Metric Characterizations of 𝛼-Well-Posedness for a System of Mixed Quasivariational-Like Inequalities in Banach Spaces

Abstract

The purpose of this paper is to investigate the problems of the well-posedness for a system of mixed quasivariational-like inequalities in Banach spaces. First, we generalize the concept of 𝛼-well-posedness to the system of mixed quasivariational-like inequalities, which includes symmetric quasi-equilibrium problems as a special case. Second, we establish some metric characterizations of 𝛼-well-posedness for the system of mixed quasivariational-like inequalities. Under some suitable conditions, we prove that the 𝛼-well-posedness is equivalent to the existence and uniqueness of solution for the system of mixed quasivariational-like inequalities. The corresponding concept of 𝛼-well-posedness in the generalized sense is also considered for the system of mixed quasivariational-like inequalities having more than one solution. The results presented in this paper generalize and improve some known results in the literature.

1. Introduction

The classical notion of well-posedness for a minimization problem (MP) is due to Tykhonov [1], which has already been known as the Tykhonov well-posedness. The so-called Tykhonov well-posedness means the existence and uniqueness of solution, and the convergence of every minimizing sequence toward the unique solution. Taking into account that in many practical situations the solution may not be unique for a minimization problem, ones naturally introduced the concept of well-posedness in the generalized sense, which means the existence of minimizers and the convergence of some subsequence of every minimizing sequence toward a minimizer. Obviously, the concept of well-posedness is inspired by numerical methods producing optimizing sequences for optimization problems. In the following years, the well-posedness has received much attention because it plays a crucial role in the stability theory for optimization problems. A large number of results about well-posedness have appeared in the literature; see, for example, [2–10], where the work in [2, 3, 5, 7, 10] is for the class of scalar optimization problems, and the work in [4, 6, 8, 11] is for the class of vector optimization problems.

On the other hand, the concept of well-posedness has been generalized to other related problems, such as variational inequalities [9, 12–22], Nash equilibrium problems [16, 23–25], inclusion problems [12, 14, 26, 27], and fixed-point problems [12, 14, 26, 28, 29]. An initial notion of well-posedness for variational inequalities is due to Lucchetti and Patrone [20]. They introduced the notion of well-posedness for variational inequalities and proved some related results by means of Ekeland’s variational principle. Since then, many authors have been devoted to generating the concept of well-posedness from the minimization problem to various variational inequalities. Lignola and Morgan [19] introduced the parametric well-posedness for a family of variational inequalities. Lignola [15] further introduced two concepts of well-posedness and 𝐿-well-posedness for quasivariational-like inequalities and derived some metric characterizations of well-posedness. At the same time, Del Prete et al. [18] introduced the concept of 𝛼-well-posedness for a class of variational inequalities. Recently, Fang et al. [14] generalized the concept of well-posedness to a class of mixed variational inequalities in Hilbert spaces. They obtained some metric characterizations of its well-posedness and established the links with the well-posedness of inclusion problems and fixed-point problems. Furthermore, Ceng and Yao [12] generalized the results of Fang et al. [14] to a class of generalized mixed variational inequalities in Hilbert spaces. Ceng et al. [13] investigated the well-posedness for a class of mixed quasivariational-like inequalities in Banach spaces. For the well-posedness of variational inequalities with functional constraints, we refer to Huang and Yang [9] and Huang et al. [17]. In 2006, Lignola and Morgan [23] presented the notion of 𝛼-well-posedness for the Nash equilibrium problem and gave some metric characterizations of this type well-posedness. Petruşel et al. [29] and Llorens-Fuster et al. [28] discussed the well-posedness of fixed-point problems for multivalued mappings in metric spaces.

It is obvious that the equilibrium problem plays a very important role in the establishment of a general mathematical model for a wide range of practical problems, which include as special cases optimization problems, Nash equilibria problems, fixed-point problems, variational inequality problems, and complementarity problems (see, e.g, [30, 31]), and has been studied extensively and intensively. It is well known that each equilibrium problem can equivalently be transformed into a minimizing problem by using gap function, and some numerical methods have been extended to solve the equilibrium problem (see, e.g., [32]). This fact motivates the researchers to study the well-posedness for equilibrium problems. Recently, Fang et al. [33] introduced the concepts of parametric well-posedness for equilibrium problems and derived some metric characterizations for these types of well-posedness. For the well-posedness of equilibrium problems with functional constraints, we refer the readers to [34]. In 2009, Long and Huang [35] generalized the concept of 𝛼-well-posedness to symmetric quasiequilibrium problems in Banach spaces, which includes equilibrium problems, Nash equilibrium problems, quasivariational inequalities, variational inequalities, and fixed-point problems as special cases. Under some suitable conditions, they established some metric characterizations of 𝛼-well-posedness for symmetric quasiequilibrium problems. Moreover, they gave some examples to illustrate their results. Their results represent the generalization and improvement of some previously known results in the literature, for instance, [12–15, 23, 33]. It is worth pointing out that up to the publication of [35] there are no results concerned with the problems of the well-posedness for symmetric quasiequilibrium problems in Banach spaces.

In this paper, we consider and study the problems of the well-posedness for a system of mixed quasivariational-like inequalities in Banach spaces. First, we generalize the concept of 𝛼-well-posedness to the system of mixed quasivariational-like inequalities, which include symmetric quasiequilibrium problems as a special case. Second, some metric characterizations of 𝛼-well-posedness for the system of mixed quasivariational-like inequalities are given under very mild conditions. Furthermore, it is also proven that under quite appropriate conditions, the 𝛼-well-posedness is equivalent to the existence and uniqueness of solution for the system of mixed quasivariational-like inequalities. At the same time, the corresponding concept of 𝛼-well-posedness in the generalized sense is also considered for the system of mixed quasivariational-like inequalities having more than one solution. In addition, we give some examples to illustrate our results. The results presented in this paper generalize and improve Long and Huang’s results in [35].

2. Preliminaries

Throughout this paper, unless specified otherwise, let 𝑋 and 𝑌 be two real Banach spaces, let their dual spaces be denoted by 𝑋^∗ and 𝑌^∗, respectively, and let the duality pairing between 𝑋 and 𝑋^∗ and the one between 𝑌 and 𝑌^∗ be denoted by the same ⟨⋅,⋅⟩. We write 𝑥_𝑛⇀𝑥 to indicate that the sequence {𝑥_𝑛} converges weakly to 𝑥. However, 𝑥_𝑛→𝑥 implies that {𝑥_𝑛} converges strongly to 𝑥. Let 𝐶⊂𝑋 and 𝐷⊂𝑌 be two nonempty closed and convex subsets. Let 𝑆∶𝐶×𝐷→2^𝐶 and 𝑇∶𝐶×𝐷→2^𝐷 be two set-valued mappings, let 𝐴∶𝐶×𝐷→𝑋^∗, 𝐵∶𝐶×𝐷→𝑌^∗, 𝜂∶𝐶×𝐶→𝑋 and 𝜂∶𝐷×𝐷→𝑌 be four single-valued mappings, and let 𝑓,𝑔∶𝐶×𝐷→𝐑 be two real-valued functions. Suppose that 𝛼 is a nonnegative real number and 𝑁={1,2,…}.

In this paper, we consider the system of mixed quasivariational-like inequalities (SMQVLIs), which is to find a point (𝑥₀,𝑦₀)∈𝐶×𝐷 such that 𝑥₀∈𝑆𝑥₀,𝑦₀,𝐴𝑥₀,𝑦₀,𝜂𝑥₀,𝑧+𝑓𝑥₀,𝑦₀−𝑓𝑧,𝑦₀≤0,∀𝑧∈𝑆𝑥₀,𝑦₀,𝑦₀∈𝑇𝑥₀,𝑦₀,𝐵𝑥₀,𝑦₀,𝜂𝑦₀,𝑤+𝑔𝑥₀,𝑦₀−𝑔𝑥₀,𝑤≤0,∀𝑤∈𝑇𝑥₀,𝑦₀. (2.1)

Remark 2.1. Whenever 𝐴=0, 𝐵=0, 𝜂=0, and 𝜂=0, the Problem (2.1) reduces to the following symmetric quasiequilibrium problem (in short, SQEP) of finding a point (𝑥₀,𝑦₀)∈𝐶×𝐷 such that 𝑥₀∈𝑆𝑥₀,𝑦₀,𝑓𝑥₀,𝑦₀≤𝑓𝑧,𝑦₀,∀𝑧∈𝑆𝑥₀,𝑦₀,𝑦₀∈𝑇𝑥₀,𝑦₀,𝑔𝑥₀,𝑦₀≤𝑔𝑥₀,𝑤,∀𝑤∈𝑇𝑥₀,𝑦₀. (2.2) This problem was first considered by Noor and Oettli [21], which includes equilibrium problems [30], Nash equilibrium problems [36], quasivariational inequalities [37], variational inequalities [38], and fixed-point problems [28, 29] as special cases. It is worth mentioning that Noor and Oettli [21] only established the existence of solutions for SQEP (2.2). Subsequently, Long and Huang [35] investigated the 𝛼-well-posedness for SQEP (2.2) in Banach spaces.

Denote by Γ the solution set of SMQVLI (2.1). In what follows, we introduce the notions of 𝛼-approximating sequence and 𝛼-well-posedness for SMQVLI (2.1).

Definition 2.2. A sequence {(𝑥_𝑛,𝑦_𝑛)}⊂𝐶×𝐷 is called an 𝛼-approximating sequence for SMQVLI (2.1) if there exists a sequence 𝜀_𝑛>0 with 𝜀_𝑛→0 such that 𝑑𝑥_𝑛,𝑆𝑥_𝑛,𝑦_𝑛≤𝜀_𝑛,thatis,𝑥_𝑛∈𝐵𝑆𝑥_𝑛,𝑦_𝑛,𝜀_𝑛𝑑𝑦,∀𝑛∈𝑁,_𝑛,𝑇𝑥_𝑛,𝑦_𝑛≤𝜀_𝑛,thatis,𝑦_𝑛∈𝐵𝑇𝑥_𝑛,𝑦_𝑛,𝜀_𝑛𝐴𝑥,∀𝑛∈𝑁,_𝑛,𝑦_𝑛,𝜂𝑥_𝑛,𝑧+𝑓𝑥_𝑛,𝑦_𝑛−𝑓𝑧,𝑦_𝑛≤𝜀_𝑛+𝛼2𝑥_𝑛−𝑧²,∀𝑧∈𝑆𝑥_𝑛,𝑦_𝑛𝐵𝑥,∀𝑛∈𝑁,_𝑛,𝑦_𝑛,𝜂𝑦_𝑛,𝑤+𝑔𝑥_𝑛,𝑦_𝑛−𝑔𝑥_𝑛,𝑤≤𝜀_𝑛+𝛼2𝑦_𝑛−𝑤²,∀𝑤∈𝑇𝑥_𝑛,𝑦_𝑛,∀𝑛∈𝑁, (2.3) where 𝐵(𝑆(𝑥,𝑦),𝜀) denotes the ball of radius 𝜀 around 𝑆(𝑥,𝑦), that is, the set {𝑚∈𝑋∶𝑑(𝑆(𝑥,𝑦),𝑚)=inf_{𝑏∈𝑆(𝑥,𝑦)}‖𝑚−𝑏‖≤𝜀_𝑛}. Whenever 𝛼=0, one says that the sequence {(𝑥_𝑛,𝑦_𝑛)} is an approximating sequence for SMQVLI (2.1).

We remark that if 𝐴=0, 𝐵=0, 𝜂=0, and 𝜂=0, the notions of 𝛼-approximating sequence and approximating sequence for SMQVLI (2.1) reduce to the ones of 𝛼-approximating sequence and approximating sequence for SQEP (2.2) in [35, Definition  2.1], respectively.

Definition 2.3. SMQVLI (2.1) is said to be 𝛼-well-posed if it has a unique solution (𝑥₀,𝑦₀) and every 𝛼-approximating sequence {(𝑥_𝑛,𝑦_𝑛)} converges strongly to (𝑥₀,𝑦₀). Whenever 𝛼=0, we say that SMQVLI (2.1) is well-posed.

We remark that if 𝐴=0, 𝐵=0, 𝜂=0, and 𝜂=0, the notions of 𝛼-well-posedness and well-posedness for SMQVLI (2.1) reduce to the ones of 𝛼-well-posedness and well-posedness for SQEP (2.2) in [35, Definition  2.2], respectively.

Definition 2.4. SMQVLI (2.1) is said to be 𝛼-well-posed in the generalized sense if the solution set Γ of SMQVLI (2.1) is nonempty and every 𝛼-approximating sequence {(𝑥_𝑛,𝑦_𝑛)} has a subsequence which converges strongly to some element of Γ. Whenever 𝛼=0, one says that SMQVLI (2.1) is well-posed in the generalized sense.

We remark that if 𝐴=0, 𝐵=0, 𝜂=0, and 𝜂=0, the notions of 𝛼-well-posedness in the generalized sense and well-posedness in the generalized sense for SMQVLI (2.1) reduce to the ones of 𝛼-well-posedness in the generalized sense and well-posedness in the generalized sense for SQEP (2.2) in [35, Definition  2.3], respectively.

In order to investigate the 𝛼-well-posedness for SMQVLI (2.1), we need the following definitions.

Definition 2.5 (see [39]). The Painleve-Kuratowski limits of a sequence {𝐻_𝑛}⊂𝑋 are defined by liminf_𝑛𝐻_𝑛=𝑦∈𝑋∶∃𝑦_𝑛∈𝐻_𝑛,𝑛∈𝑁,withlim_𝑛𝑦_𝑛=𝑦,limsup_𝑛𝐻_𝑛=𝑦∈𝑋∶∃𝑛_𝑘↑+∞,𝑛_𝑘∈𝑁,∃𝑦_𝑛𝑘∈𝐻_𝑛𝑘,𝑘∈𝑁,withlim_𝑘𝑦_𝑛𝑘=𝑦. (2.4)

Definition 2.6 (see [39]). A set-valued mapping 𝐹 from a topological space (𝑊,𝜏) to a topological space (𝑍,𝜎) is called
(i)(𝜏,𝜎)-closed if for every 𝑥∈𝐾, for every sequence {𝑥_𝑛}𝜏-converging to 𝑥, and for every sequence {𝑦_𝑛}𝜎-converging to a point 𝑦, such that 𝑦_𝑛∈𝐹(𝑥_𝑛), one has 𝑦∈𝐹(𝑥), that is, 𝐹(𝑥)⊃limsup_𝑛𝐹𝑥_𝑛, (2.5)(ii)(𝜏,𝜎)-lower semicontinuous if for every 𝑥∈𝐾, for every sequence {𝑥_𝑛}𝜏-converging to 𝑥, and for every 𝑦∈𝐹(𝑥), there exists a sequence {𝑦_𝑛}𝜎-converging to 𝑦, such that 𝑦_𝑛∈𝐹(𝑥_𝑛) for 𝑛 sufficiently large, that is, 𝐹(𝑥)⊂liminf_𝑛𝐹𝑥_𝑛, (2.6)(iii)(𝜏,𝜎)-subcontinuous on 𝐾, if for every sequence {𝑥_𝑛}𝜏-converging in 𝐾, every sequence {𝑦_𝑛}, such that 𝑦_𝑛∈𝐹(𝑥_𝑛), has a 𝜎-convergent subsequence.

Definition 2.7 (see [39]). Let 𝑉 be a nonempty subset of 𝑋. The measure of noncompactness 𝜇 of the set 𝑉 is defined by 𝜇(𝑉)=inf𝜀>0∶𝑉⊂^𝑛_𝑖=1𝑉_𝑖,diam𝑉_𝑖<𝜀,𝑖=1,2,…,𝑛, (2.7) where diam means the diameter of a set.

Definition 2.8 (see [39]). Let (𝑋,𝑑) be a metric space and let 𝑈, 𝑉 be nonempty subsets of 𝑋. The Hausdorff metric ℋ(⋅,⋅) between 𝑈 and 𝑉 is defined by ℋ(𝑈,𝑉)=max{𝑒(𝑈,𝑉),𝑒(𝑉,𝑈)}, (2.8) where 𝑒(𝑈,𝑉)=sup_𝑢∈𝑈𝑑(𝑢,𝑉) with 𝑑(𝑢,𝑉)=inf_𝑣∈𝑉‖𝑢−𝑣‖. Let {𝑈_𝑛} be a sequence of nonempty subsets of 𝑋. One says that 𝑈_𝑛 converges to 𝑈 in the sense of Hausdorff metric if ℋ(𝑈_𝑛,𝑈)→0. It is easy to see that 𝑒(𝑈_𝑛,𝑈)→0 if and only if 𝑑(𝑢_𝑛,𝑈)→0 for all section 𝑢_𝑛∈𝑈_𝑛. For more details on this topic, the readers refered one to [39].

Now, we prove the following lemma.

Lemma 2.9. Suppose that set-valued mappings 𝑆 and 𝑇 are nonempty convex-valued, the function 𝑓(⋅,𝑦) is convex on 𝐶 for any 𝑦∈𝐷, and the function 𝑔(𝑥,⋅) is convex on 𝐷 for any 𝑥∈𝐶. Then (𝑥₀,𝑦₀)∈Γ if and only if the following two conditions hold: 𝑥₀∈𝑆𝑥₀,𝑦₀,𝐴𝑥₀,𝑦₀,𝜂𝑥₀,𝑧+𝑓𝑥₀,𝑦₀−𝑓𝑧,𝑦₀≤𝛼2𝑥₀−𝑧²,∀𝑧∈𝑆𝑥₀,𝑦₀,𝑦₀∈𝑇𝑥₀,𝑦₀,𝐵𝑥₀,𝑦₀,𝜂𝑦₀,𝑤+𝑔𝑥₀,𝑦₀−𝑔𝑥₀≤𝛼,𝑤2𝑦₀−𝑤²,∀𝑤∈𝑇𝑥₀,𝑦₀, (2.9) where both 𝜂∶𝐶×𝐶→𝑋 and 𝜂∶𝐷×𝐷→𝑌 are affine in the second variable such that 𝜂(𝑥,𝑥)=0 and 𝜂(𝑦,𝑦)=0 for all (𝑥,𝑦)∈𝐶×𝐷.

Proof. The necessity is obvious. For the sufficiency, suppose that (2.9) holds. Now let us show that (𝑥₀,𝑦₀)∈Γ. Indeed, let 𝑧∈𝑆(𝑥₀,𝑦₀) and for any 𝑡∈[0,1], 𝑧_𝑡=𝑡𝑧+(1−𝑡)𝑥₀. Since 𝑆(𝑥₀,𝑦₀) is convex, 𝑧_𝑡∈𝑆(𝑥₀,𝑦₀) and so 𝐴𝑥₀,𝑦₀,𝜂𝑥₀,𝑧_𝑡+𝑓𝑥₀,𝑦₀−𝑓𝑧_𝑡,𝑦₀≤𝛼2𝑥₀−𝑧_𝑡²,∀𝑡∈(0,1]. (2.10) Also, since 𝑓(⋅,𝑦) is convex for any 𝑦∈𝐷 and 𝜂∶𝐶×𝐶→𝑋 is affine in the second variable with 𝜂(𝑥,𝑥)=0, ∀𝑥∈𝑋, we have 𝑡𝐴𝑥₀,𝑦₀,𝜂𝑥₀,𝑧+𝑓𝑥₀,𝑦₀−𝑓𝑧,𝑦₀=𝐴𝑥₀,𝑦₀,𝑡𝜂𝑥₀,𝑧+(1−𝑡)𝜂𝑥₀,𝑥₀+𝑓𝑥₀,𝑦₀−𝑡𝑓𝑧,𝑦₀+(1−𝑡)𝑓𝑥₀,𝑦₀≤𝐴𝑥₀,𝑦₀,𝜂𝑥₀,𝑧_𝑡+𝑓𝑥₀,𝑦₀−𝑓𝑧_𝑡,𝑦₀≤𝛼2𝑥₀−𝑧_𝑡²=𝛼2𝑡²𝑥₀−𝑧²,∀𝑡∈(0,1]. (2.11) Thus, dividing by 𝑡 in the above inequality, we have 𝐴𝑥₀,𝑦₀,𝜂𝑥₀,𝑧+𝑓𝑥₀,𝑦₀−𝑓𝑧,𝑦₀≤𝛼2𝑡𝑥₀−𝑧²,∀𝑡∈(0,1],𝑧∈𝑆𝑥₀,𝑦₀. (2.12) By the similar argument, 𝐵𝑥₀,𝑦₀,𝜂𝑦₀,𝑤+𝑔𝑥₀,𝑦₀−𝑔𝑥₀≤𝛼,𝑤2𝑡𝑦₀−𝑤²,∀𝑡∈(0,1],𝑤∈𝑇𝑥₀,𝑦₀. (2.13) The combination of (2.12) and (2.13) implies, for 𝑡 tending to zero, that (𝑥₀,𝑦₀) is a solution of SMQVLI (2.1). This completes the proof.

Corollary 2.10 (i.e., [35, Lemma  2.1]). Suppose that set-valued mappings 𝑆 and 𝑇 are nonempty convex-valued, the function 𝑓(⋅,𝑦) is convex on 𝐶 for any 𝑦∈𝐷, and the function 𝑔(𝑥,⋅) is convex on 𝐷 for any 𝑥∈𝐶. Then (𝑥₀,𝑦₀) solves SQEP (2.2) if and only if the following two conditions hold: 𝑥₀∈𝑆𝑥₀,𝑦₀,𝑓𝑥₀,𝑦₀≤𝑓𝑧,𝑦₀+𝛼2𝑥₀−𝑧²,∀𝑧∈𝑆𝑥₀,𝑦₀,𝑦₀∈𝑇𝑥₀,𝑦₀,𝑔𝑥₀,𝑦₀≤𝑔𝑥₀+𝛼,𝑤2𝑦₀−𝑤²,∀𝑤∈𝑇𝑥₀,𝑦₀. (2.14)

Proof. Put 𝐴=0, 𝐵=0, 𝜂=0, and 𝜂=0 in Lemma 2.9. Then, utilizing Lemma 2.9 we get the desired result.

3. Metric Characterizations of 𝛼-Well-Posedness for SMQVLI

In this section, we will investigate some metric characterizations of 𝛼-well-posedness for SMQVLI (2.1).

For any 𝜀>0, the 𝛼-approximating solution set of SMQVLI (2.1) is defined by Ω_𝛼(𝜀)=𝑥₀,𝑦₀𝑥∈𝐶×𝐷∶₀∈𝐵𝑆𝑥₀,𝑦₀,𝜀,𝐴𝑥₀,𝑦₀,𝜂𝑥₀,𝑧+𝑓𝑥₀,𝑦₀−𝑓𝑧,𝑦₀𝛼≤𝜀+2𝑥₀−𝑧²,∀𝑧∈𝑆𝑥₀,𝑦₀,𝑦₀∈𝐵𝑇𝑥₀,𝑦₀,𝜀,𝐵𝑥₀,𝑦₀,𝜂𝑦₀𝑥,𝑤+𝑔₀,𝑦₀−𝑔𝑥₀𝛼,𝑤≤𝜀+2𝑦₀−𝑤²,∀𝑤∈𝑇𝑥₀,𝑦₀. (3.1)

Theorem 3.1. SMQVLI (2.1) is 𝛼-well-posed if and only if the solution set Γ of SMQVLI (2.1) is nonempty and lim_𝜀→0diamΩ_𝛼(𝜀)=0. (3.2)

Proof . Suppose that SMQVLI (2.1) is 𝛼-well-posed. Then, Γ is a singleton, and Ω_𝛼(𝜀)≠∅ for any 𝜀>0, since Γ⊂Ω_𝛼(𝜀). Suppose by contraction that lim_𝜀→0diamΩ_𝛼(𝜀)>𝛽>0. (3.3) Then there exists 𝜀_𝑛>0 with 𝜀_𝑛→0, and (𝑥_𝑛,𝑦_𝑛),(𝑥_𝑛,𝑦_𝑛)∈Ω_𝛼(𝜀_𝑛) such that 𝑥_𝑛,𝑦_𝑛−𝑥_𝑛,𝑦_𝑛>𝛽,∀𝑛∈𝑁, (3.4) where the norm ‖⋅‖ in the product space 𝑋×𝑌 is defined as follows: ‖(𝑢,𝑣)−(𝑢,𝑣)‖=‖𝑢−𝑢‖²+‖𝑣−𝑣‖²,∀(𝑢,𝑣),(𝑢,𝑣)∈𝑋×𝑌.Itisnotdifficulttoverifythat𝑋×𝑌isaBanachspaceintermsofthelastnorm. (3.5) Since (𝑥_𝑛,𝑦_𝑛),(𝑥_𝑛,𝑦_𝑛)∈Ω_𝛼(𝜀_𝑛), and SMQVLI (2.1) is 𝛼-well-posed, the sequences {(𝑥_𝑛,𝑦_𝑛)} and {(𝑥_𝑛,𝑦_𝑛)}, which are both 𝛼-approximating sequences for SMQVLI (2.1), converge strongly to the unique solution (𝑥₀,𝑦₀), and this leads to a contraction. Therefore, (3.2) holds.
Conversely, let (3.2) hold and let {(𝑥_𝑛,𝑦_𝑛)}⊂𝐶×𝐷 be any 𝛼-approximating sequence for SMQVLI (2.1). Then, there exists a sequence 𝜀_𝑛>0 with 𝜀_𝑛→0 such that 𝑑𝑥_𝑛,𝑆𝑥_𝑛,𝑦_𝑛≤𝜀_𝑛,𝐴𝑥_𝑛,𝑦_𝑛,𝜂𝑥_𝑛,𝑧+𝑓𝑥_𝑛,𝑦_𝑛−𝑓𝑧,𝑦_𝑛≤𝜀_𝑛+𝛼2𝑥_𝑛−𝑧²,∀𝑧∈𝑆𝑥_𝑛,𝑦_𝑛,𝑑𝑦_𝑛,𝑇𝑥_𝑛,𝑦_𝑛≤𝜀_𝑛,𝐵𝑥_𝑛,𝑦_𝑛,𝜂𝑦_𝑛,𝑤+𝑔𝑥_𝑛,𝑦_𝑛−𝑔𝑥_𝑛,𝑤≤𝜀_𝑛+𝛼2𝑦_𝑛−𝑤²,∀𝑤∈𝑇𝑥_𝑛,𝑦_𝑛. (3.6) This implies that {(𝑥_𝑛,𝑦_𝑛)}⊂Ω_𝛼(𝜀_𝑛),forall𝑛∈𝑁. Since the solution set Γ of SMQVLI (2.1) is nonempty, we can take two elements in Γ arbitrarily, denoted by (𝑥₀,𝑦₀) and (𝑥₀,𝑦₀), respectively. Note that Γ⊂Ω_𝛼(𝜀) for all 𝜀>0. Hence both (𝑥₀,𝑦₀) and (𝑥₀,𝑦₀) lie in Ω_𝛼(𝜀_𝑛) for all 𝑛≥1. This fact together with (3.2) yields 𝑥_𝑛,𝑦_𝑛−𝑥₀,𝑦₀≤diamΩ_𝛼𝜀_𝑛⟶0,𝑥_𝑛,𝑦_𝑛−𝑥₀,𝑦₀≤diamΩ_𝛼𝜀_𝑛⟶0. (3.7) Utilizing (3.7) and the uniqueness of the limit, we conclude that (𝑥₀,𝑦₀)=(𝑥₀,𝑦₀). This means that Γ is a singleton. Thus, it is known that SMQVLI (2.1) has the unique solution (𝑥₀,𝑦₀) and {(𝑥_𝑛,𝑦_𝑛)} converges strongly to (𝑥₀,𝑦₀). This shows that SMQVLI (2.1) is 𝛼-well-posed. This completes the proof.

Corollary 3.2 (i.e., [35, Theorem  3.1]). SQEP (2.2) is 𝛼-well-posed if and only if the solution set Γ of SQEP (2.2) is nonempty and lim_𝜀→0diam𝑀_𝜀=0, (3.8) where 𝑀_𝜀=𝑥₀,𝑦₀∈𝐶×𝐷∶𝑥₀∈𝐵𝑆𝑥₀,𝑦₀,𝜀,𝑓𝑥₀,𝑦₀−𝑓𝑧,𝑦₀𝛼≤𝜀+2𝑥₀−𝑧²,∀𝑧∈𝑆𝑥₀,𝑦₀,𝑦₀∈𝐵𝑇𝑥₀,𝑦₀,𝜀,𝑔𝑥₀,𝑦₀−𝑔𝑥₀+𝛼,𝑤≤𝜀2𝑦₀−𝑤²,∀𝑤∈𝑇𝑥₀,𝑦₀. (3.9)

Proof. Put 𝐴=0, 𝐵=0, 𝜂=0, and 𝜂=0 in Theorem 3.1. Then, utilizing Theorem 3.1 we get the desired result.

In the sequel, the following concept will be needed to apply to our main results.

Definition 3.3. Let 𝐶 be a nonempty, closed convex subset of 𝑋. A single-valued mapping 𝜂∶𝐶×𝐶→𝑋 is said to be Lipschitz continuous if there exists a constant 𝜆>0 such that 𝜂𝑥,𝑦≤𝜆𝑥−𝑦,∀𝑥,𝑦∈𝐶. (3.10)

We remark that whenever 𝑋=𝐻 a Hilbert space and 𝐶=𝐾 a nonempty closed convex subset of 𝐻, the Lipschitz continuous mapping 𝜂∶𝐾×𝐾→𝐻 has been introduced and considered in Ansari and Yao [40]. In their main result for the existence of solutions and convergence of iterative algorithm (i.e., [40, Theorem 3.1]), the Lipschitz continuous mapping 𝜂∶𝐾×𝐾→𝐻 satisfies the following conditions:(a)𝜂(𝑥,𝑦)+𝜂(𝑦,𝑥)=0 for all 𝑥,𝑦∈𝐾,(b)𝜂(𝑥,𝑦)=𝜂(𝑥,𝑧)+𝜂(𝑧,𝑦) for all 𝑥,𝑦,𝑧∈𝐾,(c)𝜂(⋅,⋅) is affine in the first variable,(d)for each fixed 𝑦∈𝐾, 𝑥↦𝜂(𝑦,𝑥) is sequentially continuous from the weak topology to the weak topology ((𝑤,𝑤)-continuous).

Inspired by the above restrictions imposed on the Lipschitz continuous mapping 𝜂, we give the following theorem.

Theorem 3.4. Assume that the following conditions hold:(i)set-valued mappings 𝑆 and 𝑇 are nonempty convex-valued, (𝑠,𝑤)-closed, (𝑠,𝑠)-lower semicontinuous and (𝑠,𝑤)-subcontinuous on 𝐶×𝐷;(ii)single-valued mappings 𝐴 and 𝐵 are (𝑠,𝑤^∗)-continuous on 𝐶×𝐷;(iii)single-valued mappings 𝜂 and 𝜂 are Lipschitz continuous with constants 𝜆 and 𝜆 respectively, such that(a)𝜂(𝑥₁,𝑥₃)=𝜂(𝑥₁,𝑥₂)+𝜂(𝑥₂,𝑥₃) for all 𝑥₁,𝑥₂,𝑥₃∈𝐶 and 𝜂(𝑦₁,𝑦₃)=𝜂(𝑦₁,𝑦₂)+𝜂(𝑦₂,𝑦₃) for all 𝑦₁,𝑦₂,𝑦₃∈𝐷,(b)𝜂(⋅,⋅) and 𝜂(⋅,⋅) both are affine in the second variable;(iv)functions 𝑓 and 𝑔 are continuous on 𝐶×𝐷;(v)for any 𝑦∈𝐷, the function 𝑓(⋅,𝑦) is convex on 𝐶; for any 𝑥∈𝐶, the function 𝑔(𝑥,⋅) is convex on 𝐷.
Then, SMQVLI (2.1) is 𝛼-well-posed if and only if Ω_𝛼(𝜀)≠∅,∀𝜀>0,lim_𝜀→0diamΩ_𝛼(𝜀)=0. (3.11)

Proof . First, utilizing condition (iii) (a), we can readily obtain that 𝜂𝑥₁,𝑥₁=0,𝜂𝑥₁,𝑥₂=−𝜂𝑥₂,𝑥₁,∀𝑥₁,𝑥₂∈𝐶;𝜂𝑦₁,𝑦₁=0,𝜂𝑦₁,𝑦₂=−𝜂𝑦₂,𝑦₁,∀𝑦₁,𝑦₂∈𝐷. (3.12)
The necessity has been proved in Theorem 3.1. For the sufficiency, let condition (3.11) hold. Let {(𝑥_𝑛,𝑦_𝑛)}⊂𝐶×𝐷 be any 𝛼-approximating sequence for SMQVLI (2.1). Now let us show that Γ is a singleton and {(𝑥_𝑛,𝑦_𝑛)} converges strongly to the unique element of Γ. As a matter of fact, since {(𝑥_𝑛,𝑦_𝑛)} is 𝛼-approximating sequence for SMQVLI (2.1), there exists a sequence 𝜀_𝑛>0 with 𝜀_𝑛→0 such that 𝑑𝑥_𝑛,𝑆𝑥_𝑛,𝑦_𝑛≤𝜀_𝑛,𝐴𝑥_𝑛,𝑦_𝑛,𝜂𝑥_𝑛,𝑧+𝑓𝑥_𝑛,𝑦_𝑛−𝑓𝑧,𝑦_𝑛≤𝜀_𝑛+𝛼2𝑥_𝑛−𝑧²,∀𝑧∈𝑆𝑥_𝑛,𝑦_𝑛,𝑑𝑦_𝑛,𝑇𝑥_𝑛,𝑦_𝑛≤𝜀_𝑛,𝐵𝑥_𝑛,𝑦_𝑛,𝜂𝑦_𝑛,𝑤+𝑔𝑥_𝑛,𝑦_𝑛−𝑔𝑥_𝑛,𝑤≤𝜀_𝑛+𝛼2𝑦_𝑛−𝑤²,∀𝑤∈𝑇𝑥_𝑛,𝑦_𝑛. (3.13) This means {(𝑥_𝑛,𝑦_𝑛)}⊂Ω_𝛼(𝜀_𝑛),forall𝑛∈𝑁. It follows from (3.11) that {(𝑥_𝑛,𝑦_𝑛)} is a Cauchy sequence in Banach space (𝑋×𝑌,‖⋅‖) and hence converges strongly to a point (𝑥₀,𝑦₀)∈𝑋×𝑌. By the definition of the norm ‖⋅‖ in Banach space (𝑋×𝑌,‖⋅‖), we deduce that 𝑥_𝑛−𝑥₀≤𝑥_𝑛−𝑥₀²+𝑦_𝑛−𝑦₀²=𝑥_𝑛,𝑦_𝑛−𝑥₀,𝑦₀𝑦⟶0,_𝑛−𝑦₀≤𝑥_𝑛−𝑥₀²+𝑦_𝑛−𝑦₀²=𝑥_𝑛,𝑦_𝑛−𝑥₀,𝑦₀⟶0. (3.14) On account of the closedness of 𝐶 and 𝐷 we conclude from {𝑥_𝑛}⊂𝐶 and {𝑦_𝑛}⊂𝐷 that 𝑥_𝑛→𝑥₀∈𝐶 and 𝑦_𝑛→𝑦₀∈𝐷. In order to show (𝑥₀,𝑦₀)∈Γ, we start to prove that 𝑑𝑥₀,𝑆𝑥₀,𝑦₀≤liminf_𝑛𝑑𝑥_𝑛,𝑆𝑥_𝑛,𝑦_𝑛=lim_𝑛𝜀_𝑛=0. (3.15) Indeed, suppose that the left inequality does not hold. Then there exists a positive number 𝛾 such that liminf_𝑛𝑑𝑥_𝑛,𝑆𝑥_𝑛,𝑦_𝑛<𝛾<𝑑𝑥₀,𝑆𝑥₀,𝑦₀, (3.16) or equivalently, there exist an increasing sequence {𝑛_𝑘} and a sequence {𝑧_𝑘},𝑧_𝑘∈𝑆(𝑥_𝑛𝑘,𝑦_𝑛𝑘), foall𝑘∈𝑁 such that 𝑥_𝑛𝑘−𝑧_𝑘<𝛾,∀𝑘∈𝑁. (3.17) Since the set-valued mapping 𝑆 is (𝑠,𝑤)-closed and (𝑠,𝑤)-subcontinuous, the sequence {𝑧_𝑘} has a subsequence, denoted still by {𝑧_𝑘}, converging weakly to a point 𝑧₀∈𝑆(𝑥₀,𝑦₀). From the weak lower semicontinuity of the norm, it follows that 𝛾<𝑑𝑥₀,𝑆𝑥₀,𝑦₀≤𝑥₀−𝑧₀≤liminf_𝑘𝑥_𝑛𝑘−𝑧_𝑘<𝛾, (3.18) which leads to a contradiction. Thus we must have 𝑑(𝑥₀,𝑆(𝑥₀,𝑦₀))=0 and hence 𝑥₀∈𝑆(𝑥₀,𝑦₀). Similarly, we can prove 𝑦₀∈𝑇(𝑥₀,𝑦₀).
To complete the proof, we take a point 𝑧∈𝑆(𝑥₀,𝑦₀) arbitrarily. Since 𝑆 is (𝑠,𝑠)-lower semicontinuous, there exists a sequence {𝑧_𝑛} converging strongly to 𝑧, such that 𝑧_𝑛∈𝑆(𝑥_𝑛,𝑦_𝑛) for 𝑛 sufficiently large. Furthermore, utilizing condition (iii) (𝑎) and the Lipschitz continuity of 𝜂 we deduce that 𝜂𝑥_𝑛,𝑧_𝑛−𝜂𝑥₀,𝑧=𝜂𝑥_𝑛,𝑧_𝑛−𝜂𝑥₀,𝑧_𝑛+𝜂𝑥₀,𝑧_𝑛−𝜂𝑥₀≤𝑥,𝑧𝜂_𝑛,𝑧_𝑛−𝜂𝑥₀,𝑧_𝑛+𝜂𝑥₀,𝑧_𝑛−𝜂𝑥₀=𝑥,𝑧𝜂_𝑛,𝑧_𝑛+𝜂𝑧_𝑛,𝑥₀+𝜂𝑥₀,𝑧_𝑛+𝜂𝑧,𝑥₀=𝜂𝑥_𝑛,𝑥₀+𝜂𝑧,𝑧_𝑛≤𝜆𝑥_𝑛−𝑥₀+𝑧_𝑛−𝑧⟶0as𝑛⟶∞. (3.19) Since 𝐴 is (𝑠,𝑤^∗)-continuous, it is known that 𝐴(𝑥_𝑛,𝑦_𝑛) converges weakly to 𝐴(𝑥₀,𝑦₀), that is, for each 𝑥∈𝑋, the real sequence {⟨𝐴(𝑥_𝑛,𝑦_𝑛),𝑥⟩} converges to the real number ⟨𝐴(𝑥₀,𝑦₀),𝑥⟩. This implies that {⟨𝐴(𝑥_𝑛,𝑦_𝑛),𝑥⟩} is a bounded sequence of real numbers for each 𝑥∈𝑋. Thus {𝐴(𝑥_𝑛,𝑦_𝑛)} is bounded in the norm topology according to the uniform boundedness principle [41], that is, sup_𝑛≥1‖𝐴(𝑥_𝑛,𝑦_𝑛)‖<∞.
Now observe that 𝐴𝑥_𝑛,𝑦_𝑛,𝜂𝑥_𝑛,𝑧_𝑛−𝐴𝑥₀,𝑦₀,𝜂𝑥₀=𝐴𝑥,𝑧_𝑛,𝑦_𝑛,𝜂𝑥_𝑛,𝑧_𝑛−𝐴𝑥_𝑛,𝑦_𝑛,𝜂𝑥₀+𝐴𝑥,𝑧_𝑛,𝑦_𝑛,𝜂𝑥₀,𝑧−𝐴𝑥₀,𝑦₀,𝜂𝑥₀≤𝐴𝑥,𝑧_𝑛,𝑦_𝑛,𝜂𝑥_𝑛,𝑧_𝑛−𝜂𝑥₀,𝑧+𝐴𝑥_𝑛,𝑦_𝑛−𝐴𝑥₀,𝑦₀,𝜂𝑥₀≤𝐴𝑥,𝑧_𝑛,𝑦_𝑛𝜂𝑥_𝑛,𝑧_𝑛−𝜂𝑥₀+𝐴𝑥,𝑧_𝑛,𝑦_𝑛−𝐴𝑥₀,𝑦₀,𝜂𝑥₀,𝑧⟶0as𝑛⟶∞. (3.20) Consequently, it follows from condition (iv) that 𝐴𝑥₀,𝑦₀,𝜂𝑥₀,𝑧+𝑓𝑥₀,𝑦₀−𝑓𝑧,𝑦₀=lim_𝑛𝐴𝑥_𝑛,𝑦_𝑛,𝜂𝑥_𝑛,𝑧_𝑛+𝑓𝑥_𝑛,𝑦_𝑛−𝑓𝑧_𝑛,𝑦_𝑛≤lim_𝑛𝜀_𝑛+𝛼2𝑥_𝑛−𝑧_𝑛²=𝛼2𝑥₀−𝑧²,forall𝑧∈𝑆𝑥₀,𝑦₀. (3.21) Analogously, we have 𝐵𝑥₀,𝑦₀,𝜂𝑦₀,𝑤+𝑔𝑥₀,𝑦₀−𝑔𝑥₀≤𝛼,𝑤2𝑦₀−𝑤²,∀𝑤∈𝑇𝑥₀,𝑦₀. (3.22) It follows from Lemma 2.9 that (𝑥₀,𝑦₀)∈Γ. Therefore, SMQVLI (2.1) is 𝛼-well-posed. This completes the proof.

Corollary 3.5 (i.e., [35, Theorem  3.2]). Assume that the following conditions hold:(i)set-valued mappings 𝑆 and 𝑇 are nonempty convex-valued, (𝑠,𝑤)-closed, (𝑠,𝑠)-lower semicontinuous, and (𝑠,𝑤)-subcontinuous on 𝐶×𝐷;(ii)functions 𝑓 and 𝑔 are continuous on 𝐶×𝐷;(iii)for any 𝑦∈𝐷, the function 𝑓(⋅,𝑦) is convex on 𝐶; for any 𝑥∈𝐶, the function 𝑔(𝑥,⋅) is convex on 𝐷. Then, SQEP (2.2) is 𝛼-well-posed if and only if 𝑀_𝜀≠∅,∀𝜀>0,lim_𝜀→0diam𝑀_𝜀=0. (3.23)

To illustrate Theorem 3.4, we give the following two examples.

Example 3.6. Let 𝑋=𝑌=𝐑 and 𝐶=𝐷=𝐑⁺(=[0,+∞)). Let 𝑆(𝑥,𝑦)=[0,𝑥], 𝑇(𝑥,𝑦)=[0,𝑦], 𝐴(𝑥,𝑦)=𝐵(𝑥,𝑦)=−(𝑥−𝑦)², 𝜂(𝑥,𝑧)=𝑥−𝑧, 𝜂(𝑦,𝑤)=𝑦−𝑤, 𝑓(𝑥,𝑦)=𝑥²−𝑦², and 𝑔(𝑥,𝑦)=𝑦²−𝑥² for all 𝑥,𝑧∈𝐶 and 𝑦,𝑤∈𝐷. Obviously, the conditions (i)–(v) of Theorem 3.4 are satisfied. Note that 𝐴(𝛼𝑥,𝑦∈𝐶×𝐷∶𝑑𝑥,𝑆𝑥,𝑦≤𝜀,𝑥,𝑦,𝜂𝑥,𝑧)+𝑓𝑥,𝑦−𝑓𝑧,𝑦≤𝜀+2‖𝑥−𝑧‖²=,∀𝑧∈𝑆𝑥,𝑦𝑥,𝑦∈𝐶×𝐷∶𝑑𝑥,𝑆𝑥,𝑦≤𝜀,−𝑥−𝑦²(𝑥−𝑧)+𝑥²−𝑧²𝛼≤𝜀+2(𝑥−𝑧)²=,∀𝑧∈𝑆𝑥,𝑦𝑥,𝑦∈𝐶×𝐷∶𝑑𝑥,𝑆𝑥,𝑦≤𝜀,−𝑥−𝑦²()−()𝑥−𝑧2+𝛼𝑧−𝛼𝑥2+𝛼²+4𝑥2+𝛼²=−2𝜀≤0,∀𝑧∈𝑆𝑥,𝑦0,(2+𝛼)𝜀2𝐵,×𝐑,𝑥,𝑦∈𝐶×𝐷∶𝑑𝑦,𝑇𝑥,𝑦≤𝜀,𝑥,𝑦𝜂()𝛼𝑦,𝑤+𝑔𝑥,𝑦−𝑔𝑥,𝑤≤𝜀+2𝑦−𝑤²=,∀𝑤∈𝑇𝑥,𝑦𝑥,𝑦∈𝐶×𝐷∶𝑑𝑦,𝑇𝑥,𝑦≤𝜀,−𝑥−𝑦²𝑦−𝑤+𝑦²−𝑤²𝛼≤𝜀+2𝑦−𝑤²=,∀𝑤∈𝑇𝑥,𝑦𝑥,𝑦∈𝐶×𝐷∶𝑑𝑦,𝑇𝑥,𝑦≤𝜀,−𝑥−𝑦²−()𝑦−𝑤2+𝛼𝑤−𝛼𝑦2+𝛼²+4𝑦2+𝛼²−2𝜀≤0,∀𝑤∈𝑇𝑥,𝑦=𝐑×0,(2+𝛼)𝜀2. (3.24) It follows that Ω_𝛼(𝜀)=0,(2+𝛼)𝜀2×0,(2+𝛼)𝜀2 (3.25) and so diamΩ_𝛼→0 as 𝜀→0. By Theorem 3.4, SMQVLI (2.1) is 𝛼-well-posed.

Example 3.7. Let 𝑋=𝑌=𝐑 and 𝐶=𝐷=𝐑⁺(=[0,+∞)). Let 𝑆(𝑥,𝑦)=[0,𝑥], 𝑇(𝑥,𝑦)=[0,𝑦], 𝐴(𝑥,𝑦)=𝐵(𝑥,𝑦)=−(𝑥−𝑦)², 𝜂(𝑥,𝑧)=𝑥−𝑧, 𝜂(𝑦,𝑤)=𝑦−𝑤, and 𝑓(𝑥,𝑦)=𝑔(𝑥,𝑦)=−𝑥𝑦 for all 𝑥,𝑧∈𝐶 and 𝑦,𝑤∈𝐷. It is easy to see that the conditions (i)–(v) of Theorem 3.4 are satisfied, and Ω_𝛼(𝜀)=[0,+∞)×[0,+∞). But, SMQVLI (2.1) is not 𝛼-well-posed, since diamΩ_𝛼(𝜀)↛0 as 𝜀→0.

Whenever 𝛼=0, we have the following result.

Theorem 3.8. Assume that the following conditions hold:(i)set-valued mappings 𝑆 and 𝑇 are nonempty convex-valued, (𝑠,𝑤)-closed, (𝑠,𝑠)-lower semicontinuous, and (𝑠,𝑤)-subcontinuous on 𝐶×𝐷;(ii)single-valued mappings 𝐴 and 𝐵 are (𝑠,𝑤^∗)-continuous on 𝐶×𝐷;(iii)single-valued mappings 𝜂 and 𝜂 are Lipschitz continuous with constants 𝜆 and 𝜆, respectively, such that for all 𝑥₁,𝑥₂,𝑥₃∈𝐶 and 𝑦₁,𝑦₂,𝑦₃∈𝐷∶𝜂𝑥₁,𝑥₃=𝜂𝑥₁,𝑥₂+𝜂𝑥₂,𝑥₃,𝜂𝑦₁,𝑦₃=𝜂𝑦₁,𝑦₂+𝜂𝑦₂,𝑦₃; (3.26)(iv)functions 𝑓 and 𝑔 are continuous on 𝐶×𝐷. Then, SMQVLI (2.1) is well-posed if and only if Ω₀(𝜀)≠∅,∀𝜀>0,lim_𝜀→0diamΩ₀(𝜀)=0. (3.27)

Corollary 3.9 (i.e., [35, Corollary  3.1]). Assume that the following conditions hold:(i)set-valued mappings 𝑆 and 𝑇 are nonempty convex-valued, (𝑠,𝑤)-closed, (𝑠,𝑠)-lower semicontinuous, and (𝑠,𝑤)-subcontinuous on 𝐶×𝐷;(ii)functions 𝑓 and 𝑔 are continuous on 𝐶×𝐷. Then, SQEP (2.2) is well-posed if and only if 𝑀_𝜀≠∅,∀𝜀>0,lim_𝜀→0diam𝑀_𝜀=0. (3.28)

The following theorem shows that under some suitable conditions, the 𝛼-well-posedness of SMQVLI (2.1) is equivalent to the existence and uniqueness of its solutions.

Theorem 3.10. Let 𝑋 and 𝑌 be two finite-dimensional spaces. Suppose that the following conditions hold:(i)set-valued mappings 𝑆 and 𝑇 are nonempty convex-valued, closed, lower semicontinuous, and subcontinuous on 𝐶×𝐷;(ii)single-valued mappings 𝐴 and 𝐵 are continuous on 𝐶×𝐷;(iii) single-valued mappings 𝜂 and 𝜂 are Lipschitz continuous with constants 𝜆 and 𝜆 respectively, such that(a)𝜂(𝑥₁,𝑥₃)=𝜂(𝑥₁,𝑥₂)+𝜂(𝑥₂,𝑥₃) for all 𝑥₁,𝑥₂,𝑥₃∈𝐶 and 𝜂(𝑦₁,𝑦₃)=𝜂(𝑦₁,𝑦₂)+𝜂(𝑦₂,𝑦₃) for all 𝑦₁,𝑦₂,𝑦₃∈𝐷,(b)𝜂(⋅,⋅) and 𝜂(⋅,⋅) both are affine in the second variable;(iv) the functions 𝑓 and 𝑔 are continuous on 𝐶×𝐷;(v)for any 𝑦∈𝐷, the function 𝑓(⋅,𝑦) is convex on 𝐶; for any 𝑥∈𝐶, the function 𝑔(𝑥,⋅) is convex on 𝐷;(vi)Ω_𝛼(𝜀) is nonempty bounded for some 𝜀>0. Then, SMQVLI (2.1) is 𝛼-well-posed if and only if SMQVLI (2.1) has a unique solution.

Proof. The necessary of the theorem is obvious. In order to show the sufficiency, let (𝑥₀,𝑦₀) be the unique solution of SMQVLI (2.1) and let {(𝑥_𝑛,𝑦_𝑛)} be any 𝛼-approximating sequence for SMQVLI (2.1). Then there exists a sequence 𝜀_𝑛>0 with 𝜀_𝑛→0 such that 𝑑𝑥_𝑛,𝑆𝑥_𝑛,𝑦_𝑛≤𝜀_𝑛,𝐴𝑥_𝑛,𝑦_𝑛,𝜂𝑥_𝑛,𝑧+𝑓𝑥_𝑛,𝑦_𝑛−𝑓𝑧,𝑦_𝑛≤𝜀_𝑛+𝛼2𝑥_𝑛−𝑧²,∀𝑧∈𝑆𝑥_𝑛,𝑦_𝑛,𝑑𝑦_𝑛,𝑇𝑥_𝑛,𝑦_𝑛≤𝜀_𝑛,𝐵𝑥_𝑛,𝑦_𝑛,𝜂𝑦_𝑛,𝑤+𝑔𝑥_𝑛,𝑦_𝑛−𝑔𝑥_𝑛,𝑤≤𝜀_𝑛+𝛼2𝑦_𝑛−𝑤²,∀𝑤∈𝑇𝑥_𝑛,𝑦_𝑛, (3.29) which means {(𝑥_𝑛,𝑦_𝑛)}⊂Ω_𝛼(𝜀_𝑛),forall𝑛∈𝑁. Let 𝜀>0 be such that Ω_𝛼(𝜀) is nonempty bounded. Then there exists 𝑛₀∈𝑁 such that {(𝑥_𝑛,𝑦_𝑛)}⊂Ω_𝛼(𝜀_𝑛)⊂Ω_𝛼(𝜀) for all 𝑛≥𝑛₀. Thus, {(𝑥_𝑛,𝑦_𝑛)} is bounded and so the sequence {(𝑥_𝑛,𝑦_𝑛)} has a subsequence {(𝑥_𝑛𝑘,𝑦_𝑛𝑘)} which converges to (𝑥,𝑦). Reasoning as in Theorem 3.4, one can prove that (𝑥,𝑦) solves SMQVLI (2.1). The uniqueness of the solution implies that (𝑥₀,𝑦₀)=(𝑥,𝑦), and so the whole sequence {(𝑥_𝑛,𝑦_𝑛)} converges to (𝑥₀,𝑦₀). Thus, SMQVLI (2.1) is 𝛼-well-posed. This completes the proof.

Example 3.11. Let 𝑋=𝑌=𝐑 and 𝐶=𝐷=𝐑⁺(=[0,+∞)). Let 𝑆(𝑥,𝑦)=[0,𝑥], 𝑇(𝑥,𝑦)=[0,𝑦], 𝐴(𝑥,𝑦)=𝐵(𝑥,𝑦)=−(𝑥−𝑦)², 𝜂(𝑥,𝑧)=𝑥−𝑧, 𝜂(𝑦,𝑤)=𝑦−𝑤, 𝑓(𝑥,𝑦)=𝑥²−𝑦², and 𝑔(𝑥,𝑦)=𝑦²−𝑥² for all 𝑥,𝑧∈𝐶 and 𝑦,𝑤∈𝐷. Clearly, the conditions (i)–(vi) of Theorem 3.8 are satisfied, and SMQVLI (2.1) has a unique solution (𝑥₀,𝑦₀)=(0,0). By Theorem 3.8, SMQVLI (2.1) is 𝛼-well-posed.

Corollary 3.12 (i.e., [35, Theorem  3.3]). Let 𝑋 and 𝑌 be two finite-dimensional spaces. Suppose that the following conditions hold:(i)set-valued mappings 𝑆 and 𝑇 are nonempty convex-valued, closed, lower semicontinuous and subcontinuous on 𝐶×𝐷;(ii)the functions 𝑓 and 𝑔 are continuous on 𝐶×𝐷;(iii) for any 𝑦∈𝐷, the function 𝑓(⋅,𝑦) is convex on 𝐶; for any 𝑥∈𝐶, the function 𝑔(𝑥,⋅) is convex on 𝐷;(iv)𝑀_𝜀 is nonempty bounded for some 𝜀>0. Then, SQEP (2.2) is 𝛼-well-posed if and only if SQEP (2.2) has a unique solution.

4. Metric Characterizations of 𝛼-Well-Posedness in the Generalized Sense for SMQVLI

In this section, we derive some metric characterizations of 𝛼-well-posedness in the generalized sense for SMQVLI (2.1) by considering the noncompactness of approximate solution set.

Theorem 4.1. SMQVLI (2.1) is 𝛼-well-posed in the generalized sense if and only if the solution set Γ of SMQVLI (2.1) is nonempty compact and 𝑒Ω_𝛼(𝜀),Γ⟶0a𝑠𝜀⟶0. (4.1)

Proof . Suppose that SMQVLI (2.1) is 𝛼-well-posed in the generalized sense. Then Γ is nonempty. To show the compactness of Γ, let {(𝑥_𝑛,𝑦_𝑛)}⊂Γ. Clearly, if {(𝑥_𝑛,𝑦_𝑛)} is an approximation sequence of SMQVLI (2.1), then it is also 𝛼-approximation sequence. Since SMQVLI (2.1) is 𝛼-well-posed in the generalized sense, it contains a subsequence converging strongly to an element of Γ. Thus, Γ is compact. Now, we prove that (4.1) holds. Suppose by contradiction that there exist 𝛾>0, 0<𝜀_𝑛→0, and (𝑥_𝑛,𝑦_𝑛)∈Ω_𝛼(𝜀_𝑛) such that 𝑑𝑥_𝑛,𝑦_𝑛,Γ≥𝛾. (4.2) Being {(𝑥_𝑛,𝑦_𝑛)}⊂Ω_𝛼(𝜀_𝑛),{(𝑥_𝑛,𝑦_𝑛)} is an 𝛼-approximating sequence for SMQVLI (2.1). Since SMQVLI (2.1) is 𝛼-well-posed in the generalized sense, there exists a subsequence {(𝑥_𝑛𝑘,𝑦_𝑛𝑘)} of {(𝑥_𝑛,𝑦_𝑛)} converging strongly to some element of Γ. This contradicts (4.2) and so (4.1) holds.
To prove the converse, suppose that Γ is nonempty compact and (4.1) holds. Let {(𝑥_𝑛,𝑦_𝑛)} be an 𝛼-approximating sequence for SMQVLI (2.1). Then {(𝑥_𝑛,𝑦_𝑛)}⊂Ω_𝛼(𝜀_𝑛), and so 𝑒(Ω_𝛼(𝜀_𝑛),Γ)→0. This implies that there exists a sequence {(𝑧_𝑛,𝑤_𝑛)}⊂Γ such that 𝑥_𝑛,𝑦_𝑛−𝑧_𝑛,𝑤_𝑛⟶0, (4.3) where the norm ‖⋅‖ in the product space 𝑋×𝑌 is defined as follows: ‖(𝑢,𝑣)−(𝑢,𝑣)‖=‖𝑢−𝑢‖²+‖𝑣−𝑣‖²,∀(𝑢,𝑣),(𝑢,𝑣)∈𝑋×𝑌.Itisnothardtoverifythat𝑋×𝑌isaBanachspaceintermsofthelastnorm. (4.4)
Since Γ is compact, there exists a subsequence {(𝑧_𝑛𝑘,𝑤_𝑛𝑘)} of {(𝑧_𝑛,𝑤_𝑛)} converging strongly to (𝑥₀,𝑦₀)∈Γ. Hence the corresponding subsequence {(𝑥_𝑛𝑘,𝑦_𝑛𝑘)} of {(𝑥_𝑛,𝑦_𝑛)} converges strongly to (𝑥₀,𝑦₀). Therefore, SMQVLI (2.1) is 𝛼-well-posed in the generalized sense.

We give the following example to illustrate that the compactness condition of Γ is necessary.

Example 4.2. Let 𝑋=𝑌=𝐑 and 𝐶=𝐷=𝐑⁺(=[0,+∞)). Let 𝑆(𝑥,𝑦)=[𝑥,𝑥+𝑦], 𝑇(𝑥,𝑦)=[𝑦,𝑥+𝑦], 𝐴(𝑥,𝑦)=𝐵(𝑥,𝑦)=𝑥²+𝑦², 𝜂(𝑥,𝑧)=𝑥−𝑧, 𝜂(𝑦,𝑤)=𝑦−𝑤, and 𝑓(𝑥,𝑦)=𝑔(𝑥,𝑦)=𝑥𝑦 for all 𝑥,𝑧∈𝐶 and 𝑦,𝑤∈𝐷. Then Γ=Ω_𝛼(𝜀)=[0,+∞)×[0,+∞). It is clear that 𝑒(Ω_𝛼(𝜀),Γ)→0 as 𝜀→0. It is easy to see that the diverging sequence {(𝑛,𝑛)}_𝑛∈𝑁 is an 𝛼-approximating sequence, but it has no convergent subsequence. Therefore, SMQVLI (2.1) is not 𝛼-well-posed in the generalized sense.

Corollary 4.3 (i.e., [35, Theorem  4.1]). SQEP (2.2) is 𝛼-well-posed in the generalized sense if and only if the solution set Γ of SQEP (2.2) is nonempty compact and 𝑒𝑀_𝜀,Γ⟶0as𝜀⟶0. (4.5)

Theorem 4.4. Assume that the following conditions hold:(i)set-valued mappings 𝑆 and 𝑇 are nonempty convex-valued, (𝑠,𝑤)-closed, (𝑠,𝑠)-lower semicontinuous, and (𝑠,𝑤)-subcontinuous on 𝐶×𝐷;(ii)single-valued mappings 𝐴 and 𝐵 are (𝑠,𝑤^∗)-continuous on 𝐶×𝐷;(iii) single-valued mappings 𝜂 and 𝜂 are Lipschitz continuous with constants 𝜆 and 𝜆 respectively, such that(a)𝜂(𝑥₁,𝑥₃)=𝜂(𝑥₁,𝑥₂)+𝜂(𝑥₂,𝑥₃) for all 𝑥₁,𝑥₂,𝑥₃∈𝐶 and 𝜂(𝑦₁,𝑦₃)=𝜂(𝑦₁,𝑦₂)+𝜂(𝑦₂,𝑦₃) for all 𝑦₁,𝑦₂,𝑦₃∈𝐷,(b)𝜂(⋅,⋅) and 𝜂(⋅,⋅) both are affine in the second variable;(iv) functions 𝑓 and 𝑔 are continuous on 𝐶×𝐷;(v)for any 𝑦∈𝐷, the function 𝑓(⋅,𝑦) is convex on 𝐶; for any 𝑥∈𝐶, the function 𝑔(𝑥,⋅) is convex on 𝐷. Then, SMQVLI (2.1) is 𝛼-well-posed in the generalized sense if and only if Ω_𝛼(𝜀)≠∅,∀𝜀>0,lim_𝜀→0𝜇Ω_𝛼(𝜀)=0. (4.6)

Proof . Suppose that SMQVLI (2.1) is 𝛼-well-posed in the generalized sense. By the same argument as in Theorem 4.1, Γ is nonempty compact, and 𝑒(Ω_𝛼(𝜀),Γ)→0 as 𝜀→0. Clearly Ω_𝛼(𝜀)≠∅ for any 𝜀>0, because Γ⊂Ω_𝛼(𝜀). Observe that for any 𝜀>0, we have ℋΩ_𝛼(𝜀),Γ=max𝑒Ω_𝛼(𝜀),Γ,𝑒Γ,Ω_𝛼(𝜀)=𝑒Ω_𝛼(𝜀),Γ. (4.7) Since Γ is compact, 𝜇(Γ)=0 and the following relation holds (see, e.g, [2]): 𝜇Ω_𝛼(𝜀)≤2ℋΩ_𝛼(𝜀),Γ+𝜇(Γ)=2ℋΩ_𝛼(𝜀),Γ=2𝑒Ω_𝛼(𝜀),Γ. (4.8) It follows that (4.6) holds.
Conversely, suppose that (4.6) holds. It is easy to prove that Ω_𝛼(𝜀), for any 𝜀>0, is closed. Note that Ω_𝛼(𝜀)⊂Ω_𝛼(𝜀′) whenever 𝜀<𝜀^′, their intersection Ω_𝛼=⋂_𝜀>0Ω_𝛼(𝜀) is nonempty compact and satisfies lim_𝜀→0ℋ(Ω_𝛼(𝜀),Ω_𝛼)=0 [39, page 412], where Ω_𝛼=𝑥₀,𝑦₀∈𝐶×𝐷∶𝑥₀∈𝑆𝑥₀,𝑦₀,𝐴𝑥₀,𝑦₀,𝜂𝑥₀,𝑧+𝑓𝑥₀,𝑦₀−𝑓𝑧,𝑦₀≤𝛼2𝑥₀−𝑧²,∀𝑧∈𝑆𝑥₀,𝑦₀,𝑦₀∈𝑇𝑥₀,𝑦₀,𝐵𝑥₀,𝑦₀,𝜂𝑦₀𝑥,𝑤+𝑔₀,𝑦₀−𝑔𝑥₀≤𝛼,𝑤2𝑦₀−𝑤²,∀𝑤∈𝑇𝑥₀,𝑦₀. (4.9) By Lemma 2.9, we obtain that Ω_𝛼 coincides with the solution set Γ of SMQVLI (2.1). Thus, Γ is compact.
Let {(𝑥_𝑛,𝑦_𝑛)} be any 𝛼-approximating sequence for SMQVLI (2.1). Then there exists a sequence 𝜀_𝑛>0 with 𝜀_𝑛→0 such that 𝑑𝑥_𝑛,𝑆𝑥_𝑛,𝑦_𝑛≤𝜀_𝑛,𝐴𝑥_𝑛,𝑦_𝑛,𝜂𝑥_𝑛,𝑧+𝑓𝑥_𝑛,𝑦_𝑛−𝑓𝑧,𝑦_𝑛≤𝜀_𝑛+𝛼2𝑥_𝑛−𝑧²,∀𝑧∈𝑆𝑥_𝑛,𝑦_𝑛,𝑑𝑦_𝑛,𝑇𝑥_𝑛,𝑦_𝑛≤𝜀_𝑛,𝐵𝑥_𝑛,𝑦_𝑛,𝜂𝑦_𝑛,𝑤+𝑔𝑥_𝑛,𝑦_𝑛−𝑔𝑥_𝑛,𝑤≤𝜀_𝑛+𝛼2𝑦_𝑛−𝑤²,∀𝑤∈𝑇𝑥_𝑛,𝑦_𝑛, (4.10) which means {(𝑥_𝑛,𝑦_𝑛)}⊂Ω_𝛼(𝜀_𝑛),∀𝑛∈𝑁. It follows from (4.6) that there exists a sequence {(𝑧_𝑛,𝑤_𝑛)}⊂Γ such that 𝑥_𝑛,𝑦_𝑛−𝑧_𝑛,𝑤_𝑛=𝑑𝑥_𝑛,𝑦_𝑛,Γ≤𝑒Ω_𝛼𝜀_𝑛,Γ=ℋΩ_𝛼𝜀_𝑛,Γ⟶0. (4.11) Since Γ is compact, there exists a subsequence {(𝑧_𝑛𝑗,𝑤_𝑛𝑗)} of {(𝑧_𝑛,𝑤_𝑛)} converging strongly to (𝑥₀,𝑦₀)∈Γ. Hence, the corresponding subsequence {(𝑥_𝑛𝑗,𝑦_𝑛𝑗)} of {(𝑥_𝑛,𝑦_𝑛)} converges strongly to (𝑥₀,𝑦₀). Thus, SMQVLI (2.1) is 𝛼-well-posed in the generalized sense.

Example 4.5. Let 𝑋=𝑌=𝐑 and 𝐶=𝐷=[0,1]. Let 𝑆(𝑥,𝑦)=[0,𝑥], 𝑇(𝑥,𝑦)=[0,𝑦], 𝐴(𝑥,𝑦)=𝐵(𝑥,𝑦)=−(𝑥−𝑦)², 𝜂(𝑥,𝑧)=𝑥−𝑧,𝜂(𝑦,𝑤)=𝑦−𝑤, and 𝑓(𝑥,𝑦)=𝑔(𝑥,𝑦)=−𝑥𝑦 for all 𝑥,𝑧∈𝐶 and 𝑦,𝑤∈𝐷. Obviously, the conditions (i)–(v) of Theorem 4.4 are satisfied, and Ω_𝛼(𝜀)=[0,1]×[0,1]. By Theorem 4.4, SMQVLI (2.1) is 𝛼-well-posed in the generalized sense.

Corollary 4.6 (i.e., [35, Theorem  4.2]). Assume that the following conditions hold:(i)set-valued mappings 𝑆 and 𝑇 are nonempty convex-valued, (𝑠,𝑤)-closed, (𝑠,𝑠)-lower semicontinuous, and (𝑠,𝑤)-subcontinuous on 𝐶×𝐷;(ii)functions 𝑓 and 𝑔 are continuous on 𝐶×𝐷;(iii)for any 𝑦∈𝐷, the function 𝑓(⋅,𝑦) is convex on C; for any 𝑥∈𝐶, the function 𝑔(𝑥,⋅) is convex on 𝐷. Then, SQEP (2.2) is 𝛼-well-posed in the generalized sense if and only if 𝑀_𝜀≠∅,∀𝜀>0,lim_𝜀→0𝜇𝑀_𝜀=0. (4.12)

We now give a sufficient condition for the 𝛼-well-posedness in the generalized sense of SMQVLI (2.1) in finite-dimensional spaces.

Theorem 4.7. Let 𝑋 and 𝑌 be two finite-dimensional spaces. Suppose that the following conditions hold:(i)set-valued mappings 𝑆 and 𝑇 are nonempty convex-valued, closed, lower semicontinuous, and subcontinuous on 𝐶×𝐷;(ii)single-valued mappings 𝐴 and 𝐵 are continuous on 𝐶×𝐷;(iii) single-valued mappings 𝜂 and 𝜂 are Lipschitz continuous with constants 𝜆 and 𝜆 respectively, such that(a)(𝑎) 𝜂(𝑥₁,x₃)=𝜂(𝑥₁,𝑥₂)+𝜂(𝑥₂,𝑥₃) for all 𝑥₁,𝑥₂,𝑥₃∈𝐶 and 𝜂(𝑦₁,𝑦₃)=𝜂(𝑦₁,𝑦₂)+𝜂(𝑦₂,𝑦₃) for all 𝑦₁,𝑦₂,𝑦₃∈𝐷,(b)(𝑏) 𝜂(⋅,⋅) and 𝜂(⋅,⋅) both are affine in the second variable;(iv) functions 𝑓 and 𝑔 are continuous on 𝐶×𝐷;(v)for any 𝑦∈𝐷, the function 𝑓(⋅,𝑦) is convex on 𝐶; for any 𝑥∈𝐶, the function 𝑔(𝑥,⋅) is convex on 𝐷;(vi)Ω_𝛼(𝜀) is nonempty bounded for some 𝜀>0. Then, SMQVLI (2.1) is 𝛼-well-posed in the generalized sense.

Proof. Let {(𝑥_𝑛,𝑦_𝑛)} be any 𝛼-approximating sequence for SMQVLI (2.1). Then there exists a sequence 𝜀_𝑛>0 with 𝜀_𝑛→0 such that 𝑑𝑥_𝑛,𝑆𝑥_𝑛,𝑦_𝑛≤𝜀_𝑛,𝐴𝑥_𝑛,𝑦_𝑛,𝜂𝑥_𝑛,𝑧+𝑓𝑥_𝑛,𝑦_𝑛−𝑓𝑧,𝑦_𝑛≤𝜀_𝑛+𝛼2𝑥_𝑛−𝑧²,∀𝑧∈𝑆𝑥_𝑛,𝑦_𝑛,𝑑𝑦_𝑛,𝑇𝑥_𝑛,𝑦_𝑛≤𝜀_𝑛,𝐵𝑥_𝑛,𝑦_𝑛,𝜂𝑦_𝑛,𝑤+𝑔𝑥_𝑛,𝑦_𝑛−𝑔𝑥_𝑛,𝑤≤𝜀_𝑛+𝛼2𝑦_𝑛−𝑤²,∀𝑤∈𝑇𝑥_𝑛,𝑦_𝑛. (4.13) As proven in Theorem 3.8, {(𝑥_𝑛,𝑦_𝑛)} is bounded. Then there exists a subsequence {(𝑥_𝑛𝑗,𝑦_𝑛𝑗)} of {(𝑥_𝑛,𝑦_𝑛)} which converges to (𝑥₀,𝑦₀). Reasoning as in Theorem 3.4, one can prove that (𝑥₀,𝑦₀) solves SMQVLI (2.1). Therefore, SMQVLI (2.1) is 𝛼-well-posed in the generalized sense.

The following example shows that the nonempty boundedness of Ω_𝛼(𝜀) is necessary for some 𝜀>0.

Example 4.8. Let 𝑋=𝑌=𝐑 and 𝐶=𝐷=𝐑⁺(=[0,+∞)). Let 𝑆(𝑥,𝑦)=[0,𝑥], 𝑇(𝑥,𝑦)=[0,𝑦], 𝐴(𝑥,𝑦)=𝐵(𝑥,𝑦)=−(𝑥−𝑦)², 𝜂(𝑥,𝑧)=𝑥−𝑧, 𝜂(𝑦,𝑤)=𝑦−𝑤, and 𝑓(𝑥,𝑦)=𝑔(𝑥,𝑦)=−𝑥𝑦 for all 𝑥,𝑧∈𝐶 and 𝑦,𝑤∈𝐷. Clearly, the conditions (i)–(v) of Theorem 4.7 are satisfied. But Ω_𝛼(𝜀)=[0,+∞)×[0,+∞) is unbounded. Therefore, SMQVLI (2.1) is not 𝛼-well-posed in the generalized sense.

Corollary 4.9 (i.e., [35, Theorem  4.3]).. Let 𝑋 and 𝑌 be two finite-dimensional spaces. Suppose that the following conditions hold:(i)set-valued mappings 𝑆 and 𝑇 are nonempty convex-valued, closed, lower semicontinuous, and subcontinuous on 𝐶×𝐷;(ii)functions 𝑓 and 𝑔 are continuous on 𝐶×𝐷;(iii) for any 𝑦∈𝐷, the function 𝑓(⋅,𝑦) is convex on 𝐶; for any 𝑥∈𝐶, the function 𝑔(𝑥,⋅) is convex on 𝐷;(iv)𝑀_𝜀 is nonempty bounded for some 𝜀>0. Then, SQEP (2.2) is 𝛼-well-posed in the generalized sense.

5. Concluding Remarks

In this paper, we generalize the concept of 𝛼-well-posedness to the system of mixed quasivariational-like inequalities which includes as a special case symmetric quasiequilibrium problems considered by Long and Huang [35]. It is well known that symmetric quasiequilibrium problems include equilibrium problems, Nash equilibrium problems, quasivariational inequalities, variational inequalities, and fixed-point problems as special cases. It is worth emphasizing that the results presented in [35] generalize and improve some known results in the recent literature; see, for example, [12–15, 23, 33].

Further, under some suitable conditions, we obtain some metric characterizations of 𝛼-well-posedness for the system of mixed quasivariational-like inequalities in Banach spaces. The results presented in this paper represent the supplement, improvement, generalization, and development of Long and Huang’s known results [35] in the following aspects.(i)The symmetric quasiequilibrium problem (SQEP) in [35] is extended to develop the more general problem, that is, the system of mixed quasivariational-like inequalities (SMQVLIs) in the setting of Banach spaces. Moreover, the concept of 𝛼-well-posedness (resp., in the generalized sense) for SQEP is extended to develop the concept of 𝛼-well-posedness (resp., in the generalized sense) for SMQVLI.(ii)Since the system of mixed quasivariational-like inequalities (SMQVLIs) is more general and more complicated than the symmetric quasiequilibrium problem (SQEP), the assumptions in our results are very different from the corresponding ones in [35]; see, for instance, the assumptions imposed on the single-valued mappings 𝐴,𝐵 and 𝜂,𝜂.(iii)The new technique of arguments are applied to deriving our main results. As a matter of fact, in the process of proving our main results, our arguments depend on the properties of the mappings 𝜂,𝜂, the uniform boundedness principle for a family of linear continuous functionals, the completion of the Banach space (𝑋×𝑌,‖⋅‖), and so forth. For instance, see the proof of Theorem 3.4.

Acknowledgment

This research was partially supported by Grant no. NSC99-2115-M-039-001 from the National Science Council of Taiwan. This research was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), and Leading Academic Discipline Project of Shanghai Normal University (DZL707).