Strong Convergence of Viscosity Iteration Methods for Nonexpansive Mappings

Abstract

We propose a new viscosity iterative scheme for finding fixed points of nonexpansive mappings in a reflexive Banach space having a uniformly Gâteaux differentiable norm and satisfying that every weakly compact convex subset of the space has the fixed point property for nonexpansive mappings. Certain different control conditions for viscosity iterative scheme are given and strong convergence of viscosity iterative scheme to a solution of a ceratin variational inequality is established.

1. Introduction

Let 𝐸 be a real Banach space and let 𝐶 be a nonempty closed convex subset of 𝐸. Recall that a mapping 𝑓∶𝐶→𝐶 is a contraction on 𝐶 if there exists a constant 𝑘∈(0,1) such that ‖𝑓(𝑥)−𝑓(𝑦)‖≤𝑘‖𝑥−𝑦‖,𝑥,𝑦∈𝐶. We use Σ_𝐶 to denote the collection of mappings 𝑓 verifying the above inequality. That is, Σ_𝐶={𝑓∶𝐶→𝐶|𝑓isacontractionwithconstant𝑘}. Let 𝑇∶𝐶→𝐶 be a nonexpansive mapping (recall that a mapping 𝑇∶𝐶→𝐶 is nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖,𝑥,𝑦∈𝐶) and let 𝐹(𝑇) denote the set of fixed points of 𝑇; that is, 𝐹(𝑇)={𝑥∈𝐶∶𝑥=𝑇𝑥}.

We consider the iterative scheme: for 𝑇 a nonexpansive mapping, 𝑓∈Σ_C and 𝛼_𝑛∈(0,1), 𝑥_𝑛+1=𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)𝑇𝑥_𝑛,𝑛≥0.(1.1) As a special case of (1.1), the following iterative scheme: 𝑧_𝑛+1=𝛼_𝑛𝑢+(1−𝛼_𝑛)𝑇𝑧_𝑛,𝑛≥0,(1.2) where 𝑢,𝑧₀∈𝐶 are arbitrary (but fixed), has been investigated by many authors; see, for example, Browder [1], Chang [2], Cho et al. [3], Halpern [4], Lions [5], Reich [6, 7], Shioji and Takahashi [8], Wittmann [9], and Xu [10]. The authors above showed that the sequence {𝑧_𝑛} generated by (1.2) converges strongly to a point in the fixed-point set 𝐹(𝑇) under appropriate conditions on {𝛼_𝑛} in Hilbert spaces or certain Banach spaces.

The viscosity approximation method of selecting a particular fixed point of a given nonexpansive mapping in a Hilbert space was proposed by Moudafi [11] in 2000. In 2004, Xu [12] extended Theorem  2.2 of Moudafi [11] for the iterative scheme (1.1) to a Banach space setting by using the following conditions on the sequence {𝛼_𝑛}: lim_n→∞𝛼_𝑛=0,^∞_𝑛=0𝛼_𝑛=∞,^∞_𝑛=0|𝛼_𝑛+1−𝛼_𝑛|<∞orlim_𝑛→∞𝛼_n𝛼_𝑛+1=1.(1.3) For the iterative scheme (1.1) with generalized contractive mappings instead of contractions, we refer to [13].

In 2005, Kim and Xu [14] provided a simpler modification of Mann iterative scheme in a uniformly smooth Banach space as follows: 𝑥₀𝑦=𝑥∈𝐶,_𝑛=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑇𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑢+(1−𝛼_𝑛)𝑦_𝑛,𝑛≥0,(1.4) where 𝑢∈𝐶 is an arbitrary (but fixed) element, and {𝛼_𝑛} and {𝛽_𝑛} are two sequences in (0,1). They proved that the sequence {𝑥_𝑛} generated by (1.4) converges to a fixed point of 𝑇 under the following control conditions:

(i)lim_𝑛→∞𝛼_𝑛=0,lim_𝑛→∞𝛽_𝑛=0;(ii)∑^∞_𝑛=0𝛼_𝑛=∞,∑^∞_𝑛=0𝛽_𝑛=∞;(iii)∑^∞_𝑛=0|𝛼_𝑛+1−𝛼_𝑛|<∞,∑^∞_𝑛=0|𝛽_𝑛+1−𝛽_𝑛|<∞.

Recently, Yao et al. [15] considered the following modified Mann iterative scheme in a uniformly smooth Banach space as the viscosity approximation method: 𝑥₀𝑦=𝑥∈𝐶,_𝑛=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑇𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)𝑦_𝑛,𝑛≥0,(1.5) and proved strong convergence of the sequence {𝑥_𝑛} generated by (1.5) under certain different control conditions on {𝛼_𝑛} and {𝛽_𝑛}. In particular, their results remove the condition ∑^∞_𝑛=0|𝛼_𝑛+1−𝛼_𝑛|<∞ imposed on {𝛼_𝑛}.

Very recently, Qin et al. [16] proposed the composite Halpern type iterative scheme in a uniformly smooth Banach space as follows: 𝑥₀𝑧=𝑥,𝑢∈𝐶,_𝑛=𝛾_𝑛𝑥_𝑛+(1−𝛾_𝑛)𝑇𝑥_𝑛,𝑦_𝑛=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑇𝑧_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑢+(1−𝛼_𝑛)𝑦_𝑛,𝑛≥0,(1.6) and showed strong convergence of the sequence {𝑥_𝑛} generated by (1.6) under the following control conditions:

(i)∑^∞_𝑛=0𝛼_𝑛=∞;(ii)lim_𝑛→∞𝛼_𝑛=0,lim_𝑛→∞𝛽_𝑛=0and0<𝑎≤𝛾_𝑛forsome𝑎∈(0,1);(iii)∑^∞_𝑛=0|𝛼_𝑛+1−𝛼_𝑛|<∞,∑^∞_𝑛=0|𝛽_𝑛+1−𝛽_𝑛|<∞,∑^∞_𝑛=0|𝛾_𝑛+1−𝛾_𝑛|<∞.

In this paper, under the framework of a reflexive Banach space having a uniformly Gâteaux differentiable norm and satisfying that every weakly compact convex subset of 𝐸 has the fixed point property for nonexpansive mappings, we consider a new composite iterative scheme for a nonexpansive mapping 𝑇 as the viscosity approximation method: for 𝑓∈Σ_𝐶 and the initial guess 𝑥₀=𝑥∈𝐶, 𝑧_𝑛=𝛾_𝑛𝑥_𝑛+(1−𝛾_𝑛)𝑇𝑥_𝑛,𝑦_𝑛=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑇𝑧_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)𝑦_𝑛,𝑛≥0,(IS) where {𝛼_𝑛}, {𝛽_𝑛} and {𝛾_𝑛} are sequences in (0,1). First, we prove under certain control conditions on the sequences {𝛼_𝑛}, {𝛽_𝑛} and {𝛾_𝑛} different from those of Qin et al. [16] that the sequence {𝑥_𝑛} generated by (IS) converges strongly to a fixed point of 𝑇, which is a solution of a certain variational inequality. Next we study the composite iterative scheme (IS) with the weakly contractive mapping instead of the contractions. The main results develop and complement the corresponding results of [2, 3, 8, 9, 11, 12, 15, 16]. In particular, if 𝛽_𝑛=0 for all 𝑛≥0 in (IS), then (IS) reduces a new viscosity iterative scheme for finding a fixed point of 𝑇: 𝑧_𝑛=𝛾_𝑛𝑥_𝑛+(1−𝛾_𝑛)𝑇𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)𝑇𝑧_𝑛,𝑛≥0.(1.7)

2. Preliminaries and Lemmas

Let 𝐸 be a real Banach space with norm ‖⋅‖, and let 𝐸^∗ be its dual. The value of 𝑓∈𝐸^∗ at 𝑥∈𝐸 will be denoted by ⟨𝑥,𝑓⟩. When {𝑥_𝑛} is a sequence in 𝐸, then 𝑥_𝑛→𝑥 (resp., 𝑥_𝑛⇀𝑥) will denote strong (resp., weak) convergence of the sequence {𝑥_𝑛} to 𝑥.

The ( normalized) duality mapping𝐽 from 𝐸 into the family of nonempty (by Hahn-Banach theorem) weak-star compact subsets of its dual 𝐸^∗ is defined by 𝐽(𝑥)={𝑓∈𝐸^∗∶⟨𝑥,𝑓⟩=‖𝑥‖²=‖𝑓‖²},(2.1) for each 𝑥∈𝐸 [17].

The norm of 𝐸 is said to be Gâteaux differentiable (and 𝐸 is said to be smooth) if lim_𝑡→0‖𝑥+𝑡𝑦‖−‖𝑥‖𝑡(2.2) exists for each 𝑥,𝑦 in its unit sphere 𝑈={𝑥∈𝐸∶‖𝑥‖=1}. The norm is said to be uniformly Gâteaux differentiable if for 𝑦∈𝑈, the limit is attained uniformly for 𝑥∈𝑈. The space 𝐸 is said to have a uniformly Fréchet differentiable norm (and 𝐸 is said to be uniformly smooth) if the limit in (2.2) is attained uniformly for (𝑥,𝑦)∈𝑈×𝑈. It is known that 𝐸 is smooth if and only if each duality mapping 𝐽 is single-valued. It is also well-known that if 𝐸 has a uniformly Gâteaux differentiable norm, 𝐽 is uniformly norm-to-weak^∗ continuous on each bounded subsets of 𝐸 [17].

Let 𝐶 be a nonempty closed convex subset of 𝐸. 𝐶 is said to have the fixed point property for nonexpansive mappings if every nonexpansive mapping of a bounded closed convex subset 𝐷 of 𝐶 has a fixed point in 𝐷. Let 𝐷 be a subset of 𝐶. Then a mapping 𝑄∶𝐶→𝐷 is said to be a retraction from 𝐶 onto 𝐷 if 𝑄𝑥=𝑥 for all 𝑥∈𝐷. A retraction 𝑄∶𝐶→𝐷 is said to be sunny if 𝑄(𝑄𝑥+𝑡(𝑥−𝑄𝑥))=𝑄𝑥 for all 𝑥∈𝐶 and 𝑡≥0 with 𝑄𝑥+𝑡(𝑥−𝑄𝑥)∈𝐶. A subset 𝐷 of 𝐶 is said to be a sunny nonexpansive retract of 𝐶 if there exists a sunny nonexpansive retraction of 𝐶 onto 𝐷. In a smooth Banach space 𝐸, it is well-known [18, page 48] that 𝑄 is a sunny nonexpansive retraction from 𝐶 onto 𝐷 if and only if the following condition holds ⟨𝑥−𝑄𝑥,𝐽(𝑧−𝑄𝑥)⟩≤0,𝑥∈𝐶,𝑧∈𝐷.(2.3) We need the following lemmas for the proof of our main results. Lemma 2.1 was also given in Jung and Morales [19], Lemma 2.2 is Lemma  2 of Suzuki [20] and Lemma 2.3 is essentially Lemma  2 of Liu [21] (also see [10]).

Lemma 2.1. Let 𝐸 be a real Banach space and let 𝐽 be the duality mapping. Then, for any given 𝑥,𝑦∈𝐸, one has‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝑗(𝑥+𝑦)⟩,(2.4) for all 𝑗(𝑥+𝑦)∈𝐽(𝑥+𝑦).

Lemma 2.2. Let {𝑥_𝑛} and {𝑤_𝑛} be bounded sequences in a Banach space 𝐸 and let {𝛿_𝑛} be a sequence in [0,1] which satisfies the following condition:0<liminf_𝑛→∞𝛿_𝑛≤limsup_𝑛→∞𝛿_𝑛<1.(2.5) suppose that 𝑥_𝑛+1=𝛿_𝑛𝑥_𝑛+(1−𝛿_𝑛)𝑤_𝑛,𝑛≥0,limsup_𝑛→∞(‖𝑤_𝑛+1−𝑤_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0.(2.6) Then lim_𝑛→∞‖𝑤_𝑛−𝑥_𝑛‖=0.

Lemma 2.3. Let {𝑠_𝑛} be a sequence of nonnegative real numbers satisfying𝑠_𝑛+1≤(1−𝜆_𝑛)𝑠_𝑛+𝜆_𝑛𝛾_𝑛+𝛿_𝑛,𝑛≥0,(2.7) where {𝜆_𝑛}, {𝛾_𝑛}, and {𝛿_𝑛} satisfy the following conditions: (i){𝜆_𝑛}⊂[0,1] and ∑^∞_𝑛=0𝜆_𝑛=∞ or, equivalently, ∏^∞_𝑛=0(1−𝜆_𝑛)=0;(ii)limsup_𝑛→∞𝛾_𝑛≤0 or ∑^∞_𝑛=1𝜆_𝑛𝛾_𝑛<∞;(iii)𝛿_𝑛≥0(𝑛≥0),∑^∞_𝑛=0𝛿_𝑛<∞. Then lim_𝑛→∞𝑠_𝑛=0.

Recall that a mapping 𝐴∶𝐶→𝐶 is said to be weakly contractive if ‖𝐴𝑥−𝐴𝑦‖≤‖𝑥−𝑦‖−𝜓(‖𝑥−𝑦‖),∀𝑥,𝑦∈𝐶,(2.8) where 𝜓∶[0,+∞)→[0,+∞) is a continuous and strictly increasing function such that 𝜓 is positive on (0,∞) and 𝜓(0)=0. As a special case, if 𝜓(𝑡)=(1−𝑘)𝑡 for 𝑡∈[0,+∞), where 𝑘∈(0,1), then the weakly contractive mapping 𝐴 is a contraction with constant 𝑘. Rhoades [22] obtained the following result for weakly contractive mapping.

Lemma 2.4 ([22, Theorem  2]). Let (𝑋,𝑑) be a complete metric space and let 𝐴 be a weakly contractive mapping on 𝑋. Then 𝐴 has a unique fixed point 𝑝 in 𝑋. Moreover, for 𝑥∈𝑋, {𝐴^𝑛𝑥} converges strongly to 𝑝.

The following Lemma was given in [23, 24].

Lemma 2.5. Let {𝑠_𝑛} and {𝛾_𝑛} be two sequences of nonnegative real numbers and let {𝜆_𝑛} be a sequence of positive numbers satisfying the conditions:(i)∑^∞_𝑛=0𝜆_𝑛=∞;(ii)lim_𝑛→∞(𝛾_𝑛/𝜆_𝑛)=0.Let the recursive inequality, 𝑠_𝑛+1≤𝑠_𝑛−𝜆_𝑛𝜓(𝑠_𝑛)+𝛾_𝑛,𝑛≥0,(2.9) be given, where 𝜓(𝑡) is a continuous and strict increasing function on [0,+∞) with 𝜓(0)=0. Then lim_𝑛→∞𝑠_𝑛=0.

3. Main Results

First, we study a strong convergence theorem for a viscosity iterative scheme for the nonexpansive mapping with the contraction.

For 𝑇∶𝐶→𝐶 a nonexpansive mapping, 𝑡∈(0,1) and 𝑓∈Σ_𝐶, 𝑡𝑓+(1−𝑡)𝑇∶𝐶→𝐶 defines a strict contraction mapping. Thus, by the Banach contraction mapping principle, there exists a unique fixed point 𝑥^𝑓_𝑡 satisfying 𝑥^𝑓_𝑡=𝑡𝑓(𝑥^𝑓_𝑡)+(1−𝑡)𝑇𝑥^𝑓_𝑡.(R)

For simplicity we will write 𝑥_𝑡 for 𝑥^𝑓_𝑡 provided no confusion occurs.

In 2006, the following result was given by Jung [25] (see also Xu [12] for the result in uniformly smooth Banach spaces).

Theorem 3 J (see [25]). Let 𝐸 be a reflexive Banach space having a uniformly Gâteaux differentiable norm. Suppose that every weakly compact convex subset of E has the fixed point property for nonexpansive mappings. Let 𝐶 be a nonempty closed convex subset of 𝐸 and let 𝑇 be a nonexpansive mapping from 𝐶 into itself with 𝐹(𝑇)≠∅. Then {𝑥_𝑡} defined by (R) converges strongly to a point in 𝐹(𝑇). If we define 𝑄∶Σ_𝐶→𝐹(𝑇) by 𝑄(𝑓)∶=lim_𝑡→0+𝑥_𝑡,𝑓∈Σ_𝐶,(3.1) then 𝑄(𝑓) is the unique solution of the variational inequality ⟨(𝐼−𝑓)(𝑄(𝑓)),𝐽(𝑄(𝑓)−𝑝)⟩≤0,𝑓∈Σ_𝐶,𝑝∈𝐹(𝑇).(VI)

Remark 3.1. In Theorem J, if 𝑓(𝑥)=𝑢∈𝐶 is a constant, then (VI) become ⟨𝑄𝑢−𝑢,𝐽(𝑄𝑢−𝑝)⟩≤0,𝑢∈𝐶,𝑝∈𝐹(𝑇).(3.2) Hence by (2.3), 𝑄 reduces to the sunny nonexpansive retraction from 𝐶 to 𝐹(𝑇). Namely 𝐹(𝑇) is a sunny nonexpansive retraction of 𝐶.
Using Theorem J, we have the following result.

Theorem 3.2. Let 𝐸 be a reflexive Banach space having a uniformly Gâteaux differentiable norm. Suppose that every weakly compact convex subset of 𝐸 has the fixed point property for nonexpansive mappings. Let 𝐶 be a nonempty closed convex subset of 𝐸 and let 𝑇 be a nonexpansive mapping from 𝐶 into itself with 𝐹(𝑇)≠∅.Let {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛} be sequences in (0,1) which satisfy the conditions: (C1)lim_𝑛→∞𝛼_𝑛=0, ∑^∞_𝑛=0𝛼_𝑛=∞;(C2)lim_𝑛→∞𝛽_𝑛=0;(C3)0<liminf_𝑛→∞𝛾_𝑛≤limsup_𝑛→∞𝛾_𝑛<1.Let 𝑓∈Σ_𝐶 and the initial guess 𝑥₀=𝑥∈𝐶 be chosen arbitrarily. Let {𝑥_𝑛} be the sequence generated by 𝑧_𝑛=𝛾_𝑛𝑥_𝑛+(1−𝛾_𝑛)𝑇𝑥_𝑛,𝑦_𝑛=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑇𝑧_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)𝑦_𝑛,𝑛≥0.(IS) If lim_𝑛→∞‖𝑇𝑧_𝑛−𝑧_𝑛‖=0, then {𝑥_𝑛} converges strongly to 𝑄(𝑓)∈𝐹(𝑇), where 𝑄(𝑓) is the unique solution of the variational inequality ⟨(𝐼−𝑓)(𝑄(𝑓)),𝐽(𝑄(𝑓)−𝑝)⟩≤0,𝑓∈Σ_𝐶,𝑝∈𝐹(𝑇).(3.3)

Proof. We note that by Theorem J, there exists the unique solution 𝑄(𝑓) of the variational inequality ⟨(𝐼−𝑓)(𝑄(𝑓)),𝐽(𝑄(𝑓)−𝑝)⟩≤0,𝑓∈Σ_𝐶,𝑝∈𝐹(𝑇).(3.4) Namely, 𝑄(𝑓)=lim_𝑡→0⁺𝑥_𝑡 where 𝑥_𝑡 is defined by (R). We will show that 𝑥_𝑛→𝑄(𝑓).
We proceed with the following steps.
Step 1. We show that ‖𝑥_𝑛−𝑧‖≤max{‖𝑥₀−𝑧‖,(1/(1−𝑘))‖𝑓(𝑧)−𝑧‖} for all 𝑛≥0 and all 𝑧∈𝐹(𝑇) and so {𝑥_𝑛}, {𝑦_𝑛}, {𝑧_𝑛}, {𝑓(𝑥_𝑛)}, {𝑇𝑥_𝑛}, and {𝑇𝑧_𝑛} are bounded.
Indeed, let 𝑧∈𝐹(𝑇). Then, noting that
‖𝑧_𝑛−𝑧‖≤𝛾_𝑛‖𝑥_𝑛−𝑧‖+(1−𝛾_𝑛)‖𝑇𝑥_𝑛−𝑧‖≤‖𝑥_𝑛−𝑧‖,(3.5) we have ‖𝑦_𝑛−𝑧‖=‖𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)(𝑇𝑧_𝑛−𝑧)‖≤𝛽_𝑛‖𝑥_𝑛−𝑧‖+(1−𝛽_𝑛)‖𝑥_𝑛−𝑧‖≤‖𝑥_𝑛−𝑧‖,(3.6) which yields that ‖𝑥_𝑛+1−𝑧‖=‖𝛼_𝑛(𝑓(𝑥_𝑛)−𝑧)+(1−𝛼_𝑛)(𝑦_𝑛−𝑧)‖≤𝛼_𝑛(‖𝑓(𝑥_𝑛)−𝑓(𝑧)‖+‖𝑓(𝑧)−𝑧‖)+(1−𝛼_𝑛)‖𝑥_𝑛−𝑧‖≤𝛼_𝑛𝑘‖𝑥_𝑛−𝑧‖+𝛼_𝑛‖𝑓(𝑧)−𝑧‖+(1−𝛼_𝑛)‖𝑥_𝑛−𝑧‖=(1−(1−𝑘)𝛼_𝑛)‖𝑥_𝑛−𝑧‖+𝛼_𝑛‖𝑓(𝑧)−𝑧‖≤max{‖𝑥_𝑛1−𝑧‖,1−𝑘‖𝑓(𝑧)−𝑧‖}.(3.7) Using an induction, we obtain ‖𝑥_𝑛−𝑧‖≤max{‖𝑥₀1−𝑧‖,1−𝑘‖𝑓(𝑧)−𝑧‖},(3.8) for all 𝑛≥0. Hence {𝑥_𝑛} is bounded, and so are {𝑦_𝑛}, {𝑧_𝑛}, {𝑇𝑥_𝑛}, {𝑇𝑧_𝑛}, and {𝑓(𝑥_𝑛)}.Step 2. We show that lim_𝑛→∞‖𝑥_𝑛+1−𝑦_𝑛‖=0 and lim_𝑛→∞‖𝑦_𝑛−𝑇𝑧_𝑛‖=0. Indeed, it follows from condition (C1) and (C2) that ‖𝑥_𝑛+1−𝑦_𝑛‖=𝛼_𝑛‖𝑓(𝑥_𝑛)−𝑦_𝑛‖→0(as𝑛→∞),‖𝑦_𝑛−𝑇𝑧_𝑛‖=𝛽_𝑛‖𝑥_𝑛−𝑇𝑧_𝑛‖→0(as𝑛→∞).(3.9) Also from lim_𝑛→∞‖𝑇𝑧_𝑛−𝑧_𝑛‖=0, we get ‖𝑦_𝑛−𝑧_𝑛‖≤‖𝑦_𝑛−𝑇𝑧_𝑛‖+‖𝑇𝑧_𝑛−𝑧_𝑛‖→0(as𝑛→∞).(3.10)Step 3. We show that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0. To this end, set 𝛿_𝑛=(1−𝛼_𝑛)𝛾_𝑛 for 𝑛≥0. Then it follows from (C1) and (C3) that 0<liminf_𝑛→∞𝛿_𝑛≤limsup_𝑛→∞𝛿_𝑛<1.(3.11) Define 𝑥_𝑛+1=𝛿_𝑛𝑥_𝑛+(1−𝛿_𝑛)𝑤_𝑛.(3.12) Observe that 𝑤_𝑛+1−𝑤_𝑛=𝑥_𝑛+2−𝛿_𝑛+1𝑥_𝑛+11−𝛿_𝑛+1−𝑥_𝑛+1−𝛿_𝑛𝑥_𝑛1−𝛿_𝑛=𝛼_𝑛+1𝑓(𝑥_𝑛+1)+(1−𝛼_𝑛+1)𝑦_𝑛+1−𝛿_𝑛+1𝑥_𝑛+11−𝛿_𝑛+1−𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)𝑦_𝑛−𝛿_𝑛𝑥_𝑛1−𝛿_𝑛𝛼=(^𝑛+1𝑓(𝑥_𝑛+1)1−𝛿_𝑛+1−𝛼_𝑛𝑓(𝑥_𝑛)1−𝛿_𝑛)+(1−𝛼_𝑛+1)[𝛾_𝑛+1𝑥_𝑛+1+(1−𝛾_𝑛+1)𝑇𝑥_𝑛+1]−𝛿_𝑛+1𝑥_𝑛+1+(1−𝛼_𝑛+1)(𝑦_𝑛+1−𝑧_𝑛+1)1−𝛿_𝑛+1−(1−𝛼_𝑛)[𝛾_𝑛𝑥_𝑛+(1−𝛾_𝑛)𝑇𝑥_𝑛]−𝛿_𝑛𝑥_𝑛+(1−𝛼_𝑛)(𝑦_𝑛−𝑧_𝑛)1−𝛿_𝑛𝛼=(^𝑛+1𝑓(𝑥_𝑛+1)1−𝛿_𝑛+1−𝛼_𝑛𝑓(𝑥_𝑛)1−𝛿_𝑛)+(1−𝛼_𝑛+1)(1−𝛾_𝑛+1)𝑇𝑥_𝑛+1+(1−𝛼_𝑛+1)(𝑦_𝑛+1−𝑧_𝑛+1)1−𝛿_𝑛+1−(1−𝛼_𝑛)(1−𝛾_𝑛)𝑇𝑥_𝑛+(1−𝛼_𝑛)(𝑦_𝑛−𝑧_𝑛)1−𝛿_𝑛𝛼=(^𝑛+1𝑓(𝑥_𝑛+1)1−𝛿_𝑛+1−𝛼_𝑛𝑓(𝑥_𝑛)1−𝛿_𝑛)+(𝑇𝑥_𝑛+1−𝑇𝑥_𝑛𝛼)−^𝑛+11−𝛿_𝑛+1𝑇𝑥_𝑛+1+𝛼_𝑛1−𝛿_𝑛𝑇𝑥_𝑛+(1−𝛼_𝑛+1)(𝑦_𝑛+1−𝑧_𝑛+1)1−𝛿_𝑛+1−(1−𝛼_𝑛)(𝑦_𝑛−𝑧_𝑛)1−𝛿_𝑛.(3.13) It follows from (3.13) that ‖𝑤_𝑛+1−𝑤_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖≤𝛼_𝑛+11−𝛿_𝑛+1(‖𝑓(𝑥_𝑛+1)‖+‖𝑇𝑥_𝑛+1𝛼‖)+^𝑛1−𝛿_𝑛(‖𝑓(𝑥_𝑛)‖+‖𝑇𝑥_𝑛+‖)1−𝛼_𝑛+11−𝛿_𝑛+1‖𝑦_𝑛+1−𝑧_𝑛+1‖+1−𝛼_𝑛1−𝛿_𝑛‖𝑦_𝑛−𝑧_𝑛‖.(3.14) Since {𝑓(𝑥_𝑛)} and {𝑇𝑥_𝑛} are bounded, by (C1), (3.10), (3.11), and (3.14) we obtain that limsup_𝑛→∞(‖𝑤_𝑛+1−𝑤_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0.(3.15) Hence by Lemma 2.2, we have lim_𝑛→∞‖𝑤_𝑛−𝑥_𝑛‖=0.(3.16) It follows from (3.11) and (3.12) that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0.(3.17)Step 4. We show that lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0. In fact, from (IS) it follows that ‖𝑇𝑥_𝑛−𝑥_𝑛‖≤‖𝑥_𝑛−𝑥_𝑛+1‖+‖𝑥_𝑛+1−𝑦_𝑛‖+‖𝑦_𝑛−𝑇𝑧_𝑛‖+‖𝑇𝑧_𝑛−𝑇𝑥_𝑛‖≤‖𝑥_𝑛−𝑥_𝑛+1‖+‖𝑥_𝑛+1−𝑦_𝑛‖+‖𝑦_𝑛−𝑇𝑧_𝑛‖+(1−𝛾_𝑛)‖𝑇𝑥_𝑛−𝑥_𝑛‖.(3.18) So we have 𝛾_𝑛‖𝑇𝑥_𝑛−𝑥_𝑛‖≤‖𝑥_𝑛−𝑥_𝑛+1‖+‖𝑥_𝑛+1−𝑦_𝑛‖+‖𝑦_𝑛−𝑇𝑧_𝑛‖.(3.19) Thus, from condition (C3), Steps 2, and 3, we have lim_𝑛→∞‖𝑇𝑥_𝑛−𝑥_𝑛‖=0.(3.20)Step 5. We show that limsup_𝑛→∞⟨𝑄(𝑓)−𝑓(𝑄(𝑓)),𝐽(𝑄(𝑓)−𝑥_𝑛)⟩≤0. To prove this, let a subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} be such that
limsup_𝑛→∞⟨𝑄(𝑓)−𝑓(𝑄(𝑓)),𝐽(𝑄(𝑓)−𝑥_𝑛)⟩=lim_𝑗→∞⟨𝑄(𝑓)−𝑓(𝑄(𝑓)),𝐽(𝑄(𝑓)−𝑥_𝑛𝑗)⟩(3.21) and 𝑥_𝑛𝑗⇀𝑞 for some 𝑞∈𝐸. From Step 4, it follows that lim_𝑗→∞‖𝑥_𝑛𝑗−𝑇𝑥_𝑛𝑗‖=0.
Now let 𝑄(𝑓)=lim_𝑡→0⁺𝑥_𝑡, where 𝑥_𝑡=𝑡𝑓(𝑥_𝑡)+(1−𝑡)𝑇𝑥_𝑡. Then we can write
𝑥_𝑡−𝑥_𝑛𝑗=𝑡(𝑓(𝑥_𝑡)−𝑥_𝑛𝑗)+(1−𝑡)(𝑇𝑥_𝑡−𝑥_𝑛𝑗).(3.22) Putting 𝑎_𝑗(𝑡)=(1−𝑡)²‖𝑇𝑥_𝑛𝑗−𝑥_𝑛𝑗‖(2‖𝑥_𝑡−𝑥_𝑛𝑗‖+‖𝑇𝑥_𝑛𝑗−𝑥_𝑛𝑗‖)→0(𝑗→∞)(3.23) by Step 4 and using Lemma 2.1, we obtain ‖𝑥_𝑡−𝑥_𝑛𝑗‖²≤(1−𝑡)²‖𝑇𝑥_𝑡−𝑥_𝑛𝑗‖²+2𝑡⟨𝑓(𝑥_𝑡)−𝑥_𝑛𝑗,𝐽(𝑥_𝑡−𝑥_𝑛𝑗)⟩≤(1−𝑡)²(‖𝑇𝑥_𝑡−𝑇𝑥_𝑛𝑗‖+‖𝑇𝑥_𝑛𝑗−𝑥_𝑛𝑗‖)²+2𝑡⟨𝑓(𝑥_𝑡)−𝑥_𝑡,𝐽(𝑥_𝑡−𝑥_𝑛𝑗)⟩+2𝑡‖𝑥_𝑡−𝑥_𝑛𝑗‖²≤(1−𝑡)²‖𝑥_𝑡−𝑥_𝑛𝑗‖²+𝑎_𝑗(𝑡)+2𝑡⟨𝑓(𝑥_𝑡)−𝑥_𝑡,𝐽(𝑥_𝑡−𝑥_𝑛𝑗)⟩+2𝑡‖𝑥_𝑡−𝑥_𝑛𝑗‖².(3.24) The last inequality implies ⟨𝑥_𝑡−𝑓(𝑥_𝑡),𝐽(𝑥_𝑡−𝑥_𝑛𝑗𝑡)⟩≤2‖𝑥_𝑡−𝑥_𝑛𝑗‖²+1𝑎2𝑡^𝑗(𝑡).(3.25) It follows that limsup_𝑗→∞⟨𝑥_𝑡−𝑓(𝑥_𝑡),𝐽(𝑥_𝑡−𝑥_𝑛𝑗𝑡)⟩≤2𝑀,(3.26) where 𝑀>0 is a constant such that 𝑀≥‖𝑥_𝑡−𝑥_𝑛‖² for all 𝑛≥0 and 𝑡∈(0,1). Taking the limsup as 𝑡→0 in (3.26) and noticing the fact that the two limits are interchangeable due to the fact that 𝐽 is uniformly continuous on bounded subsets of 𝐸 from the strong topology of 𝐸 to the weak^∗ topology of 𝐸^∗, we have limsup_𝑗→∞⟨𝑄(𝑓)−𝑓(𝑄(𝑓)),𝐽(𝑄(𝑓)−𝑥_𝑛𝑗)⟩≤0.(3.27) Indeed, letting 𝑡→0, from (3.26) we have limsup_𝑡→0limsup_𝑗→∞⟨𝑥_𝑡−𝑓(𝑥_𝑡),𝐽(𝑥_𝑡−𝑥_𝑛𝑗)⟩≤0.(3.28) So, for any 𝜀>0, there exists a positive number 𝛿₁ such that for any 𝑡∈(0,𝛿₁), limsup_𝑗→∞⟨𝑥_𝑡−𝑓(𝑥_𝑡),𝐽(𝑥_𝑡−𝑥_𝑛𝑗𝜀)⟩≤2.(3.29) Moreover, since 𝑥_𝑡→𝑄(𝑓) as 𝑡→0, the set {𝑥_𝑡−𝑥_𝑛𝑗} is bounded and the duality mapping 𝐽 is norm-to-weak^∗ uniformly continuous on bounded subset of 𝐸, there exists 𝛿₂>0 such that, for any 𝑡∈(0,𝛿₂), |⟨𝑄(𝑓)−𝑓(𝑄(𝑓)),𝐽(𝑄(𝑓)−𝑥_𝑛𝑗)⟩−⟨𝑥_𝑡−𝑓(𝑥_𝑡),𝐽(𝑥_𝑡−𝑥_𝑛𝑗)⟩|=|⟨𝑄(𝑓)−𝑓(𝑄(𝑓)),𝐽(𝑄(𝑓)−𝑥_𝑛𝑗)−𝐽(𝑥_𝑡−𝑥_𝑛𝑗)⟩+⟨𝑄(𝑓)−𝑓(𝑄(𝑓))−(𝑥_𝑡−𝑓(𝑥_𝑡)),𝐽(𝑥_𝑡−𝑥_𝑛𝑗)⟩|≤|⟨𝑄(𝑓)−𝑓(𝑄(𝑓)),𝐽(𝑥_𝑡−𝑥_𝑛𝑗)−𝐽(𝑄(𝑓)−𝑥_𝑛𝑗)⟩|+‖𝑄(𝑓)−𝑓(𝑄(𝑓))−(𝑥_𝑡−𝑓(𝑥_𝑡))‖‖𝑥_𝑡−𝑥_𝑛𝑗𝜀‖<2.(3.30) Choose 𝛿=min{𝛿₁,𝛿₂}, we have for all 𝑡∈(0,𝛿) and 𝑗∈ℕ, ⟨𝑄(𝑓)−𝑓(𝑄(𝑓)),𝐽(𝑄(𝑓)−𝑥_𝑛𝑗)⟩<⟨𝑥_𝑡−𝑓(𝑥_𝑡),𝐽(𝑥_𝑡−𝑥_𝑛𝑗𝜀)⟩+2,(3.31) which implies that limsup_𝑗→∞⟨𝑄(𝑓)−𝑓(𝑄(𝑓)),𝐽(𝑄(𝑓)−𝑥_𝑛𝑗)⟩≤limsup_𝑗→∞⟨𝑥_𝑡−𝑓(𝑥_𝑡),𝐽(𝑥_𝑡−𝑥_𝑛𝑗𝜀)⟩+2.(3.32) Since limsup_𝑗→∞⟨𝑥_𝑡−𝑓(𝑥_𝑡),𝐽(𝑥_𝑡−𝑥_𝑛𝑗)⟩≤𝜀/2, we have limsup_𝑗→∞⟨𝑄(𝑓)−𝑓(𝑄(𝑓)),𝐽(𝑄(𝑓)−𝑥_𝑛𝑗)⟩≤𝜀.(3.33) Since 𝜀 is arbitrary, we obtain that limsup_𝑗→∞⟨𝑄(𝑓)−𝑓(𝑄(𝑓)),𝐽(𝑄(𝑓)−𝑥_𝑛𝑗)⟩≤0.(3.34)Step 6. We show that lim_𝑛→∞‖𝑥_𝑛−𝑄(𝑓)‖=0. By using (IS), we have ‖𝑥_𝑛+1−𝑄(𝑓)‖=‖𝛼_𝑛(𝑓(𝑥_𝑛)−𝑄(𝑓))+(1−𝛼_𝑛)(𝑦_𝑛−𝑄(𝑓))‖.(3.35) Applying Lemma 2.1 and (3.6), we obtain ‖𝑥_𝑛+1−𝑄(𝑓)‖²≤(1−𝛼_𝑛)²‖𝑦_𝑛−𝑄(𝑓)‖²+2𝛼_𝑛⟨𝑓(𝑥_𝑛)−𝑄(𝑓),𝐽(𝑥_𝑛+1−𝑄(𝑓))⟩≤(1−𝛼_𝑛)²‖𝑥_𝑛−𝑄(𝑓)‖²+2𝛼_𝑛⟨𝑓(𝑥_𝑛)−𝑓(𝑄(𝑓)),𝐽(𝑥_𝑛+1−𝑄(𝑓))⟩+2𝛼_𝑛⟨𝑓(𝑄(𝑓))−𝑄(𝑓),𝐽(𝑥_𝑛+1−𝑄(𝑓))⟩≤(1−𝛼_𝑛)²‖𝑥_𝑛−𝑄(𝑓)‖²+2𝑘𝛼_𝑛‖𝑥_𝑛−𝑄(𝑓)‖‖𝑥_𝑛+1−𝑄(𝑓)‖+2𝛼_𝑛⟨𝑓(𝑄(𝑓))−𝑄(𝑓),𝐽(𝑥_𝑛+1−𝑄(𝑓))⟩≤(1−2𝛼_𝑛+𝛼²_𝑛)‖𝑥_𝑛−𝑄(𝑓)‖²+𝑘𝛼_𝑛(‖𝑥_𝑛−𝑄(𝑓)‖²+‖𝑥_𝑛+1−𝑄(𝑓)‖²)+2𝛼_𝑛⟨𝑓(𝑄(𝑓))−𝑄(𝑓),𝐽(𝑥_𝑛+1−𝑄(𝑓))⟩.(3.36) It then follows that ‖𝑥_𝑛+1−𝑄(𝑓)‖²≤(1−2(1−𝑘)𝛼_𝑛1−𝑘𝛼_𝑛)‖𝑥_𝑛−𝑄(𝑓)‖²+𝛼²_𝑛1−𝑘𝛼_𝑛‖𝑥_𝑛−𝑄(𝑓)‖²+2𝛼_𝑛1−𝑘𝛼_𝑛⟨𝑄(𝑓)−𝑓(𝑄(𝑓)),𝐽(𝑄(𝑓)−𝑥_𝑛+1)⟩≤(1−2(1−𝑘)𝛼_𝑛1−𝑘𝛼_𝑛)‖𝑥_𝑛−𝑄(𝑓)‖²+𝛼²_𝑛1−𝑘𝛼_𝑛𝑀²+2𝛼_𝑛1−𝑘𝛼_𝑛⟨𝑄(𝑓)−𝑓(𝑄(𝑓)),𝐽(𝑄(𝑓)−𝑥_𝑛+1)⟩,(3.37) where 𝑀=sup_𝑛≥0‖𝑥_𝑛−𝑄(𝑓)‖. Put 𝜆_𝑛=2(1−𝑘)𝛼_𝑛1−𝑘𝛼_𝑛,𝛾_𝑛=𝛼_𝑛𝑀2(1−𝑘)²+11−𝑘⟨𝑄(𝑓)−𝑓(𝑄(𝑓)),𝐽(𝑄(𝑓)−𝑥_𝑛+1)⟩.(3.38) From the condition (C1) and Step 5, it follows that 𝜆_𝑛→0, ∑^∞_𝑛=0𝜆_𝑛=∞, and limsup_𝑛→∞𝛾_𝑛≤0. Since (3.37) reduces to ‖𝑥_𝑛+1−𝑄(𝑓)‖²≤(1−𝜆_𝑛)‖𝑥_𝑛−𝑄(𝑓)‖²+𝜆_𝑛𝛾_𝑛,(3.39) from Lemma 2.3 with 𝛿_𝑛=0, we conclude that lim_𝑛→∞‖𝑥_𝑛−𝑄(𝑓)‖=0. This completes the proof.

Corollary 3.3. Let 𝐸 be a uniformly smooth Banach space. Let 𝐶, 𝑇,{𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, 𝑓, 𝑥₀, {𝑧_𝑛}, {𝑦_𝑛}, and {𝑥_𝑛} be the same as in Theorem 3.2. Then the conclusion of Theorem 3.2 still holds.

Proof. Since 𝐸 is a uniformly smooth Banach space, 𝐸 is reflexive, the norm is uniformly Gâteaux differentiable, and every nonempty weakly compact convex subset of 𝐸 has the fixed point property for nonexpansive mappings. Thus the conclusion of Corollary 3.3 follows from Theorem 3.2 immediately.

Corollary 3.4. Let 𝐶 be a nonempty closed convex subset of a uniformly smooth Banach space 𝐸. Let 𝑇∶𝐶→𝐶 be a nonexpansive mapping with 𝐹(𝑇)≠∅. Let {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛} be three sequences in (0,1) which satisfy the control conditions (C1)–(C3)   in Theorem 3.2. Then for the initial guess 𝑥₀∈𝐶 and 𝑢∈𝐶, the sequence {𝑥_𝑛} generated by (1.6) converges strongly to a fixed point of 𝑇 under the assumption lim_𝑛→∞‖𝑇𝑧_𝑛−𝑧_𝑛‖=0.

Remark 3.5. (1) In general, the condition (C3) in Theorem 3.2 and the condition ∑^∞_𝑛=0|𝛾_𝑛+1−𝛾_𝑛|<∞ of Qin et al. [16, Theorem  2.1] are not comparable; neither of them implies the other. Theorem 3.2 (and Corollary 3.4) removes the conditions ∑^∞_𝑛=0|𝛼_𝑛+1−𝛼_𝑛|<∞ and ∑^∞_𝑛=0|𝛽_𝑛+1−𝛽_𝑛|<∞ imposed on the control parameters {𝛼_𝑛} and {𝛽_𝑛} of Qin et al. [16, Theorem  2.1].
(2) Theorem 3.2 (and Corollary 3.3) complements the corresponding results in Moudafi [11], Xu [12], and Yao et al. [15]. In particular, if 𝛽_𝑛=0 in (IS), then (IS) in Theorem 3.2 reduces a new one for finding a fixed point of 𝑇: 𝑧_𝑛=𝛾_𝑛𝑥_𝑛+(1−𝛾_𝑛)𝑇𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)𝑇𝑧_𝑛,𝑛≥0.(3.40)
(3) Corollary 3.4 with 𝛽_𝑛=0 in (IS) develops the corresponding results of Shioji and Takahashi [8], Wittmann [9] without the condition ∑^∞_𝑛=0|𝛼_𝑛+1−𝛼_𝑛|<∞ as well as the result of Chang [2] in which the condition lim_𝑛→∞‖𝑇𝑥_𝑛−𝑥_𝑛‖=0 was assumed.
Next, we consider the viscosity iterative scheme with the weakly contractive mappings instead of the contractions.

Theorem 3.6. Let 𝐸 be a reflexive Banach space having a uniformly Gâteaux differentiable norm. Suppose that every weakly compact convex subset of 𝐸 has the fixed point property for nonexpansive mappings. Let 𝐶be a nonempty closed convex subset of 𝐸and let 𝑇be a nonexpansive mapping from 𝐶into itself with 𝐹(𝑇)≠∅. Let {𝛼_𝑛},{𝛽_𝑛},and {𝛾_𝑛}be sequences in (0,1) which satisfy the conditions (C1)–(C3) in Theorem 3.2. Let 𝐴∶𝐶→𝐶be a weakly contractive mapping and let 𝑥₀∈𝐶be chosen arbitrarily. Let{𝑥_𝑛}be the sequence generated by 𝑧_𝑛=𝛾_𝑛𝑥_𝑛+(1−𝛾_𝑛)𝑇𝑥_𝑛,𝑦_𝑛=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑇𝑧_𝑛,𝑥_𝑛+1=𝛼_𝑛𝐴𝑥_𝑛+(1−𝛼_𝑛)𝑦_𝑛,𝑛≥0.(3.41) If lim_𝑛→∞‖𝑇𝑧_𝑛−𝑧_𝑛‖=0, then {𝑥_𝑛} converges strongly to 𝑄(𝐴𝑥^∗)=𝑥^∗∈𝐹(𝑇), where 𝑄 is a sunny nonexpansive retraction from 𝐶 onto 𝐹(𝑇).

Proof. It follows from Remark 3.1 that 𝐹(𝑇) is the sunny nonexpansive retract of 𝐶. Denote by 𝑄 the sunny nonexpansive retraction of 𝐶 onto 𝐹(𝑇). Then 𝑄∘𝐴 is a weakly contractive mapping of 𝐶 into itself. Indeed, ‖𝑄(𝐴𝑥)−𝑄(𝐴𝑦)‖≤‖𝐴𝑥−𝐴𝑦‖≤‖𝑥−𝑦‖−𝜓(‖𝑥−𝑦‖),∀𝑥,𝑦∈𝐶.(3.42) Lemma 2.4 assures that there exists a unique element 𝑥^∗∈𝐶 such that 𝑥^∗=𝑄(𝐴𝑥^∗). Such a 𝑥^∗∈𝐶 is an element of 𝐹(𝑇).
Now we define an iterative scheme as follows: 𝑢_𝑛=𝛾_𝑛𝑤_𝑛+(1−𝛾_𝑛)𝑇𝑤_𝑛,𝑣_𝑛=𝛽_𝑛𝑤_𝑛+(1−𝛽_𝑛)𝑇𝑢_𝑛,𝑤_𝑛+1=𝛼_𝑛𝐴𝑥^∗+(1−𝛼_𝑛)𝑣_𝑛,𝑛≥0.(3.43) Let {𝑤_𝑛} be the sequence generated by (3.37). Then, by taking 𝑢_𝑛 as 𝑧_𝑛 in lim_𝑛→∞‖𝑇𝑧_𝑛−𝑧_𝑛‖=0, Theorem 3.2 with 𝑓=𝐴𝑥^∗ a constant assures that {𝑤_𝑛} converges strongly to 𝑄(𝐴𝑥^∗)=𝑥^∗ as 𝑛→∞. For any 𝑛≥0, observe that ‖𝑦_𝑛−𝑣_𝑛‖≤𝛽_𝑛‖𝑥_𝑛−𝑤_𝑛‖+(1−𝛽_𝑛)‖𝑇𝑧_𝑛−𝑇𝑢_𝑛‖≤𝛽_𝑛‖𝑥_𝑛−𝑤_𝑛‖+(1−𝛽_𝑛)‖𝑧_𝑛−𝑢_𝑛‖≤𝛽_𝑛‖𝑥_𝑛−𝑤_𝑛‖+(1−𝛽_𝑛)‖𝛾_𝑛𝑥_𝑛+(1−𝛾_𝑛)𝑇𝑥_𝑛−(𝛾_𝑛𝑤_𝑛+(1−𝛾_𝑛)𝑇𝑤_𝑛)‖≤𝛽_𝑛‖𝑥_𝑛−𝑤_𝑛‖+(1−𝛽_𝑛)[𝛾_𝑛‖𝑥_𝑛−𝑤_𝑛‖+(1−𝛾_𝑛)‖𝑥_𝑛−𝑤_𝑛‖]=‖𝑥_𝑛−𝑤_𝑛‖(3.44) Then we have ‖𝑥_𝑛+1−𝑤_𝑛+1‖≤𝛼_𝑛‖𝐴𝑥_𝑛−𝐴𝑥^∗‖+(1−𝛼_𝑛)‖𝑦_𝑛−𝑣_𝑛‖≤𝛼_𝑛(‖𝐴𝑥_𝑛−𝐴𝑤_𝑛‖+‖𝐴𝑤_𝑛−𝐴𝑥^∗‖)+(1−𝛼_𝑛)‖𝑥_𝑛−𝑤_𝑛‖≤𝛼_𝑛‖𝑥_𝑛−𝑤_𝑛‖−𝛼_𝑛𝜓(‖𝑥_𝑛−𝑤_𝑛‖)+𝛼_𝑛(‖𝑤_𝑛−𝑥^∗‖−𝜓(‖𝑤_𝑛−𝑥^∗‖))+(1−𝛼_𝑛)‖𝑥_𝑛−𝑤_𝑛‖≤‖𝑥_𝑛−𝑤_𝑛‖−𝛼_𝑛𝜓(‖𝑥_𝑛−𝑤_𝑛‖)+𝛼_𝑛‖𝑤_𝑛−𝑥^∗‖.(3.45) Thus, for 𝑠_𝑛=‖𝑥_𝑛−𝑤_𝑛‖, we obtain the following recursive inequality: 𝑠_𝑛+1≤𝑠_𝑛−𝛼_𝑛𝜓(𝑠_𝑛)+𝛼_𝑛‖𝑤_𝑛−𝑥^∗‖.(3.46) Since ‖𝑤_𝑛−𝑥^∗‖→0, it follows from Lemma 2.5 that lim_𝑛→∞‖𝑥_𝑛−𝑤_𝑛‖=0. Hence lim_𝑛→∞‖𝑥_𝑛−𝑥^∗‖≤lim_𝑛→∞(‖𝑥_𝑛−𝑤_𝑛‖+‖𝑤_𝑛−𝑥^∗‖)=0.(3.47) This completes the proof.

Corollary 3.7. Let 𝐸 be a uniformly smooth Banach space. Let 𝐶, 𝑇, 𝐴, 𝑥₀, {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, {𝑧_𝑛}, {𝑦_𝑛}, and {𝑥_𝑛} be the same as in Theorem 3.6. Then the conclusion of Theorem 3.6 still holds.

Remark 3.8. (1) Theorem 3.6 (and Corollary 3.7) develops and complements the corresponding results in Moudafi [11], Qin et al. [16], Shioji and Takahashi [8], Wittmann [9], Xu [10, 12], and Yao et al. [15].
(2) Even in the case of 𝛽_𝑛=0 in Theorem 3.6, Theorem 3.6 appears to be independent of Theorem  5.6 of Wong et al. [13] in which the control conditions (C1) and ∑^∞_𝑛=0|𝛼_𝑛+1−𝛼_𝑛|<∞ were utilized. In fact, it appears to be unknown whether a reflexive and strictly convex space satisfies the fixed point property for nonexpansive mappings.
(3) The merits of our results in this paper are that fewer restrictions are imposed on the control parameters {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛}. All of our results can be viewed as a supplement to the results obtained by Qin et al. [16], Kim and Xu [14], and Yao et al. [15].
(4) The conclusions of Theorems 3.2 and 3.6 still hold if 𝐸 is assumed to be strictly convex instead of having the fixed point property for nonexpansive mappings.

Acknowledgments

The author thanks the editor Professor Simeon Reich and the referees for their careful reading and valuable suggestions to improve the writing of this paper.