Relaxed Viscosity Approximation Methods with Regularization for Constrained Minimization Problems

Abstract

We introduce a new relaxed viscosity approximation method with regularization and prove the strong convergence of the method to a common fixed point of finitely many nonexpansive mappings and a strict pseudocontraction that also solves a convex minimization problem and a suitable equilibrium problem.

1. Introduction

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖, 𝐶 a nonempty closed convex subset of 𝐻, and 𝑃_𝐶 the metric projection of 𝐻, onto 𝐶. Let 𝑇∶𝐶→𝐶 be self-mapping on 𝐶. We denote by Fix(𝑇) the set of fixed points of 𝑇 and by 𝐑 the set of all real numbers. A mapping 𝑇∶𝐶→𝐶 is called 𝜁-strictly pseudocontractive if there exists a constant 𝜁∈[0,1) such that 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+𝜁(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦²,∀𝑥,𝑦∈𝐶. (1) In particular, if 𝜁=0, then 𝑇 is called a nonexpansive mapping. A mapping 𝐴∶𝐶→𝐻 is called 𝛼-inverse strongly monotone, if there exists a constant 𝛼>0 such that 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥𝛼𝑥−𝑦²,∀𝑥,𝑦∈𝐶. (2)

Let 𝑓∶𝐶→𝐑 be a convex and a continuous Fréchet differentiable functional. Consider the minimization problem (MP) of minimizing 𝑓 over the constraint set 𝐶min_𝑥∈𝐶𝑓(𝑥), (3) where we assume the existence of minimizers. We denote by Γ the set of minimizers of (3). The gradient-projection algorithm (GPA) generates a sequence {𝑥_𝑛} determined by the gradient ∇𝑓 and the metric projection 𝑃_𝐶 as follows: 𝑥_𝑛+1∶=𝑃_𝐶𝑥_𝑛−𝜆∇𝑓𝑥_𝑛,∀𝑛≥0, (4) or more generally, 𝑥_𝑛+1∶=𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓𝑥_𝑛,∀𝑛≥0, (5) where, in both (4) and (5), the initial guess 𝑥₀ is taken from 𝐶 arbitrarily, the parameters 𝜆 or 𝜆_𝑛 are positive real numbers. The convergence of algorithms (4) and (5) depends on the behavior of the gradient ∇𝑓. As a matter of fact, it is known that if ∇𝑓 is strongly monotone and Lipschitz continuous, then, for 0<𝜆<2𝛼/𝐿², the operator 𝑆∶=𝑃_𝐶𝐼−𝜆∇𝑓 (6) is a contraction. Hence, the sequence {𝑥_𝑛} defined by the GPA (4) converges in norm to the unique solution of (3). More generally, if the sequence {𝜆_𝑛} is chosen to satisfy the property 0<liminf_𝑛→∞𝜆_𝑛≤limsup_𝑛→∞𝜆_𝑛<2𝛼𝐿², (7) then the sequence {𝑥_𝑛} defined by the GPA (5) converges in norm to the unique minimizer of (3). If the gradient ∇𝑓 is only assumed to be a Lipschitz continuous, then {𝑥_𝑛} can only be weakly convergent if 𝐻 is infinite dimensional. A counterexample is given by Xu in [1].

Since the Lipschitz continuity of the gradient ∇𝑓 implies that it is inverse strongly monotone (ism), it can be expressed as a proper convex combination of the identity mapping and a nonexpansive mapping. Consequently, the GPA can be rewritten as the composite of a projectionand an averaged mapping which is again an averaged mapping. This shows that averaged mappings play an important role in the GPA. Very recently, Xu [1] used averaged mappings to study the convergence analysis of the GPA which is an operator-oriented approach.

We observe that the regularization, in particular, the traditional Tikhonov regularization, is usually used to solve ill-posed optimization problems. Consider the following regularized minimization problem: min_𝑥∈𝐶𝑓_𝛼(𝑥)(𝑥)+𝛼∶=𝑓2‖𝑥‖², (8) where 𝛼>0 is the regularization parameter and again 𝑓 is convex with an 𝐿-Lipschitz continuous gradient ∇𝑓.

The advantage of a regularization method is that it is possible to get strong convergence to the minimum-norm solution of the optimization problem under investigation. The disadvantage is however its implicity, and hence explicit iterative methods seem more attractive. See, for example, [1].

Given a mapping 𝐴∶𝐶→𝐻, the classical variational inequality problem (VIP) is to find 𝑥^∗∈𝐶 such that 𝐴𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐶. (9) The solution set of VIP (9) is denoted by VI(𝐶,𝐴). It is well known that 𝑥^∗∈VI(𝐶,𝐴) if and only if 𝑥^∗=𝑃_𝐶(𝑥^∗−𝜆𝐴𝑥^∗) for some 𝜆>0. The variational inequality was first discussed by Lions [2] and now is well known. The variational inequality theory has been studied quite extensively and has emerged as an important tool in the study of a wide class of obstacle, unilateral, free, moving, and equilibrium problems arising in several branches of pure and applied sciences in a unified and general framework. See, for example, [3–10] and the references therein.

In this paper, we study the following equilibrium problem (EP) which is to find 𝑥^∗∈𝐶 such that 𝐹𝑥^∗,𝑦+ℎ𝑥^∗,𝑦≥0,∀𝑦∈𝐶. (10) The solution set of EP (10) is denoted by EP(𝐹,ℎ). We will introduce and consider a relaxed viscosity iterative scheme with regularization for finding a common element of the solution set Γ of the minimization problem (3), the solution set EP(𝐹,ℎ) of the equilibrium problem (10), and the common fixed point set Fix(𝑇)∩(⋂_𝑖Fix(𝑆_𝑖)) of finitely many nonexpansive mappings 𝑆_𝑖∶𝐶→𝐶,𝑖=1,…,𝑁, and a strictly pseudocontractive mapping 𝑇 in the setting of the infinite-dimensional Hilbert space. We will prove that this iterative scheme converges strongly to a common fixed point of the mappings 𝑇,𝑆_𝑖∶𝐶→𝐶,𝑖=1,…,𝑁, which is both a minimizer of MP (3) and an equilibrium point of EP (10).

2. Preliminaries

Let 𝐻 be a real Hilbert space whose inner product and norm are denoted by ⟨⋅,⋅⟩ and ‖⋅‖, respectively. Let 𝐾 be a nonempty closed convex subset of 𝐻. We write 𝑥_𝑛⇀𝑥 to indicate that the sequence {𝑥_𝑛} converges weakly to 𝑥 and 𝑥_𝑛→𝑥 to indicate that the sequence {𝑥_𝑛} converges strongly to 𝑥. Moreover, we use 𝜔_𝑤(𝑥_𝑛) to denote the weak 𝜔-limit set of the sequence {𝑥_𝑛} and 𝜔_𝑠(𝑥_𝑛) to denote the strong 𝜔-limit set of the sequence {𝑥_𝑛}; that is, 𝜔_𝑤𝑥_𝑛∶=𝑥∈𝐻∶𝑥_𝑛𝑖𝑥⇀𝑥forsomesubsequence_𝑛𝑖of𝑥_𝑛,𝜔_𝑠𝑥_𝑛∶=𝑥∈𝐻∶𝑥_𝑛𝑖𝑥⟶𝑥forsomesubsequence_𝑛𝑖of𝑥_𝑛. (11)

The metric (or nearest point) projection from 𝐻 onto 𝐾 is the mapping 𝑃_𝐾∶𝐻→𝐾 which assigns to each point 𝑥∈𝐻 the unique point 𝑃_𝐾𝑥∈𝐾 satisfying the property 𝑥−𝑃_𝐾𝑥=inf_𝑦∈𝐾𝑥−𝑦=∶𝑑(𝑥,𝐾). (12)

Some important properties of projections are gathered in the following.

Proposition 1. For given 𝑥∈𝐻 and 𝑧∈𝐾(i)𝑧=𝑃_𝐾𝑥⇔⟨𝑥−𝑧,𝑦−𝑧⟩≤0, for all  𝑦∈𝐾;(ii)𝑧=𝑃_𝐾𝑥⇔‖𝑥−𝑧‖²≤‖𝑥−𝑦‖²−‖𝑦−𝑧‖², for all  𝑦∈𝐾;(iii)⟨𝑃_𝐾𝑥−𝑃_𝐾𝑦,𝑥−𝑦⟩≥‖𝑃_𝐾𝑥−𝑃_𝐾𝑦‖², for all  𝑦∈𝐻, which hence implies that  𝑃_𝐾  is nonexpansive and monotone.

Definition 2. A mapping 𝑇∶𝐻→𝐻 is said to be(a) nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐻; (13)(b) firmly nonexpansive if 2𝑇−𝐼 is nonexpansive, or equivalently, 𝑥−𝑦,𝑇𝑥−𝑇𝑦≥𝑇𝑥−𝑇𝑦²,∀𝑥,𝑦∈𝐻; (14) alternatively, 𝑇 is firmly nonexpansive if and only if 𝑇 can be expressed as 1𝑇=2(𝐼+𝑆), (15) where 𝑆∶𝐻→𝐻 is nonexpansive; projections are firmly nonexpansive.

Definition 3. Let 𝑇 be a nonlinear operator with domain 𝐷(𝑇)⊆𝐻 and range 𝑅(𝑇)⊆𝐻.(a)𝑇 is said to be monotone if 𝑥−𝑦,𝑇𝑥−𝑇𝑦≥0,∀𝑥,𝑦∈𝐷(𝑇). (16)(b) Given a number 𝛽>0, 𝑇 is said to be 𝛽 strongly monotone if 𝑥−𝑦,𝑇𝑥−𝑇𝑦≥𝛽𝑥−𝑦²,∀𝑥,𝑦∈𝐷(𝑇). (17)(c) Given a number 𝜈>0, 𝑇 is said to be 𝜈-inverse strongly monotone (𝜈-ism) if 𝑥−𝑦,𝑇𝑥−𝑇𝑦≥𝜈𝑇𝑥−𝑇𝑦²,∀𝑥,𝑦∈𝐷(𝑇). (18)

It can be easily seen that if 𝑇 is nonexpansive, then 𝐼−𝑇 is monotone. It is also easy to see that a projection 𝑃_𝐾 is 1-ism. Inverse strongly monotone (also referred to as cocoercive) operators have been applied widely in solving practical problems in various fields.

Definition 4. A mapping 𝑇∶𝐻→𝐻 is said to be an averaged mapping if it can be written as the average of the identity 𝐼 and a nonexpansive mapping; that is, 𝑇≡(1−𝛼)𝐼+𝛼𝑆, (19) where 𝛼∈(0,1) and 𝑆∶𝐻→𝐻 is nonexpansive. More precisely, when the last equality holds, we say that 𝑇 is 𝛼-averaged. Thus, firmly nonexpansive mappings (in particular, projections) are (1/2)-averaged maps.

Proposition 5 (see [11]). Let 𝑇∶𝐻→𝐻 be a given mapping.(i)𝑇  is nonexpansive if and only if the complement  𝐼−𝑇  is (1⁄2)-ism. (ii)If  𝑇  is 𝜈-ism, then for  𝛾>0,   𝛾𝑇  is  (𝜈/𝛾)-ism.(iii)𝑇  is averaged if and only if the complement  𝐼−𝑇  is  𝜈-ism for some  𝜈>1/2. Indeed, for  𝛼∈(0,1),  𝑇  is 𝛼-averaged if and only if  𝐼−𝑇  is  (1/2𝛼)-ism.

Proposition 6 (see [11]). Let 𝑆,𝑇,𝑉∶𝐻→𝐻 be given operators.(i)If  𝑇=(1−𝛼)𝑆+𝛼𝑉  for some  𝛼∈(0,1)and if  𝑆  is averaged and  𝑉  is nonexpansive, then  𝑇  is averaged.(ii)𝑇  is firmly nonexpansive if and only if the complement  𝐼−𝑇  is firmly nonexpansive.(iii)If  𝑇=(1−𝛼)𝑆+𝛼𝑉  for some  𝛼∈(0,1)and if  𝑆  is firmly nonexpansive and  𝑉  is nonexpansive, then  𝑇  is averaged.(iv)The composite of finitely many averaged mappings is averaged. That is, if each of the mappings  {𝑇_𝑖}^𝑁_𝑖=1  is averaged, then so is the composite  𝑇₁⋯𝑇_𝑁. In particular, if  𝑇₁  is  𝛼₁  -averaged and  𝑇₂  is  𝛼₂  -averaged, where  𝛼₁,𝛼₂∈(0,1), then the composite  𝑇₁𝑇₂  is  𝛼-averaged, where  𝛼=𝛼₁+𝛼₂−𝛼₁𝛼₂.(v)If the mappings  {𝑇_𝑖}^𝑁_𝑖=1  are averaged and have a common fixed point, thenFix𝑇^𝑁_𝑖=1𝑖=Fix𝑇₁⋯𝑇_𝑁. (20)
The notation Fix(𝑇) denotes the set of all fixed points of the mapping 𝑇, that is, Fix(𝑇)={𝑥∈𝐻∶𝑇𝑥=𝑥}.

It is clear that, in a real Hilbert space 𝐻, 𝑇∶𝐶→𝐶 is 𝜁-strictly pseudocontractive if and only if there holds the following inequality: 𝑇𝑥−𝑇𝑦,𝑥−𝑦≤𝑥−𝑦²−1−𝜁2×(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦²,∀𝑥,𝑦∈𝐶. (21) This immediately implies that if 𝑇 is a 𝜁-strictly pseudocontractive mapping, then 𝐼−𝑇 is ((1−𝜁)/2)-inverse strongly monotone; for further detail, we refer to [12] and the references therein. It is well known that the class of strict pseudocontractions strictly includes the class of nonexpansive mappings.

Lemma 7 (see [12, Proposition 2.1]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and 𝑇∶𝐶→𝐶 be a mapping.(i) If 𝑇 is a 𝜁-strictly pseudocontractive mapping, then 𝑇 satisfies the Lipschitz condition where 𝑇𝑥−𝑇𝑦≤1+𝜁1−𝜁𝑥−𝑦,∀𝑥,𝑦∈𝐶. (22) (ii) If 𝑇 is a 𝜁-strictly pseudocontractive mapping, then the mapping 𝐼−𝑇 is semiclosed at 0; that is, if {𝑥_𝑛} is a sequence in 𝐶 such that 𝑥_𝑛→𝑥^∼ weakly and (𝐼−𝑇)𝑥_𝑛→0 strongly, then (𝐼−𝑇)𝑥^∼=0.(iii) If 𝑇 is a 𝜁-(quasi-)strict pseudocontraction, then the fixed point set Fix(𝑇) of 𝑇 is closed and convex so that the projection 𝑃_Fix(𝑇) is well defined.

The following lemma is an immediate consequence of an inner product.

Lemma 8. In a real Hilbert space 𝐻, there holds the following inequality: 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝑥+𝑦,∀𝑥,𝑦∈𝐻. (23)

The following elementary result on real sequences is quite well known.

Lemma 9 (see [13]). Let {𝑎_𝑛} be a sequence of nonnegative real numbers satisfying the property 𝑎_𝑛+1≤1−𝑠_𝑛𝑎_𝑛+𝑠_𝑛𝑡_𝑛+𝜖_𝑛,∀𝑛≥0, (24) where {𝑠_𝑛}⊂(0,1] and {𝑡_𝑛} are the real sequences such that(i)∑^∞_𝑛=0𝑠_𝑛=∞;(ii)either limsup_𝑛→∞𝑡_𝑛≤0 or ∑^∞_𝑛=0𝑠_𝑛|𝑡_𝑛|<∞;(iii)∑^∞_𝑛=0𝜖_𝑛<∞ where 𝜖_𝑛≥0, for all 𝑛≥0. Then, lim_𝑛→∞𝑎_𝑛=0.

Lemma 10 (see [14]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 be a 𝜁-strictly pseudocontractive mapping. Let 𝛾 and 𝛿 be two nonnegative real numbers such that (𝛾+𝛿)𝜁≤𝛾. Then, 𝛾𝑥−𝑦+𝛿𝑇𝑥−𝑇𝑦≤𝛾+𝛿𝑥−𝑦,∀𝑥,𝑦∈𝐶. (25)

The following lemma appears implicitly in the paper of Reinermann [15].

Lemma 11 (see [15]). Let 𝐻 be a real Hilbert space. Then, for all 𝑥,𝑦∈𝐻 and 𝜆∈[0,1], 𝜆𝑥+(1−𝜆)𝑦²=𝜆‖𝑥‖²+(1−𝜆)𝑦²−𝜆(1−𝜆)𝑥−𝑦². (26)

Lemma 12 (see [16]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐹∶𝐶×𝐶→𝐑 be a bifunction such that(f1)𝐹(𝑥,𝑥)=0 for all 𝑥∈𝐶;(f2)𝐹 is monotone and upper hemicontinuous in the first variable;(f3)𝐹 is lower semicontinuous and convex in the second variable.
Let ℎ∶𝐶×𝐶→𝐑 be a bifunction such that(h1)ℎ(𝑥,𝑥)=0 for all 𝑥∈𝐶;(h2)ℎ is monotone and weakly upper semicontinuous in the first variable;(h3)ℎ is convex in the second variable.
Moreover, let one suppose that(H) for fixed 𝑟>0 and 𝑥∈𝐶, there exists a bounded 𝐾⊂𝐶 and 𝑥∈𝐾 such that for all 𝑧∈𝐶⧵𝐾,−𝐹(𝑥,𝑧)+ℎ(𝑧,𝑥)+(1/𝑟)⟨𝑥−𝑧,𝑧−𝑥⟩<0.
For 𝑟>0 and 𝑥∈𝐻, let 𝑇_𝑟∶𝐻→2^𝐶 be a mapping defined by 𝑇_𝑟𝑥=+1𝑧∈𝐶∶𝐹𝑧,𝑦+ℎ𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶 (27) called the resolvent of 𝐹 and ℎ. Then,(1)𝑇_𝑟𝑥≠∅;(2)𝑇_𝑟𝑥 is a singleton;(3)𝑇_𝑟 is firmly nonexpansive;(4)𝐸𝑃(𝐹,ℎ)=Fix(𝑇_𝑟) and it is closed and convex.

Lemma 13 (see [16]). Let one suppose that (f1)–(f3), (h1)–(h3) and (H) hold. Let 𝑥,𝑦∈𝐻, 𝑟₁,𝑟₂>0. Then, 𝑇_𝑟2𝑦−𝑇_𝑟1𝑥≤+𝑟𝑦−𝑥²−𝑟₁𝑟₂𝑇_𝑟2𝑦−𝑦. (28)

Lemma 14 (see [17]). Suppose that the hypotheses of Lemma 12 are satisfied. Let {𝑟_𝑛} be a sequence in (0,∞) with liminf_𝑛→∞𝑟_𝑛>0. Suppose that {𝑥_𝑛} is a bounded sequence. Then, the following statements are equivalent and true.(a) if ‖𝑥_𝑛−𝑇_𝑟𝑛𝑥_𝑛‖→0 as 𝑛→∞, each weak cluster point of {𝑥_𝑛} satisfies the problem: 𝐹𝑥,𝑦+ℎ𝑥,𝑦≥0,∀𝑦∈𝐶, (29) that is, 𝜔_𝑤(𝑥_𝑛)⊆EP(𝐹,ℎ).(b) The demiclosedness principle holds in the sense that, if 𝑥_𝑛⇀𝑥^∗ and ‖𝑥_𝑛−𝑇_𝑟𝑛𝑥_𝑛‖→0 as 𝑛→∞, then (𝐼−𝑇_𝑟𝑘)𝑥^∗=0 for all 𝑘≥1.

3. Main Results

We now propose the following relaxed viscosity iterative scheme with regularization: 𝐹𝑢_𝑛,𝑦+ℎ𝑢_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑦≥0,∀𝑦∈𝐶,_𝑛,1=𝛽_𝑛,1𝑆₁𝑢_𝑛+1−𝛽_𝑛,1𝑢_𝑛,𝑦_𝑛,𝑖=𝛽_𝑛,𝑖𝑆_𝑖𝑢_𝑛+1−𝛽_𝑛,𝑖𝑦_{𝑛,𝑖−1}𝑦,𝑖=2,…,𝑁,_𝑛=𝛽_𝑛𝑄𝑦_𝑛,𝑁+1−𝛽_𝑛𝑃_𝐶𝑦_𝑛,𝑁−𝜆∇𝑓_𝛼𝑛𝑦_𝑛,𝑁,𝑥_𝑛+1=𝜎_𝑛𝑦_𝑛+𝛾_𝑛𝑃_𝐶𝑦_𝑛−𝜆∇𝑓_𝛼𝑛𝑦_𝑛+𝛿_𝑛𝑇𝑃_𝐶𝑦_𝑛−𝜆∇𝑓_𝛼𝑛𝑦_𝑛, (30) for all 𝑛≥0, where the mapping 𝑄∶𝐶→𝐶 is a 𝜌-contraction; the mapping 𝑇∶𝐶→𝐶 is a 𝜁-strict pseudocontraction;  𝑆_𝑖∶𝐶→𝐶 is a nonexpansive mapping for each 𝑖=1,…,𝑁;  ∇𝑓∶𝐶→𝐻 satisfies the Lipschitz condition (10) with 0<𝜆<(2/𝐿);   𝐹,ℎ∶𝐶×𝐶→𝐑 are two bifunctions satisfying the hypotheses of Lemma 12;  {𝛼_𝑛} is a sequence in (0,∞) with ∑^∞_𝑛=0𝛼_𝑛<∞;   {𝛽_𝑛},{𝜎_𝑛} are sequences in (0,1) with 0<liminf_𝑛→∞𝜎_𝑛≤limsup_𝑛→∞𝜎_𝑛<1;  {𝛾_𝑛},{𝛿_𝑛} are sequences in [0,1] with 𝜎_𝑛+𝛾_𝑛+𝛿_𝑛=1, for all 𝑛≥0;   {𝛽_𝑛,𝑖}^𝑁_𝑖=1 are sequences in (0,1) and (𝛾_𝑛+𝛿_𝑛)𝜁≤𝛾_𝑛, for all 𝑛≥0; {𝑟_𝑛} is a sequence in (0,∞) with liminf_𝑛→∞𝑟_𝑛>0 and liminf_𝑛→∞𝛿_𝑛>0.

Before stating and proving the main convergence results, we first establish the following lemmas.

Lemma 15. Let one suppose that Ω=Fix(𝑇)∩(⋂_𝑖Fix(𝑆_𝑖))∩EP(𝐹,ℎ)∩Γ≠∅. Then, the sequences {𝑥_𝑛}, {𝑦_𝑛}, {𝑦_𝑛,𝑖} for all 𝑖, and {𝑢_𝑛} are bounded.

Proof. First of all, we can show as in [18] that 𝑃_𝐶(𝐼−𝜆∇𝑓_𝛼) is nonexpansive for 𝜆∈(0,2/(𝛼+𝐿)), and 𝑃_𝐶(𝐼−𝜆∇𝑓_𝛼𝑛) is nonexpansive for all 𝑛≥0 and 𝜆∈(0,2/𝐿). We observe that if 𝑝∈Ω, then 𝑦_𝑛,1−𝑝≤𝑢_𝑛−𝑝≤𝑥_𝑛−𝑝. (31) For all, from 𝑖=2 to 𝑖=𝑁, by induction, one proves that 𝑦_𝑛,𝑖−𝑝≤𝛽_𝑛,𝑖𝑢_𝑛−𝑝+1−𝛽_𝑛,𝑖𝑦_{𝑛,𝑖−1}≤𝑢−𝑝_𝑛−𝑝≤𝑥_𝑛−𝑝. (32) Thus, we obtain that for every 𝑖=1,…,𝑁, 𝑦_𝑛,𝑖−𝑝≤𝑢_𝑛−𝑝≤𝑥_𝑛−𝑝. (33) For simplicity, put 𝑦^∼_𝑛,𝑁=𝑃_𝐶(𝑦_𝑛,𝑁−𝜆∇𝑓_𝛼𝑛(𝑦_𝑛,𝑁)) and 𝑦^∼_𝑛=𝑃_𝐶(𝑦_𝑛−𝜆∇𝑓_𝛼𝑛(𝑦_𝑛)) for every 𝑛≥0. Then, 𝑦_𝑛=𝛼_𝑛𝑄𝑦_𝑛,𝑁+(1−𝛼_𝑛)𝑦^∼_𝑛,𝑁 and 𝑥_𝑛+1=𝜎_𝑛𝑦_𝑛+𝛾𝑦_𝑛^∼_𝑛+𝛿_𝑛𝑇𝑦^∼_𝑛 for every 𝑛≥0. Taking into consideration that 𝑇𝑝=𝑝 and 𝑃_𝐶(𝐼−𝜆∇𝑓)𝑝=𝑝 for 𝜆∈(0,2/𝐿), we have 𝑦^∼_𝑛,𝑁=𝑃−𝑝_𝐶𝐼−𝜆∇𝑓_𝛼𝑛𝑦_𝑛,𝑁−𝑃_𝐶𝑝≤𝑃𝐼−𝜆∇𝑓_𝐶𝐼−𝜆∇𝑓_𝛼𝑛𝑦_𝑛,𝑁−𝑃_𝐶𝐼−𝜆∇𝑓_𝛼𝑛𝑝+𝑃_𝐶𝐼−𝜆∇𝑓_𝛼𝑛𝑝−𝑃_𝐶𝑝≤𝑦𝐼−𝜆∇𝑓_𝑛,𝑁−𝑝+𝑃_𝐶𝐼−𝜆∇𝑓_𝛼𝑛𝑝−𝑃_𝐶𝑝≤𝑦𝐼−𝜆∇𝑓_𝑛,𝑁−𝑝+𝜆𝛼_𝑛𝑝. (34) Similarly, we get ‖𝑦^∼_𝑛−𝑝‖≤‖𝑦_𝑛−𝑝‖+𝜆𝛼_𝑛‖𝑝‖. Thus, from (34) we have 𝑦_𝑛−𝑝=𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝+1−𝛽𝑦_𝑛^∼_𝑛,𝑁−𝑝≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝+1−𝛽𝑦_𝑛^∼_𝑛,𝑁−𝑝≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑄𝑝+𝛽_𝑛+𝑄𝑝−𝑝1−𝛽_𝑛𝑦_𝑛,𝑁−𝑝+𝜆𝛼_𝑛𝑝≤𝛽_𝑛𝜌𝑦_𝑛,𝑁−𝑝+𝛽_𝑛+𝑄𝑝−𝑝1−𝛽_𝑛𝑦_𝑛,𝑁−𝑝+𝜆𝛼_𝑛𝑝=1−𝛽_𝑛1−𝜌𝑦_𝑛,𝑁−𝑝+𝛽_𝑛𝑄𝑝−𝑝+1−𝛽_𝑛𝜆𝛼_𝑛𝑝=1−𝛽_𝑛1−𝜌𝑦_𝑛,𝑁−𝑝+𝛽_𝑛1−𝜌𝑄𝑝−𝑝+1−𝜌1−𝛽_𝑛𝜆𝛼_𝑛𝑝≤max𝑦_𝑛,𝑁−𝑝,𝑄𝑝−𝑝1−𝜌+𝜆𝛼_𝑛𝑝≤max𝑢_𝑛−𝑝,𝑄𝑝−𝑝1−𝜌+𝜆𝛼_𝑛𝑝≤max𝑥_𝑛−𝑝,𝑄𝑝−𝑝1−𝜌+𝜆𝛼_𝑛𝑝. (35) Since (𝛾_𝑛+𝛿_𝑛)𝜁≤𝛾_𝑛 for all 𝑛≥0, utilizing Lemma 10, we derive from (35) 𝑥_𝑛+1=𝜎𝑦−𝑝_𝑛−𝑝+𝛾𝑦_𝑛^∼_𝑛−𝑝+𝛿_𝑛𝑇𝑦^∼_𝑛−𝑝≤𝜎_𝑛𝑦_𝑛−𝑝+𝛾𝑦_𝑛^∼_𝑛−𝑝+𝛿_𝑛𝑇𝑦^∼_𝑛−𝑝≤𝜎_𝑛𝑦_𝑛−𝑝+𝛾_𝑛+𝛿𝑦_𝑛^∼_𝑛−𝑝≤𝜎_𝑛𝑦_𝑛−𝑝+𝛾_𝑛+𝛿_𝑛𝑦_𝑛−𝑝+𝜆𝛼_𝑛𝑝≤𝑦_𝑛−𝑝+𝜆𝛼_𝑛𝑝≤max𝑥_𝑛−𝑝,𝑄𝑝−𝑝1−𝜌+𝜆𝛼_𝑛𝑝+𝜆𝛼_𝑛𝑝=max𝑥_𝑛−𝑝,𝑄𝑝−𝑝1−𝜌+2𝜆𝛼_𝑛𝑝. (36) By induction, we get 𝑥_𝑛+1−𝑝≤max𝑥₀−𝑝,𝑄𝑝−𝑝𝑝⋅1−𝜌+2𝜆^𝑛_𝑖=0𝛼_𝑖,∀𝑛≥0. (37) This implies that {𝑥_𝑛} is bounded and so are {𝑦_𝑛}, {𝑢_𝑛}, and {𝑦_𝑛,𝑖} for each 𝑖=1,…,𝑁. It is clear that both {𝑦^∼_𝑛,𝑁} and {𝑦^∼_𝑛} are also bounded. Since ‖𝑇𝑦^∼_𝑛−𝑝‖≤((1+𝜁)/(1−𝜁))‖𝑦^∼_𝑛−𝑝‖, {𝑇𝑦^∼_𝑛} is also bounded.

Lemma 16. Let one suppose that Ω≠∅. Moreover, let one suppose that the following hold:(H1)lim_𝑛→∞𝛽_𝑛=0 and ∑^∞_𝑛=0𝛽_𝑛=∞;(H2)∑^∞_𝑛=1|𝛽_𝑛−𝛽_𝑛−1|<∞ or lim_𝑛→∞(|𝛽_𝑛−𝛽_𝑛−1|/𝛽_𝑛)=0;(H3)∑^∞_𝑛=1|𝛽_𝑛,𝑖−𝛽_{𝑛−1,𝑖}|<∞ or lim_𝑛→∞(|𝛽_𝑛,𝑖−𝛽_{𝑛−1,𝑖}|/𝛽_𝑛)=0 for each 𝑖=1,…,𝑁;(H4)∑^∞_𝑛=1|𝑟_𝑛−𝑟_𝑛−1|<∞ or lim_𝑛→∞(|𝑟_𝑛−𝑟_𝑛−1|/𝛽_𝑛)=0;(H5)∑^∞_𝑛=1|𝜎_𝑛−𝜎_𝑛−1|<∞ or lim_𝑛→∞(|𝜎_𝑛−𝜎_𝑛−1|/𝛽_𝑛)=0;(H6)∑^∞_𝑛=1|𝛾_𝑛/(1−𝜎_𝑛)−𝛾_𝑛−1/(1−𝜎_𝑛−1)|<∞ or lim_𝑛→∞(1/𝛽_𝑛)|𝛾_𝑛/(1−𝜎_𝑛)−𝛾_𝑛−1/(1−𝜎_𝑛−1)|=0. Then, lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0, that is, {𝑥_𝑛} is asymptotically regular.

Proof. Taking into account 0<liminf_𝑛→∞𝜎_𝑛≤limsup_𝑛→∞𝜎_𝑛<1, we may assume, without loss of generality, that {𝜎_𝑛}⊂[𝑐,𝑑] for some 𝑐,𝑑∈(0,1). First, we write 𝑥_𝑛=𝜎_𝑛−1𝑦_𝑛−1+(1−𝜎_𝑛−1)𝑣_𝑛−1, for all 𝑛≥1, where 𝑣_𝑛−1=(𝑥_𝑛−𝜎_𝑛−1𝑦_𝑛−1)/(1−𝜎_𝑛−1). It follows that for all 𝑛≥1𝑣_𝑛−𝑣_𝑛−1=𝑥_𝑛+1−𝜎_𝑛𝑦_𝑛1−𝜎_𝑛−𝑥_𝑛−𝜎_𝑛−1𝑦_𝑛−11−𝜎_𝑛−1=𝛾𝑦_𝑛^∼_𝑛+𝛿_𝑛𝑇𝑦^∼_𝑛1−𝜎_𝑛−𝛾^∼_𝑛−1𝑦_𝑛−1+𝛿_𝑛−1𝑇𝑦^∼_𝑛−11−𝜎_𝑛−1=𝛾𝑦_𝑛^∼_𝑛−𝑦^∼_𝑛−1+𝛿_𝑛𝑇𝑦^∼_𝑛−𝑇𝑦^∼_𝑛−11−𝜎_𝑛+𝛾_𝑛1−𝜎_𝑛−𝛾_𝑛−11−𝜎_𝑛−1^∼𝑦_𝑛−1+𝛿_𝑛1−𝜎_𝑛−𝛿_𝑛−11−𝜎_𝑛−1𝑇𝑦^∼_𝑛−1. (38) Since (𝛾_𝑛+𝛿_𝑛)𝜁≤𝛾_𝑛 for all 𝑛≥0, utilizing Lemma 10, we have 𝛾𝑦_𝑛^∼_𝑛−𝑦^∼_𝑛−1+𝛿_𝑛𝑇𝑦^∼_𝑛−𝑇𝑦^∼_𝑛−1≤𝛾_𝑛+𝛿𝑦_𝑛^∼_𝑛−𝑦^∼_𝑛−1. (39)
Next, we estimate ‖𝑦_𝑛−𝑦_𝑛−1‖. Observe that for every 𝑛≥1𝑦^∼_𝑛,𝑁−𝑦^∼_{𝑛−1,𝑁}≤𝑃_𝐶𝐼−𝜆∇𝑓_𝛼𝑛𝑦_𝑛,𝑁−𝑃_𝐶𝐼−𝜆∇𝑓_𝛼𝑛𝑦_{𝑛−1,𝑁}+𝑃_𝐶𝐼−𝜆∇𝑓_𝛼𝑛𝑦_{𝑛−1,𝑁}−𝑃_𝐶𝐼−𝜆∇𝑓_𝛼𝑛−1𝑦_{𝑛−1,𝑁}≤𝑦_𝑛,𝑁−𝑦_{𝑛−1,𝑁}+𝑃_𝐶𝐼−𝜆∇𝑓_𝛼𝑛𝑦_{𝑛−1,𝑁}−𝑃_𝐶𝐼−𝜆∇𝑓_𝛼𝑛−1𝑦_{𝑛−1,𝑁}≤𝑦_𝑛,𝑁−𝑦_{𝑛−1,𝑁}+𝐼−𝜆∇𝑓_𝛼𝑛𝑦_{𝑛−1,𝑁}−𝐼−𝜆∇𝑓_𝛼𝑛−1𝑦_{𝑛−1,𝑁}=𝑦_𝑛,𝑁−𝑦_{𝑛−1,𝑁}+𝜆∇𝑓_𝛼𝑛𝑦_{𝑛−1,𝑁}−𝜆∇𝑓_𝛼𝑛−1𝑦_{𝑛−1,𝑁}=𝑦_𝑛,𝑁−𝑦_{𝑛−1,𝑁}+𝜆𝛼_𝑛−𝛼_𝑛−1𝑦_{𝑛−1,𝑁}, (40) and similarly, 𝑦^∼_𝑛−𝑦^∼_𝑛−1≤𝑦_𝑛−𝑦_𝑛−1+𝜆𝛼_𝑛−𝛼_𝑛−1𝑦_𝑛−1. (41) Also, from (30), we have 𝑦_𝑛=𝛽_𝑛𝑄𝑦_𝑛,𝑁+1−𝛽𝑦_𝑛^∼_𝑛,𝑁,𝑦_𝑛−1=𝛽_𝑛−1𝑄𝑦_{𝑛−1,𝑁}+1−𝛽𝑦_𝑛−1^∼_{𝑛−1,𝑁},∀𝑛≥1. (42) Simple calculations show that 𝑦_𝑛−𝑦_𝑛−1=1−𝛽𝑦_𝑛^∼_𝑛,𝑁−𝑦^∼_{𝑛−1,𝑁}+𝛽_𝑛−𝛽_𝑛−1𝑄𝑦_{𝑛−1,𝑁}−𝑦^∼_{𝑛−1,𝑁}+𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑄𝑦_{𝑛−1,𝑁}. (43) Then, passing to the norm we get from (40) that 𝑦_𝑛−𝑦_𝑛−1≤1−𝛽𝑦_𝑛^∼_𝑛,𝑁−𝑦^∼_{𝑛−1,𝑁}+𝛽_𝑛−𝛽_𝑛−1𝑄𝑦_{𝑛−1,𝑁}−𝑦^∼_{𝑛−1,𝑁}+𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑄𝑦_{𝑛−1,𝑁}≤1−𝛽_𝑛𝑦_𝑛,𝑁−𝑦_{𝑛−1,𝑁}+𝜆𝛼_𝑛−𝛼_𝑛−1𝑦_{𝑛−1,𝑁}+𝛽_𝑛−𝛽_𝑛−1𝑄𝑦_{𝑛−1,𝑁}−𝑦^∼_{𝑛−1,𝑁}+𝛽_𝑛𝜌𝑦_𝑛,𝑁−𝑦_{𝑛−1,𝑁}≤1−1−𝜌𝛽_𝑛𝑦_𝑛,𝑁−𝑦_{𝑛−1,𝑁}+𝜆𝛼_𝑛−𝛼_𝑛−1𝑦_{𝑛−1,𝑁}+𝛽_𝑛−𝛽_𝑛−1𝑄𝑦_{𝑛−1,𝑁}−𝑦^∼_{𝑛−1,𝑁}≤1−1−𝜌𝛽_𝑛𝑦_𝑛,𝑁−𝑦_{𝑛−1,𝑁}+𝑀₁𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1, (44) where 𝜆‖𝑦_𝑛,𝑁‖+‖𝑄𝑦_𝑛,𝑁−𝑦^∼_𝑛,𝑁‖≤𝑀₁, for all 𝑛≥0 for some 𝑀₁≥0. Furthermore, by the definition of 𝑦_𝑛,𝑖 one obtains that, for all 𝑖=𝑁,…,2𝑦_𝑛,𝑖−𝑦_{𝑛−1,𝑖}≤𝛽_𝑛,𝑖𝑢_𝑛−𝑢_𝑛−1+𝑆_𝑖𝑢_𝑛−1−𝑦_{𝑛−1,𝑖−1}𝛽_𝑛,𝑖−𝛽_{𝑛−1,𝑖}+1−𝛽_𝑛,𝑖𝑦_{𝑛,𝑖−1}−𝑦_{𝑛−1,𝑖−1}. (45) In the case of 𝑖=1, we have 𝑦_𝑛,1−𝑦_𝑛−1,1≤𝛽_𝑛,1𝑢_𝑛−𝑢_𝑛−1+𝑆₁𝑢_𝑛−1−𝑢_𝑛−1𝛽_𝑛,1−𝛽_𝑛−1,1+1−𝛽_𝑛,1𝑢_𝑛−𝑢_𝑛−1=𝑢_𝑛−𝑢_𝑛−1+𝑆₁𝑢_𝑛−1−𝑢_𝑛−1𝛽_𝑛,1−𝛽_𝑛−1,1. (46) Substituting (46) in all (45) type one obtains for 𝑖=2,…,𝑁𝑦_𝑛,𝑖−𝑦_{𝑛−1,𝑖}≤𝑢_𝑛−𝑢_𝑛−1+^𝑖_𝑘=2𝑆_𝑘𝑢_𝑛−1−𝑦_{𝑛−1,𝑘−1}𝛽_𝑛,𝑘−𝛽_{𝑛−1,𝑘}+𝑆₁𝑢_𝑛−1−𝑢_𝑛−1𝛽_𝑛,1−𝛽_𝑛−1,1. (47) This together with (44) implies that 𝑦_𝑛−𝑦_𝑛−1≤1−1−𝜌𝛽_𝑛×𝑢_𝑛−𝑢_𝑛−1+^𝑁_𝑘=2𝑆_𝑘𝑢_𝑛−1−𝑦_{𝑛−1,𝑘−1}𝛽_𝑛,𝑘−𝛽_{𝑛−1,𝑘}+𝑆₁𝑢_𝑛−1−𝑢_𝑛−1𝛽_𝑛,1−𝛽_𝑛−1,1+𝑀₁𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1≤1−1−𝜌𝛽_𝑛𝑢_𝑛−𝑢_𝑛−1+^𝑁_𝑘=2𝑆_𝑘𝑢_𝑛−1−𝑦_{𝑛−1,𝑘−1}𝛽_𝑛,𝑘−𝛽_{𝑛−1,𝑘}+𝑆₁𝑢_𝑛−1−𝑢_𝑛−1𝛽_𝑛,1−𝛽_𝑛−1,1+𝑀₁𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1. (48) By Lemma 13, we know that 𝑢_𝑛−𝑢_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1𝑟+𝜅1−^𝑛−1𝑟_𝑛, (49) where 𝜅=sup_𝑛≥0‖𝑢_𝑛−𝑥_𝑛‖. So, substituting (49) in (48) we obtain 𝑦_𝑛−𝑦_𝑛−1≤1−1−𝜌𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1𝑟+𝜅1−^𝑛−1𝑟_𝑛+^𝑁_𝑘=2𝑆_𝑘𝑢_𝑛−1−𝑦_{𝑛−1,𝑘−1}𝛽_𝑛,𝑘−𝛽_{𝑛−1,𝑘}+𝑆₁𝑢_𝑛−1−𝑢_𝑛−1𝛽_𝑛,1−𝛽_𝑛−1,1+𝑀₁𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1≤1−1−𝜌𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1𝑟+𝜅^𝑛−𝑟_𝑛−1𝑟_𝑛+^𝑁_𝑘=2𝑆_𝑘𝑢_𝑛−1−𝑦_{𝑛−1,𝑘−1}𝛽_𝑛,𝑘−𝛽_{𝑛−1,𝑘}+𝑆₁𝑢_𝑛−1−𝑢_𝑛−1𝛽_𝑛,1−𝛽_𝑛−1,1+𝑀₁𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1≤1−1−𝜌𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑀₂𝑟_𝑛−𝑟_𝑛−1𝑟_𝑛+^𝑁_𝑘=2𝛽_𝑛,𝑘−𝛽_{𝑛−1,𝑘}+𝛽_𝑛,1−𝛽_𝑛−1,1+𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1≤1−1−𝜌𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑀₂𝑟_𝑛−𝑟_𝑛−1𝑏+^𝑁_𝑘=1𝛽_𝑛,𝑘−𝛽_{𝑛−1,𝑘}+𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1, (50) where 𝑏>0 is a minorant for {𝑟_𝑛} and 𝜅+𝑀₁+∑^𝑁_𝑘=2‖𝑆_𝑘𝑢_𝑛−𝑦_{𝑛,𝑘−1}‖+‖𝑆₁𝑢_𝑛−𝑢_𝑛‖≤𝑀₂, for all 𝑛≥0 for some 𝑀₂≥0. This together with (38)-(39), implies that 𝑣_𝑛−𝑣_𝑛−1≤𝛾𝑦_𝑛^∼_𝑛−𝑦^∼_𝑛−1+𝛿_𝑛𝑇𝑦^∼_𝑛−𝑇𝑦^∼_𝑛−11−𝜎_𝑛+𝛾_𝑛1−𝜎_𝑛−𝛾_𝑛−11−𝜎_𝑛−1^∼𝑦_𝑛−1+𝛿_𝑛1−𝜎_𝑛−𝛿_𝑛−11−𝜎_𝑛−1𝑇𝑦^∼_𝑛−1≤𝛾_𝑛+𝛿𝑦_𝑛^∼_𝑛−𝑦^∼_𝑛−11−𝜎_𝑛+𝛾_𝑛1−𝜎_𝑛−𝛾_𝑛−11−𝜎_𝑛−1^∼𝑦_𝑛−1+𝛾_𝑛1−𝜎_𝑛−𝛾_𝑛−11−𝜎_𝑛−1𝑇𝑦^∼_𝑛−1=𝑦^∼_𝑛−𝑦^∼_𝑛−1+𝛾_𝑛1−𝜎_𝑛−𝛾_𝑛−11−𝜎_𝑛−1^∼𝑦_𝑛−1+𝑇𝑦^∼_𝑛−1≤𝑦_𝑛−𝑦_𝑛−1+𝜆𝛼_𝑛−𝛼_𝑛−1𝑦_𝑛−1+𝛾_𝑛1−𝜎_𝑛−𝛾_𝑛−11−𝜎_𝑛−1^∼𝑦_𝑛−1+𝑇𝑦^∼_𝑛−1≤1−1−𝜌𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑀₂𝑟_𝑛−𝑟_𝑛−1𝑏+^𝑁_𝑘=1𝛽_𝑛,𝑘−𝛽_{𝑛−1,𝑘}+𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1+𝜆𝛼_𝑛−𝛼_𝑛−1𝑦_𝑛−1+𝛾_𝑛1−𝜎_𝑛−𝛾_𝑛−11−𝜎_𝑛−1^∼𝑦_𝑛−1+𝑇𝑦^∼_𝑛−1≤1−1−𝜌𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑀₃𝑟_𝑛−𝑟_𝑛−1𝑏+^𝑁_𝑘=1𝛽_𝑛,𝑘−𝛽_{𝑛−1,𝑘}+𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1+𝑀₃𝛼_𝑛−𝛼_𝑛−1+𝑀₃𝛾_𝑛1−𝜎_𝑛−𝛾_𝑛−11−𝜎_𝑛−1=1−1−𝜌𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑀₃𝑟_𝑛−𝑟_𝑛−1𝑏+^𝑁_𝑘=1𝛽_𝑛,𝑘−𝛽_{𝑛−1,𝑘}+2𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1+𝛾_𝑛1−𝜎_𝑛−𝛾_𝑛−11−𝜎_𝑛−1, (51) where 𝑀₂+𝜆‖𝑦_𝑛‖+‖𝑦^∼_𝑛‖+‖𝑇𝑦^∼_𝑛‖≤𝑀₃, for all 𝑛≥0 for some 𝑀₃≥0.
Further, we observe that 𝑥_𝑛+1=𝜎_𝑛𝑦_𝑛+1−𝜎_𝑛𝑣_𝑛,𝑥_𝑛=𝜎_𝑛−1𝑦_𝑛−1+1−𝛽_𝑛−1𝑣_𝑛−1,∀𝑛≥1. (52) Simple calculations show that 𝑥_𝑛+1−𝑥_𝑛=1−𝜎_𝑛𝑣_𝑛−𝑣_𝑛−1+𝜎_𝑛−𝜎_𝑛−1𝑦_𝑛−1−𝑣_𝑛−1+𝜎_𝑛𝑦_𝑛−𝑦_𝑛−1. (53) Then, passing to the norm, we get from (51) 𝑥_𝑛+1−𝑥_𝑛≤1−𝜎_𝑛𝑣_𝑛−𝑣_𝑛−1+𝜎_𝑛−𝜎_𝑛−1𝑦_𝑛−1−𝑣_𝑛−1+𝜎_𝑛𝑦_𝑛−𝑦_𝑛−1≤1−𝜎_𝑛×1−1−𝜌𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑀₃𝑟_𝑛−𝑟_𝑛−1𝑏+^𝑁_𝑘=1𝛽_𝑛,𝑘−𝛽_{𝑛−1,𝑘}+2𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1++𝛾_𝑛1−𝜎_𝑛−𝛾_𝑛−11−𝜎_𝑛−1+𝜎_𝑛−𝜎_𝑛−1𝑦_𝑛−1−𝑣_𝑛−1+𝜎_𝑛1−1−𝜌𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑀₂𝑟_𝑛−𝑟_𝑛−1𝑏+^𝑁_𝑘=1𝛽_𝑛,𝑘−𝛽_{𝑛−1,𝑘}+𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1≤1−1−𝜌𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑀₃𝑟_𝑛−𝑟_𝑛−1𝑏+^𝑁_𝑘=1𝛽_𝑛,𝑘−𝛽_{𝑛−1,𝑘}+2𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1+𝛾_𝑛1−𝜎_𝑛−𝛾_𝑛−11−𝜎_𝑛−1+𝜎_𝑛−𝜎_𝑛−1𝑦_𝑛−1−𝑣_𝑛−1≤1−1−𝜌𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1𝑟+𝑀^𝑛−𝑟_𝑛−1𝑏+^𝑁_𝑘=1𝛽_𝑛,𝑘−𝛽_{𝑛−1,𝑘}+𝛽_𝑛−𝛽_𝑛−1+𝜎_𝑛−𝜎_𝑛−1+𝛾_𝑛1−𝜎_𝑛−𝛾_𝑛−11−𝜎_𝑛−1+2𝑀𝛼_𝑛−𝛼_𝑛−1, (54) where 𝑀₃+‖𝑦_𝑛−𝑣_𝑛‖≤𝑀, for all 𝑛≥0 for some 𝑀≥0. By hypotheses (H1)–(H6) and Lemma 9, from ∑^∞_𝑛=0𝛼_𝑛<∞, we obtain the claim.

Lemma 17. Let one suppose that Ω≠∅. Let one suppose that {𝑥_𝑛} is asymptotically regular. Then, ‖𝑥_𝑛−𝑦_𝑛‖→0 and ‖𝑥_𝑛−𝑢_𝑛‖=‖𝑥_𝑛−𝑇_𝑟𝑛𝑥_𝑛‖→0 as 𝑛→∞.

Proof. We recall that, by the firm nonexpansivity of 𝑇_𝑟𝑛, a standard calculation (see [17]) shows that if 𝑣∈EP(𝐹,ℎ), then 𝑢_𝑛−𝑣²≤𝑥_𝑛−𝑣²−𝑥_𝑛−𝑢_𝑛². (55) Let 𝑝∈Ω. Then by Lemma 11, we have from (33)–(34) the following 𝑦^∼_𝑛−𝑝²=𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝+1−𝛽𝑦_𝑛^∼_𝑛,𝑁−𝑝²≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+1−𝛽𝑦_𝑛^∼_𝑛,𝑁−𝑝²≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝑦^∼_𝑛,𝑁−𝑝²≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝑦_𝑛,𝑁−𝑝+𝜆𝛼_𝑛𝑝²=𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝑦_𝑛,𝑁−𝑝²+𝜆𝛼_𝑛𝑝2𝑦_𝑛,𝑁−𝑝+𝜆𝛼_𝑛𝑝≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝑥_𝑛−𝑝²+𝜆𝛼_𝑛𝑝2𝑥_𝑛−𝑝+𝜆𝛼_𝑛𝑝,_∼𝑦_𝑛−𝑝²=𝑃_𝐶𝐼−𝜆∇𝑓_𝛼𝑛𝑦_𝑛−𝑃_𝐶𝐼−𝜆∇𝑓𝑝²≤𝑦_𝑛−𝑝+𝜆𝛼_𝑛𝑝²=𝑦_𝑛−𝑝²+𝜆𝛼_𝑛𝑝2𝑦_𝑛−𝑝+𝜆𝛼_𝑛𝑝. (56)(57) Since (𝛾_𝑛+𝛿_𝑛)𝜁≤𝛾_𝑛 for all 𝑛≥0, utilizing Lemma 10, we have 𝑥_𝑛+1−𝑝²=𝜎_𝑛𝑦_𝑛−𝑝+𝛾𝑦_𝑛^∼_𝑛−𝑝+𝛿_𝑛𝑇𝑦^∼_𝑛−𝑝²=𝜎_𝑛𝑦_𝑛−𝑝+𝛾_𝑛+𝛿_𝑛1𝛾_𝑛+𝛿_𝑛×𝛾𝑦_𝑛^∼_𝑛−𝑝+𝛿_𝑛𝑇𝑦^∼_𝑛−𝑝²=𝜎_𝑛𝑦_𝑛−𝑝²+𝛾_𝑛+𝛿_𝑛×1𝛾_𝑛+𝛿_𝑛𝛾𝑦_𝑛^∼_𝑛−𝑝+𝛿_𝑛𝑇𝑦^∼_𝑛−𝑝²−𝜎_𝑛𝛾_𝑛+𝛿_𝑛𝑦_𝑛−1−𝑝𝛾_𝑛+𝛿_𝑛×𝛾𝑦_𝑛^∼_𝑛−𝑝+𝛿_𝑛𝑇𝑦^∼_𝑛−𝑝²=𝜎_𝑛𝑦_𝑛−𝑝²+𝛾_𝑛+𝛿_𝑛×1𝛾_𝑛+𝛿_𝑛𝛾𝑦_𝑛^∼_𝑛−𝑝+𝛿_𝑛𝑇𝑦^∼_𝑛−𝑝²−𝜎_𝑛𝛾_𝑛+𝛿_𝑛1𝛾_𝑛+𝛿_𝑛𝛾𝑦_𝑛^∼_𝑛−𝑦_𝑛+𝛿_𝑛𝑇𝑦^∼_𝑛−𝑦_𝑛²=𝜎_𝑛𝑦_𝑛−𝑝²+𝛾_𝑛+𝛿_𝑛×1𝛾_𝑛+𝛿_𝑛𝛾𝑦_𝑛^∼_𝑛−𝑝+𝛿_𝑛𝑇𝑦^∼_𝑛−𝑝²−𝜎_𝑛𝛾_𝑛+𝛿_𝑛𝑥_𝑛+1−𝑦_𝑛²≤𝜎_𝑛𝑦_𝑛−𝑝²+𝛾_𝑛+𝛿𝑦_𝑛^∼_𝑛−𝑝²−𝜎_𝑛𝛾_𝑛+𝛿_𝑛𝑥_𝑛+1−𝑦_𝑛²=𝜎_𝑛𝑦_𝑛−𝑝²+1−𝜎𝑦_𝑛^∼_𝑛−𝑝²−𝜎_𝑛1−𝜎_𝑛𝑥_𝑛+1−𝑦_𝑛²≤𝜎_𝑛𝑦_𝑛−𝑝²+1−𝜎_𝑛×𝑦_𝑛−𝑝²+𝜆𝛼_𝑛𝑝2𝑦_𝑛−𝑝+𝜆𝛼_𝑛𝑝−𝜎_𝑛1−𝜎_𝑛𝑥_𝑛+1−𝑦_𝑛²≤𝑦_𝑛−𝑝²+𝜆𝛼_𝑛𝑝2𝑦_𝑛−𝑝+𝜆𝛼_𝑛𝑝−𝜎_𝑛1−𝜎_𝑛𝑥_𝑛+1−𝑦_𝑛²≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝑥_𝑛−𝑝²+𝜆𝛼_𝑛𝑝2𝑥_𝑛−𝑝+𝜆𝛼_𝑛𝑝+𝜆𝛼_𝑛𝑝2𝑦_𝑛−𝑝+𝜆𝛼_𝑛𝑝−𝜎_𝑛1−𝜎_𝑛𝑥_𝑛+1−𝑦_𝑛²=𝑥_𝑛−𝑝²+𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+2𝜆𝛼_𝑛𝑝𝑥_𝑛−𝑝+𝑦_𝑛−𝑝+𝜆𝛼_𝑛𝑝−𝜎_𝑛1−𝜎_𝑛𝑥_𝑛+1−𝑦_𝑛². (58) Taking into account 0<liminf_𝑛→∞𝜎_𝑛≤limsup_𝑛→∞𝜎_𝑛<1, we may assume that {𝜎_𝑛}⊂[𝑐,𝑑] for some 𝑐,𝑑∈(0,1). So, we deduce that 𝑐𝑥1−𝑐^𝑛+1−𝑦_𝑛²≤𝜎_𝑛1−𝜎_𝑛𝑥_𝑛+1−𝑦_𝑛²≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+2𝜆𝛼_𝑛𝑝𝑥_𝑛−𝑝+𝑦_𝑛−𝑝+𝜆𝛼_𝑛𝑝≤𝑥_𝑛−𝑝+𝑥_𝑛+1−𝑝𝑥_𝑛−𝑥_𝑛+1+𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+2𝜆𝛼_𝑛𝑝𝑥_𝑛−𝑝+𝑦_𝑛−𝑝+𝜆𝛼_𝑛𝑝. (59) Since 𝛼_𝑛→0, 𝛽_𝑛→0 and ‖𝑥_𝑛−𝑥_𝑛+1‖→0 as 𝑛→∞, we conclude from the boundedness of {𝑥_𝑛},{𝑦_𝑛}, and {𝑦_𝑛,𝑁} that ‖𝑥_𝑛+1−𝑦_𝑛‖→0 as 𝑛→∞. This together with ‖𝑥_𝑛−𝑥_𝑛+1‖→0, implies that lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=0. (60) Furthermore, from (33), (55), and (56), we have 𝑦_𝑛−𝑝²≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝑦_𝑛,𝑁−𝑝²+𝜆𝛼_𝑛𝑝2𝑦_𝑛,𝑁−𝑝+𝜆𝛼_𝑛𝑝≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝑢_𝑛−𝑝²+𝜆𝛼_𝑛𝑝2𝑦_𝑛,𝑁−𝑝+𝜆𝛼_𝑛𝑝≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝑥_𝑛−𝑝²−𝑥_𝑛−𝑢_𝑛²+𝜆𝛼_𝑛𝑝2𝑥_𝑛−𝑝+𝜆𝛼_𝑛𝑝, (61) which hence implies that 𝑥_𝑛−𝑢_𝑛²≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝑥_𝑛−𝑝²−𝑦_𝑛−𝑝²+𝜆𝛼_𝑛𝑝2𝑥_𝑛−𝑝+𝜆𝛼_𝑛𝑝≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝑥_𝑛−𝑝+𝑦_𝑛−𝑝𝑥_𝑛−𝑦_𝑛+𝜆𝛼_𝑛𝑝2𝑥_𝑛−𝑝+𝜆𝛼_𝑛𝑝. (62) Since 𝛼_𝑛→0,𝛽_𝑛→0 and ‖𝑥_𝑛−𝑦_𝑛‖→0 as 𝑛→∞, we deduce from the boundedness of {𝑥_𝑛}, {𝑦_𝑛}, and {𝑦_𝑛,𝑁} that lim_𝑛→∞𝑥_𝑛−𝑢_𝑛=0. (63)

Remark 18. By the last lemma we have 𝜔_𝑤(𝑥_𝑛)=𝜔_𝑤(𝑢_𝑛) and 𝜔_𝑠(𝑥_𝑛)=𝜔_𝑠(𝑢_𝑛); that is, the sets of strong/weak cluster points of {𝑥_𝑛} and {𝑢_𝑛} coincide.

Of course, if 𝛽_𝑛,𝑖→𝛽_𝑖≠0, as 𝑛→∞, for all index 𝑖, the assumptions of Lemma 16 are enough to assure that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛𝛽_𝑛,𝑖=0,∀𝑖∈{1,…,𝑁}. (64) In the next lemma, we examine the case in which at least one sequence {𝛽_𝑛,𝑘0} is a null sequence.

Lemma 19. Let one suppose that Ω≠∅. Let one suppose that (H1) holds. Moreover, for an index 𝑘₀∈{1,…,𝑁}, lim_𝑛→∞𝛽_𝑛,𝑘0=0, and the following hold:(H7) for all 𝑖, lim_𝑛→∞𝛽_𝑛,𝑖−𝛽_{𝑛−1,𝑖}𝛽_𝑛𝛽_𝑛,𝑘0=lim_𝑛→∞𝛼_𝑛−𝛼_𝑛−1𝛽_𝑛𝛽_𝑛,𝑘0=lim_𝑛→∞𝛽_𝑛−𝛽_𝑛−1𝛽_𝑛𝛽_𝑛,𝑘0=lim_𝑛→∞𝜎_𝑛−𝜎_𝑛−1𝛽_𝑛𝛽_𝑛,𝑘0=lim_𝑛→∞𝑟_𝑛−𝑟_𝑛−1𝛽_𝑛𝛽_𝑛,𝑘0=lim_𝑛→∞1𝛽_𝑛𝛽_𝑛,𝑘0𝛾_𝑛1−𝜎_𝑛−𝛾_𝑛−11−𝜎_𝑛−1=0; (65)(H8) there exists a constant 𝜏>0 such that (1/𝛽_𝑛)|1/𝛽_𝑛,𝑘0−1/𝛽_{𝑛−1,𝑘0}|<𝜏 for all 𝑛≥1. Then, lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛𝛽_𝑛,𝑘0=0. (66)

Proof. We start by (54). Dividing both the terms by 𝛽_𝑛,𝑘0 we have 𝑥_𝑛+1−𝑥_𝑛𝛽_𝑛,𝑘0≤1−1−𝜌𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1𝛽_𝑛,𝑘0𝑟+𝑀^𝑛−𝑟_𝑛−1𝑏𝛽_𝑛,𝑘0+∑^𝑁_𝑘=1𝛽_𝑛,𝑘−𝛽_{𝑛−1,𝑘}𝛽_𝑛,𝑘0+2𝛼_𝑛−𝛼_𝑛−1𝛽_𝑛,𝑘0+𝛽_𝑛−𝛽_𝑛−1𝛽_𝑛,𝑘0+𝜎_𝑛−𝜎_𝑛−1𝛽_𝑛,𝑘0+𝛾_𝑛/1−𝜎_𝑛−𝛾_𝑛−1/1−𝜎_𝑛−1𝛽_𝑛,𝑘0. (67) So, by (H8) we have 𝑥_𝑛+1−𝑥_𝑛𝛽_𝑛,𝑘0≤1−1−𝜌𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1𝛽_{𝑛−1,𝑘0}+1−1−𝜌𝛽_𝑛𝑥_𝑛−𝑥_𝑛−11𝛽_𝑛,𝑘0−1𝛽_{𝑛−1,𝑘0}𝑟+𝑀^𝑛−𝑟_𝑛−1𝑏𝛽_𝑛,𝑘0+∑^𝑁_𝑘=1𝛽_𝑛,𝑘−𝛽_{𝑛−1,𝑘}𝛽_𝑛,𝑘0+𝛾_𝑛/1−𝜎_𝑛−𝛾_𝑛−1/1−𝜎_𝑛−1𝛽_𝑛,𝑘0+2𝛼_𝑛−𝛼_𝑛−1𝛽_𝑛,𝑘0+𝛽_𝑛−𝛽_𝑛−1𝛽_𝑛,𝑘0+𝜎_𝑛−𝜎_𝑛−1𝛽_𝑛,𝑘0≤1−1−𝜌𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1𝛽_{𝑛−1,𝑘0}+𝑥_𝑛−𝑥_𝑛−11𝛽_𝑛,𝑘0−1𝛽_{𝑛−1,𝑘0}𝑟+𝑀^𝑛−𝑟_𝑛−1𝑏𝛽_𝑛,𝑘0+∑^𝑁_𝑘=1𝛽_𝑛,𝑘−𝛽_{𝑛−1,𝑘}𝛽_𝑛,𝑘0+𝛾_𝑛/1−𝜎_𝑛−𝛾_𝑛−1/1−𝜎_𝑛−1𝛽_𝑛,𝑘0+2𝛼_𝑛−𝛼_𝑛−1𝛽_𝑛,𝑘0+𝛽_𝑛−𝛽_𝑛−1𝛽_𝑛,𝑘0+𝜎_𝑛−𝜎_𝑛−1𝛽_𝑛,𝑘0≤1−1−𝜌𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1𝛽_{𝑛−1,𝑘0}+𝛽_𝑛𝜏𝑥_𝑛−𝑥_𝑛−1𝑟+𝑀^𝑛−𝑟_𝑛−1𝑏𝛽_𝑛,𝑘0+∑^𝑁_𝑘=1𝛽_𝑛,𝑘−𝛽_{𝑛−1,𝑘}𝛽_𝑛,𝑘0+𝛾_𝑛/1−𝜎_𝑛−𝛾_𝑛−1/1−𝜎_𝑛−1𝛽_𝑛,𝑘0+2𝛼_𝑛−𝛼_𝑛−1𝛽_𝑛,𝑘0+𝛽_𝑛−𝛽_𝑛−1𝛽_𝑛,𝑘0+𝜎_𝑛−𝜎_𝑛−1𝛽_𝑛,𝑘0=1−1−𝜌𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1𝛽_{𝑛−1,𝑘0}+1−𝜌𝛽_𝑛⋅1×𝜏𝑥1−𝜌_𝑛−𝑥_𝑛−1𝑟+𝑀^𝑛−𝑟_𝑛−1𝑏𝛽_𝑛𝛽_𝑛,𝑘0+∑^𝑁_𝑘=1𝛽_𝑛,𝑘−𝛽_{𝑛−1,𝑘}𝛽_𝑛𝛽_𝑛,𝑘0+𝛾_𝑛/1−𝜎_𝑛−𝛾_𝑛−1/1−𝜎_𝑛−1𝛽_𝑛𝛽_𝑛,𝑘0+2𝛼_𝑛−𝛼_𝑛−1𝛽_𝑛𝛽_𝑛,𝑘0+𝛽_𝑛−𝛽_𝑛−1𝛽_𝑛𝛽_𝑛,𝑘0+𝜎_𝑛−𝜎_𝑛−1𝛽_𝑛𝛽_𝑛,𝑘0. (68) Therefore, utilizing Lemma 9, from (H1), (H7), and the asymptotical regularity of {𝑥_𝑛} (due to Lemma 16), we deduce that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛𝛽_𝑛,𝑘0=0. (69)

Lemma 20. Let one suppose that Ω≠∅. Let one suppose that (H1)–(H6) hold. Then, lim𝑦_𝑛→∞^∼_𝑛,𝑁−𝑦_𝑛,𝑁=lim𝑦_𝑛→∞^∼_𝑛−𝑦_𝑛=0. (70)

Proof. Let 𝑝∈Ω. Then, by Lemma 11 we have 𝑦_𝑛−𝑝²=𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝+1−𝛽𝑦_𝑛^∼_𝑛,𝑁−𝑝²≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+1−𝛽𝑦_𝑛^∼_𝑛,𝑁−𝑝²=𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+1−𝛽_𝑛×𝑃_𝐶𝐼−𝜆∇𝑓_𝛼𝑛𝑦_𝑛,𝑁−𝑃_𝐶𝐼−𝜆∇𝑓𝑝²≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+1−𝛽_𝑛×𝐼−𝜆∇𝑓𝑦_𝑛,𝑁−𝐼−𝜆∇𝑓𝑝−𝜆𝛼_𝑛𝑦_𝑛,𝑁²≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+1−𝛽_𝑛×𝐼−𝜆∇𝑓𝑦_𝑛,𝑁−𝐼−𝜆∇𝑓𝑝²−2𝜆𝛼_𝑛𝑦_𝑛,𝑁,𝐼−𝜆∇𝑓_𝛼𝑛𝑦_𝑛,𝑁−𝐼−𝜆∇𝑓𝑝≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+1−𝛽_𝑛×𝑦_𝑛,𝑁−𝑝²2+2𝜆𝜆−𝐿∇𝑓𝑦_𝑛,𝑁−∇𝑓𝑝²+2𝜆𝛼_𝑛𝑦_𝑛,𝑁𝐼−𝜆∇𝑓_𝛼𝑛𝑦_𝑛,𝑁−𝐼−𝜆∇𝑓𝑝≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+1−𝛽_𝑛×𝑥_𝑛−𝑝²2+2𝜆𝜆−𝐿∇𝑓𝑦_𝑛,𝑁−∇𝑓𝑝²+2𝜆𝛼_𝑛𝑦_𝑛,𝑁𝐼−𝜆∇𝑓_𝛼𝑛𝑦_𝑛,𝑁−𝐼−𝜆∇𝑓𝑝. (71) So, we obtain 1−𝛽_𝑛22𝜆𝐿−𝜆∇𝑓(𝑦_𝑛,𝑁)−∇𝑓(𝑝)²≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+1−𝛽_𝑛𝑥_𝑛−𝑝²−𝑦_𝑛−𝑝²+1−𝛽_𝑛2𝜆𝛼_𝑛𝑦_𝑛,𝑁×𝐼−𝜆∇𝑓_𝛼𝑛𝑦_𝑛,𝑁−𝐼−𝜆∇𝑓𝑝≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝑥_𝑛−𝑝+𝑦_𝑛×𝑥−𝑝_𝑛−𝑝−𝑦_𝑛−𝑝+2𝜆𝛼_𝑛𝑦_𝑛,𝑁𝐼−𝜆∇𝑓_𝛼𝑛𝑦_𝑛,𝑁−𝐼−𝜆∇𝑓𝑝≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝑥_𝑛−𝑝+𝑦_𝑛−𝑝𝑥_𝑛−𝑦_𝑛+2𝜆𝛼_𝑛𝑦_𝑛,𝑁𝐼−𝜆∇𝑓_𝛼𝑛𝑦_𝑛,𝑁−𝐼−𝜆∇𝑓𝑝. (72) Since 𝛼_𝑛→0, 𝛽_𝑛→0, ‖𝑥_𝑛−𝑦_𝑛‖→0, and 0<𝜆<2/𝐿, from the boundedness of {𝑥_𝑛}, {𝑦_𝑛}, and {𝑦_𝑛,𝑁} it follows that lim_𝑛→∞‖∇𝑓(𝑦_𝑛,𝑁)−∇𝑓(𝑝)‖=0, and hence lim_𝑛→∞∇𝑓_𝛼𝑛𝑦_𝑛,𝑁−∇𝑓𝑝=0. (73) Moreover, from the firm nonexpansiveness of 𝑃_𝐶 we obtain 𝑦^∼_𝑛,𝑁−𝑝²=𝑃_𝐶𝐼−𝜆∇𝑓_𝛼𝑛𝑦_𝑛,𝑁−𝑃_𝐶𝐼−𝜆∇𝑓𝑝²≤𝐼−𝜆∇𝑓_𝛼𝑛𝑦_𝑛,𝑁−𝐼−𝜆∇𝑓𝑝,𝑦^∼_𝑛,𝑁=1−𝑝2𝐼−𝜆∇𝑓_𝛼𝑛𝑦_𝑛,𝑁−𝐼−𝜆∇𝑓𝑝²+𝑦^∼_𝑛,𝑁−𝑝²−𝐼−𝜆∇𝑓_𝛼𝑛𝑦_𝑛,𝑁−𝐼−𝜆∇𝑓𝑝−𝑦^∼_𝑛,𝑁−𝑝²≤12𝑦_𝑛,𝑁−𝑝²+2𝜆∇𝑓_𝛼𝑛𝑦_𝑛,𝑁𝑝×−∇𝑓𝐼−𝜆∇𝑓_𝛼𝑛𝑦_𝑛,𝑁−𝑝+𝐼−𝜆∇𝑓_∼𝑦_𝑛,𝑁−𝑝²−𝑦_𝑛,𝑁−𝑦^∼_𝑛,𝑁²+2𝜆𝑦_𝑛,𝑁−𝑦^∼_𝑛,𝑁,∇𝑓_𝛼𝑛𝑦_𝑛,𝑁−∇𝑓𝑝−𝜆²∇𝑓_𝛼𝑛𝑦_𝑛,𝑁−∇𝑓𝑝², (74) and so 𝑦^∼_𝑛,𝑁−𝑝²≤𝑦_𝑛,𝑁−𝑝²−𝑦_𝑛,𝑁−𝑦^∼_𝑛,𝑁²+2𝜆∇𝑓_𝛼𝑛𝑦_𝑛,𝑁𝑝×−∇𝑓𝐼−𝜆∇𝑓_𝛼𝑛𝑦_𝑛,𝑁−𝑝𝑦𝐼−𝜆∇𝑓+2𝜆_𝑛,𝑁−𝑦^∼_𝑛,𝑁,∇𝑓_𝛼𝑛𝑦_𝑛,𝑁−∇𝑓𝑝−𝜆²∇𝑓_𝛼𝑛𝑦_𝑛,𝑁−∇𝑓𝑝². (75) Thus, we have 𝑦_𝑛−𝑝²≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+1−𝛽𝑦_𝑛^∼_𝑛,𝑁−𝑝²≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝑦_𝑛,𝑁−𝑝²−1−𝛽_𝑛𝑦_𝑛,𝑁−𝑦^∼_𝑛,𝑁²+2𝜆∇𝑓_𝛼𝑛𝑦_𝑛,𝑁𝑝×−∇𝑓𝐼−𝜆∇𝑓_𝛼𝑛𝑦_𝑛,𝑁−𝐼−𝜆∇𝑓𝑝+21−𝛽_𝑛𝜆𝑦_𝑛,𝑁−𝑦^∼_𝑛,𝑁,∇𝑓_𝛼𝑛𝑦_𝑛,𝑁𝑝−−∇𝑓1−𝛽_𝑛𝜆²∇𝑓_𝛼𝑛𝑦_𝑛,𝑁−∇𝑓𝑝², (76) which implies that 1−𝛽_𝑛𝑦_𝑛,𝑁−𝑦^∼_𝑛,𝑁²≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝑦_𝑛,𝑁−𝑝²−𝑦_𝑛−𝑝²+2𝜆∇𝑓_𝛼𝑛𝑦_𝑛,𝑁𝑝×−∇𝑓𝐼−𝜆∇𝑓_𝛼𝑛𝑦_𝑛,𝑁−𝐼−𝜆∇𝑓𝑝+21−𝛽_𝑛𝜆⟨𝑦_𝑛,𝑁−𝑦^∼_𝑛,𝑁,∇𝑓_𝛼𝑛𝑦_𝑛,𝑁𝑝⟩−−∇𝑓1−𝛽_𝑛𝜆²∇𝑓_𝛼𝑛𝑦_𝑛,𝑁−∇𝑓𝑝²≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝑥_𝑛−𝑝²−𝑦_𝑛−𝑝²+2𝜆∇𝑓_𝛼𝑛𝑦_𝑛,𝑁𝑝×−∇𝑓𝐼−𝜆∇𝑓_𝛼𝑛𝑦_𝑛,𝑁−𝐼−𝜆∇𝑓𝑝+21−𝛽_𝑛𝜆𝑦_𝑛,𝑁−𝑦^∼_𝑛,𝑁,∇𝑓_𝛼𝑛𝑦_𝑛,𝑁𝑝−−∇𝑓1−𝛽_𝑛𝜆²∇𝑓_𝛼𝑛𝑦_𝑛,𝑁−∇𝑓𝑝²≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝑥_𝑛−𝑝+𝑦_𝑛−𝑝𝑥_𝑛−𝑦_𝑛+2𝜆∇𝑓_𝛼𝑛𝑦_𝑛,𝑁𝑝×−∇𝑓𝐼−𝜆∇𝑓_𝛼𝑛𝑦_𝑛,𝑁−𝑝𝑦𝐼−𝜆∇𝑓+2𝜆_𝑛,𝑁−𝑦^∼_𝑛,𝑁∇𝑓_𝛼𝑛𝑦_𝑛,𝑁−∇𝑓𝑝. (77) Since 𝛽_𝑛→0, ‖𝑥_𝑛−𝑦_𝑛‖→0, and ‖∇𝑓_𝛼𝑛(𝑦_𝑛,𝑁)−∇𝑓(𝑝)‖→0, from the boundedness of {𝑥_𝑛}, {𝑦_𝑛}, {𝑦_𝑛,𝑁}, and {𝑦^∼_𝑛,𝑁}, it follows that lim_𝑛→∞‖𝑦_𝑛,𝑁−𝑦^∼_𝑛,𝑁‖=0. Observe that 𝑦_𝑛−𝑦^∼_𝑛,𝑁=𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑦^∼_𝑛,𝑁⟶0as𝑛⟶∞, (78) and hence 𝑦^∼_𝑛−𝑦_𝑛≤𝑦^∼_𝑛−𝑦^∼_𝑛,𝑁+𝑦^∼_𝑛,𝑁−𝑦_𝑛=𝑃_𝐶𝐼−𝜆∇𝑓_𝛼𝑛𝑦_𝑛−𝑃_𝐶𝐼−𝜆∇𝑓_𝛼𝑛𝑦_𝑛,𝑁+𝑦^∼_𝑛,𝑁−𝑦_𝑛≤𝑦_𝑛−𝑦_𝑛,𝑁+𝑦^∼_𝑛,𝑁−𝑦_𝑛≤𝑦_𝑛−𝑦^∼_𝑛,𝑁+𝑦^∼_𝑛,𝑁−𝑦_𝑛,𝑁+𝑦^∼_𝑛,𝑁−𝑦_𝑛=2𝑦_𝑛−𝑦^∼_𝑛,𝑁+𝑦^∼_𝑛,𝑁−𝑦_𝑛,𝑁⟶0as𝑛⟶∞. (79) Thus, lim_𝑛→∞‖𝑦^∼_𝑛−𝑦_𝑛‖=0.

Lemma 21. Let one suppose that Ω≠∅. Let one suppose that 0<liminf_𝑛→∞𝛽_𝑛,𝑖≤limsup_𝑛→∞𝛽_𝑛,𝑖<1 for each 𝑖=1,…,𝑁. Moreover, suppose that (H1)–(H6) are satisfied. Then, lim_𝑛→∞‖𝑆_𝑖𝑢_𝑛−𝑢_𝑛‖=0 for each 𝑖=1,…,𝑁.

Proof. First of all, observe that 𝑥_𝑛+1−𝑦_𝑛=𝛾𝑦_𝑛^∼_𝑛−𝑦_𝑛+𝛿_𝑛𝑇𝑦^∼_𝑛−𝑦_𝑛=𝛾𝑦_𝑛^∼_𝑛−𝑦_𝑛+𝛿_𝑛𝑇𝑦^∼_𝑛−𝑇𝑦_𝑛+𝛿_𝑛𝑇𝑦_𝑛−𝑦_𝑛. (80) By Lemmas 16 and 20, we know that ‖𝑥_𝑛+1−𝑥_𝑛‖→0 and ‖𝑦^∼_𝑛−𝑦_𝑛‖→0 as 𝑛→∞. Hence, utilizing Lemma 7(i), we have 𝛿_𝑛𝑇𝑦_𝑛−𝑦_𝑛=𝑥_𝑛+1−𝑦_𝑛−𝛾𝑦_𝑛^∼_𝑛−𝑦_𝑛−𝛿_𝑛𝑇𝑦^∼_𝑛−𝑇𝑦_𝑛≤𝑥_𝑛+1−𝑦_𝑛+𝛾𝑦_𝑛^∼_𝑛−𝑦_𝑛+𝛿_𝑛𝑇𝑦^∼_𝑛−𝑇𝑦_𝑛≤𝑥_𝑛+1−𝑦_𝑛+𝑦^∼_𝑛−𝑦_𝑛+𝑇𝑦^∼_𝑛−𝑇𝑦_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑥_𝑛−𝑦_𝑛+𝑦^∼_𝑛−𝑦_𝑛+1+𝜁1−𝜁^∼𝑦_𝑛−𝑦_𝑛=𝑥_𝑛+1−𝑥_𝑛+𝑥_𝑛−𝑦_𝑛+21−𝜁^∼𝑦_𝑛−𝑦_𝑛, (81) which together with ‖𝑥_𝑛−𝑦_𝑛‖→0 implies that lim_𝑛→∞‖𝛿_𝑛(𝑇𝑦_𝑛−𝑦_𝑛)‖=0. Taking into account liminf_𝑛→∞𝛿_𝑛>0, we have lim_𝑛→∞𝑇𝑦_𝑛−𝑦_𝑛=0. (82)
Let us show that for each 𝑖∈{1,…,𝑁}, one has ‖𝑆_𝑖𝑢_𝑛−𝑦_{𝑛,𝑖−1}‖→0 as 𝑛→∞. Let 𝑝∈Ω. When 𝑖=𝑁, by Lemma 11, we have from (33)-(34) the following: 𝑦_𝑛−𝑝²≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+1−𝛽𝑦_𝑛^∼_𝑛,𝑁−𝑝²≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝑦^∼_𝑛,𝑁−𝑝²≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝑦_𝑛,𝑁−𝑝+𝜆𝛼_𝑛𝑝²=𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝑦_𝑛,𝑁−𝑝²+𝜆𝛼_𝑛𝑝2𝑦_𝑛,𝑁−𝑝+𝜆𝛼_𝑛𝑝=𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝛽_𝑛,𝑁𝑆_𝑁𝑢_𝑛−𝑝²+1−𝛽_𝑛,𝑁𝑦_{𝑛,𝑁−1}−𝑝²−𝛽_𝑛,𝑁1−𝛽_𝑛,𝑁𝑆_𝑁𝑢_𝑛−𝑦_{𝑛,𝑁−1}²+𝜆𝛼_𝑛𝑝2𝑦_𝑛,𝑁−𝑝+𝜆𝛼_𝑛𝑝≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝑢_𝑛−𝑝²−𝛽_𝑛,𝑁1−𝛽_𝑛,𝑁𝑆_𝑁𝑢_𝑛−𝑦_{𝑛,𝑁−1}²+𝜆𝛼_𝑛𝑝2𝑦_𝑛,𝑁−𝑝+𝜆𝛼_𝑛𝑝≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝑥_𝑛−𝑝²−𝛽_𝑛,𝑁1−𝛽_𝑛,𝑁𝑆_𝑁𝑢_𝑛−𝑦_{𝑛,𝑁−1}²+𝜆𝛼_𝑛𝑝2𝑦_𝑛,𝑁−𝑝+𝜆𝛼_𝑛𝑝. (83) So, we have 𝛽_𝑛,𝑁1−𝛽_𝑛,𝑁𝑆_𝑁𝑢_𝑛−𝑦_{𝑛,𝑁−1}²≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝑥_𝑛−𝑝²−𝑦_𝑛−𝑝²+𝜆𝛼_𝑛𝑝2𝑦_𝑛,𝑁−𝑝+𝜆𝛼_𝑛𝑝≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝑥_𝑛−𝑝+𝑦_𝑛−𝑝𝑥_𝑛−𝑦_𝑛+𝜆𝛼_𝑛𝑝2𝑦_𝑛,𝑁−𝑝+𝜆𝛼_𝑛𝑝. (84) Since 𝛼_𝑛→0, 𝛽_𝑛→0, 0<liminf_𝑛→∞𝛽_𝑛,𝑁≤limsup_𝑛→∞𝛽_𝑛,𝑁<1, and lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0, it is known that {‖𝑆_𝑁𝑢_𝑛−𝑦_{𝑛,𝑁−1}‖} is a null sequence.
Let 𝑖∈{1,…,𝑁−1}. Then, one has 𝑦_𝑛−𝑝²≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝑦^∼_𝑛,𝑁−𝑝²≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝑦_𝑛,𝑁−𝑝+𝜆𝛼_𝑛𝑝²=𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝑦_𝑛,𝑁−𝑝²+𝜆𝛼_𝑛𝑝2𝑦_𝑛,𝑁−𝑝+𝜆𝛼_𝑛𝑝≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝛽_𝑛,𝑁𝑆_𝑁𝑢_𝑛−𝑝²+1−𝛽_𝑛,𝑁𝑦_{𝑛,𝑁−1}−𝑝²+𝜆𝛼_𝑛𝑝2𝑦_𝑛,𝑁−𝑝+𝜆𝛼_𝑛𝑝≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝛽_𝑛,𝑁𝑥_𝑛−𝑝²+1−𝛽_𝑛,𝑁𝑦_{𝑛,𝑁−1}−𝑝²+𝜆𝛼_𝑛𝑝2𝑦_𝑛,𝑁−𝑝+𝜆𝛼_𝑛𝑝≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝜆𝛼_𝑛𝑝2𝑦_𝑛,𝑁−𝑝+𝜆𝛼_𝑛𝑝+𝛽_𝑛,𝑁𝑥_𝑛−𝑝²+1−𝛽_𝑛,𝑁𝛽_{𝑛,𝑁−1}𝑆_𝑁−1𝑢_𝑛−𝑝²+1−𝛽_{𝑛,𝑁−1}𝑦_{𝑛,𝑁−2}−𝑝²≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝜆𝛼_𝑛𝑝2𝑦_𝑛,𝑁−𝑝+𝜆𝛼_𝑛𝑝+𝛽_𝑛,𝑁+1−𝛽_𝑛,𝑁𝛽_{𝑛,𝑁−1}𝑥_𝑛−𝑝²+^𝑁_{𝑘=𝑁−1}1−𝛽_𝑛,𝑘𝑦_{𝑛,𝑁−2}−𝑝², (85) and so, after (𝑁−𝑖+1) iterations, 𝑦_𝑛−𝑝²≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝜆𝛼_𝑛𝑝2𝑦_𝑛,𝑁−𝑝+𝜆𝛼_𝑛𝑝+𝛽_𝑛,𝑁+^𝑁_𝑗=𝑖+2^𝑁_𝑙=𝑗1−𝛽_𝑛,𝑙𝛽_{𝑛,𝑗−1}×𝑥_𝑛−𝑝²+^𝑁_𝑘=𝑖+11−𝛽_𝑛,𝑘𝑦_𝑛,𝑖−𝑝²≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝜆𝛼_𝑛𝑝2𝑦_𝑛,𝑁−𝑝+𝜆𝛼_𝑛𝑝+𝛽_𝑛,𝑁+^𝑁_𝑗=𝑖+2^𝑁_𝑙=𝑗1−𝛽_𝑛,𝑙𝛽_{𝑛,𝑗−1}×𝑥_𝑛−𝑝²+^𝑁_𝑘=𝑖+11−𝛽_𝑛,𝑘×𝛽_𝑛,𝑖𝑆_𝑖𝑢_𝑛−𝑝²+1−𝛽_𝑛,𝑖𝑦_{𝑛,𝑖−1}−𝑝²−𝛽_𝑛,𝑖1−𝛽_𝑛,𝑖𝑆_𝑖𝑢_𝑛−𝑦_{𝑛,𝑖−1}²≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝜆𝛼_𝑛𝑝2𝑦_𝑛,𝑁−𝑝+𝜆𝛼_𝑛𝑝+𝑥_𝑛−𝑝²−𝛽^𝑁_{𝑛,𝑖𝑘=𝑖}1−𝛽_𝑛,𝑘𝑆_𝑖𝑢_𝑛−𝑦_{𝑛,𝑖−1}². (86) Again, we obtain that 𝛽^𝑁_{𝑛,𝑖𝑘=𝑖}1−𝛽_𝑛,𝑘𝑆_𝑖𝑢_𝑛−𝑦_{𝑛,𝑖−1}²≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝜆𝛼_𝑛𝑝2𝑦_𝑛,𝑁−𝑝+𝜆𝛼_𝑛𝑝+𝑥_𝑛−𝑝²−𝑦_𝑛−𝑝²≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝜆𝛼_𝑛𝑝2𝑦_𝑛,𝑁−𝑝+𝜆𝛼_𝑛𝑝+𝑥_𝑛−𝑝+𝑦_𝑛−𝑝𝑥_𝑛−𝑦_𝑛. (87) Since 𝛼_𝑛→0, 𝛽_𝑛→0, 0<liminf_𝑛→∞𝛽_𝑛,𝑖≤limsup_𝑛→∞𝛽_𝑛,𝑖<1 for each 𝑖=1,…,𝑁−1, and lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0, it is known that lim_𝑛→∞𝑆_𝑖𝑢_𝑛−𝑦_{𝑛,𝑖−1}=0. (88)
Obviously, for 𝑖=1, we have ‖𝑆₁𝑢_𝑛−𝑢_𝑛‖→0.
To conclude, we have that 𝑆₂𝑢_𝑛−𝑢_𝑛≤𝑆₂𝑢_𝑛−𝑦_𝑛,1+𝑦_𝑛,1−𝑢_𝑛=𝑆₂𝑢_𝑛−𝑦_𝑛,1+𝛽_𝑛,1𝑆₁𝑢_𝑛−𝑢_𝑛, (89) from which ‖𝑆₂𝑢_𝑛−𝑢_𝑛‖→0. Thus, by induction ‖𝑆_𝑖𝑢_𝑛−𝑢_𝑛‖→0 for all 𝑖=2,…,𝑁 since it is enough to observe that 𝑆_𝑖𝑢_𝑛−𝑢_𝑛≤𝑆_𝑖𝑢_𝑛−𝑦_{𝑛,𝑖−1}+𝑦_{𝑛,𝑖−1}−𝑆_𝑖−1𝑢_𝑛+𝑆_𝑖−1𝑢_𝑛−𝑢_𝑛≤𝑆_𝑖𝑢_𝑛−𝑦_{𝑛,𝑖−1}+1−𝛽_{𝑛,𝑖−1}𝑆_𝑖−1𝑢_𝑛−𝑦_{𝑛,𝑖−2}+𝑆_𝑖−1𝑢_𝑛−𝑢_𝑛. (90)

Remark 22. As an example, we consider 𝑁=2 and the following sequences:(a)𝜎_𝑛=1/2+2/𝑛, 𝛾_𝑛=𝛿_𝑛=1/4−1/𝑛 for all 𝑛>4;(b)𝛽_𝑛=1/√𝑛, 𝑟_𝑛=2−1/𝑛, for all 𝑛>1;(c)𝛽_𝑛,1=1/2−1/𝑛,𝛽_𝑛,2=1/2−1/𝑛², for all 𝑛>2.
Then, they satisfy the hypotheses on the parameter sequences in Lemma 21.

Lemma 23. Let one suppose that Ω≠∅ and 𝛽_𝑛,𝑖→𝛽_𝑖 for all 𝑖 as 𝑛→∞. Suppose there exists 𝑘∈{1,…,𝑁} such that 𝛽_𝑛,𝑘→0 as 𝑛→∞. Let 𝑘₀∈{1,…,𝑁} be the largest index such that 𝛽_𝑛,𝑘0→0 as 𝑛→∞. Suppose that(i)(𝛼_𝑛+𝛽_𝑛)/𝛽_𝑛,𝑘0→0 as 𝑛→∞;(ii) if 𝑖≤𝑘₀ and 𝛽_𝑛,𝑖→0, then 𝛽_𝑛,𝑘0/𝛽_𝑛,𝑖→0 as 𝑛→∞;(iii) if 𝛽_𝑛,𝑖→𝛽_𝑖≠0, then 𝛽_𝑖 lies in (0,1).
Moreover, suppose that (H1), (H7), and (H8) hold. Then, lim_𝑛→∞‖𝑆_𝑖𝑢_𝑛−𝑢_𝑛‖=0 for each 𝑖=1,…,𝑁.

Proof. First of all, we note that if (H7) holds, then also (H2)–(H6) are satisfied. So {𝑥_𝑛} is asymptotically regular. Let 𝑘₀ be as in the hypotheses. As in Lemma 21, for every index 𝑖∈{1,…,𝑁} such that 𝛽_𝑛,𝑖→𝛽_𝑖≠0 (which leads to 0<liminf_𝑛→∞𝛽_𝑛,𝑖≤limsup_𝑛→∞𝛽_𝑛,𝑖<1), one has ‖𝑆_𝑖𝑢_𝑛−𝑦_{𝑛,𝑖−1}‖→0 as 𝑛→∞.
For all the other indexes 𝑖≤𝑘₀, we can prove that ‖𝑆_𝑖𝑢_𝑛−𝑦_{𝑛,𝑖−1}‖→0 as 𝑛→∞ in a similar manner. By the following relation (due to (86)): 𝑥_𝑛+1−𝑝²≤𝜎_𝑛𝑦_𝑛−𝑝²+𝛾_𝑛+𝛿𝑦_𝑛^∼_𝑛−𝑝²=𝜎_𝑛𝑦_𝑛−𝑝²+1−𝜎𝑦_𝑛^∼_𝑛−𝑝²≤𝜎_𝑛𝑦_𝑛−𝑝²+1−𝜎_𝑛𝑦_𝑛−𝑝+𝜆𝛼_𝑛𝑝²≤𝑦_𝑛−𝑝+𝜆𝛼_𝑛𝑝²=𝑦_𝑛−𝑝²+𝜆𝛼_𝑛𝑝2𝑦_𝑛−𝑝+𝜆𝛼_𝑛𝑝≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+𝜆𝛼_𝑛𝑝2𝑦_𝑛,𝑁−𝑝+𝜆𝛼_𝑛𝑝+𝑥_𝑛−𝑝²−𝛽^𝑁_{𝑛,𝑖𝑘=𝑖}1−𝛽_𝑛,𝑘𝑆_𝑖𝑢_𝑛−𝑦_{𝑛,𝑖−1}²+𝜆𝛼_𝑛𝑝2𝑦_𝑛−𝑝+𝜆𝛼_𝑛𝑝≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑝²+2𝜆𝛼_𝑛𝑝×𝑥_𝑛−𝑝+𝑦_𝑛−𝑝+𝜆𝛼_𝑛𝑝+𝑥_𝑛−𝑝²−𝛽^𝑁_{𝑛,𝑖𝑘=𝑖}1−𝛽_𝑛,𝑘𝑆_𝑖𝑢_𝑛−𝑦_{𝑛,𝑖−1}², (91) we immediately obtain that ^𝑁_𝑘=𝑖1−𝛽_𝑛,𝑘𝑆_𝑖𝑢_𝑛−𝑦_{𝑛,𝑖−1}²≤𝛽_𝑛𝛽_𝑛,𝑖𝑄𝑦_𝑛,𝑁−𝑝²+𝛼_𝑛𝛽_𝑛,𝑖𝑝×𝑥2𝜆_𝑛−𝑝+𝑦_𝑛−𝑝+𝜆𝛼_𝑛𝑝+𝑥_𝑛−𝑝+𝑥_𝑛+1𝑥−𝑝^𝑛−𝑥_𝑛+1𝛽_𝑛,𝑖. (92) By Lemma 19 or by hypothesis (ii) of the sequences, we have 𝑥_𝑛−𝑥_𝑛+1𝛽_𝑛,𝑖=𝑥_𝑛−𝑥_𝑛+1𝛽_𝑛,𝑘0⋅𝛽_𝑛,𝑘0𝛽_𝑛,𝑖⟶0. (93) So, the thesis follows.

Remark 24. Let us consider 𝑁=3 and the following sequences:(a)𝛼_𝑛=1/𝑛^5/4, 𝛽_𝑛=1/𝑛^1/2, 𝑟_𝑛=2−1/𝑛², for all 𝑛>1;(b)𝜎_𝑛=1/2+2/𝑛², 𝛾_𝑛=𝛿_𝑛=1/4−1/𝑛², for all 𝑛>2;(c)𝛽_𝑛,1=1/𝑛^1/4, 𝛽_𝑛,2=1/2−1/𝑛², 𝛽_𝑛,3=1/𝑛^1/3, for all 𝑛>1.
It is easy to see that all hypotheses (i)–(iii), (H1), (H7), and (H8) of Lemma 23 are satisfied.

Remark 25. Under the hypotheses of Lemma 23, analogously to Lemma 21, one can see that lim_𝑛→∞𝑆_𝑖𝑢_𝑛−𝑦_{𝑛,𝑖−1}=0,∀𝑖∈{2,…,𝑁}. (94)

Corollary 26. Let one suppose that the hypotheses of either Lemma 21 or Lemma 23 are satisfied. Then, 𝜔_𝑤(𝑥_𝑛)=𝜔_𝑤(𝑢_𝑛)=𝜔_𝑤(𝑦_𝑛,1),𝜔_𝑠(𝑥_𝑛)=𝜔_𝑠(𝑢_𝑛)=𝜔_𝑠(𝑦_𝑛,1), and 𝜔_𝑤(𝑥_𝑛)⊂Ω.

Proof. By Remark 18, we have 𝜔_𝑤(𝑥_𝑛)=𝜔_𝑤(𝑢_𝑛) and 𝜔_𝑠(𝑥_𝑛)=𝜔_𝑠(𝑢_𝑛). Observe that 𝑥_𝑛−𝑦_𝑛,1≤𝑥_𝑛−𝑢_𝑛+𝑦_𝑛,1−𝑢_𝑛=𝑥_𝑛−𝑢_𝑛+𝛽_𝑛,1𝑆₁𝑢_𝑛−𝑢_𝑛. (95) By Lemmas 17 and 21, ‖𝑥_𝑛−𝑢_𝑛‖→0 and ‖𝑆₁𝑢_𝑛−𝑢_𝑛‖→0 as 𝑛→∞, and hence lim_𝑛→∞𝑥_𝑛−𝑦_𝑛,1=0. (96) So, we get 𝜔_𝑤(𝑥_𝑛)=𝜔_𝑤(𝑦_𝑛,1) and 𝜔_𝑠(𝑥_𝑛)=𝜔_𝑠(𝑦_𝑛,1).
Let 𝑝∈𝜔_𝑤(𝑥_𝑛). Since 𝑝∈𝜔_𝑤(𝑢_𝑛), by Lemma 21 and Lemma 7(ii) (demiclosedness principle), we have 𝑝∈Fix(𝑆_𝑖) for all index 𝑖, that is, 𝑝∈⋂_𝑖Fix(𝑆_𝑖). Taking into consideration that 𝑇 is 𝜁-strictly pseudocontractive, by Lemma 7(i), we get 𝑇𝑥_𝑛−𝑥_𝑛≤𝑇𝑥_𝑛−𝑇𝑦_𝑛+𝑇𝑦_𝑛−𝑦_𝑛+𝑦_𝑛−𝑥_𝑛≤1+𝜁𝑥1−𝜁^𝑛−𝑦_𝑛+𝑦_𝑛−𝑇𝑦_𝑛+𝑥_𝑛−𝑦_𝑛=2𝑥1−𝜁^𝑛−𝑦_𝑛+𝑦_𝑛−𝑇𝑦_𝑛, (97) which together with ‖𝑥_𝑛−𝑦_𝑛‖→0 (by Lemma 17) and ‖𝑦_𝑛−𝑇𝑦_𝑛‖→0 (by (82)) implies that lim_𝑛→∞𝑥_𝑛−𝑇𝑥_𝑛=0. (98)
Utilizing Lemma 7(ii) (demiclosedness principle), we have 𝑝∈Fix(𝑇). Furthermore, by Lemmas 14 and 17, we know that 𝑝∈EP(𝐹,ℎ). Finally, by similar argument as in [18], we can show that 𝑝∈Γ, and as a result 𝑝∈Ω.

Theorem 27. Let one suppose that Ω≠∅. Let {𝛽_𝑛}, {𝛽_𝑛,𝑖}, 𝑖=1,…,𝑁, be sequences in (0,1) such that 0<liminf_𝑛→∞𝛽_𝑛,𝑖≤limsup_𝑛→∞𝛽_𝑛,𝑖<1 for all index 𝑖. Moreover, Let one suppose that (H1)–(H6) hold. Then, the sequences {𝑥_𝑛}, {𝑦_𝑛}, and {𝑢_𝑛}, explicitly defined by scheme (30), all converge strongly to the unique solution 𝑥^∗∈Ω of the following variational inequality: 𝑄𝑥^∗−𝑥^∗,𝑧−𝑥^∗≤0,∀𝑧∈Ω. (99)

Proof. Since the mapping 𝑃_Ω𝑄 is a 𝜌-contraction, it has a unique fixed point 𝑥^∗; it is the unique solution of (99). Since (H1)–(H6) hold, the sequence {𝑥_𝑛} is asymptotically regular (by Lemma 16). In terms of Lemma 17, ‖𝑥_𝑛−𝑦_𝑛‖→0 and ‖𝑥_𝑛−𝑢_𝑛‖→0 as 𝑛→∞. Moreover, utilizing Lemmas 8 and 10, we have from (33)-(34) the following: 𝑥_𝑛+1−𝑥^∗2≤𝜎_𝑛𝑦_𝑛−𝑥^∗2+1−𝜎𝑦_𝑛^∼_𝑛−𝑥^∗2≤𝜎_𝑛𝑦_𝑛−𝑥^∗2+1−𝜎_𝑛𝑦_𝑛−𝑥^∗+𝜆𝛼_𝑛𝑝²≤𝑦_𝑛−𝑥^∗+𝜆𝛼_𝑛𝑝²=𝑦_𝑛−𝑥^∗2+𝜆𝛼_𝑛𝑝2𝑦_𝑛−𝑥^∗+𝜆𝛼_𝑛𝑝≤𝛽_𝑛𝑄𝑦_𝑛,𝑁−𝑄𝑥^∗+1−𝛽𝑦_𝑛^∼_𝑛,𝑁−𝑥^∗2+2𝛽_𝑛𝑄𝑥^∗−𝑥^∗,𝑦_𝑛−𝑥^∗+𝜆𝛼_𝑛𝑝2𝑦_𝑛−𝑥^∗+𝜆𝛼_𝑛𝑝≤𝛽_𝑛𝜌𝑦_𝑛,𝑁−𝑥^∗+1−𝛽_𝑛𝑦_𝑛,𝑁−𝑥^∗2+2𝛽_𝑛𝑄𝑥^∗−𝑥^∗,𝑦_𝑛−𝑥^∗+𝜆𝛼_𝑛𝑝2𝑦_𝑛−𝑥^∗+𝜆𝛼_𝑛𝑝=1−1−𝜌𝛽_𝑛²𝑦_𝑛,𝑁−𝑥^∗2+2𝛽_𝑛𝑄𝑥^∗−𝑥^∗,𝑦_𝑛−𝑥^∗+𝜆𝛼_𝑛𝑝2𝑦_𝑛−𝑥^∗+𝜆𝛼_𝑛𝑝≤1−1−𝜌𝛽_𝑛𝑢_𝑛−𝑥^∗2+2𝛽_𝑛𝑄𝑥^∗−𝑥^∗,𝑦_𝑛−𝑥^∗+𝜆𝛼_𝑛𝑝2𝑦_𝑛−𝑥^∗+𝜆𝛼_𝑛𝑝≤1−1−𝜌𝛽_𝑛𝑥_𝑛−𝑥^∗2+2𝛽_𝑛𝑄𝑥^∗−𝑥^∗,𝑦_𝑛−𝑥^∗+𝜆𝛼_𝑛𝑝2𝑦_𝑛−𝑥^∗+𝜆𝛼_𝑛𝑝=1−1−𝜌𝛽_𝑛𝑥_𝑛−𝑥^∗2+1−𝜌𝛽_𝑛⋅21−𝜌𝑄𝑥^∗−𝑥^∗,𝑦_𝑛−𝑥^∗+𝜆𝛼_𝑛𝑝2𝑦_𝑛−𝑥^∗+𝜆𝛼_𝑛𝑝. (100) Now, let {𝑥_𝑛𝑘} be a subsequence of {𝑥_𝑛} such that limsup_𝑛→∞𝑄𝑥^∗−𝑥^∗,𝑥_𝑛−𝑥^∗=lim_𝑘→∞𝑄𝑥^∗−𝑥^∗,𝑥_𝑛𝑘−𝑥^∗. (101) By the boundedness of {𝑥_𝑛}, we may assume, without loss of generality, that 𝑥_𝑛𝑘⇀𝑧∈𝜔_𝑤(𝑥_𝑛). According to Corollary 26, we know that 𝜔_𝑤(𝑥_𝑛)⊂Ω, and hence 𝑧∈Ω. Taking into consideration that 𝑥^∗=𝑃_Ω𝑄𝑥^∗ we obtain from (101) that limsup_𝑛→∞𝑄𝑥^∗−𝑥^∗,𝑦_𝑛−𝑥^∗=limsup_𝑛→∞𝑄𝑥^∗−𝑥^∗,𝑥_𝑛−𝑥^∗+𝑄𝑥^∗−𝑥^∗,𝑦_𝑛−𝑥_𝑛=limsup_𝑛→∞𝑄𝑥^∗−𝑥^∗,𝑥_𝑛−𝑥^∗=lim_𝑘→∞𝑄𝑥^∗−𝑥^∗,𝑥_𝑛𝑘−𝑥^∗=𝑄𝑥^∗−𝑥^∗,𝑧−𝑥^∗≤0. (102) Since ∑^∞_𝑛=0𝛼_𝑛<∞ and ∑^∞_𝑛=0𝛽_𝑛=∞, we deduce that ∑^∞_𝑛=0𝜆𝛼_𝑛‖𝑝‖(2‖𝑦_𝑛−𝑥^∗‖+𝜆𝛼_𝑛‖𝑝‖)<∞ and ∑^∞_𝑛=0(1−𝜌)𝛽_𝑛=∞. In terms of Lemma 9 we derive 𝑥_𝑛→𝑥^∗ as 𝑛→∞.

In a similar way, we can derive the following result.

Theorem 28. Let one suppose that Ω≠∅. Let {𝛽_𝑛}, {𝛽_𝑛,𝑖}, 𝑖=1,…,𝑁, be sequences in (0,1) such that 𝛽_𝑛,𝑖→𝛽_𝑖 for all 𝑖 as 𝑛→∞. Suppose that there exists 𝑘∈{1,…,𝑁} for which 𝛽_𝑛,𝑘→0 as 𝑛→∞. Let 𝑘₀∈{1,…,𝑁} the largest index for which 𝛽_𝑛,𝑘0→0. Moreover, let one suppose that (H1), (H7), and (H8) hold, and(i)(𝛼_𝑛+𝛽_𝑛)/𝛽_𝑛,𝑘0→0 as 𝑛→∞;(ii) if 𝑖≤𝑘₀ and 𝛽_𝑛,𝑖→0, then 𝛽_𝑛,𝑘0/𝛽_𝑛,𝑖→0 as 𝑛→∞;(iii)if 𝛽_𝑛,𝑖→𝛽_𝑖≠0, then 𝛽_𝑖 lies in (0,1).
Then, the sequences {𝑥_𝑛}, {𝑦_𝑛}, and {𝑢_𝑛} explicitly defined by scheme (30) all converge strongly to the unique solution 𝑥^∗∈Ω of the following variational inequality: ⟨𝑄𝑥^∗−𝑥^∗,𝑧−𝑥^∗⟩≤0,∀𝑧∈Ω. (103)

Remark 29. According to the above argument processes for Theorems 27 and 28, we can readily see that if in scheme (30), the iterative step 𝑦_𝑛=𝛽_𝑛𝑄𝑦_𝑛,𝑁+(1−𝛽_𝑛)𝑃_𝐶(𝑦_𝑛,𝑁−𝜆∇𝑓_𝛼𝑛(𝑦_𝑛,𝑁)) is replaced by the iterative one 𝑦_𝑛=𝛽_𝑛𝑄𝑥_𝑛+(1−𝛽_𝑛)𝑃_𝐶(𝑦_𝑛,𝑁−𝜆∇𝑓_𝛼𝑛(𝑦_𝑛,𝑁)), then Theorems 27 and 28 remain valid.

Remark 30. Theorems 27 and 28 improve, extend, supplement, and develop [17, Theorems 3.12 and 3.13] and [1, Theorems 5.2 and 6.1] in the following aspects:(a)the multistep iterative scheme (30) of [17] is extended to develop our relaxed viscosity iterative scheme (30) with regularization for MP (3), EP (10), and strict pseudocontraction 𝑇 by virtue of Xu iterative schemes in [1];(b)the argument techniques in Theorems 27 and 28 are very different from the ones in [17, Theorems 3.12 and 3.13] and the ones in [1, Theorems 5.2 and 6.1] because we use the properties of strict pseudocontractive mappings and maximal monotone mappings (see, e.g., Lemmas 7 and 10);(c)compared with the proof of Theorems 5.2 and 6.1 in [1], the proof of Theorems 27 and 28 shows that lim_𝑛→∞‖𝑦_𝑛,𝑁−𝑃_𝐶(𝐼−𝜆∇𝑓_𝛼𝑛)𝑦_𝑛,𝑁‖=lim_𝑛→∞‖𝑦_𝑛−𝑃_𝐶(𝐼−𝜆∇𝑓_𝛼𝑛)𝑦_𝑛‖=0 via the argument of lim_𝑛→∞‖∇𝑓_𝛼𝑛(𝑦_𝑛,𝑁)−∇𝑓(𝑝)‖=0, for all 𝑝∈Ω (see Lemma 20 and its proof);(d)the problem of finding an element of Fix(𝑇)∩(⋂_𝑖Fix(𝑆_𝑖))∩EP(𝐹,ℎ)∩Γ in Theorems 27 and 28 is more general than the one of finding an element of Fix(𝑇)∩(⋂_𝑖Fix(𝑆_𝑖))∩EP(𝐹,ℎ) in [17, Theorems 3.12 and 3.13] and the one of finding an element of Γ in [1, Theorems 5.2 and 6.1].

4. Applications

For a given nonlinear mapping 𝐴∶𝐶→𝐻, we consider again the variational inequality problem (VIP) of finding 𝑥∈𝐶 such that 𝐴𝑥,𝑦−𝑥≥0,∀𝑦∈𝐶. (104)

Recall that if 𝑢 is a point in 𝐶, then the following relation holds: 𝑢∈VI(𝐶,𝐴)⟺𝑢=𝑃_𝐶(𝐼−𝜆𝐴)𝑢,forsome𝜆>0, (105) from which we have the following relation: 𝑢∈Γ⟺𝑢∈VI𝐶,∇𝑓⟺𝑢=𝑃_𝐶𝐼−𝜆∇𝑓𝑢,forsome𝜆>0. (106)

An operator 𝐴∶𝐶→𝐻 is said to be an 𝛼-inverse strongly monotone operator if there exists a constant 𝛼>0 such that 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥𝛼𝐴𝑥−𝐴𝑦²,∀𝑥,𝑦∈𝐶. (107)

As an example, we recall that the 𝛼-inverse strongly monotone operators are firmly nonexpansive mappings if 𝛼≥1 and that every 𝛼-inverse strongly monotone operator is also a (1/𝛼) Lipschitz continuous (see [19]). We observe that, if 𝐴 is 𝛼-inverse strongly monotone, the mapping 𝑃_𝐶(𝐼−𝜇𝐴) is nonexpansive for all 𝜇∈(0,2𝛼] since they are compositions of nonexpansive mappings (see [19, page 419]).

Let us consider 𝑆^∼₁,…,𝑆^∼_𝑀 a finite number of nonexpansive self-mappings on 𝐶 and 𝐴₁,…,𝐴_𝑁 be a finite number of 𝛼-inverse strongly monotone operators. Let 𝑇∶𝐶→𝐶 be a 𝜁-strict pseudocontraction on 𝐶 with fixed points. Let us consider the following mixed problem of finding 𝑥^∗∈Fix(𝑇)∩EP(𝐹,ℎ)∩Γ such that 𝐼−𝑆^∼₁𝑥^∗,𝑦−𝑥^∗≥0,∀𝑦∈Fix(𝑇)∩EP(𝐹,ℎ)∩Γ,𝐼−𝑆^∼₂𝑥^∗,𝑦−𝑥^∗(𝑇)()⋮≥0,∀𝑦∈Fix∩EP𝐹,ℎ∩Γ,𝐼−𝑆^∼_𝑀𝑥^∗,𝑦−𝑥^∗(𝑇)()𝐴≥0,∀𝑦∈Fix∩EP𝐹,ℎ∩Γ,₁𝑥^∗,𝑦−𝑥^∗𝐴≥0,∀𝑦∈𝐶,₂𝑥^∗,𝑦−𝑥^∗⋮𝐴≥0,∀𝑦∈𝐶,_𝑁𝑥^∗,𝑦−𝑥^∗≥0,∀𝑦∈𝐶. (108) We denote by (SVI) the set of solutions of the above (𝑁+𝑀) system. This problem is equivalent to finding a common fixed point of 𝑇, {𝑃^∼_{Fix(𝑇)∩EP(𝐹,ℎ)∩Γ}𝑆_𝑖}^𝑁_𝑖=1, {𝑃_𝐶(𝐼−𝜇𝐴_𝑖)}^𝑀_𝑖=1. The following results are then consequences of Theorems 27 and 28.

Theorem 31. Let one suppose that Ω=Fix(𝑇)∩(SVI)∩EP(𝐹,ℎ)∩Γ≠∅. Fix 𝜇∈(0,2𝛼], and 𝜆∈(0,2/𝐿). Let {𝛼_𝑛}, {𝛽_𝑛,𝑖}, 𝑖=1,…,(𝑀+𝑁), be sequences in (0,1) such that 0<liminf_𝑛→∞𝛽_𝑛,𝑖≤limsup_𝑛→∞𝛽_𝑛,𝑖<1 for all index 𝑖. Moreover, Let one suppose that (H1)–(H6) hold. Then the sequences {𝑥_𝑛}, {𝑦_𝑛}, and {𝑢_𝑛} explicitly defined by the following scheme: 𝐹𝑢_𝑛,𝑦+ℎ𝑢_𝑛+1,𝑦𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑦⟩≥0,∀𝑦∈𝐶,_𝑛,1=𝛽_𝑛,1𝑃^∼_{Fix(𝑇)∩EP(𝐹,ℎ)∩Γ}𝑆₁𝑢_𝑛+1−𝛽_𝑛,1𝑢_𝑛,𝑦_𝑛,𝑖=𝛽_𝑛,𝑖𝑃^∼_{Fix(𝑇)∩EP(𝐹,ℎ)∩Γ}𝑆_𝑖𝑢_𝑛+1−𝛽_𝑛,𝑖𝑦_{𝑛,𝑖−1}𝑦,𝑖=2,…,𝑀,_{𝑛,𝑀+𝑗}=𝛽_{𝑛,𝑀+𝑗}𝑃_𝐶𝐼−𝜇𝐴_𝑗𝑢_𝑛+1−𝛽_{𝑛,𝑀+𝑗}𝑦_{𝑛,𝑀+𝑗−1}𝑦,𝑗=1,…,𝑁,_𝑛=𝛽_𝑛𝑄𝑦_{𝑛,𝑀+𝑁}+1−𝛽_𝑛×𝑃_𝐶𝑦_{𝑛,𝑀+𝑁}−𝜆∇𝑓_𝛼𝑛𝑦_{𝑛,𝑀+𝑁},𝑥_𝑛+1=𝜎_𝑛𝑦_𝑛+𝛾_𝑛𝑃_𝐶𝑦_𝑛−𝜆∇𝑓_𝛼𝑛𝑦_𝑛+𝛿_𝑛𝑇𝑃_𝐶𝑦_𝑛−𝜆∇𝑓_𝛼𝑛𝑦_𝑛,∀𝑛≥0, (109) all converge strongly to the unique solution 𝑥^∗∈Ω of the following variational inequality: ⟨𝑄𝑥^∗−𝑥^∗,𝑧−𝑥^∗⟩≤0,∀𝑧∈Ω. (110)

Theorem 32. Let one suppose that Ω≠∅. Fix𝜇∈(0,2𝛼] and 𝜆∈(0,2/𝐿). Let {𝛽_𝑛}, {𝛽_𝑛,𝑖}, 𝑖=1,…,(𝑀+𝑁), be sequences in (0,1) and 𝛽_𝑛,𝑖→𝛽_𝑖 for all 𝑖 as 𝑛→∞. Suppose that there exists 𝑘∈{1,…,𝑀+𝑁} such that 𝛽_𝑛,𝑘→0 as 𝑛→∞. Let 𝑘₀∈{1,…,𝑀+𝑁} be the largest index for which 𝛽_𝑛,𝑘0→0. Moreover, let one suppose that (H1), (H7), and (H8) hold, and(i)(𝛼_𝑛+𝛽_𝑛)/𝛽_𝑛,𝑘0→0 as 𝑛→∞;(ii)if 𝑖≤𝑘₀ and 𝛽_𝑛,𝑖→0, then 𝛽_𝑛,𝑘0/𝛽_𝑛,𝑖→0 as 𝑛→∞;(iii)if 𝛽_𝑛,𝑖→𝛽_𝑖≠0, then 𝛽_𝑖 lies in (0,1).
Then, the sequences {𝑥_𝑛}, {𝑦_𝑛}, and {𝑢_𝑛} explicitly defined by scheme (109) all converge strongly to the unique solution 𝑥^∗∈Ω of the following variational inequality: ⟨𝑄𝑥^∗−𝑥^∗,𝑧−𝑥^∗⟩≤0,∀𝑧∈Ω. (111)

Remark 33. If we choose ∇𝑓=𝐴₁=⋯=𝐴_𝑁=0 in system (108), we obtain a system of hierarchical fixed point problems introduced by Moudafi and Maingé [20, 21].

Further, utilizing Theorems 27 and 28, we again give the following strong convergence theorems for finding a common element of the solution set Γ of MP (3), the solution set EP(𝐹,ℎ) of EP (10), and the common fixed point set (⋂_𝑖Fix(𝑆_𝑖)) of a finite family of nonexpansive mappings 𝑆_𝑖∶𝐶→𝐶,𝑖=1,…,𝑁.

Theorem 34. Let one suppose that Ω=(⋂_𝑖Fix(𝑆_𝑖))∩EP(𝐹,ℎ)∩Γ≠∅. Let {𝛽_𝑛}, {𝛽_𝑛,𝑖}, 𝑖=1,…,𝑁, be sequences in (0,1) such that 0<liminf_𝑛→∞𝛽_𝑛,𝑖≤limsup_𝑛→∞𝛽_𝑛,𝑖<1 for all index 𝑖. Moreover, Let one suppose that there hold (H1)–(H6) with 𝛾_𝑛=0, for all 𝑛≥0. Then, the sequences {𝑥_𝑛}, {𝑦_𝑛}, and {𝑢_𝑛} generated explicitly by 𝐹𝑢_𝑛,𝑦+ℎ𝑢_𝑛+1,𝑦𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑦⟩≥0,∀𝑦∈𝐶,_𝑛,1=𝛽_𝑛,1𝑆₁𝑢_𝑛+1−𝛽_𝑛,1𝑢_𝑛,𝑦_𝑛,𝑖=𝛽_𝑛,𝑖𝑆_𝑖𝑢_𝑛+1−𝛽_𝑛,𝑖𝑦_{𝑛,𝑖−1}𝑦,𝑖=2,…,𝑁,_𝑛=𝛽_𝑛𝑄𝑦_𝑛,𝑁+1−𝛽_𝑛𝑃_𝐶𝑦_𝑛,𝑁−𝜆∇𝑓_𝛼𝑛𝑦_𝑛,𝑁,𝑥_𝑛+1=𝜎_𝑛𝑦_𝑛+1−𝜎_𝑛𝑃_𝐶𝑦_𝑛−𝜆∇𝑓_𝛼𝑛𝑦_𝑛,∀𝑛≥0, (112) all converge strongly to the unique solution 𝑥^∗∈Ω of the following variational inequality: ⟨𝑄𝑥^∗−𝑥^∗,𝑧−𝑥^∗⟩≤0,∀𝑧∈Ω. (113)

Proof. In Theorems 27, put 𝑇=𝐼 the identity mapping and 𝛾_𝑛=0, for all 𝑛≥0. Then, 𝑇 is a 𝜁-strictly pseudocontractive mapping with 𝜁=0. Hence, we deduce that Ω=Fix(𝑇)∩(⋂_𝑖Fix(𝑆_𝑖))∩EP(𝐹,ℎ)∩Γ=(⋂_𝑖Fix(𝑆_𝑖))∩EP(𝐹,ℎ)∩Γ≠∅, (𝛾_𝑛+𝛿_𝑛)𝜁≤𝛾_𝑛, for all 𝑛≥0, and 𝑥_𝑛+1=𝜎_𝑛𝑦_𝑛+𝛾_𝑛𝑃_𝐶𝑦_𝑛−𝜆∇𝑓_𝛼𝑛𝑦_𝑛+𝛿_𝑛𝑇𝑃_𝐶𝑦_𝑛−𝜆∇𝑓_𝛼𝑛𝑦_𝑛=𝜎_𝑛𝑦_𝑛+𝛿_𝑛𝑃_𝐶𝑦_𝑛−𝜆∇𝑓_𝛼𝑛𝑦_𝑛=𝜎_𝑛𝑦_𝑛+1−𝜎_𝑛𝑃_𝐶𝑦_𝑛−𝜆∇𝑓_𝛼𝑛𝑦_𝑛. (114) Thus, the conditions in Theorem 27 are all satisfied. and from which we obtain the desired result.

Theorem 35. Let one suppose that Ω=(⋂_𝑖Fix(𝑆_𝑖))∩EP(𝐹,ℎ)∩Γ≠∅. Let {𝛽_𝑛}, {𝛽_𝑛,𝑖}, 𝑖=1,…,𝑁, be sequences in (0,1) such that 𝛽_𝑛,𝑖→𝛽_𝑖 for all 𝑖 as 𝑛→∞. Suppose that there exists 𝑘∈{1,…,𝑁} for which 𝛽_𝑛,𝑘→0 as 𝑛→∞. Let 𝑘₀∈{1,…,𝑁} be the largest index for which 𝛽_𝑛,𝑘0→0. Moreover, let one suppose that there hold (H1), (H7), and (H8) with 𝛾_𝑛=0, for all 𝑛≥0, and(i)(𝛼_𝑛+𝛽_𝑛)/𝛽_𝑛,𝑘0→0 as 𝑛→∞;(ii)if 𝑖≤𝑘₀ and 𝛽_𝑛,𝑖→0, then 𝛽_𝑛,𝑘0/𝛽_𝑛,𝑖→0 as 𝑛→∞;(iii)if 𝛽_𝑛,𝑖→𝛽_𝑖≠0, then 𝛽_𝑖 lies in (0,1).
Then the sequences {𝑥_𝑛}, {𝑦_𝑛}, and {𝑢_𝑛} generated explicitly by (112) all converge strongly to the unique solution 𝑥^∗∈Ω of the following variational inequality: ⟨𝑄𝑥^∗−𝑥^∗,𝑧−𝑥^∗⟩≤0,∀𝑧∈Ω. (115)

Acknowledgments

This research was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), and Leading Academic Discipline Project of Shanghai Normal University (DZL707). This research was partially supported by the Grant from the NSC 101-2115-M-037-001.