Iterative Algorithms with Perturbations for Solving the Systems of Generalized Equilibrium Problems and the Fixed Point Problems of Two Quasi-Nonexpansive Mappings

Abstract

We introduce new iterative algorithms with perturbations for finding a common element of the set of solutions of the system of generalized equilibrium problems and the set of common fixed points of two quasi-nonexpansive mappings in a Hilbert space. Under suitable conditions, strong convergence theorems are obtained. Furthermore, we also consider the iterative algorithms with perturbations for finding a common element of the solution set of the systems of generalized equilibrium problems and the common fixed point set of the super hybrid mappings in Hilbert spaces.

1. Introduction

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖ and 𝐶 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 quasi-nonexpansive 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 closed and convex; see Itoh and Takahashi [1]. A mapping 𝑇∶𝐶→𝐻 is called nonspreading [2] if 2𝑇𝑥−𝑇𝑦²≤𝑇𝑥−𝑦²+𝑇𝑦−𝑥², (1.1) for all 𝑥,𝑦∈𝐶. We remark that nonlinear every nonspreading mappings are quasi-nonexpansive mappings if the set of fixed points is nonempty.

Recall that a mapping Ψ∶𝐶→𝐻 is said to be 𝜇-inverse strongly monotone if there exists a positive real number 𝜇 such that Ψ𝑥−Ψ𝑦,𝑥−𝑦≥𝜇Ψ𝑥−Ψ𝑦²,∀𝑥,𝑦∈𝐶. (1.2) If Ψ is a 𝜇-inverse strongly monotone mapping of 𝐶 into 𝐻, then it is obvious that Ψ is 1/𝜇-Lipschitz continuous.

Let 𝐺∶𝐶×𝐶→ℝ be a bifuction and Ψ∶𝐶→𝐻 be 𝜇-inverse strongly monotone mapping. The generalized equilibrium problem (for short, GEP) for 𝐹 and Ψ is to find 𝑧∈𝐶 such that 𝐺𝑧,𝑦+Ψ𝑧,𝑦−𝑧≥0,∀𝑦∈𝐶. (1.3) The problem (1.3) was studied by Moudafi [3]. The set of solutions for problem (1.3) is denoted by GEP(𝐹,Ψ), that is, GEP(𝐹,Ψ)=𝑧∈𝐶∶𝐺𝑧,𝑦+Ψ𝑧,𝑦−𝑧≥0,∀𝑦∈𝐶. (1.4) If Ψ≡0 in (1.3), then GEP reduces to the classical equilibrium problem and GEP(𝐺,0) is denoted by EP(𝐺), that is, EP(𝐺)=𝑧∈𝐶∶𝐺𝑧,𝑦≥0,∀𝑦∈𝐶. (1.5) If 𝐺≡0 in (1.3), then GEP reduces to the classical variational inequality and GEP(0,Ψ) is denoted by VI(Ψ,𝐶), that is, VI(Ψ,𝐶)=𝑧∈𝐶∶Ψ𝑧,𝑦−𝑧≥0,∀𝑦∈𝐶. (1.6) The problem (1.3) 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 [4] and Moudafi [3].

In 2005, Combettes and Hirstoaga [5] introduced an iterative algorithm of finding the best approximation to the initial data and proved a strong convergence theorem. In 2007, by using the viscosity approximation method, S. Takahashi and W. Takahashi [6] introduced another iterative scheme for finding a common element of the set of solutions of the equilibrium problem and the set of fixed points of a nonexpansive mapping. Subsequently, algorithms constructed for solving the equilibrium problems and fixed point problems have further developed by some authors. In particular, Ceng and Yao [7] introduced an iterative scheme for finding a common element of the set of solutions of the mixed equilibrium problem and the set of common fixed points of finitely many nonexpansive mappings. Mainge and Moudafi [8] introduced an iterative algorithm for equilibrium problems and fixed point problems. Wangkeeree [9] introduced a new iterative scheme for finding the common element of the set of common fixed points of nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of the variational inequality. Wangkeeree and Kamraksa [10] introduced an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of a general system of variational inequalities for a cocoercive mapping in a real Hilbert space. Their results extend and improve many results in the literature.

In 1967, Wittmann [11] (see also [12]) proved the strong convergence theorem of Halpern’s type [13] {𝑥_𝑛} defined by, for any 𝑥₁=𝑥∈𝐶, 𝑥_𝑛+1=𝛼_𝑛𝑥+1−𝛼_𝑛𝑇𝑥_𝑛,∀𝑛∈ℕ, (1.7) where {𝛼_𝑛}⊂(0,1) satisfies lim_𝑛→∞𝛼_𝑛=0,∑^∞_𝑛=1𝛼_𝑛=∞, and ∑^∞_𝑛=1|𝛼_𝑛−𝛼_𝑛+1|<∞. In [14], Kurokawa and Takahashi also studied the following Halpern’s type for nonspreading mappings in a Hilbert space; see also Hojo and Takahashi [15]. Let 𝑇 be a nonspreading mapping of 𝐶 into itself. Let 𝑢∈𝐶 and define two sequences {𝑥_𝑛} and {𝑧_𝑛} in 𝐶 as follows: 𝑥₁=𝑥∈𝐶 and 𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑧_𝑛,where𝑧_𝑛=1𝑛^𝑛−1_𝑘=0𝑇^𝑘𝑥_𝑛 (1.8) for all 𝑛=1,2,…, where {𝛼_𝑛}⊂(0,1), lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞. If 𝐹(𝑇) is nonempty, they proved that {𝑥_𝑛} and {𝑧_𝑛} converge strongly to 𝑃_𝐹(𝑇)𝑢, where 𝑃_𝐹(𝑇) is the metric projection of 𝐻 onto 𝐹(𝑇). Recently, Yao and Shahzad [16] gave the following iteration process for nonexpansive mappings with perturbation: 𝑥₁∈𝐶 and 𝑥_𝑛+1=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑃_𝐶𝛼_𝑛𝑢_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,∀𝑛∈ℕ, (1.9) where {𝛼_𝑛} and {𝛽_𝑛} are sequences in [0,1], and the sequence {𝑢_𝑛}⊆𝐻 is a small perturbation for the 𝑛-step iteration satisfying ‖𝑢_𝑛‖→0 as 𝑛→∞. In fact, there are perturbations always occurring in the iterative processes because the manipulations are inaccurate.

On the other hand, very recently, Chuang et al. [17] considered the following iteration process for finding a common element of the set of solutions of the equilibrium problem and the set of common fixed points for a quasi-nonexpansive mapping 𝑇∶𝐶→𝐻 with perturbation 𝑞₁𝑥∈𝐻,_𝑛∈𝐶,suchthat𝐺𝑥_𝑛+1,𝑦𝑟_𝑛𝑦−𝑥_𝑛,𝑥_𝑛−𝑞_𝑛𝑦≥0,∀𝑦∈𝐶,_𝑛=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑇𝑥_𝑛,𝑞_𝑛+1=𝛼_𝑛𝑢_𝑛+1−𝛼_𝑛𝑦_𝑛,∀𝑛∈ℕ, (1.10) where 𝐶 is a nonempty closed convex subset of 𝐻, 𝐺∶𝐶×𝐶→ℝ is a function, {𝛼_𝑛} and {𝛽_𝑛} are real sequences in (0,1), and {𝑢_𝑛}⊂𝐻 is a convergent sequence and {𝑟_𝑛}⊂[𝑎,∞) for some 𝑎>0. They obtained a strong convergence theorem for such iterations.

In this paper, inspired and motivated by Yao and Shahzad [16], S. Takahashi and W. Takahashi [18] and Chuang et al. [17], we introduce a new iterative algorithms with perturbations for finding a common element of the set of solutions of the system of generalized equilibrium problems and the set of common fixed points of two quasi-nonexpansive mappings in a Hilbert space. Under suitable conditions, strong convergence theorems are obtained. Furthermore, we also consider the iterative algorithms with perturbations for finding a common element of the solution set of the system of generalized equilibrium problems and the common fixed point set of the super hybrid mappings in a Hilbert space.

2. Preliminaries

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖. We denote the strongly convergence and the weak convergence of {𝑥_𝑛} to 𝑥∈𝐻 by 𝑥_𝑛→𝑥 and 𝑥_𝑛⇀𝑥, respectively. In a Hilbert space, it is known that 𝜆𝑥+(1−𝜆)𝑦²=𝜆‖𝑥‖²+(1−𝜆)𝑦²−𝜆(1−𝜆)𝑥−𝑦², (2.1) for all 𝑥,𝑦∈𝐻 and 𝜆∈ℝ; see [19]. Furthermore, we have that for any 𝑥,𝑦,𝑢,𝑣∈𝐻2𝑥−𝑦,𝑢−𝑣=‖𝑥−𝑣‖²+𝑦−𝑢²−‖𝑥−𝑢‖²−𝑦−𝑣². (2.2) Let 𝐶 be a nonempty closed convex subset of 𝐻 and 𝑥∈𝐻. We know that there exists a unique nearest point 𝑧∈𝐶 such that ‖𝑥−𝑧‖=inf_𝑦∈𝐶‖𝑥−𝑦‖. We denote such a correspondence by 𝑧=𝑃_𝐶𝑥. The mapping 𝑃_𝐶 is called the metric projection of 𝐻 onto 𝐶. It is known that 𝑃_𝐶 is nonexpansive and 𝑥−𝑃_𝐶𝑥,𝑃_𝐶𝑥−𝑢≥0; (2.3) for all 𝑥∈𝐻 and 𝑢∈𝐶; see [19, 20] for more details.

Let 𝐶 be a nonempty, closed and convex subset of 𝐻 and let 𝐺∶𝐶×𝐶→ℝ be a bifunction. For solving the generalized equilibrium problem, let us assume that the bifunction 𝐺∶𝐶×𝐶→ℝ satisfies the following conditions:(A1)𝐺(𝑥,𝑥)=0 for all 𝑥∈𝐶; (A2)𝐺 is monotone, that is, 𝐺(𝑥,𝑦)+𝐺(𝑦,𝑥)≤0 for any 𝑥,𝑦∈𝐶;(A3) for each 𝑥,𝑦,𝑧∈𝐶lim_𝑡↓0𝐺𝑡𝑧+(1−𝑡)𝑥,𝑦≤𝐺𝑥,𝑦; (2.4)(A4) for each 𝑥∈𝐶,𝐺(𝑥,⋅) is convex and lower semicontinuous.We know the following lemma which appears implicitly in Blum and Oettli [4].

Lemma 2.1 (see [4]). Let 𝐶 be a nonempty closed convex subset of 𝐻 and let 𝐺 be a bifunction of 𝐶×𝐶 into ℝ satisfying (A1)−(A4). Let 𝑟>0 and 𝑥∈𝐻. Then, there exists a unique 𝑧∈𝐶 such that 𝐺+1𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶. (2.5)

The following lemma was also given in Combettes and Hirstoaga [5].

Lemma 2.2 (see [5]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐺∶𝐶×𝐶→ℝ be a bifunction which satisfies conditions (A1)−(A4). For 𝑟>0 and 𝑥∈𝐻, define a mapping 𝑇_𝑟∶𝐻→𝐶 as follows: 𝑇_𝑟(𝑥)=+1𝑧∈𝐶∶𝐺𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶 (2.6) for all 𝑥∈𝐻. Then the following hold:(i)𝑇_𝑟 is single-valued;(ii)𝑇_𝑟 is firmly nonexpansive, that is, for any 𝑥,𝑦∈𝐶, 𝑇_𝑟𝑥−𝑇_𝑟𝑦²≤𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝑥−𝑦; (2.7)(iii)(𝐸𝑃) is a closed convex subset of 𝐶;(iv)𝐹(𝑇_𝑟)=𝐸𝑃(𝐺).

Remark 2.3. For any 𝑥∈𝐻 and 𝑟>0, by Lemma 2.2 (i), there exists 𝑢∈𝐻 such that 𝐺+1𝑢,𝑦𝑟𝑦−𝑢,𝑢−𝑥≥0,∀𝑦∈𝐻. (2.8) Replacing 𝑥 with 𝑥−𝑟Ψ𝑥∈𝐻 in (2.8), we have 𝐺++1𝑢,𝑦Ψ𝑥,𝑦−𝑢𝑟𝑦−𝑢,𝑢−𝑥≥0,∀𝑦∈𝐻, (2.9) where Ψ∶𝐻→𝐻 is an inverse strongly monotone mapping.

Lemma 2.4 (see [21]). Let {Γ_𝑛} be a sequence of real numbers that does not decrease at infinity in the sense that there exists a subsequence {Γ_𝑛𝑖} of {Γ_𝑛} which satisfies Γ_𝑛𝑖<Γ_𝑛𝑖+1 for all 𝑖∈ℕ. Define the sequence {𝜏(𝑛)}_𝑛≥𝑛0 of integers as follows: 𝜏(𝑛)=max𝑘≤𝑛∶Γ_𝑘<Γ_𝑘+1, (2.10) where 𝑛₀∈ℕ such that {𝑘≤𝑛₀∶Γ_𝑘<Γ_𝑘+1}≠∅. Then, the following hold:(i)𝜏(1)≤𝜏(2)≤⋯ and 𝜏(𝑛)→∞;(ii)Γ_𝜏(𝑛)≤Γ_𝜏(𝑛)+1 and Γ_𝑛≤Γ_𝜏(𝑛)+1,∀𝑛∈ℕ.

Lemma 2.5 (see [22]). Let {𝑎_𝑛}_𝑛∈ℕ be a sequence of nonnegative real numbers, {𝛼_𝑛} a sequence of real numbers in [0,1] with ∑^∞_𝑛=1𝛼_𝑛=∞,{𝑢_𝑛} a sequence of nonnegative real numbers with ∑^∞_𝑛=1𝑢_𝑛<∞,{𝑡_𝑛} a sequence of real numbers with limsup𝑡_𝑛≤0. Suppose that 𝑎_𝑛+1≤1−𝛼_𝑛𝑎_𝑛+𝛼_𝑛𝑡_𝑛+𝑢_𝑛,∀𝑛∈ℕ. (2.11) Then lim_𝑛→∞𝑎_𝑛=0.

3. Main Results

Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻. For each 𝑖=1,2,…,𝑘, let 𝐺_𝑖∶𝐶×𝐶→ℝ be a bifunction satisfying (A1)−(A4) and Ψ_𝑖 a 𝜇_𝑖-inverse strongly monotone mapping. For each 𝑗=1,2, let 𝑇_𝑗∶𝐶→𝐻 be two mappings. Let {𝑥_𝑛} be a sequence generated in the following manner: 𝑥₁𝐺∈𝐻,₁𝑢_𝑛,1,𝑦+Ψ₁𝑥_𝑛,𝑦−𝑢_𝑛,1+1𝑟_𝑛𝑦−𝑢_𝑛,1,𝑢_𝑛,1−𝑥_𝑛𝐺≥0,∀𝑦∈𝐶,₂𝑢_𝑛,2,𝑦+Ψ₂𝑥_𝑛,𝑦−𝑢_𝑛,2+1𝑟_𝑛𝑦−𝑢_𝑛,2,𝑢_𝑛,2−𝑥_𝑛⋮𝐺≥0,∀𝑦∈𝐶,_𝑘𝑢_𝑛,𝑘,𝑦+Ψ_𝑘𝑥_𝑛,𝑦−𝑢_𝑛,𝑘+1𝑟_𝑛𝑦−𝑢_𝑛,𝑘,𝑢_𝑛,𝑘−𝑥_𝑛𝜔≥0,∀𝑦∈𝐶,_𝑛=1𝑘^𝑘_𝑖=1𝑢_𝑛,𝑖,𝑦_𝑛=𝛾_𝑛𝜔_𝑛+1−𝛾_𝑛𝑇₁𝜔_𝑛,𝑧_𝑛=𝛽_𝑛𝑦_𝑛+1−𝛽_𝑛𝑇₂𝜔_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑢_𝑛+1−𝛼_𝑛𝑧_𝑛,∀𝑛∈ℕ, (3.1) where {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛} are sequences in (0, 1) and {𝑢_𝑛}⊂𝐻 is a sequence and {𝑟_𝑛}⊂[𝑎,2𝜇_𝑖) for some 𝑎>0 and for all 𝑖∈{1,2,…,𝑘}. Under certain appropriate assumptions imposed on the sequences {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛}, the strong convergence theorem of {𝑥_𝑛} defined by (3.1) is studied in the following theorem.

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻. For each 𝑖=1,2,…,𝑘, let 𝐺_𝑖∶𝐶×𝐶→ℝ be a bifunction satisfying (A1)−(A4) and Ψ_𝑖 a 𝜇_𝑖-inverse strongly monotone mapping. For each 𝑗=1,2, let 𝑇_𝑗∶𝐶→𝐻 be two quasi-nonexpansive mappings such that 𝐼−𝑇_𝑗 are demiclosed at zero with Ω∶=𝐹(𝑇₁)∩𝐹(𝑇₂)∩(∩^𝑘_𝑖=1GEP(𝐺_𝑖,Ψ_𝑖))≠∅. Let the sequences {𝑥_𝑛},  {𝑦_𝑛}, and {𝑧_𝑛} be defined by (3.1), where {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛}, and {𝑢_𝑛} satisfy the following conditions:(C1)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞;(C2)liminf_𝑛→∞𝛽_𝑛(1−𝛽_𝑛)>0;(C3)liminf_𝑛→∞𝛾_𝑛(1−𝛾_𝑛)>0; (C4)lim_𝑛→∞𝑢_𝑛=𝑢 for some 𝑢∈𝐻.Then {𝑥_𝑛} converges strongly to 𝑥^∗, where 𝑥^∗=𝑃_Ω𝑢.

Proof. We first have that for all 𝑖=1,2,…,𝑘, 𝐼−𝑟_𝑛Ψ_𝑖 is a nonexpansive mapping. Indeed, for all 𝑥,𝑦∈𝐶, we obtain 𝐼−𝑟_𝑛Ψ_𝑖𝑥−𝐼−𝑟_𝑛Ψ_𝑖𝑦²=𝑥−𝑦−𝑟_𝑛Ψ_𝑖𝑥−Ψ_𝑖𝑦²=𝑥−𝑦²−2𝑟_𝑛Ψ_𝑖𝑥−Ψ_𝑖𝑦,𝑥−𝑦+𝑟²_𝑛Ψ_𝑖𝑥−Ψ_𝑖𝑦²≤𝑥−𝑦²−𝑟_𝑛2𝜇_𝑖−𝑟_𝑛Ψ_𝑖𝑥−Ψ_𝑖𝑦²≤𝑥−𝑦². (3.2) Thus 𝐼−𝑟_𝑛Ψ_𝑖 is nonexpansive for each 𝑖∈{1,2,…,𝑘}. Now, let 𝑤∈Ω be arbitrary. By (C4), {𝑢_𝑛} is a bounded sequence, there exists 𝑀≤0 such that sup_𝑛∈ℕ𝑢_𝑛−𝑤≤𝑀. (3.3) For each 𝑖=1,2,…,𝑘 and 𝑛∈ℕ, we have from 𝑢_𝑛,𝑖=𝑇_{𝑟𝑛,𝑖}(𝑥_𝑛−𝑟_𝑛Ψ_𝑖𝑥_𝑛) that 𝑢_𝑛,𝑖−𝑤=𝑇_{𝑟𝑛,𝑖}𝑥_𝑛−𝑟_𝑛Ψ_𝑖𝑥_𝑛−𝑇_{𝑟𝑛,𝑖}𝑤−𝑟_𝑛Ψ_𝑖𝑤≤𝑥_𝑛−𝑟_𝑛Ψ_𝑖𝑥_𝑛−𝑤−𝑟_𝑛Ψ_𝑖𝑤≤𝑥_𝑛−𝑤, (3.4) which gives also that 𝜔_𝑛≤1−𝑤𝑘^𝑘_𝑖=1𝑢_𝑛,𝑖−𝑤≤𝑥_𝑛−𝑤∀𝑤∈Ω. (3.5) Since 𝑇₁ is quasi-nonexpansive, we have 𝑦_𝑛−𝑤=𝛾_𝑛𝜔_𝑛+1−𝛾_𝑛𝑇₁𝜔_𝑛=𝛾−𝑤_𝑛𝜔_𝑛−𝑤+1−𝛾_𝑛𝑇₁𝜔_𝑛−𝑤≤𝛾_𝑛𝜔_𝑛−𝑤+1−𝛾_𝑛𝑇₁𝜔_𝑛≤𝜔−𝑤_𝑛−𝑤. (3.6) So, we have from (3.5) and (3.6) and the quasi-nonexpansiveness of 𝑇₂ that 𝑥_𝑛+1−𝑤=𝛼_𝑛𝑢_𝑛−𝑤+1−𝛼_𝑛𝑧_𝑛−𝑤≤𝛼_𝑛𝑢_𝑛−𝑤+1−𝛼_𝑛𝑧_𝑛−𝑤≤𝛼_𝑛𝑢_𝑛−𝑤+1−𝛼_𝑛𝛽_𝑛𝑦_𝑛−𝑤+1−𝛽_𝑛𝑇₂𝜔_𝑛−𝑤≤𝛼_𝑛𝑢_𝑛−𝑤+1−𝛼_𝑛𝛽_𝑛𝜔_𝑛−𝑤+1−𝛽_𝑛𝜔_𝑛−𝑤≤𝛼_𝑛𝑢_𝑛−𝑤+1−𝛼_𝑛𝜔_𝑛−𝑤≤𝛼_𝑛𝑢_𝑛−𝑤+1−𝛼_𝑛𝑥_𝑛𝑥−𝑤≤max𝑀,_𝑛−𝑤. (3.7) By Induction, we have that 𝑥_𝑛−𝑤≤max𝑥₁−𝑤,𝑀,∀𝑛∈ℕ. (3.8) Thus we obtain that {‖𝑥_𝑛−𝑤‖} is bounded, so also {𝑥_𝑛},{𝑦_𝑛},{𝑧_𝑛},{𝜔_𝑛},{𝑇₁𝜔_𝑛}, and {𝑇₂𝜔_𝑛} are bounded. Since Ω is closed and convex, we can take 𝑥^∗=𝑃_Ω𝑢. It follows that 𝑦_𝑛−𝑥^∗2=𝛾_𝑛𝜔_𝑛−𝑥^∗+1−𝛾_𝑛𝑇₁𝜔_𝑛−𝑥^∗2=𝛾_𝑛𝜔_𝑛−𝑥^∗2+1−𝛾_𝑛𝑇₁𝜔_𝑛−𝑥^∗2−𝛾_𝑛1−𝛾_𝑛𝜔_𝑛−𝑇₁𝜔_𝑛²≤𝛾_𝑛𝜔_𝑛−𝑥^∗2+1−𝛾_𝑛𝜔_𝑛−𝑥^∗2−𝛾_𝑛1−𝛾_𝑛𝜔_𝑛−𝑇₁𝜔_𝑛²=𝜔_𝑛−𝑥^∗2−𝛾_𝑛1−𝛾_𝑛𝜔_𝑛−𝑇₁𝜔_𝑛²≤𝜔_𝑛−𝑥^∗2. (3.9) From (3.9), we have 𝑧_𝑛−𝑥^∗2=𝛽_𝑛𝑦_𝑛−𝑥^∗+1−𝛽_𝑛𝑇₂𝜔_𝑛−𝑥^∗2=𝛽_𝑛𝑦_𝑛−𝑥^∗2+1−𝛽_𝑛𝑇₂𝜔_𝑛−𝑥^∗2−𝛽_𝑛1−𝛽_𝑛𝑦_𝑛−𝑇₂𝜔_𝑛²≤𝛽_𝑛𝜔_𝑛−𝑥^∗2+1−𝛽_𝑛𝜔_𝑛−𝑥^∗2−𝛽_𝑛1−𝛽_𝑛𝑦_𝑛−𝑇₂𝜔_𝑛²=𝜔_𝑛−𝑥^∗2−𝛽_𝑛1−𝛽_𝑛𝑦_𝑛−𝑇₂𝜔_𝑛²≤𝜔_𝑛−𝑥^∗2. (3.10) Hence we have from (3.5), (3.9), and (3.10) that 𝜔_𝑛+1−𝑥^∗2≤𝑥_𝑛+1−𝑥^∗2=𝛼_𝑛𝑢_𝑛−𝑥^∗+1−𝛼_𝑛𝑧_𝑛−𝑥^∗2=𝛼_𝑛𝑢_𝑛−𝑥^∗2+1−𝛼_𝑛𝑧_𝑛−𝑥^∗2−𝛼_𝑛1−𝛼_𝑛𝑢_𝑛−𝑧_𝑛²≤𝛼_𝑛𝑢_𝑛−𝑥^∗2+1−𝛼_𝑛𝑧_𝑛−𝑥^∗2=𝛼_𝑛𝑢_𝑛−𝑥^∗2+1−𝛼_𝑛𝛽_𝑛𝑦_𝑛−𝑥^∗2+1−𝛽_𝑛𝑇₂𝜔_𝑛−𝑥^∗2−𝛽_𝑛1−𝛽_𝑛𝑦_𝑛−𝑇₂𝜔_𝑛²≤𝛼_𝑛𝑢_𝑛−𝑥^∗2+𝛽_𝑛𝛾_𝑛𝜔_𝑛−𝑥^∗2+1−𝛾_𝑛𝑇₁𝜔_𝑛−𝑥^∗2−𝛾_𝑛1−𝛾_𝑛𝜔_𝑛−𝑇₁𝜔_𝑛²+1−𝛽_𝑛𝑇₂𝜔_𝑛−𝑥^∗2−𝛽_𝑛1−𝛽_𝑛𝑦_𝑛−𝑇₂𝜔_𝑛²≤𝛼_𝑛𝑢_𝑛−𝑥^∗2+𝛽_𝑛𝛾_𝑛𝜔_𝑛−𝑥^∗2+1−𝛾_𝑛𝜔_𝑛−𝑥^∗2−𝛾_𝑛1−𝛾_𝑛𝜔_𝑛−𝑇₁𝜔_𝑛²+1−𝛽_𝑛𝜔_𝑛−𝑥^∗2−𝛽_𝑛1−𝛽_𝑛𝑦_𝑛−𝑇₂𝜔_𝑛²=𝛼_𝑛𝑢_𝑛−𝑥^∗2+𝜔_𝑛−𝑥^∗2−𝛾_𝑛1−𝛾_𝑛𝜔_𝑛−𝑇₁𝜔_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑦_𝑛−𝑇₂𝜔_𝑛². (3.11) We also have that 𝛾_𝑛1−𝛾_𝑛𝜔_𝑛−𝑇₁𝜔_𝑛²≤𝛼_𝑛𝑢_𝑛−𝑥^∗2+𝜔_𝑛−𝑥^∗2−𝜔_𝑛+1−𝑥^∗2,𝛽_𝑛1−𝛽_𝑛𝑦_𝑛−𝑇₂𝜔_𝑛²≤𝛼_𝑛𝑢_𝑛−𝑥^∗2+𝜔_𝑛−𝑥^∗2−𝜔_𝑛+1−𝑥^∗2. (3.12)(3.13) Furthemore, we have from 𝑦_𝑛=𝛾_𝑛𝜔_𝑛+(1−𝛾_𝑛)𝑇₁𝜔_𝑛 that 𝜔_𝑛−𝑇₂𝜔_𝑛≤𝜔_𝑛−𝑦_𝑛+𝑦_𝑛−𝑇₂𝜔_𝑛=𝜔_𝑛−𝛾_𝑛𝜔_𝑛−1−𝛾_𝑛𝑇₁𝜔_𝑛+𝑦_𝑛−𝑇₂𝑥_𝑛=1−𝛾_𝑛𝜔_𝑛−𝑇₁𝜔_𝑛+𝑦_𝑛−𝑇₂𝜔_𝑛. (3.14) On the other hand, since 𝑥_𝑛+1−𝑥^∗=𝛼_𝑛(𝑢_𝑛−𝑥^∗)+(1−𝛼_𝑛)(𝑧_𝑛−𝑥^∗), we have 𝜔_𝑛+1−𝑥^∗2≤𝑥_𝑛+1−𝑥^∗2≤1−𝛼_𝑛𝑧_𝑛−𝑥^∗2+2𝛼_𝑛𝑢_𝑛−𝑥^∗,𝑥_𝑛+1−𝑥^∗≤1−𝛼_𝑛𝜔_𝑛−𝑥^∗2+2𝛼_𝑛𝑢_𝑛−𝑥^∗,𝑥_𝑛+1−𝑥^∗=1−𝛼_𝑛𝜔_𝑛−𝑥^∗2+2𝛼_𝑛𝑢_𝑛−𝑢,𝑥_𝑛+1−𝑥^∗+2𝛼_𝑛𝑢−𝑥^∗,𝑥_𝑛+1−𝑥^∗=1−𝛼_𝑛𝜔_𝑛−𝑥^∗2+2𝛼_𝑛𝑢_𝑛−𝑢,𝑥_𝑛+1−𝑥^∗+2𝛼_𝑛𝑢−𝑥^∗,𝑥_𝑛+1−𝜔_𝑛+2𝛼_𝑛𝑢−𝑥^∗,𝜔_𝑛−𝑥^∗. (3.15) We also have that 𝑥_𝑛+1−𝜔_𝑛≤𝑥_𝑛+1−𝑦_𝑛+𝑦_𝑛−𝜔_𝑛=𝛼_𝑛𝑢_𝑛−𝑦_𝑛+1−𝛼_𝑛𝑧_𝑛−𝑦_𝑛+1−𝛾_𝑛𝜔_𝑛−𝑇₁𝜔_𝑛≤𝛼_𝑛𝑢_𝑛−𝑦_𝑛+1−𝛼_𝑛𝛽_𝑛𝑦_𝑛+1−𝛽_𝑛𝑇₂𝜔_𝑛−𝑦_𝑛+1−𝛾_𝑛𝜔_𝑛−𝑇₁𝜔_𝑛=𝛼_𝑛𝑢_𝑛−𝑦_𝑛+1−𝛼_𝑛1−𝛽_𝑛𝑦_𝑛−𝑇₂𝜔_𝑛+1−𝛾_𝑛𝜔_𝑛−𝑇₁𝜔_𝑛. (3.16) Moreover, for any 𝑖∈{1,2,…,𝑘}, we have from 𝑢_𝑛,𝑖=𝑇_{𝑟𝑛,𝑖}(𝑥_𝑛−𝑟_𝑛Ψ_𝑖𝑥_𝑛) that 𝑢_𝑛,𝑖−𝑥^∗2≤𝑥_𝑛−𝑥^∗−𝑟_𝑛Ψ_𝑖𝑥_𝑛−Ψ_𝑖𝑥^∗2=𝑥_𝑛−𝑥^∗2−2𝑟_𝑛𝑥_𝑛−𝑥^∗,Ψ_𝑖𝑥_𝑛−Ψ_𝑖𝑥^∗+𝑟²_𝑛Ψ_𝑖𝑥_𝑛−Ψ_𝑖𝑥^∗2≤𝑥_𝑛−𝑥^∗2−𝑟_𝑛2𝜇_𝑖−𝑟_𝑛Ψ_𝑖𝑥_𝑛−Ψ_𝑖𝑥^∗2. (3.17) It follows that 𝜔_𝑛−𝑥^∗2=^𝑘_𝑖=11𝑘𝑢_𝑛,𝑖−𝑥^∗2≤1𝑘^𝑘_𝑖=1𝑢_𝑛,𝑖−𝑥^∗2≤𝑥_𝑛−𝑥^∗2−1𝑘^𝑘_𝑖=1𝑟_𝑛2𝜇_𝑖−𝑟_𝑛Ψ_𝑖𝑥_𝑛−Ψ_𝑖𝑥^∗2. (3.18) This implies that 𝑥_𝑛+1−𝑥^∗2=𝛼_𝑛𝑢_𝑛−𝑥^∗+1−𝛼_𝑛𝑧_𝑛−𝑥^∗2≤𝛼_𝑛𝑢_𝑛−𝑥^∗2+1−𝛼_𝑛𝜔_𝑛−𝑥^∗2≤𝛼_𝑛𝑢_𝑛−𝑥^∗2+1−𝛼_𝑛𝑥_𝑛−𝑥^∗2−1−𝛼_𝑛1𝑘^𝑘_𝑖=1𝑟_𝑛2𝜇_𝑖−𝑟_𝑛Ψ_𝑖𝑥_𝑛−Ψ_𝑖𝑥^∗2, (3.19) and hence 1−𝛼_𝑛1𝑘^𝑘_𝑖=1𝑟_𝑛2𝜇_𝑖−𝑟_𝑛Ψ_𝑖𝑥_𝑛−Ψ_𝑖𝑥^∗2≤𝛼_𝑛𝑢_𝑛−𝑥^∗2+𝑥_𝑛−𝑥^∗2−𝑥_𝑛+1−𝑥^∗2. (3.20) Furthermore, we have from Lemma 2.2 that for any 𝑖∈1,2,…,𝑘, we have 𝑢_𝑛,𝑖−𝑥^∗2≤𝑥_𝑛−𝑟_𝑛Ψ_𝑖𝑥_𝑛−𝑥^∗−𝑟_𝑛Ψ_𝑖𝑥^∗,𝑢_𝑛,𝑖−𝑥^∗=12𝑥_𝑛−𝑟_𝑛Ψ_𝑖𝑥_𝑛−𝑥^∗−𝑟_𝑛Ψ_𝑖𝑥^∗2+𝑢_𝑛,𝑖−𝑥^∗2−𝑥_𝑛−𝑟_𝑛Ψ_𝑖𝑥_𝑛−𝑥^∗−𝑟_𝑛Ψ_𝑖𝑥^∗−𝑢_𝑛,𝑖−𝑥^∗2≤12𝑥_𝑛−𝑥^∗2+𝑢_𝑛,𝑖−𝑥^∗2−𝑥_𝑛−𝑢_𝑛,𝑖−𝑟_𝑛Ψ_𝑖𝑥_𝑛−Ψ_𝑖𝑥^∗2=12𝑥_𝑛−𝑥^∗2+𝑢_𝑛,𝑖−𝑥^∗2−𝑥_𝑛−𝑢_𝑛,𝑖²−𝑟²_𝑛Ψ_𝑖𝑥_𝑛−Ψ_𝑖𝑥^∗2+2𝑟_𝑛𝑥_𝑛−𝑢_𝑛,𝑖,Ψ_𝑖𝑥_𝑛−Ψ_𝑖𝑥^∗. (3.21) This implies that 𝑢_𝑛,𝑖−𝑥^∗2≤𝑥_𝑛−𝑥^∗2−𝑥_𝑛−𝑢_𝑛,𝑖²+2𝑟_𝑛𝑥_𝑛−𝑢_𝑛,𝑖Ψ_𝑖𝑥_𝑛−Ψ_𝑖𝑥^∗. (3.22) Then we have from (3.22) that 𝜔_𝑛−𝑥^∗2≤1𝑘^𝑘_𝑖=1𝑢_𝑛,𝑖−𝑥^∗2≤𝑥_𝑛−𝑥^∗2−1𝑘^𝑘_𝑖=1𝑢_𝑛,𝑖−𝑥_𝑛²+1𝑘^𝑘_𝑖=12𝑟_𝑛𝑥_𝑛−𝑢_𝑛,𝑖Ψ_𝑖𝑥_𝑛−Ψ_𝑖𝑥^∗. (3.23) Hence we have from (3.23) that 𝑥_𝑛+1−𝑥^∗2≤𝛼_𝑛𝑢_𝑛−𝑥^∗2+1−𝛼_𝑛𝜔_𝑛−𝑥^∗2≤𝛼_𝑛𝑢_𝑛−𝑥^∗2+1−𝛼_𝑛𝑥_𝑛−𝑥^∗2−1𝑘^𝑘_𝑖=1𝑢_𝑛,𝑖−𝑥_𝑛²+1−𝛼_𝑛1𝑘^𝑘_𝑖=12𝑟_𝑛𝑥_𝑛−𝑢_𝑛,𝑖Ψ_𝑖𝑥_𝑛−Ψ_𝑖𝑥^∗. (3.24) It follows that 1−𝛼_𝑛1𝑘^𝑘_𝑖=1𝑢_𝑛,𝑖−𝑥_𝑛²≤𝛼_𝑛𝑢_𝑛−𝑥^∗2+𝑥_𝑛−𝑥^∗2−𝑥_𝑛+1−𝑥^∗2+1−𝛼_𝑛1𝑘^𝑘_𝑖=12𝑟_𝑛𝑥_𝑛−𝑢_𝑛,𝑖Ψ_𝑖𝑥_𝑛−Ψ_𝑖𝑥^∗. (3.25) Next, we will consider the following two cases.
Case A. Put Γ_𝑛=‖𝜔_𝑛−𝑥^∗‖² for all 𝑛∈ℕ. Suppose that Γ_𝑛+1≤Γ_𝑛 for all 𝑛∈ℕ. In this case lim_𝑛→∞Γ_𝑛 exists and then lim_𝑛→∞(Γ_𝑛+1−Γ_𝑛)=0. By (C1), (C3), and (3.12), we have lim_𝑛→∞𝜔_𝑛−𝑇₁𝜔_𝑛=0. (3.26) Similarly by (C1), (C2), and (3.13), we also have lim_𝑛→∞𝑦_𝑛−𝑇₂𝜔_𝑛=0. (3.27) So, we have from (3.14), (3.26), and (3.27) that lim_𝑛→∞𝜔_𝑛−𝑇₂𝜔_𝑛=0. (3.28) Since lim_𝑛→∞‖𝜔_𝑛−𝑥^∗‖ exists, we have from (3.11) and (3.26) lim_𝑛→∞𝜔_𝑛−𝑥^∗=lim_𝑛→∞𝑥_𝑛−𝑥^∗. (3.29) We also have from (C1), (3.16), (3.26), and (3.27) that lim_𝑛→∞𝑥_𝑛+1−𝜔_𝑛=0. (3.30) Since lim_𝑛→∞‖𝑥_𝑛−𝑥^∗‖ exists we have from (C1) and (3.20) that lim_𝑛→∞Ψ_𝑖𝑥_𝑛−Ψ_𝑖𝑥^∗=0,∀𝑖=1,2,…,𝑘. (3.31) This together with (3.25) and the existence of lim_𝑛→∞‖𝑥_𝑛−𝑥^∗‖ implies that lim_𝑛→∞𝑢_𝑛,𝑖−𝑥_𝑛=0,∀𝑖=1,2,…,𝑘, (3.32) which gives that 𝜔_𝑛−𝑥_𝑛≤1𝑘^𝑘_𝑖=1𝑢_𝑛,𝑖−𝑥_𝑛→0as𝑛→∞. (3.33) So, from (3.30), lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0. Furthermore, we have from (3.33) that 𝜔_𝑛+1−𝜔_𝑛≤𝜔_𝑛+1−𝑥_𝑛+1+𝑥_𝑛+1−𝑥_𝑛+𝑥_𝑛−𝜔_𝑛→0as𝑛→∞; (3.34) that is lim_𝑛→∞𝜔_𝑛+1−𝜔_𝑛=0. (3.35) Now, since {𝜔_𝑛} is a bounded sequence, there exists a subsequence {𝜔_𝑛𝑗} of {𝜔_𝑛} such that limsup_𝑛→∞𝑢−𝑥^∗,𝜔_𝑛−𝑥^∗=lim_𝑗→∞𝑢−𝑥^∗,𝜔_𝑛𝑗−𝑥^∗. (3.36) Without loss of generality, we may assume that 𝜔_𝑛𝑗⇀𝑣. Since 𝑇₁ is demiclosed at zero and by (3.26), we conclude that 𝑣∈𝐹(𝑇₁). Similarly, since 𝑇₂ is demiclosed at zero and by (3.28), we have 𝑣∈𝐹(𝑇₂). Therefore, we get that 𝑣∈𝐹𝑇₁∩𝐹𝑇₂. (3.37) Next, we show that 𝑣∈∩^𝑘_𝑖=1GEP(𝐺_𝑖,Ψ_𝑖). For each 𝑖∈{1,2,…,𝑘}, since 𝑢_𝑛,𝑖=𝑇_{𝑟𝑛,𝑖}(𝑥_𝑛−𝑟_𝑛Ψ_𝑖𝑥_𝑛), we have 𝐺_𝑖𝑢_𝑛,𝑖,𝑦+Ψ_𝑖𝑥_𝑛,𝑦−𝑢_𝑛,𝑖+1𝑟_𝑛𝑦−𝑢_𝑛,𝑖,𝑢_𝑛,𝑖−𝑥_𝑛≥0,∀𝑦∈𝐶. (3.38) From (A2), we also have Ψ_𝑖𝑥_𝑛,𝑦−𝑢_𝑛,𝑖+1𝑟_𝑛𝑦−𝑢_𝑛,𝑖,𝑢_𝑛,𝑖−𝑥_𝑛≥𝐺_𝑖𝑦,𝑢_𝑛,𝑖. (3.39) Replacing 𝑛 by 𝑛_𝑗, we have Ψ_𝑖𝑥_𝑛𝑗,𝑦−𝑢_{𝑛𝑗,𝑖}+𝑦−𝑢_{𝑛𝑗,𝑖},𝑢_{𝑛𝑗,𝑖}−𝑥_𝑛𝑗𝑟_𝑛𝑗≥𝐺_𝑖𝑦,𝑢_{𝑛𝑗,𝑖}. (3.40) Put 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑣 for all 𝑡∈(0,1] and 𝑦∈𝐶. Since 𝑣∈𝐶, then 𝑦_𝑡∈𝐶 and 𝑦_𝑡−𝑢_{𝑛𝑗,𝑖},Ψ_𝑖𝑦_𝑡≥𝑦_𝑡−𝑢_{𝑛𝑗,𝑖},Ψ_𝑖𝑦_𝑡−𝑦_𝑡−𝑢_{𝑛𝑗,𝑖},Ψ_𝑖𝑥_𝑛𝑗−𝑦_𝑡−𝑢_{𝑛𝑗,𝑖},𝑢_{𝑛𝑗,𝑖}−𝑥_𝑛𝑗𝑟_𝑛𝑗+𝐺_𝑖𝑦_𝑡,𝑢_{𝑛𝑗,𝑖}=𝑦_𝑡−𝑢_{𝑛𝑗,𝑖},Ψ_𝑖𝑦_𝑡−Ψ_𝑖𝑢_{𝑛𝑗,𝑖}+𝑦_𝑡−𝑢_{𝑛𝑗,𝑖},Ψ_𝑖𝑢_{𝑛𝑗,𝑖}−Ψ_𝑖𝑥_𝑛𝑗−𝑦_𝑡−𝑢_{𝑛𝑗,𝑖},𝑢_{𝑛𝑗,𝑖}−𝑥_𝑛𝑗𝑟_𝑛𝑗+𝐺_𝑖𝑦_𝑡,𝑢_{𝑛𝑗,𝑖}. (3.41) Since ‖𝑢_{𝑛𝑗,𝑖}−𝑥_𝑛𝑗‖→0 as 𝑗→∞, we obtain that ‖Ψ_𝑖𝑢_{𝑛𝑗,𝑖}−Ψ_𝑖𝑥_𝑛𝑗‖→0 as 𝑗→∞. Furthermore, by the monotonicity of Ψ_𝑖, we obtain that 𝑦_𝑡−𝑢_{𝑛𝑗,𝑖},Ψ_𝑖𝑦_𝑡−Ψ_𝑖𝑢_{𝑛𝑗,𝑖}≥0. (3.42) Taking 𝑗→∞ in (3.41), we have from (A4) that 𝑦_𝑡−𝑣,Ψ_𝑖𝑦_𝑡≥𝐺_𝑖𝑦_𝑡,𝑣. (3.43) Now, from (A1), (A4), and (3.43), we also have 0=𝐺_𝑖𝑦_𝑡,𝑦_𝑡≤𝑡𝐺_𝑖𝑦_𝑡,𝑦+(1−𝑡)𝐺_𝑖𝑦_𝑡,𝑣≤𝑡𝐺_𝑖𝑦_𝑡,𝑦+(1−𝑡)𝑦_𝑡−𝑣,Ψ_𝑖𝑦_𝑡=𝑡𝐺_𝑖𝑦_𝑡,𝑦+(1−𝑡)𝑡𝑦−𝑣,Ψ_𝑖𝑦_𝑡, (3.44) which yields that 𝐺_𝑖𝑦_𝑡,𝑦+(1−𝑡)𝑦−𝑣,Ψ_𝑖𝑦_𝑡≥0. (3.45) Taking 𝑡→0, we have, for each 𝑦∈𝐶𝐺_𝑖𝑣,𝑦+𝑦−𝑣,Ψ_𝑖𝑣≥0,∀𝑖∈{1,2,…,𝑘}. (3.46) This shows 𝑣∈GEP(𝐺_𝑖,Ψ_𝑖), for all 𝑖=1,2,…,𝑘. Then, 𝑣∈∩^𝑘_𝑖=1GEP(𝐺_𝑖,Ψ_𝑖). Hence we have 𝑣∈𝐹(𝑇₁)∩𝐹(𝑇₂)∩(∩^𝑘_𝑖=1GEP(𝐺_𝑖,Ψ_𝑖))∶=Ω. So, we have from (3.36) that limsup_𝑛→∞𝑢−𝑥^∗,𝜔_𝑛−𝑥^∗=𝑢−𝑥^∗,𝑣−𝑥^∗≤0. (3.47) By (C1), (C4), (3.15), (3.30), (3.47), and Lemma 2.5, we obtain that lim_𝑛→∞‖𝜔_𝑛−𝑥^∗‖=0. Hence we have from (3.29) that {𝑥_𝑛} converges to 𝑥^∗, where 𝑥^∗=𝑃_Ω𝑢.
Case B. Assume that there exists a subsequence {Γ_𝑛𝑖}_𝑖≥0 of {Γ_𝑛}_𝑛≥0 such that Γ_𝑛𝑖<Γ_𝑛𝑖+1 for all 𝑖∈ℕ. In this case, it follows from Lemma 2.4 that there exists a subsequence {Γ_𝜏(𝑛)} of {Γ_𝑛} such that Γ_𝜏(𝑛)+1>Γ_𝜏(𝑛), where 𝜏∶ℕ→ℕ is defined by 𝜏(𝑛)=max𝑘≤𝑛∶Γ_𝑘<Γ_𝑘+1,∀𝑛∈ℕ. (3.48) So, from (3.12), that 𝜔_𝜏(𝑛)+1−𝑥^∗2−𝜔_𝜏(𝑛)−𝑥^∗2+𝛾_𝜏(𝑛)1−𝛾_𝜏(𝑛)𝜔_𝜏(𝑛)−𝑇₁𝜔_𝜏(𝑛)²≤𝛼_𝜏(𝑛)𝑢_𝜏(𝑛)−𝑥^∗2. (3.49) Since ‖𝜔_𝜏(𝑛)−𝑥^∗‖²∶=Γ_𝜏(𝑛)<Γ_𝜏(𝑛)+1∶=‖𝜔_𝜏(𝑛)+1−𝑥^∗‖², we have 𝛾_𝜏(𝑛)1−𝛾_𝜏(𝑛)𝜔_𝜏(𝑛)−𝑇₁𝜔_𝜏(𝑛)²≤𝛼_𝜏(𝑛)𝑢_𝜏(𝑛)−𝑥^∗2. (3.50) By (C1) and (C3), we have lim_𝑛→∞𝜔_𝜏(𝑛)−𝑇₁𝜔_𝜏(𝑛)=0. (3.51) By (3.15), we have 𝜔_𝜏(𝑛)+1−𝑥^∗2≤1−𝛼_𝜏(𝑛)𝜔_𝜏(𝑛)−𝑥^∗2+2𝛼_𝜏(𝑛)𝑢_𝜏(𝑛)−𝑥^∗,𝑥_𝜏(𝑛)+1−𝑥^∗. (3.52) Now, in view of Γ_𝜏(𝑛)<Γ_𝜏(𝑛)+1, we see that 𝜔_𝜏(𝑛)−𝑥^∗2≤2𝑢_𝜏(𝑛)−𝑥^∗,𝑥_𝜏(𝑛)+1−𝑥^∗=2𝑢_𝜏(𝑛)−𝑢,𝑥_𝜏(𝑛)+1−𝑥^∗+2𝑢−𝑥^∗,𝑥_𝜏(𝑛)+1−𝜔_𝜏(𝑛)+2𝑢−𝑥^∗,𝜔_𝜏(𝑛)−𝑥^∗. (3.53) Furthermore, we also have from (3.13) that 𝛽_𝜏(𝑛)1−𝛽_𝜏(𝑛)𝑦_𝜏(𝑛)−𝑇₂𝜔_𝜏(𝑛)²≤𝛼_𝜏(𝑛)𝑢_𝜏(𝑛)−𝑥^∗2+𝜔_𝜏(𝑛)−𝑥^∗2−𝜔_𝜏(𝑛)+1−𝑥^∗2≤𝛼_𝜏(𝑛)𝑢_𝜏(𝑛)−𝑥^∗2. (3.54) Applying (C1) and (C2) to the last inequality, we get that lim_𝑛→∞𝑦_𝜏(𝑛)−𝑇₂𝜔_𝜏(𝑛)=0. (3.55) By (C1), (3.16), (3.51), and (3.55), we have lim_𝑛→∞𝑥_𝜏(𝑛)+1−𝜔_𝜏(𝑛)=0. (3.56) By (3.33), we have lim_𝑛→∞𝜔_𝜏(𝑛)+1−𝑥_𝜏(𝑛)+1=0. (3.57) It follows from (3.56) and (3.57) that lim_𝑛→∞𝜔_𝜏(𝑛)+1−𝜔_𝜏(𝑛)=0. (3.58) Since {𝜔_𝜏(𝑛)} is a bounded sequence, there exists a subsequence {𝜔_{𝜏(𝑛𝑗)}} such that limsup_𝑛→∞𝑢−𝑥^∗,𝜔_𝜏(𝑛)−𝑥^∗=lim_𝑗→∞𝑢−𝑥^∗,𝜔_{𝜏(𝑛𝑗)}−𝑥^∗. (3.59) Following the same argument as the proof of Case A for {𝜔_{𝜏(𝑛𝑗)}}, we have that limsup_𝑛→∞𝑢−𝑥^∗,𝜔_𝜏(𝑛)−𝑥^∗≤0. (3.60) Using (C4), (3.53), (3.56), and (3.60), we have that lim_𝑛→∞𝜔_𝜏(𝑛)−𝑥^∗=0. (3.61) By (3.58) and (3.61), we have that lim_𝑛→∞𝜔_𝜏(𝑛)+1−𝑥^∗=0. (3.62) By Lemma 2.4 (ii), we get lim_𝑛→∞Γ_𝑛=0; that is lim_𝑛→∞‖𝜔_𝑛−𝑥^∗‖=0. We observe that 𝑥_𝑛+1−𝑥^∗2≤𝛼_𝑛𝑢_𝑛−𝑥^∗2+1−𝛼_𝑛𝜔_𝑛−𝑥^∗2. (3.63) Applying (C1), (C4), and lim_𝑛→∞‖𝜔_𝑛−𝑥^∗‖²=0, we have immediately lim_𝑛→∞𝑥_𝑛−𝑥^∗=0; (3.64) that is, {𝑥_𝑛} converges strongly to 𝑥^∗, where 𝑥^∗=𝑃_Ω𝑢. This completes the proof.

Setting Ψ_𝑖≡0 for all 𝑖=1,2,…,𝑘 in Theorem 3.1, we obtain the following result.

Corollary 3.2. Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻. For each 𝑖=1,2,…,𝑘, let 𝐺_𝑖∶𝐶×𝐶→ℝ be a bifunction satisfying(A1)−(A4). For each 𝑗=1,2, let 𝑇_𝑗∶𝐶→𝐻 be two quasi-nonexpansive mappings such that 𝐼−𝑇_𝑗 are demiclosed at zero with Ω∶=𝐹(𝑇₁)∩𝐹(𝑇₂)∩(∩^𝑘_𝑖=1𝐸𝑃(𝐺_𝑖))≠∅. Let the sequences {𝑥_𝑛},   {𝑦_𝑛}, and {𝑧_𝑛} be defined by 𝑥₁𝐺∈𝐻,₁𝑢_𝑛,1+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,1,𝑢_𝑛,1−𝑥_𝑛𝐺≥0,∀𝑦∈𝐶,₂𝑢_𝑛,2+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,2,𝑢_𝑛,2−𝑥_𝑛⋮𝐺≥0,∀𝑦∈𝐶,_𝑘𝑢_𝑛,𝑘+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑘,𝑢_𝑛,𝑘−𝑥_𝑛𝜔≥0,∀𝑦∈𝐶,_𝑛=1𝑘^𝑘_𝑖=1𝑢_𝑛,𝑖,𝑦_𝑛=𝛾_𝑛𝜔_𝑛+1−𝛾_𝑛𝑇₁𝜔_𝑛,𝑧_𝑛=𝛽_𝑛𝑦_𝑛+1−𝛽_𝑛𝑇₂𝜔_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑢_𝑛+1−𝛼_𝑛𝑧_𝑛,∀𝑛∈ℕ, (3.65) where {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛} satisfy the following conditions.(C1)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞;(C2)liminf_𝑛→∞𝛽_𝑛(1−𝛽_𝑛)>0;(C3)liminf_𝑛→∞𝛾_𝑛(1−𝛾_𝑛)>0; (C4)lim_𝑛→∞𝑢_𝑛=𝑢 for some 𝑢∈𝐻.Then {𝑥_𝑛} converges strongly to 𝑥^∗, where 𝑥^∗=𝑃_Ω𝑢.

In the next results, using Theorem 3.1, we have new strong convergence theorems for two nonexpansive mappings in a Hilbert space.

Corollary 3.3. Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻. For each 𝑖=1,2,…,𝑘, let 𝐺_𝑖∶𝐶×𝐶→ℝ be a bifunction satisfying (A1)–(A4) and Ψ_𝑖 a 𝜇_𝑖-inverse strongly monotone mapping. For each 𝑗=1,2, let 𝑇_𝑗∶𝐶→𝐻 be two nonexpansive mappings such that Ω∶=𝐹(𝑇₁)∩𝐹(𝑇₂)∩(∩^𝑘_𝑖=1𝐺𝐸𝑃(𝐺_𝑖,Ψ_𝑖))≠∅. Let the sequences {𝑥_𝑛},  {𝑦_𝑛}, and {𝑧_𝑛} be defined by (3.1), where {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛} satisfy the following conditions.(C1)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞;(C2)liminf_𝑛→∞𝛽_𝑛(1−𝛽_𝑛)>0;(C3)liminf_𝑛→∞𝛾_𝑛(1−𝛾_𝑛)>0; (C4) lim_𝑛→∞𝑢_𝑛=𝑢 for some 𝑢∈𝐻.Then {𝑥_𝑛} converges strongly to 𝑥^∗, where 𝑥^∗=𝑃_Ω𝑢.

4. Applications

In this section, we present some convergence theorems deduced from the results in the previous section. Recall that a mapping 𝑇∶𝐶→𝐻 is said to be nonspreading [2] if 2𝑇𝑥−𝑇𝑦²≤𝑇𝑥−𝑦²+𝑇𝑦−𝑥² (4.1) for all 𝑥,𝑦∈𝐶. Further, a mapping 𝑇∶𝐶→𝐻 is said to be hybrid [23] if 3𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+𝑇𝑥−𝑦²+𝑇𝑦−𝑥² (4.2) for all 𝑥,𝑦∈𝐶. These mappings are deduced from a firmly nonexpansive mapping in a Hilbert space.

A mapping 𝐹∶𝐶→𝐻 is said to be firmly nonexpansive if 𝐹𝑥−𝐹𝑦²≤𝑥−𝑦,𝐹𝑥−𝐹𝑦 (4.3) for all 𝑥,𝑦∈𝐶; see, for instance, Browder [24] and Goebel and Kirk [25]. We also know that a firmly nonexpansive mapping 𝐹 can be deduced from an equilibrium problem in a Hilbert space.

Recently, Kocourek et al. [26] introduced a more broad class of nonlinear mappings call generalized hybrid if there are 𝛼,𝛽∈ℝ such that 𝛼𝑇𝑥−𝑇𝑦²+(1−𝛼)𝑥−𝑇𝑦²≤𝛽𝑇𝑥−𝑦²+1−𝛽𝑥−𝑦² (4.4) for all 𝑥,𝑦∈𝐶. Very recently, they defined a more broad class of mappings than the class of generalized hybrid mappings in a Hilbert space. A mapping 𝑆∶𝐶→𝐻 is called super hybrid if there are 𝛼,𝛽,𝛾∈ℝ such that 𝛼𝑆𝑥−𝑆𝑦²+1−𝛼+𝛾𝑥−𝑆𝑦²≤𝛽+𝛽−𝛼𝛾𝑆𝑥−𝑦²+1−𝛽−𝛽−𝛼−1𝛾𝑥−𝑦²+𝛼−𝛽𝛾‖𝑥−𝑆𝑥‖²+𝛾𝑦−𝑆𝑦², (4.5) for all 𝑥,𝑦∈𝐶. We call such a mapping an (𝛼,𝛽,𝛾)-super hybrid mapping. We notice that an (𝛼,𝛽,0)-super hybrid mapping is (𝛼,𝛽)-generalized hybrid. So, the class of super hybrid mappings contains the class of generalized hybrid mappings. A super hybrid mapping is not quasi-nonexpansive generally. For more details, see [27]. Before proving, we need the following lemmas.

Lemma 4.1 (see [27]). Let 𝐶 be a nonempty subset of a Hilbert space 𝐻 and let 𝛼,𝛽 and 𝛾 be real numbers with 𝛾≠−1. Let 𝑆 and 𝑇 be mappings of 𝐶 into 𝐻 such that 𝑆=(1/(1+𝛾))𝑇+(𝛾/(1+𝛾))𝐼. Then, 𝑇 is (𝛼,𝛽,𝛾)-super hybrid if and only if 𝑆 is (𝛼,𝛽)-generalized hybrid. In this case, 𝐹(𝑆)=𝐹(𝑇).

Lemma 4.2 (see [27]). Let 𝐻 be a Hilbert space and let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝑆∶𝐶→𝐻 be a generalized hybrid mapping. Then 𝑆 is demiclosed on 𝐶.
Setting 𝑆_𝑗∶=(1/(1+𝛾_𝑗))𝑇_𝑗+(𝛾_𝑗/(1+𝛾_𝑗))𝐼 in Theorem 3.1, where 𝑇_𝑗 is a super hybrid mapping and 𝛾_𝑗 is a real number, we obtain the following result.

Theorem 4.3. Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻. For each 𝑖=1,2,…,𝑘, let 𝐺_𝑖∶𝐶×𝐶→ℝ be a bifunction satisfying (A1)–(A4) and Ψ_𝑖 a 𝜇_𝑖-inverse strongly monotone mapping. For each 𝑗=1,2, let 𝑇_𝑗∶𝐶→𝐻 be (𝛼_𝑗,𝛽_𝑗,𝛾_𝑗)-super hybrid mappings such that Ω∶=𝐹(𝑇₁)∩𝐹(𝑇₂)∩(∩^𝑘_𝑖=1𝐺𝐸𝑃(𝐺_𝑖,Ψ_𝑖))≠∅. Let the sequences {𝑥_𝑛},  {𝑦_𝑛}, and {𝑧_𝑛} be defined by 𝑥₁𝐺∈𝐻,₁𝑢_𝑛,1,𝑦+Ψ₁𝑥_𝑛,𝑦−𝑢_𝑛,1+1𝑟_𝑛𝑦−𝑢_𝑛,1,𝑢_𝑛,1−𝑥_𝑛𝐺≥0,∀𝑦∈𝐶,₂𝑢_𝑛,2,𝑦+⟨Ψ₂𝑥_𝑛,𝑦−𝑢_𝑛,21⟩+𝑟_𝑛⟨𝑦−𝑢_𝑛,2,𝑢_𝑛,2−𝑥_𝑛⋮𝐺⟩≥0,∀𝑦∈𝐶,_𝑘𝑢_𝑛,𝑘,𝑦+Ψ_𝑘𝑥_𝑛,𝑦−𝑢_𝑛,𝑘+1𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑘,𝑢_𝑛,𝑘−𝑥_𝑛𝜔⟩≥0,∀𝑦∈𝐶,_𝑛=1𝑘^𝑘_𝑖=1𝑢_𝑛,𝑖,𝑦_𝑛=𝛾_𝑛𝜔_𝑛+1−𝛾_𝑛11+𝛾₁𝑇₁𝜔_𝑛+𝛾₁1+𝛾₁𝜔_𝑛,𝑧_𝑛=𝛽_𝑛𝑦_𝑛+1−𝛽_𝑛11+𝛾₂𝑇₂𝜔_𝑛+𝛾₂1+𝛾₂𝜔_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑢_𝑛+1−𝛼_𝑛𝑧_𝑛,∀𝑛∈ℕ, (4.6) where {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛} are sequences in (0,1) and {𝑢_𝑛}⊂𝐻 is a sequence and {𝑟_𝑛}⊂[𝑎,2𝜇_𝑖) for some 𝑎>0 and for all 𝑖∈{1,2,…,𝑘}. Suppose the following conditions are satisfied.(C1)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞;(C2)liminf_𝑛→∞𝛽_𝑛(1−𝛽_𝑛)>0;(C3)liminf_𝑛→∞𝛾_𝑛(1−𝛾_𝑛)>0; (C4)lim_𝑛→∞𝑢_𝑛=𝑢 for some 𝑢∈𝐻.Then {𝑥_𝑛} converges strongly to 𝑥^∗, where 𝑥^∗=𝑃_Ω𝑢.

Proof. For each 𝑗=1,2, setting 𝑆_𝑗=11+𝛾_𝑗𝑇_𝑗+𝛾_𝑗1+𝛾_𝑗𝐼, (4.7) we have from Lemma 4.1 that each 𝑆_𝑗 is a generalized hybrid mapping and 𝐹(𝑆_𝑗)=𝐹(𝑇_𝑗). Since 𝐹(𝑆_𝑗)≠∅, we have that each 𝑆_𝑗 is quasi-nonexpansive. Following the proof of Theorem 3.1 and applying Lemma 4.2, we have the desired result. This completes the proof.

Setting Ψ≡0 in Theorem 4.3, we obtains the following result.

Corollary 4.4. Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻. For each 𝑖=1,2,…,𝑘, let 𝐺_𝑖∶𝐶×𝐶→ℝ be a bifunction satisfying (A1)-(A4). For each 𝑗=1,2, let 𝑇_𝑗∶𝐶→𝐻 be (𝛼_𝑗,𝛽_𝑗,𝛾_𝑗)-super hybrid mappings such that Ω∶=𝐹(𝑇₁)∩𝐹(𝑇₂)∩(∩^𝑘_𝑖=1𝐸𝑃(𝐺_𝑖))≠∅. Let the sequences {𝑥_𝑛},{𝑦_𝑛}, and {𝑧_𝑛} be defined by 𝑥₁𝐺∈𝐻,₁𝑢_𝑛,1+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,1,𝑢_𝑛,1−𝑥_𝑛𝐺≥0,∀𝑦∈𝐶,₂𝑢_𝑛,2+1,𝑦𝑟_𝑛⟨𝑦−𝑢_𝑛,2,𝑢_𝑛,2−𝑥_𝑛⋮𝐺⟩≥0,∀𝑦∈𝐶,_𝑘𝑢_𝑛,𝑘+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑘,𝑢_𝑛,𝑘−𝑥_𝑛𝜔≥0,∀𝑦∈𝐶,_𝑛=1𝑘^𝑘_𝑖=1𝑢_𝑛,𝑖,𝑦_𝑛=𝛾_𝑛𝜔_𝑛+1−𝛾_𝑛11+𝛾₁𝑇₁𝜔_𝑛+𝛾₁1+𝛾₁𝜔_𝑛,𝑧_𝑛=𝛽_𝑛𝑦_𝑛+1−𝛽_𝑛11+𝛾₂𝑇₂𝜔_𝑛+𝛾₂1+𝛾₂𝜔_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑢_𝑛+1−𝛼_𝑛𝑧_𝑛,∀𝑛∈ℕ, (4.8) where {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛} are sequences in (0, 1) and {𝑢_𝑛}⊂𝐻 is a sequence and {𝑟_𝑛}⊂[𝑎,∞) for some 𝑎>0. Suppose the following conditions are satisfied.(C1)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞;(C2)liminf_𝑛→∞𝛽_𝑛(1−𝛽_𝑛)>0;(C3)liminf_𝑛→∞𝛾_𝑛(1−𝛾_𝑛)>0; (C4)lim_𝑛→∞𝑢_𝑛=𝑢 for some 𝑢∈𝐻.Then {𝑥_𝑛} converges strongly to 𝑥^∗, where 𝑥^∗=𝑃_Ω𝑢.

In Corollary 4.4, put 𝐺_𝑖(𝑥,𝑦)=0 for all 𝑥,𝑦∈𝐶 and 𝑟_𝑛=1 for all 𝑛∈ℕ. Then we have that 𝑢_𝑛,𝑖=𝑥_𝑛 for all 𝑖=1,2,…,𝑘, which gives that 𝜔_𝑛=(1/𝑘)∑^𝑘_𝑖=1𝑢_𝑛,𝑖=𝑥_𝑛. Thus we obtain the following results from Corollary 4.4.

Corollary 4.5. Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻. For each 𝑗=1,2, let 𝑇_𝑗∶𝐶→𝐻 be (𝛼_𝑗,𝛽_𝑗,𝛾_𝑗)-super hybrid mappings such that 𝐹(𝑇₁)∩𝐹(𝑇₂)≠∅. Let the sequences {𝑥_𝑛},   {𝑦_𝑛}, and {𝑧_𝑛} be defined by 𝑥₁𝑦∈𝐻,_𝑛=𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛11+𝛾₁𝑇₁𝑥_𝑛+𝛾₁1+𝛾₁𝑥_𝑛,𝑧_𝑛=𝛽_𝑛𝑦_𝑛+1−𝛽_𝑛11+𝛾₂𝑇₂𝑥_𝑛+𝛾₂1+𝛾₂𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑢_𝑛+1−𝛼_𝑛𝑧_𝑛,∀𝑛∈ℕ, (4.9) where {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛} are sequences in (0,1) and {𝑢_𝑛}⊂𝐻 is a sequence. Suppose the following conditions are satisfied.(C1)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞;(C2)liminf_𝑛→∞𝛽_𝑛(1−𝛽_𝑛)>0;(C3) liminf_𝑛→∞𝛾_𝑛(1−𝛾_𝑛)>0; (C4)lim_𝑛→∞𝑢_𝑛=𝑢 for some 𝑢∈𝐻.Then {𝑥_𝑛} converges strongly to 𝑥^∗, where 𝑥^∗=𝑃_{𝐹(𝑇1)∩𝐹(𝑇2)}𝑢.

In Corollary 4.5, put 𝑇₁=𝐼, the identity mapping, and 𝑇₂∶=𝑇, an (𝛼,𝛽,𝛾)-super hybrid mapping. Thus we obtain the following results.

Corollary 4.6. Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻. Let 𝑇 be an (𝛼,𝛽,𝛾)-super hybrid mapping such that 𝐹(𝑇)≠∅. Let the sequences {𝑥_𝑛},{𝑦_𝑛}, and {𝑧_𝑛} be defined by 𝑥₁𝑧∈𝐻,_𝑛=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛11+𝛾𝑇𝑥_𝑛+𝛾𝑥1+𝛾^𝑛,𝑥_𝑛+1=𝛼_𝑛𝑢_𝑛+1−𝛼_𝑛𝑧_𝑛,∀𝑛∈ℕ, (4.10)(4.11)(4.12) where {𝛼_𝑛} and {𝛽_𝑛} are sequences in (0,1) and {𝑢_𝑛}⊂𝐻 is a sequence. Suppose the following conditions are satisfied.(C1)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞;(C2)liminf_𝑛→∞𝛽_𝑛(1−𝛽_𝑛)>0; (C3)lim_𝑛→∞𝑢_𝑛=𝑢 for some 𝑢∈𝐻.Then {𝑥_𝑛} converges strongly to 𝑥^∗, where 𝑥^∗=𝑃_𝐹(𝑇)𝑢.

Acknowledgments

The first author is supported by the National Research Council of Thailand. The authors would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper.