Hybrid Gradient-Projection Algorithm for Solving Constrained Convex Minimization Problems with Generalized Mixed Equilibrium Problems

Abstract

It is well known that the gradient-projection algorithm (GPA) for solving constrained convex minimization problems has been proven to have only weak convergence unless the underlying Hilbert space is finite dimensional. In this paper, we introduce a new hybrid gradient-projection algorithm for solving constrained convex minimization problems with generalized mixed equilibrium problems in a real Hilbert space. It is proven that three sequences generated by this algorithm converge strongly to the unique solution of some variational inequality, which is also a common element of the set of solutions of a constrained convex minimization problem, the set of solutions of a generalized mixed equilibrium problem, and the set of fixed points of a strict pseudocontraction 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 𝑃_𝐶 be the metric projection of 𝐻 onto 𝐶. Recall that a 𝜌-Lipschitz continuous mapping 𝑇∶𝐶→𝐻 is a mapping on 𝐶 such that 𝑇𝑥−𝑇𝑦≤𝜌𝑥−𝑦,∀𝑥,𝑦∈𝐶, (1.1) where 𝜌≥0 is a constant. In particular, if 𝜌∈[0,1) then 𝑇 is called a contraction on 𝐶; if 𝜌=1 then 𝑇 is called a nonexpansive mapping on 𝐶. A mapping 𝐴∶𝐶→𝐻 is called monotone if 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥0,∀𝑥,𝑦∈𝐶. (1.2) A mapping 𝐴∶𝐶→𝐻 is called 𝛼-inverse strongly monotone if there exists a constant 𝛼>0 such that 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥𝛼𝐴𝑥−𝐴𝑦²,∀𝑥,𝑦∈𝐶; (1.3) see, for example, [1]. A self-mapping 𝑆∶𝐶→𝐶 is called 𝑘-strictly pseudocontractive if there exists a constant 𝑘∈[0,1) such that 𝑆𝑥−𝑆𝑦²≤𝑥−𝑦²+𝑘(𝐼−𝑆)𝑥−(𝐼−𝑆)𝑦²,∀𝑥,𝑦∈𝐶; (1.4) see, for example, [2]. In particular, if 𝑘=0, then 𝑆 reduces to a nonexpansive self-mapping on 𝐶.

Consider the following constrained convex minimization problem: minimize𝑓(𝑥)∶𝑥∈𝐶, (1.5) where 𝑓∶𝐶→𝐑 is a real-valued convex function. If 𝑓 is (Frechet) differentiable, then the gradient-projection method (for short, GPM) generates a sequence {𝑥_𝑛} via the recursive formula 𝑥_𝑛+1=𝑃_𝐶𝑥_𝑛−𝜆∇𝑓𝑥_𝑛,∀𝑛≥0, (1.6) or more generally, 𝑥_𝑛+1=𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓𝑥_𝑛,∀𝑛≥0, (1.7) where in both (1.6) and (1.7), the initial guess 𝑥₀ is taken from 𝐶 arbitrarily, the parameters, 𝜆 or 𝜆_𝑛, are positive real numbers, and 𝑃_𝐶 is the metric projection from 𝐻 onto 𝐶. The convergence of the algorithms (1.6) and (1.7) depends on the behavior of the gradient ∇𝑓. As a matter of fact, it is known that if ∇𝑓 is strongly monotone and Lipschitzian; namely, there are constants 𝜂,𝐿>0 satisfying the properties ∇𝑓(𝑥)−∇𝑓𝑦,𝑥−𝑦≥𝜂𝑥−𝑦²,∇𝑓(𝑥)−∇𝑓𝑦≤𝐿𝑥−𝑦 (1.8)(1.9) for all 𝑥,𝑦∈𝐶, then, for 0<𝜆<2𝜂/𝐿², the operator 𝑇∶=𝑃_𝐶𝐼−𝜆∇𝑓 (1.10) is a contraction; hence, the sequence {𝑥_𝑛} defined by algorithm (1.6) converges in norm to the unique solution of the minimization (1.5). More generally, if the sequence {𝜆_𝑛} is chosen to satisfy the property 0<liminf_𝑛→∞𝜆_𝑛≤limsup_𝑛→∞𝜆_𝑛<2𝜂𝐿² (1.11) then the sequence {𝑥_𝑛} defined by algorithm (1.7) converges in norm to the unique minimizer of (1.5). However, if the gradient ∇𝑓 fails to be strongly monotone, the operator 𝑇 defined in (1.10) would fail to be contractive; consequently, the sequence {𝑥_𝑛} generated by algorithm (1.6) may fail to converge strongly (see Section 4 in Xu [3]). The following theorem states that if the Lipschitz condition (1.9) holds, then the algorithms (1.6) and (1.7) can still converge in the weak topology.

Theorem 1.1 (see [3, Theorem 3.2]). Assume the minimization (1.5) is consistent and let Ω denote its solution set. Assume the gradient ∇𝑓 satisfies the Lipschitz condition (1.9). Let the sequence of parameters, {𝜆_𝑛}, satisfy the condition 0<liminf_𝑛→∞𝜆_𝑛≤limsup_𝑛→∞𝜆_𝑛<2𝐿. (1.12) Then the sequence {𝑥_𝑛} generated by the gradient-projection algorithm (1.7) converges weakly to a minimizer of (1.5).

From the above theorem, it is known that the gradient-projection algorithm has weak convergence, in general, unless the underlying Hilbert space is finite dimensional. This gives naturally rise to a question how to appropriately modify the gradient-projection algorithm so as to have strong convergence. Xu [3] gave two such modifications, one of which is simply a convex combination of a contraction with the point generated by the projected gradient algorithm.

Theorem 1.2 (see [3, Theorem 5.2]). Assume the minimization (1.5) is consistent and let Ω denote its solution set. Assume the gradient ∇𝑓 satisfies the Lipschitz condition (1.9). Let 𝑄∶𝐶→𝐶 be a 𝜌-contraction with 𝜌∈[0,1). Let a sequence {𝑥_𝑛} be generated by the following hybrid gradient-projection algorithm: 𝑥_𝑛+1=𝛼_𝑛𝑄𝑥_𝑛+1−𝛼_𝑛𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓𝑥_𝑛,∀𝑛≥0. (1.13) Assume the sequence {𝜆_𝑛} satisfies the condition (1.12) and, in addition, the following conditions are satisfied for {𝜆_𝑛} and {𝛼_𝑛}⊂[0,1]:(i)𝛼_𝑛→0;(ii)∑^∞_𝑛=0𝛼_𝑛=∞;(iii)∑^∞_𝑛=0|𝛼_𝑛+1−𝛼_𝑛|<∞;(iv)∑^∞_𝑛=0|𝜆_𝑛+1−𝜆_𝑛|<∞.
Then the sequence {𝑥_𝑛} converges in norm to a minimizer of (1.5) which is also the unique solution of the variational inequality of finding 𝑥^∗∈Ω such that (𝐼−𝑄)𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈Ω. (1.14) In other words, 𝑥^∗ is the unique fixed point of the contraction 𝑃_Ω𝑄, 𝑥^∗=𝑃_Ω𝑄𝑥^∗.

On the other hand, Peng and Yao [4] recently introduced the following generalized mixed equilibrium problem of finding 𝑥∈𝐶 such that Θ𝑥,𝑦+𝜑𝑦−𝜑(𝑥)+𝐹𝑥,𝑦−𝑥≥0,∀𝑦∈C, (1.15) where 𝐹∶𝐶→𝐻 is a nonlinear mapping and 𝜑∶𝐶→𝐑 is a function and Θ∶𝐶×𝐶→𝐑 is a bifunction. The set of solutions of problem (1.15) is denoted by GMEP. Subsequently, Yao et al. [5] and Ceng and Yao [6] also considered problem (1.15).

The problem (1.15) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problems in noncooperative games, and others; see, for example, [7–15]. Here some special cases of problem (1.15) are stated as follows.

If 𝐹=0, then problem (1.15) reduces to the following mixed equilibrium problem of finding 𝑥∈𝐶 such that Θ𝑥,𝑦+𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐶, (1.16) which was considered by Ceng and Yao [7] and Bigi et al. [16]. Very recently, Peng [10] further discussed this problem. The set of solutions of this problem is denoted by MEP.

If 𝜑=0, then problem (1.15) reduces to the following generalized equilibrium problem of finding 𝑥∈𝐶 such that Θ𝑥,𝑦+𝐹𝑥,𝑦−𝑥≥0,∀𝑦∈𝐶, (1.17) which was studied by S. Takahashi and W. Takahashi [8].

If 𝜑=0 and 𝐹=0, then problem (1.15) reduces to the following equilibrium problem of finding 𝑥∈𝐶 such that Θ𝑥,𝑦≥0,∀𝑦∈𝐶. (1.18)

If Θ=0,  𝜑=0 and 𝐹=𝐴, then problem (1.15) reduces to the following classical variational inequality of finding 𝑥∈𝐶 such that 𝐴𝑥,𝑦−𝑥≥0,∀𝑦∈𝐶, (1.19) whose solution set is denoted by VI(𝐶,𝐴).

The variational inequalities have been extensively studied in the literature; see [14, 17–27] and the references therein. In 2006, Nadezhkina and Takahashi [22, 25] and Zeng and Yao [18] proposed some variants of Korpelevič's extragradient method [17] for finding an element of Fix(𝑆)∩VI(𝐶,𝐴), where 𝑆 is a nonexpansive self-mapping on 𝐶.

Very recently, Peng [10] also introduced a variant of Korpelevič's extragradient method [17] for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a strict pseudocontraction, and the set of solutions of a variational inequality for a monotone, Lipschitz continuous mapping.

Theorem 1.3 (see [10, Theorem 3.1]). Let C be a nonempty closed convex subset of a real Hilbert space 𝐻. Let Θ∶𝐶×𝐶→𝐑 be a bifunction satisfying conditions (H1)–(H4) and 𝜑∶𝐶→𝐑 a lower semicontinuous and convex function with assumptions (A1) or (A2), where(H1)Θ(𝑥,𝑥)=0, for all 𝑥∈𝐶;(H2)Θ is monotone, that is, Θ(𝑥,𝑦)+Θ(𝑦,𝑥)≤0, for all 𝑥,𝑦∈𝐶;(H3) for each 𝑦∈𝐶,  𝑥↦Θ(𝑥,𝑦) is weakly upper semicontinuous;(H4) for each 𝑥∈𝐶,  𝑦↦Θ(𝑥,𝑦) is convex and lower semicontinuous;(A1) for each 𝑥∈𝐻 and 𝑟>0, there exists a bounded subset 𝐷_𝑥⊂𝐶 and 𝑦_𝑥∈𝐶 such that for any 𝑧∈𝐶⧵𝐷_𝑥, Θ𝑧,𝑦_𝑥+𝜑𝑦_𝑥(𝑧)+1−𝜑𝑟𝑦_𝑥−𝑧,𝑧−𝑥<0; (1.20)(A2)𝐶 is a bounded set.
Let 𝐴∶𝐶→𝐻 be a monotone and 𝐿-Lipschitz-continuous mapping and 𝑆∶𝐶→𝐶 be a 𝑘-strict pseudocontraction for some 0≤𝑘<1 such that Fix(𝑆)∩VI(𝐶,𝐴)∩MEP≠∅. For given 𝑥₀∈𝐻 arbitrarily, let {𝑥_𝑛}, {𝑡_𝑛}, {𝑦_𝑛}, {𝑢_𝑛}, {𝑧_𝑛} be sequences generated by Θ𝑡_𝑛,𝑦+𝜑𝑦−𝜑𝑡_𝑛+1𝑟_𝑛𝑦−𝑡_𝑛,𝑡_𝑛−𝑥_𝑛𝑦≥0,∀𝑦∈𝐶,_𝑛=𝑃_𝐶𝑡_𝑛−𝜆_𝑛𝐴𝑡_𝑛,𝑢_𝑛=𝑃_𝐶𝑡_𝑛−𝜆_𝑛𝐴𝑦_𝑛,𝑧_𝑛=𝛼_𝑛𝑢_𝑛+1−𝛼_𝑛𝑆𝑢_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝑧_𝑛−𝑧²≤𝑥_𝑛−𝑧²−1−𝛼_𝑛𝛼_𝑛−𝜀𝑡_𝑛−𝑆𝑡_𝑛²,𝑄_𝑛=𝑧∈𝐻∶𝑥_𝑛−𝑧,𝑥−𝑥_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥,∀𝑛≥0. (1.21) Assume that {𝜆_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1/𝐿), {𝛼_𝑛}⊂[𝑐,𝑑] for some 𝑐,𝑑∈(𝑘,1) and let {𝑟_𝑛}⊂(0,∞) satisfy liminf_𝑛→∞𝑟_𝑛>0. Then, {𝑥_𝑛}, {𝑡_𝑛}, {𝑦_𝑛}, {𝑢_𝑛}, {𝑧_𝑛} converge strongly to 𝑤=𝑃_{Fix(𝑆)∩VI(𝐶,𝐴)∩MEP}𝑥.

Furthermore, related iterative methods for solving fixed point problems, variational inequalities, equilibrium problems, and optimization problems can be found in [1, 2, 6, 11, 13–16, 19, 20, 24, 26–35].

In this paper, let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let Θ∶𝐶×𝐶→𝐑 be a bifunction satisfying conditions (H1)–(H4) and 𝜑∶𝐶→𝐑 a lower semicontinuous and convex function with assumptions (A1) or (A2). Suppose the minimization (1.5) is consistent and let Ω denote its solution set. Let the gradient ∇𝑓 be 𝐿-Lipschitzian with constant 𝐿>0 and 𝐹∶𝐶→𝐻 be an 𝛼-inverse strongly monotone mapping. Let 𝑆∶𝐶→𝐶 be a 𝑘-strictly pseudocontractive mapping such that Fix(𝑆)∩Ω∩GMEP≠∅. Let 𝑄∶𝐶→𝐶 be a 𝜌-contraction with 𝜌∈[0,1/2). For given 𝑥₀∈𝐶 arbitrarily, let the sequences {𝑥_𝑛}, {𝑦_𝑛} and {𝑧_𝑛} be generated iteratively by Θ𝑧_𝑛,𝑦+𝜑𝑦−𝜑𝑧_𝑛+𝐹𝑥_𝑛,𝑦−𝑧_𝑛+1𝑟_𝑛𝑦−𝑧_𝑛,𝑧_𝑛−𝑥_𝑛𝑦≥0,∀𝑦∈𝐶,_𝑛=𝛼_𝑛𝑄𝑥_𝑛+1−𝛼_𝑛𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛+𝛿_𝑛𝑆𝑦_𝑛,∀𝑛≥0, (1.22) where {𝜆_𝑛}⊂(0,2/𝐿],{𝑟_𝑛}⊂(0,2𝛼] and {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, {𝛿_𝑛} are four sequences in [0,1] such that 𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 for all 𝑛≥0. It is proven that under very mild conditions, the sequences {𝑥_𝑛}, {𝑦_𝑛} and {𝑧_𝑛} converge strongly to the unique solution of the variational inequality of finding 𝑥^∗∈Fix(𝑆)∩Ω∩GMEP such that (𝐼−𝑄)𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈Fix(𝑆)∩Ω∩GMEP. (1.23) In other words, 𝑥^∗ is the unique fixed point of the contraction 𝑃_{Fix(𝑆)∩Ω∩GMEP}𝑄,  𝑥^∗=𝑃_{Fix(𝑆)∩Ω∩GMEP}𝑄𝑥^∗. The result presented in this paper generalizes and improves some well-known results in the literature. Indeed, compared with some well-known results in the literature, our result improves and extends them in the following aspects.(i) Compared with Xu [3, Theorem 3.2], a weak convergence result, our result is a strong convergence result.(ii) Our problem of finding an element of Fix(𝑆)∩Ω∩GMEP is more general than the problem of finding an element of Fix(𝑆)∩VI(𝐶,𝐴) in [14, 18, 22, 23, 25].(iii) In our algorithm (1.22), Xu's modified gradient-projection algorithm in [3, Theorem 5.2] is rewritten as the second iteration step 𝑦_𝑛=𝛼_𝑛𝑄𝑥_𝑛+1−𝛼_𝑛𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛. (1.24) Here the main purpose of the reason why we use such an iteration step is to play a convenience and efficiency role in the computation of an element of Ω. Therefore, Xu's algorithm (1.13) is extended to develop our algorithm (1.22).(iv) Our problem of finding an element of Fix(𝑆)∩Ω∩GMEP is more general than the problem of finding an element of Ω in Xu [3]. In addition, it is worth pointing out that Xu's conditions ∑^∞_𝑛=0|𝛼_𝑛+1−𝛼_𝑛|<∞ and ∑^∞_𝑛=0|𝜆_𝑛+1−𝜆_𝑛|<∞ in the above Theorem 1.2 are replaced by the weaker conditions lim_𝑛→∞(𝛼_𝑛−𝛼_𝑛+1)=0 and lim_𝑛→∞(𝜆_𝑛−𝜆_𝑛+1)=0 in our result (see Theorem 3.2 in Section 3).

2. Preliminaries

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖ and 𝐶 a nonempty closed convex subset of 𝐻. We write → to indicate that the sequence {𝑥_𝑛} converges strongly to 𝑥 and ⇀ to indicate that the sequence {𝑥_𝑛} converges weakly to 𝑥. Moreover, we use 𝜔_𝑤(𝑥_𝑛) to denote the weak 𝜔-limit set of the sequence {𝑥_𝑛}, that is, 𝜔_𝑤𝑥_𝑛∶=𝑥∶𝑥_𝑛𝑖⇀𝑥forsomesubsequence𝑥_𝑛𝑖of𝑥_𝑛. (2.1)

For every point 𝑥∈𝐻, there exists a unique nearest point in 𝐶, denoted by 𝑃_𝐶𝑥, such that 𝑥−𝑃_𝐶𝑥≤𝑥−𝑦,∀𝑥∈𝐶. (2.2)𝑃_𝐶 is called the metric projection of 𝐻 onto 𝐶. We know that 𝑃_𝐶 is a firmly nonexpansive mapping of 𝐻 onto 𝐶; that is, there holds the following relation: 𝑃_𝐶𝑥−𝑃_𝐶𝑦,𝑥−𝑦≥𝑃_𝐶𝑥−𝑃_𝐶𝑦²,∀𝑥,𝑦∈𝐻. (2.3) Consequently, 𝑃_𝐶 is nonexpansive and monotone. It is also known that 𝑃_𝐶 is characterized by the following properties: 𝑃_𝐶𝑥∈𝐶 and 𝑥−𝑃_𝐶𝑥,𝑃_𝐶𝑥−𝑦≥0,𝑥−𝑦²≥𝑥−𝑃_𝐶𝑥²+𝑦−𝑃_𝐶𝑥², (2.4)(2.5) for all 𝑥∈𝐻,  𝑦∈𝐶; see [36] for more details. Let 𝐴∶𝐶→𝐻 be a monotone mapping. In the context of the variational inequality, this implies that 𝑥∈VI(𝐶,𝐴)⟺𝑥=𝑃_C(𝑥−𝜆𝐴𝑥)∀𝜆>0. (2.6)

A set-valued mapping 𝑇∶𝐻→2^𝐻 is called monotone if for all 𝑥,𝑦∈𝐻,  𝑓∈𝑇𝑥 and 𝑔∈𝑇𝑦 imply ⟨𝑓−𝑔,𝑥−𝑦⟩≥0. A monotone mapping 𝑇∶𝐻→2^𝐻 is called maximal if its graph 𝐺(𝑇) is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping 𝑇 is maximal if and only if for (𝑥,𝑓)∈𝐻×𝐻, ⟨𝑓−𝑔,𝑥−𝑦⟩≥0 for every (𝑦,𝑔)∈𝐺(𝑇) implies 𝑓∈𝑇𝑥.

Let 𝐴∶𝐶→𝐻 be a monotone, 𝑘-Lipschitz-continuous mapping and let 𝑁_𝐶𝑣 be the normal cone to 𝐶 at 𝑣∈𝐶, that is, 𝑁_𝐶𝑣={𝑤∈𝐻∶⟨𝑣−𝑢,𝑤⟩≥0, for all 𝑢∈𝐶}. Define 𝑇𝑣=𝐴𝑣+𝑁_𝐶𝑣,if𝑣∈𝐶,∅,if𝑣∉𝐶. (2.7) Then, 𝑇 is maximal monotone and 0∈𝑇𝑣 if and only if 𝑣∈VI(𝐶,𝐴); see [37].

Recall that a mapping 𝑆∶𝐶→𝐶 is called a strict pseudocontraction if there exists a constant 0≤𝑘<1 such that 𝑆𝑥−𝑆𝑦²≤𝑥−𝑦²+𝑘(𝐼−𝑆)𝑥−(𝐼−𝑆)𝑦²,∀𝑥,𝑦∈𝐶. (2.8) In this case, we also say that 𝑆 is a 𝑘-strict pseudocontraction. A mapping 𝐴∶𝐶→𝐻 is called 𝛼-inverse strongly monotone if there exists a constant 𝛼>0 such that 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥𝛼𝐴𝑥−𝐴𝑦²,∀𝑥,𝑦∈𝐶. (2.9) It is obvious that any 𝛼-inverse strongly monotone mapping is Lipschitz continuous. Meantime, observe that (2.8) is equivalent to 𝑆𝑥−𝑆𝑦,𝑥−𝑦≤𝑥−𝑦²−1−𝑘2(𝐼−𝑆)𝑥−(𝐼−𝑆)𝑦²,∀𝑥,𝑦∈𝐶. (2.10) It is easy to see that if 𝑆 is a 𝑘-strictly pseudocontractive mapping, then 𝐼−𝑆 is ((1−𝑘)/2)-inverse strongly monotone and hence (2/(1−𝑘))-Lipschitz continuous. Thus, 𝑆 is Lipschitz continuous with constant (1+𝑘)/(1−𝑘). We denote by Fix(𝑆) the set of fixed points of 𝑆. It is clear that the class of strict pseudocontractions strictly includes the one of nonexpansive mappings which are mappings 𝑆∶𝐶→𝐶 such that ‖𝑆𝑥−𝑆𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶.

In order to prove our main result in the next section, we need the following lemmas and propositions.

Lemma 2.1 (see [7]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let Θ∶𝐶×𝐶→𝐑 be a bifunction satisfying conditions (H1)–(H4) and let 𝜑∶𝐶→𝐑 be a lower semicontinuous and convex function. For 𝑟>0 and 𝑥∈𝐻, define a mapping 𝑇_𝑟^(Θ,𝜑)∶𝐻→𝐶 as follows: 𝑇_𝑟^(Θ,𝜑)(𝑥)=𝑦(𝑧)+1𝑧∈𝐶∶Θ𝑧,𝑦+𝜑−𝜑𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶 (2.11) for all 𝑥∈𝐻. Assume that either (A1) or (A2) holds. Then the following conclusions hold:(i)𝑇_𝑟^(Θ,𝜑)(𝑥)≠∅ for each 𝑥∈𝐻 and 𝑇_𝑟^(Θ,𝜑) is single-valued;(ii)𝑇_𝑟^(Θ,𝜑) is firmly nonexpansive, that is, for any 𝑥,𝑦∈𝐻, 𝑇_𝑟^(Θ,𝜑)𝑥−𝑇_𝑟^(Θ,𝜑)𝑦²≤𝑇_𝑟^(Θ,𝜑)𝑥−𝑇_𝑟^(Θ,𝜑)𝑦,𝑥−𝑦; (2.12)(iii)Fix(𝑇_𝑟^(Θ,𝜑))=MEP(Θ,𝜑);(iv)MEP(Θ,𝜑) is closed and convex.

Remark 2.2. If 𝜑=0, then 𝑇_𝑟^(Θ,𝜑) is rewritten as 𝑇^Θ_𝑟.

The following lemma is an immediate consequence of an inner product.

Lemma 2.3. In a real Hilbert space 𝐻, there holds the inequality 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝑥+𝑦,∀𝑥,𝑦∈𝐻. (2.13)

Proposition 2.4 (see [6, Proposition 2.1]). Let 𝐶, 𝐻, Θ, 𝜑, and 𝑇_𝑟^(Θ,𝜑) be as in Lemma 2.1. Then the following relation holds: 𝑇_𝑠^(Θ,𝜑)𝑥−𝑇_𝑡^(Θ,𝜑)𝑥²≤𝑠−𝑡𝑠𝑇_𝑠^(Θ,𝜑)𝑥−𝑇_𝑡^(Θ,𝜑)𝑥,𝑇_𝑠^(Θ,𝜑)𝑥−𝑥 (2.14) for all 𝑠,𝑡>0 and 𝑥∈𝐻.

Recall that 𝑆∶𝐶→𝐶 is called a quasi-strict pseudocontraction if the fixed point set of 𝑆, Fix(𝑆), is nonempty and if there exists a constant 0≤𝑘<1 such that 𝑆𝑥−𝑝²≤𝑥−𝑝²+𝑘‖𝑥−𝑆𝑥‖²∀𝑥∈𝐶,𝑝∈Fix(𝑆). (2.15) We also say that 𝑆 is a 𝑘-quasi-strict pseudocontraction if condition (2.15) holds.

Proposition 2.5 (see [2, Proposition 2.1]). Assume 𝐶 is a nonempty closed convex subset of a real Hilbert space 𝐻 and let 𝑆∶𝐶→𝐶 be a self-mapping on 𝐶.(i)If 𝑆 is a 𝑘-strict pseudocontraction, then 𝑆 satisfies the Lipschitz condition 𝑆𝑥−𝑆𝑦≤1+𝑘1−𝑘𝑥−𝑦,∀𝑥,𝑦∈𝐶. (2.16)(ii)If 𝑆 is a 𝑘-strict pseudocontraction, then the mapping 𝐼−𝑆 is demiclosed (at 0). That is, if {𝑥_𝑛} is a sequence in 𝐶 such that 𝑥_𝑛⇀𝑥 and (𝐼−𝑆)𝑥_𝑛→0, then (𝐼−𝑆)𝑥=0, that is, 𝑥∈Fix(𝑆).(iii)If 𝑆 is a 𝑘-quasi-strict pseudocontraction, then the fixed point set Fix(𝑆) of 𝑆 is closed and convex so that the projection 𝑃_Fix(𝑆) is well defined.

The following lemma was proved by Suzuki [30].

Lemma 2.6 (see [30]). Let {𝑥_𝑛} and {𝑦_𝑛} be bounded sequences in a Banach space 𝑋 and let {𝛽_𝑛} be a sequence in [0,1] with 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1. Suppose 𝑥_𝑛+1=(1−𝛽_𝑛)𝑦_𝑛+𝛽_𝑛𝑥_𝑛 for all integers 𝑛≥0 and limsup_𝑛→∞(‖𝑦_𝑛+1−𝑦_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0. Then, lim_𝑛→∞‖𝑦_𝑛−𝑥_𝑛‖=0.

Lemma 2.7 (see [34]). Let {𝑎_𝑛} be a sequence of nonnegative numbers satisfying the condition 𝑎_𝑛+1≤1−𝛿_𝑛𝑎_𝑛+𝛿_𝑛𝜎_𝑛,∀𝑛≥0, (2.17) where {𝛿_𝑛}, {𝜎_𝑛} are sequences of real numbers such that(i){𝛿_𝑛}⊂[0,1] and ∑^∞_𝑛=0𝛿_𝑛=∞, or equivalently, ^∞_𝑛=01−𝛿_𝑛∶=lim^𝑛_{𝑛→∞𝑗=0}1−𝛿_𝑗=0; (2.18)(ii)limsup_𝑛→∞𝜎_𝑛≤0, or,(iii)∑^∞_𝑛=0𝛿_𝑛𝜎_𝑛 is convergent.
Then lim_𝑛→∞𝑎_𝑛=0.

3. Strong Convergence Theorem

In order to prove our main result, we shall need the following lemma given in [21].

Lemma 3.1. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑆∶𝐶→𝐶 be a 𝑘-strictly pseudocontractive mapping. Let 𝛾 and 𝛿 be two nonnegative real numbers. Assume (𝛾+𝛿)𝑘≤𝛾. Then 𝛾𝑥−𝑦+𝛿𝑆𝑥−𝑆𝑦≤𝛾+𝛿𝑥−𝑦,∀𝑥⋅𝑦∈𝐶. (3.1)

We are now in a position to state and prove our main result.

Theorem 3.2. Let 𝐶 be a nonempty bounded closed convex subset of a real Hilbert space 𝐻. Let Θ∶𝐶×𝐶→𝐑 be a bifunction satisfying conditions (H1)–(H4) and 𝜑∶𝐶→𝐑 a lower semicontinuous and convex function with assumptions (A1) or (A2). Suppose the minimization (1.5) is consistent and let Ω denote its solution set. Assume the gradient ∇𝑓 is 𝐿-Lipschitzian with constant 𝐿>0 and 𝐹∶𝐶→𝐻 is an 𝛼-inverse strongly monotone mapping. Let 𝑆∶𝐶→𝐶 be a 𝑘-strictly pseudocontractive mapping such that Fix(𝑆)∩Ω∩GMEP≠∅. Let 𝑄∶𝐶→𝐶 be a 𝜌-contraction with 𝜌∈[0,1/2). For given 𝑥₀∈𝐶 arbitrarily, let the sequences {𝑥_𝑛}, {𝑦_𝑛}, and {𝑧_𝑛} be generated iteratively by Θ𝑧_𝑛,𝑦+𝜑𝑦−𝜑𝑧_𝑛+𝐹𝑥_𝑛,𝑦−𝑧_𝑛+1𝑟_𝑛𝑦−𝑧_𝑛,𝑧_𝑛−𝑥_𝑛𝑦≥0,∀𝑦∈𝐶,_𝑛=𝛼_𝑛𝑄𝑥_𝑛+1−𝛼_𝑛𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛+𝛿_𝑛𝑆𝑦_𝑛,∀𝑛≥0, (3.2) where {𝜆_𝑛}⊂(0,2/𝐿],  {𝑟_𝑛}⊂(0,2𝛼], and {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, {𝛿_𝑛} are four sequences in [0,1] such that(i)0<liminf_𝑛→∞𝜆_𝑛≤limsup_𝑛→∞𝜆_𝑛<2/𝐿 and lim_𝑛→∞(𝜆_𝑛−𝜆_𝑛+1)=0;(ii)0<liminf_𝑛→∞𝑟_𝑛≤limsup_𝑛→∞𝑟_𝑛<2𝛼 and lim_𝑛→∞(𝑟_𝑛−𝑟_𝑛+1)=0;(iii)𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 and (𝛾_𝑛+𝛿_𝑛)𝑘≤𝛾_𝑛<(1−2𝜌)𝛿_𝑛 for all 𝑛≥0;(iv)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞;(v)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and liminf_𝑛→∞𝛿_𝑛>0;(vi)lim_𝑛→∞(𝛾_𝑛+1/(1−𝛽_𝑛+1)−𝛾_𝑛/(1−𝛽_𝑛))=0.
Then the sequences {𝑥_𝑛},  {𝑦_𝑛}, and {𝑧_𝑛} converge strongly to the unique solution of the variational inequality of finding 𝑥^∗∈Fix(𝑆)∩Ω∩GMEP such that (𝐼−𝑄)𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈Fix(𝑆)∩Ω∩GMEP. (3.3) In other words, 𝑥^∗ is the unique fixed point of the contraction 𝑃_{Fix(𝑆)∩Ω∩GMEP}𝑄, 𝑥^∗=𝑃_{Fix(𝑆)∩Ω∩GMEP}𝑄𝑥^∗.

Proof. First it is obvious that there hold the following assertions:(a)𝑥^∗∈𝐶 solves the minimization (1.5);(b)𝑥^∗ solves the fixed point equation 𝑥^∗=𝑃_𝐶𝐼−𝜆∇𝑓𝑥^∗, (3.4) where 𝜆>0 is any fixed positive number;(c)𝑥^∗ solves the variational inequality of finding 𝑥^∗∈𝐶 such that ∇𝑓𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐶, (3.5)
where its solution set is denoted by VI(𝐶,∇𝑓).
We divide the proof into several steps.
Step  1. We claim that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0.
Indeed, first, we can write (3.2) as 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑢_𝑛, for all 𝑛≥0, where 𝑢_𝑛=(𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛)/(1−𝛽_𝑛). It follows that 𝑢_𝑛+1−𝑢_𝑛=𝑥_𝑛+2−𝛽_𝑛+1𝑥_𝑛+11−𝛽_𝑛+1−𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛1−𝛽_𝑛=𝛾_𝑛+1𝑃_𝐶𝑧_𝑛+1−𝜆_𝑛+1∇𝑓𝑧_𝑛+1+𝛿_𝑛+1𝑆𝑦_𝑛+11−𝛽_𝑛+1−𝛾_𝑛𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛+𝛿_𝑛𝑆𝑦_𝑛1−𝛽_𝑛=𝛾_𝑛+1𝑃_𝐶𝑧_𝑛+1−𝜆_𝑛+1∇𝑓𝑧_𝑛+1−𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛+𝛿_𝑛+1𝑆𝑦_𝑛+1−𝑆𝑦_𝑛1−𝛽_𝑛+1+𝛾_𝑛+11−𝛽_𝑛+1−𝛾_𝑛1−𝛽_𝑛𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛+𝛿_𝑛+11−𝛽_𝑛+1−𝛿_𝑛1−𝛽_𝑛𝑆𝑦_𝑛. (3.6) From Lemma 3.1 and (3.2), we get 𝛾_𝑛+1𝑃_𝐶𝑧_𝑛+1−𝜆_𝑛+1∇𝑓𝑧_𝑛+1−𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛+𝛿_𝑛+1𝑆𝑦_𝑛+1−𝑆𝑦_𝑛≤𝛾_𝑛+1𝑦_𝑛+1−𝑦_𝑛+𝛿_𝑛+1𝑆𝑦_𝑛+1−𝑆𝑦_𝑛+𝛾_𝑛+1𝑃_𝐶𝑧_𝑛+1−𝜆_𝑛+1∇𝑓𝑧_𝑛+1−𝑦_𝑛+1+𝑦_𝑛−𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛≤𝛾_𝑛+1+𝛿_𝑛+1𝑦_𝑛+1−𝑦_𝑛+𝛾_𝑛+1𝛼_𝑛+1𝑄𝑥_𝑛+1−𝑃_𝐶𝑧_𝑛+1−𝜆_𝑛+1∇𝑓𝑧_𝑛+1+𝛾_𝑛+1𝛼_𝑛𝑄𝑥_𝑛−𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛. (3.7) Let {𝑇_𝑟^(Θ,𝜑)_𝑛} be a sequence of mappings defined as in Lemma 2.1. Note that the 𝐿-Lipschitz continuity of ∇𝑓 implies that the gradient ∇𝑓 is (1/𝐿)-ism [31]. Since ∇𝑓 and 𝐹 are (1/𝐿)-inverse strongly monotone mapping and 𝛼-inverse strongly monotone mapping, respectively, then we have (𝐼−𝜆∇𝑓)𝑥−(𝐼−𝜆∇𝑓)𝑦²=𝑥−𝑦²−2𝜆∇𝑓(𝑥)−∇𝑓𝑦,𝑥−𝑦+𝜆²∇𝑓(𝑥)−∇𝑓(𝑦)²≤𝑥−𝑦²2+𝜆𝜆−𝐿∇𝑓(𝑥)−∇𝑓(𝑦)²,(𝐼−𝜇𝐹)𝑥−(𝐼−𝜇𝐹)𝑦²≤𝑥−𝑦²+𝜇𝜇−2𝛼𝐹𝑥−𝐹𝑦². (3.8) It is clear that if 0≤𝜆≤2/𝐿 and 0≤𝜇≤2𝛼, then (𝐼−𝜆∇𝑓) and (𝐼−𝜇𝐹) are nonexpansive. It follows from that 𝑃_𝐶𝑧_𝑛+1−𝜆_𝑛+1∇𝑓𝑧_𝑛+1−𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛≤𝑧_𝑛+1−𝜆_𝑛+1∇𝑓𝑧_𝑛+1−𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛≤𝑧_𝑛+1−𝜆_𝑛+1∇𝑓𝑧_𝑛+1−𝑧_𝑛−𝜆_𝑛+1∇𝑓𝑧_𝑛+𝜆_𝑛+1−𝜆_𝑛∇𝑓𝑧_𝑛≤𝑧_𝑛+1−𝑧_𝑛+𝜆_𝑛+1−𝜆_𝑛∇𝑓𝑧_𝑛=𝑇_𝑟^(Θ,𝜑)_𝑛+1𝑥_𝑛+1−𝑟_𝑛+1𝐹𝑥_𝑛+1−𝑇_𝑟^(Θ,𝜑)_𝑛𝑥_𝑛−𝑟_𝑛𝐹𝑥_𝑛+𝜆_𝑛+1−𝜆_𝑛∇𝑓𝑧_𝑛≤𝑇_𝑟^(Θ,𝜑)_𝑛+1𝑥_𝑛+1−𝑟_𝑛+1𝐹𝑥_𝑛+1−𝑇_𝑟^(Θ,𝜑)_𝑛+1𝑥_𝑛−𝑟_𝑛𝐹𝑥_𝑛+𝑇_𝑟^(Θ,𝜑)_𝑛+1𝑥_𝑛−𝑟_𝑛𝐹𝑥_𝑛−𝑇_𝑟^(Θ,𝜑)_𝑛𝑥_𝑛−𝑟_𝑛𝐹𝑥_𝑛+𝜆_𝑛+1−𝜆_𝑛∇𝑓𝑧_𝑛≤𝑥_𝑛+1−𝑟_𝑛+1𝐹𝑥_𝑛+1−𝑥_𝑛−𝑟_𝑛𝐹𝑥_𝑛+𝑇_𝑟^(Θ,𝜑)_𝑛+1𝑥_𝑛−𝑟_𝑛𝐹𝑥_𝑛−𝑇_𝑟^(Θ,𝜑)_𝑛𝑥_𝑛−𝑟_𝑛𝐹𝑥_𝑛+𝜆_𝑛+1−𝜆_𝑛∇𝑓𝑧_𝑛≤𝑥_𝑛+1−𝑟_𝑛+1𝐹𝑥_𝑛+1−𝑥_𝑛−𝑟_𝑛+1𝐹𝑥_𝑛+𝑟_𝑛+1−𝑟_𝑛𝐹𝑥_𝑛+𝑇_𝑟^(Θ,𝜑)_𝑛+1𝑥_𝑛−𝑟_𝑛𝐹𝑥_𝑛−𝑇_𝑟^(Θ,𝜑)_𝑛𝑥_𝑛−𝑟_𝑛𝐹𝑥_𝑛+𝜆_𝑛+1−𝜆_𝑛∇𝑓𝑧_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑇_𝑟^(Θ,𝜑)_𝑛+1𝑥_𝑛−𝑟_𝑛𝐹𝑥_𝑛−𝑇_𝑟^(Θ,𝜑)_𝑛𝑥_𝑛−𝑟_𝑛𝐹𝑥_𝑛+𝑟_𝑛+1−𝑟_𝑛𝐹𝑥_𝑛+𝜆_𝑛+1−𝜆_𝑛∇𝑓𝑧_𝑛. (3.9) Then, 𝑦_𝑛+1−𝑦_𝑛≤𝑃_𝐶𝑧_𝑛+1−𝜆_𝑛+1∇𝑓𝑧_𝑛+1−𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛+𝛼_𝑛+1𝑄𝑥_𝑛+1−𝑃_𝐶𝑧_𝑛+1−𝜆_𝑛+1∇𝑓𝑧_𝑛+1+𝛼_𝑛𝑄𝑥_𝑛−𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑇_𝑟^(Θ,𝜑)_𝑛+1𝑥_𝑛−𝑟_𝑛𝐹𝑥_𝑛−𝑇_𝑟^(Θ,𝜑)_𝑛𝑥_𝑛−𝑟_𝑛𝐹𝑥_𝑛+𝑟_𝑛+1−𝑟_𝑛𝐹𝑥_𝑛+𝜆_𝑛+1−𝜆_𝑛∇𝑓𝑧_𝑛+𝛼_𝑛𝑄𝑥_𝑛−𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛+𝛼_𝑛+1𝑄𝑥_𝑛+1−𝑃_𝐶𝑧_𝑛+1−𝜆_𝑛+1∇𝑓𝑧_𝑛+1. (3.10) So, from (3.6), (3.7), and (3.10), we have 𝑢_𝑛+1−𝑢_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝛾1+^𝑛+11−𝛽_𝑛+1𝛼_𝑛𝑄𝑥_𝑛−𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛+𝛾1+^𝑛+11−𝛽_𝑛+1𝛼_𝑛+1𝑄𝑥_𝑛+1−𝑃_𝐶𝑧_𝑛+1−𝜆_𝑛+1∇𝑓𝑧_𝑛+1+𝛾_𝑛+11−𝛽_𝑛+1−𝛾_𝑛1−𝛽_𝑛𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛+𝑆𝑦_𝑛+𝑇_𝑟^(Θ,𝜑)_𝑛+1𝑥_𝑛−𝑟_𝑛𝐹𝑥_𝑛−𝑇_𝑟^(Θ,𝜑)_𝑛𝑥_𝑛−𝑟_𝑛𝐹𝑥_𝑛+𝑟_𝑛+1−𝑟_𝑛𝐹𝑥_𝑛+𝜆_𝑛+1−𝜆_𝑛∇𝑓𝑧_𝑛. (3.11) Utilizing Proposition 2.4 and condition (ii), we have lim_𝑛→∞𝑇_𝑟^(Θ,𝜑)_𝑛+1𝑥_𝑛−𝑟_𝑛𝐹𝑥_𝑛−𝑇_𝑟^(Θ,𝜑)_𝑛𝑥_𝑛−𝑟_𝑛𝐹𝑥_𝑛=0. (3.12) This implies that limsup_𝑛→∞𝑢_𝑛+1−𝑢_𝑛−𝑥_𝑛+1−𝑥_𝑛≤0. (3.13) Hence by Lemma 2.6, we get lim_𝑛→∞‖𝑢_𝑛−𝑥_𝑛‖=0. Consequently, lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=lim_𝑛→∞1−𝛽_𝑛𝑢_𝑛−𝑥_𝑛=0. (3.14)Step  2. We claim that lim_𝑛→∞‖∇𝑓(𝑧_𝑛)−∇𝑓(𝑥^∗)‖=0 and lim_𝑛→∞‖𝐹𝑥_𝑛−𝐹𝑥^∗‖=0.
Indeed, let 𝑥^∗∈Fix(𝑆)∩Ω∩GMEP. Then we have 𝑥^∗=𝑆𝑥^∗,  𝑥^∗=𝑃_𝐶(𝑥^∗−𝜆_𝑛∇𝑓(𝑥^∗)) and 𝑥^∗=𝑇_𝑟^(Θ,𝜑)_𝑛𝑥^∗−𝑟_𝑛𝐹𝑥^∗. (3.15) Hence from (3.8), we have 𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛−𝑃_𝐶𝑥^∗−𝜆_𝑛∇𝑓𝑥^∗2≤(𝑧_𝑛−𝜆_𝑛∇𝑓(𝑧_𝑛))−(𝑥^∗−𝜆_𝑛∇𝑓(𝑥^∗))²≤𝑧_𝑛−𝑥^∗2+𝜆_𝑛𝜆_𝑛−2𝐿∇𝑓(𝑧_𝑛)−∇𝑓(𝑥^∗)²,𝑧_𝑛−𝑥^∗2=𝑇_𝑟^(Θ,𝜑)_𝑛(𝑥_𝑛−𝑟_𝑛𝐹𝑥_𝑛)−𝑇_𝑟^(Θ,𝜑)_𝑛(𝑥^∗−𝑟_𝑛𝐹𝑥^∗)²≤(𝑥_𝑛−𝑟_𝑛𝐹𝑥_𝑛)−(𝑥^∗−𝑟_𝑛𝐹𝑥^∗)²≤𝑥_𝑛−𝑥^∗2+𝑟_𝑛𝑟_𝑛−2𝛼𝐹𝑥_𝑛−𝐹𝑥^∗2. (3.16)(3.17) It follows from (3.2), (3.16), and (3.17) that 𝑦_𝑛−𝑥^∗2≤1−𝛼_𝑛𝑃_𝐶(𝑧_𝑛−𝜆_𝑛∇𝑓(𝑧_𝑛))−𝑃_𝐶(𝑥^∗−𝜆_𝑛∇𝑓(𝑥^∗))²+𝛼_𝑛𝑄𝑥_𝑛−𝑥^∗2≤𝛼_𝑛𝑄𝑥_𝑛−𝑥^∗2+𝑧_𝑛−𝑥^∗2+𝜆_𝑛𝜆_𝑛−2𝐿∇𝑓(𝑧_𝑛)−∇𝑓(𝑥^∗)²≤𝛼_𝑛𝑄𝑥_𝑛−𝑥^∗2+𝑥_𝑛−𝑥^∗2+𝑟_𝑛𝑟_𝑛−2𝛼𝐹𝑥_𝑛−𝐹𝑥^∗2+𝜆_𝑛𝜆_𝑛−2𝐿∇𝑓(𝑧_𝑛)−∇𝑓(𝑥^∗)². (3.18) Utilizing the convexity of ‖⋅‖, we have 𝑥_𝑛+1−𝑥^∗2=𝛽_𝑛(𝑥_𝑛−𝑥^∗)+(1−𝛽_𝑛)11−𝛽_𝑛[𝛾_𝑛(𝑃_𝐶(𝑧_𝑛−𝜆_𝑛∇𝑓(𝑧_𝑛))−𝑥^∗)+𝛿_𝑛(𝑆𝑦_𝑛−𝑥^∗)]²≤𝛽_𝑛𝑥_𝑛−𝑥^∗2+1−𝛽_𝑛𝛾_𝑛1−𝛽_𝑛(𝑃_𝐶(𝑧_𝑛−𝜆_𝑛∇𝑓(𝑧_𝑛))−𝑥^∗𝛿)+^𝑛1−𝛽_𝑛(𝑆𝑦_𝑛−𝑥^∗)²=𝛽_𝑛𝑥_𝑛−𝑥^∗2+1−𝛽_𝑛𝛾_𝑛(𝑦_𝑛−𝑥^∗)+𝛿_𝑛(𝑆𝑦_𝑛−𝑥^∗)1−𝛽_𝑛+𝛼_𝑛𝛾_𝑛1−𝛽_𝑛(𝑃_𝐶(𝑧_𝑛−𝜆_𝑛∇𝑓(𝑧_𝑛))−𝑄𝑥_𝑛)²≤𝛽_𝑛𝑥_𝑛−𝑥^∗2+1−𝛽_𝑛𝛾_𝑛𝑦_𝑛−𝑥^∗+𝛿_𝑛𝑆𝑦_𝑛−𝑥^∗1−𝛽_𝑛²+𝑀𝛼_𝑛≤𝛽_𝑛𝑥_𝑛−𝑥^∗2+1−𝛽_𝑛𝑦_𝑛−𝑥^∗2+𝑀𝛼_𝑛, (3.19) where 𝑀>0 is some appropriate constant. So, from (3.18) and (3.19), it follows that 𝑥_𝑛+1−𝑥^∗2≤𝑥_𝑛−𝑥^∗2+𝑟_𝑛𝑟_𝑛−2𝛼1−𝛽_𝑛𝐹𝑥_𝑛−𝐹𝑥^∗2+𝜆_𝑛𝜆_𝑛−2𝐿1−𝛽_𝑛∇𝑓(𝑧_𝑛)−∇𝑓(𝑥^∗)²+𝑀+𝑄𝑥_𝑛−𝑥^∗2𝛼_𝑛. (3.20) Therefore, 𝜆_𝑛2𝐿−𝜆_𝑛1−𝛽_𝑛∇𝑓(𝑧_𝑛)−∇𝑓(𝑥^∗)²+𝑟_𝑛2𝛼−𝑟_𝑛1−𝛽_𝑛𝐹𝑥_𝑛−𝐹𝑥^∗2≤𝑥_𝑛−𝑥^∗2−𝑥_𝑛+1−𝑥^∗2+𝑀+𝑄𝑥_𝑛−𝑥^∗2𝛼_𝑛≤𝑥_𝑛−𝑥^∗+𝑥_𝑛+1−𝑥^∗𝑥_𝑛−𝑥_𝑛+1+𝑀+𝑄𝑥_𝑛−𝑥^∗2𝛼_𝑛. (3.21) Since liminf_𝑛→∞𝜆_𝑛(2/𝐿−𝜆_𝑛)(1−𝛽_𝑛)>0,  liminf_𝑛→∞𝑟_𝑛(2𝛼−𝑟_𝑛)(1−𝛽_𝑛)>0,  ‖𝑥_𝑛−𝑥_𝑛+1‖→0 and 𝛼_𝑛→0, we have lim_𝑛→∞∇𝑓𝑧_𝑛−∇𝑓𝑥^∗=0,lim_𝑛→∞𝐹𝑥_𝑛−𝐹𝑥^∗=0. (3.22)Step  3. We claim that lim_𝑛→∞‖𝑆𝑦_𝑛−𝑦_𝑛‖=0.
Indeed, set 𝑣_𝑛=𝑃_𝐶(𝑧_𝑛−𝜆_𝑛∇𝑓(𝑧_𝑛)). Noticing the firm nonexpansivity of 𝑇_𝑟^(Θ,𝜑)_𝑛, we have 𝑧_𝑛−𝑥^∗2=𝑇_𝑟^(Θ,𝜑)_𝑛𝑥_𝑛−𝑟_𝑛𝐹𝑥_𝑛−𝑇_𝑟^(Θ,𝜑)_𝑛𝑥^∗−𝑟_𝑛𝐹𝑥^∗2≤𝑥_𝑛−𝑟_𝑛𝐹𝑥_𝑛−𝑥^∗−𝑟_𝑛𝐹𝑥^∗,𝑧_𝑛−𝑥^∗=12𝑥_𝑛−𝑥^∗−𝑟_𝑛𝐹𝑥_𝑛−𝐹𝑥^∗2+𝑧_𝑛−𝑥^∗2−𝑥_𝑛−𝑥^∗−𝑟_𝑛𝐹𝑥_𝑛−𝐹𝑥^∗−𝑧_𝑛−𝑥^∗2≤12𝑥_𝑛−𝑥^∗2+𝑧_𝑛−𝑥^∗2−𝑥_𝑛−𝑧_𝑛−𝑟_𝑛𝐹𝑥_𝑛−𝐹𝑥^∗2=12𝑥_𝑛−𝑥^∗2+𝑧_𝑛−𝑥^∗2−𝑥_𝑛−𝑧_𝑛²+2𝑟_𝑛𝑥_𝑛−𝑧_𝑛,𝐹𝑥_𝑛−𝐹𝑥^∗−𝑟²_𝑛𝐹𝑥_𝑛−𝐹𝑥^∗2,𝑣_𝑛−𝑥^∗2=𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛−𝑃_𝐶𝑥^∗−𝜆_𝑛∇𝑓𝑥^∗2≤𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛−𝑥^∗−𝜆_𝑛∇𝑓𝑥^∗,𝑣_𝑛−𝑥^∗=12𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛−𝑥^∗−𝜆_𝑛∇𝑓𝑥^∗2+𝑣_𝑛−𝑥^∗2−𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛−𝑥^∗−𝜆_𝑛∇𝑓𝑥^∗−𝑣_𝑛−𝑥^∗2≤12𝑧_𝑛−𝑥^∗2+𝑣_𝑛−𝑥^∗2−𝑧_𝑛−𝑣_𝑛²+2𝜆_𝑛∇𝑓𝑧_𝑛−∇𝑓𝑥^∗,𝑧_𝑛−𝑣_𝑛−𝜆²_𝑛∇𝑓𝑧_𝑛−∇𝑓𝑥^∗2≤12𝑥_𝑛−𝑥^∗2+𝑣_𝑛−𝑥^∗2−𝑧_𝑛−𝑣_𝑛²+2𝜆_𝑛∇𝑓𝑧_𝑛−∇𝑓𝑥^∗,𝑧_𝑛−𝑣_𝑛. (3.23)(3.24) Thus, we have 𝑧_𝑛−𝑥^∗2≤𝑥_𝑛−𝑥^∗2−𝑥_𝑛−𝑧_𝑛²+2𝑟_𝑛𝑥_𝑛−𝑧_𝑛,𝐹𝑥_𝑛−𝐹𝑥^∗−𝑟²_𝑛𝐹𝑥_𝑛−𝐹𝑥^∗2,𝑣_𝑛−𝑥^∗2≤𝑥_𝑛−𝑥^∗2−𝑧_𝑛−𝑣_𝑛²+2𝜆_𝑛∇𝑓𝑧_𝑛−∇𝑓𝑥^∗𝑧_𝑛−𝑣_𝑛. (3.25)(3.26) It follows that 𝑦_𝑛−𝑥^∗2≤𝛼_𝑛𝑄𝑥_𝑛−𝑥^∗2+1−𝛼_𝑛𝑣_𝑛−𝑥^∗2≤𝛼_𝑛𝑄𝑥_𝑛−𝑥^∗2+𝑣_𝑛−𝑥^∗2≤𝛼_𝑛𝑄𝑥_𝑛−𝑥^∗2+𝑥_𝑛−𝑥^∗2−𝑧_𝑛−𝑣_𝑛²+2𝜆_𝑛∇𝑓𝑧_𝑛−∇𝑓𝑥^∗𝑧_𝑛−𝑣_𝑛. (3.27) From (3.18), (3.19), and (3.25), we have 𝑥_𝑛+1−𝑥^∗2≤𝛽_𝑛𝑥_𝑛−𝑥^∗2+1−𝛽_𝑛𝛼_𝑛𝑄𝑥_𝑛−𝑥^∗2+1−𝛽_𝑛𝑧_𝑛−𝑥^∗2+𝑀𝛼_𝑛≤𝑥_𝑛−𝑥^∗2−1−𝛽_𝑛𝑥_𝑛−𝑧_𝑛²+21−𝛽_𝑛𝑟_𝑛𝑥_𝑛−𝑧_𝑛𝐹𝑥_𝑛−𝐹𝑥^∗+𝑀+𝑄𝑥_𝑛−𝑥^∗2𝛼_𝑛. (3.28) It follows that 1−𝛽_𝑛𝑥_𝑛−𝑧_𝑛²≤𝑥_𝑛−𝑥^∗+𝑥_𝑛+1−𝑥^∗𝑥_𝑛+1−𝑥_𝑛+𝑀+𝑄𝑥_𝑛−𝑥^∗2𝛼_𝑛+21−𝛽_𝑛𝑟_𝑛𝑥_𝑛−𝑧_𝑛𝐹𝑥_𝑛−𝐹𝑥^∗. (3.29) Note that ‖𝑥_𝑛+1−𝑥_𝑛‖→0,  𝛼_𝑛→0 and ‖𝐹𝑥_𝑛−𝐹𝑥^∗‖→0. Then we immediately deduce that lim_𝑛→∞𝑥_𝑛−𝑧_𝑛=0. (3.30) From (3.19) and (3.27), we have 𝑥_𝑛+1−𝑥^∗2≤𝑥_𝑛−𝑥^∗2−1−𝛽_𝑛𝑧_𝑛−𝑣_𝑛²+2𝜆_𝑛1−𝛽_𝑛∇𝑓𝑧_𝑛−∇𝑓𝑥^∗𝑧_𝑛−𝑣_𝑛+𝑀+𝑄𝑥_𝑛−𝑥^∗2𝛼_𝑛. (3.31) So, we obtain 1−𝛽_𝑛𝑧_𝑛−𝑣_𝑛²≤𝑥_𝑛−𝑥^∗+𝑥_𝑛+1−𝑥^∗𝑥_𝑛+1−𝑥_𝑛+𝑀+𝑄𝑥_𝑛−𝑥^∗2𝛼_𝑛+2𝜆_𝑛1−𝛽_𝑛∇𝑓𝑧_𝑛−∇𝑓𝑥^∗𝑧_𝑛−𝑣_𝑛. (3.32) Note that ‖𝑥_𝑛+1−𝑥_𝑛‖→0,  𝛼_𝑛→0 and ‖∇𝑓(𝑧_𝑛)−∇𝑓(𝑥^∗)‖→0. Then we immediately conclude that lim_𝑛→∞𝑧_𝑛−𝑣_𝑛=0. (3.33) This together with ‖𝑦_𝑛−𝑣_𝑛‖≤𝛼_𝑛‖𝑄𝑥_𝑛−𝑣_𝑛‖→0, implies that lim_𝑛→∞𝑧_𝑛−𝑦_𝑛=0. (3.34) Thus, from (3.30) and (3.34), we deduce that lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=0. (3.35) Since 𝛿_𝑛𝑆𝑦_𝑛−𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝛾_𝑛𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛−𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝛾_𝑛𝑦_𝑛−𝑥_𝑛+𝛾_𝑛𝛼_𝑛𝑄𝑥_𝑛−𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛. (3.36) Therefore, lim_𝑛→∞𝑆𝑦_𝑛−𝑥_𝑛=0,lim_𝑛→∞𝑆𝑦_𝑛−𝑦_𝑛=0. (3.37)Step  4. We claim that limsup_𝑛→∞⟨𝑄𝑥^∗−𝑥^∗,𝑥_𝑛−𝑥^∗⟩≤0 where 𝑥^∗=𝑃_{Fix(𝑆)∩Ω∩GMEP}𝑄𝑥^∗.
Indeed, since {𝑥_𝑛} is bounded, there exists a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that 𝑥_𝑛𝑖⇀𝑢 and limsup_𝑛→∞𝑄𝑥^∗−𝑥^∗,𝑥_𝑛−𝑥^∗=lim_𝑖→∞𝑄𝑥^∗−𝑥^∗,𝑥_𝑛𝑖−𝑥^∗=𝑄𝑥^∗−𝑥^∗,𝑢−𝑥^∗. (3.38) We can obtain that 𝑢∈Fix(𝑆)∩Ω∩GMEP. First, we show 𝑢∈Ω(=VI(𝐶,∇𝑓)). Since 𝑥_𝑛−𝑧_𝑛→0 and 𝑣_𝑛−𝑧_𝑛→0, we conclude that 𝑧_𝑛𝑖⇀𝑢 and 𝑣_𝑛𝑖⇀𝑢. Let (𝑣)𝑇𝑣=∇𝑓+𝑁_𝐶𝑣,if𝑣∈𝐶,∅,if𝑣∉𝐶, (3.39) where 𝑁_𝐶𝑣 is the normal cone to 𝐶 at 𝑣∈𝐶. We have already mentioned that in this case, the mapping 𝑇 is maximal monotone, and 0∈𝑇𝑣 if and only if 𝑣∈VI(𝐶,∇𝑓)(=Ω); see [37]. Let 𝐺(𝑇) be the graph of 𝑇 and let (𝑣,𝑤)∈𝐺(𝑇). Then, we have 𝑤∈𝑇𝑣=∇𝑓(𝑣)+𝑁_𝐶𝑣 and hence 𝑤−∇𝑓(𝑣)∈𝑁_𝐶𝑣. So, we have ⟨𝑣−𝑡,𝑤−∇𝑓(𝑣)⟩≥0 for all 𝑡∈𝐶. On the other hand, from 𝑣_𝑛=𝑃_𝐶(𝑧_𝑛−𝜆_𝑛∇𝑓(𝑧_𝑛)) and 𝑣∈𝐶, we have 𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛−𝑣_𝑛,𝑣_𝑛−𝑣≥0 (3.40) and hence 𝑣−𝑣_𝑛,𝑣_𝑛−𝑧_𝑛𝜆_𝑛+∇𝑓𝑧_𝑛≥0. (3.41) From ⟨𝑣−𝑡,𝑤−∇𝑓(𝑣)⟩≥0 for all 𝑡∈𝐶 and 𝑣_𝑛𝑖∈𝐶, we have 𝑣−𝑣_𝑛𝑖≥,𝑤𝑣−𝑣_𝑛𝑖(𝑣≥,∇𝑓)𝑣−𝑣_𝑛𝑖,∇𝑓(𝑣)−𝑣−𝑣_𝑛𝑖,𝑣_𝑛𝑖−𝑧_𝑛𝑖𝜆_𝑛𝑖+∇𝑓𝑧_𝑛𝑖=𝑣−𝑣_𝑛𝑖,∇𝑓(𝑣)−∇𝑓𝑣_𝑛𝑖+𝑣−𝑣_𝑛𝑖,∇𝑓𝑣_𝑛𝑖−∇𝑓𝑧_𝑛𝑖−𝑣−𝑣_𝑛𝑖,𝑣_𝑛𝑖−𝑧_𝑛𝑖𝜆_𝑛𝑖≥𝑣−𝑣_𝑛𝑖,∇𝑓𝑣_𝑛𝑖−∇𝑓𝑧_𝑛𝑖−𝑣−𝑣_𝑛𝑖,𝑣_𝑛𝑖−𝑧_𝑛𝑖𝜆_𝑛𝑖. (3.42) Hence, we obtain ⟨𝑣−𝑢,𝑤⟩≥0 as 𝑖→∞. Since 𝑇 is maximal monotone, we have 𝑢∈𝑇⁻¹0 and hence 𝑢∈VI(𝐶,∇𝑓)(=Ω).
Secondly, let us show 𝑢∈Fix(𝑆). Since 𝑥_𝑛−𝑦_𝑛→0 and 𝑥_𝑛𝑖⇀𝑢, we have 𝑦_𝑛𝑖⇀𝑢. Also, since 𝑦_𝑛−𝑆𝑦_𝑛→0, it follows that 𝑦_𝑛𝑖−𝑆𝑦_𝑛𝑖→0 as 𝑖→∞. So, in terms of Proposition 2.5(ii) we obtain 𝑢∈Fix(𝑆).
Next, let us show 𝑢∈GMEP. From 𝑧_𝑛=𝑇_𝑟^(Θ,𝜑)_𝑛(𝑥_𝑛−𝑟_𝑛𝐹𝑥_𝑛), we know that Θ𝑧_𝑛,𝑦+𝜑𝑦−𝜑𝑧_𝑛+𝐹𝑥_𝑛,𝑦−𝑧_𝑛+1𝑟_𝑛𝑦−𝑧_𝑛,𝑧_𝑛−𝑥_𝑛≥0,∀𝑦∈𝐶. (3.43) From (H2), it follows that 𝜑𝑦−𝜑𝑧_𝑛+𝐹𝑥_𝑛,𝑦−𝑧_𝑛+1𝑟_𝑛𝑦−𝑧_𝑛,𝑧_𝑛−𝑥_𝑛≥Θ𝑦,𝑧_𝑛,∀𝑦∈𝐶. (3.44) Replacing 𝑛 by 𝑛_𝑖, we have 𝜑𝑦−𝜑𝑧_𝑛𝑖+𝐹𝑥_𝑛𝑖,𝑦−𝑧_𝑛𝑖+𝑦−𝑧_𝑛𝑖,𝑧_𝑛𝑖−𝑥_𝑛𝑖𝑟_𝑛𝑖≥Θ𝑦,𝑧_𝑛𝑖,∀𝑦∈𝐶. (3.45) Put 𝑧_𝑠=𝑠𝑦+(1−𝑠)𝑢 for all 𝑠∈(0,1] and 𝑦∈𝐶. Then, we have 𝑧_𝑠∈𝐶. So, from (3.45), we have 𝑧_𝑠−𝑧_𝑛𝑖,𝐹𝑧_𝑠≥𝑧_𝑠−𝑧_𝑛𝑖,𝐹𝑧_𝑠−𝜑𝑧_𝑠+𝜑𝑧_𝑛𝑖−𝑧_𝑠−𝑧_𝑛𝑖,𝐹𝑥_𝑛𝑖−𝑧_𝑠−𝑧_𝑛𝑖,𝑧_𝑛𝑖−𝑥_𝑛𝑖𝑟_𝑛𝑖+Θ𝑧_𝑠,𝑧_𝑛𝑖=𝑧_𝑠−𝑧_𝑛𝑖,𝐹𝑧_𝑠−𝐹𝑧_𝑛𝑖+𝑧_𝑠−𝑧_𝑛𝑖,𝐹𝑧_𝑛𝑖−𝐹𝑥_𝑛𝑖−𝜑𝑧_𝑠+𝜑𝑧_𝑛𝑖−𝑧_𝑠−𝑧_𝑛𝑖,𝑧_𝑛𝑖−𝑥_𝑛𝑖𝑟_𝑛𝑖+Θ𝑧_𝑠,𝑧_𝑛𝑖. (3.46) Since ‖𝑧_𝑛𝑖−𝑥_𝑛𝑖‖→0, we have ‖𝐹𝑧_𝑛𝑖−𝐹𝑥_𝑛𝑖‖→0. Further, from the monotonicity of 𝐹, we have ⟨𝑧_𝑠−𝑧_𝑛𝑖,𝐹𝑧_𝑠−𝐹𝑧_𝑛𝑖⟩≥0. So, from (H4), the weakly lower semicontinuity of 𝜑,(𝑧_𝑛𝑖−𝑥_𝑛𝑖)/𝑟_𝑛𝑖→0 and 𝑧_𝑛𝑖⇀𝑢, we have 𝑧_𝑠−𝑧_𝑛𝑖,𝐹𝑧_𝑠≥−𝜑𝑧_𝑠+𝜑(𝑢)+Θ𝑧_𝑠,𝑢, (3.47) as 𝑖→∞. From (H1), (H4), and (3.47), we also have 0=Θ𝑧_𝑠,𝑧_𝑠+𝜑𝑧_𝑠+𝜑𝑧_𝑠≤𝑠Θ𝑧_𝑠,𝑦+(1−𝑠)Θ𝑧_𝑠,𝑢+𝑠𝜑𝑦+(1−𝑠)𝜑(𝑢)−𝜑𝑧_𝑠=𝑠Θ𝑧_𝑠,𝑦+𝜑𝑦−𝜑𝑧_𝑠+(1−𝑠)Θ𝑧_𝑠,𝑢+𝜑(𝑢)−𝜑𝑧_𝑠≤𝑠Θ𝑧_𝑠,𝑦+𝜑𝑦−𝜑𝑧_𝑠+(1−𝑠)𝑧_𝑠−𝑢,𝐹𝑧_𝑠=𝑠Θ𝑧_𝑠,𝑦+𝜑𝑦−𝜑𝑧_𝑠+(1−𝑠)𝑠𝑦−𝑢,𝐹𝑧_𝑠, (3.48) and hence 0≤Θ𝑧_𝑠,𝑦+𝜑𝑦−𝜑𝑧_𝑠+(1−𝑠)𝑦−𝑢,𝐹𝑧_𝑠. (3.49) Letting 𝑠→0, we have, for each 𝑦∈𝐶, 0≤Θ𝑢,𝑦+𝜑𝑦−𝜑(𝑢)+𝑦−𝑢,𝐹𝑢. (3.50) This shows that 𝑢∈GMEP. Therefore, 𝑢∈Fix(𝑆)∩Ω∩GMEP. Hence, it follows from (2.4) that limsup_𝑛→∞𝑄𝑥^∗−𝑥^∗,𝑥_𝑛−𝑥^∗=lim_𝑖→∞𝑄𝑥^∗−𝑥^∗,𝑥_𝑛𝑖−𝑥^∗=𝑄𝑥^∗−𝑥^∗,𝑢−𝑥^∗≤0. (3.51)Step  5. We claim that lim_𝑛→∞‖𝑥_𝑛−𝑥^∗‖=0.
Indeed, from (3.2) and the convexity of ‖⋅‖, we have 𝑥_𝑛+1−𝑥^∗2=𝛽_𝑛(𝑥_𝑛−𝑥^∗)+𝛾_𝑛(𝑦_𝑛−𝑥^∗)+𝛿_𝑛(𝑆𝑦_𝑛−𝑥^∗)+𝛾_𝑛𝛼_𝑛(𝑃_𝐶(𝑧_𝑛−𝜆_𝑛∇𝑓(𝑧_𝑛))−𝑄𝑥_𝑛)²≤𝛽_𝑛𝑥_𝑛−𝑥^∗+𝛾_𝑛𝑦_𝑛−𝑥^∗+𝛿_𝑛𝑆𝑦_𝑛−𝑥^∗2+2𝛾_𝑛𝛼_𝑛𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛−𝑄𝑥_𝑛,𝑥_𝑛+1−𝑥^∗≤𝛽_𝑛𝑥_𝑛−𝑥^∗2+1−𝛽_𝑛11−𝛽_𝑛[𝛾_𝑛(𝑦_𝑛−𝑥^∗)+𝛿_𝑛(𝑆𝑦_𝑛−𝑥^∗)]²+2𝛾_𝑛𝛼_𝑛𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛−𝑥^∗,𝑥_𝑛+1−𝑥^∗+2𝛾_𝑛𝛼_𝑛𝑥^∗−𝑄𝑥_𝑛,𝑥_𝑛+1−𝑥^∗. (3.52) Utilizing Lemma 3.1, we get from (3.52) 𝑥_𝑛+1−𝑥^∗2≤𝛽_𝑛𝑥_𝑛−𝑥^∗2+1−𝛽_𝑛𝑦_𝑛−𝑥^∗2+2𝛾_𝑛𝛼_𝑛𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛−𝑥^∗𝑥_𝑛+1−𝑥^∗+2𝛾_𝑛𝛼_𝑛𝑥^∗−𝑄𝑥_𝑛,𝑥_𝑛+1−𝑥^∗≤𝛽_𝑛𝑥_𝑛−𝑥^∗2+1−𝛽_𝑛1−𝛼_𝑛𝑧_𝑛−𝑥^∗2+2𝛼_𝑛𝑄𝑥_𝑛−𝑥^∗,𝑦_𝑛−𝑥^∗+2𝛾_𝑛𝛼_𝑛𝑧_𝑛−𝑥^∗𝑥_𝑛+1−x^∗+2𝛾_𝑛𝛼_𝑛𝑥^∗−𝑄𝑥_𝑛,𝑥_𝑛+1−𝑥^∗. (3.53) From (3.17), we note that ‖𝑧_𝑛−𝑥^∗‖≤‖𝑥_𝑛−𝑥^∗‖. Hence we have 𝑥_𝑛+1−𝑥^∗2≤𝛽_𝑛𝑥_𝑛−𝑥^∗2+1−𝛽_𝑛1−𝛼_𝑛𝑥_𝑛−𝑥^∗2+2𝛼_𝑛1−𝛽_𝑛𝑄𝑥_𝑛−𝑥^∗,𝑦_𝑛−𝑥^∗+2𝛾_𝑛𝛼_𝑛𝑥_𝑛−𝑥^∗𝑥_𝑛+1−𝑥^∗+2𝛾_𝑛𝛼_𝑛𝑥^∗−𝑄𝑥_𝑛,𝑥_𝑛+1−𝑥^∗≤1−1−𝛽_𝑛𝛼_𝑛𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝛾_𝑛𝑄𝑥_𝑛−𝑥^∗,𝑦_𝑛−𝑥_𝑛+1+2𝛼_𝑛𝛿_𝑛𝑄𝑥_𝑛−𝑥^∗,𝑦_𝑛−𝑥^∗+2𝛼_𝑛𝛾_𝑛𝑥_𝑛−𝑥^∗𝑥_𝑛+1−𝑥^∗≤1−1−𝛽_𝑛𝛼_𝑛𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝛾_𝑛𝑄𝑥_𝑛−𝑥^∗𝑦_𝑛−𝑥_𝑛+1+2𝛼_𝑛𝛿_𝑛𝑄𝑥_𝑛−𝑥^∗,𝑥_𝑛−𝑥^∗+2𝛼_𝑛𝛿_𝑛𝑄𝑥_𝑛−𝑥^∗,𝑦_𝑛−𝑥_𝑛+2𝛼_𝑛𝛾_𝑛𝑥_𝑛−𝑥^∗𝑥_𝑛+1−𝑥^∗≤1−1−𝛽_𝑛𝛼_𝑛𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝛾_𝑛𝑄𝑥_𝑛−𝑥^∗𝑦_𝑛−𝑥_𝑛+1+2𝛼_𝑛𝛿_𝑛𝜌𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝛿_𝑛𝑄𝑥^∗−𝑥^∗,𝑥_𝑛−𝑥^∗+2𝛼_𝑛𝛿_𝑛𝑄𝑥_𝑛−𝑥^∗𝑦_𝑛−𝑥_𝑛+2𝛼_𝑛𝛾_𝑛𝑥_𝑛−𝑥^∗𝑥_𝑛+1−𝑥^∗≤1−1−𝛽_𝑛𝛼_𝑛𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝛾_𝑛𝑄𝑥_𝑛−𝑥^∗𝑦_𝑛−𝑥_𝑛+1+2𝛼_𝑛𝛿_𝑛𝜌𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝛿_𝑛𝑄𝑥^∗−𝑥^∗,𝑥_𝑛−𝑥^∗+2𝛼_𝑛𝛿_𝑛𝑄𝑥_𝑛−𝑥^∗𝑦_𝑛−𝑥_𝑛+𝛼_𝑛𝛾_𝑛𝑥_𝑛−𝑥^∗2+𝑥_𝑛+1−𝑥^∗2, (3.54) that is, 𝑥_𝑛+1−𝑥^∗2≤𝛿1−1−2𝜌_𝑛−𝛾_𝑛1−𝛼_𝑛𝛾_𝑛𝛼_𝑛𝑥_𝑛−𝑥^∗2+1−2𝜌𝛿_𝑛−𝛾_𝑛𝛼_𝑛1−𝛼_𝑛𝛾_𝑛×2𝛾_𝑛1−2𝜌𝛿_𝑛−𝛾_𝑛𝑄𝑥_𝑛−𝑥^∗𝑦_𝑛−𝑥_𝑛+1+2𝛿_𝑛1−2𝜌𝛿_𝑛−𝛾_𝑛𝑄𝑥_𝑛−𝑥^∗𝑦_𝑛−𝑥_𝑛+2𝛿_𝑛1−2𝜌𝛿_𝑛−𝛾_𝑛𝑄𝑥^∗−𝑥^∗,𝑥_𝑛−𝑥^∗. (3.55) Note that liminf_𝑛→∞((1−2𝜌)𝛿_𝑛−𝛾_𝑛)/(1−𝛼_𝑛𝛾_𝑛)>0. It follows that (∑^∞_𝑛=0((1−2𝜌)𝛿_𝑛−𝛾_𝑛)/(1−𝛼_𝑛𝛾_𝑛))𝛼_𝑛=∞. It is clear that limsup_𝑛→∞2𝛾_𝑛1−2𝜌𝛿_𝑛−𝛾_𝑛𝑄𝑥_𝑛−𝑥^∗𝑦_𝑛−𝑥_𝑛+1+2𝛿_𝑛1−2𝜌𝛿_𝑛−𝛾_𝑛𝑄𝑥_𝑛−𝑥^∗𝑦_𝑛−𝑥_𝑛+2𝛿_𝑛1−2𝜌𝛿_𝑛−𝛾_𝑛𝑄𝑥^∗−𝑥^∗,𝑥_𝑛−𝑥^∗≤0. (3.56) Therefore, all conditions of Lemma 2.7 are satisfied. This immediately implies that 𝑥_𝑛→𝑥^∗. It is readily seen that both {𝑦_𝑛} and {𝑧_𝑛} converge strongly to the same point 𝑥^∗. The proof is complete.

Utilizing Theorem 3.2, we establish the following corollaries.

Corollary 3.3. Let 𝐶 be a nonempty bounded closed convex subset of a real Hilbert space 𝐻. Let Θ∶𝐶×𝐶→𝐑 be a bifunction satisfying conditions (H1)–(H4) and 𝜑∶𝐶→𝐑 be a lower semicontinuous and convex function with assumptions (A1) or (A2). Suppose the minimization (1.5) is consistent and let Ω denote its solution set. Assume the gradient ∇𝑓 is 𝐿-Lipschitzian with constant 𝐿>0 and 𝐹∶𝐶→𝐻 is an 𝛼-inverse strongly monotone mapping. Let 𝑆∶𝐶→𝐶 be a 𝑘-strictly pseudocontractive mapping such that Fix(𝑆)∩Ω∩GMEP≠∅. For fixed 𝑢∈𝐶 and given 𝑥₀∈𝐶 arbitrarily, let the sequences {𝑥_𝑛}, {𝑦_𝑛} and {𝑧_𝑛} be generated iteratively by Θ𝑧_𝑛,𝑦+𝜑𝑦−𝜑𝑧_𝑛+𝐹𝑥_𝑛,𝑦−𝑧_𝑛+1𝑟_𝑛𝑦−𝑧_𝑛,𝑧_𝑛−𝑥_𝑛𝑦≥0,∀𝑦∈𝐶,_𝑛=𝛼_𝑛𝑢+1−𝛼_𝑛𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛+𝛿_𝑛𝑆𝑦_𝑛,∀𝑛≥0, (3.57) where {𝜆_𝑛}⊂(0,2/𝐿],  {𝑟_𝑛}⊂(0,2𝛼], and {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, {𝛿_𝑛} are four sequences in [0,1] such that:(i)0<liminf_𝑛→∞𝜆_𝑛≤limsup_𝑛→∞𝜆_𝑛<2/𝐿 and lim_𝑛→∞(𝜆_𝑛−𝜆_𝑛+1)=0;(ii)0<liminf_𝑛→∞𝑟_𝑛≤limsup_𝑛→∞𝑟_𝑛<2𝛼 and lim_𝑛→∞(𝑟_𝑛−𝑟_𝑛+1)=0;(iii)𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 and (𝛾_𝑛+𝛿_𝑛)𝑘≤𝛾_𝑛<𝛿_𝑛 for all 𝑛≥0;(iv)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞;(v)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and liminf_𝑛→∞𝛿_𝑛>0;(vi)lim_𝑛→∞(𝛾_𝑛+1/(1−𝛽_𝑛+1)−𝛾_𝑛/(1−𝛽_𝑛))=0.
Then, {𝑥_𝑛}, {𝑦_𝑛} and {𝑧_𝑛} converge strongly to the same point 𝑥^∗=𝑃_{Fix(𝑆)∩Ω∩GMEP}𝑢.

Corollary 3.4. Let 𝐶 be a nonempty bounded closed convex subset of a real Hilbert space 𝐻. Let Θ∶𝐶×𝐶→𝐑 be a bifunction satisfying conditions (H1)–(H4) and 𝜑∶𝐶→𝐑 be a lower semicontinuous and convex function with assumptions (A1) or (A2). Suppose the minimization (1.5) is consistent and let Ω denote its solution set. Assume the gradient ∇𝑓 is 𝐿-Lipschitzian with constant 𝐿>0 and 𝐹∶𝐶→𝐻 is an 𝛼-inverse strongly monotone mapping. Let 𝑆∶𝐶→𝐶 be a nonexpansive mapping such that Fix(𝑆)∩Ω∩GMEP≠∅. Let 𝑄∶𝐶→𝐶 be a 𝜌-contraction with 𝜌∈[0,1/2). For given 𝑥₀∈𝐶 arbitrarily, let the sequences {𝑥_𝑛}, {𝑦_𝑛} and {𝑧_𝑛} be generated iteratively by Θ𝑧_𝑛,𝑦+𝜑𝑦−𝜑𝑧_𝑛+𝐹𝑥_𝑛,𝑦−𝑧_𝑛+1𝑟_𝑛𝑦−𝑧_𝑛,𝑧_𝑛−𝑥_𝑛𝑦≥0,∀𝑦∈𝐶,_𝑛=𝛼_𝑛𝑄𝑥_𝑛+1−𝛼_𝑛𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛+𝛿_𝑛𝑆𝑦_𝑛,∀𝑛≥0, (3.58) where {𝜆_𝑛}⊂(0,2/𝐿], {𝑟_𝑛}⊂(0,2𝛼] and {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, {𝛿_𝑛} are four sequences in [0,1] such that(i)0<liminf_𝑛→∞𝜆_𝑛≤limsup_𝑛→∞𝜆_𝑛<2/𝐿 and lim_𝑛→∞(𝜆_𝑛−𝜆_𝑛+1)=0;(ii)0<liminf_𝑛→∞𝑟_𝑛≤limsup_𝑛→∞𝑟_𝑛<2𝛼 and lim_𝑛→∞(𝑟_𝑛−𝑟_𝑛+1)=0;(iii)𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 and 𝛾_𝑛<(1−2𝜌)𝛿_𝑛 for all 𝑛≥0;(iv)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞;(v)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and liminf_𝑛→∞𝛾_𝑛>0;(vi)lim_𝑛→∞(𝛾_𝑛+1/(1−𝛽_𝑛+1)−𝛾_𝑛/(1−𝛽_𝑛))=0.
Then {𝑥_𝑛}, {𝑦_𝑛}, and {𝑧_𝑛} converge strongly to the same point 𝑥^∗=𝑃_{Fix(𝑆)∩Ω∩GMEP}𝑄𝑥^∗.

Corollary 3.5. Let 𝐶 be a nonempty bounded closed convex subset of a real Hilbert space 𝐻. Let Θ∶𝐶×𝐶→𝐑 be a bifunction satisfying conditions (H1)–(H4) and 𝜑∶𝐶→𝐑 a lower semicontinuous and convex function with assumptions (A1) or (A2). Suppose the minimization (1.5) is consistent and let Ω denote its solution set. Assume the gradient ∇𝑓 is 𝐿-Lipschitzian with constant 𝐿>0 and 𝐹∶𝐶→𝐻 is an 𝛼-inverse strongly monotone mapping. Let 𝑆∶𝐶→𝐶 be a nonexpansive mapping such that Fix(𝑆)∩Ω∩GMEP≠∅. For fixed 𝑢∈𝐶 and given 𝑥₀∈𝐶 arbitrarily, let the sequences {𝑥_𝑛}, {𝑦_𝑛}, and {𝑧_𝑛} be generated iteratively by Θ𝑧_𝑛,𝑦+𝜑𝑦−𝜑𝑧_𝑛+𝐹𝑥_𝑛,𝑦−𝑧_𝑛+1𝑟_𝑛𝑦−𝑧_𝑛,𝑧_𝑛−𝑥_𝑛𝑦≥0,∀𝑦∈𝐶,_𝑛=𝛼_𝑛𝑢+1−𝛼_𝑛𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛+𝛿_𝑛𝑆𝑦_𝑛,∀𝑛≥0, (3.59) where {𝜆_𝑛}⊂(0,2/𝐿], {𝑟_𝑛}⊂(0,2𝛼], and {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, {𝛿_𝑛} are four sequences in [0,1] such that:(i)0<liminf_𝑛→∞𝜆_𝑛≤limsup_𝑛→∞𝜆_𝑛<2/𝐿 and lim_𝑛→∞(𝜆_𝑛−𝜆_𝑛+1)=0;(ii)0<liminf_𝑛→∞𝑟_𝑛≤limsup_𝑛→∞𝑟_𝑛<2𝛼 and lim_𝑛→∞(𝑟_𝑛−𝑟_𝑛+1)=0;(iii)𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 and 𝛾_𝑛<𝛿_𝑛 for all 𝑛≥0;(iv)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞;(v)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and liminf_𝑛→∞𝛾_𝑛>0;(vi)lim_𝑛→∞(𝛾_𝑛+1/(1−𝛽_𝑛+1)−𝛾_𝑛/(1−𝛽_𝑛))=0.
Then, {𝑥_𝑛}, {𝑦_𝑛}, and {𝑧_𝑛} converge strongly to the same point 𝑥^∗=𝑃_{Fix(𝑆)∩Ω∩GMEP}𝑢.

Corollary 3.6. Let 𝐶 be a nonempty bounded closed convex subset of a real Hilbert space 𝐻. Suppose the minimization (1.5) is consistent and let Ω denote its solution set. Assume the gradient ∇𝑓 is 𝐿-Lipschitzian with constant 𝐿>0 and 𝐴∶𝐶→𝐻 is an 𝛼-inverse strongly monotone mapping. Let 𝑆∶𝐶→𝐶 be a 𝑘-strictly pseudocontractive mapping such that Fix(𝑆)∩Ω∩VI(𝐶,𝐴)≠∅. For fixed 𝑢∈𝐶 and given 𝑥₀∈𝐶 arbitrarily, let the sequences {𝑥_𝑛}, {𝑦_𝑛}, and {𝑧_𝑛} be generated iteratively by 𝑧_𝑛=𝑃_𝐶𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛,𝑦_𝑛=𝛼_𝑛𝑢+1−𝛼_𝑛𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛+𝛿_𝑛𝑆𝑦_𝑛,∀𝑛≥0, (3.60) where {𝜆_𝑛}⊂(0,2/𝐿],  {𝑟_𝑛}⊂(0,2𝛼], and {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, {𝛿_𝑛} are four sequences in [0,1] such that:(i)0<liminf_𝑛→∞𝜆_𝑛≤limsup_𝑛→∞𝜆_𝑛<2/𝐿 and lim_𝑛→∞(𝜆_𝑛−𝜆_𝑛+1)=0;(ii)0<liminf_𝑛→∞𝑟_𝑛≤limsup_𝑛→∞𝑟_𝑛<2𝛼 and lim_𝑛→∞(𝑟_𝑛−𝑟_𝑛+1)=0;(iii)𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 and (𝛾_𝑛+𝛿_𝑛)𝑘≤𝛾_𝑛<𝛿_𝑛 for all 𝑛≥0;(iv)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞;(v)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and liminf_𝑛→∞𝛿_𝑛>0;(vi)lim_𝑛→∞(𝛾_𝑛+1/(1−𝛽_𝑛+1)−𝛾_𝑛/(1−𝛽_𝑛))=0.
Then, {𝑥_𝑛}, {𝑦_𝑛} and {𝑧_𝑛} converge strongly to the same point 𝑥^∗=𝑃_{Fix(𝑆)∩Ω∩VI(𝐶,𝐴)}𝑢.

Proof. In Theorem 3.2, putting Θ=0,  𝜑=0 and 𝐹=𝐴, the following relation Θ𝑧_𝑛,𝑦+𝜑𝑦−𝜑𝑧_𝑛+𝐹𝑥_𝑛,𝑦−𝑧_𝑛+1𝑟_𝑛𝑦−𝑧_𝑛,𝑧_𝑛−𝑥_𝑛≥0,∀𝑦∈𝐶, (3.61) is reduced to 𝐴𝑥_𝑛,𝑦−𝑧_𝑛+1𝑟_𝑛𝑦−𝑧_𝑛,𝑧_𝑛−𝑥_𝑛≥0,∀𝑦∈𝐶. (3.62) This implies that 𝑦−𝑧_𝑛,𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛−𝑧_𝑛≤0,∀𝑦∈𝐶. (3.63) So, it follows that 𝑧_𝑛=𝑃_𝐶(𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛) for all 𝑛≥0. Then, by Theorem 3.2, we obtain the desired result.

Let 𝑇∶𝐶→𝐶 be a 𝑘-strictly pseudocontractive mapping. For recent convergence result for strictly pseudocontractive mappings, we refer to Zeng et al. [38]. Putting 𝐹=𝐼−𝑇, we know that (𝐼−𝐹)𝑥−(𝐼−𝐹)𝑦²≤𝑥−𝑦²+𝑘𝐹𝑥−𝐹𝑦²,∀𝑥,𝑦∈𝐶. (3.64) Note that (𝐼−𝐹)𝑥−(𝐼−𝐹)𝑦²=𝑥−𝑦²+𝐹𝑥−𝐹𝑦²−2𝑥−𝑦,𝐹𝑥−𝐹𝑦. (3.65) Hence ≥𝑘𝑥−𝑦,𝐹𝑥−𝐹𝑦1−2𝐹𝑥−𝐹𝑦²,∀𝑥,𝑦∈𝐶. (3.66) This implies that the mapping 𝐹=𝐼−𝑇 is ((1−𝑘)/2)-inverse-strongly monotone.

Corollary 3.7. Let 𝐶 be a nonempty bounded closed convex subset of a real Hilbert space 𝐻. Let Θ∶𝐶×𝐶→𝐑 be a bifunction satisfying conditions (H1)–(H4) and 𝜑∶𝐶→𝐑 a lower semicontinuous and convex function with assumptions (A1) or (A2). Suppose the minimization (1.5) is consistent and let Ω denote its solution set. Assume the gradient ∇𝑓 is 𝐿-Lipschitzian with constant 𝐿>0 and 𝑇∶𝐶→𝐶 is a 𝑘-strictly pseudocontractive mapping. Let 𝑆∶𝐶→𝐶 be a 𝑘-strictly pseudocontractive mapping such that Fix(𝑆)∩Ω∩GMEP≠∅, where 𝐹=𝐼−𝑇. For fixed 𝑢∈𝐶 and given 𝑥₀∈𝐶 arbitrarily, let the sequences {𝑥_𝑛}, {𝑦_𝑛}, and {𝑧_𝑛} be generated iteratively by 𝑧_𝑛=𝑇_𝑟^(Θ,𝜑)_𝑛1−𝑟_𝑛𝑥_𝑛+𝑟_𝑛𝑇𝑥_𝑛,𝑦_𝑛=𝛼_𝑛𝑢+1−𝛼_𝑛𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑃_𝐶𝑧_𝑛−𝜆_𝑛∇𝑓𝑧_𝑛+𝛿_𝑛𝑆𝑦_𝑛,∀𝑛≥0, (3.67) where {𝜆_𝑛}⊂(0,2/𝐿],  {𝑟_𝑛}⊂(0,1−𝑘] and {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, {𝛿_𝑛} are four sequences in [0,1] such that(i)0<liminf_𝑛→∞𝜆_𝑛≤limsup_𝑛→∞𝜆_𝑛<2/𝐿 and lim_𝑛→∞(𝜆_𝑛−𝜆_𝑛+1)=0;(ii)0<liminf_𝑛→∞𝑟_𝑛≤limsup_𝑛→∞𝑟_𝑛<1−𝑘 and lim_𝑛→∞(𝑟_𝑛−𝑟_𝑛+1)=0;(iii)𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 and (𝛾_𝑛+𝛿_𝑛)𝑘≤𝛾_𝑛<𝛿_𝑛 for all 𝑛≥0;(iv)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞;(v)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and liminf_𝑛→∞𝛿_𝑛>0;(vi)lim_𝑛→∞(𝛾_𝑛+1/(1−𝛽_𝑛+1)−𝛾_𝑛/(1−𝛽_𝑛))=0.
Then, {𝑥_𝑛}, {𝑦_𝑛} and {𝑧_𝑛} converge strongly to the same point 𝑥^∗=𝑃_{Fix(𝑆)∩Ω∩GMEP}𝑢.

Proof. Since 𝑇 is a 𝑘-strictly pseudocontractive mapping, the mapping 𝐹=𝐼−𝑇 is (1−𝑘)/2-inverse-strongly monotone. In this case, put 𝛼=(1−𝑘)/2. Then, we conclude that 𝑧_𝑛=𝑇_𝑟^(Θ,𝜑)_𝑛𝑥_𝑛−𝑟_𝑛𝐹𝑥_𝑛=𝑇_𝑟^(Θ,𝜑)_𝑛𝑥_𝑛−𝑟_𝑛(𝐼−𝑇)𝑥_𝑛=𝑇_𝑟^(Θ,𝜑)_𝑛1−𝑟_𝑛𝑥_𝑛+𝑟_𝑛𝑇𝑥_𝑛. (3.68) So, by Theorem 3.2, we obtain the desired result.

Acknowledgments

This research was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), Leading Academic Discipline Project of Shanghai Normal University (DZL707), Ph.D. Program Foundation of Ministry of Education of China (20070270004), Science and Technology Commission of Shanghai Municipality Grant (075105118), and Shanghai Leading Academic Discipline Project (S30405). This research was partially supported by a Grant from NSC 101-2115-M-037-001.