Well-Posedness by Perturbations of Generalized Mixed Variational Inequalities in Banach Spaces

Abstract

We consider an extension of the notion of well-posedness by perturbations, introduced by Zolezzi (1995, 1996) for a minimization problem, to a class of generalized mixed variational inequalities in Banach spaces, which includes as a special case the class of mixed variational inequalities. We establish some metric characterizations of the well-posedness by perturbations. On the other hand, it is also proven that, under suitable conditions, the well-posedness by perturbations of a generalized mixed variational inequality is equivalent to the well-posedness by perturbations of the corresponding inclusion problem and corresponding fixed point problem. Furthermore, we derive some conditions under which the well-posedness by perturbations of a generalized mixed variational inequality is equivalent to the existence and uniqueness of its solution.

1. Introduction

Let 𝑋 be a real Banach space and 𝑓∶𝑋→𝐑∪{+∞} a real-valued functional on 𝑋. In 1966, Tikhonov [1] first introduced the classical notion of well-posedness for a minimization problem min_𝑥∈𝑋𝑓(𝑥), which has been known as the Tikhonov well-posedness. A minimization problem is said to be Tikhonov well-posed if it has a unique solution toward which every minimizing sequence of the problem converges. It is obvious that the notion of Tikhonov well-posedness is inspired by the numerical methods producing optimizing sequences for optimization problems and plays a crucial role in the optimization theory. The notion of generalized Tikhonov well-posedness is also introduced for a minimization problem having more than one solution, which requires the existence of solutions and the convergence of some subsequence of every minimizing sequence toward some solution. Another important notion of well-posedness for a minimization problem is the well-posedness by perturbations or extended well-posedness due to Zolezzi [2, 3]. The notion of well-posedness by perturbations establishes a form of continuous dependence of the solutions upon a parameter. There are many other notions of well-posedness in optimization problems. For more details, we refer the readers to [1–7] and the references therein.

On the other hand, the concept of well-posedness has been generalized to other variational problems, such as variational inequalities [4, 8–14], saddle point problems [15], Nash equilibrium problems [14, 16–18], equilibrium problems [19], inclusion problems [20, 21], and fixed point problems [20–22]. An initial notion of well-posedness for a variational inequality is due to Lucchetti and Patrone [4]. 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 papers have been devoted to the extensions of well-posedness of minimization problems to various variational inequalities. Lignola and Morgan [12] generalized the notion of well-posedness by perturbations to a variational inequality and established the equivalence between the well-posedness by perturbations of a variational inequality and the well-posedness by perturbations of the corresponding minimization problem. Lignola and Morgan [14] introduced the concepts of 𝛼-well-posedness for variational inequalities. Del Prete et al. [13] further proved that the 𝛼-well-posedness of variational inequalities is closely related to the well-posedness of minimization problems. Recently, Fang et al. [9] generalized the notions of well-posedness and 𝛼-well-posedness to a mixed variational inequality. In the setting of Hilbert spaces, Fang et al. [9] proved that under suitable conditions the well-posedness of a mixed variational inequality is equivalent to the existence and uniqueness of its solution. They also showed that the well-posedness of a mixed variational inequality has close links with the well-posedness of the corresponding inclusion problem and corresponding fixed point problem in the setting of Hilbert spaces. Subsequently, the notions of well-posedness and 𝛼-well-posedness for a mixed variational inequality in [9] are extended by Ceng and Yao [11] to a generalized mixed variational inequality in the setting of Hilbert spaces. Very recently, Fang et al. [10] generalized the notion of well-posedness by perturbations to a mixed variational inequality in Banach spaces. In the setting of Banach spaces, they established some metric characterizations and showed that the well-posedness by perturbations of a mixed variational inequality is closely related to the well-posedness by perturbations of the corresponding inclusion problem and corresponding fixed point problem. They also derived some conditions under which the well-posedness by perturbations of the mixed variational inequality is equivalent to the existence and uniqueness of its solution.

In this paper, we further extend the notion of well-posedness by perturbations to a class of generalized mixed variational inequalities in Banach spaces, which includes as a special case the class of mixed variational inequalities in [10]. Under very mild conditions, we establish some metric characterizations for the well-posed generalized mixed variational inequality and show that the well-posedness by perturbations of a generalized mixed variational inequality is closely related to the well-posedness by perturbations of the corresponding inclusion problem and corresponding fixed point problem. We also derive some conditions under which the well-posedness by perturbations of the generalized mixed variational inequality is equivalent to the existence and uniqueness of its solution.

2. Preliminaries

Throughout this paper, unless stated otherwise, we always suppose that 𝑋 is a real reflexive Banach space with its dual 𝑋^∗ and the duality pairing ⟨⋅,⋅⟩ between 𝑋 and 𝑋^∗. For convenience, we denote strong (resp., weak) convergence by → (resp., ⇀). Let 𝐹∶𝑋→2^𝑋 be a nonempty-valued multifunction, 𝐴∶𝑋→𝑋^∗ a single-valued mapping, and 𝑓∶𝑋→𝐑∪{+∞} a proper, convex, and lower semicontinuous functional. Denote by dom𝑓 the domain of 𝑓, that is,dom𝑓∶=𝑥∈𝑋∶𝑓(𝑥)<+∞. (2.1)

The generalized mixed variational inequality associated with (𝐴,𝐹,𝑓) is formulated as follows:(𝑥),(𝑥)𝑦GMVI𝐴,𝐹,𝑓:find𝑥∈𝑋suchthat,forsome𝑢∈𝐹𝐴𝑢,𝑥−𝑦+𝑓−𝑓≤0,∀𝑦∈𝑋, (2.2) which has been studied intensively (see, e.g., [11, 23–25]).

In the following, we give some special cases of GMVI(𝐴,𝐹,𝑓).(i) Whenever 𝐹=𝐼, the identity mapping of 𝑋, GMVI(𝐴,𝐹,𝑓) reduces to the following mixed variational inequality associated with (𝐴,𝑓): MVI𝐴,𝑓:find𝑥∈𝑋suchthat𝐴𝑥,𝑥−𝑦+𝑓(𝑥)−𝑓𝑦≤0,∀𝑦∈𝑋, (2.3) which has been considered in [8–11, 26].(ii) Whenever 𝑓=𝛿_𝐾, MVI(𝐴,𝑓) reduces to the following classical variational inequality: VI(𝐴,𝐾):find𝑥∈𝐾suchthat𝐴𝑥,𝑥−𝑦≤0,∀𝑦∈𝐾, (2.4) where 𝛿_𝐾 denotes the indicator functional of a convex subset 𝐾 of 𝑋.(iii) Whenever 𝐴=0, MVI(𝐴,𝑓) reduces to the global minimization problem: MP𝑓,𝑋:min_𝑥∈𝑋𝑓(𝑥). (2.5)

Suppose that 𝐿 is a parametric normed space, 𝑃⊂𝐿 is a closed ball with positive radius, and 𝑝^∗∈𝑃 is a fixed point. The perturbed problem of GMVI(𝐴,𝐹,𝑓) is always given byGMVI_𝑝(𝑥),𝐴+𝑓−𝑓𝐴,𝐹,𝑓:find𝑥∈𝑋suchthatforsome𝑢∈𝐹𝑝,𝑢,𝑥−𝑦𝑝,𝑥𝑝,𝑦≤0,∀𝑦∈𝑋, (2.6) where 𝐴∶𝑃×𝑋→𝑋^∗ is such that 𝐴(𝑝^∗,⋅)=𝐴 and 𝑓∶𝑃×𝑋→𝐑∪{+∞} is such that 𝑓(𝑝^∗,⋅)=𝑓.

Now we recall some concepts and results.

Definition 2.1 (see [26]). A mapping 𝑀∶𝑋→2^𝑋∗ is said to be(i) monotone if 𝑔−ℎ,𝑥−𝑦≥0,∀𝑥,𝑦∈Dom𝑀,𝑔∈𝑀(𝑥),ℎ∈𝑀𝑦; (2.7)(ii) maximal monotone if 𝑀 is monotone and 𝑔−𝑤,𝑥−𝑧≥0,∀𝑥∈Dom𝑀,𝑔∈𝑀(𝑥)⟹𝑤∈𝑀(𝑧), (2.8) where 2^𝑋∗ denotes the family of all subsets of 𝑋^∗ and Dom𝑀={𝑥∈𝑋∶𝑀(𝑥)≠∅}.

Definition 2.2 (see [11]). A nonempty-valued multifunction 𝐹∶𝑋→2^𝑋 is said to be monotone with respect to a single-valued mapping 𝐴∶𝑋→𝑋^∗ if, for all 𝑥,𝑦∈𝑋, 𝐴𝑢−𝐴𝑣,𝑥−𝑦≥0,∀𝑢∈𝐹(𝑥),𝑣∈𝐹𝑦. (2.9)

Proposition 2.3 (Nadler’s Theorem [27]). Let (𝑋,‖⋅‖) be a normed vector space and ℋ(⋅,⋅) the Hausdorff metric on the collection CB(𝑋) of all nonempty, closed, and bounded subsets of 𝑋, induced by a metric 𝑑 in terms of 𝑑(𝑥,𝑦)=‖𝑥−𝑦‖, which is defined by ℋ(𝑈,𝑉)=max{𝑒(𝑈,𝑉),𝑒(𝑉,𝑈)} for 𝑈 and 𝑉 in CB(𝑋), where 𝑒(𝑈,𝑉)=sup_𝑥∈𝑈𝑑(𝑥,𝑉) with 𝑑(𝑥,𝑉)=inf_𝑦∈𝑉‖𝑥−𝑦‖. If 𝑈 and 𝑉 lie in CB(𝑋), then, for any 𝜀>0 and any 𝑢∈𝑈, there exists 𝑣∈𝑉 such that ‖𝑢−𝑣‖≤(1+𝜀)ℋ(𝑈,𝑉). In particular, whenever 𝑈 and 𝑉 are compact subsets in 𝑋, one has ‖𝑢−𝑣‖≤ℋ(𝑈,𝑉).

Definition 2.4. 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(𝑥_𝑛,𝑈)→0 for all selection 𝑥_𝑛∈𝑈_𝑛. For more details on this topic, the reader is referred to [28].

Definition 2.5 (see [29]). A mapping 𝐴∶𝑋→𝑋^∗ is said to be(i) coercive if lim_{‖𝑥‖→∞}⟨𝐴𝑥,𝑥⟩‖𝑥‖=+∞; (2.10)(ii) bounded if 𝐴(𝐵) is bounded for every bounded subset 𝐵 of 𝑋;(iii) hemicontinuous if, for any 𝑥,𝑦∈𝑋, the function 𝑡↦⟨𝐴(𝑥+𝑡(𝑦−𝑥)),𝑦−𝑥⟩ from [0,1] into 𝐑 is continuous at 0⁺;(iv) uniformly continuous if, for any neighborhood 𝑉 of 0 in 𝑋^∗, there exists a neighborhood 𝑈 of 0 in 𝑋 such that 𝐴𝑥−𝐴𝑦∈𝑉 for all 𝑥,𝑦∈𝑈.

Clearly, the uniform continuity implies the continuity, and the continuity implies the hemicontinuity, but the converse is not true in general.

Definition 2.6. (i) A nonempty weakly compact-valued multifunction 𝐹∶𝑋→2^𝑋 is said to be ℋ-hemicontinuous if, for any 𝑥,𝑦∈𝑋, the function 𝑡↦ℋ(𝐹(𝑥+𝑡(𝑦−𝑥)),𝐹(𝑥)) from [0,1] into 𝐑⁺=[0,∞) is continuous at 0⁺, where ℋ is the Hausdorff metric defined on CB(𝑋).
(ii) A nonempty weakly compact-valued multifunction 𝐹∶𝑋→2^𝑋 is said to be ℋ-continuous at a point 𝑥∈𝑋 if, for any 𝜀>0, there exists 𝛿>0 such that, for all 𝑦∈𝑋 with ‖𝑥−𝑦‖<𝛿, one has ℋ(𝐹(𝑥),𝐹(𝑦))<𝜀, where ℋ is the Hausdorff metric defined on CB(𝑋). If this multifunction 𝐹∶𝑋→2^𝑋 is ℋ-continuous at each 𝑥∈𝑋, then one says that 𝐹∶𝑋→2^𝑋 is ℋ-continuous.
(iii) A nonempty weakly compact-valued multifunction 𝐹∶𝑋→2^𝑋 is said to be ℋ-uniformly continuous if, for any 𝜀>0, there exists 𝛿>0 such that, for all 𝑥,𝑦∈𝑋 with ‖𝑥−𝑦‖<𝛿, one has ℋ(𝐹(𝑥),𝐹(𝑦))<𝜀, where ℋ is the Hausdorff metric defined on CB(𝑋).

Remark 2.7. If 𝑋=𝐻 a real Hilbert space, then Definition 2.6(i)–(iii) reduce to Definition  2.3 (ii)–(iv) in [11], respectively.

Lemma 2.8. Let 𝐴∶𝑋→𝑋^∗ be weakly continuous (i.e., continuous from the weak topology of 𝑋 to the weak topology of 𝑋^∗), let 𝐹∶𝑋→2^𝑋 be a nonempty weakly compact-valued multifunction which is ℋ-hemicontinuous and monotone with respect to 𝐴, and let 𝑓∶𝑋→𝐑∪{+∞} be proper and convex. Then, for a given 𝑥∈𝑋, the following statements are equivalent:(i) there exists 𝑢∈𝐹(𝑥) such that ⟨𝐴𝑢,𝑥−𝑦⟩+𝑓(𝑥)−𝑓(𝑦)≤0, for all 𝑦∈𝑋;(ii)⟨𝐴𝑣,𝑥−𝑦⟩+𝑓(𝑥)−𝑓(𝑦)≤0, for all 𝑦∈𝑋,𝑣∈𝐹(𝑦).

Proof. Suppose that, for some 𝑢∈𝐹(𝑥), 𝐴𝑢,𝑥−𝑦+𝑓(𝑥)−𝑓𝑦≤0,∀𝑦∈𝑋. (2.11) Since 𝐹 is monotone with respect to 𝐴, one has 𝐴𝑣,𝑥−𝑦+𝑓(𝑥)−𝑓𝑦≤𝐴𝑢,𝑥−𝑦+𝑓(𝑥)−𝑓𝑦≤0,∀𝑦∈𝑋,𝑣∈𝐹𝑦. (2.12) Consequently, 𝐴𝑣,𝑥−𝑦+𝑓(𝑥)−𝑓𝑦≤0,∀𝑦∈𝑋,𝑣∈𝐹𝑦. (2.13)
Conversely, suppose that the last inequality is valid. Given any 𝑦∈𝑋, we define 𝑦_𝑡=𝑥+𝑡(𝑦−𝑥) for all 𝑡∈(0,1). Replacing 𝑦 by 𝑦_𝑡 in the left-hand side of the last inequality, one derives, for each 𝑣_𝑡∈𝐹(𝑦_𝑡), 0≥𝐴𝑣_𝑡,𝑥−𝑦_𝑡+𝑓(𝑥)−𝑓𝑦_𝑡≥𝑡𝐴𝑣_𝑡,𝑥−𝑦+𝑓(𝑥)−𝑓𝑦, (2.14) which hence implies that 𝐴𝑣_𝑡,𝑥−𝑦+𝑓(𝑥)−𝑓𝑦≤0,∀𝑣_𝑡∈𝐹𝑦_𝑡,𝑡∈(0,1). (2.15) Since 𝐹∶𝑋→2^𝑋 is a nonempty weakly compact-valued multifunction, both 𝐹(𝑦_𝑡) and 𝐹(𝑥) are nonempty weakly compact and hence are nonempty, weakly closed, and weakly bounded. Note that the weak closedness of sets in 𝑋 implies the strong closedness and that the weak boundedness of sets in 𝑋 is equivalent to the strong boundedness. Thus, it is known that both 𝐹(𝑦_𝑡) and 𝐹(𝑥) lie in CB(𝑋). From Proposition 2.3, it follows that, for each 𝑡∈(0,1) and each fixed 𝑣_𝑡∈𝐹(𝑦_𝑡), there exists a 𝑢_𝑡∈𝐹(𝑥) such that 𝑣_𝑡−𝑢_𝑡≤(1+𝑡)ℋ𝐹𝑦_𝑡,𝐹(𝑥). (2.16) Since 𝐹(𝑥) is weakly compact, it follows from the net {𝑢_𝑡∶𝑡∈(0,1)}⊂𝐹(𝑥) that there exists some subnet which converges weakly to a point of 𝐹(𝑥). Without loss of generality, we may assume that 𝑢_𝑡⇀𝑢∈𝐹(𝑥) as 𝑡→0⁺. Since 𝐹 is ℋ-hemicontinuous, one deduces that as 𝑡→0⁺𝑣_𝑡−𝑢_𝑡≤(1+𝑡)ℋ𝐹𝑦_𝑡,𝐹(𝑥)=(1+𝑡)ℋ𝐹𝑥+𝑡𝑦−𝑥,𝐹(𝑥)⟶0. (2.17) Observe that, for each 𝜑∈𝑋^∗, 𝜑,𝑣_𝑡−𝑢=𝜑,𝑣_𝑡−𝜑,𝑢_𝑡+𝜑,𝑢_𝑡−≤𝜑𝑣𝜑,𝑢_𝑡−𝑢_𝑡+𝜑,𝑢_𝑡−𝑢⟶0as𝑡⟶0⁺, (2.18) that is, 𝑣_𝑡⇀𝑢 as 𝑡→0⁺. Since 𝐴 is weakly continuous, 𝐴𝑣_𝑡⇀𝐴𝑢 and hence, for 𝑦∈𝑋, 𝐴𝑣_𝑡,𝑥−𝑦⟶𝐴𝑢,𝑥−𝑦as𝑡⟶0⁺. (2.19) Thus, letting 𝑡→0⁺ in the left-hand side of (2.15), we obtain that 𝐴𝑢,𝑥−𝑦+𝑓(𝑥)−𝑓𝑦≤0,∀𝑦∈𝑋. (2.20) Finally let us show that the vector 𝑢 in the last inequality is not dependent on 𝑦, that is, ⟨𝐴𝑢,𝑥−𝑧⟩+𝑓(𝑥)−𝑓(𝑧)≤0,∀𝑧∈𝑋. (2.21) Indeed, take a fixed 𝑧∈𝑋 arbitrarily, and define 𝑧_𝑡=𝑥+𝑡(𝑧−𝑥) for all 𝑡∈(0,1). Utilizing Proposition 2.3, for each 𝑡∈(0,1) and 𝑢_𝑡∈𝐹(𝑥), there exists 𝑤_𝑡∈𝐹(𝑧_𝑡) such that 𝑢_𝑡−𝑤_𝑡≤(1+𝑡)ℋ𝐹(𝑥),𝐹𝑧_𝑡. (2.22) Since 𝐹 is ℋ-hemicontinuous, we deduce that as 𝑡→0⁺𝑢_𝑡−𝑤_𝑡≤(1+𝑡)ℋ𝐹(𝑥),𝐹𝑧_𝑡=(1+𝑡)ℋ(𝐹(𝑥),𝐹(𝑥+𝑡(𝑧−𝑥)))⟶0. (2.23) Thus, one has, for each 𝜑∈𝑋^∗, 𝜑,𝑤_𝑡−𝑢=𝜑,𝑤_𝑡−𝜑,𝑢_𝑡+𝜑,𝑢_𝑡−≤𝜑𝑢𝜑,𝑢_𝑡−𝑤_𝑡+𝜑,𝑢_𝑡−𝑢⟶0as𝑡⟶0⁺. (2.24) This shows that 𝑤_𝑡⇀𝑢 as 𝑡→0⁺. Since 𝐴 is weakly continuous, 𝐴𝑤_𝑡⇀𝐴𝑢 and hence, for 𝑧∈𝑋, 𝐴𝑤_𝑡,𝑥−𝑧⟶⟨𝐴𝑢,𝑥−𝑧⟩as𝑡⟶0⁺. (2.25) Replacing 𝑦, 𝑦_𝑡, and 𝑣_𝑡 in (2.15) by 𝑧, 𝑧_𝑡, and 𝑤_𝑡, respectively, one concludes that 𝐴𝑤_𝑡,𝑥−𝑧+𝑓(𝑥)−𝑓(𝑧)≤0,∀𝑡∈(0,1). (2.26) This immediately implies that inequality (2.21) is valid. This completes the proof.

Corollary 2.9 (see [11, Lemma  2.2]). Let 𝐻 be a real Hilbert space. Let 𝐴∶𝐻→𝐻 be weakly continuous (i.e., continuous from the weak topology of 𝐻 to the weak topology of 𝐻), let 𝐹∶𝐻→2^𝐻 be a nonempty weakly compact-valued multifunction which is ℋ-hemicontinuous and monotone with respect to 𝐴, and let 𝑓∶𝐻→𝐑∪{+∞} be proper and convex. Then, for a given 𝑥∈𝐻, the following statements are equivalent:(i) there exists 𝑢∈𝐹(𝑥) such that ⟨𝐴𝑢,𝑥−𝑦⟩+𝑓(𝑥)−𝑓(𝑦)≤0, for all 𝑦∈𝐻;(ii)⟨𝐴𝑣,𝑥−𝑦⟩+𝑓(𝑥)−𝑓(𝑦)≤0, for all 𝑦∈𝐻,𝑣∈𝐹(𝑦).

Definition 2.10 (see [30]). Let 𝐾 be a nonempty, closed, and convex subset of 𝑋. One says that 𝐾 is well-positioned if there exist 𝑥₀∈𝑋 and 𝑔∈𝑋^∗ such that 𝑔,𝑥−𝑥₀≥𝑥−𝑥₀,∀𝑥∈𝐾. (2.27)

Remark 2.11 (see [10, Remark  2.1]). (i) If 𝐾 is well-positioned, then 𝐾+𝑥^∗ is well-positioned for all 𝑥^∗∈𝑋.
(ii) As pointed out in [30, Remark  2.2], every nonempty compact convex set of a finite-dimensional space is well-positioned. Some useful properties and interesting applications have been discussed in [30, 31]. The following result is exacted from Proposition  2.1 of [30]. Also see [31, Proposition  2.1].

Lemma 2.12. Let 𝐾 be a nonempty, closed, and convex subset of a reflexive Banach space 𝑋. If 𝐾 is well-positioned, then there is no sequence {𝑥_𝑛}⊂𝐾 with ‖𝑥_𝑛‖→+∞ such that origin is a weak limit of {𝑥_𝑛/‖𝑥_𝑛‖}.

Definition 2.13 (see [28]). Let 𝐵 be a nonempty subset of 𝑋. The measure of noncompactness 𝜇 of the set 𝐵 is defined by 𝜇(𝐵)=inf𝜀>0:𝐴⊂^𝑛_𝑖=1𝐵_𝑖,diam𝐵_𝑖<𝜀,𝑖=1,2,…,𝑛,forsomeinteger𝑛≥1, (2.28) where diam means the diameter of a set.

Lemma 2.14 (see [10, Lemma  2.3]). Let 𝐴_𝑛, 𝐴 be nonempty, closed, and convex subsets of a real reflexive Banach space 𝑋, and let 𝐴 be well-positioned. Suppose that 𝑒(𝐴_𝑛,𝐴)→0 as 𝑛→∞ and 𝑥∈𝑋. Then, there is no sequence 𝑎_𝑛∈𝐴_𝑛 with ‖𝑎_𝑛‖→+∞ such that origin is a weak limit of {(𝑎_𝑛−𝑥)/‖𝑎_𝑛−𝑥‖}.

3. Well-Posedness by Perturbations and Metric Characterizations

In this section, we generalize the concepts of well-posedness by perturbations to the generalized mixed variational inequality and establish their metric characterizations. In the sequel we always denote by → and ⇀ the strong convergence and weak convergence, respectively. Let 𝛼≥0 be a fixed number.

Definition 3.1. Let {𝑝_𝑛}⊂𝑃 be with 𝑝_𝑛→𝑝^∗. A sequence {𝑥_𝑛}⊂𝑋 is called an 𝛼-approximating sequence corresponding to {𝑝_𝑛} for GMVI(𝐴,𝐹,𝑓) if there exists a sequence {𝑢_𝑛}⊂𝑋 with 𝑢_𝑛∈𝐹(𝑥_𝑛) (for all 𝑛≥1) and a sequence {𝜀_𝑛} of nonnegative numbers with 𝜀_𝑛→0 such that 𝑥_𝑛∈dom𝑓𝑝_𝑛,𝐴𝑝,⋅_𝑛,𝑢_𝑛,𝑥_𝑛−𝑦+𝑓𝑝_𝑛,𝑥_𝑛−𝑓𝑝_𝑛≤𝛼,𝑦2𝑥_𝑛−𝑦²+𝜀_𝑛,∀𝑦∈𝑋,𝑛≥1. (3.1) Whenever 𝛼=0, we say that {𝑥_𝑛} is an approximating sequence corresponding to {𝑝_𝑛} for GMVI(𝐴,𝐹,𝑓). Clearly, every 𝛼₂-approximating sequence corresponding to {𝑝_𝑛} is 𝛼₁-approximating corresponding to {𝑝_𝑛} provided 𝛼₁>𝛼₂≥0.

Definition 3.2. One says that GMVI(𝐴,𝐹,𝑓) is strongly (resp., weakly) 𝛼-well-posed by perturbations if GMVI(𝐴,𝐹,𝑓) has a unique solution and, for any {𝑝_𝑛}⊂𝑃 with 𝑝_𝑛→𝑝^∗, every 𝛼-approximating sequence corresponding to {𝑝_𝑛} converges strongly (resp., weakly) to the unique solution. In the sequel, strong (resp., weak) 0-well-posedness by perturbations is always called strong (resp., weak) well-posedness by perturbations. If 𝛼₁>𝛼₂≥0, then strong (resp., weak) 𝛼₁-well-posedness by perturbations implies strong (resp., weak) 𝛼₂-well-posedness by perturbations.

Remark 3.3. (i) When 𝑋 is a Hilbert space and 𝑝_𝑛=𝑝^∗ (for all 𝑛≥1), Definitions  3.1 and 3.2 coincide with Definitions  3.1 and 3.2 of [11], respectively. (ii) When 𝑓=𝛿_𝐾 and 𝐹=𝐼 the identity mapping of 𝑋, Definitions 3.1 and 3.2 reduce to the definitions of approximating sequences of the classical variational inequality (see [12, 13]).

Definition 3.4. One says that GMVI(𝐴,𝐹,𝑓) is strongly (resp., weakly) generalized 𝛼-well-posed by perturbations if GMVI(𝐴,𝐹,𝑓) has a nonempty solution set 𝑆 and, for any {𝑝_𝑛}⊂𝑃 with 𝑝_𝑛→𝑝^∗, every 𝛼-approximating sequence corresponding to {𝑝_𝑛} has some subsequence which converges strongly (resp., weakly) to some point of 𝑆. Strong (resp., weak) generalized 0-well-posedness by perturbations is always called strong (resp., weak) generalized well-posedness by perturbations. Clearly, if 𝛼₁>𝛼₂≥0, then strong (resp., weak) generalized 𝛼₁-well-posedness by perturbations implies strong (resp., weak) generalized 𝛼₂-well-posedness by perturbations.

Remark 3.5. (i) When 𝑋 is a Hilbert space and 𝑝_𝑛=𝑝^∗ (for all 𝑛≥1), Definition 3.4 coincides with Definition  3.3 of [11]. (ii) When 𝑓=𝛿_𝐾 and 𝐹=𝐼 the identity mapping of 𝑋, Definition 3.4 reduces to the definition of strong (resp., weak) parametric 𝛼-well-posedness in the generalized sense for the classical variational inequality (see [12–14]). (iii) When 𝐴=0 and 𝛼=0, Definition 3.4 coincides with the definition of well-posedness by perturbations introduced for a minimization problem [2, 3].

To derive the metric characterizations of 𝛼-well-posedness by perturbations, we consider the following approximating solution set of GMVI(𝐴,𝐹,𝑓):Ω_𝛼(𝜀)=_{𝑝∈𝐵(𝑝∗,𝜀)}𝑓(𝑥)𝐴+𝑓−𝑓≤𝛼𝑥∈dom𝑝,⋅:thereexists𝑢∈𝐹suchthat𝑝,𝑢,𝑥−𝑦𝑝,𝑥𝑝,𝑦2𝑥−𝑦²+𝜀,∀𝑦∈𝑋,∀𝜀≥0, (3.2) where 𝐵(𝑝^∗,𝜀) denotes the closed ball centered at 𝑝^∗ with radius 𝜀. In this section, we always suppose that 𝑥^∗ is a fixed solution of GMVI(𝐴,𝐹,𝑓). Define𝜃(𝜀)=sup𝑥−𝑥^∗∶𝑥∈Ω_𝛼(𝜀),∀𝜀≥0. (3.3) It is easy to see that 𝜃(𝜀) is the radius of the smallest closed ball centered at 𝑥^∗ containing Ω_𝛼(𝜀). Now, we give a metric characterization of strong 𝛼-well-posedness by perturbations by considering the behavior of 𝜃(𝜀) when 𝜀→0.

Theorem 3.6. GMVI(𝐴,𝐹,𝑓) is strongly 𝛼-well-posed by perturbations if and only if 𝜃(𝜀)→0 as 𝜀→0.

Proof. Repeating almost the same argument as in the proof of [10, Theorem  3.1], we can easily obtain the desired result.

Remark 3.7. Theorem 3.6 improves Proposition  2.2 of [13], Theorem  3.1 of [9], and Theorem  3.1 of [10].

Now, we give an example to illustrate Theorem 3.6.

Example 3.8. Let 𝑋=𝐑, 𝑃=[−1,1], 𝑝^∗=0, 𝛼=2, 𝐴(𝑝,𝑥)=𝑥(𝑝²+1), 𝐹(𝑥)={𝑥,0}, and 𝑓(𝑝,𝑥)=𝑥² for all 𝑥∈𝑋, 𝑝∈𝑃. Clearly, 𝑥^∗=0 is a solution of GMVI(𝐴,𝐹,𝑓). For any 𝜀>0, it follows that Ω^𝑝_𝛼(𝜀)=𝑥∈𝐑:𝑢𝑝²+1𝑥−𝑦+𝑥²−𝑦²≤𝑥−𝑦²+𝜀,∀𝑦∈𝐑,forsome𝑢∈𝐹(𝑥)=Δ∪Λ, (3.4) where Δ=𝑥∈𝐑:𝑥𝑝²+1𝑥−𝑦+𝑥²−𝑦²≤𝑥−𝑦²,𝑝+𝜀,∀𝑦∈𝐑Λ=𝑥∈𝐑:0²+1𝑥−𝑦+𝑥²−𝑦²≤𝑥−𝑦²+𝜀,∀𝑦∈𝐑. (3.5) Observe that 𝑝Δ=𝑥∈𝐑:−2𝑦+²−14𝑥²+𝑝²+3²8𝑥²=−2√≤𝜀,∀𝑦∈𝐑2𝜀𝑝²,2√+32𝜀𝑝²,𝑥+3Λ=𝑥∈𝐑:𝑦−2²−𝑥²4+𝜀2≥0,∀𝑦∈𝐑=−√2𝜀,√2𝜀. (3.6) Thus, we obtain Ω^𝑝_𝛼(𝜀)−2√=Δ∪Λ=2𝜀𝑝²,2√+32𝜀𝑝²∪−√+32𝜀,√2𝜀=−√2𝜀,√2𝜀. (3.7) Therefore, Ω_𝛼(𝜀)=_{𝑝∈𝐵(0,𝜀)}Ω^𝑝_𝛼(𝜀)=−√2𝜀,√2𝜀 (3.8) for sufficiently small 𝜀>0. By trivial computation, we have 𝜃(𝜀)=sup𝑥−𝑥^∗∶𝑥∈Ω_𝛼(𝜀)=√2𝜀⟶0as𝜀⟶0. (3.9) By Theorem 3.6, GMVI(𝐴,𝐹,𝑓) is 2-well-posed by perturbations.

To derive a characterization of strong generalized 𝛼-well-posedness by perturbations, we need another function 𝑞 which is defined by𝑞(𝜀)=𝑒Ω_𝛼(𝜀),𝑆,∀𝜀≥0, (3.10) where 𝑆 is the solution set of GMVI(𝐴,𝐹,𝑓) and 𝑒 is defined as in Proposition 2.3.

Theorem 3.9. GMVI(𝐴,𝐹,𝑓) is strongly generalized 𝛼-well-posed by perturbations if and only if 𝑆 is nonempty compact and 𝑞(𝜀)→0 as 𝜀→0.

Proof. Repeating almost the same argument as in the proof of [10, Theorem  3.2], we can readily derive the desired result.

Example 3.10. Let 𝑋=𝐑, 𝑃=[−1,1], 𝑝^∗=0, 𝛼=2, 𝐴(𝑝,𝑥)=𝑥(𝑝²+1), 𝐹(𝑥)={𝑥,0}, and 𝑓(𝑝,𝑥)=𝑥² for all 𝑥∈𝑋, 𝑝∈𝑃. Clearly, 𝑥^∗=0 is a solution of GMVI(𝐴,𝐹,𝑓). Repeating the same argument as in Example 3.8, we obtain that, for any 𝜀>0, Ω_𝛼(𝜀)=_{()𝑝∈𝐵0,𝜀}Ω^𝑝_𝛼(𝜀)=−√2𝜀,√2𝜀 (3.11) for sufficiently small 𝜀>0. By trivial computation, we have 𝑞(𝜀)=𝑒Ω_𝛼(𝜀),𝑆=sup_{𝑥(𝜀)∈Ω𝛼(𝜀)}𝑑(𝑥(𝜀),𝑆)⟶0as𝜀⟶0. (3.12) By Theorem 3.9, GMVI(𝐴,𝐹,𝑓) is generalized 𝛼-well-posed by perturbations.

The strong generalized 𝛼-well-posedness by perturbations can be also characterized by the behavior of the noncompactness measure 𝜇(Ω_𝛼(𝜀)).

Theorem 3.11. Let 𝐿 be finite dimensional, 𝐴∶𝑃×𝑋→𝑋^∗ weakly continuous (i.e., continuous from the product of the norm topology of 𝑃 and weak topology of 𝑋 to the weak topology of 𝑋^∗), 𝐹∶𝑋→2^𝑋 a nonempty weakly compact-valued multifunction which is ℋ-continuous, and 𝑓∶𝑃×𝑋→𝐑∪{+∞} a continuous functional such that 𝑓(𝑝,⋅) is proper and convex. Then, GMVI(𝐴,𝐹,𝑓) is strongly generalized 𝛼-well-posed by perturbations if and only if Ω_𝛼(𝜀)≠∅, for all 𝜀>0 and 𝜇(Ω_𝛼(𝜀))→0 as 𝜀→0.

Proof. First, we will prove that Ω_𝛼(𝜀) is closed for all 𝜀≥0. Let {𝑥_𝑛}⊂Ω_𝛼(𝜀) with 𝑥_𝑛→𝑥. Then, there exist {𝑝_𝑛}⊂𝐵(𝑝^∗,𝜀) and {𝑢_𝑛}⊂𝑋 with 𝑢_𝑛∈𝐹(𝑥_𝑛) (for all 𝑛≥1) such that 𝐴𝑝_𝑛,𝑢_𝑛,𝑥_𝑛−𝑦+𝑓𝑝_𝑛,𝑥_𝑛−𝑓𝑝_𝑛≤𝛼,𝑦2𝑥_𝑛−𝑦²+𝜀,∀𝑦∈𝑋,𝑛≥1. (3.13) Without loss of generality, we may assume 𝑝_𝑛→𝑝∈𝐵(𝑝^∗,𝜀) since 𝐿 is finite dimensional. Since 𝐹∶𝑋→2^𝑋 is a nonempty weakly compact-valued multifunction, 𝐹(𝑥_𝑛) and 𝐹(𝑥) are nonempty weakly compact and hence are nonempty, weakly closed, and weakly bounded. Note that the weak closedness of sets in 𝑋 implies the strong closedness and that the weak boundedness of sets in 𝑋 is equivalent to the strong boundedness. Thus, it is known that 𝐹(𝑥_𝑛) and 𝐹(𝑥) lie in CB(𝑋). According to Proposition 2.3, for each 𝑛≥1 and 𝑢_𝑛∈𝐹(𝑥_𝑛), there exists 𝑣_𝑛∈𝐹(𝑥) such that 𝑢_𝑛−𝑣_𝑛≤11+𝑛ℋ𝐹𝑥_𝑛,𝐹(𝑥). (3.14) Since 𝐹 is ℋ-continuous, one deduces that 𝑢_𝑛−𝑣_𝑛≤11+𝑛ℋ𝐹𝑥_𝑛,𝐹(𝑥)⟶0as𝑛⟶∞. (3.15) Also, since 𝐹(𝑥) is weakly compact, it follows from {𝑣_𝑛}⊂𝐹(𝑥) that there exists some subsequence of {𝑣_𝑛} which converges weakly to a point of 𝐹(𝑥). Without loss of generality, we may assume that 𝑣_𝑛⇀𝑢∈𝐹(𝑥)as𝑛⟶∞. (3.16) Consequently, one has, for each 𝜑∈𝑋^∗, 𝜑,𝑢_𝑛−𝑢≤𝜑,𝑢_𝑛−𝑣_𝑛+𝜑,𝑣_𝑛−𝑢≤𝜑𝑢_𝑛−𝑣_𝑛+𝜑,𝑣_𝑛−𝑢⟶0as𝑛⟶∞. (3.17) This implies that 𝑢_𝑛⇀𝑢 as 𝑛→∞. Taking into account the weak continuity of 𝐴, we immediately obtain that 𝐴𝑝_𝑛,𝑢_𝑛⇀𝐴𝑝,𝑢as𝑛⟶∞, (3.18) and hence, for each 𝑦∈𝑋, 𝐴𝑝_𝑛,𝑢_𝑛,𝑥_𝑛−𝑦−𝐴𝑝,𝑢,≤𝐴𝑝𝑥−𝑦_𝑛,𝑢_𝑛,𝑥_𝑛−𝑦−𝐴𝑝_𝑛,𝑢_𝑛,+𝐴𝑝𝑥−𝑦_𝑛,𝑢_𝑛,𝑥−𝑦−𝐴𝑝,𝑢,≤𝐴𝑝𝑥−𝑦_𝑛,𝑢_𝑛𝑥_𝑛−𝑥+𝐴𝑝_𝑛,𝑢_𝑛−𝐴𝑝,𝑢,𝑥−𝑦⟶0as𝑛⟶∞, (3.19) that is, 𝐴𝑝_𝑛,𝑢_𝑛,𝑥_𝑛−𝑦⟶𝐴𝑝,𝑢,𝑥−𝑦as𝑛⟶∞. (3.20) Therefore, it follows from (3.13) and the continuity of 𝑓 that 𝐴𝑝,𝑢,𝑥−𝑦+𝑓𝑝,𝑥−𝑓≤𝛼𝑝,𝑦2𝑥−𝑦²+𝜀,∀𝑦∈𝑋. (3.21) This shows that 𝑥∈Ω_𝛼(𝜀) and so Ω_𝛼(𝜀) is closed.
Second, we show that 𝑆=_𝜀>0Ω_𝛼(𝜀). (3.22) It is obvious that 𝑆⊂⋂_𝜀>0Ω_𝛼(𝜀). Let 𝑥^∗∈⋂_𝜀>0Ω_𝛼(𝜀). Let {𝜀_𝑛} be a sequence of positive numbers such that 𝜀_𝑛→0. Then, 𝑥^∗∈Ω_𝛼(𝜀_𝑛) and so there exist 𝑝_𝑛∈𝐵(𝑝^∗,𝜀_𝑛) and 𝑢^∗_𝑛∈𝐹(𝑥^∗) such that 𝐴𝑝_𝑛,𝑢^∗_𝑛,𝑥^∗−𝑦+𝑓𝑝_𝑛,𝑥^∗−𝑓𝑝_𝑛≤𝛼,𝑦2𝑥^∗−𝑦²+𝜀_𝑛,∀𝑦∈𝑋,𝑛≥1. (3.23) It is clear that 𝑝_𝑛→𝑝^∗ as 𝑛→∞. Since 𝐹(𝑥^∗) is weakly compact, it follows from {𝑢^∗_𝑛}⊂𝐹(𝑥^∗) that there exists some subsequence of {𝑢^∗_𝑛} which converges weakly to a point of 𝐹(𝑥^∗). Without loss of generality, we may assume that 𝑢^∗_𝑛⇀𝑢^∗∈𝐹𝑥^∗as𝑛⟶∞. (3.24) Note that 𝐴 is weakly continuous. Thus, 𝐴𝑝_𝑛,𝑢^∗_𝑛⇀𝐴𝑝^∗,𝑢^∗, (3.25) and hence, letting 𝑛→∞ in the last inequality, we get 𝐴𝑢^∗,𝑥^∗−𝑦+𝑓𝑥^∗−𝑓𝑦=𝐴𝑝^∗,𝑢^∗,𝑥^∗−𝑦+𝑓𝑝^∗,𝑥^∗−𝑓𝑝^∗≤𝛼,𝑦2𝑥^∗−𝑦²,∀𝑦∈𝑋. (3.26) For any 𝑧∈𝑋 and 𝑡∈(0,1), putting 𝑦=𝑥^∗+𝑡(𝑧−𝑥^∗) in (3.26), we have 𝑡𝐴𝑢^∗,𝑥^∗−𝑧+𝑓𝑥^∗−𝑓(𝑧)≤𝑡𝐴𝑢^∗,𝑥^∗−𝑧+𝑓𝑥^∗−𝑓𝑥^∗+𝑡𝑧−𝑥^∗≤𝛼𝑡²2𝑥^∗−𝑧². (3.27) This implies that 𝐴𝑢^∗,𝑥^∗−𝑧+𝑓𝑥^∗−𝑓(𝑧)≤𝛼𝑡2𝑥^∗−𝑧²,∀𝑧∈𝑋. (3.28) Letting 𝑡→0 in the last inequality, we get 𝐴𝑢^∗,𝑥^∗−𝑧+𝑓𝑥^∗−𝑓(𝑧)≤0,∀𝑧∈𝑋. (3.29) Consequently, 𝑥^∗∈𝑆 and so (3.22) is proved.
Now, we suppose that GMVI(𝐴,𝐹,𝑓) is strongly generalized 𝛼-well-posed by perturbations. By Theorem 3.9, 𝑆 is nonempty compact and 𝑞(𝜀)→0. Then, Ω_𝛼(𝜀)≠∅ since 𝑆⊂Ω_𝛼(𝜀) for all 𝜀>0. Observe that, for all 𝜀>0, ℋΩ_𝛼(𝜀),𝑆=max𝑒Ω_𝛼(𝜀),𝑆,𝑒𝑆,Ω_𝛼(𝜀)=𝑒Ω_𝛼(𝜀),𝑆. (3.30) Taking into account the compactness of 𝑆, we get 𝜇Ω_𝛼(𝜀)≤2ℋΩ_𝛼(𝜀),𝑆=2𝑒Ω_𝛼(𝜀),𝑆=2𝑞(𝜀)⟶0. (3.31)
Conversely, we suppose that Ω_𝛼(𝜀)≠∅, for all 𝜀>0, and 𝜇(Ω_𝛼(𝜀))→0 as 𝜀→0. Since Ω_𝛼(𝜀) is increasing with respect to 𝜀>0, by the Kuratowski theorem [28, page 318], we have from (3.22) 𝑞(𝜀)=ℋΩ_𝛼(𝜀),𝑆⟶0a𝑠𝜀⟶0 (3.32) and 𝑆 is nonempty compact. By Theorem 3.9, GMVI(𝐴,𝐹,𝑓) is strongly generalized 𝛼-well-posed by perturbations.

Remark 3.12. Theorem  3.3 of [10] generalizes Theorem  3.2 of [9]. Theorem 3.11 generalizes Theorem  3.2 of [11] from the case of strong generalized 𝛼-well-posedness in the setting of Hilbert spaces to the case of strong generalized 𝛼-well-posedness by perturbations in the setting of Banach spaces. Furthermore, Theorem 3.11 improves, extends, and develops [10, Theorem  3.3] in the following aspects.(i) The mixed variational inequality problem (MVI) in [10, Theorem  3.3] is extended to develop the more general problem, that is, the generalized mixed variational inequality problem (GMVI) with a nonempty weakly compact-valued multifunction in the setting of Banach spaces. Moreover, the concept of strong generalized 𝛼-well-posedness by perturbations for MVI in [10, Theorem  3.3] is extended to develop the concept of strong generalized 𝛼-well-posedness by perturbations for GMVI.(ii) Since the generalized mixed variational inequality problem (GMVI) is more general and more complicated than the mixed variational inequality problem (MVI), the assumptions in Theorem 3.11 are very different from the ones in [10, Theorem  3.3]; for instance, in Theorem 3.11, let 𝐿 be finite dimensional, 𝐴∶𝑃×𝑋→𝑋^∗ weakly continuous, and 𝐹∶𝑋→2^𝑋 a nonempty weakly compact-valued multifunction which is ℋ-continuous, but, in [10, Theorem  3.3], let 𝐿 be finite dimensional, 𝐴∶𝑃×𝑋→𝑋^∗ a continuous mapping.(iii) The technique of proving strong generalized 𝛼-well-posedness by perturbations for GMVI in Theorem 3.11 is very different from the one for MVI in [10, Theorem  3.3] because our technique depends on the well-known Nadler’s Theorem [27], the ℋ-continuity of nonempty weakly compact-valued multifunction 𝐹 and the property of the Hausdorff metric ℋ.

Remark 3.13. Clearly, any solution of GMVI(𝐴,𝐹,𝑓) is a solution of the 𝛼 problem: find 𝑥∈𝑋 such that, for some 𝑢∈𝐹(𝑥), 𝐴(𝑢)(𝑥)𝑦≤𝛼,𝑥−𝑦+𝑓−𝑓2𝑥−𝑦²,∀𝑦∈𝑋, (3.33) but the converse is not true in general. To show this, let 𝑋=𝐑, 𝐴(𝑥)=𝑥, 𝐹(𝑥)={𝑥,0}, and 𝑓(𝑥)=−𝑥² for all 𝑥∈𝑋. It is easy to verify that the solution set of GMVI(𝐴,𝐹,𝑓) is empty and 0 is the unique solution of the corresponding 𝛼 problem with 𝛼=2. If 𝑓 is proper and convex, then GMVI(𝐴,𝐹,𝑓) and 𝛼 problem have the same solution (this fact has been shown in the proof of Theorem 3.11).

4. Links with the Well-Posedness by Perturbations of Inclusion Problems

Lemaire et al. [20] introduced the concept of well-posedness by perturbations for an inclusion problem. In this section, we will show that the well-posedness by perturbations of a generalized mixed variational inequality is closely related to the well-posedness by perturbations of the corresponding inclusion problem. Let us recall some concepts. Let 𝑀∶𝑋→2^𝑋∗. The inclusion problem associated with 𝑀 is defined byIP(𝑀):find𝑥∈𝑋suchthat0∈𝑀(𝑥). (4.1) The perturbed problem of IP(𝑀) is given byIP_𝑝(𝑀):find𝑥∈𝑋suchthat0∈𝑀𝑝,𝑥, (4.2) where 𝑀∶𝑃×𝑋→2^𝑋∗ is such that 𝑀(𝑝^∗,⋅)=𝑀.

Definition 4.1 (see [20]). Let {𝑝_𝑛}⊂𝑃 be with 𝑝_𝑛→𝑝^∗. A sequence {𝑥_𝑛}⊂𝑋 is called an approximating sequence corresponding to {𝑝_𝑛} for IP(𝑀) if 𝑥_𝑛∈Dom𝑀(𝑝_𝑛,⋅) for all 𝑛≥1 and 𝑑(0,𝑀(𝑝_𝑛,𝑥_𝑛))→0, or, equivalently, there exists 𝑦_𝑛∈𝑀(𝑝_𝑛,𝑥_𝑛) such that ‖𝑦_𝑛‖→0 as 𝑛→∞.

Definition 4.2 (see [20]). One says that IP(𝑀) is strongly (resp., weakly) well-posed by perturbations if it has a unique solution and, for any {𝑝_𝑛}⊂𝑃 with 𝑝_𝑛→𝑝^∗, every approximating sequence corresponding to {𝑝_𝑛} converges strongly (resp., weakly) to the unique solution of IP(𝑀). IP(𝑀) is said to be strongly (resp., weakly) generalized well posed by perturbations if the solution set 𝑆 of IP(𝑀) is nonempty and, for any {𝑝_𝑛}⊂𝑃 with 𝑝_𝑛→𝑝^∗, every approximating sequence corresponding to {𝑝_𝑛} has a subsequence which converges strongly (resp., weakly) to a point of 𝑆.

Let 𝑓∶𝑋→𝐑∪{+∞} be a proper, convex, and lower semicontinuous functional. Denote by 𝜕𝑓 and 𝜕_𝜀𝑓 the subdifferential and 𝜀-subdifferential of 𝑓, respectively, that is,𝜕𝑓(𝑥)=𝑥^∗∈𝑋^∗:𝑓𝑦−𝑓(𝑥)≥𝑥^∗𝜕,𝑦−𝑥,∀𝑦∈𝑋,∀𝑥∈dom𝑓,_𝜀𝑓(𝑥)=𝑥^∗∈𝑋^∗:𝑓𝑦−𝑓(𝑥)≥𝑥^∗,𝑦−𝑥−𝜀,∀𝑦∈𝑋,∀𝑥∈dom𝑓. (4.3) It is known that 𝜕𝑓 is maximal monotone and 𝜕_𝜀𝑓(𝑥)⊃𝜕𝑓(𝑥)≠∅ for all 𝑥∈dom𝑓 and for all 𝜀>0. In terms of 𝜕𝑓, GMVI(𝐴,𝐹,𝑓) is equivalent to the following inclusion problem:IP𝐴𝐹+𝜕𝑓:find𝑥∈𝑋suchthat0∈𝐴𝐹(𝑥)+𝜕𝑓(𝑥). (4.4)

In other words, we have the following lemma.

Lemma 4.3. Let 𝑥∈𝑋 be a fixed point, and let 𝑓∶𝑋→𝐑∪{+∞} be a proper, convex, and lower semicontinuous functional. Then, the following statements are equivalent:(i)⟨𝐴𝑢,𝑥−𝑦⟩+𝑓(𝑥)−𝑓(𝑦)≤0 for all 𝑦∈𝑋 and some 𝑢∈𝐹(𝑥);(ii)0∈𝐴𝐹(𝑥)+𝜕𝑓(𝑥).

Proof. Observe that (𝑥)𝑦𝑦(𝑥)≥(𝑥)(𝑥)(𝑥)(𝑥),𝐴𝑢,𝑥−𝑦+𝑓−𝑓≤0⟺𝑓−𝑓−𝐴𝑢,𝑦−𝑥⟺−𝐴𝑢∈𝜕𝑓⟺0∈𝐴𝑢+𝜕𝑓⟺0∈𝐴𝐹+𝜕𝑓 (4.5) for all 𝑦∈𝑋 and some 𝑢∈𝐹(𝑥). The desired result follows immediately from the above relations.

Naturally, we consider the perturbed problem of IP(𝐴𝐹+𝜕𝑓) as follows:IP_𝑝𝐴𝐹+𝜕𝑓:find𝑥∈𝑋suchthat0∈𝐴𝑝,𝐹(𝑥)+𝜕𝑓𝑝,⋅(𝑥), (4.6) where 𝑓∶𝑃×𝑋→𝐑∪{+∞} is such that 𝑓(𝑝,⋅) is proper, convex, and lower semicontinuous for all 𝑝∈𝑃, and 𝑓(𝑝^∗,⋅)=𝑓.

The following theorems establish the relations between the strong (resp., weak) well-posedness by perturbations of generalized mixed variational inequalities and the strong (resp., weak) well-posedness by perturbations of inclusion problems.

Theorem 4.4. Let 𝐴(⋅,𝑦)∶𝑃→𝑋^∗ be continuous for each 𝑦∈𝑋, let 𝐴(=𝐴(𝑝^∗,⋅))∶𝑋→X^∗ be weakly continuous, let 𝐹∶𝑋→2^𝑋 be a nonempty weakly compact-valued multifunction which is ℋ-hemicontinuous and monotone with respect to 𝐴(𝑝,⋅) for each 𝑝∈𝑃, and let 𝑓∶𝑃×𝑋→𝐑∪{+∞} be a continuous functional with respect to the product of the norm topology of 𝑃 and weak topology of 𝑋 such that the following conditions hold:(i)  𝑓(𝑝,⋅) is proper and convex for all 𝑝∈𝑃;(ii)dom𝑓 is well-positioned and dom𝑓⊂⋂_𝑝∈𝑃dom𝑓(𝑝,⋅);(iii)𝑒(dom𝑓(𝑝_𝑛,⋅),dom𝑓)→0 whenever 𝑝_𝑛→𝑝^∗, where 𝑒 is defined as in Proposition 2.3.
Then, IP(𝐴𝐹+𝜕𝑓) is weakly well-posed by perturbations whenever GMVI(𝐴,𝐹,𝑓) has a unique solution.

Proof. Suppose that GMVI(𝐴,𝐹,𝑓) has a unique solution 𝑥^∗. Let {𝑝_𝑛}⊂𝑃 be with 𝑝_𝑛→𝑝^∗, and let {𝑥_𝑛} be an approximating sequence corresponding to {𝑝_𝑛} for IP(𝐴𝐹+𝜕𝑓). Then, there exists 𝑤_𝑛∈𝐴(𝑝_𝑛,𝐹(𝑥_𝑛))+𝜕𝑓(𝑝_𝑛,⋅)(𝑥_𝑛) such that ‖𝑤_𝑛‖→0. Further, there exists 𝑢_𝑛∈𝐹(𝑥_𝑛) such that 𝑤_𝑛∈𝐴(𝑝_𝑛,𝑢_𝑛)+𝜕𝑓(𝑝_𝑛,⋅)(𝑥_𝑛) with ‖𝑤_𝑛‖→0. It follows that 𝑓𝑝_𝑛,𝑦−𝑓𝑝_𝑛,𝑥_𝑛≥𝑤_𝑛−𝐴𝑝_𝑛,𝑢_𝑛,𝑦−𝑥_𝑛,∀𝑦∈𝑋,𝑛≥1. (4.7) We claim that {𝑥_𝑛} is bounded. Indeed, if {𝑥_𝑛} is unbounded, without loss of generality, we may assume that ‖𝑥_𝑛‖→+∞. Let 𝑡_𝑛=1𝑥_𝑛−𝑥^∗,𝑧_𝑛=𝑥^∗+𝑡_𝑛𝑥_𝑛−𝑥^∗. (4.8) By conditions (i)-(ii), we get 𝑧_𝑛∈dom𝑓(𝑝_𝑛,⋅). Note that 𝑧_𝑛≤𝑥^∗+𝑡_𝑛𝑥_𝑛−𝑥^∗=𝑥^∗+1,∀𝑛≥1. (4.9) So, {𝑧_𝑛} is bounded. Since 𝑋 is reflexive, it follows from the boundedness of {𝑧_𝑛} that there exists some subsequence of {𝑧_𝑛} which converges weakly to a point of 𝑋. Hence, without loss of generality, we may assume that 𝑡_𝑛∈(0,1) and 𝑧_𝑛⇀𝑧. It follows from Lemma 2.14 and conditions (ii)-(iii) that 𝑧≠𝑥^∗. For any 𝑦∈𝑋, observe that, for all 𝑣∈𝐹(𝑦), 𝐴𝑝_𝑛,𝑣,𝑧−𝑦=𝐴𝑝_𝑛,𝑣,𝑧−𝑧_𝑛+𝐴𝑝_𝑛,𝑣,𝑧_𝑛−𝑥^∗+𝐴𝑝_𝑛,𝑣,𝑥^∗=𝐴𝑝−𝑦_𝑛,𝑣,𝑧−𝑧_𝑛+𝑡_𝑛𝐴𝑝_𝑛,𝑣,𝑥_𝑛−𝑥^∗+𝐴𝑝_𝑛,𝑣,𝑥^∗=𝐴𝑝−𝑦_𝑛,𝑣,𝑧−𝑧_𝑛+𝑡_𝑛𝐴𝑝_𝑛,𝑣,𝑥_𝑛−𝑦+1−𝑡_𝑛𝐴𝑝_𝑛,𝑣,𝑥^∗−𝑦. (4.10) Since 𝑥^∗ is the unique solution of GMVI(𝐴,𝐹,𝑓), there exists some 𝑢^∗∈𝐹(𝑥^∗) such that 𝐴𝑢^∗,𝑥^∗−𝑦+𝑓𝑥^∗−𝑓𝑦≤0,∀𝑦∈𝑋. (4.11) Also, since 𝐹 is monotone with respect to 𝐴(𝑝_𝑛,⋅), we deduce that, for 𝑢^∗∈𝐹(𝑥^∗), 𝑣∈𝐹(𝑦), and 𝑢_𝑛∈𝐹(𝑥_𝑛), 𝐴𝑝_𝑛,𝑣,𝑥^∗−𝑦≤𝐴𝑝_𝑛,𝑢^∗,𝑥^∗−𝑦,𝐴𝑝_𝑛,𝑣,𝑥_𝑛−𝑦≤𝐴𝑝_𝑛,𝑢_𝑛,𝑥_𝑛−𝑦. (4.12) In addition, we have 𝑓𝑝_𝑛,𝑧_𝑛≤𝑡_𝑛𝑓𝑝_𝑛,𝑥_𝑛+1−𝑡_𝑛𝑓𝑝_𝑛,𝑥^∗ (4.13) by virtue of the convexity of 𝑓(𝑝_𝑛,⋅). It follows from (4.7)–(4.13) that 𝐴𝑝_𝑛,𝑣,𝑧−𝑦≤𝐴𝑝_𝑛,𝑣,𝑧−𝑧_𝑛+𝑡_𝑛𝐴𝑝_𝑛,𝑢_𝑛,𝑥_𝑛+−𝑦1−𝑡_𝑛𝐴𝑝_𝑛,𝑢^∗,𝑥^∗≤𝐴𝑝−𝑦_𝑛,𝑣,𝑧−𝑧_𝑛+1−𝑡_𝑛𝐴𝑝_𝑛,𝑢^∗,𝑥^∗−𝑦+𝑡_𝑛𝑓𝑝_𝑛,𝑦−𝑓𝑝_𝑛,𝑥_𝑛+𝑡_𝑛𝑤_𝑛,𝑥_𝑛≤𝐴𝑝−𝑦_𝑛,𝑣,𝑧−𝑧_𝑛+1−𝑡_𝑛𝐴𝑝_𝑛,𝑢^∗,𝑥^∗−𝑦+𝑡_𝑛𝑤_𝑛,𝑥_𝑛−𝑦+𝑡_𝑛𝑓𝑝_𝑛,𝑦+1−𝑡_𝑛𝑓𝑝_𝑛,𝑥^∗−𝑓𝑝_𝑛,𝑧_𝑛. (4.14) Moreover, it is easy to see from 𝑤_𝑛→0 that 𝑡_𝑛⟨𝑤_𝑛,𝑥_𝑛−𝑦⟩→0. Further, since 𝐴(⋅,𝑦)∶𝑃→𝑋^∗ is continuous for each 𝑦∈𝑋 and 𝑧_𝑛⇀𝑧, it is known that 𝐴(𝑝_𝑛,𝑣)→𝐴(𝑝^∗,𝑣) and {𝑧_𝑛} is bounded. Consequently, 𝐴𝑝_𝑛,𝑣,𝑧−𝑧_𝑛=𝐴𝑝_𝑛,𝑣,𝑧−𝑧_𝑛−𝐴𝑝^∗,𝑣,𝑧−𝑧_𝑛+𝐴𝑝^∗,𝑣,𝑧−𝑧_𝑛≤𝐴𝑝_𝑛,𝑣−𝐴𝑝^∗,𝑣,𝑧−𝑧_𝑛+𝐴𝑝^∗,𝑣,𝑧−𝑧_𝑛≤𝐴𝑝_𝑛,𝑣−𝐴𝑝^∗,𝑣𝑧_𝑛+𝐴𝑝−𝑧_∗,𝑣,𝑧_𝑛−𝑧⟶0as𝑛⟶∞. (4.15) In the meantime, since 𝑓∶𝑃×𝑋→𝐑∪{+∞}is a continuous functional with respect to the product of the norm topology of 𝑃 and weak topology of 𝑋, we conclude from 𝑝_𝑛→𝑝^∗ and 𝑧_𝑛⇀𝑧 that 𝑓(𝑝_𝑛,𝑧_𝑛)→𝑓(𝑝^∗,𝑧) and 𝑓(𝑝_𝑛,𝑥^∗)→𝑓(𝑝^∗,𝑥^∗) as 𝑛→∞. Now, letting 𝑛→∞ in (4.14) we get 𝐴𝑣,𝑧−𝑦=𝐴𝑝^∗≤𝐴𝑝,𝑣,𝑧−𝑦_∗,𝑢^∗,𝑥^∗−𝑦+𝑓𝑝^∗,𝑥^∗−𝑓𝑝^∗=,𝑧𝐴𝑢^∗,𝑥^∗−𝑦+𝑓𝑥^∗−𝑓(𝑧),∀𝑦∈𝑋. (4.16) Since 𝑥^∗ is the unique solution of GMVI(𝐴,𝐹,𝑓), from (4.16) we get 𝐴𝑣,𝑧−𝑦≤𝐴𝑢^∗,𝑥^∗−𝑦+𝑓𝑥^∗(𝑧)=−𝑓𝐴𝑢^∗,𝑥^∗−𝑦+𝑓𝑥^∗𝑦𝑦(𝑧)𝑦(𝑧)−𝑓+𝑓−𝑓≤𝑓−𝑓,∀𝑦∈𝑋, (4.17) which implies that 𝐴𝑣,𝑧−𝑦+𝑓(𝑧)−𝑓𝑦≤0,∀𝑦∈𝑋,𝑣∈𝐹𝑦. (4.18) Note that 𝐴 is weakly continuous, that 𝑓 is proper and convex, and that 𝐹 is a nonempty weakly compact-valued multifunction which is ℋ-hemicontinuous and monotone with respect to 𝐴. Hence, all conditions of Lemma 2.8 are satisfied. Thus, it follows from Lemma 2.8 that there exists 𝑤∈𝐹(𝑧) such that 𝐴𝑤,𝑧−𝑦+𝑓(𝑧)−𝑓𝑦≤0,∀𝑦∈𝑋. (4.19) Therefore, 𝑧 is a solution of GMVI(𝐴,𝐹,𝑓), a contradiction. This shows that {𝑥_𝑛} is bounded.
Let {𝑥_𝑛𝑘} be any subsequence of {𝑥_𝑛} such that 𝑥_𝑛𝑘⇀𝑥 as 𝑘→∞. It follows from (4.7) that 𝐴𝑝_𝑛𝑘,𝑢_𝑛𝑘,𝑥_𝑛𝑘−𝑦+𝑓𝑝_𝑛𝑘,𝑥_𝑛𝑘−𝑓𝑝_𝑛𝑘,𝑦≤𝑤_𝑛𝑘,𝑥_𝑛𝑘−𝑦,∀𝑦∈𝑋,𝑘≥1. (4.20) Since 𝐴(⋅,𝑦)∶𝑃→𝑋^∗ is continuous for each 𝑦∈𝑋 and 𝑥_𝑛𝑘⇀𝑥, it is known that 𝐴(𝑝_𝑛𝑘,𝑣)→𝐴(𝑝^∗,𝑣) and {𝑥_𝑛𝑘} is bounded. Consequently, 𝐴𝑝_𝑛𝑘,𝑣,𝑥_𝑛𝑘−𝑦−𝐴𝑝^∗,𝑣,=𝐴𝑝𝑥−𝑦_𝑛𝑘,𝑣,𝑥_𝑛𝑘−𝑦−𝐴𝑝^∗,𝑣,𝑥_𝑛𝑘−𝑦+𝐴𝑝^∗,𝑣,𝑥_𝑛𝑘−𝑦−𝐴𝑝^∗,𝑣,≤𝐴𝑝𝑥−𝑦_𝑛𝑘,𝑣−𝐴𝑝^∗,𝑣,𝑥_𝑛𝑘−𝑦+𝐴𝑝^∗,𝑣,𝑥_𝑛𝑘−𝑥≤𝐴𝑝_𝑛𝑘,𝑣−𝐴𝑝^∗,𝑣𝑥_𝑛𝑘−𝑦+𝐴𝑝^∗,𝑣,𝑥_𝑛𝑘−𝑥⟶0as𝑛⟶∞. (4.21) Moreover, since 𝑓∶𝑃×𝑋→𝐑∪{+∞}is a continuous functional with respect to the product of the norm topology of 𝑃 and weak topology of 𝑋, we conclude from 𝑝_𝑛𝑘→𝑝^∗ and 𝑥_𝑛𝑘⇀𝑥 that 𝑓(𝑝_𝑛𝑘,𝑥_𝑛𝑘)→𝑓(𝑝^∗,𝑥) and 𝑓(𝑝_𝑛𝑘,𝑦)→𝑓(𝑝^∗,𝑦) as 𝑘→∞. Note that 𝐹 is monotone with respect to 𝐴(𝑝_𝑛𝑘,⋅). Hence, it follows that for, 𝑢_𝑛𝑘∈𝐹(𝑥_𝑛𝑘) and 𝑣∈𝐹(𝑦), 𝐴𝑣,𝑥−𝑦+𝑓(𝑥)−𝑓𝑦=𝐴𝑝^∗,𝑣,𝑥−𝑦+𝑓𝑝^∗,𝑥−𝑓𝑝^∗,𝑦=lim_𝑘→∞𝐴𝑝_𝑛𝑘,𝑣,𝑥_𝑛𝑘−𝑦+𝑓𝑝_𝑛𝑘,𝑥_𝑛𝑘−𝑓𝑝_𝑛𝑘,𝑦≤lim_𝑘→∞𝐴𝑝_𝑛𝑘,𝑢_𝑛𝑘,𝑥_𝑛𝑘−𝑦+𝑓𝑝_𝑛𝑘,𝑥_𝑛𝑘−𝑓𝑝_𝑛𝑘,𝑦≤lim_𝑘→∞𝑤_𝑛𝑘,𝑥_𝑛𝑘−𝑦=0,∀𝑦∈𝑋. (4.22) This together with Lemma 2.8 yields that there exists 𝑢∈𝐹(𝑥) such that 𝐴𝑢,𝑥−𝑦+𝑓(𝑥)−𝑓𝑦≤0,∀𝑦∈𝑋. (4.23) Consequently, 𝑥 solves GMVI(𝐴,𝐹,𝑓). We must have 𝑥=𝑥^∗ since GMVI(𝐴,𝐹,𝑓) has a unique solution 𝑥^∗. Therefore, {𝑥_𝑛} converges weakly to 𝑥^∗ and so IP(𝐴𝐹+𝜕𝑓) is weakly well-posed by perturbations.

Remark 4.5. Theorem 4.4 improves, extends, and develops [10, Theorem  4.1] in the following aspects. (i) The mixed variational inequality problem (MVI) in [10, Theorem  4.1] is extended to develop the more general problem, that is, the generalized mixed variational inequality problem (GMVI) with a nonempty weakly compact-valued multifunction in the setting of Banach spaces. Moreover, the inclusion problem corresponding to MVI in [10, Theorem  4.1] is extended to develop the more general problem, that is, the inclusion problem corresponding to GMVI.(ii) Since the generalized mixed variational inequality problem (GMVI) is more general and more complicated than the mixed variational inequality problem (MVI), the assumptions in Theorem 4.4 are very different from the ones in [10, Theorem  4.1], for instance, in Theorem 4.4, let 𝐴(⋅,𝑦)∶𝑃→𝑋^∗ be continuous for each 𝑦∈𝑋, let 𝐴(=𝐴(𝑝^∗,⋅))∶𝑋→𝑋^∗ be weakly continuous, and let 𝐹∶𝑋→2^𝑋 be a nonempty weakly compact-valued multifunction which is ℋ-hemicontinuous and monotone with respect to 𝐴(𝑝,⋅) for each 𝑝∈𝑃, but, in [10, Theorem  4.1], let 𝐴∶𝑃×𝑋→𝑋^∗ be a continuous mapping such that 𝐴(𝑝,⋅)∶𝑋→𝑋^∗ is monotone for all 𝑝∈𝑃.(iii) The technique of proving weak well-posedness by perturbations for inclusion problem IP(𝐴𝐹+𝜕𝑓) in Theorem 4.4 is very different from the one for inclusion problem IP(𝐴+𝜕𝑓) in [10, Theorem  4.1] because our technique depends on Lemma 2.8. Note that 𝐴 is weakly continuous, that 𝑓 is proper and convex, and that 𝐹 is a nonempty weakly compact-valued multifunction which is ℋ-hemicontinuous and monotone with respect to 𝐴. Hence, all the conditions of Lemma 2.8 are satisfied. Recall that the proof of Lemma 2.8 depends on the well-known Nadler’s Theorem [27]. Thus, our technique depends essentially on the well-known Nadler’s Theorem [27], the ℋ-hemicontinuity of nonempty weakly compact-valued multifunction 𝐹 and the monotonicity of 𝐹 with respect to 𝐴(𝑝,⋅) for each 𝑝∈𝑃.

Theorem 4.6. Let 𝐴(⋅,𝑦)∶𝑃→𝑋^∗ be continuous for each 𝑦∈𝑋, let 𝐴∶𝑋→𝑋^∗ be weakly continuous, let 𝐹∶𝑋→2^𝑋 be a nonempty weakly compact-valued multifunction which is ℋ-hemicontinuous and monotone with respect to 𝐴(𝑝,⋅) for each 𝑝∈𝑃, and let 𝑓∶𝑃×𝑋→𝐑∪{+∞} be a continuous functional with respect to the product of the norm topology of 𝑃 and weak topology of 𝑋 such that the following conditions hold:(i)  𝑓(𝑝,⋅) is proper and convex for all 𝑝∈𝑃;(ii)dom𝑓 is well-positioned and dom𝑓⊂⋂_𝑝∈𝑃dom𝑓(𝑝,⋅);(iii)𝑒(dom𝑓(𝑝_𝑛,⋅),dom𝑓)→0 whenever 𝑝_𝑛→𝑝^∗.
Then, GMVI(𝐴,𝐹,𝑓) is weakly well-posed by perturbations whenever IP(𝐴𝐹+𝜕𝑓) has a unique solution.

Proof. Let IP(𝐴𝐹+𝜕𝑓) have a unique solution 𝑥^∗. By Lemma 4.3, 𝑥^∗ is also the unique solution of GMVI(𝐴,𝐹,𝑓). Let {𝑝_𝑛}⊂𝑃 be with 𝑝_𝑛→𝑝^∗, and let {𝑥_𝑛} be an approximating sequence corresponding to {𝑝_𝑛} for GMVI(𝐴,𝐹,𝑓). Then, there exist {𝑢_𝑛}⊂𝑋 with 𝑢_𝑛∈𝐹(𝑥_𝑛) (for all 𝑛≥1) and 0<𝜀_𝑛→0 such that 𝐴𝑝_𝑛,𝑢_𝑛,𝑥_𝑛−𝑦+𝑓𝑝_𝑛,𝑥_𝑛−𝑓𝑝_𝑛,𝑦≤𝜀_𝑛,∀𝑦∈𝑋,𝑛≥1. (4.24) We claim that {𝑥_𝑛} is bounded. Indeed, if {𝑥_𝑛} is unbounded, without loss of generality, we may assume that ‖𝑥_𝑛‖→+∞. Let 𝑡_𝑛=1𝑥_𝑛−𝑥^∗,𝑧_𝑛=𝑥^∗+𝑡_𝑛𝑥_𝑛−𝑥^∗. (4.25) By conditions (i)-(ii), we get 𝑧_𝑛∈dom𝑓(𝑝_𝑛,⋅). Without loss of generality we may assume that 𝑡_𝑛∈(0,1) and 𝑧_𝑛⇀𝑧. From Lemma 2.14 and conditions (ii)-(iii) we obtain that 𝑧≠𝑥^∗. For any 𝑦∈𝑋, observe that, for all 𝑣∈𝐹(𝑦), 𝐴𝑝_𝑛,𝑣,𝑧−𝑦=𝐴𝑝_𝑛,𝑣,𝑧−𝑧_𝑛+𝐴𝑝_𝑛,𝑣,𝑧_𝑛−𝑥^∗+𝐴𝑝_𝑛,𝑣,𝑥^∗=𝐴𝑝−𝑦_𝑛,𝑣,𝑧−𝑧_𝑛+𝑡_𝑛𝐴𝑝_𝑛,𝑣,𝑥_𝑛−𝑦+1−𝑡_𝑛𝐴𝑝_𝑛,𝑣,𝑥^∗−𝑦. (4.26) Since 𝑥^∗ is the unique solution of GMVI(𝐴,𝐹,𝑓), there exists some 𝑢^∗∈𝐹(𝑥^∗) such that 𝐴𝑢^∗,𝑥^∗−𝑦+𝑓𝑥^∗−𝑓𝑦≤0,∀𝑦∈𝑋. (4.27) Also, since 𝐹 is monotone with respect to 𝐴(𝑝_𝑛,⋅), we deduce that, for 𝑢^∗∈𝐹(𝑥^∗), 𝑣∈𝐹(𝑦), and 𝑢_𝑛∈𝐹(𝑥_𝑛), 𝐴𝑝_𝑛,𝑣,𝑥^∗−𝑦≤𝐴𝑝_𝑛,𝑢^∗,𝑥^∗−𝑦,𝐴𝑝_𝑛,𝑣,𝑥_𝑛−𝑦≤𝐴𝑝_𝑛,𝑢_𝑛,𝑥_𝑛−𝑦. (4.28) In addition, since 𝑓(𝑝_𝑛,⋅) is convex, we get 𝑓𝑝_𝑛,𝑧_𝑛≤𝑡_𝑛𝑓𝑝_𝑛,𝑥_𝑛+1−𝑡_𝑛𝑓𝑝_𝑛,𝑥^∗. (4.29) It follows from (4.24)–(4.29) that 𝐴𝑝_𝑛,𝑣,𝑧−𝑦≤𝐴𝑝_𝑛,𝑣,𝑧−𝑧_𝑛+𝑡_𝑛𝐴𝑝_𝑛,𝑢_𝑛,𝑥_𝑛+−𝑦1−𝑡_𝑛𝐴𝑝_𝑛,𝑢^∗,𝑥^∗≤𝐴𝑝−𝑦_𝑛,𝑣,𝑧−𝑧_𝑛+1−𝑡_𝑛𝐴𝑝_𝑛,𝑢^∗,𝑥^∗−𝑦+𝑡_𝑛𝑓𝑝_𝑛,𝑦−𝑓𝑝_𝑛,𝑥_𝑛+𝑡_𝑛𝜀_𝑛≤𝐴𝑝_𝑛,𝑣,𝑧−𝑧_𝑛+1−𝑡_𝑛𝐴𝑝_𝑛,𝑢^∗,𝑥^∗−𝑦+𝑡_𝑛𝜀_𝑛+𝑡_𝑛𝑓𝑝_𝑛,𝑦+1−𝑡_𝑛𝑓𝑝_𝑛,𝑥^∗−𝑓𝑝_𝑛,𝑧_𝑛. (4.30) Letting 𝑛→∞ in the last inequality we get 𝐴𝑣,𝑧−𝑦=𝐴𝑝^∗≤𝐴𝑝,𝑣,𝑧−𝑦_∗,𝑢^∗,𝑥^∗−𝑦+𝑓𝑝^∗,𝑥^∗−𝑓𝑝^∗=,𝑧𝐴𝑢^∗,𝑥^∗−𝑦+𝑓𝑥^∗−𝑓(𝑧),∀𝑦∈𝑋. (4.31) By using (4.31) and the same argument as in the proof of Theorem 4.4, we can prove that 𝑧 is a solution of GMVI(𝐴,𝐹,𝑓), a contradiction. Thus, {𝑥_𝑛} is bounded.
The rest follows from the similar argument to that in the proof of Theorem 4.4 and so is omitted.

By Lemma 4.3 and Theorems 4.4 and 4.6, we have the following result.

Theorem 4.7. Let 𝐴(⋅,𝑦)∶𝑃→𝑋^∗ be continuous for each 𝑦∈𝑋, let 𝐴∶𝑋→𝑋^∗ be weakly continuous, let 𝐹∶𝑋→2^𝑋 be a nonempty weakly compact-valued multifunction which is ℋ-hemicontinuous and monotone with respect to 𝐴(𝑝,⋅) for each 𝑝∈𝑃, and let 𝑓∶𝑃×𝑋→𝐑∪{+∞} be a continuous functional with respect to the product of the norm topology of 𝑃 and weak topology of 𝑋 such that the following conditions hold:(i)𝑓(𝑝,⋅) is proper and convex for all 𝑝∈𝑃;(ii)dom𝑓 is well-positioned and dom𝑓⊂⋂_𝑝∈𝑃dom𝑓(𝑝,⋅);(iii)𝑒(dom𝑓(𝑝_𝑛,⋅),dom𝑓)→0 whenever 𝑝_𝑛→𝑝^∗.Then, the following statements are equivalent:(i)GMVI(𝐴,𝐹,𝑓) is weakly well posed by perturbations;(ii)IP(𝐴𝐹+𝜕𝑓) is weakly well posed by perturbations;(iii)GMVI(𝐴,𝐹,𝑓) has a unique solution;(iv)IP(𝐴𝐹+𝜕𝑓) has a unique solution.

Remark 4.8. Theorem 4.7 improves Theorems  4.1, 4.2, and 6.1 of [9], Theorems 4.1–4.3 of [10], and Theorems 4.1, 4.2, and 6.1 of [11].

Now we give the following example as an application of Theorem 4.7.

Example 4.9. Let 𝑋=𝐑, 𝑃=[−1,1], and 𝑝^∗=0. Let 𝐴(𝑝,𝑥)=𝑥+𝑝, 𝐹(𝑥)=𝑥³ for all 𝑥∈𝑋, 𝑝∈𝑃, and 𝑓=𝑝𝑝,𝑥²+1𝑥²,𝑥∈[−1,1],+∞,otherwise, (4.32) for all 𝑝∈𝑃. Clearly, Dom𝐴(𝑝,⋅)=𝐑 and Dom𝑓=Dom𝑓(𝑝,⋅)=[−1,1] for all 𝑝∈𝑃. It is easy to see that 𝐴 and 𝑓 are continuous, 𝑓(𝑝,⋅) is proper and convex, and 𝐹 is ℋ-hemicontinuous and monotone with respect to 𝐴(𝑝,⋅) for each 𝑝∈𝑃. By (ii) of Remark 2.7, Dom𝑓 is well-positioned. Hence, all the assumptions of Theorem 4.7 are satisfied. Let 𝑆 be the solution set of GMVI(𝐴,𝐹,𝑓). It follows that 𝑆=𝑥∈[−1,1]∶𝑥³𝑥−𝑦+𝑥²−𝑦²[=[]𝑥≤0,∀𝑦∈−1,1]𝑥∈−1,1∶−𝑦+³2²+𝑥⁶4+𝑥⁴+𝑥²[={0}.≤0,∀𝑦∈−1,1] (4.33) So 𝑥^∗=0 is the unique solution of GMVI(𝐴,𝐹,𝑓). By Theorem 4.7, GMVI(𝐴,𝐹,𝑓) is well-posed by perturbations.

Next, we discuss the relationships between the generalized well-posedness by perturbations of GMVI(𝐴,𝐹,𝑓) and the generalized well-posedness by perturbations of IP(𝐴𝐹+𝜕𝑓).

Theorem 4.10. Let 𝐴∶𝑃×𝑋→𝑋^∗ be a uniformly continuous mapping, let 𝐹∶𝑋→2^𝑋 be a nonempty weakly compact-valued multifunction which is ℋ-uniformly continuous, and let 𝑓∶𝑃×𝑋→𝐑∪{+∞} be a functional such that 𝑓(𝑝,⋅) is proper, convex, and lower semicontinuous for each 𝑝∈𝑃. Then, GMVI(𝐴,𝐹,𝑓) is strongly (resp., weakly) generalized well-posed by perturbations whenever IP(𝐴𝐹+𝜕𝑓) is strongly (resp., weakly) generalized well-posed by perturbations.

Proof. Let {𝑝_𝑛}⊂𝑃 be with 𝑝_𝑛→𝑝^∗, and let {𝑥_𝑛} be an approximating sequence corresponding to {𝑝_𝑛} for GMVI(𝐴,𝐹,𝑓). Then, there exist {𝑢_𝑛}⊂𝑋 with 𝑢_𝑛∈𝐹(𝑥_𝑛) (for all 𝑛≥1) and 0<𝜀_𝑛→0 such that 𝐴𝑝_𝑛,𝑢_𝑛,𝑥_𝑛−𝑦+𝑓𝑝_𝑛,𝑥_𝑛−𝑓𝑝_𝑛,𝑦≤𝜀_𝑛,∀𝑦∈𝑋,𝑛≥1. (4.34) Define 𝜑_𝑛∶𝑋→𝐑∪{+∞} as follows: 𝜑_𝑛𝑦=𝑓𝑝_𝑛,𝑦+𝐴𝑝_𝑛,𝑢_𝑛,𝑦−𝑥_𝑛,∀𝑦∈𝑋,𝑛≥1. (4.35) Clearly, 𝜑_𝑛 is proper, convex, and lower semicontinuous and 0∈𝜕_𝜀𝑛𝜑_𝑛(𝑥_𝑛) for all 𝑛≥1. By the Brondsted-Rockafellar theorem [32], there exist 𝑥_𝑛∈𝑋 and 𝑤_𝑛∈𝜕𝜑_𝑛𝑥_𝑛=𝐴𝑝_𝑛,𝑢_𝑛+𝜕𝑓𝑝_𝑛,⋅𝑥_𝑛 (4.36) such that 𝑥_𝑛−𝑥_𝑛≤√𝜀_𝑛,𝑤_𝑛≤√𝜀_𝑛. (4.37) Since 𝐹∶𝑋→2^𝑋 is a nonempty weakly compact-valued multifunction, both 𝐹(𝑥_𝑛) and 𝐹(𝑥_𝑛) are nonempty weakly compact and hence are nonempty, weakly closed, and weakly bounded. Note that the weak closedness of sets in 𝑋 implies the strong closedness and that the weak boundedness of sets in 𝑋 is equivalent to the strong boundedness. Thus, it is known that both 𝐹(𝑥_𝑛) and 𝐹(𝑥_𝑛) lie in CB(𝑋) for each 𝑛≥1. By Proposition 2.3, for each 𝑛≥1 and 𝑢_𝑛∈𝐹(𝑥_𝑛), there exists 𝑢_𝑛∈𝐹(𝑥_𝑛) such that 𝑢_𝑛−𝑢_𝑛≤11+𝑛ℋ𝐹𝑥_𝑛,𝐹𝑥_𝑛. (4.38) Since 𝐹 is ℋ-uniformly continuous, we have from (4.37) 𝑢_𝑛−𝑢_𝑛≤11+𝑛ℋ𝐹𝑥_𝑛,𝐹𝑥_𝑛⟶0as𝑛⟶∞. (4.39) It follows from (4.36) that 𝑤_𝑛+𝐴𝑝_𝑛,𝑢_𝑛−𝐴𝑝_𝑛,𝑢_𝑛∈𝐴𝑝_𝑛,𝑢_𝑛+𝜕𝑓𝑝_𝑛,⋅𝑥_𝑛. (4.40) Since 𝐴 is a uniformly continuous, from (4.37) we get 𝑤_𝑛+𝐴𝑝_𝑛,𝑢_𝑛−𝐴𝑝_𝑛,𝑢_𝑛≤𝑤_𝑛+𝐴𝑝_𝑛,𝑢_𝑛−𝐴𝑝_𝑛,𝑢_𝑛⟶0. (4.41) So {𝑥_𝑛} is an approximating sequence corresponding to {𝑝_𝑛} for IP(𝐴𝐹+𝜕𝑓).
By the strong (resp., weak) generalized well-posedness by perturbations of IP(𝐴𝐹+𝜕𝑓), there exists some subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that 𝑥_𝑛𝑘→𝑥^∗ (resp., 𝑥_𝑛𝑘⇀𝑥^∗), where 𝑥^∗ is some solution of IP(𝐴𝐹+𝜕𝑓). By Lemma 4.3, 𝑥^∗ is also a solution of GMVI(𝐴,𝐹,𝑓).
Case i. IP(𝐴𝐹+𝜕𝑓) is strongly generalized well-posed by perturbations. It follows from (4.37) that 𝑥_𝑛𝑘−𝑥^∗≤𝑥_𝑛𝑘−𝑥_𝑛𝑘+𝑥_𝑛𝑘−𝑥^∗⟶0 (4.42) and so GMVI(𝐴,𝐹,𝑓) is strongly generalized well-posed by perturbations.Case ii. IP(𝐴𝐹+𝜕𝑓) is weakly generalized well-posed by perturbations. For any 𝜑∈𝑋^∗, from (4.37) we have 𝜑,𝑥_𝑛𝑘−𝑥^∗≤𝜑,𝑥_𝑛𝑘−𝑥_𝑛𝑘+𝜑,𝑥_𝑛𝑘−𝑥^∗≤𝜑𝜀_𝑛𝑘+𝜑,𝑥_𝑛𝑘−𝑥^∗⟶0. (4.43) Thus, GMVI(𝐴,𝐹,𝑓) is weakly generalized well-posed by perturbations.

Theorem 4.11. Let 𝐴∶𝑃×𝑋→𝑋^∗ and 𝐹∶𝑋→2^𝑋, and let 𝑓∶𝑃×𝑋→𝐑∪{+∞} be a functional such that 𝑓(𝑝,⋅) is proper, convex, and lower semicontinuous for each 𝑝∈𝑃. Then, IP(𝐴𝐹+𝜕𝑓) is strongly (resp., weakly) generalized well-posed by perturbations whenever GMVI(𝐴,𝐹,𝑓) is strongly (resp., weakly) generalized 1-well-posed by perturbations.

Proof. Let {𝑝_𝑛}⊂𝑃 be with 𝑝_𝑛→𝑝^∗, and let {𝑥_𝑛} be an approximating sequence corresponding to {𝑝_𝑛} for IP(𝐴𝐹+𝜕𝑓). Then, there exists 𝑤_𝑛∈𝐴(𝑝_𝑛,𝐹(𝑥_𝑛))+𝜕𝑓(𝑝_𝑛,⋅)(𝑥_𝑛) such that ‖𝑤_𝑛‖→0. It follows that there exists a sequence {𝑢_𝑛}⊂𝑋 with 𝑢_𝑛∈𝐹(𝑥_𝑛) (for all 𝑛≥1) such that 𝑤_𝑛∈𝐴𝑝_𝑛,𝑢_𝑛+𝜕𝑓𝑝_𝑛,⋅𝑥_𝑛,∀𝑛≥1, (4.44) and hence 𝐴𝑝_𝑛,𝑢_𝑛,𝑥_𝑛−𝑦+𝑓𝑝_𝑛,𝑥_𝑛−𝑓𝑝_𝑛,𝑦≤𝑤_𝑛,𝑥_𝑛≤1−𝑦2𝑥_𝑛−𝑦²+12𝑤_𝑛²,∀𝑦∈𝑋,𝑛≥1. (4.45) This together with ‖𝑤_𝑛‖→0 implies that {𝑥_𝑛} is a 1-approximating sequence corresponding to {𝑝_𝑛} for GMVI(𝐴,𝐹,𝑓). Since GMVI(𝐴,𝐹,𝑓) is strongly (resp., weakly) generalized 1-well-posed by perturbations, {𝑥_𝑛} converges strongly (resp., weakly) to some solution 𝑥^∗ of GMVI(𝐴,𝐹,𝑓). By Lemma 4.3, 𝑥^∗ is also a solution of IP(𝐴𝐹+𝜕𝑓). So IP(𝐴𝐹+𝜕𝑓) is strongly (resp., weakly) generalized well-posed by perturbations.

Remark 4.12. When 𝑝_𝑛=𝑝^∗ (for all 𝑛≥1) and 𝐹=𝐼 the identity mapping of 𝑋, Theorems 4.10 and 4.11 coincide with Theorems  4.3 and 4.4 of [9], respectively. Also, when 𝐹=𝐼, Theorems 4.10 and 4.11 coincide with Theorems  4.4 and 4.5 of [10], respectively. Furthermore, it can be found that Theorems 4.10 and 4.11 also improve and extend Theorems  4.3 and 4.4 of [11], respectively. In the meantime, Theorems 4.10 and 4.11 partially generalize Theorem 2.1 of Lemaire et al. [20].

5. Links with the Well-Posedness by Perturbations of Fixed Point Problems

Lemaire et al. [20] also considered the concepts of well-posedness by perturbations for a (single-valued) fixed point problem. In this section, we consider the concepts of well-posedness by perturbations for a (set-valued) fixed point problem. Let 𝑇∶𝑋→2^𝑋 be a set-valued mapping. The fixed point problem associated with 𝑇 is defined byFP(𝑇):find𝑥∈𝑋suchthat𝑥∈𝑇(𝑥). (5.1)

The perturbed problem of FP(𝑇) is given byFP_𝑝(𝑇):find𝑥∈𝑋suchthat𝑥∈𝑇𝑝,𝑥, (5.2) where 𝑇∶𝑃×𝑋→2^𝑋 is such that 𝑇(𝑝^∗,⋅)=𝑇.

Definition 5.1. Let {𝑝_𝑛}⊂𝑃 be with 𝑝_𝑛→𝑝^∗. A sequence {𝑥_𝑛}⊂𝑋 is called an approximating sequence corresponding to {𝑝_𝑛} for FP(𝑇) if there exists a sequence {𝑦_𝑛}⊂𝑋 with 𝑦_𝑛∈𝑇(𝑥_𝑛) (for all 𝑛≥1) such that ‖𝑥_𝑛−𝑦_𝑛‖→0 as 𝑛→∞.

Definition 5.2. One says that FP(𝑇) is strongly (resp., weakly) well-posed by perturbations if FP(𝑇) has a unique solution and, for any {𝑝_𝑛}⊂𝑃 with 𝑝_𝑛→𝑝^∗, every approximating sequence corresponding to {𝑝_𝑛} for FP(𝑇) converges strongly (resp., weakly) to the unique solution. FP(𝑇) is said to be strongly (resp., weakly) generalized well-posed by perturbations if FP(𝑇) has a nonempty solution set 𝑆 and, for any {𝑝_𝑛}⊂𝑃 with 𝑝_𝑛→𝑝^∗, every approximating sequence corresponding to {𝑝_𝑛} for FP(𝑇) has a subsequence which converges strongly (resp., weakly) to some point of 𝑆.

In particular, whenever 𝑇 is a single-valued mapping, we can readily see that Definitions  5.1 and 5.2 reduce to the corresponding definitions in [20]. It is known that in the setting of Hilbert spaces a generalized mixed variational inequality can be transformed into a fixed point problem (see [11, Proposition  2.1]). Utilizing this result, Ceng and Yao [11] proved that in the setting of Hilbert spaces the well-posedness of a generalized mixed variational inequality is equivalent to the well-posedness of the corresponding fixed point problem. In this section, we will further show that the well-posedness by perturbations of a generalized mixed variational inequality is closely related to the well-posedness by perturbations of the corresponding fixed point problem in the setting of Banach spaces. Let us first recall some concepts.

Let 𝑈={𝑥∈𝑋∶‖𝑥‖=1} be the unit sphere. A Banach space 𝑋 is said to be(a) strictly convex if, for any 𝑥,𝑦∈𝑈,𝑥≠𝑦⟹𝑥+𝑦2<1, (5.3)(b) smooth if the limit lim_𝑡→0𝑥+𝑡𝑦−‖𝑥‖𝑡 (5.4) exists for all 𝑥,𝑦∈𝑈.

The modulus of convexity of 𝑋 is defined by𝛿_𝑋(𝜀)=inf1−𝑥+𝑦2∶𝑥,𝑦∈𝑈,𝑥+𝑦≥𝜀, (5.5) and the modulus of smoothness of 𝑋 is defined by𝜌_𝑋(𝜏)1=sup2𝑥+𝑦+𝑥−𝑦−1∶𝑥∈𝑈,𝑦≤𝜏. (5.6) In the sequel we always suppose that 𝑞>1 and 𝑠>1 are fixed numbers. A Banach space 𝑋 is said to be(c) uniformly convex if 𝛿_𝑋(𝜀)>0 for all 𝜀∈(0,2),(d)𝑞-uniformly convex if there exists a constant 𝑐>0 such that 𝛿_𝑋(𝜀)≥𝑐𝜀^𝑞 for all 𝜀∈[0,2],(e) uniformly smooth if𝜌_𝑋(𝜏)𝜏⟶0as𝜏⟶0, (5.7)(f)𝑞-uniformly smooth if there exists a constant 𝑘>0 such that𝜌_𝑋(𝜏)≤𝑘𝜏^𝑞. (5.8)

It is well known that the Lebesgue 𝐿^𝑞(𝑞≥2) spaces are 𝑞-uniformly convex and 2-uniformly smooth and 𝐿^𝑞(1<𝑞<2) is 2-uniformly convex and 𝑞-uniformly smooth.

The generalized duality mapping 𝐽_𝑞∶𝑋→2^𝑋∗ is defined by𝐽_𝑞(𝑥)=𝑗_𝑞(𝑥)∈𝑋^∗∶𝑗_𝑞(𝑥),𝑥=‖𝑥‖^𝑞,𝑗_𝑞(𝑥)=‖𝑥‖^𝑞−1. (5.9) In particular, 𝐽=𝐽₂ is called the normalized duality mapping. 𝐽_𝑞 has the following properties:(i)  𝐽_𝑞 is bounded;(ii) if 𝑋 is smooth, then 𝐽_𝑞 is single-valued;(iii) if 𝑋 is strictly convex, then 𝐽_𝑞 is one-to-one and strictly monotone.

For more details, we refer the readers to [29, 33] and the references therein.

Lemma 5.3 (see [34]). Let 𝑋 be a 𝑞-uniformly smooth Banach space. Then, there exists a constant 𝐿_𝑞>0 such that 𝐽_𝑞(𝑥)−𝐽_𝑞𝑦≤𝐿_𝑞𝑥−𝑦^𝑞−1,∀𝑥,𝑦∈𝑋. (5.10)

Lemma 5.4 (see [34]). Let 𝑋 be a 𝑞-uniformly convex Banach space. Then, there exists a constant 𝑘_𝑞>0 such that 𝐽_𝑞(𝑥)−𝐽_𝑞𝑦,𝑥−𝑦≤𝑘_𝑞𝑥−𝑦^𝑞,∀𝑥,𝑦∈𝑋. (5.11)

Lemma 5.5 (see [10, Lemma  5.3]). Let 𝑋 be a 𝑞-uniformly convex Banach space and 𝑀∶𝑋→2^𝑋∗ a maximal monotone operator. Then, for every 𝜆>0 and 𝑠>1, (𝑀+𝜆𝐽_𝑠)⁻¹ is well-defined and single-valued.

The following result indicates that, under suitable conditions, the mapping Π^𝑓_𝜆∶𝑋^∗→𝑋 is Lipschitz continuous, where Π^𝑓_𝜆=(𝐽_𝑞+𝜆𝜕𝑓)⁻¹.

Lemma 5.6 (see [10, Lemma  5.4]). Let 𝑋 be a 𝑞-uniformly convex Banach space and 𝑀∶𝑋→2^𝑋∗ a maximal monotone operator. Then, for every 𝜆>0, Π^𝑓_𝜆𝑤₁−Π^𝑓_𝜆𝑤₂≤1𝜆𝑘_𝑞^1/(𝑞−1)𝑤₁−𝑤₂^1/(𝑞−1),∀𝑤₁,𝑤₂∈𝑋^∗. (5.12)

By means of Lemma 5.5, we can transform GMVI(𝐴,𝐹,𝑓) into a (set-valued) fixed point problem.

Lemma 5.7. Let 𝑋 be a 𝑞-uniformly convex Banach space, and let 𝐴∶𝑋→𝑋^∗ and 𝐹∶𝑋→2^𝑋. Let 𝑓∶𝑋→𝐑∪{+∞} be a proper, convex, and lower semicontinuous functional. Then, 𝑥∈𝑋 is a solution of GMVI(𝐴,𝐹,𝑓) if and only if it is a solution of the following fixed point problem: FPΠ^𝑓_𝜆𝐽_𝑞−𝜆𝐴𝐹:find𝑥∈𝑋suchthat𝑥∈Π^𝑓_𝜆𝐽_𝑞(𝑥)−𝜆𝐴𝐹(𝑥). (5.13)

Proof. The conclusion follows directly from the definitions of 𝜕𝑓 and Π^𝑓_𝜆 and Lemma 5.5.

Naturally, the perturbed problem of FP(Π^𝑓_𝜆(𝐽_𝑞−𝜆𝐴𝐹)) is given byFP_𝑝Π^𝑓_𝜆𝐽_𝑞𝑓−𝜆𝐴𝐹:find𝑥∈𝑋suchthat𝑥∈Π_𝜆^(𝑝,⋅)𝐽_𝑞(𝑥)−𝜆𝐴𝑝,𝐹(𝑥). (5.14)

Theorem 5.8. Let 𝑋 be an 𝑠-uniformly convex and 𝑞-uniformly smooth Banach space. Let 𝐴∶𝑃×𝑋→𝑋^∗ be uniformly continuous, and let 𝐹∶𝑋→2^𝑋 be a nonempty weakly compact-valued multifunction which is ℋ-uniformly continuous and monotone with respect to 𝐴(𝑝,⋅) for each 𝑝∈𝑃. Let 𝑓∶𝑃×𝑋→𝐑∪{+∞} be a continuous functional with respect to the product of the norm topology of 𝑃 and weak topology of 𝑋 such that the following conditions hold:(i)  𝑓(𝑝,⋅) is proper and convex for all 𝑝∈𝑃;(ii)dom𝑓 is well-positioned and dom𝑓⊂⋂_𝑝∈𝑃dom𝑓(𝑝,⋅);(iii)𝑒(dom𝑓(𝑝_𝑛,⋅),dom𝑓)→0 whenever 𝑝_𝑛→𝑝^∗.
Then, FP(Π^𝑓_𝜆(𝐽_𝑞−𝜆𝐴𝐹)) is weakly well-posed by perturbations whenever GMVI(𝐴,𝐹,𝑓) has a unique solution.

Proof. By Lemma 5.5, Π𝑓_{(𝑝𝑛𝜆,⋅)}=(𝐽_𝑞+𝜆𝜕𝑓(𝑝_𝑛,⋅))⁻¹ is well-defined and single-valued. Suppose that GMVI(𝐴,𝐹,𝑓) has a unique solution 𝑥^∗. Then, by Lemma 5.7,  𝑥^∗ is also the unique solution of FP(Π^𝑓_𝜆(𝐽_𝑞−𝜆𝐴𝐹)). Let {𝑝_𝑛}⊂𝑃 be with 𝑝_𝑛→𝑝^∗, and let {𝑥_𝑛} be an approximating sequence corresponding to {𝑝_𝑛} for FP(Π^𝑓_𝜆(𝐽_𝑞−𝜆𝐴𝐹)). Then, there exists a sequence {𝑦_𝑛}⊂𝑋 with 𝑦_𝑛𝑓∈Π^(𝑝𝑛_𝜆^,⋅)(𝐽_𝑞(𝑥_𝑛)−𝜆𝐴(𝑝_𝑛,𝐹(𝑥_𝑛))) (for all 𝑛≥1) such that ‖𝑥_𝑛−𝑦_𝑛‖→0. Further, it is known that there exists a sequence {𝑢_𝑛}⊂𝑋 with 𝑢_𝑛∈𝐹(𝑥_𝑛) (for all 𝑛≥1) such that 𝑦_𝑛𝑓=Π^(𝑝𝑛_𝜆^,⋅)𝐽_𝑞𝑥_𝑛−𝜆𝐴𝑝_𝑛,𝑢_𝑛,∀𝑛≥1. (5.15) By the definition of Π𝑓_{(𝑝𝑛𝜆,⋅)}, 𝐽_𝑞𝑥_𝑛−𝐽_𝑞𝑦_𝑛𝜆−𝐴𝑝_𝑛,𝑢_𝑛∈𝜕𝑓𝑝_𝑛,⋅𝑦_𝑛. (5.16) It follows that 𝑓𝑝_𝑛,𝑦−𝑓𝑝_𝑛,𝑦_𝑛≥𝐽_𝑞𝑥_𝑛−𝐽_𝑞𝑦_𝑛𝜆−𝐴𝑝_𝑛,𝑢_𝑛,𝑦−𝑦_𝑛,∀𝑦∈𝑋,𝑛≥1. (5.17) From (5.17) we get 𝑦_𝑛∈dom𝑓(𝑝_𝑛,⋅). We claim that {𝑦_𝑛} is bounded. Indeed, if {𝑦_𝑛} is unbounded, without loss of generality, we may assume that ‖𝑦_𝑛‖→+∞. Let 𝑡_𝑛=1𝑦_𝑛−𝑥^∗,𝑧_𝑛=𝑥^∗+𝑡_𝑛𝑦_𝑛−𝑥^∗. (5.18) From conditions (i)-(ii), we have 𝑧_𝑛∈dom𝑓(𝑝_𝑛,⋅). Note that 𝑧_𝑛≤𝑥^∗+𝑡_𝑛𝑦_𝑛−𝑥^∗=𝑥^∗+1,∀𝑛≥1. (5.19) So, {𝑧_𝑛} is bounded. Since 𝑋 is reflexive, it follows from the boundedness of {𝑧_𝑛} that there exists some subsequence of {𝑧_𝑛} which converges weakly to a point of 𝑋. Hence, without loss of generality, we may assume that 𝑡_𝑛∈(0,1) and 𝑧_𝑛⇀𝑧. By Lemma 2.14 and conditions (ii)-(iii), we get 𝑧≠𝑥^∗. For any 𝑦∈𝑋, observe that, for all 𝑣∈𝐹(𝑦), 𝐴𝑝_𝑛,𝑣,𝑧−𝑦=𝐴𝑝_𝑛,𝑣,𝑧−𝑧_𝑛+𝐴𝑝_𝑛,𝑣,𝑧_𝑛−𝑥^∗+𝐴𝑝_𝑛,𝑣,𝑥^∗=𝐴𝑝−𝑦_𝑛,𝑣,𝑧−𝑧_𝑛+𝑡_𝑛𝐴𝑝_𝑛,𝑣,𝑦_𝑛−𝑦+1−𝑡_𝑛𝐴𝑝_𝑛,𝑣,𝑥^∗−𝑦. (5.20) Since 𝑥^∗ is the unique solution of GMVI(𝐴,𝐹,𝑓), there exists some 𝑢^∗∈𝐹(𝑥^∗) such that 𝐴𝑢^∗,𝑥^∗−𝑦+𝑓𝑥^∗−𝑓𝑦≤0,∀𝑦∈𝑋. (5.21) Also, since 𝐹∶𝑋→2^𝑋 is a nonempty weakly compact-valued multifunction, both 𝐹(𝑥_𝑛) and 𝐹(𝑦_𝑛) are nonempty weakly compact and hence are nonempty, weakly closed, and weakly bounded. Note that the weak closedness of sets in 𝑋 implies the strong closedness and that the weak boundedness of sets in 𝑋 is equivalent to the strong boundedness. So, it is known that both 𝐹(𝑥_𝑛) and 𝐹(𝑦_𝑛) lie in CB(𝑋). According to Proposition 2.3, for each 𝑛≥1 and 𝑢_𝑛∈𝐹(𝑥_𝑛), there exists 𝑣_𝑛∈𝐹(𝑦_𝑛) such that 𝑢_𝑛−𝑣_𝑛≤11+𝑛ℋ𝐹𝑥_𝑛,𝐹𝑦_𝑛. (5.22) Note that 𝐹 is ℋ-uniformly continuous. Thus, it follows that 𝑢_𝑛−𝑣_𝑛≤11+𝑛ℋ𝐹𝑥_𝑛,𝐹𝑦_𝑛⟶0as𝑛⟶∞. (5.23) Furthermore, since 𝐹 is monotone with respect to 𝐴(𝑝_𝑛,⋅), one concludes that, for 𝑢^∗∈𝐹(𝑥^∗), 𝑣∈𝐹(𝑦), and 𝑣_𝑛∈𝐹(𝑦_𝑛), 𝐴𝑝_𝑛,𝑣,𝑥^∗−𝑦≤𝐴𝑝_𝑛,𝑢^∗,𝑥^∗−𝑦,𝐴𝑝_𝑛,𝑣,𝑦_𝑛−𝑦≤𝐴𝑝_𝑛,𝑣_𝑛,𝑦_𝑛−𝑦. (5.24) It follows from (5.20) and (5.24) that 𝐴𝑝_𝑛,𝑣,𝑧−𝑦≤𝐴𝑝_𝑛,𝑣,𝑧−𝑧_𝑛+𝑡_𝑛𝐴𝑝_𝑛,𝑣_𝑛,𝑦_𝑛−𝑦+1−𝑡_𝑛𝐴𝑝_𝑛,𝑢^∗,𝑥^∗=𝐴𝑝−𝑦_𝑛,𝑣,𝑧−𝑧_𝑛+𝑡_𝑛𝐴𝑝_𝑛,𝑣_𝑛−𝐴𝑝_𝑛,𝑢_𝑛,𝑦_𝑛−𝑦+𝑡_𝑛𝐴𝑝_𝑛,𝑢_𝑛,𝑦_𝑛−𝑦+1−𝑡_𝑛𝐴𝑝_𝑛,𝑢^∗,𝑥^∗−𝑦. (5.25) In addition, we have 𝑓𝑝_𝑛,𝑧_𝑛≤𝑡_𝑛𝑓𝑝_𝑛,𝑦_𝑛+1−𝑡_𝑛𝑓𝑝_𝑛,𝑥^∗ (5.26) since 𝑓(𝑝_𝑛,⋅) is convex. It follows from (5.17), (5.25), and (5.26) that 𝐴𝑝_𝑛,𝑣,𝑧−𝑦≤𝐴𝑝_𝑛,𝑣,𝑧−𝑧_𝑛+𝑡_𝑛𝐴𝑝_𝑛,𝑣_𝑛−𝐴𝑝_𝑛,𝑢_𝑛,𝑦_𝑛+𝑡−𝑦_𝑛𝜆𝐽_𝑞𝑥_𝑛−𝐽_𝑞𝑦_𝑛,𝑦_𝑛−𝑦+𝑡_𝑛𝑓𝑝_𝑛,𝑦−𝑡_𝑛𝑓𝑝_𝑛,𝑦_𝑛+1−𝑡_𝑛𝐴𝑝_𝑛,𝑢^∗,𝑥^∗=𝐴𝑝−𝑦_𝑛,𝑣,𝑧−𝑧_𝑛+𝑡_𝑛𝐴𝑝_𝑛,𝑣_𝑛−𝐴𝑝_𝑛,𝑢_𝑛,𝑦_𝑛+𝑡−𝑦_𝑛𝜆𝐽_𝑞𝑥_𝑛−𝐽_𝑞𝑦_𝑛,𝑦_𝑛−𝑦+1−𝑡_𝑛𝐴𝑝_𝑛,𝑢^∗,𝑥^∗−𝑦+𝑡_𝑛𝑓𝑝_𝑛,𝑦+1−𝑡_𝑛𝑓𝑝_𝑛,𝑥^∗−𝑡_𝑛𝑓𝑝_𝑛,𝑦_𝑛+1−𝑡_𝑛𝑓𝑝_𝑛,𝑥^∗≤𝐴𝑝_𝑛,𝑣,𝑧−𝑧_𝑛+𝑡_𝑛𝐴𝑝_𝑛,𝑣_𝑛−𝐴𝑝_𝑛,𝑢_𝑛,𝑦_𝑛+𝑡−𝑦_𝑛𝜆𝐽_𝑞𝑥_𝑛−𝐽_𝑞𝑦_𝑛,𝑦_𝑛−𝑦+1−𝑡_𝑛𝐴𝑝_𝑛,𝑢^∗,𝑥^∗−𝑦+𝑡_𝑛𝑓𝑝_𝑛,𝑦+1−𝑡_𝑛𝑓𝑝_𝑛,𝑥^∗−𝑓𝑝_𝑛,𝑧_𝑛. (5.27) Note that ‖𝑦_𝑛‖→+∞ and 𝑡_𝑛𝑦_𝑛≤𝑦−𝑦^𝑛−𝑥^∗+𝑥^∗−𝑦𝑦_𝑛−𝑥^∗𝑥=1+^∗−𝑦𝑦_𝑛−𝑥^∗. (5.28) It is easy to see that {𝑡_𝑛(𝑦_𝑛−𝑦)} is bounded. Since ‖𝑥_𝑛−𝑦_𝑛‖→0, by Lemma 5.3, 𝐽_𝑞𝑥_𝑛−𝐽_𝑞𝑦_𝑛≤𝐿_𝑞𝑥_𝑛−𝑦_𝑛^𝑞−1⟶0. (5.29) In the meantime, on account of ‖𝑢_𝑛−𝑣_𝑛‖→0, we have 𝐴𝑝_𝑛,𝑣_𝑛−𝐴𝑝_𝑛,𝑢_𝑛⟶0 (5.30) by means of the uniformly continuity of 𝐴. Consequently, letting 𝑛→∞ we obtain that 𝑡_𝑛𝐴𝑝_𝑛,𝑣_𝑛−𝐴𝑝_𝑛,𝑢_𝑛,𝑦_𝑛−𝑦≤𝐴𝑝_𝑛,𝑣_𝑛−𝐴𝑝_𝑛,𝑢_𝑛𝑡_𝑛𝑦_𝑛𝑡−𝑦⟶0,_𝑛𝜆𝐽_𝑞𝑥_𝑛−𝐽_𝑞𝑦_𝑛,𝑦_𝑛≤𝐽−𝑦^𝑞𝑥_𝑛−𝐽_𝑞𝑦_𝑛𝜆𝑡_𝑛𝑦_𝑛−𝑦⟶0,1−𝑡_𝑛𝐴𝑝_𝑛,𝑢^∗,𝑥^∗−𝑦⟶𝐴𝑝^∗,𝑢^∗,𝑥^∗−𝑦. (5.31) Moreover, also observe that 𝐴𝑝_𝑛,𝑣,𝑧−𝑧_𝑛=𝐴𝑝_𝑛,𝑣−𝐴𝑝^∗,𝑣,𝑧−𝑧_𝑛+𝐴𝑝^∗,𝑣,𝑧−𝑧_𝑛≤𝐴𝑝_𝑛,𝑣−𝐴𝑝^∗,𝑣𝑧_𝑛+𝐴𝑝−𝑧_∗,𝑣,𝑧_𝑛−𝑧⟶0as𝑛⟶∞. (5.32) Further, since 𝑓∶𝑃×𝑋→𝐑∪{+∞} is a continuous functional with respect to the product of the norm topology of 𝑃 and weak topology of 𝑋, we get that, as 𝑛→∞, 𝑡_𝑛𝑓𝑝_𝑛,𝑦⟶0,1−𝑡_𝑛𝑓𝑝_𝑛,𝑥^∗⟶𝑓𝑝^∗,𝑥^∗,𝑓𝑝_𝑛,𝑧_𝑛⟶𝑓𝑝^∗,𝑧. (5.33) Therefore, letting 𝑛→∞ in (5.27) we conclude that 𝐴𝑣,𝑧−𝑦=𝐴𝑝^∗≤𝐴𝑝,𝑣,𝑧−𝑦_∗,𝑢^∗,𝑥^∗−𝑦+𝑓𝑝^∗,𝑥^∗−𝑓𝑝^∗=,𝑧𝐴𝑢^∗,𝑥^∗−𝑦+𝑓𝑥^∗−𝑓(𝑧),∀𝑦∈𝑋,𝑣∈𝐹𝑦. (5.34) By using (5.34) and the same argument as in the proof of Theorem 4.4, we can prove that 𝑧 is a solution of GMVI(𝐴,𝐹,𝑓), a contradiction. Thus, {𝑦_𝑛} is bounded and so is {𝑥_𝑛}.
By using (5.17) and the similar argument to that in the proof of Theorem 4.4, we can prove that {𝑥_𝑛} converges weakly to 𝑥^∗. Since 𝑥^∗ is the unique solution of FP(Π^𝑓_𝜆(𝐽_𝑞−𝜆𝐴𝐹)), FP(Π^𝑓_𝜆(𝐽_𝑞−𝜆𝐴𝐹)) is weakly well-posed by perturbations.

Remark 5.9. Theorem 5.8 generalizes Theorem  5.1 of [9] and Theorem  5.1 of [11] since every Hilbert space is 2-uniformly convex and 2-uniformly smooth. Theorem 5.8 improves, extends, and develops [10, Theorem  5.1] in the following aspects.(i) The mixed variational inequality problem (MVI) in [10, Theorem  5.1] is extended to develop the more general problem, that is, the generalized mixed variational inequality problem (GMVI) with a nonempty weakly compact-valued multifunction in the setting of Banach spaces. In the meantime, the (single-valued) fixed point problem corresponding to MVI in [10, Theorem  5.1] is extended to develop the more general problem, that is, the (set-valued) fixed point problem corresponding to GMVI. Furthermore, the concept of weak well-posedness by perturbations for the (single-valued) fixed point problem corresponding to MVI is extended to develop the concept of weak well-posedness by perturbations for the (set-valued) fixed point problem corresponding to GMVI.(ii) Since the generalized mixed variational inequality problem (GMVI) is more general and more complicated than the mixed variational inequality problem (MVI), the assumptions in Theorem 5.8 are very different from the ones in [10, Theorem  5.1]; for instance, in Theorem 5.8, let 𝐴∶𝑃×𝑋→𝑋^∗ be uniformly continuous, and let 𝐹∶𝑋→2^𝑋 be a nonempty weakly compact-valued multifunction which is ℋ-uniformly continuous and monotone with respect to 𝐴(𝑝,⋅) for each 𝑝∈𝑃, but, in [10, Theorem  5.1], let 𝐴∶𝑃×𝑋→𝑋^∗ be a uniformly continuous mapping such that 𝐴(𝑝,⋅)∶𝑋→𝑋^∗ is monotone for all 𝑝∈𝑃.(iii) The technique of proving weak well-posedness by perturbations for (set-valued) fixed point problem FP(Π^𝑓_𝜆(𝐽_𝑞−𝜆𝐴𝐹)) in Theorem 5.8 is very different from the one for (single-valued) fixed point problem FP(Π^𝑓_𝜆(𝐽_𝑞−𝜆𝐴)) in [10, Theorem  5.1] because our technique depends on the well-known Nadler’s Theorem [27], the ℋ-uniformly continuity of nonempty weakly compact-valued multifunction 𝐹 and the monotonicity of 𝐹 with respect to 𝐴(𝑝,⋅) for each 𝑝∈𝑃.

Now we give the following example as an application of Theorem 5.8.

Example 5.10. Let 𝑋=𝐑, 𝑃=[−1,1], and 𝑝^∗=0. Let 𝐴(𝑝,𝑥)=𝑥+𝑝, 𝐹(𝑥)=𝑥−sin𝑥 for all 𝑥∈𝑋, 𝑝∈𝑃, and let 𝑓=𝑝,𝑝,𝑥0,𝑥∈𝐾+∞,otherwise, (5.35) for all 𝑝∈𝑃, where 𝐾(𝑝)=[−1,1]+{𝑝}. Then, Dom𝐴(𝑝,⋅)=𝐑, Dom𝑓=[−1,1], Dom𝑓(𝑝,⋅)=𝐾(𝑝) for all 𝑝∈𝑃. Clearly, if 𝑝_𝑛→𝑝^∗, then 𝑒(Dom𝑓(𝑝_𝑛,⋅),Dom𝑓)→0. By the above definitions, it is easy to see that 𝐴 is uniformly continuous, 𝐹 is ℋ-uniformly continuous and monotone with respect to 𝐴(𝑝,⋅) for each 𝑝∈𝑃, and 𝑓(𝑝,⋅) is proper and convex. By (ii) of Remark 2.7, Dom𝑓 is well-positioned. It is known that 𝐑 is 2-uniformly convex and 2-uniformly smooth. Hence, all the assumptions of Theorem 5.8 are satisfied. Let 𝑆 be the solution set of GMVI(𝐴,𝐹,𝑓). It follows that 𝑆=𝑥∈[−1,1]∶(𝑥−sin𝑥)𝑥−𝑦≤0,∀𝑦∈[−1,1]={0}. (5.36) So 𝑥^∗=0 is the unique solution of GMVI(𝐴,𝐹,𝑓). By Theorem 5.8, FP(Π^𝑓_𝜆(𝐽_𝑞−𝜆𝐴𝐹)) is well-posed by perturbations.

Based on Theorems 4.7 and 5.8 and Lemma 5.7, we have the following result.

Theorem 5.11. Let 𝑋 be an 𝑠-uniformly convex and 𝑞-uniformly smooth Banach space. Let 𝐴∶𝑃×𝑋→𝑋^∗ be uniformly continuous, let 𝐴∶𝑋→𝑋^∗ be weakly continuous, and let 𝐹∶𝑋→2^𝑋 be a nonempty weakly compact-valued multifunction which is ℋ-uniformly continuous and monotone with respect to 𝐴(𝑝,⋅) for each 𝑝∈𝑃. Let 𝑓∶𝑃×𝑋→𝐑∪{+∞} be a continuous functional with respect to the product of the norm topology of 𝑃 and weak topology of 𝑋 such that the following conditions hold:(i)  𝑓(𝑝,⋅) is proper and convex for all 𝑝∈𝑃;(ii)dom𝑓 is well-positioned and dom𝑓⊂⋂_𝑝∈𝑃dom𝑓(𝑝,⋅);(iii)𝑒(dom𝑓(𝑝_𝑛,⋅),dom𝑓)→0 whenever 𝑝_𝑛→𝑝^∗. Then, the following statements are equivalent:(i)  GMVI(𝐴,𝐹,𝑓) is weakly well-posed by perturbations;(ii)IP(𝐴𝐹+𝜕𝑓) is weakly well-posed by perturbations;(iii)FP(Π^𝑓_𝜆(𝐽_𝑞−𝜆𝐴𝐹)) is weakly well-posed by perturbations;(iv)GMVI(𝐴,𝐹,𝑓) has a unique solution;(v)IP(𝐴𝐹+𝜕𝑓) has a unique solution;(vi)FP(Π^𝑓_𝜆(𝐽_𝑞−𝜆𝐴𝐹)) has a unique solution.

Next we consider the case of generalized well-posedness by perturbations.

Theorem 5.12. Let 𝑋 be an 𝑠-uniformly convex and 𝑞-uniformly smooth Banach space. Let 𝐴∶𝑃×𝑋→𝑋^∗ be uniformly continuous, and let 𝐹∶𝑋→2^𝑋 be a nonempty weakly compact-valued multifunction which is ℋ-uniformly continuous and monotone with respect to 𝐴(𝑝,⋅) for each 𝑝∈𝑃. Let 𝑓∶𝑃×𝑋→𝐑∪{+∞} be a continuous functional with respect to the product of the norm topology of 𝑃 and weak topology of 𝑋 such that 𝑓(𝑝,⋅) is proper and convex. Then, FP(Π^𝑓_𝜆(𝐽_𝑞−𝜆𝐴𝐹)) is strongly (resp., weakly) generalized well-posed by perturbations whenever GMVI(𝐴,𝐹,𝑓) is strongly (resp., weakly) generalized (1+1/𝜆)-well-posed by perturbations.

Proof. Suppose that GMVI(𝐴,𝐹,𝑓) is strongly (resp., weakly) generalized (1+1/𝜆)-well-posed by perturbations. Let {𝑝_𝑛}⊂𝑃 be with 𝑝_𝑛→𝑝^∗, and let {𝑥_𝑛} be an approximating sequence corresponding to {𝑝_𝑛} for FP(Π^𝑓_𝜆(𝐽_𝑞−𝜆𝐴𝐹)). Then, there exists a sequence {𝑦_𝑛}⊂𝑋 with 𝑦_𝑛𝑓∈Π^(𝑝𝑛_𝜆^,⋅)(𝐽_𝑞(𝑥_𝑛)−𝜆𝐴(𝑝_𝑛,𝐹(𝑥_𝑛))) (for all 𝑛≥1) such that ‖𝑥_𝑛−𝑦_𝑛‖→0. Further, it is known that there exists a sequence {𝑢_𝑛}⊂𝑋 with 𝑢_𝑛∈𝐹(𝑥_𝑛) (for all 𝑛≥1) such that 𝑦_𝑛𝑓=Π^(𝑝𝑛_𝜆^,⋅)𝐽_𝑞𝑥_𝑛−𝜆𝐴𝑝_𝑛,𝑢_𝑛,∀𝑛≥1. (5.37) By the definition of Π𝑓_{(𝑝𝑛𝜆,⋅)}, 𝐽_𝑞𝑥_𝑛−𝐽_𝑞𝑦_𝑛𝜆−𝐴𝑝_𝑛,𝑢_𝑛∈𝜕𝑓𝑝_𝑛,⋅𝑦_𝑛. (5.38) It follows that 𝑓𝑝_𝑛,𝑦−𝑓𝑝_𝑛,𝑦_𝑛≥𝐽_𝑞𝑥_𝑛−𝐽_𝑞𝑦_𝑛𝜆−𝐴𝑝_𝑛,𝑢_𝑛,𝑦−𝑦_𝑛,∀𝑦∈𝑋,𝑛≥1. (5.39) Furthermore, since 𝐹∶𝑋→2^𝑋 is a nonempty weakly compact-valued multifunction, both 𝐹(𝑥_𝑛) and 𝐹(𝑦_𝑛) are nonempty weakly compact and hence are nonempty, weakly closed, and weakly bounded. Note that the weak closedness of sets in 𝑋 implies the strong closedness and that the weak boundedness of sets in 𝑋 is equivalent to the strong boundedness. So, it is known that both 𝐹(𝑥_𝑛) and 𝐹(𝑦_𝑛) lie in CB(𝑋). According to Proposition 2.3, for each 𝑛≥1 and 𝑢_𝑛∈𝐹(𝑥_𝑛) there exists 𝑣_𝑛∈𝐹(𝑦_𝑛) such that 𝑢_𝑛−𝑣_𝑛≤11+𝑛ℋ𝐹𝑥_𝑛,𝐹𝑦_𝑛. (5.40) Note that 𝐹 is ℋ-uniformly continuous. Thus, one deduces that 𝑢_𝑛−𝑣_𝑛≤11+𝑛ℋ𝐹𝑥_𝑛,𝐹𝑦_𝑛⟶0as𝑛⟶∞. (5.41)
Now utilizing (5.39) we have 𝐴𝑝_𝑛,𝑣_𝑛,𝑦_𝑛−𝑦+𝑓𝑝_𝑛,𝑦_𝑛−𝑓𝑝_𝑛≤𝐽,𝑦_𝑞𝑥_𝑛−𝐽_𝑞𝑦_𝑛𝜆,𝑦_𝑛−𝑦+𝐴𝑝_𝑛,𝑣_𝑛−𝐴𝑝_𝑛,𝑢_𝑛,𝑦_𝑛≤1−𝑦211+𝜆𝑦_𝑛−𝑦²+12𝜀_𝑛,∀𝑦∈𝑋,𝑛≥1, (5.42) where 𝜀_𝑛=1𝜆𝐽_𝑞𝑥_𝑛−𝐽_𝑞𝑦_𝑛²+𝐴𝑝_𝑛,𝑣_𝑛−𝐴𝑝_𝑛,𝑢_𝑛²≤𝐿_𝑞𝜆𝑥_𝑛−𝑦_𝑛^2(𝑞−1)+𝐴𝑝_𝑛,𝑣_𝑛−𝐴𝑝_𝑛,𝑢_𝑛²⟶0. (5.43) Therefore, {𝑦_𝑛} is a (1+1/𝜆)-approximating sequence corresponding to {𝑝_𝑛} for GMVI(𝐴,𝐹,𝑓). By the strong (resp., weak) generalized (1+1/𝜆)-well-posedness by perturbations, {𝑦_𝑛} has some subsequence which converges strongly (resp., weakly) to a solution 𝑥^∗ of GMVI(𝐴,𝐹,𝑓). By Lemma 5.7, 𝑥^∗ is also a solution of FP(Π^𝑓_𝜆(𝐽_𝑞−𝜆𝐴𝐹)). Consequently, FP(Π^𝑓_𝜆(𝐽_𝑞−𝜆𝐴𝐹)) is strongly (resp., weakly) generalized well-posed by perturbations.

Remark 5.13. Theorem 5.12 generalizes Theorem  5.3 of [9] and Theorem  5.3 of [11]. In addition, whenever 𝐹=𝐼 the identity mapping of 𝑋, Theorem 5.12 reduces to Theorem  5.3 of [10].

Theorem 5.14. Let 𝑋 be a 𝑞-uniformly convex Banach space. Let 𝐴∶𝑃×𝑋→𝑋^∗ be uniformly continuous, and let 𝐹∶𝑋→2^𝑋 be a nonempty weakly compact-valued multifunction which is ℋ-uniformly continuous. Let 𝑓∶𝑃×𝑋→𝐑∪{+∞} be a functional such that 𝑓(𝑝,⋅) is proper, convex, and lower semicontinuous for each 𝑝∈𝑃. Then, GMVI(𝐴,𝐹,𝑓) is strongly (resp., weakly) generalized well-posed by perturbations whenever FP(Π^𝑓_𝜆(𝐽_𝑞−𝜆𝐴𝐹)) is strongly (resp., weakly) generalized well-posed by perturbations.

Proof. Let {𝑝_𝑛}⊂𝑃 be with 𝑝_𝑛→𝑝^∗, and let {𝑥_𝑛} be an approximating sequence corresponding to {𝑝_𝑛} for GMVI(𝐴,𝐹,𝑓). Then, there exist {𝑢_𝑛}⊂𝑋 with 𝑢_𝑛∈𝐹(𝑥_𝑛) (for all 𝑛≥1) and 0<𝜀_𝑛→0 such that 𝐴𝑝_𝑛,𝑢_𝑛,𝑥_𝑛−𝑦+𝑓𝑝_𝑛,𝑥_𝑛−𝑓𝑝_𝑛,𝑦≤𝜀_𝑛,∀𝑦∈𝑋,𝑛≥1. (5.44) Define 𝜑_𝑛∶𝑋→𝐑∪{+∞} as follows: 𝜑_𝑛𝑦=𝑓𝑝_𝑛,𝑦+𝐴𝑝_𝑛,𝑢_𝑛,𝑦−𝑥_𝑛,∀𝑦∈𝑋,𝑛≥1. (5.45) Clearly, 𝜑_𝑛 is proper, convex, and lower semicontinuous and 0∈𝜕_𝜀𝑛𝜑_𝑛(𝑥_𝑛) for all 𝑛≥1. By the Brondsted-Rockafellar theorem [32], there exist 𝑥_𝑛∈𝑋 and 𝑤_𝑛∈𝜕𝜑_𝑛𝑥_𝑛=𝐴𝑝_𝑛,𝑢_𝑛+𝜕𝑓𝑝_𝑛,⋅𝑥_𝑛 (5.46) such that 𝑥_𝑛−𝑥_𝑛≤√𝜀_𝑛,𝑤_𝑛≤√𝜀_𝑛. (5.47) Since 𝐹 is a nonempty weakly compact-valued multifunction, both 𝐹(𝑥_𝑛) and 𝐹(𝑥_𝑛) lie in CB(𝑋) for each 𝑛≥1. By Proposition 2.3, for each 𝑛≥1 and 𝑢_𝑛∈𝐹(𝑥_𝑛) there exists 𝑢_𝑛∈𝐹(𝑥_𝑛) such that 𝑢_𝑛−𝑢_𝑛≤11+𝑛ℋ𝐹𝑥_𝑛,𝐹𝑥_𝑛. (5.48) Since 𝐹 is ℋ-uniformly continuous, we obtain from (5.47) that 𝑢_𝑛−𝑢_𝑛≤11+𝑛ℋ𝐹𝑥_𝑛,𝐹𝑥_𝑛⟶0as𝑛⟶∞. (5.49) Utilizing (5.46) we have 𝑥_𝑛𝑓=Π^(𝑝𝑛_𝜆^,⋅)𝜆𝑤_𝑛+𝐽_𝑞𝑥_𝑛−𝜆𝐴𝑝_𝑛,𝑢_𝑛. (5.50) Since 𝐴 is uniformly continuous, it follows from (5.47) and (5.50) that 𝜆𝑤_𝑛+𝐽_𝑞𝑥_𝑛−𝜆𝐴𝑝_𝑛,𝑢_𝑛−𝐽_𝑞𝑥_𝑛−𝜆𝐴𝑝_𝑛,𝑢_𝑛≤𝜆𝑤_𝑛+𝜆𝐴𝑝_𝑛,𝑢_𝑛−𝐴𝑝_𝑛,𝑢_𝑛⟶0. (5.51) By Lemma 5.6 and (5.50) and (5.51), 𝑥_𝑛𝑓−Π^(𝑝𝑛_𝜆^,⋅)𝐽_𝑞𝑥_𝑛−𝜆𝐴𝑝_𝑛,𝑢_𝑛⟶0. (5.52) Thus, {𝑥_𝑛} is an approximating sequence corresponding to {𝑝_𝑛} for FP(Π^𝑓_𝜆(𝐽_𝑞−𝜆𝐴𝐹)). Repeating the same argument as in the proof of Theorem 4.10, we can deduce that {𝑥_𝑛} has some subsequence {𝑥_𝑛𝑘} which converges strongly (resp., weakly) to some solution 𝑥^∗ of FP(Π^𝑓_𝜆(𝐽_𝑞−𝜆𝐴𝐹)). By Lemma 5.7, 𝑥^∗ is also a solution of GMVI(𝐴,𝐹,𝑓). Thus, GMVI(𝐴,𝐹,𝑓) is strongly (resp., weakly) generalized well-posed by perturbations.

Remark 5.15. Theorem 5.14 generalizes Theorem  5.4 of [9] and Theorem  5.4 of [11]. In addition, whenever 𝐹=𝐼 the identity mapping of 𝑋, Theorem 5.14 reduces to Theorem  5.4 of [10].

Acknowledgments

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) and by a grant from the National Science Council of ROC (NSC 100-2115-M-037-001).