Relaxed Extragradient Methods with Regularization for General System of Variational Inequalities with Constraints of Split Feasibility and Fixed Point Problems

Abstract

We suggest and analyze relaxed extragradient iterative algorithms with regularization for finding a common element of the solution set of a general system of variational inequalities, the solution set of a split feasibility problem, and the fixed point set of a strictly pseudocontractive mapping defined on a real Hilbert space. Here the relaxed extragradient methods with regularization are based on the well-known successive approximation method, extragradient method, viscosity approximation method, regularization method, and so on. Strong convergence of the proposed algorithms under some mild conditions is established. Our results represent the supplementation, improvement, extension, and development of the corresponding results in the very recent literature.

1. Introduction

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. The (nearest point or metric) projection from ℋ onto 𝐾is denoted by 𝑃_𝐾. 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.

Let 𝐶 and 𝑄 be nonempty closed convex subsets of infinite-dimensional real Hilbert spaces ℋ₁ and ℋ₂, respectively. The split feasibility problem (SFP) is to find a point 𝑥^∗ with the property: 𝑥^∗∈𝐶,𝐴𝑥^∗∈𝑄, (1) where 𝐴∈𝐵(ℋ₁,ℋ₂) and 𝐵(ℋ₁,ℋ₂) denotes the family of all bounded linear operators from ℋ₁ to ℋ₂.

In 1994, the SFP was first introduced by Censor and Elfving [1], in finite-dimensional Hilbert spaces, for modeling inverse problems which arise from phase retrievals and in medical image reconstruction. A number of image reconstruction problems can be formulated as the SFP; see, for example, [2] and the references therein. Recently, it was found that the SFP can also be applied to study intensity-modulated radiation therapy; see, for example, [3–5] and the references therein. In the recent past, a wide variety of iterative methods have been used in signal processing and image reconstruction and for solving the SFP; see, for example, [2–12] and the references therein. A special case of the SFP is the following convex constrained linear inverse problem [13] of finding an element 𝑥 such that 𝑥∈𝐶,𝐴𝑥=𝑏. (2) It has been extensively investigated in the literature using the projected Landweber iterative method [14]. Comparatively, the SFP has received much less attention so far, due to the complexity resulting from the set 𝑄. Therefore, whether various versions of the projected Landweber iterative method [14] can be extended to solve the SFP remains an interesting open topics. For example, it is yet not clear whether the dual approach to (1.2) of [15] can be extended to the SFP. The original algorithm given in [1] involves the computation of the inverse 𝐴⁻¹ (assuming the existence of the inverse of 𝐴) and thus has not become popular. A seemingly more popular algorithm that solves the SFP is the 𝐶𝑄 algorithm of Byrne [2, 7] which is found to be a gradient-projection method (GPM) in convex minimization. It is also a special case of the proximal forward-backward splitting method [16]. The 𝐶𝑄 algorithm only involves the computation of the projections 𝑃_𝐶 and 𝑃_𝑄 onto the sets 𝐶 and 𝑄, respectively, and is therefore implementable in the case where 𝑃_𝐶 and 𝑃_𝑄 have closed-form expressions, for example, 𝐶 and 𝑄 are closed balls or half-spaces. However, it remains a challenge how to implement the 𝐶𝑄 algorithm in the case where the projections 𝑃_𝐶 and/or 𝑃_𝑄 fail to have closed-form expressions, though theoretically we can prove the (weak) convergence of the algorithm.

Very recently, Xu [6] gave a continuation of the study on the 𝐶𝑄 algorithm and its convergence. He applied Mann’s algorithm to the SFP and purposed an averaged 𝐶𝑄 algorithm which was proved to be weakly convergent to a solution of the SFP. He also established the strong convergence result, which shows that the minimum-norm solution can be obtained.

Furthermore, Korpelevič [17] introduced the so-called extragradient method for finding a solution of a saddle point problem. He proved that the sequences generated by the proposed iterative algorithm converge to a solution of the saddle point problem.

Throughout this paper, assume that the SFP is consistent, that is, the solution set Γ of the SFP is nonempty. Let 𝑓∶ℋ₁→𝑅 be a continuous differentiable function. The minimization problem min_𝑥∈𝐶𝑓(𝑥)1∶=2𝐴𝑥−𝑃_𝑄𝐴𝑥² (3) is ill posed. Therefore, Xu [6] considered the following Tikhonov regularization problem: min_𝑥∈𝐶𝑓_𝛼(𝑥)1∶=2𝐴𝑥−𝑃_𝑄𝐴𝑥²+12𝛼‖𝑥‖², (4) where 𝛼>0 is the regularization parameter. The regularized minimization (4) has a unique solution which is denoted by 𝑥_𝛼. The following results are easy to prove.

Proposition 1 (see [18, Proposition  3.1]). Given 𝑥^∗∈ℋ₁, the following statements are equivalent:(i)𝑥^∗ solves the SFP; (ii)𝑥^∗ solves the fixed point equation 𝑃_𝐶𝐼−𝜆∇𝑓𝑥^∗=𝑥^∗, (5) where 𝜆>0, ∇𝑓=𝐴^∗(𝐼−𝑃_𝑄)𝐴 and 𝐴^∗ is the adjoint of 𝐴;(iii)𝑥^∗ solves the variational inequality problem (VIP) of finding 𝑥^∗∈𝐶 such that ∇𝑓𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐶. (6)

It is clear from Proposition 1 that Γ=Fix𝑃_𝐶𝐼−𝜆∇𝑓=VI𝐶,∇𝑓, (7) for all 𝜆>0, where Fix(𝑃_𝐶(𝐼−𝜆∇𝑓)) and VI(𝐶,∇𝑓) denote the set of fixed points of 𝑃_𝐶(𝐼−𝜆∇𝑓) and the solution set of VIP (6), respectively.

Proposition 2 (see [18]). There hold the following statements:(i) the gradient ∇𝑓_𝛼=∇𝑓+𝛼𝐼=𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼𝐼 (8) is (𝛼+‖𝐴‖²)-Lipschitz continuous and 𝛼-strongly monotone; (ii) the mapping 𝑃_𝐶(𝐼−𝜆∇𝑓_𝛼) is a contraction with coefficient 1−𝜆2𝛼−𝜆‖𝐴‖²+𝛼²≤√11−𝛼𝜆≤1−2𝛼𝜆, (9) where 0<𝜆≤𝛼/(‖𝐴‖²+𝛼)²;(iii) if the SFP is consistent, then the strong lim_𝛼→0𝑥_𝛼 exists and is the minimum-norm solution of the SFP.

Very recently, by combining the regularization method and extragradient method due to Nadezhkina and Takahashi [19], Ceng et al. [18] proposed an extragradient algorithm with regularization and proved that the sequences generated by the proposed algorithm converge weakly to an element of Fix(𝑆)∩Γ, where 𝑆∶𝐶→𝐶 is a nonexpansive mapping.

Theorem 3 (see [18, Theorem  3.1]). Let 𝑆∶𝐶→𝐶 be a nonexpansive mapping such that Fix(𝑆)∩Γ≠∅. Let {𝑥_𝑛} and {𝑦_𝑛} the sequences in 𝐶 generated by the following extragradient algorithm: 𝑥₀𝑦=𝑥∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑆𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛,∀𝑛≥0, (10) where ∑^∞_𝑛=0𝛼_𝑛<∞,{𝜆_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1/‖𝐴‖²) and {𝛽_𝑛}⊂[𝑐,𝑑] for some 𝑐,𝑑∈(0,1). Then, both the sequences {𝑥_𝑛} and {𝑦_𝑛} converge weakly to an element 𝑥∈Fix(𝑆)∩Γ.

On the other hand, assume that 𝐶 is a nonempty closed convex subset of ℋ and 𝐴∶𝐶→ℋ is a mapping. The classical variational inequality problem (VIP) is to find 𝑥^∗∈𝐶 such that 𝐴𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐶. (11) It is now well known that the variational inequalities are equivalent to the fixed point problems, the origin of which can be traced back to Lions and Stampacchia [20]. This alternative formulation has been used to suggest and analyze projection iterative method for solving variational inequalities under the conditions that the involved operator must be strongly monotone and Lipschitz continuous. Related to the variational inequalities, we have the problem of finding the fixed points of nonexpansive mappings or strict pseudocontraction mappings, which is the current interest in functional analysis. Several people considered a unified approach to solve variational inequality problems and fixed point problems; see, for example, [21–28] and the references therein.

A mapping 𝐴∶𝐶→ℋ is said to be an 𝛼-inverse strongly monotone if there exists 𝛼>0 such that 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥𝛼𝐴𝑥−𝐴𝑦²,∀𝑥,𝑦∈𝐶. (12)

A mapping 𝑆∶𝐶→𝐶 is said to be 𝑘-strictly pseudocontractive if there exists 0≤𝑘<1 such that 𝑆𝑥−𝑆𝑦²≤𝑥−𝑦²+𝑘(𝐼−𝑆)𝑥−(𝐼−𝑆)𝑦²,∀𝑥,𝑦∈𝐶. (13) In this case, we also say that 𝑆 is a 𝑘-strict pseudo-contraction. In particular, whenever 𝑘=0, 𝑆 becomes a nonexpansive mapping from 𝐶 into itself. It is clear that every inverse strongly monotone mapping is a monotone and Lipschitz continuous mapping. We denote by VI(𝐶,𝐴) and Fix(𝑆) the solution set of problem (11) and the set of all fixed points of 𝑆, respectively.

For finding an element of Fix(𝑆)∩VI(𝐶,𝐴) under the assumption that a set 𝐶⊂ℋ is nonempty, closed, and convex, a mapping 𝑆∶𝐶→𝐶 is nonexpansive, and a mapping 𝐴∶𝐶→ℋ is 𝛼-inverse strongly monotone, Takahashi and Toyoda [29] introduced an iterative scheme and studied the weak convergence of the sequence generated by the proposed scheme to a point of Fix(𝑆)∩VI(𝐶,𝐴). Recently, Iiduka and Takahashi [30] presented another iterative scheme for finding an element of Fix(𝑆)∩VI(𝐶,𝐴) and showed that the sequence generated by the scheme converges strongly to 𝑃_{Fix(𝑆)∩VI(𝐶,𝐴)}𝑢, where 𝑢 is the initially chosen point in the iterative scheme and 𝑃_𝐾 denotes the metric projection of ℋ onto 𝐾.

Based on Korpelevič’s extragradient method [17], Nadezhkina and Takahashi [19] introduced an iterative process for finding an element of Fix(𝑆)∩VI(𝐶,𝐴) and proved the weak convergence of the sequence to a point of Fix(𝑆)∩VI(𝐶,𝐴). Zeng and Yao [27] presented an iterative scheme for finding an element of Fix(𝑆)∩VI(𝐶,𝐴) and proved that two sequences generated by the method converges strongly to an element of Fix(𝑆)∩VI(𝐶,𝐴). Recently, Bnouhachem et al. [31] suggested and analyzed an iterative scheme for finding a common element of the fixed point set Fix(𝑆) of a nonexpansive mapping 𝑆 and the solution set VI(𝐶,𝐴) of the variational inequality (11) for an inverse strongly monotone mapping 𝐴∶𝐶→ℋ.

Furthermore, as a much more general generalization of the classical variational inequality problem (11), Ceng et al. [23] introduced and considered the following problem of finding (𝑥^∗,𝑦^∗)∈𝐶×𝐶 such that 𝜇₁𝐵₁𝑦^∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗𝜇≥0,∀𝑥∈𝐶,₂𝐵₂𝑥^∗+𝑦^∗−𝑥^∗,𝑥−𝑦^∗≥0,∀𝑥∈𝐶, (14) which is called a general system of variational inequalities (GSVI), which 𝜇₁>0 and 𝜇₂>0 are two constants. The set of solutions of problem (14) is denoted by GSVI(𝐶,𝐵₁,𝐵₂). In particular, if 𝐵₁=𝐵₂, then problem (14) reduces to the new system of variational inequalities, introduced and studied by Verma [32]. Recently, Ceng et al. [23] transformed problem (14) into a fixed point problem in the following way.

Lemma 4 (see [23]). For given 𝑥,𝑦∈𝐶,(𝑥,𝑦) is a solution of problem (14) if and only if 𝑥 is a fixed point of the mapping 𝐺∶𝐶→𝐶 defined by 𝐺(𝑥)=𝑃_𝐶𝑃_𝐶𝑥−𝜇₂𝐵₂𝑥−𝜇₁𝐵₁𝑃_𝐶𝑥−𝜇₂𝐵₂𝑥,∀𝑥∈𝐶, (15) where 𝑦=𝑃_𝐶(𝑥−𝜇₂𝐵₂𝑥).
In particular, if the mapping 𝐵_𝑖∶𝐶→ℋ is 𝛽_𝑖-inverse strongly monotone for 𝑖=1,2, then the mapping 𝐺 is nonexpansive provided 𝜇_𝑖∈(0,2𝛽_𝑖) for 𝑖=1,2.

Utilizing Lemma 4, they introduced and studied a relaxed extragradient method for solving problem (14). Throughout this paper, the set of fixed points of the mapping 𝐺 is denoted by Ξ. Based on the relaxed extragradient method and viscosity approximation method, Yao et al. [26] proposed and analyzed an iterative algorithm for finding a common solution of the GSVI (14) and the fixed point problem of a strictly pseudo-contractive mapping 𝑆∶𝐶→𝐶. Subsequently, Ceng et al. [33] further presented and analyzed an iterative scheme for finding a common element of the solution set of the VIP (11), the solution set of the GSVI (14), and the fixed point set of a strictly pseudo-contractive mapping 𝑆∶𝐶→𝐶.

Theorem 5 (see [33, Theorem  3.1]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space ℋ. Let 𝐴∶𝐶→ℋ be 𝛼-inverse strongly monotone and 𝐵_𝑖∶𝐶→ℋ be 𝛽_𝑖-inverse strongly monotone for 𝑖=1,2. Let 𝑆∶𝐶→𝐶 be a 𝑘-strictly pseudo-contractive mapping such that Fix(𝑆)∩Ξ∩V𝐼(𝐶,𝐴)≠∅. Let 𝑄∶𝐶→𝐶 be a 𝜌-contraction with 𝜌∈[0,1/2). For given 𝑥₀∈𝐶 arbitrarily, let the sequences {𝑥_𝑛},{𝑢_𝑛},{𝑢_𝑛} be generated by the relaxed extragradient iterative scheme: 𝑢_𝑛=𝑃_𝐶𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛,𝑢_𝑛=𝑃_𝐶𝑢_𝑛−𝜆_𝑛𝐴𝑢_𝑛,𝑦_𝑛=𝛼_𝑛𝑄𝑥_𝑛+1−𝛼_𝑛𝑃_𝐶𝑢_𝑛−𝜆_𝑛𝐴𝑢_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑦_𝑛+𝛿_𝑛𝑆𝑦_𝑛,∀𝑛≥0, (16) where 𝜇_𝑖∈(0,2𝛽_𝑖) for 𝑖=1,2, and the following conditions hold for five sequences {𝜆_𝑛}⊂(0,𝛼] and {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛},{𝛿_𝑛}⊂[0,1]:(i)𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 and (𝛾_𝑛+𝛿_𝑛)𝑘≤𝛾_𝑛 for all 𝑛≥0;(ii)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞;(iii)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and liminf_𝑛→∞𝛿_𝑛>0;(iv)lim_𝑛→∞(𝛾_𝑛+1/(1−𝛽_𝑛+1)−𝛾_𝑛/(1−𝛽_𝑛))=0;(v)0<liminf_𝑛→∞𝜆_𝑛≤limsup_𝑛→∞𝜆_𝑛<𝛼 and lim_𝑛→∞|𝜆_𝑛+1−𝜆_𝑛|=0.
Then the sequences {𝑥_𝑛},{𝑢_𝑛},{𝑢_𝑛} converge strongly to the same point 𝑥=𝑃_{Fix(𝑆)∩Ξ∩V𝐼(𝐶,𝐴)}𝑄𝑥 if and only if lim_𝑛→∞‖𝑢_𝑛+1−𝑢_𝑛‖=0. Furthermore, (𝑥,𝑦) is a solution of the GSVI (14), where 𝑦=𝑃_𝐶(𝑥−𝜇₂𝐵₂𝑥).

Motivated and inspired by the research going on this area, we propose and analyze the following relaxed extragradient iterative algorithms with regularization for finding a common element of the solution set of the GSVI (14), the solution set of the SFP (1), and the fixed point set of a strictly pseudo-contractive mapping 𝑆∶𝐶→𝐶.

Algorithm 6. Let 𝜇_𝑖∈(0,2𝛽_𝑖) for 𝑖=1,2,{𝛼_𝑛}⊂(0,∞),{𝜆_𝑛}⊂(0,1/‖𝐴‖²) and {𝜎_𝑛},{𝛽_𝑛},{𝛾_𝑛},{𝛿_𝑛}⊂[0,1] such that 𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 for all 𝑛≥0. For given 𝑥₀∈𝐶 arbitrarily, let {𝑥_𝑛},{𝑢_𝑛},{𝑢_𝑛} be the sequences generated by the following relaxed extragradient iterative scheme with regularization: 𝑢_𝑛=𝑃_𝐶𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛,𝑢_𝑛=𝑃_𝐶𝑢_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛,𝑦_𝑛=𝜎_𝑛𝑄𝑥_𝑛+1−𝜎_𝑛𝑃_𝐶𝑢_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑦_𝑛+𝛿_𝑛𝑆𝑦_𝑛,∀𝑛≥0. (17)
Under mild assumptions, it is proven that the sequences {𝑥_𝑛},{𝑢_𝑛},{𝑢_𝑛} converge strongly to the same point 𝑥=𝑃_{Fix(𝑆)∩Ξ∩Γ}𝑄𝑥 if and only if lim_𝑛→∞‖𝑢_𝑛+1−𝑢_𝑛‖=0. Furthermore, (𝑥,𝑦) is a solution of the GSVI (14), where 𝑦=𝑃_𝐶(𝑥−𝜇₂𝐵₂𝑥).

Algorithm 7. Let 𝜇_𝑖∈(0,2𝛽_𝑖) for 𝑖=1,2,{𝛼_𝑛}⊂(0,∞),{𝜆_𝑛}⊂(0,1/‖𝐴‖²) and {𝜎_𝑛},{𝜏_𝑛},{𝛽_𝑛},{𝛾_𝑛},{𝛿_𝑛}⊂[0,1] such that 𝜎_𝑛+𝜏_𝑛≤1 and 𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 for all 𝑛≥0. For given 𝑥₀∈𝐶 arbitrarily, let {𝑥_𝑛},{𝑦_𝑛},{𝑧_𝑛} be the sequences generated by the following relaxed extragradient iterative scheme with regularization: 𝑧_𝑛=𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑥_𝑛,𝑦_𝑛=𝜎_𝑛𝑄𝑥_𝑛+𝜏_𝑛𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑧_𝑛+1−𝜎_𝑛−𝜏_𝑛𝑃_𝐶𝑃_𝐶𝑧_𝑛−𝜇₂𝐵₂𝑧_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑧_𝑛−𝜇₂𝐵₂𝑧_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑦_𝑛+𝛿_𝑛𝑆𝑦_𝑛,∀𝑛≥0, (18)
Also, under appropriate conditions, it is shown that the sequences {𝑥_𝑛},{𝑦_𝑛},{𝑧_𝑛} converge strongly to the same point 𝑥=𝑃_{Fix(𝑆)∩Ξ∩Γ}𝑄𝑥 if and only if lim_𝑛→∞‖𝑧_𝑛+1−𝑧_𝑛‖=0. Furthermore, (𝑥,𝑦) is a solution of the GSVI (14), where 𝑦=𝑃_𝐶(𝑥−𝜇₂𝐵₂𝑥).
Note that both [6, Theorem  5.7] and [18, Theorem  3.1] are weak convergence results for solving the SFP (1). Beyond question our strong convergence results are very interesting and quite valuable. Because our relaxed extragradient iterative schemes (17) and (18) with regularization involve a contractive self-mapping 𝑄, a 𝑘-strictly pseudo-contractive self-mapping 𝑆 and several parameter sequences, they are more flexible and more subtle than the corresponding ones in [6, Theorem  5.7] and [18, Theorem  3.1], respectively. Furthermore, the relaxed extragradient iterative scheme (16) is extended to develop our relaxed extragradient iterative schemes (17) and (18) with regularization. All in all, our results represent the modification, supplementation, extension, and improvement of [6, Theorem  5.7], [18, Theorem  3.1], and [33, Theorem  3.1].

2. Preliminaries

Let 𝐾 be a nonempty, closed, and convex subset of a real Hilbert space ℋ. Now we present some known results and definitions which will be used in the sequel.

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). (19)

The following properties of projections are useful and pertinent to our purpose.

Proposition 8 (see [34]). For given 𝑥∈ℋ and 𝑧∈𝐾:(i)𝑧=𝑃_𝐾𝑥⇔⟨𝑥−𝑧,𝑦−𝑧⟩≤0,∀𝑦∈𝐾;(ii)𝑧=𝑃_𝐾𝑥⇔‖𝑥−𝑧‖²≤‖𝑥−𝑦‖²−‖𝑦−𝑧‖²,∀𝑦∈𝐾;(iii)⟨𝑃_𝐾𝑥−𝑃_𝐾𝑦,𝑥−𝑦⟩≥‖𝑃_𝐾𝑥−𝑃_𝐾𝑦‖²,∀𝑦∈ℋ, which hence implies that 𝑃_𝐾 is nonexpansive and monotone.

Definition 9. A mapping 𝑇∶ℋ→ℋ is said to be(a)nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈ℋ; (20)(b)firmly nonexpansive if 2𝑇−𝐼 is nonexpansive, or equivalently, 𝑥−𝑦,𝑇𝑥−𝑇𝑦≥𝑇𝑥−𝑇𝑦²,∀𝑥,𝑦∈ℋ; (21)alternatively, 𝑇 is firmly nonexpansive if and only if 𝑇 can be expressed as 1𝑇=2(𝐼+𝑆), (22) where 𝑆∶ℋ→ℋ is nonexpansive; projections are firmly nonexpansive.

Definition 10. 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). (23)(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). (24)(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). (25)
It can be easily seen that if 𝑆 is nonexpansive, then 𝐼−𝑆 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, for instance, in traffic assignment problems; see, for example, [35, 36].

Definition 11. 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, (26) where 𝛼∈(0,1) and 𝑆∶ℋ→ℋis 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 12 (see [7]). 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 13 (see [7, 37]). 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∘𝑇₂∘⋯∘𝑇_𝑁. In particular, if 𝑇₁ is 𝛼₁-averaged and 𝑇₂ is 𝛼₂-averaged, where 𝛼₁,𝛼₂∈(0,1), then the composite 𝑇₁∘𝑇₂ 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, then Fix𝑇^𝑁_𝑖=1𝑖=Fix𝑇₁⋯𝑇_𝑁. (27)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, 𝑆∶𝐶→𝐶 is 𝑘-strictly pseudo-contractive if and only if there holds the following inequality: ≤𝑆𝑥−𝑆𝑦,𝑥−𝑦𝑥−𝑦²−1−𝑘2(𝐼−𝑆)𝑥−(𝐼−𝑆)𝑦²,∀𝑥,𝑦∈𝐶. (28) This immediately implies that if 𝑆 is a 𝑘-strictly pseudo-contractive mapping, then 𝐼−𝑆 is (1−𝑘)/2-inverse strongly monotone; for further detail, we refer to [38] and the references therein. It is well known that the class of strict pseudo-contractions strictly includes the class of nonexpansive mappings.

In order to prove the main result of this paper, the following lemmas will be required.

Lemma 14 (see [39]). Let {𝑥_𝑛} and {𝑦_𝑛} be bounded sequences in a Banach space 𝑋 and let {𝛽_𝑛} be a sequence in [0,1] with 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1. Suppose 𝑥_𝑛+1=(1−𝛽_𝑛)𝑦_𝑛+𝛽_𝑛𝑥_𝑛 for all integers 𝑛≥0 and limsup_𝑛→∞(‖𝑦_𝑛+1−𝑦_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0. Then, lim_𝑛→∞‖𝑦_𝑛−𝑥_𝑛‖=0.

Lemma 15 (see [38, Proposition  2.1]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space ℋ and 𝑆∶𝐶→𝐶 a mapping.(i)If 𝑆 is a 𝑘-strict pseudo-contractive mapping, then 𝑆 satisfies the Lipschitz condition 𝑆𝑥−𝑆𝑦≤1+𝑘1−𝑘𝑥−𝑦,∀𝑥,𝑦∈𝐶. (29)(ii)If 𝑆 is a 𝑘-strict pseudo-contractive mapping, then the mapping 𝐼−𝑆 is semiclosed at 0, that is, if {𝑥_𝑛} is a sequence in 𝐶 such that 𝑥_𝑛→𝑥 weakly and (𝐼−𝑆)𝑥_𝑛→0 strongly, then (𝐼−𝑆)𝑥=0.(iii)If 𝑆 is 𝑘-(quasi-)strict pseudo-contraction, then the fixed point set Fix(𝑆) of 𝑆 is closed and convex so that the projection 𝑃_Fix(𝑆) is well defined.

The following lemma plays a key role in proving strong convergence of the sequences generated by our algorithms.

Lemma 16 (see [34]). Let {𝑎_𝑛} be a sequence of nonnegative real numbers satisfying the property 𝑎_𝑛+1≤1−𝑠_𝑛𝑎_𝑛+𝑠_𝑛𝑡_𝑛+𝑟_𝑛,∀𝑛≥0, (30) where {𝑠_𝑛}⊂(0,1] and {𝑡_𝑛} are such that(i)∑^∞_𝑛=0𝑠_𝑛=∞;(ii)either limsup_𝑛→∞𝑡_𝑛≤0 or ∑^∞_𝑛=0|𝑠_𝑛𝑡_𝑛|<∞;(iii)∑^∞_𝑛=0𝑟_𝑛<∞ where 𝑟_𝑛≥0,∀𝑛≥0. Then, lim_𝑛→∞𝑎_𝑛=0.

Lemma 17 (see [26]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space ℋ. Let 𝑆∶𝐶→𝐶 be a 𝑘-strictly pseudo-contractive mapping. Let 𝛾 and 𝛿 be two nonnegative real numbers such that (𝛾+𝛿)𝑘≤𝛾. Then 𝛾𝑥−𝑦+𝛿𝑆𝑥−𝑆𝑦≤𝛾+𝛿𝑥−𝑦,∀𝑥,𝑦∈𝐶. (31)

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

Lemma 18. In a real Hilbert space ℋ, there holds the inequality 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝑥+𝑦,∀𝑥,𝑦∈ℋ. (32)

Let 𝐾 be a nonempty closed convex subset of a real Hilbert space ℋ and let 𝐹∶𝐾→ℋ be a monotone mapping. The variational inequality problem (VIP) is to find 𝑥∈𝐾 such that 𝐹𝑥,𝑦−𝑥≥0,∀𝑦∈𝐾. (33) The solution set of the VIP is denoted by VI(𝐾,𝐹). It is well known that 𝑥∈VI(𝐾,𝐹)⟺𝑥=𝑃_𝐾(𝑥−𝜆𝐹𝑥),∀𝜆>0. (34)

A set-valued mapping 𝑇∶ℋ→2^ℋ is called monotone if for all 𝑥,𝑦∈ℋ,𝑓∈𝑇𝑥 and 𝑔∈𝑇𝑦 imply that ⟨𝑥−𝑦,𝑓−𝑔⟩≥0. A monotone set-valued mapping 𝑇∶ℋ→2^ℋ is called maximal if its graph Gph(𝑇) is not properly contained in the graph of any other monotone set-valued mapping. It is known that a monotone set-valued mapping 𝑇∶ℋ→2^ℋ is maximal if and only if for (𝑥,𝑓)∈ℋ×ℋ,⟨𝑥−𝑦,𝑓−𝑔⟩≥0 for every (𝑦,𝑔)∈Gph(𝑇) implies that 𝑓∈𝑇𝑥. Let 𝐹∶𝐾→ℋ be a monotone and Lipschitz continuous mapping and let 𝑁_𝐾𝑣 be the normal cone to 𝐾 at 𝑣∈𝐾, that is, 𝑁_𝐾𝑣={𝑤∈ℋ∶⟨𝑣−𝑢,𝑤⟩≥0,∀𝑢∈𝐾}. (35) Define 𝑇𝑣=𝐹𝑣+𝑁_𝐾𝑣,if𝑣∈𝐾,∅,if𝑣∉𝐾. (36) It is known that in this case the mapping 𝑇 is maximal monotone, and 0∈𝑇𝑣 if and only if 𝑣∈VI(𝐾,𝐹); see further details, one refers to [40] and the references therein.

3. Main Results

In this section, we first prove the strong convergence of the sequences generated by the relaxed extragradient iterative algorithm (17) with regularization.

Theorem 19. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space ℋ₁. Let 𝐴∈𝐵(ℋ₁,ℋ₂), and let 𝐵_𝑖∶𝐶→ℋ₁ be 𝛽_𝑖-inverse strongly monotone for 𝑖=1,2. Let 𝑆∶𝐶→𝐶 be a 𝑘-strictly pseudo-contractive mapping such that Fix(𝑆)∩Ξ∩Γ≠∅. Let 𝑄∶𝐶→𝐶 be a 𝜌-contraction with 𝜌∈[0,1/2). For given 𝑥₀∈𝐶 arbitrarily, let the sequences {𝑥_𝑛},{𝑢_𝑛},{𝑢_𝑛} be generated by the relaxed extragradient iterative algorithm (17) with regularization, where 𝜇_𝑖∈(0,2𝛽_𝑖) for 𝑖=1,2, {𝛼_𝑛}⊂(0,∞),{𝜆_𝑛}⊂(0,1/‖𝐴‖²) and {𝜎_𝑛},{𝛽_𝑛},{𝛾_𝑛},{𝛿_𝑛}⊂[0,1] such that(i)∑^∞_𝑛=0𝛼_𝑛<∞;(ii)𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 and (𝛾_𝑛+𝛿_𝑛)𝑘≤𝛾_𝑛 for all 𝑛≥0;(iii)lim_𝑛→∞𝜎_𝑛=0 and ∑^∞_𝑛=0𝜎_𝑛=∞;(iv)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and liminf_𝑛→∞𝛿_𝑛>0;(v)lim_𝑛→∞(𝛾_𝑛+1/(1−𝛽_𝑛+1)−𝛾_𝑛/(1−𝛽_𝑛))=0;(vi)0<liminf_𝑛→∞𝜆_𝑛≤limsup_𝑛→∞𝜆_𝑛<1/‖𝐴‖² and lim_𝑛→∞|𝜆_𝑛+1−𝜆_𝑛|=0.
Then the sequences {𝑥_𝑛},{𝑢_𝑛},{𝑢_𝑛} converge strongly to the same point 𝑥=𝑃_{Fix(𝑆)∩Ξ∩Γ}𝑄𝑥 if and only if lim_𝑛→∞‖𝑢_𝑛+1−𝑢_𝑛‖=0. Furthermore, (𝑥,𝑦) is a solution of the GSVI (14), where 𝑦=𝑃_𝐶(𝑥−𝜇₂𝐵₂𝑥).

Proof. First, taking into account 0<liminf_𝑛→∞𝜆_𝑛≤limsup_𝑛→∞𝜆_𝑛<1/‖𝐴‖², without loss of generality we may assume that {𝜆_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1/‖𝐴‖²).
Now, let us show that 𝑃_𝐶(𝐼−𝜆∇𝑓_𝛼) is 𝜁-averaged for each 𝜆∈(0,2/(𝛼+‖𝐴‖²)), where ‖𝐴‖𝜁=2+𝜆𝛼+²4. (37)
Indeed, it is easy to see that ∇𝑓=𝐴^∗(𝐼−𝑃_𝑄)𝐴 is 1/‖𝐴‖²-ism, that is, (𝑥)𝑦≥1∇𝑓−∇𝑓,𝑥−𝑦‖𝐴‖²∇𝑓(𝑥)−∇𝑓𝑦². (38) Observe that 𝛼+‖𝐴‖²∇𝑓_𝛼(𝑥)−∇𝑓_𝛼𝑦=‖𝐴‖,𝑥−𝑦𝛼+²𝛼𝑥−𝑦²+∇𝑓(𝑥)−∇𝑓𝑦,𝑥−𝑦=𝛼²𝑥−𝑦²+𝛼∇𝑓(𝑥)−∇𝑓𝑦,𝑥−𝑦+𝛼‖𝐴‖²𝑥−𝑦²+‖𝐴‖²∇𝑓(𝑥)−∇𝑓𝑦,𝑥−𝑦≥𝛼²𝑥−𝑦²(𝑥)𝑦+(𝑥)𝑦+2𝛼∇𝑓−∇𝑓,𝑥−𝑦∇𝑓−∇𝑓²=𝛼𝑥−𝑦+∇𝑓(𝑥)−∇𝑓𝑦²=∇𝑓_𝛼(𝑥)−∇𝑓_𝛼𝑦². (39) Hence, it follows that ∇𝑓_𝛼=𝛼𝐼+𝐴^∗(𝐼−𝑃_𝑄)𝐴 is 1/(𝛼+‖𝐴‖²)-ism. Thus, 𝜆∇𝑓_𝛼 is 1/𝜆(𝛼+‖𝐴‖²)-ism according to Proposition 12(ii). By Proposition 12(iii) the complement 𝐼−𝜆∇𝑓_𝛼 is 𝜆(𝛼+‖𝐴‖²)/2-averaged. Therefore, noting that 𝑃_𝐶 is 1/2-averaged, and utilizing Proposition 13(iv), we know that for each 𝜆∈(0,2/(𝛼+‖𝐴‖²)),𝑃_𝐶(𝐼−𝜆∇𝑓_𝛼) is 𝜁-averaged with 1𝜁=2+𝜆𝛼+‖𝐴‖²2−12⋅𝜆𝛼+‖𝐴‖²2=2+𝜆𝛼+‖𝐴‖²4∈(0,1). (40) This shows that 𝑃_𝐶(𝐼−𝜆∇𝑓_𝛼) is nonexpansive. Furthermore, for {𝜆_𝑛}⊂[𝑎,𝑏] with 𝑎,𝑏∈(0,1/‖𝐴‖²), we have 𝑎≤inf_𝑛≥0𝜆_𝑛≤sup_𝑛≥0𝜆_𝑛1≤𝑏<‖𝐴‖²=lim_𝑛→∞1𝛼_𝑛+‖𝐴‖². (41) Without loss of generality we may assume that 𝑎≤inf_𝑛≥0𝜆_𝑛≤sup_𝑛≥0𝜆_𝑛1≤𝑏<𝛼_𝑛+‖𝐴‖²,∀𝑛≥0. (42) Consequently, it follows that for each integer 𝑛≥0,𝑃_𝐶(𝐼−𝜆_𝑛∇𝑓_𝛼𝑛) is 𝜁_𝑛-averaged with 𝜁_𝑛=12+𝜆_𝑛𝛼_𝑛+‖𝐴‖²2−12⋅𝜆_𝑛𝛼_𝑛+‖𝐴‖²2=2+𝜆_𝑛𝛼_𝑛+‖𝐴‖²4∈(0,1). (43) This immediately implies that 𝑃_𝐶(𝐼−𝜆_𝑛∇𝑓_𝛼𝑛) is nonexpansive for all 𝑛≥0.
Next we divide the remainder of the proof into several steps.
Step  1. {𝑥_𝑛} is bounded.
Indeed, take an arbitrary 𝑝∈Fix(𝑆)∩Ξ∩Γ. Then, we get 𝑆𝑝=𝑝,  𝑃_𝐶(𝐼−𝜆∇𝑓)𝑝=𝑝 for 𝜆∈(0,2/‖𝐴‖²), and 𝑝=𝑃_𝐶𝑃_𝐶𝑝−𝜇₂𝐵₂𝑝−𝜇₁𝐵₁𝑃_𝐶𝑝−𝜇₂𝐵₂𝑝. (44) From (17) it follows that 𝑢_𝑛−𝑝=𝑃_𝐶𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛−𝑃_𝐶𝐼−𝜆_𝑛𝑝≤𝑃∇𝑓_𝐶𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛−𝑃_𝐶𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑝+𝑃_𝐶𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑝−𝑃_𝐶𝐼−𝜆_𝑛𝑝≤𝑢∇𝑓_𝑛−𝑝+𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑝−𝐼−𝜆_𝑛𝑝≤𝑢∇𝑓_𝑛−𝑝+𝜆_𝑛𝛼_𝑛𝑝. (45) Utilizing Lemma 18 we also have 𝑢_𝑛−𝑝²=𝑃_𝐶𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛−𝑃_𝐶𝐼−𝜆_𝑛∇𝑓𝑝²=𝑃_𝐶𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛−𝑃_𝐶𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑝+𝑃_𝐶𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑝−𝑃_𝐶𝐼−𝜆_𝑛∇𝑓𝑝²≤𝑃_𝐶𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛−𝑃_𝐶𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑝²+2𝑃_𝐶𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑝−𝑃_𝐶𝐼−𝜆_𝑛∇𝑓𝑝,𝑢_𝑛≤𝑢−𝑝_𝑛−𝑝²+2𝑃_𝐶𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑝−𝑃_𝐶𝐼−𝜆_𝑛∇𝑓𝑝𝑢_𝑛≤𝑢−𝑝_𝑛−𝑝²+2𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑝−𝐼−𝜆_𝑛∇𝑓𝑝𝑢_𝑛=𝑢−𝑝_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑢_𝑛−𝑝. (46) For simplicity, we write 𝑞=𝑃_𝐶𝑝−𝜇₂𝐵₂𝑝,𝑥_𝑛=𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛,𝑢_𝑛=𝑃_𝐶𝑢_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛, (47) for each 𝑛≥0. Then 𝑦_𝑛=𝜎_𝑛𝑄𝑥_𝑛+(1−𝜎_𝑛)𝑢_𝑛 for each 𝑛≥0. Since 𝐵_𝑖∶𝐶→ℋ₁ is 𝛽_𝑖-inverse strongly monotone for 𝑖=1,2 and 0<𝜇_𝑖<2𝛽_𝑖 for 𝑖=1,  2, we know that for all 𝑛≥0, 𝑢_𝑛−𝑝²=𝑃_𝐶𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝑝²=𝑃_𝐶𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝑃_𝐶𝑃_𝐶𝑝−𝜇₂𝐵₂𝑝−𝜇₁𝐵₁𝑃_𝐶𝑝−𝜇₂𝐵₂𝑝²≤𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝑃_𝐶𝑝−𝜇₂𝐵₂𝑝−𝜇₁𝐵₁𝑃_𝐶𝑝−𝜇₂𝐵₂𝑝²=𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝑃_𝐶𝑝−𝜇₂𝐵₂𝑝−𝜇₁𝐵₁𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝐵₁𝑃_𝐶𝑝−𝜇₂𝐵₂𝑝²≤𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝑃_𝐶𝑝−𝜇₂𝐵₂𝑝²−𝜇₁2𝛽₁−𝜇₁𝐵₁𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝐵₁𝑃_𝐶𝑝−𝜇₂𝐵₂𝑝²≤𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝑝−𝜇₂𝐵₂𝑝²−𝜇₁2𝛽₁−𝜇₁𝐵₁𝑥_𝑛−𝐵₁𝑞²=𝑥_𝑛−𝑝−𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²−𝜇₁2𝛽₁−𝜇₁𝐵₁𝑥_𝑛−𝐵₁𝑞²≤𝑥_𝑛−𝑝²−𝜇₂2𝛽₂−𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²−𝜇₁2𝛽₁−𝜇₁𝐵₁𝑥_𝑛−𝐵₁𝑞²≤𝑥_𝑛−𝑝². (48) Furthermore, by Proposition 8(ii), we have 𝑢_𝑛−𝑝²≤𝑢_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛−𝑝²−𝑢_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛−𝑢_𝑛²=𝑢_𝑛−𝑝²−𝑢_𝑛−𝑢_𝑛²+2𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛,𝑝−𝑢_𝑛=𝑢_𝑛−𝑝²−𝑢_𝑛−𝑢_𝑛²+2𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛−∇𝑓_𝛼𝑛𝑝,𝑝−𝑢_𝑛+∇𝑓_𝛼𝑛𝑝,𝑝−𝑢_𝑛+∇𝑓_𝛼𝑛𝑢_𝑛,𝑢_𝑛−𝑢_𝑛≤𝑢_𝑛−𝑝²−𝑢_𝑛−𝑢_𝑛²+2𝜆_𝑛∇𝑓_𝛼𝑛𝑝,𝑝−𝑢_𝑛+∇𝑓_𝛼𝑛𝑢_𝑛,𝑢_𝑛−𝑢_𝑛=𝑢_𝑛−𝑝²−𝑢_𝑛−𝑢_𝑛²+2𝜆_𝑛𝛼_𝑛𝐼+∇𝑓𝑝,𝑝−𝑢_𝑛+∇𝑓_𝛼𝑛𝑢_𝑛,𝑢_𝑛−𝑢_𝑛≤𝑢_𝑛−𝑝²−𝑢_𝑛−𝑢_𝑛²+2𝜆_𝑛𝛼_𝑛𝑝,𝑝−𝑢_𝑛+∇𝑓_𝛼𝑛𝑢_𝑛,𝑢_𝑛−𝑢_𝑛=𝑢_𝑛−𝑝²−𝑢_𝑛−𝑢_𝑛²−2𝑢_𝑛−𝑢_𝑛,𝑢_𝑛−𝑢_𝑛−𝑢_𝑛−𝑢_𝑛²+2𝜆_𝑛𝛼_𝑛𝑝,𝑝−𝑢_𝑛+∇𝑓_𝛼𝑛𝑢_𝑛,𝑢_𝑛−𝑢_𝑛=𝑢_𝑛−𝑝²−𝑢_𝑛−𝑢_𝑛²−𝑢_𝑛−𝑢_𝑛²+2𝑢_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛−𝑢_𝑛,𝑢_𝑛−𝑢_𝑛+2𝜆_𝑛𝛼_𝑛𝑝,𝑝−𝑢_𝑛. (49) Further, by Proposition 8(i), we have 𝑢_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛−𝑢_𝑛,𝑢_𝑛−𝑢_𝑛=𝑢_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛−𝑢_𝑛,𝑢_𝑛−𝑢_𝑛+𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛,𝑢_𝑛−𝑢_𝑛≤𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛,𝑢_𝑛−𝑢_𝑛≤𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛−∇𝑓_𝛼𝑛𝑢_𝑛𝑢_𝑛−𝑢_𝑛≤𝜆_𝑛𝛼_𝑛+‖𝐴‖²𝑢_𝑛−𝑢_𝑛𝑢_𝑛−𝑢_𝑛. (50) So, from (45) we obtain 𝑢_𝑛−𝑝²≤𝑢_𝑛−𝑝²−𝑢_𝑛−𝑢_𝑛²−𝑢_𝑛−𝑢_𝑛²+2𝑢_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛−𝑢_𝑛,𝑢_𝑛−𝑢_𝑛+2𝜆_𝑛𝛼_𝑛𝑝,𝑝−𝑢_𝑛≤𝑢_𝑛−𝑝²−𝑢_𝑛−𝑢_𝑛²−𝑢_𝑛−𝑢_𝑛²+2𝜆_𝑛𝛼_𝑛+‖𝐴‖²×𝑢_𝑛−𝑢_𝑛𝑢_𝑛−𝑢_𝑛+2𝜆_𝑛𝛼_𝑛𝑝𝑢_𝑛≤𝑢−𝑝_𝑛−𝑝²−𝑢_𝑛−𝑢_𝑛²−𝑢_𝑛−𝑢_𝑛²+𝜆²_𝑛𝛼_𝑛+‖𝐴‖²²𝑢_𝑛−𝑢_𝑛²+𝑢_𝑛−𝑢_𝑛²+2𝜆_𝑛𝛼_𝑛𝑝𝑢_𝑛=𝑢−𝑝_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑢_𝑛+𝜆−𝑝_2𝑛𝛼_𝑛+‖𝐴‖²²−1𝑢_𝑛−𝑢_𝑛²≤𝑢_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑢_𝑛≤𝑢−𝑝_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑢_𝑛−𝑝+𝜆_𝑛𝛼_𝑛𝑝≤𝑢_𝑛−𝑝²+4𝜆_𝑛𝛼_𝑛𝑝𝑢_𝑛−𝑝+4𝜆²_𝑛𝛼²_𝑛𝑝²=𝑢_𝑛−𝑝+2𝜆_𝑛𝛼_𝑛𝑝². (51) Hence it follows from (48) and (51) that 𝑦_𝑛=𝜎−𝑝_𝑛𝑄𝑥_𝑛−𝑝+1−𝜎_𝑛𝑢_𝑛−𝑝≤𝜎_𝑛𝑄𝑥_𝑛−𝑝+1−𝜎_𝑛𝑢_𝑛−𝑝≤𝜎_𝑛𝑄𝑥_𝑛++−𝑄𝑝𝑄𝑝−𝑝1−𝜎_𝑛𝑢_𝑛−𝑝+2𝜆_𝑛𝛼_𝑛𝑝≤𝜎_𝑛𝜌𝑥_𝑛++−𝑝𝑄𝑝−𝑝1−𝜎_𝑛𝑥_𝑛−𝑝+2𝜆_𝑛𝛼_𝑛𝑝≤1−1−𝜌𝜎_𝑛𝑥_𝑛−𝑝+𝜎_𝑛𝑄𝑝−𝑝+2𝜆_𝑛𝛼_𝑛𝑝=1−1−𝜌𝜎_𝑛𝑥_𝑛+𝜎−𝑝1−𝜌_𝑛𝑄𝑝−𝑝1−𝜌+2𝜆_𝑛𝛼_𝑛𝑝≤max𝑥_𝑛−𝑝,𝑄𝑝−𝑝1−𝜌+2𝜆_𝑛𝛼_𝑛𝑝. (52) Since (𝛾_𝑛+𝛿_𝑛)𝑘≤𝛾_𝑛 for all 𝑛≥0, utilizing Lemma 17 we obtain from (52) 𝑥_𝑛+1=𝛽−𝑝_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑦_𝑛−𝑝+𝛿_𝑛𝑆𝑦_𝑛−𝑝≤𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑦_𝑛−𝑝+𝛿_𝑛𝑆𝑦_𝑛−𝑝≤𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛+𝛿_𝑛𝑦_𝑛−𝑝≤𝛽_𝑛𝑥_𝑛+𝛾−𝑝_𝑛+𝛿_𝑛max𝑥_𝑛−𝑝,𝑄𝑝−𝑝1−𝜌+2𝜆_𝑛𝛼_𝑛𝑝≤𝛽_𝑛𝑥_𝑛+𝛾−𝑝_𝑛+𝛿_𝑛max𝑥_𝑛−𝑝,𝑄𝑝−𝑝1−𝜌+2𝜆_𝑛𝛼_𝑛𝑝≤max𝑥_𝑛−𝑝,𝑄𝑝−𝑝𝑝𝛼1−𝜌+2𝑏_𝑛. (53) Now, we claim that 𝑥_𝑛+1−𝑝≤max𝑥₀−𝑝,𝑄𝑝−𝑝𝑝1−𝜌+2𝑏^𝑛_𝑗=0𝛼_𝑗. (54) As a matter of fact, if 𝑛=0, then it is clear that (54) is valid, that is, 𝑥₁−𝑝≤max𝑥₀−𝑝,𝑄𝑝−𝑝𝑝1−𝜌+2𝑏⁰_𝑗=0𝛼_𝑗. (55) Assume that (54) holds for 𝑛≥1, that is, 𝑥_𝑛−𝑝≤max𝑥₀−𝑝,𝑄𝑝−𝑝𝑝1−𝜌+2𝑏^𝑛−1_𝑗=0𝛼_𝑗. (56) Then, we conclude from (53) and (56) that 𝑥_𝑛+1𝑥−𝑝≤max_𝑛−𝑝,𝑄𝑝−𝑝𝑝𝛼1−𝜌+2𝑏_𝑛≤maxmax𝑥₀−𝑝,𝑄𝑝−𝑝𝑝1−𝜌+2𝑏^𝑛−1_𝑗=0𝛼_𝑗,𝑄𝑝−𝑝𝑝𝛼1−𝜌+2𝑏_𝑛≤max𝑥₀−𝑝,𝑄𝑝−𝑝𝑝1−𝜌+2𝑏^𝑛−1_𝑗=0𝛼_𝑗+2𝑏𝑝𝛼_𝑛=max𝑥₀−𝑝,𝑄𝑝−𝑝𝑝1−𝜌+2𝑏^𝑛_𝑗=0𝛼_𝑗. (57) By induction, we conclude that (54) is valid. Hence, {𝑥_𝑛} is bounded. Since 𝑃_𝐶,∇𝑓_𝛼𝑛,𝐵₁ and 𝐵₂ are Lipschitz continuous, it is easy to see that {𝑢_𝑛},{𝑢_𝑛},{𝑢_𝑛},{𝑦_𝑛} and {𝑥_𝑛} are bounded, where 𝑥_𝑛=𝑃_𝐶(𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛) for all 𝑛≥0.
Step  2. lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0.
Indeed, define 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑤_𝑛 for all 𝑛≥0. It follows that 𝑤_𝑛+1−𝑤_𝑛=𝑥_𝑛+2−𝛽_𝑛+1𝑥_𝑛+11−𝛽_𝑛+1−𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛1−𝛽_𝑛=𝛾_𝑛+1𝑦_𝑛+1+𝛿_𝑛+1𝑆𝑦_𝑛+11−𝛽_𝑛+1−𝛾_𝑛𝑦_𝑛+𝛿_𝑛𝑆𝑦_𝑛1−𝛽_𝑛=𝛾_𝑛+1𝑦_𝑛+1−𝑦_𝑛+𝛿_𝑛+1𝑆𝑦_𝑛+1−𝑆𝑦_𝑛1−𝛽_𝑛+1+𝛾_𝑛+11−𝛽_𝑛+1−𝛾_𝑛1−𝛽_𝑛𝑦_𝑛+𝛿_𝑛+11−𝛽_𝑛+1−𝛿_𝑛1−𝛽_𝑛𝑆𝑦_𝑛. (58) Since (𝛾_𝑛+𝛿_𝑛)𝑘≤𝛾_𝑛 for all 𝑛≥0, utilizing Lemma 17 we have 𝛾_𝑛+1𝑦_𝑛+1−𝑦_𝑛+𝛿_𝑛+1𝑆𝑦_𝑛+1−𝑆𝑦_𝑛≤𝛾_𝑛+1+𝛿_𝑛+1𝑦_𝑛+1−𝑦_𝑛. (59) Next, we estimate ‖𝑦_𝑛+1−𝑦_𝑛‖. Observe that 𝑢_𝑛+1−𝑢_𝑛=𝑃_𝐶𝑢_𝑛+1−𝜆_𝑛+1∇𝑓_𝛼𝑛+1𝑢_𝑛+1−𝑃_𝐶𝑢_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛≤𝑃_𝐶𝐼−𝜆_𝑛+1∇𝑓_𝛼𝑛+1𝑢_𝑛+1−𝑃_𝐶𝐼−𝜆_𝑛+1∇𝑓_𝛼𝑛+1𝑢_𝑛+𝑃_𝐶𝐼−𝜆_𝑛+1∇𝑓_𝛼𝑛+1𝑢_𝑛−𝑃_𝐶𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛≤𝑢_𝑛+1−𝑢_𝑛+𝐼−𝜆_𝑛+1∇𝑓_𝛼𝑛+1𝑢_𝑛−𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛=𝑢_𝑛+1−𝑢_𝑛+𝜆_𝑛+1𝛼_𝑛+1𝐼+∇𝑓𝑢_𝑛−𝜆_𝑛𝛼_𝑛𝐼+∇𝑓𝑢_𝑛≤𝑢_𝑛+1−𝑢_𝑛+𝜆_𝑛+1−𝜆_𝑛∇𝑓𝑢_𝑛+𝜆_𝑛+1𝛼_𝑛+1−𝜆_𝑛𝛼_𝑛𝑢_𝑛,𝑢_𝑛+1−𝑢_𝑛²=𝑃_𝐶𝑃_𝐶𝑥_𝑛+1−𝜇₂𝐵₂𝑥_𝑛+1−𝜇₁𝐵₁𝑃_𝐶𝑥_𝑛+1−𝜇₂𝐵₂𝑥_𝑛+1−𝑃_𝐶𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛²≤𝑃_𝐶𝑥_𝑛+1−𝜇₂𝐵₂𝑥_𝑛+1−𝜇₁𝐵₁𝑃_𝐶𝑥_𝑛+1−𝜇₂𝐵₂𝑥_𝑛+1−𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛²=𝑃_𝐶𝑥_𝑛+1−𝜇₂𝐵₂𝑥_𝑛+1−𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑥_𝑛+1−𝜇₂𝐵₂𝑥_𝑛+1−𝐵₁𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛²≤𝑃_𝐶𝑥_𝑛+1−𝜇₂𝐵₂𝑥_𝑛+1−𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛²−𝜇₁2𝛽₁−𝜇₁×𝐵₁𝑃_𝐶𝑥_𝑛+1−𝜇₂𝐵₂𝑥_𝑛+1−𝐵₁𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛²≤𝑃_𝐶𝑥_𝑛+1−𝜇₂𝐵₂𝑥_𝑛+1−𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛²≤𝑥_𝑛+1−𝜇₂𝐵₂𝑥_𝑛+1−𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛²=𝑥_𝑛+1−𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛+1−𝐵₂𝑥_𝑛²≤𝑥_𝑛+1−𝑥_𝑛²−𝜇₂2𝛽₂−𝜇₂𝐵₂𝑥_𝑛+1−𝐵₂𝑥_𝑛²≤𝑥_𝑛+1−𝑥_𝑛², (60)(61) and hence 𝑢_𝑛+1−𝑢_𝑛=𝑃_𝐶𝑢_𝑛+1−𝜆_𝑛+1∇𝑓_𝛼𝑛+1𝑢_𝑛+1−𝑃_𝐶𝑢_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛≤𝑢_𝑛+1−𝜆_𝑛+1∇𝑓_𝛼𝑛+1𝑢_𝑛+1−𝑢_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛=𝐼−𝜆_𝑛+1∇𝑓_𝛼𝑛+1𝑢_𝑛+1−𝐼−𝜆_𝑛+1∇𝑓_𝛼𝑛+1𝑢_𝑛+𝜆_𝑛+1∇𝑓_𝛼𝑛+1𝑢_𝑛+1−∇𝑓_𝛼𝑛+1𝑢_𝑛−𝜆_𝑛+1∇𝑓_𝛼𝑛+1𝑢_𝑛+1+𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛≤𝐼−𝜆_𝑛+1∇𝑓_𝛼𝑛+1𝑢_𝑛+1−𝐼−𝜆_𝑛+1∇𝑓_𝛼𝑛+1𝑢_𝑛+𝜆_𝑛+1∇𝑓_𝛼𝑛+1𝑢_𝑛+1−∇𝑓_𝛼𝑛+1𝑢_𝑛+𝜆_𝑛+1∇𝑓_𝛼𝑛+1𝑢_𝑛+1−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛≤𝑢_𝑛+1−𝑢_𝑛+𝜆_𝑛+1𝛼_𝑛+1+‖𝐴‖²𝑢_𝑛+1−𝑢_𝑛+𝜆_𝑛+1𝛼_𝑛+1𝐼+∇𝑓𝑢_𝑛+1−𝜆_𝑛𝛼_𝑛𝐼+∇𝑓𝑢_𝑛≤𝑢_𝑛+1−𝑢_𝑛+𝜆_𝑛+1𝛼_𝑛+1+‖𝐴‖²𝑢_𝑛+1−𝑢_𝑛+𝜆_𝑛+1𝛼_𝑛+1𝑢_𝑛+1−𝜆_𝑛𝛼_𝑛𝑢_𝑛+𝜆_𝑛+1∇𝑓𝑢_𝑛+1−𝜆_𝑛∇𝑓𝑢_𝑛≤𝑢_𝑛+1−𝑢_𝑛+𝜆_𝑛+1𝛼_𝑛+1+‖𝐴‖²𝑢_𝑛+1−𝑢_𝑛+𝜆_𝑛+1𝛼_𝑛+1𝑢_𝑛+1−𝜆_𝑛𝛼_𝑛𝑢_𝑛+𝜆_𝑛+1∇𝑓𝑢_𝑛+1−∇𝑓𝑢_𝑛+𝜆_𝑛+1−𝜆_𝑛∇𝑓𝑢_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝜆_𝑛+1𝛼_𝑛+1+‖𝐴‖²𝑢_𝑛+1−𝑢_𝑛+𝜆_𝑛+1𝛼_𝑛+1𝑢_𝑛+1−𝜆_𝑛𝛼_𝑛𝑢_𝑛+𝜆_𝑛+1−𝜆_𝑛∇𝑓𝑢_𝑛+𝜆_𝑛+1‖𝐴‖²𝑢_𝑛+1−𝑢_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝜆_𝑛+1𝛼_𝑛+1+‖𝐴‖²×𝑢_𝑛+1−𝑢_𝑛+𝑢_𝑛+1−𝑢_𝑛+𝜆_𝑛+1𝛼_𝑛+1𝑢_𝑛+1−𝜆_𝑛𝛼_𝑛𝑢_𝑛+𝜆_𝑛+1−𝜆_𝑛∇𝑓𝑢_𝑛. (62) Combining (60) with (61), we get 𝑢_𝑛+1−𝑢_𝑛≤𝑢_𝑛+1−𝑢_𝑛+𝜆_𝑛+1−𝜆_𝑛∇𝑓𝑢_𝑛+𝜆_𝑛+1𝛼_𝑛+1−𝜆_𝑛𝛼_𝑛𝑢_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝜆_𝑛+1−𝜆_𝑛∇𝑓𝑢_𝑛+𝜆_𝑛+1𝛼_𝑛+1−𝜆_𝑛𝛼_𝑛𝑢_𝑛. (63) This together with (62) implies that 𝑦_𝑛+1−𝑦_𝑛=𝑢_𝑛+1+𝜎_𝑛+1𝑄𝑥_𝑛+1−𝑢_𝑛+1−𝑢_𝑛−𝜎_𝑛𝑄𝑥_𝑛−𝑢_𝑛≤𝑢_𝑛+1−𝑢_𝑛+𝜎_𝑛+1𝑄𝑥_𝑛+1−𝑢_𝑛+1+𝜎_𝑛𝑄𝑥_𝑛−𝑢_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝜆_𝑛+1𝛼_𝑛+1+‖𝐴‖²𝑢_𝑛+1−𝑢_𝑛+𝑢_𝑛+1−𝑢_𝑛+𝜆_𝑛+1𝛼_𝑛+1𝑢_𝑛+1−𝜆_𝑛𝛼_𝑛𝑢_𝑛+𝜆_𝑛+1−𝜆_𝑛∇𝑓𝑢_𝑛+𝜎_𝑛+1𝑄𝑥_𝑛+1−𝑢_𝑛+1+𝜎_𝑛𝑄𝑥_𝑛−𝑢_𝑛. (64) Hence it follows from (58), (59), and (64) that 𝑤_𝑛+1−𝑤_𝑛≤𝛾_𝑛+1𝑦_𝑛+1−𝑦_𝑛+𝛿_𝑛+1𝑆𝑦_𝑛+1−𝑆𝑦_𝑛1−𝛽_𝑛+1+𝛾_𝑛+11−𝛽_𝑛+1−𝛾_𝑛1−𝛽_𝑛𝑦_𝑛+𝛿_𝑛+11−𝛽_𝑛+1−𝛿_𝑛1−𝛽_𝑛𝑆𝑦_𝑛≤𝛾_𝑛+1+𝛿_𝑛+11−𝛽_𝑛+1𝑦_𝑛+1−𝑦_𝑛+𝛾_𝑛+11−𝛽_𝑛+1−𝛾_𝑛1−𝛽_𝑛𝑦_𝑛+𝑆𝑦_𝑛=𝑦_𝑛+1−𝑦_𝑛+𝛾_𝑛+11−𝛽_𝑛+1−𝛾_𝑛1−𝛽_𝑛𝑦_𝑛+𝑆𝑦_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝜆_𝑛+1𝛼_𝑛+1+‖𝐴‖²𝑢_𝑛+1−𝑢_𝑛+𝑢_𝑛+1−𝑢_𝑛+𝜆_𝑛+1𝛼_𝑛+1𝑢_𝑛+1−𝜆_𝑛𝛼_𝑛𝑢_𝑛+𝜆_𝑛+1−𝜆_𝑛∇𝑓𝑢_𝑛+𝜎_𝑛+1𝑄𝑥_𝑛+1−𝑢_𝑛+1+𝜎_𝑛𝑄𝑥_𝑛−𝑢_𝑛+𝛾_𝑛+11−𝛽_𝑛+1−𝛾_𝑛1−𝛽_𝑛𝑦_𝑛+𝑆𝑦_𝑛. (65) From (60), we deduce from condition (vi) that lim_𝑛→∞‖𝑢_𝑛+1−𝑢_𝑛‖=0. Since {𝑥_𝑛},{𝑢_𝑛}, {𝑢_𝑛},{𝑦_𝑛} and {𝑢_𝑛} are bounded, it follows from conditions (i), (iii), (v), and (vi) that limsup_𝑛→∞𝑤_𝑛+1−𝑤_𝑛−𝑥_𝑛+1−𝑥_𝑛≤limsup_𝑛→∞𝜆_𝑛+1𝛼_𝑛+1+‖𝐴‖²×𝑢_𝑛+1−𝑢_𝑛+𝑢_𝑛+1−𝑢_𝑛+𝜆_𝑛+1𝛼_𝑛+1𝑢_𝑛+1−𝜆_𝑛𝛼_𝑛𝑢_𝑛+𝜆_𝑛+1−𝜆_𝑛∇𝑓𝑢_𝑛+𝜎_𝑛+1𝑄𝑥_𝑛+1−𝑢_𝑛+1+𝜎_𝑛𝑄𝑥_𝑛−𝑢_𝑛+𝛾_𝑛+11−𝛽_𝑛+1−𝛾_𝑛1−𝛽_𝑛𝑦_𝑛+𝑆𝑦_𝑛=0. (66) Hence by Lemma 14 we get lim_𝑛→∞‖𝑤_𝑛−𝑥_𝑛‖=0. Thus, lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=lim_𝑛→∞1−𝛽_𝑛𝑤_𝑛−𝑥_𝑛=0. (67)
Step  3. lim_𝑛→∞‖𝐵₂𝑥_𝑛−𝐵₂𝑝‖=0,  lim_𝑛→∞‖𝐵₁𝑥_𝑛−𝐵₁𝑞‖=0, and lim_𝑛→∞‖𝑢_𝑛−𝑢_𝑛‖=0, where 𝑞=𝑃_𝐶(𝑝−𝜇₂𝐵₂𝑝).
Indeed, utilizing Lemma 17 and the convexity of ‖⋅‖², we obtain from (17), (48), and (51) that 𝑥_𝑛+1−𝑝²=𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑦_𝑛−𝑝+𝛿_𝑛𝑆𝑦_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛+𝛿_𝑛1𝛾_𝑛+𝛿_𝑛𝛾_𝑛𝑦_𝑛−𝑝+𝛿_𝑛𝑆𝑦_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛+𝛿_𝑛𝑦_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛+𝛿_𝑛𝜎_𝑛𝑄𝑥_𝑛−𝑝²+1−𝜎_𝑛𝑢_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝜎_𝑛𝑄𝑥_𝑛−𝑝²+𝛾_𝑛+𝛿_𝑛𝑢_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝜎_𝑛𝑄𝑥_𝑛−𝑝²+𝛾_𝑛+𝛿_𝑛𝑢_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑢_𝑛+𝜆−𝑝_2𝑛𝛼_𝑛+‖𝐴‖²²−1𝑢_𝑛−𝑢_𝑛²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝜎_𝑛𝑄𝑥_𝑛−𝑝²+𝛾_𝑛+𝛿_𝑛×𝑥_𝑛−𝑝²−𝜇₂2𝛽₂−𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²−𝜇₁2𝛽₁−𝜇₁𝐵₁𝑥_𝑛−𝐵₁𝑞²+2𝜆_𝑛𝛼_𝑛𝑝𝑢_𝑛+𝜆−𝑝_2𝑛𝛼_𝑛+‖𝐴‖²²−1𝑢_𝑛−𝑢_𝑛²≤𝑥_𝑛−𝑝²+𝜎_𝑛𝑄𝑥_𝑛−𝑝²−𝛾_𝑛+𝛿_𝑛×𝜇₂2𝛽₂−𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²+𝜇₁2𝛽₁−𝜇₁𝐵₁𝑥_𝑛−𝐵₁𝑞²+1−𝜆²_𝑛𝛼_𝑛+‖𝐴‖²²𝑢_𝑛−𝑢_𝑛²+2𝜆_𝑛𝛼_𝑛𝑝𝑢_𝑛−𝑝. (68) Therefore, 𝛾_𝑛+𝛿_𝑛𝜇₂2𝛽₂−𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²+𝜇₁2𝛽₁−𝜇₁𝐵₁𝑥_𝑛−𝐵₁𝑞²+1−𝜆²_𝑛𝛼_𝑛+‖𝐴‖²²𝑢_𝑛−𝑢_𝑛²≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+𝜎_𝑛𝑄𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑢_𝑛≤𝑥−𝑝_𝑛−𝑝+𝑥_𝑛+1−𝑝𝑥_𝑛−𝑥_𝑛+1+𝜎_𝑛𝑄𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑢_𝑛−𝑝. (69) Since 𝛼_𝑛→0,𝜎_𝑛→0,  ‖𝑥_𝑛−𝑥_𝑛+1‖→0,  liminf_𝑛→∞(𝛾_𝑛+𝛿_𝑛)>0 and {𝜆_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1/‖𝐴‖²), it follows that lim_𝑛→∞𝑢_𝑛−𝑢_𝑛=0,lim_𝑛→∞𝐵₁𝑥_𝑛−𝐵₁𝑞=0,lim_𝑛→∞𝐵₂𝑥_𝑛−𝐵₂𝑝=0. (70)
Step  4. lim_𝑛→∞‖𝑆𝑦_𝑛−𝑦_𝑛‖=0.
Indeed, observe that 𝑢_𝑛−𝑢_𝑛=𝑃_𝐶𝑢_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛−𝑃_𝐶𝑢_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛≤𝑢_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛−𝑢_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛=𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛−∇𝑓_𝛼𝑛𝑢_𝑛≤𝜆_𝑛𝛼_𝑛+‖𝐴‖²𝑢_𝑛−𝑢_𝑛. (71) This together with ‖𝑢_𝑛−𝑢_𝑛‖→0 implies that lim_𝑛→∞‖𝑢_𝑛−𝑢_𝑛‖=0 and hence lim_𝑛→∞‖𝑢_𝑛−𝑢_𝑛‖=0. By firm nonexpansiveness of 𝑃_𝐶, we have 𝑥_𝑛−𝑞²=𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝑃_𝐶𝑝−𝜇₂𝐵₂𝑝²≤𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝑝−𝜇₂𝐵₂𝑝,𝑥_𝑛=1−𝑞2𝑥_𝑛−𝑝−𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²+𝑥_𝑛−𝑞²−𝑥_𝑛−𝑝−𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝−𝑥_𝑛−𝑞²≤12𝑥_𝑛−𝑝²+𝑥_𝑛−𝑞²−𝑥_𝑛−𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝−𝑝−𝑞²=12𝑥_𝑛−𝑝²+𝑥_𝑛−𝑞²−𝑥_𝑛−𝑥_𝑛−𝑝−𝑞²+2𝜇₂𝑥_𝑛−𝑥_𝑛−𝑝−𝑞,𝐵₂𝑥_𝑛−𝐵₂𝑝−𝜇²₂𝐵₂𝑥_𝑛−𝐵₂𝑝²≤12𝑥_𝑛−𝑝²+𝑥_𝑛−𝑞²−𝑥_𝑛−𝑥_𝑛−𝑝−𝑞²+2𝜇₂𝑥_𝑛−𝑥_𝑛−𝑝−𝑞𝐵₂𝑥_𝑛−𝐵₂𝑝, (72) that is, 𝑥_𝑛−𝑞²≤𝑥_𝑛−𝑝²−𝑥_𝑛−𝑥_𝑛−𝑝−𝑞²+2𝜇₂𝑥_𝑛−𝑥_𝑛−𝑝−𝑞𝐵₂𝑥_𝑛−𝐵₂𝑝. (73) Moreover, using the argument technique similar to the above one, we derive 𝑢_𝑛−𝑝²=𝑃_𝐶𝑥_𝑛−𝜇₁𝐵₁𝑥_𝑛−𝑃_𝐶𝑞−𝜇₁𝐵₁𝑞²≤𝑥_𝑛−𝜇₁𝐵₁𝑥_𝑛−𝑞−𝜇₁𝐵₁𝑞,𝑢_𝑛=1−𝑝2𝑥_𝑛−𝑞−𝜇₁𝐵₁𝑥_𝑛−𝐵₁𝑞²+𝑢_𝑛−𝑝²−𝑥_𝑛−𝑞−𝜇₁𝐵₁𝑥_𝑛−𝐵₁𝑞−𝑢_𝑛−𝑝²≤12𝑥_𝑛−𝑞²+𝑢_𝑛−𝑝²−𝑥_𝑛−𝑢_𝑛−𝜇₁𝐵₁𝑥_𝑛−𝐵₁𝑞+𝑝−𝑞²=12𝑥_𝑛−𝑞²+𝑢_𝑛−𝑝²−𝑥_𝑛−𝑢_𝑛+𝑝−𝑞²+2𝜇₁𝑥_𝑛−𝑢_𝑛+𝑝−𝑞,𝐵₁𝑥_𝑛−𝐵₁𝑞−𝜇²₁𝐵₁𝑥_𝑛−𝐵₁𝑞²≤12𝑥_𝑛−𝑞²+𝑢_𝑛−𝑝²−𝑥_𝑛−𝑢_𝑛+𝑝−𝑞²+2𝜇₁𝑥_𝑛−𝑢_𝑛+𝑝−𝑞𝐵₁𝑥_𝑛−𝐵₁𝑞, (74) that is, 𝑢_𝑛−𝑝²≤𝑥_𝑛−𝑞²−𝑥_𝑛−𝑢_𝑛+𝑝−𝑞²+2𝜇₁𝑥_𝑛−𝑢_𝑛+𝑝−𝑞𝐵₁𝑥_𝑛−𝐵₁𝑞. (75) Utilizing (46), (73), and (75), we have 𝑢_𝑛−𝑝²=𝑢_𝑛−𝑝+𝑢_𝑛−𝑢_𝑛²≤𝑢_𝑛−𝑝²+2𝑢_𝑛−𝑢_𝑛,𝑢_𝑛≤𝑢−𝑝_𝑛−𝑝²+2𝑢_𝑛−𝑢_𝑛𝑢_𝑛≤𝑢−𝑝_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑢_𝑛−𝑝+2𝑢_𝑛−𝑢_𝑛𝑢_𝑛≤𝑥−𝑝_𝑛−𝑞²−𝑥_𝑛−𝑢_𝑛+𝑝−𝑞²+2𝜇₁𝑥_𝑛−𝑢_𝑛+𝑝−𝑞𝐵₁𝑥_𝑛−𝐵₁𝑞+2𝜆_𝑛𝛼_𝑛𝑝𝑢_𝑛−𝑝+2𝑢_𝑛−𝑢_𝑛𝑢_𝑛≤𝑥−𝑝_𝑛−𝑝²−𝑥_𝑛−𝑥_𝑛−𝑝−𝑞²+2𝜇₂𝑥_𝑛−𝑥_𝑛−𝑝−𝑞𝐵₂𝑥_𝑛−𝐵₂𝑝−𝑥_𝑛−𝑢_𝑛+𝑝−𝑞²+2𝜇₁𝑥_𝑛−𝑢_𝑛+𝑝−𝑞𝐵₁𝑥_𝑛−𝐵₁𝑞+2𝜆_𝑛𝛼_𝑛𝑝𝑢_𝑛−𝑝+2𝑢_𝑛−𝑢_𝑛𝑢_𝑛−𝑝. (76) Thus from (17) and (76) it follows that 𝑥_𝑛+1−𝑝²=𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑦_𝑛−𝑝+𝛿_𝑛𝑆𝑦_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛+𝛿_𝑛𝑦_𝑛−𝑝²=𝛽_𝑛𝑥_𝑛−𝑝²+1−𝛽_𝑛𝑦_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+1−𝛽_𝑛𝜎_𝑛𝑄𝑥_𝑛−𝑝²+1−𝜎_𝑛𝑢_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝜎_𝑛𝑄𝑥_𝑛−𝑝²+1−𝛽_𝑛𝑢_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝜎_𝑛𝑄𝑥_𝑛−𝑝²+1−𝛽_𝑛𝑥_𝑛−𝑝²−𝑥_𝑛−𝑥_𝑛−𝑝−𝑞²+2𝜇₂𝑥_𝑛−𝑥_𝑛−𝑝−𝑞𝐵₂𝑥_𝑛−𝐵₂𝑝−𝑥_𝑛−𝑢_𝑛+𝑝−𝑞²+2𝜇₁𝑥_𝑛−𝑢_𝑛+𝑝−𝑞𝐵₁𝑥_𝑛−𝐵₁𝑞+2𝜆_𝑛𝛼_𝑛𝑝𝑢_𝑛−𝑝+2𝑢_𝑛−𝑢_𝑛𝑢_𝑛−𝑝, (77) which hence implies that 1−𝛽_𝑛𝑥_𝑛−𝑥_𝑛−𝑝−𝑞²+𝑥_𝑛−𝑢_𝑛+𝑝−𝑞²≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+𝜎_𝑛𝑄𝑥_𝑛−𝑝²+1−𝛽_𝑛2𝜇₂𝑥_𝑛−𝑥_𝑛−𝑝−𝑞𝐵₂𝑥_𝑛−𝐵₂𝑝+2𝜇₁𝑥_𝑛−𝑢_𝑛+𝑝−𝑞𝐵₁𝑥_𝑛−𝐵₁𝑞+2𝜆_𝑛𝛼_𝑛𝑝𝑢_𝑛−𝑝+2𝑢_𝑛−𝑢_𝑛𝑢_𝑛≤𝑥−𝑝_𝑛−𝑝+𝑥_𝑛+1×𝑥−𝑝_𝑛−𝑥_𝑛+1+𝜎_𝑛𝑄𝑥_𝑛−𝑝²+2𝜇₂𝑥_𝑛−𝑥_𝑛−𝑝−𝑞𝐵₂𝑥_𝑛−𝐵₂𝑝+2𝜇₁𝑥_𝑛−𝑢_𝑛+𝑝−𝑞𝐵₁𝑥_𝑛−𝐵₁𝑞+2𝜆_𝑛𝛼_𝑛𝑝𝑢_𝑛−𝑝+2𝑢_𝑛−𝑢_𝑛𝑢_𝑛−𝑝. (78) Since limsup_𝑛→∞𝛽_𝑛<1,  {𝜆_𝑛}⊂[𝑎,𝑏],  𝛼_𝑛→0,  𝜎_𝑛→0,  ‖𝐵₂𝑥_𝑛−𝐵₂𝑝‖→0,  ‖𝐵₁𝑥_𝑛−𝐵₁𝑞‖→0,  ‖𝑢_𝑛−𝑢_𝑛‖→0 and ‖𝑥_𝑛+1−𝑥_𝑛‖→0, it follows from the boundedness of {𝑥_𝑛},{𝑥_𝑛},{𝑢_𝑛},{𝑢_𝑛}, and {𝑢_𝑛} that lim_𝑛→∞𝑥_𝑛−𝑥_𝑛−𝑝−𝑞=0,lim_𝑛→∞𝑥_𝑛−𝑢_𝑛+𝑝−𝑞=0. (79) Consequently, it immediately follows that lim_𝑛→∞𝑥_𝑛−𝑢_𝑛=0,lim_𝑛→∞𝑥_𝑛−𝑢_𝑛=0. (80) This together with ‖𝑦_𝑛−𝑢_𝑛‖≤𝜎_𝑛‖𝑄𝑥_𝑛−𝑢_𝑛‖→0 implies that lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=0. (81) Since 𝛿_𝑛𝑆𝑦_𝑛−𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝛾_𝑛𝑦_𝑛−𝑥_𝑛, (82) it follows that lim_𝑛→∞𝑆𝑦_𝑛−𝑥_𝑛=0,lim_𝑛→∞𝑆𝑦_𝑛−𝑦_𝑛=0. (83)
Step  5. limsup_𝑛→∞⟨𝑄𝑥−𝑥,𝑥_𝑛−𝑥⟩≤0 where 𝑥=𝑃_{Fix(𝑆)∩Ξ∩Γ}𝑄𝑥.
Indeed, since {𝑥_𝑛} is bounded, there exists a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that limsup_𝑛→∞𝑄𝑥−𝑥,𝑥_𝑛−𝑥=lim_𝑖→∞𝑄𝑥−𝑥,𝑥_𝑛𝑖−𝑥. (84) Also, since 𝐻 is reflexive and {𝑦_𝑛} is bounded, without loss of generality we may assume that 𝑦_𝑛𝑖→𝑝 weakly for some 𝑝∈𝐶. First, it is clear from Lemma 15 that 𝑝∈Fix(𝑆). Now let us show that 𝑝∈Ξ. We note that 𝑥_𝑛−𝐺𝑥_𝑛=𝑥_𝑛−𝑃_𝐶𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛=𝑥_𝑛−𝑢_𝑛⟶0(𝑛⟶∞), (85) where 𝐺∶𝐶→𝐶 is defined as such that in Lemma 4. According to Lemma 15 we obtain 𝑝∈Ξ. Further, let us show that 𝑝∈Γ. As a matter of fact, since ‖𝑥_𝑛−𝑢_𝑛‖→0,‖𝑢_𝑛−𝑢_𝑛‖→0 and ‖𝑥_𝑛−𝑦_𝑛‖→0, we deduce that 𝑥_𝑛𝑖→𝑝 weakly and 𝑢_𝑛𝑖→𝑝 weakly. Let (𝑣)𝑇𝑣=∇𝑓+𝑁_𝐶𝑣,if𝑣∈𝐶,∅,if𝑣∉𝐶, (86) where 𝑁_𝐶𝑣={𝑤∈ℋ₁∶⟨𝑣−𝑢,𝑤⟩≥0,∀𝑢∈𝐶}. Then, 𝑇 is maximal monotone and 0∈𝑇𝑣 if and only if 𝑣∈VI(𝐶,∇𝑓); see [40] for more details. Let (𝑣,𝑤)∈Gph(𝑇). Then, we have 𝑤∈𝑇𝑣=∇𝑓(𝑣)+𝑁_𝐶𝑣, (87) and hence 𝑤−∇𝑓(𝑣)∈𝑁_𝐶𝑣. (88) So, we have 𝑣−𝑢,𝑤−∇𝑓(𝑣)≥0,∀𝑢∈𝐶. (89) On the other hand, from 𝑢_𝑛=𝑃_𝐶𝑢_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛,𝑣∈𝐶, (90) we have 𝑢_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛−𝑢_𝑛,𝑢_𝑛−𝑣≥0, (91) and, hence, 𝑣−𝑢_𝑛,𝑢_𝑛−𝑢_𝑛𝜆_𝑛+∇𝑓_𝛼𝑛𝑢_𝑛≥0. (92) Therefore, from 𝑤−∇𝑓(𝑣)∈𝑁_𝐶𝑣,𝑢_𝑛𝑖∈𝐶, (93) we have 𝑣−𝑢_𝑛𝑖,𝑤≥𝑣−𝑢_𝑛𝑖(𝑣≥𝑢,∇𝑓)𝑣−_𝑛𝑖(𝑣−𝑢,∇𝑓)𝑣−_𝑛𝑖,𝑢_𝑛𝑖−𝑢_𝑛𝑖𝜆_𝑛𝑖+∇𝑓_𝛼𝑛𝑖𝑢_𝑛𝑖=𝑣−𝑢_𝑛𝑖(𝑣−𝑢,∇𝑓)𝑣−_𝑛𝑖,𝑢_𝑛𝑖−𝑢_𝑛𝑖𝜆_𝑛𝑖+∇𝑓𝑢_𝑛𝑖−𝛼_𝑛𝑖𝑣−𝑢_𝑛𝑖,𝑢_𝑛𝑖=𝑣−𝑢_𝑛𝑖,∇𝑓(𝑣)−∇𝑓𝑢_𝑛𝑖+𝑣−𝑢_𝑛𝑖,∇𝑓𝑢_𝑛𝑖−∇𝑓𝑢_𝑛𝑖−𝑣−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑢_𝑛𝑖𝜆_𝑛𝑖−𝛼_𝑛𝑖𝑣−𝑢_𝑛𝑖,𝑢_𝑛𝑖≥𝑣−𝑢_𝑛𝑖,∇𝑓𝑢_𝑛𝑖−∇𝑓𝑢_𝑛𝑖−𝑣−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑢_𝑛𝑖𝜆_𝑛𝑖−𝛼_𝑛𝑖𝑣−𝑢_𝑛𝑖,𝑢_𝑛𝑖. (94) Hence, we get 𝑣−𝑝,𝑤≥0,as𝑖⟶∞. (95) Since 𝑇 is maximal monotone, we have 𝑝∈𝑇⁻¹0, and, hence, 𝑝∈VI(𝐶,∇𝑓). Thus it is clear that 𝑝∈Γ. Therefore, 𝑝∈Fix(𝑆)∩Ξ∩Γ. Consequently, in terms of Proposition 8(i) we obtain from (84) that limsup_𝑛→∞𝑄𝑥−𝑥,𝑥_𝑛−𝑥=lim_𝑖→∞𝑄𝑥−𝑥,𝑥_𝑛𝑖−𝑥=𝑄𝑥−𝑥,𝑝−𝑥≤0. (96)
Step 6. lim_𝑛→∞‖𝑥_𝑛−𝑥‖=0.
Indeed, from (48) and (51) it follows that 𝑢_𝑛−𝑥²≤𝑢_𝑛−𝑥²+2𝜆_𝑛𝛼_𝑛‖𝑥‖𝑢_𝑛−𝑥≤𝑥_𝑛−𝑥²+2𝜆_𝑛𝛼_𝑛‖𝑥‖𝑢_𝑛−𝑥. (97) Note that 𝑄𝑥_𝑛−𝑥,𝑦_𝑛−𝑥=𝑄𝑥_𝑛−𝑥,𝑥_𝑛−𝑥+𝑄𝑥_𝑛−𝑥,𝑦_𝑛−𝑥_𝑛=𝑄𝑥_𝑛−𝑄𝑥,𝑥_𝑛−𝑥+𝑄𝑥−𝑥,𝑥_𝑛−𝑥+𝑄𝑥_𝑛−𝑥,𝑦_𝑛−𝑥_𝑛≤𝜌𝑥_𝑛−𝑥²+𝑄𝑥−𝑥,𝑥_𝑛−𝑥+𝑄𝑥_𝑛−𝑥𝑦_𝑛−𝑥_𝑛. (98) Utilizing Lemmas 17 and 18, we obtain from (48) and the convexity of ‖⋅‖²𝑥_𝑛+1−𝑥²=𝛽_𝑛𝑥_𝑛−𝑥+𝛾_𝑛𝑦_𝑛−𝑥+𝛿_𝑛𝑆𝑦_𝑛−𝑥²≤𝛽_𝑛𝑥_𝑛−𝑥²+𝛾_𝑛+𝛿_𝑛1𝛾_𝑛+𝛿_𝑛𝛾_𝑛𝑦_𝑛−𝑥+𝛿_𝑛𝑆𝑦_𝑛−𝑥²≤𝛽_𝑛𝑥_𝑛−𝑥²+𝛾_𝑛+𝛿_𝑛𝑦_𝑛−𝑥²≤𝛽_𝑛𝑥_𝑛−𝑥²+𝛾_𝑛+𝛿_𝑛1−𝜎_𝑛²𝑢_𝑛−𝑥²+2𝜎_𝑛𝑄𝑥_𝑛−𝑥,𝑦_𝑛−𝑥≤𝛽_𝑛𝑥_𝑛−𝑥²+𝛾_𝑛+𝛿_𝑛1−𝜎_𝑛×𝑥_𝑛−𝑥²+2𝜆_𝑛𝛼_𝑛‖𝑥‖𝑢_𝑛−𝑥+2𝜎_𝑛𝑄𝑥_𝑛−𝑥,𝑦_𝑛−𝑥=1−𝛾_𝑛+𝛿_𝑛𝜎_𝑛𝑥_𝑛−𝑥²+𝛾_𝑛+𝛿_𝑛2𝜎_𝑛𝑄𝑥_𝑛−𝑥,𝑦_𝑛−𝑥+𝛾_𝑛+𝛿_𝑛2𝜆_𝑛𝛼_𝑛‖𝑥‖𝑢_𝑛−𝑥≤1−𝛾_𝑛+𝛿_𝑛𝜎_𝑛𝑥_𝑛−𝑥²+𝛾_𝑛+𝛿_𝑛2𝜎_𝑛𝑄𝑥_𝑛−𝑥,𝑦_𝑛−𝑥+2𝜆_𝑛𝛼_𝑛‖𝑥‖𝑢_𝑛−𝑥≤1−𝛾_𝑛+𝛿_𝑛𝜎_𝑛𝑥_𝑛−𝑥²+𝛾_𝑛+𝛿_𝑛2𝜎_𝑛×𝜌𝑥_𝑛−𝑥²+𝑄𝑥−𝑥,𝑥_𝑛−𝑥+𝑄𝑥_𝑛−𝑥𝑦_𝑛−𝑥_𝑛+2𝜆_𝑛𝛼_𝑛‖𝑥‖𝑢_𝑛−𝑥=1−1−2𝜌𝛾_𝑛+𝛿_𝑛𝜎_𝑛𝑥_𝑛−𝑥²+𝛾_𝑛+𝛿_𝑛2𝜎_𝑛×𝑄𝑥−𝑥,𝑥_𝑛−𝑥+𝑄𝑥_𝑛−𝑥𝑦_𝑛−𝑥_𝑛+2𝜆_𝑛𝛼_𝑛‖𝑥‖𝑢_𝑛−𝑥=1−1−2𝜌𝛾_𝑛+𝛿_𝑛𝜎_𝑛𝑥_𝑛−𝑥²+1−2𝜌×𝜎_𝑛2𝑄𝑥−𝑥,𝑥_𝑛−𝑥+𝑄𝑥_𝑛−𝑥𝑦_𝑛−𝑥_𝑛1−2𝜌+2𝜆_𝑛𝛼_𝑛‖𝑥‖𝑢_𝑛−𝑥. (99) Note that liminf_𝑛→∞(1−2𝜌)(𝛾_𝑛+𝛿_𝑛)>0. It follows that ∑^∞_𝑛=0(1−2𝜌)(𝛾_𝑛+𝛿_𝑛)𝜎_𝑛=∞. It is clear that limsup_𝑛→∞2𝑄𝑥−𝑥,𝑥_𝑛−𝑥+𝑄𝑥_𝑛−𝑥𝑦_𝑛−𝑥_𝑛1−2𝜌≤0, (100) because limsup_𝑛→∞⟨𝑄𝑥−𝑥,𝑥_𝑛−𝑥⟩≤0 and lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0. In addition, note also that {𝜆_𝑛}⊂[𝑎,𝑏],  ∑^∞_𝑛=0𝛼_𝑛<∞ and {𝑢_𝑛} is bounded. Hence we get ∑^∞_𝑛=02𝜆_𝑛𝛼_𝑛‖𝑥‖‖𝑢_𝑛−𝑥‖<∞. Therefore, all conditions of Lemma 16 are satisfied. Consequently, we immediately deduce that ‖𝑥_𝑛−𝑥‖→0 as 𝑛→∞. This completes the proof.

Corollary 20. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space ℋ₁. Let 𝐴∈𝐵(ℋ₁,ℋ₂) and 𝐵_𝑖∶𝐶→ℋ₁ be 𝛽_𝑖-inverse strongly monotone for 𝑖=1,2. Let 𝑆∶𝐶→𝐶 be a 𝑘-strictly pseudo-contractive mapping such that Fix(𝑆)∩Ξ∩Γ≠∅. For fixed 𝑢∈𝐶 and given 𝑥₀∈𝐶 arbitrarily, let the sequences {𝑥_𝑛},{𝑢_𝑛},{𝑢_𝑛} be generated iteratively by 𝑢_𝑛=𝑃_𝐶𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛,𝑢_𝑛=𝑃_𝐶𝑢_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛,𝑦_𝑛=𝜎_𝑛𝑢+1−𝜎_𝑛𝑃_𝐶𝑢_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑦_𝑛+𝛿_𝑛𝑆𝑦_𝑛,∀𝑛≥0, (101) where 𝜇_𝑖∈(0,2𝛽_𝑖) for 𝑖=1,2, and the following conditions hold for six sequences {𝛼_𝑛}⊂(0,∞),{𝜆_𝑛}⊂(0,1/‖𝐴‖²) and {𝜎_𝑛},{𝛽_𝑛},{𝛾_𝑛},{𝛿_𝑛}⊂[0,1]:(i)∑^∞_𝑛=0𝛼_𝑛<∞;(ii)𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 and (𝛾_𝑛+𝛿_𝑛)𝑘≤𝛾_𝑛 for all 𝑛≥0;(iii)lim_𝑛→∞𝜎_𝑛=0 and ∑^∞_𝑛=0𝜎_𝑛=∞;(iv)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and liminf_𝑛→∞𝛿_𝑛>0;(v)lim_𝑛→∞(𝛾_𝑛+1/(1−𝛽_𝑛+1)−𝛾_𝑛/(1−𝛽_𝑛))=0;(vi)0<liminf_𝑛→∞𝜆_𝑛≤limsup_𝑛→∞𝜆_𝑛<1/‖𝐴‖²  and lim_𝑛→∞|𝜆_𝑛+1−𝜆_𝑛|=0. Then the sequences {𝑥_𝑛},{𝑢_𝑛},{𝑢_𝑛} converge strongly to the same point 𝑥=𝑃_{Fix(𝑆)∩Ξ∩Γ}𝑢 if and only if lim_𝑛→∞‖𝑢_𝑛+1−𝑢_𝑛‖=0. Furthermore, (𝑥,𝑦) is a solution of the GSVI (14), where 𝑦=𝑃_𝐶(𝑥−𝜇₂𝐵₂𝑥).

Next, utilizing Corollary 20 we give the following improvement and extension of the main result in [18] (i.e., [18, Theorem  3.1]).

Corollary 21. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space ℋ₁. Let 𝐴∈𝐵(ℋ₁,ℋ₂) and 𝑆∶𝐶→𝐶 be a nonexpansive mapping such that Fix(𝑆)∩Γ≠∅. For fixed 𝑢∈𝐶 and given 𝑥₀∈𝐶 arbitrarily, let the sequences {𝑢_𝑛},{𝑢_𝑛} be generated iteratively by 𝑢_𝑛=𝑃_𝐶𝑢_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛,𝑢_𝑛+1=𝛽_𝑛𝑢_𝑛+1−𝛽_𝑛×𝑆𝜎_𝑛𝑢+1−𝜎_𝑛𝑃_𝐶×𝑢_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛,∀𝑛≥0, (102) where the following conditions hold for four sequences {𝛼_𝑛}⊂(0,∞),{𝜆_𝑛}⊂(0,1/‖𝐴‖²) and {𝜎_𝑛},{𝛽_𝑛}⊂[0,1]:(i)∑^∞_𝑛=0𝛼_𝑛<∞;(ii)lim_𝑛→∞𝜎_𝑛=0 and ∑^∞_𝑛=0𝜎_𝑛=∞;(iii)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1;(iv)0<liminf_𝑛→∞𝜆_𝑛≤limsup_𝑛→∞𝜆_𝑛<1/‖𝐴‖² and lim_𝑛→∞|𝜆_𝑛+1−𝜆_𝑛|=0.
Then the sequences {𝑢_𝑛},{𝑢_𝑛} converge strongly to the same point 𝑥=𝑃_{Fix(𝑆)∩Γ}𝑢 if and only if lim_𝑛→∞‖𝑢_𝑛+1−𝑢_𝑛‖=0.

Proof. In Corollary 20, put 𝐵₁=𝐵₂=0 and 𝛾_𝑛=0. Then, Ξ=𝐶,  𝛽_𝑛+𝛿_𝑛=1 for all 𝑛≥0, and the iterative scheme (101) is equivalent to 𝑢_𝑛=𝑥_𝑛,𝑢_𝑛=𝑃_𝐶𝑢_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛,𝑦_𝑛=𝜎_𝑛𝑢+1−𝜎_𝑛𝑃_𝐶𝑢_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑢_𝑛,𝑢_𝑛+1=𝛽_𝑛𝑥_𝑛+𝛿_𝑛𝑆𝑦_𝑛,∀𝑛≥0. (103) This is is equivalent to (102). Since 𝑆 is a nonexpansive mapping, 𝑆 must be a 𝑘-strictly pseudo-contractive mapping with 𝑘=0. In this case, it is easy to see that conditions (i)–(vi) in Corollary 20 all are satisfied. Therefore, in terms of Corollary 20, we obtain the desired result.

Now, we are in a position to prove the strong convergence of the sequences generated by the relaxed extragradient iterative algorithm (18) with regularization.

Theorem 22. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space ℋ₁. Let 𝐴∈𝐵(ℋ₁,ℋ₂) and 𝐵_𝑖∶𝐶→ℋ₁ be 𝛽_𝑖-inverse strongly monotone for 𝑖=1,2. Let 𝑆∶𝐶→𝐶 be a 𝑘-strictly pseudocontractive mapping such that Fix(𝑆)∩Ξ∩Γ≠∅. Let 𝑄∶𝐶→𝐶 be a 𝜌-contraction with 𝜌∈[0,1/2). For given 𝑥₀∈𝐶 arbitrarily, let the sequences {𝑥_𝑛},{𝑦_𝑛},{𝑧_𝑛} be generated by the relaxed extragradient iterative algorithm (18) with regularization, where 𝜇_𝑖∈(0,2𝛽_𝑖) for 𝑖=1,2,{𝛼_𝑛}⊂(0,∞),{𝜆_𝑛}⊂(0,1/‖𝐴‖²) and {𝜎_𝑛},{𝜏_𝑛},{𝛽_𝑛},{𝛾_𝑛},{𝛿_𝑛}⊂[0,1] such that(i)∑^∞_𝑛=0𝛼_𝑛<∞;(ii)𝜎_𝑛+𝜏_𝑛≤1, 𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 and (𝛾_𝑛+𝛿_𝑛)𝑘≤𝛾_𝑛 for all 𝑛≥0;(iii)lim_𝑛→∞𝜎_𝑛=0 and ∑^∞_𝑛=0𝜎_𝑛=∞;(iv)0<liminf_𝑛→∞𝜏_𝑛≤limsup_𝑛→∞𝜏_𝑛<1 and lim_𝑛→∞|𝜏_𝑛+1−𝜏_𝑛|=0;(v)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and liminf_𝑛→∞𝛿_𝑛>0;(vi)lim_𝑛→∞(𝛾_𝑛+1/(1−𝛽_𝑛+1)−𝛾_𝑛/(1−𝛽_𝑛))=0;(vii)0<liminf_𝑛→∞𝜆_𝑛≤limsup_𝑛→∞𝜆_𝑛<1/‖𝐴‖² and lim_𝑛→∞|𝜆_𝑛+1−𝜆_𝑛|=0. Then the sequences {𝑥_𝑛},{𝑦_𝑛},{𝑧_𝑛} converge strongly to the same point 𝑥=𝑃_{Fix(𝑆)∩Ξ∩Γ}𝑄𝑥 if and only if lim_𝑛→∞‖𝑧_𝑛+1−𝑧_𝑛‖=0. Furthermore, (𝑥,𝑦) is a solution of the GSVI (14), where 𝑦=𝑃_𝐶(𝑥−𝜇₂𝐵₂𝑥).

Proof. First, taking into account 0<liminf_𝑛→∞𝜆_𝑛≤limsup_𝑛→∞𝜆_𝑛<1/‖𝐴‖², without loss of generality we may assume that {𝜆_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1/‖𝐴‖²). Repeating the same argument as that in the proof of Theorem 19, we can show that 𝑃_𝐶(𝐼−𝜆∇𝑓_𝛼) is 𝜁-averaged for each 𝜆∈(0,2/(𝛼+‖𝐴‖²)), where 𝜁=(2+𝜆(𝛼+‖𝐴‖²))/4. Further, repeating the same argument as that in the proof of Theorem 19, we can also show that for each integer 𝑛≥0,𝑃_𝐶(𝐼−𝜆_𝑛∇𝑓_𝛼𝑛) is 𝜁_𝑛-averaged with 𝜁_𝑛=(2+𝜆_𝑛(𝛼_𝑛+‖𝐴‖²))/4∈(0,1).
Next we divide the remainder of the proof into several steps.
Step  1. {𝑥_𝑛} is bounded.
Indeed, take 𝑝∈Fix(𝑆)∩Ξ∩Γ arbitrarily. Then 𝑆𝑝=𝑝,𝑃_𝐶(𝐼−𝜆∇𝑓)𝑝=𝑝 for 𝜆∈(0,2/‖𝐴‖²), and 𝑝=𝑃_𝐶𝑃_𝐶𝑝−𝜇₂𝐵₂𝑝−𝜇₁𝐵₁𝑃_𝐶𝑝−𝜇₂𝐵₂𝑝. (104) Utilizing the arguments similar to those of (45) and (46) in the proof of Theorem 19, from (18) we can obtain 𝑧_𝑛−𝑝≤𝑥_𝑛−𝑝+𝜆_𝑛𝛼_𝑛𝑝,𝑧_𝑛−𝑝²≤𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑧_𝑛−𝑝. (105)(106) For simplicity, we write 𝑞=𝑃_𝐶(𝑝−𝜇₂𝐵₂𝑝),  𝑧_𝑛=𝑃_𝐶(𝑧_𝑛−𝜇₂𝐵₂𝑧_𝑛), 𝑢_𝑛=𝑃_𝐶𝑃_𝐶𝑧_𝑛−𝜇₂𝐵₂𝑧_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑧_𝑛−𝜇₂𝐵₂𝑧_𝑛,𝑢_𝑛=𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑧_𝑛, (107) for each 𝑛≥0. Then 𝑦_𝑛=𝜎_𝑛𝑄𝑥_𝑛+𝜏_𝑛𝑢_𝑛+(1−𝜎_𝑛−𝜏_𝑛)𝑢_𝑛 for each 𝑛≥0. Since 𝐵_𝑖∶𝐶→ℋ₁ is 𝛽_𝑖-inverse strongly monotone and 0<𝜇_𝑖<2𝛽_𝑖 for 𝑖=1,2, utilizing the argument similar to that of (48) in the proof of Theorem 19, we can obtain that for all 𝑛≥0, 𝑢_𝑛−𝑝²≤𝑧_𝑛−𝑝²−𝜇₂2𝛽₂−𝜇₂𝐵₂𝑧_𝑛−𝐵₂𝑝²−𝜇₁2𝛽₁−𝜇₁𝐵₁𝑧_𝑛−𝐵₁𝑞²≤𝑧_𝑛−𝑝². (108) Furthermore, utilizing Proposition 8(i)-(ii) and the argument similar to that of (51) in the proof of Theorem 19, from (105) we obtain 𝑢_𝑛−𝑝²≤𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑧_𝑛+𝜆−𝑝_2𝑛𝛼_𝑛+‖𝐴‖²²−1𝑥_𝑛−𝑧_𝑛²≤𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑧_𝑛≤𝑥−𝑝_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑥_𝑛−𝑝+𝜆_𝑛𝛼_𝑛𝑝≤𝑥_𝑛−𝑝²+4𝜆_𝑛𝛼_𝑛𝑝𝑥_𝑛−𝑝+4𝜆²_𝑛𝛼²_𝑛𝑝²=𝑥_𝑛−𝑝+2𝜆_𝑛𝛼_𝑛𝑝². (109) Hence it follows from (105), (108), and (109) that 𝑦_𝑛=𝜎−𝑝_𝑛𝑄𝑥_𝑛−𝑝+𝜏_𝑛𝑢_𝑛−𝑝+1−𝜎_𝑛−𝜏_𝑛𝑢_𝑛−𝑝≤𝜎_𝑛𝑄𝑥_𝑛−𝑝+𝜏_𝑛𝑢_𝑛+−𝑝1−𝜎_𝑛−𝜏_𝑛𝑢_𝑛−𝑝≤𝜎_𝑛𝜌𝑥_𝑛−𝑝+𝑄𝑝−𝑝+𝜏_𝑛𝑢_𝑛−𝑝+1−𝜎_𝑛−𝜏_𝑛𝑧_𝑛−𝑝≤𝜎_𝑛𝜌𝑥_𝑛−𝑝+𝑄𝑝−𝑝+𝜏_𝑛𝑥_𝑛−𝑝+2𝜆_𝑛𝛼_𝑛𝑝+1−𝜎_𝑛−𝜏_𝑛𝑥_𝑛−𝑝+𝜆_𝑛𝛼_𝑛𝑝=1−1−𝜌𝜎_𝑛𝑥_𝑛−𝑝+𝜎_𝑛+𝑄𝑝−𝑝2𝜏_𝑛+1−𝜎_𝑛−𝜏_𝑛𝜆_𝑛𝛼_𝑛𝑝≤1−1−𝜌𝜎_𝑛𝑥_𝑛−𝑝+𝜎_𝑛𝑄𝑝−𝑝+2𝜆_𝑛𝛼_𝑛𝑝=1−1−𝜌𝜎_𝑛𝑥_𝑛+𝜎−𝑝1−𝜌_𝑛𝑄𝑝−𝑝1−𝜌+2𝜆_𝑛𝛼_𝑛𝑝≤max𝑥_𝑛−𝑝,𝑄𝑝−𝑝1−𝜌+2𝜆_𝑛𝛼_𝑛𝑝. (110) Since (𝛾_𝑛+𝛿_𝑛)𝑘≤𝛾_𝑛 for all 𝑛≥0, utilizing Lemma 17 we obtain from (110) 𝑥_𝑛+1=𝛽−𝑝_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑦_𝑛−𝑝+𝛿_𝑛𝑆𝑦_𝑛−𝑝≤𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑦_𝑛−𝑝+𝛿_𝑛𝑆𝑦_𝑛−𝑝≤𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛+𝛿_𝑛𝑦_𝑛−𝑝≤𝛽_𝑛𝑥_𝑛+𝛾−𝑝_𝑛+𝛿_𝑛max𝑥_𝑛−𝑝,𝑄𝑝−𝑝1−𝜌+2𝜆_𝑛𝛼_𝑛𝑝≤𝛽_𝑛𝑥_𝑛+𝛾−𝑝_𝑛+𝛿_𝑛max𝑥_𝑛−𝑝,𝑄𝑝−𝑝1−𝜌+2𝜆_𝑛𝛼_𝑛𝑝≤max𝑥_𝑛−𝑝,𝑄𝑝−𝑝𝑝𝛼1−𝜌+2𝑏_𝑛. (111) Repeating the same argument as that of (54) in the proof of Theorem 19, by induction we can prove that 𝑥_𝑛+1−𝑝≤max𝑥₀−𝑝,𝑄𝑝−𝑝𝑝1−𝜌+2𝑏^𝑛_𝑗=0𝛼_𝑗. (112) Thus, {𝑥_𝑛} is bounded. Since 𝑃_𝐶,∇𝑓_𝛼𝑛,𝐵₁ and 𝐵₂ are Lipschitz continuous, it is easy to see that {𝑧_𝑛},{𝑢_𝑛},{𝑢_𝑛},{𝑦_𝑛}, and {𝑧_𝑛} are bounded, where 𝑧_𝑛=𝑃_𝐶(𝑧_𝑛−𝜇₂𝐵₂𝑧_𝑛) for all 𝑛≥0.
Step  2. lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0.
Indeed, define 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑤_𝑛 for all 𝑛≥0. Then, utilizing the arguments similar to those of (58)–(61) in the proof of Theorem 19, we can obtain that 𝑤_𝑛+1−𝑤_𝑛=𝛾_𝑛+1𝑦_𝑛+1−𝑦_𝑛+𝛿_𝑛+1𝑆𝑦_𝑛+1−𝑆𝑦_𝑛1−𝛽_𝑛+1+𝛾_𝑛+11−𝛽_𝑛+1−𝛾_𝑛1−𝛽_𝑛𝑦_𝑛+𝛿_𝑛+11−𝛽_𝑛+1−𝛿_𝑛1−𝛽_𝑛𝑆𝑦_𝑛,𝛾_𝑛+1𝑦_𝑛+1−𝑦_𝑛+𝛿_𝑛+1𝑆𝑦_𝑛+1−𝑆𝑦_𝑛≤𝛾_𝑛+1+𝛿_𝑛+1𝑦_𝑛+1−𝑦_𝑛, (113) (due to Lemma 17) 𝑧_𝑛+1−𝑧_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝜆_𝑛+1−𝜆_𝑛∇𝑓𝑥_𝑛+𝜆_𝑛+1𝛼_𝑛+1−𝜆_𝑛𝛼_𝑛𝑥_𝑛,𝑢_𝑛+1−𝑢_𝑛²≤𝑃_𝐶𝑧_𝑛+1−𝜇₂𝐵₂𝑧_𝑛+1−𝑃_𝐶𝑧_𝑛−𝜇₂𝐵₂𝑧_𝑛²−𝜇₁2𝛽₁−𝜇₁𝐵₁𝑃_𝐶𝑧_𝑛+1−𝜇₂𝐵₂𝑧_𝑛+1−𝐵₁𝑃_𝐶𝑧_𝑛−𝜇₂𝐵₂𝑧_𝑛²≤𝑧_𝑛+1−𝜇₂𝐵₂𝑧_𝑛+1−𝑧_𝑛−𝜇₂𝐵₂𝑧_𝑛²≤𝑧_𝑛+1−𝑧_𝑛²−𝜇₂2𝛽₂−𝜇₂𝐵₂𝑧_𝑛+1−𝐵₂𝑧_𝑛²≤𝑧_𝑛+1−𝑧_𝑛². (114)(115) So, we have 𝑢_𝑛+1−𝑢_𝑛=𝑃_𝐶𝑥_𝑛+1−𝜆_𝑛+1∇𝑓_𝛼𝑛+1𝑧_𝑛+1−𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑧_𝑛≤𝑥_𝑛+1−𝜆_𝑛+1∇𝑓_𝛼𝑛+1𝑧_𝑛+1−𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑧_𝑛=𝑥_𝑛+1−𝑥_𝑛−𝜆_𝑛+1∇𝑓_𝛼𝑛+1𝑧_𝑛+1−∇𝑓_𝛼𝑛+1𝑧_𝑛−𝜆_𝑛+1∇𝑓_𝛼𝑛+1𝑧_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑧_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝜆_𝑛+1∇𝑓_𝛼𝑛+1𝑧_𝑛+1−∇𝑓_𝛼𝑛+1𝑧_𝑛+𝜆_𝑛+1∇𝑓_𝛼𝑛+1𝑧_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑧_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝜆_𝑛+1𝛼_𝑛+1+‖𝐴‖²𝑧_𝑛+1−𝑧_𝑛+𝜆_𝑛+1𝛼_𝑛+1𝐼+∇𝑓𝑧_𝑛−𝜆_𝑛𝛼_𝑛𝐼+∇𝑓𝑧_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝜆_𝑛+1𝛼_𝑛+1+‖𝐴‖²𝑧_𝑛+1−𝑧_𝑛+𝜆_𝑛+1𝛼_𝑛+1−𝜆_𝑛𝛼_𝑛𝑧_𝑛+𝜆_𝑛+1−𝜆_𝑛∇𝑓𝑧_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑧_𝑛+1−𝑧_𝑛+𝜆_𝑛+1𝛼_𝑛+1−𝜆_𝑛𝛼_𝑛𝑧_𝑛+𝜆_𝑛+1−𝜆_𝑛∇𝑓𝑧_𝑛. (116) This together with (114) implies that 𝑦_𝑛+1−𝑦_𝑛=𝜎_𝑛+1𝑄𝑥_𝑛+1−𝑢_𝑛+1+𝜏_𝑛+1𝑢_𝑛+1+1−𝜏_𝑛+1𝑢_𝑛+1−𝜎_𝑛𝑄𝑥_𝑛−𝑢_𝑛−𝜏_𝑛𝑢_𝑛−1−𝜏_𝑛𝑢_𝑛≤𝜏_𝑛+1𝑢_𝑛+1−𝜏_𝑛𝑢_𝑛+1−𝜏_𝑛+1𝑢_𝑛+1−1−𝜏_𝑛𝑢_𝑛+𝜎_𝑛+1𝑄𝑥_𝑛+1−𝑢_𝑛+1+𝜎_𝑛𝑄𝑥_𝑛−𝑢_𝑛≤𝜏_𝑛+1−𝜏_𝑛𝑢_𝑛+1+𝜏_𝑛𝑢_𝑛+1−𝑢_𝑛+𝜏_𝑛+1−𝜏_𝑛𝑢_𝑛+1+1−𝜏_𝑛𝑢_𝑛+1−𝑢_𝑛+𝜎_𝑛+1𝑄𝑥_𝑛+1−𝑢_𝑛+1+𝜎_𝑛𝑄𝑥_𝑛−𝑢_𝑛≤𝜏_𝑛𝑥_𝑛+1−𝑥_𝑛+𝑧_𝑛+1−𝑧_𝑛+𝜆_𝑛+1𝛼_𝑛+1−𝜆_𝑛𝛼_𝑛𝑧_𝑛+𝜆_𝑛+1−𝜆_𝑛∇𝑓𝑧_𝑛+1−𝜏_𝑛𝑧_𝑛+1−𝑧_𝑛+𝜏_𝑛+1−𝜏_𝑛𝑢_𝑛+1+𝑢_𝑛+1+𝜎_𝑛+1𝑄𝑥_𝑛+1−𝑢_𝑛+1+𝜎_𝑛𝑄𝑥_𝑛−𝑢_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑧_𝑛+1−𝑧_𝑛+𝜆_𝑛+1𝛼_𝑛+1−𝜆_𝑛𝛼_𝑛𝑧_𝑛+𝜆_𝑛+1−𝜆_𝑛∇𝑓𝑧_𝑛+𝜏_𝑛+1−𝜏_𝑛𝑢_𝑛+1+𝑢_𝑛+1+𝜎_𝑛+1𝑄𝑥_𝑛+1−𝑢_𝑛+1+𝜎_𝑛𝑄𝑥_𝑛−𝑢_𝑛. (117) Hence it follows from (113), and (117) that 𝑤_𝑛+1−𝑤_𝑛≤𝛾_𝑛+1𝑦_𝑛+1−𝑦_𝑛+𝛿_𝑛+1𝑆𝑦_𝑛+1−𝑆𝑦_𝑛1−𝛽_𝑛+1+𝛾_𝑛+11−𝛽_𝑛+1−𝛾_𝑛1−𝛽_𝑛𝑦_𝑛+𝛿_𝑛+11−𝛽_𝑛+1−𝛿_𝑛1−𝛽_𝑛𝑆𝑦_𝑛≤𝛾_𝑛+1+𝛿_𝑛+11−𝛽_𝑛+1𝑦_𝑛+1−𝑦_𝑛+𝛾_𝑛+11−𝛽_𝑛+1−𝛾_𝑛1−𝛽_𝑛𝑦_𝑛+𝑆𝑦_𝑛=𝑦_𝑛+1−𝑦_𝑛+𝛾_𝑛+11−𝛽_𝑛+1−𝛾_𝑛1−𝛽_𝑛𝑦_𝑛+𝑆𝑦_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑧_𝑛+1−𝑧_𝑛+𝜆_𝑛+1𝛼_𝑛+1−𝜆_𝑛𝛼_𝑛𝑧_𝑛+𝜆_𝑛+1−𝜆_𝑛∇𝑓𝑧_𝑛+𝜏_𝑛+1−𝜏_𝑛𝑢_𝑛+1+𝑢_𝑛+1+𝜎_𝑛+1𝑄𝑥_𝑛+1−𝑢_𝑛+1+𝜎_𝑛𝑄𝑥_𝑛−𝑢_𝑛+𝛾_𝑛+11−𝛽_𝑛+1−𝛾_𝑛1−𝛽_𝑛𝑦_𝑛+𝑆𝑦_𝑛. (118) Since {𝑥_𝑛},{𝑦_𝑛},{𝑧_𝑛},{𝑢_𝑛}, and {𝑢_𝑛} are bounded, it follows from conditions (i), (iii), (iv), (vi), and (vii) that limsup_𝑛→∞𝑤_𝑛+1−𝑤_𝑛−𝑥_𝑛+1−𝑥_𝑛≤limsup_𝑛→∞𝑧_𝑛+1−𝑧_𝑛+𝜆_𝑛+1𝛼_𝑛+1−𝜆_𝑛𝛼_𝑛𝑧_𝑛+𝜆_𝑛+1−𝜆_𝑛∇𝑓𝑧_𝑛+𝜏_𝑛+1−𝜏_𝑛𝑢_𝑛+1+𝑢_𝑛+1+𝜎_𝑛+1𝑄𝑥_𝑛+1−𝑢_𝑛+1+𝜎_𝑛𝑄𝑥_𝑛−𝑢_𝑛+𝛾_𝑛+11−𝛽_𝑛+1−𝛾_𝑛1−𝛽_𝑛𝑦_𝑛+𝑆𝑦_𝑛=0. (119) Hence by Lemma 14 we get lim_𝑛→∞‖𝑤_𝑛−𝑥_𝑛‖=0. Thus, lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=lim_𝑛→∞1−𝛽_𝑛𝑤_𝑛−𝑥_𝑛=0. (120)Step  3. lim_𝑛→∞‖𝐵₂𝑧_𝑛−𝐵₂𝑝‖=0,  lim_𝑛→∞‖𝐵₁𝑧_𝑛−𝐵₁𝑞‖=0, and lim_𝑛→∞‖𝑥_𝑛−𝑧_𝑛‖=0, where 𝑞=𝑃_𝐶(𝑝−𝜇₂𝐵₂𝑝).
Indeed, utilizing Lemma 17 and the convexity of ‖⋅‖², we obtain from (18) and (106)–(109) that 𝑥_𝑛+1−𝑝²=𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑦_𝑛−𝑝+𝛿_𝑛𝑆𝑦_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛+𝛿_𝑛×1𝛾_𝑛+𝛿_𝑛𝛾_𝑛𝑦_𝑛−𝑝+𝛿_𝑛𝑆𝑦_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛+𝛿_𝑛𝑦_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛+𝛿_𝑛×𝜎_𝑛𝑄𝑥_𝑛−𝑝²+𝜏_𝑛𝑢_𝑛−𝑝²+1−𝜎_𝑛−𝜏_𝑛𝑢_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝜎_𝑛𝑄𝑥_𝑛−𝑝²+𝛾_𝑛+𝛿_𝑛𝜏_𝑛𝑢_𝑛−𝑝²+1−𝜏_𝑛𝑢_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝜎_𝑛𝑄𝑥_𝑛−𝑝²+𝛾_𝑛+𝛿_𝑛×𝜏_𝑛𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑧_𝑛+𝜆−𝑝_2𝑛𝛼_𝑛+‖𝐴‖²²−1𝑥_𝑛−𝑧_𝑛²+1−𝜏_𝑛𝑧_𝑛−𝑝²−𝜇₂2𝛽₂−𝜇₂𝐵₂𝑧_𝑛−𝐵₂𝑝²−𝜇₁2𝛽₁−𝜇₁𝐵₁𝑧_𝑛−𝐵₁𝑞²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝜎_𝑛𝑄𝑥_𝑛−𝑝²+𝛾_𝑛+𝛿_𝑛×𝜏_𝑛𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑧_𝑛+𝜆−𝑝_2𝑛𝛼_𝑛+‖𝐴‖²²−1𝑥_𝑛−𝑧_𝑛²+1−𝜏_𝑛𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑧_𝑛−𝑝−𝜇₂2𝛽₂−𝜇₂𝐵₂𝑧_𝑛−𝐵₂𝑝²−𝜇₁2𝛽₁−𝜇₁𝐵₁𝑧_𝑛−𝐵₁𝑞²=𝛽_𝑛𝑥_𝑛−𝑝²+𝜎_𝑛𝑄𝑥_𝑛−𝑝²+𝛾_𝑛+𝛿_𝑛×𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑧_𝑛−𝑝−𝜏_𝑛1−𝜆²_𝑛𝛼_𝑛+‖𝐴‖²²𝑥_𝑛−𝑧_𝑛²−1−𝜏_𝑛×𝜇₂2𝛽₂−𝜇₂𝐵₂𝑧_𝑛−𝐵₂𝑝²+𝜇₁2𝛽₁−𝜇₁𝐵₁𝑧_𝑛−𝐵₁𝑞²≤𝑥_𝑛−𝑝²+𝜎_𝑛𝑄𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑧_𝑛−𝛾−𝑝_𝑛+𝛿_𝑛𝜏_𝑛1−𝜆²_𝑛𝛼_𝑛+‖𝐴‖²²𝑥_𝑛−𝑧_𝑛²+1−𝜏_𝑛𝜇₂2𝛽₂−𝜇₂𝐵₂𝑧_𝑛−𝐵₂𝑝²+𝜇₁2𝛽₁−𝜇₁𝐵₁𝑧_𝑛−𝐵₁𝑞². (121) Therefore, 𝛾_𝑛+𝛿_𝑛𝜏_𝑛1−𝜆²_𝑛𝛼_𝑛+‖𝐴‖²²𝑥_𝑛−𝑧_𝑛²+1−𝜏_𝑛𝜇₂2𝛽₂−𝜇₂𝐵₂𝑧_𝑛−𝐵₂𝑝²+𝜇₁2𝛽₁−𝜇₁𝐵₁𝑧_𝑛−𝐵₁𝑞²≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+𝜎_𝑛𝑄𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑧_𝑛≤𝑥−𝑝_𝑛−𝑝+𝑥_𝑛+1−𝑝𝑥_𝑛−𝑥_𝑛+1+𝜎_𝑛𝑄𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑧_𝑛−𝑝. (122) Since 𝛼_𝑛→0,𝜎_𝑛→0,‖𝑥_𝑛−𝑥_𝑛+1‖→0,liminf_𝑛→∞(𝛾_𝑛+𝛿_𝑛)>0,{𝜆_𝑛}⊂[𝑎,𝑏], and 0<liminf_𝑛→∞𝜏_𝑛≤limsup_𝑛→∞𝜏_𝑛<1, it follows that lim_𝑛→∞𝑥_𝑛−𝑧_𝑛=0,lim_𝑛→∞𝐵₁𝑧_𝑛−𝐵₁𝑞=0,lim_𝑛→∞𝐵₂𝑧_𝑛−𝐵₂𝑝=0. (123)
Step  4. lim_𝑛→∞‖𝑆𝑦_𝑛−𝑦_𝑛‖=0.
Indeed, observe that 𝑢_𝑛−𝑧_𝑛=𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑧_𝑛−𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑥_𝑛≤𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑧_𝑛−𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑥_𝑛=𝜆_𝑛∇𝑓_𝛼𝑛𝑧_𝑛−∇𝑓_𝛼𝑛𝑥_𝑛≤𝜆_𝑛𝛼_𝑛+‖𝐴‖²𝑧_𝑛−𝑥_𝑛. (124) This together with ‖𝑧_𝑛−𝑥_𝑛‖→0 implies that lim_𝑛→∞‖𝑢_𝑛−𝑧_𝑛‖=0 and hence lim_𝑛→∞‖𝑢_𝑛−𝑥_𝑛‖=0. Utilizing the arguments similar to those of (73) and (75) in the proof of Theorem 19 we can prove that 𝑧_𝑛−𝑞²≤𝑧_𝑛−𝑝²−𝑧_𝑛−𝑧_𝑛−𝑝−𝑞²+2𝜇₂𝑧_𝑛−𝑧_𝑛−𝑝−𝑞𝐵₂𝑧_𝑛−𝐵₂𝑝,𝑢_𝑛−𝑝²≤𝑧_𝑛−𝑞²−𝑧_𝑛−𝑢_𝑛+𝑝−𝑞²+2𝜇₁𝑧_𝑛−𝑢_𝑛+𝑝−𝑞𝐵₁𝑧_𝑛−𝐵₁𝑞. (125) Utilizing (106), (109), (125), we have 𝑦_𝑛−𝑝²=𝜎_𝑛𝑄𝑥_𝑛−𝑝+𝜏_𝑛𝑢_𝑛−𝑝+1−𝜎_𝑛−𝜏_𝑛𝑢_𝑛−𝑝²≤𝜎_𝑛𝑄𝑥_𝑛−𝑝²+𝜏_𝑛𝑢_𝑛−𝑝²+1−𝜎_𝑛−𝜏_𝑛𝑢_𝑛−𝑝²≤𝜎_𝑛𝑄𝑥_𝑛−𝑝²+𝜏_𝑛𝑢_𝑛−𝑝²+1−𝜏_𝑛𝑢_𝑛−𝑝²≤𝜎_𝑛𝑄𝑥_𝑛−𝑝²+𝜏_𝑛𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑧_𝑛−𝑝+1−𝜏_𝑛×𝑧_𝑛−𝑞²−𝑧_𝑛−𝑢_𝑛+𝑝−𝑞²+2𝜇₁𝑧_𝑛−𝑢_𝑛+𝑝−𝑞𝐵₁𝑧_𝑛−𝐵₁𝑞≤𝜎_𝑛𝑄𝑥_𝑛−𝑝²+𝜏_𝑛𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑧_𝑛−𝑝+1−𝜏_𝑛×𝑧_𝑛−𝑝²−𝑧_𝑛−𝑧_𝑛−𝑝−𝑞²+2𝜇₂𝑧_𝑛−𝑧_𝑛−𝑝−𝑞𝐵₂𝑧_𝑛−𝐵₂𝑝−𝑧_𝑛−𝑢_𝑛+𝑝−𝑞²+2𝜇₁𝑧_𝑛−𝑢_𝑛+𝑝−𝑞𝐵₁𝑧_𝑛−𝐵₁𝑞≤𝜎_𝑛𝑄𝑥_𝑛−𝑝²+𝜏_𝑛𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑧_𝑛−𝑝+1−𝜏_𝑛×𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑧_𝑛−𝑧−𝑝_𝑛−𝑧_𝑛−𝑝−𝑞²+2𝜇₂𝑧_𝑛−𝑧_𝑛−𝑝−𝑞𝐵₂𝑧_𝑛−𝐵₂𝑝−𝑧_𝑛−𝑢_𝑛+𝑝−𝑞²+2𝜇₁𝑧_𝑛−𝑢_𝑛+𝑝−𝑞𝐵₁𝑧_𝑛−𝐵₁𝑞≤𝑥_𝑛−𝑝²+𝜎_𝑛𝑄𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑧_𝑛−𝑝+2𝜇₂𝑧_𝑛−𝑧_𝑛−𝑝−𝑞𝐵₂𝑧_𝑛−𝐵₂𝑝+2𝜇₁𝑧_𝑛−𝑢_𝑛+𝑝−𝑞𝐵₁𝑧_𝑛−𝐵₁𝑞−1−𝜏_𝑛𝑧_𝑛−𝑧_𝑛−𝑝−𝑞²+𝑧_𝑛−𝑢_𝑛+𝑝−𝑞². (126)
Thus from (18) and (126) it follows that 𝑥_𝑛+1−𝑝²=𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑦_𝑛−𝑝+𝛿_𝑛𝑆𝑦_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛+𝛿_𝑛𝑦_𝑛−𝑝²=𝛽_𝑛𝑥_𝑛−𝑝²+1−𝛽_𝑛𝑦_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+1−𝛽_𝑛×𝑥_𝑛−𝑝²+𝜎_𝑛𝑄𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑧_𝑛−𝑝+2𝜇₂𝑧_𝑛−𝑧_𝑛−𝑝−𝑞𝐵₂𝑧_𝑛−𝐵₂𝑝+2𝜇₁𝑧_𝑛−𝑢_𝑛+𝑝−𝑞𝐵₁𝑧_𝑛−𝐵₁𝑞−1−𝜏_𝑛𝑧_𝑛−𝑧_𝑛−𝑝−𝑞²+𝑧_𝑛−𝑢_𝑛+𝑝−𝑞²≤𝑥_𝑛−𝑝²+𝜎_𝑛𝑄𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑧_𝑛−𝑝+2𝜇₂𝑧_𝑛−𝑧_𝑛−𝑝−𝑞𝐵₂𝑧_𝑛−𝐵₂𝑝+2𝜇₁𝑧_𝑛−𝑢_𝑛+𝑝−𝑞𝐵₁𝑧_𝑛−𝐵₁𝑞−1−𝛽_𝑛1−𝜏_𝑛×𝑧_𝑛−𝑧_𝑛−𝑝−𝑞²+𝑧_𝑛−𝑢_𝑛+𝑝−𝑞², (127) which hence implies that 1−𝛽_𝑛1−𝜏_𝑛×𝑧_𝑛−𝑧_𝑛−𝑝−𝑞²+𝑧_𝑛−𝑢_𝑛+𝑝−𝑞²≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+𝜎_𝑛𝑄𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑧_𝑛−𝑝+2𝜇₂𝑧_𝑛−𝑧_𝑛−𝑝−𝑞𝐵₂𝑧_𝑛−𝐵₂𝑝+2𝜇₁𝑧_𝑛−𝑢_𝑛+𝑝−𝑞𝐵₁𝑧_𝑛−𝐵₁𝑞≤𝑥_𝑛−𝑝+𝑥_𝑛+1−𝑝𝑥_𝑛−𝑥_𝑛+1+𝜎_𝑛𝑄𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑧_𝑛−𝑝+2𝜇₂𝑧_𝑛−𝑧_𝑛−𝑝−𝑞𝐵₂𝑧_𝑛−𝐵₂𝑝+2𝜇₁𝑧_𝑛−𝑢_𝑛+𝑝−𝑞𝐵₁𝑧_𝑛−𝐵₁𝑞. (128) Since limsup_𝑛→∞𝛽_𝑛<1,  limsup_𝑛→∞𝜏_𝑛<1,  {𝜆_𝑛}⊂[𝑎,𝑏],  𝛼_𝑛→0,  𝜎_𝑛→0,  ‖𝐵₂𝑧_𝑛−𝐵₂𝑝‖→0,  ‖𝐵₁𝑧_𝑛−𝐵₁𝑞‖→0 and ‖𝑥_𝑛+1−𝑥_𝑛‖→0, it follows from the boundedness of {𝑥_𝑛},{𝑢_𝑛},{𝑧_𝑛}, and {𝑧_𝑛} that lim_𝑛→∞𝑧_𝑛−𝑧_𝑛−𝑝−𝑞=0,lim_𝑛→∞𝑧_𝑛−𝑢_𝑛+𝑝−𝑞=0. (129) Consequently, it immediately follows that lim_𝑛→∞𝑧_𝑛−𝑢_𝑛=0,lim_𝑛→∞𝑢_𝑛−𝑢_𝑛=0. (130) Also, note that 𝑦_𝑛−𝑢_𝑛≤𝜎_𝑛𝑄𝑥_𝑛−𝑢_𝑛+1−𝜎_𝑛−𝜏_𝑛𝑢_𝑛−𝑢_𝑛⟶0. (131) This together with ‖𝑥_𝑛−𝑢_𝑛‖→0 implies that lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=0. (132) Since 𝛿_𝑛𝑆𝑦_𝑛−𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝛾_𝑛𝑦_𝑛−𝑥_𝑛, (133) it follows that lim_𝑛→∞𝑆𝑦_𝑛−𝑥_𝑛=0,lim_𝑛→∞𝑆𝑦_𝑛−𝑦_𝑛=0. (134)
Step  5. limsup_𝑛→∞⟨𝑄𝑥−𝑥,𝑥_𝑛−𝑥⟩≤0 where 𝑥=𝑃_{Fix(𝑆)∩Ξ∩Γ}𝑄𝑥.
Indeed, since {𝑥_𝑛} is bounded, there exists a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that limsup_𝑛→∞𝑄𝑥−𝑥,𝑥_𝑛−𝑥=lim_𝑖→∞𝑄𝑥−𝑥,𝑥_𝑛𝑖−𝑥. (135) Also, since 𝐻 is reflexive and {𝑥_𝑛} is bounded, without loss of generality we may assume that 𝑥_𝑛𝑖→𝑝 weakly for some 𝑝∈𝐶. Taking into account that ‖𝑥_𝑛−𝑦_𝑛‖→0 and ‖𝑥_𝑛−𝑧_𝑛‖→0 as 𝑛→∞, we deduce that 𝑦_𝑛𝑖→𝑝 weakly and 𝑧_𝑛𝑖→𝑝 weakly.
First, it is clear from Lemma 15 and ‖𝑆𝑦_𝑛−𝑦_𝑛‖→0 that 𝑝∈Fix(𝑆). Now let us show that 𝑝∈Ξ. Note that 𝑧_𝑛−𝐺𝑧_𝑛=𝑧_𝑛−𝑃_𝐶𝑃_𝐶𝑧_𝑛−𝜇₂𝐵₂𝑧_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑧_𝑛−𝜇₂𝐵₂𝑧_𝑛=𝑧_𝑛−𝑢_𝑛⟶0(𝑛⟶∞), (136) where 𝐺∶𝐶→𝐶 is defined as that in Lemma 4. According to Lemma 15 we get 𝑝∈Ξ. Further, let us show that 𝑝∈Γ. As a matter of fact, define (𝑣)𝑇𝑣=∇𝑓+𝑁_𝐶𝑣,if𝑣∈𝐶,∅,if𝑣∉𝐶, (137) where 𝑁_𝐶𝑣={𝑤∈ℋ₁∶⟨𝑣−𝑢,𝑤⟩≥0,∀𝑢∈𝐶}. Then, 𝑇 is maximal monotone and 0∈𝑇𝑣 if and only if 𝑣∈VI(𝐶,∇𝑓); see [40] for more details. Let (𝑣,𝑤)∈Gph(𝑇). Then, we have 𝑣−𝑢,𝑤−∇𝑓(𝑣)≥0,∀𝑢∈𝐶. (138) Observe that 𝑧_𝑛=𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑥_𝑛,𝑣∈𝐶. (139) Utilizing the arguments similar to those of Step  5 in the proof of Theorem 19 we can prove that 𝑣−𝑝,𝑤≥0. (140) Since 𝑇 is maximal monotone, we have 𝑝∈𝑇⁻¹0, and, hence, 𝑝∈VI(𝐶,∇𝑓). Thus it is clear that 𝑝∈Γ. Therefore, 𝑝∈Fix(𝑆)∩Ξ∩Γ. Consequently, in terms of Proposition 8 (i) we obtain from (135) that limsup_𝑛→∞𝑄𝑥−𝑥,𝑥_𝑛−𝑥=lim_𝑖→∞𝑄𝑥−𝑥,𝑥_𝑛𝑖−𝑥=𝑄𝑥−𝑥,𝑝−𝑥≤0. (141)
Step  6. lim_𝑛→∞‖𝑥_𝑛−𝑥‖=0.
Indeed, observe that 𝑄𝑥_𝑛−𝑥,𝑦_𝑛−𝑥=𝑄𝑥_𝑛−𝑥,𝑥_𝑛−𝑥+𝑄𝑥_𝑛−𝑥,𝑦_𝑛−𝑥_𝑛=𝑄𝑥_𝑛−𝑄𝑥,𝑥_𝑛−𝑥+𝑄𝑥−𝑥,𝑥_𝑛−𝑥+𝑄𝑥_𝑛−𝑥,𝑦_𝑛−𝑥_𝑛≤𝜌𝑥_𝑛−𝑥²+𝑄𝑥−𝑥,𝑥_𝑛−𝑥+𝑄𝑥_𝑛−𝑥𝑦_𝑛−𝑥_𝑛. (142) Utilizing Lemmas 17 and 18, we obtain from (106)–(109) and the convexity of ‖⋅‖² that 𝑥_𝑛+1−𝑥²=𝛽_𝑛𝑥_𝑛−𝑥+𝛾_𝑛𝑦_𝑛−𝑥+𝛿_𝑛𝑆𝑦_𝑛−𝑥²≤𝛽_𝑛𝑥_𝑛−𝑥²+𝛾_𝑛+𝛿_𝑛1𝛾_𝑛+𝛿_𝑛𝛾_𝑛𝑦_𝑛−𝑥+𝛿_𝑛𝑆𝑦_𝑛−𝑥²≤𝛽_𝑛𝑥_𝑛−𝑥²+𝛾_𝑛+𝛿_𝑛𝑦_𝑛−𝑥²≤𝛽_𝑛𝑥_𝑛−𝑥²+𝛾_𝑛+𝛿_𝑛×𝜏_𝑛𝑢_𝑛−𝑥+1−𝜎_𝑛−𝜏_𝑛𝑢_𝑛−𝑥²+2𝜎_𝑛𝑄𝑥_𝑛−𝑥,𝑦_𝑛−𝑥≤𝛽_𝑛𝑥_𝑛−𝑥²+𝛾_𝑛+𝛿_𝑛×𝜏_𝑛𝑢_𝑛−𝑥²+1−𝜎_𝑛−𝜏_𝑛𝑢_𝑛−𝑥²+2𝜎_𝑛𝑄𝑥_𝑛−𝑥,𝑦_𝑛−𝑥≤𝛽_𝑛𝑥_𝑛−𝑥²+𝛾_𝑛+𝛿_𝑛×𝜏_𝑛𝑢_𝑛−𝑥²+1−𝜎_𝑛−𝜏_𝑛𝑧_𝑛−𝑥²+2𝜎_𝑛𝑄𝑥_𝑛−𝑥,𝑦_𝑛−𝑥≤𝛽_𝑛𝑥_𝑛−𝑥²+𝛾_𝑛+𝛿_𝑛×𝜏_𝑛𝑥_𝑛−𝑥²+2𝜆_𝑛𝛼_𝑛‖𝑥‖𝑧_𝑛−𝑥+1−𝜎_𝑛−𝜏_𝑛×𝑥_𝑛−𝑥²+2𝜆_𝑛𝛼_𝑛‖𝑥‖𝑧_𝑛−𝑥+2𝜎_𝑛𝑄𝑥_𝑛−𝑥,𝑦_𝑛−𝑥=𝛽_𝑛𝑥_𝑛−𝑥²+𝛾_𝑛+𝛿_𝑛×1−𝜎_𝑛𝑥_𝑛−𝑥²+2𝜆_𝑛𝛼_𝑛‖𝑥‖𝑧_𝑛−𝑥+2𝜎_𝑛𝑄𝑥_𝑛−𝑥,𝑦_𝑛−𝑥=1−𝛾_𝑛+𝛿_𝑛𝜎_𝑛𝑥_𝑛−𝑥²+𝛾_𝑛+𝛿_𝑛1−𝜎_𝑛2𝜆_𝑛𝛼_𝑛‖𝑥‖𝑧_𝑛−𝑥+𝛾_𝑛+𝛿_𝑛2𝜎_𝑛𝑄𝑥_𝑛−𝑥,𝑦_𝑛−𝑥≤1−𝛾_𝑛+𝛿_𝑛𝜎_𝑛𝑥_𝑛−𝑥²+𝛾_𝑛+𝛿_𝑛2𝜎_𝑛×𝜌𝑥_𝑛−𝑥²+𝑄𝑥−𝑥,𝑥_𝑛−𝑥+𝑄𝑥_𝑛−𝑥𝑦_𝑛−𝑥_𝑛+2𝜆_𝑛𝛼_𝑛‖𝑥‖𝑧_𝑛−𝑥=1−1−2𝜌𝛾_𝑛+𝛿_𝑛𝜎_𝑛𝑥_𝑛−𝑥²+1−2𝜌𝛾_𝑛+𝛿_𝑛𝜎_𝑛×2𝑄𝑥−𝑥,𝑥_𝑛−𝑥+𝑄𝑥_𝑛−𝑥𝑦_𝑛−𝑥_𝑛1−2𝜌+2𝜆_𝑛𝛼_𝑛‖𝑥‖𝑧_𝑛−𝑥. (143) Note that liminf_𝑛→∞(1−2𝜌)(𝛾_𝑛+𝛿_𝑛)>0. It follows that ∑^∞_𝑛=0(1−2𝜌)(𝛾_𝑛+𝛿_𝑛)𝜎_𝑛=∞. It is clear that limsup_𝑛→∞2𝑄𝑥−𝑥,𝑥_𝑛−𝑥+𝑄𝑥_𝑛−𝑥𝑦_𝑛−𝑥_𝑛1−2𝜌≤0, (144) because limsup_𝑛→∞⟨𝑄𝑥−𝑥,𝑥_𝑛−𝑥⟩≤0 and lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0. In addition, note also that {𝜆_𝑛}⊂[𝑎,𝑏],∑^∞_𝑛=0𝛼_𝑛<∞ and {𝑧_𝑛} is bounded. Hence we get ∑^∞_𝑛=02𝜆_𝑛𝛼_𝑛‖𝑥‖‖𝑧_𝑛−𝑥‖<∞. Therefore, all conditions of Lemma 16 are satisfied. Consequently, we immediately deduce that ‖𝑥_𝑛−𝑥‖→0 as 𝑛→∞. In the meantime, taking into account that ‖𝑥_𝑛−𝑦_𝑛‖→0 and ‖𝑥_𝑛−𝑧_𝑛‖→0 as 𝑛→∞, we infer that lim_𝑛→∞𝑦_𝑛−𝑥=lim_𝑛→∞𝑧_𝑛−𝑥=0. (145) This completes the proof.

Corollary 23. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space ℋ₁. Let 𝐴∈𝐵(ℋ₁,ℋ₂) and let 𝐵_𝑖∶𝐶→ℋ₁ be 𝛽_𝑖-inverse strongly monotone for 𝑖=1,2. Let 𝑆∶𝐶→𝐶 be a 𝑘-strictly pseudocontractive mapping such that Fix(𝑆)∩Ξ∩Γ≠∅. For fixed 𝑢∈𝐶 and given 𝑥₀∈𝐶 arbitrarily, let the sequences {𝑥_𝑛},{𝑦_𝑛},{𝑧_𝑛} be generated iteratively by 𝑧_𝑛=𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑥_𝑛,𝑦_𝑛=𝜎_𝑛𝑢+𝜏_𝑛𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑧_𝑛+1−𝜎_𝑛−𝜏_𝑛×𝑃_𝐶𝑃_𝐶𝑧_𝑛−𝜇₂𝐵₂𝑧_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑧_𝑛−𝜇₂𝐵₂𝑧_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑦_𝑛+𝛿_𝑛𝑆𝑦_𝑛,∀𝑛≥0, (146) where 𝜇_𝑖∈(0,2𝛽_𝑖) for 𝑖=1,2,{𝛼_𝑛}⊂(0,∞),{𝜆_𝑛}⊂(0,1/‖𝐴‖²) and {𝜎_𝑛},{𝜏_𝑛},{𝛽_𝑛},{𝛾_𝑛}, {𝛿_𝑛}⊂[0,1] such that(i)∑^∞_𝑛=0𝛼_𝑛<∞;(ii)𝜎_𝑛+𝜏_𝑛≤1, 𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 and (𝛾_𝑛+𝛿_𝑛)𝑘≤𝛾_𝑛 for all 𝑛≥0;(iii)lim_𝑛→∞𝜎_𝑛=0 and ∑^∞_𝑛=0𝜎_𝑛=∞;(iv)0<liminf_𝑛→∞𝜏_𝑛≤limsup_𝑛→∞𝜏_𝑛<1 and lim_𝑛→∞|𝜏_𝑛+1−𝜏_𝑛|=0;(v)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and liminf_𝑛→∞𝛿_𝑛>0;(vi)lim_𝑛→∞(𝛾_𝑛+1/(1−𝛽_𝑛+1)−𝛾_𝑛/(1−𝛽_𝑛))=0;(vii)0<liminf_𝑛→∞𝜆_𝑛≤limsup_𝑛→∞𝜆_𝑛<1/‖𝐴‖² and lim_𝑛→∞|𝜆_𝑛+1−𝜆_𝑛|=0. Then the sequences {𝑥_𝑛},{𝑦_𝑛},{𝑧_𝑛} converge strongly to the same point 𝑥=𝑃_{Fix(𝑆)∩Ξ∩Γ}𝑢 if and only if lim_𝑛→∞‖𝑧_𝑛+1−𝑧_𝑛‖=0. Furthermore, (𝑥,𝑦) is a solution of the GSVI (14), where 𝑦=𝑃_𝐶(𝑥−𝜇₂𝐵₂𝑥).

Corollary 24. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space ℋ₁. Let 𝐴∈𝐵(ℋ₁,ℋ₂) and let 𝐵_𝑖∶𝐶→ℋ₁ be 𝛽_𝑖-inverse strongly monotone for 𝑖=1,2. Let 𝑆∶𝐶→𝐶 be a nonexpansive mapping such that Fix(𝑆)∩Ξ∩Γ≠∅. Let 𝑄∶𝐶→𝐶 be a 𝜌-contraction with 𝜌∈[0,1/2). For given 𝑥₀∈𝐶 arbitrarily, let the sequences {𝑥_𝑛},{𝑦_𝑛},{𝑧_𝑛} be generated iteratively by 𝑧_𝑛=𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑥_𝑛,𝑦_𝑛=𝜎_𝑛𝑄𝑥_𝑛+𝜏_𝑛𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑧_𝑛+1−𝜎_𝑛−𝜏_𝑛×𝑃_𝐶𝑃_𝐶𝑧_𝑛−𝜇₂𝐵₂𝑧_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑧_𝑛−𝜇₂𝐵₂𝑧_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑦_𝑛+𝛿_𝑛𝑆𝑦_𝑛,∀𝑛≥0, (147) where 𝜇_𝑖∈(0,2𝛽_𝑖) for 𝑖=1,2,{𝛼_𝑛}⊂(0,∞),{𝜆_𝑛}⊂(0,1/‖𝐴‖²) and {𝜎_𝑛},{𝜏_𝑛},{𝛽_𝑛},{𝛾_𝑛}, {𝛿_𝑛}⊂[0,1] such that(i)∑^∞_𝑛=0𝛼_𝑛<∞;(ii)𝜎_𝑛+𝜏_𝑛≤1, 𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 and (𝛾_𝑛+𝛿_𝑛)𝑘≤𝛾_𝑛 for all 𝑛≥0;(iii)lim_𝑛→∞𝜎_𝑛=0 and ∑^∞_𝑛=0𝜎_𝑛=∞;(iv)0<liminf_𝑛→∞𝜏_𝑛≤limsup_𝑛→∞𝜏_𝑛<1 and lim_𝑛→∞|𝜏_𝑛+1−𝜏_𝑛|=0;(v)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and liminf_𝑛→∞𝛿_𝑛>0;(vi)lim_𝑛→∞(𝛾_𝑛+1/(1−𝛽_𝑛+1)−𝛾_𝑛/(1−𝛽_𝑛))=0;(vii)0<liminf_𝑛→∞𝜆_𝑛≤limsup_𝑛→∞𝜆_𝑛<1/‖𝐴‖² and lim_𝑛→∞|𝜆_𝑛+1−𝜆_𝑛|=0. Then the sequences {𝑥_𝑛},{𝑦_𝑛},{𝑧_𝑛} converge strongly to the same point 𝑥=𝑃_{Fix(𝑆)∩Ξ∩Γ}𝑄𝑥 if and only if lim_𝑛→∞‖𝑧_𝑛+1−𝑧_𝑛‖=0. Furthermore, (𝑥,𝑦) is a solution of the GSVI (14), where 𝑦=𝑃_𝐶(𝑥−𝜇₂𝐵₂𝑥).

Next, utilizing Corollary 23 one gives the following improvement and extension of the main result in [18] (i.e., [18, Theorem  3.1]).

Corollary 25. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space ℋ₁. Let 𝐴∈𝐵(ℋ₁,ℋ₂) and let 𝑆∶𝐶→𝐶 be a nonexpansive mapping such that Fix(𝑆)∩Γ≠∅. For fixed 𝑢∈𝐶 and given 𝑥₀∈𝐶 arbitrarily, let the sequences {𝑥_𝑛},{𝑧_𝑛} be generated iteratively by 𝑧_𝑛=𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛×𝑆𝜎_𝑛𝑢+𝜏_𝑛𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑧_𝑛+1−𝜎_𝑛−𝜏_𝑛𝑧_𝑛,∀𝑛≥0, (148) where {𝛼_𝑛}⊂(0,∞),{𝜆_𝑛}⊂(0,1/‖𝐴‖²) and {𝜎_𝑛},{𝜏_𝑛},{𝛽_𝑛}⊂[0,1] such that(i)∑^∞_𝑛=0𝛼_𝑛<∞;(ii)𝜎_𝑛+𝜏_𝑛≤1 for all 𝑛≥0;(iii)lim_𝑛→∞𝜎_𝑛=0 and ∑^∞_𝑛=0𝜎_𝑛=∞;(iv)0<liminf_𝑛→∞𝜏_𝑛≤limsup_𝑛→∞𝜏_𝑛<1 and lim_𝑛→∞|𝜏_𝑛+1−𝜏_𝑛|=0;(v)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1;(vi)0<liminf_𝑛→∞𝜆_𝑛≤limsup_𝑛→∞𝜆_𝑛<1/‖𝐴‖² and lim_𝑛→∞|𝜆_𝑛+1−𝜆_𝑛|=0. Then the sequences {𝑥_𝑛},{𝑧_𝑛} converge strongly to the same point 𝑥=𝑃_{Fix(𝑆)∩Γ}𝑢 if and only if lim_𝑛→∞‖𝑧_𝑛+1−𝑧_𝑛‖=0.

Proof. In Corollary 23, put 𝐵₁=𝐵₂=0 and 𝛾_𝑛=0. Then, Ξ=𝐶,𝛽_𝑛+𝛿_𝑛=1, 𝑃_𝐶[𝑃_𝐶(𝑧_𝑛−𝜇₂𝐵₂𝑧_𝑛)−𝜇₁𝐵₁𝑃_𝐶(𝑧_𝑛−𝜇₂𝐵₂𝑧_𝑛)]=𝑧_𝑛, and the iterative scheme (146) is equivalent to 𝑧_𝑛=𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑥_𝑛,𝑦_𝑛=𝜎_𝑛𝑢+𝜏_𝑛𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑧_𝑛+1−𝜎_𝑛−𝜏_𝑛𝑧_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+𝛿_𝑛𝑆𝑦_𝑛,∀𝑛≥0. (149) This is equivalent to (148). Since 𝑆 is a nonexpansive mapping, 𝑆 must be a 𝑘-strictly pseudocontractive mapping with 𝑘=0. In this case, it is easy to see that conditions (i)–(vii) in Corollary 23 all are satisfied. Therefore, in terms of Corollary 23, we obtain the desired result.

Remark 26. Our Theorems 19 and 22 improve, extend, and develop [6, Theorem  5.7], [18, Theorem  3.1], and [33, Theorem  3.1] in the following aspects.(i)Because both [6, Theorem  5.7] and [18, Theorem  3.1] are weak convergence results for solving the SFP, beyond question, our Theorems 19 and 22 as strong convergence results are very interesting and quite valuable.(ii)The problem of finding an element of Fix(𝑆)∩Ξ∩Γ in our Theorems 19 and 22 is more general than the corresponding problems in [6, Theorem  5.7] and [18, Theorem  3.1], respectively.(iii)The relaxed extragradient iterative method for finding an element of Fix(𝑆)∩Ξ∩VI(𝐶,𝐴) in [33, Theorem  3.1] is extended to develop the relaxed extragradient method with regularization for finding an element of Fix(𝑆)∩Ξ∩Γ in our Theorem 19.(iv)The proof of our Theorems 19 and 22 is very different from that of [33, Theorem  3.1] because our argument technique depends on Lemma 16, the restriction on the regularization parameter sequence {𝛼_𝑛}, and the properties of the averaged mappings 𝑃_𝐶(𝐼−𝜆_𝑛∇𝑓_𝛼𝑛) to a great extent.(v)Because our iterative schemes (17) and (18) involve a contractive self-mapping 𝑄, a 𝑘-strictly pseudo-contractive self-mapping 𝑆, and several parameter sequences, they are more flexible and more subtle than the corresponding ones in [6, Theorem  5.7] and [18, Theorem  3.1], respectively.

Acknowledgment

The work of L. C. Ceng was partially supported by the National Science Foundation of China (11071169), Ph. D. Program Foundation of Ministry of Education of China (20123127110002). The work of J. C. Yao was partially supported by the Grant NSC 99-2115-M-037-002-MY3 of Taiwan. For A. Petruşsel, this work was possible with the nancial support of a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0094.