≤‖𝑥_𝑛−𝑧‖ for all 𝑛≥𝑁. It follows then that lim_𝑛→∞‖𝑥_𝑛−𝑧‖ exists and hence liminf_𝑛→∞(‖𝑥_𝑛+1−𝑧‖−‖𝑥_𝑛−𝑧‖)=0. This implies that limsup_𝑛→∞⟨𝑓(𝑧)−𝑧,𝑥_𝑛+1−𝑧⟩≤0. By Lemma 2.5, for all 𝑛≥𝑁, 𝑥_𝑛+1−𝑧²≤1−𝛼_𝑛²𝑥_𝑛−𝑧²+2𝛼_𝑛𝛼𝑥_𝑛−𝑧𝑥_𝑛+1−𝑧+2𝛼_𝑛𝑓(𝑧)−𝑧,𝑥_𝑛+1≤−𝑧1−2𝛼_𝑛+2𝛼_𝑛𝛼𝑥_𝑛−𝑧²+𝛼²_𝑛𝑥_𝑛−𝑧²+2𝛼_𝑛𝑓(𝑧)−𝑧,𝑥_𝑛+1=−𝑧1−2𝛼_𝑛(1−𝛼)𝑥_𝑛−𝑧²+2𝛼_𝑛()𝛼1−𝛼^𝑛𝑥_𝑛−𝑧²2()+𝑓(𝑧)1−𝛼−𝑧,𝑥_𝑛+1−𝑧.1−𝛼 (2.14) Notice that ∑^∞_𝑛=𝑁2𝛼_𝑛(1−𝛼)=∞ and limsup_𝑛→∞𝛼_𝑛𝑥_𝑛−𝑧²2()+𝑓(𝑧)1−𝛼−𝑧,𝑥_𝑛+1−𝑧1−𝛼≤0. (2.15) By Lemma 2.1, we have lim_𝑛→∞‖𝑥_𝑛−𝑧‖²=0.Case 2. There exists a subsequence {‖𝑥_𝑛𝑗−𝑧‖} of {‖𝑥_𝑛−𝑧‖} such that ‖𝑥_𝑛𝑗−𝑧‖<‖𝑥_𝑛𝑗+1−𝑧‖ for all 𝑗∈ℕ. In this case, it follows from Lemma 2.2 that there exists a subsequence {‖𝑥_𝑚𝑘−𝑧‖} of {‖𝑥_𝑛−𝑧‖} such that 𝑥_𝑚𝑘−𝑧≤𝑥_𝑚𝑘+1−𝑧,𝑥_𝑘−𝑧≤𝑥_𝑚𝑘+1−𝑧 (2.16) for all 𝑘∈ℕ. It follows from liminf_𝑘→∞(‖𝑥_𝑚𝑘+1−𝑧‖−‖𝑥_𝑚𝑘−𝑧‖)≥0 that limsup_𝑘→∞⟨𝑓(𝑧)−𝑧,𝑥_𝑚𝑘+1−𝑧⟩≤0. Moreover, by Lemma 2.5, we have 𝑥_𝑚𝑘+1−𝑧²≤1−𝛼_𝑚𝑘²𝑥_𝑚𝑘−𝑧²+2𝛼_𝑚𝑘𝛼𝑥_𝑚𝑘−𝑧𝑥_𝑚𝑘+1−𝑧+2𝛼_𝑚𝑘𝑓(𝑧)−𝑧,𝑥_𝑚𝑘+1≤−𝑧1−𝛼_𝑚𝑘²𝑥_𝑚𝑘+1−𝑧²+2𝛼_𝑚𝑘𝛼𝑥_𝑚𝑘+1−𝑧²+2𝛼_𝑚𝑘𝑓(𝑧)−𝑧,𝑥_𝑚𝑘+1−𝑧. (2.17) In particular, it follows that 2−𝛼_𝑚𝑘−2𝛼𝑥_𝑚𝑘+1−𝑧²≤2𝑓(𝑧)−𝑧,𝑥_𝑚𝑘+1−𝑧. (2.18) This implies that (2−2𝛼)limsup_𝑘→∞𝑥_𝑚𝑘+1−𝑧²≤limsup_𝑘→∞2𝑓(𝑧)−𝑧,𝑥_𝑚𝑘+1−𝑧≤0. (2.19) Hence limsup_𝑘→∞𝑥_𝑘−𝑧²≤limsup_𝑘→∞𝑥_𝑚𝑘+1−𝑧²=0. (2.20) Then lim_𝑘→∞‖𝑥_𝑘−𝑧‖²=0. This completes the proof.

Remark 2.7. If 𝐶 is a convex subset of a Hilbert space and 𝑇∶𝐶→𝐶 is a quasinonexpansive mapping, then the mapping 𝑇_𝜔∶=(1−𝜔)𝐼+𝜔𝑇 is strongly quasinonexpansive whenever 𝜔∈(0,1) (see [10]). This means that Maingé's result is included in ours as a special case.

Remark 2.8. There is a strongly quasinonexpansive mapping 𝑆 such that 𝑆 is not of the form (1−𝜔)𝐼+𝜔𝑇 where 𝛼∈(0,1/2) and 𝑇 is a quasinonexpansive mapping. This means that there is an example which is beyond the scope of Maingé's result (see Remark 1.3, Theorem 1.1 with his old proof is valid for only 𝛼∈(0,1/2)).

Example 2.9. Let 𝐴={(𝑥,𝑥)∶𝑥∈ℝ}. It is clear that 𝐴 is a closed and convex subset of ℝ². Notice that 𝑆∶=𝑃_𝐴 is a strongly quasinonexpansive mapping and (0,0)∈Fix(𝑆). Suppose that 𝑆=(1−𝜔)𝐼+𝜔𝑇 where 𝜔∈(0,1/2) and 𝑇 is a quasinonexpansive mapping. Then, by Proposition 1.2, we have )(),()−(≥𝜔⟨(2,0−𝑆2,02,00,0)⟩2‖(2,0)−𝑇(2,0)‖²=1)(2𝜔‖(2,0−𝑆2,0)‖². (2.21) It is easy to see that 𝑆(2,0)=(1,1). In particular, )−(),(≥12=⟨(2,01,12,0)⟩‖‖2𝜔(2,0)−(1,1)²=1𝜔. (2.22) That is 𝜔≥1/2, a contradiction.

3. Conclusion

We propose a viscosity approximation process for approximation of a fixed point of a quasinonexpansive mapping. This not only corrects Maingé's result but also essentially improves his result to a more general relaxation.

Acknowledgments

The first 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 Commission on Higher Education. The second author is supported by Thailand Research Fund, the Commission on Higher Education and Khon Kaen University under Grant RMU5380039.