Weak Convergence Theorems for Strictly Pseudocontractive Mappings and Generalized Mixed Equilibrium Problems

Abstract

We introduce a new iterative method for finding a common element of the set of fixed points of a strictly pseudocontractive mapping, the set of solutions of a generalized mixed equilibrium problem, and the set of solutions of a variational inequality problem for an inverse-strongly-monotone mapping in Hilbert spaces and then show that the sequence generated by the proposed iterative scheme converges weakly to a common element of the above three sets under suitable control conditions. The results in this paper substantially improve, develop, and complement the previous well-known results in this area.

1. Introduction

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and induced norm ‖⋅‖. Let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝐵∶𝐶→𝐻 be a nonlinear mapping and 𝜑∶𝐶→ℝ be a function, and Θ be a bifunction of 𝐶×𝐶 into ℝ, where ℝ is the set of real numbers.

Then, we consider the following generalized mixed equilibrium problem of finding 𝑥∈𝐶 such thatΘ𝑥,𝑦+𝐵𝑥,𝑦−𝑥+𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐶, (1.1) which was introduced by Peng and Yao [1] recently. The set of solutions of the problem (1.1) is denoted by GMEP(Θ,𝜑,𝐵). Here some special cases of the problem (1.1) are stated as follows.

If 𝜑=0, then the problem (1.1) reduced the following generalized equilibrium problem (GEP) of finding 𝑥∈𝐶 such thatΘ𝑥,𝑦+𝐵𝑥,𝑦−𝑥≥0,∀𝑦∈𝐶, (1.2) which was studied by S. Takahashi and M. Takahashi [2]. The set of solutions of the problem (1.2) is denoted by GEP(Θ,𝐵).

If 𝐵=0, then the problem (1.1) reduces the following mixed equilibrium problem of finding 𝑥∈𝐶 such thatΘ𝑥,𝑦+𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐶, (1.3) which was studied by Ceng and Yao [3] (see also [4]). The set of solutions of the problem (1.3) is denoted by MEP(Θ,𝜑).

If 𝜑=0 and 𝐵=0, then the problem (1.1) reduces the following equilibrium problem of finding 𝑥∈𝐶 such thatΘ𝑥,𝑦≥0,∀𝑦∈𝐶. (1.4) The set of solutions of the problem (1.4) is denoted by EP(Θ).

If 𝜑=0 and Θ(𝑥,𝑦)=0 for all 𝑥,𝑦∈𝐶, the problem (1.1) reduces the following variational inequality problem of finding 𝑥∈𝐶 such that𝐵𝑥,𝑦−𝑥≥0,∀𝑦∈𝐶. (1.5) The set of solutions of the problem (1.5) is denoted by VI(𝐶,𝐵).

The problem (1.1) is very general in the sense that it includes, as special cases, fixed point problems, optimization problems, variational inequality problems, minimax problems, Nash equilibrium problems in noncooperative games, and others; see, for example, [3, 5–7].

The class of pseudocontractive mappings is one of the most important classes of mappings among nonlinear mappings. We recall that a mapping 𝑆∶𝐶→𝐻 is said to be 𝑘-strictly pseudocontractive if there exists a constant 𝑘∈[0,1) such that𝑆𝑥−𝑆𝑦²≤𝑥−𝑦²+𝑘(𝐼−𝑆)𝑥−(𝐼−𝑆)𝑦²,∀𝑥,𝑦∈𝐶. (1.6) Note that the class of 𝑘-strictly pseudocontractive mappings includes the class of nonexpansive mappings as a subclass. That is, 𝑆 is nonexpansive (i.e., ‖𝑆𝑥−𝑆𝑦‖≤‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶) if and only if 𝑆 is 0-strictly pseudocontractive. The mapping 𝑆 is also said to be pseudocontractive if 𝑘=1 and 𝑆 is said to be strongly pseudocontractive if there exists a constant 𝜆∈(0,1) such that 𝑆−𝜆𝐼 is pseudocontractive. Clearly, the class of 𝑘-strictly pseudocontractive mappings falls into the one between classes of nonexpansive mappings and pseudocontractive mappings. Also we remark that the class of strongly pseudocontractive mappings is independent of the class of 𝑘-strictly pseudocontractive mappings (see [8, 9]). Recently, many authors have been devoting the studies on the problems of finding fixed points to the class of pseudocontractive mappings; see, for example, [10–15] and the references therein.

Recently, in order to study the problems (1.1)–(1.5) coupled with the fixed point problem, many authors have introduced some iterative schemes for finding a common element of the set of the solutions of the problem (1.1)–(1.5) and the set of fixed points of a countable family of nonexpansive mappings and have studied strong convergence of the sequences generated by the proposed schemes; see [1–4, 16–18] and the references therein. Also we refer to [19–21] for the problems (1.1), (1.3), and (1.5) combined to the fixed point problem for nonexpansive semigroups and strictly pseudocontractrive mappings.

In this paper, inspired and motivated by [18, 22–27], we introduce a new iterative method for finding a common element of the set of fixed points of a 𝑘-strictly pseudocontractive mapping, the set of solutions of a generalized mixed equilibrium problem (1.1), and the set of solutions of the variational inequality problem (1.5) for an inverse-strongly monotone mapping in a Hilbert space. We show that, under suitable conditions, the sequence generated by the proposed iterative scheme converges weakly to a common element of the above three sets. The results in this paper can be viewed as an improvement and complement of the recent results in this direction.

2. Preliminaries and Lemmas

Let 𝐻 be a real Hilbert space and let 𝐶 be a nonempty closed convex subset of 𝐻. In the following, we write 𝑥_𝑛⇀𝑥 to indicate that the sequence {𝑥_𝑛} converges weakly to 𝑥. 𝑥_𝑛→𝑥 implies that {𝑥_𝑛} converges strongly to 𝑥. We denote by 𝐹(𝑇) the set of fixed points of the mapping 𝑇.

In a real Hilbert space 𝐻, we have𝜆𝑥+(1−𝜆)𝑦²=𝜆‖𝑥‖²+(1−𝜆)𝑦²−𝜆(1−𝜆)𝑥−𝑦² (2.1) for all 𝑥,𝑦∈𝐻 and 𝜆∈ℝ. For every point 𝑥∈𝐻, there exists a unique nearest point in 𝐶, denoted by 𝑃_𝐶(𝑥), such that𝑥−𝑃_𝐶(𝑥)≤𝑥−𝑦 (2.2) for all 𝑦∈𝐶. 𝑃_𝐶 is called the metric projection of 𝐻 onto 𝐶. It is well known that 𝑃_𝐶 is nonexpansive and 𝑃_𝐶 satisfies𝑥−𝑦,𝑃_𝐶(𝑥)−𝑃_𝐶𝑦≥𝑃_𝐶(𝑥)−𝑃_𝐶𝑦² (2.3) for every 𝑥,𝑦∈𝐻. Moreover, 𝑃_𝐶(𝑥) is characterized by the properties:𝑥−𝑦²≥𝑥−𝑃_𝐶(𝑥)²+𝑦−𝑃_𝐶(𝑥)²,𝑢=𝑃_𝐶(𝑥)⟺𝑥−𝑢,𝑢−𝑦≥0∀𝑥∈𝐻,𝑦∈𝐶. (2.4) In the context of the variational inequality problem for a nonlinear mapping 𝐹, this implies that𝑢∈VI(𝐶,𝐹)⟺𝑢=𝑃_𝐶(𝑢−𝜆𝐹𝑢)forany𝜆>0. (2.5) It is also well known that 𝐻 satisfies the Opial condition, that is, for any sequence {𝑥_𝑛} with 𝑥_𝑛⇀𝑥, the inequalityliminf_𝑛→∞𝑥_𝑛−𝑥<liminf_𝑛→∞𝑥_𝑛−𝑦 (2.6) holds for every 𝑦∈𝐻 with 𝑦≠𝑥.

A mapping 𝐹 of 𝐶 into 𝐻 is called 𝛼-inverse-strongly monotone if there exists a constant 𝛼>0 such that𝑥−𝑦,𝐹𝑥−𝐹𝑦≥𝛼𝐹𝑥−𝐹𝑦²,∀𝑥,𝑦∈𝐶. (2.7) We know that if 𝐹=𝐼−𝑇, where 𝑇 is a nonexpansive mapping of 𝐶 into itself and 𝐼 is the identity mapping of 𝐻, then 𝐹 is 1/2-inverse-strongly monotone and VI(𝐶,𝐹)=𝐹(𝑇). A mapping 𝐹 of 𝐶 into 𝐻 is called strongly monotone if there exists a positive real number 𝜂 such that𝑥−𝑦,𝐹𝑥−𝐹𝑦≥𝜂𝑥−𝑦²,∀𝑥,𝑦∈𝐶. (2.8) In such a case, we say 𝐹 is 𝜂-strongly monotone. If 𝐹 is 𝜂-strongly monotone and 𝜅-Lipschitzian, continuous, that is, ‖𝐹𝑥−𝐹𝑦‖≤𝐿‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶, then 𝐹 is 𝜂/𝜅²-inverse-strongly monotone. If 𝐹 is an 𝛼-inverse-strongly monotone mapping of 𝐶 into 𝐻, then it is obvious that 𝐹 is 1/𝛼-Lipschitzian. We also have that for all 𝑥,𝑦∈𝐶 and 𝜆>0,(𝐼−𝜆𝐹)𝑥−(𝐼−𝜆𝐹)𝑦²=𝑥−𝑦−𝜆𝐹𝑥−𝐹𝑦²=𝑥−𝑦²−2𝜆𝑥−𝑦,𝐹𝑥−𝐹𝑦+𝜆²𝐹𝑥−𝐹𝑦²≤𝑥−𝑦²+𝜆(𝜆−2𝛼)𝐹𝑥−𝐹𝑦². (2.9) So, if 𝜆≤2𝛼, then 𝐼−𝜆𝐹 is a nonexpansive mapping of 𝐶 into 𝐻. The following result for the existence of solutions of the variational inequality problem for inverse-strongly monotone mappings was given in Takahashi and Toyoda [27].

Proposition 2.1. Let 𝐶 be a bounded closed convex subset of a real Hilbert space and let 𝐹 be an 𝛼-inverse-strongly monotone mapping of 𝐶 into 𝐻. Then, VI(𝐶,𝐹) is nonempty.

A set-valued mapping 𝑄∶𝐻→2^𝐻 is called monotone if for all 𝑥,𝑦∈𝐻, 𝑓∈𝑄𝑥 and 𝑔∈𝑄𝑦 imply ⟨𝑥−𝑦,𝑓−𝑔⟩≥0. A monotone mapping 𝑄∶𝐻→2^𝐻 is maximal if the graph 𝐺(𝑄) of 𝑄 is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping 𝑄 is maximal if and only if for (𝑥,𝑓)∈𝐻×𝐻, ⟨𝑥−𝑦,𝑓−𝑔⟩≥0 for every (𝑦,𝑔)∈𝐺(𝑄) implies 𝑓∈𝑄𝑥. Let 𝐹 be an inverse-strongly monotone mapping of 𝐶 into 𝐻 and let 𝑁_𝐶𝑣 be the normal cone to 𝐶 at 𝑣, that is, 𝑁_𝐶𝑣={𝑤∈𝐻∶⟨𝑣−𝑢,𝑤⟩≥0,forall𝑢∈𝐶}, and define𝑄𝑣=𝐹𝑣+𝑁_𝐶𝑣,𝑣∈𝐶,∅,𝑣∉𝐶. (2.10) Then 𝑄 is maximal monotone and 0∈𝑄𝑣 if and only if 𝑣∈VI(𝐶,𝐹); see [28, 29].

For solving the equilibrium problem for a bifunction Θ∶𝐶×𝐶→ℝ, let us assume that Θ and 𝜑 satisfy the following conditions:

(A1) Θ(𝑥,𝑥)=0 for all 𝑥∈𝐶,

(A2) Θ is monotone, that is, Θ(𝑥,𝑦)+Θ(𝑦,𝑥)≤0 for all 𝑥,𝑦∈𝐶,

(A3) for each 𝑥,𝑦,𝑧∈𝐶, lim_𝑡↓0Θ𝑡𝑧+(1−𝑡)𝑥,𝑦≤Θ𝑥,𝑦, (2.11)

(A4) for each 𝑥∈𝐶,𝑦↦Θ(𝑥,𝑦) is convex and lower semicontinuous,

(A5) for each 𝑦∈𝐶, 𝑥↦Θ(𝑥,𝑦) is weakly upper semicontiunuous,

(B1) for each 𝑥∈𝐻 and 𝑟>0, there exist 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, (2.12)

(B2) 𝐶 is a bounded set.

The following lemmas were given in [1, 5].

Lemma 2.2 (see [5]). Let 𝐶 be a nonempty closed convex subset of 𝐻 and Θ a bifunction of 𝐶×𝐶 into ℝ satisfying (A1)–(A4). Let 𝑟>0 and 𝑥∈𝐻. Then, there exists 𝑧∈𝐶 such that Θ+1𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶. (2.13)

Lemma 2.3 (see [1]). Let 𝐶 be a nonempty closed convex subset of 𝐻. Let Θ be a bifunction form 𝐶×𝐶 to ℝ which satisfies (A1)–(A5) and 𝜑∶𝐶→ℝ a proper lower semicontinuous and convex function. For 𝑟>0 and 𝑥∈𝐻, define a mapping 𝑇_𝑟∶𝐻→𝐶 as follows: 𝑇_𝑟(𝑥)=𝑦(𝑧)+1𝑧∈𝐶∶Θ𝑧,𝑦+𝜑−𝜑𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶 (2.14) for all 𝑧∈𝐻. Assume that either (B1) or (B2) holds. Then, the following hold:
(1) for each 𝑥∈𝐻,𝑇_𝑟(𝑥)≠∅,
(2)𝑇_𝑟 is single-valued,
(3)𝑇_𝑟 is firmly nonexpansive, that is, for any 𝑥,𝑦∈𝐻, 𝑇_𝑟𝑥−𝑇_𝑟𝑦²≤𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝑥−𝑦, (2.15)
(4)𝐹(𝑇_𝑟)=MEP(Θ,𝜑),
(5)MEP(Θ,𝜑) is closed and convex.

We also need the following lemmas for the proof of our main results.

Lemma 2.4 (see [30]). Let 𝐻 be a real Hilbert space, let {𝛼_𝑛} be a sequence of real numbers such that 0<𝑎≤𝛼_𝑛≤𝑏<1 for all 𝑛≥1 and let {𝑣_𝑛} and {𝑤_𝑛} be sequences in 𝐻 such that, for some 𝑐limsup_𝑛→∞𝑣_𝑛≤𝑐,limsup_𝑛→∞𝑤_𝑛≤𝑐,limsup_𝑛→∞𝛼_𝑛𝑣_𝑛+1−𝛼_𝑛𝑤_𝑛=𝑐. (2.16) Then lim_𝑛→∞‖𝑣_𝑛−𝑤_𝑛‖=0.

Lemma 2.5 (see [27]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert spaces 𝐻 and let {𝑥_𝑛} be a sequence in 𝐻. If 𝑥_𝑛+1−𝑥≤𝑥_𝑛−𝑥,∀𝑥∈𝐶,∀𝑛≥1, (2.17) then {𝑃_𝐶𝑥_𝑛} converges strongly to some 𝑧∈𝐶, where 𝑃_𝐶 stands for the metric projection of 𝐻 onto 𝐶.

Lemma 2.6 (see [31]). Let 𝐻 be a Hilbert space, 𝐶 a closed convex subset of 𝐻. If 𝑇 is a 𝑘-strictly pseudocontractive mapping on 𝐶, then the fixed point set 𝐹(𝑇) is closed convex, so that the projection 𝑃_𝐹(𝑇) is well defined.

Lemma 2.7 (see [31]). Let 𝐻 be a Hilbert space, 𝐶 a closed convex subset of 𝐻, and 𝑇∶𝐶→𝐻 a 𝑘-strictly pseudocontractive mapping. Define a mapping 𝑆∶𝐶→𝐻 by 𝑆𝑥=𝜆𝑥+(1−𝜆)𝑇𝑥 for all 𝑥∈𝐶. Then, as 𝜆∈[𝑘,1), 𝑆 is a nonexpansive mapping such that 𝐹(𝑆)=𝐹(𝑇).

3. Main Results

In this section, we introduce a new iterative scheme for finding a common point of the set of fixed points of a 𝑘-strictly pseudocontractive mapping, the set of solutions of the problem (1.1), and the set of solutions of the problem (1.5) for an inverse-strongly monotone mapping.

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let Θ be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A5) and 𝜑∶𝐶→ℝ a lower semicontinuous and convex function. Let 𝐹,𝐵 be two 𝛼, 𝛽-inverse-strongly monotone mappings of 𝐶 into 𝐻, respectively. Let 𝑇 be a 𝑘-strictly pseudocontractive mapping of 𝐶 into itself for some 𝑘∈[0,1) such that Ω₁∶=𝐹(𝑇)⋂GMEP(Θ,𝜑,𝐵)⋂VI(𝐶,𝐹)≠∅. Assume that either (B1) or (B2) holds. Let {𝑥_𝑛} and {𝑢_𝑛} be sequences generated by 𝑥₁∈𝐶 and Θ𝑢_𝑛,𝑦+𝐵𝑥_𝑛,𝑦−𝑢_𝑛+𝜑𝑦−𝜑𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=𝑆𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑃_𝐶𝑢_𝑛−𝜆_𝑛𝐹𝑢_𝑛,∀𝑛≥1, (3.1) where 𝑆∶𝐶→𝐶 is a mapping defined by 𝑆𝑥=𝑘𝑥+(1−𝑘)𝑇𝑥 for 𝑥∈𝐶,{𝛼_𝑛}⊂[0,1] and {𝑟_𝑛}⊂(0,∞). Assume that {𝛼_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1),𝜆_𝑛∈[𝑐,𝑑]⊂(0,2𝛼) and 𝑟_𝑛∈[𝑒,𝑓]⊂(0,2𝛽). Then {𝑥_𝑛} and {𝑢_𝑛} converge weakly to 𝑧∈Ω₁, where 𝑧=lim_𝑛→∞𝑃_Ω1(𝑥_𝑛).

Proof. From now, we put 𝑧_𝑛=𝑃_𝐶(𝑢_𝑛−𝜆_𝑛𝐹𝑢_𝑛).
We divide the proof into several steps.
Step 1. We show that {𝑥_𝑛} is bounded. To this end, let 𝑝∈Ω₁∶=𝐹(𝑇)⋂GMEP(Θ,𝜑,𝐵)⋂VI(𝐶,𝐹) and {𝑇_𝑟𝑛} be a sequence of mappings defined as in Lemma 2.3. Then, since 𝐹(𝑆)=𝐹(𝑇) by Lemma 2.7, 𝑝=𝑆𝑝. Also, from (4) in Lemma 2.3 and (2.5), it follows that 𝑝=𝑇_𝑟𝑛(𝑝−𝑟_𝑛𝐵𝑝) and 𝑝=𝑃_𝐶(𝑝−𝜆_𝑛𝐹𝑝). From 𝑧_𝑛=𝑃_𝐶(𝑢_𝑛−𝜆_𝑛𝐹𝑢_𝑛) and the fact that 𝑃_𝐶 and 𝐼−𝜆_𝑛𝐹 are nonexpansive, it follows that 𝑧_𝑛−𝑝≤𝐼−𝜆_𝑛𝐹𝑢_𝑛−𝐼−𝜆_𝑛𝐹𝑝≤𝑢_𝑛−𝑝. (3.2) Also, by 𝑢_𝑛=𝑇_𝑟𝑛(𝑥_𝑛−𝑟_𝑛𝐵𝑥_𝑛)∈𝐶 and the 𝛽-inverse-strongly monotonicity of 𝐵, we have with 𝑟_𝑛∈(0,2𝛽), 𝑢_𝑛−𝑝²≤𝑥_𝑛−𝑟_𝑛𝐵𝑥_𝑛−𝑝−𝑟_𝑛𝐵𝑝²≤𝑥_𝑛−𝑝²−2𝑟_𝑛𝑥_𝑛−𝑝,𝐵𝑥_𝑛−𝐵𝑝+𝑟²_𝑛𝐵𝑥_𝑛−𝐵𝑝²≤𝑥_𝑛−𝑝²+𝑟_𝑛𝑟_𝑛−2𝛽𝐵𝑥_𝑛−𝐵𝑝²≤𝑥_𝑛−𝑝², (3.3) that is, ‖𝑢_𝑛−𝑝‖≤‖𝑥_𝑛−𝑝‖, and so 𝑧_𝑛−𝑝≤𝑥_𝑛−𝑝. (3.4) So, by using the convexity of ‖⋅‖², (3.2) and (3.3), we have 𝑥_𝑛+1−𝑝²=𝑆𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑧_𝑛−𝑆𝑝²≤𝛼_𝑛𝑥_𝑛−𝑝+1−𝛼_𝑛𝑧_𝑛−𝑝²≤𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛𝑧_𝑛−𝑝²≤𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛𝑥_𝑛−𝑝²+𝑟_𝑛𝑟_𝑛−2𝛽𝐵𝑥_𝑛−𝐵𝑝²≤𝑥_𝑛−𝑝²+(1−𝑏)𝑒𝑓−2𝛽𝐵𝑥_𝑛−𝐵𝑝²≤𝑥_𝑛−𝑝². (3.5) So, there exists 𝑟∈ℝ such that 𝑟=lim_𝑛→∞𝑥_𝑛−𝑝. (3.6) Therefore, {𝑥_𝑛} is bounded, and so are {𝑢_𝑛} and {𝑧_𝑛} by (3.2) and (3.4). Moreover, from (3.5), it follows that (1−𝑏)𝑒2𝛽−𝑓𝐵𝑥_𝑛−𝐵𝑝²≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝², (3.7) which implies that lim_𝑛→∞𝐵𝑥_𝑛−𝐵𝑝=0. (3.8)Step 2. We show that lim_𝑛→∞‖𝑥_𝑛−𝑢_𝑛‖=0. To this end, let 𝑝∈Ω₁. Since 𝑇_𝑟𝑛 is firmly nonexpansive and 𝑢_𝑛=𝑇_𝑟𝑛(𝑥_𝑛−𝑟_𝑛𝐵𝑥_𝑛), we have 𝑢_𝑛−𝑝²=𝑇_𝑟𝑛𝑥_𝑛−𝑟_𝑛𝐵𝑥_𝑛−𝑇_𝑟𝑛𝑝−𝑟_𝑛𝐵𝑝²≤𝑇_𝑟𝑛𝑥_𝑛−𝑟_𝑛𝐵𝑥_𝑛−𝑇_𝑟𝑛𝑝−𝑟_𝑛𝐵𝑝,𝑥_𝑛−𝑟_𝑛𝐵𝑥_𝑛−𝑝−𝑟_𝑛=𝑥𝐵𝑝_𝑛−𝑟_𝑛𝐵𝑥_𝑛−𝑝−𝑟_𝑛𝐵𝑝,𝑢_𝑛=1−𝑝2𝑢_𝑛−𝑝²+𝑥_𝑛−𝑟_𝑛𝐵𝑥_𝑛−𝑝−𝑟_𝑛𝐵𝑝²−12𝑥_𝑛−𝑟_𝑛𝐵𝑥_𝑛−𝑝−𝑟_𝑛𝐵𝑝−𝑢_𝑛−𝑝²≤12𝑢_𝑛−𝑝²+𝑥_𝑛−𝑝²−𝑥_𝑛−𝑢_𝑛−𝑟_𝑛𝐵𝑥_𝑛−𝐵𝑝²≤12𝑢_𝑛−𝑝²+𝑥_𝑛−𝑝²−𝑥_𝑛−𝑢_𝑛²+2𝑟_𝑛𝐵𝑥_𝑛−𝐵𝑝,𝑥_𝑛−𝑢_𝑛−𝑟²_𝑛𝐵𝑥_𝑛−𝐵𝑝², (3.9) and hence 𝑢_𝑛−𝑝²≤𝑥_𝑛−𝑝²−𝑥_𝑛−𝑢_𝑛²+2𝑟_𝑛𝐵𝑥_𝑛−𝐵𝑝,𝑥_𝑛−𝑢_𝑛−𝑟²_𝑛𝐵𝑥_𝑛−𝐵𝑝²≤𝑥_𝑛−𝑝²−𝑥_𝑛−𝑢_𝑛²+2𝑟_𝑛𝐵𝑥_𝑛−𝐵𝑝,𝑥_𝑛−𝑢_𝑛≤𝑥_𝑛−𝑝²−𝑥_𝑛−𝑢_𝑛²+2𝑟_𝑛𝐵𝑥_𝑛−𝐵𝑝𝑥_𝑛−𝑢_𝑛. (3.10) On the other hand, by using the convexity of ‖⋅‖², (3.2) and (3.10), we obtain 𝑥_𝑛+1−𝑝²=𝑆𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑧_𝑛−𝑆𝑝²≤𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛𝑧_𝑛−𝑝²≤𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛𝑢_𝑛−𝑝²≤𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛𝑥_𝑛−𝑝²−𝑥_𝑛−𝑢_𝑛²+2𝑟_𝑛𝐵𝑥_𝑛−𝐵𝑝𝑥_𝑛−𝑢_𝑛, (3.11) and hence (1−𝑏)𝑥_𝑛−𝑢_𝑛²≤1−𝛼_𝑛𝑥_𝑛−𝑢_𝑛²≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+2𝑟_𝑛𝐵𝑥_𝑛−𝐵𝑝𝑥_𝑛−𝑢_𝑛≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+2𝑓𝐵𝑥_𝑛−𝐵𝑝𝑀₁, (3.12) where 𝑀₁=sup{‖𝑥_𝑛‖+‖𝑢_𝑛‖∶𝑛≥1}. Since lim_𝑛→∞‖𝑥_𝑛+1−𝑝‖²=lim_𝑛→∞‖𝑥_𝑛−𝑝‖² and lim_𝑛→∞‖𝐵𝑥_𝑛−𝐵𝑝‖=0 in (3.8), we obtain lim_𝑛→∞𝑥_𝑛−𝑢_𝑛=0 (3.13) and so is the limit of {𝐵𝑥_𝑛−𝐵𝑢_𝑛} since 𝐵 is Lipschitz.Step 3. We show that lim_𝑛→∞‖𝑥_𝑛−𝑆𝑧_𝑛‖=0. Indeed, let 𝑝∈Ω₁ and set 𝑟=lim_𝑛→∞‖𝑥_𝑛−𝑝‖. Being 𝑆 nonexpansive and 𝐹(𝑇)=𝐹(𝑆), from (3.4) we can write 𝑆𝑧_𝑛−𝑝≤𝑧_𝑛−𝑝≤𝑥_𝑛−𝑝, (3.14) and hence limsup_𝑛→∞‖𝑆𝑧_𝑛−𝑝‖≤𝑟. By (3.4), we also have limsup_𝑛→∞𝛼_𝑛𝑥_𝑛−𝑝+1−𝛼_𝑛𝑆𝑧_𝑛−𝑝=limsup_𝑛→∞𝛼_𝑛𝑥_𝑛−𝑝+1−𝛼_𝑛𝑧_𝑛−𝑝≤limsup_𝑛→∞𝑥_𝑛−𝑝=lim_𝑛→∞𝑥_𝑛−𝑝=𝑟. (3.15) By Lemma 2.4, we obtain lim_𝑛→∞‖𝑆𝑧_𝑛−𝑥_𝑛‖=0.Step 4. We show that lim_𝑛→∞‖𝑢_𝑛−𝑧_𝑛‖=0. Using 𝑧_𝑛=𝑃_𝐶(𝑢_𝑛−𝜆_𝑛𝐹𝑢_𝑛), 𝑝=𝑃_𝐶(𝑝−𝜆_𝑛𝐹𝑝), we compute 𝑧_𝑛−𝑝²≤𝑢_𝑛−𝜆_𝑛𝐹𝑢_𝑛−𝑝−𝜆_𝑛𝐹𝑝²≤𝑢_𝑛−𝑝²−2𝜆_𝑛𝑢_𝑛−𝑝,𝐹𝑢_𝑛−𝐹𝑝+𝜆²_𝑛𝐹𝑢_𝑛−𝐹𝑝²≤𝑥_𝑛−𝑝²+𝜆_𝑛𝜆_𝑛−2𝛼𝐹𝑢_𝑛−𝐹𝑝². (3.16) So, we get 𝑥_𝑛+1−𝑝²=𝑆𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑧_𝑛−𝑆𝑝²≤𝛼𝑥_𝑛−𝑝²+1−𝛼_𝑛𝑧_𝑛−𝑝²≤𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛𝜆_𝑛𝜆_𝑛−2𝛼𝐹𝑢_𝑛−𝐹𝑝²=𝑥_𝑛−𝑝²+1−𝛼_𝑛𝜆_𝑛𝜆_𝑛−2𝛼𝐹𝑢_𝑛−𝐹𝑝². (3.17) From conditions 𝛼_𝑛∈[𝑎,𝑏]⊂(0,1) and 𝜆_𝑛∈[𝑐,𝑑]⊂(0,2𝛼), it follows that (1−𝑏)𝑐(2𝛼−𝑑)𝐹𝑢_𝑛−𝐹𝑝²≤1−𝛼_𝑛𝜆_𝑛2𝛼−𝜆_𝑛𝐹𝑢_𝑛−𝐹𝑝²≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝². (3.18) By 𝑟=lim_𝑛→∞‖𝑥_𝑛−𝑝‖, we obtain lim_𝑛→∞𝐹𝑢_𝑛−𝐹𝑝=0. (3.19) On the other hand, using 𝑧_𝑛=𝑃_𝐶(𝑢_𝑛−𝜆_𝑛𝐹𝑢_𝑛) and (2.3), we observe that 𝑧_𝑛−𝑝²=𝑃_𝐶𝑢_𝑛−𝜆_𝑛𝐹𝑢_𝑛−𝑃_𝐶𝑝−𝜆_𝑛𝐹𝑝²≤𝑢_𝑛−𝜆_𝑛𝐹𝑢_𝑛−𝑝−𝜆_𝑛𝐹𝑝,𝑧_𝑛≤1−𝑝2𝑢_𝑛−𝑝²+𝑧_𝑛−𝑝²−𝑢_𝑛−𝑧_𝑛−𝜆_𝑛𝐹𝑢_𝑛−𝐹𝑝²≤12𝑥_𝑛−𝑝²+𝑧_𝑛−𝑝²−𝑢_𝑛−𝑧_𝑛²+2𝜆_𝑛𝑢_𝑛−𝑧_𝑛,𝐹𝑢_𝑛−𝐹𝑝−𝜆²_𝑛𝐹𝑢_𝑛−𝐹𝑝², (3.20) that is, 𝑧_𝑛−𝑝²≤𝑥_𝑛−𝑝²−𝑢_𝑛−𝑧_𝑛²+2𝜆_𝑛𝑢_𝑛−𝑧_𝑛,𝐹𝑢_𝑛−𝐹𝑝−𝜆²_𝑛𝐹𝑢_𝑛−𝐹𝑝². (3.21) Thus, by (3.21), we have 𝑥_𝑛+1−𝑝²=𝑆𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑧_𝑛−𝑆𝑝²≤𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛𝑧_𝑛−𝑝²≤𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛𝑥_𝑛−𝑝²−1−𝛼_𝑛𝑢_𝑛−𝑧_𝑛²+1−𝛼_𝑛2𝜆_𝑛𝑢_𝑛−𝑧_𝑛,𝐹𝑢_𝑛−𝐹𝑝−𝜆²_𝑛𝐹𝑢_𝑛−𝐹𝑝², (3.22) which implies that (1−𝑏)𝑢_𝑛−𝑧_𝑛²≤1−𝛼_𝑛𝑢_𝑛−𝑧_𝑛²≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+2𝑑(1−𝑎)𝑀₂𝐹𝑢_𝑛−𝐹𝑝, (3.23) where 𝑀₂=sup{‖𝑧_𝑛‖+‖𝑢_𝑛‖∶𝑛≥1}. From lim_𝑛→∞‖𝐹𝑢_𝑛−𝐹𝑝‖=0 in (3.19) and 𝑟=lim_𝑛→∞‖𝑥_𝑛−𝑝‖, we conclude that lim_𝑛→∞‖𝑢_𝑛−𝑧_𝑛‖=0.Step 5. We show that lim_𝑛→∞‖𝑥_𝑛−𝑆𝑥_𝑛‖=0. Indeed, since 𝑆𝑥_𝑛−𝑥_𝑛≤𝑆𝑥_𝑛−𝑆𝑢_𝑛+𝑆𝑢_𝑛−𝑆𝑧_𝑛+𝑆𝑧_𝑛−𝑥_𝑛, (3.24) by Step 2, Step 3, and Step 4, we have lim_𝑛→∞𝑥_𝑛−𝑆𝑥_𝑛=0. (3.25)Step 6. We show that any of its weak cluster point 𝑧 of {𝑥_𝑛} belongs in Ω₁. In this case, there exists a subsequence {𝑥_𝑛𝑖} which converges weakly to 𝑧. By Step 2 and Step 4, without loss of generality, we may assume that {𝑧_𝑛𝑖} converges weakly to 𝑧∈𝐶. Since 𝑆𝑧_𝑛−𝑧_𝑛≤𝑆𝑧_𝑛−𝑥_𝑛+𝑥_𝑛−𝑢_𝑛+𝑢_𝑛−𝑧_𝑛, (3.26) from Step 2, Step 3, and Step 4, it follows that ‖𝑆𝑧_𝑛−𝑧_𝑛‖→0 and 𝑆𝑧_𝑛𝑖⇀𝑧.
We will show that 𝑧∈Ω₁. First we show that 𝑧∈𝐹(𝑆)=𝐹(𝑇). Assume that 𝑧∉𝐹(𝑆). Since 𝑧_𝑛𝑖⇀𝑧 and 𝑆𝑧≠𝑧, by the Opial condition, we obtain liminf_𝑖→∞𝑧_𝑛𝑖−𝑧<liminf_𝑖→∞𝑧_𝑛𝑖−𝑆𝑧≤liminf_𝑖→∞𝑧_𝑛𝑖−𝑆𝑧_𝑛𝑖+𝑆𝑧_𝑛𝑖−𝑆𝑧≤liminf_𝑖→∞𝑧_𝑛𝑖−𝑧, (3.27) which is a contradiction. Thus we have 𝑧∈𝐹(𝑆)=𝐹(𝑇).
Next we prove that 𝑧∈VI(𝐶,𝐹). Let 𝑄𝑣=𝐹𝑣+𝑁_𝐶𝑣,𝑣∈𝐶,∅,𝑣∉𝐶, (3.28) where 𝑁_𝐶𝑣 is normal cone to 𝐶 at 𝑣. We have already known that in this case the mapping 𝑄 is maximal monotone, and 0∈𝑄𝑣 if and only if 𝑣∈VI(𝐶,𝐹). Let (𝑣,𝑤)∈𝐺(𝑄). Since 𝑤−𝐹𝑣∈𝑁_𝐶𝑣 and 𝑧_𝑛∈𝐶, we have 𝑣−𝑧_𝑛,𝑤−𝐹𝑣≥0. (3.29) On the other hand, from 𝑧_𝑛=𝑃_𝐶(𝑢_𝑛−𝜆_𝑛𝐹𝑢_𝑛), we have 𝑣−𝑧_𝑛,𝑧_𝑛−𝑢_𝑛−𝜆_𝑛𝐹𝑢_𝑛≥0, (3.30) that is, 𝑣−𝑧_𝑛,𝑧_𝑛−𝑢_𝑛𝜆_𝑛+𝐹𝑢_𝑛≥0. (3.31) Thus, we obtain ⟨𝑣−𝑧_𝑛𝑖,𝑤⟩≥𝑣−𝑧_𝑛𝑖≥,𝐹𝑣𝑣−𝑧_𝑛𝑖,𝐹𝑣−𝑣−𝑧_𝑛𝑖,𝑧_𝑛𝑖−𝑢_𝑛𝑖𝜆_𝑛𝑖+𝐹𝑢_𝑛𝑖=𝑣−𝑧_𝑛𝑖,𝐹𝑣−𝐹𝑧_𝑛𝑖+𝑣−𝑧_𝑛𝑖,𝐹𝑧_𝑛𝑖−𝐹𝑢_𝑛𝑖−𝑣−𝑧_𝑛𝑖,𝑧_𝑛𝑖−𝑢_𝑛𝑖𝜆_𝑛𝑖≥𝑣−𝑧_𝑛𝑖,𝐹𝑧_𝑛𝑖−𝐹𝑢_𝑛𝑖−𝑣−𝑧_𝑛𝑖,𝑧_𝑛𝑖−𝑢_𝑛𝑖𝜆_𝑛𝑖. (3.32) Since ‖𝑧_𝑛−𝑢_𝑛‖→0 in Step 4 and 𝐹 is 𝛼-inverse-strongly monotone, it follows from (3.32) that ⟨𝑣−𝑧,𝑤⟩≥0,as𝑖⟶∞. (3.33) Since 𝑄 is maximal monotone, we have 𝑧∈𝑄⁻¹0 and hence 𝑧∈VI(𝐶,𝐹).
Finally, we show that 𝑧∈GMEP(Θ,𝜑,𝐵). By 𝑢_𝑛=𝑆_𝑟𝑛(𝑥_𝑛−𝑟_𝑛𝐵𝑥_𝑛), we know that Θ𝑢_𝑛,𝑦+𝐵𝑥_𝑛,𝑦−𝑢_𝑛+𝜑𝑦−𝜑𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛≥0,∀𝑦∈𝐶. (3.34) It follows from (A2) that 𝐵𝑥_𝑛,𝑦−𝑢_𝑛+𝜑𝑦−𝜑𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛≥Θ𝑦,𝑢_𝑛,∀𝑦∈𝐶. (3.35) Hence 𝐵𝑥_𝑛𝑖,𝑦−𝑢_𝑛𝑖+𝜑𝑦−𝜑𝑢_𝑛𝑖+1𝑟_𝑛𝑖𝑦−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖≥Θ𝑦,𝑢_𝑛𝑖,∀𝑦∈𝐶. (3.36) For 𝑡 with 0<𝑡≤1 and 𝑦∈𝐶, let 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑧. Since 𝑦∈𝐶 and 𝑧∈𝐶, we have 𝑦_𝑡∈𝐶 and hence Θ(𝑦_𝑡,𝑧)≤0. So, from (3.36), we have 𝑦_𝑡−𝑢_𝑛𝑖,𝐵𝑦_𝑡≥𝑦_𝑡−𝑢_𝑛𝑖,𝐵𝑦_𝑡−𝜑𝑦_𝑡+𝜑𝑢_𝑛𝑖−𝑦_𝑡−𝑢_𝑛𝑖,𝐵𝑥_𝑛𝑖−𝑦_𝑡−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖𝑟_𝑛𝑖+Θ𝑦_𝑡,𝑢_𝑛𝑖=𝑦_𝑡−𝑢_𝑛𝑖,𝐵𝑦_𝑡−𝐵𝑢_𝑛𝑖+𝑦_𝑡−𝑢_𝑛𝑖,𝐵𝑢_𝑛𝑖−𝐵𝑥_𝑛𝑖−𝜑𝑦_𝑡+𝜑𝑢_𝑛𝑖−𝑦_𝑡−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖𝑟_𝑛𝑖+Θ𝑦_𝑡,𝑢_𝑛𝑖. (3.37) Since ‖𝑢_𝑛𝑖−𝑥_𝑛𝑖‖→0 by Step 2, we have ‖𝐵𝑢_𝑛𝑖−𝐵𝑥_𝑛𝑖‖→0 and ‖(𝑢_𝑛𝑖−𝑥_𝑛𝑖)/𝑟_𝑛𝑖‖≤‖(𝑢_𝑛𝑖−𝑥_𝑛𝑖)/𝑒‖→0, that is, (𝑢_𝑛𝑖−𝑥_𝑛𝑖)/𝑟_𝑛𝑖→0. Also by ‖𝑢_𝑛−𝑧_𝑛‖→0 in Step 4, we have 𝑢_𝑛𝑖⇀𝑧. Moreover, from the inverse-strongly monotonicity of 𝐵, we have ⟨𝑦_𝑡−𝑢_𝑛𝑖,𝐵𝑦_𝑡−𝐵𝑢_𝑛𝑖⟩≥0. So, from (A4) and the weak lower semicontinuity of 𝜑, if follows that 𝑦_𝑡−𝑧,𝐵𝑦_𝑡≥−𝜑𝑦_𝑡+𝜑(𝑧)+Θ𝑦_𝑡,𝑧as𝑖⟶∞. (3.38) By (A1), (A4), and (3.38), we also obtain 0=Θ𝑦_𝑡,𝑦_𝑡+𝜑𝑦_𝑡−𝜑𝑦_𝑡≤𝑡Θ𝑦_𝑡,𝑦+(1−𝑡)Θ𝑦_𝑡,𝑧+𝑡𝜑𝑦_𝑡+(1−𝑡)𝜑(𝑧)−𝜑𝑦_𝑡≤𝑡Θ𝑦_𝑡,𝑦+𝜑𝑦−𝜑𝑦_𝑡+(1−𝑡)𝑦_𝑡−𝑧,𝐵𝑦_𝑡=𝑡Θ𝑦_𝑡,𝑦+𝜑𝑦−𝜑𝑦_𝑡+(1−𝑡)𝑡𝑦−𝑧,𝐵𝑦_𝑡, (3.39) and hence 0≤Θ𝑦_𝑡,𝑦+𝜑𝑦−𝜑𝑦_𝑡+(1−𝑡)𝑦−𝑧,𝐵𝑦_𝑡. (3.40) Letting 𝑡→0 in (3.40), we have for each 𝑦∈𝐶Θ𝑧,𝑦+𝐵𝑧,𝑦−𝑧+𝜑𝑦−𝜑(𝑧)≥0. (3.41) This implies that 𝑧∈GMEP(Θ,𝜑,𝐵). Therefore, we have 𝑧∈Ω₁.
Let {𝑥_𝑛𝑗} be another subsequence of {𝑥_𝑛} such that 𝑥_𝑛𝑗⇀𝑧^′. Then, we have 𝑧^′∈Ω₁. If 𝑧≠𝑧^′, from the Opial condition, we have lim_𝑛→∞𝑥_𝑛−𝑧=liminf_𝑖→∞𝑥_𝑛𝑖−𝑧<liminf_𝑖→∞𝑥_𝑛𝑖−𝑧^′=lim_𝑛→∞𝑥_𝑛−𝑧^′=liminf_𝑗→∞𝑥_𝑛𝑗−𝑧^′<liminf_𝑗→∞𝑥_𝑛𝑗−𝑧=lim_𝑛→∞𝑥_𝑛−𝑧. (3.42) This is a contradiction. So, we have 𝑧=𝑧^′. This implies that 𝑥_𝑛⇀𝑧∈Ω₁. (3.43) Also from Step 2, it follows that 𝑢_𝑛⇀𝑧∈Ω₁.
Let 𝑤_𝑛=𝑃_Ω1(𝑥_𝑛). Since 𝑧∈Ω₁, we have 𝑥_𝑛−𝑤_𝑛,𝑤_𝑛−𝑧≥0. (3.44) Since ‖𝑥_𝑛+1−𝑝‖≤‖𝑥_𝑛−𝑝‖ for 𝑝∈Ω₁, by Lemma 2.5, we have that {𝑤_𝑛} converges strongly to some 𝑧₀∈Ω₁. Since {𝑥_𝑛} converges weakly to 𝑧, we have 𝑧−𝑧₀,𝑧₀−𝑧≥0. (3.45) Therefore, we obtain 𝑧=𝑧₀=lim_𝑛→∞𝑃_Ω1𝑥_𝑛. (3.46) This completes the proof.

As direct consequences of Theorem 3.1, we also obtain the following new weak convergence theorems for the problems (1.2) and (1.3) and fixed point problem of a strict pseudocontractive mapping.

Corollary 3.2. Let 𝐻,𝐶,Θ,𝐵, and 𝐹 be as in Theorem 3.1. Let 𝑇 be a 𝑘-strictly pseudocontractive mapping of 𝐶 into itself for some 𝑘∈[0,1) such that Ω₂∶=𝐹(𝑇)⋂GEP(Θ,𝐵)⋂VI(𝐶,𝐹)≠∅. Assume that either (B1) or (B2) holds. Let {𝑥_𝑛} and {𝑢_𝑛} be sequences generated by 𝑥₁∈𝐶 and Θ𝑢_𝑛,𝑦+𝐵𝑥_𝑛,𝑦−𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=𝑆𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑃_𝐶𝑢_𝑛−𝜆_𝑛𝐹𝑢_𝑛,∀𝑛≥1, (3.47) where 𝑆∶𝐶→𝐶 is a mapping defined by 𝑆𝑥=𝑘𝑥+(1−𝑘)𝑇𝑥 for 𝑥∈𝐶,{𝛼_𝑛}⊂[0,1] and {𝑟_𝑛}⊂(0,∞). Assume that {𝛼_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1),𝜆_𝑛∈[𝑐,𝑑]⊂(0,2𝛼) and 𝑟_𝑛∈[𝑒,𝑓]⊂(0,2𝛽). Then {𝑥_𝑛} and {𝑢_𝑛} converge weakly to 𝑧∈Ω₂, where 𝑧=lim_𝑛→∞𝑃_Ω2(𝑥_𝑛).

Proof. Putting 𝜑≡0 in Theorem 3.1, we obtain the desired result.

Corollary 3.3. Let 𝐻,𝐶,Θ, and 𝐵 be as in Corollary 3.2. Let 𝑇 be a 𝑘-strictly pseudocontractive mapping of 𝐶 into itself for some 𝑘∈[0,1) such that Ω₃∶=𝐹(𝑇)⋂GEP(Θ,𝐵)≠∅. Assume that either (B1) or (B2) holds. Let {𝑥_𝑛} and {𝑢_𝑛} be sequences generated by 𝑥₁∈𝐶 and Θ𝑢_𝑛,𝑦+𝐵𝑥_𝑛,𝑦−𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=𝑆𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑢_𝑛,∀𝑛≥1, (3.48) where 𝑆∶𝐶→𝐶 is a mapping defined by 𝑆𝑥=𝑘𝑥+(1−𝑘)𝑇𝑥 for 𝑥∈𝐶,{𝛼_𝑛}⊂[0,1] and {𝑟_𝑛}⊂(0,∞). Assume that {𝛼_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1) and 𝑟_𝑛∈[𝑒,𝑓]⊂(0,2𝛽). Then {𝑥_𝑛} and {𝑢_𝑛} converge weakly to 𝑧∈Ω₃, where 𝑧=lim_𝑛→∞𝑃_Ω3(𝑥_𝑛).

Proof. Putting 𝐹≡0 in Corollary 3.2, we obtain the desired result.

Corollary 3.4. Let 𝐻,𝐶,Θ,𝜑, and 𝐹 be as in Theorem 3.1. Let 𝑇 be a 𝑘-strictly pseudocontractive mapping of 𝐶 into itself for some 𝑘∈[0,1) such that Ω₄∶=𝐹(𝑇)⋂MEP(Θ,𝜑)⋂VI(𝐶,𝐹)≠∅. Assume that either (B1) or (B2) holds. Let {𝑥_𝑛} and {𝑢_𝑛} be sequences generated by 𝑥₁∈𝐶 and Θ𝑢_𝑛,𝑦+𝜑𝑦−𝜑𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=𝑆𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑃_𝐶𝑢_𝑛−𝜆_𝑛𝐹𝑢_𝑛,∀𝑛≥1, (3.49) where 𝑆∶𝐶→𝐶 is a mapping defined by 𝑆𝑥=𝑘𝑥+(1−𝑘)𝑇𝑥 for 𝑥∈𝐶,{𝛼_𝑛}⊂[0,1] and {𝑟_𝑛}⊂(0,∞). Assume that {𝛼_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1),𝜆_𝑛∈[𝑐,𝑑]⊂(0,2𝛼) and 𝑟_𝑛∈[𝑒,𝑓]⊂(0,∞). Then {𝑥_𝑛} and {𝑢_𝑛} converge weakly to 𝑧∈Ω₄, where 𝑧=lim_𝑛→∞𝑃_Ω4(𝑥_𝑛).

Proof. Putting 𝐵≡0 in Theorem 3.1, we obtain the desired result.

Corollary 3.5. Let 𝐻,𝐶,Θ and 𝜑 be as in Theorem 3.1. Let 𝑇 be a 𝑘-strictly pseudocontractive mapping of 𝐶 into itself for some 𝑘∈[0,1) such that Ω₅∶=𝐹(𝑇)⋂MEP(Θ,𝜑)≠∅. Assume that either (B1) or (B2) holds. Let {𝑥_𝑛} and {𝑢_𝑛} be sequences generated by 𝑥₁∈𝐶 and Θ𝑢_𝑛,𝑦+𝜑𝑦−𝜑𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=𝑆𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑢_𝑛,∀𝑛≥1, (3.50) where 𝑆∶𝐶→𝐶 is a mapping defined by 𝑆𝑥=𝑘𝑥+(1−𝑘)𝑇𝑥 for 𝑥∈𝐶,{𝛼_𝑛}⊂[0,1] and {𝑟_𝑛}⊂(0,∞). Assume that {𝛼_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1) and 𝑟_𝑛∈[𝑒,𝑓]⊂(0,∞). Then {𝑥_𝑛} and {𝑢_𝑛} converge weakly to 𝑧∈Ω₅, where 𝑧=lim_𝑛→∞𝑃_Ω5(𝑥_𝑛).

Proof. Putting 𝐹≡0 in Corollary 3.4, we obtain the desired result.

Remark 3.6. (1)As a new result for a new iterative scheme, Theorem 3.1 develops and complements the corresponding results, which were obtained recently by many authors in references and others; for example, see [22–24, 26]. In particular, even though 𝐹≡0 in Theorem 3.1, Theorem 3.1 develops and complements Theorem 3.1 of Ceng et al. [22] in the following aspects:(a)the iterative scheme (3.1) in Theorem 3.1 is a new one different from those in Theorem 3.1 of [22].(b)the equilibrium problem in Theorem 3.1 of [22] is extended to the case of generalized mixed equilibrium problem.(2)We point out that our iterative schemes in Corollaries 3.2, 3.3, 3.4 and 3.5 are new ones different from those in the literature (see [22–24, 26] and others in references).

Acknowledgments

The author thanks the referees for their valuable comments and suggests, which improved the presentation of this paper, and for providing some recent related papers. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0003901).