Weak Convergence Theorem for Finding Fixed Points and Solution of Split Feasibility and Systems of Equilibrium Problems

Abstract

The purpose of this paper is to introduce an iterative algorithm for finding a common element of the set of fixed points of quasi-nonexpansive mappings and the solution of split feasibility problems (SFP) and systems of equilibrium problems (SEP) in Hilbert spaces. We prove that the sequences generated by the proposed algorithm converge weakly to a common element of the fixed points set of quasi-nonexpansive mappings and the solution of split feasibility problems and systems of equilibrium problems under mild conditions. Our main result improves and extends the recent ones announced by Ceng et al. (2012) and many others.

1. Introduction

Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. A mapping 𝑇∶𝐶→𝐶 is said to be nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶. Denote the set of fixed points of 𝑇 by 𝐹(𝑇). On the other hand, a mapping 𝑇∶𝐶→𝐶 is said to be quasi-nonexpansive if 𝐹(𝑇)≠∅ and ‖𝑇𝑥−𝑞‖≤‖𝑥−𝑞‖ for all 𝑥∈𝐶 and 𝑞∈𝐹(𝑇). If 𝑇∶𝐶→𝐶 is nonexpansive and the set 𝐹(𝑇) of fixed points of 𝑇 is nonempty, then 𝑇 is quasi-nonexpansive. Fixed point iterations process for nonexpansive mappings and quasi-nonexpansive mappings in Banach spaces including Mann and Ishikawa iterations process have been studied extensively by many authors to solve the nonlinear operator equations (see [1–4]).

Let 𝐹 be a bifunction of 𝐶×𝐶 into ℝ, where ℝ is the set of real numbers. The equilibrium problem for 𝐹∶𝐶×𝐶→ℝ is to find 𝑥∈𝐶 such that 𝐹𝑥,𝑦≥0∀𝑦∈𝐶. (1) The set of solutions of (1) is denoted by EP(𝐹). Numerous problems in physics, optimization, and economics reduce to find a solution of (1) in Hilbert spaces; see, for instance, Blum and Oettli [5], Flam and Antipin [6], and Moudafi [7]. Moreover, Flam and Antipin [6] introduced an iterative scheme of finding the best approximation to the solution of equilibrium problem, when EP(𝐹) is nonempty, and proved a strong convergence theorem (see also in [8–11]). Let 𝐹₁,𝐹₂∶𝐶×𝐶→ℝ be two-monotone bifunction and 𝜆>0 is a constant. Recently, Moudafi [12] considered the following of a system of equilibrium problem, denoting the set of solution of SEP by Ω, for finding (𝑥^∗,𝑦^∗)∈𝐶×𝐶 such that 𝜆𝐹₁𝑥^∗,𝑧+𝑦^∗−𝑥^∗,𝑥^∗−𝑧≥0,∀𝑧∈𝐶,𝜆𝐹₂𝑦^∗,𝑧+𝑥^∗−𝑦^∗,𝑦^∗−𝑧≥0,∀𝑧∈𝐶. (2) He also proved the weak convergence theorem of this problem (some related work can be found in [13, 14]).

The split feasibility problem (SFP) in Hilbert spaces for modeling inverse problems which arise from phase retrievals and in medical image reconstruction was first introduced by Censor and Elfving [15] (see, e.g. [16, 17]). It has been found that the SFP can also be used to model the intensity-modulated radiation therapy (see [18, 19]). In this work, the SFP is formulated as finding a point 𝑥^∗ with the property 𝑥^∗∈𝐶,𝐴𝑥^∗∈𝑄, (3) where 𝐶 and 𝑄 are the nonempty closed convex subsets of the infinite-dimensional real Hilbert spaces 𝐻₁ and 𝐻₂, respectively, and 𝐴∈𝐵(𝐻₁,𝐻₂) (i.e., 𝐴 is a bounded linear operator from 𝐻₁ to 𝐻₂). Very recently, there are related works which we can find in [16, 18, 20–26] and the references therein.

A special case of the SFP is called the convex constrained linear inverse problem (see [27]), that is, the problem to finding an element 𝑥 such that 𝑥∈𝐶,𝐴𝑥=𝑏∈𝑄. (4) In fact, it has been extensively investigated in the literature using the projected Landweber iterative method [27, 28]. Throughout this paper, we assume that the solution set Γ of the SFP is nonempty.

Motivated and inspired by the regularization method and extragradient method due to Ceng et al. [29], we introduce and analyze an extragradient method with regularization for finding a common element of the fixed points set of quasi-nonexpansive mappings and the solution of split feasibility problems (SFP) and systems of equilibrium problems (SEP) in Hilbert spaces. Our results represent the improvement, extension, and development of the corresponding results in [14, 29].

2. Preliminaries

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖, and let 𝐶 be a closed convex subset of 𝐻. We write 𝑥_𝑛⇀𝑥 to indicate that the sequence {𝑥_𝑛} converges weakly to 𝑥 and 𝑥_𝑛→𝑥 to indicate that the sequence {𝑥_𝑛} converges strongly to 𝑥. For every point 𝑥∈𝐻, there exists a unique nearest point in 𝐶, denoted by 𝑃_𝐶𝑥, such that 𝑥−𝑃_𝐶𝑥=inf_𝑦∈𝐶𝑥−𝑦≤𝑥−𝑦∀𝑦∈𝐶. (5)𝑃_𝐶 is called the metric projection of 𝐻 onto 𝐶.

Some important properties of projections are gathered in the following proposition.

Proposition 1 (see [29]). For given 𝑥∈𝐻 and 𝑧∈𝐶: (i)𝑧=𝑃_𝐶𝑥𝑖𝑓𝑎𝑛𝑑𝑜𝑛𝑙𝑦𝑖𝑓⟨𝑥−𝑧,𝑦−𝑧⟩≤0,𝑓𝑜𝑟𝑎𝑙𝑙𝑦∈𝐶; (ii)𝑧=𝑃_𝐶𝑥𝑖𝑓𝑎𝑛𝑑𝑜𝑛𝑙𝑦𝑖𝑓‖𝑥−𝑧‖²≤‖𝑥−𝑦‖²−‖𝑦−𝑧‖²,𝑓𝑜𝑟𝑎𝑙𝑙𝑦∈𝐶.

Definition 2 (see [30, 31]). Let 𝑇 be a nonlinear operator with domain 𝐷(𝑇)⊆𝐻 and range 𝑅(𝑇)⊆𝐻, and let 𝛽>0 and 𝑣>0 be given constants. The operator 𝑇 is called(a)monotone if 𝑥−𝑦,𝑇𝑥−𝑇𝑦≥0,∀𝑥,𝑦∈𝐷(𝑇); (6)(b)𝛽-strongly  monotone if 𝑥−𝑦,𝑇𝑥−𝑇𝑦≥𝛽𝑥−𝑦²,∀𝑥,𝑦∈𝐷(𝑇); (7)(c)𝑣-inverse  strongly  monotone (𝑣-ism) if 𝑥−𝑦,𝑇𝑥−𝑇𝑦≥𝑣𝑇𝑥−𝑇𝑦²,∀𝑥,𝑦∈𝐷(𝑇). (8)

We can easily see that if 𝑆 is nonexpansive, then 𝐼−𝑆 is monotone. It is also easy to see that a projection 𝑃_𝐶 is a 1-ism.

Definition 3 (see [29]). A mapping 𝑇∶𝐻→𝐻 is said to be an averaged mapping if it can be written as the average of the identity 𝐼 and a nonexpansive mapping, that is, 𝑇≡(1−𝛼)𝐼+𝛼𝑆, (9) where 𝛼∈(0,1) and 𝑆∶𝐻→𝐻 is nonexpansive. More precisely, when (9) holds, we say that 𝑇 is 𝛼-𝑎𝑣𝑒𝑟𝑎𝑔𝑒𝑑. It is easly to see that if 𝑇 is an averaged mapping, then 𝑇 is nonexpansive.

Proposition 4 (see [20]). Let 𝑇∶𝐻→𝐻 be a given mapping. Then consider the following.(i)𝑇 is nonexpansive if and only if the complement 𝐼−𝑇 is (1/2)-ism. (ii)𝑇 is averaged if and only if the complement 𝐼−𝑇 is 𝑣-ism for some 𝑣>1/2. Indeed, for 𝛼∈(0,1),𝑇 is 𝛼-averaged if and only if 𝐼−𝑇 is (1/2𝛼)-ism. (iii)The composite of finitely many averaged mappings is averaged. That is, if each of the mappings {𝑇_𝑖}_𝑁^𝑖=1 is averaged, then so is the composite 𝑇₁∘𝑇₂∘⋯∘𝑇_𝑁. In particular, if 𝑇₁ is 𝛼₁-averaged and 𝑇₂𝑖𝑠𝛼₂-𝑎𝑣𝑒𝑟𝑎𝑔𝑒𝑑, where 𝛼₁,𝛼₂∈(0,1), then the composite 𝑇₁∘𝑇₂ is 𝛼-averaged, where 𝛼=𝛼₁+𝛼₂−𝛼₁𝛼₂.

In this paper, we use an equilibrium bifunction 𝐹∶𝐶×𝐶→ℝ for solving the equilibrium problems, let us assume that 𝐹 satisfies the following conditions: (A1)𝐹(𝑥,𝑥)=0forall𝑥∈𝐶; (A2)𝐹 is monotone, that is, 𝐹(𝑥,𝑦)+𝐹(𝑦,𝑥)≤0forall𝑥,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; semicontinuous.

Lemma 5 (see [6]). Assume that 𝐹∶𝐶×𝐶→𝐑 satisfies (A1)–(A4). For 𝑟>0 and 𝑥∈𝐻, define a mapping 𝑇_𝑟∶𝐻→𝐶 as follows: 𝑇_𝑟(𝑥)=+1𝑧∈𝐶∶𝐹𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶, (10) for all 𝑧∈𝐻. Then, the following hold: (i)𝑇_𝑟 is single-valued; (ii)𝑇_𝑟 is firmly nonexpansive, that is, for any 𝑥,𝑦∈𝐻,‖𝑇_𝑟𝑥−𝑇_𝑟𝑦‖²≤⟨𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝑥−𝑦⟩;(iii)𝐹(𝑇_𝑟)=𝐸𝑃(𝐹); (iv)𝐸𝑃(𝐹) is closed and convex.

Lemma 6 (see [32]). 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 𝑇_1,𝜆 and 𝑇_2,𝜇 are defined as in Lemma 5 associated to 𝐹₁ and 𝐹₂, respectively. For given 𝑥^∗,𝑦^∗∈𝐶,(𝑥^∗,𝑦^∗) is a solution of problem (2) if and only if 𝑥^∗ is a fixed point of the mapping 𝐺∶𝐶→𝐶 defined by 𝐺(𝑥)=𝑇_1,𝜆𝑇_2,𝜇𝑥,∀𝑥∈𝐶, (11) where 𝑦^∗=𝑇_2,𝜇𝑥^∗.

Lemma 7 (see [33]). Let {𝑎_𝑛}^∞_𝑛=1,{𝑏_𝑛}^∞_𝑛=1 and {𝛿_𝑛}^∞_𝑛=1 be sequences of nonnegative real numbers satisfying the inequality 𝑎_𝑛+1≤1+𝛿_𝑛𝑎_𝑛+𝑏_𝑛,∀𝑛≥1. (12) If ∑^∞_𝑛=1𝛿_𝑛<∞ and ∑^∞_𝑛=1𝑏_𝑛<∞, then lim_𝑛→∞𝑎_𝑛 exists. If, in addition, {𝑎_𝑛}^∞_𝑛=1 has a subsequence which converges to zero, then lim_𝑛→∞𝑎_𝑛=0.

3. Weak Convergence Theorem

In this section, we prove a weak convergence theorem by an extragradient methods for finding a common element of the fixed points set of quasi-nonexpansive mappings and the solution of split feasibility problems and systems of equilibrium problems in Hilbert spaces. The function 𝑓∶𝐻→ℝ is a continuous differentiable function with the minimization problem given by min_𝑥∈𝐶𝑓(𝑥)1∶=2𝐴𝑥−𝑃_𝑄𝐴𝑥². (13) In 2010, Xu [17] considered the following Tikhonov regularized problem: min_𝑥∈𝐶𝑓_𝛼(𝑥)1∶=2𝐴𝑥−𝑃_𝑄𝐴𝑥²+12𝛼‖𝑥‖², (14) where 𝛼>0 is the regularization parameter. The gradient given by ∇𝑓_𝛼(𝑥)=∇𝑓(𝑥)+𝛼𝐼=𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼𝐼 (15) is (𝛼+‖𝐴‖²)-Lipschitz continuous and 𝛼-strongly monotone (see [29] for the details).

Lemma 8 (see [17, 29]). The following hold: (i)Γ=𝐹(𝑃_𝐶(𝐼−𝜆∇𝑓))=𝑉𝐼(𝐶,∇𝑓) for any 𝜆>0, where 𝐹(𝑃_𝐶(𝐼−𝜆∇𝑓)) and 𝑉𝐼(𝐶,∇𝑓) denote the set of fixed points of 𝑃_𝐶(𝐼−𝜆∇𝑓) and the solution set of VIP; (ii)𝑃_𝐶(𝐼−𝜆∇𝑓_𝛼) is 𝜉-averaged for each 𝜆∈(0,2/(𝛼+‖𝐴‖²)), where 𝜉=(2+𝜆(𝛼+‖𝐴‖²))/4.

Theorem 9. Let 𝐶 be a nonempty closed convex subset in a real Hilbert space 𝐻. Let 𝜇>0, 𝐹₁ and 𝐹₂ be two bifunctions from 𝐶×𝐶→ℝ satisfying (A1)–(A4). Let 𝑆 be a quasi-nonexpansive mapping of 𝐶 into itself such that 𝐼−𝑆 be demiclosed at zero, that is, if {𝑤_𝑛}⊂𝐶,𝑤_𝑛⇀𝑤 and (𝐼−𝑆)𝑤_𝑛→0, then 𝑤∈𝐹(𝑆), with 𝐹(𝑆)∩Γ∩Ω≠∅. Let {𝑥_𝑛}, {𝑦_𝑛}, {𝑧_𝑛}, and {𝑤_𝑛} be the sequence in 𝐶 generated by the following extragradient algorithm: 𝑥₀𝑤=𝑥∈𝐶,_𝑛∈𝐶;𝐹₂𝑤_𝑛,𝑧+𝜑(𝑧)−𝜑𝑤_𝑛+1𝜇𝑧−𝑤_𝑛,𝑤_𝑛−𝑥_𝑛𝑧≥0,∀𝑧∈𝐶,_𝑛∈𝐶;𝐹₁𝑧_𝑛,𝑧+𝜑(𝑧)−𝜑𝑧_𝑛+1𝜇𝑧−𝑧_𝑛,𝑧_𝑛−𝑤_𝑛𝑦≥0,∀𝑧∈𝐶,_𝑛=𝑃_𝐶𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑧_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑆𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛, (16) where ∑^∞_𝑛=0𝛼_𝑛<∞, {𝜆_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1/‖𝐴‖²) and {𝛽_𝑛}⊂[𝑐,𝑑] for some 𝑐,𝑑∈(0,1). Then, the sequences {𝑥_𝑛} and {𝑦_𝑛} converge weakly to an element 𝑥∈𝐹(𝑆)∩Γ∩Ω.

Proof. By Lemma 8, we have that 𝑃_𝐶(𝐼−𝜆∇𝑓_𝛼) is 𝜉-averaged for each 𝜆∈(0,2/(𝛼+‖𝐴‖²)), where 𝜉=(2+𝜆(𝛼+‖𝐴‖²))/4. Hence, by Proposition 2.4, 𝑃_𝐶(𝐼−𝜆∇𝑓_𝛼) is nonexpansive. From {𝜆_𝑛}⊂[𝑎,𝑏] and 𝑎,𝑏∈(0,1/‖𝐴‖²), we have 𝑎≤inf_𝑛≥0𝜆_𝑛≤sup_𝑛≥0𝜆_𝑛≤𝑏<1/‖𝐴‖²=lim_𝑛→∞1/(𝛼_𝑛+‖𝐴‖²). Without loss of generality, we assume that 𝑎≤inf_𝑛≥0𝜆_𝑛≤sup_𝑛≥0𝜆_𝑛≤𝑏<1/(𝛼_𝑛+‖𝐴‖²),forall𝑛≥0. Hence, for each 𝑛≥0,𝑃_𝐶(𝐼−𝜆_𝑛∇𝑓_𝛼𝑛) is 𝜉_𝑛-averaged with 𝜉_𝑛=12+𝜆_𝑛𝛼_𝑛+‖𝐴‖²2−12⋅𝜆_𝑛𝛼_𝑛+‖𝐴‖²2=2+𝜆_𝑛𝛼_𝑛+‖𝐴‖²4∈(0,1). (17) This implies that 𝑃_𝐶(𝐼−𝜆_𝑛∇𝑓_𝛼𝑛) is nonexpansive for all 𝑛≥0.
Next, we show that the sequence {𝑥_𝑛} is bounded. Indeed, take a fixed 𝑝∈𝐹(𝑆)∩Γ∩Ω arbitrarily. Let 𝑇_1,𝜇 and 𝑇_2,𝜇 be defined as in Lemma 5 associated to 𝐹₁ and 𝐹₂, respectively. Thus, we get 𝑝=𝑆𝑝=𝑆𝑃_𝐶(𝑝),forall𝑛≥0,𝑝=𝑃_𝐶(𝐼−𝜆∇𝑓)𝑝,forall𝜆∈(0,2/‖𝐴‖²) and 𝑝=𝑇_1,𝜇(𝑇_2,𝜇𝑝),forall𝜇>0. Put 𝑦^∗=𝑇_2,𝜇𝑝,𝑧_𝑛=𝑇_1,𝜇𝑤_𝑛 and 𝑤_𝑛=𝑇_2,𝜇𝑥_𝑛. From (29), we have 𝑦_𝑛−𝑝=𝑃_𝐶𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑧_𝑛−𝑃_𝐶𝐼−𝜆_𝑛𝑝≤𝑃∇𝑓_𝐶𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑧_𝑛−𝑃_𝐶𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑝+𝑃_𝐶𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑝−𝑃_𝐶𝐼−𝜆_𝑛𝑝≤𝑧∇𝑓_𝑛−𝑝+𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑝−𝐼−𝜆_𝑛𝑝=𝑧∇𝑓_𝑛−𝑝+𝜆_𝑛𝑝∇𝑓−∇𝑓_𝛼𝑛=𝑧_𝑛−𝑝+𝜆_𝑛𝛼_𝑛𝑝=𝑧_𝑛−𝑝+𝜆_𝑛𝛼_𝑛𝑝. This implies that ‖𝑧_𝑛−𝑝‖=‖𝑇_1,𝜇𝑤_𝑛−𝑇_1,𝜇𝑦^∗‖≤‖𝑤_𝑛−𝑦^∗‖=‖𝑇_2,𝜇𝑥_𝑛−𝑇_2,𝜇𝑝‖≤‖𝑥_𝑛−𝑝‖. Thus, we obtain ‖𝑦_𝑛−𝑝‖≤‖𝑥_𝑛−𝑝‖+𝜆_𝑛𝛼_𝑛‖𝑝‖. Put 𝑙_𝑛=𝑃_𝐶(𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛(𝑦_𝑛)) for each 𝑛≥0. Then, by Proposition 1(ii), we have 𝑙_𝑛−𝑝²≤𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−𝑝²−𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−𝑙_𝑛²=𝑥_𝑛−𝑝²−𝑥_𝑛−𝑙_𝑛²+2𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛,𝑝−𝑙_𝑛≤𝑥_𝑛−𝑝²−𝑥_𝑛−𝑙_𝑛²+2𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛,𝑦_𝑛−𝑙_𝑛+∇𝑓_𝛼𝑛𝑝,𝑝−𝑦_𝑛≤𝑥_𝑛−𝑝²−𝑥_𝑛−𝑙_𝑛²+2𝜆_𝑛𝛼_𝑛𝑝,𝑝−𝑦_𝑛+∇𝑓_𝛼𝑛𝑦_𝑛,𝑦_𝑛−𝑙_𝑛≤𝑥_𝑛−𝑝²−𝑥_𝑛−𝑦_𝑛²−2𝑥_𝑛−𝑦_𝑛,𝑦_𝑛−𝑙_𝑛−𝑦_𝑛−𝑙_𝑛²+2𝜆_𝑛𝛼_𝑛𝑝,𝑝−𝑦_𝑛+∇𝑓_𝛼𝑛𝑦_𝑛,𝑦_𝑛−𝑙_𝑛=𝑥_𝑛−𝑝²−𝑥_𝑛−𝑦_𝑛²−𝑦_𝑛−𝑙_𝑛²+2𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−𝑦_𝑛,𝑙_𝑛−𝑦_𝑛+2𝜆_𝑛𝛼_𝑛𝑝,𝑝−𝑦_𝑛. (19) Hence, by Proposition 1(i), we have 𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−𝑦_𝑛,𝑙_𝑛−𝑦_𝑛=𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑥_𝑛−𝑦_𝑛,𝑙_𝑛−𝑦_𝑛+𝜆_𝑛∇𝑓_𝛼𝑛𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛,𝑙_𝑛−𝑦_𝑛≤𝜆_𝑛∇𝑓_𝛼𝑛𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛,𝑙_𝑛−𝑦_𝑛≤𝜆_𝑛∇𝑓_𝛼𝑛𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛𝑙_𝑛−𝑦_𝑛≤𝜆_𝑛𝛼_𝑛+‖𝐴‖²𝑥_𝑛−𝑦_𝑛𝑙_𝑛−𝑦_𝑛. (20) So, we have 𝑙_𝑛−𝑝²≤𝑥_𝑛−𝑝²−𝑥_𝑛−𝑦_𝑛²−𝑦_𝑛−𝑙_𝑛²+2𝜆_𝑛𝛼_𝑛+‖𝐴‖²𝑥_𝑛−𝑦_𝑛𝑙_𝑛−𝑦_𝑛+2𝜆_𝑛𝛼_𝑛𝑝𝑝−𝑦_𝑛≤𝑥_𝑛−𝑝²−𝑥_𝑛−𝑦_𝑛²−𝑦_𝑛−𝑙_𝑛²+𝜆²_𝑛𝛼_𝑛+‖𝐴‖²²𝑥_𝑛−𝑦_𝑛²+𝑙_𝑛−𝑦_𝑛²+2𝜆_𝑛𝛼_𝑛𝑝𝑝−𝑦_𝑛=𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑝−𝑦_𝑛+𝜆²_𝑛𝛼_𝑛+‖𝐴‖²²−1𝑥_𝑛−𝑦_𝑛²≤𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑝−𝑦_𝑛≤𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑧_𝑛−𝑝+𝜆_𝑛𝛼_𝑛𝑝≤𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑥_𝑛−𝑝+𝜆_𝑛𝛼_𝑛𝑝. (21) Then, from the last inequality we conclude that 𝑥_𝑛+1−𝑝²=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑆𝑙_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+1−𝛽_𝑛𝑆𝑙_𝑛−𝑝²−𝛽_𝑛1−𝛽_𝑛𝑥_𝑛−𝑆𝑙_𝑛²≤𝛽_𝑛𝑥_𝑛−𝑝²+1−𝛽_𝑛𝑙_𝑛−𝑝²−𝛽_𝑛1−𝛽_𝑛𝑥_𝑛−𝑆𝑙_𝑛²≤𝛽_𝑛𝑥_𝑛−𝑝²+1−𝛽_𝑛𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑝−𝑦_𝑛+𝜆²_𝑛𝛼_𝑛+‖𝐴‖²²−1𝑥_𝑛−𝑦_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑥_𝑛−𝑆𝑙_𝑛²≤𝑥_𝑛−𝑝²+2𝜆_𝑛𝛼_𝑛𝑝𝑝−𝑦_𝑛+1−𝛽_𝑛𝜆²_𝑛𝛼_𝑛+‖𝐴‖²²−1𝑥_𝑛−𝑦_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑥_𝑛−𝑆𝑙_𝑛²≤𝑥_𝑛−𝑝²+𝛼_𝑛𝜆²_𝑛𝑝²+𝑝−𝑦_𝑛²+1−𝛽_𝑛𝜆²_𝑛𝛼_𝑛+‖𝐴‖²²−1𝑥_𝑛−𝑦_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑥_𝑛−𝑆𝑙_𝑛²≤𝑥_𝑛−𝑝²+𝛼_𝑛𝜆²_𝑛𝑝²+𝑥_𝑛−𝑝+𝜆_𝑛𝛼_𝑛𝑝²+1−𝛽_𝑛𝜆²_𝑛𝛼_𝑛+‖𝐴‖²²−1𝑥_𝑛−𝑦_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑥_𝑛−𝑆𝑙_𝑛²≤𝑥_𝑛−𝑝²+𝛼_𝑛𝜆²_𝑛𝑝²+2𝑥_𝑛−𝑝²+2𝜆²_𝑛𝛼²_𝑛𝑝²+1−𝛽_𝑛𝜆²_𝑛𝛼_𝑛+‖𝐴‖²²−1𝑥_𝑛−𝑦_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑥_𝑛−𝑆𝑙_𝑛²=1+2𝛼_𝑛𝑥_𝑛−𝑝²+𝛼_𝑛𝜆²_𝑛𝑝²1+2𝛼²_𝑛+1−𝛽_𝑛𝜆²_𝑛𝛼_𝑛+‖𝐴‖²²−1𝑥_𝑛−𝑦_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑥_𝑛−𝑆𝑙_𝑛²≤1+2𝛼_𝑛𝑥_𝑛−𝑝²+𝛼_𝑛𝜆²_𝑛𝑝²1+2𝛼²_𝑛=1+𝛿_𝑛𝑥_𝑛−𝑝²+𝑏_𝑛, where 𝛿_𝑛=2𝛼_𝑛 and 𝑏_𝑛=𝛼_𝑛𝜆²_𝑛‖𝑝‖²(1+2𝛼²_𝑛). Since ∑^∞_𝑛=0𝛼_𝑛<∞ and {𝜆_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1/‖𝐴‖²), we conclude that ∑^∞_𝑛=0𝛿_𝑛<∞ and ∑^∞_𝑛=0𝑏_𝑛<∞. Therefore, by Lemma 7, we note that lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists for each 𝑝∈𝐹(𝑆)∩Γ∩Ω and hence the sequences {𝑥_𝑛},{𝑙_𝑛},{𝑦_𝑛},{𝑧_𝑛}, and {𝑤_𝑛} are bounded. From (22), we also obtain (1−𝑑)1−𝑏²𝛼_𝑛+‖𝐴‖²²𝑥_𝑛−𝑦_𝑛²+𝑐(1−𝑑)𝑥_𝑛−𝑆𝑙_𝑛²≤1−𝛽_𝑛1−𝜆²_𝑛𝛼_𝑛+‖𝐴‖²²𝑥_𝑛−𝑦_𝑛²+𝛽_𝑛1−𝛽_𝑛𝑥_𝑛−𝑆𝑙_𝑛²≤1+2𝛼_𝑛𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+𝛼_𝑛𝜆²_𝑛𝑝²1+2𝛼²_𝑛, (23) where {𝜆_𝑛}⊂[𝑎,𝑏] and {𝛽_𝑛}⊂[𝑐,𝑑]. Since lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists and 𝛼_𝑛→0, it follows that lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=lim_𝑛→∞𝑥_𝑛−𝑆𝑙_𝑛≤lim_𝑛→∞𝑥_𝑛−𝑝+𝑝−𝑆𝑙_𝑛≤lim_𝑛→∞𝑥_𝑛−𝑝+𝑝−𝑙_𝑛=0. (24) Similarly, from inequality (22), we have (1−𝑑)1−𝑏²𝛼_𝑛+‖𝐴‖²²𝑥_𝑛−𝑧_𝑛²+𝑐(1−𝑑)𝑥_𝑛−𝑆𝑙_𝑛²≤1−𝛽_𝑛1−𝜆²_𝑛𝛼_𝑛+‖𝐴‖²²𝑥_𝑛−𝑧_𝑛²+𝛽_𝑛1−𝛽_𝑛𝑥_𝑛−𝑆𝑙_𝑛²≤1+𝛼_𝑛𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+𝛼_𝑛𝜆²_𝑛𝑝², (25) where {𝜆_𝑛}⊂[𝑎,𝑏] and {𝛽_𝑛}⊂[𝑐,𝑑]. Since lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists and 𝛼_𝑛→0, we obtain lim_𝑛→∞𝑥_𝑛−𝑧_𝑛=lim_𝑛→∞𝑥_𝑛−𝑆𝑙_𝑛=0. (26) Moreover, we note that 𝑦_𝑛−𝑙_𝑛=𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑧_𝑛−𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛≤𝑧_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑧_𝑛−𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛=𝑧_𝑛−𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑧_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛≤𝑧_𝑛−𝑥_𝑛+𝜆_𝑛∇𝑓_𝛼𝑛𝑧_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛≤𝑧_𝑛−𝑥_𝑛+𝜆_𝑛∇𝑓_𝛼𝑛𝑧_𝑛−∇𝑓_𝛼𝑛𝑥_𝑛+∇𝑓_𝛼𝑛𝑥_𝑛−∇𝑓_𝛼𝑛𝑦_𝑛≤𝑧_𝑛−𝑥_𝑛+𝜆_𝑛𝛼_𝑛+‖𝐴‖²𝑥_𝑛−𝑦_𝑛+∇𝑓_𝛼𝑛𝑧_𝑛−∇𝑓_𝛼𝑛𝑥_𝑛. (27) From (26) and 𝛼_𝑛→0, it is implied that lim_𝑛→∞𝑦_𝑛−𝑙_𝑛=0. (28) Note that ‖𝑙_𝑛−𝑆(𝑙_𝑛)‖≤‖𝑙_𝑛−𝑦_𝑛‖+‖𝑦_𝑛−𝑥_𝑛‖+‖𝑥_𝑛−𝑆(𝑙_𝑛)‖. This together with (24) and (28) implies that lim_𝑛→∞‖𝑙_𝑛−𝑆(𝑙_𝑛)‖=0. Also, from ‖𝑥_𝑛−𝑙_𝑛‖≤‖𝑥_𝑛−𝑦_𝑛‖+‖𝑦_𝑛−𝑙_𝑛‖, it follows that lim_𝑛→∞‖𝑥_𝑛−𝑙_𝑛‖=0. Since ∇𝑓=𝐴^∗(𝐼−𝑃_𝐶)𝐴 is a Lipschitz condition, where 𝐴^∗ is the adjoint of 𝐴, we have lim_𝑛→∞‖∇𝑓(𝑦_𝑛)−∇𝑓(𝑙_𝑛)‖=0. Since {𝑥_𝑛} is a bounded sequence, there exists a subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} that converges weakly to some 𝑥.
Next, we show that 𝑥∈Γ. Since ‖𝑥_𝑛−𝑙_𝑛‖→0 and ‖𝑦_𝑛−𝑙_𝑛‖→0, it is known that 𝑙_𝑛𝑖⇀𝑥 and 𝑦_𝑛𝑖⇀𝑥. Let 𝑇∶𝐻→2^𝐻 be a set value mappings defined by (𝑣)𝑇𝑣=∇𝑓+𝑁_𝐶𝑣if𝑣∈𝐶,∅if𝑣∉𝐶, (29) where 𝑁_𝐶𝑣={𝑤∈𝐻₁∶⟨𝑣−𝑢,𝑤⟩≥0,forall𝑢∈𝐶}. Hence, by [34], 𝑇 is maximal monotone and 0∈𝑇𝑣 if and only if 𝑣∈VI(𝐶,∇𝑓). Let (𝑣,𝑤)∈𝐺(𝑇). Then we have 𝑤∈𝑇𝑣=∇𝑓(𝑣)+𝑁_𝐶𝑣 and hence 𝑤−∇𝑓(𝑣)∈𝑁_𝐶𝑣. So, we obtain ⟨𝑣−𝑢,𝑤−∇𝑓(𝑣)⟩≥0,forall𝑢∈𝐶. On the other hand, from 𝑙_𝑛=𝑃_𝐶(𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛(𝑦_𝑛)) and 𝑣∈𝐶, we have 𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−𝑙_𝑛,𝑙_𝑛−𝑣≥0, (30) and hence 𝑣−𝑙_𝑛,𝑙_𝑛−𝑥_𝑛𝜆_𝑛+∇𝑓_𝛼𝑛𝑦_𝑛≥0. (31) Therefore, from 𝑤−∇𝑓(𝑣)∈𝑁_𝐶𝑣 and 𝑙_𝑛𝑖∈𝐶, we get 𝑣−𝑙_𝑛𝑖≥,𝑤𝑣−𝑙_𝑛𝑖(𝑣≥,∇𝑓)𝑣−𝑙_𝑛𝑖(𝑣−,∇𝑓)𝑣−𝑙_𝑛𝑖,𝑙_𝑛𝑖−𝑥_𝑛𝑖𝜆_𝑛𝑖+∇𝑓_𝛼𝑛𝑖𝑦_𝑛𝑖=𝑣−𝑙_𝑛𝑖(𝑣−,∇𝑓)𝑣−𝑙_𝑛𝑖,𝑙_𝑛𝑖−𝑥_𝑛𝑖𝜆_𝑛𝑖+∇𝑓𝑦_𝑛𝑖−𝛼_𝑛𝑖𝑣−𝑙_𝑛𝑖,𝑦_𝑛𝑖=𝑣−𝑙_𝑛𝑖,∇𝑓(𝑣)−∇𝑓𝑙_𝑛𝑖+𝑣−𝑙_𝑛𝑖,∇𝑓𝑙_𝑛𝑖−∇𝑓𝑦_𝑛𝑖−𝑣−𝑙_𝑛𝑖,𝑙_𝑛𝑖−𝑥_𝑛𝑖𝜆_𝑛𝑖−𝛼_𝑛𝑖𝑣−𝑙_𝑛𝑖,𝑦_𝑛𝑖≥𝑣−𝑙_𝑛𝑖,∇𝑓𝑙_𝑛𝑖−∇𝑓𝑦_𝑛𝑖−𝑣−𝑙_𝑛𝑖,𝑙_𝑛𝑖−𝑥_𝑛𝑖𝜆_𝑛𝑖−𝛼_𝑛𝑖𝑣−𝑙_𝑛𝑖,𝑦_𝑛𝑖. (32) By taking 𝑖→∞, we obtain ⟨𝑣−𝑥,𝑤⟩≥0. Since 𝑇 is maximal monotone, it follows that 𝑥∈𝑇⁻¹0 and hence 𝑥∈VI(𝐶,∇𝑓). Therefore, by Lemma 8, 𝑥∈Γ.
Next, we show that 𝑥∈𝐹(𝑆). Since 𝑙_𝑛𝑖⇀𝑥 and ‖𝑙_𝑛𝑖−𝑆(𝑙_𝑛𝑖)‖→0, it follows by the demiclosed principle that 𝑥∈𝐹(𝑆). Hence, we have 𝑥∈𝐹(𝑆)∩Γ.
Next, we show that 𝑥∈Ω. Let 𝐺 be a mapping which is defined as in Lemma 6, thus we have 𝑧_𝑛−𝐺𝑧_𝑛=𝑇_1,𝜇𝑇_2,𝜇𝑥_𝑛−𝐺𝑧_𝑛=𝐺𝑥_𝑛−𝐺𝑧_𝑛≤𝑥_𝑛−𝑧_𝑛, (33) and hence 𝑥_𝑛−𝐺𝑥_𝑛≤𝑥_𝑛−𝑧_𝑛+𝑧_𝑛−𝐺𝑧_𝑛+𝐺𝑧_𝑛−𝐺𝑥_𝑛≤𝑥_𝑛−𝑧_𝑛+𝑥_𝑛−𝑧_𝑛+𝑧_𝑛−𝑥_𝑛=3𝑥_𝑛−𝑧_𝑛. (34) By taking 𝑛→∞, we have ‖𝑥_𝑛−𝐺(𝑥_𝑛)‖→0. From lim_𝑛→∞‖𝑥_𝑛−𝑧_𝑛‖=0 and 𝑧_𝑛𝑗⇀𝑥, we obtain 𝑥_𝑛𝑗⇀𝑥. According to demiclosedness and Lemma 6, we have 𝑥∈Ω. Therefore, we have 𝑥∈𝐹(𝑆)∩Γ∩Ω. Let {𝑥_𝑛𝑗} be another subsequence of {𝑥_𝑛} such that 𝑥_𝑛𝑗⇀𝑥. We show that 𝑥=𝑥, suppose that 𝑥=𝑥. Since lim_𝑛→∞‖𝑥_𝑛−𝑥‖ exists for all 𝑥∈𝐹(𝑆)∩Γ∩Ω, it follows by the Opial's condition that 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. (35) It is a contradiction. Thus, we have 𝑥=𝑥 and so 𝑥_𝑛⇀𝑥∈𝐹(𝑆)∩Γ∩Ω. Further, from ‖𝑥_𝑛−𝑦_𝑛‖→0, it follows that 𝑦_𝑛⇀𝑥 and hence 𝑧_𝑛, 𝑤_𝑛. This completes the proof.

Theorem 9 extends the extragradient method according to Nadezhkina and Takahashi [35].

Corollary 10. Let 𝐶 be a nonempty closed convex subset in a real Hilbert space 𝐻. Let 𝐹₁ and 𝐹₂ be two bifunctions from 𝐶×𝐶→ℝ satisfying (A1)–(A4). Let 𝜇>0 and let 𝑇_1,𝜇 and 𝑇_2,𝜇 be defined as in Lemma 5 associated to 𝐹₁ and 𝐹₂, respectively. Let 𝑆 be a quasi-nonexpansive mapping of 𝐶 into itself such that 𝐹(𝑆)∩Γ∩Ω≠∅. Let {𝑥_𝑛}, {𝑦_𝑛}, {𝑧_𝑛}, and {𝑤_𝑛} be the sequence in 𝐶 generated by the following extragradient algorithm: 𝑥₀𝑤=𝑥∈𝐶,_𝑛∈𝐶;𝐹₂𝑤_𝑛+1,𝑧𝜇𝑧−𝑤_𝑛,𝑤_𝑛−𝑥_𝑛𝑧≥0,∀𝑧∈𝐶,_𝑛∈𝐶;𝐹₁𝑧_𝑛+1,𝑧𝜇𝑧−𝑧_𝑛,𝑧_𝑛−𝑤_𝑛𝑦≥0,∀𝑧∈𝐶,_𝑛=𝑃_𝐶𝐼−𝜆_𝑛∇𝑓𝑧_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑆𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓𝑦_𝑛, (36) where ∑^∞_𝑛=0𝛼_𝑛<∞, {𝜆_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1/‖𝐴‖²) and {𝛽_𝑛}⊂[𝑐,𝑑] for some 𝑐,𝑑∈(0,1). Then, the sequences {𝑥_𝑛} and {𝑦_𝑛} converge weakly to an element 𝑥∈𝐹(𝑆)∩Γ∩Ω.

Corollary 11. Let 𝐶 be a nonempty closed convex subset in a real Hilbert space 𝐻. Let 𝐹₁ and 𝐹₂ be two bifunctions from 𝐶×𝐶→ℝ satisfying (A1)–(A4). Let 𝜇>0 and let 𝑇_1,𝜇 and 𝑇_2,𝜇 be defined as in Lemma 5 associated to 𝐹₁ and 𝐹₂, respectively. Let 𝑆 be a quasi-nonexpansive mapping of 𝐶 into itself such that 𝐹(𝑆)∩Γ∩Ω≠∅. Let {𝑥_𝑛}, {𝑦_𝑛}, and {𝑧_𝑛} be the sequence in 𝐶 generated by the following extragradient algorithm: 𝑥₀𝑧=𝑥∈𝐶,_𝑛∈𝐶;𝐹₁𝑧_𝑛,𝑧+𝜑(𝑧)−𝜑𝑧_𝑛+1𝜇𝑧−𝑧_𝑛,𝑧_𝑛−𝑥_𝑛𝑦≥0,∀𝑧∈𝐶,_𝑛=𝑃_𝐶𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑧_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑆𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛, (37) where ∑^∞_𝑛=0𝛼_𝑛<∞, {𝜆_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1/‖𝐴‖²) and {𝛽_𝑛}⊂[𝑐,𝑑] for some 𝑐,𝑑∈(0,1). Then, the sequences {𝑥_𝑛} and {𝑦_𝑛} converge weakly to an element 𝑥∈𝐹(𝑆)∩Γ∩Ω.

Proof. Setting 𝐹₂=0 in Theorem 9, we obtain the desired result.

Corollary 12. Let 𝐶 be a nonempty closed convex subset in a real Hilbert space 𝐻. Let 𝑆 be a quasi-nonexpansive mapping of 𝐶 into itself such that 𝐹(𝑆)∩Γ≠∅. Let {𝑥_𝑛}, {𝑦_𝑛}, and {𝑧_𝑛} be the sequence in 𝐶 generated by the following extragradient algorithm: 𝑥₀𝑦=𝑥∈𝐶,_𝑛=𝑃_𝐶𝐼−𝜆_𝑛∇𝑓_𝛼𝑛𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑆𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛, (38) where ∑^∞_𝑛=0𝛼_𝑛<∞, {𝜆_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1/‖𝐴‖²) and {𝛽_𝑛}⊂[𝑐,𝑑] for some 𝑐,𝑑∈(0,1). Then, the sequences {𝑥_𝑛} and {𝑦_𝑛} converge weakly to an element 𝑥∈𝐹(𝑆)∩Γ.

Proof. Setting 𝐹₁=𝐹₂=0 in Theorem 9, we obtain the desired result.

Acknowledgments

The authors would like to thank The Commission on Higher Education for financial support. Moreover, K. Sombut is also supported by The Strategic Scholarships Fellowships Frontier Research Networks under Grant CHE-Ph.D-THA-SUP/86/2550, Thailand.