On Mann’s Method with Viscosity for Nonexpansive and Nonspreading Mappings in Hilbert Spaces

Abstract

In the setting of Hilbert spaces, inspired by Iemoto and Takahashi (2009), we study a Mann’s method with viscosity to approximate strongly (common) fixed points of a nonexpansive mapping and a nonspreading mapping. A crucial tool in our results is the nonspreading-average type mapping.

Introduction

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ that induces the norm ‖⋅‖=⟨⋅,⋅⟩. Let 𝑇∶Dom(𝑇)→𝐻 be a mapping. We denote by Fix(𝑇) the set of fixed points of 𝑇, Fix(𝑇)={𝑥∈Dom(𝑇)∶𝑇𝑥=𝑥}.

A mapping 𝑇 is said to be(i)nonexpansive [1] (1967) if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈Dom(𝑇);(ii)firmly nonexpansive [1] (1967) if ⟨𝑥−𝑦,𝑇𝑥−𝑇𝑦⟩≥‖𝑇𝑥−𝑇𝑦‖² for all 𝑥,𝑦∈Dom(𝑇);(iii)firmly type nonexpansive [2] (2009) if 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²−𝑘(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦² (1) for all 𝑥,𝑦∈Dom(𝑇) for all 𝑥,𝑦∈Dom(𝑇);(iv)strongly nonexpansive [3] (1977) if  𝑇  is nonexpansive and (𝑥_𝑛)_𝑛∈ℕ,(𝑦_𝑛)_𝑛∈ℕ⊂Dom(𝑇), ‖𝑥_𝑛−𝑦_𝑛‖ bounded and (‖𝑥_𝑛−𝑦_𝑛‖−‖𝑇𝑥_𝑛−𝑇𝑦_𝑛‖)→0 then ‖(𝐼−𝑇)𝑥_𝑛−(𝐼−𝑇)𝑦_𝑛‖→0 for all 𝑥,𝑦∈Dom(𝑇);(v)nonspreading [4] (2008) if 2‖𝑇𝑥−𝑇𝑦‖²≤‖𝑇𝑥−𝑦‖²+‖𝑥−𝑇𝑦‖² for all 𝑥,𝑦∈Dom(𝑇);(vi)𝑘-strict nonspreading [5] (2011) if 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²)()𝑦+2𝑥−𝑇𝑥,𝑦−𝑇𝑦+𝑘(𝐼−𝑇𝑥−𝐼−𝑇²; (2)(vii)quasi-nonexpansive if ‖𝑇𝑥−𝑦‖≤‖𝑥−𝑦‖ for all 𝑥∈Dom(𝑇) and for all 𝑦∈Fix(𝑇).

Of course,

firmly nonexpansive ⇒ firmly type nonexpansive ⇒ strongly nonexpansive ⇒ nonexpansive ⇒ quasi-nonexpansive ⇐ nonspreading ⇐  𝑘-strict pseudononspreading.

If 𝐶 is a nonempty, closed, and convex subset of 𝐻, we denote by 𝑃_𝐶∶𝐻→𝐶 the metric projection on 𝐶; that is, for any 𝑥∈𝐻, 𝑃_𝐶𝑥 is the unique element in 𝐶 such that 𝑥−𝑃_𝐶𝑥≤𝑥−𝑦,∀𝑦∈𝐶. (3) It is well known (see [1]) that 𝑃_𝐶 is a firmly nonexpansive mapping and that 𝑃_𝐶 is characterized by the variational inequality 𝑃_𝐶𝑥−𝑥,𝑧−𝑃_𝐶𝑥≥0,∀𝑧∈𝐶. (4) The firm nonexpansivity has many equivalent formulations.

Theorem 1. Let 𝑇∶𝐶→𝐶 a mapping. There are equivalents.(1)𝑇 is firmly nonexpansive; that is, ⟨𝑥−𝑦,𝑇𝑥−𝑇𝑦⟩≥‖𝑇𝑥−𝑇𝑦‖² for all 𝑥,𝑦∈𝐷𝑜𝑚(𝑇).(2)For each 𝑥,𝑦∈𝐶, the convex function Φ_𝑥,𝑦∶[0,1]→ℝ defined by Φ_𝑥,𝑦(𝑡)=(1−𝑡)𝑥−𝑦+𝑡𝑇𝑥−𝑇𝑦 (5) is nonincreasing on [0,1].(3)The mapping (2𝑡−𝐼) is nonexpansive.(4)𝑇=(1/2)(𝐼+𝑁) with 𝑁 is nonexpansive.(5)‖𝑇𝑥−𝑇𝑦‖≤Φ_𝑥,𝑦(𝑡) for all 𝑡∈[0,1].(6)The mapping (𝐼−𝑇) is firmly nonexpansive.(7)‖𝑇𝑥−𝑇𝑦‖²≤‖𝑥−𝑦‖²−‖(𝐼−𝑡)𝑥−(𝐼−𝑇)𝑦‖² holds.(8)One has 2‖𝑇𝑥−𝑇𝑦‖²+‖𝑇𝑥−𝑥‖²+‖𝑇𝑦−𝑦‖²≤‖𝑇𝑥−𝑦‖²+‖𝑇𝑦−𝑥‖².

Proof. The equivalences (1) to (5) are proved in [6]. The equivalences (1) and (6) are proved in [7]. Let us prove that (1) is equivalent to (8): 2𝑇𝑥−𝑇𝑦²+‖𝑇𝑥−𝑥‖²+𝑇𝑦−𝑦²≤𝑇𝑥−𝑦²+𝑇𝑦−𝑥²⟺2𝑇𝑥−𝑇𝑦²−2⟨𝑇𝑥,𝑥⟩−2𝑇𝑦,𝑦+‖𝑇𝑥‖²+𝑦²≤‖𝑇𝑥‖²+𝑦²+𝑇𝑦²+‖𝑥‖²−2𝑇𝑥,𝑦−2𝑇𝑦,𝑥⟺2𝑇𝑥−𝑇𝑦²−2⟨𝑇𝑥,𝑥⟩−2𝑇𝑦,𝑦≤−2𝑇𝑥,𝑦−2𝑇𝑦,𝑥⟺2𝑇𝑥−𝑇𝑦²⟨⟩⟺≤2𝑇𝑥,𝑥+2𝑇𝑦,𝑦−2𝑇𝑥,𝑦−2𝑇𝑦,𝑥=2𝑇𝑥,𝑥−𝑦−2𝑇𝑦,𝑥−𝑦=2𝑇𝑥−𝑇𝑦,𝑥−𝑦𝑇𝑥−𝑇𝑦²≤𝑇𝑥−𝑇𝑦,𝑥−𝑦. (6) Finally, we prove that (1) is equivalent to (7). (𝐼−𝑇) is firmly nonexpansive ⇔⟨𝑥−𝑦,(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦⟩  ≥  ‖(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦‖²⇔‖(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦‖²  ≤  ‖𝑥−𝑦‖²−⟨𝑥−𝑦,𝑇𝑥−𝑇𝑦⟩  ≤  ‖𝑥−𝑦‖²−‖𝑇𝑥−𝑇𝑦‖².

Two important classes of mappings containing the firmly nonexpansive mappings are the average mappings and the nonspreading mappings.

After [7], we say that 𝑇 is an nonexpansive-average mappings if 𝑇=(1−𝛼)𝐼+𝛼𝑁 (7) for some 𝛼∈(0,1), and 𝑁 is a nonexpansive mapping.

Definition 2. Let 𝔐 be a class of mappings. One says that 𝑇 is a 𝔐-average mapping if 𝑇=𝑁_𝛼=(1−𝛼)𝐼+𝛼𝑁 (8) for some 𝛼∈(0,1) where 𝑁 is a mapping belonging to the class 𝔐.

Of course Fix(𝑁_𝛼)=Fix(𝑁).

The nonexpansive-average mapping regularizes a nonexpansive mapping 𝑁 according to the celebrated Schaefer’s result [8].

Theorem 3. Any orbit (𝑁^𝑘_𝛼𝑥)_𝑘∈ℕ of a nonexpansive-average mapping 𝑁_𝛼=(1−𝛼)𝐼−𝛼𝑁 converges weakly to a fixed point of  𝑁 whenever such points exist.

Here we are interested in nonspreading and nonspreading-average mappings.

Theorem 4. Let 𝑆∶𝐶→𝐶 be a mapping. The following are equivalent.(1)𝑆 is nonspreading; that is, 2‖𝑇𝑥−𝑇𝑦‖²≤‖𝑇𝑥−𝑦‖²+‖𝑥−𝑇𝑦‖²;(2)One has ‖𝑆𝑥−𝑆𝑦‖²≤‖𝑥−𝑦‖²+2⟨𝑥−𝑆𝑥,𝑦−𝑆𝑦⟩;(3)‖𝑆𝑥−𝑆𝑦‖²≤‖𝑥−𝑦‖²+‖𝑥−𝑆𝑥‖²+‖𝑦−𝑆𝑦‖²−‖(𝐼−𝑆)𝑥−(𝐼−𝑆)𝑦‖².
Moreover, let 𝑆 be a nonspreading mapping. Then(a)Fix(𝑆) is closed and convex;(b)(𝐼−𝑆) is demiclosed;(c)One has ‖(𝐼−𝑆)𝑥−(𝐼−𝑆)𝑦‖²≤⟨𝑥−𝑦,(𝐼−𝑆)𝑥−(𝐼−𝑆)𝑦⟩+(1/2)‖(𝐼−𝑆)𝑥‖²+(1/2)‖(𝐼−𝑆)𝑦‖².
If 𝑆_𝜔=(1−𝜔)𝐼+𝜔𝑆 is a nonspreading-average mapping, then one has the following.(i)𝑆_𝜔𝑥−𝑆_𝜔𝑦²≤𝑥−𝑦²+2𝜔𝑥−𝑆_𝜔𝑥,𝑦−𝑆_𝜔𝑦−(1−𝜔)𝑥−𝑆_𝜔𝑥−𝑦−𝑆_𝜔𝑦². (9) In particular 𝑆_𝜔 is quasi firmly nonexpansive; that is, 𝑆_𝜔𝑥−𝑞²≤𝑥−𝑞²−(1−𝜔)𝑥−𝑆_𝜔𝑥². (10)(ii)(𝜔/2)‖𝑥−𝑆𝑥‖²≤⟨𝑥−𝑆_𝜔𝑥,𝑥−𝑞⟩, for all 𝑞∈Fix(𝑆).

Proof. The equivalence of (1) and (2) is proved in Lemma 3.2 of [9].
The equivalence of (1) and (3) follows by the fact that 2𝑥−𝑆_𝜔𝑥,𝑥−𝑞=−(𝑥−𝑆𝑥)−𝑦−𝑆𝑦²+‖𝑥−𝑆𝑥‖²+𝑦−𝑆𝑦². (11) The item (a) is proved in [4], while (b) and (c) are proved in [9].
The item (i) is proved in Theorem 3.1 of [5].
Now we prove (ii). Since 𝑥−𝑆_𝜔𝑥,𝑥−𝑞=𝑥−(1−𝜔)𝑥−𝑆𝑥,𝑥−𝑞=𝜔𝑥−𝑆𝑥,𝑥−𝑞 (12) thus we need to show that 12‖𝑥−𝑆𝑥‖²≤𝑥−𝑆𝑥,𝑥−𝑞. (13) This follows by quasi-nonexpansivity of 𝑆. Indeed 12‖𝑥−𝑆𝑥‖²=12𝑥−𝑞+𝑞−𝑆𝑥²=12𝑥−𝑞²+𝑞−𝑆𝑥²≤+2𝑥−𝑞,𝑞−𝑆𝑥bythequasi-nonexpansivityof𝑆𝑥−𝑞²+=+=.𝑥−𝑞,𝑞−𝑆𝑥𝑥−𝑞,𝑥−𝑞𝑥−𝑞,𝑞−𝑆𝑥𝑥−𝑞,𝑥−𝑆𝑥 (14)

Recently, Song and Chai [2] in the general setting of Banach spaces obtained strong convergence of Halpern’s iteration 𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇𝑥_𝑛 (15) for firmly type nonexpansive mapping 𝑇. (Saejung in [10] noted that their proof seems to be questionable, but the result is true as a consequence of a more general result proved in [10]). Indeed, in [10] is proved the strong convergence of Halpern iteration for strongly nonexpansive mappings (and it is easy to see that the class of strongly nonexpansive mappings contains the class of firmly type nonexpansive mappings).

Osilike and Isiogugu [5] studied the Halpern iteration for 𝑘-strict pseudo-non-spreading mappings. They showed that if one considers the 𝑘-strict pseudo-non-spreading-average mapping, then Halpern’s iteration converges strongly to a fixed point of such a mapping.

On the other hand, Iemoto and Takahashi [9] approximated weakly fixed points of nonexpansive mappings and/or a nonspreading mapping in a Hilbert space using Moudafi’s iteration scheme [11]. Specifically, they proved the following result.

Theorem 5. Let 𝐻 be a Hilbert space and let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝑆 be a nonspreading mapping on 𝐶 into itself and let 𝑇 be a nonexpansive mappings on 𝐶 into itself such that Fix(𝑇)∩Fix(𝑆)≠∅. Define a sequence (𝑥_𝑛)_𝑛∈ℕ as follows: 𝑥₁𝑥∈𝐶,_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑇𝑥_𝑛 (16) for all 𝑛∈ℕ, where (𝛼_𝑛)_𝑛∈ℕ, (𝛽_𝑛)_𝑛∈ℕ, are in [0,1]. Then the following holds.(I)if liminf𝛼_𝑛(1−𝛼_𝑛)>0 and ∑(1−𝛽_𝑛)<∞, then (𝑥_𝑛)_𝑛 weakly converges to 𝑣∈Fix(𝑆).(II)if ∑𝛼_𝑛(1−𝛼_𝑛)=∞ and ∑𝛽_𝑛<∞, then (𝑥_𝑛)_𝑛 weakly converges to 𝑣∈Fix(𝑇).(III)if liminf𝛼_𝑛(1−𝛼_𝑛)>0 and liminf𝛽_𝑛(1−𝛽_𝑛)>0, then (𝑥_𝑛)_𝑛 weakly converges to 𝑣∈Fix(𝑇)∩Fix(𝑆).

In [12], the authors obtained strong convergence for the Halpern method by using type average mappings, with assumptions on the coefficients very similar to Theorem 5.

So one can ask if this result holds for Moudafi’s viscosity method [13]. We cannot take advantage of using the above positive results on Halpern’s iteration and invoke Suzuki’s result [14] that affirms that Halpern’s approximation convergence implies Moudafi’s viscosity approximation convergence. Indeed, as proved by Suzuki, this is true for nonexpansive mappings not for nonspreading mappings.

In spite of this we obtain the affirmative answer in our main result.

Our proofs took inspiration by [5, 12, 15, 16]. Related papers in which there are not nonspreading but other types of mappings or semigroups of nonexpansive mappings are [17–23].

2. Main Results

In this section, we always will assume the following.(i)𝐻 is a Hilbert space.(ii)𝐶 is a closed and convex subset of 𝐻.(iii)𝑇∶𝐶→𝐶 is a nonexpansive mapping.(iv)𝑇_𝜔∶𝐶→𝐶 is an average mapping of 𝑇, 𝑇_𝜔=(1−𝜔)𝐼+𝜔𝑇.(v)𝑆∶𝐶→𝐶 is a nonspreading mapping.(vi)𝑆_𝜔∶𝐶→𝐶 is a nonspreading-average mapping of 𝑆, 𝑆_𝜔=(1−𝜔)𝐼+𝜔𝑆.(vii)𝑈_𝑛∶𝐶→𝐶 is a convex combination of 𝑇_𝜔 and 𝑆_𝜔, 𝑈_𝑛=(1−𝛽_𝑛)𝑆_𝜔+𝛽_𝑛𝑇_𝜔.(viii)Fix(𝑇)∩Fix(𝑆)≠∅.(ix)𝑓∶𝐶→𝐶 is a 𝜌-contraction; that is, ‖𝑓(𝑥)−𝑓(𝑦)‖≤𝜌‖𝑥−𝑦‖, 0<𝜌<1.(x)(𝛼_𝑛)_𝑛∈ℕ⊂[0,1] is a real sequence satisfying 𝛼_𝑛→0 and ∑^∞_𝑛=1𝛼_𝑛=∞.(xi)𝑂(1) denote any bounded real sequence (so 𝑂(1)+𝑂(1)=𝑂(1)).

The following lemmas are the keys to obtain our main result.

Lemma 6 (see [24]). Assume that (𝑎_𝑛)_𝑛∈ℕ is a sequence of nonnegative numbers such that 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛾_𝑛𝛿_𝑛+𝜎_𝑛,𝑛≥0, (17) where (𝛾_𝑛)_𝑛 is a sequence in (0,1) and (𝛿_𝑛)_𝑛 is a sequence in ℝ and (𝜎_𝑛)_𝑛⊂ℝ⁺ such that, (1)∑^∞_𝑛=1𝛾_𝑛=∞; (2)limsup_𝑛→∞𝛿_𝑛≤0 and ∑^∞_𝑛=1|𝜎_𝑛|<∞.Then, lim_𝑛→∞𝑎_𝑛=0.

Lemma 7. Let (𝑥_𝑛)_𝑛 be the sequence defined by 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑈_𝑛𝑥_𝑛. (18) Then, (i) 𝑈_𝑛 is quasi nonexpansive; (ii) (𝑥_𝑛)_𝑛,(𝑆𝑥_𝑛)_𝑛,(𝑇𝑥_𝑛)_𝑛,(𝑆_𝜔𝑥_𝑛)_𝑛,(𝑇_𝜔𝑥_𝑛)_𝑛, and (𝑈_𝑛𝑥_𝑛)_𝑛 are bounded sequences.

Proof. (i) Any convex combination of quasi nonexpansive mappings is, in turn, quasi nonexpansive. So is 𝑈_𝑛, since 𝑇_𝜔 and 𝑆_𝜔 are quasi nonexpansive (see Theorem 4, (i)).
(ii) We see that the boundedness of (𝑥_𝑛)_𝑛 follows by the quasi nonexpansivity of 𝑈_𝑛. For this let 𝑞∈Fix(𝑇)∩Fix(𝑆). Then 𝑥_𝑛+1−𝑞≤𝛼_𝑛𝑓𝑥_𝑛−𝑞+1−𝛼_𝑛𝑈_𝑛𝑥_𝑛−𝑞≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑞+𝛼_𝑛𝑓𝑞+−𝑞1−𝛼_𝑛𝑈_𝑛𝑥_𝑛−𝑞≤𝛼_𝑛𝜌𝑥_𝑛−𝑧+𝛼_𝑛𝑓𝑞+−𝑞1−𝛼_𝑛𝑥_𝑛≤−𝑞1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑞+𝛼_𝑛𝑓𝑞=−𝑞1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑞+𝛼_𝑛𝑓𝑞1−𝜌−𝑞‖⋅‖𝑥1−𝜌byconvexityof≤max_𝑛,𝑓𝑞−𝑞−𝑞⟹𝑥1−𝜌^𝑛+1𝑥−𝑞≤max₁,𝑓𝑞−𝑞−𝑞.1−𝜌 (19) The boundedness of (𝑥_𝑛)_𝑛 is proved. The boundedness of the other sequences in (ii) follows by this last (since Fix(𝑇)∩Fix(𝑆)≠∅).

Lemma 8. Let (𝑦_𝑛)_𝑛 be a bounded sequence in 𝐶. Then one has the following. (i)If ‖𝑦_𝑛−𝑇𝑦_𝑛‖→0, then limsup𝐼−𝑓𝑝,𝑦_𝑛−𝑝≥0, (20) where 𝑝=𝑃_Fix(𝑇)𝑓(𝑝) is the unique point in Fix(𝑇) that satisfies the variational inequality 𝐼−𝑓𝑝,𝑥−𝑝≥0,∀𝑥∈Fix(𝑇). (21)(ii)If ‖𝑦_𝑛−𝑆𝑦_𝑛‖→0, then limsup𝐼−𝑓𝑝,𝑦_𝑛−𝑝≥0, (22) where 𝑝=𝑃_Fix(𝑆)𝑓(𝑝) is the unique point in Fix(𝑆) that satisfies the variational inequality 𝐼−𝑓𝑝,𝑥−𝑝≥0,∀𝑥∈Fix(𝑇). (23)(iii)If both ‖𝑦_𝑛−𝑆𝑦_𝑛‖→0 and ‖𝑦_𝑛−𝑇𝑦_𝑛‖→0, then limsup𝐼−𝑓𝑝₀,𝑦_𝑛−𝑝₀≥0, (24) where 𝑝₀=𝑃_{Fix(𝑇)∩Fix(𝑆)}𝑓(𝑝₀) is the unique point in Fix(𝑇)∩Fix(𝑆) that satisfies the variational inequality 𝐼−𝑓𝑝₀,𝑥−𝑝₀≥0,∀𝑥∈Fix(𝑇)∩Fix(𝑆). (25)

Proof. (i) Let 𝑝 satisfy (21). Let (𝑦_𝑛𝑘) be a subsequence of (𝑦_𝑛) for which limsup_𝑛𝐼−𝑓𝑝,𝑦_𝑛−𝑝=lim_𝑘𝐼−𝑓𝑝,𝑦_𝑛𝑘−𝑝. (26) Select a subsequence (𝑦_𝑛𝑘𝑗)_𝑗  of (𝑦_𝑛𝑘)_𝑘  such that 𝑦_𝑛𝑘𝑗⇀𝑣 (this, of course, is possible by boundedness of (𝑦_𝑛)_𝑛). From the assumption ‖𝑦_𝑛−𝑇𝑦_𝑛‖→0 and demiclosedness of 𝑇 (see [1]) we have 𝑣∈Fix(𝑇), and limsup_𝑛𝐼−𝑓𝑝,𝑦_𝑛−𝑝=lim_𝑗𝐼−𝑓𝑝,𝑦_𝑛𝑘−𝑝=𝐼−𝑓𝑝,𝑥−𝑝 (27) so the claim follows by (21).
(ii) It follows as in (i) since 𝑆 is demicloded too (see Theorem 4, (b)).
(iii) Select a subsequence (𝑦_𝑛𝑘)_𝑘 of (𝑦_𝑛)_𝑛 such that limsup_𝑛𝐼−𝑓𝑝₀,𝑦_𝑛−𝑝₀=lim_𝑘𝐼−𝑓𝑝₀,𝑦_𝑛𝑘−𝑝₀, (28) where 𝑝₀ satisfies (25). Now select a subsequence (𝑦_𝑛𝑘𝑗)_𝑗 of (𝑦_𝑛𝑘)_𝑘 such that 𝑦_𝑛𝑘𝑗⇀𝑤. Then, by demiclosedness of both 𝑇 and 𝑆, and by the hypotheses ‖𝑦_𝑛−𝑇𝑦_𝑛‖→0 and ‖𝑦_𝑛−𝑆𝑦_𝑛‖→0, we obtain that 𝑇𝑤=𝑆𝑤=𝑤; that is, 𝑤∈Fix(𝑇)∩Fix(𝑆). So the claim follows by (25) and limsup_𝑛𝐼−𝑓𝑝₀,𝑦_𝑛−𝑝₀=lim_𝑗𝐼−𝑓𝑝₀,𝑦_𝑛𝑘−𝑝₀=𝐼−𝑓𝑝₀,𝑥−𝑝₀. (29)

Lemma 9 (see [6]). Let 𝐶 be a nonempty closed convex subset of 𝐻 and let 𝑇∶𝐶→𝐶 be a nonexpansive mapping. Then (𝐼−𝑇)∶𝐶→𝐻 is 1/2-inverse strongly monotone; that is, 12(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦²≤𝑥−𝑦,(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦. (30)

Lemma 10 (Maingé [25]). Let (𝜏_𝑛)_𝑛 be real sequence that has a subsequence (𝜏_𝑛𝑗) which satisfies 𝜏_𝑛𝑗<𝜏_𝑛𝑗+1 for all 𝑗. Then the sequence of integers (𝛿(𝑛))_𝑛 defined by 𝛿(𝑛)=max{𝑘≤𝑛∶𝜏_𝑘<𝜏_𝑘+1} has the following properties:(1)𝛿(𝑛)≤𝛿(𝑛+1);(2)𝛿(𝑛)→∞ as 𝑛→∞;(3)𝜏_𝛿(𝑛)<𝜏_𝛿(𝑛)+1;(4)𝜏_𝑛<𝜏_𝛿(𝑛)+1.

Theorem 11. Let 𝑥₁𝑥∈𝐶,_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝛽_𝑛𝑇_𝜔𝑥_𝑛+1−𝛽_𝑛𝑆_𝜔𝑥_𝑛,𝑛≥1 (31) with (𝛼_𝑛)_𝑛⊂(0,1),𝛼_𝑛→0 and ∑_𝑛𝛼_𝑛=∞. Then one has the following.(i)𝐼𝑓∑_𝑛(1−𝛽_𝑛)<∞ and ∑_𝑛|𝛼_𝑛−𝛼_𝑛+1|<∞, then (𝑥_𝑛) strongly converges to 𝑝=𝑃_Fix(𝑇)𝑓(𝑝) that is the unique point in Fix(𝑇) that satisfies the variational inequality 𝐼−𝑓𝑝,𝑥−𝑝≥0,∀𝑥∈Fix(𝑇). (32)(ii)𝐼𝑓∑_𝑛𝛽_𝑛<∞, then (𝑥_𝑛)_𝑛 converges strongly to 𝑝=𝑃_Fix(𝑆)𝑓(𝑝) that is the unique point in Fix(𝑆) that satisfies the variational inequality ⟨𝐼−𝑓𝑝,𝑥−𝑝⟩≥0,∀𝑥∈Fix(𝑆). (33)(iii)𝐼𝑓liminf_𝑛𝛽_𝑛(1−𝛽_𝑛)>0, then (𝑥_𝑛) strongly converges to 𝑝₀=𝑃_{Fix(𝑇)∩Fix(𝑆)}𝑓(𝑝₀) which is the unique point in Fix(𝑇)∩Fix(𝑆) that satisfies the variational inequality ⟨𝐼−𝑓𝑝₀,𝑥−𝑝₀⟩≥0,∀𝑥∈Fix(𝑇)∩Fix(𝑆). (34)

Proof. By Lemma 7, we obtain that (𝑥_𝑛)_𝑛 is bounded.
Proof of (i). Let 𝑝 be as in (i) of Lemma 8; that is,𝐼−𝑓𝑝,𝑥−𝑝≥0,∀𝑥∈Fix(𝑇). (35)
Step 1.  One has limsup_𝑛(‖𝑥_𝑛−𝑝‖−‖𝑥_𝑛+1−𝑝‖)=0.
Proof of Step 1. This immediately follows by the asymptotic regularity of (𝑥_𝑛)_𝑛. So we prove that (𝑥_𝑛) is asymptotically regular; that is, ‖𝑥_𝑛−𝑥_𝑛+1‖→0: 𝑥_𝑛+1−𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑈_𝑛𝑥_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1−1−𝛼_𝑛−1𝑈_𝑛−1𝑥_𝑛−1±1−𝛼_𝑛𝑈_𝑛−1𝑥_𝑛−1=1−𝛼_𝑛𝑈_𝑛𝑥_𝑛−𝑈_𝑛−1𝑥_𝑛−1+𝛼_𝑛−1−𝛼_𝑛𝑈_𝑛−1𝑥_𝑛−1−𝛼_𝑛−1𝑓𝑥_𝑛−1+𝛼_𝑛𝑓𝑥_𝑛−𝛼_𝑛𝑓𝑥_𝑛−1+𝛼_𝑛𝑓𝑥_𝑛−1=1−𝛼_𝑛𝑈_𝑛𝑥_𝑛−𝑈_𝑛−1𝑥_𝑛−1+𝛼_𝑛−1−𝛼_𝑛𝑈_𝑛−1𝑥_𝑛−1−𝑓𝑥_𝑛−1+𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑥_𝑛−1=𝛼_𝑛−1−𝛼_𝑛𝑈_𝑛−1𝑥_𝑛−1−𝑓𝑥_𝑛−1+𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑥_𝑛−1+1−𝛼_𝑛𝛽_𝑛𝑇_𝜔𝑥_𝑛+1−𝛽_𝑛𝑆_𝜔𝑥_𝑛−𝛽_𝑛−1𝑇_𝜔𝑥_𝑛−1−1−𝛽_𝑛−1𝑆_𝜔𝑥_𝑛−1±𝛽_𝑛𝑇_𝜔𝑥_𝑛−1±1−𝛽_𝑛𝑆_𝜔𝑥_𝑛−1=𝛼_𝑛−1−𝛼_𝑛𝑈_𝑛−1𝑥_𝑛−1−𝑓𝑥_𝑛−1+𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑥_𝑛−1+1−𝛼_𝑛𝛽_𝑛𝑇_𝜔𝑥_𝑛−𝑇_𝜔𝑥_𝑛−1+1−𝛽_𝑛𝑆_𝜔𝑥_𝑛−𝑆_𝜔𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑇_𝜔𝑥_𝑛−1−𝑆𝜔𝑥_𝑛−1. (36) So 𝑥_𝑛+1−𝑥_𝑛≤𝛼_𝑛−1−𝛼_𝑛𝑂(1)+𝛼_𝑛𝜌𝑥_𝑛−𝑥_𝑛−1+1−𝛼_𝑛𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+1−𝛽_𝑛𝑂(1)+𝛽_𝑛−𝛽_𝑛−1𝑂(1=)1−𝛼_𝑛𝛽_𝑛+𝛼_𝑛𝜌𝑥_𝑛−𝑥_𝑛−1+𝛾_𝑛=1−1−1−𝛼_𝑛𝛽_𝑛+𝛼_𝑛𝜌𝑥_𝑛−𝑥_𝑛−1+𝛾_𝑛, (37) where 𝛾_𝑛=|𝛼_𝑛−𝛼_𝑛−1|𝑂(1)+(1−𝛼_𝑛)(1−𝛽_𝑛)𝑂(1)+|𝛽_𝑛−𝛽_𝑛−1|𝑂(1) is such that ∑_𝑛𝛾_𝑛<∞, thanks to the assumptions ∑_𝑛|𝛼_𝑛−𝛼_𝑛−1|<∞ and ∑_𝑛(1−𝛽_𝑛)<∞.
So if we put 𝛿_𝑛=1−𝛽_𝑛+𝛼_𝑛𝛽_𝑛−𝛼_𝑛𝜌 we have 𝑥_𝑛+1−𝑥_𝑛≤1−𝛿_𝑛𝑥_𝑛−1−𝑥_𝑛+𝛾_𝑛. (38) From the assumption ∑_𝑛𝛼_𝑛=∞ we deduce immediately ∑_𝑛𝛿_𝑛=∞. This is sufficient for Xu’s Lemma 6, to conclude that (𝑥_𝑛)_𝑛 is asymptotically regular.
Step 2. One has ‖𝑥_𝑛−𝑇_𝜔𝑥_𝑛‖→0,and‖𝑥_𝑛−𝑇𝑥_𝑛‖=(1/𝜔)‖𝑥_𝑛−𝑇_𝜔𝑥_𝑛‖→0.
Proof of Step 2. We define an auxiliary sequence (𝑧_𝑛)_𝑛 by 𝑧_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑇_𝜔𝑥_𝑛. (39) Observe that 𝑧_𝑛+1−𝑇_𝜔𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛−𝑇_𝜔𝑥_𝑛⟶0 (40) and so 𝑧_𝑛+1−𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑇_𝜔𝑥_𝑛−𝛼_𝑛𝑓𝑥_𝑛−1−𝛼_𝑛𝑈_𝑛𝑥_𝑛=1−𝛼_𝑛𝑇_𝜔𝑥_𝑛−𝑈𝑥_𝑛=1−𝛼_𝑛𝑇_𝜔𝑥_𝑛−𝛽_𝑛𝑇_𝜔𝑥_𝑛−1−𝛽_𝑛𝑆_𝜔𝑥_𝑛=1−𝛼_𝑛1−𝛽_𝑛𝑇_𝜔𝑥_𝑛−𝑆_𝜔𝑥_𝑛⟶0; (41) hence we get _𝑛𝑥_𝑛−𝑧_𝑛<∞. (42) Now 𝑧_𝑛+1−𝑝²=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛(1−𝜔)𝑥_𝑛+1−𝛼_𝑛𝜔𝑇𝑥_𝑛−𝑝²=1−𝛼_𝑛𝜔𝑇𝑥_𝑛−𝑥_𝑛+𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛²bythewellknowninequality𝑥+𝑦²≤‖𝑥‖²≤+2𝑦,𝑥+𝑦1−𝛼_𝑛𝜔𝑇𝑥_𝑛−𝑥_𝑛+𝑥_𝑛−𝑝²+2𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛,𝑧_𝑛+1−𝑝≤1−𝛼_𝑛²𝜔²𝑇𝑥_𝑛−𝑥_𝑛²+𝑥_𝑛−𝑝²−21−𝛼_𝑛𝜔𝑥_𝑛−𝑝,𝑥_𝑛−𝑇𝑥_𝑛+2𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛𝑧_𝑛+1−𝑝≤1−𝛼_𝑛²𝜔²𝑇𝑥_𝑛−𝑥_𝑛²+𝑥_𝑛−𝑝²−21−𝛼_𝑛𝜔𝑥_𝑛−𝑝,(𝐼−𝑇)𝑥_𝑛−(𝐼−𝑇)𝑝+2𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛𝑧_𝑛+1−𝑝())≤bytheinversestrongmonotonicityof𝐼−𝑇,Lemma91−𝛼_𝑛²𝜔²𝑇𝑥_𝑛−𝑥_𝑛²+𝑥_𝑛−𝑝²−1−𝛼_𝑛𝜔(𝐼−𝑇)𝑥_𝑛−(𝐼−𝑇)𝑝²+2𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛𝑧_𝑛+1−𝑝=𝑥_𝑛−𝑝²−1−𝛼_𝑛𝜔1−1−𝛼_𝑛𝜔×𝑥_𝑛−𝑇𝑥_𝑛²+2𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛𝑧_𝑛+1−𝑝 (43) and hence 0≤1−𝛼_𝑛𝜔1−1−𝛼_𝑛𝜔𝑥_𝑛−𝑇𝑥_𝑛²≤−𝑧_𝑛+1−𝑝²+𝑥_𝑛−𝑝²+2𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛𝑧_𝑛+1−𝑝. (44) Passing to limsup_𝑛, the last member goes to zero thanks to Step 1, to boundedness of (𝑥_𝑛)_𝑛 and (41). So we obtain 𝜔(1−𝜔)lim_𝑛𝑥_𝑛−𝑇𝑥_𝑛=0. (45) From this immediately we have also ‖𝑧_𝑛−𝑇𝑧_𝑛‖→0.
From Step 2 and Lemma 8(i) we obtain limsup_𝑛𝐼−𝑓𝑝,𝑥_𝑛−𝑝≥0. (46) Moreover, from Step 2 and Lemma 8 we also havelimsup_𝑛𝐼−𝑓𝑝,𝑧_𝑛−𝑝≥0. (47)Step 3. One has 𝑧_𝑛→𝑝.
Proof of Step 3. By using the auxiliary sequence (𝑧_𝑛)_𝑛, we can write 𝑥_𝑛 as 𝑥_𝑛+1=𝑧_𝑛+1+1−𝛽_𝑛𝐸_𝑛, (48) where 𝐸_𝑛=(1−𝛼_𝑛)(𝑆_𝜔𝑥_𝑛−𝑇_𝜔𝑥_𝑛) is a bounded sequence and so 𝑥_𝑛−𝑧_𝑛≤1−𝛽_𝑛𝑂(1),𝑧_𝑛+1−𝑝²=𝛼_𝑛𝑓𝑥_𝑛−𝑝+1−𝛼_𝑛𝑇_𝜔𝑥_𝑛−𝑝²=1−𝛼_𝑛²𝑇_𝜔𝑥_𝑛−𝑝²+𝛼²_𝑛𝑓(𝑥_𝑛)−𝑝²+2𝛼_𝑛1−𝛼_𝑛𝑓𝑥_𝑛−𝑝,𝑇_𝜔𝑥_𝑛−𝑝≤1−2𝛼_𝑛+𝛼²_𝑛𝑥_𝑛−𝑝²+𝛼²_𝑛𝑓𝑥_𝑛−𝑝²+2𝛼_𝑛1−𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑝,𝑇_𝜔𝑥_𝑛−𝑝+2𝛼_𝑛1−𝛼_𝑛𝑓𝑝−𝑝,𝑇_𝜔𝑥_𝑛−𝑝≤1−2𝛼_𝑛+𝛼²_𝑛𝑥_𝑛−𝑝²+𝛼²_𝑛𝑓𝑥_𝑛−𝑝²+2𝛼_𝑛1−𝛼_𝑛𝜌𝑥_𝑛−𝑝²+2𝛼_𝑛1−𝛼_𝑛𝑓𝑝−𝑝,𝑇_𝜔𝑥_𝑛−𝑥_𝑛+2𝛼_𝑛1−𝛼_𝑛𝑓𝑝−𝑝,𝑥_𝑛−𝑝≤1−2𝛼_𝑛+𝛼²_𝑛+2𝛼_𝑛1−𝛼_𝑛𝜌×𝑥_𝑛−𝑧_𝑛+𝑧_𝑛−𝑝²+2𝛼_𝑛1−𝛼_𝑛𝑓𝑝−𝑝,𝑇_𝜔𝑥_𝑛−𝑥_𝑛+2𝛼_𝑛1−𝛼_𝑛𝑓𝑝−𝑝,𝑥_𝑛−𝑝+𝛼²_𝑛𝑓(𝑥_𝑛)−𝑝²=1−2𝛼_𝑛+𝛼²_𝑛+2𝛼_𝑛1−𝛼_𝑛𝜌𝑧_𝑛−𝑝²+1−𝛽_𝑛𝑂(1)+2𝛼_𝑛1−𝛼_𝑛𝑓𝑝−𝑝,𝑇_𝜔𝑥_𝑛−𝑥_𝑛×2𝛼_𝑛1−𝛼_𝑛𝑓𝑝−𝑝,𝑥_𝑛−𝑝+𝛼²_𝑛𝑓𝑥_𝑛−𝑝². (49)(50) So putting 𝛿_𝑛=𝛼_𝑛[2−𝛼_𝑛−2𝜌(1−𝛼_𝑛)], 𝛾_𝑛=(1−𝛽_𝑛)𝑂(1), and 𝜎_𝑛=21−𝛼_𝑛𝑓𝑝−𝑝,𝑇_𝜔𝑥_𝑛−𝑥_𝑛+21−𝛼_𝑛𝑓𝑝−𝑝,𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑥_𝑛−𝑝² (51) one has easily that 𝛿_𝑛∈(0,1), ∑_𝑛𝛿_𝑛=∞, ∑_𝑛𝛾_𝑛<∞, and limsup_𝑛𝜎_𝑛=limsup_𝑛𝑥_𝑛−𝑝,𝑓𝑝−𝑝≤0. (52) Thus, we can rewrite (50) as 𝑧_𝑛+1−𝑝²≤1−𝛿_𝑛𝑧_𝑛−𝑝²+𝛼_𝑛𝜎_𝑛+𝛾_𝑛. (53) This is sufficient, for Xu’s Lemma 6, to conclude that 𝑧_𝑛→𝑝. Lastly, by (49) immediately follows 𝑥_𝑛→𝑝.
Proof of (ii). Rewrite 𝑥_𝑛+1 as 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑆_𝜔𝑥_𝑛+𝛽_𝑛𝐸_𝑛, (54) where 𝐸_𝑛=(1−𝛼_𝑛)(𝑇_𝜔𝑥_𝑛+𝑆_𝜔𝑥_𝑛) is bounded (i.e., ‖𝐸_𝑛‖≤𝑂(1)).
Now, 𝑥_𝑛+1−𝑝²=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛(1−𝜔)𝑥_𝑛+1−𝛼_𝑛𝜔𝑆𝑥_𝑛+𝛽𝐸_𝑛−𝑝²=1−𝛼_𝑛𝜔𝑆𝑥_𝑛−𝑥_𝑛+𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛+𝛽_𝑛𝐸_𝑛²bythewellknowninequality𝑥+𝑦²≤‖𝑥‖²≤+2𝑦,𝑥+𝑦1−𝛼_𝑛𝜔𝑆𝑥_𝑛−𝑥_𝑛+𝑥_𝑛−𝑝²+2𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛,𝑥_𝑛+1−𝑝+𝛽_𝑛𝑂(1)≤1−𝛼_𝑛²𝜔²𝑆𝑥_𝑛−𝑥_𝑛²+𝑥_𝑛−𝑝²−21−𝛼_𝑛𝜔𝑥_𝑛−𝑆𝑥_𝑛,𝑥_𝑛−𝑝+2𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛𝑥_𝑛+1−𝑝+𝛽_𝑛𝑂(1)(c)≤byofTheorem41−𝛼_𝑛²𝜔²𝑆𝑥_𝑛−𝑥_𝑛²+𝑥_𝑛−𝑝²−21−𝛼_𝑛𝜔𝑥_𝑛−𝑆𝑥_𝑛²−12𝑥_𝑛−𝑆𝑥_𝑛²+2𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛𝑥_𝑛+1−𝑝+𝛽_𝑛𝑂(1)=𝑥_𝑛−𝑝²+1−𝛼_𝑛²𝜔²−1−𝛼_𝑛𝜔𝑥_𝑛−𝑆𝑥_𝑛²+2𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛𝑥_𝑛+1−𝑝+𝛽_𝑛𝑂(1)=𝑥_𝑛−𝑝²−1−𝛼_𝑛𝜔1−1−𝛼_𝑛𝜔𝑥_𝑛−𝑆𝑥_𝑛²+2𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛𝑥_𝑛+1−𝑝+𝛽_𝑛𝑂(1) (55) and hence 1−𝛼_𝑛𝜔1−1−𝛼_𝑛𝜔𝑥_𝑛−𝑆𝑥_𝑛²−2𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛𝑥_𝑛+1−𝑝−𝛽_𝑛𝑂(1)≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝². (56) Now we distinguish two alternatives.
Alternative 1. (‖𝑥_𝑛−𝑝‖)_𝑛 is definitively nonincreasing.
Then there exists lim_𝑛‖𝑥_𝑛−𝑝‖² and so, passing to the limsup in (56), we obtain 𝑥_𝑛−𝑆𝑥_𝑛⟶0,𝑥_𝑛−𝑆_𝜔𝑥_𝑛=𝜔𝑥_𝑛−𝑆𝑥_𝑛⟶0. (57) By Lemma 8(ii) it follows that 𝐼−𝑓𝑝,𝑥−𝑝≥0. (58) So 𝑥_𝑛+1−𝑝²=𝛼_𝑛𝑓𝑥_𝑛−𝑝+(1−𝛼_𝑛)(𝑆_𝜔𝑥_𝑛−𝑝)+𝛽_𝑛𝐸_𝑛²≤𝛼_𝑛𝑓𝑥_𝑛−𝑝+1−𝛼_𝑛𝑆_𝜔𝑥_𝑛−𝑝²+𝛽_𝑛𝑂(1)=1−𝛼_𝑛²𝑆_𝜔𝑥_𝑛−𝑝²+𝛼²_𝑛𝑂(1)+2𝛼_𝑛1−𝛼_𝑛𝑓𝑥_𝑛−𝑝,𝑆_𝜔𝑥_𝑛−𝑝+𝛽_𝑛𝑂(1)bythequasi-nonexpansivityof𝑆_𝜔≤1−2𝛼_𝑛+𝛼²_𝑛𝑥_𝑛−𝑝²+𝛼²_𝑛𝑂(1)+𝛽_𝑛𝑂(1)+2𝛼_𝑛1−𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑝,𝑆_𝜔𝑥_𝑛−𝑝+2𝛼_𝑛1−𝛼_𝑛𝑓𝑝−𝑝,𝑆_𝜔𝑥_𝑛−𝑝≤1−2𝛼_𝑛+𝛼²_𝑛+2𝛼_𝑛1−𝛼_𝑛𝜌𝑥_𝑛−𝑝²+𝛼²_𝑛𝑂(1)+𝛽_𝑛𝑂(1)+2𝛼_𝑛1−𝛼_𝑛𝑓𝑝−𝑝,𝑆_𝜔𝑥_𝑛−𝑥_𝑛+2𝛼_𝑛1−𝛼_𝑛𝑓𝑝−𝑝,𝑥_𝑛−𝑝=1−2𝛼_𝑛+𝛼²_𝑛+2𝛼_𝑛1−𝛼_𝑛𝜌𝑥_𝑛−𝑝²+𝛼²_𝑛𝑂(1)+𝛼_𝑛𝑆_𝜔𝑥_𝑛−𝑥_𝑛𝑂(1)+(1−𝛼)𝑓𝑝−𝑝,𝑥_𝑛−𝑝+𝛽_𝑛𝑂(1). (59) So, as in Step 1, thanks to (57), (58), and Xu’s Lemma, we obtain 𝑥_𝑛→𝑝.
Alternative 2. (‖𝑥_𝑛−𝑝‖)_𝑛 is not definitively nonincreasing.
This means that there exists a subsequence (𝑥_𝑛𝑘)_𝑘 such that 𝑥_𝑛𝑘−𝑝²≤𝑥_𝑛𝑘+1−𝑝². (60) Then, thanks to Maingé’s Lemma, we know that there exists a sequence of integers (𝛿(𝑛))_𝑛 that satisfies the following.
(i) 𝛿(𝑛) is nondecreasing, (ii) 𝛿(𝑛)→∞, (iii) ‖𝑥_𝛿(𝑛)−𝑝‖²<‖𝑥_𝛿(𝑛)+1−𝑝‖², and (iv) ‖𝑥_𝑛−𝑝‖<‖𝑥_𝛿(𝑛)+1−𝑝‖, for all 𝑛≥𝑛₁.
Consequently, 0≤liminf_𝑛𝑥_𝛿(𝑛)+1−𝑝−𝑥_𝛿(𝑛)−𝑝≤limsup_𝑛𝑥_𝛿(𝑛)+1−𝑝−𝑥_𝛿(𝑛)−𝑝≤limsup_𝑛𝑥_𝑛+1−𝑝−𝑥_𝑛−𝑝≤limsup_𝑛𝛼_𝑛𝑓𝑥_𝑛−𝑆_𝜔𝑥_𝑛+𝑆_𝜔𝑥_𝑛−𝑝+𝛽_𝑛𝐸_𝑛−𝑥_𝑛−𝑝bythequasi-nonexpansivityof𝑆_𝜔≤limsup𝛼_𝑛𝑂(1)+𝑥_𝑛−𝑝+𝛽_𝑛𝑂(1)−𝑥_𝑛−𝑝=0. (61) So lim_𝑛𝑥_𝛿(𝑛)+1−𝑝−𝑥_𝑛−𝑝=0. (62) If we rewrite (56) as 1−𝛼_𝑛𝜔1−1−𝛼_𝑛𝜔𝑥_𝑛−𝑆𝑥_𝑛²≤𝛼_𝑛𝑂(1)+𝛽_𝑛𝑂(1)+𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²≤𝛼_𝑛𝑂(1)+𝛽_𝑛𝑂(1)+𝑥_𝑛−𝑝−𝑥_𝑛+1−𝑝𝑂(1), (63) then (62) implies that 𝑥_𝛿(𝑛)−𝑆𝑥_𝛿(𝑛)⟶0 (64) and this, in turn, by using Lemma 8(ii), means that limsup𝐼−𝑓𝑝,𝑦_𝑛−𝑝≥0. (65) At this point it is clear that we can continue as in Alternative 1 and we obtain ‖𝑥_𝛿(𝑛)−𝑝‖→0.
Then (62) furnishes 𝑥_𝛿(𝑛)+1−𝑝⟶0, (66) and finally by property (iv) of Maingé’s Lemma, that is, ‖𝑥_𝑛−𝑝‖<‖𝑥_𝛿(𝑛)+1−𝑝‖, for all 𝑛≥𝑛₁, we point out that 𝑥_𝑛→𝑝.
Proof of (iii). Let be 𝑝₀ as in (iii) of Lemma 8; that is, 𝐼−𝑓𝑝₀,𝑥−𝑝₀≥0,∀𝑥∈Fix(𝑇)∩Fix(𝑆). (67) Now, 𝑈_𝑛𝑥_𝑛−𝑝₀²=𝛽_𝑛𝑇𝜔𝑥_𝑛−𝑝₀+1−𝛽_𝑛𝑆_𝜔𝑥_𝑛−𝑝₀²=𝛽_𝑛𝑇𝜔𝑥_𝑛−𝑝₀²+1−𝛽_𝑛𝑆_𝜔𝑥_𝑛−𝑝₀²−𝛽_𝑛1−𝛽_𝑛𝑇_𝜔𝑥_𝑛−𝑆_𝜔𝑥_𝑛²bythequasi-nonexpansivityof𝑇_𝜔,byquasifirmlynonexpansivityof𝑆_𝜔andTheorem4(i)≤𝛽_𝑛𝑥_𝑛−𝑝₀²+1−𝛽_𝑛𝑥_𝑛−𝑝₀²−1−𝛽_𝑛(1−𝜔)𝑥_𝑛−𝑆_𝜔𝑥_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑇_𝜔𝑥_𝑛−𝑆_𝜔𝑥_𝑛²=𝑥_𝑛−𝑝₀²−1−𝛽_𝑛(1−𝜔)𝑥_𝑛−𝑆_𝜔𝑥_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑇_𝜔𝑥_𝑛−𝑆_𝜔𝑥_𝑛² (68)(69) so 𝑥_𝑛+1−𝑝₀²=𝛼_𝑛𝑓𝑥_𝑛−𝑝₀+1−𝛼_𝑛(𝑈_𝑛𝑥_𝑛−𝑝₀)²≤𝑈_𝑛𝑥_𝑛−𝑝₀²+𝛼_𝑛𝛼_𝑛𝑂(1)(1)()≤𝑥+𝑂≤by68_𝑛−𝑝₀²−1−𝛽_𝑛(1−𝜔)𝑥_𝑛−𝑆_𝜔𝑥_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑇_𝜔𝑥_𝑛−𝑆_𝜔𝑥_𝑛². (70) From this we derive the following inequalities: 1−𝛽_𝑛(1−𝜔)𝑥_𝑛−𝑆_𝜔𝑥_𝑛²≤𝑥_𝑛−𝑝₀²−𝑥_𝑛+1−𝑝₀+𝛼_𝑛𝑂(1),𝛽_𝑛1−𝛽_𝑛𝑇_𝜔𝑥_𝑛−𝑆_𝜔𝑥_𝑛²≤𝑥_𝑛−𝑝₀²−𝑥_𝑛+1−𝑝₀²+𝛼_𝑛𝑂(1). (71) Now, also here consider two alternatives.
Alternative 1. (‖𝑥_𝑛−𝑝₀‖)_𝑛 is definitively nonincreasing.
Then there exist lim‖𝑥_𝑛−𝑝₀‖ and (|‖𝑥_𝑛+1−𝑝₀‖−‖𝑥_𝑛−𝑝₀‖|)→0.
So, passing to the limsup in (71), the assumption liminf_𝑛𝛽_𝑛(1−𝛽_𝑛)>0 yields: 𝑥_𝑛−𝑆_𝜔𝑥_𝑛𝑆⟶0,_𝜔𝑥_𝑛−𝑇_𝜔𝑥_𝑛⟶0. (72) Moreover, 𝑥_𝑛−𝑆_𝜔𝑥_𝑛=𝜔(𝑥_𝑛−𝑆𝑥_𝑛) so, 𝑥_𝑛−𝑆𝑥_𝑛⟶0 (73) and 𝑇_𝜔𝑥_𝑛−𝑆_𝜔𝑥_𝑛=𝜔(𝑇𝑥_𝑛−𝑆𝑥_𝑛) so 𝑇𝑥_𝑛−𝑆𝑥_𝑛⟶0 (74) so that from 𝑥_𝑛−𝑇𝑥_𝑛=𝑥_𝑛−𝑆𝑥_𝑛+𝑆𝑥_𝑛−𝑇𝑥_𝑛 it follows at once that 𝑥_𝑛−𝑇𝑥_𝑛⟶0. (75) From Lemma 8(iii) we obtain limsup𝐼−𝑓𝑝₀,𝑥_𝑛−𝑝₀≥0. (76) Further, from 𝑈_𝑛𝑥_𝑛−𝑥_𝑛=𝛽_𝑛(𝑇_𝜔𝑥_𝑛−𝑥_𝑛)+(1−𝛽_𝑛)(𝑆_𝜔𝑥_𝑛−𝑥_𝑛), we get 𝑈_𝑛𝑥_𝑛−𝑥_𝑛⟶0. (77) Now we are able to show that 𝑥_𝑛→𝑝₀.
Indeed, 𝑥_𝑛+1−𝑝₀²=1−𝛼_𝑛𝑈_𝑛𝑥_𝑛−𝑝₀+𝛼_𝑛𝑓𝑥_𝑛−𝑝₀²=1−𝛼_𝑛²𝑈_𝑛𝑥_𝑛−𝑝₀²+𝛼²_𝑛𝑓𝑥_𝑛−𝑝₀²+21−𝛼_𝑛𝛼_𝑛𝑈_𝑛𝑥_𝑛−𝑝₀,𝑓𝑥_𝑛−𝑝₀≤1−2𝛼_𝑛+𝛼²_𝑛𝑥_𝑛−𝑝₀²+𝛼²_𝑛𝑓𝑥_𝑛−𝑝₀²+21−𝛼_𝑛𝛼_𝑛𝑈_𝑛𝑥_𝑛−𝑝₀,𝑓𝑥_𝑛−𝑓𝑝₀+21−𝛼_𝑛𝛼_𝑛𝑈_𝑛𝑥_𝑛−𝑝₀,𝑓𝑝₀−𝑝₀≤1−2𝛼_𝑛+𝛼²_𝑛+2𝛼_𝑛1−𝛼_𝑛𝜌𝑥_𝑛−𝑝₀²+𝛼²_𝑛𝑓𝑥_𝑛−𝑝₀²+21−𝛼_𝑛𝛼_𝑛𝑈_𝑛𝑥_𝑛−𝑥_𝑛,𝑓𝑝₀−𝑝₀+21−𝛼_𝑛𝛼_𝑛𝑥_𝑛−𝑝₀,𝑓𝑝₀−𝑝₀. (78) So, put 𝛿_𝑛=𝛼_𝑛[2−𝛼_𝑛−2𝜌(1−𝛼_𝑛)], 𝜎_𝑛=𝛼_𝑛‖𝑓(𝑥_𝑛)−𝑝₀‖²  +  2(1−𝛼_𝑛)⟨𝑈_𝑛𝑥_𝑛−𝑥_𝑛,𝑓(𝑝₀)−𝑝₀⟩  +  2(1−𝛼_𝑛)⟨𝑥_𝑛−𝑝₀,𝑓(𝑝₀)−𝑝₀⟩ one easily has 𝛿_𝑛∈(0,1),   ∑_𝑛𝛿_𝑛=∞, limsup_𝑛𝜎_𝑛  =  limsup_𝑛⟨𝑥_𝑛−𝑝₀  ,  𝑓(𝑝₀)−𝑝₀⟩  ≤  0, and 𝑥_𝑛+1−𝑝₀²≤1−𝛿_𝑛𝑥_𝑛−𝑝₀²+𝛼_𝑛𝜎_𝑛. (79) This is sufficient, for Xu’s Lemma 6, to ensure that 𝑥_𝑛→𝑝₀.
Alternative 2. (‖𝑥_𝑛−𝑝₀‖)_𝑛 is not definitively nonincreasing.
This means that there exists a subsequence (‖𝑥_𝑛𝑘−𝑝₀‖)_𝑘 such that 𝑥_𝑛𝑘−𝑝₀≤𝑥_𝑛𝑘+1−𝑝₀. (80) Then, thanks to Maingé’s Lemma, we know that there exists a sequence of integers (𝛿_𝑛) that satisfies the following.
(i) 𝛿(𝑛) is nondecreasing, (ii) 𝛿(𝑛)→∞, (iii) ‖𝑥_𝛿(𝑛)−𝑝₀‖²<‖𝑥_𝛿(𝑛)+1−𝑝₀‖², and (iv) ‖𝑥_𝑛−𝑝₀‖  <  ‖𝑥_𝛿(𝑛)+1−𝑝₀‖², for all 𝑛≥𝑛₁.
Consequently, 0≤liminf_𝑛𝑥_𝛿(𝑛)+1−𝑝₀−𝑥_𝛿(𝑛)−𝑝₀≤limsup_𝑛𝑥_𝛿(𝑛)+1−𝑝₀−𝑥_𝛿(𝑛)−𝑝₀≤limsup_𝑛𝑥_𝑛+1−𝑝−𝑥_𝑛−𝑝≤limsup_𝑛𝛼_𝑛𝑓𝑥_𝑛−𝑝₀+1−𝛼_𝑛𝑈_𝑛𝑥_𝑛−𝑝₀−𝑥_𝑛−𝑝₀≤limsup𝛼_𝑛𝑓𝑥_𝑛−𝑝₀−𝑥_𝑛−𝑝₀=0. (81) Hence lim_𝑛𝑥_𝛿(𝑛)+1−𝑝₀−𝑥_𝛿(𝑛)−𝑝₀=0. (82) So, passing to limsup on 𝛿(𝑛) in (71), one obtains, as in the Alternative 1, 𝑥_𝛿(𝑛)−𝑆_𝜔𝑥_𝛿(𝑛)𝑇⟶0,_𝜔𝑥_𝛿(𝑛)−𝑆_𝜔𝑥_𝛿(𝑛)𝑥⟶0,_𝛿(𝑛)−𝑆𝑥_𝛿(𝑛)⟶0,𝑇𝑥_𝛿(𝑛)−𝑆𝑥_𝛿(𝑛)𝑥⟶0,_𝛿(𝑛)−𝑇𝑥_𝛿(𝑛)𝑈⟶0,_𝛿(𝑛)𝑥_𝛿(𝑛)−𝑥_𝛿(𝑛)⟶0. (83) Again from Lemma 8(iii) it follows that limsup𝐼−𝑓𝑝₀,𝑥_𝛿(𝑛)−𝑝₀≥0. (84) Following the reasoning of Alternative 1, one obtains that 𝑥_𝛿(𝑛)−𝑝₀⟶0. (85) Then (82) furnishes ‖𝑥_𝛿(𝑛)+1−𝑝₀‖→0 and finally by the property (iv) of Maingé Lemma; that is, 𝑥_𝑛−𝑝₀<𝑥_𝛿(𝑛)+1−𝑝₀, (86) we obtain 𝑥_𝑛→𝑝₀ as required.

Remark 12. The main result of this paper contains as a particular case the positive answer to the question raised by Kurokawa and Takahashi page  1567 in [26].

Remark 13. Our reasoning, different from that of Tian and Jin [27] and Deng et al. [28], has allowed us to prove our results without having 𝜔<1/2.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University under Grant no. 29-130-35-HiCi. The authors, therefore, acknowledge technical and financial support of KAU.