Generalized Systems of Variational Inequalities and Projection Methods for Inverse-Strongly Monotone Mappings

Abstract

We introduce an iterative sequence for finding a common element of the set of fixed points of a nonexpansive mapping and the solutions of the variational inequality problem for three inverse-strongly monotone mappings. Under suitable conditions, some strong convergence theorems for approximating a common element of the above two sets are obtained. Moreover, using the above theorem, we also apply to find solutions of a general system of variational inequality and a zero of a maximal monotone operator in a real Hilbert space. As applications, at the end of the paper we utilize our results to study some convergence problem for strictly pseudocontractive mappings. Our results include the previous results as special cases extend and improve the results of Ceng et al., (2008) and many others.

1. Introduction

Variational inequalities are known to play a crucial role in mathematics as a unified framework for studying a large variety of problems arising, for instance, in structural analysis, engineering sciences and others. Roughly speaking, they can be recast as fixed-point problems, and most of the numerical methods related to this topic are based on projection methods. Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and ‖⋅‖, and let 𝐸 be a nonempty, closed, convex subset of 𝐻. A mapping 𝐴∶𝐸→𝐻 is called 𝛼-inverse-strongly monotone if there exists a positive real number 𝛼>0 such that ⟨𝐴𝑥−𝐴𝑦,𝑥−𝑦⟩≥𝛼‖𝐴𝑥−𝐴𝑦‖²,∀𝑥,𝑦∈𝐸(1.1) (see [1, 2]). It is obvious that every 𝛼-inverse-strongly monotone mapping 𝐴 is monotone and Lipschitz continuous. A mapping 𝑆∶𝐸→𝐸 is called nonexpansive if ‖𝑆𝑥−𝑆𝑦‖≤‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐸.(1.2) We denote by 𝐹(𝑆) the set of fixed points of 𝑆 and by 𝑃_𝐸 the metric projection of 𝐻 onto 𝐸. Recall that the classical variational inequality, denoted by 𝑉𝐼(𝐴,𝐸), is to find an 𝑥^∗∈𝐸 such that ⟨𝐴𝑥^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈𝐸.(1.3) The set of solutions of 𝑉𝐼(𝐴,𝐸) is denoted by Γ. The variational inequality has been widely studied in the literature; see, for example, [3–6] and the references therein.

For finding an element of 𝐹(𝑆)∩Γ, Takahashi and Toyoda [7] introduced the following iterative scheme:𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑆𝑃_𝐸𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛,(1.4) for every 𝑛=0,1,2,…, where 𝑥₀=𝑥∈𝐸,{𝛼_𝑛} is a sequence in (0,1), and {𝜆_𝑛} is a sequence in (0,2𝛼). On the other hand, for solving the variational inequality problem in the finite-dimensional Euclidean space 𝐑^𝑛, Korpelevich (1976) [8] introduced the following so-called extragradient method:𝑦𝑥₀=𝑥∈𝐸,_𝑛𝑥_𝑛+1=𝑃_𝐸𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛,=𝑃_𝐸𝑥_n−𝜆_𝑛𝐴𝑦_𝑛,(1.5) for every 𝑛=0,1,2,…, where 𝜆_𝑛∈(0,1/𝑘). Many authors using extragradient method for approximating common fixed points and variational inequality problems (see also [9, 10]). Recently, Nadezhkina and Takahashi [11] and Zeng et al. [12] proposed some iterative schemes for finding elements in 𝐹(𝑆)∩Γ by combining (1.4) and (1.5). Further, these iterative schemes are extended in Y. Yao and J. C. Yao [13] to develop a new iterative scheme for finding elements in 𝐹(𝑆)∩Γ.

Consider the following problem of finding (𝑥^∗,𝑦^∗)∈𝐸×𝐸 such that (see cf. Ceng et al. [14]):⟨⟨𝜇𝜆𝐴𝐵𝑥𝑦^∗_∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈𝐸,+𝑦^∗−𝑥^∗,𝑥−𝑦^∗⟩≥0,∀𝑥∈𝐸,(1.6) which is called general system of variational inequalities (GSVI), where 𝜆>0 and 𝜇>0 are two constants. In particular, if 𝐴=𝐵, then problem (1.6) reduces to finding (𝑥^∗,𝑦^∗)∈𝐸×𝐸 such that⟨⟨𝜇𝜆𝐴𝐴𝑥𝑦^∗_∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈𝐸,+𝑦^∗−𝑥^∗,𝑥−𝑦^∗⟩≥0,∀𝑥∈𝐸,(1.7) which is defined by [15, 16], and is called the new system of variational inequalities. Further, if 𝑥^∗=𝑦^∗, then problem (1.7) reduces to the classical variational inequality VI(𝐴,𝐸), that is, find 𝑥^∗∈𝐸 such that ⟨𝐴𝑥^∗,𝑥−𝑥^∗⟩≥0, forall𝑥∈𝐸.

We can characteristic problem, if 𝑥^∗∈𝐹(𝑆)∩𝑉𝐼(𝐴,𝐸), then it follows that 𝑥^∗=𝑆𝑥^∗=𝑃_𝐸[𝑥^∗−𝜌𝐴𝑥^∗], where 𝜌>0 is a constant.

In 2008, Ceng et al. [14] introduced a relaxed extragradient method for finding solutions of problem (1.6). Let the mappings 𝐴,𝐵∶𝐸→𝐻 be 𝛼-inverse-strongly monotone and 𝛽-inverse-strongly monotone, respectively. Let 𝑆∶𝐸→𝐸 be a nonexpansive mapping. Suppose 𝑥₁=𝑢∈𝐸 and {𝑥_𝑛} is generated by𝑥_𝑛+1𝑦_𝑛=𝑃_𝐸𝑥_𝑛−𝜇𝐵𝑥_𝑛,=𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑆𝑃_𝐸𝑦_𝑛−𝜆_𝑛A𝑦_𝑛,(1.8) where 𝜆∈(0,2𝛼),𝜇∈(0,2𝛽), and {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛} are three sequences in [0,1] such that 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1,forall𝑛≥1. First, problem (1.6) is proven to be equivalent to a fixed point problem of a nonexpansive mapping.

In this paper, motivated by what is mentioned above, we consider generalized system of variational inequalities as follows.

Let 𝐸 be a nonempty, closed, convex subset of a real Hilbert space 𝐻. Let 𝐴,𝐵,𝐶∶𝐸→𝐻 be three mappings. We consider the following problem of finding (𝑥^∗,𝑦^∗,𝑧^∗)∈𝐸×𝐸×𝐸 such that⟨⟨𝜆𝜇𝐴𝐵𝑦𝑧^∗_∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈𝐸,⟨𝜏𝐶𝑥_∗+𝑦^∗−𝑧^∗,𝑥−𝑦^∗⟩≥0,∀𝑥∈𝐸,+𝑧^∗−𝑥^∗,𝑥−𝑧^∗⟩≥0,∀𝑥∈𝐸,(1.9) which is called a general system of variational inequalities where 𝜆>0,𝜇>0 and 𝜏>0 are three constants.

In particular, if 𝐴=𝐵=𝐶, then problem (1.9) reduces to finding (𝑥^∗,𝑦^∗,𝑧^∗)∈𝐸×𝐸×𝐸 such that⟨⟨𝜆𝜇𝐴𝐴𝑦𝑧^∗_∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈𝐸,⟨𝜏𝐴𝑥_∗+𝑦^∗−𝑧^∗,𝑥−𝑦^∗⟩≥0,∀𝑥∈𝐸,+𝑧^∗−𝑥^∗,𝑥−𝑧^∗⟩≥0,∀𝑥∈𝐸.(1.10) Next, we consider some special classes of the GSVI problem (1.9) reduce to the following GSVI.(i)If 𝜏=0, then the GSVI problems (1.9) reduce to GSVI problem (1.6).(ii)If 𝜏=𝜇=0, then the GSVI problems (1.9) reduce to classical variational inequality VI(A,E) problem.

The above system enters a class of more general problems which originated mainly from the Nash equilibrium points and was treated from a theoretical viewpoint in [17, 18]. Observe at the same time that, to construct a mathematical model which is as close as possible to a real complex problem, we often have to use constraints which can be expressed as one several subproblems of a general problem. These constrains can be given, for instance, by variational inequalities, by fixed point problems, or by problems of different types.

This paper deals with a relaxed extragradient approximation method for solving a system of variational inequalities over the fixed-point sets of nonexpansive mapping. Under classical conditions, we prove a strong convergence theorem for this method. Moreover, the proposed algorithm can be applied for instance to solving the classical variational inequality problems.

2. Preliminaries

Let 𝐸 be a nonempty, closed, convex subset of a real Hilbert space 𝐻. For every point 𝑥∈𝐻, there exists a unique nearest point in 𝐸, denoted by 𝑃_𝐸𝑥, such that ‖‖𝑥−𝑃_𝐸𝑥‖‖≤‖𝑥−𝑦‖,∀𝑦∈𝐸.(2.1)𝑃_𝐸 is call the metric projection of 𝐻 onto 𝐸.

Recall that, 𝑃_𝐸𝑥 is characterized by following properties: 𝑃_𝐸𝑥∈𝐸 and ‖𝑥−𝑦‖⟨₂𝑥−𝑃_𝐸𝑥,𝑦−𝑃_𝐸𝑥⟩≤0,≥‖‖𝑥−𝑃_𝐸𝑥‖‖²+‖‖𝑃_𝐸𝑥−𝑦‖‖²,(2.2) for all 𝑥∈𝐻 and 𝑦∈𝐸.

Lemma 2.1 (see cf. Zhang et al. [19]). The metric projection 𝑃_𝐸 has the following properties: (i)𝑃_𝐸∶𝐻→𝐸 is nonexpansive;(ii)𝑃_𝐸∶𝐻→𝐸 is firmly nonexpansive, that is, ‖‖𝑃_𝐸𝑥−𝑃_𝐸𝑦‖‖²≤⟨𝑃_𝐸𝑥−𝑃_𝐸𝑦,𝑥−𝑦⟩,∀𝑥,𝑦∈𝐻;(2.3)(iii)for each 𝑥∈𝐻, 𝑧=𝑃_𝐸(𝑥)⟺⟨𝑥−𝑧,𝑧−𝑦⟩≥0,∀𝑦∈𝐸.(2.4)

Lemma 2.2 (see Osilike and Igbokwe [20]). Let (𝐸,⟨⋅,⋅⟩) be an inner product space. Then for all 𝑥,𝑦,𝑧∈𝐸 and 𝛼,𝛽,𝛾∈[0,1] with 𝛼+𝛽+𝛾=1, one has ‖𝛼𝑥+𝛽𝑦+𝛾𝑧‖²=𝛼‖𝑥‖²+𝛽‖𝑦‖²+𝛾‖𝑧‖²−𝛼𝛽‖𝑥−𝑦‖²−𝛼𝛾‖𝑥−𝑧‖²−𝛽𝛾‖𝑦−𝑧‖².(2.5)

Lemma 2.3 (see Suzuki [21]). 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 2.4 (see Xu [22]). Assume {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛼_𝑛𝑎_𝑛+𝛿_𝑛,𝑛≥0,(2.6) where {𝛼_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence in 𝐑 such that (i)∑^∞𝑛=1𝛼_𝑛=∞, (ii)limsup_𝑛→∞(𝛿_𝑛/𝛼_𝑛)≤0 or ∑^∞𝑛=1|𝛿_𝑛|<∞. Then, lim_𝑛→∞𝑎_𝑛=0.

Lemma 2.5 (Goebel and Kirk [23]). Demiclosedness Principle. Assume that 𝑇 is a nonexpansive self-mapping of a nonempty, closed, convex subset 𝐸 of a real Hilbert space 𝐻. If 𝑇 has a fixed point, then 𝐼−𝑇 is demiclosed; that is, whenever {𝑥_𝑛} is a sequence in 𝐸 converging weakly to some 𝑥∈𝐸 (for short, 𝑥_𝑛⇀𝑥∈𝐸), and the sequence {(𝐼−𝑇)𝑥_𝑛} converges strongly to some 𝑦 (for short, (𝐼−𝑇)𝑥_𝑛→𝑦), it follows that (𝐼−𝑇)𝑥=𝑦. Here, 𝐼 is the identity operator of 𝐻.

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

Lemma 2.6. In a real Hilbert space 𝐻, there holds the inequality ‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝑥+𝑦⟩,∀𝑥,𝑦∈𝐻.(2.7)

Remark 2.7. We also have that, for all 𝑢,𝑣∈𝐸 and 𝜆>0, ‖(𝐼−𝜆𝐴)𝑢−(𝐼−𝜆𝐴)𝑣‖²=‖(𝑢−𝑣)−𝜆(𝐴𝑢−𝐴𝑣)‖²=+‖𝑢𝜆−₂𝑣‖²−2𝜆⟨𝑢−𝑣,𝐴𝑢−𝐴𝑣⟩‖𝐴𝑢−𝐴𝑣‖²≤‖𝑢−𝑣‖²+𝜆(𝜆−2𝛼)‖𝐴𝑢−𝐴𝑣‖².(2.8) So, if 𝜆≤2𝛼, then 𝐼−𝜆𝐴 is a nonexpansive mapping from 𝐸 to 𝐻.

3. Main Results

In this section, we introduce an iterative precess by the relaxed extragradient approximation method for finding a common element of the set of fixed points of a nonexpansive mapping and the solution set of the variational inequality problem for three inverse-strongly monotone mappings in a real Hilbert space. We prove that the iterative sequence converges strongly to a common element of the above two sets.

In order to prove our main result, the following lemmas are needed.

Lemma 3.1. For given 𝑥^∗,𝑦^∗,𝑧^∗∈𝐸×𝐸×𝐸,(𝑥^∗,y^∗,𝑧^∗) is a solution of problem (1.9) if and only if 𝑥^∗ is a fixed point of the mapping 𝐺∶𝐸→𝐸 defined by 𝐺(𝑥)=𝑃_𝐸𝑃−_𝐸𝜆𝐴𝑃_𝐸𝑃_𝐸(𝑥−𝜏𝐶𝑥)−𝜇𝐵𝑃_𝐸(𝑥−𝜏𝐶𝑥)𝑃_𝐸(𝑥−𝜏𝐶𝑥)−𝜇𝐵𝑃_𝐸(𝑥−𝜏𝐶𝑥),∀𝑥∈𝐸,(3.1) where 𝑦^∗=𝑃_𝐸(𝑧^∗−𝜇𝐵𝑧^∗) and 𝑧^∗=𝑃_𝐸(𝑥^∗−𝜏𝐶𝑥^∗).

Proof. ⟨⟨𝜆𝜇𝐴𝐵𝑦𝑧^∗_∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈𝐸,⟨𝜏𝐶𝑥_∗+𝑦^∗−𝑧^∗,𝑥−𝑦^∗⟩≥0,∀𝑥∈𝐸,+𝑧^∗−𝑥^∗,𝑥−𝑧^∗⟩≥0,∀𝑥∈𝐸,(3.2)⇔−−𝑦𝑧^∗_∗+𝜆𝐴𝑦^∗+𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐸,−𝑥_∗+𝜇𝐵𝑧^∗+𝑦^∗,𝑥−𝑦^∗≥0,∀𝑥∈𝐸,+𝜏𝐶𝑥^∗+𝑧^∗,𝑥−𝑧^∗≥0,∀𝑥∈𝐸,(3.3)⇔𝑦𝑧^∗_∗−𝜆𝐴𝑦^∗−𝑥^∗,𝑥^∗−𝑥≥0,∀𝑥∈𝐸,𝑥_∗−𝜇𝐵𝑧^∗−𝑦^∗,𝑦^∗−𝑥≥0,∀𝑥∈𝐸,−𝜏𝐶𝑥^∗−𝑧^∗,𝑧^∗−𝑥≥0,∀𝑥∈𝐸,(3.4)⇔𝑦x^∗_∗=𝑃_𝐸𝑦^∗−𝜆𝐴𝑦^∗,𝑧_∗=𝑃_𝐸𝑧^∗−𝜇𝐵𝑧^∗,=𝑃_𝐸𝑥^∗−𝜏𝐶𝑥^∗,(3.5)⇔𝑥^∗=𝑃_𝐸[𝑃_𝐸(𝑧^∗−𝜇𝐵𝑧^∗)−𝜆𝐴𝑃_𝐸(𝑧^∗−𝜇𝐵𝑧^∗)].
Thus, 𝑥^∗=𝑃_𝐸𝑃_𝐸𝑃_𝐸𝑥^∗−𝜏𝐶𝑥^∗−𝜇𝐵𝑃_𝐸𝑥^∗−𝜏𝐶𝑥^∗−𝜆𝐴𝑃_𝐸𝑃_𝐸𝑥^∗−𝜏𝐶𝑥^∗−𝜇𝐵𝑃_𝐸𝑥^∗−𝜏𝐶𝑥^∗(3..6)

Lemma 3.2. The mapping 𝐺 defined by Lemma 3.1 is nonexpansive mappings.

Proof. For all 𝑥,𝑦∈𝐸, ‖𝐺(𝑥)−𝐺(𝑦)‖=‖‖𝑃_𝐸−𝑃_𝐸𝜆𝐴𝑃𝑃_𝐸𝐸(𝑥−𝜏𝐶𝑥)−𝜇𝐵𝑃_𝐸(𝑥−𝜏𝐶𝑥)−𝑃_𝐸𝑃_𝐸(𝑥−𝜏𝐶𝑥)−𝜇𝐵𝑃_𝐸(𝑥−𝜏𝐶𝑥)−𝑃𝜆_𝐸𝐴𝑃𝑃_𝐸𝐸(𝑦−𝜏𝐶𝑦)−𝜇𝐵𝑃_𝐸(𝑦−𝜏𝐶𝑦)≤‖‖𝑃𝑃_𝐸(𝑦−𝜏𝐶𝑦)−𝜇𝐵𝑃_𝐸(𝑦−𝜏𝐶𝑦)_𝐸−𝜆𝐴(𝑃𝑥_𝐸−𝜏𝐶𝑥)−𝜇𝐵𝑃_𝐸(𝑥−𝜏𝐶𝑥)−𝑃𝑃_𝐸(𝑥−𝜏𝐶𝑥)−𝜇𝐵𝑃_𝐸(𝑥−𝜏𝐶𝑥)_𝐸−𝜆𝐴𝑃_𝐸(𝑦−𝜏𝐶𝑦)−𝜇𝐵𝑃_𝐸(𝑦−𝜏𝐶𝑦)=‖‖(𝐼−𝜆𝐴)𝑃𝑃_𝐸𝐸(𝑦−𝜏𝐶𝑦)−𝜇𝐵𝑃_𝐸(𝑦−𝜏𝐶𝑦)‖‖−(𝐼−𝜆𝐴)𝑃(_𝐸𝑥−𝜏𝐶𝑥)−𝜇𝐵𝑃_𝐸(𝑥−𝜏𝐶𝑥)≤‖‖𝑃(𝑦−𝜏𝐶𝑦)−𝜇𝐵𝑃_𝐸(𝑦−𝜏𝐶𝑦)_𝐸‖‖=‖‖(𝐼−(𝜇𝑥𝐵−)𝜏𝑃𝐶_𝐸𝑥)−𝜇𝐵𝑃_𝐸(𝑥−𝜏𝐶𝑥)−𝑃_𝐸(𝑦−𝜏𝐶𝑦)−𝜇𝐵𝑃_𝐸(𝑦−𝜏𝐶𝑦)‖‖≤‖‖𝑃(𝑥−𝜏𝐶𝑥)−(𝐼−𝜇𝐵)𝑃_𝐸(𝑦−𝜏𝐶𝑦)_𝐸‖‖≤=≤‖‖‖(𝑥(𝐼𝑥−−−(𝑥𝑦𝜏‖𝜏−𝐶𝐶.)𝜏𝑥(𝐶)𝑥−𝑥))−(−𝑦(𝐼−𝑃−_𝐸𝜏𝐶(𝜏𝑦𝐶𝑦−))‖(𝜏𝑦𝐶)‖𝑦)‖‖(3.7) This shows that 𝐺∶𝐸→𝐸 is a nonexpansive mapping.

Throughout this paper, the set of fixed points of the mapping 𝐺 is denoted by Υ.

Now, we are ready to proof our main results in this paper.

Theorem 3.3. Let 𝐸 be a nonempty, closed, convex subset of a real Hilbert space 𝐻. Let the mapping 𝐴,𝐵,𝐶∶𝐸→𝐻 be 𝛼-inverse-strongly monotone, 𝛽-inverse-strongly monotone, and 𝛾-inverse-strongly monotone, respectively. Let 𝑆 be a nonexpansive mapping of 𝐸 into itself such that 𝐹(𝑆)∩Υ≠∅. Let 𝑓 be a contraction of 𝐻 into itself and given 𝑥₁∈𝐻 arbitrarily and {𝑥_𝑛} is generated by 𝑦𝑧_𝑛𝑛=𝑃_𝐸𝑥_𝑛−𝜏𝐶𝑥_𝑛,𝑥_𝑛+1=𝑃_𝐸𝑧_𝑛−𝜇𝐵𝑧_𝑛,=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑆𝑃_𝐸𝑦_𝑛−𝜆𝐴𝑦_𝑛,𝑛≥0,(3.8) where 𝜆∈(0,2𝛼),𝜇∈(0,2𝛽),𝜏∈(0,2𝛾), and {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛} are three sequences in [0,1] such that (i)𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1,(ii)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞𝑛=1𝛼_𝑛=∞,(iii)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1. Then, {𝑥_𝑛} converges strongly to 𝑥∈𝐹(𝑆)∩Γ, where 𝑥=𝑃_{𝐹(𝑆)∩Γ}𝑓(𝑥) and (𝑥,𝑦,𝑧) is a solution of problem (1.9), where 𝑦=𝑃_𝐸𝑧−𝜇𝐵𝑧,𝑧=𝑃_𝐸𝑥−𝜏𝐶𝑥.(3.9)

Proof. Let 𝑥^∗∈𝐹(𝑆)∩Γ. Then, 𝑥^∗=𝑆𝑥^∗ and 𝑥^∗=𝐺𝑥^∗, that is, 𝑥^∗=𝑃_𝐸𝑃_𝐸𝑃_𝐸𝑥^∗−𝜏𝐶𝑥^∗−𝜇𝐵𝑃_𝐸𝑥^∗−𝜏𝐶𝑥^∗−𝜆𝐴𝑃_𝐸𝑃_𝐸𝑥^∗−𝜏𝐶𝑥^∗−𝜇𝐵𝑃_𝐸𝑥^∗−𝜏𝐶𝑥^∗(3..10) Put 𝑥^∗=𝑃_𝐸(y^∗−𝜆𝐴𝑦^∗) and 𝑡_𝑛=𝑃_𝐸(𝑦_𝑛−𝜆𝐴𝑦_𝑛). Then, 𝑥^∗=𝑃_𝐸[𝑃_𝐸(𝑧^∗−𝜇𝐵𝑧^∗)−𝜆𝐴𝑃_𝐸(𝑧^∗−𝜇𝐵𝑧^∗)] implies that 𝑦^∗=𝑃_𝐸(𝑧^∗−𝜇𝐵𝑧^∗), where 𝑧^∗=𝑃_𝐸(𝑥^∗−𝜏𝐶𝑥^∗). Since 𝐼−𝜆𝐴, 𝐼−𝜇𝐵 and 𝐼−𝜏𝐶 are nonexpansive mappings. We obtain that ‖‖𝑡_𝑛−𝑥^∗‖‖==‖‖‖‖𝑃𝑃_𝐸𝐸𝑦_𝑛−𝜆𝐴𝑦_𝑛−𝑥^∗‖‖≤‖‖𝑦_𝑛𝑦_𝑛−𝜆𝐴𝑦_𝑛−𝑃_𝐸𝑦^∗−𝜆𝐴𝑦^∗‖‖=‖‖(𝐼−−𝜆𝐴𝜆𝐴)𝑦𝑦_𝑛𝑛−𝑦^∗−𝜆𝐴𝑦^∗‖‖≤‖‖𝑦_𝑛−(𝐼−𝜆𝐴)𝑦^∗‖‖=‖‖𝑦−𝑦^∗‖‖(3.11)_𝑛=‖‖𝑃_𝐸−𝑃_𝐸𝑧^∗−𝜇𝐵𝑧^∗‖‖≤‖‖(𝐼−𝑧𝜇_𝑛𝐵−)𝜇𝑧𝐵_𝑛𝑧_𝑛−𝑃_𝐸𝑧^∗−𝜇𝐵𝑧^∗‖‖≤‖‖𝑧_𝑛−(𝐼−𝜇𝐵)𝑧^∗‖‖‖‖𝑧−𝑧^∗‖‖,(3.12)_𝑛−𝑧^∗‖‖=≤‖‖‖‖𝑃_𝐸𝑥_𝑛𝑥_𝑛−𝜏𝐶𝑥_𝑛−𝑃_𝐸𝑥^∗−𝜏𝐶𝑥^∗‖‖=‖‖(𝐼−−𝜏𝐶𝜏)𝐶𝑥𝑥_𝑛𝑛−𝑥^∗−𝜏𝐶𝑥^∗‖‖≤‖‖𝑥_𝑛−(𝐼−𝜏𝐶)𝑥^∗‖‖−𝑥^∗‖‖.(3.13) Substituting (3.13) into (3.12), we have ‖‖𝑡_𝑛−𝑥^∗‖‖≤‖‖𝑥_𝑛−𝑥^∗‖‖,(3.14) and by (3.11) we also have ‖‖𝑦_𝑛−𝑦^∗‖‖≤‖‖𝑥_𝑛−𝑥^∗‖‖.(3.15) Since 𝑥_𝑛+1=𝛼_𝑛𝑓(𝑥_𝑛)+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑆𝑡_𝑛 and by Lemma 2.2, we compute ‖‖𝑥_𝑛+1−𝑥^∗‖‖==‖‖‖‖𝛼𝛼_𝑛𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑆𝑡_𝑛−𝑥^∗‖‖≤𝛼_𝑛𝑓𝑥_𝑛−𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑥^∗+𝛾_𝑛𝑆𝑡_𝑛−𝑥^∗‖‖≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖+𝛾_𝑛‖‖𝑆𝑡_𝑛−𝑥^∗‖‖≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖+𝛾_𝑛‖‖𝑡_𝑛−𝑥^∗‖‖=𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖+𝛾_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖=𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖+1−𝛼_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑓𝑥^∗+𝑓𝑥^∗−𝑥^∗‖‖+1−𝛼_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑓𝑥^∗‖‖+𝛼_𝑛‖‖𝑓𝑥^∗−𝑥^∗‖‖+1−𝛼_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖=𝛼𝑘_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖+𝛼_𝑛‖‖𝑓𝑥^∗−𝑥^∗‖‖+1−𝛼_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖=1−𝑘𝛼+_𝑛1−𝛼_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖+𝛼_𝑛‖‖𝑓𝑥^∗−𝑥^∗‖‖=1−𝛼_𝑛(1−𝑘)‖‖𝑥_𝑛−𝑥^∗‖‖+𝛼_𝑛‖‖𝑓𝑥^∗−𝑥^∗‖‖(1−𝑘)‖‖𝑥_𝑛−𝑥^∗‖‖+𝛼_𝑛(1−𝑘)‖‖𝑓𝑥^∗−𝑥^∗‖‖(1−𝑘).(3.16) By induction, we get ‖‖𝑥_𝑛+1−𝑥^∗‖‖≤𝑀,(3.17) where 𝑀=max{‖𝑥₀−𝑥^∗‖+(1/(1−𝑘))‖𝑓(𝑥^∗)−𝑥^∗‖},𝑛≥0. Therefore, {𝑥_𝑛}is bounded. Consequently, by (3.11), (3.12) and (3.13), the sequences {𝑡_𝑛},{𝑆𝑡_𝑛},{𝑦_𝑛},{𝐴𝑦_𝑛},{𝑧_𝑛},{𝐵𝑧_𝑛},{𝐶𝑥_𝑛}, and {𝑓(𝑥_𝑛)} are also bounded. Also, we observe that ‖‖𝑧_𝑛+1−𝑧_𝑛‖‖≤=‖‖‖‖𝑃(_𝐸𝐼−𝑥𝜏𝐶_𝑛+1)−𝑥𝜏𝐶_𝑛+1𝑥_𝑛+1−𝑃_𝐸𝑥_𝑛−𝜏𝐶𝑥_𝑛‖‖≤‖‖𝑥_𝑛+1−(𝐼−𝜏𝐶)𝑥_𝑛‖‖‖‖𝑡−𝑥_𝑛‖‖,(3.18)_𝑛+1−𝑡_𝑛‖‖≤=‖‖‖‖𝑃_𝐸𝑦𝑦_𝑛+1𝑛+1−𝜆𝐴𝑦_𝑛+1−𝑃_𝐸𝑦_𝑛−𝜆𝐴𝑦_𝑛‖‖=‖‖(𝐼−𝜆𝐴−)𝜆𝑦𝐴𝑦_𝑛+1𝑛+1−𝑦_𝑛−𝜆𝐴𝑦_𝑛‖‖≤‖‖𝑦_𝑛+1−(𝐼−𝜆𝐴)𝑦_𝑛‖‖=‖‖𝑃_𝐸−𝑦_𝑛‖‖≤‖‖𝑧_𝑛+1𝑧_𝑛+1−𝜇𝐵𝑧_𝑛+1−𝑃_𝐸𝑧_𝑛−𝜇𝐵𝑧_𝑛‖‖≤‖‖𝑥_𝑛+1−𝑧_𝑛‖‖−𝑥_𝑛‖‖.(3.19) Let 𝑥_𝑛+1=(1−𝛽_𝑛)𝑤_𝑛+𝛽_𝑛𝑥_𝑛. Thus, we get 𝑤_𝑛=𝑥^𝑛+1−𝛽_𝑛𝑥_𝑛1−𝛽_𝑛=𝛼^𝑛𝑓𝑥_𝑛+𝛾_𝑛𝑆𝑃_𝐶𝑦_𝑛−𝜆_𝑛𝐴𝑦_𝑛1−𝛽_𝑛=𝛼^𝑛𝑢+𝛾_𝑛𝑆𝑡_𝑛1−𝛽_𝑛(3.20) It follows that 𝑤_𝑛+1−𝑤_𝑛=𝛼^𝑛+1𝑓𝑥_𝑛+1+𝛾_𝑛+1𝑆𝑡_𝑛+11−𝛽_𝑛+1−𝛼^𝑛𝑓𝑥_𝑛+𝛾_𝑛𝑆𝑡_𝑛1−𝛽_𝑛=𝛼^𝑛+1𝑓𝑥_𝑛+11−𝛽_𝑛+1+𝛾^𝑛+1𝑆𝑡_𝑛+11−𝛽_𝑛+1−𝛼^𝑛+1𝑓𝑥_𝑛1−𝛽_𝑛+1+𝛼^𝑛+1𝑓𝑥_𝑛1−𝛽_𝑛+1−𝛼^𝑛𝑓𝑥_𝑛1−𝛽_𝑛−𝛾^𝑛𝑆𝑡_𝑛1−𝛽_𝑛=𝛼^𝑛+11−𝛽_𝑛+1𝑓𝑥_𝑛+1−𝑓𝑥_𝑛+1−𝛼𝛽^𝑛+1_𝑛+1−𝛼^𝑛1−𝛽_𝑛𝑓𝑥^𝑛+𝛾^𝑛+1𝑆𝑡_𝑛+11−𝛽_𝑛+1−𝛾^𝑛𝑆𝑡_𝑛1−𝛽_𝑛=𝛼^𝑛+11−𝛽_𝑛+1𝑓𝑥_𝑛+1−𝑓𝑥_𝑛+1−𝛼𝛽^𝑛+1_𝑛+1−𝛼^𝑛+𝛾_𝑛+11−𝛽_𝑛𝑓𝑥^𝑛𝑆𝑡_𝑛+11−𝛽_𝑛+1−𝛾^𝑛+1𝑆𝑡_𝑛1−𝛽_𝑛+1+𝛾^𝑛+1𝑆𝑡_𝑛1−𝛽_𝑛+1−𝛾^𝑛𝑆𝑡_𝑛1−𝛽_𝑛=𝛼^𝑛+11−𝛽_𝑛+1𝑓𝑥_𝑛+1−𝑓𝑥_𝑛+1−𝛼𝛽^𝑛+1_𝑛+1−𝛼^𝑛+𝛾_𝑛+11−𝛽_𝑛𝑓𝑥^𝑛1−𝛽_𝑛+1𝑆𝑡_𝑛+1−𝑆𝑡_𝑛+𝛾^𝑛+11−𝛽_𝑛+1−𝛾^𝑛1−𝛽_𝑛𝑆𝑡^𝑛=𝛼^𝑛+11−𝛽_𝑛+1𝑓𝑥_𝑛+1−𝑓𝑥_𝑛+1−𝛼𝛽^𝑛+1_𝑛+1−𝛼^𝑛+𝛼_𝑛+11−𝛽_𝑛𝑓𝑥^𝑛1−𝛽_𝑛+1−𝛼^𝑛1−𝛽_𝑛𝑆𝑡^𝑛+𝛾^𝑛+11−𝛽_𝑛+1𝑆𝑡_𝑛+1=𝛼_𝑛+1−𝑆𝑡_𝑛1−𝛽_𝑛+1𝑓𝑥_𝑛+1−𝑓𝑥_𝑛+1−𝛼𝛽^𝑛+1_𝑛+1−𝛼^𝑛+𝛾_𝑛+11−𝛽_𝑛𝑓𝑥^𝑛+𝑆𝑡_𝑛1−𝛽_𝑛+1𝑆𝑡_𝑛+1−𝑆𝑡_𝑛.(3.21) Combining (3.19) and (3.21), we obtain ‖‖𝑤_𝑛+1−𝑤_𝑛‖‖−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤||||𝛼^𝑛+11−𝛽_𝑛+1||||‖‖𝑓𝑥_𝑛+1−𝑓𝑥_𝑛‖‖+||||1−𝛼𝛽^𝑛+1_𝑛+1−1−𝛼^𝑛𝛽_𝑛||||‖‖𝑓𝑥_𝑛+||||𝛾_𝑛+1+𝑆𝑡_𝑛‖‖1−𝛽_𝑛+1||||‖‖𝑆𝑡_𝑛+1≤||||𝛼_𝑛+1−𝑆𝑡_𝑛‖‖−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖1−𝛽_𝑛+1||||𝑘‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+||||𝛼^𝑛+11−𝛽_𝑛+1−𝛼^𝑛1−𝛽_𝑛||||‖‖𝑓𝑥_𝑛+||||𝛾_𝑛+1+𝑆𝑡_𝑛‖‖1−𝛽_𝑛+1||||‖‖𝑡_𝑛+1≤||||𝛼_𝑛+1−𝑡_𝑛‖‖−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖1−𝛽_𝑛+1||||𝑘‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+||||𝛼^𝑛+11−𝛽_𝑛+1−𝛼^𝑛1−𝛽_𝑛||||‖‖𝑓𝑥_𝑛+||||𝛾_𝑛+1+𝑆𝑡_𝑛‖‖1−𝛽_𝑛+1||||‖‖𝑥_𝑛+1=||||𝛼_𝑛+1−𝑥_𝑛‖‖−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖1−𝛽_𝑛+1||||𝑘‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+||||𝛼^𝑛+11−𝛽_𝑛+1−𝛼^𝑛1−𝛽_𝑛||||‖‖𝑓𝑥_𝑛+||||𝛾_𝑛+1+𝑆𝑡_𝑛‖‖−1+𝛽_𝑛+11−𝛽_𝑛+1||||‖‖𝑥_𝑛+1=||||𝛼_𝑛+1−𝑥_𝑛‖‖1−𝛽_𝑛+1||||𝑘‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+||||𝛼^𝑛+11−𝛽_𝑛+1−𝛼^𝑛1−𝛽_𝑛||||‖‖𝑓𝑥_𝑛+||||𝛼_𝑛+1+𝑆𝑡_𝑛‖‖1−𝛽_𝑛+1||||‖‖𝑥_𝑛+1−𝑥_𝑛‖‖.(3.22) This together with (i), (ii), and (iii) implies that limsup_𝑛→∞‖‖𝑤_𝑛+1−𝑤_𝑛‖‖−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤0.(3.23) Hence, by Lemma 2.3, we have lim_𝑛→∞‖‖𝑤_𝑛−𝑥_𝑛‖‖=0.(3.24) Consequently, lim_𝑛→∞‖‖𝑥_𝑛+1−𝑥_𝑛‖‖=lim_𝑛→∞1−𝛽_𝑛‖‖𝑤_𝑛−𝑥_𝑛‖‖=0.(3.25) From (3.18) and (3.19), we also have ‖𝑧_𝑛+1−𝑧_𝑛‖→0‖𝑡_𝑛+1−𝑡_𝑛‖→0 and ‖𝑦_𝑛+1−𝑦_𝑛‖→0 as 𝑛→∞. Since 𝑥_𝑛+1−𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑆𝑡_𝑛−𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛+𝛾_𝑛𝑆𝑡_𝑛−𝑥_𝑛,(3.26) it follows by (ii) and (3.25) that lim_𝑛→∞‖‖𝑥_𝑛−𝑆𝑡_𝑛‖‖=0.(3.27) Since 𝑥^∗∈𝐹(𝑆)∩Γ, from (3.15) and Lemma 2.2, we get ‖‖𝑥_𝑛+1−𝑥^∗‖‖²=‖‖𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑆𝑡_𝑛−𝑥^∗‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛾_𝑛‖‖𝑆𝑡_𝑛−𝑥^∗‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛾_𝑛‖‖𝑡_𝑛−𝑥^∗‖‖²=𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛾_𝑛‖‖𝑃_𝐸𝑦_𝑛−𝜆𝐴𝑦_𝑛−𝑃_𝐸𝑦^∗−𝜆𝐴𝑦^∗‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛾_𝑛‖‖𝑦_𝑛−𝜆𝐴𝑦_𝑛−𝑦^∗−𝜆𝐴𝑦^∗‖‖²=𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛾_𝑛‖‖𝑦_𝑛−𝑦^∗−𝜆𝐴𝑦_𝑛−𝐴𝑦^∗‖‖²=𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²≤+𝛼_𝑛𝛾_𝑛‖𝑦^𝑛−𝑦^∗‖²−2𝜆⟨𝑦_𝑛−𝑦^∗,𝐴𝑦_𝑛−𝐴𝑦^∗⟩+𝜆²‖‖𝐴𝑦_𝑛−𝐴𝑦^∗‖‖²‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²=+𝛼_𝑛𝛾_𝑛‖‖𝑦^𝑛−𝑦^∗‖‖²−2𝜆𝛼‖‖𝐴𝑦_𝑛−𝐴𝑦^∗‖‖²+𝜆²‖‖𝐴𝑦_𝑛−𝐴𝑦^∗‖‖²‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛾_𝑛≤𝛼_𝑛‖‖𝑦^𝑛−𝑦^∗‖‖²+𝜆(𝜆−2𝛼)‖‖𝐴𝑦_𝑛−𝐴𝑦^∗‖‖²‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛾_𝑛=𝛼_𝑛‖‖𝑥^𝑛−𝑥^∗‖‖²+𝜆(𝜆−2𝛼)‖‖𝐴𝑦_𝑛−𝐴𝑦^∗‖‖²‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛾_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛾_𝑛𝜆(𝜆−2𝛼)‖‖𝐴𝑦_𝑛−𝐴𝑦^∗‖‖²=𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+𝛽_𝑛+𝛾_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛾_𝑛𝜆(𝜆−2𝛼)‖‖𝐴𝑦_𝑛−𝐴𝑦^∗‖‖²=𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+1−𝛼_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛾_𝑛𝜆(𝜆−2𝛼)‖‖𝐴𝑦_𝑛−𝐴𝑦^∗‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛾_𝑛𝜆(𝜆−2𝛼)‖‖𝐴𝑦_𝑛−𝐴𝑦^∗‖‖².(3.28) Therefore, we have −𝛾_𝑛𝜆(𝜆−2𝛼)‖‖𝐴𝑦_𝑛−𝐴𝑦^∗‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝑥_𝑛+1−𝑥^∗‖‖²=×𝛼_𝑛‖‖‖‖𝑓𝑥_𝑛𝑥_𝑛−𝑥^∗‖‖²+‖‖𝑥_𝑛−𝑥^∗‖‖+‖‖𝑥_𝑛+1−𝑥^∗‖‖≤𝛼_𝑛−𝑥^∗‖‖−‖‖𝑥_𝑛+1−𝑥^∗‖‖‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+‖‖𝑥_𝑛−𝑥^∗‖‖+‖‖𝑥_𝑛+1−𝑥^∗‖‖‖‖𝑥_𝑛−𝑥_𝑛+1‖‖.(3.29) From (ii), (iii), and ‖𝑥_𝑛+1−𝑥_𝑛‖→0, as 𝑛→∞, we get ‖𝐴𝑦_𝑛−𝐴𝑦^∗‖→0 as 𝑛→∞.
Since 𝑥^∗∈𝐹(𝑆)∩Υ, from (3.11) and Lemma 2.2, we get ‖‖𝑥_𝑛+1−𝑥^∗‖‖²=‖‖𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑆𝑡_𝑛−𝑥^∗‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛾_𝑛‖‖𝑡_𝑛−𝑥^∗‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛾_𝑛‖‖𝑦_𝑛−𝑦^∗‖‖²=𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛾_𝑛‖‖𝑃_𝐸𝑧_𝑛−𝜇𝐵𝑧_𝑛−𝑃_𝐸𝑧^∗−𝜇𝐵𝑧^∗‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛾_𝑛‖‖𝑧_𝑛−𝜇𝐵𝑧_𝑛−𝑧^∗−𝜇𝐵𝑧^∗‖‖²=𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛾_𝑛‖‖𝑧_𝑛−𝑧^∗−𝜇𝐵𝑧_𝑛−𝜇𝐵𝑧^∗‖‖²≤≤𝛼𝛼_𝑛𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛾_𝑛‖‖𝑧^𝑛−𝑧^∗‖‖²+𝜇(𝜇−2𝛽)‖‖𝐵𝑧_𝑛−𝐵𝑧^∗‖‖²‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛾_𝑛𝜇(𝜇−2𝛽)‖‖𝐵𝑧_𝑛−𝐵𝑧^∗‖‖².(3.30)
Thus, we also have −𝛾_𝑛𝜇(𝜇−2𝛽)‖‖𝐵𝑧_𝑛−𝐵𝑧^∗‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝑥_𝑛+1−𝑥^∗‖‖²=×𝛼_𝑛‖‖‖‖𝑓𝑥_𝑛𝑥_𝑛−𝑥^∗‖‖²+‖‖𝑥_𝑛−𝑥^∗‖‖+‖‖𝑥_𝑛+1−𝑥^∗‖‖≤𝛼_𝑛−𝑥^∗‖‖−‖‖𝑥_𝑛+1−𝑥^∗‖‖‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+‖‖𝑥_𝑛−𝑥^∗‖‖+‖‖𝑥_𝑛+1−𝑥^∗‖‖‖‖𝑥_𝑛−𝑥_𝑛+1‖‖.(3.31) By again (ii), (iii), and (3.25), we also get ‖𝐵𝑧_𝑛−𝐵𝑧^∗‖→0 as 𝑛→∞.
Let 𝑥^∗∈𝐹(𝑆)∩Υ; again from (3.12), (3.13) and Lemma 2.2, we get ‖‖𝑥_𝑛+1−𝑥^∗‖‖²=‖‖𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑆𝑡_𝑛−𝑥^∗‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛾_𝑛‖‖𝑡_𝑛−𝑥^∗‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛾_𝑛‖‖𝑧_𝑛−𝑧^∗‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛾_𝑛‖‖𝑥_𝑛−𝜏𝐶𝑥_𝑛−𝑥^∗−𝜏𝐶𝑥^∗‖‖²≤≤𝛼𝛼^𝑛_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛾_𝑛‖‖𝑥^𝑛−𝑥^∗‖‖²+𝜏(𝜏−2𝛾)‖‖𝐶𝑥_𝑛−𝐶𝑥^∗‖‖²‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛾_𝑛𝜏(𝜏−2𝛾)‖‖𝐶𝑥_𝑛−𝐶𝑥^∗‖‖².(3.32)
Again, we have −𝛾_𝑛𝜏(𝜏−2𝛾)‖‖𝐶𝑥_𝑛−𝐶𝑥^∗‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝑥_𝑛+1−𝑥^∗‖‖²=×𝛼_𝑛‖‖‖‖𝑓𝑥_𝑛𝑥_𝑛−𝑥^∗‖‖²+‖‖𝑥_𝑛−𝑥^∗‖‖+‖‖𝑥_𝑛+1−𝑥^∗‖‖≤𝛼_𝑛−𝑥^∗‖‖−‖‖𝑥_𝑛+1−𝑥^∗‖‖‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+‖‖𝑥_𝑛−𝑥^∗‖‖+‖‖𝑥_𝑛+1−𝑥^∗‖‖‖‖𝑥_𝑛−𝑥_𝑛+1‖‖.(3.33) Similarly again by (ii), (iii), and ‖𝑥_𝑛−𝑥_𝑛+1‖→0 as 𝑛→∞, and from (3.33), we also that ‖𝐶𝑥_𝑛−𝐶𝑥^∗‖→0.
On the other hand, we compute that ‖‖𝑧_𝑛−𝑧^∗‖‖²=‖‖𝑃_𝐸𝑥_𝑛−𝜏𝐶𝑥_𝑛−𝑃_𝐸𝑥^∗−𝜏𝐶𝑥^∗‖‖²=≤𝑥𝑥_𝑛𝑛−𝜏𝐶𝑥_𝑛−𝑥^∗−𝜏𝐶𝑥^∗,𝑃_𝐸𝑥_𝑛−𝜏𝐶𝑥_𝑛−𝑃_𝐸𝑥^∗−𝜏𝐶𝑥^∗=1−𝜏𝐶𝑥_𝑛−𝑥^∗−𝜏𝐶𝑥^∗,𝑧_𝑛−𝑧^∗2‖‖𝑥^𝑛−𝜏𝐶𝑥_𝑛−𝑥^∗−𝜏𝐶𝑥^∗‖‖²+‖‖𝑧_𝑛−𝑧^∗‖‖²=1−‖‖𝑥_𝑛−𝜏𝐶𝑥_𝑛−𝑥^∗−𝜏𝐶𝑥^∗−𝑧_𝑛−𝑧^∗‖‖²2‖‖(𝐼−𝜏𝐶)𝑥^𝑛−(𝐼−𝜏𝐶)𝑥^∗‖‖²+‖‖𝑧_𝑛−𝑧^∗‖‖²≤1−‖‖𝑥_𝑛−𝜏𝐶𝑥_𝑛−𝑥^∗−𝜏𝐶𝑥^∗−𝑧_𝑛−𝑧^∗‖‖²2‖𝑥^𝑛=1−𝑥^∗‖²+‖𝑧_𝑛−𝑧^∗‖²−‖𝑥_𝑛−𝑧_𝑛−𝜏𝐶𝑥_𝑛−𝐶𝑥^∗−𝑥^∗−𝑧^∗‖²2‖‖𝑥^𝑛=1−𝑥^∗‖‖²+‖‖𝑧_𝑛−𝑧^∗‖‖²−‖‖𝑥_𝑛−𝑧_𝑛−𝑥^∗−𝑧^∗−𝜏𝐶𝑥_𝑛−𝐶𝑥^∗‖‖²2‖‖𝑥^𝑛−𝑥^∗‖‖²+‖‖𝑧_𝑛−𝑧^∗‖‖²−‖‖𝑥_𝑛−𝑧_𝑛−𝑥^∗−𝑧^∗‖‖²+2𝜏𝑥_𝑛−𝑧_𝑛−𝑥^∗−𝑧^∗,𝐶𝑥_𝑛−𝐶𝑥^∗−𝜏²‖‖𝐶𝑥_𝑛−𝐶𝑥^∗‖‖².(3.34)
So, we obtain ‖‖𝑧_𝑛−𝑧^∗‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝑥_𝑛−z_𝑛−𝑥^∗−𝑧^∗‖‖²+2𝜏𝑥_𝑛−𝑧_𝑛−𝑥^∗−𝑧^∗,𝐶𝑥_𝑛−𝐶𝑥^∗−𝜏²‖‖𝐶𝑥_𝑛−𝐶𝑥^∗‖‖².(3.35) Hence, by (3.12), it follows that ‖‖𝑥_𝑛+1−𝑥^∗‖‖²=‖‖𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑆𝑡_𝑛−𝑥^∗‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛾_𝑛‖‖𝑆𝑡_𝑛−𝑥^∗‖‖²≤𝛼_𝑛𝑘‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛾_𝑛‖‖𝑡_𝑛−𝑥^∗‖‖²≤𝛼_𝑛𝑘‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛾_𝑛‖‖𝑧_𝑛−𝑧^∗‖‖²≤𝛼_𝑛𝑘‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛾_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²−𝛾_𝑛‖‖𝑥_𝑛−𝑧_𝑛−𝑥^∗−𝑧^∗‖‖²+2𝜏𝛾_𝑛𝑥_𝑛−𝑧_𝑛−𝑥^∗−𝑧^∗,𝐶𝑥_𝑛−𝐶𝑥^∗−𝜏²𝛾_𝑛‖‖𝐶𝑥_𝑛−𝐶𝑥^∗‖‖²=𝛼_𝑛𝑘‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝛼_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²−𝛾_𝑛‖‖𝑥_𝑛−𝑧_𝑛−𝑥^∗−𝑧^∗‖‖²+2𝜏𝛾_𝑛𝑥_𝑛−𝑧_𝑛−𝑥^∗−𝑧^∗,𝐶𝑥_𝑛−𝐶𝑥^∗−𝜏²𝛾_𝑛‖‖𝐶𝑥_𝑛−𝐶𝑥^∗‖‖²≤𝛼_𝑛𝑘‖‖𝑥_𝑛−𝑥^∗‖‖²+‖‖𝑥_𝑛−𝑥^∗‖‖²−𝛾_𝑛‖‖𝑥_𝑛−𝑧_𝑛−𝑥^∗−𝑧^∗‖‖²≤+𝛼_𝑛2𝜏𝛾_𝑛𝑥_𝑛−𝑧_𝑛−𝑥^∗−𝑧^∗,𝐶𝑥_𝑛−𝐶𝑥^∗𝑘‖‖𝑥_𝑛−𝑥^∗‖‖²+‖‖𝑥_𝑛−𝑥^∗‖‖²−𝛾_𝑛‖‖𝑥_𝑛−𝑧_𝑛−𝑥^∗−𝑧^∗‖‖²+2𝜏𝛾_𝑛‖‖𝑥_𝑛−𝑧_𝑛−𝑥^∗−𝑧^∗‖‖‖‖𝐶𝑥_𝑛−𝐶𝑥^∗‖‖,(3.36) which implies that 𝛾_𝑛‖‖𝑥_𝑛−𝑧_𝑛−𝑥^∗−𝑧^∗‖‖²≤𝛼_𝑛𝑘‖‖𝑥_𝑛−𝑥^∗‖‖²+‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝑥_𝑛+1−𝑥^∗‖‖²≤+𝛼_𝑛2𝜏𝛾_𝑛‖‖𝑥_𝑛−𝑧_𝑛−𝑥^∗−𝑧^∗‖‖‖‖𝐶𝑥_𝑛−𝐶𝑥^∗‖‖+‖‖𝑘𝑥‖‖_𝑛𝑥_𝑛+1−𝑥^∗‖‖²+2𝛾_𝑛𝜏‖‖𝑥_𝑛−𝑧_𝑛−𝑥^∗−𝑧^∗‖‖‖‖𝐶𝑥_𝑛−𝐶𝑥^∗‖‖−𝑥_𝑛+1‖‖‖‖𝑥_𝑛−𝑥^∗‖‖+‖‖𝑥_𝑛+1−𝑥^∗‖‖.(3.37) By (ii), (iii), ‖𝑥_𝑛−𝑥_𝑛+1‖→0, and ‖𝐶𝑥_𝑛−𝐶𝑥^∗‖→0 as 𝑛→∞, from (3.37) and we get ‖(𝑥_𝑛−𝑧_𝑛)−(𝑥^∗−𝑧^∗)‖→0 as 𝑛→∞. Now, observe that ‖‖𝑧_𝑛−𝑡_𝑛+𝑥^∗−𝑧^∗‖‖²=‖‖𝑧_𝑛−𝑃_𝐸𝑦_𝑛−𝜆𝐴𝑦_𝑛+𝑃_𝐸𝑦^∗−𝜆𝐴y^∗−𝑧^∗‖‖²=‖‖−𝑧𝑧_𝑛∗−𝑃_𝐸𝑦_𝑛−𝜆𝐴𝑦_𝑛+𝑃_𝐸𝑦^∗−𝜆𝐴𝑦^∗+𝜇𝐵𝑧_𝑛−𝜇𝐵𝑧_𝑛+𝜇𝐵𝑧^∗−𝜇𝐵𝑧^∗‖‖²=‖‖−𝑧_𝑛𝑃−_𝐸𝜇𝐵𝑧_𝑛−𝑧^∗−𝜇𝐵𝑧^∗𝑦_𝑛−𝜆𝐴𝑦_𝑛−𝑃_𝐸𝑦^∗−𝜆𝐴𝑦^∗+𝜇𝐵𝑧_𝑛−𝐵𝑧^∗‖‖²≤‖‖𝑧_𝑛−𝜇𝐵𝑧_𝑛−𝑧^∗−𝜇𝐵𝑧^∗−𝑃_𝐸𝑦_𝑛−𝜆𝐴𝑦_𝑛−𝑃_𝐸𝑦^∗−𝜆𝐴𝑦^∗‖‖²+2𝜇−𝐵𝑧𝑃_𝑛𝐸−𝐵𝑧^∗,𝑧_𝑛−𝜇𝐵𝑧_𝑛−𝑧^∗−𝜇𝐵𝑧^∗=‖‖𝑧𝑦_𝑛−𝜆𝐴𝑦_𝑛−𝑃_𝐸𝑦^∗−𝜆𝐴𝑦^∗+𝜇𝐵𝑧_𝑛−𝐵𝑧^∗_𝑛−𝜇𝐵𝑧_𝑛−𝑧^∗−𝜇𝐵𝑧^∗−𝑃_𝐸𝑦_𝑛−𝜆𝐴𝑦_𝑛−𝑃_𝐸𝑦^∗−𝜆𝐴𝑦^∗‖‖²≤+‖‖𝑧2𝜇𝐵𝑧_𝑛−𝐵𝑧^∗,𝑧_𝑛−𝑡_𝑛+𝑥^∗−𝑧^∗_𝑛−𝜇𝐵𝑧_𝑛−𝑧^∗−𝜇𝐵𝑧^∗‖‖²−‖‖𝑃_𝐸𝑦_𝑛−𝜆_𝑛𝐴𝑦_𝑛−𝑃_𝐸𝑦^∗−𝜆_𝑛𝐴𝑦^∗‖‖²≤+‖‖𝑧2_𝑛𝜇‖‖𝐵𝑧_𝑛−𝐵𝑧^∗‖‖‖‖𝑧_𝑛−𝑡_𝑛+𝑥^∗−𝑧^∗‖‖−𝜇𝐵𝑧_𝑛−𝑧^∗−𝜇𝐵𝑧^∗‖‖²−‖‖𝑆𝑃_𝐸𝑦_𝑛−𝜆_𝑛𝐴𝑦_𝑛−𝑆𝑃_𝐸𝑦^∗−𝜆_𝑛𝐴𝑦^∗‖‖²=+‖‖𝑧2_𝑛𝜇‖‖𝐵𝑧_𝑛−𝐵𝑧^∗‖‖‖‖𝑧_𝑛−𝑡_𝑛+𝑥^∗−𝑧^∗‖‖−𝜇𝐵𝑧_𝑛−𝑧^∗−𝜇𝐵𝑧^∗‖‖²−‖‖𝑆𝑡_𝑛−𝑆𝑥^∗‖‖²≤+‖‖2𝑧𝜇_𝑛‖‖𝐵𝑧_𝑛−𝐵𝑧^∗‖‖‖‖𝑧_𝑛−𝑡_𝑛+𝑥^∗−𝑧^∗‖‖×‖𝑧−_𝑛𝜇𝐵𝑧_𝑛−𝑧^∗−𝜇𝐵𝑧^∗−𝑆𝑡_𝑛−𝑥^∗‖‖+2𝜇‖‖𝐵−𝑧_𝑛𝜇𝐵𝑧_𝑛−𝑧^∗−𝜇𝐵𝑧^∗‖+‖‖𝑆𝑡_𝑛−𝑥^∗‖‖−𝐵𝑧^∗‖‖‖‖𝑧_𝑛−𝑡_𝑛+𝑥^∗−𝑧^∗‖‖.(3.38) Since ‖𝑆𝑡_𝑛−𝑥_𝑛‖→0, ‖(𝑥_𝑛−𝑧_𝑛)−(𝑥^∗−𝑧^∗)‖→0, and ‖𝐵𝑧_𝑛−𝐵𝑧^∗‖→0,as𝑛→∞, it follows that ‖𝑧_𝑛−𝑡_𝑛+𝑥^∗−𝑧^∗‖→0,as𝑛→∞.(3.39) Since ‖‖𝑆𝑡_𝑛−𝑡_𝑛‖‖≤‖‖𝑆𝑡_𝑛−𝑥_𝑛‖‖+‖‖𝑥_𝑛−𝑧_𝑛−𝑥^∗−𝑧^∗‖‖+‖‖𝑧_𝑛−𝑡_𝑛+𝑥^∗−𝑧^∗‖‖,(3.40) we obtain lim_𝑛→∞‖‖𝑆𝑡_𝑛−𝑡_𝑛‖‖=0.(3.41) Next, we show that limsup_𝑛→∞𝑓𝑥−𝑥,𝑥_𝑛−𝑥≤0,(3.42) where 𝑥=𝑃_{𝐹(𝑆)∩Υ}𝑓(𝑥).
Indeed, since {𝑡_𝑛} and {𝑆𝑡_𝑛} are two bounded sequence in 𝐸, we can choose a subsequence {𝑡_𝑛𝑖} of {𝑡_𝑛} such that 𝑡_𝑛𝑖 of 𝑡_𝑛 such that 𝑡_𝑛𝑖⇀𝑧∈𝐸 and limsup_𝑛→∞𝑓𝑥−𝑥,𝑆𝑡_𝑛−𝑥=lim_𝑖→∞𝑓𝑥−𝑥,𝑆𝑡_𝑛𝑖−𝑥.(3.43) Since lim_𝑛→∞‖𝑆𝑡_𝑛−𝑡_𝑛‖=0, we obtain 𝑆𝑡_𝑛𝑖⇀𝑧 as 𝑖→∞. Now, we claim that 𝑧∈𝐹(𝑆)∩Υ. First by Lemma 2.5, it is easy to see that 𝑧∈𝐹(𝑆).
Since ‖𝑆𝑡_𝑛−𝑡_𝑛‖→0, ‖𝑆𝑡_𝑛−𝑥_𝑛‖→0, and ‖‖𝑡_𝑛−𝑥_𝑛‖‖=≤‖‖‖‖𝑡𝑡_𝑛𝑛−𝑆𝑡_𝑛+𝑆𝑡_𝑛−𝑥_𝑛‖‖=‖‖𝑆𝑡_𝑛−𝑆𝑡_𝑛‖‖+‖‖𝑆𝑡_𝑛−𝑥_𝑛‖‖−𝑡_𝑛‖‖+‖‖𝑆𝑡_𝑛−𝑥_𝑛‖‖,(3.44)
we conclude that ‖𝑡_𝑛−𝑥_𝑛‖→0 as 𝑛→∞. Furthermore, by Lemma 3.2 that 𝐺 is nonexpansive, then ‖‖𝑡_𝑛−𝐺𝑡_𝑛‖‖=≤‖‖‖‖𝐺𝑥_𝑛𝑥_𝑛−𝐺𝑡_𝑛‖‖−𝑡_𝑛‖‖.(3.45) Thus lim_𝑛→∞‖𝑡_𝑛−𝐺(𝑡_𝑛)‖=0. According to Lemma 2.5, we obtain 𝑧∈Υ. Therefore, there holds 𝑧∈𝐹(𝑆)∩Υ.
On the other hand, it follows from (2.2) that limsup_𝑛→∞𝑓𝑥−𝑥,𝑥_𝑛−𝑥==lliimm_𝑛→∞sup_𝑛→∞𝑓𝑥−𝑥,𝑆𝑡_𝑛−𝑥=𝑓𝑓𝑥−𝑥,𝑆𝑡_𝑛𝑖−𝑥≤0.𝑥−𝑥,𝑧−𝑥(3.46)
Finally, we show that 𝑥_𝑛→𝑥, and by (3.14) that ‖‖𝑥_𝑛+1−𝑥‖‖²=‖‖𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑆𝑡_𝑛−𝑥‖‖²≤≤‖‖‖‖𝛽𝛽_𝑛𝑛𝑥_𝑛−𝑥+𝛾_𝑛𝑆𝑡_𝑛−𝑥‖‖²+2𝛼_𝑛𝑓𝑥_𝑛−𝑥,𝑥_𝑛+1−𝑥+2𝛼_𝑛𝑥_𝑛−𝑥+𝛾_𝑛𝑆𝑡_𝑛−𝑥‖‖²+2𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑥,𝑥_𝑛+1−𝑥≤𝛽_𝑛𝑓𝑥−𝑥,𝑥_𝑛+1−𝑥+2𝛼‖‖_𝑛𝑥_𝑛−𝑥‖‖²+𝛾_𝑛‖‖𝑆𝑡_𝑛−𝑥‖‖²+2𝛼^𝑛‖‖𝑓𝑥_𝑛−𝑓𝑥‖‖‖‖𝑥_𝑛+1−𝑥‖‖≤𝛽_𝑛𝑓𝑥−𝑥,𝑥_𝑛+1−𝑥+2𝛼‖‖_𝑛𝑥_𝑛−𝑥‖‖²+𝛾_𝑛‖‖𝑡_𝑛−𝑥‖‖²+2𝛼^𝑛𝛼‖‖𝑥_𝑛−𝑥‖‖‖‖𝑥_𝑛+1−𝑥‖‖≤1−𝛼_𝑛𝑓𝑥−𝑥,𝑥_𝑛+1−𝑥+2𝛼_𝑛²‖‖𝑥_𝑛−𝑥‖‖²+𝛼_𝑛𝛼‖‖𝑥_𝑛−𝑥‖‖²+‖‖𝑥_𝑛+1−𝑥‖‖²𝑓𝑥−𝑥,𝑥_𝑛+1−𝑥(3.47) which implies that ‖‖𝑥_𝑛+1−𝑥‖‖²≤1−2(1−𝛼)𝛼^𝑛1−𝛼𝛼_𝑛‖‖𝑥^𝑛−𝑥‖‖²+𝛼^2𝑛1−𝛼𝛼_𝑛‖‖𝑥_𝑛−𝑥‖‖²+2𝛼^𝑛1−𝛼𝛼_𝑛𝑓∶=1−𝜎_𝑛𝑥−𝑥,𝑥_𝑛+1−𝑥‖‖𝑥_𝑛−𝑥‖‖²+𝛿_𝑛,𝑛≥0,(3.48) where 𝜎_𝑛=(2(1−𝛼)𝛼_𝑛)/1−𝛼𝛼_𝑛 and 𝛿_𝑛=𝛼^2𝑛/(1−𝛼𝛼_𝑛)‖𝑥_𝑛−𝑥‖²+2𝛼_𝑛/(1−𝛼𝛼_𝑛)⟨𝑓(𝑥)−𝑥,𝑥_𝑛+1−𝑥⟩. Therefore, by (3.46) and Lemma 2.4, we get that {𝑥_𝑛} converges to 𝑥, where 𝑥=𝑃_{𝐹(𝑆)∩Υ}𝑓(𝑥). This completes the proof.

Setting 𝐴=𝐵=𝐶, we obtain the following corollary.

Corollary 3.4. Let 𝐸 be a nonempty, closed, convex subset of a real Hilbert space 𝐻. Let the mapping 𝐴∶𝐸→𝐻 be 𝛼-inverse-strongly monotone. Let 𝑆 be a nonexpansive mapping of 𝐸 into itself such that 𝐹(𝑆)∩Υ≠∅. Let 𝑓 be a contraction of 𝐻 into itself and given 𝑥₀∈𝐻 arbitrarily and {𝑥_𝑛} is generated by 𝑧𝑦_𝑛𝑛=𝑃_𝐸𝑥_𝑛−𝜏𝐴𝑥_𝑛,𝑥_𝑛+1=𝑃_𝐸𝑧_𝑛−𝜇𝐴𝑧_𝑛,=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑆𝑃_𝐸𝑦_𝑛−𝜆𝐴𝑦_𝑛,𝑛≥1,(3.49) where 𝜆,𝜇,𝜏∈(0,2𝛼) and {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛} are three sequences in [0,1] such that (i)𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1,(ii)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞𝑛=1𝛼_𝑛=∞,(iii)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1. Then, {𝑥_𝑛} converges strongly to 𝑥∈𝐹(𝑆)∩Υ, where 𝑥=𝑃_{𝐹(𝑆)∩Υ}𝑓(𝑥) and (𝑥,𝑦,𝑧) is a solution of problem (1.10), where 𝑦=𝑃_𝐸𝑧−𝜇𝐴𝑧,z=𝑃_𝐸𝑥−𝜏𝐴𝑥.(3.50)

Setting 𝐴≡𝐵≡0 (the zero operators), we obtain the following corollary for solving the foxed points problem and the classical variational inequality problems.

Corollary 3.5. Let 𝐸 be a nonempty, closed, convex subset of a real Hilbert space 𝐻. Let the mapping 𝐴∶𝐸→𝐻 be 𝛼-inverse-strongly monotone. Let 𝑆 be a nonexpansive mapping of 𝐸 into itself such that 𝐹(𝑆)∩𝑉𝐼(𝐴,𝐸)≠∅. Let 𝑓 be a contraction of 𝐻 into itself and given 𝑥₀∈𝐻 arbitrarily and {𝑥_𝑛} is generated by 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑆𝑃_𝐸𝑥_𝑛−𝜆𝐴𝑥_𝑛,𝑛≥1,(3.51) where 𝜆∈(0,2𝛼) and {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛} are three sequences in [0,1] such that (i)𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1,(ii)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞𝑛=1𝛼_𝑛=∞,(iii)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1.

Then, {𝑥_𝑛} converges strongly to 𝑥∈𝐹(𝑆)∩𝑉𝐼(𝐴,𝐸), where 𝑥=𝑃_{𝐹(𝑆)∩𝑉𝐼(𝐴,𝐸)}𝑓(𝑥).

4. Some Applications

We recall that a mapping 𝑇∶𝐸→𝐸 is called strictly pseudocontractive if there exists some 𝑘 with 0≤𝑘<1 such that ‖𝑇𝑥−𝑇𝑦‖²≤‖𝑥−𝑦‖²+𝑘‖(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦‖²,∀𝑥,𝑦∈𝐸.(4.1) For recent convergence result for strictly pseudocontractive mappings, put 𝐴=𝐼−𝑇. Then, we have ‖(𝐼−𝐴)𝑥−(𝐼−𝐴)𝑦‖²≤‖𝑥−𝑦‖²+𝑘‖𝐴𝑥−𝐴𝑦‖².(4.2) On the other hand, ‖(𝐼−𝐴)𝑥−(𝐼−𝐴)𝑦‖²≤‖𝑥−𝑦‖²+‖𝐴𝑥−𝐴𝑦‖²−2⟨𝑥−𝑦,𝐴𝑥−𝐴𝑦⟩.(4.3) Hence, we have ⟨𝑥−𝑦,𝐴𝑥−𝐴𝑦⟩≥1−𝑘2‖𝐴𝑥−𝐴𝑦‖².(4.4) Consequently, if 𝑇∶𝐸→𝐸 is a strictly pseudocontractive mapping with constant 𝑘, then the mapping 𝐴=𝐼−𝑇 is (1−𝑘)/2-inverse-strongly monotone.

Setting 𝐴=𝐼−𝑇, 𝐵=𝐼−𝑉, and 𝐶=𝐼−𝑊, we obtain the following corollary.

Theorem 4.1. Let 𝐸 be a nonempty, closed, convex subset of a real Hilbert space 𝐻. Let 𝑇,𝑉,𝑊 be strictly pseudocontractive mappings with constant 𝑘 of 𝐶 into itself, and let 𝑆 be a nonexpansive mapping of 𝐸 into itself such that 𝐹(𝑆)∩Υ≠∅. Let 𝑓 be a contraction of 𝐻 into itself and given 𝑥₀∈𝐻 arbitrarily and {𝑥_𝑛} is generated by 𝑧𝑦_𝑛𝑛=(𝐼−𝜏)𝑥_𝑛+𝜏𝑊𝑥_𝑛,𝑥_𝑛+1=(𝐼−𝜇)𝑧_𝑛+𝜇𝑉𝑧_𝑛,=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑆(1−𝜆)𝑦_𝑛+𝜆𝑇𝑦_𝑛,𝑛≥1,(4.5) where 𝜆∈(0,2𝛼),𝜇∈(0,2𝛽),and𝜏∈(0,2𝛾) and {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛} are three sequences in [0,1] such that (i)𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1,(ii)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞𝑛=1𝛼_𝑛=∞,(iii)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1. Then {𝑥_𝑛} converges strongly to 𝑥∈𝐹(𝑆)∩Υ, where 𝑥=𝑃_{𝐹(𝑆)∩Υ}𝑓(𝑥) and (𝑥,𝑦,𝑧) is a solution of problem (1.9), where 𝑦=𝑃_𝐸𝑧−𝜇𝐵𝑧,𝑧=𝑃_𝐸𝑥−𝜏𝐶𝑥.(4.6)

Proof. Since 𝐴=𝐼−𝑇, 𝐵=𝐼−𝑉, and 𝐶=𝐼−𝑊, we have 𝑃𝑃_𝐸𝐸𝑥_𝑛−𝜏𝐶𝑥_𝑛=(𝐼−𝜏)𝑥_𝑛+𝜏𝑊𝑥_𝑛,𝑃_𝐸𝑦_𝑛−𝜆𝐴𝑦_𝑛=(𝐼−𝜆)𝑦_𝑛+𝜆𝑇𝑦_𝑛,𝑧_𝑛−𝜇𝐵𝑧_𝑛=(𝐼−𝜇)𝑧_𝑛+𝜇𝑉𝑧_𝑛.(4.7) Thus, the conclusion follows immediately from Theorem 3.3.

If 𝑓(𝑥)=𝑥₀,forall𝑥∈𝐸, and 𝑇=𝑉=𝑊 in Theorem 4.1, we obtain the following corollary.

Corollary 4.2. Let 𝐸 be a nonempty, closed, convex subset of a real Hilbert space 𝐻. Let 𝑇 be strictly pseudocontractive mappings with constant 𝑘 of 𝐶 into itself, and let 𝑆 be a nonexpansive mapping of 𝐸 into itself such that 𝐹(𝑆)∩Υ≠∅. Given 𝑥₀∈𝐻 arbitrarily and {𝑥_𝑛} is generated by 𝑦𝑧_𝑛𝑛=(𝐼−𝜏)𝑥_𝑛+𝜏𝑇𝑥_𝑛,𝑥_𝑛+1=(𝐼−𝜇)𝑧_𝑛+𝜇𝑇𝑧_𝑛,=𝛼_𝑛𝑥₀+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑆(1−𝜆)𝑦_𝑛+𝜆𝑇𝑦_𝑛,𝑛≥1,(4.8) where 𝜆∈(0,2𝛼),𝜇∈(0,2𝛽),and𝜏∈(0,2𝛾) and {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛} are three sequences in [0,1] such that lim_𝑛→∞𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1,𝛼_𝑛=0and^∞_𝑛=10<liminf_𝑛→∞𝛼_𝑛=∞,𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1.(4.9) Then, {𝑥_𝑛} converges strongly to 𝑥∈𝐹(𝑆)∩Υ, where 𝑥=𝑃_{𝐹(𝑆)∩Υ}𝑥 and (𝑥,𝑦,𝑧) is a solution of problem (1.10), where 𝑦=𝑃_𝐸𝑧−𝜇𝐴𝑧𝑧=𝑃_𝐸𝑥−𝜏𝐴𝑥.(4.10)

Acknowledgments

W. Kumam would like to give thanks to the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand, for their financial support for Ph.D. program at KMUTT. Moreover, the authors also would like to thank the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission for financial support (under CSEC Project no. 54000267) and P. Kumam was supported by the Commission on Higher Education and the Thailand Research Fund under Grant no. MRG5380044.