Some Modified Extragradient Methods for Solving Split Feasibility and Fixed Point Problems

Abstract

We consider and study the modified extragradient methods for finding a common element of the solution set Γ of a split feasibility problem (SFP) and the fixed point set Fix(𝑆) of a strictly pseudocontractive mapping 𝑆 in the setting of infinite-dimensional Hilbert spaces. We propose an extragradient algorithm for finding an element of Fix(𝑆)∩Γ where 𝑆 is strictly pseudocontractive. It is proven that the sequences generated by the proposed algorithm converge weakly to an element of Fix(𝑆)∩Γ. We also propose another extragradient-like algorithm for finding an element of Fix(𝑆)∩Γ where 𝑆∶𝐶→𝐶 is nonexpansive. It is shown that the sequences generated by the proposed algorithm converge strongly to an element of Fix(𝑆)∩Γ.

1. Introduction

Let ℋ be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖. Let 𝐶 be a nonempty closed convex subset of ℋ and let 𝑃_𝐶 be the metric projection from ℋ onto 𝐶. Let 𝑆∶𝐶→𝐶 be a self-mapping on 𝐶. We denote by Fix(𝑆) the set of fixed points of 𝑆 and by 𝐑 the set of all real numbers.

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. (1.1) For a given mapping 𝐴∶𝐶→ℋ, we consider the following variational inequality (VI) of finding 𝑥^∗∈𝐶 such that 𝐴𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐶. (1.2) The solution set of the VI (1.2) is denoted by VI(𝐶,𝐴). The variational inequality was first discussed by Lions [1] and now is well known. 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, equilibrium problems; see, for example, [2–4].

A mapping 𝑆∶𝐶→𝐶 is called 𝑘-strictly pseudocontractive if there exists a constant 𝑘∈[0,1) such that 𝑆𝑥−𝑆𝑦²≤𝑥−𝑦²+𝑘(𝐼−𝑆)𝑥−(𝐼−𝑆)𝑦²,∀𝑥,𝑦∈𝐶; (1.3) see [5]. We denote by Fix(𝑆) the fixed point set of 𝑆; that is, Fix(𝑆)={𝑥∈𝐶∶𝑆𝑥=𝑥}. In particular, if 𝑘=0, then 𝑆 is called a nonexpansive mapping. In 2003, for finding an element of Fix(𝑆)∩VI(𝐶,𝐴) when 𝐶⊂ℋ is nonempty, closed and convex, 𝑆∶𝐶→𝐶 is nonexpansive and 𝐴∶𝐶→ℋ is 𝛼-inverse strongly monotone, Takahashi and Toyoda [6] introduced the following Mann’s type iterative algorithm: 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑆𝑃_𝐶𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛,∀𝑛≥0, (1.4) where 𝑥₀∈𝐶 chosen arbitrarily, {𝛼_𝑛} is a sequence in (0,1), and {𝜆_𝑛} is a sequence in (0,2𝛼). They showed that if Fix(𝑆)∩VI(𝐶,𝐴)≠∅, then the sequence {𝑥_𝑛} converges weakly to some 𝑧∈Fix(𝑆)∩VI(𝐶,𝐴). Further, motivated by the idea of Korpelevich’s extragradient method [7], Nadezhkina and Takahashi [8] introduced an iterative algorithm for finding a common element of the fixed point set of a nonexpansive mapping and the solution set of a variational inequality problem for a monotone, Lipschitz continuous mapping in a real Hilbert space. They obtained a weak convergence theorem for two sequences generated by the proposed algorithm. Here the so-called extragradient method was first introduced by Korpelevich [7]. In 1976, She applied this method for finding a solution of a saddle point problem and proved the convergence of the proposed algorithm to a solution of this saddle point problem. Very recently, Jung [9] introduced a new composite iterative scheme by the viscosity approximation method and proved the strong convergence of the proposed scheme to a common element of the fixed point set of a nonexpansive mapping and the solution set of a variational inequality for an inverse-strongly monotone mapping in a Hilbert space.

On the other hand, 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 following property: 𝑥^∗∈𝐶,𝐴𝑥^∗∈𝑄, (1.5) where 𝐴∈𝐵(ℋ₁,ℋ₂) and 𝐵(ℋ₁,ℋ₂) denote the family of all bounded linear operators from ℋ₁ to ℋ₂.

In 1994, the SFP was first introduced by Censor and Elfving [10], 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, [11] and the references therein. Recently, it is found that the SFP can also be applied to study intensity-modulated radiation therapy (IMRT) [12–14]. 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, [11, 13, 15–19] and the references therein (see also [20] for relevant projection methods for solving image recovery problems). A special case of the SFP is the following convex constrained linear inverse problem [21] of finding an element 𝑥 such that 𝑥∈𝐶,𝐴𝑥=𝑏. (1.6) It has been extensively investigated in the literature using the projected Landweber iterative method [22]. 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 [23] can be extended to solve the SFP remains an interesting open topics. The original algorithm given in [10] involves the computation of the inverse 𝐴⁻¹ (assuming the existence of the inverse of 𝐴), and thus, did not become popular. A seemingly more popular algorithm that solves the SFP is the 𝐶𝑄 algorithm of Byrne [11, 15] 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 [24]. 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.

In 2010, Xu [25] gave a continuation of the study on the 𝐶𝑄 algorithm and its convergence. He applied Mann’s algorithm to the SFP and proposed an averaged 𝐶𝑄 algorithm which was proved to be weakly convergent to a solution of the SFP. He derived a weak convergence result, which shows that for suitable choices of iterative parameters (including the regularization), the sequence of iterative solutions can converge weakly to an exact solution of the SFP.

Very recently, Ceng et al. [26] introduced and studied an extragradient method with regularization for finding a common element of the solution set Γ of the SFP and the set Fix(𝑆) of fixed points of a nonexpansive mapping 𝑆 in the setting of infinite-dimensional Hilbert spaces. By combining the regularization method and extragradient method due to Nadezhkina and Takahashi [8], the authors proposed an iterative algorithm for finding an element of Fix(𝑆)∩Γ. The authors proved that the sequences generated by the proposed algorithm converge weakly to an element 𝑧∈Fix(𝑆)∩Γ.

The purpose of this paper is to investigate modified extragradient methods for finding a common element of the solution set Γ of the SFP and the fixed point set Fix(𝑆) of a strictly pseudocontractive mapping 𝑆 in the setting of infinite-dimensional Hilbert spaces. Assume that Fix(𝑆)∩Γ≠∅. By combining the regularization method and Nadezhkina and Takahashi’s extragradient method [8], we propose an extragradient algorithm for finding an element of Fix(𝑆)∩Γ. It is proven that the sequences generated by the proposed algorithm converge weakly to an element of Fix(𝑆)∩Γ. This result represents the supplementation, improvement, and extension of the corresponding results in [25, 26]; for example, [25, Theorem 5.7] and [26, Theorem 3.1]. On the other hand, by combining the regularization method and Jung’s composite viscosity approximation method [9], we also propose another extragradient-like algorithm for finding an element of Fix(𝑆)∩Γ where 𝑆∶𝐶→𝐶 is nonexpansive. It is shown that the sequences generated by the proposed algorithm converge strongly to an element of Fix(𝑆)∩Γ. Such a result substantially develops and improves the corresponding results in [9, 25, 26]; for example, [25, Theorem 5.7], [26, Theorem 3.1], and [9, Theorem 3.1]. It is worth pointing out that our results are new and novel in the Hilbert spaces setting. Essentially new approaches for finding the fixed points of strictly pseudocontractive mappings (including nonexpansive mappings) and solutions of the SFP are provided.

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}, that is, 𝜔_𝑤𝑥_𝑛∶=𝑥∶𝑥_𝑛𝑖⇀𝑥forsomesubsequence𝑥_𝑛𝑖of𝑥_𝑛. (2.1)

Recall that 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). (2.2)

Some important properties of projections are gathered in the following proposition.

Proposition 2.1. 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 2.2. A mapping 𝑇∶ℋ→ℋ is said to be(a)nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈ℋ; (2.3)(b)firmly nonexpansive if 2𝑇−𝐼 is nonexpansive, or equivalently, 𝑥−𝑦,𝑇𝑥−𝑇𝑦≥𝑇𝑥−𝑇𝑦²,∀𝑥,𝑦∈ℋ; (2.4)alternatively, 𝑇 is firmly nonexpansive if and only if 𝑇 can be expressed as 1𝑇=2(𝐼+𝑆), (2.5) where 𝑆∶ℋ→ℋ is nonexpansive, projections are firmly nonexpansive.

Definition 2.3. Let 𝑇 be a nonlinear operator with domain 𝐷(𝑇)⊆ℋ and range 𝑅(𝑇)⊆ℋ, and let 𝛽>0 and 𝜈>0 be given constants. The operator 𝑇 is called:(a)monotone if 𝑥−𝑦,𝑇𝑥−𝑇𝑦≥0,∀𝑥,𝑦∈𝐷(𝑇). (2.6)(b)𝛽-strongly monotone if 𝑥−𝑦,𝑇𝑥−𝑇𝑦≥𝛽𝑥−𝑦²,∀𝑥,𝑦∈𝐷(𝑇). (2.7)(c)𝜈-inverse strongly monotone (𝜈-ism) if 𝑥−𝑦,𝑇𝑥−𝑇𝑦≥𝜈𝑇𝑥−𝑇𝑦²,∀𝑥,𝑦∈𝐷(𝑇). (2.8)

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 to solve practical problems in various fields, for instance, in traffic assignment problems; see, for example, [27, 28].

Definition 2.4. 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, (2.9) 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 2.5 (see [15]). 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 complement𝐼−𝑇 is 𝜈-ism for some𝜈>1/2. Indeed, for 𝛼∈(0,1),𝑇 is 𝛼-averaged if and only if 𝐼−𝑇 is (1/2𝛼)-ism.

Proposition 2.6 (see [15, 29]). Let 𝑆,𝑇,𝑉∶ℋ→ℋ be given operators.(i)If 𝑇=(1−𝛼)𝑆+𝛼𝑉 for some 𝛼∈(0,1), 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), 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 𝑇₂is 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, thenFix𝑇^𝑁_𝑖=1𝑖=Fix𝑇₁⋯𝑇_𝑁. (2.10)The notation Fix(𝑇) denotes the set of all fixed points of the mapping 𝑇, that is, Fix(𝑇)={𝑥∈ℋ∶𝑇𝑥=𝑥}.

On the other hand, it is clear that, in a real Hilbert space ℋ, 𝑆∶𝐶→𝐶 is 𝑘-strictly pseudocontractive if and only if there holds the following inequality: 𝑆𝑥−𝑆𝑦,𝑥−𝑦≤𝑥−𝑦²−1−𝑘2(𝐼−𝑆)𝑥−(𝐼−𝑆)𝑦²,∀𝑥,𝑦∈𝐶. (2.11) This immediately implies that if 𝑆 is a 𝑘-strictly pseudocontractive mapping, then 𝐼−𝑆 is ((1−𝑘)/2)-inverse strongly monotone; for further detail, we refer to [5] and the references therein. It is well known that the class of strict pseudocontractions strictly includes the class of nonexpansive mappings. The so-called demiclosedness principle for strict pseudocontractive mappings in the following lemma will often be used.

Lemma 2.7 (see [5, Proposition 2.1]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space ℋ and 𝑆∶𝐶→𝐶 be a mapping.(i)If 𝑆 is a 𝑘-strict pseudocontractive mapping, then 𝑆 satisfies the Lipschitz condition 𝑆𝑥−𝑆𝑦≤1+𝑘1−𝑘𝑥−𝑦,∀𝑥,𝑦∈𝐶. (2.12)(ii)If 𝑆 is a 𝑘-strict pseudocontractive 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 and (𝐼−𝑆)𝑥_𝑛→0, 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 elementary result on real sequences is quite well known.

Lemma 2.8 (see [30, page 80]). Let {𝑎_𝑛}^∞_𝑛=1,{𝑏_𝑛}^∞_𝑛=1 and {𝜎_𝑛}^∞_𝑛=1 be sequences of nonnegative real numbers satisfying the inequality 𝑎_𝑛+1≤1+𝜎_𝑛𝑎_𝑛+𝑏_𝑛,∀𝑛≥1. (2.13) If ∑^∞_𝑛=1𝜎_𝑛<∞ and ∑^∞_𝑛=1𝑏_𝑛<∞, then lim_𝑛→∞𝑎_𝑛 exists. If, in addition, {𝑎_𝑛}^∞_𝑛=1 has a subsequence which converges to zero, then lim_𝑛→∞𝑎_𝑛=0.

Corollary 2.9 (see [31, page 303]). Let {𝑎_𝑛}^∞_𝑛=0 and {𝑏_𝑛}^∞_𝑛=0 be two sequences of nonnegative real numbers satisfying the inequality 𝑎_𝑛+1≤𝑎_𝑛+𝑏_𝑛,∀𝑛≥1. (2.14) If ∑^∞_𝑛=0𝑏_𝑛 converges, then lim_𝑛→∞𝑎_𝑛 exists.

It is easy to see that the following lemma holds.

Lemma 2.10 (see [32]). 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. (2.15)

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

Lemma 2.11 (see [33]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space ℋ. Let 𝑆∶𝐶→𝐶 be a 𝑘-strictly pseudocontractive 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. (2.16)

The following result is useful when we prove the weak convergence of a sequence.

Lemma 2.12 (see [25, Proposition 2.6]). Let 𝐾 be a nonempty closed convex subset of a real Hilbert space ℋ. Let {𝑥_𝑛} be a bounded sequence which satisfies the following properties:(i)every weak limit point of {𝑥_𝑛} lies in 𝐾;(ii)lim_𝑛→∞‖𝑥_𝑛−𝑥‖ exists for every 𝑥∈𝐾. Then {𝑥_𝑛} converges weakly to a point in 𝐾.

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 (VI) is to find 𝑥∈𝐾 such that 𝐹𝑥,𝑦−𝑥≥0,∀𝑦∈𝐾. (2.17) 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. (2.18)

A set-valued mapping 𝑇∶ℋ→2^ℋ is called monotone if for all 𝑥,𝑦∈ℋ,𝑓∈𝑇𝑥 and 𝑔∈𝑇𝑦 imply 𝑥−𝑦,𝑓−𝑔≥0. (2.19) A monotone mapping 𝑇∶ℋ→2^ℋ is called maximal if its graph 𝐺(𝑇) is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping 𝑇 is maximal if and only if, for (𝑥,𝑓)∈ℋ×ℋ,⟨𝑥−𝑦,𝑓−𝑔⟩≥0 for every (𝑦,𝑔)∈𝐺(𝑇) implies 𝑓∈𝑇𝑥. Let 𝐹∶𝐾→ℋ be a monotone and 𝐿-Lipschitz continuous mapping and let 𝑁_𝐾𝑣 be the normal cone to 𝐾 at 𝑣∈𝐾, that is, 𝑁_𝐾𝑣=𝑤∈ℋ∶𝑣−𝑦,𝑤≥0,∀𝑦∈𝐾. (2.20) Define 𝑇𝑣=𝐹𝑣+𝑁_𝐾𝑣,if𝑣∈𝐾,∅,if𝑣∉𝐾. (2.21) Then, 𝑇 is maximal monotone and 0∈𝑇𝑣 if and only if 𝑣∈VI(𝐾,𝐹); see [34] for more details.

3. Some Modified Extragradient Methods

Throughout the paper, we 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 (3.1) is ill posed. Therefore, Xu [25] considered the following Tikhonov regularized problem: min_𝑥∈𝐶𝑓_𝛼(𝑥)1∶=2𝐴𝑥−𝑃_𝑄𝐴𝑥²+12𝛼‖𝑥‖², (3.2) where 𝛼>0 is the regularization parameter.

We observe that the gradient ∇𝑓_𝛼(𝑥)=∇𝑓(𝑥)+𝛼𝐼=𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼𝐼 (3.3) is (𝛼+‖𝐴‖²)-Lipschitz continuous and 𝛼-strongly monotone.

We can use fixed point algorithms to solve the SFP on the basis of the following observation.

Let 𝜆>0 and assume that 𝑥^∗∈Γ. Then 𝐴𝑥^∗∈𝑄, which implies that (𝐼−𝑃_𝑄)𝐴𝑥^∗=0, and thus, 𝜆𝐴^∗(𝐼−𝑃_𝑄)𝐴𝑥^∗=0. Hence, we have the fixed point equation (𝐼−𝜆𝐴^∗(𝐼−𝑃_𝑄)𝐴)𝑥^∗=𝑥^∗. Requiring that 𝑥^∗∈𝐶, we consider the fixed point equation 𝑃_𝐶𝐼−𝜆∇𝑓𝑥^∗=𝑃_𝐶𝐼−𝜆𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗=𝑥^∗. (3.4) It is proven in [25, Proposition 3.2] that the solutions of the fixed point equation (3.4) are exactly the solutions of the SFP; namely, for given 𝑥^∗∈ℋ₁,𝑥^∗ solves the SFP if and only if 𝑥^∗ solves the fixed point equation (3.4).

Proposition 3.1 (see [26, proposition 3.1]). Given 𝑥^∗∈ℋ₁, the following statements are equivalent:(i)𝑥^∗ solves the SFP;(ii)𝑥^∗ solves the fixed point equation (3.4);(iii)𝑥^∗ solves the variational inequality problem (VIP) of finding 𝑥^∗∈𝐶such that⟨∇𝑓𝑥^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈𝐶. (3.5)

Remark 3.2. It is clear from Proposition 3.1 that Γ=Fix𝑃_𝐶𝐼−𝜆∇𝑓=VI𝐶,∇𝑓 (3.6) 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 (3.4), respectively.

We are now in a position to propose a modified extragradient method for solving the SFP and the fixed point problem of a 𝑘-strictly pseudocontractive mapping 𝑆∶𝐶→𝐶 and prove that the sequences generated by the proposed method converge weakly to an element of Fix(𝑆)∩Γ.

Theorem 3.3. Let 𝑆∶𝐶→𝐶 be a 𝑘-strictly pseudocontractive mapping such that Fix(𝑆)∩Γ≠∅. Let {𝑥_𝑛} and {𝑦_𝑛} be the sequences in 𝐶 generated by the following modified extragradient algorithm: 𝑥₀𝑦=𝑥∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝑃_𝐶𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛+𝛿_𝑛𝑆𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛,∀𝑛≥0, (3.7) where {𝛼_𝑛}⊂(0,∞),{𝜆_𝑛}⊂(0,1/‖𝐴‖²) and {𝛽_𝑛},{𝛾_𝑛},{𝛿_𝑛}⊂[0,1] such that(i)∑^∞_𝑛=0𝛼_𝑛<∞;(ii)0<liminf_𝑛→∞𝜆_𝑛≤limsup_𝑛→∞𝜆_𝑛<1/‖𝐴‖²;(iii)𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 and (𝛾_𝑛+𝛿_𝑛)𝑘≤𝛾_𝑛 for all 𝑛≥0;(iv)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and liminf_𝑛→∞𝛿_𝑛>0. Then, both the sequences {𝑥_𝑛} and {𝑦_𝑛} converge weakly to an element 𝑥∈Fix(𝑆)∩Γ.

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).
We observe that 𝑃_𝐶(𝐼−𝜆∇𝑓_𝛼) is 𝜁-averaged for each 𝜆∈(0,2/(𝛼+‖𝐴‖²)), where ‖𝐴‖𝜁=2+𝜆𝛼+²4∈(0,1). (3.8) See, for example, [35] and from which it follows that 𝑃_𝐶(𝐼−𝜆∇𝑓_𝛼) and 𝑃_𝐶(𝐼−𝜆_𝑛∇𝑓_𝛼𝑛) are nonexpansive for all 𝑛≥0.
Next, we show that the sequence {𝑥_𝑛} is bounded. Indeed, take a fixed 𝑝∈Fix(𝑆)∩Γ arbitrarily. Then, we get 𝑆𝑝=𝑝 and 𝑃_𝐶(𝐼−𝜆∇𝑓)𝑝=𝑝 for 𝜆∈(0,2/‖𝐴‖²). For simplicity, we write 𝑣_𝑛=𝑃_𝐶(𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛(𝑦_𝑛)) for all 𝑛≥0. Then we get 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑣_𝑛+𝛿_𝑛𝑆𝑣_𝑛 for all 𝑛≥0. From (3.7), it follows that 𝑦_𝑛−𝑝=𝑃_𝐶𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑥_𝑛−𝑃_𝐶𝐼−𝜆_𝑛𝑝≤𝑃∇𝑓_𝐶𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑥_𝑛−𝑃_𝐶𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑝+𝑃_𝐶𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑝−𝑃_𝐶𝐼−𝜆_𝑛𝑝≤𝑥∇𝑓_𝑛−𝑝+𝑃_𝐶𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑝−𝑃_𝐶𝐼−𝜆_𝑛𝑝≤𝑥∇𝑓_𝑛−𝑝+𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑝−𝐼−𝜆_𝑛𝑝=𝑥∇𝑓_𝑛−𝑝+𝜆_𝑛𝛼_𝑛𝑝. (3.9) Also, by Proposition 2.1(ii), we have 𝑣_𝑛−𝑝²≤𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−𝑝²−𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−𝑣_𝑛²=𝑥_𝑛−𝑝²−𝑥_𝑛−𝑣_𝑛²+2𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛,𝑝−𝑣_𝑛=𝑥_𝑛−𝑝²−𝑥_𝑛−𝑣_𝑛²+2𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−∇𝑓_𝛼𝑛𝑝,𝑝−𝑦_𝑛+∇𝑓_𝛼𝑛𝑝,𝑝−𝑦_𝑛+∇𝑓_𝛼𝑛𝑦_𝑛,𝑦_𝑛−𝑣_𝑛≤𝑥_𝑛−𝑝²−𝑥_𝑛−𝑣_𝑛²+2𝜆_𝑛∇𝑓_𝛼𝑛𝑝,𝑝−𝑦_𝑛+∇𝑓_𝛼𝑛𝑦_𝑛,𝑦_𝑛−𝑣_𝑛=𝑥_𝑛−𝑝²−𝑥_𝑛−𝑣_𝑛²+2𝜆_𝑛𝛼_𝑛𝐼+∇𝑓𝑝,𝑝−𝑦_𝑛+∇𝑓_𝛼𝑛𝑦_𝑛,𝑦_𝑛−𝑣_𝑛≤𝑥_𝑛−𝑝²−𝑥_𝑛−𝑣_𝑛²+2𝜆_𝑛𝛼_𝑛𝑝,𝑝−𝑦_𝑛+∇𝑓_𝛼𝑛𝑦_𝑛,𝑦_𝑛−𝑣_𝑛=𝑥_𝑛−𝑝²−𝑥_𝑛−𝑦_𝑛²−2𝑥_𝑛−𝑦_𝑛,𝑦_𝑛−𝑣_𝑛−𝑦_𝑛−𝑣_𝑛²+2𝜆_𝑛𝛼_𝑛𝑝,𝑝−𝑦_𝑛+∇𝑓_𝛼𝑛𝑦_𝑛,𝑦_𝑛−𝑣_𝑛=𝑥_𝑛−𝑝²−𝑥_𝑛−𝑦_𝑛²−𝑦_𝑛−𝑣_𝑛²+2𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−𝑦_𝑛,𝑣_𝑛−𝑦_𝑛+2𝜆_𝑛𝛼_𝑛𝑝,𝑝−𝑦_𝑛. (3.10) Further, by Proposition 2.1(i), we have 𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−𝑦_𝑛,𝑣_𝑛−𝑦_𝑛=𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑥_𝑛−𝑦_𝑛,𝑣_𝑛−𝑦_𝑛+𝜆_𝑛∇𝑓_𝛼𝑛𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛,𝑣_𝑛−𝑦_𝑛≤𝜆_𝑛∇𝑓_𝛼𝑛𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛,𝑣_𝑛−𝑦_𝑛≤𝜆_𝑛∇𝑓_𝛼𝑛𝑥_𝑛−∇𝑓_𝛼𝑛𝑦_𝑛𝑣_𝑛−𝑦_𝑛≤𝜆_𝑛𝛼_𝑛+‖𝐴‖²𝑥_𝑛−𝑦_𝑛𝑣_𝑛−𝑦_𝑛. (3.11) So, we obtain 𝑣_𝑛−𝑝²≤𝑥_𝑛−𝑝²−𝑥_𝑛−𝑦_𝑛²−𝑦_𝑛−𝑣_𝑛²+2𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−𝑦_𝑛,𝑣_𝑛−𝑦_𝑛+2𝜆_𝑛𝛼_𝑛𝑝,𝑝−𝑦_𝑛≤𝑥_𝑛−𝑝²−𝑥_𝑛−𝑦_𝑛²−𝑦_𝑛−𝑣_𝑛²+2𝜆_𝑛𝛼_𝑛+‖𝐴‖²𝑥_𝑛−𝑦_𝑛𝑣_𝑛−𝑦_𝑛+2𝜆_𝑛𝛼_𝑛𝑝𝑝−𝑦_𝑛≤𝑥_𝑛−𝑝²−𝑥_𝑛−𝑦_𝑛²−𝑦_𝑛−𝑣_𝑛²+𝜆²_𝑛𝛼_𝑛+‖𝐴‖²²𝑥_𝑛−𝑦_𝑛²−𝑦_𝑛−𝑣_𝑛²+2𝜆_𝑛𝛼_𝑛𝑝𝑝−𝑦_𝑛=𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑝−𝑦_𝑛+𝜆²_𝑛𝛼_𝑛+‖𝐴‖²²−1𝑥_𝑛−𝑦_𝑛²≤𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑝−𝑦_𝑛. (3.12) Since (𝛾_𝑛+𝛿_𝑛)𝑘≤𝛾_𝑛, utilizing Lemmas 2.10 and 2.11, from (3.9) and the last inequality, we conclude that 𝑥_𝑛+1−𝑝²=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑣_𝑛+𝛿_𝑛𝑆𝑣_𝑛−𝑝²=𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛+𝛿_𝑛1𝛾_𝑛+𝛿_𝑛𝛾_𝑛𝑣_𝑛−𝑝+𝛿_𝑛𝑆𝑣_𝑛−𝑝²=𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛+𝛿_𝑛1𝛾_𝑛+𝛿_𝑛𝛾_𝑛𝑣_𝑛−𝑝+𝛿_𝑛𝑆𝑣_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛+𝛿_𝑛1𝛾_𝑛+𝛿_𝑛𝛾_𝑛𝑣_𝑛−𝑥_𝑛+𝛿_𝑛𝑆𝑣_𝑛−𝑥_𝑛²≤𝛽_𝑛𝑥_𝑛−𝑝²+1−𝛽_𝑛𝑣_𝑛−𝑝²−𝛽_𝑛1−𝛽_𝑛𝑥_𝑛+1−𝑥_𝑛²≤𝛽_𝑛𝑥_𝑛−𝑝²+1−𝛽_𝑛𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑝−𝑦_𝑛+𝜆²_𝑛𝛼_𝑛+‖𝐴‖²²−1𝑥_𝑛−𝑦_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑥_𝑛+1−𝑥_𝑛²≤𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑝−𝑦_𝑛+1−𝛽_𝑛𝜆²_𝑛𝛼_𝑛+‖𝐴‖²²−1𝑥_𝑛−𝑦_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑥_𝑛+1−𝑥_𝑛²≤𝑥_𝑛−𝑝²+𝛼_𝑛𝜆²_𝑛𝑝²+𝑝−𝑦_𝑛²+1−𝛽_𝑛𝜆²_𝑛𝛼_𝑛+‖𝐴‖²²−1𝑥_𝑛−𝑦_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑥_𝑛+1−𝑥_𝑛²≤𝑥_𝑛−𝑝²+𝛼_𝑛𝜆²_𝑛𝑝²+𝑥_𝑛−𝑝+𝜆_𝑛𝛼_𝑛𝑝²+1−𝛽_𝑛𝜆²_𝑛𝛼_𝑛+‖𝐴‖²²−1𝑥_𝑛−𝑦_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑥_𝑛+1−𝑥_𝑛²≤𝑥_𝑛−𝑝²+𝛼_𝑛𝜆²_𝑛𝑝²+2𝑥_𝑛−𝑝²+2𝜆²_𝑛𝛼²_𝑛𝑝²+1−𝛽_𝑛𝜆²_𝑛𝛼_𝑛+‖𝐴‖²²−1𝑥_𝑛−𝑦_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑥_𝑛+1−𝑥_𝑛²=1+2𝛼_𝑛𝑥_𝑛−𝑝²+𝛼_𝑛𝜆²_𝑛𝑝²1+2𝛼²_𝑛+1−𝛽_𝑛𝜆²_𝑛𝛼_𝑛+‖𝐴‖²²−1𝑥_𝑛−𝑦_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑥_𝑛+1−𝑥_𝑛²≤1+2𝛼_𝑛𝑥_𝑛−𝑝²+𝛼_𝑛𝜆²_𝑛𝑝²1+2𝛼²_𝑛=1+𝜎_𝑛𝑥_𝑛−𝑝²+𝑏_𝑛, (3.13) where 𝜎_𝑛=2𝛼_𝑛 and 𝑏_𝑛=𝛼_𝑛𝜆²_𝑛‖𝑝‖²(1+2𝛼²_𝑛). Since ∑^∞_𝑛=0𝛼_𝑛<∞ and {𝜆_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1/‖𝐴‖²), we conclude that ∑^∞_𝑛=0𝜎_𝑛<∞ and ∑^∞_𝑛=0𝑏_𝑛<∞. Therefore, by Lemma 2.8, we deduce that lim_𝑛→∞𝑥_𝑛−𝑝existsforeach𝑝∈Fix(𝑆)∩Γ, (3.14) and the sequence {𝑥_𝑛} is bounded and so are {𝑦_𝑛} and {𝑣_𝑛}. From the last relations, we also obtain 1−𝛽_𝑛1−𝜆²_𝑛𝛼_𝑛+‖𝐴‖²²𝑥_𝑛−𝑦_𝑛²+𝛽_𝑛1−𝛽_𝑛𝑥_𝑛+1−𝑥_𝑛²≤1+2𝛼_𝑛𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+𝛼_𝑛𝜆²_𝑛𝑝²1+2𝛼²_𝑛. (3.15) Since {𝜆_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1/‖𝐴‖²),0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and lim_𝑛→∞𝛼_𝑛=0, we have lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (3.16) Furthermore, we obtain 𝑦_𝑛−𝑣_𝑛=𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑥_𝑛−𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛≤𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑥_𝑛−𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛=𝜆_𝑛∇𝑓_𝛼𝑛𝑥_𝑛−∇𝑓_𝛼𝑛𝑦_𝑛≤𝜆_𝑛𝛼_𝑛+‖𝐴‖²𝑥_𝑛−𝑦_𝑛. (3.17) This together with (3.16) implies that lim_𝑛→∞𝑦_𝑛−𝑣_𝑛=0. (3.18) Note that 𝑣_𝑛−𝑥_𝑛≤𝑣_𝑛−𝑦_𝑛+𝑦_𝑛−𝑥_𝑛,𝛿_𝑛𝑆𝑣_𝑛−𝑥_𝑛=𝑥_𝑛+1−𝑥_𝑛−𝛾_𝑛𝑣_𝑛−𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝛾_𝑛𝑣_𝑛−𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑣_𝑛−𝑥_𝑛. (3.19) This together with (3.16), (3.18), and liminf𝛿_𝑛>0 implies that lim_𝑛→∞𝑣_𝑛−𝑥_𝑛=lim_𝑛→∞𝑆𝑣_𝑛−𝑥_𝑛=0. (3.20) So, we derive lim_𝑛→∞𝑆𝑣_𝑛−𝑣_𝑛=0. (3.21) Since ∇𝑓=𝐴^∗(𝐼−𝑃_𝑄)𝐴 is Lipschitz continuous, from (3.18), we have lim_𝑛→∞∇𝑓𝑦_𝑛−∇𝑓𝑣_𝑛=0. (3.22)
As {𝑥_𝑛} is bounded, there is a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} that converges weakly to some 𝑥. We obtain that 𝑥∈Fix(𝑆)∩Γ. First, we show that 𝑥∈Γ. Since ‖𝑥_𝑛−𝑣_𝑛‖→0 and ‖𝑦_𝑛−𝑣_𝑛‖→0, it is known that 𝑣_𝑛𝑖⇀𝑥 and 𝑦_𝑛𝑖⇀𝑥. Let (𝑣)𝑇𝑣=∇𝑓+𝑁_𝐶𝑣,if𝑣∈𝐶,∅,if𝑣∉𝐶, (3.23) where 𝑁_𝐶𝑣={𝑤∈ℋ₁∶⟨𝑣−𝑢,𝑤⟩≥0,forall𝑢∈𝐶}. Then, 𝑇 is maximal monotone and 0∈𝑇𝑣 if and only if 𝑣∈VI(𝐶,∇𝑓); see [34] for more details. Let (𝑣,𝑤)∈𝐺(𝑇). Then, we have 𝑤∈𝑇𝑣=∇𝑓(𝑣)+𝑁_𝐶𝑣 (3.24) and hence, 𝑤−∇𝑓(𝑣)∈𝑁_𝐶𝑣. (3.25) So, we have 𝑣−𝑢,𝑤−∇𝑓(𝑣)≥0,∀𝑢∈𝐶. (3.26) On the other hand, from 𝑦_𝑛=𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑥_𝑛,𝑣∈𝐶, (3.27) we have ⟨𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑥_𝑛−𝑦_𝑛,𝑦_𝑛−𝑣⟩≥0, (3.28) and hence, 𝑣−𝑦_𝑛,𝑦_𝑛−𝑥_𝑛𝜆_𝑛+∇𝑓_𝛼𝑛𝑥_𝑛≥0. (3.29) Therefore, from 𝑤−∇𝑓(𝑣)∈𝑁_𝐶𝑣,𝑦_𝑛𝑖∈𝐶, (3.30) we have 𝑣−𝑦_𝑛𝑖,𝑤≥𝑣−𝑦_𝑛𝑖(𝑣≥,∇𝑓)𝑣−𝑦_𝑛𝑖,∇𝑓(𝑣)−𝑣−𝑦_𝑛𝑖,𝑦_𝑛𝑖−𝑥_𝑛𝑖𝜆_𝑛𝑖+∇𝑓_𝛼𝑛𝑖𝑥_𝑛𝑖=𝑣−𝑦_𝑛𝑖,∇𝑓(𝑣)−𝑣−𝑦_𝑛𝑖,𝑦_𝑛𝑖−𝑥_𝑛𝑖𝜆_𝑛𝑖+∇𝑓𝑥_𝑛𝑖−𝛼_𝑛𝑖𝑣−𝑦_𝑛𝑖,𝑥_𝑛𝑖=𝑣−𝑦_𝑛𝑖,∇𝑓(𝑣)−∇𝑓𝑦_𝑛𝑖+𝑣−𝑦_𝑛𝑖,∇𝑓𝑦_𝑛𝑖−∇𝑓𝑥_𝑛𝑖−𝑣−𝑦_𝑛𝑖,𝑦_𝑛𝑖−𝑥_𝑛𝑖𝜆_𝑛𝑖−𝛼_𝑛𝑖𝑣−𝑦_𝑛𝑖,𝑥_𝑛𝑖≥𝑣−𝑦_𝑛𝑖,∇𝑓𝑦_𝑛𝑖−∇𝑓𝑥_𝑛𝑖−𝑣−𝑦_𝑛𝑖,𝑦_𝑛𝑖−𝑥_𝑛𝑖𝜆_𝑛𝑖−𝛼_𝑛𝑖𝑣−𝑦_𝑛𝑖,𝑥_𝑛𝑖. (3.31) Hence, we obtain ⟨𝑣−𝑥,𝑤⟩≥0,as𝑖→∞. (3.32) Since 𝑇 is maximal monotone, we have 𝑥∈𝑇⁻¹0, and hence, 𝑥∈VI(𝐶,∇𝑓). Thus it is clear that 𝑥∈Γ.
We show that 𝑥∈Fix(𝑆). Indeed, since 𝑣_𝑛𝑖⇀𝑥 and ‖𝑣_𝑛𝑖−𝑆𝑣_𝑛𝑖‖→0 by (3.21), by Lemma 2.7(ii), we get 𝑥∈Fix(𝑆). Therefore, we have 𝑥∈Fix(𝑆)∩Γ. This shows that 𝜔_𝑤(𝑥_𝑛)⊂Fix(𝑆)∩Γ, where 𝜔_𝑤𝑥_𝑛∶=𝑥∶𝑥_𝑛𝑖⇀𝑥forsomesubsequence𝑥_𝑛𝑖of𝑥_𝑛. (3.33) Since the limit lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists for every 𝑝∈Fix(𝑆)∩Γ, by Lemma 2.12, we know that 𝑥_𝑛⇀𝑥∈Fix(𝑆)∩Γ. (3.34) Further, from ‖𝑥_𝑛−𝑦_𝑛‖→0, it follows that 𝑦_𝑛⇀𝑥. This shows that both sequences {𝑥_𝑛} and {𝑦_𝑛} converge weakly to 𝑥∈Fix(𝑆)∩Γ.

Remark 3.4. It is worth emphasizing that the modified extragradient algorithm in Theorem 3.3 is essentially the predictor-corrector algorithm. Indeed, the first iterative step 𝑦_𝑛=𝑃_𝐶(𝐼−𝜆_𝑛∇𝑓_𝛼𝑛)𝑥_𝑛 is the predictor one, and the second iterative step 𝑥_𝑛+1 = 𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑃_𝐶(𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛(𝑦_𝑛))+𝛿_𝑛𝑆𝑃_𝐶(𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛(𝑦_𝑛)) is actually the corrector one. In addition, Theorem 3.3 extends the extragradient method due to Nadezhkina and Takahashi [8, Theorem 3.1].

Corollary 3.5. Let 𝑆∶𝐶→𝐶 be a nonexpansive mapping such that Fix(𝑆)∩Γ≠∅. Let {𝑥_𝑛} and {𝑦_𝑛} be the sequences in 𝐶 generated by the following extragradient algorithm: 𝑥₀𝑦=𝑥∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝑃_𝐶𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑆𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛,∀𝑛≥0, (3.35) where {𝛼_𝑛}⊂(0,∞),{𝜆_𝑛}⊂(0,1/‖𝐴‖²), and {𝛽_𝑛}⊂[0,1] such that(i)∑^∞_𝑛=0𝛼_𝑛<∞;(ii)0<liminf_𝑛→∞𝜆_𝑛≤limsup_𝑛→∞𝜆_𝑛<1/‖𝐴‖²;(iii)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1. Then, both the sequences {𝑥_𝑛} and {𝑦_𝑛} converge weakly to an element 𝑥∈Fix(𝑆)∩Γ.

Proof. In Theorem 3.3, putting 𝛾_𝑛=0 for every 𝑛≥0, we obtain that 𝛽_𝑛+𝛿_𝑛=𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 and𝑥₀𝑦=𝑥∈𝐶chosenarbitrarily,_𝑛=𝑃_𝐶𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛+𝛿_𝑛𝑆𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛=𝛽_𝑛𝑥_𝑛+𝛿_𝑛𝑆𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛,∀𝑛≥0. (3.36) Since 𝑆∶𝐶→𝐶 is a nonexpansive mapping, 𝑆∶𝐶→𝐶 must be a 𝑘-strictly pseudocontractive mapping with coefficient 𝑘=0. It is clear that (𝛾_𝑛+𝛿_𝑛)𝑘≤𝛾_𝑛 for every 𝑛≥0 and liminf_𝑛→∞𝛿_𝑛=1−limsup_𝑛→∞𝛽_𝑛>0. In this case, all conditions in Theorem 3.3 are satisfied. Therefore, by Theorem 3.3, we derive the desired result.

Remark 3.6. Compared with [26, Theorem 3.1], Corollary 3.5 is essentially coincident with [26, Theorem 3.1]. Hence our Theorem 3.3 includes [26, Theorem 3.1] as a special case. Utilizing [8, Theorem 3.1], Ceng et al. gave the following weak convergence result [26, Theorem3.2].

Let 𝑆∶𝐶→𝐶 be a nonexpansive mapping such that Fix(𝑆)∩Γ≠∅. Let {𝑥_𝑛} and {𝑦_𝑛} be the sequences in 𝐶 generated by the following Nadezhkina and Takahashi extragradient algorithm 𝑥₀𝑦=𝑥∈𝐶chosenarbitrarily,_𝑛=𝑃_𝐶𝐼−𝜆_𝑛∇𝑓𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑆𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓𝑦_𝑛,∀𝑛≥0, (3.37) where {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.

Obviously, there is no doubt that [26, Theorem 3.2] is a weak convergence result for {𝛼_𝑛} satisfying 𝛼_𝑛=0, for  all 𝑛≥0. However, Corollary 3.5 is another weak convergence one for the sequence of regularization parameters {𝛼_𝑛}⊂(0,∞).

Remark 3.7. Theorem 3.3 improves, extends, and develops [25, theorem 5.7] and [26, Theorem 3.1] in the following aspects.(a)The corresponding iterative algorithms in [25, Theorem 5.7] and [26, Theorem 3.1] are extended for developing our modified extragradient algorithm with regularization in Theorem 3.3.(b)The technique of proving weak convergence in Theorem 3.3 is different from those in [25, Theorem 5.7] and [26, Theorem 3.1] because our technique depends on the properties of maximal monotone mappings and strictly pseudocontractive mappings (e.g., Lemma 2.11) and the demiclosedness principle for strictly pseudocontractive mappings (e.g., Lemma 2.7) in Hilbert spaces.(c)The problem of finding an element of Fix(𝑆)∩Γ with 𝑆∶𝐶→𝐶 being strictly pseudocontractive is more general than the problem of finding a solution of the SFP in [25, Theorem 5.7] and the problem of finding an element of Fix(𝑆)∩Γ with 𝑆∶𝐶→𝐶 being nonexpansive in [26, Theorem 3.1].(d)The second iterative step 𝑥_𝑛+1 = 𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑃_𝐶(𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛(𝑦_𝑛))+𝛿_𝑛𝑆𝑃_𝐶(𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛(𝑦_𝑛)) in our algorithm reduces to the the second iterative one 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑆𝑃_𝐶(𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛(𝑦_𝑛)) in the algorithm of [26, Theorem 3.1] whenever 𝛾_𝑛=0 for all 𝑛≥0.

Utilizing Theorem 3.3, we have the following two new results in the setting of real Hilbert spaces.

Corollary 3.8. Let 𝑆∶ℋ₁→ℋ₁ be a 𝑘-strictly pseudocontractive mapping such that Fix(𝑆)∩(∇𝑓)⁻¹0≠∅. Let {𝑥_𝑛} and {𝑦_𝑛} be two sequences generated by 𝑥₀𝑦=𝑥∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛+𝛿_𝑛𝑆𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛,∀𝑛≥0, (3.38) where {𝛼_𝑛}⊂(0,∞),{𝜆_𝑛}⊂(0,1/‖𝐴‖²) and {𝛽_𝑛},{𝛾_𝑛},{𝛿_𝑛}⊂[0,1] such that(i)∑^∞_𝑛=0𝛼_𝑛<∞;(ii)0<liminf_𝑛→∞𝜆_𝑛≤limsup_𝑛→∞𝜆_𝑛<1/‖𝐴‖²;(iii)𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 and (𝛾_𝑛+𝛿_𝑛)𝑘≤𝛾_𝑛 for all 𝑛≥0;(iv)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and liminf_𝑛→∞𝛿_𝑛>0. Then, both the sequences {𝑥_𝑛} and {𝑦_𝑛} converge weakly to an element 𝑥∈Fix(𝑆)∩(∇𝑓)⁻¹0.

Proof. In Theorem 3.3, putting 𝐶=ℋ₁, we have ∇𝑓⁻¹0=VIℋ₁,∇𝑓=Γ (3.39) and 𝑃_ℋ1=𝐼 the identity mapping. By Theorem 3.3, we obtain the desired result.

Remark 3.9. In Corollary 3.8, putting 𝛾_𝑛=0 for every 𝑛≥0 and letting 𝑆∶ℋ₁→ℋ₁ be a nonexpansive mapping, Corollary 3.8 essentially reduces to [26, Corollary 3.2]. Hence, Corollary 3.8 includes [26, Corollary 3.2] as a special case.

Corollary 3.10. Let 𝐵∶ℋ₁→2^ℋ1 be a maximal monotone mapping such that 𝐵⁻¹0∩(∇𝑓)⁻¹0≠∅. Let 𝐽^𝐵_𝑟 be the resolvent of 𝐵 for each 𝑟>0. Let {𝑥_𝑛} and {𝑦_𝑛} be the sequences generated by 𝑥₀𝑦=𝑥∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛+𝛿_𝑛𝐽^𝐵_𝑟𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛,∀𝑛≥0, (3.40) where {𝛼_𝑛}⊂(0,∞),{𝜆_𝑛}⊂(0,1/‖𝐴‖²) and {𝛽_𝑛},{𝛾_𝑛},{𝛿_𝑛}⊂[0,1] such that(i)∑^∞_𝑛=0𝛼_𝑛<∞;(ii)0<liminf_𝑛→∞𝜆_𝑛≤limsup_𝑛→∞𝜆_𝑛<1/‖𝐴‖²;(iii)𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 and (𝛾_𝑛+𝛿_𝑛)𝑘≤𝛾_𝑛 for all 𝑛≥0;(iv)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and liminf_𝑛→∞𝛿_𝑛>0. Then, both the sequences {𝑥_𝑛} and {𝑦_𝑛} converge weakly to an element 𝑥∈𝐵⁻¹0∩(∇𝑓)⁻¹0.

Proof. In Theorem 3.3, putting 𝐶=ℋ₁ and 𝑆=𝐽^𝐵_𝑟 the resolvent of 𝐵, we know that 𝑃_ℋ1=𝐼 the identity mapping and 𝑆 is nonexpansive. In this case, we get Fix(𝑆)=Fix(𝐽^𝐵_𝑟)=𝐵⁻¹0 and ∇𝑓⁻¹0=VIℋ₁,∇𝑓=Γ. (3.41) By Theorem 3.3, we obtain the desired result.

Remark 3.11. In Corollary 3.10, putting 𝛾_𝑛=0 for every 𝑛≥0, Corollary 3.10 essentially reduces to [26, Corollary 3.3]. Hence, Corollary 3.10 includes [26, Corollary 3.3] as a special case.

On the other hand, by combining the regularization method and Jung’s composite viscosity approximation method [9], we introduce another new composite iterative scheme for finding an element of Fix(𝑆)∩Γ, where 𝑆∶𝐶→𝐶 is nonexpansive, and prove strong convergence of this scheme. To attain this object, we need to use the following lemmas.

Lemma 3.12 (see [36]). 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, (3.42) 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,forall𝑛≥0. Then, lim_𝑛→∞𝑎_𝑛=0.

Lemma 3.13. In a real Hilbert space ℋ, there holds the following inequality: 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝑥+𝑦,∀𝑥,𝑦∈ℋ. (3.43)

Theorem 3.14. Let 𝑄∶𝐶→𝐶 be a contractive mapping with coefficient 𝜌∈[0,1) and 𝑆∶𝐶→𝐶 be a nonexpansive mapping such that Fix(𝑆)∩Γ≠∅. Assume that 0<𝜆<2/‖𝐴‖², and let {𝑥_𝑛} and {𝑦_𝑛} be the sequences in 𝐶 generated by the following composite extragradient-like algorithm: 𝑥₀𝑦=𝑥∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝛽_𝑛𝑄𝑥_𝑛+1−𝛽_𝑛𝑆𝑃_𝐶𝑥_𝑛−𝜆∇𝑓_𝛼𝑛𝑥_𝑛,𝑥_𝑛+1=1−𝛾_𝑛𝑦_𝑛+𝛾_𝑛𝑆𝑃_𝐶𝑦_𝑛−𝜆∇𝑓_𝛼𝑛𝑦_𝑛,∀𝑛≥0, (3.44) where the sequences of parameters {𝛼_𝑛}⊂(0,∞) and {𝛽_𝑛},{𝛾_𝑛}⊂[0,1] satisfy the following conditions:(i)∑^∞_𝑛=0𝛼_𝑛<∞;(ii)lim_𝑛→∞𝛽_𝑛=0,∑^∞_𝑛=0𝛽_𝑛=∞ and ∑^∞_𝑛=0|𝛽_𝑛+1−𝛽_𝑛|<∞;(iii)limsup_𝑛→∞𝛾_𝑛<1 and ∑^∞_𝑛=0|𝛾_𝑛+1−𝛾_𝑛|<∞. Then, both the sequences {𝑥_𝑛} and {𝑦_𝑛} converge strongly to 𝑞∈Fix(𝑆)∩Γ, which is a unique solution of the following variational inequality: (𝐼−𝑄)𝑞,𝑞−𝑝≤0,∀𝑝∈Fix(𝑆)∩Γ. (3.45)

Proof. Repeating the same argument as in the proof of Theorem 3.3, we obtain 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). (3.46) This shows that 𝑃_𝐶(𝐼−𝜆∇𝑓_𝛼) is nonexpansive. Furthermore, for 𝜆∈(0,2/‖𝐴‖²), utilizing the fact that lim_𝑛→∞(2/(𝛼_𝑛+‖𝐴‖²))=2/‖𝐴‖² we may assume that 20<𝜆<𝛼_𝑛+‖𝐴‖²,∀𝑛≥0. (3.47) Consequently, it follows that for each integer 𝑛≥0, 𝑃_𝐶(𝐼−𝜆∇𝑓_𝛼𝑛) is 𝜁_𝑛-averaged with 𝜁_𝑛=12+𝜆𝛼_𝑛+‖𝐴‖²2−12⋅𝜆𝛼_𝑛+‖𝐴‖²2=2+𝜆𝛼_𝑛+‖𝐴‖²4∈(0,1). (3.48) This immediately implies that 𝑃_𝐶(𝐼−𝜆∇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, put 𝑢_𝑛=𝑃_𝐶(𝑥_𝑛−𝜆∇𝑓_𝛼𝑛(𝑥_𝑛)) and 𝑣_𝑛=𝑃_𝐶(𝑦_𝑛−𝜆∇𝑓_𝛼𝑛(𝑦_𝑛)) for every 𝑛≥0. Take a fixed 𝑝∈Fix(𝑆)∩Γ arbitrarily. Then, we get 𝑆𝑝=𝑝 and 𝑃_𝐶(𝐼−𝜆∇𝑓)𝑝=𝑝 for 𝜆∈(0,2/‖𝐴‖²). Hence, we have 𝑢_𝑛−𝑝=𝑃_𝐶𝐼−𝜆∇𝑓_𝛼𝑛𝑥_𝑛−𝑃_𝐶𝑝≤𝑃𝐼−𝜆∇𝑓_𝐶𝐼−𝜆∇𝑓_𝛼𝑛𝑥_𝑛−𝑃_𝐶𝐼−𝜆∇𝑓_𝛼𝑛𝑝+𝑃_𝐶𝐼−𝜆∇𝑓_𝛼𝑛𝑝−𝑃_𝐶𝑝≤𝑥𝐼−𝜆∇𝑓_𝑛−𝑝+𝑃_𝐶𝐼−𝜆∇𝑓_𝛼𝑛𝑝−𝑃_𝐶𝑝≤𝑥𝐼−𝜆∇𝑓_𝑛−𝑝+𝜆𝛼_𝑛𝑝. (3.49) Similarly we get ‖𝑣_𝑛−𝑝‖≤‖𝑦_𝑛−𝑝‖+𝜆𝛼_𝑛‖𝑝‖. Thus, from (3.44), we have 𝑦_𝑛−𝑝=𝛽_𝑛𝑄𝑥_𝑛−𝑝+1−𝛽_𝑛𝑆𝑢_𝑛−𝑝≤𝛽_𝑛𝑄𝑥_𝑛−𝑝+1−𝛽_𝑛𝑢_𝑛−𝑝≤𝛽_𝑛𝑄𝑥_𝑛−𝑄𝑝+𝛽_𝑛𝑄𝑝−𝑝+1−𝛽_𝑛𝑥_𝑛−𝑝+𝜆𝛼_𝑛𝑝≤𝛽_𝑛𝜌𝑥_𝑛−𝑝+𝛽_𝑛𝑄𝑝−𝑝+1−𝛽_𝑛𝑥_𝑛−𝑝+𝜆𝛼_𝑛𝑝=1−𝛽_𝑛1−𝜌𝑥_𝑛−𝑝+𝛽_𝑛𝑄𝑝−𝑝+1−𝛽_𝑛𝜆𝛼_𝑛𝑝=1−𝛽_𝑛1−𝜌𝑥_𝑛−𝑝+𝛽_𝑛1−𝜌𝑄𝑝−𝑝+1−𝜌1−𝛽_𝑛𝜆𝛼_𝑛𝑝≤max𝑥_𝑛−𝑝,𝑄𝑝−𝑝1−𝜌+𝜆𝛼_𝑛𝑝, (3.50) and hence, 𝑥_𝑛+1−𝑝=1−𝛾_𝑛𝑦_𝑛−𝑝+𝛾_𝑛𝑆𝑣_𝑛≤−𝑝1−𝛾_𝑛𝑦_𝑛−𝑝+𝛾_𝑛𝑣_𝑛≤−𝑝1−𝛾_𝑛𝑦_𝑛−𝑝+𝛾_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. (3.51) By induction, we get 𝑥_𝑛+1−𝑝≤max𝑥₀−𝑝,𝑄𝑝−𝑝𝑝⋅1−𝜌+2𝜆^𝑛_𝑖=0𝛼_𝑖,∀𝑛≥0. (3.52) This implies that {𝑥_𝑛} is bounded and so are {𝑦_𝑛},{𝑢_𝑛},{𝑣_𝑛}. It is clear that both {𝑆𝑢_𝑛} and {𝑆𝑣_𝑛} are also bounded. By condition (ii), we also obtain 𝑦_𝑛−𝑆𝑢_𝑛=𝛽_𝑛𝑄𝑥_𝑛−𝑆𝑢_𝑛→0(𝑛→∞). (3.53)Step 2. lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0.
Indeed, from (3.44), we have 𝑦_𝑛=𝛽_𝑛𝑄𝑥_𝑛+1−𝛽_𝑛𝑆𝑢_𝑛,𝑦_𝑛−1=𝛽_𝑛−1𝑄𝑥_𝑛−1+1−𝛽_𝑛−1𝑆𝑢_𝑛−1,∀𝑛≥1. (3.54) Simple calculations show that 𝑦_𝑛−𝑦_𝑛−1=1−𝛽_𝑛𝑆𝑢_𝑛−𝑆𝑢_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑄𝑥_𝑛−1−𝑆𝑢_𝑛−1+𝛽_𝑛𝑄𝑥_𝑛−𝑄𝑥_𝑛−1. (3.55) Since 𝑢_𝑛−𝑢_𝑛−1≤𝑃_𝐶𝐼−𝜆∇𝑓_𝛼𝑛𝑥_𝑛−𝑃_𝐶𝐼−𝜆∇𝑓_𝛼𝑛𝑥_𝑛−1+𝑃_𝐶𝐼−𝜆∇𝑓_𝛼𝑛𝑥_𝑛−1−𝑃_𝐶𝐼−𝜆∇𝑓_𝛼𝑛−1𝑥_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+𝐼−𝜆∇𝑓_𝛼𝑛𝑥_𝑛−1−𝐼−𝜆∇𝑓_𝛼𝑛−1𝑥_𝑛−1=𝑥_𝑛−𝑥_𝑛−1+𝜆∇𝑓_𝛼𝑛𝑥_𝑛−1−𝜆∇𝑓_𝛼𝑛−1𝑥_𝑛−1=𝑥_𝑛−𝑥_𝑛−1+𝜆𝛼_𝑛−𝛼_𝑛−1𝑥_𝑛−1 (3.56) for every 𝑛≥1, we have 𝑦_𝑛−𝑦_𝑛−1≤1−𝛽_𝑛𝑆𝑢_𝑛−𝑆𝑢_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑄𝑥_𝑛−1−𝑆𝑢_𝑛−1+𝛽_𝑛𝑄𝑥_𝑛−𝑄𝑥_𝑛−1≤1−𝛽_𝑛𝑢_𝑛−𝑢_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑄𝑥_𝑛−1−𝑆𝑢_𝑛−1+𝛽_𝑛𝜌𝑥_𝑛−𝑥_𝑛−1≤1−𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+𝜆𝛼_𝑛−𝛼_𝑛−1𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑄𝑥_𝑛−1−𝑆𝑢_𝑛−1+𝛽_𝑛𝜌𝑥_𝑛−𝑥_𝑛−1≤1−𝛽_𝑛1−𝜌𝑥_𝑛−𝑥_𝑛−1+𝜆𝛼_𝑛−𝛼_𝑛−1𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑄𝑥_𝑛−1−𝑆𝑢_𝑛−1≤1−𝛽_𝑛1−𝜌𝑥_𝑛−𝑥_𝑛−1+𝑀₁𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1 (3.57) for every 𝑛≥1, where 𝑀₁=sup{𝜆‖𝑥_𝑛−1‖+‖𝑄𝑥_𝑛−1−𝑆𝑢_𝑛−1‖∶𝑛≥1}.
On the other hand, from (3.44), we have 𝑥_𝑛+1=1−𝛾_𝑛𝑦_𝑛+𝛾_𝑛𝑆𝑣_𝑛,𝑥_𝑛=1−𝛾_𝑛−1𝑦_𝑛−1+𝛾_𝑛−1𝑆𝑣_𝑛−1. (3.58) Also, simple calculations show that 𝑥_𝑛+1−𝑥_𝑛=1−𝛾_𝑛𝑦_𝑛−𝑦_𝑛−1+𝛾_𝑛𝑆𝑣_𝑛−𝑆𝑣_𝑛−1+𝛾_𝑛−𝛾_𝑛−1𝑆𝑣_𝑛−1−𝑦_𝑛−1. (3.59) Since 𝑣_𝑛−𝑣_𝑛−1≤𝑃_𝐶𝐼−𝜆∇𝑓_𝛼𝑛𝑦_𝑛−𝑃_𝐶𝐼−𝜆∇𝑓_𝛼𝑛𝑦_𝑛−1+𝑃_𝐶𝐼−𝜆∇𝑓_𝛼𝑛𝑦_𝑛−1−𝑃_𝐶𝐼−𝜆∇𝑓_𝛼𝑛−1𝑦_𝑛−1≤𝑦_𝑛−𝑦_𝑛−1+𝐼−𝜆∇𝑓_𝛼𝑛𝑦_𝑛−1−𝐼−𝜆∇𝑓_𝛼𝑛−1𝑦_𝑛−1=𝑦_𝑛−𝑦_𝑛−1+𝜆∇𝑓_𝛼𝑛𝑦_𝑛−1−𝜆∇𝑓_𝛼𝑛−1𝑦_𝑛−1=𝑦_𝑛−𝑦_𝑛−1+𝜆𝛼_𝑛−𝛼_𝑛−1𝑦_𝑛−1 (3.60) for every 𝑛≥1, it follows from (3.57) that 𝑥_𝑛+1−𝑥_𝑛≤1−𝛾_𝑛𝑦_𝑛−𝑦_𝑛−1+𝛾_𝑛𝑆𝑣_𝑛−𝑆𝑣_𝑛−1+𝛾_𝑛−𝛾_𝑛−1𝑆𝑣_𝑛−1−𝑦_𝑛−1≤1−𝛾_𝑛𝑦_𝑛−𝑦_𝑛−1+𝛾_𝑛𝑣_𝑛−𝑣_𝑛−1+𝛾_𝑛−𝛾_𝑛−1𝑆𝑣_𝑛−1−𝑦_𝑛−1≤1−𝛾_𝑛𝑦_𝑛−𝑦_𝑛−1+𝛾_𝑛𝑦_𝑛−𝑦_𝑛−1+𝜆𝛼_𝑛−𝛼_𝑛−1𝑦_𝑛−1+𝛾_𝑛−𝛾_𝑛−1𝑆𝑣_𝑛−1−𝑦_𝑛−1≤𝑦_𝑛−𝑦_𝑛−1+𝜆𝛼_𝑛−𝛼_𝑛−1𝑦_𝑛−1+𝛾_𝑛−𝛾_𝑛−1𝑆𝑣_𝑛−1−𝑦_𝑛−1≤𝑦_𝑛−𝑦_𝑛−1+𝑀₂𝛼_𝑛−𝛼_𝑛−1+𝛾_𝑛−𝛾_𝑛−1≤1−𝛽_𝑛1−𝜌𝑥_𝑛−𝑥_𝑛−1+𝑀₁𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1+𝑀₂𝛼_𝑛−𝛼_𝑛−1+𝛾_𝑛−𝛾_𝑛−1=1−𝛽_𝑛1−𝜌𝑥_𝑛−𝑥_𝑛−1+𝑀₁+𝑀₂𝛼_𝑛−𝛼_𝑛−1+𝑀₁𝛽_𝑛−𝛽_𝑛−1+𝑀₂𝛾_𝑛−𝛾_𝑛−1 (3.61) for every 𝑛≥1, where 𝑀₂=sup{𝜆‖𝑦_𝑛−1‖+‖𝑆𝑣_𝑛−1−𝑦_𝑛−1‖∶𝑛≥1}. From conditions (i), (ii), (iii), it is easy to see that lim_𝑛→∞𝛽_𝑛1−𝜌=0,^∞_𝑛=0𝛽_𝑛1−𝜌=∞,_∞𝑛=0𝑀₁+𝑀₂𝛼_𝑛−𝛼_𝑛−1+𝑀₁𝛽_𝑛−𝛽_𝑛−1+𝑀₂𝛾_𝑛−𝛾_𝑛−1<∞. (3.62) Applying Lemma 3.12 to (3.61), we have lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (3.63) From (3.57), we also have that ‖𝑦_𝑛+1−𝑦_𝑛‖→0 as 𝑛→∞.
Step 3. lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=lim_𝑛→∞‖𝑥_𝑛−𝑆𝑢_𝑛‖=0.
Indeed, it follows that 𝑥_𝑛+1−𝑦_𝑛=𝛾_𝑛𝑆𝑣_𝑛−𝑦_𝑛≤𝛾_𝑛𝑆𝑣_𝑛−𝑆𝑢_𝑛+𝑆𝑢_𝑛−𝑦_𝑛≤𝛾_𝑛𝑣_𝑛−𝑢_𝑛+𝑆𝑢_𝑛−𝑦_𝑛=𝛾_𝑛𝑃_𝐶𝐼−𝜆∇𝑓_𝛼𝑛𝑦_𝑛−𝑃_𝐶𝐼−𝜆∇𝑓_𝛼𝑛𝑥_𝑛+𝑆𝑢_𝑛−𝑦_𝑛≤𝛾_𝑛𝑦_𝑛−𝑥_𝑛+𝑆𝑢_𝑛−𝑦_𝑛≤𝛾_𝑛𝑦_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑥_𝑛+𝑆𝑢_𝑛−𝑦_𝑛, (3.64) which implies that 1−𝛾_𝑛𝑦_𝑛−𝑥_𝑛+1≤𝛾_𝑛𝑥_𝑛+1−𝑥_𝑛+𝑆𝑢_𝑛−𝑦_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑆𝑢_𝑛−𝑦_𝑛. (3.65) Obviously, utilizing (3.53), ‖𝑥_𝑛+1−𝑥_𝑛‖→0 and limsup_𝑛→∞𝛾_𝑛<1, we have ‖𝑥_𝑛+1−𝑦_𝑛‖→0 as 𝑛→∞. This implies that 𝑥_𝑛−𝑦_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑦_𝑛→0as𝑛→∞. (3.66) From (3.53) and (3.66), we also have 𝑥_𝑛−𝑆𝑢_𝑛≤𝑥_𝑛−𝑦_𝑛+𝑦_𝑛−𝑆𝑢_𝑛→0as𝑛→∞. (3.67)
Step 4. lim_𝑛→∞‖𝑥_𝑛−𝑢_𝑛‖=0.
Indeed, take a fixed 𝑝∈Fix(𝑆)∩Γ arbitrarily. Then, by the convexity of ‖⋅‖², we have 𝑦_𝑛−𝑝²=𝛽_𝑛𝑄𝑥_𝑛−𝑝+1−𝛽_𝑛𝑆𝑢_𝑛−𝑝²≤𝛽_𝑛𝑄𝑥_𝑛−𝑝²+1−𝛽_𝑛𝑆𝑢_𝑛−𝑝²≤𝛽_𝑛𝑄𝑥_𝑛−𝑝²+1−𝛽_𝑛𝑢_𝑛−𝑝²≤𝛽_𝑛𝑄𝑥_𝑛−𝑝²+1−𝛽_𝑛𝐼−𝜆∇𝑓_𝛼𝑛𝑥_𝑛−𝐼−𝜆∇𝑓𝑝²=𝛽_𝑛𝑄𝑥_𝑛−𝑝²+1−𝛽_𝑛𝐼−𝜆∇𝑓𝑥_𝑛−𝐼−𝜆∇𝑓𝑝−𝜆𝛼_𝑛𝑥_𝑛²≤𝛽_𝑛𝑄𝑥_𝑛−𝑝²+1−𝛽_𝑛𝐼−𝜆∇𝑓𝑥_𝑛−𝐼−𝜆∇𝑓𝑝²−2𝜆𝛼_𝑛𝑥_𝑛,𝐼−𝜆∇𝑓_𝛼𝑛𝑥_𝑛−𝐼−𝜆∇𝑓𝑝≤𝛽_𝑛𝑄𝑥_𝑛−𝑝²+1−𝛽_𝑛𝑥_𝑛−𝑝²2+2𝜆𝜆−‖𝐴‖²∇𝑓𝑥_𝑛−∇𝑓𝑝²+2𝜆𝛼_𝑛𝑥_𝑛𝐼−𝜆∇𝑓_𝛼𝑛𝑥_𝑛−𝐼−𝜆∇𝑓𝑝≤𝛽_𝑛𝑄𝑥_𝑛−𝑝²+1−𝛽_𝑛𝑥_𝑛−𝑝²+2𝜆𝛼_𝑛𝑥_𝑛𝐼−𝜆∇𝑓_𝛼𝑛𝑥_𝑛−𝐼−𝜆∇𝑓𝑝. (3.68) So we obtain 1−𝛽_𝑛22𝜆‖𝐴‖²−𝜆∇𝑓𝑥_𝑛−∇𝑓𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+1−𝛽_𝑛𝑥_𝑛−𝑝²−𝑦_𝑛−𝑝²+1−𝛽_𝑛2𝜆𝛼_𝑛𝑥_𝑛𝐼−𝜆∇𝑓_𝛼𝑛𝑥_𝑛−𝐼−𝜆∇𝑓𝑝≤𝛽_𝑛𝑥_𝑛−𝑝²+𝑥_𝑛−𝑝+𝑦_𝑛−𝑝𝑥_𝑛−𝑝−𝑦_𝑛−𝑝+2𝜆𝛼_𝑛𝑥_𝑛𝐼−𝜆∇𝑓_𝛼𝑛𝑥_𝑛−𝐼−𝜆∇𝑓𝑝≤𝛽_𝑛𝑥_𝑛−𝑝²+𝑥_𝑛−𝑝+𝑦_𝑛−𝑝𝑥_𝑛−𝑦_𝑛+2𝜆𝛼_𝑛𝑥_𝑛𝐼−𝜆∇𝑓_𝛼𝑛𝑥_𝑛−𝐼−𝜆∇𝑓𝑝. (3.69) Since 𝛼_𝑛→0,𝛽_𝑛→0,‖𝑥_𝑛−𝑦_𝑛‖→0, and 0<𝜆<2/‖𝐴‖², from the boundedness of {𝑥_𝑛} and {𝑦_𝑛}, it follows that lim_𝑛→∞‖∇𝑓(𝑥_𝑛)−∇𝑓(𝑝)‖=0, and hence lim_𝑛→∞∇𝑓_𝛼𝑛𝑥_𝑛−∇𝑓𝑝=0. (3.70) Moreover, from the firm nonexpansiveness of 𝑃_𝐶, we obtain 𝑢_𝑛−𝑝²=𝑃_𝐶𝐼−𝜆∇𝑓_𝛼𝑛𝑥_𝑛−𝑃_𝐶𝐼−𝜆∇𝑓𝑝²≤𝐼−𝜆∇𝑓_𝛼𝑛𝑥_𝑛−𝐼−𝜆∇𝑓𝑝,𝑢_𝑛=1−𝑝2𝐼−𝜆∇𝑓_𝛼𝑛𝑥_𝑛−𝐼−𝜆∇𝑓𝑝²+𝑢_𝑛−𝑝²−𝐼−𝜆∇𝑓_𝛼𝑛𝑥_𝑛−𝐼−𝜆∇𝑓𝑝−𝑢_𝑛−𝑝²≤12𝑥_𝑛−𝑝²+2𝜆𝛼_𝑛𝑥_𝑛𝐼−𝜆∇𝑓_𝛼𝑛𝑥_𝑛−𝐼−𝜆∇𝑓𝑝+𝑢_𝑛−𝑝²−𝑥_𝑛−𝑢_𝑛²+2𝜆𝑥_𝑛−𝑢_𝑛,∇𝑓_𝛼𝑛𝑥_𝑛−∇𝑓𝑝−𝜆²∇𝑓_𝛼𝑛𝑥_𝑛−∇𝑓𝑝², (3.71) and so 𝑢_𝑛−𝑝²≤𝑥_𝑛−𝑝²−𝑥_𝑛−𝑢_𝑛²+2𝜆𝛼_𝑛𝑥_𝑛𝐼−𝜆∇𝑓_𝛼𝑛𝑥_𝑛−𝑝𝑥𝐼−𝜆∇𝑓+2𝜆_𝑛−𝑢_𝑛,∇𝑓_𝛼𝑛𝑥_𝑛−∇𝑓𝑝−𝜆²∇𝑓_𝛼𝑛𝑥_𝑛−∇𝑓𝑝². (3.72) Thus, we have 𝑦_𝑛−𝑝²≤𝛽_𝑛𝑄𝑥_𝑛−𝑝²+1−𝛽_𝑛𝑢_𝑛−𝑝²≤𝛽_𝑛𝑄𝑥_𝑛−𝑝²+𝑥_𝑛−𝑝²−1−𝛽_𝑛𝑥_𝑛−𝑢_𝑛²+2𝜆𝛼_𝑛𝑥_𝑛𝐼−𝜆∇𝑓_𝛼𝑛𝑥_𝑛−𝐼−𝜆∇𝑓𝑝+21−𝛽_𝑛𝜆𝑥_𝑛−𝑢_𝑛,∇𝑓_𝛼𝑛𝑥_𝑛−∇𝑓𝑝−1−𝛽_𝑛𝜆²∇𝑓_𝛼𝑛𝑥_𝑛−∇𝑓𝑝², (3.73) which implies that 1−𝛽_𝑛𝑥_𝑛−𝑢_𝑛²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝑥_𝑛−𝑝+𝑦_𝑛−𝑝𝑥_𝑛−𝑝−𝑦_𝑛−𝑝+2𝜆𝛼_𝑛𝑥_𝑛𝐼−𝜆∇𝑓_𝛼𝑛𝑥_𝑛−𝐼−𝜆∇𝑓𝑝+21−𝛽_𝑛𝜆𝑥_𝑛−𝑢_𝑛,∇𝑓_𝛼𝑛𝑥_𝑛−∇𝑓𝑝−1−𝛽_𝑛𝜆²∇𝑓_𝛼𝑛𝑥_𝑛−∇𝑓𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝑥_𝑛−𝑝+𝑦_𝑛−𝑝𝑥_𝑛−𝑦_𝑛+2𝜆𝛼_𝑛𝑥_𝑛𝐼−𝜆∇𝑓_𝛼𝑛𝑥_𝑛−𝐼−𝜆∇𝑓𝑝+21−𝛽_𝑛𝜆𝑥_𝑛−𝑢_𝑛,∇𝑓_𝛼𝑛𝑥_𝑛−∇𝑓𝑝−1−𝛽_𝑛𝜆²∇𝑓_𝛼𝑛(𝑥_𝑛)−∇𝑓(𝑝)². (3.74) Since 𝛼_𝑛→0,𝛽_𝑛→0,‖𝑥_𝑛−𝑦_𝑛‖→0, and ‖∇𝑓_𝛼𝑛(𝑥_𝑛)−∇𝑓(𝑝)‖→0, from the boundedness of {𝑥_𝑛},{𝑦_𝑛}, and {𝑢_𝑛}, it follows that lim_𝑛→∞‖𝑥_𝑛−𝑢_𝑛‖=0 and hence lim_𝑛→∞𝑦_𝑛−𝑢_𝑛=0. (3.75)
Step 5. limsup_𝑛→∞⟨𝑄𝑞−𝑞,𝑦_𝑛−𝑞⟩≤0 for 𝑞∈Fix(𝑆)∩Γ, where 𝑞 is a unique solution of the variational inequality (𝐼−𝑄)𝑞,𝑞−𝑝≤0,∀𝑝∈Fix(𝑆)∩Γ. (3.76)
Indeed, we choose a subsequence {𝑢_𝑛𝑖} of {𝑢_𝑛} such that limsup_𝑛→∞𝑄𝑞−𝑞,𝑆𝑢_𝑛−𝑞=lim_𝑖→∞𝑄𝑞−𝑞,𝑆𝑢_𝑛𝑖−𝑞. (3.77) Since {𝑢_𝑛𝑖} is bounded, there exists a subsequence {𝑢_𝑛𝑖𝑗} of {𝑢_𝑛𝑖} which converges weakly to 𝑢. Without loss of generality we may assume that 𝑢_𝑛𝑖⇀𝑢. Then we can obtain 𝑢∈Fix(𝑆)∩Γ. Let us first show that 𝑢∈Γ. Define (𝑣)𝑇𝑣=∇𝑓+𝑁_𝐶𝑣,if𝑣∈𝐶,∅,if𝑣∉𝐶, (3.78) where 𝑁_𝐶𝑣={𝑤∈ℋ₁∶⟨𝑣−𝑢,𝑤⟩≥0,forall𝑢∈𝐶}. Then, 𝑇 is maximal monotone and 0∈𝑇𝑣 if and only if 𝑣∈VI(𝐶,∇𝑓); see [34] for more details. Let (𝑣,𝑤)∈𝐺(𝑇). Then, we have 𝑤∈𝑇𝑣=∇𝑓(𝑣)+𝑁_𝐶𝑣 (3.79) and hence, 𝑤−∇𝑓(𝑣)∈𝑁_𝐶𝑣. (3.80) So, we have 𝑣−𝑢,𝑤−∇𝑓(𝑣)≥0,∀𝑢∈𝐶. (3.81) On the other hand, from 𝑢_𝑛=𝑃_𝐶𝑥_𝑛−𝜆∇𝑓_𝛼𝑛𝑥_𝑛,𝑣∈𝐶, (3.82) we have 𝑥_𝑛−𝜆∇𝑓_𝛼𝑛𝑥_𝑛−𝑢_𝑛,𝑢_𝑛−𝑣≥0, (3.83) and hence, 𝑣−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝜆+∇𝑓_𝛼𝑛𝑥_𝑛≥0. (3.84) Therefore, from 𝑤−∇𝑓(𝑣)∈𝑁_𝐶𝑣,𝑢_𝑛𝑖∈𝐶, (3.85) we have 𝑣−𝑢_𝑛𝑖,𝑤≥𝑣−𝑢_𝑛𝑖(𝑣≥,∇𝑓)𝑣−𝑢_𝑛𝑖,∇𝑓(𝑣)−𝑣−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖𝜆+∇𝑓_𝛼𝑛𝑖𝑥_𝑛𝑖=𝑣−𝑢_𝑛𝑖,∇𝑓(𝑣)−𝑣−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖𝜆+∇𝑓𝑥_𝑛𝑖−𝛼_𝑛𝑖𝑣−𝑢_𝑛𝑖,𝑥_𝑛𝑖=𝑣−𝑢_𝑛𝑖,∇𝑓(𝑣)−∇𝑓𝑢_𝑛𝑖+𝑣−𝑢_𝑛𝑖,∇𝑓𝑢_𝑛𝑖−∇𝑓𝑥_𝑛𝑖−𝑣−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖𝜆−𝛼_𝑛𝑖𝑣−𝑢_𝑛𝑖,𝑥_𝑛𝑖≥𝑣−𝑢_𝑛𝑖,∇𝑓𝑢_𝑛𝑖−∇𝑓𝑥_𝑛𝑖−𝑣−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖𝜆−𝛼_𝑛𝑖𝑣−𝑢_𝑛𝑖,𝑥_𝑛𝑖. (3.86) Hence, we obtain ⟨𝑣−𝑢,𝑤⟩≥0,as𝑖→∞. (3.87) Since 𝑇 is maximal monotone, we have 𝑢∈𝑇⁻¹0, and hence, 𝑢∈VI(𝐶,∇𝑓). Thus it is clear that 𝑢∈Γ.
On the other hand, by Steps 3 and 4, ‖𝑢_𝑛−𝑆𝑢_𝑛‖≤‖𝑢_𝑛−𝑥_𝑛‖+‖𝑥_𝑛−𝑆𝑢_𝑛‖→0. So, by Lemma 2.7(ii), we derive 𝑢∈Fix(𝑆) and hence 𝑢∈Fix(𝑆)∩Γ. Then from (3.76) and (3.77), we have limsup_𝑛→∞𝑄𝑞−𝑞,𝑆𝑢_𝑛−𝑞=lim_𝑖→∞𝑄𝑞−𝑞,𝑆𝑢_𝑛𝑖−𝑞=𝑄𝑞−𝑞,=)𝑢−𝑞(𝐼−𝑄𝑞,𝑞−𝑢≤0. (3.88) Thus, from (3.53), we obtain limsup_𝑛→∞𝑄𝑞−𝑞,𝑦_𝑛−𝑞≤limsup_𝑛→∞𝑄𝑞−𝑞,𝑦_𝑛−𝑆𝑢_𝑛+limsup_𝑛→∞𝑄𝑞−𝑞,𝑆𝑢_𝑛−𝑞≤limsup_𝑛→∞𝑄𝑞−𝑞𝑦_𝑛−𝑆𝑢_𝑛+limsup_𝑛→∞𝑄𝑞−𝑞,𝑆𝑢_𝑛−𝑞≤0. (3.89)Step 6. lim_𝑛→∞‖𝑥_𝑛−𝑞‖=0 for 𝑞∈Fix(𝑆)∩Γ, where 𝑞 is a unique solution of the variational inequality (𝐼−𝑄)𝑞,𝑞−𝑝≤0,∀𝑝∈Fix(𝑆)∩Γ. (3.90)
Indeed, utilizing (3.49), (3.51), and Lemma 3.13, we have 𝑥_𝑛+1−𝑞²≤𝑦_𝑛−𝑞+𝜆𝛼_𝑛𝑞²=𝑦_𝑛−𝑞²+𝜆𝛼_𝑛𝑞2𝑦_𝑛−𝑞+𝜆𝛼_𝑛𝑞≤𝑦_𝑛−𝑞²+𝑀₃𝛼_𝑛=𝛽_𝑛𝑄𝑥_𝑛−𝑞+1−𝛽_𝑛𝑆𝑢_𝑛−𝑞²+𝑀₃𝛼_𝑛≤1−𝛽_𝑛²𝑆𝑢_𝑛−𝑞²+2𝛽_𝑛𝑄𝑥_𝑛−𝑞,𝑦_𝑛−𝑞+𝑀₃𝛼_𝑛≤1−𝛽_𝑛²𝑢_𝑛−𝑞²+2𝛽_𝑛𝑄𝑥_𝑛−𝑄𝑞,𝑦_𝑛−𝑞+2𝛽_𝑛𝑄𝑞−𝑞,𝑦_𝑛−𝑞+𝑀₃𝛼_𝑛≤1−𝛽_𝑛²𝑥_𝑛−𝑞+𝜆𝛼_𝑛𝑞²+2𝛽_𝑛𝜌𝑥_𝑛−𝑞𝑦_𝑛−𝑞+2𝛽_𝑛𝑄𝑞−𝑞,𝑦_𝑛−𝑞+𝑀₃𝛼_𝑛≤1−𝛽_𝑛²𝑥_𝑛−𝑞²+𝑀₃𝛼_𝑛+2𝛽_𝑛𝜌𝑥_𝑛−𝑞𝑦_𝑛−𝑥_𝑛+𝑥_𝑛−𝑞+2𝛽_𝑛𝑄𝑞−𝑞,𝑦_𝑛−𝑞+𝑀₃𝛼_𝑛≤1−𝛽_𝑛²𝑥_𝑛−𝑞²+2𝛽_𝑛𝜌𝑥_𝑛−𝑞𝑦_𝑛−𝑥_𝑛+𝑥_𝑛−𝑞+2𝛽_𝑛𝑄𝑞−𝑞,𝑦_𝑛−𝑞+2𝑀₃𝛼_𝑛=1−2𝛽_𝑛1−𝜌𝑥_𝑛−𝑞²+𝛽_𝑛2𝜌𝑥_𝑛−𝑞𝑦_𝑛−𝑥_𝑛+𝛽_𝑛𝑥_𝑛−𝑞²+2𝑄𝑞−𝑞,𝑦_𝑛−𝑞+2𝑀₃𝛼_𝑛 (3.91) for every 𝑛≥0, where 𝑀₃=sup{𝜆‖𝑞‖[2(‖𝑥_𝑛−𝑞‖+‖𝑦_𝑛−𝑞‖)+𝜆𝛼_𝑛‖𝑞‖]∶𝑛≥0}. Now, put 𝑎_𝑛=‖𝑥_𝑛−𝑞‖²,𝑠_𝑛=2𝛽_𝑛(1−𝜌),𝑡_𝑛=(1/2(1−𝜌))[2𝜌‖𝑥_𝑛−𝑞‖‖𝑦_𝑛−𝑥_𝑛‖+𝛽_𝑛‖𝑥_𝑛−𝑞‖²+2⟨𝑄𝑞−𝑞,𝑦_𝑛−𝑞⟩], and 𝑟_𝑛=2𝑀₃𝛼_𝑛. Then (3.91) is rewritten as 𝑎_𝑛+1≤1−𝑠_𝑛𝑎_𝑛+𝑠_𝑛𝑡_𝑛+𝑟_𝑛. (3.92) It is easy to see that ∑^∞_𝑛=0𝑠_𝑛=∞,limsup_𝑛→∞𝑡_𝑛≤0 and ∑^∞_𝑛=0𝑟_𝑛<∞. Thus by Lemma 3.12, we obtain 𝑥_𝑛→𝑞. This completes the proof.

Corollary 3.15. Let 𝑄∶ℋ₁→ℋ₁ be a contractive mapping with coefficient 𝜌∈[0,1) and 𝑆∶ℋ₁→ℋ₁ be a nonexpansive mapping such that Fix(𝑆)∩(∇𝑓)⁻¹0≠∅. Assume that 0<𝜆<2/‖𝐴‖², and let {𝑥_𝑛} and {𝑦_𝑛} be the sequences in ℋ₁ generated by 𝑥₀=𝑥∈ℋ₁𝑦𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝛽_𝑛𝑄𝑥_𝑛+1−𝛽_𝑛𝑆𝑥_𝑛−𝜆∇𝑓_𝛼𝑛𝑥_𝑛,𝑥_𝑛+1=1−𝛾_𝑛𝑦_𝑛+𝛾_𝑛𝑆𝑦_𝑛−𝜆∇𝑓_𝛼𝑛𝑦_𝑛,∀𝑛≥0, (3.93) where the sequences of parameters {𝛼_𝑛}⊂(0,∞) and {𝛽_𝑛},{𝛾_𝑛}⊂[0,1] satisfy the following conditions:(i)∑^∞_𝑛=0𝛼_𝑛<∞;(ii)lim_𝑛→∞𝛽_𝑛=0,∑^∞_𝑛=0𝛽_𝑛=∞, and ∑^∞_𝑛=0|𝛽_𝑛+1−𝛽_𝑛|<∞;(iii)limsup_𝑛→∞𝛾_𝑛<1 and ∑^∞_𝑛=0|𝛾_𝑛+1−𝛾_𝑛|<∞. Then, both the sequences {𝑥_𝑛} and {𝑦_𝑛} converge strongly to 𝑞∈Fix(𝑆)∩(∇𝑓)⁻¹0, which is a unique solution of the following variational inequality (𝐼−𝑄)𝑞,𝑞−𝑝≤0,∀𝑝∈Fix(𝑆)∩∇𝑓⁻¹0. (3.94)

Proof. In Theorem 3.14, putting 𝐶=ℋ₁, we deduce that 𝑃_ℋ1=𝐼 the identity mapping, Γ=VI(ℋ₁,∇𝑓)=(∇𝑓)⁻¹0 and 𝑥₀=𝑥∈𝐶=ℋ₁𝑦chosenarbitrarily,_𝑛=𝛽_𝑛𝑄𝑥_𝑛+1−𝛽_𝑛𝑆𝑃_𝐶𝑥_𝑛−𝜆∇𝑓_𝛼𝑛𝑥_𝑛=𝛽_𝑛𝑄𝑥_𝑛+1−𝛽_𝑛𝑆𝑥_𝑛−𝜆∇𝑓_𝛼𝑛𝑥_𝑛,𝑥_𝑛+1=1−𝛾_𝑛𝑦_𝑛+𝛾_𝑛𝑆𝑃_𝐶𝑦_𝑛−𝜆∇𝑓_𝛼𝑛𝑦_𝑛=1−𝛾_𝑛𝑦_𝑛+𝛾_𝑛𝑆𝑦_𝑛−𝜆∇𝑓_𝛼𝑛𝑦_𝑛, (3.95) for every 𝑛≥0. Then, by Theorem 3.14, we obtain the desired result.

Remark 3.16. Theorem 3.14 improves and develops [25, Theorem5.7], [26, Theorem 3.1], and [9, Theorem 3.1] in the following aspects.(a)The corresponding iterative algorithm in [9, Theorem 3.1] is extended for developing our composite extragradient-like algorithm with regularization in Theorem 3.14.(b)The technique of proving strong convergence in Theorem 3.14 is very different from those in [25, Theorem 5.7], [26, Theorem 3.1], and [9, Theorem 3.1] because our technique depends on Lemmas 3.12 and 3.13.(c)Compared with [25, Theorem 5.7], and [26, Theorem 3.1], two weak convergence results, Theorem 3.14 is a strong convergence result. Thus, Theorem 3.14 is quite interesting and very valuable.(d)In [9, Theorem 3.1], Jung actually introduced the following composite iterative algorithm: 𝑥₀𝑦=𝑥∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝛽_𝑛𝑄𝑥_𝑛+1−𝛽_𝑛𝑆𝑃_𝐶𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛,𝑥_𝑛+1=1−𝛾_𝑛𝑦_𝑛+𝛾_𝑛𝑆𝑃_𝐶𝑦_𝑛−𝜆_𝑛𝐴𝑦_𝑛,∀𝑛≥0, (3.96)where 𝐴 is inverse-strongly monotone and 𝑆 is nonexpansive. Now, via replacing 𝜆_𝑛𝐴 by 𝜆∇𝑓_𝛼𝑛, we obtain the composite extragradient-like algorithm in Theorem 3.14. Consequently, this algorithm is very different from Jung’s algorithm.

Furthermore, utilizing Jung [9, Theorem 3.1], we can immediately obtain the following strong convergence result.

Theorem 3.17. Let 𝑄∶𝐶→𝐶 be a contractive mapping with coefficient 𝜌∈[0,1) and 𝑆∶𝐶→𝐶 be a nonexpansive mapping such that Fix(𝑆)∩Γ≠∅. Let {𝑥_𝑛} and {𝑦_𝑛} be the sequences in 𝐶 generated by the following composite extragradient-like algorithm: 𝑥₀𝑦=𝑥∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝛽_𝑛𝑄𝑥_𝑛+1−𝛽_𝑛𝑆𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓𝑥_𝑛,𝑥_𝑛+1=1−𝛾_𝑛𝑦_𝑛+𝛾_𝑛𝑆𝑃_𝐶𝑦_𝑛−𝜆_𝑛∇𝑓𝑦_𝑛,∀𝑛≥0, (3.97) where the sequences of parameters {𝜆_𝑛}⊂(0,2/‖𝐴‖²) and {𝛽_𝑛},{𝛾_𝑛}⊂[0,1] satisfy the following conditions:(i)lim_𝑛→∞𝛽_𝑛=0,∑^∞_𝑛=0𝛽_𝑛=∞, and ∑^∞_𝑛=0|𝛽_𝑛+1−𝛽_𝑛|<∞;(ii)0<liminf_𝑛→∞𝜆_𝑛≤limsup_𝑛→∞𝜆_𝑛<2/‖𝐴‖² and ∑^∞_𝑛=0|𝜆_𝑛+1−𝜆_𝑛|<∞;(iii)limsup_𝑛→∞𝛾_𝑛<1 and ∑^∞_𝑛=0|𝛾_𝑛+1−𝛾_𝑛|<∞. Then, both the sequences {𝑥_𝑛} and {𝑦_𝑛} converge strongly to 𝑞∈Fix(𝑆)∩Γ, which is a unique solution of the following variational inequality: (𝐼−𝑄)𝑞,𝑞−𝑝≤0,∀𝑝∈Fix(𝑆)∩Γ. (3.98)

Remark 3.18. It is not hard to see that ∇𝑓 is (1/‖𝐴‖²)-ism. Thus, Theorem 3.17 is an immediate consequence of Jung [9, Theorem 3.1].

Acknowledgments

In this research, the first and second authors were 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). Third author was partially supported by the Grant NSC 101–2115-M-037-001.