𝛼-Well-Posedness for Mixed Quasi Variational-Like Inequality Problems

Abstract

The concepts of 𝛼-well-posedness, 𝛼-well-posedness in the generalized sense, L-𝛼-well-posedness and L-𝛼-well-posedness in the generalized sense for mixed quasi variational-like inequality problems are investigated. We present some metric characterizations for these well-posednesses.

1. Introduction

Well-posedness plays a crucial role in the stability theory for optimization problems, which guarantees that, for an approximating solution sequence, there exists a subsequence which converges to a solution. The study of well-posedness for scalar minimization problems started from Tykhonov [1] and Levitin and Polyak [2]. Since then, various notions of well-posedness for scalar minimization problems have been defined and studied in [3–8] and the references therein. It is worth noting that the recent study for various types of well-posedness has been generalized to variational inequality problems [9–13], generalized variational inequality problems [14, 15], quasi variational inequality problems [16], generalized quasi variational inequality problems [17], generalized vector variational inequality problems [18], vector quasi variational inequality problems [19], mixed quasi variational-like inequality problems [20], and many other problems.

In this paper, we are interested in investigating four classes of well-posednesses for a mixed quasi variational-like inequality problem. The paper is organized as follows. In Section 2, we introduce the definitions of 𝛼-well-posedness, 𝛼-well-posedness in the generalized sense, L-𝛼-well-posedness and L-𝛼-well-posedness in the generalized sense for a mixed quasi variational-like inequality problem. In Section 3, some characterizations of 𝛼-well-posedness, and L-𝛼-well-posedness for a mixed quasi variational-like inequality problem are obtained. In Section 4, some characterizations of 𝛼-well-posedness in the generalized sense and L-𝛼-well-posedness in the generalized sense for a mixed quasi variational-like inequality problem are presented.

2. Preliminaries

Throughout this paper, without other specification, let 𝐸 be a real Banach space with the dual 𝐸^∗, let 𝐾 be a nonempty closed convex subset of 𝐸, and let 𝑆∶𝐾→2^𝐾 be a set-valued map. Let 𝐹∶𝐾→2^𝐸∗ be a set-valued map with nonempty values, let 𝜂∶𝐾×𝐾→𝐸 be a single-valued map, and let 𝑓∶𝐾→𝑅 be a real-valued function. Ceng et al. [20] introduced the following mixed quasi variational-like inequality problem, which is to find a point 𝑥₀∈𝐾 such that, for some 𝑢₀∈𝐹(𝑥₀), (MQVLI)𝑥₀∈𝑆𝑥₀,𝑢₀,𝜂𝑥₀,𝑦+𝑓𝑥₀−𝑓(𝑦)≤0,∀𝑦∈𝑆𝑥₀.(2.1)

Denote by Γ the solution set of (MQVLI). Let 𝛼>0; we introduce the notions of several classes of 𝛼-well-posednesses for (MQVLI).

Definition 2.1. A sequence (𝑥_𝑛)_𝑛 in 𝐾 is an 𝛼-approximating sequence for (MQVLI) if(i)there exists a sequence (𝑢_𝑛)_𝑛 in 𝐸^∗, with 𝑢_𝑛∈𝐹(𝑥_𝑛),forall𝑛∈𝑁;(ii)there exists a sequence (𝜀_𝑛)_𝑛,𝜀_𝑛>0, 𝜀_𝑛→0 such that 𝑢𝑑𝑥_𝑛,𝑆𝑥_𝑛≤𝜀_𝑛,∀𝑛∈𝑁,_𝑛,𝜂𝑥_𝑛,𝑦+𝑓𝑥_𝑛−𝑓(𝑦)−𝛼2‖‖𝑥^𝑛−𝑦‖‖²≤𝜀_𝑛,∀𝑦∈𝑆𝑥_𝑛,∀𝑛∈𝑁.(2.2)

Definition 2.2. (MQVLI) is said to be 𝛼-well-posed (resp., 𝛼-well-posed in the generalized sense) if it has a unique solution 𝑥₀ and every 𝛼-approximating sequence (𝑥_𝑛)_𝑛 strongly converges to 𝑥₀ (resp., if the solution set Γ of (MQVLI) is nonempty and for every 𝛼-approximating sequence (𝑥_𝑛)_𝑛 has a subsequence which strongly converges to a point of Γ).

Definition 2.3. A sequence (𝑥_𝑛)_𝑛 is an L-𝛼-approximating sequence for (MQVLI) if there exists a real number sequence (𝜀_𝑛)_𝑛,𝜀_𝑛>0, 𝜀_𝑛→0 such that 𝑣,𝜂𝑥_𝑛𝑑𝑥_𝑛,𝑆𝑥_𝑛≤𝜀_𝑛,∀𝑛∈𝑁,,𝑦+𝑓𝑥_𝑛−𝑓(𝑦)−𝛼2‖‖𝑥^𝑛−𝑦‖‖²≤𝜀_𝑛,∀𝑦∈𝑆𝑥_𝑛,𝑣∈𝐹(𝑦),𝑛∈𝑁.(2.3)

Definition 2.4. (MQVLI) is said to be L-𝛼-well-posed (resp., L-𝛼-well-posed in the generalized sense) if it has a unique solution 𝑥₀ and every L-𝛼-approximating sequence (𝑥_𝑛)_𝑛 strongly converges to 𝑥₀ (resp., if the solution set Γ of (MQVLI) is nonempty and for every L-𝛼-approximating sequence (𝑥_𝑛)_𝑛 has a subsequence which strongly converges to a point of Γ).

It is worth noting that if 𝛼=0, then the definitions of 𝛼-well-posedness, 𝛼-well-posedness in the generalized sense, L-𝛼-well-posedness, and L-𝛼-well-posedness in the generalized sense for (MQVLI), respectively, reduce to those of the well-posedness, well-posedness in the generalized sense, L-well-posedness, and L-well-posedness in the generalized sense for (MQVLI) in [20]. We also note that Definition 2.2 generalizes and extends 𝛼-well-posedness and 𝛼-well-posedness in the generalized sense of variational inequalities in [10] which are related to the continuously differentiable gap function of variational inequalities introduced by Fukushima [21].

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

We recall the notion of Mosco convergence [22]. A sequence (𝐻_𝑛)_𝑛 of subsets of 𝐸 Mosco converges to a set 𝐻 if 𝐻=liminf_𝑛𝐻_𝑛=𝑤−limsup_𝑛𝐻_𝑛,(2.4) where liminf_𝑛𝐻_𝑛 and 𝑤−limsup_𝑛𝐻_𝑛 are, respectively, the Painlevé-Kuratowski strong limit inferior and weak limit superior of a sequence (𝐻_𝑛)_𝑛, that is, 𝑤−limsup_𝑛liminf_𝑛𝐻_𝑛=𝑦∈𝐸∶∃𝑦_𝑛∈𝐻_𝑛,𝑛∈𝑁,with𝑦_𝑛→𝑦,𝐻_𝑛=𝑦∈𝐸∶∃𝑛_𝑘↑+∞,𝑛_𝑘∈𝑁,∃𝑦_𝑛𝑘∈𝐻_𝑛𝑘,𝑘∈𝑁,with𝑦_𝑛𝑘⇀𝑦,(2.5) where “⇀” means weak convergence, and “→” means strong convergence.

If 𝐻=liminf_𝑛𝐻_𝑛, we call the sequence (𝐻_𝑛)_𝑛 of subsets of 𝐸 Lower Semi-Mosco convergent to a set 𝐻.

It is easy to see that a sequence (𝐻_𝑛)_𝑛 of subsets of 𝐸 Mosco converges to a set 𝐻 implies that the sequence (𝐻_𝑛)_𝑛 also Lower Semi-Mosco converges to the set 𝐻, but the converse is not true in general.

We will use the usual abbreviations usc and lsc for “upper semicontinuous” and “lower semicontinuous”, respectively. For any 𝑥,𝑦∈𝐸, [𝑥,𝑦] will denote the line segment {𝑡𝑥+(1−𝑡)𝑦∶𝑡∈[0,1]}, while [𝑥,𝑦) and (𝑥,𝑦) are defined analogously. We will frequently use 𝑠,𝑤, and 𝑤^∗ to denote, respectively, the norm topology on 𝐸, the weak topology on 𝐸, and the weak* topology on 𝐸. Given a convex set 𝐾, a multivalued map 𝐹∶𝐾→2^𝐸∗ will be called upper hemicontinuous, if its restriction on any line segment [𝑥,𝑦]⊆𝐾 is usc with respect to the 𝑤^∗ topology on 𝐸^∗. 𝐹∶𝐾→2^𝐸∗ will be called 𝜂-monotone if, for any 𝑥,𝑦∈𝐾, for all 𝑢∈𝐹(𝑥),𝑣∈𝐹(𝑦), ⟨𝑢−𝑣,𝜂(𝑥,𝑦)⟩≥0. We refer the reader to [23, 24] for basic facts about multivalued maps.

Lemma 2.5 (see [25]). Let (𝐻_𝑛)_𝑛 be a sequence of nonempty subsets of a Banach space 𝐸 such that (i)𝐻_𝑛 is convex for every 𝑛∈𝑁;(ii)𝐻₀⊆liminf_𝑛𝐻_𝑛;(iii)there exists 𝑚∈𝑁 such that int⋂_𝑛≥𝑚𝐻_𝑛≠∅. Then, for every 𝑢₀∈int𝐻₀, there exists a positive real number 𝛿 such that int𝐵𝑢₀,𝛿⊆𝐻_𝑛,∀𝑛≥𝑚,(2.6) where 𝐵(𝑢₀,𝛿) is a closed ball with a center 𝑢₀ and radius 𝛿. If 𝐸 is a finite dimensional space, then assumption (iii) can be replaced by (iii)′int𝐻₀≠∅.

The following lemmas play important role in this paper.

Lemma 2.6. Let 𝐸 be a real separable Banach space with the dual 𝐸^∗, let 𝑆₀ be a nonempty convex subset of 𝐸, and let 𝐹∶𝑆₀→2^𝐸∗ be a set-valued map with nonempty, weakly* compact convex valued, 𝜂-monotone, and upper hemicontinuous. Let 𝜂∶𝑆₀×𝑆₀→𝐸 be a single-valued map with 𝜂(𝑥,𝑥)=0,forall𝑥∈𝑆₀, and let 𝑓∶𝑆₀→𝑅 be a convex lsc function. Assume that the map 𝑦↦⟨𝑢,𝜂(𝑥,𝑦)⟩ is concave for each (𝑢,𝑥)∈𝐹(𝑆₀)×𝑆₀ and usc. If 𝑆₁ is a convex subset of 𝑆₀ with the property that, for each 𝑥∈𝑆₀ and each 𝑦∈𝑆₁,(𝑥,𝑦]⊆𝑆₁, then for each 𝑥₀∈𝑆₀, the following conditions are equivalent: (i)Thereexists𝑢₀∈𝐹(𝑥₀), such that forall𝑦∈𝑆₀,⟨𝑢₀,𝜂(𝑥₀,𝑦)⟩+𝑓(𝑥₀)−𝑓(𝑦)−(𝛼/2)‖𝑥₀−𝑦‖²≤0,(ii)forall𝑦∈𝑆₁,thereexists𝑣∈𝐹(𝑦), such that ⟨𝑣,𝜂(𝑥₀,𝑦)⟩+𝑓(𝑥₀)−𝑓(𝑦)−(𝛼/2)‖𝑥₀−𝑦‖²≤0.

Proof. According to the 𝜂-monotonicity of 𝐹, (i) ⇒ (ii) is obvious.
Next prove (ii) ⇒ (i). Suppose that (ii) holds. Given any 𝑦∈𝑆₁, let 𝑦_𝑛=(1/𝑛)𝑦+(1−(1/𝑛))𝑥₀, for 𝑛∈𝑁. By the assumptions of 𝑆₁,𝑦_𝑛∈𝑆₁ for each 𝑛∈𝑁. It follows from the condition (ii) that for each 𝑛∈𝑁, there exists 𝑣_𝑛∈𝐹(𝑦_𝑛) such that 𝑣_𝑛,𝜂𝑥₀,𝑦_𝑛+𝑓𝑥₀−𝑓𝑦_𝑛−𝛼2‖‖𝑥⁰−𝑦_𝑛‖‖²≤0.(2.7) Then 0≥𝑣_𝑛,𝜂𝑥₀,𝑦_𝑛+𝑓𝑥₀−𝑓𝑦_𝑛−𝛼2‖‖𝑥⁰−𝑦_𝑛‖‖²=𝑣_𝑛,𝜂𝑥₀,1𝑛𝑦+1−1𝑛𝑥⁰+𝑓𝑥⁰−𝑓1𝑛𝑦+1−1𝑛𝑥⁰−𝛼2‖‖‖𝑥⁰−1𝑛𝑦−1−1𝑛𝑥⁰‖‖‖²≥1𝑛𝑣^𝑛,𝜂𝑥₀,𝑦+1−1𝑛𝑣^𝑛,𝜂𝑥₀,𝑥₀+𝑓𝑥₀−1𝑛𝑓(𝑦)−1−1𝑛𝑓𝑥⁰−𝛼2𝑛1²‖‖𝑥₀−𝑦‖‖²=1𝑛𝑣^𝑛,𝜂𝑥₀,𝑦+1𝑛𝑓𝑥⁰−1𝑛𝑓(𝑦)−2𝛼𝑛²‖‖𝑥₀−𝑦‖‖²,(2.8) which implies that 𝑣_𝑛,𝜂𝑥₀,𝑦+𝑓𝑥₀−𝑓(𝑦)−2𝛼𝑛‖‖𝑥⁰−𝑦‖‖²≤0,∀𝑛∈𝑁.(2.9) It follows that for each 𝑛∈𝑁, ∃𝑣_𝑛∈𝐹𝑦_𝑛,𝑣_𝑛,𝜂𝑥₀,𝑦+𝑓𝑥₀−𝑓(𝑦)−𝛼2‖‖𝑥⁰−𝑦‖‖²≤0.(2.10) Since 𝐹 is a weak* compact valued and (𝑠,𝑤^∗)-usc on the line segment [𝑥₀,𝑦], 𝐹 is (𝑠,𝑤^∗)-closed, and (𝑠,𝑤^∗)-subcontinuous on [𝑥₀,𝑦], it follows from lim_𝑛𝑦_𝑛=𝑥₀ and 𝑣_𝑛∈𝐹(𝑦_𝑛) that {𝑣_𝑛} has a subsequence weak* converging to some 𝑣∈𝐹(𝑥₀). By taking the limit of subsequence in (2.10) we get ∀𝑦∈𝑆₁,∃𝑣∈𝐹𝑥₀,𝑣,𝜂𝑥₀,𝑦+𝑓𝑥₀−𝑓(𝑦)−𝛼2‖‖𝑥⁰−𝑦‖‖²≤0.(2.11) Define the bifunction 𝜙(𝑣,𝑦) on 𝐹(𝑥₀)×𝑆₀ by 𝜙(𝑣,𝑦)=𝑣,𝜂𝑥₀,𝑦+𝑓𝑥₀−𝑓(𝑦)−𝛼2‖‖𝑥⁰−𝑦‖‖².(2.12) For each 𝑦∈𝑆₀,𝜙(⋅,𝑦) is weakly* lsc and quasiconvex on the weakly* compact convex set 𝐹(𝑥₀) while for each 𝑣∈𝐹(𝑥₀),𝜙(𝑣,⋅) is usc and quasiconcave on the convex set S₁. Hence, according to the Sion Minimax Theorem [26], sup_𝑦∈𝑆1min_{𝑥𝑣∈𝐹0}𝜙(𝑣,𝑦)=min_{𝑥𝑣∈𝐹0}sup_𝑦∈𝑆1𝜙(𝑣,𝑦).(2.13) By (2.11), we have sup_𝑦∈𝑆1min_{𝑣∈𝐹(𝑥0)}𝜙(𝑣,𝑦)≤0; hence, min_{𝑣∈𝐹(𝑥0)}sup_𝑦∈𝑆1𝜙(𝑣,𝑦)≤0, which implies that there exists 𝑣₀∈𝐹(𝑥₀), such that 𝑣₀,𝜂𝑥₀,𝑦+𝑓𝑥₀−𝑓(𝑦)−𝛼2‖‖𝑥⁰−𝑦‖‖²≤0,∀𝑦∈𝑆₁.(2.14)
Finally, for each 𝑦∈𝑆₀, choose 𝑧∈𝑆₁, and a sequence (𝑦_𝑛)_𝑛 in (𝑦,𝑧]⊆𝑆₁ converging to 𝑦. The function 𝜙(𝑣,⋅) is usc and concave on 𝑆₀; hence its restriction on any line segment is continuous [27, Theorem  2.35]. Accordingly, (2.14) implies thereexists𝑣₀∈𝐹(𝑥₀),forall𝑦∈𝑆₀, 𝑣₀,𝜂𝑥₀,𝑦+𝑓𝑥₀−𝑓(𝑦)−𝛼2‖‖𝑥⁰−𝑦‖‖²=lim_𝑛𝑣⁰,𝜂𝑥₀,𝑦_𝑛+𝑓𝑥₀−𝑓𝑦_𝑛−𝛼2‖‖𝑥⁰−𝑦_𝑛‖‖²≤0.(2.15) Hence, (i) holds.

Lemma 2.7. Let 𝐸 be a real Banach space with the dual 𝐸^∗, let 𝐾 be a nonempty convex subset of 𝐸, and let 𝑆 be a convex-valued set-valued map from 𝐾 to 2^𝐾. Let 𝐹∶𝐾→2^𝐸∗ be a set-valued map with nonempty values, let 𝜂∶𝐾×𝐾→𝐸 be a single-valued map with 𝜂(𝑥,𝑥)=0,forall𝑥∈𝐾, and let 𝑓∶𝐾→𝑅 be a convex function. Assume that the function 𝑦↦⟨𝑢,𝜂(𝑥,𝑦)⟩ is concave, for each (𝑢,𝑥)∈𝐹(𝐾)×𝐾. Then 𝑥₀∈Γ if and only if the following condition holds: ∃𝑢₀∈𝐹𝑥₀,𝑥₀∈𝑆𝑥₀,𝑢₀,𝜂𝑥₀,𝑦+𝑓𝑥₀−𝑓(𝑦)−𝛼2‖‖𝑥⁰−𝑦‖‖²≤0,∀𝑦∈𝑆𝑥₀.(2.16)

Proof. The necessity is easy to get; next we start to prove the sufficiency. Let forall𝑦∈𝑆(𝑥₀),forall𝑡∈(0,1),𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑥₀. Since 𝑢₀∈𝐹(𝑥₀),𝑥₀∈𝑆(𝑥₀), and 𝑆 is convex-valued, 𝑦_𝑡∈𝑆(𝑥₀), it follows that 𝑢₀,𝜂𝑥₀,𝑦_𝑡+𝑓𝑥₀−𝑓𝑦_𝑡−𝛼2‖‖𝑥⁰−𝑦_𝑡‖‖²≤0,∀𝑡∈(0,1).(2.17) Thus, 0≥𝑢₀,𝜂𝑥₀,𝑦_𝑡+𝑓𝑥₀−𝑓𝑦_𝑡−𝛼2‖‖𝑥⁰−𝑦_𝑡‖‖²≥𝑡𝑢₀,𝜂𝑥₀,𝑦+(1−𝑡)𝑢₀,𝜂𝑥₀,𝑥₀+𝑓𝑥₀−𝑡𝑓(𝑦)−(1−𝑡)𝑓𝑥₀−𝛼2𝑡²‖‖𝑥₀−𝑦‖‖²=𝑡𝑢₀,𝜂𝑥₀,𝑦+𝑓𝑥₀−𝑓(𝑦)−𝛼2𝑡‖‖𝑥⁰−𝑦‖‖²,(2.18) which implies that 𝑢₀∈𝐹𝑥₀,𝑥₀∈𝑆𝑥₀,𝑢₀,𝜂𝑥₀,𝑦+𝑓𝑥₀−𝑓(𝑦)−𝛼2𝑡‖‖𝑥⁰−𝑦‖‖²≤0∀𝑦∈𝑆𝑥₀,∀𝑡∈(0,1).(2.19) The above inequality implies, for 𝑡 converging to zero, that 𝑥₀ is a solution of (MQVLI). This completes the proof.

3. The Characterizations of Well-Posedness for (MQVLI)

In this section, we investigate some metric characterizations of 𝛼-well-posedness and L-𝛼-well-posedness for (MQVLI).

For any 𝜀>0, we consider the sets 𝑄_𝜀=𝑥∈𝐾∶𝑑(𝑥,𝑆(𝑥))≤𝜀,∃𝑢∈𝐹(𝑥)∶⟨𝑢,𝜂(𝑥,𝑦)⟩+𝑓(𝑥)−𝑓(𝑦)−𝛼2‖𝑥−𝑦‖²𝐿≤𝜀,∀𝑦∈𝑆(𝑥)_𝜀,=𝑥∈𝐾∶𝑑(𝑥,𝑆(𝑥))≤𝜀,⟨𝑣,𝜂(𝑥,𝑦)⟩+𝑓(𝑥)−𝑓(𝑦)−𝛼2‖𝑥−𝑦‖²≤𝜀,∀𝑦∈𝑆(𝑥),∀𝑣∈𝐹(𝑦).(3.1)

Theorem 3.1. Let the same assumptions be as in Lemma 2.7. Then, one has the following. (a)(MQVLI) is 𝛼-well-posed if and only if the solution set Γ of (MQVLI) is nonempty and lim_𝜀→0diam𝑄_𝜀=0.(b)Moreover, if 𝐹 is 𝜂-monotone, then (MQVLI) is L-𝛼-well-posed if and only if the solution set Γ of (MQVLI) is nonempty and lim_𝜀→0diam𝐿_𝜀=0.

Proof. We only prove (a). The proof of (b) is similar and is omitted here. Suppose that (MQVLI) is 𝛼-well-posed; then Γ≠∅. It follows from Lemma 2.7 that 𝑄_𝜀≠∅. Suppose by contradiction that exists a real number 𝛽, such that lim_𝜀→0diam𝑄_𝜀>𝛽>0; then there exists 𝜀_𝑛>0, with 𝜀_𝑛↘0, and (𝑤_𝑛)_𝑛,(𝑧_𝑛)_𝑛∈𝑄_𝜀𝑛, such that ‖𝑤_𝑛−𝑧_𝑛‖>𝛽,forall𝑛∈𝑁. Since the sequences (𝑤_𝑛)_𝑛, and (𝑧_𝑛)_𝑛 are both 𝛼-approximating sequences for (MQVLI), (𝑤_𝑛)_𝑛 and (𝑧_𝑛)_𝑛 strongly converge to the unique solution 𝑢₀, and this gives a contradiction. Therefore, lim_𝜀→0diam𝑄_𝜀=0.
Conversely, let (𝑥_𝑛)_𝑛⊂𝐾 be an 𝛼-approximating sequence for (MQVLI). Then there exists a sequence (𝑢_𝑛)_𝑛 in 𝐸^∗ with 𝑢_𝑛∈𝐹(𝑥_𝑛) and a sequence (𝜀_𝑛)_𝑛 in 𝑅⁺ with 𝜀_𝑛→0, such that 𝑑𝑥_𝑛,𝑆𝑥_𝑛≤𝜀_𝑛,𝑢_𝑛,𝜂𝑥_𝑛,𝑦+𝑓𝑥_𝑛−𝑓(𝑦)−𝛼2‖‖𝑥^𝑛−𝑦‖‖²≤𝜀_𝑛,∀𝑦∈𝑆𝑥_𝑛,∀𝑛∈𝑁.(3.2) That is, 𝑥_𝑛∈𝑄_𝜀𝑛,forall𝑛∈𝑁. It is easy to see lim_𝜀→0diam𝑄_𝜀=0 and Γ≠∅ imply that Γ is a singleton point set. Indeed, if there exist two different solutions 𝑧₁,𝑧₂, then from Lemma 2.7, we know that 𝑧₁,𝑧₂∈𝑄_𝜀,forall𝜀>0. Thus, lim_𝜀→0diam𝑄_𝜀≥‖𝑧₁−𝑧₂‖≠0, a contradiction. Let 𝑥₀ be the unique solution of (MQVLI). It follows from Lemma 2.7 that 𝑥₀∈𝑄_𝜀𝑛. Thus, lim_𝑛→∞‖𝑥_𝑛−𝑥₀‖≤lim_𝑛→∞diam𝑄_𝜀𝑛=0. So (𝑥_𝑛)_𝑛 strongly converges to 𝑥₀. Therefore, (MQVLI) is 𝛼-well-posed.

Theorem 3.2. Let 𝐸 be a real separable Banach space with the dual 𝐸^∗, let 𝐾 be a nonempty closed convex subset of 𝐸, and let 𝜂∶𝐾×𝐾→𝐸 be a single-valued map with 𝜂(𝑥,𝑥)=0,forall𝑥∈𝐾, which is (𝑠,𝑤)-continuous in each of its variables separately. And let 𝑓∶𝐾→𝑅 be a convex lsc function; let 𝑆∶𝐾→2^𝐾 and 𝐹∶𝐾→2^𝐸∗ be two set-valued maps. Assume the following conditions hold: (i)𝑆 is nonempty convex-valued and, for each sequence (𝑥_𝑛)_𝑛 in 𝐾 converging to 𝑥₀, the sequence (𝑆(𝑥_𝑛))_𝑛 Lower Semi-Mosco converging to 𝑆(𝑥₀);(ii)for every converging sequence (𝑤_𝑛)_𝑛, there exists 𝑚∈𝑁, such that int⋂_𝑛≥𝑚𝑆(𝑤_𝑛)≠∅;(iii)𝐹∶𝐾→2^𝐸∗ is nonempty, weak* compact convex valued, 𝜂-monotone, and upper hemicontinuous;(iv)the map 𝑦↦⟨𝑢,𝜂(𝑥,𝑦)⟩ is concave for each (𝑢,𝑥)∈𝐹(𝐾)×𝐾.Then, (MQVLI) is 𝛼-well-posed if and only if 𝑄_𝜀≠∅,∀𝜀≥0,lim_𝜀→0diam𝑄_𝜀=0.(3.3)

The proof of the above theorem relies on the following lemma.

Lemma 3.3. Let the same assumptions be made as in Theorem 3.2. Let (𝑥_𝑛)_𝑛 in 𝐾 be an 𝛼-approximating sequence. If (𝑥_𝑛)_𝑛 converges to some 𝑥₀∈𝐾, then 𝑥₀ is a solution of (MQVLI).

Proof. Since (𝑥_𝑛)_𝑛 is an 𝛼-approximating sequence for (MQVLI), there exists a sequence (𝑢_𝑛)_𝑛 in 𝐸^∗ with 𝑢_𝑛∈𝐹(𝑥_𝑛) and a sequence (𝜀_𝑛)_𝑛 in 𝑅⁺ with 𝜀_𝑛→0, such that 𝑑𝑥_𝑛,𝑆𝑥_𝑛≤𝜀_𝑛,𝑢_𝑛,𝜂𝑥_𝑛,𝑦+𝑓𝑥_𝑛−𝑓(𝑦)−𝛼2‖‖𝑥^𝑛−𝑦‖‖²≤𝜀_𝑛,∀𝑦∈𝑆𝑥_𝑛,∀𝑛∈𝑁.(3.4)
For each 𝑛∈𝑁, choose 𝑥^𝑛∈𝑆(𝑥_𝑛), such that ‖𝑥_𝑛−𝑥^𝑛‖<𝑑(𝑥_𝑛,𝑆(𝑥_𝑛))+𝜀_𝑛≤2𝜀_𝑛. It follows from 𝑥_𝑛→𝑥₀ and 𝜀_𝑛→0 that 𝑥^𝑛→𝑥₀. It follows from the assumption (i) that liminf_𝑛𝑆(𝑥_𝑛)=𝑆(𝑥₀). Thus, 𝑥₀∈𝑆(𝑥₀).
Assumption (ii) applied to the constant sequence 𝑤_𝑛=𝑥₀,forall𝑛∈𝑁, implies that int𝑆(𝑥₀)≠∅. For every 𝑦∈int𝑆(𝑥₀), it follows from assumptions (i) and (ii) and Lemma 2.5 that there exist 𝑚∈𝑁 and 𝛿>0 such that int𝐵(𝑦,𝛿)⊆𝑆(𝑥_𝑛),forall𝑛>𝑚. Therefore, for 𝑛 sufficiently large, we have 𝑦∈𝑆(𝑥_𝑛). Notice that 𝜂(⋅,𝑦) is (𝑠,𝑤)-continuous, 𝑓 is lsc, 𝐹 is 𝜂-monotone, and (𝑥_𝑛)_𝑛 is an approximating sequence; we have, for every 𝑣∈𝐹(𝑦)𝑣,𝜂𝑥₀,𝑦+𝑓𝑥₀−𝑓(𝑦)≤≤lliimmiinnf_𝑛f_𝑛𝑣,𝜂𝑥_𝑛,𝑦+𝑓𝑥_𝑛−𝑓(𝑦)≤liminf_𝑛𝑢_𝑛,𝜂𝑥_𝑛,𝑦+𝑓𝑥_𝑛−𝑓(𝑦)=𝛼𝜀^𝑛+𝛼2‖‖𝑥^𝑛−𝑦‖‖²2‖‖𝑥⁰−𝑦‖‖².(3.5)
Thus, for every 𝑦∈int𝑆(𝑥₀) and every 𝑣∈𝐹(𝑦), we get ⟨𝑣,𝜂(𝑥₀,𝑦)⟩+𝑓(𝑥₀)−𝑓(𝑦)−(𝛼/2)‖𝑥₀−𝑦‖²≤0. Let 𝑆₀=𝑆(𝑥₀) and 𝑆₁=int𝑆(𝑥₀); it follows from Lemma 2.6 that there exists 𝑢₀∈𝐹(𝑥₀) such that for all 𝑦∈𝑆(𝑥₀),⟨𝑢₀,𝜂(𝑥₀,𝑦)⟩+𝑓(𝑥₀)−𝑓(𝑦)−(𝛼/2)‖𝑥₀−𝑦‖²≤0. According to Lemma 2.7, 𝑥₀ is a solution of (MQVLI).

Proof of Theorem 3.2. The necessity follows from Theorem 3.1 and Lemma 2.7. Now we prove the sufficiency. Suppose that (3.3) holds. Let us show that there exists at most one solution of (MQVLI). Indeed, if there exist two different solutions 𝑧₁,𝑧₂, then from Lemma 2.7, we know that 𝑧₁,𝑧₂∈𝑄_𝜀,forall𝜀>0. Thus, lim_𝜀→0diam𝑄_𝜀≥‖𝑧₁−𝑧₂‖≠0, a contradiction. Note also that there exist 𝛼-approximate sequences for (MQVLI); indeed, for any sequence (𝜀_𝑛)_𝑛 in 𝑅⁺ with 𝜀_𝑛→0, and any choice of 𝑥_𝑛∈𝑄_𝜀𝑛 (which is nonempty by assumption), (𝑥_𝑛)_𝑛 is an 𝛼-approximate sequence.
Let (𝑥_𝑛)_𝑛 be an 𝛼-approximating sequence for (MQVLI); then 𝑥_𝑛∈𝑄_𝜀𝑛,forall𝑛∈𝑁. In light of (3.3), (𝑥_𝑛)_𝑛 is a Cauchy sequence and strongly converging to a point 𝑥₀∈𝐾. Applying Lemma 3.3, we get that 𝑥₀ is a solution of (MQVLI) and so (MQVLI) is 𝛼-well-posed.

Now, we present a result in which assumption (ii) and the monotonicity of 𝐹 are dropped, while the continuity requirements are strengthened.

Theorem 3.4. Let 𝐸 be a real separable Banach space with the dual 𝐸^∗, let 𝐾 be a nonempty closed convex subset of 𝐸, and let 𝜂∶𝐾×𝐾→𝐸 be a single-valued map with 𝜂(𝑥,𝑥)=0,forall𝑥∈𝐾, which is (𝑠,𝑠)-continuous. And let 𝑓∶𝐾→𝑅 be a convex and continuous function, let 𝑆∶𝐾→2^𝐾 and 𝐹∶𝐾→2^𝐸∗ be two set-valued maps. Assume the following assumptions hold: (i)the multifunction 𝑆 is nonempty convex-valued and for each sequence (𝑥_𝑛)_𝑛 in 𝐾 converging to 𝑥₀, the sequence (𝑆(𝑥_𝑛))_𝑛 Lower Semi-Mosco converging to 𝑆(𝑥₀);(ii)𝐹∶𝐾→2^𝐸∗ is nonempty, weak* compact, and convex valued, (𝑠,𝑤^∗)-usc;(iii)the map 𝑦↦⟨𝑢,𝜂(𝑥,𝑦)⟩ is concave for each (𝑢,𝑥)∈𝐹(𝐾)×𝐾.Then, (MQVLI) is 𝛼-well-posed if and only if (3.3) holds.

The proof of the above theorem relies on the following lemma.

Lemma 3.5. Let the assumptions be as in Theorem 3.4. Let (𝑥_𝑛)_𝑛 in 𝐾 be an 𝛼-approximating sequence. If (𝑥_𝑛)_𝑛 converges to some 𝑥₀∈𝐾, then 𝑥₀ is a solution of (MQVLI).

Proof. Since (𝑥_𝑛)_𝑛 is an 𝛼-approximating sequence for (MQVLI), there exist a sequence (𝑢_𝑛)_𝑛 in 𝐸^∗ with 𝑢_𝑛∈𝐹(𝑥_𝑛) and a sequence (𝜀_𝑛)_𝑛 in 𝑅⁺,𝜀_𝑛→0, such that 𝑑𝑥_𝑛,𝑆𝑥_𝑛≤𝜀_𝑛,𝑢_𝑛,𝜂𝑥_𝑛,𝑦+𝑓𝑥_𝑛−𝑓(𝑦)−𝛼2‖‖𝑥^𝑛−𝑦‖‖²≤𝜀_𝑛,∀𝑦∈𝑆𝑥_𝑛,∀𝑛∈𝑁.(3.6)
As in Lemma 3.3, we infer 𝑥₀∈𝑆(𝑥₀). Since 𝑆(𝑥_𝑛) Lower Semi-Mosco converges to 𝑆(𝑥₀), for every 𝑦∈𝑆(𝑥₀), there exists a sequence 𝑦_𝑛∈𝑆(𝑥_𝑛),forall𝑛∈𝑁, such that lim_𝑛𝑦_𝑛=𝑦 in the strongly topology. Since 𝜂 is (𝑠,𝑠)-continuous, the sequence (𝜂(𝑥_𝑛,𝑦_𝑛))_𝑛 converges strongly to 𝜂(𝑥₀,𝑦). It follows from (ii) and Proposition  2.19 in [24] that there exists a subsequence (𝑢_𝑛𝑗)_𝑗 of (𝑢_𝑛)_𝑛 weak* converging to some 𝑢₀∈𝐸^∗. It follows from (ii) and Proposition  2.17 in [24] that 𝐹 is (𝑠,𝑤^∗)-closed, and so 𝑢₀∈𝐹(𝑥₀). Thus, we have |||𝑢_𝑛𝑗≤,𝜂|||𝑥𝑢_𝑛𝑗,𝑦_𝑛𝑗−𝑢⁰,𝜂𝑥₀,𝑦_𝑛𝑗|||≤‖‖𝑢,𝜂𝑥_𝑛𝑗,𝑦_𝑛𝑗−𝜂𝑥⁰,𝑦|||+|||𝑢₀−𝑢_𝑛𝑗,𝜂𝑥₀,𝑦_𝑛𝑗|||‖‖‖‖𝜂𝑥_𝑛𝑗,𝑦_𝑛𝑗⟶0.−𝜂𝑥⁰,𝑦‖‖−|||𝑢₀−𝑢_𝑛𝑗,𝜂𝑥₀,𝑦|||(3.7) Hence, ⟨𝑢_𝑛𝑗,𝜂(𝑥_𝑛𝑗,𝑦_𝑛𝑗)⟩→⟨𝑢₀,𝜂(𝑥₀,𝑦)⟩ and so 𝑢₀,𝜂𝑥₀,𝑦+𝑓𝑥₀−𝑓(𝑦)−𝛼2‖‖𝑥⁰−𝑦‖‖²=lim_𝑗𝑢^𝑛_𝑗,𝜂𝑥_𝑛𝑗,𝑦_𝑛𝑗+𝑓𝑥^𝑛_𝑗−𝑓𝑦^𝑛_𝑗−𝛼2‖‖𝑥^𝑛𝑗≤lim_𝑗−𝑦_𝑛𝑗‖‖²𝜀_𝑛𝑗=0.(3.8) Applying Lemma 2.7, 𝑥₀ is a solution of (MQVLI).

Proof of Theorem 3.4. The necessity follows from Theorem 3.1 and Lemma 2.7. Now we prove the sufficiency. Suppose that (3.3) holds. It follows from the proof of Theorem 3.2 that there exists at most one solution of (MQVLI) and there exist 𝛼-approximate sequences for (MQVLI). Let (𝑥_𝑛)_𝑛 be an 𝛼-approximating sequence for (MQVLI); then 𝑥_𝑛∈𝑄_𝜀𝑛,forall𝑛∈𝑁. In light of (3.3), (𝑥_𝑛)_𝑛 is a Cauchy sequence and strongly converging to a point 𝑥₀∈𝐾. Applying Lemma 3.5, we get that 𝑥₀ is a solution of (MQVLI) and so (MQVLI) is 𝛼-well-posed.

We have analogous results for L-𝛼-well-posedness.

Theorem 3.6. Let 𝐸 be a real separable Banach space with the dual 𝐸^∗, let 𝐾 be a nonempty closed convex subset of 𝐸, and let 𝜂∶𝐾×𝐾→𝐸 be a single-valued map with 𝜂(𝑥,𝑥)=0,forall𝑥∈𝐾, which is (𝑠,𝑤)-continuous in each of its variables separately. And let 𝑓∶𝐾→𝑅 be a convex lsc function; let 𝑆∶𝐾→2^𝐾 and 𝐹∶𝐾→2^𝐸∗ be two set-valued maps. Assume the following assumptions hold: (i)the multifunction 𝑆 is nonempty convex-valued and for each sequence (𝑥_𝑛)_𝑛 in 𝐾 converging to 𝑥₀, the sequence (𝑆(𝑥_𝑛))_𝑛 Lower Semi-Mosco converging to 𝑆(𝑥₀);(ii)for every converging sequence (𝑤_𝑛)_𝑛, there exists 𝑚∈𝑁, such that int⋂_𝑛≥𝑚𝑆(𝑤_𝑛)≠∅;(iii)𝐹∶𝐾→2^𝐸∗ is a set-valued map with nonempty, weak* compact convex valued, 𝜂-monotone and upper hemicontinuous;(iv)the map 𝑦↦⟨𝑢,𝜂(𝑥,𝑦)⟩ is concave for each (𝑢,𝑥)∈𝐹(𝐾)×𝐾.Then (MQVLI) is L-𝛼-well-posed if and only if 𝐿_𝜀≠∅,∀𝜀≥0,lim_𝜀→0diam𝐿_𝜀=0.(3.9)

Lemma 3.7. Let the same assumptions be as in Theorem 3.6. Let (𝑥_𝑛)_𝑛 in 𝐾 be an L-𝛼-approximating sequence. If (𝑥_𝑛)_𝑛 converges to some 𝑥₀∈𝐾, then 𝑥₀ is a solution of (MQVLI).

Proof. Since (𝑥_𝑛)_𝑛 is an L-𝛼-approximating sequence for (MQVLI), there exists a sequence (𝜀_𝑛)_𝑛 in 𝑅⁺, 𝜀_𝑛→0, such that 𝑑(𝑥_𝑛,𝑆(𝑥_𝑛))≤𝜀_𝑛, and 𝑣,𝜂𝑥_𝑛,𝑦+𝑓𝑥_𝑛−𝑓(𝑦)−𝛼2‖‖𝑥^𝑛−𝑦‖‖²≤𝜀_𝑛,∀𝑦∈𝑆𝑥_𝑛,𝑣∈𝐹(𝑦),𝑛∈𝑁.(3.10) From the proof of Lemma 3.3, (i) and (ii), we can obtain 𝑥₀∈𝑆(𝑥₀), int𝑆(𝑥₀)≠∅, and for each 𝑦∈int𝑆(𝑥₀), one has 𝑦∈𝑆(𝑥_𝑛) for 𝑛 sufficiently large. It follows from (iii) that for every 𝑦∈int𝑆(𝑥₀) and every 𝑣∈𝐹(𝑦), we have 𝑣,𝜂𝑥₀,𝑦+𝑓𝑥₀−𝑓(𝑦)−𝛼2‖‖𝑥⁰−𝑦‖‖²≤liminf_𝑛𝑣,𝜂𝑥^𝑛,𝑦+𝑓𝑥_𝑛−𝑓(𝑦)−𝛼2‖‖𝑥^𝑛≤liminf_𝑛−𝑦‖‖²𝜀_𝑛=0.(3.11)
Let 𝑆₀=𝑆(𝑥₀) and 𝑆₁=int𝑆(𝑥₀); it follows from Lemma 2.6 that there exists 𝑢₀∈𝐹(𝑥₀) such that for all 𝑦∈𝑆(𝑥₀),⟨𝑢₀,𝜂(𝑥₀,𝑦)⟩+𝑓(𝑥₀)−𝑓(𝑦)−(𝛼/2)‖𝑥₀−𝑦‖²≤0. According to Lemma 2.7, 𝑥₀ is a solution of (MQVLI).

Proof of Theorem 3.6. Assume that (3.9) holds. Let (𝑥_𝑛)_𝑛 in 𝐾 be an L-𝛼-approximating sequence for (MQVLI); then there exists a sequence (𝜀_𝑛)_𝑛 in 𝑅⁺, such that 𝑥_𝑛∈𝐿_𝜀𝑛. It is easy to see that lim_𝜀→0diam𝐿_𝜀=0 and Γ≠∅ imply that Γ is a singleton point set. Indeed, if there exist two different solutions 𝑧₁,𝑧₂, then from Lemma 2.7 and the 𝜂-monotonicity of 𝐹, we know that 𝑧₁,𝑧₂∈𝐿_𝜀,forall𝜀>0. Thus, lim_𝜀→0diam𝐿_𝜀≥‖𝑧₁−𝑧₂‖≠0, a contradiction. Let 𝑥₀ be the unique solution of (MQVLI). It follows from Lemma 2.7 and the 𝜂-monotonicity of 𝐹 that 𝑥₀∈𝐿_𝜀𝑛. Thus, lim_𝑛→∞‖𝑥_𝑛−𝑥₀‖≤lim_𝑛→∞diam𝐿_𝜀𝑛=0. So (𝑥_𝑛)_𝑛 strongly converge to 𝑥₀. It follows from Lemma 3.7 that 𝑥₀∈Γ. Therefore, (MQVLI) is L-𝛼-well-posed.
Conversely, assume that the problem is L-𝛼-well-posed, It follows from the 𝜂-monotonicity of 𝐹 that ∅≠Γ⊂𝐿_𝜀,∀𝜀>0. Suppose by contradiction that a real number 𝛽 exists, such that lim_𝜀→0diam𝐿_𝜀>𝛽>0; then there exists 𝜀_𝑛>0, with 𝜀_𝑛→0, and (𝑤_𝑛)_𝑛,(𝑧_𝑛)_𝑛∈𝐿_𝜀𝑛, such that ‖𝑤_𝑛−𝑧_𝑛‖>𝛽,∀𝑛∈𝑁. Since the sequences (𝑤_𝑛)_𝑛 and (𝑧_𝑛)_𝑛 are both L-𝛼-approximating sequences for (MQVLI), (𝑤_𝑛)_𝑛 and (𝑧_𝑛)_𝑛 strongly converge to the unique solution 𝑢₀, and this gives a contradiction. Therefore, lim_𝜀→0diam𝐿_𝜀=0.

Theorem 3.8. Let 𝐸 be a real separable Banach space, let 𝐾 be a nonempty closed convex subset of 𝐸, and let 𝜂∶𝐾×𝐾→𝐸 be a single-valued map with 𝜂(𝑥,𝑥)=0,forall𝑥∈𝐾, which is (𝑠,𝑠)-continuous. And let 𝑓∶𝐾→𝑅 be a convex continuous function; 𝑆 be a set-valued map from 𝐾 to 2^𝐾. Assume the following assumptions hold: (i)the multifunction 𝑆 is nonempty convex-valued and for each sequence (𝑥_𝑛)_𝑛 in 𝐾 converging to 𝑥₀, the sequence (𝑆(𝑥_𝑛))_𝑛 Lower Semi-Mosco converging to 𝑆(𝑥₀);(ii)𝐹∶𝐾→2^𝐸∗ is a set-valued map with nonempty, weak* compact convex-valued, (𝑠,𝑤^∗)-usc, and 𝜂-monotone;(iii)the map 𝑦↦⟨𝑢,𝜂(𝑥,𝑦)⟩ is concave for each (𝑢,𝑥)∈𝐹(𝐾)×𝐾.
Then (MQVLI) is L-𝛼-well-posed if and only if (3.9) holds.

Lemma 3.9. Let the same assumptions be as in Theorem 3.8. Let (𝑥_𝑛)_𝑛 in 𝐾 be an L-𝛼-approximating sequence. If (𝑥_𝑛)_𝑛 converges to some 𝑥₀∈𝐾, then 𝑥₀ is a solution of (MQVLI).

Proof. Since (𝑥_𝑛)_𝑛 is an L-𝛼-approximating sequence for (MQVLI), there exists a sequence (𝜀_𝑛)_𝑛 in 𝑅⁺, 𝜀_𝑛→0, such that 𝑑(𝑥_𝑛,𝑆(𝑥_𝑛))≤𝜀_𝑛, and 𝑣,𝜂𝑥_𝑛,𝑦+𝑓𝑥_𝑛−𝑓(𝑦)−𝛼2‖‖𝑥^𝑛−𝑦‖‖²≤𝜀_𝑛,∀𝑦∈𝑆𝑥_𝑛,∀𝑣∈𝐹(𝑦),𝑛∈𝑁.(3.12)
It follows from the Lower Semi-Mosco convergence of 𝑆 and the proof of Lemma 3.3 that 𝑥₀∈𝑆(𝑥₀). Since 𝑆(𝑥_𝑛) Lower Semi-Mosco converges to 𝑆(𝑥₀), for every 𝑦∈𝑆(𝑥₀), there exists a sequence 𝑦_𝑛∈𝑆(𝑥_𝑛),forall𝑛∈𝑁, strongly converging to 𝑦. For each 𝑛∈𝑁 select 𝑣_𝑛∈𝐹(𝑦_𝑛). It follows from (ii) and Proposition  2.19 in [24] that there exists a subsequence (𝑣_𝑛𝑗)_𝑗 of (𝑣_𝑛)_𝑛 weak* converging to some 𝑣∈𝐸^∗. It follows from (ii) and Proposition  2.17 in [24] that 𝐹 is (𝑠,𝑤^∗)-closed, and so 𝑣∈𝐹(𝑦). By the continuity of 𝜂 and similar argument with the proof of Lemma 3.5, we know that 𝑣^𝑛_𝑗,𝜂𝑥_𝑛𝑗,𝑦_𝑛𝑗⟶𝑣,𝜂𝑥⁰,𝑦.(3.13) It follows from (3.12) that 𝑣^𝑛_𝑗,𝜂𝑥_𝑛𝑗,𝑦_𝑛𝑗+𝑓𝑥^𝑛_𝑗−𝑓𝑦^𝑛_𝑗−𝛼2‖‖𝑥^𝑛𝑗−𝑦_𝑛𝑗‖‖²≤𝜀_𝑛𝑗.(3.14) We deduce from the above inequality that ∀𝑦∈𝑆𝑥₀,∃𝑣∈𝐹(𝑦),𝑣,𝜂𝑥₀,𝑦+𝑓𝑥₀−𝑓(𝑦)−𝛼2‖‖𝑥⁰−𝑦‖‖²≤0.(3.15) Let 𝑆₀=𝑆₁=𝑆(𝑥₀); by Lemma 2.6 we know that there exists 𝑢₀∈𝐹(𝑥₀), such that ∀𝑦∈𝑆𝑥₀,𝑢₀,𝜂𝑥₀,𝑦+𝑓𝑥₀−𝑓(𝑦)−𝛼2‖‖𝑥⁰−𝑦‖‖²≤0.(3.16) Then using Lemma 2.7, 𝑥₀ is a solution of (MQVLI).

Proof of Theorem 3.8. Assume that (3.9) holds. If (𝑥_𝑛)_𝑛 in 𝐾 is an L-𝛼-approximating sequence, then from the proof of Theorem 3.6, we know that (𝑥_𝑛)_𝑛 converges to some 𝑥₀∈𝐾. By Lemma 3.9, 𝑥₀ is a solution of (MQVLI) and so (MQVLI) is L-𝛼-well-posed. The converse is exactly same as that in the proof of Theorem 3.6.

4. The Characterizations of 𝛼-Well-Posed in the Generalized Sense for (MQVLI)

In this section, we investigate some metric characterizations of 𝛼-well-posedness in the generalized sense for (MQVLI).

Definition 4.1 (see [8]). Let 𝐴 be a nonempty subset of 𝑋. The measure of noncompactness 𝜇 of the set 𝐴 is defined by 𝜇(𝐴)=inf𝜀>0∶𝐴⊆^𝑛_𝑖=1𝐴_𝑖,diam𝐴_𝑖<𝜀,𝑖=1,2,…,𝑛.(4.1)

Definition 4.2 (see [8]). Let (𝑋,𝑑) be a metric space and let 𝐴 and 𝐵 be nonempty subsets of 𝑋. The Hausdorff distance 𝐻(⋅,⋅) between 𝐴 and 𝐵 is defined by 𝐻(𝐴,𝐵)=max{𝑒(𝐴,𝐵),𝑒(𝐵,𝐴)},(4.2) where 𝑒(𝐴,𝐵)=sup_𝑎∈𝐴𝑑(𝑎,𝐵) with 𝑑(𝑎,𝐵)=inf_𝑏∈𝐵‖𝑎−𝑏‖.

Theorem 4.3. Let the same assumptions be as in Lemma 2.7. Then, one has the following. (a)(MQVLI) is 𝛼-well-posed in the generalized sense if and only if the solution set Γ of (MQVLI) is nonempty compact and 𝑒(𝑄_𝜀,Γ)→0,𝑎𝑠𝜀→0.(b)Moreover, if 𝐹 is 𝜂-monotone, then (MQVLI) is 𝐿-𝛼-well-posed in the generalized sense if and only if the solution set Γ of (MQVLI) is nonempty compact and 𝑒(𝐿_𝜀,Γ)→0,𝑎𝑠𝜀→0.

Proof. We only prove (a). The proof of (b) is similar and is omitted here. Assume that (MQVLI) is 𝛼-well-posed in the generalized sense; then Γ is nonempty and compact. It follows from Lemma 2.7 that 𝑄_𝜀≠∅. Now we show that 𝑒(𝑄_𝜀,Γ)→0,𝑎𝑠𝜀→0. Suppose by contradiction that there exists 𝛽>0,𝜀_𝑛→0 and 𝑤_𝑛∈𝑄_𝜀𝑛, such that 𝑑(𝑤_𝑛,Γ)>𝛽. It follows from 𝑤_𝑛∈𝑄_𝜀𝑛 that (𝑤_𝑛)_𝑛 is an 𝛼-approximating sequence for (MQVLI). Since (MQVLI) is 𝛼-well-posedness in the generalized sense, there exists a subsequence (𝑤_𝑛𝑘)_𝑘 of (𝑤_𝑛)_𝑛 strongly converging to a point of Γ. This contradicts 𝑑(𝑤_𝑛,Γ)>𝛽. Thus 𝑒(𝑄_𝜀,Γ)→0,𝑎𝑠𝜀→0.
For the converse, let (𝑥_𝑛)_𝑛 be an 𝛼-approximating sequence for (MQVLI); then 𝑥_𝑛∈𝑄_𝜀𝑛. It follows from 𝑒(𝑄_𝜀𝑛,Γ)→0 that there exists a sequence 𝑧_𝑛⊂Γ, such that 𝑑(𝑥_𝑛,𝑧_𝑛)→0. Since Γ is compact, there exists a subsequence (𝑧_𝑛𝑘)_𝑘 of (𝑧_𝑛)_𝑛 strongly converging to 𝑥₀∈Γ. Thus the corresponding subsequence (𝑥_𝑛𝑘)_𝑘 of (𝑥_𝑛)_𝑛 is strongly converging to 𝑥₀. Therefore, (MQVLI) is 𝛼-well-posed in the generalized sense.

Theorem 4.4. Let the same assumptions be as in Theorem 3.4. Then (MQVLI) is 𝛼-well-posed in the generalized sense if and only if 𝑄_𝜀≠∅,∀𝜀>0,lim_𝜀→∞𝜇𝑄_𝜀=0.(4.3)

Proof. Assume that (MQVLI) is 𝛼-well-posed in the generalized sense; so 𝑄_𝜀≠∅,forall𝜀>0. By Theorem 4.3(a), Γ is nonempty compact and lim_𝜀→0𝑒(𝑄_𝜀,Γ)→0. For any 𝜀>0, we have 𝐻𝑄_𝜀,Γ=max𝑒𝑄_𝜀,Γ,𝑒Γ,𝑄_𝜀=𝑒𝑄_𝜀,Γ,(4.4) since Γ is compact, 𝜇(Γ)=0. For every 𝜀>0, the following relation holds (see, e.g., [13]) 𝜇𝑄_𝜀≤2𝐻𝑄_𝜀,Γ+𝜇(Γ)=2𝐻𝑄_𝜀,Γ=2𝑒𝑄_𝜀,Γ.(4.5) It follows from lim_𝜀→0𝑒(𝑄_𝜀,Γ)→0 that lim_𝜀→0𝜇(𝑄_𝜀)=0.
Conversely, assume that (4.3) holds. Then, for any 𝜀>0, cl(𝑄_𝜀) is nonempty closed and increasing with 𝜀>0. By (4.3), lim_𝜀→0𝜇(cl(𝑄_𝜀))=lim_𝜀→0𝜇(𝑄_𝜀)=0, where cl(𝑄_𝜀) is the closure of 𝑄_𝜀. By the generalized Cantor theorem [23, page 412], we know that lim_𝜀→0𝐻cl𝑄_𝜀,Δ=0,as𝜀⟶0,(4.6) where Δ=⋂_𝜀>0cl(𝑄_𝜀) is nonempty compact.
Now we show that Γ=Δ.(4.7) It follows from Lemma 2.7 that Γ⊆Δ. So we need to prove that Δ⊆Γ. Indeed, let 𝑥₀∈Δ. Then 𝑑(𝑥₀,𝑄_𝜀)=0 for every 𝜀>0. Given 𝜀_𝑛>0, 𝜀_𝑛→0, for every 𝑛 there exists 𝑥_𝑛∈𝑄_𝜀𝑛 such that 𝑑(𝑥₀,𝑥_𝑛)<𝜀_𝑛. Hence, 𝑥_𝑛→𝑥₀ and 𝑑𝑥_𝑛,𝑆𝑥_𝑛≤𝜀_𝑛,(4.8)∃𝑢_𝑛∈𝐹𝑥_𝑛,𝑢_𝑛,𝜂𝑥_𝑛,𝑦+𝑓𝑥_𝑛−𝑓(𝑦)≤𝜀_𝑛+𝛼2‖‖𝑥^𝑛−𝑦‖‖²,∀𝑦∈𝑆𝑥_𝑛.(4.9)
It follows from (4.8), 𝑥_𝑛→𝑥₀, and the proof of Lemma 3.3 that 𝑥₀∈𝑆(𝑥₀).
Since 𝑆(𝑥_𝑛) Lower Semi-Mosco converges to 𝑆(𝑥₀), for every 𝑦∈𝑆(𝑥₀), there exists a sequence 𝑦_𝑛∈𝑆(𝑥_𝑛),forall𝑛∈𝑁, such that lim_𝑛𝑦_𝑛=𝑦 in the strong topology.
Since 𝜂 is (𝑠,𝑠)-continuous, the sequence (𝜂(𝑥_𝑛,𝑦_𝑛))_𝑛 converges strongly to 𝜂(𝑥₀,𝑦). It follows from (ii) and Proposition  2.19 in [24] that there exists a subsequence (𝑢_𝑛𝑗)_𝑗 of (𝑢_𝑛)_𝑛 weak* converging to some 𝑢₀∈𝐸^∗. It follows from (ii) and Proposition  2.17 in [24] that 𝐹 is (𝑠,𝑤^∗)-closed, and so 𝑢₀∈𝐹(𝑥₀). It follows from the proof of Lemma 3.5 that 𝑢^𝑛_𝑗,𝜂𝑥_𝑛𝑗,𝑦_𝑛𝑗⟶𝑢⁰,𝜂𝑥₀,𝑦.(4.10) Hence, 𝑢₀,𝜂𝑥₀,𝑦+𝑓𝑥₀−𝑓(𝑦)−𝛼2‖‖𝑥⁰−𝑦‖‖²=lim_𝑗𝑢^𝑛_𝑗,𝜂𝑥_𝑛𝑗,𝑦_𝑛𝑗+𝑓𝑥^𝑛_𝑗−𝑓𝑦^𝑛_𝑗−𝛼2‖‖𝑥^𝑛𝑗≤lim_𝑗−𝑦_𝑛𝑗‖‖²𝜀_𝑛𝑗=0,(4.11) that is, ∃𝑢₀∈𝐹𝑥₀,𝑢₀,𝜂𝑥₀,𝑦+𝑓𝑥₀−𝑓(𝑦)−𝛼2‖‖𝑥⁰−𝑦‖‖²≤0,∀𝑦∈𝑆𝑥₀.(4.12) By Lemma 2.7, we know that 𝑥₀∈Γ. Thus, Δ⊆Γ. It follows from (4.6) and (4.7) that lim_𝜀→0𝑒(𝑄_𝜀,Γ)=0. It follows from the compactness of Γ and Theorem 4.3(a) that (MQVLI) is 𝛼-well-posed in the generalized sense. The proof is completed.

Theorem 4.5. Let the same assumptions be as in Theorem 3.8. Then (MQVLI) is L-𝛼-well-posed in the generalized sense if and only if 𝐿_𝜀≠∅,∀𝜀>0,lim_𝜀→0𝜇𝐿_𝜀=0.(4.13)

Proof. Assume that (MQVLI) is L-𝛼-well-posed in the generalized sense. It follows from Lemma 2.7 and the 𝜂-monotonicity of 𝐹 that Γ⊂𝐿_𝜀,forall𝜀>0. And so 𝐿_𝜀≠∅, for each 𝜀>0. By similar argument with that in the proof of Theorem 4.3(a), we can get 𝑒(𝐿_𝜀,Γ)→0 as 𝜀→0. From the proof of Theorem 4.4, we also obtain 𝜇𝐿_𝜀≤2𝐻𝐿_𝜀,Γ+𝜇(Γ)=2𝐻𝐿_𝜀,Γ=2𝑒𝐿_𝜀,Γ.(4.14) Thus, lim_𝜀→0𝜇(𝐿_𝜀)=0.
Conversely, assume that (4.13) holds. Then, for any 𝜀>0, cl(𝐿_𝜀) is nonempty closed and increasing with 𝜀>0. By (4.13), lim_𝜀→0𝜇(cl(𝐿_𝜀))=lim_𝜀→0𝜇(𝐿_𝜀)=0, where cl(𝐿_𝜀) is the closure of 𝐿_𝜀. By the generalized Cantor theorem [23, Page 412], we know that lim_𝜀→0𝐻cl𝐿_𝜀,Δ=0,as𝜀⟶0,(4.15) where Δ=⋂_𝜀>0cl(𝐿_𝜀) is nonempty compact.
Now we show that Γ=Δ.(4.16) It follows from Lemma 2.7 and the monotonicity of 𝐹 that Γ⊆Δ. So we need to prove that Δ⊆Γ. Indeed, let 𝑥₀∈Δ. Then 𝑑(𝑥₀,𝐿_𝜀)=0 for every 𝜀>0. Given 𝜀_𝑛>0, 𝜀_𝑛→0, for every 𝑛 there exists 𝑥_𝑛∈𝐿_𝜀𝑛 such that 𝑑(𝑥₀,𝑥_𝑛)<𝜀_𝑛. Hence, 𝑥_𝑛→𝑥₀ and 𝑑𝑥_𝑛,𝑆𝑥_𝑛≤𝜀_𝑛,(4.17)𝑣,𝜂𝑥_𝑛,𝑦+𝑓𝑥_𝑛−𝑓(𝑦)≤𝜀_𝑛+𝛼2‖‖𝑥^𝑛−𝑦‖‖²,∀𝑦∈𝑆𝑥_𝑛,∀𝑣∈𝐹(𝑦).(4.18)
It follows from (4.17), 𝑥_𝑛→𝑥₀, and the proof of Lemma 3.3 that 𝑥₀∈𝑆(𝑥₀).
Since 𝑆(𝑥_𝑛) Lower Semi-Mosco converges to 𝑆(𝑥₀), for every 𝑦∈𝑆(𝑥₀), there exists a sequence 𝑦_𝑛∈𝑆(𝑥_𝑛),forall𝑛∈𝑁, such that lim_𝑛𝑦_𝑛=𝑦 in the strong topology.
For each 𝑛∈𝑁 select 𝑣_𝑛∈𝐹(𝑦_𝑛). Since 𝐹 is (𝑠,𝑤^∗)-usc with weak* compact convex values, we can find a subsequence (𝑣_𝑛𝑗)_𝑗 of (𝑣_𝑛)_𝑛 weak* converging to some 𝑣∈𝐹(𝑦). By the continuity of 𝜂 and similar argument with the proof of Lemma 3.5, we know that 𝑣^𝑛_𝑗,𝜂𝑥_𝑛𝑗,𝑦_𝑛𝑗⟶𝑣,𝜂𝑥⁰,𝑦.(4.19) Hence, 𝑣^𝑛_𝑗,𝜂𝑥_𝑛𝑗,𝑦_𝑛𝑗+𝑓𝑥^𝑛_𝑗−𝑓𝑦^𝑛_𝑗−𝛼2‖‖𝑥^𝑛𝑗−𝑦_𝑛𝑗‖‖²≤𝜀_𝑛𝑗.(4.20) We deduce from the above inequality that ∀𝑦∈𝑆𝑥₀,∃𝑣∈𝐹(𝑦),𝑣,𝜂𝑥₀,𝑦+𝑓𝑥₀−𝑓(𝑦)−𝛼2‖‖𝑥⁰−𝑦‖‖²≤0.(4.21) By Lemma 2.6 we know that there exist 𝑢₀∈𝐹(𝑥₀), such that ∀𝑦∈𝑆𝑥₀,𝑢₀,𝜂𝑥₀,𝑦+𝑓𝑥₀−𝑓(𝑦)−𝛼2‖‖𝑥⁰−𝑦‖‖²≤0.(4.22) It follows from Lemma 2.7 that 𝑥₀∈Γ. Thus, Δ⊆Γ. It follows from (4.15) and (4.16) that lim_𝜀→0𝑒(𝐿_𝜀,Γ)=0. It follows from the compactness of Γ and Theorem 4.3(b) that (MQVLI) is L-𝛼-well-posed in the generalized sense. The problem is completed.

Remark 4.6. (i) It is easy to see that if 𝛼=0, then by the main results in our paper, we can recover the corresponding results in [20] with the weaker condition 𝑆(𝑥_𝑛) Lower Semi-Mosco converging to 𝑆(𝑥₀) instead of the condition 𝑆(𝑥_𝑛) Mosco converging to 𝑆(𝑥₀).
(ii) The proof methods of Theorems 4.4 and 4.5 are different from those ofTheorems  4.1 and  4.2in [20].

Acknowledgments

The authors would like to express their thanks to the referee for helpful suggestions. This research was supported by the National Natural Science Foundation of China (Grants 10771228 and 10831009), the Natural Science Foundation of Chongqing (Grant no. CSTC, 2009BB8240), and the Research Project of Chongqing Normal University (Grant 08XLZ05).