Viscosity Approximations by the Shrinking Projection Method of Quasi-Nonexpansive Mappings for Generalized Equilibrium Problems

Abstract

We introduce viscosity approximations by using the shrinking projection method established by Takahashi, Takeuchi, and Kubota, for finding a common element of the set of solutions of the generalized equilibrium problem and the set of fixed points of a quasi-nonexpansive mapping. Furthermore, we also consider the viscosity shrinking projection method for finding a common element of the set of solutions of the generalized equilibrium problem and the set of fixed points 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}. Recall that a mapping Ψ∶𝐶→𝐻 is said to be 𝛿-inverse strongly monotone if there exists a positive real number 𝛿 such that Ψ𝑥−Ψ𝑦,𝑥−𝑦≥𝛿Ψ𝑥−Ψ𝑦²,∀𝑥,𝑦∈𝐶. (1.1) If Ψ is an 𝛿-inverse strongly monotone mapping of 𝐶 into 𝐻, then it is obvious that Ψ is 1/𝛿-Lipschitz continuous.

Let 𝐹∶𝐶×𝐶→ℝ be a bifunction and Ψ∶𝐶→𝐻 be 𝛿-inverse strongly monotone mapping. The generalized equilibrium problem (for short, GEP) for 𝐹 and Ψ is to find 𝑧∈𝐶 such that 𝐹𝑧,𝑦+Ψ𝑧,𝑦−𝑧≥0,∀𝑦∈𝐶. (1.2) The problem (1.2) was studied by Moudafi [1]. The set of solutions for the problem (1.2) is denoted by GEP(𝐹,Ψ), that is, GEP(𝐹,Ψ)=𝑧∈𝐶∶𝐹𝑧,𝑦+Ψ𝑧,𝑦−𝑧≥0,∀𝑦∈𝐶. (1.3) If Ψ≡0 in (1.2), then GEP reduces to the classical equilibrium problem and GEP(𝐹,0) is denoted by EP(𝐹), that is, EP(𝐹)=𝑧∈𝐶∶𝐹𝑧,𝑦≥0,∀𝑦∈𝐶. (1.4) If 𝐹≡0 in (1.2), 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. (1.5) The problem (1.2) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, min-max problems, and the Nash equilibrium problems in noncooperative games, see, for example, Blum and Oettli [2] and Moudafi [3].

In 2005, Combettes and Hirstoaga [4] 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 [5] 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 [6] 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. Maingé and Moudafi [7] introduced an iterative algorithm for equilibrium problems and fixed point problems. Wangkeeree [8] 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 [9] 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 1953, Mann [10] introduced the following iterative procedure to approximate a fixed point of a nonexpansive mapping 𝑇 in a Hilbert space 𝐻 as follows: 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,∀𝑛∈ℕ, (1.6) where the initial point 𝑥₁ is taken in 𝐶 arbitrarily and {𝛼_𝑛} is a sequence in [0,1]. Wittmann [11] obtained the strong convergence results of the sequence {𝑥_𝑛} defined by (1.6) to 𝑃_𝐹𝑥₁ under the following assumptions: (C1)lim_𝑛→∞𝛼_𝑛=0; (C2)∑^∞_𝑛=1𝛼_𝑛=∞; (C3)∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞, where 𝑃_𝐹(𝑇) is the metric projection of 𝐻 onto 𝐹(𝑇). In 2000, Moudafi [12] introduced the viscosity approximation method for nonexpansive mappings (see [13] for further developments in both Hilbert and Banach spaces). Let 𝑓 be a contraction on 𝐻. Starting with an arbitrary initial 𝑥₁∈𝐻, define a sequence {𝑥_𝑛} recursively by 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,𝑛≥1, (1.7) where {𝛼_𝑛} is a sequence in (0,1). It is proved [12, 13] that under conditions (C1), (C2), and (C3) imposed on {𝛼_𝑛}, the sequence {𝑥_𝑛} generated by (1.7) strongly converges to the unique fixed point 𝑥^∗ of 𝑃_𝐹(𝑇)𝑓 which is a unique solution of the variational inequality 𝐼−𝑓𝑥^∗,𝑥−𝑥^∗≥0,𝑥∈𝐶. (1.8) Suzuki [14] considered the Meir-Keeler contractions, which is extended notion of contractions and studied equivalency of convergence of these approximation methods.

Using the viscosity approximation method, in 2007, S. Takahashi and W. Takahashi [5] introduced an iterative scheme for finding a common element of the solution set of the classical equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. Let 𝑇∶𝐶→𝐻 be a nonexpansive mapping. Starting with arbitrary initial 𝑥₁∈𝐻, define sequences {𝑥_𝑛} and {𝑢_𝑛} recursively by 𝐹𝑢_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑇𝑢_𝑛,∀𝑛∈ℕ. (1.9) They proved that under certain appropriate conditions imposed on {𝛼_𝑛} and {𝑟_𝑛}, the sequences {𝑥_𝑛} and {𝑢_𝑛} converge strongly to 𝑧∈𝐹(𝑇)∩EP(𝐹), where 𝑧=𝑃_{𝐹(𝑇)∩EP(𝐹)}𝑓(𝑧).

On the other hand, in 2008, Takahashi et al. [15] has adapted Nakajo and Takahashi’s [16] idea to modify the process (1.6) so that strong convergence has been guaranteed. They proposed the following modification for a family of nonexpansive mappings in a Hilbert space: 𝑥₀∈𝐻, 𝐶₁=𝐶, 𝑢₁=𝑃_𝐶1𝑥₀ and 𝑦_𝑛=𝛼_𝑛𝑢_𝑛+1−𝛼_𝑛𝑇_𝑛𝑢_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶𝑦_𝑛−𝑧≤𝑢_𝑛,𝑢−𝑧_𝑛+1=𝑃_𝐶𝑛+1𝑥₀,𝑛∈ℕ, (1.10) where 0≤𝛼_𝑛≤𝑎<1 for all 𝑛∈ℕ. They proved that if {𝑇_𝑛} satisfies the appropriate conditions, then {𝑢_𝑛} generated by (1.10) converges strongly to a common fixed point of 𝑇_𝑛.

Very recently, Kimura and Nakajo [17] considered viscosity approximations by using the shrinking projection method established by Takahashi et al. [15] and the modified shrinking projection method proposed by Qin et al. [18], for finding a common fixed point of countably many nonlinear mappings, and they obtained some strong convergence theorems.

Motivated by these results, we introduce the viscosity shrinking projection method for finding a common element of the set of solutions of the generalized equilibrium problem and the set of fixed points of a quasi-nonexpansive mapping. Furthermore, we also consider the viscosity shrinking projection method for finding a common element of the set of solutions of the generalized equilibrium problem and the set of fixed points of the super hybrid mappings in Hilbert spaces.

2. Preliminaries

Throughout this paper, we denote by ℕ the set of positive integers and by ℝ the set of real numbers. Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖. 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 [19], we know the following basic properties. For 𝑥,𝑦∈𝐻 and 𝜆∈ℝ we have 𝜆𝑥+(1−𝜆)𝑦²=𝜆‖𝑥‖²+(1−𝜆)𝑦²−𝜆(1−𝜆)𝑥−𝑦². (2.1) We also know that for 𝑢,𝑣,𝑥,𝑦∈𝐻, we have 2𝑢−𝑣,𝑥−𝑦=𝑢−𝑦²+‖𝑣−𝑥‖²−‖𝑢−𝑥‖²−𝑣−𝑦². (2.2)

For every point 𝑥∈𝐻, there exists a unique nearest point of 𝐶, denoted by 𝑃_𝐶𝑥, such that ‖𝑥−𝑃_𝐶𝑥‖≤‖𝑥−𝑦‖ for all 𝑦∈𝐶. 𝑃_𝐶 is called the metric projection from 𝐻 onto 𝐶. It is well known that 𝑧=𝑃_𝐶𝑥⇔⟨𝑥−𝑧,𝑧−𝑦⟩≥0, for all 𝑥∈𝐻 and 𝑧,𝑦∈𝐶. We also know that 𝑃_𝐶 is firmly nonexpansive mapping from 𝐻 onto 𝐶, that is, 𝑃_𝐶𝑥−𝑃_𝐶𝑦²≤𝑃_𝐶𝑥−𝑃_𝐶𝑦,𝑥−𝑦,∀𝑥,𝑦∈𝐻, (2.3) and so is nonexpansive mapping.

For solving the generalized equilibrium problem, 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 𝑥,𝑦,𝑧∈𝐶,lim_𝑡↓0𝐹(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝐹(𝑥,𝑦);(A4) for each 𝑥∈𝐶,𝑦↦𝐹(𝑥,𝑦) is convex and lower semicontinuous. In order to prove our main results, we also need the following lemmas.

Lemma 2.1 (see [2]). Let 𝐶 be a nonempty closed convex subset of 𝐻 and let 𝐹 be a bifunction from 𝐶×𝐶 into ℝ satisfying (A1), (A2), (A3), and (A4). Then, for any 𝑟>0 and 𝑥∈𝐻, there exists a unique 𝑧∈𝐶 such that 𝐹+1𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶. (2.4)

Lemma 2.2 (see [4]). Let 𝐶 be a nonempty closed convex subset of 𝐻 and let 𝐹 be a bifunction from 𝐶×𝐶 into ℝ satisfying (A1), (A2), (A3), and (A4). Then, for any 𝑟>0 and 𝑥∈𝐻, define a mapping 𝑇_𝑟𝑥∶𝐻→𝐶 as follows: 𝑇_𝑟+1𝑥=𝑧∈𝐶∶𝐹𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶∀𝑥∈𝐻,𝑟∈ℝ. (2.5) 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; (2.6)(iii)𝐹(𝑇_𝑟)=EP(𝐹); (iv)EP(𝐹) is closed and convex.

Remark 2.3 (see [20]). Using (ii) in Lemma 2.2 and (2.2), we have 2𝑇_𝑟𝑥−𝑇_𝑟𝑦²≤2𝑇_𝑟𝑥−𝑇_𝑟=𝑇𝑦,𝑥−𝑦_𝑟𝑥−𝑦²+𝑇_𝑟𝑦−𝑥²−𝑇_𝑟𝑥−𝑥²−𝑇_𝑟𝑦−𝑦². (2.7) So, for 𝑦∈𝐹(𝑇_𝑟) and 𝑥∈𝐻, we have 𝑇_𝑟𝑥−𝑢²+𝑇_𝑟𝑥−𝑥²≤‖𝑥−𝑢‖². (2.8)

Remark 2.4. For any 𝑥∈𝐻 and 𝑟>0, by Lemma 2.1, there exists 𝑧∈𝐶 such that 𝐹+1𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶. (2.9) Replacing 𝑥 with 𝑥−𝑟Ψ𝑥∈𝐻 in (2.9), we have 𝐹++1𝑧,𝑦Ψ𝑥,𝑦−𝑧𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶, (2.10) where Ψ∶𝐶→𝐻 is an inverse-strongly monotone mapping.

For a sequence {𝐶_𝑛} of nonempty closed convex subsets of a Hilbert space 𝐻, define 𝑠−Li_𝑛𝐶_𝑛 and 𝑤−Ls_𝑛𝐶_𝑛 as follows.

𝑥∈𝑠−Li_𝑛𝐶_𝑛 if and only if there exists {𝑥_𝑛}⊂𝐻 such that 𝑥_𝑛→𝑥 and that 𝑥_𝑛∈𝐶_𝑛 for all 𝑛∈ℕ.

𝑥∈𝑤−Ls_𝑛𝐶_𝑛 if and only if there exists a subsequence {𝐶_𝑛𝑖} of {𝐶_𝑛} and a subsequence {𝑦_𝑖}⊂𝐻 such that 𝑦_𝑖⇀𝑦 and that 𝑦_𝑖∈𝐶_𝑛𝑖 for all 𝑖∈ℕ.

If 𝐶₀ satisfies 𝐶₀=𝑠−Li_𝑛𝐶_𝑛=𝑤−Ls_𝑛𝐶_𝑛, (2.11) it is said that {𝐶_𝑛} converges to 𝐶₀ in the sense of Mosco [21] and we write 𝐶₀=𝑀−lim_𝑛→∞𝐶_𝑛. It is easy to show that if {𝐶_𝑛} is nonincreasing with respect to inclusion, then {𝐶_𝑛} converges to ⋂^∞_𝑛=1𝐶_𝑛 in the sense of Mosco. For more details, see [21]. Tsukada [22] proved the following theorem for the metric projection.

Theorem 2.5 (see Tsukada [22]). Let 𝐻 be a Hilbert space. Let {𝐶_𝑛} be a sequence of nonempty closed convex subsets of 𝐻. If 𝐶₀=𝑀−lim_𝑛→∞𝐶_𝑛 exists and is nonempty, then for each 𝑥∈𝐻, {𝑃_𝐶𝑛𝑥} converges strongly to 𝑃_𝐶0𝑥, where 𝑃_𝐶𝑛 and 𝑃_𝐶0 are the metric projections of 𝐻 onto 𝐶_𝑛 and 𝐶₀, respectively.

On the other hand, a mapping 𝑓 of a complete metric space (𝑋,𝑑) into itself is said to be a contraction with coefficient 𝑟∈(0,1) if 𝑑(𝑓(𝑥),𝑓(𝑦))≤𝑟𝑑(𝑥,𝑦) for all 𝑥,𝑦∈𝐶. It is well known that 𝑓 has a unique fixed point [23]. Meir-Keeler [24] defined the following mapping called Meir-Keeler contraction. Let (𝑋,𝑑) be a complete metric space. A mapping 𝑓∶𝑋→𝑋 is called a Meir-Keeler contraction if for all 𝜀>0, there exists 𝛿>0 such that 𝜀≤𝑑(𝑥,𝑦)<𝜀+𝛿 implies 𝑑(𝑓(𝑥),𝑓(𝑦))<𝜀 for all 𝑥,𝑦∈𝑋. It is well known that Meir-Keeler contraction is a generalization of contraction and the following result is proved in [24].

Theorem 2.6 (see Meir-Keeler [24]). A Meir-Keeler contraction defined on a complete metric space has a unique fixed point.

We have the following results for Meir-Keeler contractions defined on a Banach space by Suzuki [14].

Theorem 2.7 (see Suzuki [14]). Let 𝑓 be a Meir-Keeler contraction on a convex subset 𝐶 of a Banach space 𝐸. Then, for every 𝜀>0, there exists 𝑟∈(0,1) such that ‖𝑥−𝑦‖≥𝜀 implies ‖𝑓(𝑥)−𝑓(𝑦)‖≤𝑟‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶.

Lemma 2.8 (see Suzuki [14]). Let 𝐶 be a convex subset of a Banach space 𝐸. Let 𝑇 be a nonexpansive mapping on 𝐶, and let 𝑓 be a Meir-Keeler contraction on 𝐶. Then the following hold. (i)𝑇∘𝑓 is a Meir-Keeler contraction on 𝐶.(ii)For each 𝛼∈(0,1), a mapping 𝑥↦(1−𝛼)𝑇𝑥+𝛼𝑓𝑥 is a Meir-Keeler contraction on 𝐶.

3. Main Results

In this section, using the shrinking projection method by Takahashi et al. [15], we prove a strong convergence theorem for a quasi-nonexpansive mapping with a generalized equilibrium problem in a Hilbert space. Before proving it, we need the following lemmas.

Lemma 3.1. Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻 and 𝛿>0 and let Ψ∶𝐶→𝐻 be 𝛿-inverse strongly monotone. If 0<𝜆≤2𝛿, then 𝐼−𝜆Ψ is a nonexpansive mapping.

Proof. For 𝑥,𝑦∈𝐶, we can calculate (𝐼−𝜆Ψ)𝑥−(𝐼−𝜆Ψ)𝑦²=𝑥−𝑦−𝜆(Ψ𝑥−Ψ𝑦)²=𝑥−𝑦²−2𝜆𝑥−𝑦,Ψ𝑥−Ψ𝑦+𝜆²Ψ𝑥−Ψ𝑦²≤𝑥−𝑦²−2𝜆𝛿Ψ𝑥−Ψ𝑦²+𝜆²Ψ𝑥−Ψ𝑦²=𝑥−𝑦²+𝜆(𝜆−2𝛿)Ψ𝑥−Ψ𝑦²≤𝑥−𝑦². (3.1) Therefore 𝐼−𝜆Ψ is nonexpansive. This completes the proof.

Lemma 3.2. Let 𝐶 be a nonempty closed convex subset of 𝐻, and let 𝑇 be a quasi-nonexpansive mapping of 𝐶 into 𝐻. Then, 𝐹(𝑇) is closed and convex.

Proof. We first show that 𝐹(𝑇) is closed. Let {𝑧_𝑛} be any sequence in 𝐹(𝑇) with 𝑧_𝑛→𝑧. We claim that 𝑧∈𝐹(𝑇). Since 𝐶 is closed, we have 𝑧∈𝐶. We observe that ‖𝑧−𝑇𝑧‖≤𝑧−𝑧_𝑛+𝑧_𝑛≤−𝑇𝑧𝑧−𝑧_𝑛+𝑧−𝑧_𝑛=2𝑧−𝑧_𝑛. (3.2) Since 𝑧_𝑛→𝑧, we obtain that ‖𝑧−𝑇𝑧‖≤0 and hence 𝑧=𝑇𝑧. This show that 𝑧∈𝐹(𝑇).
Next, we show that 𝐹(𝑇) is convex. Let 𝑥,𝑦∈𝐹(𝑇) and 𝛼∈[0,1]. We claim that 𝛼𝑥+(1−𝛼)𝑦∈𝐹(𝑇). Putting 𝑧=𝛼𝑥+(1−𝛼)𝑦, we have ‖𝑧−𝑇𝑧‖²=𝛼𝑥+(1−𝛼)𝑦−𝑇𝑧²=𝛼(𝑥−𝑇𝑧)+(1−𝛼)𝑦−𝑇𝑧²=𝛼‖𝑥−𝑇𝑧‖²+(1−𝛼)𝑦−𝑇𝑧²−𝛼(1−𝛼)𝑥−𝑦²≤𝛼‖𝑥−𝑧‖²+(1−𝛼)𝑦−𝑧²−𝛼(1−𝛼)𝑥−𝑦²=𝛼(1−𝛼)𝑥−𝑦²+(1−𝛼)𝛼𝑦−𝑥²−𝛼(1−𝛼)𝑥−𝑦²=𝛼(1−𝛼)²+𝛼²(1−𝛼)−𝛼(1−𝛼)𝑥−𝑦²=𝛼(1−𝛼)(1−𝛼+𝛼−1)𝑥−𝑦²=0. (3.3) Hence 𝐹(𝑇) is convex. This completes the proof.

Theorem 3.3. Let 𝐻 be a Hilbert space and let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝐹∶𝐶×𝐶→ℝ be a bifunction satisfying (A1), (A2), (A3), and (A4) and let Ψ be a 𝛿-inverse strongly monotone mapping from 𝐶 into 𝐻. Let 𝑇∶𝐶→𝐶 be a quasi-nonexpansive mapping which is demiclosed on 𝐶, that is, if {𝑤_𝑘}⊂𝐶,𝑤_𝑘⇀𝑤 and (𝐼−𝑇)𝑤_𝑘→0, then 𝑤∈𝐹(𝑇). Assume that Ω∶=GEP(𝐹,Ψ)∩𝐹(𝑇)≠∅ and 𝑓 is a Meir-Keeler contraction of 𝐶 into itself. Let the sequence {𝑥_𝑛}⊂𝐶 be defined by𝐶₁=𝐶,𝑥₁𝐹𝑧=𝑥∈𝐶,_𝑛,𝑦+Ψ𝑥_𝑛,𝑦−𝑧_𝑛+1𝜆_𝑛𝑧_𝑛−𝑥_𝑛,𝑦−𝑧_𝑛𝑦≥0,∀𝑦∈𝐶,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑧_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶𝑦_𝑛−𝑧≤𝑥_𝑛,𝑥−𝑧_𝑛+1=𝑃_𝐶𝑛+1𝑓𝑥_𝑛,∀𝑛∈ℕ, (3.4) where 𝑃_𝐶𝑛+1 is the metric projection of 𝐻 onto 𝐶_𝑛+1 and {𝛼_𝑛}⊂[0,1] and {𝜆_𝑛}⊂(0,2𝛿) are real sequences satisfying liminf_𝑛→∞𝛼_𝑛<1,0<𝑎≤𝜆_𝑛≤𝑏<2𝛿, (3.5) for some 𝑎,𝑏∈ℝ. Then, {𝑥_𝑛} converges strongly to 𝑧₀∈Ω, which satisfies 𝑧₀=𝑃_Ω𝑓(𝑧₀).

Proof. Since Ω is a closed convex subset of 𝐶, we have that 𝑃_Ω is well defined and nonexpansive. Furthermore, we know that 𝑓 is Meir-Keeler contraction and we know from Lemma 2.8 (i) that 𝑃_Ω𝑓 of 𝐶 onto Ω is a Meir-Keeler contraction on 𝐶. By Theorem 2.6, there exists a unique fixed point 𝑧₀∈𝐶 such that 𝑧₀=𝑃_Ω𝑓(𝑧₀). Next, we observe that 𝑧_𝑛=𝑇_𝜆𝑛(𝑥_𝑛−𝜆_𝑛Ψ𝑥_𝑛) for each 𝑛∈ℕ and take 𝑧∈𝐹(𝑇)∩GEP(𝐹,Ψ). From 𝑧=𝑇_𝜆𝑛(𝑧−𝜆_𝑛Ψ𝑧) and Lemma 2.2, we have that for any 𝑛∈ℕ, 𝑧_𝑛−𝑧²=𝑇_𝜆𝑛𝑥_𝑛−𝜆_𝑛Ψ𝑥_𝑛−𝑇_𝜆𝑛𝑧−𝜆_𝑛Ψ𝑧²≤𝑥_𝑛−𝜆_𝑛Ψ𝑥_𝑛−𝑧−𝜆_𝑛Ψ𝑧²≤𝑥_𝑛−𝑧². (3.6) Next, we divide the proof into several steps.
Step 1. 𝐶_𝑛 is closed convex and {𝑥_𝑛} is well defined for every 𝑛∈ℕ.
It is obvious from the assumption that 𝐶₁∶=𝐶 is closed convex and Ω⊂𝐶₁. For any 𝑘∈ℕ, suppose that 𝐶_𝑘 is closed and convex, and Ω⊂𝐶_𝑘. Note that for all 𝑧∈𝐶_𝑘, 𝑦_𝑘−𝑧²≤𝑥_𝑘−𝑧²⟺𝑦_𝑘−𝑧²−𝑥_𝑘−𝑧²⟺𝑦≤0_𝑘2+‖𝑧‖²−2⟨𝑦_𝑘,𝑧⟩−𝑥_𝑘²−‖𝑧‖²+2⟨𝑥_𝑘⟺𝑦,𝑧⟩≤0_𝑘2−2⟨𝑦_𝑘−𝑥_𝑘,𝑧⟩−𝑥_𝑘²≤0. (3.7) It is easy to see that 𝐶_𝑘+1 is closed. Next, we prove that 𝐶_𝑘+1 is convex. For any 𝑢,𝑣∈𝐶_𝑘+1 and 𝛼∈[0,1], we claim that 𝑧∶=𝛼𝑢+(1−𝛼)𝑣∈𝐶_𝑘+1. Since 𝑢∈𝐶_𝑘+1, we have ‖𝑦_𝑘−𝑢‖≤‖𝑥_𝑘−𝑢‖ and so ‖𝑦_𝑘−𝑢‖²≤‖𝑥_𝑘−𝑢‖², that is, ‖𝑦_𝑘‖²−2⟨𝑦_𝑘−𝑥_𝑘,𝑢⟩−‖𝑥_𝑘‖²≤0. Similarly, 𝑣∈𝐶_𝑘+1, we get ‖𝑦_𝑘‖²−2⟨𝑦_𝑘−𝑥_𝑘,𝑣⟩−‖𝑥_𝑘‖²≤0.
Thus, 𝛼𝑦_𝑘²−2𝑦_𝑘−𝑥_𝑘,𝛼𝑢−𝛼𝑥_𝑘²(𝑦≤0,1−𝛼)_𝑘²−2𝑦_𝑘−𝑥_𝑘,(1−𝛼)𝑣−(1−𝛼)𝑥_𝑘²≤0. (3.8) Combining the above inequalities, we obtain 𝑦_𝑘²−2𝑦_𝑘−𝑥_𝑘,𝛼𝑢+(1−𝛼)𝑣−𝑥_𝑘²≤0. (3.9) Therefore ‖𝑦_𝑘−𝑧‖≤‖𝑥_𝑘−𝑧‖. This shows that 𝑧∈𝐶_𝑘+1 and hence 𝐶_𝑘+1 is convex. Therefore 𝐶_𝑛 is closed and convex for all 𝑛∈ℕ.
Next, we show that Ω⊂𝐶_𝑛, for all 𝑛∈ℕ. For any 𝑘∈ℕ, suppose that 𝑣∈Ω⊂𝐶_𝑘. Since 𝑇 is quasi-nonexpansive and from (3.6), we have 𝑦_𝑘−𝑣²=𝛼_𝑘𝑥_𝑘+1−𝛼_𝑘𝑇𝑧_𝑘−𝑣²=𝛼_𝑘(𝑥_𝑘−𝑣)+1−𝛼_𝑘(𝑇𝑧_𝑘−𝑣)²≤𝛼_𝑘𝑥_𝑘−𝑣²+1−𝛼_𝑘𝑇𝑧_𝑘−𝑣²≤𝛼_𝑘𝑥_𝑘−𝑣²+1−𝛼_𝑘𝑧_𝑘−𝑣²≤𝛼_𝑘𝑥_𝑘−𝑣²+1−𝛼_𝑘𝑥_𝑘−𝑣²=𝑥_𝑘−𝑣². (3.10) So, we have 𝑣∈𝐶_𝑘+1. By principle of mathematical induction, we can conclude that 𝐶_𝑛 is closed and convex, and Ω⊂𝐶_𝑛, for all 𝑛∈ℕ. Hence, we have ∅≠Ω⊂𝐶_𝑛+1⊂𝐶_𝑛, (3.11) for all 𝑛∈ℕ. Therefore {𝑥_𝑛} is well defined.
Step 2. lim_𝑛→∞‖𝑥_𝑛−𝑢‖=0 for some 𝑢∈⋂^∞_𝑛=1𝐶_𝑛 and ⟨𝑓(𝑢)−𝑢,𝑢−𝑦⟩≥0 for all 𝑦∈Ω.
Since ⋂^∞_𝑛=1𝐶_𝑛 is closed convex, we also have that 𝑃_⋂^∞_{𝑛=1𝐶𝑛} is well defined and so 𝑃_⋂^∞_{𝑛=1𝐶𝑛}𝑓 is a Meir-Keeler contraction on 𝐶. By Theorem 2.6, there exists a unique fixed point 𝑢∈⋂^∞_𝑛=1𝐶_𝑛 of 𝑃_⋂^∞_{𝑛=1𝐶𝑛}𝑓. Since 𝐶_𝑛 is a nonincreasing sequence of nonempty closed convex subsets of 𝐻 with respect to inclusion, it follows that ∅≠Ω⊂^∞_𝑛=1𝐶_𝑛=𝑀−lim_𝑛→∞𝐶_𝑛. (3.12) Setting 𝑢_𝑛∶=𝑃_𝐶𝑛𝑓(𝑢) and applying Theorem 2.5, we can conclude that lim_𝑛→∞𝑢_𝑛=𝑃_⋂^∞_{𝑛=1𝐶𝑛}𝑓(𝑢)=𝑢. (3.13) Next, we will prove that lim_𝑛→∞‖𝑥_𝑛−𝑢‖=0. Assume to contrary that limsup_𝑛→∞‖𝑥_𝑛−𝑢‖≠0, there exists 𝜀>0 and a subsequence {‖𝑥_𝑛𝑗−𝑢‖} of {‖𝑥_𝑛−𝑢‖} such that 𝑥_𝑛𝑗−𝑢≥𝜀,∀𝑗∈ℕ, (3.14) which gives that limsup_𝑗→∞𝑥_𝑛𝑗−𝑢≥𝜀>0. (3.15) We choose a positive number 𝜀^′>0 such that limsup_𝑗→∞𝑥_𝑛𝑗−𝑢>𝜀^′>0. (3.16) For such 𝜀^′, by the definition of Meir-Keeler contraction, there exists 𝛿_𝜀′>0 with 𝜀^′+𝛿_𝜀′<limsup_𝑗→∞𝑥_𝑛𝑗−𝑢, (3.17) such that 𝑥−𝑦<𝜀^′+𝛿_𝜀′implies𝑓(𝑥)−𝑓𝑦<𝜀^′, (3.18) for all 𝑥,𝑦∈𝐶. Again for such 𝜀′, by Theorem 2.7, there exists 𝑟_𝜀′∈(0,1) such that 𝑥−𝑦≥𝜀^′+𝛿_𝜀′implies𝑓(𝑥)−𝑓𝑦<𝑟_𝜀′𝑥−𝑦. (3.19) Since 𝑢_𝑛→𝑢, there exists 𝑛₀∈ℕ such that 𝑢_𝑛−𝑢<𝛿_𝜀′,∀𝑛≥𝑛₀. (3.20) By the idea of Suzuki [14] and Kimura and Nakajo [17], we consider the following two cases.
Case 𝐼. Assume that there exists 𝑛₁≥𝑛₀ such that 𝑥_𝑛1−𝑢<𝜀^′+𝛿_𝜀′. (3.21) Thus, we get 𝑥_𝑛1+1−𝑢≤𝑥_𝑛1+1−𝑢_𝑛1+1+𝑢_𝑛1+1=𝑃−𝑢_𝐶𝑛1+1𝑓𝑥_𝑛1−𝑃_𝐶𝑛1+1𝑓(𝑢)+𝑢_𝑛1+1≤𝑓𝑥−𝑢_𝑛1−𝑓(𝑢)+𝑢_𝑛1+1−𝑢<𝜀^′+𝛿_𝜀′. (3.22) By induction on {𝑛}, we can obtain that 𝑥_𝑛−𝑢<𝜀^′+𝛿_𝜀′, (3.23) for all 𝑛≥𝑛₀. In particular, for all 𝑗≥𝑛₀, we have 𝑛_𝑗≥𝑗≥𝑛₀ and 𝑥_𝑛𝑗−𝑢<𝜀^′+𝛿_𝜀′. (3.24) This implies that limsup_𝑗→∞𝑥_𝑛𝑗−𝑢≤𝜀^′+𝛿_𝜀′<limsup_𝑗→∞𝑥_𝑛𝑗−𝑢, (3.25) which is a contradiction. Therefore, we conclude that ‖𝑥_𝑛−𝑢‖→0 as 𝑛→∞.
Case 𝐼𝐼. Assume that 𝑥_𝑛−𝑢≥𝜀^′+𝛿_𝜀′,∀𝑛≥𝑛₀. (3.26) By (3.19), we have 𝑓𝑥_𝑛−𝑓(𝑢)<𝑟_𝜀′𝑥_𝑛−𝑢,∀𝑛≥𝑛₀. (3.27) Thus, we have 𝑥_𝑛+1−𝑢_𝑛+1=𝑃_𝐶𝑛+1𝑓𝑥_𝑛−𝑃_𝐶𝑛+1𝑓(𝑢≤𝑓𝑥_𝑛)−𝑓(𝑢)≤𝑟_𝜀′𝑥_𝑛−𝑢≤𝑟_𝜀′𝑥_𝑛−𝑢_𝑛+𝑢_𝑛−𝑢, (3.28) for every 𝑛≥𝑛₀. In particular, we have 𝑥_𝑛𝑗+1−𝑢_𝑛𝑗+1≤𝑟_𝜀′𝑥_𝑛𝑗−𝑢_𝑛𝑗+𝑢_𝑛𝑗−𝑢, (3.29) for every 𝑗≥𝑛₀(𝑛_𝑗≥𝑗≥𝑛₀). Let us consider limsup_𝑛→∞𝑥_𝑛𝑗−𝑢_𝑛𝑗=limsup_𝑗→∞𝑥_𝑛𝑗+1−𝑢_𝑛𝑗+1≤𝑟_𝜀′limsup_𝑗→∞𝑥_𝑛𝑗−𝑢_𝑛𝑗+𝑢_𝑛𝑗−𝑢≤𝑟_𝜀′limsup_𝑗→∞𝑥_𝑛𝑗−𝑢_𝑛𝑗+𝑟_𝜀′limsup_𝑗→∞𝑢_𝑛𝑗−𝑢=𝑟_𝜀′limsup_𝑗→∞𝑥_𝑛𝑗−𝑢_𝑛𝑗<limsup_𝑗→∞𝑥_𝑛𝑗−𝑢_𝑛𝑗, (3.30) which gives a contradiction. Hence, we obtain that lim_𝑛→∞𝑥_𝑛−𝑢=0, (3.31) and therefore {𝑥_𝑛} is bounded. Moreover, {𝑓(𝑥_𝑛)},{𝑧_𝑛}, and {𝑦_𝑛} are also bounded. Since 𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑓(𝑥_𝑛), we have 𝑓𝑥_𝑛−𝑥_𝑛+1,𝑥_𝑛+1−𝑦≥0,∀𝑦∈𝐶_𝑛+1. (3.32) Since Ω⊂𝐶_𝑛+1, we get 𝑓𝑥_𝑛−𝑥_𝑛+1,𝑥_𝑛+1−𝑦≥0,∀𝑛∈ℕ,𝑦∈Ω. (3.33) We have from 𝑥_𝑛→𝑢 that 𝑓(𝑢)−𝑢,𝑢−𝑦≥0,∀𝑦∈Ω. (3.34)
Step 3. There exists a subsequence {‖𝑥_𝑛𝑖−𝑧_𝑛𝑖‖} of {‖𝑥_𝑛−𝑧_𝑛‖} such that ‖𝑥_𝑛𝑖−𝑧_𝑛𝑖‖→0 as 𝑖→∞.
We have from (3.13) and (3.31) that 𝑥_𝑛−𝑥_𝑛+1≤𝑥_𝑛−𝑢+𝑢−𝑢_𝑛+1+𝑢_𝑛+1−𝑥_𝑛+1=𝑥_𝑛−𝑢+𝑢−𝑢_𝑛+1+𝑃_𝐶𝑛+1𝑓𝑥_𝑛−𝑃_𝐶𝑛+1𝑓(𝑢≤𝑥_𝑛)−𝑢+𝑢−𝑢_𝑛+1+𝑓𝑥_𝑛−𝑓(𝑢)→0. (3.35) From 𝑥_𝑛+1∈𝐶_𝑛+1, we have that 𝑦_𝑛−𝑥_𝑛+1≤𝑥_𝑛−𝑥_𝑛+1, (3.36) and so ‖𝑦_𝑛−𝑥_𝑛+1‖→0. We also have 𝑦_𝑛−𝑥_𝑛≤𝑦_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑥_𝑛→0. (3.37) From liminf_𝑛→∞𝛼_𝑛<1, there exists a subsequence {𝛼_𝑛𝑖} of {𝛼_𝑛} and 𝛼₀ with 0≤𝛼₀<1 such that 𝛼_𝑛𝑖→𝛼₀. Since ‖𝑥_𝑛−𝑦_𝑛‖=‖𝑥_𝑛−𝛼_𝑛𝑥_𝑛−(1−𝛼)𝑇𝑧_𝑛‖=(1−𝛼_𝑛)‖𝑥_𝑛−𝑇𝑧_𝑛‖, we have 𝑇𝑧_𝑛𝑖−𝑥_𝑛𝑖→0as𝑖→∞. (3.38) Using Lemma 2.2 (ii) and (3.6), we have 𝑧_𝑛−𝑧²=𝑇_𝜆𝑛(𝑥_𝑛−𝜆_𝑛Ψ𝑥_𝑛)−𝑇_𝜆𝑛(𝑧−𝜆_𝑛Ψ𝑧)²≤𝑥_𝑛−𝜆_𝑛Ψ𝑥_𝑛−𝑧−𝜆_𝑛Ψ𝑧,𝑧_𝑛𝑥−𝑧=−_𝑛−𝜆_𝑛Ψ𝑥_𝑛−𝑧−𝜆_𝑛Ψ𝑧,𝑧−𝑧_𝑛=12(𝑥_𝑛−𝜆_𝑛Ψ𝑥_𝑛)−(𝑧−𝜆_𝑛Ψ𝑧)²+𝑧_𝑛−𝑧²−(𝑥_𝑛−𝜆_𝑛Ψ𝑥_𝑛)−(𝑧−𝜆_𝑛Ψ𝑧)+(𝑧−𝑧_𝑛)²≤12𝑥_𝑛−𝑧²+𝑧_𝑛−𝑧²−𝑥_𝑛−𝑧_𝑛−𝜆_𝑛Ψ𝑥_𝑛−Ψ𝑧²=12𝑥_𝑛−𝑧²+𝑧_𝑛−𝑧²−𝑥_𝑛−𝑧_𝑛²+2𝜆_𝑛𝑥_𝑛−𝑧_𝑛,Ψ𝑥_𝑛−Ψ𝑧−𝜆²_𝑛Ψ𝑥_𝑛−Ψ𝑧². (3.39) So, we have 𝑧_𝑛−𝑧²≤𝑥_𝑛−𝑧²−𝑥_𝑛−𝑧_𝑛²+2𝜆_𝑛𝑥_𝑛−𝑧_𝑛,Ψ𝑥_𝑛−Ψ𝑧−𝜆²_𝑛Ψ𝑥_𝑛−Ψ𝑧². (3.40) Let us consider 𝑦_𝑛−𝑧²=𝛼_𝑛(𝑥_𝑛−𝑧)+(1−𝛼_𝑛)(𝑇𝑧_𝑛−𝑧)²≤𝛼_𝑛𝑥_𝑛−𝑧²+1−𝛼_𝑛𝑇𝑧_𝑛−𝑧²≤𝛼_𝑛𝑥_𝑛−𝑧²+1−𝛼_𝑛𝑧_𝑛−𝑧²=𝛼_𝑛𝑥_𝑛−𝑧²+1−𝛼_𝑛𝑇_𝜆𝑛(𝐼−𝜆_𝑛Ψ)𝑥_𝑛−𝑇_𝜆𝑛(𝐼−𝜆_𝑛Ψ)𝑧²≤𝛼_𝑛𝑥_𝑛−𝑧²+1−𝛼_𝑛(𝐼−𝜆_𝑛Ψ)𝑥_𝑛−(𝐼−𝜆_𝑛Ψ)𝑧²=𝛼_𝑛𝑥_𝑛−𝑧²+1−𝛼_𝑛(𝑥_𝑛−𝑧)−𝜆_𝑛(Ψ𝑥_𝑛−Ψ𝑧)²=𝛼_𝑛𝑥_𝑛−𝑧²+1−𝛼_𝑛𝑥_𝑛−𝑧²+1−𝛼_𝑛𝜆²_𝑛Ψ𝑥_𝑛−Ψ𝑧²−21−𝛼_𝑛𝜆_𝑛⟨𝑥_𝑛−𝑧,Ψ𝑥_𝑛≤𝑥−Ψ𝑧⟩_𝑛−𝑧²+1−𝛼_𝑛𝜆²_𝑛Ψ𝑥_𝑛−Ψ𝑧²−21−𝛼_𝑛𝜆_𝑛𝛿Ψ𝑥_𝑛−Ψ𝑧²=𝑥_𝑛−𝑧²+1−𝛼_𝑛𝜆_𝑛−2𝛿𝜆_𝑛Ψ𝑥_𝑛−Ψ𝑧²≤𝑥_𝑛−𝑧²+1−𝛼_𝑛(𝑏−2𝛿)𝑏Ψ𝑥_𝑛−Ψ𝑧². (3.41) In particular, we have 1−𝛼_𝑛𝑖(2𝛿−𝑏)𝑏Ψ𝑥_𝑛𝑖−Ψ𝑧²≤𝑥_𝑛𝑖−𝑧²−𝑦_𝑛𝑖−𝑧²≤𝑥_𝑛𝑖−𝑦_𝑛𝑖²+2𝑥_𝑛𝑖−𝑦_𝑛𝑖𝑦_𝑛𝑖−𝑧. (3.42) Since 𝛼_𝑛𝑖→𝛼₀ with 𝛼₀<1 and ‖𝑥_𝑛𝑖−𝑦_𝑛𝑖‖→0, we obtain that Ψ𝑥_𝑛𝑖−Ψ𝑧→0. (3.43) Using (3.40), we have 𝑦_𝑛−𝑧²≤𝛼_𝑛𝑥_𝑛−𝑧²+1−𝛼_𝑛𝑧_𝑛−𝑧²≤𝛼_𝑛𝑥_𝑛−𝑧²+1−𝛼_𝑛×𝑥_𝑛−𝑧²−𝑥_𝑛−𝑧_𝑛²+2𝜆_𝑛𝑥_𝑛−𝑧_𝑛,Ψ𝑥_𝑛−Ψ𝑧−𝜆²_𝑛Ψ𝑥_𝑛−Ψ𝑧²≤𝛼_𝑛𝑥_𝑛−𝑧²+1−𝛼_𝑛𝑥_𝑛−𝑧²−𝑥_𝑛−𝑧_𝑛²+2𝜆_𝑛𝑥_𝑛−𝑧_𝑛Ψ𝑥_𝑛≤𝑥−Ψ𝑧_𝑛−𝑧²−1−𝛼_𝑛𝑥_𝑛−𝑧_𝑛²+21−𝛼_𝑛𝜆_𝑛𝑥_𝑛+𝑧_𝑛Ψ𝑥_𝑛≤𝑥−Ψ𝑧_𝑛−𝑧²−1−𝛼_𝑛𝑥_𝑛−𝑧_𝑛²+21−𝛼_𝑛𝑏𝑀Ψ𝑥_𝑛−Ψ𝑧, (3.44) where 𝑀∶=sup{‖𝑥_𝑛‖+‖𝑦_𝑛‖∶𝑛∈ℕ}.
So, we have 1−𝛼_𝑛𝑖𝑥_𝑛𝑖−𝑧_𝑛𝑖²≤𝑥_𝑛𝑖−𝑧²−𝑦_𝑛𝑖−𝑧²+21−𝛼_𝑛𝑖𝑏𝑀Ψ𝑥_𝑛𝑖≤𝑥−Ψ𝑧_𝑛𝑖−𝑦_𝑛𝑖²+2𝑥_𝑛𝑖−𝑦_𝑛𝑖𝑦_𝑛𝑖−𝑧+21−𝛼_𝑛𝑖𝑏𝑀Ψ𝑥_𝑛𝑖−Ψ𝑧. (3.45) We have from 𝛼_𝑛𝑖→𝛼₀, (3.37), and (3.43) that 𝑥_𝑛𝑖−𝑧_𝑛𝑖→0. (3.46)
Step 4. Finally, we prove that 𝑢∈Ω∶=𝐹(𝑇)∩GEP(𝐹,Ψ).
Since 𝑦_𝑛=𝛼_𝑛𝑥_𝑛+(1−𝛼_𝑛)𝑇𝑧_𝑛, we have 𝑦_𝑛−𝑇𝑧_𝑛=𝛼_𝑛(𝑥_𝑛−𝑇𝑧_𝑛). So, from (3.38) we have 𝑦_𝑛𝑖−𝑇𝑧_𝑛𝑖=𝛼_𝑛𝑖𝑥_𝑛𝑖−𝑇𝑧_𝑛𝑖→0. (3.47) Since ‖𝑧_𝑛𝑖−𝑇𝑧_𝑛𝑖‖≤‖𝑧_𝑛𝑖−𝑥_𝑛𝑖‖+‖𝑥_𝑛𝑖−𝑦_𝑛𝑖‖+‖𝑦_𝑛𝑖−𝑇𝑧_𝑛𝑖‖, from (3.37), (3.46), and (3.47) we have 𝑧_𝑛𝑖−𝑇𝑧_𝑛𝑖→0. (3.48) Since 𝑥_𝑛𝑖→𝑢, we have 𝑧_𝑛𝑖→𝑢. So, from (3.48) and the demiclosed property of 𝑇, we have 𝑢∈𝐹(𝑇). (3.49) We next show that 𝑢∈GEP(𝐹,Ψ). Since 𝑧_𝑛=𝑇_𝜆𝑛(𝑥_𝑛−𝜆_𝑛Ψ𝑥_𝑛), for any 𝑦∈𝐶 we have 𝐹𝑧_𝑛,𝑦+Ψ𝑥_𝑛,𝑦−𝑧_𝑛+1𝜆_𝑛𝑦−𝑧_𝑛,𝑧_𝑛−𝑥_𝑛≥0. (3.50) From (A2), we have −𝐹𝑦,𝑧_𝑛+Ψ𝑥_𝑛,𝑦−𝑧_𝑛+1𝜆_𝑛𝑦−𝑧_𝑛,𝑧_𝑛−𝑥_𝑛≥0 (3.51) and so Ψ𝑥_𝑛,𝑦−𝑧_𝑛+1𝜆_𝑛𝑦−𝑧_𝑛,𝑧_𝑛−𝑥_𝑛≥𝐹𝑦,𝑧_𝑛. (3.52) Replacing 𝑛 by 𝑛_𝑖, we have Ψ𝑥_𝑛𝑖,𝑦−𝑧_𝑛𝑖+𝑦−𝑧_𝑛𝑖,𝑧_𝑛𝑖−𝑥_𝑛𝑖𝜆_𝑛𝑖≥𝐹𝑦,𝑧_𝑛𝑖. (3.53) Note that Ψ is 1/𝛿-Lipschitz continuous, and from (3.46), we have Ψ𝑧_𝑛𝑖−Ψ𝑥_𝑛𝑖→0. (3.54) For 𝑡∈(0,1] and 𝑦∈𝐶, let 𝑧^∗_𝑡=𝑡𝑦+(1−𝑡)𝑢. Since 𝐶 is convex, we have 𝑧^∗_𝑡∈𝐶. So, from (3.53) we have 𝑧^∗_𝑡−𝑧_𝑛𝑖,Ψ𝑧^∗_𝑡≥𝑧^∗_𝑡−𝑧_𝑛𝑖,Ψ𝑧^∗_𝑡−𝑧^∗_𝑡−𝑧_𝑛𝑖,Ψ𝑥_𝑛𝑖−𝑧^∗_𝑡−𝑧_𝑛𝑖,𝑧_𝑛𝑖−𝑥_𝑛𝑖𝜆_𝑛𝑖+𝐹𝑧^∗_𝑡,𝑧_𝑛𝑖=𝑧^∗_𝑡−𝑧_𝑛𝑖,Ψ𝑧^∗_𝑡−Ψ𝑥_𝑛𝑖−𝑧^∗_𝑡−𝑧_𝑛𝑖,𝑧_𝑛𝑖−𝑥_𝑛𝑖𝜆_𝑛𝑖+𝐹𝑧^∗_𝑡,𝑧_𝑛𝑖=𝑧^∗_𝑡−𝑧_𝑛𝑖,Ψ𝑧^∗_𝑡−Ψ𝑧_𝑛𝑖+𝑧^∗_𝑡−𝑧_𝑛𝑖,Ψ𝑧_𝑛𝑖−Ψ𝑥_𝑛𝑖−𝑧^∗_𝑡−𝑧_𝑛𝑖,𝑧_𝑛𝑖−𝑥_𝑛𝑖𝜆_𝑛𝑖+𝐹𝑧^∗_𝑡,𝑧_𝑛𝑖. (3.55) From ⟨𝑧^∗_𝑡−𝑧_𝑛𝑖,Ψ𝑧^∗_𝑡−Ψ𝑧_𝑛𝑖⟩≥0, we have 𝑧^∗_𝑡−𝑧_𝑛𝑖,Ψ𝑧^∗_𝑡≥𝑧^∗_𝑡−𝑧_𝑛𝑖,Ψ𝑧_𝑛𝑖−Ψ𝑥_𝑛𝑖−𝑧^∗_𝑡−𝑧_𝑛𝑖,𝑧_𝑛𝑖−𝑥_𝑛𝑖𝜆_𝑛𝑖+𝐹𝑧^∗_𝑡,𝑧_𝑛𝑖. (3.56) Thus, 𝑧^∗_𝑡−𝑧_𝑛𝑖,Ψ𝑧^∗_𝑡−𝑧^∗_𝑡−𝑧_𝑛𝑖Ψ𝑧_𝑛𝑖−Ψ𝑥_𝑛𝑖≥−𝑧^∗_𝑡−𝑧_𝑛𝑖𝑧_𝑛𝑖−𝑥_𝑛𝑖𝜆_𝑛𝑖+𝐹𝑧^∗_𝑡,𝑧_𝑛𝑖. (3.57) From Step  3 and (3.54), we obtain 𝑧^∗_𝑡−𝑢,Ψ𝑧^∗_𝑡≥𝐹𝑧^∗_𝑡,𝑢. (3.58) From (A1), (A4), and (3.58), we have 0=𝐹𝑧^∗_𝑡,𝑧^∗_𝑡=𝐹𝑧^∗_𝑡()𝑢𝑧,𝑡𝑦+1−𝑡≤𝑡𝐹^∗_𝑡,𝑦+(1−𝑡)𝐹𝑧^∗_𝑡𝑧,𝑢≤𝑡𝐹^∗_𝑡,𝑦+(1−𝑡)⟨𝑧^∗_𝑡−𝑢,Ψ𝑧^∗_𝑡⟩≤𝑡𝐹𝑧^∗_𝑡,𝑦+(1−𝑡)𝑡⟨𝑦−𝑢,Ψ𝑧^∗_𝑡⟩, (3.59) and hence 0≤𝐹𝑧^∗_𝑡,𝑦+(1−𝑡)𝑦−𝑢,Ψ𝑧^∗_𝑡. (3.60) Letting 𝑡↓0 and from (A3), we have that for each 𝑦∈𝐶, 0≤lim_𝑡↓0𝐹𝑧^∗_𝑡,𝑦+(1−𝑡)𝑦−𝑢,Ψ𝑧^∗_𝑡=lim_𝑡↓0𝐹()+(()+.𝑡𝑦+1−𝑡𝑢,𝑦1−𝑡)𝑦−𝑢,𝑡Ψ𝑦+1−𝑡Ψ𝑢≤𝐹𝑢,𝑦𝑦−𝑢,Ψ𝑢 (3.61) This implies that 𝑢∈GEP(𝐹,Ψ). So, we have 𝑢∈𝐹(𝑇)∩GEP(𝐹,Ψ). We obtain from (3.34) that 𝑢=𝑧₀ and hence, {𝑥_𝑛} converges strongly to 𝑧₀. This completes the proof.

By Theorem 3.3, we can obtain some new and interesting strong convergence theorems. Now we give some examples as follows.

Setting 𝑓(𝑥_𝑛)=𝑥,∀𝑛∈ℕ in Theorem 3.3, we obtain the following result.

Corollary 3.4. Let 𝐻 be a Hilbert space and let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝐹∶𝐶×𝐶→ℝ be a bifunction satisfying (A1), (A2), (A3), and (A4) and let Ψ be a 𝛿-inverse strongly monotone mapping from 𝐶 into 𝐻. Let 𝑇∶𝐶→𝐶 be a quasi-nonexpansive mapping which is demiclosed on 𝐶. Assume that Ω≠∅ and let 𝐶₁=𝐶 and {𝑥_𝑛}⊂𝐶 be a sequence generated by 𝑥₁=𝑥∈𝐶 and 𝐹𝑧_𝑛,𝑦+Ψ𝑥_𝑛,𝑦−𝑧_𝑛+1𝜆_𝑛𝑧_𝑛−𝑥_𝑛,𝑦−𝑧_𝑛𝑦≥0,∀𝑦∈𝐶,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑧_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶𝑦_𝑛−𝑧≤𝑥_𝑛,𝑥−𝑧_𝑛+1=𝑃_𝐶𝑛+1𝑥,∀𝑛∈ℕ, (3.62) where 𝑃_𝐶𝑛+1 is the metric projection of 𝐻 onto 𝐶_𝑛+1 and {𝛼_𝑛}⊂[0,1] and {𝜆_𝑛}⊂[0,2𝛿) are sequences such that liminf_𝑛→∞𝛼_𝑛<1,0<𝑎≤𝜆_𝑛≤𝑏<2𝛿, (3.63) for some 𝑎∈ℝ. Then {𝑥_𝑛} converges strongly to 𝑧₀=𝑃_Ω𝑧₀.

Setting Ψ≡0 in Theorem 3.3, we obtain the following result.

Corollary 3.5. Let 𝐻 be a Hilbert space and let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝐹∶𝐶×𝐶→ℝ be a bifunction satisfying (A1), (A2), (A3), and (A4). Let 𝑇∶𝐶→𝐶 be a quasi-nonexpansive mapping which is demiclosed on 𝐶. Assume that EP(𝐹)∩𝐹(𝑇)≠∅ and 𝑓 is a Meir-Keeler contraction of 𝐶 into itself. Let 𝐶₁=𝐶 and {𝑥_𝑛}⊂𝐶 be a sequence generated by 𝑥₁=𝑥∈𝐶 and 𝐹𝑧_𝑛+1,𝑦𝜆_𝑛𝑧_𝑛−𝑥_𝑛,𝑦−𝑧_𝑛𝑦≥0,∀𝑦∈𝐶,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑧_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶𝑦_𝑛−𝑧≤𝑥_𝑛,𝑥−𝑧_𝑛+1=𝑃_𝐶𝑛+1𝑓𝑥_𝑛,∀𝑛∈ℕ, (3.64) where 𝑃_𝐶𝑛+1 is the metric projection of 𝐻 onto 𝐶_𝑛+1 and {𝛼_𝑛}⊂[0,1] and {𝜆_𝑛}⊂[0,∞) are sequences such that liminf_𝑛→∞𝛼_𝑛<1,0<𝑎≤𝜆_𝑛, (3.65) for some 𝑎∈ℝ. Then {𝑥_𝑛} converges strongly to 𝑧₀∈𝐹(𝑇)∩EP(𝐹).

Setting Ψ≡0 and 𝑓(𝑥_𝑛)=𝑥 for all 𝑛∈ℕ in Theorem 3.3, we obtain the following result.

Corollary 3.6. Let 𝐻 be a Hilbert space and let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝐹∶𝐶×𝐶→ℝ be a bifunction satisfying (A1), (A2), (A3), and (A4). Let 𝑇∶𝐶→𝐶 be an quasi-nonexpansive mapping which is demiclosed on 𝐶 and assume that EP(𝐹)∩𝐹(𝑇)≠∅. Let 𝐶₁=𝐶 and {𝑥_𝑛}⊂𝐶 be a sequence generated by 𝑥₁=𝑥∈𝐶 and 𝐹𝑧_𝑛+1,𝑦𝜆_𝑛𝑧_𝑛−𝑥_𝑛,𝑦−𝑧_𝑛𝑦≥0,∀𝑦∈𝐶,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑧_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶𝑦_𝑛−𝑧≤𝑥_𝑛,𝑥−𝑧_𝑛+1=𝑃_𝐶𝑛+1𝑥,∀𝑛∈ℕ, (3.66) where 𝑃_𝐶𝑛+1 is the metric projection of 𝐻 onto 𝐶_𝑛+1 and {𝛼_𝑛}⊂[0,1] and {𝜆_𝑛}⊂[0,∞) are sequences such that liminf_𝑛→∞𝛼_𝑛<1,0<𝑎≤𝜆_𝑛, (3.67) for some 𝑎∈ℝ. Then {𝑥_𝑛} converges strongly to 𝑧₀∈𝐹(𝑇)∩EP(𝐹).

Next, using the CQ hybrid method introduced by Nakajo and Takahashi [16], we prove a strong convergence theorem of a quasi-nonexpansive mapping for solving the generalized equilibrium problem.

Theorem 3.7. Let 𝐻 be a Hilbert space and let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝐹∶𝐶×𝐶→ℝ be a bifunction satisfying (A1), (A2), (A3), and (A4) and let Ψ be a 𝛿-inverse strongly monotone mapping from 𝐶 into 𝐻. Let 𝑇∶𝐶→𝐶 be an quasi-nonexpansive mapping which is demiclosed on 𝐶. Assume that Ω≠∅ and 𝑓 is a Meir-Keeler contraction of 𝐶 into itself. Let 𝑄₁=𝐶 and {𝑥_𝑛}⊂𝐶 be a sequence generated by 𝑥₁=𝑥∈𝐶 and 𝐹𝑧_𝑛,𝑦+Ψ𝑥_𝑛,𝑦−𝑧_𝑛+1𝜆_𝑛𝑧_𝑛−𝑥_𝑛,𝑦−𝑧_𝑛𝑦≥0,∀𝑦∈𝐶,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑧_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝑦_𝑛−𝑧≤𝑥_𝑛,𝑄−𝑧_𝑛=𝑧∈𝑄_𝑛−1∶𝑓𝑥_𝑛−1−𝑥_𝑛,𝑥_𝑛,𝑥−𝑧≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑓𝑥_𝑛,∀𝑛∈ℕ, (3.68) where 𝑃_{𝐶𝑛∩𝑄𝑛} is the metric projection of 𝐻 onto 𝐶_𝑛∩𝑄_𝑛 and {𝛼_𝑛}⊂(0,1) and {𝜆_𝑛}⊂(0,2𝛿) satisfy 0≤𝛼_𝑛≤𝑏<1,0<𝑐≤𝜆_𝑛≤𝑑<2𝛿, (3.69) for some 𝑏,𝑐,𝑑∈ℝ. Then, {𝑥_𝑛} converges strongly to 𝑧₀∈Ω, which satisfies 𝑧₀=𝑃_Ω𝑓(𝑧₀).

Proof. As in the proof of Theorem 3.3, we have that the mapping 𝑃_Ω𝑓 of 𝐶 onto Ω is a Meir-Keeler contraction on 𝐶. By Theorem 2.6, there exists a unique fixed point 𝑧₀∈𝐶 such that 𝑧₀=𝑃_Ω𝑓(𝑧₀). Next, it is clear that 𝐶_𝑛 is closed and convex. Next, we will show that 𝑄_𝑛 is closed and convex for all 𝑛∈ℕ. For any 𝑛∈ℕ, let {𝑧_𝑘} be a sequence in 𝑄_𝑛 such that 𝑧_𝑘→𝑧. For each 𝑘∈ℕ, we observe that 0≤𝑥_𝑛−𝑧_𝑘,𝑓𝑥_𝑛−1−𝑥_𝑛=12𝑓𝑥_𝑛−1−𝑧_𝑘²−𝑥_𝑛−𝑧_𝑘²−𝑓𝑥_𝑛−1−𝑥_𝑛². (3.70) Taking 𝑘→∞, we get ⟨𝑥_𝑛−𝑧,𝑓(𝑥_𝑛−1)−𝑥_𝑛⟩≥0 and then 𝑧∈𝑄_𝑛. Therefore 𝑄_𝑛 is closed.
Next, we will show that 𝑄_𝑛 is convex. For any 𝑢,𝑣∈𝑄_𝑛, and 𝛼∈[0,1], put 𝑧=𝛼𝑢+(1−𝛼)𝑣. We claim that 𝑧∈𝑄_𝑛. Since 𝑢∈𝑄_𝑛, we have ⟨𝛼𝑥_𝑛−𝛼𝑢,𝑓(𝑥_𝑛−1)−𝑥_𝑛⟩≥0. Similarly, since 𝑣∈𝑄_𝑛, we have ⟨(1−𝛼)𝑥_𝑛−(1−𝛼)𝑣,𝑓(𝑥_𝑛−1)−𝑥_𝑛⟩≥0. Thus, 0≤⟨𝛼𝑥_𝑛−𝛼𝑢+(1−𝛼)𝑥_𝑛−(1−𝛼)𝑣,𝑓𝑥_𝑛−1−𝑥_𝑛⟩=⟨𝑥_𝑛−𝛼𝑢−(1−𝛼)𝑣,𝑓𝑥_𝑛−1−𝑥_𝑛⟩=⟨𝑥_𝑛−𝑧,𝑓𝑥_𝑛−1−𝑥_𝑛⟩. (3.71) It follows that 𝑧∈𝑄_𝑛, and therefore we have that 𝑄_𝑛 is convex. We obtain from both 𝐶_𝑛 and 𝑄_𝑛 which are closed convex sets for every 𝑛∈ℕ that 𝐶_𝑛∩𝑄_𝑛 is closed and convex for every 𝑛∈ℕ.
Next, we will show that 𝐶_𝑛∩𝑄_𝑛 is nonempty. Let 𝑧∈𝐹(𝑇)∩GEP(𝐹,Ψ). We will show that 𝑧∈𝐶_𝑛 for any 𝑛∈ℕ. We notice that 𝑧_𝑛=𝑇_𝜆𝑛(𝑥_𝑛−𝜆_𝑛Ψ𝑥_𝑛) for each 𝑛∈ℕ and 𝑧=𝑇_𝜆𝑛(𝑧−𝜆_𝑛Ψ𝑧). From Ψ which is an inverse strongly monotone mapping Lemma 2.2 (ii), Lemma 3.1, we obtain 𝑧_𝑛−𝑧≤𝑥_𝑛−𝑧,forany𝑛∈ℕ. (3.72) Since 𝑇 is quasi-nonexpansive with the fixed point 𝑧 and from (3.72), we have 𝑦_𝑛−𝑧²≤𝑥_𝑛−𝑧². (3.73) So, we have 𝑧∈𝐶_𝑛. Therefore 𝐹(𝑇)∩GEP(𝐹,Ψ)⊂𝐶_𝑛, for all 𝑛∈ℕ.
Next, we will show that 𝐹(𝑇)∩GEP(𝐹,Ψ)⊂𝐶_𝑛∩𝑄_𝑛,∀𝑛∈ℕ. (3.74) It is obvious that 𝐹(𝑇)∩GEP(𝐹,Ψ)⊂𝐶=𝑄₁. Hence 𝐹(𝑇)∩GEP(𝐹,Ψ)⊂𝐶₁∩𝑄₁. (3.75) For any 𝑘∈ℕ, suppose that 𝐹(𝑇)∩GEP(𝐹,Ψ)⊂𝐶_𝑘∩𝑄_𝑘. (3.76) Since 𝑥_𝑘+1=𝑃_{𝐶𝑘∩𝑄𝑘}𝑓(𝑥_𝑘), we have 𝑓𝑥_𝑘−𝑥_𝑘+1,𝑥_𝑘+1−𝑧≥0,∀𝑧∈𝐶_𝑘∩𝑄_𝑘. (3.77) In particular, for any 𝑧∈𝐹(𝑇)∩GEP(𝐹,Ψ), we obtain that 𝑓𝑥_𝑘−𝑥_𝑘+1,𝑥_𝑘+1−𝑧≥0. (3.78) This shows that 𝑧∈𝑄_𝑘+1. Hence 𝐹(𝑇)∩GEP(𝐹,Ψ)⊂𝑄_𝑘+1. Therefore, we conclude that 𝐹(𝑇)∩GEP(𝐹,Ψ)⊂𝐶_𝑘+1∩𝑄_𝑘+1. (3.79) By principle of mathematical induction, we can conclude that 𝐹(𝑇)∩GEP(𝐹,Ψ)⊂𝐶_𝑛∩𝑄_𝑛,∀𝑛∈ℕ. (3.80) Hence {𝑥_𝑛} is well defined. Since 𝑃_⋂^∞_{𝑛=1𝑄𝑛}𝑓 is a Meir-Keeler contraction on 𝐶, there exists a unique element 𝑢∈𝐶 such that 𝑢=𝑃_⋂^∞_{𝑛=1𝑄𝑛}𝑓(𝑢). For each 𝑛, let 𝑢_𝑛=𝑃_𝑄𝑛𝑓(𝑢). Since 𝐹(𝑇)∩GEP(𝐹,Ψ)⊂𝑄_𝑛+1⊂𝑄_𝑛, we have from Theorem 2.5 that 𝑢_𝑛→𝑢. Notice that 𝑥_𝑛=𝑃_𝑄𝑛𝑓(𝑥_𝑛−1). Thus, as in the proof of Theorem 3.3, we get 𝑥_𝑛→𝑢 and hence {𝑥_𝑛} is bounded. Moreover, lim_𝑛→∞𝑥_𝑛−𝑥_𝑛+1=0,lim_𝑛→∞𝑦_𝑛−𝑥_𝑛=0. (3.81) As the proof of Theorem 3.3, we have that 𝑦_𝑛−𝑧²≤𝑥_𝑛−𝑧²+1−𝛼_𝑛𝜆_𝑛−2𝛿𝜆_𝑛Ψ𝑥_𝑛−Ψ𝑧². (3.82) Thus, 𝑦_𝑛−𝑧²≤𝑥_𝑛−𝑧²+(1−𝑎)(𝑑−2𝛿)𝑑Ψ𝑥_𝑛−Ψ𝑧² (3.83) and so (1−𝑎)(2𝛿−𝑑)𝑑Ψ𝑥_𝑛−Ψ𝑧²≤𝑥_𝑛−𝑧²−𝑦_𝑛−𝑧²≤𝑥_𝑛−𝑦_𝑛²+2𝑥_𝑛−𝑦_𝑛𝑦_𝑛−𝑧. (3.84) Since (1−𝑎)(2𝛿−𝑑)𝑑>0 and ‖𝑥_𝑛−𝑦_𝑛‖→0, we obtain that Ψ𝑥_𝑛−Ψ𝑧→0. (3.85) Using (3.40) in Theorem 3.3, we have 𝑦_𝑛−𝑧²≤𝑥_𝑛−𝑧²−(1−𝑏)𝑥_𝑛−𝑧_𝑛²+2(1−𝑎)𝑑𝑀Ψ𝑥_𝑛−Ψ𝑧, (3.86) where 𝑀∶=sup{‖𝑥_𝑛‖+‖𝑦_𝑛‖}. So, we have (1−𝑏)𝑥_𝑛−𝑧_𝑛²≤𝑥_𝑛−𝑧²−𝑦_𝑛−𝑧²+2(1−𝑎)𝑑𝑀Ψ𝑥_𝑛≤𝑥−Ψ𝑧_𝑛−𝑦_𝑛²+2𝑥_𝑛−𝑦_𝑛𝑦_𝑛−𝑧+2(1−𝑎)𝑑𝑀Ψ𝑥_𝑛−Ψ𝑧. (3.87) We have from 1−𝑏>0, ‖𝑥_𝑛−𝑦_𝑛‖→0 and (3.85) that 𝑥_𝑛−𝑧_𝑛→0, (3.88) which implies that 𝑧_𝑛→𝑢. (3.89) Notice that 𝑥_𝑛−𝑦_𝑛=𝑥_𝑛−𝛼_𝑛𝑥_𝑛−(1−𝛼)𝑇𝑧_𝑛=1−𝛼_𝑛𝑥_𝑛−𝑇𝑧_𝑛, (3.90) and from limsup_𝑛→∞𝛼_𝑛≤𝑏<1, we have 𝑇𝑧_𝑛−𝑥_𝑛→0. (3.91) Since 𝑦_𝑛=𝛼_𝑛𝑥_𝑛+(1−𝛼_𝑛)𝑇𝑧_𝑛, we have 𝑦_𝑛−𝑇𝑧_𝑛=𝛼_𝑛(𝑥_𝑛−𝑇𝑧_𝑛). So, from 0<𝑎≤𝛼_𝑛≤𝑏<1 and (3.91), we obtain 𝑦_𝑛−𝑇𝑧_𝑛→0 (3.92) and hence 𝑧_𝑛−𝑇𝑧_𝑛→0. (3.93) From (3.89), (3.93), and the demiclosed property of 𝑇, we have 𝑢∈𝐹(𝑇). As in the proof of Theorem 3.3 we have that 𝑢∈𝐹(𝑇)∩GEP(𝐹,Ψ). Since 𝐹(𝑇)∩GEP(𝐹,Ψ)⊂𝑄_𝑛+1, we get 𝑓𝑥_𝑛−𝑥_𝑛+1,𝑥_𝑛+1−𝑦≥0, (3.94) for all 𝑛∈ℕ and 𝑦∈𝐹(𝑇)∩GEP(𝐹,Ψ). We have from 𝑥_𝑛→𝑢 that 𝑓(𝑢)−𝑢,𝑢−𝑦≥0, (3.95) for all 𝑦∈𝐹(𝑇)∩GEP(𝐹,Ψ), which implies that 𝑢=𝑃_Ω𝑓(𝑢). It follows that 𝑢=𝑧₀, since 𝑧₀∈𝐹(𝑇)∩GEP(𝐹,Ψ) of 𝑃_Ω𝑓 is unique. Hence, {𝑥_𝑛} converges strongly to 𝑧₀. This completes the proof.

Setting 𝑓(𝑥_𝑛)=𝑥, for all 𝑛∈ℕ in Theorem 3.7, we obtain the following result.

Corollary 3.8. Let 𝐻 be a Hilbert space and let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝐹∶𝐶×𝐶→ℝ be a bifunction satisfying (A1), (A2), (A3), and (A4) and let Ψ be a 𝛿-inverse strongly monotone mapping from 𝐶 into 𝐻 and let 𝑇∶𝐶→𝐶 be a quasi-nonexpansive mapping which is demiclosed on 𝐶. Assume that Ω∶=GEP(𝐹,Ψ)∩𝐹(𝑇)≠∅. Let 𝑄₁=𝐶 and {𝑥_𝑛}⊂𝐶 be a sequence generated by 𝑥₁=𝑥∈𝐶 and 𝐹𝑧_𝑛,𝑦+Ψ𝑥_𝑛,𝑦−𝑧_𝑛+1𝜆_𝑛𝑧_𝑛−𝑥_𝑛,𝑦−𝑧_𝑛𝑦≥0,∀𝑦∈𝐶,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑧_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝑦_𝑛−𝑧≤𝑥_𝑛,𝑄−𝑧_𝑛=𝑧∈𝑄_𝑛−1∶𝑥−𝑥_𝑛,𝑥_𝑛,𝑥−𝑧≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥,∀𝑛∈ℕ, (3.96) where 𝑃_{𝐶𝑛∩𝑄𝑛} is the metric projection of 𝐻 onto 𝐶_𝑛∩𝑄_𝑛 and {𝛼_𝑛}⊂(0,1) and {𝜆_𝑛}⊂(0,2𝛿) satisfy 0≤𝛼_𝑛≤𝑏<1,0<𝑐≤𝜆_𝑛≤𝑑<2𝛿, (3.97) for some 𝑎,𝑏,𝑐,𝑑∈ℝ. Then {𝑥_𝑛} converges strongly to 𝑧₀=𝑃_Ω𝑓(𝑧₀).

Setting Ψ≡0 in Theorem 3.7, we obtain the following result.

Corollary 3.9. Let 𝐻 be a Hilbert space and let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝐹∶𝐶×𝐶→ℝ be a bifunction satisfying (A1), (A2), (A3), and (A4) and let 𝑇∶𝐶→𝐶 be an quasi-nonexpansive mapping which is demiclosed on 𝐶. Assume that EP(𝐹)∩𝐹(𝑇)≠∅ and 𝑓 is a Meir-Keeler contraction of 𝐶 into itself. Let 𝑄₁=𝐶 and {𝑥_𝑛}⊂𝐶 be a sequence generated by 𝑥₁=𝑥∈𝐶 and 𝐹𝑧_𝑛+1,𝑦𝜆_𝑛𝑧_𝑛−𝑥_𝑛,𝑦−𝑧_𝑛𝑦≥0,∀𝑦∈𝐶,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑧_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝑦_𝑛−𝑧≤𝑥_𝑛,𝑄−𝑧_𝑛=𝑧∈𝑄_𝑛−1∶𝑓𝑥_𝑛−1−𝑥_𝑛,𝑥_𝑛,𝑥−𝑧≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑓𝑥_𝑛,∀𝑛∈ℕ, (3.98) where 𝑃_{𝐶𝑛∩𝑄𝑛} is the metric projection of 𝐻 onto 𝐶_𝑛∩𝑄_𝑛 and {𝛼_𝑛}⊂(0,1) and {𝜆_𝑛}⊂(0,∞) satisfy 0≤𝛼_𝑛≤𝑏<1,0<𝑐≤𝜆_𝑛, (3.99) for some 𝑎,𝑏,𝑐∈ℝ. Then {𝑥_𝑛} converges strongly to 𝑧₀∈𝐹(𝑇)∩EP(𝐹).

Setting Ψ≡0 and 𝑓(𝑥_𝑛)=𝑥,∀𝑛∈ℕ in Theorem 3.7, we obtain the following result.

Corollary 3.10. Let 𝐻 be a Hilbert space and let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝐹∶𝐶×𝐶→ℝ be a bifunction satisfying (A1), (A2), (A3), and (A4) and let 𝑇∶𝐶→𝐶 be a quasi-nonexpansive mapping which is demiclosed on 𝐻. Assume that EP(𝐹)∩𝐹(𝑇)≠∅. Let 𝑄₁=𝐶 and {𝑥_𝑛}⊂𝐶 be a sequence generated by 𝑥₁=𝑥∈𝐶 and 𝐹𝑧_𝑛+1,𝑦𝜆_𝑛⟨𝑧_𝑛−𝑥_𝑛,𝑦−𝑧_𝑛𝑦⟩≥0,∀𝑦∈𝐶,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑧_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝑦_𝑛−𝑧≤𝑥_𝑛,𝑄−𝑧_𝑛=𝑧∈𝑄_𝑛−1∶𝑥−𝑥_𝑛,𝑥_𝑛,𝑥−𝑧≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥,∀𝑛∈ℕ, (3.100) where 𝑃_{𝐶𝑛∩𝑄𝑛} is the metric projection of 𝐻 onto 𝐶_𝑛∩𝑄_𝑛 and {𝛼_𝑛}⊂(0,1) and {𝜆_𝑛}⊂(0,∞) satisfy 0<𝑎≤𝛼_𝑛≤𝑏<1,0<𝑐≤𝜆_𝑛, (3.101) for some 𝑎,𝑏,𝑐∈ℝ. Then {𝑥_𝑛} converges strongly to 𝑧₀∈𝐹(𝑇)∩EP(𝐹).

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 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 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 [25] and Goebel and Kirk [26]. We also know that a firmly nonexpansive mapping 𝐹 can be deduced from an equilibrium problem in a Hilbert space.

Recently, Kocourek et al. [27] introduced a more broad class of nonlinear mappings called 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 [20]. Before proving, we need the following lemmas.

Lemma 4.1 (see [20]). 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 𝑇=(1/(1+𝛾))𝑆+(𝛾/(1+𝛾))𝐼. Then, 𝑆 is (𝛼,𝛽,𝛾)-super hybrid if and only if 𝑇 is (𝛼,𝛽)-generalized hybrid. In this case, 𝐹(𝑆)=𝐹(𝑇).

Lemma 4.2 (see [20]). Let 𝐻 be a Hilbert space and let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝑇∶𝐶→𝐻 be a generalized hybrid mapping. Then 𝑇 is demiclosed on 𝐶.

Setting 𝑇∶=(1/(1+𝛾))𝑆+(𝛾/(1+𝛾))𝐼 in Theorem 3.3, where 𝑆 is a super hybrid mapping and 𝛾 is a real number, we obtain the following result.

Theorem 4.3. Let 𝐻 be a Hilbert space and let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝐹∶𝐶×𝐶→ℝ be a bifunction satisfying (A1), (A2), (A3), and (A4) and let Ψ be a 𝛿-inverse strongly monotone mapping from 𝐶 into 𝐻. Let 𝛼,𝛽 and 𝛾 be real numbers with 𝛾≠−1 and let 𝑆∶𝐶→𝐻 be an (𝛼,𝛽,𝛾)-super hybrid mapping such that GEP(𝐹,Ψ)∩𝐹(𝑆)≠∅ and let 𝑓 be a Meir-Keeler contraction of 𝐶 into itself. Let 𝐶₁=𝐶 and {𝑥_𝑛}⊂𝐶 be a sequence generated by 𝑥₁=𝑥∈𝐶 and 𝐹𝑧_𝑛,𝑦+Ψ𝑥_𝑛,𝑦−𝑧_𝑛+1𝜆_𝑛𝑧_𝑛−𝑥_𝑛,𝑦−𝑧_𝑛𝑦≥0,∀𝑦∈𝐶,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛11+𝛾𝑆𝑧_𝑛+𝛾𝑧1+𝛾^𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶𝑦_𝑛−𝑧≤𝑥_𝑛,𝑥−𝑧_𝑛+1=𝑃_𝐶𝑛+1𝑓𝑥_𝑛,∀𝑛∈ℕ, (4.6) where 𝑃_𝐶𝑛+1 is the metric projection of 𝐻 onto 𝐶_𝑛+1 and {𝛼_𝑛}⊂(0,1) and {𝜆_𝑛}⊂(0,2𝛿) are sequences such that liminf_𝑛→∞𝛼_𝑛<1,0<𝑎≤𝜆_𝑛≤𝑏<2𝛿, (4.7) for some 𝑎,𝑏∈ℝ. Then {𝑥_𝑛} converges strongly to 𝑧₀=𝑃_{𝐹(𝑆)∩GEP(𝐹,Ψ)}𝑓(𝑧₀).

Proof. Put 𝑇=(1/(1+𝛾))𝑆+(𝛾/(1+𝛾))𝐼; we have from Lemma 4.1 that 𝑇 is a generalized hybrid mapping and 𝐹(𝑇)=𝐹(𝑆). Since 𝐹(𝑇)≠∅, we have that 𝑇 is quasi-nonexpansive. Following the proof of Theorem 3.3 and applying Lemma 4.2, we have the following result.

Setting 𝑓(𝑥_𝑛)=𝑥,∀𝑛∈ℕ in Theorem 4.3, we obtain the following result.

Corollary 4.4. Let 𝐻 be a Hilbert space and let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝐹∶𝐶×𝐶→ℝ be a bifunction satisfying (A1), (A2), (A3), and (A4) and let Ψ be a 𝛿-inverse strongly monotone mapping from 𝐶 into 𝐻. Let 𝛼,𝛽, and 𝛾 be real numbers with 𝛾≠−1 and let 𝑆∶𝐶→𝐻 be an (𝛼,𝛽,𝛾)-super hybrid mapping such that GEP(𝐹,Ψ)∩𝐹(𝑆)≠∅. Let 𝐶₁=𝐶 and {𝑥_𝑛}⊂𝐶 be a sequence generated by 𝑥₁=𝑥∈𝐶 and 𝐹𝑧_𝑛,𝑦+Ψ𝑥_𝑛,𝑦−𝑧_𝑛+1𝜆_𝑛𝑧_𝑛−𝑥_𝑛,𝑦−𝑧_𝑛𝑦≥0,∀𝑦∈𝐶,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛11+𝛾𝑆𝑧_𝑛+𝛾𝑧1+𝛾^𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶𝑦_𝑛−𝑧≤𝑥_𝑛,𝑥−𝑧_𝑛+1=𝑃_𝐶𝑛+1𝑥,∀𝑛∈ℕ, (4.8) where 𝑃_𝐶𝑛+1 is the metric projection of 𝐻 onto 𝐶_𝑛+1 and {𝛼_𝑛}⊂(0,1) and {𝜆_𝑛}⊂(0,2𝛿) are sequences such that liminf_𝑛→∞𝛼_𝑛<1,0<𝑎≤𝜆_𝑛≤𝑏<2𝛿, (4.9) for some 𝑎,𝑏∈ℝ. Then {𝑥_𝑛} converges strongly to 𝑧₀∈𝐹(𝑆)∩GEP(𝐹,Ψ).

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

Corollary 4.5. Let 𝐻 be a Hilbert space and let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝐹∶𝐶×𝐶→ℝ be a bifunction satisfying (A1), (A2), (A3), and (A4). Let 𝛼,𝛽, and 𝛾 be real numbers with 𝛾≠−1 and let 𝑆∶𝐶→𝐻 be an (𝛼,𝛽,𝛾)-super hybrid mapping such that EP(𝐹)∩𝐹(𝑆)≠∅ and let 𝑓 be a Meir-Keeler contraction of 𝐶 into itself. Let 𝐶₁=𝐶 and {𝑥_𝑛}⊂𝐶 be a sequence generated by 𝑥₁=𝑥∈𝐶 and 𝐹𝑧_𝑛+1,𝑦𝜆_𝑛⟨𝑧_𝑛−𝑥_𝑛,𝑦−𝑧_𝑛𝑦⟩≥0,∀𝑦∈𝐶,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛11+𝛾𝑆𝑧_𝑛+𝛾𝑧1+𝛾^𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶𝑦_𝑛−𝑧≤𝑥_𝑛,𝑥−𝑧_𝑛+1=𝑃_𝐶𝑛+1𝑓𝑥_𝑛,∀𝑛∈ℕ, (4.10) where 𝑃_𝐶𝑛+1 is the metric projection of 𝐻 onto 𝐶_𝑛+1 and {𝛼_𝑛}⊂(0,1) and {𝜆_𝑛}⊂(0,∞) are sequences such that liminf_𝑛→∞𝛼_𝑛<1,0<𝑎≤𝜆_𝑛, (4.11) for some 𝑎∈ℝ. Then {𝑥_𝑛} converges strongly to 𝑧₀∈𝐹(𝑆)∩EP(𝐹).

Setting Ψ≡0 and 𝑓(𝑥_𝑛)=𝑥, for all 𝑛∈ℕ in Theorem 4.3, we obtain the following result.

Corollary 4.6 (see [20], Theorem  5.2). Let 𝐻 be a Hilbert space and let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝐹∶𝐶×𝐶→ℝ be a bifunction satisfying (A1), (A2), (A3), and (A4). Let 𝛼,𝛽, and 𝛾 be real numbers with 𝛾≠−1 and let 𝑆∶𝐶→𝐻 be an (𝛼,𝛽,𝛾)-super hybrid mapping such that EP(𝐹)∩𝐹(𝑆)≠∅. Let 𝐶₁=𝐶 and let {𝑥_𝑛}⊂𝐶 be a sequence generated by 𝑥₁=𝑥∈𝐶 and 𝐹𝑧_𝑛+1,𝑦𝜆_𝑛𝑧_𝑛−𝑥_𝑛,𝑦−𝑧_𝑛𝑦≥0,∀𝑦∈𝐶,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛11+𝛾𝑆𝑧_𝑛+𝛾𝑧1+𝛾^𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶𝑦_𝑛−𝑧≤𝑥_𝑛,𝑥−𝑧_𝑛+1=𝑃_𝐶𝑛+1𝑥,∀𝑛∈ℕ, (4.12) where 𝑃_𝐶𝑛+1 is the metric projection of 𝐻 onto 𝐶_𝑛+1 and {𝛼_𝑛}⊂(0,1) and {𝜆_𝑛}⊂(0,∞) are sequences such that liminf_𝑛→∞𝛼_𝑛<1,0≤𝜆_𝑛, (4.13) for some 𝑎∈ℝ. Then {𝑥_𝑛} converges strongly to 𝑧₀∈𝐹(𝑆)∩EP(𝐹).

Setting 𝑇∶=(1/(1+𝛾))𝑆+(𝛾/(1+𝛾))𝐼 in Theorem 3.7, where 𝑆 is an super hybrid mapping and 𝛾 is a real number, we obtain the following result.

Theorem 4.7. Let 𝐻 be a Hilbert space and let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝐹∶𝐶×𝐶→ℝ be a bifunction satisfying (A1), (A2), (A3), and (A4) and let Ψ be a 𝛿-inverse strongly monotone mapping from 𝐶 into 𝐻. Let 𝛼,𝛽, and 𝛾 be real numbers with 𝛾≠−1 and let 𝑆∶𝐶→𝐻 be an (𝛼,𝛽,𝛾)-super hybrid mapping such that GEP(𝐹,Ψ)∩𝐹(𝑆)≠∅ and let 𝑓 be a Meir-Keeler contraction of 𝐶 into itself. Let 𝑄₁=𝐶 and {𝑥_𝑛}⊂𝐶 be a sequence generated by 𝑥₁=𝑥∈𝐶 and 𝐹𝑧_𝑛,𝑦+Ψ𝑥_𝑛,𝑦−𝑧_𝑛+1𝜆_𝑛𝑧_𝑛−𝑥_𝑛,𝑦−𝑧_𝑛𝑦≥0,∀𝑦∈𝐶,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛11+𝛾𝑆𝑧_𝑛+𝛾𝑧1+𝛾^𝑛,𝐶_𝑛=𝑧∈𝐶∶𝑦_𝑛−𝑧≤𝑥_𝑛,𝑄−𝑧_𝑛=𝑧∈𝑄_𝑛−1∶𝑓𝑥_𝑛−1−𝑥_𝑛,𝑥_𝑛,𝑥−𝑧≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑓𝑥_𝑛,∀𝑛∈ℕ, (4.14) where 𝑃_{𝐶𝑛∩𝑄𝑛} is the metric projection of 𝐻 onto 𝐶_𝑛∩𝑄_𝑛 and {𝛼_𝑛}⊂(0,1) and {𝜆_𝑛}⊂(0,2𝛿) satisfy 0≤𝛼_𝑛≤𝑏<1,0<𝑐≤𝜆_𝑛≤𝑑<2𝛿, (4.15) for some 𝑏,𝑐,𝑑∈ℝ. Then {𝑥_𝑛} converges strongly to 𝑧₀=𝑃_{𝐹(𝑆)∩GEP(𝐹,Ψ)}𝑓(𝑧₀).

Proof. Put 𝑇=(1/(1+𝛾))𝑆+(𝛾/(1+𝛾))𝐼; we have from Lemma 4.1 that 𝑇 is a generalized hybrid mapping and 𝐹(𝑇)=𝐹(𝑆). Since 𝐹(𝑇)≠∅, we have that 𝑇 is quasi-nonexpansive. Following the proof of Theorem 3.7 and applying Lemma 4.2, we obtain the following result.

Setting 𝑓(𝑥_𝑛)=𝑥,∀𝑛∈ℕ in Theorem 4.7, we obtains the following result.

Corollary 4.8. Let 𝐻 be a Hilbert space and let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝐹∶𝐶×𝐶→ℝ be a bifunction satisfying (A1), (A2), (A3), and (A4) and let Ψ be a 𝛿-inverse strongly monotone mapping from 𝐶 into 𝐻. Let 𝛼,𝛽, and 𝛾 be real numbers with 𝛾≠−1 and let 𝑆∶𝐶→𝐻 be an (𝛼,𝛽,𝛾)-super hybrid mapping such that GEP(𝐹,Ψ)∩𝐹(𝑆)≠∅ and let 𝑓 be a Meir-Keeler contraction of 𝐶 into itself. Let {𝑥_𝑛}⊂𝐶 be a sequence generated by 𝑥₁=𝑥∈𝐶 and 𝐹𝑧_𝑛,𝑦+Ψ𝑥_𝑛,𝑦−𝑧_𝑛+1𝜆_𝑛𝑧_𝑛−𝑥_𝑛,𝑦−𝑧_𝑛𝑦≥0,∀𝑦∈𝐶,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛11+𝛾𝑆𝑧_𝑛+𝛾𝑧1+𝛾^𝑛,𝐶_𝑛=𝑧∈𝐶∶𝑦_𝑛−𝑧≤𝑥_𝑛,𝑄−𝑧_𝑛=𝑧∈𝐶∶𝑥−𝑥_𝑛,𝑥_𝑛,𝑥−𝑧≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥,∀𝑛∈ℕ, (4.16) where 𝑃_{𝐶𝑛∩𝑄𝑛} is the metric projection of 𝐻 onto 𝐶_𝑛∩𝑄_𝑛 and {𝛼_𝑛}⊂(0,1) and {𝜆_𝑛}⊂(0,2𝛿) satisfy 0≤𝛼_𝑛≤𝑏<1,0<𝑐≤𝜆_𝑛≤𝑑<2𝛿, (4.17) for some 𝑏,𝑐,𝑑∈ℝ. Then {𝑥_𝑛} converges strongly to 𝑧₀∈𝐹(𝑆)∩GEP(𝐹,Ψ).

Setting Ψ≡0 in Theorem 4.7, we obtain the following result.

Corollary 4.9. Let 𝐻 be a Hilbert space and let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝐹∶𝐶×𝐶→ℝ be a bifunction satisfying (A1), (A2), (A3), and (A4). Let 𝛼,𝛽, and 𝛾 be real numbers with 𝛾≠−1 and let 𝑆∶𝐶→𝐻 be an (𝛼,𝛽,𝛾)-super hybrid mapping such that EP(𝐹)∩𝐹(𝑆)≠∅ and let 𝑓 be a Meir-Keeler contraction of 𝐶 into itself. Let 𝑄₁=𝐶 and {𝑥_𝑛}⊂𝐶 be a sequence generated by 𝑥₁=𝑥∈𝐶 and 𝐹𝑧_𝑛+1,𝑦𝜆_𝑛𝑧_𝑛−𝑥_𝑛,𝑦−𝑧_𝑛𝑦≥0,∀𝑦∈𝐶,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛11+𝛾𝑆𝑧_𝑛+𝛾𝑧1+𝛾^𝑛,𝐶_𝑛=𝑧∈𝐶∶𝑦_𝑛−𝑧≤𝑥_𝑛,𝑄−𝑧_𝑛=𝑧∈𝑄_𝑛−1∶𝑓𝑥_𝑛−1−𝑥_𝑛,𝑥_𝑛,𝑥−𝑧≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑓𝑥_𝑛,∀𝑛∈ℕ, (4.18) where 𝑃_{𝐶𝑛∩𝑄𝑛} is the metric projection of 𝐻 onto 𝐶_𝑛∩𝑄_𝑛 and {𝛼_𝑛}⊂(0,1) and {𝜆_𝑛}⊂(0,∞) satisfy 0≤𝛼_𝑛≤𝑏<1,0<𝑐≤𝜆_𝑛, (4.19) for some 𝑏,𝑐∈ℝ. Then {𝑥_𝑛} converges strongly to 𝑧₀∈𝐹(𝑆)∩EP(𝐹).

Setting Ψ≡0 and 𝑓(𝑥_𝑛)=𝑥,∀𝑛∈ℕ in Theorem 4.7, we obtain the following result.

Corollary 4.10 (see [20], Theorem  5.1). Let 𝐻 be a Hilbert space and let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝐹∶𝐶×𝐶→ℝ be a bifunction satisfying (A1), (A2), (A3), and (A4). Let 𝛼,𝛽, and 𝛾 be real numbers with 𝛾≠−1 and let 𝑆∶𝐶→𝐻 be an (𝛼,𝛽,𝛾)-super hybrid mapping such that EP(𝐹)∩𝐹(𝑆)≠∅. Let {𝑥_𝑛}⊂𝐶 be a sequence generated by 𝑥₁=𝑥∈𝐶 and 𝐹𝑧_𝑛+1,𝑦𝜆_𝑛𝑧_𝑛−𝑥_𝑛,𝑦−𝑧_𝑛𝑦≥0,∀𝑦∈𝐶,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛11+𝛾𝑆𝑧_𝑛+𝛾𝑧1+𝛾^𝑛,𝐶_𝑛=𝑧∈𝐶∶𝑦_𝑛−𝑧≤𝑥_𝑛,𝑄−𝑧_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑥−𝑧_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥,∀𝑛∈ℕ, (4.20) where 𝑃_{𝐶𝑛∩𝑄𝑛} is the metric projection of 𝐻 onto 𝐶_𝑛∩𝑄_𝑛 and {𝛼_𝑛}⊂(0,1) and {𝜆_𝑛}⊂(0,∞) satisfy 0≤𝛼_𝑛≤𝑏<1,0<𝑐≤𝜆_𝑛, (4.21) for some 𝑏,𝑐∈ℝ. Then {𝑥_𝑛} converges strongly to 𝑧₀∈𝐹(𝑆)∩EP(𝐹).

Acknowledgment

The authors would like to thank Naresuan University for financial support.