Strong Convergence to Solutions of Generalized Mixed Equilibrium Problems with Applications

Abstract

We introduce a Halpern-type iteration for a generalized mixed equilibrium problem in uniformly smooth and uniformly convex Banach spaces. Strong convergence theorems are also established in this paper. As applications, we apply our main result to mixed equilibrium, generalized equilibrium, and mixed variational inequality problems in Banach spaces. Finally, examples and numerical results are also given.

1. Introduction

Let 𝐸 be a real Banach space, 𝐶 a nonempty, closed, and convex subset of 𝐸, and 𝐸^∗ the dual space of 𝐸. Let 𝑇∶𝐶→𝐶 be a nonlinear mapping. The fixed points set of 𝑇 is denoted by 𝐹(𝑇), that is, 𝐹(𝑇)={𝑥∈𝐶∶𝑥=𝑇𝑥}.

One classical way often used to approximate a fixed point of a nonlinear self-mapping 𝑇 on 𝐶 was firstly introduced by Halpern [1] which is defined by 𝑥₁=𝑥∈𝐶 and 𝑥_𝑛+1=𝛼_𝑛𝑥+1−𝛼_𝑛𝑇𝑥_𝑛,∀𝑛≥1, (1.1) where {𝛼_𝑛} is a real sequence in [0,1]. He proved, in a real Hilbert space, a strong convergence theorem for a nonexpansive mapping 𝑇 when 𝛼_𝑛=𝑛^−𝑎 for any 𝑎∈(0,1).

Subsequently, motivated by Halpern [1], many mathematicians devoted time to study algorithm (1.1) in different styles. Several strong convergence results for nonlinear mappings were also continuously established in some certain Banach spaces (see also [2–9]).

Let 𝑓∶𝐶×𝐶→ℝ be a bifunction, 𝐴∶𝐶→𝐸^∗ a mapping, and 𝜑∶𝐶→ℝ a real-valued function. The generalized mixed equilibrium problem is to find 𝑥∈𝐶 such that 𝑓𝑥,𝑦+𝐴𝑥,𝑦−𝑥+𝜑𝑦≥𝜑(𝑥),∀𝑦∈𝐶. (1.2) The solutions set of (1.2) is denoted by GMEP(𝑓,𝐴,𝜑) (see Peng and Yao [10]).

If 𝐴≡0, then the generalized mixed equilibrium problem (1.2) reduces to the following mixed equilibrium problem: finding 𝑥∈𝐶 such that 𝑓𝑥,𝑦+𝜑𝑦≥𝜑(𝑥),∀𝑦∈𝐶. (1.3) The solutions set of (1.3) is denoted by MEP(𝑓,𝜑) (see Ceng and Yao [11]).

If 𝑓≡0, then the generalized mixed equilibrium problem (1.2) reduces to the following mixed variational inequality problem: finding 𝑥∈𝐶 such that 𝐴𝑥,𝑦−𝑥+𝜑𝑦≥𝜑(𝑥),∀𝑦∈𝐶. (1.4) The solutions set of (1.4) is denoted by VI(𝐶,𝐴,𝜑) (see Noor [12]).

If 𝜑≡0, then the generalized mixed equilibrium problem (1.2) reduces to the following generalized equilibrium problem: finding 𝑥∈𝐶 such that 𝑓𝑥,𝑦+𝐴𝑥,𝑦−𝑥≥0,∀𝑦∈𝐶. (1.5) The solutions set of (1.5) is denoted by GEP(𝑓,𝐴) (see Moudafi [13]).

If 𝜑≡0, then the mixed equilibrium problem (1.3) reduces to the following equilibrium problem: finding 𝑥∈𝐶 such that 𝑓𝑥,𝑦≥0,∀𝑦∈𝐶. (1.6) The solutions set of (1.6) is denoted by EP(𝑓) (see Combettes and Hirstoaga [14]).

If 𝑓≡0, then the mixed equilibrium problem (1.3) reduces to the following convex minimization problem: finding 𝑥∈𝐶 such that 𝜑𝑦≥𝜑(𝑥),∀𝑦∈𝐶. (1.7) The solutions set of (1.7) is denoted by CMP(𝜑).

If 𝜑≡0, then the mixed variational inequality problem (1.4) reduces to the following variational inequality problem: finding 𝑥∈𝐶 such that 𝐴𝑥,𝑦−𝑥≥0,∀𝑦∈𝐶. (1.8) The solutions set of (1.8) is denoted by VI(𝐶,𝐴) (see Stampacchia [7]).

The problem (1.2) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others. For more details on these topics, see, for instance, [14–34].

For solving the generalized mixed equilibrium problem, let us assume the following [25]:(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 all 𝑥,𝑦,𝑧∈𝐶, limsup_𝑡↓0𝑓(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝑓(𝑥,𝑦);(A4) for all 𝑥∈𝐶,𝑓(𝑥,⋅) is convex and lower semicontinuous.

The purpose of this paper is to investigate strong convergence of Halpern-type iteration for a generalized mixed equilibrium problem in uniformly smooth and uniformly convex Banach spaces. As applications, our main result can be deduced to mixed equilibrium, generalized equilibrium, mixed variational inequality problems, and so on. Examples and numerical results are also given in the last section.

2. Preliminaries and Lemmas

In this section, we need the following preliminaries and lemmas which will be used in our main theorem.

Let 𝐸 be a real Banach space and let 𝑈={𝑥∈𝐸∶‖𝑥‖=1} be the unit sphere of 𝐸. A Banach space 𝐸 is said to be strictly convex if, for any 𝑥,𝑦∈𝑈, 𝑥≠𝑦implies𝑥+𝑦2<1. (2.1) It is also said to be uniformly convex if, for any 𝜀∈(0,2], there exists 𝛿>0 such that, for any 𝑥,𝑦∈𝑈, 𝑥−𝑦≥𝜀implies𝑥+𝑦2<1−𝛿. (2.2) It is known that a uniformly convex Banach space is reflexive and strictly convex. Define a function 𝛿∶[0,2]→[0,1] called the modulus of convexity of 𝐸 as follows: 𝛿(𝜀)=inf1−𝑥+𝑦2∶𝑥,𝑦∈𝐸,‖𝑥‖=𝑦=1,𝑥−𝑦≥𝜀. (2.3) Then 𝐸 is uniformly convex if and only if 𝛿(𝜀)>0 for all 𝜀∈(0,2]. A Banach space 𝐸 is said to be smooth if the limit lim_𝑡→0𝑥+𝑡𝑦−‖𝑥‖𝑡 (2.4) exists for all 𝑥,𝑦∈𝑈. It is also said to be uniformly smooth if the limit (2.4) is attained uniformly for 𝑥,𝑦∈𝑈. The normalized duality mapping 𝐽∶𝐸→2^𝐸∗ is defined by 𝐽(𝑥)=𝑥^∗∈𝐸^∗∶𝑥,𝑥^∗=‖𝑥‖²=𝑥^∗2 (2.5) for all 𝑥∈𝐸. It is also known that if 𝐸 is uniformly smooth, then 𝐽 is uniformly norm-to-norm continuous on each bounded subset of 𝐸 (see [35]).

Let 𝐸 be a smooth Banach space. The function 𝜙∶𝐸×𝐸→ℝ is defined by 𝜙𝑥,𝑦=‖𝑥‖²−2𝑥,𝐽𝑦+𝑦²,∀𝑥,𝑦∈𝐸. (2.6)

Remark 2.1. We know the following: for any 𝑥,𝑦,𝑧∈𝐸,(1)(‖𝑥‖−‖𝑦‖)²≤𝜙(𝑥,𝑦)≤(‖𝑥‖+‖𝑦‖)²;(2)𝜙(𝑥,𝑦)=𝜙(𝑥,𝑧)+𝜙(𝑧,𝑦)+2⟨𝑥−𝑧,𝐽𝑧−𝐽𝑦⟩;(3)𝜙(𝑥,𝑦)=‖𝑥−𝑦‖² in a real Hilbert space.

Lemma 2.2 (see [36]). Let 𝐸 be a uniformly convex and smooth Banach space and let {𝑥_𝑛} and {𝑦_𝑛} be sequences of 𝐸 such that {𝑥_𝑛} or {𝑦_𝑛} is bounded and lim_𝑛→∞𝜙(𝑥_𝑛,𝑦_𝑛)=0. Then lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0.

Let 𝐸 be a reflexive, strictly convex, and smooth Banach space and let 𝐶 be a nonempty closed and convex subset of 𝐸. The generalized projection mapping, introduced by Alber [37], is a mapping Π_𝐶∶𝐸→𝐶, that assigns to an arbitrary point 𝑥∈𝐸 the minimum point of the functional 𝜙(𝑦,𝑥), that is, Π_𝐶𝑥=𝑥, where 𝑥 is the solution to the minimization problem: 𝜙(𝑥,𝑥)=min𝜙𝑦,𝑥∶𝑦∈𝐶. (2.7)

In fact, we have the following result.

Lemma 2.3 (see [37]). Let 𝐶 be a nonempty, closed, and convex subset of a reflexive, strictly convex, and smooth Banach space 𝐸 and let 𝑥∈𝐸. Then there exists a unique element 𝑥₀∈𝐶 such that 𝜙(𝑥₀,𝑥)=min{𝜙(𝑧,𝑥)∶𝑧∈𝐶}.

Lemma 2.4 (see [36, 37]). Let 𝐶 be a nonempty closed and convex subset of a reflexive, strictly convex, and smooth Banach space 𝐸, 𝑥∈𝐸, and 𝑧∈𝐶. Then 𝑧=Π_𝐶𝑥 if and only if 𝐽𝑥−𝐽𝑧,𝑦−𝑧≤0,∀𝑦∈𝐶. (2.8)

Lemma 2.5 (see [36, 37]). Let 𝐶 be a nonempty closed and convex subset of a reflexive, strictly convex, and smooth Banach space 𝐸 and let 𝑥∈𝐸. Then 𝜙𝑦,Π_𝐶𝑥+𝜙Π_𝐶𝑥,𝑥≤𝜙𝑦,𝑥,∀𝑦∈𝐶. (2.9)

Lemma 2.6 (see [38]). Let 𝐸 be a uniformly convex and uniformly smooth Banach space and 𝐶 a nonempty, closed, and convex subset of E. Then Π_𝐶 is uniformly norm-to-norm continuous on every bounded set.

We make use of the following mapping 𝑉 studied in Alber [37]: 𝑉𝑥,𝑥^∗=‖𝑥‖²−2𝑥,𝑥^∗+𝑥^∗2 (2.10) for all 𝑥∈𝐸 and 𝑥^∗∈𝐸^∗, that is, 𝑉(𝑥,𝑥^∗)=𝜙(𝑥,𝐽⁻¹(𝑥^∗)).

Lemma 2.7 (see [39]). Let 𝐸 be a reflexive, strictly convex, smooth Banach space. Then 𝑉𝑥,𝑥^∗+2𝐽⁻¹𝑥^∗−𝑥,𝑦^∗≤𝑉𝑥,𝑥^∗+𝑦^∗ (2.11) for all 𝑥∈𝐸 and 𝑥^∗,𝑦^∗∈𝐸^∗.

Lemma 2.8 (see [25]). Let 𝐶 be a closed and convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸, let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ which satisfies conditions (A1)–(A4), and let 𝑟>0 and 𝑥∈𝐸. Then there exists 𝑧∈𝐶 such that 𝑓+1𝑧,𝑦𝑟𝐽𝑧−𝐽𝑥,𝑦−𝑧≥0,∀𝑦∈𝐶. (2.12)

Following [25, 40], we know the following lemma.

Lemma 2.9 (see [41]). Let 𝐶 be a nonempty closed and convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸. Let 𝐴∶𝐶→𝐸^∗ be a continuous and monotone mapping, let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4), and let 𝜑 be a lower semicontinuous and convex function from 𝐶 to ℝ. For all 𝑟>0 and 𝑥∈𝐸, there exists 𝑧∈𝐶 such that 𝑓+𝑦+1𝑧,𝑦𝐴𝑧,𝑦−𝑧+𝜑𝑟𝐽𝑧−𝐽𝑥,𝑦−𝑧≥𝜑(𝑧),∀𝑦∈𝐶. (2.13) Define the mapping 𝑇_𝑟∶𝐸→2^𝐶 as follows: 𝑇_𝑟(𝑥)=+𝑦+1𝑧∈𝐶∶𝑓𝑧,𝑦𝐴𝑧,𝑦−𝑧+𝜑𝑟𝐽𝑧−𝐽𝑥,𝑦−𝑧≥𝜑(𝑧),∀𝑦∈𝐶. (2.14) Then, the followings hold:(1)𝑇_𝑟 is single-valued;(2)𝑇_𝑟 is firmly nonexpansive-type mapping [42], that is, for all 𝑥,𝑦∈𝐸, 𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝐽𝑇_𝑟𝑥−𝐽𝑇_𝑟𝑦≤𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝐽𝑥−𝐽𝑦; (2.15)(3)𝐹(𝑇_𝑟)=GMEP(𝑓,𝐴,𝜑);(4)GMEP(𝑓,𝐴,𝜑) is closed and convex.

Remark 2.10. It is known that 𝑇 is of firmly nonexpansive type if and only if 𝜙𝑇𝑥,𝑇𝑦+𝜙𝑇𝑦,𝑇𝑥+𝜙(𝑇𝑥,𝑥)+𝜙𝑇𝑦,𝑦≤𝜙𝑇𝑥,𝑦+𝜙𝑇𝑦,𝑥 (2.16) for all 𝑥,𝑦∈ dom 𝑇 (see [42]).

The following lemmas give us some nice properties of real sequences.

Lemma 2.11 (see [43]). Assume that {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛼_𝑛𝑎_𝑛+𝑏_𝑛,∀𝑛≥1, (2.17) where {𝛼_𝑛} is a sequence in (0,1) and {𝑏_𝑛} is a sequence such that(a)∑^∞_𝑛=1𝛼_𝑛=+∞;(b)limsup_𝑛→∞𝑏_𝑛/𝛼_𝑛≤0 or ∑^∞_𝑛=1|𝑏_𝑛|<+∞.Then lim_𝑛→∞𝑎_𝑛=0.

Lemma 2.12 (see [44]). Let {𝛾_𝑛} be a sequence of real numbers such that there exists a subsequence {𝛾_𝑛𝑗} of {𝛾_𝑛} such that 𝛾_𝑛𝑗<𝛾_𝑛𝑗+1 for all 𝑗≥1. Then there exists a nondecreasing sequence {𝑚_𝑘} of ℕ such that lim_𝑘→∞𝑚_𝑘=∞ and the following properties are satisfied by all (sufficiently large) numbers 𝑘≥1: 𝛾_𝑚𝑘≤𝛾_𝑚𝑘+1,𝛾_𝑘≤𝛾_𝑚𝑘+1. (2.18) In fact, 𝑚_𝑘 is the largest number 𝑛 in the set {1,2,…,𝑘} such that the condition 𝛾_𝑛<𝛾_𝑛+1 holds.

3. Main Results

In this section, we prove our main theorem in this paper. To this end, we need the following proposition.

Proposition 3.1. Let 𝐶 be a nonempty closed and convex subset of a reflexive, strictly convex, and uniformly smooth Banach space 𝐸. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4), 𝐴∶𝐶→𝐸^∗ a continuous and monotone mapping, and 𝜑 a lower semicontinuous and convex function from 𝐶 to ℝ such that GMEP(𝑓,𝐴,𝜑)≠∅. Let {𝑟_𝑛}⊂(0,∞) be such that liminf_𝑛→∞𝑟_𝑛>0. For each 𝑛≥1, let 𝑇_𝑟𝑛 be defined as in Lemma 2.9. Suppose that 𝑥∈𝐶 and {𝑥_𝑛} is a bounded sequence in 𝐶 such that lim_𝑛→∞‖𝑥_𝑛−𝑇_𝑟𝑛𝑥_𝑛‖=0. Then limsup_𝑛→∞𝐽𝑥−𝐽𝑝,𝑥_𝑛−𝑝≤0, (3.1) where 𝑝=Π_{GMEP(𝑓,𝐴,𝜑)}𝑥 and Π_{GMEP(𝑓,𝐴,𝜑)} is the generalized projection of 𝐶 onto GMEP(𝑓,𝐴,𝜑).

Proof. Let 𝑥∈𝐶 and put 𝑝=Π_{GMEP(𝑓,𝐴,𝜑)}𝑥. Since 𝐸 is reflexive and {𝑥_𝑛} is bounded, there exists a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that 𝑥_𝑛𝑘⇀𝑣∈𝐶 and limsup_𝑛→∞𝐽𝑥−𝐽𝑝,𝑥_𝑛−𝑝=𝐽𝑥−𝐽𝑝,𝑣−𝑝. (3.2) Put 𝑦_𝑛=𝑇_𝑟𝑛𝑥_𝑛. Since lim_𝑘→∞‖𝑥_𝑛𝑘−𝑦_𝑛𝑘‖=0, we have 𝑦_𝑛𝑘⇀𝑣. On the other hand, since 𝐸 is uniformly smooth, 𝐽 is uniformly norm-to-norm continuous on bounded subsets of 𝐸. So we have lim_𝑘→∞𝐽𝑥_𝑛𝑘−𝐽𝑦_𝑛𝑘=0. (3.3) Since liminf_𝑘→∞𝑟_𝑛𝑘>0, lim_𝑘→∞𝐽𝑥_𝑛𝑘−𝐽𝑦_𝑛𝑘𝑟_𝑛𝑘=0. (3.4) By the definition of 𝑇_𝑟𝑛𝑘, for any 𝑦∈𝐶, we see that 𝑓𝑦_𝑛𝑘,𝑦+𝐴𝑦_𝑛𝑘,𝑦−𝑦_𝑛𝑘𝑦+1+𝜑𝑟_𝑛𝑘𝐽𝑦_𝑛𝑘−𝐽𝑥_𝑛𝑘,𝑦−𝑦_𝑛𝑘≥𝜑𝑦_𝑛𝑘. (3.5) By (A2), for each 𝑦∈𝐶, we obtain 𝑓𝑦,𝑦_𝑛𝑘+𝜑𝑦_𝑛𝑘≤−𝑓𝑦_𝑛𝑘,𝑦+𝜑𝑦_𝑛𝑘≤𝐴𝑦_𝑛𝑘,𝑦−𝑦_𝑛𝑘𝑦+1+𝜑𝑟_𝑛𝑘𝐽𝑦_𝑛𝑘−𝐽𝑥_𝑛𝑘,𝑦−𝑦_𝑛𝑘. (3.6) For any 𝑡∈(0,1) and 𝑦∈𝐶, we define 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑣. Then 𝑦_𝑡∈𝐶. It follows by the monotonicity of 𝐴 that 𝑓𝑦_𝑡,𝑦_𝑛𝑘+𝜑𝑦_𝑛𝑘≤𝐴𝑦_𝑛𝑘−𝐴𝑦_𝑡,𝑦_𝑡−𝑦_𝑛𝑘+𝐴𝑦_𝑡,𝑦_𝑡−𝑦_𝑛𝑘+𝜑𝑦_𝑡+1𝑟_𝑛𝑘𝐽𝑦_𝑛𝑘−𝐽𝑥_𝑛𝑘,𝑦_𝑡−𝑦_𝑛𝑘≤𝐴𝑦_𝑡,𝑦_𝑡−𝑦_𝑛𝑘+𝜑𝑦_𝑡+1𝑟_𝑛𝑘𝐽𝑦_𝑛𝑘−𝐽𝑥_𝑛𝑘,𝑦_𝑡−𝑦_𝑛𝑘. (3.7) By (A4), (3.4), and the weakly lower semicontinuity of 𝜑, letting 𝑘→∞, we obtain 𝑓𝑦_𝑡,𝑣+𝜑(𝑣)≤𝐴𝑦_𝑡,𝑦_𝑡−𝑣+𝜑𝑦_𝑡. (3.8) By (A1), (A4), and the convexity of 𝜑, we have 0=𝑓𝑦_𝑡,𝑦_𝑡+𝜑𝑦_𝑡−𝜑𝑦_𝑡≤𝑡𝑓𝑦_𝑡,𝑦+(1−𝑡)𝑓𝑦_𝑡,𝑣+𝑡𝜑𝑦+(1−𝑡)𝜑(𝑣)−𝜑𝑦_𝑡=𝑡𝑓𝑦_𝑡,𝑦+𝜑𝑦−𝜑𝑦_𝑡+(1−𝑡)𝑓𝑦_𝑡,𝑣+𝜑(𝑣)−𝜑𝑦_𝑡≤𝑡𝑓𝑦_𝑡,𝑦+𝜑𝑦−𝜑𝑦_𝑡+(1−𝑡)⟨𝐴𝑦_𝑡,𝑦_𝑡𝑓𝑦−𝑣⟩=𝑡_𝑡,𝑦+𝜑𝑦−𝜑𝑦_𝑡+(1−𝑡)𝑡⟨𝐴𝑦_𝑡,𝑦−𝑣⟩. (3.9) It follows that 𝑓𝑦_𝑡,𝑦+𝜑𝑦−𝜑𝑦_𝑡+(1−𝑡)𝐴𝑦_𝑡,𝑦−𝑣≥0. (3.10) By (A3), the weakly lower semicontinuity of 𝜑, and the continuity of 𝐴, letting 𝑡→0, we obtain 𝑓𝑣,𝑦+𝜑𝑦−𝜑(𝑣)+𝐴𝑣,𝑦−𝑣≥0,∀𝑦∈𝐶. (3.11) This shows that 𝑣∈GMEP(𝑓,𝐴,𝜑). By Lemma 2.4, we have limsup_𝑛→∞𝐽𝑥−𝐽𝑝,𝑥_𝑛−𝑝=𝐽𝑥−𝐽𝑝,𝑣−𝑝≤0. (3.12) This completes the proof.

Theorem 3.2. Let 𝐶 be nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach space 𝐸. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4), 𝐴∶𝐶→𝐸^∗ a continuous and monotone mapping, and 𝜑 a lower semicontinuous and convex function from 𝐶 to ℝ such that GMEP(𝑓,𝐴,𝜑)≠∅. Define the sequence {𝑥_𝑛} as follows: 𝑥₁=𝑥∈𝐶 and 𝑓𝑦_𝑛,𝑦+𝐴𝑦_𝑛,𝑦−𝑦_𝑛𝑦+1+𝜑𝑟_𝑛𝐽𝑦_𝑛−𝐽𝑥_𝑛,𝑦−𝑦_𝑛≥𝜑𝑦_𝑛𝑥,∀𝑦∈𝐶,_𝑛+1=Π_𝐶𝐽⁻¹𝛼_𝑛𝐽𝑥+1−𝛼_𝑛𝐽𝑦_𝑛,∀𝑛≥1, (3.13) where {𝛼_𝑛}⊂(0,1) and {𝑟_𝑛}⊂(0,∞) satisfy the following conditions:(a)lim_𝑛→∞𝛼_𝑛=0;(b)∑^∞_𝑛=1𝛼_𝑛=∞;(c)liminf_𝑛→∞𝑟_𝑛>0.Then {𝑥_𝑛} converges strongly to Π_{GMEP(𝑓,𝐴,𝜑)}𝑥, where Π_{GMEP(𝑓,𝐴,𝜑)} is the generalized projection of 𝐶 onto GMEP(𝑓,𝐴,𝜑).

Proof. From Lemma 2.9(4), we know that GMEP(𝑓,𝐴,𝜑) is closed and convex. Let 𝑝=Π_{GMEP(𝑓,𝐴,𝜑)}𝑥. Put 𝑦_𝑛=𝑇_𝑟𝑛𝑥_𝑛 and 𝑧_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥+(1−𝛼_𝑛)𝐽𝑦_𝑛) for all 𝑛∈ℕ. So, by Lemma 2.5, we have 𝜙𝑝,𝑥_𝑛+1≤𝜙𝑝,𝑧_𝑛≤𝛼_𝑛𝜙𝑝,𝑥+1−𝛼_𝑛𝜙𝑝,𝑦_𝑛≤𝛼_𝑛𝜙𝑝,𝑥+1−𝛼_𝑛𝜙𝑝,𝑥_𝑛. (3.14) By induction, we can show that 𝜙(𝑝,𝑥_𝑛)≤𝜙(𝑝,𝑥) for each 𝑛∈ℕ. Hence {𝜙(𝑝,𝑥_𝑛)} is bounded and thus {𝑥_𝑛} is also bounded.
We next show that if there exists a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that lim_𝑘→∞𝜙𝑝,𝑥_𝑛𝑘+1−𝜙𝑝,𝑥_𝑛𝑘=0, (3.15) then lim_𝑘→∞𝜙𝑝,𝑦_𝑛𝑘−𝜙𝑝,𝑥_𝑛𝑘=0. (3.16) Since 𝛼_𝑛𝑘→0, lim_𝑘→∞𝐽𝑧_𝑛𝑘−𝐽𝑦_𝑛𝑘=lim_𝑘→∞𝛼_𝑛𝑘𝐽𝑥−𝐽𝑦_𝑛𝑘=0. (3.17) Since 𝐽 is uniformly norm-to-norm continuous on bounded subsets of 𝐸, so is 𝐽⁻¹. It follows that lim_𝑘→∞𝑧_𝑛𝑘−𝑦_𝑛𝑘=0. (3.18) Since 𝐸 is uniformly smooth and uniformly convex, by Lemma 2.6, Π_𝐶 is uniformly norm-to-norm continuous on bounded sets. So we obtain lim_𝑘→∞𝑥_𝑛𝑘+1−𝑦_𝑛𝑘=lim_𝑘→∞Π_𝐶𝑧_𝑛𝑘−Π_𝐶𝑦_𝑛𝑘=0, (3.19) and hence lim_𝑘→∞𝐽𝑥_𝑛𝑘+1−𝐽𝑦_𝑛𝑘=0. (3.20) Furthermore, lim_𝑘→∞𝜙(𝑥_𝑛𝑘+1,𝑦_𝑛𝑘)=0. Indeed, by the definition of 𝜙, we observe that 𝜙𝑥_𝑛𝑘+1,𝑦_𝑛𝑘=𝑥_𝑛𝑘+1²−2𝑥_𝑛𝑘+1,𝐽𝑦_𝑛𝑘+𝑦_𝑛𝑘²=𝑥_𝑛𝑘+1,𝐽𝑥_𝑛𝑘+1−𝐽𝑦_𝑛𝑘+𝑦_𝑛𝑘−𝑥_𝑛𝑘+1,𝐽𝑦_𝑛𝑘. (3.21) It follows from (3.19) and (3.20) that lim_𝑘→∞𝜙(𝑥_𝑛𝑘+1,𝑦_𝑛𝑘)=0. On the other hand, from Remark 2.1(2), we have 𝜙𝑝,𝑦_𝑛𝑘−𝜙𝑝,𝑥_𝑛𝑘=𝜙𝑝,𝑥_𝑛𝑘+1−𝜙𝑝,𝑥_𝑛𝑘+𝜙𝑝,𝑦_𝑛𝑘−𝜙𝑝,𝑥_𝑛𝑘+1=𝜙𝑝,𝑥_𝑛𝑘+1−𝜙𝑝,𝑥_𝑛𝑘+𝜙𝑥_𝑛𝑘+1,𝑦_𝑛𝑘+2𝑝−𝑥_𝑛𝑘+1,𝐽𝑥_𝑛𝑘+1−𝐽𝑦_𝑛𝑘. (3.22) It follows from (3.20) and (3.21) thatlim_𝑘→∞(𝜙(𝑝,𝑦_𝑛𝑘)−𝜙(𝑝,𝑥_𝑛𝑘))=0.
We next consider the following two cases.
Case 1. 𝜙(𝑝,𝑥_𝑛+1)≤𝜙(𝑝,𝑥_𝑛) for all sufficiently large 𝑛. Hence the sequence {𝜙(𝑝,𝑥_𝑛)} is bounded and nonincreasing. So lim_𝑛→∞𝜙(𝑝,𝑥_𝑛) exists. This shows that lim_𝑛→∞(𝜙(𝑝,𝑥_𝑛+1)−𝜙(𝑝,𝑥_𝑛))=0 and hence lim_𝑛→∞𝜙𝑝,𝑦_𝑛−𝜙𝑝,𝑥_𝑛=0. (3.23) Since 𝑇_𝑟𝑛 is of firmly nonexpansive type, by Remark 2.10, we have 𝜙𝑦_𝑛,𝑝+𝜙𝑝,𝑦_𝑛+𝜙𝑦_𝑛,𝑥_𝑛+𝜙𝑇_𝑟𝑛𝑝,𝑝≤𝜙𝑦_𝑛,𝑝+𝜙𝑝,𝑥_𝑛, (3.24) which implies 𝜙𝑝,𝑦_𝑛+𝜙𝑦_𝑛,𝑥_𝑛≤𝜙𝑝,𝑥_𝑛. (3.25) Hence 𝜙𝑦_𝑛,𝑥_𝑛≤𝜙𝑝,𝑥_𝑛−𝜙𝑝,𝑦_𝑛⟶0 (3.26) as 𝑛→∞. By Lemma 2.2, we obtain lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=0. (3.27) Proposition 3.1 yields that limsup_𝑛→∞𝐽𝑥−𝐽𝑝,𝑥_𝑛−𝑝≤0. (3.28) It also follows that limsup_𝑛→∞𝐽𝑥−𝐽𝑝,𝑦_𝑛−𝑝≤0. (3.29)
Finally, we show that 𝑥_𝑛→𝑝. Using Lemma 2.7, we see that 𝜙𝑝,𝑥_𝑛+1≤𝜙𝑝,𝑧_𝑛=𝑉𝑝,𝛼_𝑛𝐽𝑥+1−𝛼_𝑛𝐽𝑦_𝑛≤𝑉𝑝,𝛼_𝑛𝐽𝑥+1−𝛼_𝑛𝐽𝑦_𝑛−𝛼_𝑛𝐽𝑥−𝐽𝑝+𝛼_𝑛𝐽𝑥−𝐽𝑝,𝑧_𝑛−𝑝=𝑉𝑝,𝛼_𝑛𝐽𝑝+1−𝛼_𝑛𝐽𝑦_𝑛+𝛼_𝑛𝐽𝑥−𝐽𝑝,𝑧_𝑛−𝑝≤𝛼_𝑛𝑉𝑝,𝐽𝑝+1−𝛼_𝑛𝑉𝑝,𝐽𝑦_𝑛+𝛼_𝑛𝐽𝑥−𝐽𝑝,𝑧_𝑛=−𝑝1−𝛼_𝑛𝜙𝑝,𝑦_𝑛+𝛼_𝑛𝐽𝑥−𝐽𝑝,𝑧_𝑛≤−𝑝1−𝛼_𝑛𝜙𝑝,𝑥_𝑛+𝛼_𝑛𝐽𝑥−𝐽𝑝,𝑧_𝑛=−𝑝1−𝛼_𝑛𝜙𝑝,𝑥_𝑛+𝛼_𝑛𝐽𝑥−𝐽𝑝,𝑧_𝑛−𝑦_𝑛+𝐽𝑥−𝐽𝑝,𝑦_𝑛−𝑝. (3.30) Set 𝑎_𝑛=𝜙(𝑝,𝑥_𝑛) and 𝑏_𝑛=𝛼_𝑛(⟨𝐽𝑥−𝐽𝑝,𝑧_𝑛−𝑦_𝑛⟩+⟨𝐽𝑥−𝐽𝑝,𝑦_𝑛−𝑝⟩). We see that limsup_𝑛→∞𝑏_𝑛/𝛼_𝑛≤0. By Lemma 2.11, since ∑^∞_𝑛=1𝛼_𝑛=+∞, we conclude that lim_𝑛→∞𝜙(𝑝,𝑥_𝑛)=0. Hence 𝑥_𝑛→𝑝 as 𝑛→∞.Case 2. There exists a subsequence {𝜙(𝑝,𝑥_𝑛𝑗)} of {𝜙(𝑝,𝑥_𝑛)} such that 𝜙(𝑝,𝑥_𝑛𝑗)<𝜙(𝑝,𝑥_𝑛𝑗+1) for all 𝑗∈ℕ. By Lemma 2.12, there exists a strictly increasing sequence {𝑚_𝑘} of positive integers such that the following properties are satisfied by all numbers 𝑘∈ℕ: 𝜙𝑝,𝑥_𝑚𝑘≤𝜙𝑝,𝑥_𝑚𝑘+1,𝜙𝑝,𝑥_𝑘≤𝜙𝑝,𝑥_𝑚𝑘+1. (3.31) So we have 0≤lim_𝑘→∞𝜙𝑝,𝑥_𝑚𝑘+1−𝜙𝑝,𝑥_𝑚𝑘≤limsup_𝑛→∞𝜙𝑝,𝑥_𝑛+1−𝜙𝑝,𝑥_𝑛≤limsup_𝑛→∞𝜙𝑝,𝑧_𝑛−𝜙𝑝,𝑥_𝑛≤limsup_𝑛→∞𝛼_𝑛𝜙𝑝,𝑥+1−𝛼_𝑛𝜙𝑝,𝑦_𝑛−𝜙𝑝,𝑥_𝑛=limsup_𝑛→∞𝛼_𝑛𝜙𝑝,𝑥−𝜙𝑝,𝑦_𝑛+𝜙𝑝,𝑦_𝑛−𝜙𝑝,𝑥_𝑛≤limsup_𝑛→∞𝛼_𝑛𝜙𝑝,𝑥−𝜙𝑝,𝑦_𝑛=0. (3.32) This shows that lim_𝑘→∞𝜙𝑝,𝑥_𝑚𝑘+1−𝜙𝑝,𝑥_𝑚𝑘=0. (3.33) Following the proof line in Case 1, we can show that limsup_𝑘→∞𝐽𝑥−𝐽𝑝,𝑦_𝑚𝑘𝜙−𝑝≤0,𝑝,𝑥_𝑚𝑘+1≤1−𝛼_𝑚𝑘𝜙𝑝,𝑥_𝑚𝑘+𝛼_𝑚𝑘𝐽𝑥−𝐽𝑝,𝑧_𝑚𝑘−𝑦_𝑚𝑘+𝐽𝑥−𝐽𝑝,𝑦_𝑚𝑘−𝑝. (3.34) This implies 𝛼_𝑚𝑘𝜙𝑝,𝑥_𝑚𝑘≤𝜙𝑝,𝑥_𝑚𝑘−𝜙𝑝,𝑥_𝑚𝑘+1+𝛼_𝑚𝑘𝐽𝑥−𝐽𝑝,𝑧_𝑚𝑘−𝑦_𝑚𝑘+𝐽𝑥−𝐽𝑝,𝑦_𝑚𝑘−𝑝≤𝛼_𝑚𝑘𝐽𝑥−𝐽𝑝,𝑧_𝑚𝑘−𝑦_𝑚𝑘+𝐽𝑥−𝐽𝑝,𝑦_𝑚𝑘−𝑝. (3.35) Hence lim_𝑘→∞𝜙(𝑝,𝑥_𝑚𝑘)=0. Using this and (3.33) together, we conclude that limsup_𝑘→∞𝜙𝑝,𝑥_𝑘≤lim_𝑘→∞𝜙𝑝,𝑥_𝑚𝑘+1=0. (3.36) This completes the proof.

As a direct consequence of Theorem 3.2, we obtain the following results.

Corollary 3.3. Let 𝐶 be nonempty closed and convex subset of a uniformly smooth and uniformly convex Banach space 𝐸. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4) and 𝜑 a lower semicontinuous and convex function from 𝐶 to ℝ such that MEP(𝑓,𝜑)≠∅. Define the sequence {𝑥_𝑛} as follows: 𝑥₁=𝑥∈𝐶 and 𝑓𝑦_𝑛𝑦+1,𝑦+𝜑𝑟_𝑛𝐽𝑦_𝑛−𝐽𝑥_𝑛,𝑦−𝑦_𝑛≥𝜑𝑦_𝑛𝑥,∀𝑦∈𝐶,_𝑛+1=Π_𝐶𝐽⁻¹𝛼_𝑛𝐽𝑥+1−𝛼_𝑛𝐽𝑦_𝑛,∀𝑛≥1, (3.37) where {𝛼_𝑛}⊂(0,1) and {𝑟_𝑛}⊂(0,∞) satisfy the following conditions:(a)lim_𝑛→∞𝛼_𝑛=0;(b)∑^∞_𝑛=1𝛼_𝑛=∞;(c)liminf_𝑛→∞𝑟_𝑛>0.Then {𝑥_𝑛} converges strongly to Π_{MEP(𝑓,𝜑)}𝑥, where Π_{MEP(𝑓,𝜑)} is the generalized projection of 𝐶 onto MEP(𝑓,𝜑).

Corollary 3.4. Let 𝐶 be nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach space 𝐸. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4), and 𝐴∶𝐶→𝐸^∗ a continuous and monotone mapping such that GEP(𝑓,𝐴)≠∅. Define the sequence {𝑥_𝑛} as follows: 𝑥₁=𝑥∈𝐶 and 𝑓𝑦_𝑛,𝑦+𝐴𝑦_𝑛,𝑦−𝑦_𝑛+1𝑟_𝑛𝐽𝑦_𝑛−𝐽𝑥_𝑛,𝑦−𝑦_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=Π_𝐶𝐽⁻¹𝛼_𝑛𝐽𝑥+1−𝛼_𝑛𝐽𝑦_𝑛,∀𝑛≥1, (3.38) where {𝛼_𝑛}⊂(0,1) and {𝑟_𝑛}⊂(0,∞) satisfy the following conditions:(a)lim_𝑛→∞𝛼_𝑛=0;(b)∑^∞_𝑛=1𝛼_𝑛=∞;(c)liminf_𝑛→∞𝑟_𝑛>0.Then {𝑥_𝑛} converges strongly to Π_{GEP(𝑓,𝐴)}𝑥, where Π_{GEP(𝑓,𝐴)} is the generalized projection of 𝐶 onto GEP(𝑓,𝐴).

Corollary 3.5. Let 𝐶 be nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach space 𝐸. Let 𝐴∶𝐶→𝐸^∗ be a continuous and monotone mapping, and 𝜑 a lower semicontinuous and convex function from 𝐶 to ℝ such that VI(𝐶,𝐴,𝜑)≠∅. Define the sequence {𝑥_𝑛} as follows: 𝑥₁=𝑥∈𝐶 and 𝐴𝑦_𝑛,𝑦−𝑦_𝑛𝑦+1+𝜑𝑟_𝑛𝐽𝑦_𝑛−𝐽𝑥_𝑛,𝑦−𝑦_𝑛≥𝜑𝑦_𝑛𝑥,∀𝑦∈𝐶,_𝑛+1=Π_𝐶𝐽⁻¹𝛼_𝑛𝐽𝑥+1−𝛼_𝑛𝐽𝑦_𝑛,∀𝑛≥1, (3.39) where {𝛼_𝑛}⊂(0,1) and {𝑟_𝑛}⊂(0,∞) satisfy the following conditions:(a)lim_𝑛→∞𝛼_𝑛=0;(b)∑^∞_𝑛=1𝛼_𝑛=∞;(c)liminf_𝑛→∞𝑟_𝑛>0.Then {𝑥_𝑛} converges strongly to Π_{VI(𝐶,𝐴,𝜑)}𝑥, where Π_{VI(𝐶,𝐴,𝜑)} is the generalized projection of 𝐶 onto VI(𝐶,𝐴,𝜑).

4. Examples and Numerical Results

In this section, we give examples and numerical results for our main theorem.

Example 4.1. Let 𝐸=ℝ and 𝐶=[−1,1]. Let 𝑓(𝑥,𝑦)=−9𝑥²+𝑥𝑦+8𝑦², 𝜑(𝑥)=3𝑥², and 𝐴𝑥=2𝑥. Find 𝑥∈[−1,1] such that 𝑓𝑥,𝑦+𝐴𝑥,𝑦−𝑥+𝜑𝑦≥𝜑(𝑥),∀𝑦∈[−1,1]. (4.1)

Solution 4. It is easy to check that 𝑓, 𝜑, and 𝐴 satisfy all conditions in Theorem 3.2. For each 𝑟>0 and 𝑥∈[−1,1], Lemma 2.9 ensures that there exists 𝑧∈[−1,1] such that, for any 𝑦∈[−1,1], 𝑓𝑦+1𝑧,𝑦+⟨𝐴𝑧,𝑦−𝑧⟩+𝜑𝑟⟨𝑧−𝑥,𝑦−𝑧⟩≥𝜑(𝑧)⟺−9𝑧²+𝑦𝑧+8𝑦²+2𝑧𝑦−𝑧+3𝑦²+1𝑟(𝑧−𝑥)𝑦−𝑧≥3𝑧²⟺11𝑟𝑦²+(3𝑟𝑧+𝑧−𝑥)𝑦−14𝑟𝑧²+𝑧²−𝑥𝑧≥0. (4.2) Put 𝐺(𝑦)=11𝑟𝑦²+(3𝑟𝑧+𝑧−𝑥)𝑦−(14𝑟𝑧²+𝑧²−𝑥𝑧). Then 𝐺 is a quadratic function of 𝑦 with coefficient 𝑎=11𝑟, 𝑏=(3𝑟𝑧+𝑧−𝑥), and 𝑐=−(14𝑟𝑧²+𝑧²−𝑥𝑧). We next compute the discriminant Δ of 𝐺 as follows: Δ=𝑏²=)]−4𝑎𝑐[(3𝑟+1𝑧−𝑥²+44𝑟14𝑟𝑧²+𝑧²−𝑥𝑧=𝑥²−2(3𝑟+1)𝑥𝑧+(3𝑟+1)²𝑧²+616𝑟²𝑧²+44𝑟𝑧²−44𝑟𝑥𝑧=𝑥²−50𝑟𝑥𝑧−2𝑥𝑧+625𝑟²𝑧²+50𝑟𝑧²+𝑧²=𝑥²−2(25𝑟𝑧+𝑧)𝑥+625𝑟²𝑧²+50𝑟𝑧²+𝑧²=[𝑥−(25𝑟𝑧+𝑧)]². (4.3) We know that 𝐺(𝑦)≥0 for all 𝑦∈[−1,1] if it has at most one solution in [−1,1]. So Δ≤0 and hence 𝑥=25𝑟𝑧+𝑧. Now we have 𝑧=𝑇_𝑟𝑥=𝑥/(25𝑟+1).
Let {𝑥_𝑛}^∞_𝑛=1 be the sequence generated by 𝑥₁=𝑥∈[−1,1] and 𝑓𝑦_𝑛,𝑦+𝐴𝑦_𝑛,𝑦−𝑦_𝑛𝑦+1+𝜑𝑟_𝑛𝑦_𝑛−𝑥_𝑛,𝑦−𝑦_𝑛≥𝜑𝑦_𝑛[],𝑥,∀𝑦∈−1,1_𝑛+1=𝛼_𝑛𝑥+1−𝛼_𝑛𝑦_𝑛,∀𝑛≥1, (4.4) and, equivalently, 𝑥_𝑛+1=𝛼_𝑛𝑥+1−𝛼_𝑛𝑇_𝑟𝑛𝑥_𝑛,∀𝑛≥1. (4.5)

We next give two numerical results for algorithm (4.5).

Algorithm 4.2. Let 𝛼_𝑛=1/80𝑛 and 𝑟_𝑛=𝑛/(𝑛+1). Choose 𝑥₁=𝑥=1. Then algorithm (4.5) becomes 𝑥_𝑛+1=1+180𝑛1−80𝑛𝑛+1𝑥26𝑛+1^𝑛,∀𝑛≥1. (4.6)

Numerical Result I
See Table 1.

Table 1 

Algorithm 4.3. Let 𝛼_𝑛=1/100𝑛 and 𝑟_𝑛=(𝑛+1)/2𝑛. Choose 𝑥₁=𝑥=−1. Then algorithm (4.5) becomes 𝑥_𝑛+11=−+1100𝑛1−100𝑛2𝑛𝑥27𝑛+25^𝑛,∀𝑛≥1. (4.7)

Numerical Result II
See Table 2.

Table 2 

5. Conclusion

Tables 1 and 2 show that the sequence {𝑥_𝑛} converges to 0 which solves the generalized mixed equilibrium problem. On the other hand, using Lemma 2.9(3), we can check that GMEP(𝑓,𝐴,𝜑)=𝐹(𝑇_𝑟)={0}.

Remark 5.1. In the view of computation, our algorithm is simple in order to get strong convergence for generalized mixed equilibrium problems.

Acknowledgments

The first and the second authors wish to thank the Thailand Research Fund and the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. The third author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no. 2011-0021821).