Some Algorithms for Finding Fixed Points and Solutions of Variational Inequalities

Abstract

We introduce new implicit and explicit algorithms for finding the fixed point of a k-strictly pseudocontractive mapping and for solving variational inequalities related to the Lipschitzian and strongly monotone operator in Hilbert spaces. We establish results on the strong convergence of the sequences generated by the proposed algorithms to a fixed point of a k-strictly pseudocontractive mapping. Such a point is also a solution of a variational inequality defined on the set of fixed points. As direct consequences, we obtain the unique minimum-norm fixed point of a k-strictly pseudocontractive mapping.

1. Introduction

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and induced norm ‖⋅‖. Let 𝐶 be a nonempty closed convex subset of 𝐻 and 𝑆∶𝐶→𝐶 a self-mapping on 𝐶. We denote by Fix(𝑆) the set of fixed points of 𝑆. We recall that a mapping 𝑇∶𝐶→𝐻 is said to be 𝑘-strictly pseudocontractive if there exists a constant 𝑘∈[0,1) such that𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+𝑘(𝐼−T)𝑥−(𝐼−𝑇)𝑦²,∀𝑥,𝑦∈𝐶. (1.1)

Note that the class of 𝑘-strictly pseudocontractive mappings includes the class of nonexpansive mappings 𝑇 on 𝐶 (i.e., ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖,forall𝑥,𝑦∈𝐶) as a subclass. That is, 𝑇 is nonexpansive if and only if 𝑇 is 0-strictly pseudocontractive. Recently, many authors have been devoting their studies to the problems of finding fixed points for 𝑘-strictly pseudocontractive mappings; see [1–5] and the references therein.

Variational inequalities have been studied widely and are being used as a mathematical programming tool in modeling a wide class of problems arising in several branches of pure and applied sciences; see [6–8]. For general variational inequalities and extended general variational inequalities, we can refer to [9–13] and references therein.

A variational inequality (VI) is formulated as finding a point 𝑥 with the property𝑥∈𝐶suchthat𝐹𝑥,𝑝−𝑥≥0,∀𝑝∈𝐶, (1.2) where 𝐹∶𝐶→𝐻 is a nonlinear mapping. It is well known that VI (1.2) is equivalent to the fixed point equation𝑥=𝑃_𝐶(𝐼−𝜆𝐹)𝑥, (1.3) where 𝜆>0 and 𝑃_𝐶 is the the metric projection of 𝐻 onto 𝐶, which assigns, to each 𝑥∈𝐻, the unique point in 𝐶, denoted by 𝑃_𝐶𝑥, such that𝑥−𝑃_𝐶𝑥=inf𝑥−𝑦∶𝑦∈𝐶. (1.4)

Therefore, fixed point algorithms can be applied to solve VI (1.2). It is also well known that if 𝐹 is 𝜌-Lipschitzian and 𝜂-strongly monotone with constants 𝜌,𝜂>0 (i.e., there exist 𝜌,𝜂≥0 such that ‖𝐹𝑥−𝐹𝑦‖≤𝜌‖𝑥−𝑦‖ and ⟨𝐹𝑥−𝐹𝑦,𝑥−𝑦⟩≥𝜂‖𝑥−𝑦‖²,𝑥,𝑦∈𝐶, resp.), then, for small enough 𝜆>0, the mapping 𝑃_𝐶(𝐼−𝜆𝐹) is a contractive mapping on 𝐶 and so the sequence {𝑥_𝑛} of Picard iterates, given by 𝑥_𝑛=𝑃_𝐶(𝐼−𝜆𝐹)𝑥_𝑛−1(𝑛≥1), converges strongly to the unique solution of VI (1.2).

This sort of VI (1.2) where 𝐹 is 𝜌-Lipschitzian and 𝜂-strongly monotone and where solutions are sought from the set of fixed points of a nonexpansive mapping is originated from Yamada [14], who provided the hybrid method for solving VI (1.2). In order to find solutions of ceratin variational inequality problems defined on the set of fixed points of nonexpansive mappings, several iterative algorithms were studied by many authors; see [15–24] and the references therein.

In this paper, we investigate the following variational inequality (VI) as a special form of VI (1.2), where the constraint set is the fixed points of a 𝑘-strictly pseudocontractive mapping 𝑇: finding a point 𝑥 with property𝑥∈Fix(𝑇)suchthat𝜇𝐹−𝛾𝑉𝑥,𝑝−𝑥≥0,∀𝑝∈Fix(𝑇), (1.5) where 𝑇∶𝐶→𝐶 is a 𝑘-strictly pseudocontractive mapping with Fix(𝑇)≠∅ for some 𝑘∈[0,1), 𝐹∶𝐶→𝐻 is a 𝜌-Lipschitzian and 𝜂-strongly monotone mapping with constants 𝜌,𝜂>0, and 𝑉∶𝐶→𝐻 is an 𝑙-Lipschitzian mapping with constant 𝑙≥0 and 𝜇,𝛾>0. Indeed, variational inequalities of form (1.5) cover several topics recently considered in the literature, including monotone inclusions, convex optimization, and quadratic minimization over fixed point sets; see [2–4, 15, 18, 19, 21, 22, 24] and the references therein. For some iterative methods and some results related to our approach about VI (1.5), we can refer to [25–31] and references therein.

The main purpose of the present paper is to further study the hierarchical fixed point approach to the VI of form (1.5). First, we introduce new implicit and explicit algorithms for finding the fixed point of the 𝑘-strictly pseudocontractive mapping 𝑇. Then, we establish results on the strong convergence of the sequences generated by the proposed algorithms to a fixed point of the mapping 𝑇, which is also a solution of VI (1.5) defined on the set of fixed points of 𝑇. As direct consequences, we obtain the unique minimum-norm fixed point of 𝑇. Namely, we find the unique solution of the quadratic minimization problem: ‖𝑥‖²=min{‖𝑥‖²∶𝑥∈Fix(𝑇)}.

2. Preliminaries and Lemmas

Throughout this paper, when {𝑥_𝑛} is a sequence in 𝐻, 𝑥_𝑛→𝑥 (resp., 𝑥_𝑛⇀𝑥) denotes strong (resp., weak) convergence of the sequence {𝑥_𝑛} to 𝑥.

Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Recall that 𝑓∶𝐶→𝐻 is called a contractive mapping with constant 𝛼∈(0,1) if there exists a constant 𝛼∈(0,1) such that ‖𝑓(𝑥)−𝑓(𝑦)‖≤𝛼‖𝑥−𝑦‖,forall𝑥,𝑦∈𝐶.

For every point 𝑥∈𝐻, there exists a unique nearest point in 𝐶, denoted by 𝑃_𝐶𝑥, such that𝑥−𝑃_𝐶𝑥≤𝑥−𝑦,∀𝑦∈𝐶. (2.1)𝑃_𝐶 is called the metric projection of 𝐻 to 𝐶. It is well known that 𝑃_𝐶 is nonexpansive and that, for 𝑥∈𝐻,𝑧=𝑃_𝐶𝑥⟺𝑥−𝑧,𝑦−𝑧≤0,∀𝑦∈𝐶. (2.2)

In a Hilbert space 𝐻, we have𝑥−𝑦²=‖𝑥‖²+𝑦²−2𝑥,𝑦,∀𝑥,𝑦∈𝐻. (2.3)

We need the following lemmas for the proof of our main results.

Lemma 2.1. In a real Hilbert space 𝐻, the following inequality holds: 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝑥+𝑦,∀𝑥,𝑦∈𝐻. (2.4)

Lemma 2.2 (see [32]). Let {𝑠_𝑛} be a sequence of nonnegative real numbers satisfying 𝑠_𝑛+1≤1−𝜆_𝑛𝑠_𝑛+𝜆_𝑛𝛿_𝑛,∀𝑛≥0, (2.5) where {𝜆_𝑛}⊂(0,1) and {𝛿_𝑛} satisfy the following conditions:(i)lim_𝑛→∞𝜆_𝑛=0,(ii)∑^∞_𝑛=0𝜆_𝑛=∞,(iii)limsup_𝑛→∞𝛿_𝑛≤0 or ∑^∞_𝑛=0𝜆_𝑛𝛿_𝑛<∞.Then, lim_𝑛→∞𝑠_𝑛=0.

Lemma 2.3 (see [33]). Let {𝑥_𝑛} and {𝑧_𝑛} be bounded sequences in a Banach space 𝐸 and {𝛾_𝑛} a sequence in [0,1] that satisfies the following condition: 0<liminf_𝑛→∞𝛾_𝑛≤limsup_𝑛→∞𝛾_𝑛<1. (2.6) Suppose that 𝑥_𝑛+1=𝛾_𝑛𝑥_𝑛+(1−𝛾_𝑛)𝑧_𝑛  for all 𝑛≥0 and limsup_𝑛→∞𝑧_𝑛+1−𝑧_𝑛−𝑥_𝑛+1−𝑥_𝑛≤0. (2.7) Then, lim_𝑛→∞‖𝑧_𝑛−𝑥_𝑛‖=0.

Lemma 2.4 (Demiclosedness principle [34]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and 𝑆∶𝐶→𝐶 a nonexpansive mapping with Fix(𝑆)≠∅. If {𝑥_𝑛} is a sequence in 𝐶 weakly converging to 𝑥 and {(𝐼−𝑆)𝑥_𝑛} converges strongly to 𝑦, then (𝐼−𝑆)𝑥=𝑦; in particular, if 𝑦=0, then 𝑥∈Fix(𝑆).

Lemma 2.5 (see [14, 16]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Assume that the mapping 𝐺∶𝐶→𝐻 is monotone and weakly continuous along segments, that is, 𝐺(𝑥+𝑡𝑦)→𝐺(𝑥) weakly as 𝑡→0. Then, the variational inequality 𝑥∈𝐶,𝐺𝑥,𝑝−𝑥≥0,∀𝑝∈𝐶, (2.8) is equivalent to the dual variational inequality 𝑥∈𝐶,𝐺𝑝,𝑝−𝑥≥0,∀𝑝∈𝐶. (2.9)

Lemma 2.6 (see [5]). Let 𝐻 be a real Hilbert space and 𝐶 a closed convex subset of 𝐻. If 𝑇 is a 𝑘- strictly pseudocontractive mapping on 𝐶, then the fixed point set Fix(𝑇) is closed convex, so that the projection 𝑃_Fix(𝑇) is well defined.

Lemma 2.7 (see [5]). Let 𝐻 be a Hilbert space, 𝐶 a closed convex subset of 𝐻, and 𝑇∶𝐶→𝐻 a 𝑘- strictly pseudocontractive mapping. Define a mapping 𝑆∶𝐶→𝐻 by 𝑆𝑥=𝜆𝑥+(1−𝜆)𝑇𝑥 for all 𝑥∈𝐶. Then, as 𝜆∈[𝑘,1), 𝑆 is a nonexpansive mapping such that Fix(𝑆)=Fix(𝑇).

The following lemma can be easily proven, and, therefore, we omit the proof.

Lemma 2.8. Let 𝐻 be a real Hilbert space. Let 𝑉∶𝐻→𝐻 be an l-Lipschitzian mapping with constant 𝑙≥0 and 𝐹∶𝐻→𝐻 a 𝜌-Lipschitzian and 𝜂-strongly monotone mapping with constants 𝜌,𝜂>0. Then, for 0≤𝛾𝑙<𝜇𝜂, 𝜇𝐹−𝛾𝑉𝑥−𝜇𝐹−𝛾𝑉𝑦,𝑥−𝑦≥𝜇𝜂−𝛾𝑙𝑥−𝑦²,∀𝑥,𝑦∈𝐶. (2.10) That is, 𝜇𝐹−𝛾𝑉 is strongly monotone with constant 𝜇𝜂−𝛾𝑙.

The following lemma is an improvement of Lemma  2.9 in [4] (see also [14]).

Lemma 2.9. Let 𝐻 be a real Hilbert space 𝐻. Let 𝐹∶𝐻→𝐻 be a 𝜌-Lipschitzian and 𝜂-strongly monotone mapping with constants 𝜌,𝜂>0. Let 0<𝜇<2𝜂/𝜌² and 0<𝑡<𝜉≤1. Then, 𝑅∶=𝜌𝐼−𝑡𝜇𝐹∶𝐻→𝐻 is a contractive mapping with constant 𝜉−𝑡𝜏, where 𝜏=1−1−𝜇(2𝜂−𝜇𝜌²)<1.

Proof. First we show that 𝐼−𝜇𝐹 is strictly contractive. In fact, by applying the 𝜌-Lipschitz continuity and 𝜂-strongly monotonicity of 𝐹 and (2.3), we obtain, for 𝑥,𝑦∈𝐻, 𝐼−𝜇𝐹𝑥−𝐼−𝜇𝐹𝑦²=𝑥−𝑦−𝜇𝐹𝑥−𝐹𝑦²=𝑥−𝑦²−2𝜇𝐹𝑥−𝐹𝑦,𝑥−𝑦+𝜇²𝐹𝑥−𝐹𝑦²≤𝑥−𝑦²−2𝜇𝜂𝑥−𝑦²+𝜇²𝜌²𝑥−𝑦²=1−𝜇2𝜂−𝜇𝜌²𝑥−𝑦², (2.11) and so 𝐼−𝜇𝐹𝑥−𝐼−𝜇𝐹𝑦≤1−𝜇2𝜂−𝜇𝜌²𝑥−𝑦. (2.12) Now, noting that 𝑅∶=𝜉𝐼−𝑡𝜇𝐹=(𝜉−𝑡)𝐼−𝑡(𝜇𝐹−𝐼), from (2.12), we have, for 𝑥,𝑦∈𝐻, =)𝑦≤(𝑦≤(𝑅𝑥−𝑅𝑦(𝜉−𝑡𝑥−𝑦−𝑡𝜇𝐹−𝐼𝑥−𝜇𝐹−𝐼𝜉−𝑡)𝑥−𝑦+𝑡𝜇𝐹−𝐼𝑥−𝜇𝐹−𝐼𝜉−𝑡)𝑥−𝑦+𝑡1−𝜇2𝜂−𝜇𝜌²=𝑥−𝑦𝜉−𝑡1−1−𝜇2𝜂−𝜇𝜌²=(.𝑥−𝑦𝜉−𝑡𝜏)𝑥−𝑦 (2.13) Hence, 𝑅 is a contractive mapping with constant 𝜉−𝑡𝜏.

3. Iterative Algorithms

Let 𝐻 be a real Hilbert space and 𝐶 a nonempty closed convex subset of 𝐻. Let 𝑇∶𝐶→𝐶 be a 𝑘-strictly pseudocontractive mapping with Fix(𝑇)≠∅ for some 0≤𝑘<1, 𝐹∶𝐶→𝐻 a 𝜌-Lipschitzian and 𝜂-strongly monotone mapping with constants 𝜌,𝜂>0, and 𝑉∶𝐶→𝐻 an 𝑙-Lipschitzian mapping with constant 𝑙≥0. Let 0<𝜇<2𝜂/𝜌² and 0<𝛾𝑙<𝜏, where 𝜏=1−1−𝜇(2𝜂−𝜇𝜌²)<1. Let 𝑆∶𝐶→𝐶 be a mapping defined by 𝑆𝑥=𝑘𝑥+(1−𝑘)𝑇𝑥 and 𝑃_𝐶 a metric projection of 𝐻 onto 𝐶.

In this section, we introduce the following algorithm that generates a net {𝑥_𝑡}_𝑡∈(0,1) in an implicit way:𝑥_𝑡=𝑆𝑃_𝐶𝑡𝛾𝑉𝑥_𝑡+𝐼−𝑡𝜇𝐹𝑥_𝑡. (3.1) We prove the strong convergence of {𝑥_𝑡} as 𝑡→0 to a fixed point 𝑥 of 𝑇, which is a solution of the following variational inequality:𝜇𝐹−𝛾𝑉𝑥,𝑝−𝑥≥0,∀𝑝∈Fix(𝑇). (3.2) We also propose the following explicit algorithm, which generates a sequence in an explicit way:𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑆𝑃_𝐶𝛼_𝑛𝛾𝑉𝑥_𝑛+𝐼−𝛼_𝑛𝜇𝐹𝑥_𝑛,∀𝑛≥0, (3.3) where {𝛼_𝑛}, {𝛽_𝑛}⊂(0,1) and 𝑥₀∈𝐶 is an arbitrary initial guess, and we establish the strong convergence of this sequence to a fixed point 𝑥 of 𝑇, which is also a solution of the variational inequality (3.2).

3.1. Strong Convergence of the Implicit Algorithm

Now, for 𝑡∈(0,1), consider a mapping 𝑄_𝑡∶𝐶→𝐶 defined by𝑄_𝑡𝑥=𝑆𝑃_𝐶𝑡𝛾𝑉𝑥+𝐼−𝑡𝜇𝐹𝑥,∀𝑥∈𝐶. (3.4) It is easy to see that 𝑄_𝑡 is a contractive mapping with constant 1−𝑡(𝜏−𝛾𝑙). Indeed, by Lemma 2.9, we have𝑄_𝑡𝑥−𝑄_𝑡𝑦+𝑦+(=.≤𝑡𝛾𝑉𝑥−𝑉𝑦𝐼−𝑡𝜇𝐹𝑥−𝐼−𝑡𝜇𝐹≤𝑡𝛾𝑙𝑥−𝑦1−𝑡𝜏)𝑥−𝑦1−𝑡𝜏−𝛾𝑙𝑥−𝑦 (3.5) Hence, 𝑄_𝑡 has a unique fixed point, denoted by 𝑥_𝑡, which uniquely solves the fixed point equation (3.1).

We summarize the basic properties of {𝑥_𝑡}.

Proposition 3.1. Let 𝐻 be a real Hilbert space and 𝐶 a nonempty closed convex subset of 𝐻. Let 𝑇∶𝐶→𝐶 be a k-strictly pseudocontractive mapping with Fix(𝑇)≠∅ for some 0≤𝑘<1, 𝐹∶𝐶→𝐻 a 𝜌-Lipschitzain and 𝜂-strongly monotone mapping with constants 𝜌,𝜂>0, and 𝑉∶𝐶→𝐻 an 𝑙-Lipschitzian mapping with constant 𝑙≥0. Let 0<𝜇<2𝜂/𝜌² and 0<𝛾𝑙<𝜏, where 𝜏=1−1−𝜇(2𝜂−𝜇𝜌²)<1. Let 𝑆∶𝐶→𝐶 be a mapping defined by 𝑆𝑥=𝑘𝑥+(1−𝑘)𝑇𝑥 and 𝑃_𝐶 a metric projection of 𝐻 onto 𝐶. Let {𝑥_𝑡} be defined via (3.1). Then,(i){𝑥_𝑡} is bounded for 𝑡∈(0,1),(ii)lim_𝑡→0‖𝑥_𝑡−𝑆𝑥_𝑡‖=0,(iii)𝑥_𝑡 defines a continuous path from (0,1) in 𝐶.

Proof. (i) Let 𝑝∈Fix(𝑇). Observing Fix(𝑇)=Fix(𝑆) by Lemma 2.7, we have 𝑥_𝑡−𝑝=𝑆𝑃_𝐶𝑡𝛾𝑉𝑥_𝑡+𝐼−𝑡𝜇𝐹𝑥_𝑡−𝑆𝑃_𝐶𝑝≤𝑡𝛾𝑉𝑥_𝑡−𝜇𝐹𝑝+𝐼−𝑡𝜇𝐹𝑥_𝑡−𝑝≤(𝑥𝐼−𝑡𝜇𝐹1−𝑡𝜏)_𝑡−𝑝+𝑡𝛾𝑙𝑥_𝑡−𝑝+𝛾𝑉−𝜇𝐹𝑝. (3.6) So, it follows that 𝑥_𝑡≤𝑝−𝑝𝛾𝑉−𝜇𝐹.𝜏−𝛾𝑙 (3.7) Hence, {𝑥_𝑡} is bounded and so are {𝑉𝑥_𝑡}, {𝑆𝑥_𝑡}, and {𝐹𝑥_𝑡}.
(ii) By the boundedness of {𝑉𝑥_𝑡} and {𝐹𝑥_𝑡} in (i), we have 𝑥_𝑡−𝑆𝑥_𝑡=𝑆𝑃_𝐶𝑡𝛾𝑉𝑥_𝑡+𝐼−𝑡𝜇𝐹𝑥_𝑡−𝑆𝑃_𝐶𝑥_𝑡≤𝑡𝛾𝑉−𝜇𝐹𝑥_𝑡⟶0as𝑡⟶0. (3.8)
(iii) Let 𝑡,𝑡₀∈(0,1), and calculate 𝑥_𝑡−𝑥_𝑡0=𝑆𝑃_𝐶𝑡𝛾𝑉𝑥_𝑡+𝐼−𝑡𝜇𝐹𝑥_𝑡−𝑆𝑃_𝐶𝑡₀𝛾𝑉𝑥_𝑡0+𝐼−𝑡₀𝜇𝐹𝑥_𝑡0≤𝑡−𝑡₀𝛾𝑉𝑥_𝑡+𝑡₀𝛾𝑉𝑥_𝑡−𝑉𝑥_𝑡0−𝑡−𝑡₀𝜇𝐹𝑥_𝑡+𝐼−𝑡₀𝜇𝐹𝑥_𝑡−𝐼−𝑡₀𝜇𝐹𝑥_𝑡0≤𝑡−𝑡₀𝛾𝑉𝑥_𝑡+𝑡₀𝛾𝑙𝑥_𝑡−𝑥_𝑡0+𝑡−𝑡₀𝜇𝐹𝑥_𝑡+1−𝑡₀𝜏𝑥_𝑡−𝑥_𝑡0. (3.9) It follows that 𝑥_𝑡−𝑥_𝑡0≤𝛾𝑉𝑥_𝑡+𝜇𝐹𝑥_𝑡𝑡₀𝜏−𝛾𝑙𝑡−𝑡₀. (3.10) This shows that 𝑥_𝑡 is locally Lipschitzian and hence continuous.

We establish the strong convergence of the net {𝑥_𝑡} as 𝑡→0, which guarantees the existence of solutions of the variational inequality (3.2).

Theorem 3.2. Let 𝐻 be a real Hilbert space and 𝐶 a nonempty closed convex subset of 𝐻. Let 𝑇∶𝐶→𝐶 be a 𝑘-strictly pseudocontractive mapping with Fix(𝑇)≠∅ for some 0≤𝑘<1, 𝐹∶𝐶→𝐻 a 𝜌-Lipschitzain and 𝜂-strongly monotone mapping with constants 𝜌,𝜂>0, and 𝑉∶𝐶→𝐻 an 𝑙-Lipschitzian mapping with constant 𝑙≥0. Let 0<𝜇<2𝜂/𝜌² and 0<𝛾𝑙<𝜏, where 𝜏=1−1−𝜇(2𝜂−𝜇𝜌²)<1. Let 𝑆∶𝐶→𝐶 be a mapping defined by 𝑆𝑥=𝑘𝑥+(1−𝑘)𝑇𝑥 and 𝑃_𝐶 a metric projection of 𝐻 onto 𝐶. The net {𝑥_𝑡} defined via (3.1) converges strongly to a fixed point 𝑥 of 𝑇 as 𝑡→0, which solves the variational inequality (3.2), or, equivalently, one has 𝑃_𝐹(𝑇)(𝐼−𝜇𝐹+𝛾𝑉)𝑥=𝑥.

Proof. We first show the uniqueness of a solution of the variational inequality (3.2), which is indeed a consequence of the strong monotonicity of 𝜇𝐹−𝛾𝑉. In fact, noting that 0≤𝛾𝑙<𝜏 and 𝜇𝜂≥𝜏⇔𝜌≥𝜂, it follows from Lemma 2.8 that 𝜇𝐹−𝛾𝑉𝑥−𝜇𝐹−𝛾𝑉𝑦,𝑥−𝑦≥𝜇𝜂−𝛾𝑙𝑥−𝑦². (3.11) That is, 𝜇𝐹−𝛾𝑉 is strongly monotone for 0≤𝛾𝑙<𝜏≤𝜇𝜂. Suppose that 𝑥∈Fix(𝑇) and 𝑥∈Fix(𝑇) both are solutions to (3.2). Then, we have 𝜇𝐹−𝛾𝑉𝑥,𝑥−𝑥≥0,𝜇𝐹−𝛾𝑉𝑥,𝑥−𝑥≥0. (3.12) Adding up (3.12) yields 𝜇𝐹−𝛾𝑉𝑥−𝜇𝐹−𝛾𝑉𝑥,𝑥−𝑥≤0. (3.14) The strong monotonicity of 𝜇𝐹−𝛾𝑉 implies that 𝑥=𝑥 and the uniqueness is proved.
Next, we prove that 𝑥_𝑡→𝑥 as 𝑡→0. To this end, set 𝑦_𝑡=𝑃_𝐶[𝑡𝛾𝑉𝑥_𝑡+(𝐼−𝑡𝜇𝐹)𝑥_𝑡] for all 𝑡∈(0,1). Then, observing Fix(𝑇)=Fix(𝑆) by Lemma 2.7, we have 𝑥_𝑡=𝑆𝑦_𝑡 and for any 𝑝∈Fix(𝑇)𝑥_𝑡−𝑝≤𝑦_𝑡−𝑝. (3.15) Also it follows that 𝑦_𝑡−𝑥_𝑡=𝑥_𝑡−𝑃_𝐶𝑡𝛾𝑉𝑥_𝑡+𝐼−𝑡𝜇𝐹𝑥_𝑡≤𝑡𝛾𝑥_𝑡+𝜇𝐹𝑥_𝑡⟶0as𝑡⟶0. (3.16) Since 𝑃_𝐶 is the metric projection from 𝐻 onto 𝐶, we have, for given 𝑝∈Fix(𝑇), 𝑦_𝑡−𝑝²=𝑃_𝐶𝑡𝛾𝑉𝑥_𝑡+𝐼−𝑡𝜇𝐹𝑥_𝑡−𝑡𝛾𝑉𝑥_𝑡+𝐼−𝑡𝜇𝐹𝑥_𝑡,𝑦_𝑡+−𝑝𝑡𝛾𝑉𝑥_𝑡+𝐼−𝑡𝜇𝐹𝑥_𝑡−𝑝,𝑦_𝑡≤−𝑝𝑡𝛾𝑉𝑥_𝑡+𝐼−𝑡𝜇𝐹𝑥_𝑡−𝑝,𝑦_𝑡=−𝑝𝑡𝛾𝑉𝑥_𝑡−𝑡𝜇𝐹𝑝,𝑦_𝑡−𝑝+𝐼−𝑡𝜇𝐹𝑥_𝑡−𝐼−𝑡𝜇𝐹𝑝,𝑦_𝑡≤(𝑥−𝑝1−𝑡𝜏)_𝑡−𝑝𝑦_𝑡−𝑝+𝑡𝛾𝑉𝑥_𝑡−𝜇𝐹𝑝,𝑦_𝑡−𝑝. (3.17) It follows that 𝑦_𝑡−𝑝²≤1𝜏𝛾𝑉𝑥_𝑡−𝜇𝐹𝑝,𝑦_𝑡=1−𝑝𝜏𝛾𝑉𝑥_𝑡−𝑉𝑝,𝑦_𝑡−𝑝+𝛾𝑉−𝜇𝐹𝑝,𝑦_𝑡≤1−𝑝𝜏𝛾𝑙𝑥_𝑡−𝑝𝑦_𝑡−𝑝+𝛾𝑉−𝜇𝐹𝑝,𝑥_𝑡−𝑝. (3.18) By (3.15), this implies 𝑦_𝑡−𝑝²≤1𝜏−𝛾𝑙𝛾𝑉−𝜇𝐹𝑝,𝑦_𝑡=1−𝑝𝜏−𝛾𝑙𝜇𝐹−𝛾𝑉𝑝,𝑝−𝑦_𝑡, (3.19) and hence 𝑥_𝑡−𝑝²≤1𝜏−𝛾𝑙𝜇𝐹−𝛾𝑉𝑝,𝑝−𝑦_𝑡. (3.20) Since {𝑥_𝑡} is bounded as 𝑡→0 (by Proposition 3.1 (i)), we see that if {𝑡_𝑛} is a subsequence in (0,1) such that 𝑡_𝑛→0 and 𝑥_𝑡𝑛⇀𝑥, then, from (3.16), we have 𝑦_𝑡𝑛⇀𝑥. By Proposition 3.1 (ii), lim_𝑛→∞(𝐼−𝑆)𝑥_𝑡𝑛=0. By Lemmas 2.4 and 2.7, 𝑥∈Fix(𝑇). Therefore, we can substitute 𝑥 for 𝑝 in (3.20) to obtain 𝑥_𝑡𝑛−𝑥²≤1𝜏−𝛾𝑙𝜇𝐹−𝛾𝑉𝑥,𝑥−𝑦_𝑡𝑛. (3.21) Consequently, the weak convergence of {𝑦_𝑡𝑛} to 𝑥 yields that {𝑥_𝑡𝑛}→𝑥 strongly. Now we show that 𝑥 solves the variational inequality (3.2). Again, observe (3.20) and take the limit as 𝑛→∞ to obtain 𝑥−𝑝²≤1(𝑇).𝜏−𝛾𝑙𝜇𝐹−𝛾𝑉𝑝,𝑝−𝑥,∀𝑝∈Fix (3.22) Hence 𝑥 solves the following variational inequality: 𝜇𝐹−𝛾𝑉𝑝,𝑝−𝑥≥0,∀𝑝∈Fix(𝑇), (3.23) or the equivalent dual variational inequality (see Lemmas 2.5 and 2.8) 𝜇𝐹−𝛾𝑉𝑥,𝑝−𝑥≥0,∀𝑝∈Fix(𝑇). (3.24) Moreover, if {𝑡_𝑗} is another subsequence in (0,1) such that 𝑡_𝑗→0 and 𝑥_𝑡𝑗⇀𝑥, then we also have 𝑦_𝑡𝑗⇀𝑥 from (3.16). By the same argument, we can show that 𝑥∈Fix(𝑇) and 𝑥 solves the variational inequality (3.2); hence 𝑥=𝑥 by uniqueness. In sum, we have shown that each cluster point of {𝑥_𝑡} (at 𝑡→0) equals 𝑥. Therefore 𝑥_𝑡→𝑥 as 𝑡→0.
The variational inequality (3.2) can be rewritten as 𝐼−𝜇𝐹+𝛾𝑉𝑥−𝑥,𝑝−𝑥≤0,∀𝑝∈Fix(𝑇). (3.25) By reminding the reader of (2.2) and Lemma 2.6, this is equivalent to the fixed point equation 𝑃_𝐹(𝑇)𝐼−𝜇𝐹+𝛾𝑉𝑥=𝑥. (3.26)

From Theorem 3.2, we can deduce the following result.

Corollary 3.3. Let 𝐻 be a real Hilbert space and 𝐶 a nonempty closed convex subset of 𝐻. Let 𝑇∶𝐶→𝐶 be a 𝑘-strictly pseudocontractive mapping with Fix(𝑇)≠∅ for some 0≤𝑘<1. For each 𝑡∈(0,1), let the net {𝑥_𝑡} be defined by 𝑥_𝑡=𝑆𝑃_𝐶(1−𝑡)𝑥_𝑡,∀𝑡∈(0,1), (3.27) where 𝑆∶𝐶→𝐶 is a mapping defined by 𝑆𝑥=𝑘𝑥+(1−𝑘)𝑇𝑥 and 𝑃_𝐶 is a metric projection of 𝐻 onto 𝐶. Then the net {𝑥_𝑡} defined via (3.27) converges strongly, as 𝑡→0, to the minimum-norm point 𝑥∈Fix(𝑇).

Proof. In (3.20) with 𝐹≡𝐼, 𝑉≡0, 𝑙=0, 𝜇=1, and 𝜏=1, letting 𝑡→0 yields 𝑥−𝑝²≤𝑝,𝑝−𝑥,∀𝑝∈Fix(𝑇). (3.28) Equivalently, 𝑥,𝑝−𝑥≥0,∀𝑝∈Fix(𝑇). (3.29) This obviously implies that ‖𝑥‖²≤𝑝,𝑥≤𝑝‖𝑥‖,∀𝑝∈Fix(𝑇). (3.30) It turns out that ‖𝑥‖≤‖𝑝‖ for all 𝑝∈Fix(𝑇). Therefore, 𝑥 is minimum-norm point of Fix(𝑇).

3.2. Strong Convergence of the Explicit Algorithm

Now, using Theorem 3.2, we show the strong convergence of the sequence generated by the explicit algorithm (3.3) to a fixed point 𝑥 of 𝑇, which is also a solution of the variational inequality (3.2).

Theorem 3.4. Let 𝐻 be a real Hilbert space and 𝐶 a nonempty closed convex subset of 𝐻. Let 𝑇∶𝐶→𝐶 be a 𝑘-strictly pseudocontractive mapping with Fix(𝑇)≠∅ for some 0≤𝑘<1, 𝐹∶𝐶→𝐻 a 𝜌-Lipschitzain and 𝜂-strongly monotone mapping with constants 𝜌,𝜂>0, and 𝑉∶𝐶→𝐻 an 𝑙-Lipschitzian mapping with constant 𝑙≥0. Let 0<𝜇<2𝜂/𝜌² and 0<𝛾𝑙<𝜏, where 𝜏=1−1−𝜇(2𝜂−𝜇𝜌²)<1. Let 𝑆∶𝐶→𝐻 be a mapping defined by 𝑆𝑥=𝑘𝑥+(1−𝑘)𝑇𝑥 and 𝑃_𝐶 a metric projection of 𝐻 onto 𝐶. For any given 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence generated by the explicit algorithm (3.3), where {𝛼_𝑛} and {𝛽_𝑛} satisfy the following conditions:(C1)lim_𝑛→∞𝛼_𝑛=0,(C2)∑^∞_𝑛=0𝛼_𝑛=∞,(C3)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1.
Then, {𝑥_𝑛} converges strongly to 𝑥∈Fix(𝑇), which is the unique solution of the variational inequality (3.2).

Proof. First, from condition (C1), without loss of generality, we assume that 𝛼_𝑛(𝜏−𝛾𝑙)<1 and 𝛼_𝑛<(1−𝛽_𝑛) for 𝑛≥0. From now, we put 𝑦_𝑛=𝑆𝑃_𝐶𝑢_𝑛 and 𝑢_𝑛=𝛼_𝑛𝛾𝑉𝑥_𝑛+(𝐼−𝛼_𝑛𝜇𝐹)𝑥_𝑛.
We divide the proof several steps.
Step 1. We show that ‖𝑥_𝑛−𝑝‖≤max{‖𝑥₀−𝑝‖,‖(𝛾𝑉−𝜇𝐹)𝑝‖/(𝜏−𝛾𝑙)} for all 𝑛≥0 and all 𝑝∈Fix(𝑇)=Fix(𝑆). Indeed, let 𝑝∈Fix(𝑇). Then, from Lemma 2.9, we have 𝑦_𝑛−𝑝=𝑆𝑃_𝐶𝑢_𝑛−𝑆𝑃_𝐶𝑝≤𝛼_𝑛𝛾𝑉𝑥_𝑛−𝜇𝐹𝑝+𝛽_𝑛𝑥_𝑛+−𝑝1−𝛽_𝑛𝐼−𝛼_𝑛𝜇𝐹𝑥_𝑛−1−𝛽_𝑛𝐼−𝛼_𝑛𝑝≤𝜇𝐹1−𝛽_𝑛−𝛼_𝑛𝜏𝑥_𝑛−𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝛾𝑉𝑥_𝑛≤−𝜇𝐹𝑝1−𝛼_𝑛𝜏𝑥_𝑛−𝑝+𝛼_𝑛𝛾𝑉𝑥_𝑛+𝑝≤𝛼−𝛾𝑉𝑝𝛾𝑉−𝜇𝐹1−𝜏−𝛾𝑙_𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝛾𝑉−𝜇𝐹𝑝. (3.31) Thus, it follows that 𝑥_𝑛+1−𝑝≤𝛽_𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛𝑦_𝑛−𝑝≤𝛽_𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛1−𝛼_𝑛𝜏−𝛾𝑙𝑥_𝑛−𝑝+1−𝛽_𝑛𝛼_𝑛𝑝=𝛾𝑉−𝜇𝐹1−𝛼_𝑛1−𝛽_𝑛𝜏−𝛾𝑙𝑥_𝑛−𝑝+𝛼_𝑛1−𝛽_𝑛𝑝𝜏−𝛾𝑙𝛾𝑉−𝜇𝐹𝑥𝜏−𝛾𝑙≤max_𝑛,𝑝−𝑝𝛾𝑉−𝜇𝐹.𝜏−𝛾𝑙 (3.32) Using an induction, we have ‖𝑥_𝑛−𝑝‖≤max{‖𝑥₀−𝑝‖,‖(𝛾𝑉−𝜇𝐹)𝑝‖/(𝜏−𝛾𝑙)}. Hence, {𝑥_𝑛} is bounded, and so are {𝑦_𝑛}, {𝑢_𝑛}, {𝑉𝑥_𝑛}, {𝑆𝑥_𝑛}, and {𝐹𝑥_𝑛}.Step 2. We show that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0. Indeed, from (3.3), we observe 𝑦_𝑛+1−𝑦_𝑛=𝑆𝑃_𝐶𝑢_𝑛+1−𝑆𝑃_𝐶𝑢_𝑛≤𝛼_𝑛+1𝛾𝑉𝑥_𝑛+1+𝐼−𝛼_𝑛+1𝜇𝐹𝑥_𝑛+1−𝛼_𝑛𝛾𝑉𝑥_𝑛+𝐼−𝛼_𝑛𝜇𝐹𝑥_𝑛=𝛼_𝑛+1𝛾𝑉𝑥_𝑛+1−𝑉𝑥_𝑛+𝛾𝛼_𝑛+1−𝛼_𝑛𝑉𝑥_𝑛+𝐼−𝛼_𝑛+1𝜇𝐹𝑥_𝑛+1−𝐼−𝛼_𝑛+1𝜇𝐹𝑥_𝑛+𝜇𝛼_𝑛+1−𝛼_𝑛𝐹𝑥_𝑛≤1−𝛼_𝑛+1𝜏−𝛾𝑙𝑥_𝑛+1−𝑥_𝑛+𝑀𝛼_𝑛+1−𝛼_𝑛, (3.33) where 𝑀=sup{𝛾‖𝑉𝑥_𝑛‖+𝜇‖𝐹𝑥_𝑛‖∶𝑛≥0}. Thus, it follows that 𝑦_𝑛+1−𝑦_𝑛−𝑥_𝑛+1−𝑥_𝑛≤−𝛼_𝑛+1𝜏−𝛾𝑙𝑥_𝑛+1−𝑥_𝑛+𝑀𝛼_𝑛+1−𝛼_𝑛, (3.34) which implies, from condition (C1), that limsup_𝑛→∞𝑦_𝑛+1−𝑦_𝑛−𝑥_𝑛+1−𝑥_𝑛≤0. (3.35) Hence, by Lemma 2.3, we have lim_𝑛→∞𝑦_𝑛−𝑥_𝑛=0. (3.36) Consequently, from condition (C3), it follows that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=lim_𝑛→∞1−𝛽_𝑛𝑦_𝑛−𝑥_𝑛=0. (3.37)Step 3. We show that lim_𝑛→∞‖𝑥_𝑛−𝑆𝑥_𝑛‖=0. Indeed, we have 𝑥_𝑛−𝑆𝑥_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑆𝑥_𝑛=𝑥_𝑛+1−𝑥_𝑛+𝛽_𝑛𝑥_𝑛−𝑆𝑥_𝑛+1−𝛽_𝑛𝑦_𝑛−𝑆𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝛽_𝑛𝑥_𝑛−𝑆𝑥_𝑛+1−𝛽_𝑛𝑦_𝑛−𝑆𝑃_𝐶𝑥_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝛽_𝑛𝑥_𝑛−𝑆𝑥_𝑛+1−𝛽_𝑛𝛼_𝑛𝛾𝑉−𝜇𝐹𝑥_𝑛, (3.38) that is, 𝑥_𝑛−𝑆𝑥_𝑛≤11−𝛽_𝑛𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛𝛾𝑉−𝜇𝐹𝑥_𝑛. (3.39) This together with conditions (C1) and (C3) and Step 2 implies lim_𝑛→∞𝑥_𝑛−𝑆𝑥_𝑛=0. (3.40)Step 4. We show that limsup_𝑛→∞𝛾𝑉−𝜇𝐹𝑥,𝑢_𝑛−𝑥≤0, (3.41) where 𝑥=lim_𝑡→0𝑥_𝑡 with 𝑥_𝑡 being defined by (3.1). (We note that, from Theorem 3.2, 𝑥∈Fix(𝑇) and 𝑥 is the unique solution of the variational inequality (3.2)). To show this, we can choose a subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} such that lim_𝑗→∞𝛾𝑉−𝜇𝐹𝑥,𝑥_𝑛𝑗−𝑥=limsup_𝑛→∞𝛾𝑉−𝜇𝐹𝑥,𝑥_𝑛−𝑥. (3.42) Since {𝑥_𝑛} is bounded, there exists a subsequence {𝑥_𝑛𝑗𝑖} of {𝑥_𝑛𝑗}, which converges weakly to 𝑤. Without loss of generality, we can assume that 𝑥_𝑛𝑗⇀𝑤. Since ‖𝑥_𝑛−𝑆𝑥_𝑛‖→0 by Step 3, we obtain 𝑤=𝑆𝑤 by virtue of Lemma 2.4. From Lemma 2.7, we have 𝑤∈Fix(𝑇). Therefore, from (3.2), it follows that limsup_𝑛→∞𝛾𝑉−𝜇𝐹𝑥,𝑥_𝑛−𝑥=lim_𝑗→∞𝛾𝑉−𝜇𝐹𝑥,𝑥_𝑛𝑗=−𝑥𝛾𝑉−𝜇𝐹𝑥,𝑤−𝑥≤0. (3.43) We notice that, by condition (C1), 𝑢_𝑛−𝑥_𝑛≤𝛼_𝑛𝜇𝐹−𝛾𝑉𝑥_𝑛⟶0as𝑛⟶∞. (3.44) Hence, from (3.43), we obtain limsup_𝑛→∞𝛾𝑉−𝜇𝐹𝑥,𝑢_𝑛−𝑥≤0. (3.45)Step 5. We show that lim_𝑛→∞‖𝑥_𝑛−𝑥‖=0, where 𝑥=lim_𝑡→0𝑥_𝑡 with 𝑥_𝑡 being defined by (3.1), and 𝑥 is the unique solution of the variational inequality (3.2). Indeed, we observe that 𝑢_𝑛−𝑥≤𝑥_𝑛−𝑥+𝛼_𝑛𝛾𝑉−𝜇𝐹𝑥_𝑛. (3.46) Therefore, from the convexity of ‖⋅‖², (3.3), and Lemma 2.1, we have 𝑥_𝑛+1−𝑥²≤𝛽_𝑛𝑥_𝑛−𝑥²+1−𝛽_𝑛𝑦_𝑛−𝑥²≤𝛽_𝑛𝑥_𝑛−𝑥²+1−𝛽_𝑛𝑢_𝑛−𝑥²≤𝛽_𝑛𝑥_𝑛−𝑥²+1−𝛽_𝑛𝐼−𝛼_𝑛𝜇𝐹𝑥_𝑛−𝐼−𝛼_𝑛𝜇𝐹𝑥²+21−𝛽_𝑛𝛼_𝑛𝛾𝑉𝑥_𝑛−𝜇𝐹𝑥,𝑢_𝑛−𝑥≤𝛽_𝑛𝑥_𝑛−𝑥²+1−𝛽_𝑛1−𝛼_𝑛𝜏²𝑥_𝑛−𝑥²+21−𝛽_𝑛𝛼_𝑛𝛾𝑉𝑥_𝑛−𝛾𝑉𝑥,𝑢_𝑛−𝑥+𝛾𝑉−𝜇𝐹𝑥,𝑢_𝑛≤𝛽−𝑥_𝑛+1−𝛽_𝑛1−2𝛼_𝑛𝜏+𝛼²_𝑛𝜏²𝑥_𝑛−𝑥²+21−𝛽_𝑛𝛼_𝑛𝛾𝑙𝑥_𝑛−𝑥𝑢_𝑛−𝑥+21−𝛽_𝑛𝛼_𝑛𝛾𝑉𝑥−𝜇𝐹𝑥,𝑢_𝑛≤−𝑥1−1−𝛽_𝑛2𝛼_𝑛𝜏−𝛼²_𝑛𝜏²𝑥_𝑛−𝑥²+21−𝛽_𝑛𝛼_𝑛𝛾𝑙𝑥_𝑛−𝑥²+21−𝛽_𝑛𝛼²_𝑛𝛾𝑙𝑥_𝑛−𝑥𝛾𝑉−𝜇𝐹𝑥_𝑛+21−𝛽_𝑛𝛼_𝑛𝛾𝑉−𝜇𝐹𝑥,𝑢_𝑛≤−𝑥1−21−𝛽_𝑛𝜏−𝛾𝑙𝛼_𝑛𝑥_𝑛−𝑥²+𝛼²_𝑛𝜏²𝑥_𝑛−𝑥²+2𝛼²_𝑛𝛾𝑙𝑥_𝑛−𝑥𝛾𝑉−𝜇𝐹𝑥_𝑛+21−𝛽_𝑛𝛼_𝑛𝛾𝑉−𝜇𝐹𝑥,𝑢_𝑛≤()𝛼−𝑥1−21−𝑏𝜏−𝛾𝑙_𝑛𝑥_𝑛−𝑥²+𝛼_𝑛𝛼_𝑛𝜏²𝑀²₁+2𝛼_𝑛𝛾𝑙𝑀₁𝑀₂+21−𝛽_𝑛𝛼_𝑛𝛾𝑉−𝜇𝐹𝑥,𝑢_𝑛=−𝑥1−𝜆_𝑛𝑥_𝑛−𝑥²+𝜆_𝑛𝛿_𝑛, (3.47) where 0<𝑏=limsup_𝑛→∞𝛽_𝑛<1, 𝑀₁=sup{‖𝑥_𝑛−𝑥‖∶𝑛≥0}, 𝑀₂=sup{‖𝛾𝑉𝑥_𝑛−𝜇𝐹𝑥_𝑛‖∶𝑛≥0}, 𝜆_𝑛=2(1−𝑏)(𝜏−𝛾𝑙)𝛼_𝑛, and 𝛿_𝑛=1()𝛼1−𝑏𝜏−𝛾𝑙^𝑛𝜏²𝑀²₁2+𝛼_𝑛𝛾𝑙𝑀₁𝑀₂+1−𝛽_𝑛𝛾𝑉−𝜇𝐹𝑥,𝑢_𝑛−𝑥. (3.48) From conditions (C1), (C2), and (C3) and Step 4, it is easy to see that 𝜆_𝑛→0, ∑^∞_𝑛=0𝜆_𝑛=∞ and limsup_𝑛→∞𝛿_𝑛≤0. Hence, by Lemma 2.2, we conclude that 𝑥_𝑛→𝑥 as 𝑛→∞. This completes the proof.

From Theorem 3.4, we can also deduce the following result.

Corollary 3.5. Let 𝐻 be a real Hilbert space and 𝐶 a nonempty closed convex subset of 𝐻. Let 𝑇∶𝐶→𝐶 be a 𝑘-strictly pseudocontractive mapping with Fix(𝑇)≠∅ for some 0≤𝑘<1. For each 𝑥₀∈𝐶, let the sequence {𝑥_𝑛} be defined by 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_n𝑆𝑃_𝐶1−𝛼_𝑛𝑥_𝑛,∀𝑛≥0, (3.49) where 𝑆∶𝐶→𝐶 is a mapping defined by 𝑆𝑥=𝑘𝑥+(1−𝑘)𝑇𝑥 and 𝑃_𝐶 is a metric projection of 𝐻 onto 𝐶. If {𝛼_𝑛} and {𝛽_𝑛} satisfy conditions (C1), (C2), and (C3) in Theorem 3.4, then the sequence {𝑥_𝑛} defined via (3.49) converges strongly, as 𝑛→∞, to the minimum-norm point 𝑥∈Fix(𝑇).

Proof. VI (3.2) is reduced to the inequality 𝑥,𝑝−𝑥≥0,∀𝑝∈Fix(𝑇). (3.50) This is equivalent to ‖𝑥‖²≤⟨𝑝,𝑥⟩≤‖𝑝‖‖𝑥‖ for all 𝑝∈Fix(𝑇). It turns out that ‖𝑥‖≤‖𝑝‖ for all 𝑝∈Fix(𝑇) and 𝑥 is the minimum-norm point of Fix(𝑇).

Remark 3.6. We point out that our algorithms (3.1) and (3.3) are new ones different from those in the literature (see [2–4, 15, 18, 21, 22] and references therein).

4. Conclusion and Future Directions

In this paper, we have introduced new implicit and explicit algorithms for finding fixed points of a 𝑘-strictly pseudocontractive mapping and for solving a certain variational inequality and have established strong convergence of the proposed algorithms to a fixed point of the mapping, which is a solution of a certain variational inequality, where the constraint set is the fixed points of the mapping. As direct consequences, we have considered the quadratic minimization problem on the set of fixed points of the mapping.

In forthcoming studies, we will consider implicit and explicit algorithms for solving some variational inequalities, where the constraint set is the common set of the set of fixed points of the mapping and the set of solutions of the equilibrium problem.

We hope that the ideas and techniques of this paper may stimulate further research in this field.

Acknowledgments

The author thanks the referees for their valuable comments and suggests, which improved the presentation of this paper, and for providing some recent related papers. This study was supported by research funds from Dong-A University.