The System of Mixed Equilibrium Problems for Quasi-Nonexpansive Mappings in Hilbert Spaces

Abstract

We first introduce the iterative procedure to approximate a common element of the fixed-point set of two quasinonexpansive mappings and the solution set of the system of mixed equilibrium problem (SMEP) in a real Hilbert space. Next, we prove the weak convergence for the given iterative scheme under certain assumptions. Finally, we apply our results to approximate a common element of the set of common fixed points of asymptotic nonspreading mapping and asymptotic TJ mapping and the solution set of SMEP in a real Hilbert space.

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 𝑇 be a mapping of 𝐶 into 𝐻, then 𝑇∶𝐶→𝐻 is said to be nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶. A mapping 𝑇∶𝐶→𝐻 is said to be quasinonexpansive if ‖𝑇𝑥−𝑦‖≤‖𝑥−𝑦‖ for all 𝑥∈𝐶 and 𝑦∈𝐹(𝑇)∶={𝑥∈𝐶∶𝑇𝑥=𝑥}. It is well known that the set 𝐹(𝑇) of fixed points of a quasi-nonexpansive mapping 𝑇 is a closed and convex set [1]. A mapping 𝑇∶𝐶→𝐶 is said to be firmly nonexpansive [2] if 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦,𝑇𝑥−𝑇𝑦, (1.1) for all 𝑥,𝑦∈𝐶, and it is an important example of nonexpansive mappings in a Hilbert space.

Let 𝜑∶𝐶→ℝ be a real-valued function, and let 𝐹∶𝐶×𝐶→ℝ be an equilibrium bifunction, that is, 𝐹(𝑢,𝑢)=0 for each 𝑢∈𝐶. The mixed equilibrium problem is to find 𝑥∈𝐶 such that𝐹𝑥,𝑦+𝜑𝑦−𝜑(𝑥)≥0∀𝑦∈𝐶. (1.2) Denote the set of solution of (1.2) by MEP(𝐹,𝜑). In particular, if 𝜑=0, this problem reduces to the equilibrium problem, which is to find 𝑥∈𝐶 such that𝐹𝑥,𝑦≥0∀𝑦∈𝐶. (1.3) The set of solution of (1.3) is denoted by EP(𝐹).

The problem (1.2) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, Min-Max problems, the Nash equilibrium problems in noncooperative games, and others; see, for example, Blum and Oettli [3] and Moudafi [4]. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.3).

Let 𝐹₁,𝐹₂∶𝐶×𝐶→ℝ be two monotone bifunctions and 𝜇>0 is constant. In 2009, Moudafi [5] introduced an alternating algorithm for approximating a solution of the system of equilibrium problems, finding (𝑥^′,𝑦^′)∈𝐶×𝐶 such that()𝐹SEP¹𝑥^′+1,𝑧𝜇𝑦^′−𝑥^′,𝑥^′𝐹−𝑧≥0,∀𝑧∈𝐶,₂𝑦^′+1,𝑧𝜇𝑥^′−𝑦^′,𝑦^′−𝑧≥0,∀𝑧∈𝐶. (1.4) For such mappings 𝐹₁ and 𝐹₂ and two given positive constants 𝜆,𝜇>0, Plubtieng and Sombut [6] considered the following system of mixed equilibrium problem, finding (𝑥′,𝑦′)∈𝐶×𝐶 such that()𝐹SMEP¹𝑥^′,𝑧+𝜑(𝑧)−𝜑𝑥^′+1𝜆𝑦^′−𝑥^′,𝑥^′𝐹−𝑧≥0,∀𝑧∈𝐶,₂𝑦^′,𝑧+𝜑(𝑧)−𝜑𝑦^′+1𝜇𝑥^′−𝑦^′,𝑦^′−𝑧≥0,∀𝑧∈𝐶. (1.5) In particular, if 𝜆=𝜇 and 𝜑≡0, then problem (SMEP) reduces to (SEP). Furthermore, Plubtieng and Sombut [6] introduced the following iterative procedure to approximate a common element of the fixed-point set of a quasi-nonexpansive mapping 𝑇 and the solution set of (SMEP) in a Hilbert space 𝐻. Let {𝑥_𝑛},{𝑦_𝑛}, and {𝑢_𝑛} be given by 𝑥₁𝑢∈𝐶chosenarbitrary,_𝑛∈𝐶,𝐹₂𝑢_𝑛,𝑧+𝜑(𝑧)−𝜑𝑢_𝑛+1𝜇𝑧−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑦≥0,∀𝑧∈𝐶,_𝑛∈𝐶,𝐹₁𝑦_𝑛,𝑧+𝜑(𝑧)−𝜑𝑦_𝑛+1𝜆𝑧−𝑦_𝑛,𝑦_𝑛−𝑢_𝑛𝑥≥0,∀𝑧∈𝐶,_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑦_𝑛,∀𝑛∈ℕ, (1.6) where {𝛼_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1) and satisfying appropriate conditions. The weak convergence theorems are obtained in a real Hilbert space.

On the other hand, in 1953, Mann [7] introduced the following iterative procedure to approximate a fixed point of a nonexpansive mapping 𝑇 in a Hilbert space 𝐻:𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,∀𝑛∈ℕ, (1.7) where the initial point 𝑥₀ is taken in 𝐶 arbitrarily, and {𝛼_𝑛} is a sequence in [0,1].

For two nonexpansive mappings 𝑇₁,𝑇₂ of 𝐶 into itself, Moudafi [4] studied weak convergence theorems in the following iterative process:𝑥₀𝑥∈𝐶chosenarbitrary,_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝛽_𝑛𝑇₁𝑥_𝑛+1−𝛽_𝑛𝑇₂𝑥_𝑛, (1.8) for all 𝑛∈ℕ, where {𝛼_𝑛} and {𝛽_𝑛} are appropriate sequences in [0,1] and 𝐹(𝑇₁)∩𝐹(𝑇₂)≠∅. Recently, Iemoto and Takahashi [8] also considered this iterative procedure for 𝑇₁ is a nonexpansive mapping and 𝑇₂∶𝐶→𝐶 is a nonspreading mapping. Very recently, Kim [9] studied the weak and strong convergence for the Moudafi’s iterative scheme (1.8) of two quasi-nonexpansive mappings.

In this paper, inspired and motivated by Plubtieng and Sombut [6], Moudafi [4], Iemoto and Takahashi [8], and Kim [9], we first introduce the iterative procedure to approximate a common element of the common fixed point set of two quasi-nonexpansive mappings and the solution set of SMEP in a real Hilbert space. Next, we prove the weak convergence theorem for the given iterative scheme under certain assumptions. Finally, we apply our results to approximate a common element of the set of common fixed point of asymptotic nonspreading mapping and asymptotic 𝑇𝐽 mapping and the solution set of SMEP in a real Hilbert space.

2. Preliminaries

Throughout this paper, let ℕ be the set of positive integers, and let ℝ be the set of real numbers. Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖, respectively, and let 𝐶 be a closed convex subset of 𝐻. We denote the strong convergence and the weak convergence of {𝑥_𝑛} to 𝑥∈𝐻 by 𝑥_𝑛→𝑥 and 𝑥_𝑛⇀𝑥, respectively.

From [10], for each 𝑥,𝑦∈𝐻 and 𝜆∈[0,1], we have𝜆𝑥+(1−𝜆)𝑦²=𝜆‖𝑥‖²+(1−𝜆)𝑦²−𝜆(1−𝜆)𝑥−𝑦². (2.1) 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. (2.2)𝑃_𝐶 is called the metric projection of 𝐻 onto 𝐶. It is well know that 𝑃_𝐶 is a nonexpansive mapping of 𝐻 onto 𝐶 and satisfies𝑃_𝐶𝑥−𝑃_𝐶𝑦²≤𝑃_𝐶𝑥−𝑃_𝐶𝑦,𝑥−𝑦∀𝑥,𝑦∈𝐻. (2.3) Moreover, 𝑃_𝐶𝑥 is characterized by the following properties: 𝑃_𝐶𝑥∈𝐶, ⟨𝑥−𝑃_𝐶𝑥,𝑃_𝐶𝑦−𝑦⟩≥0,𝑥−𝑦²≥𝑥−𝑃_𝐶𝑥²+𝑦−𝑃_𝐶𝑦²∀𝑥∈𝐻,𝑦∈𝐶. (2.4) Further, for all 𝑥∈𝐻 and 𝑦∈𝐶,𝑦=𝑃_𝐶𝑥 if and only if ⟨𝑥−𝑦,𝑦−𝑧⟩≥0, for all 𝑧∈𝐶.

Lemma 2.1 (see [11]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑃 be the metric projection of 𝐻 onto 𝐶, and let {𝑥_𝑛}_𝑛∈ℕ be in 𝐻. If 𝑥_𝑛+1−𝑢≤𝑥_𝑛−𝑢, (2.5) for all 𝑢∈𝐶 and 𝑛∈ℕ. Then, {𝑃_𝐶𝑥_𝑛} converges strongly to an element of 𝐶.

Theorem 2.2 (Opial’s theorem, [10]). Let 𝐻 be a real Hilbert space, and suppose that 𝑥_𝑛⇀𝑥, then liminf_𝑛→∞𝑥_𝑛−𝑥<liminf_𝑛→∞𝑥_𝑛−𝑦, (2.6) for all 𝑦∈𝐻 with 𝑥≠𝑦.

All Hilbert space and 𝑙^𝑝(1<𝑝<∞) satisfy Opial’s condition, while 𝐿^𝑝 with 1<𝑝≠2<∞ do not.

For solving the mixed equilibrium problem for an equilibrium bifunction 𝐹∶𝐶×𝐶→ℝ, let us assume that 𝐹 satisfies the following conditions:(A1)𝐹(𝑥,𝑥)=0 for all 𝑥∈𝐶,(A2)𝐹 is monotone, that is, 𝐹(𝑥,𝑦)+𝐹(𝑦,𝑥)≤0 for all 𝑥,𝑦∈𝐶,(A3)for each 𝑦∈𝐶,𝑥↦𝐹(𝑥,𝑦) is weakly upper semicontinuous,(A4)for each 𝑥∈𝐶,𝑦↦𝐹(𝑥,𝑦) is convex and semicontinuous.

The following lemma appears implicitly in [3, 12].

Lemma 2.3 (see [3]). Let 𝐶 be a nonempty closed convex subset of 𝐻, and let 𝐹∶𝐶×𝐶→ℝ be a bifunction satisfying (A1)–(A4). Let 𝑟>0 and 𝑥∈𝐻, then there exists 𝑧∈𝐶 such that 𝐹+1𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0∀𝑦∈𝐶. (2.7)

The following lemma was also given in [12].

Lemma 2.4 (see [12]). Let 𝐶 be a nonempty closed closed convex subset of 𝐻 and let 𝐹∶𝐶×𝐶→ℝ be a bifunction satisfying (A1)–(A4), then, for any 𝑟>0 and 𝑥∈𝐻, define a mapping 𝑇_𝑟𝑥∶𝐻→𝐶 as follows: 𝑇_𝑟(𝑥)=+1𝑧∈𝐶∶𝐹𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶, (2.8) for all𝑧∈𝐻,𝑟∈ℝ. Then the following hold:(i)𝑇_𝑟 is single valued,(ii)𝑇_𝑟 is firmly nonexpansive, that is, 𝑇_𝑟𝑥−𝑇_𝑟𝑦²≤⟨𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝑥−𝑦⟩,∀𝑥,𝑦∈𝐻; (2.9)(iii)𝐹(𝑇_𝑟)=EP(𝐹),(iv)EP(𝐹) is closed and convex.

We note that Lemma 2.4 is equivalent to the following lemma.

Lemma 2.5 (see [6]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐹∶𝐶×𝐶→ℝ be an equilibrium bifunction satisfying (A1)–(A4) and let 𝜑∶𝐶→ℝ be a lower semicontinuous and convex functional. For each 𝑟>0 and 𝑥∈𝐻, define a mapping 𝑆_𝑟(𝑥)=(𝑧)𝑦+1𝑦∈𝐶∶𝐹𝑦,𝑧+𝜑−𝜑𝑟𝑦−𝑥,𝑧−𝑦≥0,∀𝑧∈𝐶,∀𝑥∈𝐻. (2.10) Then, the following results hold:(i)for each 𝑥∈𝐶, 𝑆_𝑟(𝑥)≠∅,(ii)𝑆_𝑟 is single valued,(iii)𝑆_𝑟 is firmly nonexpansive, that is, for any 𝑥,𝑦∈𝐻, 𝑆_𝑟(𝑥)−𝑆_𝑟𝑦²≤𝑆_𝑟(𝑥)−𝑆_𝑟𝑦,𝑥−𝑦, (2.11)(iv)𝐹(𝑆_𝑟)=MEP(𝐹,𝜑),(v)MEP(𝐹,𝜑) is closed and convex.

3. Main Results

In this section, we prove the weak convergence for approximating a common element of the common fixed point set of two quasi-nonexpansive mappings and the solution set of the system of mixed equilibrium problems in a Hilbert space.

To begin with, let us state and proof the following characterizations of the solution set of GMEP.

Lemma 3.1. Let 𝐶 be a closed convex subset of a real Hilbert space 𝐻. Let 𝐹₁ and 𝐹₂ be two mappings from 𝐶×𝐶→ℝ satisfying (A1)–(A4), and let 𝑆_1,𝜆 and S_2,𝜇 be defined as in Lemma 2.5 associated to 𝐹₁ and 𝐹₂, respectively. For given 𝑥^′,𝑦^′∈𝐶,(𝑥^′,𝑦^′) is a solution of problem (1.5) if and only if 𝑥′ is a fixed point of the mapping 𝐺∶𝐶→𝐶 defined by 𝐺(𝑥)=𝑆_1,𝜆𝑆_2,𝜇𝑥,∀𝑥∈𝐶, (3.1) where 𝑦^′=𝑆_2,𝜇𝑥^′.

Proof. For given 𝑥′,𝑦′∈𝐶, we observe the following equivalency: 𝑥^′,𝑦^′()⟺𝐹isasolutionofproblem1.5¹𝑥^′,𝑧+𝜑(𝑧)−𝜑𝑥^′+1𝜆𝑦^′−𝑥^′,𝑥^′𝐹−𝑧≥0∀𝑧∈𝐶,₂𝑦^′,𝑧+𝜑(𝑧)−𝜑𝑦^′+1𝜇𝑥^′−𝑦^′,𝑦^′−𝑧≥0∀𝑧∈𝐶,⟺𝑥^′=𝑆_1,𝜆𝑦^′𝑦^′=𝑆_2,𝜇𝑥^′,⟺𝑥^′=𝑆_1,𝜆𝑆_2,𝜇𝑥^′,⟺𝑥′=𝐺𝑥′. (3.2) This completes the proof.

We note from Lemma 3.1 that the mapping 𝐺 is nonexpansive. Moreover, if 𝐶 is a closed bounded convex subset of 𝐻, then the solution of problem (1.5) always exists. Throughout this paper, we denote the set of fixed points of 𝐺 by Ω.

Theorem 3.2. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐹₁ and 𝐹₂ be two bifunctions from 𝐶×𝐶→ℝ satisfying (A1)–(A4). Let 𝑟,𝜆>0 and 𝑆_1,𝑟 and 𝑆_2,𝜆 be defined as in Lemma 2.5 associated to 𝐹₁ and 𝐹₂, respectively. Let 𝑇_𝑖∶𝐶→𝐶, 𝑖=1,2, be two quasi-nonexpansive mappings such that 𝐼−𝑇_𝑖 are demiclosed at zero, that is, if {𝑤_𝑛}⊂𝐶, 𝑤_𝑛⇀𝑤, and (𝐼−𝑇_𝑖)𝑤_𝑛→0, then 𝑤∈𝐹(𝑇_𝑖), with 𝐹(𝑇₁)∩𝐹(𝑇₂)∩Ω≠∅. Let the sequences {𝑥_𝑛},{𝑦_𝑛}, and {𝑢_𝑛}, be given by 𝑥₁𝑢∈𝐶chosenarbitrary,_𝑛∈𝐶,𝐹₂𝑢_𝑛,𝑧+𝜑(𝑧)−𝜑𝑢_𝑛+1𝜆𝑧−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑦≥0,∀𝑧∈𝐶,_𝑛∈𝐶,𝐹₁𝑦_𝑛,𝑧+𝜑(𝑧)−𝜑𝑦_𝑛+1𝑟𝑧−𝑦_𝑛,𝑦_𝑛−𝑢_𝑛𝑥≥0,∀𝑧∈𝐶,_𝑛+1=𝑎_𝑛𝑥_𝑛+1−𝑎_𝑛𝑏_𝑛𝑇₁𝑦_𝑛+1−𝑏_𝑛𝑇₂𝑦_𝑛,∀𝑛∈ℕ, (3.3) where {𝑎_𝑛}, {𝑏_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1), and satisfy liminf_𝑛→∞𝑎_𝑛1−𝑎_𝑛>0,liminf_𝑛→∞𝑏_𝑛1−𝑏_𝑛>0, (3.4) then 𝑥_𝑛⇀𝑥∶=lim_𝑛→∞𝑃_{𝐹(𝑇1)∩𝐹(𝑇2)∩Ω}𝑥_𝑛 and (𝑥,𝑦) is a solution of problem (1.5), where 𝑦=𝑆_2,𝜆𝑥.

Proof. Let 𝑥^∗∈𝐹(𝑇₁)∩𝐹(𝑇₂)∩Ω, then 𝑥^∗=𝑇₁𝑥^∗=𝑇₂𝑥^∗ and 𝑥^∗=𝑆_1,𝑟(𝑆_2,𝜆𝑥^∗).
Putting 𝑦^∗=𝑆_2,𝜆𝑥^∗,𝑦_𝑛=𝑆_1,𝑟𝑢_𝑛, and 𝑢_𝑛=𝑆_2,𝜆𝑥_𝑛, we have 𝑦_𝑛−𝑥^∗=𝑆_1,𝑟𝑢_𝑛−𝑆_1,𝑟𝑦^∗≤𝑢_𝑛−𝑦^∗=𝑆_2,𝜆𝑥_𝑛−𝑆_2,𝜆𝑥^∗≤𝑥_𝑛−𝑥^∗. (3.5) Next, we prove that lim_𝑛→∞𝑥_𝑛−𝑥^∗exists. (3.6) Since 𝑇₁ and 𝑇₂ are quasi-nonexpansive, we obtain that 𝑏_𝑛𝑇₁𝑦_𝑛+(1−𝑏_𝑛)𝑇₂𝑦_𝑛−𝑥^∗2=𝑏_𝑛(𝑇₁𝑦_𝑛−𝑥^∗)+(1−𝑏_𝑛)(𝑇₂𝑦_𝑛−𝑥^∗)²=𝑏_𝑛𝑇₁𝑦_𝑛−𝑥^∗2+1−𝑏_𝑛𝑇₂𝑦_𝑛−𝑥^∗2−𝑏_𝑛1−𝑏_𝑛𝑇₁𝑦_𝑛−𝑇₂𝑦_𝑛²≤𝑏_𝑛𝑇₁𝑦_𝑛−𝑥^∗2+1−𝑏_𝑛𝑇₂𝑦_𝑛−𝑥^∗2≤𝑏_𝑛𝑦_𝑛−𝑥^∗2+1−𝑏_𝑛𝑦_𝑛−𝑥^∗2=𝑦_𝑛−𝑥^∗2≤𝑥_𝑛−𝑥^∗2, (3.7) which gives that 𝑥_𝑛+1−𝑥^∗2=𝑎_𝑛𝑥_𝑛+(1−𝑎_𝑛)(𝑏_𝑛𝑇₁𝑦_𝑛+(1−𝑏_𝑛)𝑇₂𝑦_𝑛)−𝑥^∗2=𝑎_𝑛(𝑥_𝑛−𝑥^∗)+(1−𝑎_𝑛)(𝑏_𝑛𝑇₁𝑦_𝑛+(1−𝑏_𝑛)𝑇₂𝑦_𝑛−𝑥^∗)²=𝑎_𝑛𝑥_𝑛−𝑥^∗2+1−𝑎_𝑛𝑏_𝑛𝑇₁𝑦_𝑛+(1−𝑏_𝑛)𝑇₂𝑦_𝑛−𝑥^∗2−𝑎_𝑛1−𝑎_𝑛𝑥_𝑛−(𝑏_𝑛𝑇₁𝑦_𝑛+(1−𝑏_𝑛)𝑇₂𝑦_𝑛)²≤𝑎_𝑛𝑥_𝑛−𝑥^∗2+1−𝑎_𝑛𝑥_𝑛−𝑥^∗2−𝑎_𝑛1−𝑎_𝑛𝑥_𝑛−(𝑏_𝑛𝑇₁𝑦_𝑛+(1−𝑏_𝑛)𝑇₂𝑦_𝑛)²=𝑥_𝑛−𝑥^∗2−𝑎_𝑛1−𝑎_𝑛𝑥_𝑛−(𝑏_𝑛𝑇₁𝑦_𝑛+(1−𝑏_𝑛)𝑇₂𝑦_𝑛)²≤𝑥_𝑛−𝑥^∗2. (3.8) Hence, {‖𝑥_𝑛−𝑥^∗‖} is a nonincreasing sequence, and hence, lim_𝑛→∞‖𝑥_𝑛−𝑥^∗‖ exists. This implies that {𝑥_𝑛},{𝑦_𝑛},{𝑢_𝑛},{𝑇₁𝑦_𝑛}, and {𝑇₂𝑦_𝑛} are bounded. From (3.8), we have 𝑎_𝑛1−𝑎_𝑛𝑥_𝑛−(𝑏_𝑛𝑇₁𝑦_𝑛+(1−𝑏_𝑛)𝑇₂𝑦_𝑛)²≤𝑥_𝑛−𝑥^∗2−𝑥_𝑛+1−𝑥^∗2. (3.9) Since liminf_𝑛→∞𝑎_𝑛(1−𝑎_𝑛)>0, this implies that lim_𝑛→∞𝑥_𝑛−𝑏_𝑛𝑇₁𝑦_𝑛+1−𝑏_𝑛𝑇₂𝑦_𝑛=0. (3.10) Furthermore, since 0<𝑎≤𝑎_𝑛≤𝑏<1, we have 𝑥_𝑛+1−𝑥_𝑛=𝑎_𝑛𝑥_𝑛+1−𝑎_𝑛𝑏_𝑛𝑇₁𝑦_𝑛+1−𝑏_𝑛𝑇₂𝑦_𝑛−𝑥_𝑛=1−𝑎_𝑛𝑏_𝑛𝑇₁𝑦_𝑛+1−𝑏_𝑛𝑇₂𝑦_𝑛−1−𝑎_𝑛𝑥_𝑛=1−𝑎_𝑛𝑏_𝑛𝑇₁𝑦_𝑛+1−𝑏_𝑛𝑇₂𝑦_𝑛−𝑥_𝑛≤(1−𝑎)𝑏_𝑛𝑇₁𝑦_𝑛+1−𝑏_𝑛𝑇₂𝑦_𝑛−𝑥_𝑛. (3.11) From (3.10), we conclude that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (3.12) From (3.7), we have 𝑏_𝑛1−𝑏_𝑛𝑇₁𝑦_𝑛−𝑇₂𝑦_𝑛²=𝑏_𝑛𝑇₁𝑦_𝑛−𝑥^∗2+1−𝑏_𝑛𝑇₂𝑦_𝑛−𝑥^∗2−𝑏_𝑛𝑇₁𝑦_𝑛+(1−𝑏_𝑛)𝑇₂𝑦_𝑛−𝑥^∗2≤𝑏_𝑛𝑦_𝑛−𝑥^∗2+1−𝑏_𝑛𝑦_𝑛−𝑥^∗2−𝑏_𝑛𝑇₁𝑦_𝑛+(1−𝑏_𝑛)𝑇₂𝑦_𝑛−𝑥^∗2=𝑦_𝑛−𝑥^∗2−𝑏_𝑛𝑇₁𝑦_𝑛+(1−𝑏_𝑛)𝑇₂𝑦_𝑛−𝑥^∗2≤𝑥_𝑛−𝑥^∗2−𝑏_𝑛𝑇₁𝑦_𝑛+(1−𝑏_𝑛)𝑇₂𝑦_𝑛−𝑥^∗2=𝑥_𝑛−𝑥^∗−𝑏_𝑛𝑇₁𝑦_𝑛+1−𝑏_𝑛𝑇₂𝑦_𝑛−𝑥^∗×𝑥_𝑛−𝑥^∗+𝑏_𝑛𝑇₁𝑦_𝑛+1−𝑏_𝑛𝑇₂𝑦_𝑛−𝑥^∗≤𝑀𝑥_𝑛−𝑥^∗−𝑏_𝑛𝑇₁𝑦_𝑛+1−𝑏_𝑛𝑇₂𝑦_𝑛−𝑥^∗≤𝑀𝑥_𝑛−𝑏_𝑛𝑇₁𝑦_𝑛+1−𝑏_𝑛𝑇₂𝑦_𝑛, (3.13) where 𝑀 is a constant satisfying 𝑀≥sup_𝑛≥1[‖𝑥_𝑛−𝑥^∗‖+‖𝑏_𝑛𝑇₁𝑦_𝑛+(1−𝑏_𝑛)𝑇₂𝑦_𝑛−𝑥^∗‖]. Again from (3.10), we conclude that lim_𝑛→∞𝑏_𝑛1−𝑏_𝑛𝑇₁𝑦_𝑛−𝑇₂𝑦_𝑛=0. (3.14) Using liminf_𝑛→∞𝑏_𝑛(1−𝑏_𝑛)>0, we have lim_𝑛→∞𝑇₁𝑦_𝑛−𝑇₂𝑦_𝑛=0. (3.15) Now, we prove that lim_𝑛→∞𝑥_𝑛+1−𝑇₁𝑦_𝑛=0,lim_𝑛→∞𝑥_𝑛+1−𝑇₂𝑦_𝑛=0. (3.16) We observe that 𝑥_𝑛+1−𝑇₁𝑦_𝑛=𝑎_𝑛𝑥_𝑛+1−𝑎_𝑛𝑏_𝑛𝑇₁𝑦_𝑛+1−𝑏_𝑛𝑇₂𝑦_𝑛−𝑇₁𝑦_𝑛=𝑎_𝑛𝑥_𝑛+1−𝑎_𝑛𝑏_𝑛𝑇₁𝑦_𝑛+1−𝑏_𝑛𝑇₂𝑦_𝑛−𝑎_𝑛+1−𝑎_𝑛𝑇₁𝑦_𝑛=𝑎_𝑛𝑥_𝑛−𝑇₁𝑦_𝑛+1−𝑎_𝑛𝑏_𝑛𝑇₁𝑦_𝑛+1−𝑏_𝑛𝑇₂𝑦_𝑛−𝑇₁𝑦_𝑛=𝑎_𝑛𝑥_𝑛−𝑇₁𝑦_𝑛+1−𝑎_𝑛𝑏_𝑛𝑇₁𝑦_𝑛+𝑇₂𝑦_𝑛−𝑏_𝑛𝑇₂𝑦_𝑛−𝑇₁𝑦_𝑛=𝑎_𝑛𝑥_𝑛−𝑇₁𝑦_𝑛+1−𝑎_𝑛1−𝑏_𝑛𝑇₂𝑦_𝑛−1−𝑏_𝑛𝑇₁𝑦_𝑛=𝑎_𝑛𝑥_𝑛−𝑇₁𝑦_𝑛+1−𝑎_𝑛1−𝑏_𝑛𝑇₂𝑦_𝑛−𝑇₁𝑦_𝑛≤𝑎_𝑛𝑥_𝑛−𝑇₁𝑦_𝑛+1−𝑎_𝑛1−𝑏_𝑛𝑇₂𝑦_𝑛−𝑇₁𝑦_𝑛=𝑎_𝑛𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑇₁𝑦_𝑛+1−𝑎_𝑛1−𝑏_𝑛𝑇₂𝑦_𝑛−𝑇₁𝑦_𝑛≤𝑎_𝑛𝑥_𝑛−𝑥_𝑛+1+𝑎_𝑛𝑥_𝑛+1−𝑇₁𝑦_𝑛+1−𝑎_𝑛1−𝑏_𝑛𝑇₂𝑦_𝑛−𝑇₁𝑦_𝑛, (3.17) which gives that 1−𝑎_𝑛𝑥_𝑛+1−𝑇₁𝑦_𝑛≤𝑎_𝑛𝑥_𝑛−𝑥_𝑛+1+1−𝑎_𝑛1−𝑏_𝑛𝑇₂𝑦_𝑛−𝑇₁𝑦_𝑛. (3.18) Since 0<𝑎≤𝑎_𝑛≤𝑏<1, we have 𝑎_𝑛1−𝑎_𝑛𝑥_𝑛+1−𝑇₁𝑦_𝑛≤1−𝑎_𝑛𝑥_𝑛+1−𝑇₁𝑦_𝑛≤𝑎_𝑛𝑥_𝑛−𝑥_𝑛+1+1−𝑎_𝑛1−𝑏_𝑛𝑇₂𝑦_𝑛−𝑇₁𝑦_𝑛. (3.19) Using (3.12) and (3.15), we conclude that lim_𝑛→∞𝑎_𝑛1−𝑎_𝑛𝑥_𝑛+1−𝑇₁𝑦_𝑛=0, (3.20) which gives that lim_𝑛→∞𝑥_𝑛+1−𝑇₁𝑦_𝑛=0, (3.21) since liminf_𝑛→∞𝑎_𝑛(1−𝑎_𝑛)>0. Similarly, we have 𝑥_𝑛+1−𝑇₂𝑦_𝑛=𝑎_𝑛𝑥_𝑛+1−𝑎_𝑛𝑏_𝑛𝑇₁𝑦_𝑛+1−𝑏_𝑛𝑇₂𝑦_𝑛−𝑇₂𝑦_𝑛=𝑎_𝑛𝑥_𝑛−𝑇₂𝑦_𝑛+1−𝑎_𝑛𝑏_𝑛𝑇₁𝑦_𝑛+1−𝑏_𝑛𝑇₂𝑦_𝑛−𝑇₂𝑦_𝑛=𝑎_𝑛𝑥_𝑛−𝑇₂𝑦_𝑛+1−𝑎_𝑛𝑏_𝑛𝑇₁𝑦_𝑛−𝑇₂𝑦_𝑛≤𝑎_𝑛𝑥_𝑛−𝑇₂𝑦_𝑛+1−𝑎_𝑛𝑏_𝑛𝑇₁𝑦_𝑛−𝑇₂𝑦_𝑛=𝑎_𝑛𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑇₂𝑦_𝑛+1−𝑎_𝑛𝑏_𝑛𝑇₂𝑦_𝑛−𝑇₁𝑦_𝑛≤𝑎_𝑛𝑥_𝑛−𝑥_𝑛+1+𝑎_𝑛𝑥_𝑛+1−𝑇₂𝑦_𝑛+1−𝑎_𝑛𝑏_𝑛𝑇₂𝑦_𝑛−𝑇₁𝑦_𝑛, (3.22) which implies that 1−𝑎_𝑛𝑥_𝑛+1−𝑇₂𝑦_𝑛≤𝑎_𝑛𝑥_𝑛−𝑥_𝑛+1+1−𝑎_𝑛𝑏_𝑛𝑇₂𝑦_𝑛−𝑇₁𝑦_𝑛. (3.23) Thus, we have 𝑎_𝑛1−𝑎_𝑛𝑥_𝑛+1−𝑇₂𝑦_𝑛≤1−𝑎_𝑛𝑥_𝑛+1−𝑇₂𝑦_𝑛≤𝑎_𝑛𝑥_𝑛−𝑥_𝑛+1+1−𝑎_𝑛𝑏_𝑛𝑇₂𝑦_𝑛−𝑇₁𝑦_𝑛≤𝑏𝑥_𝑛−𝑥_𝑛+1+1−𝑎_𝑛𝑇₂𝑦_𝑛−𝑇₁𝑦_𝑛≤𝑏𝑥_𝑛−𝑥_𝑛+1+(1−𝑎)𝑇₂𝑦_𝑛−𝑇₁𝑦_𝑛. (3.24) Hence, lim_𝑛→∞𝑎_𝑛(1−𝑎_𝑛)‖𝑥_𝑛+1−𝑇₂𝑦_𝑛‖=0. Since liminf_𝑛→∞𝑎_𝑛(1−𝑎_𝑛)>0, we have lim_𝑛→∞𝑥_𝑛+1−𝑇₂𝑦_𝑛=0. (3.25) Next, we prove that lim_𝑛→∞𝑦_𝑛−𝑇₁𝑦_𝑛=0,lim_𝑛→∞𝑦_𝑛−𝑇₂𝑦_𝑛=0. (3.26) Since 𝑆_1,𝑟 and 𝑆_2,𝜆 are firmly nonexpansive, it follows that 𝑢_𝑛−𝑦^∗2=𝑆_2,𝜆𝑥_𝑛−𝑆_2,𝜆𝑥^∗2≤⟨𝑆_2,𝜆𝑥_𝑛−𝑆_2,𝜆𝑥^∗,𝑥_𝑛−𝑥^∗⟩=⟨𝑢_𝑛−𝑦^∗,𝑥_𝑛−𝑥^∗⟩, (3.27) which gives that (𝑢_𝑛−𝑦^∗)−(𝑥_𝑛−𝑥^∗)²=𝑢_𝑛−𝑦^∗2−2⟨𝑢_𝑛−𝑦^∗,𝑥_𝑛−𝑥^∗⟩+𝑥_𝑛−𝑥^∗2≤𝑢_𝑛−𝑦^∗2−2𝑢_𝑛−𝑦^∗2+𝑥_𝑛−𝑥^∗2=−𝑢_𝑛−𝑦^∗2+𝑥_𝑛−𝑥^∗2. (3.28) This implies that 𝑢_𝑛−𝑦^∗2≤𝑥_𝑛−𝑥^∗2−(𝑢_𝑛−𝑦^∗)−(𝑥_𝑛−𝑥^∗)². (3.29) By the convexity of ‖⋅‖², we have 𝑥_𝑛+1−𝑥^∗2=𝑎_𝑛𝑥_𝑛+(1−𝑎_𝑛)(𝑏_𝑛𝑇₁𝑦_𝑛+(1−𝑏_𝑛)𝑇₂𝑦_𝑛)−𝑥^∗2=𝑎_𝑛(𝑥_𝑛−𝑥^∗)+(1−𝑎_𝑛)(𝑏_𝑛𝑇₁𝑦_𝑛+(1−𝑏_𝑛)𝑇₂𝑦_𝑛−𝑥^∗)²≤𝑎_𝑛𝑥_𝑛−𝑥^∗2+1−𝑎_𝑛𝑏_𝑛𝑇₁𝑦_𝑛+(1−𝑏_𝑛)𝑇₂𝑦_𝑛−𝑥^∗2≤𝑎_𝑛𝑥_𝑛−𝑥^∗2+1−𝑎_𝑛𝑦_𝑛−𝑥^∗2≤𝑎_𝑛𝑥_𝑛−𝑥^∗2+1−𝑎_𝑛𝑢_𝑛−𝑦^∗2≤𝑎_𝑛𝑥_𝑛−𝑥^∗2+1−𝑎_𝑛𝑥_𝑛−𝑥^∗2−(𝑢_𝑛−𝑦^∗)−(𝑥_𝑛−𝑥^∗)²=𝑥_𝑛−𝑥^∗2−1−𝑎_𝑛(𝑢_𝑛−𝑦^∗)−(𝑥_𝑛−𝑥^∗)². (3.30) Thus, 1−𝑎_𝑛(𝑢_𝑛−𝑦^∗)−(𝑥_𝑛−𝑥^∗)²≤𝑥_𝑛−𝑥^∗2−𝑥_𝑛+1−𝑥^∗2. (3.31) Since 0<𝑎≤𝑎_𝑛≤𝑏<1, we have (1−𝑏)(𝑢_𝑛−𝑦^∗)−(𝑥_𝑛−𝑥^∗)²≤1−𝑎_𝑛(𝑢_𝑛−𝑦^∗)−(𝑥_𝑛−𝑥^∗)²≤𝑥_𝑛−𝑥^∗2−𝑥_𝑛+1−𝑥^∗2. (3.32) Since lim_𝑛→∞‖𝑥_𝑛−𝑥^∗‖ exists, we have lim_𝑛→∞𝑢_𝑛−𝑥_𝑛+𝑥^∗−𝑦^∗=0. (3.33) Similarly, we have 𝑦_𝑛−𝑥^∗2=𝑆_1,𝑟𝑛𝑢_𝑛−𝑆_1,𝑟𝑛𝑦^∗2≤⟨𝑆_1,𝑟𝑛𝑢_𝑛−𝑆_1,𝑟𝑛𝑦^∗,𝑢_𝑛−𝑦^∗⟩=⟨𝑦_𝑛−𝑥^∗,𝑢_𝑛−𝑥^∗⟩, (3.34) which gives that (𝑦_𝑛−𝑥^∗)−(𝑢_𝑛−𝑦^∗)²=𝑦_𝑛−𝑥^∗2−2⟨𝑦_𝑛−𝑥^∗,𝑢_𝑛−𝑥^∗⟩+𝑢_𝑛−𝑦^∗2≤𝑦_𝑛−𝑥^∗2−2𝑦_𝑛−𝑥^∗2+𝑢_𝑛−𝑦^∗2=−𝑦_𝑛−𝑥^∗2+𝑢_𝑛−𝑦^∗2. (3.35) This implies that 𝑦_𝑛−𝑥^∗2≤𝑢_𝑛−𝑦^∗2−𝑦_𝑛−𝑢_𝑛−𝑥^∗+𝑦^∗2. (3.36) By the convexity of ‖⋅‖², we have 𝑥_𝑛+1−𝑥^∗2=𝑎_𝑛𝑥_𝑛+(1−𝑎_𝑛)(𝑏_𝑛𝑇₁𝑦_𝑛+(1−𝑏_𝑛)𝑇₂𝑦_𝑛)−𝑥^∗2≤𝑎_𝑛𝑥_𝑛−𝑥^∗2+1−𝑎_𝑛𝑦_𝑛−𝑥^∗2≤𝑎_𝑛𝑥_𝑛−𝑥^∗2+1−𝑎_𝑛𝑢_𝑛−𝑦^∗2−𝑦_𝑛−𝑢_𝑛−𝑥^∗+𝑦^∗2≤𝑎_𝑛𝑥_𝑛−𝑥^∗2+1−𝑎_𝑛𝑥_𝑛−𝑥^∗2−1−𝑎_𝑛𝑦_𝑛−𝑢_𝑛−𝑥^∗+𝑦^∗2=𝑥_𝑛−𝑥^∗2−1−𝑎_𝑛(𝑦_𝑛−𝑥^∗)−(𝑢_𝑛−𝑦^∗)². (3.37) Thus, 1−𝑎_𝑛𝑦_𝑛−𝑢_𝑛−𝑥^∗+𝑦^∗2≤𝑥_𝑛−𝑥^∗2−𝑥_𝑛+1−𝑥^∗2. (3.38) Since 0<𝑎≤𝑎_𝑛≤𝑏<1, we have (1−𝑏)𝑦_𝑛−𝑢_𝑛−𝑥^∗+𝑦^∗2≤1−𝑎_𝑛𝑦_𝑛−𝑢_𝑛−𝑥^∗+𝑦^∗2≤𝑥_𝑛−𝑥^∗2−𝑥_𝑛+1−𝑥^∗2. (3.39) Since lim_𝑛→∞‖𝑥_𝑛−𝑥^∗‖ exists, we have lim_𝑛→∞𝑦_𝑛−𝑢_𝑛−𝑥^∗+𝑦^∗=0. (3.40) Hence, 𝑏_𝑛𝑇₁𝑦_𝑛+1−𝑏_𝑛𝑇₂𝑦_𝑛−𝑦_𝑛=𝑏_𝑛𝑇₁𝑦_𝑛+1−𝑏_𝑛𝑇₂𝑦_𝑛−𝑥_𝑛+𝑥_𝑛−𝑢_𝑛+𝑢_𝑛−𝑦^∗+𝑦^∗−𝑥^∗+𝑥^∗−𝑦_𝑛≤𝑏_𝑛𝑇₁𝑦_𝑛+1−𝑏_𝑛𝑇₂𝑦_𝑛−𝑥_𝑛+𝑥_𝑛−𝑢_𝑛−𝑥^∗+𝑦^∗+𝑢_𝑛−𝑦_𝑛+𝑥^∗−𝑦^∗. (3.41) It follow from (3.10), (3.33), and (3.40) that lim_𝑛→∞𝑏_𝑛𝑇₁𝑦_𝑛+1−𝑏_𝑛𝑇₂𝑦_𝑛−𝑦_𝑛=0, (3.42) from which it follows that 𝑦_𝑛−𝑥_𝑛=𝑦_𝑛−𝑏_𝑛𝑇₁𝑦_𝑛+1−𝑏_𝑛𝑇₂𝑦_𝑛+𝑏_𝑛𝑇₁𝑦_𝑛+1−𝑏_𝑛𝑇₂𝑦_𝑛−𝑥_𝑛≤𝑦_𝑛−𝑏_𝑛𝑇₁𝑦_𝑛+1−𝑏_𝑛𝑇₂𝑦_𝑛+𝑏_𝑛𝑇₁𝑦_𝑛+1−𝑏_𝑛𝑇₂𝑦_𝑛−𝑥_𝑛⟶0as𝑛⟶∞, (3.43) that is, lim_𝑛→∞𝑦_𝑛−𝑥_𝑛=0. (3.44) Thus, 𝑦_𝑛−𝑇₁𝑦_𝑛≤𝑦_𝑛−𝑥_𝑛+𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑇₁𝑦_𝑛⟶0. (3.45) Similarly, we have ‖𝑦_𝑛−𝑇₂𝑦_𝑛‖→0. Since {𝑦_𝑛} is bounded sequence, there exists a subsequence {𝑦_𝑛𝑖} of {𝑦_𝑛} such that 𝑦_𝑛𝑖⇀𝑥 as 𝑖→∞. Since 𝑇₁ and 𝑇₂ are demiclosed at 0, we conclude that 𝑥∈𝐹(𝑇₁)∩𝐹(𝑇₂). Let 𝐺 be a mapping which is defined as in Lemma 3.1. Thus, we have 𝑦_𝑛−𝐺𝑦_𝑛=𝑆_1,𝑟𝑆_2,𝜆𝑥_𝑛−𝐺𝑦_𝑛=𝐺𝑥_𝑛−𝐺𝑦_𝑛≤𝑥_𝑛−𝑦_𝑛, (3.46) and hence, 𝑥_𝑛−𝐺𝑥_𝑛=𝑥_𝑛−𝑦_𝑛+𝑦_𝑛−𝐺𝑦_𝑛+𝐺𝑦_𝑛−𝐺𝑥_𝑛≤𝑥_𝑛−𝑦_𝑛+𝑦_𝑛−𝐺𝑦_𝑛+𝐺𝑦_𝑛−𝐺𝑥_𝑛≤𝑥_𝑛−𝑦_𝑛+𝑥_𝑛−𝑦_𝑛+𝑦_𝑛−𝑥_𝑛=3𝑥_𝑛−𝑦_𝑛⟶0. (3.47) This together with 𝑥_𝑛𝑖⇀𝑥 implies that 𝑥∈𝐹(𝐺)∶=Ω, if {𝑦_𝑛𝑗} is another subsequence of {𝑦_𝑛} such that 𝑦_𝑛𝑗⇀𝑥 as 𝑖→∞. Since 𝑇₁ and 𝑇₁ are demiclosed at 0, we conclude that 𝑥∈𝐹(𝑇₁)∩𝐹(𝑇₂)∩Ω. From 𝑥_𝑛𝑖⇀𝑥 and 𝑥_𝑛𝑗⇀𝑥, we will show that 𝑥=𝑥. Assume that 𝑥≠𝑥. Since lim_𝑛→∞‖𝑥_𝑛−𝑥^∗‖ exists for all 𝑥^∗∈𝐹(𝑇₁)∩𝐹(𝑇₂)∩Ω, by Opial’s Theorem 2.2, we have lim_𝑛→∞𝑥_𝑛−𝑥=liminf_𝑖→∞𝑥_𝑛𝑖−𝑥<liminf_𝑖→∞𝑥_𝑛𝑖−𝑥=lim_𝑛→∞𝑥_𝑛−𝑥=liminf_𝑗→∞𝑥_𝑛𝑗−𝑥<liminf_𝑗→∞𝑥_𝑛𝑗−𝑥=lim_𝑛→∞𝑥_𝑛−𝑥. (3.48) This is a contradiction. Thus, we have 𝑥=𝑥. This implies that 𝑦_𝑛⇀𝑥∈𝐹(𝑇₁)∩𝐹(𝑇₂)∩Ω. Since ‖𝑥_𝑛−𝑦_𝑛‖→0, we have 𝑥_𝑛⇀𝑥. Put 𝑧_𝑛=𝑃_{𝐹(𝑇1)∩𝐹(𝑇2)∩Ω}𝑥_𝑛. Finally, we show that 𝑥=lim_𝑛→∞𝑧_𝑛. Now from (2.4) and 𝑥∈𝐹(𝑇₁)∩𝐹(𝑇₂)∩Ω, we have ⟨𝑥−𝑧_𝑛,𝑧_𝑛−𝑥_𝑛⟩≥0. (3.49) Since {‖𝑥_𝑛−𝑥^∗‖} is nonnegative and nonincreasing for all 𝑥^∗∈𝐹(𝑇₁)∩𝐹(𝑇₂)∩Ω, it follows by Lemma 2.1 that {𝑧_𝑛} converges strongly to some 𝑥∈𝐹(𝑇₁)∩𝐹(𝑇₂)∩Ω. By (3.49), we have ⟨𝑥−𝑥,𝑥−𝑥⟩≥0. (3.50) Therefore, 𝑥=𝑥.

Setting 𝑇∶=𝑇₁=𝑇₂ in Theorem 3.2, we have the following result.

Corollary 3.3 (see [6]). Let 𝐶 be a closed convex subset of a real Hilbert space 𝐻. Let 𝐹₁ and 𝐹₂ be two bifunctions from 𝐶×𝐶→ℝ satisfying (A1)–(A4). Let 𝜆,𝜇>0, and let 𝑆_1,𝜆 and 𝑆_2,𝜇 be defined as in Lemma 2.5 associated to 𝐹₁ and 𝐹₂, respectively. Let 𝑇∶𝐶→𝐶 be a quasi-nonexpansive mapping such that 𝐼−𝑇 is demiclosed at zero and 𝐹(𝑇)∩Ω≠∅. Suppose that 𝑥₀=𝑥∈𝐶 and {𝑥_𝑛},{𝑦_𝑛},and{𝑧_𝑛} are given by 𝑧_𝑛∈𝐶,𝐹₂𝑧_𝑛,𝑧+𝜑(𝑧)−𝜑𝑧_𝑛+1𝜇𝑧−𝑧_𝑛,𝑧_𝑛−𝑥_𝑛𝑦≥0,∀𝑧∈𝐶,_𝑛∈𝐶,𝐹₁𝑦_𝑛,𝑧+𝜑(𝑧)−𝜑𝑦_𝑛+1𝜆𝑧−𝑦_𝑛,𝑦_𝑛−𝑧_𝑛𝑥≥0,∀𝑧∈𝐶,_𝑛+1=𝑎_𝑛𝑥_𝑛+1−𝑎_𝑛𝑇𝑦_𝑛, (3.51) for all 𝑛∈ℕ, where {𝑎_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1), and satisfy liminf_𝑛→∞𝑎_𝑛(1−𝑎_𝑛)>0, then {𝑥_𝑛} converges weakly to 𝑥=lim_𝑛→∞𝑃_{𝐹(𝑇)∩Ω}𝑥_𝑛 and (𝑥,𝑦) is a solution of problem (1.5), where 𝑦=𝑆_2,𝜇𝑥.

Setting 𝐹₁=𝐹₂≡0, 𝜑≡0 in Theorem 3.2, we have the following result.

Corollary 3.4 (see [9]). Let 𝐻 be a Hilbert space, let 𝐶 be a nonempty, closed, and convex subset of 𝐻, and let 𝑇₁,𝑇₂ be two quasi-nonexpansive mappings of 𝐶 into itself such that 𝐼−𝑇₁, 𝐼−𝑇₂ are demiclosed at zero with 𝐹(𝑇₁)∩𝐹(𝑇₂)≠∅. For any 𝑥₁ in C, let {𝑥_𝑛} be defined by 𝑥_𝑛+1=1−𝑎_𝑛𝑥_𝑛+𝑎_𝑛𝑏_𝑛𝑇₁𝑥_𝑛+1−𝑏_𝑛𝑇₂𝑥_𝑛, (3.52) where {𝑎_𝑛} and {𝑏_𝑛} are chosen so that liminf_𝑛→∞𝑎_𝑛1−𝑎_𝑛>0liminf_𝑛→∞𝑏_𝑛1−𝑏_𝑛>0, (3.53) then 𝑥_𝑛⇀𝑝∈𝐹(𝑇₁)∩𝐹(𝑇₂).

4. Applications

In this section, we apply our results to approximate a common element of the set of common fixed points of an asymptotic nonspreading mapping and an asymptotic 𝑇𝐽 mapping and the solution set of SMEP in a real Hilbert space. We recall the following definitions. A mapping 𝑇∶𝐶→𝐶 is called nonspreading [13] if2𝑇𝑥−𝑇𝑦²≤𝑇𝑥−𝑦²+𝑇𝑦−𝑥²∀𝑥,𝑦∈𝐶. (4.1) Furthermore, Takahashi and Yao [14] also introduced two nonlinear mappings in Hilbert spaces. A mapping 𝑇∶𝐶→𝐶 is called a 𝑇𝐽−1 mapping [14] if2𝑇𝑥−𝑇𝑦²=𝑥−𝑦²+𝑇𝑥−𝑦², (4.2) for all 𝑥,𝑦∈𝐶. A mapping 𝑇∶𝐶→𝐶 is called a 𝑇𝐽−2 [14] mapping if3𝑇𝑥−𝑇𝑦²=2𝑇𝑥−𝑦²+𝑇𝑦−𝑥², (4.3) for all 𝑥,𝑦∈𝐶. For these two nonlinear mappings, 𝑇𝐽−1 and 𝑇𝐽−2 mappings, Takahashi and Yao [14] studied the existence results of fixed points in Hilbert spaces. Very recently, Lin et al. [15] introduced the following definitions of new mappings.

Definition 4.1. Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻. We say that 𝑇∶𝐶→𝐶 is an asymptotic nonspreading mapping if there exist two functions 𝛼∶𝐶→[0,2) and 𝛽∶𝐶→[0,𝑘],𝑘<2, such that(A1)2‖𝑇𝑥−𝑇𝑦‖²≤𝛼(𝑥)‖𝑇𝑥−𝑦‖²+𝛽(𝑥)‖𝑇𝑦−𝑥‖² for all 𝑥,𝑦∈𝐶,(A2)0<𝛼(𝑥)+𝛽(𝑥)≤2 for all 𝑥∈𝐶.

Remark 4.2. The class of asymptotic nonspreading mappings contains the class of nonspreading mappings and the class of 𝑇𝐽−2 mappings in a Hilbert space. Indeed, in Definition 4.1, we know that(i)if 𝛼(𝑥)=𝛽(𝑥)=1 for all 𝑥∈𝐶, then 𝑇 is a nonspreading mapping,(ii)if 𝛼(𝑥)=4/3 and 𝛽(𝑥)=2/3 for all 𝑥∈𝐶, then 𝑇 is a 𝑇𝐽−2 mapping.

Definition 4.3. Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻. We say 𝑇∶𝐶→𝐶 is an asymptotic 𝑇𝐽 mapping if there exists two functions 𝛼∶𝐶→[0,2] and 𝛽∶𝐶→[0,𝑘],𝑘<2, such that(B1)2‖𝑇𝑥−𝑇𝑦‖²≤𝛼(𝑥)‖𝑥−𝑦‖²+𝛽(𝑥)‖𝑇𝑥−𝑦‖² for all 𝑥,𝑦∈𝐶,(B2)𝛼(𝑥)+𝛽(𝑥)≤2 for all 𝑥∈𝐶.

Remark 4.4. The class of asymptotic 𝑇𝐽 mappings contains the class of 𝑇𝐽−1 mappings and the class of nonexpansive mappings in a Hilbert space. Indeed, in Definition 4.3, we know that(i)if 𝛼(𝑥)=2 and 𝛽(𝑥)=0 for each 𝑥∈𝐶, then 𝑇 is a nonexpansive mapping,(ii)if 𝛼(𝑥)=𝛽(𝑥)=1 for each 𝑥∈𝐶, then 𝑇 is a 𝑇𝐽−1 mapping.

It is well known that the set 𝐹(𝑇) of fixed points of a quasi-nonexpansive mapping 𝑇 is a closed and convex set [1]. Hence, if 𝑇∶𝐶→𝐶 is an asymptotic nonspreading mapping (resp., asymptotic 𝑇𝐽 mapping) with 𝐹(𝑇)≠∅, then 𝑇 is a quasi-nonexpansive mapping, and this implies that 𝐹(𝑇) is a nonempty closed convex subset of 𝐶.

Theorem 4.5 (see [15]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻, and let 𝑇∶𝐶→𝐶 be an asymptotic nonspreading mapping, then 𝐼−𝑇 is demiclosed at 0.

Theorem 4.6 (see [15]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻, and let 𝑇∶𝐶→𝐶 be an asymptotic 𝑇𝐽 mapping, then 𝐼−𝑇 is demiclosed at 0.

Applying the above results, we have the following theorem.

Theorem 4.7. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐹₁ and 𝐹₂ be two bifunctions from 𝐶×𝐶→ℝ satisfying (A1)–(A4). Let 𝑟,𝜆>0 and 𝑆_1,𝑟 and 𝑆_2,𝜆 be defined as in Lemma 2.5 associated to 𝐹₁ and 𝐹₂, respectively. Let 𝑇_𝑖∶𝐶→𝐶, 𝑖=1,2, be any one of asymptotic nonspreading mapping and asymptotic 𝑇𝐽 mapping such that 𝐹(𝑇₁)∩𝐹(𝑇₂)∩Ω≠∅. Let {𝑥_𝑛},{𝑦_𝑛}, and {𝑢_𝑛} be given by 𝑥₁𝑢∈𝐶chosenarbitrary,_𝑛∈𝐶,𝐹₂𝑢_𝑛,𝑧+𝜑(𝑧)−𝜑𝑢_𝑛+1𝜆𝑧−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑦≥0,∀𝑧∈𝐶,_𝑛∈𝐶,2𝐹₁𝑦_𝑛,𝑧+𝜑(𝑧)−𝜑𝑦_𝑛+1𝑟𝑧−𝑦_𝑛,𝑦_𝑛−𝑢_𝑛𝑥≥0,∀𝑧∈𝐶,_𝑛+1=𝑎_𝑛𝑥_𝑛+1−𝑎_𝑛𝑏_𝑛𝑇₁𝑦_𝑛+1−𝑏_𝑛𝑇₂𝑦_𝑛,∀𝑛∈ℕ, (4.4) where {𝑎_𝑛},{𝑏_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1) and satisfying liminf_𝑛→∞𝑎_𝑛1−𝑎_𝑛>0,liminf_𝑛→∞𝑏_𝑛1−𝑏_𝑛>0, (4.5) then 𝑥_𝑛⇀𝑥∶=lim_𝑛→∞𝑃_{𝐹(𝑇1)∩𝐹(𝑇2)∩Ω}𝑥_𝑛, and (𝑥,𝑦) is a solution of problem (1.5), where 𝑦=𝑆_2,𝜆𝑛𝑥.

Setting 𝐹₁=𝐹₂∶=𝐹 and 𝜑≡0 in the above theorem, we have the following result.

Corollary 4.8. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐹 be the bifunctions from 𝐶×𝐶→ℝ satisfying (A1)–(A4). Let 𝑇_𝑖∶𝐶→𝐶, 𝑖=1,2, be any one of asymptotic nonspreading mapping and asymptotic 𝑇𝐽 mapping such that ℱ∶=𝐹(𝑇₁)∩𝐹(𝑇₂)∩EP(𝐹)≠∅. For given 𝑢∈𝐶 and 𝑟>0, let the sequences {𝑥_𝑛} and {𝑢_𝑛} be defined by 𝑥₁𝑢∈𝐶chosenarbitrary,_𝑛∈𝐶,𝐹𝑢_𝑛+1,𝑦𝑟𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥≥0∀𝑦∈𝐶,_𝑛+1=𝑎_𝑛𝑥_𝑛+1−a_𝑛𝑏_𝑛𝑇₁𝑢_𝑛+1−𝑏_𝑛𝑇₂𝑢_𝑛∀𝑛∈ℕ, (4.6) where {𝑎_𝑛}, {𝑏_𝑛} are two sequences in (0,1) satisfying liminf_𝑛→∞𝑎_𝑛1−𝑎_𝑛>0,liminf_𝑛→∞𝑏_𝑛1−𝑏_𝑛>0, (4.7) then 𝑥_𝑛⇀𝑤 for some 𝑤∈ℱ.

Setting 𝐹₁=𝐹₂≡0 and 𝜑≡0 in Theorem 4.7, we have the following result.

Corollary 4.9 (see [15]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻, and let 𝑇_𝑖∶𝐶→𝐶, 𝑖=1,2, be any one of asymptotic nonspreading mapping and asymptotic 𝑇𝐽 mapping. Let ℱ∶=𝐹(𝑇₁)∩𝐹(𝑇₂)≠∅. Let {𝑎_𝑛} and {𝑏_𝑛} be two sequences in (0,1). Let {𝑥_𝑛} be defined by 𝑥₁𝑥∈𝐶chosenarbitrary,_𝑛+1=𝑎_𝑛𝑥_𝑛+1−𝑎_𝑛𝑏_𝑛𝑇₁𝑥_𝑛+1−𝑏_𝑛𝑇₂𝑥_𝑛. (4.8) Assume that liminf_𝑛→∞𝑎_𝑛(1−𝑎_𝑛)>0 and liminf_𝑛→∞𝑏_𝑛(1−𝑏_𝑛)>0, then 𝑥_𝑛⇀𝑤 for some 𝑤∈ℱ.

Acknowledgment

This paper is supported by the Centre of Excellence in Mathematics under the Commission on Higher Education, Ministry of Education, Thailand.