Well-Posedness for Generalized Set Equilibrium Problems

Abstract

We study the well-posedness for generalized set equilibrium problems (GSEP) and propose two types of the well-posed concepts for these problems in topological vector space settings. These kinds of well-posedness arise from some well-posedness in the vector settings. We also study the relationship between these well-posedness concepts and present several criteria for the well-posedness of GSEP. Our results are new or include as special cases recent existing results.

1. Introduction and Preliminaries

Let 𝑋, 𝑌, 𝑍 be three topological vector spaces, 𝐾 a nonempty closed convex subset of 𝑋,𝐶⊂𝑌 a closed convex and pointed cone with apex at the origin, and int𝐶≠∅; that is, 𝐶 is properly closed with nonempty interior and satisfies 𝜆𝐶⊆𝐶, for all 𝜆>0; 𝐶+𝐶⊆𝐶; and 𝐶∩(−𝐶)={0}.

The set-valued mapping 𝑓∶𝑍×𝐾×𝐾⇉𝑌 satisfies 𝑓(𝑠,𝑥,𝑥)={0} for all 𝑥∈𝐾 and for all 𝑠∈𝑍, and a set-valued mapping 𝑇∶𝐾⇉𝑍 is given. The generalized set equilibrium problem (GSEP) is to find an 𝑥∈𝐾 with some 𝑠∈𝑇(𝑥) such that 𝑓𝑠,𝑥,𝑦∩(−int𝐶)=∅∀𝑦∈𝐾. (1) We denote the set of all solutions for (GSEP) by Ω.

The concept of well-posedness is inspired by numerical methods producing optimizing sequences for optimization problems [1]. There are many cases so that the solutions may not be unique for a minimization problem. A naturally generalized concept of well-posedness which permits the existence but not uniqueness of minimizers and the convergence of some subsequence of every minimizing sequence toward a minimizer. Other more general notions of well-posedness have been introduced in [2] and there are many others in the literature; see, for example, [3–15]. Our main purpose is to derive some properties of well-posedness for the generalized set equilibrium problems. We also study the relations between these properties.

A minimizing mapping Φ∶𝑍×𝑋⇉𝑌 is defined by Φ(𝑠,𝑥)=Min_𝑤𝑓(𝑠,𝑥,𝐾) (2) for all (𝑠,𝑥)∈𝑍×𝑋, where Min_𝑤𝐴={𝜂∈𝐴∶𝐴∩(𝜂−int𝐶)=∅} and 𝑓(𝑠,𝑥,𝐾)=⋃_𝑦∈𝐾{𝑓(𝑠,𝑥,𝑦)} for all (𝑠,𝑥)∈𝑍×𝑋. Assume that Dom(Φ)≠∅. We note that 0∈𝑓(𝑠,𝑥,𝐾) for all (𝑠,𝑥)∈𝑍×𝑋 since 𝑓(𝑠,𝑥,𝑥)={0} for all 𝑥∈𝐾 and for all 𝑠∈𝑍. 𝔑_𝑋(𝑥₀) denotes the collection of neighborhoods around 𝑥₀ in 𝑋, similar notations for 𝔑_𝑌(𝑦₀) and 𝔑_𝑍(𝑠₀). For any mapping 𝐹, 𝐹(𝐴) denotes the union ⋃_𝑥∈𝐴𝐹(𝑥).

We propose some properties that can be easily derived from the definition. For the sake of clarity, we give the following proof.

Lemma 1. (i) Φ(𝑠,𝑥)∩int𝐶=∅ for all 𝑥∈𝐾 and for all 𝑠∈𝑇(𝑥).
(ii) 𝑥∈Ω with 𝑠∈𝑇(𝑥) if and only if 0∈Φ(𝑠,𝑥).
(iii) 𝑥∈Ω with 𝑠∈𝑇(𝑥) if and only if Φ(𝑠,𝑥)∩𝐶≠∅.

Proof. (i) If not, there exists 𝜏∈Φ(𝑠,𝑥)∩int𝐶 for some 𝑥∈𝐾 and 𝑠∈𝑇(𝑥). Then 𝑓(𝑠,𝑥,𝐾)∩(𝜏−int𝐶)=∅ and 𝜏∈int𝐶. Hence, 0∉𝑓(𝑠,𝑥,𝐾) which contradicts the fact that 0∈𝑓(𝑠,𝑥,𝐾) for all 𝑥∈𝐾 and for all 𝑠∈𝑇(𝑥).
(ii) 0∈Φ(𝑠,𝑥) if and only if 𝑓(𝑠,𝑥,𝐾)∩(−int𝐶)=∅ if and only if 𝑥∈Ω.
(iii) By (i) and (ii), we have 𝑥∈Ω with 𝑠∈𝑇(𝑥) if and only if 0∈Φ(𝑠,𝑥) if and only if Φ(𝑠,𝑥)∩𝐶≠∅.

Definition 2. A sequence {(𝑠_𝑛,𝑥_𝑛)∈𝑍×𝐾∶𝑠_𝑛∈𝑇(𝑥_𝑛)} is a minimizing sequence for Φ if for every neighborhood 𝑈_𝑌∈𝔑_𝑌(0) of 0, there is 𝑛₀∈ℕ, such that Φ(𝑠_𝑛,𝑥_𝑛)∩𝑈_𝑌≠∅ for all 𝑛≥𝑛₀.

Definition 3. (GSEP) is 𝑀-well-posed if it satisfies the following conditions: (i) there exists at least one solution, that is, the set Ω≠∅; (ii) for every minimizing sequence {(𝑠_𝑛,𝑥_𝑛)} and for every 𝑈_𝑋∈𝔑_𝑋(0), there is 𝑛₀∈ℕ such that 𝑥_𝑛∈Ω+𝑈_𝑋 for all 𝑛≥𝑛₀.

Definition 4. For 𝜖∈𝐶, the 𝜖-approximate solution set of (GSEP) is defined by Ω(𝜖)={𝑥∈𝐾∶Φ(𝑠,𝑥)∩(𝐶−𝜖)≠∅ for some 𝑠∈𝑇(𝑥)}.

We can easily see that Ω(0)=Ω in Definition 4. Indeed, 𝑥∈Ω(0) with some 𝑠∈𝑇(𝑥) if and only if Φ(𝑠,𝑥)∩𝐶≠∅ if and only if 𝑥∈Ω from Lemma 1(iii).

Definition 5. (GSEP) is 𝐵-well-posed if it satisfies the following conditions: (i) there exists at least one solution, that is, the set Ω≠∅; (ii) the mapping Ω∶𝐶⇉𝑋 is upper Hausdorff continuous at 𝜖=0; that is, for every 𝑈_𝑋∈𝔑_𝑋(0), there exists 𝑈_𝑌∈𝔑_𝑌(0) such that Ω(𝜖)⊂Ω+𝑈_𝑋 for every 𝜖∈𝑈_𝑌∩𝐶.

Definition 3 arises from [8], and Definition 5 is originally proposed by [9].

Definition 6 (see [16, 17]). A set-valued mapping 𝑇∶𝑋⇉𝑍 is (i) upper semicontinuous if for every 𝑥∈𝑋 and every open set 𝑉 in 𝑌 with 𝑇(𝑥)⊂𝑉, there exists a neighborhood 𝑊(𝑥) of 𝑥 such that 𝑇(𝑊(𝑥))⊂𝑉; (ii) lower semicontinuous if for every 𝑥∈𝑋 and every open neighborhood 𝑉(𝑦) of every 𝑦∈𝑇(𝑥), there exists a neighborhood 𝑊(𝑥) of 𝑥 such that 𝑇(𝑢)∩𝑉(𝑦)≠∅ for all 𝑢∈𝑊(𝑥); (iii) continuous if it is both upper semicontinuous and lower semicontinuous.

We note that 𝑇 is upper semicontinuous at 𝑥₀ and 𝑇(𝑥₀) is compact; then for any net {𝑥_𝜈}⊂𝑋, 𝑥_𝜈→𝑥₀, and for any net 𝑦_𝜈∈𝑇(𝑥_𝜈) for each 𝜈, there exists 𝑦₀∈𝑇(𝑥₀) and a subnet {𝑦_𝜈𝛼} such that 𝑦_𝜈𝛼→𝑦₀. We can refer to [18] for more details. We also note that 𝑇 is lower semicontinuous at 𝑥₀ if for any net {𝑥_𝜈}⊂𝑋, 𝑥_𝜈→𝑥₀, 𝑦₀∈𝑇(𝑥₀) implies that there exists net 𝑦_𝜈∈𝑇(𝑥_𝜈) such that 𝑦_𝜈→𝑦₀. For more details, we refer the reader to [16] or [17]. Another more weaker upper semi-continuity is said above 𝐶-upper Hausdorff semicontinuous [9]. A mapping 𝑇∶𝑋⇉𝑍 is above 𝐶-upper Hausdorff semicontinuous if for every 𝑥∈𝑋 and every open set 𝑊_𝑌∈𝔑_𝑌(0), there exists a neighborhood 𝑉_𝑋∈𝔑_𝑋(0) such that 𝑇(𝑥+𝑉_𝑌)⊂𝑇(𝑥)+𝑊_𝑌−𝐶. Obviously, the upper Hausdorff continuity is weaker than the upper semi-continuity, and an upper Hausdorff continuous mapping is an above 𝐶-upper Hausdorff semi-continuous mapping.

2. 𝐵-Well-Posed and 𝑀-Well-Posed

In this section, we will discuss the relationship between these two kinds of well-posedness. The first one is given as follows.

Example 7. (i) There is an example that satisfies 𝑀-well-posed, but not 𝐵-well-posed. (ii) There is an example that satisfies both 𝐵-well-posed and 𝑀-well-posed.
Solution. (i) The first one is inspired by the example of [10]. Let 𝑋=𝑍=ℝ, 𝑌=ℝ², 𝐶=ℝ²₊, 𝐾=ℝ₊, 𝑇∶𝐾⇉𝑍 satisfy 𝑇(𝑥)=[𝑥/2,𝑥] for all 𝑥∈𝐾. The set-valued mapping 𝑓∶𝑍×𝐾×𝐾⇉𝑌 is defined by 𝑓=𝑠,𝑥,𝑦{(0,0)},if𝑥=0,𝑦≥0;𝑦−𝑠,0,if𝑥>0,0≤𝑦<2𝑠;𝑠,−8𝑠𝑦³,−𝑠,−8𝑠𝑦³,otherwise, (3) for all (𝑠,𝑥,𝑦)∈𝑍×𝐾×𝐾 with 𝑠∈𝑇(𝑥).
Then the set of all solutions for (GSEP) is Ω={0}. For any 𝑥∈𝐾 and any 𝑠∈𝑇(𝑥), the minimizing mapping is Φ()=}×−𝑠,𝑥1{(0,0)}if𝑥=0;{−𝑠𝑠²∪1,0𝑠,−𝑠²,if𝑥>0. (4)
If we choose 𝑈_𝑋=(−1/2,1/2), a neighborhood of 0, and 𝑥_𝑛=𝑛, 𝜖_𝑛=(0,1/𝑛²) for all 𝑛∈ℕ, we can easily see that 𝜖_𝑛→(0,0) as 𝑛→∞, and 𝑛∈Ω(𝜖_𝑛)⧵(Ω+𝑈_𝑋) for all 𝑛∈ℕ. Thus, (GSEP) is not 𝐵-well-posed.
Nevertheless, for any minimizing sequence {(𝑠_𝑛,𝑥_𝑛)} for Φ with 𝑠_𝑛∈𝑇(𝑥_𝑛) for all 𝑛 and for every 𝑈_𝑌∈𝔑_𝑌(0), there exists 𝑛₀∈ℕ such that Φ(𝑠_𝑛,𝑥_𝑛)∩𝑈_𝑌≠∅ for 𝑛≥𝑛₀. This will force that 𝑥_𝑛→0 as 𝑛→∞, and hence 𝑥_𝑛∈Ω+𝑈_𝑋 for all 𝑛≥𝑛₀. Therefore, (GSEP) is 𝑀-well-posed.
(ii) We modify the above example as follows. Let 𝑋, 𝑍, 𝑌, 𝐶, 𝐾, 𝑇 be given the same as in (i). The set-valued mapping 𝑓∶𝑍×𝐾×𝐾⇉𝑌 is defined by 𝑓𝑠,𝑥,𝑦={(0,0)},if𝑥=0,𝑦≥0;𝑦−𝑠,0,if𝑥>0,0≤𝑦<2𝑠;𝑠,−8𝑠𝑦³,otherwise, (5) for all (𝑠,𝑥,𝑦)∈𝑍×𝐾×𝐾 with 𝑠∈𝑇(𝑥).
Then the set of all solutions for (GSEP) is Ω=𝐾. For any 𝑥∈𝐾 and any 𝑠∈𝑇(𝑥), the minimizing mapping is Φ(𝑠,𝑥)=1{(0,0)}if𝑥=0;𝑠,−𝑠²,if𝑥>0. (6)
Since for any minimizing sequence {(𝑠_𝑛,𝑥_𝑛)}, and for every 𝑈_𝑋∈𝔑_𝑋(0), we always have 𝑥_𝑛∈Ω+𝑈_𝑋. Thus, (GSEP) is 𝑀-well-posed. Furthermore, since Ω+𝑈_𝑋=𝐾+𝑈_𝑋, for all 𝜖∈ℝ²₊, we always have Ω(𝜖)⊂Ω+𝑈_𝑋. Hence, (GSEP) is 𝐵-well-posed.

From the above observation, the 𝑀-well-posed is weaker than 𝐵-well-posed for (GSEP). What conditions need to be added so that the converse statement can be valid? The following results will be one of the answers.

Proposition 8. (a) If (GSEP) is 𝐵-well-posed, then it is 𝑀-well-posed. (b) If (GSEP) is 𝑀-well-posed, and for every 𝑊_𝑌∈𝔑_𝑌(0), there exists 𝑈_𝑌∈𝔑_𝑌(0) such that Φ(ℎ(𝐾⧵cl(Ω)))∩𝐶+𝑈_𝑌⊂𝑊_𝑌, (7) where ℎ(𝑥)=𝑇(𝑥)×{𝑥} for all 𝑥∈𝐾⧵cl(Ω), then (GSEP) is 𝐵-well-posed.

Proof. For the idea of the proof, we can use the similar direction of [10, Propositions 3 and 4]. For the sake of clarity, we give the proof of (b) as follows. Suppose that (GSEP) is not 𝐵-well-posed. Then there is a neighborhood 𝑈_𝑋 of 0, and sequences {𝜖_𝑛}⊂𝐶 with 𝜖_𝑛→0 and 𝑥_𝑛∈Ω(𝜖_𝑛) such that 𝑥_𝑛∉Ω+𝑈_𝑋,∀𝑛∈ℕ. (8) This means 𝑥_𝑛∉cl(Ω),∀𝑛∈ℕ. (9) Since 𝑥_𝑛∈Ω(𝜖_𝑛), there exists 𝑠_𝑛∈𝑇(𝑥_𝑛) such that Φ𝑠_𝑛,𝑥_𝑛∩𝐶−𝜖_𝑛≠∅,∀𝑛∈ℕ. (10) Now, we separate into two cases.
Case 1. If the sequence {(𝑠_𝑛,𝑥_𝑛)} is a minimizing sequence, then by 𝑀-well-posedness, for this 𝑈_𝑋, there is a 𝑛₀∈ℕ such that 𝑥_𝑛∈Ω+𝑈_𝑋, for all 𝑛≥𝑛₀, which contradicts (8).
Case 2. If the sequence {(𝑠_𝑛,𝑥_𝑛)} is not a minimizing sequence, then there is a 𝑊_𝑌∈𝔑_𝑌(0) and a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} with a corresponding subsequence {𝑠_𝑛𝑘}∈𝑇({𝑥_𝑛𝑘}) such that Φ𝑠_𝑛𝑘,𝑥_𝑛𝑘∩𝑊_𝑌=∅,∀𝑘∈ℕ. (11) By relation (10), we have Φ(𝑠_𝑛𝑘,𝑥_𝑛𝑘)∩(𝐶−𝜖_𝑛𝑘)≠∅, for all 𝑘∈ℕ. For this 𝑊_𝑌 and condition (7), there is a symmetric neighborhood 𝑈_𝑌∈𝔑_𝑌(0) such that Φ(𝑠_𝑛𝑘,𝑥_𝑛𝑘)∩(𝐶+𝑈_𝑌)⊂𝑊_𝑌, for all 𝑘∈ℕ. For 𝑘 large enough, −𝜖_𝑛𝑘∈𝑈_𝑌. Taking 𝜂_𝑛𝑘∈Φ(𝑠_𝑛𝑘,𝑥_𝑛𝑘)∩(𝐶+𝑈_𝑌) for all 𝑘∈ℕ. This implies that, for 𝑘 large enough, 𝜂_𝑛𝑘∈Φ(𝑠_𝑛𝑘,𝑥_𝑛𝑘)∩𝑊_𝑌 which contradicts (11). This completes the proof.

We need the following lemma for the next criterion for 𝑀-well-posedness of (GSEP).

Lemma 9. Let 𝑌 be a regular topological vector spaces, and let 𝐴 be a nonempty compact subset of 𝑌. Suppose that 𝐴∩(−𝐶)=∅; then there is a neighborhood 𝑊_𝑌 of 0 such that (𝐴+𝑊_𝑌)∩(𝑊_𝑌−𝐶)=∅. In particular, (𝐴+𝑊_𝑌)∩𝑊_𝑌=∅.

Proof. Suppose that 𝐴∩(−𝐶)=∅. For all 𝜂∈𝐴, we have 𝜂∉−𝐶. Since 𝑌 is regular, there is a neighborhood 𝑈^𝜂_𝑌 of 0 such that 𝜂+𝑈^𝜂_𝑌𝑈^𝜂_𝑌−𝐶=∅. (12) Since 𝐴 is a nonempty compact subset, the set 𝐴⊂_𝜂∈𝐴𝜂+𝑈^𝜂_𝑌. (13) There exist 𝜂₁,𝜂₂,…,𝜂_𝑛∈𝐴, such that 𝐴⊂^𝑛_𝑖=1𝜂_𝑖+𝑈^𝜂𝑖_𝑌⊂𝐴+^𝑛_𝑖=1𝑈^𝜂𝑖_𝑌. (14) Let 𝑊_𝑌=⋂^𝑛_𝑖=1𝑈^𝜂𝑖_𝑌. Then 𝐴+^𝑛_𝑖=1𝑈^𝜂𝑖_𝑌𝑊_𝑌−𝐶=∅. (15) Since 𝑊_𝑌⊂⋃^𝑛_𝑖=1𝑈^𝜂𝑖_𝑌, we have 𝐴+𝑊_𝑌∩𝑊_𝑌−𝐶=∅. (16)

We note that, although every compact regular space is a normal space, Lemma 9 is not so intuitive. Furthermore, if the set 𝐴 is not compact, the conclusion may not hold. For example, we choose 𝐴={(𝑥,𝑦)∈ℝ²∶𝑦≥𝑥⁻²,𝑥<0} and 𝐶=ℝ²₊.

Now, we present first criterion of 𝑀-well-posedness for (GSEP).

Theorem 10. Let 𝑋, 𝑌, 𝑍 be three Hausdorff topological vector spaces where 𝑋 is a finite dimensional space and 𝑌 is regular, let 𝐾 be a nonempty closed convex subset of 𝑋, and let 𝐶⊂𝑌 be a closed convex and pointed cone with apex at the origin and int𝐶≠∅. The mapping 𝑓∶𝑍×𝐾×𝐾⇉𝑌 is upper semi-continuous with nonempty compact values and satisfies 𝑓(𝑠,𝑥,𝑥)={0} for all 𝑥∈𝐾 and for all 𝑠∈𝑇(𝑥), and the mapping 𝑇∶𝐾⇉𝑍 is upper semi-continuous with nonempty compact values, such that (i) the solution set Ω of (GSEP) is nonempty and bounded; (ii) the minimizing mapping Φ is upper Hausdorff continuous on 𝑇(𝐾)×𝐾; (iii)𝑓(𝑠,𝑥,𝑦)∩(−𝐶)=∅ for all 𝑥∈Ω, for all 𝑠∈𝑇(𝐾) and for all 𝑦∈𝐾⧵Ω; (iv) the mapping (𝑠,𝑥)→𝑓(𝑠,𝑥,𝑦) is above-C-upper Hausdorff continuous on 𝑍×𝐾 for every 𝑦∈𝐾, and the mapping 𝑥→𝑓(𝑠,𝑥,𝑦) is above-𝐶-concave [19] on 𝐾 for every 𝑠∈𝑇(𝐾) and 𝑦∈𝐾; (v) for every minimizing sequence {(𝑠_𝑛,𝑥_𝑛)}⊂𝑇(𝐾)×𝐾, and for each (𝑠,𝑦)∈𝑇(𝐾)×𝐾, there is a sequence {𝜁_𝑛} with 𝜁_𝑛∈𝑓(𝑠,𝑥_𝑛,𝑦) for each 𝑛∈ℕ is a bounded sequence in 𝑌. Then (GSEP) is 𝑀-well-posedness.

Proof. We prove it by contradiction. Suppose that (GSEP) is not 𝑀-well-posedness. Then there exists a minimizing sequence {(𝑠_𝑛,𝑥_𝑛)}⊂𝑇𝐾×𝐾 and 𝜖>0 such that 𝑥_𝑛∉Ω+𝜖𝐵, (17) for infinitely many 𝑛, where 𝐵 denotes the unit open ball in 𝑋. Let us choose a subsequence from {(𝑠_𝑛,𝑥_𝑛)} so that the relation (17) holds for all elements of the subsequence. Such a subsequence is still a minimizing sequence, and we still denote it by {(𝑠_𝑛,𝑥_𝑛)} if there is no any confusion. Now, we separate our discussion into two cases.
Case 1. If the sequence {𝑥_𝑛} is bounded, then it has a convergent subsequence {𝑥_𝑛𝑘} that converges to some point 𝑥^⋆∈𝑋 with a corresponding subsequence {𝑠_𝑛𝑘} with 𝑠_𝑛𝑘∈𝑇(𝑥_𝑛𝑘) for every 𝑘∈ℕ. By the upper semi-continuity of 𝑇, there exists a convergent subsequence of {𝑠_𝑛𝑘} (without any confuse, we still denote it by {𝑠_𝑛𝑘}) converges to some point 𝑠^⋆∈𝑇(𝑥^⋆). From (17), 𝑥_𝑛𝑘∉Ω+𝜖𝐵 for every 𝑘∈ℕ. Hence, 𝑥^⋆∉Ω, and by Lemma 1, we have 0∉Φ(𝑠^⋆,𝑥^⋆). By Lemma 9, there is a neighborhood 𝑈_𝑌∈𝔑_𝑌(0) such that Φ𝑠^⋆,𝑥^⋆+𝑊_𝑌∩𝑊_𝑌=∅. (18) Since {(𝑠_𝑛𝑘,𝑥_𝑛𝑘)} is a minimizing sequence, for each 𝑘, we can choose 𝜂_𝑛𝑘∈Φ(𝑠_𝑛𝑘,𝑥_𝑛𝑘) such that 𝜂_𝑛𝑘→0. Since Φ is upper Hausdorff continuity of Φ at (𝑠^⋆,𝑥^⋆), we have Φ𝑠_𝑛𝑘,𝑥_𝑛𝑘⊂Φ𝑠^⋆,𝑥^⋆+𝑊_𝑌. (19) Hence, for 𝑘 large enough, 𝜂_𝑛𝑘∈(Φ(𝑠^⋆,𝑥^⋆)+𝑊_𝑌)∩𝑊_𝑌 which contradicts (18).
Case 2. If the sequence {𝑥_𝑛} is unbounded. Since Ω is bounded, so are Ω+𝜖𝐵 and cl(Ω+𝜖𝐵). Then the set cl(Ω+𝜖𝐵) is compact. We denote that Ω∩𝜕(Ω+𝜖𝐵)=∅, where 𝜕(Ω+𝜖𝐵) means the boundary of Ω+𝜖𝐵. Since the sequence {𝑥_𝑛} is unbounded, there is a subsequence {𝑥_𝑛𝑘} with ‖𝑥_𝑛𝑘‖→∞ as 𝑘→∞. Without loss of generality, we may assume that 𝑥_𝑛∉𝜕(Ω+𝜖𝐵) for all 𝑛∈ℕ. Fix any 𝑥∈Ω and let 𝑣_𝑛=𝜆_𝑛𝑥+(1−𝜆_𝑛)𝑥_𝑛∈𝜕(Ω+𝜖𝐵) for any 𝑛∈ℕ, where 𝜆_𝑛=sup{𝜆∈[0,1]∶𝜆𝑥+(1−𝜆)𝑥_𝑛∉Ω+𝜖𝐵}. After a simple calculation, we can see that 𝜆_𝑛→1 as 𝑛→∞. Hence, {𝑣_𝑛} has a subsequence {𝑣_𝑛𝑘} that converges to some point 𝑣^⋆∈𝜕(Ω+𝜖𝐵). For the similar process in Case 1, we have a subsequence {𝑡_𝑛𝑘} of the corresponding sequence {𝑡_𝑛} with 𝑡_𝑛∈𝑇(𝑣_𝑛) converges to some point 𝑡^⋆∈𝑇(𝑣^⋆). By the above 𝐶-concavity of 𝑓 in 𝑥, we have 𝜆_𝑛𝑘𝑓𝑡_𝑛𝑘,𝑥,𝑣^⋆+1−𝜆_𝑛𝑘𝑓𝑡_𝑛𝑘,𝑥_𝑛𝑘,𝑣^⋆⊂𝑓𝑡_𝑛𝑘,𝑣_𝑛𝑘,𝑣^⋆−𝐶. (20) By condition (v), there is a bounded sequence {𝜁_𝑛𝑘} with 𝜁_𝑛𝑘∈𝑓(𝑡_𝑛𝑘,𝑥_𝑛𝑘,𝑣^⋆) for each 𝑘∈ℕ in 𝑌. Hence, by (20), we have 𝜆_𝑛𝑘𝑓𝑡_𝑛𝑘,𝑥,𝑣^⋆+1−𝜆_𝑛𝑘𝜁_𝑛𝑘⊂𝑓𝑡_𝑛𝑘,𝑣_𝑛𝑘,𝑣^⋆−𝐶. (21) Next, we claim that 𝑓𝑡^⋆,𝑥,𝑣^⋆∩(−𝐶)≠∅. (22) Indeed, if 𝑓(𝑡^⋆,𝑥,𝑣^⋆)∩(−𝐶)=∅. By Lemma 9, there is a neighborhood 𝑈_𝑌∈𝔑_𝑌(0) such that 𝑓𝑡^⋆,𝑥,𝑣^⋆+𝑈_𝑌∩𝑈_𝑌−𝐶=∅. (23) For this 𝑈_𝑌, by the above 𝐶-upper Hausdorff continuity of 𝑓 and the fact that 𝑓(𝑡^⋆,𝑣^⋆,𝑣^⋆)={0}, we have 𝑓𝑡_𝑛𝑘,𝑣_𝑛𝑘,𝑣^⋆⊂𝑈_𝑌𝑓𝑡−𝐶,_𝑛𝑘,𝑥,𝑣^⋆⊂𝑓𝑡^⋆,𝑥,𝑣^⋆+𝑈_𝑌 (24) for 𝑘 large enough. Since the sequence {𝜁_𝑛𝑘} is bounded, the left-hand side of (21) will fell into 𝑓(𝑡^⋆,𝑥,𝑣^⋆)+𝑈_𝑌 for 𝑘 large enough. But in this situation, we can see that (𝑓(𝑡^⋆,𝑥,𝑣^⋆)+𝑈_𝑌)∩(𝑈_𝑌−𝐶)≠∅ which contradicts (23). Thus, the relation (22) holds. Since 𝑥∈Ω and 𝑡^⋆∈𝑇(𝐾), by condition (iii) we have 𝑣^⋆∈Ω which contradicts the fact that 𝑣^⋆∈𝜕(Ω+𝜖𝐵).
From the discussions of above two cases, (GSEP) is 𝑀-well-posedness.

Remark 11. Theorem 10 generalize the Theorem 1 of [10] to (GSEP).

Lemma 12. Suppose that 𝐹∶𝐾⇉𝑌 and 𝑓∶𝐾×𝐾⇉𝑌 satisfy 𝑓(𝑥,𝑦)=𝐹(𝑦)−𝐹(𝑥) for all 𝑥,𝑦∈𝐾, then Min_𝑤𝑓(𝑥,𝐾)⊂Min_𝑤𝐹(𝐾)−𝐹(𝑥) (25) for all 𝑥∈𝐾.

Proof. For any fixed 𝑥∈𝐾. Choose any 𝜉∈Min_𝑤𝑓(𝑥,𝐾); we have 𝜉∈𝐹(𝐾)−𝐹(𝑥) and (𝐹(𝐾)−𝐹(𝑥))∩(𝜉−int𝐶)=∅. There exist 𝜉₁∈𝐹(𝐾) and 𝜉₂∈𝐹(𝑥) such that 𝜉=𝜉₁−𝜉₂ and (𝐹(𝐾)−𝐹(𝑥))∩(𝜉₁−𝜉₂−int𝐶)=∅. That is, (𝐹(𝐾)+𝜉₂−𝐹(𝑥))∩(𝜉₁−int𝐶)=∅. Since 0∈𝜉₂−𝐹(𝑥) and 𝐹(𝐾)⊂𝐹(𝐾)+𝜉₂−𝐹(𝑥), we know that 𝐹(𝐾)∩(𝜉₁−int𝐶)=∅. Thus, 𝜉₁∈Min_𝑤𝐹(𝐾), and hence, 𝜉∈Min_𝑤𝐹(𝐾)−𝐹(𝑥). Therefore, Min_𝑤𝑓(𝑥,𝐾)⊂Min_𝑤𝐹(𝐾)−𝐹(𝑥). (26)

Lemma 13. Let 𝑋, 𝑌, 𝐶, 𝐾 be given the same as in Theorem 10, let 𝑓 be as given in Lemma 12, and let 𝐹∶𝐾⇉𝑌 be upper semi-continuous with nonempty compact values such that the set Ω=𝑤−Eff(𝐹,𝐾) is bounded, where 𝑤−Eff(𝐹,𝐾)={𝑥∈𝐾∶𝑦∈𝐹(𝑥),𝐹(𝐾)∩(𝑦−int𝐶)=∅}. Any sequence {𝑥_𝑛} satisfies for every neighborhood 𝑈_𝑌∈𝔑_𝑌(0) of 0; there is 𝑛₀∈ℕ, such that Min_𝑤𝑓(𝑥_𝑛,𝐾)∩𝑈_𝑌≠∅ for all 𝑛≥𝑛₀. Then there exists a bounded sequence {𝜂_𝑛} in 𝑌 with 𝜂_𝑛∈𝐹(𝑥_𝑛), for all 𝑛∈ℕ.

Proof. Since Ω is bounded, its closure cl(Ω) is compact. By the upper semi-continuity of 𝐹, 𝐹(cl(Ω)) is compact. Hence it is bounded, so is 𝐹(Ω). Fixed 𝑊_𝑌∈𝔑_𝑌(0), a symmetric neighborhood of 0. Since the sequence {𝑥_𝑛} satisfies for every neighborhood 𝑈_𝑌∈𝔑_𝑌(0) of 0, there is 𝑛₀∈ℕ, such that Min_𝑤𝑓(𝑥_𝑛,𝐾)∩𝑈_𝑌≠∅ for all 𝑛≥𝑛₀. From Lemma 12, we have Min_𝑤𝐹(𝐾)−𝐹𝑥_𝑛∩𝑊_𝑌≠∅ (27) for all 𝑛≥𝑛₀. We can pick some points 𝜉_𝑛∈Min_𝑤𝐹(𝐾), and 𝜂_𝑛∈𝐹(𝑥_𝑛) such that 𝜉_𝑛−𝜂_𝑛∈𝑊_𝑌 (28) for all 𝑛≥𝑛₀. Since 𝑊_𝑌 is symmetric, we have 𝜂_𝑛∈𝜉_𝑛+𝑊_𝑌⊂Min_𝑤𝐹(𝐾)+𝑊_𝑌=𝐹(𝑤−Eff(𝐹,𝐾))+𝑊_𝑌=𝐹(Ω)+𝑊_𝑌 (29) for all 𝑛≥𝑛₀. Since 𝐹(Ω) is bounded, so is 𝐹(Ω)+𝑊_𝑌. Therefore, the sequence {𝜂_𝑛} is bounded.

Now, let us present another criterion for 𝑀-well-posedness of (GSEP).

Theorem 14. Let 𝑋, 𝑌, 𝑍, 𝐶, 𝐾, 𝑓, 𝑇 be given the same as in Theorem 10. Suppose that (i) the solution set Ω of (GSEP) is nonempty and compact; (ii) for every 𝑥,𝑧∈𝐾 and 𝑠∈𝑇(𝐾), if 𝑓(𝑠,𝑥,𝑧)∩𝐶≠∅, then 𝑓(𝑠,𝑧,𝑥)∩(−𝐶)≠∅; (iii)𝑓(𝑠,𝑥,𝑧)∩(−𝐶)=∅ for all 𝑥∈Ω, for all 𝑠∈𝑇(𝐾) and for all 𝑧∈𝐾⧵Ω; (iv) the mapping (𝑠,𝑥)→𝑓(𝑠,𝑥,𝑦) is above 𝐶-upper Hausdorff continuous on 𝑍×𝐾 for every 𝑦∈𝐾, and the mapping 𝑥→𝑓(𝑠,𝑥,𝑦) is above 𝐶-concave on 𝐾 for every 𝑠∈𝑇(𝐾) and 𝑦∈𝐾; (v) for each minimizing sequence {(𝑠_𝑛,𝑥_𝑛)}⊂𝑇(𝐾)×𝐾, and for each (𝑦,𝑠)∈𝐾×𝑍, there is a sequence {𝜁_𝑛} with 𝜁_𝑛∈𝑓(𝑠,𝑥_𝑛,𝑦) for each 𝑛∈ℕ is a bounded sequence in 𝑌; (vi) for every 𝑥∈𝐾⧵Ω and for every 𝑠∈𝑇(𝑥), Min_𝑤𝑓(𝑠,𝑥,𝐾)⊂𝑓(𝑠,𝑥,Ω). Then (GSEP) is 𝑀-well-posedness.

Proof. We prove it by contradiction. Suppose that (GSEP) is not 𝑀-well-posedness. Hence there exists a minimizing sequence {(𝑠_𝑛,𝑥_𝑛)}⊂𝑇(𝐾)×𝐾 and 𝜖>0 such that the relation (17) holds. If the sequence {𝑥_𝑛} is unbounded, by a similar process in Case 2 of Theorem 10 we know that (GSEP) is 𝑀-well-posedness. If the sequence {𝑥_𝑛} is bounded, then it has a convergent subsequence that converges to some point 𝑥^⋆∈𝑋 with a corresponding subsequence {𝑠_𝑛} with 𝑠_𝑛∈𝑇(𝑥_𝑛) for every 𝑛∈ℕ. We still denote it by {𝑥_𝑛} if there is no confusion. The relation (17) tells us that 𝑥^⋆∉Ω. (30) Since {(𝑠_𝑛,𝑥_𝑛)} is a minimizing sequence, we can choose a sequence {𝜏_𝑛} with 𝜏_𝑛→0, where 𝜏_𝑛∈Φ(𝑥_𝑛)=Min_𝑤𝑓(𝑠_𝑛,𝑥_𝑛,𝐾) for all 𝑛∈ℕ. For the same process as in Case 1 of Theorem 10, by the upper semi-continuity of 𝑇, there exists a convergent subsequence of {𝑠_𝑛} (without any confusion, we still denote it by {𝑠_𝑛}) that converges to some point 𝑠^⋆∈𝑇(𝑥^⋆). Since 𝑥_𝑛∈𝐾⧵Ω, by condition (vi), 𝜏_𝑛∈Min_𝑤𝑓(𝑠_𝑛,𝑥_𝑛,𝐾)⊂𝑓(𝑠_𝑛,𝑥_𝑛,Ω). Then, for each 𝑛∈ℕ, there is a 𝑧_𝑛∈Ω such that 𝜏_𝑛∈𝑓(𝑠_𝑛,𝑥_𝑛,𝑧_𝑛). Since Ω is compact, there is a subsequence {𝑧_𝑛𝑘} of {𝑧_𝑛} that converges to some point 𝑧^⋆∈Ω. Now we claim that 𝑓(𝑠^⋆,𝑥^⋆,𝑧^⋆)∩𝐶≠∅. Indeed, suppose that 𝑓(𝑠^⋆,𝑥^⋆,𝑧^⋆)∩𝐶=∅. By Lemma 9, there is 𝑊_𝑌∈ℕ_𝑌(0) such that 𝑓𝑠^⋆,𝑥^⋆,𝑧^⋆+𝑊_𝑌∩𝐶+𝑊_𝑌=∅, (31) or 𝑓𝑠^⋆,𝑥^⋆,𝑧^⋆+𝑊_𝑌−𝐶∩𝑊_𝑌=∅. (32) By the above 𝐶-upper Hausdorff continuity of 𝑓, for this 𝑊_𝑌, there is 𝑘₀∈ℕ such that 𝜏_𝑛𝑘∈𝑓𝑠_𝑛𝑘,𝑥_𝑛𝑘,𝑧_𝑛𝑘⊂𝑓𝑠^⋆,𝑥^⋆,𝑧^⋆+𝑊_𝑌−𝐶, (33) for all 𝑘≥𝑘₀. Thus, from (32), 𝜏_𝑛𝑘∉𝑊_𝑌 which contradicts the fact that 𝜏_𝑛𝑘→0. Hence, 𝑓(𝑠^⋆,𝑥^⋆,𝑧^⋆)∩𝐶≠∅. By condition (ii), 𝑓(𝑠^⋆,𝑧^⋆,𝑥^⋆)∩(−𝐶)≠∅. Since 𝑧^⋆∈Ω, by condition (iii), 𝑥^⋆∈Ω which contradicts (30). Hence (GSEP) is 𝑀-well-posedness.

Let us present the third criterion for 𝑀-well-posedness of (GSEP) as follows.

Theorem 15. Let 𝑋, 𝑌, 𝑍, 𝐶, 𝐾, 𝑇, 𝑓 be given the same as in Theorem 10. Suppose that (i) the solution set Ω of (GSEP) is nonempty and bounded; (ii)𝑓(𝑠,𝑥,𝑧)∩(−𝐶)=∅ and 𝑓(𝑠,𝑧,𝑥)∩(−𝐶)=∅ for all 𝑧∈cl(Ω), for all 𝑠∈𝑇(𝐾) and for all 𝑥∈𝐾⧵cl(Ω); (iii) the mapping 𝑥→𝑓(𝑠,𝑥,𝑦) is above 𝐶-concave on 𝐾 for every (𝑠,𝑦)∈𝑇(𝐾)×𝐾, and the mapping 𝑠→𝑓(𝑠,𝑥,𝑦) is lower semi-continuous on 𝑇(𝐾), for every (𝑥,𝑦)∈𝐾×𝐾; (iv) for every 𝑥∈𝐾⧵cl(Ω) and for every 𝑠∈𝑇(𝑥), Min_𝑤𝑓(𝑠,𝑥,𝐾)⊂𝑓(𝑠,𝑥,cl(Ω)). Then (GSEP) is 𝑀-well-posedness.

Proof. Suppose that (GSEP) is not 𝑀-well-posedness. Then we have a minimizing sequence {(𝑠_𝑛,𝑥_𝑛)}⊂𝑇(𝐾)×𝐾 and 𝜖>0 such that the relation (17) holds for all 𝑛∈ℕ. We can use the same process as in the proof of Theorem 10 under the situation when we replace Ω by cl(Ω). If the sequence {𝑥_𝑛} is unbounded, combining Lemmas 12 and 13, we have the sequences {𝑥_𝑛𝑘}, {𝑣_𝑛}, {𝑣_𝑛𝑘}, {𝜆_𝑛}, {𝑡_𝑛}, {𝑡_𝑛𝑘}, {𝜁_𝑛𝑘}, and points 𝑥, 𝑣^⋆, 𝑡^⋆ with the same properties as in the proof (Case 2) of Theorem 10. By condition (iii), we have the mapping 𝑥→𝑓(𝑠,𝑥,𝑦) which is above 𝐶-concave on 𝐾 for every (𝑠,𝑦)∈𝑇(𝐾)×𝐾 and the relations (20) and (21) hold. Since the mapping 𝑠→𝑓(𝑠,𝑥,𝑦) is lower semi-continuous on 𝑇(𝐾) for every (𝑥,𝑦)∈𝐾×𝐾 and fix any 𝜉∈𝑓(𝑡^⋆,𝑥,𝑣^⋆), there exists 𝜉_𝑛𝑘∈𝑓(𝑡_𝑛𝑘,𝑥,𝑣^⋆) such that 𝜉_𝑛𝑘→𝜉. For this 𝜉_𝑛𝑘, by (21), there exists 𝜈_𝑛𝑘∈𝑓(𝑡_𝑛𝑘,𝑣_𝑛𝑘,𝑣^⋆) such that 𝜆_𝑛𝑘𝜉_𝑛𝑘+1−𝜆_𝑛𝑘𝜁_𝑛𝑘∈𝜈_𝑛𝑘−𝐶. (34) Since the mapping (𝑠,𝑥)→𝑓(𝑠,𝑥,𝑦) is upper semi-continuous, hence it is upper Hausdorff continuous on 𝑍×𝐾 for every 𝑦∈𝐾, and (𝑡_𝑛𝑘,𝑣_𝑛𝑘)→(𝑡^⋆,𝑥), for any given 𝑊_𝑌∈𝔑_𝑌(0), 𝑓𝑡_𝑛𝑘,𝑣_𝑛𝑘,𝑣^⋆⊂𝑓𝑡^⋆,𝑣^⋆,𝑣^⋆+𝑊_𝑌=𝑊_𝑌 (35) for 𝑘 large enough. From (34) and (35) and the fact that {𝜆_𝑛𝑘}→1, we have 𝜉∈𝑊_𝑌−𝐶. (36) Next, we claim that 𝑓𝑡^⋆,𝑥,𝑣^⋆∩(−𝐶)≠∅. (37) Indeed, if not, 𝑓(𝑡^⋆,𝑥,𝑣^⋆)∩(−𝐶)=∅. By Lemma 9, there exists 𝑁_𝑌∈𝔑_𝑌(0) such that (𝑓(𝑡^⋆,𝑥,𝑣^⋆)+𝑁_𝑌)∩(𝑁_𝑌−𝐶)=∅. But this contradicts (36), and hence (37) holds. Since 𝑥∈cl(Ω), by condition (ii), 𝑣^⋆∈cl(Ω) which contradicts the fact 𝑣^⋆∈𝜕(cl(Ω)+𝜖𝐵). Hence, (GSEP) is 𝑀-well-posed. On the other hand, if the sequence {𝑥_𝑛} is bounded, the sequences {𝑥_𝑛𝑘}, {𝑠_𝑛}, {𝑠_𝑛𝑘}, the points 𝑥^⋆, 𝑠^⋆, and the number 𝜖 are the same as in the Case 1 of Theorem 10, so that 𝑥_𝑛𝑘∉cl(Ω)+𝜖𝐵 for all 𝑘∈ℕ. Hence 𝑥^⋆∉cl(Ω). Since {(𝑠_𝑛,𝑥_𝑛)} is minimizing, we can choose a sequence 𝜂_𝑛∈Φ(𝑠_𝑛,𝑥_𝑛)=Min_𝑤𝑓(𝑠_𝑛,𝑥_𝑛,𝐾) with 𝜂_𝑛→0. By condition (v), 𝜂_𝑛∈Min_𝑤𝑓(𝑠_𝑛,𝑥_𝑛,𝐾)⊂𝑓(𝑠_𝑛,𝑥_𝑛,cl(Ω)). For each 𝑛∈ℕ, there is 𝑧_𝑛∈cl(Ω) such that 𝜂_𝑛∈𝑓(𝑠_𝑛,𝑥_𝑛,𝑧_𝑛). Since cl(Ω) is compact, there is a subsequence {𝑧_𝑛𝑘} of {𝑧_𝑛} that converges to some point 𝑧^⋆∈cl(Ω). Since 𝑓∶𝑍×𝐾×𝐾⇉𝑌 is upper semi-continuous with nonempty compact values, we have 0∈𝑓𝑠^⋆,𝑥^⋆,𝑧^⋆. (38) From condition (ii) and the fact that 𝑧^⋆∈cl(Ω), we have 𝑥^⋆∈Ω which contradicts the fact 𝑥^⋆∉Ω. Hence, (GSEP) is 𝑀-well-posed.

Example 7 tells us that if (GSEP) is 𝑀-well-posed, then it is 𝐵-well-posed. But the converse is not true. Proposition 8 proposes a possible condition so that the converse holds. To the end, we state this result as follows.

Corollary 16. Under the framework of Theorem 14 (resp., Theorem 15) the following condition (A) holds: (A) for every 𝑊_𝑌∈𝔑_𝑌(0), there is a 𝑈_𝑌∈𝔑(0) such that 𝑓𝑔(𝐾⧵cl(Ω))∩𝐶+𝑈_𝑌⊂𝑊_𝑌, (39)where 𝑔(𝑥)=ℎ(𝑥)×cl(Ω) and ℎ is given in (7).
Then (GSEP) is 𝐵-well-posed.

Proof. From Theorem 14 (resp., Theorem 15), (GSEP) is 𝑀-well-posed. By condition (vi) of Theorem 14 (resp., condition (iv) of Theorem 15), we have Φ(𝑠,𝑥)⊂𝑓(𝑠,𝑥,cl(Ω)) (40) for every 𝑠∈𝑇(𝑥) and 𝑥∈𝐾⧵cl(Ω). Hence, Φ(ℎ(𝐾⧵cl(Ω)))⊂𝑓𝑔(𝐾⧵cl(Ω)). (41) Combining this with condition (A), the condition (7) holds. Hence, by Proposition 8, (GSEP) is 𝐵-well-posed.

Acknowledgments

This work was supported by Grant no. NSC101-2115-M-039-001 of the National Science Council of Taiwan which is gratefully acknowledged. The author would like to thank the referees for their useful comments which have helped to improve some results and the presentation of the paper.