Hybrid Extragradient Iterative Algorithms for Variational Inequalities, Variational Inclusions, and Fixed-Point Problems

Abstract

We investigate the problem of finding a common solution of a general system of variational inequalities, a variational inclusion, and a fixed-point problem of a strictly pseudocontractive mapping in a real Hilbert space. Motivated by Nadezhkina and Takahashi's hybrid-extragradient method, we propose and analyze new hybrid-extragradient iterative algorithm for finding a common solution. It is proven that three sequences generated by this algorithm converge strongly to the same common solution under very mild conditions. Based on this result, we also construct an iterative algorithm for finding a common fixed point of three mappings, such that one of these mappings is nonexpansive, and the other two mappings are strictly pseudocontractive mappings.

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 monotone if 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥0,∀𝑥,𝑦∈𝐶. (1.1) A mapping 𝐴∶𝐶→𝐻 is called 𝐿-Lipschitz continuous if there exists a constant 𝐿>0, such that 𝐴𝑥−𝐴𝑦≤𝐿𝑥−𝑦,∀𝑥,𝑦∈𝐶. (1.2) 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. (1.3) The solution set of the VI (1.3) 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, and equilibrium problems; see, for example, [2–4]. To construct a mathematical model which is as close as possible to a real complex problem, we often have to use more than one constraint. Solving such problems, we have to obtain some solution which is simultaneously the solution of two or more subproblem or the solution of one subproblem on the solution set of another subproblem. Actually, these subproblems can be given by problems of different types. For example, Antipin considered a finite-dimensional variant of the variational inequality, where the solution should satisfy some related constraint in inequality form [5] or some system of constraints in inequality and equality form [6]. Yamada [7] considered an infinite-dimensional variant of the solution of the variational inequality on the fixed-point set of some mapping.

A mapping 𝐴∶𝐶→𝐻 is called 𝛼-inverse strongly monotone if there exists a constant 𝛼>0, such that 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥𝛼𝐴𝑥−𝐴𝑦²,∀𝑥,𝑦∈𝐶; (1.4) see [8, 9]. It is obvious that an 𝛼-inverse strongly monotone mapping 𝐴 is monotone and Lipschitz continuous. A self-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; (1.5) see [10]. In particular, if 𝑘=0, then 𝑆 is called a nonexpansive mapping; see [11].

A set-valued mapping 𝑀 with domain 𝐷(𝑀) and range 𝑅(𝑀) in 𝐻 is called monotone if its graph 𝐺(𝑀)={(𝑥,𝑓)∈𝐻×𝐻∶𝑥∈𝐷(𝑀),𝑓∈𝑀𝑥} is a monotone set in 𝐻×𝐻; that is, 𝑀 is monotone if and only if 𝑥,𝑓,𝑦,𝑔∈𝐺(𝑀)⇒𝑥−𝑦,𝑓−𝑔≥0. (1.6) A monotone set-valued mapping 𝑀 is called maximal if its graph 𝐺(𝑀) is not properly contained in the graph of any other monotone mapping in 𝐻.

Let Φ be a single-valued mapping of 𝐶 into 𝐻, and let 𝑀 be a multivalued mapping with 𝐷(𝑀)=𝐶. Consider the following variational inclusion: find 𝑥^∗∈𝐶, such that 0∈Φ𝑥^∗+𝑀𝑥^∗. (1.7) We denote by Ω the solution set of the variational inclusion (1.7). In particular, if Φ=𝑀=0, then Ω=𝐶.

In 1998, Huang [12] studied problem (1.7) in the case where 𝑀 is maximal monotone, and Φ is strongly monotone and Lipschitz continuous with 𝐷(𝑀)=𝐶=𝐻. Subsequently, Zeng et al. [13] further studied problem (1.7) in the case which is more general than Huang's one [12]. Moreover, the authors [13] obtained the same strong convergence conclusion as in Huang's result [12]. In addition, the authors also gave the geometric convergence rate estimate for approximate solutions.

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 [14] introduced the following iterative algorithm: 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑆𝑃_𝐶𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛,∀𝑛≥0, (1.8) 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). In 2006, to solve this problem (i.e., to find an element of Fix(𝑆)∩VI(𝐶,𝐴)), Nadezhkina and Takahashi [15] introduced an iterative algorithm by a hybrid method. Generally speaking, the suggested algorithm is based on two well-known types of methods, that is, on the extragradient-type method due to Korpelevich [16] for solving variational inequality and so-called hybrid or outer-approximation method due to Haugazeau (see [15]) for solving fixed point problem. It is worth emphasizing that the idea of “hybrid” or “outer-approximation” types of methods was successfully generalized and extended in many papers; see, for example, [17–23]. In addition, the idea of the extragradient iterative algorithm introduced by Korpelevich [16] was successfully generalized and extended not only in Euclidean but also in Hilbert and Banach spaces; see, for example, [24–29].

Theorem NT (see [15, Theorem 3.1]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐴∶𝐶→𝐻 be a monotone and 𝑘-Lipschitz-continuous mapping, and let 𝑆∶𝐶→𝐶 be a nonexpansive mapping such that Fix(𝑆)∩VI(𝐶,𝐴)≠∅. Let {𝑥_𝑛}, {𝑦_𝑛} and {𝑧_𝑛} be the sequences generated by 𝑦_𝑛=𝑃_𝐶𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛,𝑧_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑆𝑃_𝐶𝑥_𝑛−𝜆_𝑛𝐴𝑦_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝑧_𝑛−𝑧≤𝑥_𝑛,𝑄−𝑧_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑥₀−𝑥_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀,∀𝑛≥0, (1.9) where 𝑥₀∈𝐶 is chosen arbitrarily, {𝜆_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1/𝑘), and {𝛼_𝑛}⊂[0,𝑐] for some 𝑐∈[0,1). Then the sequences {𝑥_𝑛}, {𝑦_𝑛}, and {𝑧_𝑛} converge strongly to 𝑃_{Fix(𝑆)∩VI(𝐶,𝐴)}𝑥₀.

It is easy to see that the class of 𝛼-inverse strongly monotone mappings in the above mentioned problem of Takahashi and Toyoda [14] is the quite important class of mappings in various classes of well-known mappings. It is also easy to see that while 𝛼-inverse strongly monotone mappings are tightly connected with the important class of nonexpansive mappings, 𝛼-inverse strongly monotone mappings are also tightly connected with the more general and also quite important class of strictly pseudocontractive mappings. That is, if a mapping 𝑆∶𝐶→𝐶 is nonexpansive, then the mapping 𝐼−𝑆 is (1/2-) inverse strongly monotone; moreover, Fix(𝑆)=VI(𝐶,𝐼−𝑆) (see, e.g., [14]). The construction of fixed points of nonexpansive mappings via Mann's algorithm has extensively been investigated in the literature (see, e.g., [30, 31] and references therein). At the same time, if a mapping 𝑆∶𝐶→𝐶 is 𝑘-strictly pseudocontractive, then the mapping 𝐼−𝑆 is (1−𝑘)/2-inverse strongly monotone and 2/(1−𝑘)-Lipschitz continuous.

Let 𝐵₁,𝐵₂∶𝐶→𝐻 be two mappings. Recently, Ceng et al. [32] introduced and considered the following problem of finding (𝑥^∗,𝑦^∗)∈𝐶×𝐶, such that 𝜇₁𝐵₁𝑦^∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗𝜇≥0,∀𝑥∈𝐶,₂𝐵₂𝑥^∗+𝑦^∗−𝑥^∗,𝑥−𝑦^∗≥0,∀𝑥∈𝐶, (1.10) which is called a general system of variational inequalities (GSVI), where 𝜇₁>0 and 𝜇₂>0 are two constants. The set of solutions of problem (1.10) is denoted by GSVI(𝐶,𝐵₁,𝐵₂). In particular, if 𝐵₁=𝐵₂=𝐴, then problem (1.10) reduces to the new system of variational inequalities (NSVI), introduced and studied by Verma [33]. Further, if 𝑥^∗=𝑦^∗ additionally, then the NSVI reduces to the VI (1.3).

In particular, if 𝐵₁=𝐴 and 𝐵₂=0, then the GSVI (1.10) is equivalent to the VI (1.3).

Indeed, in this case, the GSVI (1.10) is equivalent to the following problem of finding (𝑥^∗,𝑦^∗)∈𝐶×𝐶, such that 𝜇₁𝐵₁𝑦^∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗𝑦≥0,∀𝑥∈𝐶,_∗−𝑥^∗,𝑥−𝑦^∗≥0,∀𝑥∈𝐶. (1.11) Thus we must have 𝑥^∗=𝑦^∗. As a matter of fact, if 𝑥^∗≠𝑦^∗, then by setting 𝑥=𝑥^∗ we have 0>−𝑥^∗−𝑦^∗2=𝑦^∗−𝑥^∗,𝑥^∗−𝑦^∗≥0, (1.12) which hence leads to a contradiction. Therefore, the GSVI (1.10) coincides with the VI (1.3).

Recently, Ceng at al. [32] transformed problem (1.10) into a fixed-point problem in the following way.

Lemma 1.1 (see [32]). For given 𝑥,𝑦∈𝐶,(𝑥,𝑦) is a solution of problem (1.10) if and only if 𝑥 is a fixed point of the mapping 𝐺∶𝐶→𝐶 defined by 𝐺(𝑥)=𝑃_𝐶𝑃_𝐶𝑥−𝜇₂𝐵₂𝑥−𝜇₁𝐵₁𝑃_𝐶𝑥−𝜇₂𝐵₂𝑥,∀𝑥∈𝐶, (1.13) where 𝑦=𝑃_𝐶(𝑥−𝜇₂𝐵₂𝑥).

In particular, if the mapping 𝐵_𝑖∶𝐶→𝐻 is 𝛽_𝑖-inverse strongly monotone for 𝑖=1,2, then the mapping 𝐺 is nonexpansive provided 𝜇_𝑖∈(0,2𝛽_𝑖] for 𝑖=1,2.

Utilizing Lemma 1.1, they introduced and studied a relaxed extragradient method for solving the GSVI (1.10). Throughout this paper, the set of fixed points of the mapping 𝐺 is denoted by Ξ. Based on the relaxed extragradient method and viscosity approximation method, Yao et al. [34] proposed and analyzed an iterative algorithm for finding a common solution of the GSVI (1.10) and the fixed point problem of a strictly pseudocontractive mapping 𝑆∶𝐶→𝐶.

Subsequently, Ceng et al. [35] further presented and analyzed an iterative scheme for finding a common element of the solution set of the VI (1.3), the solution set of the GSVI (1.10), and the fixed point set of a strictly pseudo-contractive mapping 𝑆∶𝐶→𝐶.

Theorem CGY (see [35, Theorem 3.1]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐴∶𝐶→𝐻 be 𝛼-inverse strongly monotone, and let 𝐵_𝑖∶𝐶→𝐻 be 𝛽_𝑖-inverse strongly monotone for 𝑖=1,2. Let 𝑆∶𝐶→𝐶 be a 𝑘-strictly pseudocontractive mapping such that Fix(𝑆)∩Ξ∩VI(𝐶,𝐴)≠∅. Let 𝑄∶𝐶→𝐶 be a 𝜌-contraction with 𝜌∈[0,1/2). For given 𝑥₀∈𝐶 arbitrarily, let the sequences {𝑥_𝑛}, {𝑦_𝑛}, and {𝑧_𝑛} be generated iteratively by 𝑧_𝑛=𝑃_𝐶𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛,𝑦_𝑛=𝛼_𝑛𝑄𝑥_𝑛+1−𝛼_𝑛𝑃_𝐶𝑃_𝐶𝑧_𝑛−𝜇₂𝐵₂𝑧_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑧_𝑛−𝜇₂𝐵₂𝑧_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑦_𝑛+𝛿_𝑛𝑆𝑦_𝑛,∀𝑛≥0, (1.14) where 𝜇_𝑖∈(0,2𝛽_𝑖) for 𝑖=1,2, {𝜆_𝑛}⊂(0,2𝛼] and {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛},{𝛿_𝑛}⊂[0,1], such that(i)𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 and (𝛾_𝑛+𝛿_𝑛)𝑘≤𝛾_𝑛, for all 𝑛≥0;(ii)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞;(iii)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and liminf_𝑛→∞𝛿_𝑛>0;(iv)lim_𝑛→∞(𝛾_𝑛+1/(1−𝛽_𝑛+1)−𝛾_𝑛/(1−𝛽_𝑛))=0;(v)0<liminf_𝑛→∞𝜆_𝑛≤limsup_𝑛→∞𝜆_𝑛<2𝛼 and lim_𝑛→∞|𝜆_𝑛+1−𝜆_𝑛|=0.

Then the sequence {𝑥_𝑛} generated by (1.14) converges strongly to 𝑥=𝑃_{Fix(𝑆)∩Ξ∩VI(𝐶,𝐴)}𝑄𝑥, and (𝑥,𝑦) is a solution of the GSVI (1.10), where 𝑦=𝑃_𝐶(𝑥−𝜇₂𝐵₂𝑥).

On the other hand, let 𝐴∶𝐶→𝐻 be a monotone, and let 𝐿-Lipschitz-continuous mapping, Φ∶𝐶→𝐻 be an 𝛼-inverse strongly monotone mapping. Let 𝑀 be a maximal monotone mapping with 𝐷(𝑀)=𝐶, and let 𝑆∶𝐶→𝐶 be a nonexpansive mapping such that Fix(𝑆)∩Ω∩VI(𝐶,𝐴)≠∅. Motivated Nadezhkina and Takahashi's hybrid-extragradient algorithm (1.9), Ceng et al. [36, Theorem 3.1] introduced another modified hybrid-extragradient algorithm 𝑦_𝑛=𝑃_𝐶𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛,𝑡_𝑛=𝑃_𝐶𝑥_𝑛−𝜆_𝑛𝐴𝑦_𝑛,𝑡_𝑛̂=𝐽_{𝑀,𝜇𝑛}𝑡_𝑛−𝜇_𝑛Φ𝑡_𝑛,𝑧_𝑛=1−𝛼_𝑛−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑡_𝑛̂+𝛼_𝑛𝑆𝑡_𝑛̂,𝐶_𝑛=𝑧∈𝐶∶𝑧_𝑛−𝑧≤𝑥_𝑛,𝑄−𝑧_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑥₀−𝑥_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀,∀𝑛≥0, (1.15) where 𝐽_{𝑀,𝜇𝑛}=(𝐼+𝜇_𝑛𝑀)⁻¹,𝑥₀∈𝐶 chosen arbitrarily, {𝜆_𝑛}⊂(0,1/𝐿),{𝜇_𝑛}⊂(0,2𝛼], and {𝛼_𝑛},{𝛼_𝑛}⊂(0,1] such that 𝛼_𝑛+𝛼_𝑛≤1. It was proven in [36] that under very mild conditions three sequences {𝑥_𝑛}, {𝑦_𝑛}, and {𝑧_𝑛} generated by (1.15) converge strongly to the same point 𝑃_{Fix(𝑆)∩Ω∩VI(𝐶,𝐴)}𝑥₀.

Inspired by the research going on this area, we propose and analyze the following hybrid extragradient iterative algorithm for finding a common element of the solution set Ξ of the GSVI (1.10), the solution set Ω of the variational inclusion (1.7), and the fixed point set Fix(𝑆) of a strictly pseudo-contractive mapping 𝑆∶𝐶→𝐶.

Algorithm 1.2. Assume that Fix(𝑆)∩Ω∩Ξ≠∅. Let 𝜇_𝑖∈(0,2𝛽_𝑖) for 𝑖=1,2,{𝜇_𝑛}⊂(0,2𝛼], and {𝜎_𝑛},{𝛽_𝑛},{𝛾_𝑛},{𝛿_𝑛}⊂[0,1] such that 𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1, for all 𝑛≥0. For given 𝑥₀∈𝐶 arbitrarily, let {𝑥_𝑛},{𝑦_𝑛}, and {𝑧_𝑛} be the sequences generated by the hybrid extragradient iterative scheme 𝑦_𝑛=𝑃_𝐶𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛,𝑡_𝑛=𝑃_𝐶𝑃_𝐶𝑦_𝑛−𝜇₂𝐵₂𝑦_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑦_𝑛−𝜇₂𝐵₂𝑦_𝑛,𝑡_𝑛̂=𝜎_𝑛𝑡_𝑛+1−𝜎_𝑛𝐽_{𝑀,𝜇𝑛}𝑡_𝑛−𝜇_𝑛Φ𝑡_𝑛,𝑧_𝑛=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑡_𝑛̂+𝛿_𝑛𝑆𝑡_𝑛̂,𝐶_𝑛=𝑧∈𝐶∶𝑧_𝑛−𝑧≤𝑥_𝑛,𝑄−𝑧_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑥₀−𝑥_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀,∀𝑛≥0, (1.16) where 𝐽_{𝑀,𝜇𝑛}=(𝐼+𝜇_𝑛𝑀)⁻¹, for all 𝑛≥0.

Under very appropriate assumptions, it is proven that all the sequences {𝑥_𝑛},{𝑦_𝑛}, and {𝑧_𝑛} converge strongly to the same point 𝑥=𝑃_{Fix(𝑆)∩Ω∩Ξ}𝑥₀. Furthermore, (𝑥,𝑦) is a solution of the GSVI (1.10), where 𝑦=𝑃_𝐶(𝑥−𝜇₂𝐵₂𝑥).

Let 𝑇∶𝐶→𝐶 be a 𝑘-strictly pseudocontractive mapping, let Γ∶𝐶→𝐶 be a 𝜅-strictly pseudocontractive mapping, and let 𝑆∶𝐶→𝐶 be a nonexpansive mapping. Putting 𝐵₁=𝐼−𝑇,𝐵₂=0,Φ=𝐼−Γ,𝑀=0, and 𝜎_𝑛=0, for all 𝑛≥0 in Algorithm 1.2, we consider and analyze the following hybrid extragradient iterative algorithm for finding a common fixed point of three mappings 𝑆, Γ, and 𝑇.

Algorithm 1.3. Assume that Fix(𝑆)∩Fix(Γ)∩Fix(𝑇)≠∅. Let 𝜇₁∈(0,1−𝑘), {𝜇_𝑛}⊂(0,1−𝜅], and {𝛽_𝑛},{𝛾_𝑛},{𝛿_𝑛}⊂[0,1] such that 𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1, for all 𝑛≥0. For given 𝑥₀∈𝐶 arbitrarily, let {𝑥_𝑛},{𝑦_𝑛}, and {𝑧_𝑛} be the sequences generated by the hybrid extragradient iterative scheme 𝑦_𝑛=𝑥_𝑛−𝜇₁𝑥_𝑛−𝑇𝑥_𝑛,𝑡_𝑛=𝑦_𝑛−𝜇₁𝑦_𝑛−𝑇𝑦_𝑛,𝑡_𝑛̂=𝑡_𝑛−𝜇_𝑛𝑡_𝑛−Γ𝑡_𝑛,𝑧_𝑛=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑡_𝑛̂+𝛿_𝑛𝑆𝑡_𝑛̂,𝐶_𝑛=𝑧∈𝐶∶𝑧_𝑛−𝑧≤𝑥_𝑛,𝑄−𝑧_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑥₀−𝑥_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀,∀𝑛≥0. (1.17)
Under quite mild conditions, it is shown that all the sequences {𝑥_𝑛}, {𝑦_𝑛}, and {𝑧_𝑛} converge strongly to the same point 𝑃_{Fix(𝑆)∩Fix(Γ)∩Fix(𝑇)}𝑥₀.

Observe that Ceng et al. [36, Theorem 3.1] considered the problem of finding an element of Fix(𝑆)∩Ω∩VI(𝐶,𝐴) where 𝑆∶𝐶→𝐶 is nonexpansive, Nadezhkina and Takahashi [15, Theorem 3.1] studied the problem of finding an element of Fix(𝑆)∩VI(𝐶,𝐴) where 𝑆∶𝐶→𝐶 is nonexpansive, and Ceng et al. [35, Theorem 3.1] investigated the problem of finding an element of Fix(𝑆)∩Ξ∩VI(𝐶,𝐴) where 𝑆∶𝐶→𝐶 is strictly pseudocontractive. It is clear that every one of these three problems is very different from our problem of finding an element of Fix(𝑆)∩Ω∩Ξ where 𝑆∶𝐶→𝐶 is strictly pseudocontractive. Hence there is no doubt that the strong convergence results for solving our problem are very interesting and quite valuable. Because our hybrid extragradient iterative algorithms involve two inverse strongly monotone mappings 𝐵₁ and 𝐵₂, a strictly pseudo-contractive self-mapping 𝑆, and several parameter sequences, they are more flexible and more subtle than the corresponding ones in [36, Theorem 3.1] and [15, Theorem 3.1], respectively. Furthermore, the relaxed extragradient iterative scheme in Yao et al. [34, Theorem 3.2] is extended to develop our hybrid extragradient iterative algorithms. In our results, the hybrid extragradient iterative algorithms drop the requirements that 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and lim_𝑛→∞(𝛾_𝑛+1/(1−𝛽_𝑛+1)−𝛾_𝑛/(1−𝛽_𝑛))=0 in [34, Theorem 3.2] and [35, Theorem 3.1]. Therefore, our results represent the modification, supplementation, extension, and improvement of [36, Theorem 3.1], [15, Theorem 3.1], [34, Theorem 3.2], and [35, Theorem 3.1] to a great extent.

2. Preliminaries

Let 𝐻 be a real Hilbert space, whose inner product and norm are denoted by ⟨⋅,⋅⟩ and ‖⋅‖, respectively. Let 𝐶 be a nonempty closed convex subset of 𝐻. We write → to indicate that the sequence {𝑥_𝑛} converges strongly to 𝑥 and ⇀ to indicate that the sequence {𝑥_𝑛} converges weakly 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)

For every point 𝑥∈𝐻, there exists a unique nearest point in 𝐶, denoted by 𝑃_𝐶𝑥, such that 𝑥−𝑃_𝐶𝑥≤𝑥−𝑦,∀𝑦∈𝐶. (2.2)𝑃_𝐶 is called the metric projection of 𝐻 onto 𝐶. We know that 𝑃_𝐶 is a firmly nonexpansive mapping of 𝐻 onto 𝐶; that is, there holds the following relation 𝑃_𝐶𝑥−𝑃_𝐶𝑦,𝑥−𝑦≥𝑃_𝐶𝑥−𝑃_𝐶𝑦²,∀𝑥,𝑦∈𝐻. (2.3) Consequently, 𝑃_𝐶 is nonexpansive and monotone. It is also known that 𝑃_𝐶 is characterized by the following properties: 𝑃_𝐶𝑥∈𝐶 and 𝑥−𝑃_𝐶𝑥,𝑃_𝐶𝑥−𝑦≥0,𝑥−𝑦²≥𝑥−𝑃_𝐶𝑥²+𝑦−𝑃_𝐶𝑥², (2.4)(2.5) for all 𝑥∈𝐻,𝑦∈𝐶; see [11, 37] for more details. Let 𝐴∶𝐶→𝐻 be a monotone mapping. In the context of the variational inequality, this implies that 𝑥∈VI(𝐶,𝐴)⟺𝑥=𝑃_𝐶(𝑥−𝜆𝐴𝑥),∀𝜆>0. (2.6)

It is also known that the norm of every Hilbert space 𝐻 satisfies the weak lower semicontinuity [4]. That is, for any sequence {𝑥_𝑛} with 𝑥_𝑛⇀𝑥, the inequality liminf_𝑛→∞𝑥_𝑛≥‖𝑥‖ (2.7) holds.

Recall that a set-valued mapping 𝑀∶𝐷(𝑀)⊂𝐻→2^𝐻 is called maximal monotone if 𝑀 is monotone and (𝐼+𝜆𝑀)𝐷(𝑀)=𝐻 for each 𝜆>0, where 𝐼 is the identity mapping of 𝐻. We denote by 𝐺(𝑀) the graph of 𝑀. It is known that a monotone mapping 𝑀 is maximal if and only if, for (𝑥,𝑓)∈𝐻×𝐻,⟨𝑓−𝑔,𝑥−𝑦⟩≥0 for every (𝑦,𝑔)∈𝐺(𝑀) implies 𝑓∈𝑀𝑥. Here the following example illustrates the concept of maximal monotone mappings in the setting of Hilbert spaces.

Let 𝐴∶𝐶→𝐻 be a monotone, 𝐿-Lipschitz-continuous mapping, and let 𝑁_𝐶𝑣 be the normal cone to 𝐶 at 𝑣∈𝐶, that is, 𝑁_𝐶𝑣=𝐴𝑣+𝑁_𝐶𝑣,if𝑣∈𝐶,∅,if𝑣∉𝐶. (2.8) Then, 𝑇 is maximal monotone and 0∈𝑇𝑣 if and only if 𝑣∈VI(𝐶,𝐴); see [38].

Assume that 𝑀∶𝐷(𝑀)⊂𝐻→2^𝐻 is a maximal monotone mapping. Then, for 𝜆>0, associated with 𝑀, the resolvent operator 𝐽_𝑀,𝜆 can be defined as 𝐽_𝑀,𝜆𝑥=(𝐼+𝜆𝑀)⁻¹𝑥,∀𝑥∈𝐻. (2.9) In terms of Huang [12] (see also [13]), there holds the following property for the resolvent operator 𝐽_𝑀,𝜆∶𝐻→𝐻.

Lemma 2.1. 𝐽_𝑀,𝜆 is single valued and firmly nonexpansive, that is, 𝐽_𝑀,𝜆𝑥−𝐽_𝑀,𝜆𝑦,𝑥−𝑦≥𝐽_𝑀,𝜆𝑥−𝐽_𝑀,𝜆𝑦²,∀𝑥,𝑦∈𝐻. (2.10) Consequently, 𝐽_𝑀,𝜆 is nonexpansive and monotone.

Lemma 2.2 (see [39]). There holds the relation: 𝜆𝑥+𝜇𝑦+𝜈𝑧²=𝜆‖𝑥‖²+𝜇𝑦²+𝜈‖𝑧‖²−𝜆𝜇𝑥−𝑦²−𝜇𝜈𝑦−𝑧²−𝜆𝜈‖𝑥−𝑧‖², (2.11) for all 𝑥,𝑦,𝑧∈𝐻 and 𝜆,𝜇,𝜈∈[0,1] with 𝜆+𝜇+𝜈=1.

Lemma 2.3 (see [36]). Let 𝑀 be a maximal monotone mapping with 𝐷(𝑀)=𝐶. Then for any given 𝜆>0, 𝑥^∗∈𝐶 is a solution of problem (1.7) if and only if 𝑥^∗∈𝐶 satisfies 𝑥^∗=𝐽_𝑀,𝜆𝑥^∗−𝜆Φ𝑥^∗. (2.12)

Lemma 2.4 (see [13]). Let 𝑀 be a maximal monotone mapping with 𝐷(𝑀)=𝐶, and let 𝑉∶𝐶→𝐻 be a strong monotone, continuous, and single-valued mapping. Then for each 𝑧∈𝐻, the equation 𝑧∈𝑉𝑥+𝜆𝑀𝑥 has a unique solution 𝑥_𝜆 for 𝜆>0.

Lemma 2.5 (see [36]). Let 𝑀 be a maximal monotone mapping with 𝐷(𝑀)=𝐶, and let 𝐴∶𝐶→𝐻 be a monotone, continuous, and single-valued mapping. Then (𝐼+𝜆(𝑀+𝐴))𝐶=𝐻 for each 𝜆>0. In this case, 𝑀+𝐴 is maximal monotone.

It is clear that, in a real Hilbert space 𝐻, 𝑆∶𝐶→𝐶 is 𝑘-strictly pseudo-contractive if and only if there holds the following inequality: 𝑆𝑥−𝑆𝑦,𝑥−𝑦≤𝑥−𝑦²−1−𝑘2(𝐼−𝑆)𝑥−(𝐼−𝑆)𝑦²,∀𝑥,𝑦∈𝐶. (2.13) 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 [10] and the references therein. It is well known that the class of strict pseudocontractions strictly includes the class of nonexpansive mappings.

Lemma 2.6 (see [10, Proposition 2.1]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻, and let 𝑆∶𝐶→𝐶 be a mapping.(i)If 𝑆 is a 𝑘-strict pseudo-contractive mapping, then 𝑆 satisfies the Lipschitz condition 𝑆𝑥−𝑆𝑦≤1+𝑘1−𝑘𝑥−𝑦,∀𝑥,𝑦∈𝐶. (2.14)(ii)If 𝑆 is a 𝑘-strict pseudo-contractive mapping, then the mapping 𝐼−𝑆 is semiclosed at 0; that is, if {𝑥_𝑛} is a sequence in 𝐶 such that 𝑥_𝑛→𝑥 weakly and (𝐼−𝑆)𝑥_𝑛→0 strongly, then (𝐼−𝑆)𝑥=0.(iii)If 𝑆 is 𝑘-quasistrict 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.

Lemma 2.7 (see [34]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑆∶𝐶→𝐶 be a 𝑘-strictly pseudo-contractive mapping. Let 𝛾 and 𝛿 be two nonnegative real numbers such that (𝛾+𝛿)𝑘≤𝛾. Then 𝛾𝑥−𝑦+𝛿𝑆𝑥−𝑆𝑦≤𝛾+𝛿𝑥−𝑦,∀𝑥,𝑦∈𝐶. (2.15)

The following lemma is well known to us.

Lemma 2.8 (see [11]). Every Hilbert space 𝐻 has the Kadec-Klee property; that is, for given 𝑥∈𝐻 and {𝑥_𝑛}⊂𝐻, we have 𝑥_𝑛𝑥⇀𝑥_𝑛→‖𝑥‖⇒𝑥_𝑛→𝑥. (2.16)

3. Main Results

In this section, we first prove the strong convergence of the sequences generated by our hybrid extragradient iterative algorithm for finding a common solution of a general system of variational inequalities, a variational inclusion, and a fixed problem of a strictly pseudocontractive self-mapping.

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐵_𝑖∶𝐶→𝐻 be 𝛽_𝑖-inverse strongly monotone for 𝑖=1,2, let Φ∶𝐶→𝐻 be an 𝛼-inverse strongly monotone mapping, let 𝑀 be a maximal monotone mapping with 𝐷(𝑀)=𝐶, and let 𝑆∶𝐶→𝐶 be a 𝑘-strictly pseudocontractive mapping such that Fix(𝑆)∩Ω∩Ξ≠∅. For given 𝑥₀∈𝐶 arbitrarily, let {𝑥_𝑛},{𝑦_𝑛}, and {𝑧_𝑛} be the sequences generated by 𝑦_𝑛=𝑃_𝐶𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛,𝑡_𝑛=𝑃_𝐶𝑃_𝐶𝑦_𝑛−𝜇₂𝐵₂𝑦_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑦_𝑛−𝜇₂𝐵₂𝑦_𝑛,𝑡_𝑛̂=𝜎_𝑛𝑡_𝑛+1−𝜎_𝑛𝐽_{𝑀,𝜇𝑛}𝑡_𝑛−𝜇_𝑛Φ𝑡_𝑛,𝑧_𝑛=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑡_𝑛̂+𝛿_𝑛𝑆𝑡_𝑛̂,𝐶_𝑛=𝑧∈𝐶∶𝑧_𝑛−𝑧≤𝑥_𝑛,𝑄−𝑧_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑥₀−𝑥_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀,∀𝑛≥0, (3.1) where 𝜇_𝑖∈(0,2𝛽_𝑖) for 𝑖=1,2, {𝜇_𝑛}⊂[𝜖,2𝛼] for some 𝜖∈(0,2𝛼], and {𝜎_𝑛},{𝛽_𝑛},{𝛾_𝑛},{𝛿_𝑛}⊂[0,1] such that {𝜎_𝑛}⊂[0,𝑐] for some 𝑐∈[0,1), {𝛿_𝑛}⊂[𝑑,1] for some 𝑑∈(0,1], 𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 and (𝛾_𝑛+𝛿_𝑛)𝑘≤𝛾_𝑛, for all 𝑛≥0. Then the sequences {𝑥_𝑛}, {𝑦_𝑛}, and {𝑧_𝑛} converge strongly to the same point 𝑥=𝑃_{Fix(𝑆)∩Ω∩Ξ}𝑥₀ if and only if ‖𝑆𝑡_𝑛̂−𝑡_𝑛̂‖→0. Furthermore, (𝑥,𝑦) is a solution of the GSVI (1.10), where 𝑦=𝑃_𝐶(𝑥−𝜇₂𝐵₂𝑥).

Proof. It is obvious that 𝐶_𝑛 is closed and 𝑄_𝑛 is closed and convex for every 𝑛=0,1,2,…. As 𝐶_𝑛=𝑧∈𝐶∶𝑧_𝑛−𝑥_𝑛²+2𝑧_𝑛−𝑥_𝑛,𝑥_𝑛−𝑧≤0, (3.2) we also know that 𝐶_𝑛 is convex for every 𝑛=0,1,2,…. As 𝑄_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑥−𝑥_𝑛≥0, (3.3) we have ⟨𝑥_𝑛−𝑧,𝑥−𝑥_𝑛⟩≥0, for all 𝑧∈𝑄_𝑛, and hence 𝑥_𝑛=𝑃_𝑄𝑛𝑥₀ by (2.4).
First of all, assume that the sequences {𝑥_𝑛},{𝑦_𝑛}, and {𝑧_𝑛} converge strongly to the same point 𝑥=𝑃_{Fix(𝑆)∩Ω∩Ξ}𝑥₀. Then it is clear that ‖𝑥_𝑛−𝑦_𝑛‖→0 and ‖𝑥_𝑛−𝑧_𝑛‖→0. Observe that from the nonexpansiveness of the mappings 𝑃_𝐶(𝐼−𝜇₁𝐵₁) and 𝑃_𝐶(𝐼−𝜇₂𝐵₂) (due to 𝜇_𝑖∈(0,2𝛽_𝑖) for 𝑖=1,2), we have 𝑦_𝑛−𝑡_𝑛=𝑃_𝐶𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝑃_𝐶𝑃_𝐶𝑦_𝑛−𝜇₂𝐵₂𝑦_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑦_𝑛−𝜇₂𝐵₂𝑦_𝑛=𝑃_𝐶𝐼−𝜇₁𝐵₁𝑃_𝐶𝐼−𝜇₂𝐵₂𝑥_𝑛−𝑃_𝐶𝐼−𝜇₁𝐵₁𝑃_𝐶𝐼−𝜇₂𝐵₂𝑦_𝑛≤𝑃_𝐶𝐼−𝜇₂𝐵₂𝑥_𝑛−𝑃_𝐶𝐼−𝜇₂𝐵₂𝑦_𝑛≤𝑥_𝑛−𝑦_𝑛. (3.4) Hence, we conclude that ‖𝑦_𝑛−𝑡_𝑛‖→0 and 𝑡_𝑛→𝑥. Since 𝑥∈Fix(𝑆)∩Ω∩Ξ, we obtain that 𝑆𝑥=𝑥 and 𝑥=𝐽_{𝑀,𝜇𝑛}(𝑥−𝜇_𝑛Φ(𝑥)). Thus, from the nonexpansiveness of the mapping 𝐽_{𝑀,𝜇𝑛}(𝐼−𝜇_𝑛Φ), we have 𝑡_𝑛̂−𝑡_𝑛=𝜎_𝑛𝑡_𝑛+1−𝜎_𝑛𝐽_{𝑀,𝜇𝑛}𝑡_𝑛−𝜇_𝑛Φ𝑡_𝑛−𝑡_𝑛=1−𝜎_𝑛𝐽_{𝑀,𝜇𝑛}𝑡_𝑛−𝜇_𝑛Φ𝑡_𝑛−𝑡_𝑛≤𝐽_{𝑀,𝜇𝑛}𝑡_𝑛−𝜇_𝑛Φ𝑡_𝑛−𝑥+𝑥−𝑡_𝑛=𝐽_{𝑀,𝜇𝑛}𝐼−𝜇_𝑛Φ𝑡_𝑛−𝐽_{𝑀,𝜇𝑛}𝐼−𝜇_𝑛Φ𝑥+𝑡_𝑛−𝑥≤𝑡_𝑛−𝑥+𝑡_𝑛−𝑥=2𝑡_𝑛−𝑥. (3.5) So, we deduce that ‖𝑡_𝑛̂−𝑡_𝑛‖→0 and 𝑡_𝑛̂→𝑥. Note that 𝑆𝑡_𝑛̂−𝑡_𝑛̂≤𝑆𝑡_𝑛̂−𝑥+𝑥−𝑡_𝑛̂=𝑆𝑡_𝑛̂−𝑆𝑥+𝑥−𝑡_𝑛̂≤1+𝑘̂𝑡1−𝑘+1_𝑛−𝑥=2̂𝑡1−𝑘^𝑛−𝑥. (3.6) This implies that ‖𝑆𝑡_𝑛̂−𝑡_𝑛̂‖→0 as 𝑛→∞.
For the remainder of the proof, we divide it into several steps.
Step 1. We claim that Fix(𝑆)∩Ω∩Ξ⊂𝐶_𝑛∩𝑄_𝑛 for every 𝑛=0,1,2,….
Indeed, take a fixed 𝑝∈Fix(𝑆)∩Ω∩Ξ arbitrarily. Then 𝑆𝑝=𝑝,𝐽_{𝑀,𝜇𝑛}(𝑝−𝜇_𝑛Φ(𝑝))=𝑝, for all 𝑛≥0, and 𝑝=𝑃_𝐶𝑃_𝐶𝑝−𝜇₂𝐵₂𝑝−𝜇₁𝐵₁𝑃_𝐶𝑝−𝜇₂𝐵₂𝑝. (3.7) For simplicity, we write 𝑞=𝑃_𝐶(𝑝−𝜇₂𝐵₂𝑝),𝑥_𝑛=𝑃_𝐶(𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛), and 𝑦_𝑛=𝑃_𝐶(𝑦_𝑛−𝜇₂𝐵₂𝑦_𝑛), 𝑦_𝑛=𝑃_𝐶𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛=𝑃_𝐶𝑥_𝑛−𝜇₁𝐵₁𝑥_𝑛,𝑡_𝑛=𝑃_𝐶𝑃_𝐶𝑦_𝑛−𝜇₂𝐵₂𝑦_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑦_𝑛−𝜇₂𝐵₂𝑦_𝑛=𝑃_𝐶𝑦_𝑛−𝜇₁𝐵₁𝑦_𝑛, (3.8) for each 𝑛≥0. Since 𝐵_𝑖∶𝐶→𝐻 is 𝛽_𝑖-inverse strongly monotone, and 0<𝜇_𝑖<2𝛽_𝑖 for 𝑖=1,2, we know that for all 𝑛≥0, 𝑦_𝑛−𝑝²=𝑃_𝐶𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝑝²=𝑃_𝐶𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝑃_𝐶𝑃_𝐶𝑝−𝜇₂𝐵₂𝑝−𝜇₁𝐵₁𝑃_𝐶𝑝−𝜇₂𝐵₂𝑝²≤𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝑃_𝐶𝑝−𝜇₂𝐵₂𝑝−𝜇₁𝐵₁𝑃_𝐶𝑝−𝜇₂𝐵₂𝑝²=𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝑃_𝐶𝑝−𝜇₂𝐵₂𝑝−𝜇₁𝐵₁𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝐵₁𝑃_𝐶𝑝−𝜇₂𝐵₂𝑝²≤𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝑃_𝐶𝑝−𝜇₂𝐵₂𝑝²−𝜇₁2𝛽₁−𝜇₁𝐵₁𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝐵₁𝑃_𝐶𝑝−𝜇₂𝐵₂𝑝²≤𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝑝−𝜇₂𝐵₂𝑝²−𝜇₁2𝛽₁−𝜇₁𝐵₁𝑥_𝑛−𝐵₁𝑞²=𝑥_𝑛−𝑝−𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²−𝜇₁2𝛽₁−𝜇₁𝐵₁𝑥_𝑛−𝐵₁𝑞²≤𝑥_𝑛−𝑝²−𝜇₂2𝛽₂−𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²−𝜇₁2𝛽₁−𝜇₁𝐵₁𝑥_𝑛−𝐵₁𝑞²≤𝑥_𝑛−𝑝². (3.9) Repeating the same argument, we can obtain that for all 𝑛≥0, 𝑡_𝑛−𝑝²≤𝑦_𝑛−𝑝²−𝜇₂2𝛽₂−𝜇₂𝐵₂𝑦_𝑛−𝐵₂𝑝²−𝜇₁2𝛽₁−𝜇₁𝐵₁𝑦_𝑛−𝐵₁𝑞²≤𝑦_𝑛−𝑝². (3.10) Furthermore, by Lemma 2.1 we derive from (3.9) and (3.10) 𝑡_𝑛̂−𝑝²=𝜎_𝑛𝑡_𝑛−𝑝+1−𝜎_𝑛𝐽_{𝑀,𝜇𝑛}𝑡_𝑛−𝜇_𝑛Φ𝑡_𝑛−𝑝²≤𝜎_𝑛𝑡_𝑛−𝑝²+1−𝜎_𝑛𝐽_{𝑀,𝜇𝑛}𝑡_𝑛−𝜇_𝑛Φ𝑡_𝑛−𝑝²=𝜎_𝑛𝑡_𝑛−𝑝²+1−𝜎_𝑛𝐽_{𝑀,𝜇𝑛}𝑡_𝑛−𝜇_𝑛Φ𝑡_𝑛−𝐽_{𝑀,𝜇𝑛}𝑝−𝜇_𝑛Φ𝑝²≤𝜎_𝑛𝑡_𝑛−𝑝²+1−𝜎_𝑛𝑡_𝑛−𝜇_𝑛Φ𝑡_𝑛−𝑝−𝜇_𝑛Φ𝑝²≤𝜎_𝑛𝑡_𝑛−𝑝²+1−𝜎_𝑛𝑡_𝑛−𝑝²+𝜇_𝑛𝜇_𝑛−2𝛼Φ𝑡_𝑛−Φ𝑝²≤𝑡_𝑛−𝑝²≤𝑦_𝑛−𝑝²−𝜇₂2𝛽₂−𝜇₂𝐵₂𝑦_𝑛−𝐵₂𝑝²−𝜇₁2𝛽₁−𝜇₁𝐵₁𝑦_𝑛−𝐵₁𝑞²≤𝑥_𝑛−𝑝²−𝜇₂2𝛽₂−𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²−𝜇₁2𝛽₁−𝜇₁𝐵₁𝑥_𝑛−𝐵₁𝑞²−𝜇₂2𝛽₂−𝜇₂𝐵₂𝑦_𝑛−𝐵₂𝑝²−𝜇₁2𝛽₁−𝜇₁𝐵₁𝑦_𝑛−𝐵₁𝑞²=𝑥_𝑛−𝑝²−𝜇₂2𝛽₂−𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²+𝐵₂𝑦_𝑛−𝐵₂𝑝²−𝜇₁2𝛽₁−𝜇₁𝐵₁𝑥_𝑛−𝐵₁𝑞²+𝐵₁𝑦_𝑛−𝐵₁𝑞². (3.11) Since (𝛾_𝑛+𝛿_𝑛)𝑘≤𝛾_𝑛, for all 𝑛≥0, utilizing Lemmas 2.2 and 2.7, we get from (3.11) 𝑧_𝑛−𝑝²=𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑡_𝑛̂−𝑝+𝛿_𝑛𝑆𝑡_𝑛̂−𝑝²=𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛+𝛿_𝑛1𝛾_𝑛+𝛿_𝑛𝛾_𝑛𝑡_𝑛̂−𝑝+𝛿_𝑛𝑆𝑡_𝑛̂−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛+𝛿_𝑛1𝛾_𝑛+𝛿_𝑛𝛾_𝑛𝑡_𝑛̂−𝑝+𝛿_𝑛𝑆𝑡_𝑛̂−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛+𝛿_𝑛𝑡_𝑛̂−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛+𝛿_𝑛𝑥_𝑛−𝑝²−𝜇₂2𝛽₂−𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²+𝐵₂𝑦_𝑛−𝐵₂𝑝²−𝜇₁2𝛽₁−𝜇₁𝐵₁𝑥_𝑛−𝐵₁𝑞²+𝐵₁𝑦_𝑛−𝐵₁𝑞²=𝑥_𝑛−𝑝²−𝛾_𝑛+𝛿_𝑛𝜇₂2𝛽₂−𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²+𝐵₂𝑦_𝑛−𝐵₂𝑝²+𝜇₁2𝛽₁−𝜇₁𝐵₁𝑥_𝑛−𝐵₁𝑞²+𝐵₁𝑦_𝑛−𝐵₁𝑞²≤𝑥_𝑛−𝑝², (3.12) for every 𝑛=0,1,2,…, and hence 𝑝∈𝐶_𝑛. So, Fix(𝑆)∩Ω∩Ξ⊂𝐶_𝑛 for every 𝑛=0,1,2,…. Next, let us show by mathematical induction that {𝑥_𝑛} is well defined and Fix(𝑆)∩Ω∩Ξ⊂𝐶_𝑛∩𝑄_𝑛 for every 𝑛=0,1,2,…. For 𝑛=0, we have 𝑄₀=𝐶. Hence we obtain Fix(𝑆)∩Ω∩Ξ⊂𝐶₀∩𝑄₀. Suppose that 𝑥_𝑛 is given and Fix(𝑆)∩Ω∩Ξ⊂𝐶_𝑛∩𝑄_𝑛 for some integer 𝑛≥0. Since Fix(𝑆)∩Ω∩Ξ is nonempty, 𝐶_𝑛∩𝑄_𝑛 is a nonempty closed convex subset of 𝐶. So, there exists a unique element 𝑥_𝑛+1∈𝐶_𝑛∩𝑄_𝑛 such that 𝑥_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀. It is also obvious that there holds ⟨𝑥_𝑛+1−𝑧,𝑥₀−𝑥_𝑛+1⟩≥0 for 𝑧∈Fix(𝑆)∩Ω∩Ξ, and hence Fix(𝑆)∩Ω∩Ξ⊂𝑄_𝑛+1. Therefore, we derive Fix(𝑆)∩Ω∩Ξ∩𝐶_𝑛+1∩𝑄_𝑛+1.
Step 2. We claim that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=lim_𝑛→∞𝑥_𝑛−𝑧_𝑛=0. (3.13)
Indeed, let 𝑙₀=𝑃_{Fix(𝑆)∩Ω∩Ξ}𝑥₀. From 𝑥_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀, and 𝑙₀∈Fix(𝑆)∩Ω∩Ξ⊂𝐶_𝑛∩𝑄_𝑛, we have 𝑥_𝑛+1−𝑥₀≤𝑥_𝑛+1−𝑥₀ (3.14) for every 𝑛=0,1,2,…. Therefore, {𝑥_𝑛} is bounded. From (3.9)–(3.12), we also obtain that {𝑥_𝑛}, {𝑦_𝑛}, {𝑦_𝑛}, {𝑡_𝑛}, {𝑡_𝑛̂}, and {𝑧_𝑛} all are bounded. Since 𝑥_𝑛+1∈𝐶_𝑛∩𝑄_𝑛⊂𝑄_𝑛 and 𝑥_𝑛=𝑃_𝑄𝑛𝑥₀, we have 𝑥_𝑛−𝑥₀≤𝑥_𝑛+1−𝑥₀ (3.15) for every 𝑛=0,1,2,…. Therefore, there exists lim_𝑛→∞‖𝑥_𝑛−𝑥₀‖. Since 𝑥_𝑛=𝑃_𝑄𝑛𝑥₀ and 𝑥_𝑛+1∈𝑄_𝑛, utilizing (2.5), we have 𝑥_𝑛+1−𝑥_𝑛²≤𝑥_𝑛+1−𝑥₀²−𝑥_𝑛−𝑥₀² (3.16) for every 𝑛=0,1,2,…. This implies that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (3.17) Since 𝑥_𝑛+1∈𝐶_𝑛, we have ‖𝑧_𝑛−𝑥_𝑛+1‖≤‖𝑥_𝑛−𝑥_𝑛+1‖, and hence 𝑥_𝑛−𝑧_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑧_𝑛≤2𝑥_𝑛+1−𝑥_𝑛 (3.18) for every 𝑛=0,1,2,…. From ‖𝑥_𝑛+1−𝑥_𝑛‖→0 it follows that lim_𝑛→∞𝑥_𝑛−𝑧_𝑛=0. (3.19)
Step 3. We claim that lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=lim_𝑛→∞𝑥_𝑛−𝑡_𝑛=lim_𝑛→∞𝑡_𝑛̂−𝑡_𝑛=0. (3.20) Indeed, for 𝑝∈Fix(𝑆)∩Ω∩Ξ, we obtain from (3.12) 𝑧_𝑛−𝑝²≤𝑥_𝑛−𝑝²−𝛾_𝑛+𝛿_𝑛𝜇₂2𝛽₂−𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²+𝐵₂𝑦_𝑛−𝐵₂𝑝²+𝜇₁2𝛽₁−𝜇₁𝐵₁𝑥_𝑛−𝐵₁𝑞²+𝐵₁𝑦_𝑛−𝐵₁𝑞². (3.21) Therefore, we have 𝛾_𝑛+𝛿_𝑛𝜇₂2𝛽₂−𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²+𝐵₂𝑦_𝑛−𝐵₂𝑝²+𝜇₁2𝛽₁−𝜇₁𝐵₁𝑥_𝑛−𝐵₁𝑞²+𝐵₁𝑦_𝑛−𝐵₁𝑞²≤𝑥_𝑛−𝑝²−𝑧_𝑛−𝑝²=𝑥_𝑛−𝑝−𝑧_𝑛−𝑝𝑥_𝑛−𝑝+𝑧_𝑛≤𝑥−𝑝_𝑛−𝑧_𝑛𝑥_𝑛−𝑝+𝑧_𝑛−𝑝. (3.22) Since {𝛿_𝑛}⊂[𝑑,1] for some 𝑑∈(0,1],‖𝑥_𝑛−𝑧_𝑛‖→0, and the sequences {𝑥_𝑛} and {𝑧_𝑛} are bounded, we deduce that lim_𝑛→∞𝐵₂𝑥_𝑛−𝐵₂𝑝=lim_𝑛→∞𝐵₂𝑦_𝑛−𝐵₂𝑝=lim_𝑛→∞𝐵₁𝑥_𝑛−𝐵₁𝑞=lim_𝑛→∞𝐵₁𝑦_𝑛−𝐵₁𝑞=0. (3.23) On the other hand, by firm nonexpansiveness of 𝑃_𝐶, we have 𝑥_𝑛−𝑞²=𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝑃_𝐶𝑝−𝜇₂𝐵₂𝑝²≤𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝑝−𝜇₂𝐵₂𝑝,𝑥_𝑛=1−𝑞2𝑥_𝑛−𝑝−𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²+𝑥_𝑛−𝑞²−𝑥_𝑛−𝑝−𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝−𝑥_𝑛−𝑞²≤12𝑥_𝑛−𝑝²+𝑥_𝑛−𝑞²−𝑥_𝑛−𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝−𝑝−𝑞²=12𝑥_𝑛−𝑝²+𝑥_𝑛−𝑞²−𝑥_𝑛−𝑥_𝑛−𝑝−𝑞²+2𝜇₂𝑥_𝑛−𝑥_𝑛−𝑝−𝑞,𝐵₂𝑥_𝑛−𝐵₂𝑝−𝜇²₂𝐵₂𝑥_𝑛−𝐵₂𝑝²≤12𝑥_𝑛−𝑝²+𝑥_𝑛−𝑞²−𝑥_𝑛−𝑥_𝑛−𝑝−𝑞²+2𝜇₂𝑥_𝑛−𝑥_𝑛−𝑝−𝑞𝐵₂𝑥_𝑛−𝐵₂𝑝; (3.24) that is, 𝑥_𝑛−𝑞²≤𝑥_𝑛−𝑝²−𝑥_𝑛−𝑥_𝑛−𝑝−𝑞²+2𝜇₂𝑥_𝑛−𝑥_𝑛−𝑝−𝑞𝐵₂𝑥_𝑛−𝐵₂𝑝. (3.25) Repeating the same argument, we can also obtain 𝑦_𝑛−𝑞²≤𝑦_𝑛−𝑝²−𝑦_𝑛−𝑦_𝑛−𝑝−𝑞²+2𝜇₂𝑦_𝑛−𝑦_𝑛−𝑝−𝑞𝐵₂𝑦_𝑛−𝐵₂𝑝. (3.26) Moreover, using the argument technique similar to the above one, we derive 𝑦_𝑛−𝑝²=𝑃_𝐶𝑥_𝑛−𝜇₁𝐵₁𝑥_𝑛−𝑃_𝐶𝑞−𝜇₁𝐵₁𝑞²≤𝑥_𝑛−𝜇₁𝐵₁𝑥_𝑛−𝑞−𝜇₁𝐵₁𝑞,𝑦_𝑛=1−𝑝2𝑥_𝑛−𝑞−𝜇₁𝐵₁𝑥_𝑛−𝐵₁𝑞²+𝑦_𝑛−𝑝²−𝑥_𝑛−𝑞−𝜇₁𝐵₁𝑥_𝑛−𝐵₁𝑞−𝑦_𝑛−𝑝²≤12𝑥_𝑛−𝑞²+𝑦_𝑛−𝑝²−𝑥_𝑛−𝑦_𝑛−𝜇₁𝐵₁𝑥_𝑛−𝐵₁𝑞+𝑝−𝑞²=12𝑥_𝑛−𝑞²+𝑦_𝑛−𝑝²−𝑥_𝑛−𝑦_𝑛+𝑝−𝑞²+2𝜇₁𝑥_𝑛−𝑦_𝑛+𝑝−𝑞,𝐵₁𝑥_𝑛−𝐵₁𝑞−𝜇²₁𝐵₁𝑥_𝑛−𝐵₁𝑞²≤12𝑥_𝑛−𝑞²+𝑦_𝑛−𝑝²−𝑥_𝑛−𝑦_𝑛+𝑝−𝑞²+2𝜇₁𝑥_𝑛−𝑦_𝑛+𝑝−𝑞𝐵₁𝑥_𝑛−𝐵₁𝑞; (3.27) that is, 𝑦_𝑛−𝑝²≤𝑥_𝑛−𝑞²−𝑥_𝑛−𝑦_𝑛+𝑝−𝑞²+2𝜇₁𝑥_𝑛−𝑦_𝑛+𝑝−𝑞𝐵₁𝑥_𝑛−𝐵₁𝑞. (3.28) Repeating the same argument, we can also obtain 𝑡_𝑛−𝑝²≤𝑦_𝑛−𝑞²−𝑦_𝑛−𝑡_𝑛+𝑝−𝑞²+2𝜇₁𝑦_𝑛−𝑡_𝑛+𝑝−𝑞𝐵₁𝑦_𝑛−𝐵₁𝑞. (3.29) Utilizing (3.11), (3.25)–(3.29), we have 𝑧_𝑛−𝑝²=𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑡_𝑛̂−𝑝+𝛿_𝑛𝑆𝑡_𝑛̂−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛+𝛿_𝑛𝑡_𝑛̂−𝑝²=𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛+𝛿_𝑛𝑡_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛+𝛿_𝑛𝑦_𝑛−𝑞²−𝑦_𝑛−𝑡_𝑛+𝑝−𝑞²+2𝜇₁𝑦_𝑛−𝑡_𝑛+𝑝−𝑞𝐵₁𝑦_𝑛−𝐵₁𝑞≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛+𝛿_𝑛𝑦_𝑛−𝑝²−𝑦_𝑛−𝑦_𝑛−𝑝−𝑞²+2𝜇₂𝑦_𝑛−𝑦_𝑛−𝑝−𝑞𝐵₂𝑦_𝑛−𝐵₂𝑝−𝑦_𝑛−𝑡_𝑛+𝑝−𝑞²+2𝜇₁𝑦_𝑛−𝑡_𝑛+𝑝−𝑞𝐵₁𝑦_𝑛−𝐵₁𝑞≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛+𝛿