Strong Convergence Theorems for Family of Nonexpansive Mappings and System of Generalized Mixed Equilibrium Problems and Variational Inequality Problems

Abstract

We introduce a new iterative scheme by hybrid method for finding a common element of the set of common fixed points of infinite family of nonexpansive mappings, the set of common solutions to a system of generalized mixed equilibrium problems, and the set of solutions to a variational inequality problem in a real Hilbert space. We then prove strong convergence of the scheme to a common element of the three sets. We give some applications of our results. Our results extend important recent results.

1. Introduction

Let 𝐾 be a nonempty closed and convex subset of a real Hilbert space 𝐻. A mapping 𝐴∶𝐾→𝐻 is called monotone if𝐴𝑥−𝐴𝑦,𝑥−𝑦≥0,∀𝑥,𝑦∈𝐾. (1.1) The variational inequality problem is to find an 𝑥^∗∈𝐾 such that𝑦−𝑥^∗,𝐴𝑥^∗≥0,∀𝑦∈𝐾 (1.2) (see, e.g., [1]). We will denote the set of solutions to the variational inequality problem (1.2) by VI(𝐾,𝐴).

A mapping 𝐴∶𝐾→𝐻 is called inverse-strongly monotone (see, e.g., [2, 3]) if there exists a positive real number 𝛼 such that ⟨𝐴𝑥−𝐴𝑦,𝑥−𝑦⟩≥𝛼‖𝐴𝑥−𝐴𝑦‖², for all 𝑥,𝑦∈𝐾. For such a case, 𝐴 is called 𝛼-inverse-strongly monotone.

A mapping 𝑇∶𝐾→𝐾 is said to be nonexpansive if𝑇𝑥−𝑇𝑦≤𝑥−𝑦, (1.3) for all 𝑥,𝑦∈𝐾. A point 𝑥∈𝐾 is called a fixed point of 𝑇 if 𝑇𝑥=𝑥. The set of fixed points of 𝑇 is the set 𝐹(𝑇)∶={𝑥∈𝐾∶𝑇𝑥=𝑥}.

Let 𝜑∶𝐾→ℝ be a real-valued function and 𝐴∶𝐾→𝐻 a nonlinear mapping. Suppose that 𝐹∶𝐾×𝐾 into ℝ is an equilibrium bifunction. That is, 𝐹(𝑢,𝑢)=0, for all 𝑢∈𝐾. The generalized mixed equilibrium problem is to find 𝑥∈𝐾 (see, e.g., [4–6]) such that𝐹𝑥,𝑦+𝜑𝑦−𝜑(𝑥)+𝐴𝑥,𝑦−𝑥≥0, (1.4) for all 𝑦∈𝐾. We will denote the set of solutions of this generalized mixed equilibrium problem by GMEP(𝐹,𝐴,𝜑). Thus GMEP𝐹,𝐴,𝜑∶=𝑥^∗∈𝐾∶𝐹𝑥^∗,𝑦+𝜑𝑦−𝜑𝑥^∗+𝐴𝑥^∗,𝑦−𝑥^∗≥0,∀𝑦∈𝐾. (1.5) If 𝜑=0, 𝐴=0, then problem (1.4) reduces to equilibrium problem studied by many authors (see, e.g., [7–14]), which is to find 𝑥^∗∈𝐾 such that𝐹𝑥^∗,𝑦≥0, (1.6) for all 𝑦∈𝐾. The set of solutions of (1.6) is denoted by EP(𝐹).

If 𝜑=0, then problem (1.4) reduces to generalized equilibrium problem studied by many authors (see, e.g., [15–18]), which is to find 𝑥^∗∈𝐾 such that𝐹𝑥^∗,𝑦+𝐴𝑥^∗,𝑦−𝑥^∗≥0, (1.7) for all 𝑦∈𝐾. The set of solutions of (1.7) is denoted by GEP(𝐹,𝐴).

If 𝐴=0, then problem (1.4) reduces to mixed equilibrium problem considered by many authors (see, e.g., [19–21]), which is to find 𝑥^∗∈𝐾 such that𝐹𝑥^∗,𝑦+𝜑𝑦−𝜑𝑥^∗≥0, (1.8) for all 𝑦∈𝐾. The set of solutions of (1.8) is denoted by MEP(𝐹,𝜑).

The generalized mixed equilibrium problems include fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and equilibrium problems as special cases (see, e.g., [22]). Numerous problems in physics, optimization, and economics reduce to find a solution of problem (1.4). Several methods have been proposed to solve the fixed point problems, variational inequality problems, and equilibrium problems in the literature. See, for example, [23–33].

One of the iterative processes (see Halpern [34]) which is often used to approximate a fixed point of a nonexpansive mapping 𝑇 is defined as follows. Take an initial guess 𝑥₀∈𝐾 arbitrarily and define {𝑥_𝑛} recursively by𝑥_𝑛+1=𝛼_𝑛𝑥₀+1−𝛼_𝑛𝑇𝑥_𝑛,𝑛≥0, (1.9) where {𝛼_𝑛} is a sequence in [0, 1]. The iteration process (1.9) has been proved to be strongly convergent both in Hilbert spaces [34–36] and uniformly smooth Banach spaces [37, 38] when the sequence {𝛼_𝑛} satisfies the conditions(i)lim_𝑛→∞𝛼_𝑛=0,(ii)∑^∞_𝑛=1𝛼_𝑛=∞,(iii)either ∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞ or lim_𝑛→∞𝛼_𝑛/𝛼_𝑛+1=1.

Motivated by (1.9), Martinez-Yanes and Xu [39] introduced the following iterative scheme for a single nonexpansive mapping 𝑇 in a Hilbert space:𝑥₀𝑦∈𝐾,_𝑛=𝛼_𝑛𝑥₀+1−𝛼_𝑛𝑇𝑥_𝑛,𝐶_𝑛=𝑧∈𝐾∶𝑦_𝑛−𝑧²≤𝑥_𝑛−𝑧²+𝛼_𝑛𝑥₀²+2𝑥_𝑛−𝑥₀,𝑄,𝑧_𝑛=𝑧∈𝐾∶𝑥_𝑛−𝑧,𝑥_𝑛−𝑥₀,𝑥≤0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀, (1.10) where 𝑃_𝐾 denotes the metric projection of 𝐻 onto a closed and convex subset 𝐾 of 𝐻. They proved that if {𝛼_𝑛}⊂(0,1) and lim_𝑛→∞𝛼_𝑛=0, then the sequence {𝑥_𝑛} converges strongly to 𝑃_𝐹(𝑇)𝑥₀.

Furthermore, algorithm (1.10) has been modified by many authors for relatively nonexpansive mappings and quasi-𝜙-nonexpansive mappings in Banach spaces (see, e.g., [40–43]).

Recently, Ceng and Yao [44] introduced a new iterative scheme of approximating a common element of the set of solutions to mixed equilibrium problem and set of common fixed points of finite family of nonexpansive mappings in a real Hilbert space 𝐻. In the proof process of their results, they imposed the following condition on a nonempty closed and convex subset 𝐾 of 𝐻:(E)   𝐴∶𝐾→ℝ is 𝜂-strongly convex and its derivative 𝐴^′ is sequentially continuous from weak topology to the strong topology.

We remark here that this condition (E) has been used by many authors for approximation of solution to mixed equilibrium problem in a real Hilbert space (see, e.g., [45, 46]). However, it is observed that condition (E) does not include the case 𝐴(𝑥)=‖𝑥‖²/2 and 𝜂(𝑥,𝑦)=𝑥−𝑦. Furthermore, Peng and Yao [19], Wangkeeree and Wangkeeree [47], and other authors replaced condition (E) with these conditions:(B1)  for each 𝑥∈𝐻 and 𝑟>0, there exist a bounded subset 𝐷_𝑥⊆𝐾 and 𝑦_𝑥∈𝐾 such that, for any 𝑧∈𝐾⧵𝐷_𝑥, 𝐹𝑧,𝑦_𝑥+𝜑𝑦_𝑥(𝑧)+1−𝜑𝑟𝑦_𝑥−𝑧,𝑧−𝑥<0, (1.11) or(B2)  𝐾 is a bounded set.

Consequently, conditions (B1) and (B2) have been used by many authors in approximating solution to generalized mixed equilibrium (mixed equilibrium) problems in a real Hilbert space (see, e.g., [19, 47]).

In [48], Takahashi et al. proved the following convergence theorem using hybrid method.

Theorem 1.1 (Takahashi et al. [48]). Let 𝐾 be a nonempty closed and convex subset of a real Hilbert space 𝐻. Let 𝑇 be a nonexpansive mapping of 𝐾 into itself such that 𝐹(𝑇)≠∅. For 𝐶₁=𝐾, 𝑥₁=𝑃_𝐶1𝑥₀, define sequences {𝑥_𝑛}^∞_𝑛=0 and {𝑦_𝑛}^∞_𝑛=1 of 𝐾 as follows: 𝑦_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛𝐶,𝑛≥1,_𝑛+1=𝑧∈𝐶_𝑛∶𝑦_𝑛−𝑧≤𝑥_𝑛𝑥−𝑧,𝑛≥1,_𝑛+1=𝑃_𝐶𝑛+1𝑥₀,𝑛≥1. (1.12) Assume that {𝛼_𝑛}^∞_𝑛=1⊂[0,1) satisfies 0≤𝛼_𝑛<𝛼<1. Then, {𝑥_𝑛}^∞_𝑛=0 converges strongly to 𝑃_𝐹(𝑇)𝑥₀.

Motivated by the results of Takahashi et al. [48], Kumam [49] studied the problem of approximating a common element of set of solutions to an equilibrium problem, set of solutions to variational inequality problem, and set of fixed points of a nonexpansive mapping in a real Hilbert space. In particular, he proved the following theorem.

Theorem 1.2 (Kumam [49]). Let 𝐾 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐹 be a bifunction from 𝐾×𝐾 satisfying (A1)–(A4), and let 𝐵 be a 𝛽-inverse-strongly monotone mapping of 𝐾 into 𝐻. Let 𝑇 be a nonexpansive mapping of 𝐾 into 𝐻 such that 𝐹(𝑇)∩EP(𝐹)∩VI(𝐾,𝐵)≠∅. For 𝐶₁=𝐾, 𝑥₁=𝑃_𝐶1𝑥₀, define sequences {𝑥_𝑛}^∞_𝑛=0 and {𝑧_𝑛}^∞_𝑛=1 of 𝐾 as follows: 𝐹𝑧_𝑛+1,𝑦𝑟_𝑛𝑦−𝑧_𝑛,𝑧_𝑛−𝑥_𝑛𝑦≥0,∀𝑦∈𝐾,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑃_𝐾𝑧_𝑛−𝜆_𝑛𝐵𝑧_𝑛𝐶,𝑛≥1,_𝑛+1=𝑧∈𝐶_𝑛∶𝑦_𝑛−𝑧≤𝑥_𝑛𝑥−𝑧,𝑛≥1,_𝑛+1=𝑃_𝐶𝑛+1𝑥₀,𝑛≥1. (1.13) Assume that {𝛼_𝑛}^∞_𝑛=1⊂[0,1), {𝑟_𝑛}^∞_𝑛=1⊂(0,∞), and {𝜆_𝑛}^∞_𝑛=1⊂[0,2𝛽] satisfy liminf_𝑛→∞𝑟_𝑛>0,0<𝑐≤𝜆_𝑛≤𝑓<2𝛽,lim_𝑛→∞𝛼_𝑛=0. (1.14) Then, {𝑥_𝑛}^∞_𝑛=0 converges strongly to 𝑃_{𝐹(𝑇)∩EP(𝐹)∩VI(𝐾,𝐵)}𝑥₀.

Quite recently, Chantarangsi et al. [50] proved the following convergence theorem for approximation of fixed point of a nonexpansive mapping which is also a common solution to a system of generalized mixed equilibrium problems and variational inequality problem in a real Hilbert space.

Theorem 1.3 (Chantarangsi et al. [50]). Let 𝐾 be a nonempty closed and convex subset of a real Hilbert space 𝐻. Let 𝜃₁, 𝜃₂ be bifunctions from 𝐾×𝐾 satisfying (A1)–(A4), Ψ₁ an 𝜉-inverse-strongly monotone mapping of 𝐾 into 𝐻, Ψ₂ a 𝛽-inverse-strongly monotone mapping of 𝐾 into 𝐻 with assumption (B1) or (B2), and 𝑇 : 𝐾→𝐾 a nonexpansive mapping. Let 𝐵 be an 𝜔-Lipschitz continuous and relaxed (𝜐,𝜈) co-coercive mapping of 𝐾 into 𝐻, 𝑓∶𝐾→𝐾 a contraction mapping with coefficient 𝜂∈(0,1), and 𝐴 a strongly positive linear bounded selfadjoint operator with coefficient 𝛾>0 and 0<𝛾<𝛾/𝜂. Suppose that 𝐹∶=𝐹(𝑇)∩GMEP(𝜃₁,𝜑,Ψ₁)∩GMEP(𝜃₂,𝜑,Ψ₂)∩VI(𝐾,𝐵)≠∅. Let {𝑧_𝑛}^∞_𝑛=1, {𝑢_𝑛}^∞_𝑛=1, {𝑦_𝑛}^∞_𝑛=1, and {𝑥_𝑛}^∞_𝑛=1 be generated by 𝑢_𝑛=𝑇^(𝜃1_𝑟^,𝜑)_𝑛𝑥_𝑛−𝑟_𝑛Ψ₁𝑥_𝑛,𝜈_𝑛=𝑇^(𝜃2_𝑠^,𝜑)_𝑛𝑢_𝑛−𝑠_𝑛Ψ₁𝑢_𝑛,𝑧_𝑛=𝑃_𝐾𝜈_𝑛−𝛼_𝑛𝐵𝑆𝜈_𝑛,𝑦_𝑛=𝜀_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑧_𝑛,𝑥_𝑛+1=𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝑦_𝑛,∀𝑛≥1, (1.15) where {𝑟_𝑛}⊂[𝑎,𝑏]⊂[0,2𝜉], {𝑠_𝑛}⊂[𝑐,𝑑]⊂[0,2𝛽], {𝛾_𝑛}⊂[ℎ,𝑗]⊂(0,1), {𝛾_𝑛}, {𝜀_𝑛}, {𝛽_𝑛} are three sequences in (0, 1) satisfying the following conditions:(C1)  lim_𝑛→∞𝜀_𝑛=0 and ∑^∞_𝑛=1𝜀_𝑛=∞,(C2)  0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1,(C3)  0<liminf_𝑛→∞𝑟_𝑛≤limsup_𝑛→∞𝑟_𝑛<2𝜉 and lim_𝑛→∞|𝑟_𝑛+1−𝑟_𝑛|=0,(C4)  0<liminf_𝑛→∞𝑠_𝑛≤limsup_𝑛→∞𝑠_𝑛<2𝛽 and lim_𝑛→∞|𝑠_𝑛+1−𝑠_𝑛|=0,(C5)  {𝛼_𝑛}⊂[𝑒,𝑔]⊂(0,(2(𝜈−𝜐𝜔²))/𝜔²),𝜈>𝜐𝜔² and lim_𝑛→∞|𝛼_n+1−𝛼_𝑛|=0.Then, {𝑥_𝑛} converges strongly to 𝑧=𝑃_𝐹(𝛾𝑓+(𝐼−𝐴))(𝑧).

Motivated by the ongoing research and the above-mentioned results, we modify algorithm (1.10) and introduce a new iterative scheme for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings, the set of common solutions to a system of generalized mixed equilibrium problems, and the set of solutions to a variational inequality problem in a real Hilbert space. Furthermore, we show that our new iterative scheme converges strongly to a common element of the three sets. In the proof process of our results, we use conditions (B1) and (B2) mentioned above. Our result extends many important recent results. Finally, we give some applications of our results.

2. Preliminaries

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖, and let 𝐾 be a nonempty closed and convex subset of 𝐻. The strong convergence of {𝑥_𝑛}^∞_𝑛=1 to 𝑥 is written 𝑥_𝑛→𝑥 as 𝑛→∞.

For any point 𝑢∈𝐻, there exists a unique point 𝑃_𝐾𝑢∈𝐾 such that𝑢−𝑃_𝐾𝑢≤𝑢−𝑦,∀𝑦∈𝐾. (2.1)𝑃_𝐾 is called the metric projection of 𝐻 onto 𝐾. We know that 𝑃_𝐾 is a nonexpansive mapping of 𝐻 onto 𝐾. It is also known that 𝑃_𝐾 satisfies𝑥−𝑦,𝑃_𝐾𝑥−𝑃_𝐾𝑦≥𝑃_𝐾𝑥−𝑃_𝐾𝑦², (2.2) for all 𝑥,𝑦∈𝐻. Furthermore, 𝑃_𝐾𝑥 is characterized by the properties 𝑃_𝐾𝑥∈𝐾 and𝑥−𝑃_𝐾𝑥,𝑃_𝐾𝑥−𝑦≥0, (2.3) for all 𝑦∈𝐾 and𝑥−𝑃_𝐾𝑥²≤𝑥−𝑦²−𝑦−𝑃_𝐾𝑥²,∀𝑥∈𝐻,𝑦∈𝐾. (2.4) In the context of the variational inequality problem, (2.3) implies that 𝑥^∗∈VI(𝐾,𝐴)⟺𝑥^∗=𝑃_𝐾𝑥^∗−𝜆𝐴𝑥^∗,∀𝜆≥0. (2.5) If 𝐴 is an 𝛼-inverse-strongly monotone mapping of 𝐾 into 𝐻, then it is obvious that 𝐴 is a (1/𝛼)-Lipschitz continuous. We also have that, for all 𝑥,𝑦∈𝐾 and 𝑟>0,(𝐼−𝑟𝐴)𝑥−(𝐼−𝑟𝐴)𝑦²=𝑥−𝑦−𝑟𝐴𝑥−𝐴𝑦²=𝑥−𝑦²−2𝑟𝐴𝑥−𝐴𝑦,𝑥−𝑦+𝑟²𝐴𝑥−𝐴𝑦²≤𝑥−𝑦²+𝑟(𝑟−2𝛼)𝐴𝑥−𝐴𝑦². (2.6) So, if 𝑟≤2𝛼, then 𝐼−𝑟𝐴 is a nonexpansive mapping of 𝐾 into 𝐻.

For solving the generalized mixed equilibrium problem for a bifunction 𝐹 : 𝐾×𝐾→ℝ, let us assume that 𝐹, 𝜑, and 𝐾 satisfy the following conditions:(A1)𝐹(𝑥,𝑥)=0 for all 𝑥∈𝐾,(A2)𝐹 is monotone, that is, 𝐹(𝑥,𝑦)+𝐹(𝑦,𝑥)≤0 for all 𝑥,𝑦,∈𝐾,(A3) for each 𝑥,𝑦,𝑧∈𝐾, lim_𝑡→0𝐹(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝐹(𝑥,𝑦),(A4) for each 𝑥∈𝐾, 𝑦↦𝐹(𝑥,𝑦) is convex and lower semicontinuous,(B1) for each 𝑥∈𝐻 and 𝑟>0 there exist a bounded subset 𝐷_𝑥⊆𝐾 and 𝑦_𝑥∈𝐾 such that for any 𝑧∈𝐾⧵𝐷_𝑥,𝐹𝑧,𝑦_𝑥+𝜑𝑦_𝑥(𝑧)+1−𝜑𝑟𝑦_𝑥−𝑧,𝑧−𝑥<0, (2.7)(B2)𝐾 is a bounded set.

Then, we have the following lemma.

Lemma 2.1 (Wangkeeree and Wangkeeree [47]). Assume that 𝐹∶𝐾×𝐾→ℝ satisfies (A1)–(A4), and let 𝜑∶𝐾→ℝ be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For 𝑟>0 and 𝑥∈𝐻, define a mapping 𝑇_𝑟^(𝐹,𝜑)∶𝐻→𝐾 as follows: 𝑇_𝑟^(𝐹,𝜑)(𝑥)=𝑦(𝑧)+1𝑧∈𝐾∶𝐹𝑧,𝑦+𝜑−𝜑𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐾, (2.8) for all 𝑧∈𝐻. Then, the following hold:(1)for each 𝑥∈𝐻, 𝑇_𝑟^(𝐹,𝜑)≠∅,(2)𝑇_𝑟^(𝐹,𝜑) is single-valued,(3)𝑇_𝑟^(𝐹,𝜑) is firmly nonexpansive, that is, for any 𝑥,𝑦∈𝐻, 𝑇_𝑟^(𝐹,𝜑)𝑥−𝑇_𝑟^(𝐹,𝜑)𝑦²≤𝑇_𝑟^(𝐹,𝜑)𝑥−𝑇_𝑟^(𝐹,𝜑)𝑦,𝑥−𝑦, (2.9)(4)𝐹(𝑇_𝑟^(𝐹,𝜑))=GMEP(𝐹),(5)GMEP(𝐹) is closed and convex.

We will also use the following lemma in our results.

Lemma 2.2 (Baillon and Haddad [51]). Let 𝐸 be a Banach space, let 𝑓 be a continuously Fréchet differentiable convex functional on 𝐸, and let ∇𝑓 be the gradient of 𝑓. If ∇𝑓 is (1/𝛼)-Lipschitz continuous, then ∇𝑓 is 𝛼-inverse-strongly monotone.

3. Main Results

Theorem 3.1. Let 𝐾 be a nonempty closed and convex subset of a real Hilbert space 𝐻. For each 𝑚=1,2, let 𝐹_𝑚 be a bifunction from 𝐾×𝐾 satisfying (A1)–(A4), 𝜑_𝑚 : 𝐾→ℝ∪{+∞} a proper lower semicontinuous and convex function with assumption (B1) or (B2), 𝐴 be an 𝛼-inverse-strongly monotone mapping of 𝐾 into 𝐻, and 𝐵 a 𝛽-inverse-strongly monotone mapping of 𝐾 into 𝐻, and, for each 𝑖=1,2,…, let 𝑇_𝑖∶𝐾→𝐾 be a nonexpansive mapping such that ⋂^∞_𝑖=1𝐹(𝑇_𝑖)≠∅. Let 𝐷 be a 𝛾-inverse-strongly monotone mapping of 𝐾 into 𝐻. Suppose that 𝐹∶=⋂^∞_𝑖=1𝐹(𝑇_𝑖)∩GMEP(𝐹₁,𝐴,𝜑₁)∩GMEP(𝐹₂,𝐵,𝜑₂)∩VI(𝐾,𝐷)≠∅. Let {𝑧_𝑛}^∞_𝑛=1, {𝑢_𝑛}^∞_𝑛=1, {𝑤_𝑛}^∞_𝑛=1, {𝑦_𝑛,𝑖}^∞_𝑛=1 (𝑖=1,2,…), and {𝑥_𝑛}^∞_𝑛=0 be generated by 𝑥₀∈𝐾, 𝐶_1,𝑖=𝐾, 𝐶₁=⋂^∞_𝑖=1𝐶_1,𝑖, 𝑥₁=𝑃_𝐶1𝑥₀, 𝑧_𝑛=𝑇^{(𝐹1,𝜑1)}_𝑟𝑛𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛,𝑢_𝑛=𝑇^{(𝐹2,𝜑2)}_𝜆𝑛𝑧_𝑛−𝜆_𝑛𝐵𝑧_𝑛,𝑤_𝑛=𝑃_𝐾𝑢_𝑛−𝑠_𝑛𝐷𝑢_𝑛,𝑦_𝑛,𝑖=𝛼_𝑛,𝑖𝑥₀+1−𝛼_𝑛,𝑖𝑇_𝑖𝑤_𝑛,𝐶_𝑛+1,𝑖=𝑧∈𝐶_𝑛,𝑖∶𝑦_𝑛,𝑖−𝑧²≤𝑥_𝑛−𝑧²+𝛼_𝑛,𝑖𝑥₀²+2𝑥_𝑛−𝑥₀,𝐶,𝑧_𝑛+1=^∞_𝑖=1𝐶_𝑛+1,𝑖,𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥₀,𝑛≥1. (3.1) Assume that {𝛼_𝑛,𝑖}^∞_𝑛=1⊂(0,1) (𝑖=1,2,…), {𝑟_𝑛}^∞_𝑛=1⊂[0,2𝛼], and {𝜆_𝑛}^∞_𝑛=1⊂[0,2𝛽] satisfy(i)0<𝑎≤𝑟_𝑛≤𝑏<2𝛼,(ii)0<𝑐≤𝜆_𝑛≤𝑓<2𝛽,(iii)lim_𝑛→∞𝛼_𝑛,𝑖=0,(iv)0<ℎ≤𝑠_𝑛≤𝑗<2𝛾. Then, {𝑥_𝑛}^∞_𝑛=0 converges strongly to 𝑃_𝐹𝑥₀.

Proof. Let 𝑥^∗∈𝐹. Then, 𝑤_𝑛−𝑥^∗2=𝑃_𝐾𝑢_𝑛−𝑠_𝑛𝐷𝑢_𝑛−𝑃_𝐾𝑥^∗−𝑠_𝑛𝐷𝑥^∗2≤𝑢_𝑛−𝑠_𝑛𝐷𝑢_𝑛−𝑥^∗−𝑠_𝑛𝐷𝑥^∗2≤𝑢_𝑛−𝑥^∗2+𝑠_𝑛𝑠_𝑛−2𝛾𝐷𝑢_𝑛−𝐷𝑥^∗2≤𝑢_𝑛−𝑥^∗2. (3.2) Since both 𝐼−𝑟_𝑛𝐴 and 𝐼−𝜆_𝑛𝐵 are nonexpansive for each 𝑛≥1 and 𝑥^∗=𝑇^{(𝐹1,𝜑1)}_𝑟𝑛(𝑥^∗−𝑟_𝑛𝐴𝑥^∗), 𝑥^∗=𝑇^{(𝐹2,𝜑2)}_𝜆𝑛(𝑥^∗−𝜆_𝑛𝐵𝑥^∗), from (2.6), we have that 𝑢_𝑛−𝑥^∗2=𝑇^{(𝐹2,𝜑2)}_𝜆𝑛𝑧_𝑛−𝜆_𝑛𝐵𝑧_𝑛−𝑥^∗2=𝑇^{(𝐹2,𝜑2)}_𝜆𝑛𝑧_𝑛−𝜆_𝑛𝐵𝑧_𝑛−𝑇^{(𝐹2,𝜑2)}_𝜆𝑛𝑥^∗−𝜆_𝑛𝐵𝑥^∗2≤𝐼−𝜆_𝑛𝐵𝑧_𝑛−𝐼−𝜆_𝑛𝐵𝑥^∗2≤𝑧_𝑛−𝑥^∗2+𝜆_𝑛𝜆_𝑛−2𝛽𝐵𝑧_𝑛−𝐵𝑥^∗2≤𝑧_𝑛−𝑥^∗2since𝜆_𝑛,𝑧<2𝛽,∀𝑛≥1_𝑛−𝑥^∗2=𝑇^{(𝐹1,𝜑1)}_𝑟𝑛𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛−𝑥^∗2=𝑇^{(𝐹1,𝜑1)}_𝑟𝑛𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛−𝑇^{(𝐹1,𝜑1)}_𝑟𝑛𝑥^∗−𝑟_𝑛𝐴𝑥^∗2≤𝐼−𝑟_𝑛𝐴𝑥_𝑛−𝐼−𝑟_𝑛𝐴𝑥^∗2≤𝑥_𝑛−𝑥^∗2+𝑟_𝑛𝑟_𝑛−2𝛼𝐴𝑥_𝑛−𝐴𝑥^∗2≤𝑥_𝑛−𝑥^∗2. (3.3) Therefore, 𝑢_𝑛−𝑥^∗≤𝑥_𝑛−𝑥^∗. (3.4) Let 𝑛=1, then 𝐶_1,𝑖=𝐾 is closed convex for each 𝑖=1,2,…. Now assume that 𝐶_𝑛,𝑖 is closed convex for some 𝑛>1. Then, from definition of 𝐶_𝑛+1,𝑖, we know that 𝐶_𝑛+1,𝑖 is closed convex for the same 𝑛>1. Hence, 𝐶_𝑛,𝑖 is closed convex for 𝑛≥1 and for each 𝑖=1,2,…. This implies that 𝐶_𝑛 is closed convex for 𝑛≥1. Furthermore, we show that 𝐹⊂𝐶_𝑛. For 𝑛=1, 𝐹⊂𝐾=𝐶_1,𝑖. For 𝑛≥2, let 𝑥^∗∈𝐹. Then, 𝑦_𝑛,𝑖−𝑥^∗2=𝛼_𝑛,𝑖𝑥₀−𝑥^∗+1−𝛼_𝑛,𝑖𝑇_𝑖𝑤_𝑛−𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+1−𝛼_𝑛,𝑖𝑤_𝑛−𝑥^∗2≤𝑤_𝑛−𝑥^∗2+𝛼_𝑛,𝑖𝑥₀−𝑥^∗2−𝑤_𝑛−𝑥^∗2≤𝑥_𝑛−𝑥^∗2+𝛼_𝑛,𝑖𝑥₀²+2𝑥_𝑛−𝑥₀,𝑥^∗, (3.5) which shows that 𝑥^∗∈𝐶_𝑛,𝑖, for all 𝑛≥2, for all 𝑖=1,2,…. Thus, 𝐹⊂𝐶_𝑛,𝑖, for all 𝑛≥1, for all 𝑖=1,2,…. Hence, it follows that 𝐹⊂𝐶_𝑛, for all 𝑛≥1. Since 𝑥_𝑛=𝑃_𝐶𝑛𝑥₀, for all 𝑛≥1, and 𝑥_𝑛+1∈𝐶_𝑛+1⊂𝐶_𝑛, for all 𝑛≥1, we have that 𝑥_𝑛−𝑥₀≤𝑥_𝑛+1−𝑥₀,∀𝑛≥1. (3.6) Also, as 𝐹⊂𝐶_𝑛 by (2.1), it follows that 𝑥_𝑛−𝑥₀≤𝑧−𝑥₀,𝑧∈𝐹,∀𝑛≥1. (3.7) From (3.6) and (3.7), we have that lim_𝑛→∞‖𝑥_𝑛−𝑥₀‖ exists. Hence, {𝑥_𝑛}^∞_𝑛=0 is bounded and so are {𝑧_𝑛}^∞_𝑛=1, {𝐴𝑥_𝑛}^∞_𝑛=1, {𝑢_𝑛}^∞_𝑛=1, {𝐷𝑢_𝑛}^∞_𝑛=1, {𝐵𝑧_𝑛}^∞_𝑛=1, {𝑤_𝑛}^∞_𝑛=1, {𝑇_𝑖𝑤_𝑛}^∞_𝑛=1, and {𝑦_𝑛,𝑖}^∞_𝑛=1, 𝑖=1,2,…. For 𝑚>𝑛≥1, we have that 𝑥_𝑚=𝑃_𝐶𝑚𝑥₀∈𝐶_𝑚⊂𝐶_𝑛. By (2.4), we obtain 𝑥_𝑚−𝑥_𝑛²≤𝑥_𝑛−𝑥₀²−𝑥_𝑚−𝑥₀². (3.8) Letting 𝑚,𝑛→∞ and taking the limit in (3.8), we have that 𝑥_𝑚−𝑥_𝑛→0, 𝑚,𝑛→∞, which shows that {𝑥_𝑛}^∞_𝑛=0 is Cauchy. In particular, lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0. Since, {𝑥_𝑛}^∞_𝑛=0 is Cauchy, we assume that 𝑥_𝑛→𝑧∈𝐾.
Since 𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥₀∈𝐶_𝑛+1, then 𝑦_𝑛,𝑖−𝑥_𝑛+1²≤𝑥_𝑛−𝑥_𝑛+1²+𝛼_𝑛,𝑖𝑥₀²+2𝑥_𝑛−𝑥₀,𝑥_𝑛+1⟶0, (3.9) and it follows that 𝑦_𝑛,𝑖−𝑥_𝑛≤𝑦_𝑛,𝑖−𝑥_𝑛+1+𝑥_𝑛−𝑥_𝑛+1. (3.10) Thus, lim_𝑛→∞𝑦_𝑛,𝑖−𝑥_𝑛=0,𝑖=1,2,…. (3.11) Furthermore, 𝑦_𝑛,𝑖−𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+1−𝛼_𝑛,𝑖𝑇_𝑖𝑤_𝑛−𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+1−𝛼_𝑛,𝑖𝑤_𝑛−𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+1−𝛼_𝑛,𝑖𝑢_𝑛−𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+1−𝛼_𝑛,𝑖𝑇^{(𝐹2,𝜑2)}_𝜆𝑛𝑧_𝑛−𝜆_𝑛𝐵𝑧_𝑛−𝑇^{(𝐹2,𝜑2)}_𝜆𝑛𝑥^∗−𝜆_𝑛𝐵𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+1−𝛼_𝑛,𝑖𝑧_𝑛−𝜆_𝑛𝐵𝑧_𝑛−𝑥^∗−𝜆_𝑛𝐵𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+1−𝛼_𝑛,𝑖𝑧_𝑛−𝑥^∗2+𝜆_𝑛𝜆_𝑛−2𝛽𝐵𝑧_𝑛−𝐵𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+𝑥_𝑛−𝑥^∗2+𝜆_𝑛𝜆_𝑛−2𝛽𝐵𝑧_𝑛−𝐵𝑥^∗2. (3.12) Since 0<𝑐≤𝜆_𝑛≤𝑓<2𝛽, we have that 𝑐2𝛽−𝑓𝐵𝑧_𝑛−𝐵𝑥^∗2≤𝑥_𝑛−𝑥^∗2−𝑦_𝑛,𝑖−𝑥^∗2+𝛼_𝑛,𝑖𝑥₀−𝑥^∗2≤𝑦_𝑛,𝑖−𝑥_𝑛𝑥_𝑛−𝑥^∗+𝑦_𝑛,𝑖−𝑥^∗+𝛼_𝑛,𝑖𝑥₀−𝑥^∗2. (3.13) Hence, lim_𝑛→∞‖𝐵𝑧_𝑛−𝐵𝑥^∗‖=0. From (3.1), we have that 𝑦_𝑛,𝑖−𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+1−𝛼_𝑛,𝑖𝑇_𝑖𝑤_𝑛−𝑥^∗2=𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+1−𝛼_𝑛,𝑖𝑢_𝑛−𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+𝑢_𝑛−𝑥^∗2. (3.14) On the other hand, 𝑢_𝑛−𝑥^∗2≤𝑇^{(𝐹2,𝜑2)}_𝜆𝑛𝑧_𝑛−𝜆_𝑛𝐵𝑧_𝑛−𝑇^{(𝐹2,𝜑2)}_𝜆𝑛𝑥^∗−𝜆_𝑛𝐵𝑥^∗2≤𝑧_𝑛−𝜆_𝑛𝐵𝑧_𝑛−𝑥^∗−𝜆_𝑛𝐵𝑥^∗,𝑢_𝑛−𝑥^∗=12𝑧_𝑛−𝜆_𝑛𝐵𝑧_𝑛−𝑥^∗−𝜆_𝑛𝐵𝑥^∗2+𝑢_𝑛−𝑥^∗2−𝑧_𝑛−𝜆_𝑛𝐵𝑧_𝑛−𝑥^∗−𝜆_𝑛𝐵𝑥^∗−𝑢_𝑛−𝑥^∗2≤12𝑧_𝑛−𝑥^∗2+𝑢_𝑛−𝑥^∗2−𝑧_𝑛−𝜆_𝑛𝐵𝑧_𝑛−𝑥^∗−𝜆_𝑛𝐵𝑥^∗−𝑢_𝑛−𝑥^∗2=12𝑧_𝑛−𝑥^∗2+𝑢_𝑛−𝑥^∗2−𝑢_𝑛−𝑧_𝑛²+2𝜆_𝑛𝑧_𝑛−𝑢_𝑛,𝐵𝑧_𝑛−𝐵𝑥^∗−𝜆²_𝑛𝐵𝑧_𝑛−𝐵𝑥^∗2, (3.15) and, hence, 𝑢_𝑛−𝑥^∗2≤𝑧_𝑛−𝑥^∗2−𝑢_𝑛−𝑧_𝑛²+2𝜆_𝑛𝑧_𝑛−𝑢_𝑛,𝐵𝑧_𝑛−𝐵𝑥^∗−𝜆²_𝑛𝐵𝑧_𝑛−𝐵𝑥^∗2≤𝑧_𝑛−𝑥^∗2−𝑢_𝑛−𝑧_𝑛²+2𝜆_𝑛𝑧_𝑛−𝑢_𝑛𝐵𝑧_𝑛−𝐵𝑥^∗. (3.16) Putting (3.16) into (3.14), we have that 𝑦_𝑛,𝑖−𝑥^∗2≤𝑧_𝑛−𝑥^∗2−𝑢_𝑛−𝑧_𝑛²+2𝜆_𝑛𝑧_𝑛−𝑢_𝑛𝐵𝑧_𝑛−𝐵𝑥^∗+𝛼_𝑛,𝑖𝑥₀−𝑥^∗2. (3.17) It follows that 𝑧_𝑛−𝑢_𝑛²≤𝑧_𝑛−𝑥^∗2−𝑦_𝑛,𝑖−𝑥^∗2+2𝜆_𝑛𝑧_𝑛−𝑢_𝑛𝐵𝑧_𝑛−𝐵𝑥^∗+𝛼_𝑛,𝑖𝑥₀−𝑥^∗2≤𝑥_𝑛−𝑥^∗2−𝑦_𝑛,𝑖−𝑥^∗2+2𝜆_𝑛𝑧_𝑛−𝑢_𝑛𝐵𝑧_𝑛−𝐵𝑥^∗+𝛼_𝑛,𝑖𝑥₀−𝑥^∗2≤𝑦_𝑛,𝑖−𝑥_𝑛𝑥_𝑛−𝑥^∗+𝑦_𝑛,𝑖−𝑥^∗+2𝜆_𝑛𝑧_𝑛−𝑢_𝑛𝐵𝑧_𝑛−𝐵𝑥^∗+𝛼_𝑛,𝑖𝑥₀−𝑥^∗2. (3.18) Therefore, lim_𝑛→∞‖𝑧_𝑛−𝑢_𝑛‖=0. Furthermore, 𝑦_𝑛,𝑖−𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+1−𝛼_𝑛,𝑖𝑇_𝑖𝑤_𝑛−𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+1−𝛼_𝑛,i𝑢_𝑛−𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+1−𝛼_𝑛,𝑖𝑧_𝑛−𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+1−𝛼_𝑛,𝑖𝑇^{(𝐹1,𝜑1)}_𝑟𝑛𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛−𝑇^{(𝐹1,𝜑1)}_𝑟𝑛𝑥^∗−𝑟_𝑛𝐴𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+1−𝛼_𝑛,𝑖𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛−𝑥^∗−𝑟_𝑛𝐴𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+1−𝛼_𝑛,𝑖𝑥_𝑛−𝑥^∗2+𝑟_𝑛𝑟_𝑛−2𝛼𝐴𝑥_𝑛−𝐴𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+𝑥_𝑛−𝑥^∗2+𝑟_𝑛𝑟_𝑛−2𝛼𝐴𝑥_𝑛−𝐴𝑥^∗2. (3.19) Since 0<𝑎≤𝑟_𝑛≤𝑏<2𝛼, we have that 𝑎(2𝛼−𝑏)𝐴𝑥_𝑛−𝐴𝑥^∗2≤𝑥_𝑛−𝑥^∗2−𝑦_𝑛,𝑖−𝑥^∗2+𝛼_𝑛,𝑖𝑥₀−𝑥^∗2≤𝑦_𝑛,𝑖−𝑥_𝑛𝑥_𝑛−𝑥^∗+𝑦_𝑛,𝑖−𝑥^∗+𝛼_𝑛,𝑖𝑥₀−𝑥^∗2. (3.20) Hence, lim_𝑛→∞‖𝐴𝑥_𝑛−𝐴𝑥^∗‖=0. From (3.1), we have that 𝑦_𝑛,𝑖−𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+1−𝛼_𝑛,𝑖𝑇_𝑖𝑤_𝑛−𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+1−𝛼_𝑛,𝑖𝑢_𝑛−𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+𝑧_𝑛−𝑥^∗2. (3.21) On the other hand, 𝑧_𝑛−𝑥^∗2≤𝑇^{(𝐹1,𝜑1)}_𝑟𝑛𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛−𝑇^{(𝐹1,𝜑1)}_𝑟𝑛𝑥^∗−𝑟_𝑛𝐴𝑥^∗2≤𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛−𝑥^∗−𝑟_𝑛𝐴𝑥^∗,𝑧_𝑛−𝑥^∗=12𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛−𝑥^∗−𝑟_𝑛𝐴𝑥^∗2+𝑧_𝑛−𝑥^∗2−𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛−𝑥^∗−𝑟_𝑛𝐴𝑥^∗−𝑧_𝑛−𝑥^∗2≤12𝑥_𝑛−𝑥^∗2+𝑧_𝑛−𝑥^∗2−𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛−𝑥^∗−𝑟_𝑛𝐴𝑥^∗−𝑧_𝑛−𝑥^∗2=12𝑥_𝑛−𝑥^∗2+𝑧_𝑛−𝑥^∗2−𝑧_𝑛−𝑥_𝑛²+2𝑟_𝑛𝑥_𝑛−𝑧_𝑛,𝐴𝑥_𝑛−𝐴𝑥^∗−𝑟²_𝑛𝐴𝑥_𝑛−𝐴𝑥^∗2, (3.22) and, hence, 𝑧_𝑛−𝑥^∗2≤𝑥_𝑛−𝑥^∗2−𝑧_𝑛−𝑥_𝑛²+2𝑟_𝑛𝑥_𝑛−𝑧_𝑛,𝐴𝑥_𝑛−𝐴𝑥^∗−𝑟²_𝑛𝐴𝑥_𝑛−𝐴𝑥^∗2≤𝑥_𝑛−𝑥^∗2−𝑧_𝑛−𝑥_𝑛²+2𝑟_𝑛𝑥_𝑛−𝑧_𝑛𝐴𝑥_𝑛−𝐴𝑥^∗. (3.23) Putting (3.23) into (3.21), we have that 𝑦_𝑛,𝑖−𝑥^∗2≤𝑥_𝑛−𝑥^∗2−𝑧_𝑛−𝑥_𝑛²+2𝑟_𝑛𝑥_𝑛−𝑧_𝑛𝐴𝑥_𝑛−𝐴𝑥^∗+𝛼_𝑛,𝑖𝑥₀−𝑥^∗2. (3.24) It follows that 𝑥_𝑛−𝑧_𝑛²≤𝑥_𝑛−𝑥^∗2−𝑦_𝑛,𝑖−𝑥^∗2+2𝑟_𝑛𝑥_𝑛−𝑧_𝑛𝐴𝑥_𝑛−𝐴𝑥^∗+𝛼_𝑛,𝑖𝑥₀−𝑥^∗2≤𝑦_𝑛,𝑖−𝑥_𝑛𝑥_𝑛−𝑥^∗+𝑦_𝑛,𝑖−𝑥^∗+2𝑟_𝑛𝑥_𝑛−𝑧_𝑛𝐴𝑥_𝑛−𝐴𝑥^∗+𝛼_𝑛,𝑖𝑥₀−𝑥^∗2. (3.25) Therefore, lim_𝑛→∞‖𝑥_𝑛−𝑧_𝑛‖=0. But 𝑦_𝑛,𝑖=𝛼_𝑛,𝑖𝑥₀+(1−𝛼_𝑛,𝑖)𝑇_𝑖𝑤_𝑛 implies that 𝑦_𝑛,𝑖−𝑇_𝑖𝑤_𝑛=𝛼_𝑛,𝑖𝑥₀−𝑇_𝑖𝑤_𝑛⟶0. (3.26) Furthermore, we have that 𝑥_𝑛−𝑇_𝑖𝑤_𝑛≤𝑦_𝑛,𝑖−𝑇_𝑖𝑤_𝑛+𝑦_𝑛,𝑖−𝑥_𝑛⟶0. (3.27) Furthermore, 𝑦_𝑛,𝑖−𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+1−𝛼_𝑛,𝑖𝑇_𝑖𝑤_𝑛−𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+1−𝛼_𝑛,𝑖𝑤_𝑛−𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+1−𝛼_𝑛,𝑖𝑃_𝐾𝑢_𝑛−𝑠_𝑛𝐷𝑢_𝑛−𝑃_𝐾𝑥^∗−𝑠_𝑛𝐷𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+1−𝛼_𝑛,𝑖𝑢_𝑛−𝑠_𝑛𝐷𝑢_𝑛−𝑥^∗−𝑠_𝑛𝐷𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+1−𝛼_𝑛,𝑖𝑢_𝑛−𝑥^∗2+𝑠_𝑛𝑠_𝑛−2𝛾𝐷𝑢_𝑛−𝐷𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+𝑥_𝑛−𝑥^∗2+1−𝛼_𝑛,𝑖𝑠_𝑛𝑠_𝑛−2𝛾𝐷𝑢_𝑛−𝐷𝑥^∗2. (3.28) Thus, 1−𝑑_𝑖ℎ2𝛾−𝑗𝐷𝑢_𝑛−𝐷𝑥^∗2≤1−𝛼_𝑛,𝑖𝑠_𝑛2𝛾−𝑠_𝑛𝐷𝑢_𝑛−𝐷𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+𝑥_𝑛−𝑥^∗2−𝑦_𝑛,𝑖−𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+𝑦_𝑛,𝑖−𝑥_𝑛𝑥_𝑛−𝑥^∗+𝑦_𝑛,𝑖−𝑥^∗. (3.29) Since 0<ℎ≤𝑠_𝑛≤𝑗<2𝛾, condition (iii) and ‖𝑦_𝑛,𝑖−𝑥_𝑛‖→0 as 𝑛→∞, we have that lim_𝑛→∞‖𝐷𝑢_𝑛−𝐷𝑥^∗‖=0. Now, using (2.2), we obtain 𝑤_𝑛−𝑥^∗2≤𝑃_𝐾𝑢_𝑛−𝑠_𝑛𝐷𝑢_𝑛−𝑃_𝐾𝑥^∗−𝑠_𝑛𝐷𝑥^∗2≤𝑢_𝑛−𝑠_𝑛𝐷𝑢_𝑛−𝑥^∗−𝑠_𝑛𝐷𝑥^∗,𝑤_𝑛−𝑥^∗=12𝑢_𝑛−𝑠_𝑛𝐷𝑢_𝑛−𝑥^∗−𝑠_𝑛𝐷𝑥^∗2+𝑤_𝑛−𝑥^∗2−𝑢_𝑛−𝑠_𝑛𝐷𝑢_𝑛−𝑥^∗−𝑠_𝑛𝐷𝑥^∗−𝑤_𝑛−𝑥^∗2≤12𝑢_𝑛−𝑥^∗2+𝑤_𝑛−𝑥^∗2−𝑢_𝑛−𝑠_𝑛𝐷𝑢_𝑛−𝑥^∗−𝑠_𝑛𝐷𝑥^∗−𝑤_𝑛−𝑥^∗2=12𝑥_𝑛−𝑥^∗2+𝑤_𝑛−𝑥^∗2−𝑤_𝑛−𝑢_𝑛²+2𝑠_𝑛𝑢_𝑛−𝑤_𝑛,𝐷𝑢_𝑛−𝐷𝑥^∗−𝑠²_𝑛𝐷𝑢_𝑛−𝐷𝑥^∗2. (3.30) Thus, 𝑤_𝑛−𝑥^∗2≤𝑥_𝑛−𝑥^∗2−𝑤_𝑛−𝑢_𝑛²+2𝑠_𝑛𝑤_𝑛−𝑢_𝑛𝐷𝑢_𝑛−𝐷𝑥^∗. (3.31) Using this last inequality, we obtain from (3.1) 𝑦_𝑛,𝑖−𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+1−𝛼_𝑛,𝑖𝑇_𝑖𝑤_𝑛−𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+1−𝛼_𝑛,𝑖𝑤_𝑛−𝑥^∗2≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+𝑥_𝑛−𝑥^∗2−1−𝛼_𝑛,𝑖𝑤_𝑛−𝑢_𝑛²+2𝑠_𝑛1−𝛼_𝑛,𝑖𝑤_𝑛−𝑢_𝑛𝐷𝑢_𝑛−D𝑥^∗. (3.32) This implies that 1−𝛼_𝑛,𝑖𝑤_𝑛−𝑢_𝑛²≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+𝑥_𝑛−𝑥^∗2−𝑦_𝑛,𝑖−𝑥^∗2+2𝑠_𝑛1−𝛼_𝑛,𝑖𝑤_𝑛−𝑢_𝑛𝐷𝑢_𝑛−𝐷𝑥^∗≤𝛼_𝑛,𝑖𝑥₀−𝑥^∗2+𝑦_𝑛,𝑖−𝑥_𝑛𝑥_𝑛−𝑥^∗+𝑦_𝑛,𝑖−𝑥^∗+2𝑠_𝑛1−𝛼_𝑛,𝑖𝑤_𝑛−𝑢_𝑛𝐷𝑢_𝑛−𝐷𝑥^∗. (3.33) Since lim_𝑛→∞𝛼_𝑛,𝑖=0, ‖𝑦_𝑛,𝑖−𝑥_𝑛‖→0 as 𝑛→∞, and ‖𝐷𝑢_𝑛−𝐷𝑥^∗‖→0 as 𝑛→∞, we have that lim_𝑛→∞‖𝑤_𝑛−𝑢_𝑛‖=0. Also since lim_𝑛→∞‖𝑤_𝑛−𝑥_𝑛‖=0 and lim_𝑛→∞‖𝑥_𝑛−𝑧‖=0, we have that lim_𝑛→∞‖𝑤_𝑛−𝑧‖=0. Now, 𝑤_𝑛−𝑇_𝑖𝑤_𝑛≤𝑥_𝑛−𝑇_𝑖𝑤_𝑛+𝑤_𝑛−𝑥_𝑛. (3.34) Hence, lim_𝑛→∞‖𝑤_𝑛−𝑇_𝑖𝑤_𝑛‖=0, 𝑖=1,2,…. By lim_𝑛→∞‖𝑤_𝑛−𝑧‖=0 and lim_𝑛→∞‖𝑤_𝑛−𝑇_𝑖𝑤_𝑛‖=0, 𝑖=1,2,…, we have that 𝑧∈⋂^∞_𝑖=1𝐹(𝑇_𝑖).
Since 𝑧_𝑛∶=𝑇^{(𝐹1,𝜑1)}_𝑟𝑛(𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛), 𝑛≥1, we have, for any 𝑦∈𝐾, that 𝐹₁𝑧_𝑛,𝑦+𝜑₁𝑦−𝜑₁𝑧_𝑛+𝐴𝑥_𝑛,𝑦−𝑧_𝑛+1𝑟_𝑛𝑦−𝑧_𝑛,𝑧_𝑛−𝑥_𝑛≥0. (3.35) Furthermore, replacing 𝑛 by 𝑛_𝑗 in the last inequality and using (A2), we obtain 𝜑₁𝑦−𝜑₁𝑧_𝑛𝑗+𝐴𝑥_𝑛𝑗,𝑦−𝑧_𝑛𝑗+1𝑟_𝑛𝑗𝑦−𝑧_𝑛𝑗,𝑧_𝑛𝑗−𝑥_𝑛𝑗≥𝐹₁𝑦,𝑧_𝑛𝑗. (3.36) Let 𝑧_𝑡∶=𝑡𝑦+(1−𝑡)𝑧 for all 𝑡∈(0,1] and 𝑦∈𝐾. This implies that 𝑧_𝑡∈𝐾. Then, we have that 𝑧_𝑡−𝑧_𝑛𝑗,𝐴𝑧_𝑡≥𝜑₁𝑧_𝑛𝑗−𝜑₁𝑧_𝑡+𝑧_𝑡−𝑧_𝑛𝑗,𝐴𝑧_𝑡−𝑧_𝑡−𝑧_𝑛𝑗,𝐴𝑥_𝑛𝑗−𝑧_𝑡−𝑧_𝑛𝑗,𝑧_𝑛𝑗−𝑥_𝑛𝑗𝑟_𝑛𝑗+𝐹₁𝑧_𝑡,𝑧_𝑛𝑗=𝜑₁𝑧_𝑛𝑗−𝜑₁𝑧_𝑡+𝑧_𝑡−𝑧_𝑛𝑗,𝐴𝑧_𝑡−𝐴𝑧_𝑛𝑗+𝑧_𝑡−𝑧_𝑛𝑗,𝐴𝑧_𝑛𝑗−𝐴𝑥_𝑛𝑗−𝑧_𝑡−𝑧_𝑛𝑗,𝑧_𝑛𝑗−𝑥_𝑛𝑗𝑟_𝑛𝑗+𝐹₁𝑧_𝑡,𝑧_𝑛𝑗. (3.37) Since ‖𝑥_𝑛𝑗−𝑧_𝑛𝑗‖→0, 𝑗→∞, we obtain ‖𝐴𝑥_𝑛𝑗−𝐴𝑧_𝑛𝑗‖→0, 𝑗→∞. Furthermore, by the monotonicity of 𝐴, we obtain ⟨𝑧_𝑡−𝑧_𝑛𝑗,𝐴𝑧_𝑡−𝐴𝑧_𝑛𝑗⟩≥0. Then, by (A4), we obtain (noting that 𝑧_𝑛𝑗→𝑧) 𝑧_𝑡−𝑧,𝐴𝑧_𝑡≥𝜑₁(𝑧)−𝜑₁𝑧_𝑡+𝐹₁𝑧_𝑡,𝑧,𝑗⟶∞. (3.38) Using (A1), (A4), and (3.38), we also obtain 0=𝐹₁𝑧_𝑡,𝑧_𝑡+𝜑₁𝑧_𝑡−𝜑₁𝑧_𝑡≤𝑡𝐹₁𝑧_𝑡,𝑦+(1−𝑡)𝐹₁𝑧_𝑡,𝑧+𝑡𝜑₁𝑦+(1−𝑡)𝜑₁(𝑧)−𝜑₁𝑧_𝑡≤𝑡𝐹₁𝑧_𝑡,𝑦+𝜑₁𝑦−𝜑₁𝑧_𝑡+(1−𝑡)𝑧_𝑡−𝑧,𝐴𝑧_𝑡=𝑡𝐹₁𝑧_𝑡,𝑦+𝜑₁𝑦−𝜑₁𝑧_𝑡+(1−𝑡)𝑡𝑦−𝑧,𝐴𝑧_𝑡, (3.39) and, hence, 0≤𝐹₁𝑧_𝑡,𝑦+𝜑₁𝑦−𝜑₁𝑧_𝑡+(1−𝑡)𝑦−𝑧,𝐴𝑧_𝑡. (3.40) Letting 𝑡→0, we have, for each 𝑦∈𝐾, that 0≤𝐹₁𝑧,𝑦+𝜑₁𝑦−𝜑₁(𝑧)+𝑦−𝑧,𝐴𝑧. (3.41) This implies that 𝑧∈GMEP(𝐹₁,𝐴,𝜑₁). By following the same arguments, we can show that 𝑧∈GMEP(𝐹₂,𝐵,𝜑₂).
Following the arguments of [3, Theorem 3.1, pages 346-347], we can show that 𝑧∈VI(𝐾,𝐷). Therefore, 𝑧∈⋂^∞_𝑖=1𝐹(𝑇_𝑖)∩GMEP(𝐹₁,𝐴,𝜑₁)∩GMEP(𝐹₂,𝐵,𝜑₂)∩VI(𝐾,𝐷).
Noting that 𝑥_𝑛=𝑃_𝐶𝑛𝑥₀, we have by (2.3), 𝑥₀−𝑥_𝑛,𝑦−𝑥_𝑛≤0, (3.42) for all 𝑦∈𝐶_𝑛. Since 𝐹⊂𝐶_𝑛 and by the continuity of inner product, we obtain, from the above inequality, 𝑥₀−𝑧,𝑦−𝑧≤0, (3.43) for all 𝑦∈𝐹. By (2.3) again, we conclude that 𝑧=𝑃_𝐹𝑥₀. This completes the proof.

Corollary 3.2. Let 𝐾 be a nonempty closed and convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐾→𝐾 be a nonexpansive mapping such that 𝐹(𝑇)≠∅. Let {𝑥_𝑛}^∞_𝑛=0 be generated by 𝑥₀𝑦∈𝐾,_𝑛=𝛼_𝑛𝑥₀+1−𝛼_𝑛𝑇𝑥_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶𝑦_𝑛−𝑧²≤𝑥_𝑛−𝑧²+𝛼_𝑛𝑥₀²+2𝑥_𝑛−𝑥₀,𝑥,𝑧_𝑛+1=𝑃_𝐶𝑛+1𝑥₀,𝑛≥1. (3.44) Assume that {𝛼_𝑛}^∞_𝑛=1⊂(0,1) such that lim_𝑛→∞𝛼_𝑛=0. Then, {𝑥_𝑛}^∞_𝑛=0 converges strongly to 𝑃_𝐹(𝑇)𝑥₀.

Remark 3.3. Corollary 3.2 can be viewed as an improvement of Theorem 3.1 of Martinez-Yanes and Xu [39] because we relax the iterative step Q_𝑛 in the algorithm of Theorem 3.1 of [39].

4. Applications

Let 𝐶 be a nonempty closed and convex cone in 𝐻 and 𝐷 an operator of 𝐶 into 𝐻. We define the polar of 𝐶 in 𝐻 to be the set 𝐾^∗∶=𝑦^∗∈𝐻∶𝑥,𝑦^∗≥0,∀𝑥∈𝐶. (4.1) Then, the element 𝑢∈𝐶 is called a solution of the complementarity problem if 𝐷𝑢∈𝐾^∗,⟨𝑢,𝐷𝑢⟩=0. (4.2) The set of solutions of the complementarity problem is denoted by 𝐶(𝐶,𝐷). We will assume that 𝐷 satisfies the following conditions:(E1)  𝐷 is 𝛾-inverse strongly monotone,(E2)  𝐶(𝐶,𝐷)≠∅.

Also, we replace conditions (B1) and (B2) with (D1)  for each 𝑥∈𝐻 and 𝑟>0, there exist a bounded subset 𝐷_𝑥⊆𝐶 and 𝑦_𝑥∈𝐶 such that, for any 𝑧∈𝐶⧵𝐷_𝑥, 𝐹𝑧,𝑦_𝑥+𝜑𝑦_𝑥(𝑧)+1−𝜑𝑟𝑦_𝑥−𝑧,𝑧−𝑥<0, (4.3)(D2)  𝐶 is a bounded set.

Theorem 4.1. Let 𝐶 be a nonempty closed and convex subset of a real Hilbert space 𝐻. For each 𝑚=1,2, let 𝐹_𝑚 be a bifunction from 𝐶×𝐶 satisfying (A1)–(A4), 𝜑_𝑚∶𝐶→ℝ∪{+∞} a proper lower semicontinuous and convex function with assumption (B1) or (B2), 𝐴 an 𝛼-inverse-strongly monotone mapping of 𝐶 into 𝐻, and 𝐵 a 𝛽-inverse-strongly monotone mapping of 𝐶 into 𝐻, and, for each 𝑖=1,2,…, let 𝑇_𝑖∶𝐶→𝐶 be a nonexpansive mapping such that ⋂^∞_𝑖=1𝐹(𝑇_𝑖)≠∅. Let 𝐷 be a 𝛾-inverse-strongly monotone mapping of 𝐾 into 𝐻. Suppose that 𝐹∶=⋂^∞_𝑖=1𝐹(𝑇_𝑖)∩GMEP(𝐹₁,𝐴,𝜑₁)∩GMEP(𝐹₂,𝐵,𝜑₂)∩𝐶(𝐶,𝐷)≠∅. Let {𝑧_𝑛}^∞_𝑛=1, {𝑢_𝑛}^∞_𝑛=1, {𝑤_𝑛}^∞_𝑛=1, {y_𝑛,𝑖}^∞_𝑛=1 (𝑖=1,2,…), and {𝑥_𝑛}^∞_𝑛=0 be generated by 𝑥₀∈𝐶, 𝐶_1,𝑖=𝐶, 𝐶₁=⋂^∞_𝑖=1𝐶_1,𝑖, 𝑥₁=𝑃_𝐶1𝑥₀, 𝑧_𝑛=𝑇^{(𝐹1,𝜑1)}_𝑟𝑛𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛,𝑢_𝑛=𝑇^{(𝐹2,𝜑2)}_𝜆𝑛𝑧_𝑛−𝜆_𝑛𝐵𝑧_𝑛,𝑤_𝑛=𝑃_𝐶𝑢_𝑛−𝑠_𝑛𝐷𝑢_𝑛,𝑦_𝑛,𝑖=𝛼_𝑛,𝑖𝑥₀+1−𝛼_𝑛,𝑖𝑇_𝑖𝑤_𝑛,𝐶_𝑛+1,𝑖=𝑧∈𝐶_𝑛,𝑖∶𝑦_𝑛,𝑖−𝑧²≤𝑥_𝑛−𝑧²+𝛼_𝑛,𝑖𝑥₀²+2𝑥_𝑛−𝑥₀,𝐶,𝑧_𝑛+1=^∞_𝑖=1𝐶_𝑛+1,𝑖,𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥₀,𝑛≥1. (4.4) Assume that {𝛼_𝑛,𝑖}^∞_𝑛=1⊂(0,1) (𝑖=1,2,…), {𝑟_𝑛}^∞_𝑛=1⊂[0,2𝛼], and {𝜆_𝑛}^∞_𝑛=1⊂[0,2𝛽] satisfy(i)0<𝑎≤𝑟_𝑛≤𝑏<2𝛼,(ii)0<𝑐≤𝜆_n≤𝑓<2𝛽,(iii)lim_𝑛→∞𝛼_𝑛,𝑖=0,(iv)0<ℎ≤𝑠_𝑛≤𝑗<2𝛾. Then, {𝑥_𝑛}^∞_𝑛=0 converges strongly to 𝑃_𝐹𝑥₀.

Proof. Using Lemma 7.1.1 of [52], we have that VI(𝐶,𝐷)=𝐶(𝐶,𝐷). Hence, by Theorem 3.1 we obtain the desired conclusion.

Next we study the problem of finding a minimizer of a continuously Fréchet differentiable convex functional in a Hilbert space.

Theorem 4.2. For each 𝑚=1,2, let 𝐹_𝑚 be a bifunction from 𝐻×𝐻 satisfying (A1)–(A4), 𝜑_𝑚∶𝐻→ℝ∪{+∞} a proper lower semicontinuous and convex function with assumption (B1) or (B2), 𝐴 an 𝛼-inverse-strongly monotone mapping of 𝐻 into itself, and 𝐵 a 𝛽-inverse-strongly monotone mapping of 𝐻 into itself, and, for each 𝑖=1,2,…, let 𝑇_𝑖∶𝐻→𝐻 be a nonexpansive mapping such that ⋂^∞_𝑖=1𝐹(𝑇_𝑖)≠∅. Suppose that 𝑓 is a functional on 𝐻 which satisfies the following conditions:(1)𝑓 is a continuously Fréchet differentiable convex functional on 𝐻 and ∇𝑓 is (1/𝛾)-Lipschitz continuous,(2)(∇𝑓)⁻¹0={𝑧∈𝐻∶𝑓(𝑧)=min_𝑦∈𝐻𝑓(𝑦)}≠∅.Suppose that 𝐹∶=⋂^∞_𝑖=1𝐹(𝑇_𝑖)∩GMEP(𝐹₁,𝐴,𝜑₁)∩GMEP(𝐹₂,𝐵,𝜑₂)∩(∇𝑓)⁻¹0≠∅. Let {𝑧_𝑛}^∞_𝑛=1, {𝑢_𝑛}^∞_𝑛=1, {𝑤_𝑛}^∞_𝑛=1, {𝑦_𝑛,𝑖}^∞_𝑛=1 (𝑖=1,2,…), and {𝑥_𝑛}^∞_𝑛=0 be generated by 𝑥₀∈𝐾, 𝐶_1,𝑖=𝐾, 𝐶₁=⋂^∞_𝑖=1𝐶_1,𝑖, 𝑥₁=𝑃_𝐶1𝑥₀, 𝑧_𝑛=𝑇^{(𝐹1,𝜑1)}_𝑟𝑛𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛,𝑢_𝑛=𝑇^{(𝐹2,𝜑2)}_𝜆𝑛𝑧_𝑛−𝜆_𝑛𝐵𝑧_𝑛,𝑤_𝑛=𝑢_𝑛−𝑠_𝑛∇𝑓𝑢_𝑛,𝑦_𝑛,𝑖=𝛼_𝑛,𝑖𝑤_𝑛+1−𝛼_𝑛,𝑖𝑇_𝑖𝑤_𝑛,𝐶_𝑛+1,𝑖=𝑧∈𝐶_𝑛,𝑖∶𝑦_𝑛,𝑖−𝑧²≤𝑥_𝑛−𝑧²+𝛼_𝑛,𝑖𝑥₀²+2𝑥_𝑛−𝑥₀,𝐶,z_𝑛+1=^∞_𝑖=1𝐶_𝑛+1,𝑖,𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥₀,𝑛≥1. (4.5) Assume that {𝛼_𝑛,𝑖}^∞_𝑛=1⊂(0,1) (𝑖=1,2,…), {𝑟_𝑛}^∞_𝑛=1⊂[0,2𝛼], and {𝜆_𝑛}^∞_𝑛=1⊂[0,2𝛽] satisfy(i)0<𝑎≤𝑟_𝑛≤𝑏<2𝛼,(ii)0<𝑐≤𝜆_𝑛≤𝑓<2𝛽,(iii)lim_𝑛→∞𝛼_𝑛,𝑖=0,(iv)0<ℎ≤𝑠_𝑛≤𝑗<2𝛾. Then, {𝑥_𝑛}^∞_𝑛=0 converges strongly to 𝑃_𝐹𝑥₀.

Proof. We know from condition (i) and Lemma 2.2 that ∇𝑓 is an 𝛾-inverse-strongly monotone operator from 𝐻 into 𝐻. Using Theorem 3.1, we have the desired conclusion.

We now study a kind of multiobjective optimization problem with nonempty set of solutions:minℎ₁(𝑥),minℎ₂(𝑥),𝑥∈𝐾, (4.6) where 𝐾 is a nonempty closed convex subset of a real Hilbert space 𝐻, and ℎ_𝑖∶𝐾→ℝ, 𝑖=1,2, is a convex and a lower semicontinuous functional. Let us denote the set of solutions to (4.6) by Ω and assume that Ω≠∅.

We will denote the set of solutions of the following two optimization problems by Ω₁ and Ω₂, respectively: min_𝑥∈𝐾ℎ₁(𝑥),min_𝑥∈𝐾ℎ₂(𝑥). (4.7) Clearly, if we find a solution 𝑥∈Ω₁∩Ω₂, then one must have 𝑥∈Ω.

Now, for each 𝑖=1,2, let 𝐹_𝑖∶𝐾×𝐾→ℝ be defined by 𝐹_𝑖(𝑥,𝑦)∶=ℎ_𝑖(𝑦)−ℎ_𝑖(𝑥). Let us now find the following equilibrium problem: find 𝑥∈𝐾 such that𝐹_𝑖𝑥,𝑦≥0,𝑖=1,2, (4.8) for all 𝑦∈𝐾. It is obvious that 𝐹_𝑖 satisfies conditions (A1)–(A4) and EP(𝐹_𝑖)=Ω_𝑖, 𝑖=1,2, where EP(𝐹_𝑖) is the set of solutions to (4.8). By Theorem 3.1, we have the following theorem.

Theorem 4.3. Let 𝐾 be a nonempty closed and convex subset of a real Hilbert space 𝐻. For each 𝑖=1,2, let ℎ_𝑖 be a lower semicontinuous and convex function such that Ω₁∩Ω₂≠∅. Let {𝑧_𝑛}^∞_𝑛=1, {𝑢_𝑛}^∞_𝑛=1, {𝑦_𝑛}^∞_𝑛=1, and {𝑥_𝑛}^∞_𝑛=0 be generated by 𝑥₀∈𝐾, 𝐶₁=𝐾, 𝑥₁=𝑃_𝐶1𝑥₀, ℎ₁𝑦−ℎ₁𝑧_𝑛+1𝑟_𝑛𝑦−𝑧_𝑛,𝑧_𝑛−𝑥_𝑛ℎ𝑦𝑢≥0,∀𝑦∈𝐾,−ℎ_𝑛+1𝜆_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑧_𝑛𝑦≥0,∀𝑦∈𝐾,_𝑛=𝛼_𝑛𝑥₀+1−𝛼_𝑛𝑢_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶𝑦_𝑛−𝑧²≤𝑥_𝑛−𝑧²+𝛼_𝑛𝑥₀²+2𝑥_𝑛−𝑥₀,𝑥,𝑧_𝑛+1=𝑃_𝐶𝑛+1𝑥₀,𝑛≥1. (4.9) Assume that {𝛼_𝑛}^∞_𝑛=1⊂(0,1), {𝑟_n}^∞_𝑛=1⊂(0,∞), and {𝜆_𝑛}^∞_𝑛=1⊂(0,∞) satisfy(i)liminf_𝑛→∞𝑟_𝑛>0,(ii)liminf_𝑛→∞𝜆_𝑛>0,(iii)lim_𝑛→∞𝛼_𝑛=0. Then, {𝑥_𝑛}^∞_𝑛=0 converges strongly to 𝑃_Ω1∩Ω2𝑥₀.

Remark 4.4. Our results in this paper also hold for infinite family of uniformly continuous quasi-nonexpansive mappings in a real Hilbert space.