# Well-Posedness for Generalized Set Equilibrium Problems

Authors: Yen-Cherng Lin
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 1. Introduction and Preliminaries

LetX, 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 mappingf: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
(1)f(s-,x-,y)∩(-intC)=∅∀y∈K.
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 mappingΦ:Z×X⇉Y is defined by
(2)Φ(s,x)=Minwf(s,x,K)
for all (s,x)∈Z×X, where MinwA={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. 𝔑X(x0) denotes the collection of neighborhoods around x0 in X, similar notations for 𝔑Y(y0) and 𝔑Z(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 existsτ∈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 havex-∈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{(sn,xn)∈Z×K:sn∈T(xn)} is a minimizing sequence for Φ if for every neighborhood UY∈𝔑Y(0) of 0, there is n0∈ℕ, such that Φ(sn,xn)∩UY≠∅ for all n≥n0.Definition 3.
(GSEP) isM-well-posed if it satisfies the following conditions:  (i)
there exists at least one solution, that is, the setΩ≠∅;(ii)
for every minimizing sequence{(sn,xn)} and for every UX∈𝔑X(0), there is n0∈ℕ such that xn∈Ω+UX for all n≥n0.Definition 4.
Forϵ∈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 thatΩ(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) isB-well-posed if it satisfies the following conditions:  (i)
there exists at least one solution, that is, the setΩ≠∅;(ii)
the mappingΩ:C⇉X is upper Hausdorff continuous at ϵ=0; that is, for every UX∈𝔑X(0), there exists UY∈𝔑Y(0) such that Ω(ϵ)⊂Ω+UX for every ϵ∈UY∩C.Definition3 arises from [8], and Definition 5 is originally proposed by [9].Definition 6 (see [16, 17]).
A set-valued mappingT:X⇉Z is  (i)
upper semicontinuous if for everyx∈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 everyx∈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 thatT is upper semicontinuous at x0 and T(x0) is compact; then for any net {xν}⊂X, xν→x0, and for any net yν∈T(xν) for each ν, there exists y0∈T(x0) and a subnet {yνα} such that yνα→y0. We can refer to [18] for more details. We also note that T is lower semicontinuous at x0 if for any net {xν}⊂X, xν→x0, y0∈T(x0) implies that there exists net yν∈T(xν) such that yν→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 WY∈𝔑Y(0), there exists a neighborhood VX∈𝔑X(0) such that T(x+VY)⊂T(x)+WY-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 satisfiesM-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=ℝ+2, 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
(3)f(s,x,y)={{(0,0)},ifx=0,y≥0;{(y-s,0)},ifx>0,0≤y<2s;{(s,-8sy3),(-s,-8sy3)},otherwise,
for all (s,x,y)∈Z×K×K with s∈T(x).
Then the set of all solutions for (GSEP) isΩ={0}. For any x∈K and any s∈T(x), the minimizing mapping is
(4)Φ(s,x)={{(0,0)}ifx=0;{({-s}×[-1s2,0])∪{(s,-1s2)}},ifx>0.
If we chooseUX=(-1/2,1/2), a neighborhood of 0, and xn=n, ϵn=(0,1/n2) for all n∈ℕ, we can easily see that ϵn→(0,0) as n→∞, and n∈Ω(ϵn)∖(Ω+UX) for all n∈ℕ. Thus, (GSEP) is not B-well-posed.
Nevertheless, for any minimizing sequence{(sn,xn)} for Φ with sn∈T(xn) for all n and for every UY∈𝔑Y(0), there exists n0∈ℕ such that Φ(sn,xn)∩UY≠∅ for n≥n0. This will force that xn→0 as n→∞, and hence xn∈Ω+UX for all n≥n0. Therefore, (GSEP) is M-well-posed.
(ii) We modify the above example as follows. LetX, Z, Y, C, K, T be given the same as in (i). The set-valued mapping f:Z×K×K⇉Y is defined by
(5)f(s,x,y)={{(0,0)},ifx=0,y≥0;{(y-s,0)},ifx>0,0≤y<2s;{(s,-8sy3)},otherwise,
for all (s,x,y)∈Z×K×K with s∈T(x).
Then the set of all solutions for (GSEP) isΩ=K. For any x∈K and any s∈T(x), the minimizing mapping is
(6)Φ(s,x)={{(0,0)}ifx=0;{(s,-1s2)},ifx>0.
Since for any minimizing sequence{(sn,xn)}, and for every UX∈𝔑X(0), we always have xn∈Ω+UX. Thus, (GSEP) is M-well-posed. Furthermore, since Ω+UX=K+UX, for all ϵ∈ℝ+2, we always have Ω(ϵ)⊂Ω+UX. Hence, (GSEP) is B-well-posed.From the above observation, theM-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) isB-well-posed, then it is M-well-posed. (b) If (GSEP) is M-well-posed, and for every WY∈𝔑Y(0), there exists UY∈𝔑Y(0) such that
(7)Φ(h(K∖cl(Ω)))∩(C+UY)⊂WY,
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 {ϵn}⊂C with ϵn→0 and xn∈Ω(ϵn) such that
(8)xn∉Ω+U-X,∀n∈ℕ.
This means
(9)xn∉cl(Ω),∀n∈ℕ.
Since xn∈Ω(ϵn), there exists sn∈T(xn) such that
(10)Φ(sn,xn)∩(C-ϵn)≠∅,∀n∈ℕ.
Now, we separate into two cases.
Case
1. If the sequence {(sn,xn)} is a minimizing sequence, then by M-well-posedness, for this U-X, there is a n0∈ℕ such that xn∈Ω+U-X, for all n≥n0, which contradicts (8).
Case
2. If the sequence {(sn,xn)} is not a minimizing sequence, then there is a WY∈𝔑Y(0) and a subsequence {xnk} of {xn} with a corresponding subsequence {snk}∈T({xnk}) such that
(11)Φ(snk,xnk)∩WY=∅,∀k∈ℕ.
By relation (10), we have Φ(snk,xnk)∩(C-ϵnk)≠∅, for all k∈ℕ. For this WY and condition (7), there is a symmetric neighborhood UY∈𝔑Y(0) such that Φ(snk,xnk)∩(C+UY)⊂WY, for all k∈ℕ. For k large enough, -ϵnk∈UY. Taking ηnk∈Φ(snk,xnk)∩(C+UY) for all k∈ℕ. This implies that, for k large enough, ηnk∈Φ(snk,xnk)∩WY which contradicts (11). This completes the proof.We need the following lemma for the next criterion forM-well-posedness of (GSEP).Lemma 9.
LetY 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 WY of 0 such that (A+WY)∩(WY-C)=∅. In particular, (A+WY)∩WY=∅.Proof.
Suppose thatA∩(-C)=∅. For all η∈A, we have η∉-C. Since Y is regular, there is a neighborhood UYη of 0 such that
(12)(η+UYη)⋂‍(UYη-C)=∅.
Since A is a nonempty compact subset, the set
(13)A⊂⋃η∈A‍(η+UYη).
There exist η1,η2,…,ηn∈A, such that
(14)A⊂⋃i=1n‍(ηi+UYηi)⊂A+⋃i=1n‍UYηi.
Let WY=⋂i=1n‍UYηi. Then
(15)(A+⋃i=1n‍UYηi)⋂‍(WY-C)=∅.
Since WY⊂⋃i=1n‍UYηi, we have
(16)(A+WY)∩(WY-C)=∅.We note that, although every compact regular space is a normal space, Lemma9 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=ℝ+2.Now, we present first criterion ofM-well-posedness for (GSEP).Theorem 10.
LetX, 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 setΩ of (GSEP) is nonempty and bounded;(ii)
the minimizing mappingΦ 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{(sn,xn)}⊂T(K)×K, and for each (s,y)∈T(K)×K, there is a sequence {ζn} with ζn∈f(s,xn,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 notM-well-posedness. Then there exists a minimizing sequence {(sn,xn)}⊂TK×K and ϵ>0 such that
(17)xn∉Ω+ϵB,
for infinitely many n, where B denotes the unit open ball in X. Let us choose a subsequence from {(sn,xn)} 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 {(sn,xn)} if there is no any confusion. Now, we separate our discussion into two cases.
Case
1. If the sequence {xn} is bounded, then it has a convergent subsequence {xnk} that converges to some point x⋆∈X with a corresponding subsequence {snk} with snk∈T(xnk) for every k∈ℕ. By the upper semi-continuity of T, there exists a convergent subsequence of {snk} (without any confuse, we still denote it by {snk}) converges to some point s⋆∈T(x⋆). From (17), xnk∉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 UY∈𝔑Y(0) such that
(18)(Φ(s⋆,x⋆)+WY)∩WY=∅.
Since {(snk,xnk)} is a minimizing sequence, for each k, we can choose ηnk∈Φ(snk,xnk) such that ηnk→0. Since Φ is upper Hausdorff continuity of Φ at (s⋆,x⋆), we have
(19)Φ(snk,xnk)⊂Φ(s⋆,x⋆)+WY.
Hence, for k large enough, ηnk∈(Φ(s⋆,x⋆)+WY)∩WY which contradicts (18).
Case
2. If the sequence {xn} 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 {xn} is unbounded, there is a subsequence {xnk} with ∥xnk∥→∞ as k→∞. Without loss of generality, we may assume that xn∉∂(Ω+ϵB) for all n∈ℕ. Fix any x-∈Ω and let vn=λnx-+(1-λn)xn∈∂(Ω+ϵB) for any n∈ℕ, where λn=sup{λ∈[0,1]:λx-+(1-λ)xn∉Ω+ϵB}. After a simple calculation, we can see that λn→1 as n→∞. Hence, {vn} has a subsequence {vnk} that converges to some point v⋆∈∂(Ω+ϵB). For the similar process in Case 1, we have a subsequence {tnk} of the corresponding sequence {tn} with tn∈T(vn) converges to some point t⋆∈T(v⋆). By the above C-concavity of f in x, we have
(20)λnkf(tnk,x-,v⋆)+(1-λnk)f(tnk,xnk,v⋆)⊂f(tnk,vnk,v⋆)-C.
By condition (v), there is a bounded sequence {ζnk} with ζnk∈f(tnk,xnk,v⋆) for each k∈ℕ in Y. Hence, by (20), we have
(21)λnkf(tnk,x-,v⋆)+(1-λnk)ζnk⊂f(tnk,vnk,v⋆)-C.
Next, we claim that
(22)f(t⋆,x-,v⋆)∩(-C)≠∅.
Indeed, if f(t⋆,x-,v⋆)∩(-C)=∅. By Lemma 9, there is a neighborhood UY∈𝔑Y(0) such that
(23)(f(t⋆,x-,v⋆)+UY)∩(UY-C)=∅.
For this UY, by the above C-upper Hausdorff continuity of f and the fact that f(t⋆,v⋆,v⋆)={0}, we have
(24)f(tnk,vnk,v⋆)⊂UY-C,f(tnk,x-,v⋆)⊂f(t⋆,x-,v⋆)+UY
for k large enough. Since the sequence {ζnk} is bounded, the left-hand side of (21) will fell into f(t⋆,x-,v⋆)+UY for k large enough. But in this situation, we can see that (f(t⋆,x-,v⋆)+UY)∩(UY-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) isM-well-posedness.Remark 11.
Theorem10 generalize the Theorem 1 of [10] to (GSEP).Lemma 12.
Suppose thatF:K⇉Y and f:K×K⇉Y satisfy f(x,y)=F(y)-F(x) for all x,y∈K, then
(25)Minwf(x,K)⊂MinwF(K)-F(x)
for all x∈K.Proof.
For any fixedx∈K. Choose any ξ∈Minwf(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∈MinwF(K), and hence, ξ∈MinwF(K)-F(x). Therefore,
(26)Minwf(x,K)⊂MinwF(K)-F(x).Lemma 13.
LetX, 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 {xn} satisfies for every neighborhood UY∈𝔑Y(0) of 0; there is n0∈ℕ, such that Minwf(xn,K)∩UY≠∅ for all n≥n0. Then there exists a bounded sequence {ηn} in Y with ηn∈F(xn), for all n∈ℕ.Proof.
SinceΩ 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 WY∈𝔑Y(0), a symmetric neighborhood of 0. Since the sequence {xn} satisfies for every neighborhood UY∈𝔑Y(0) of 0, there is n0∈ℕ, such that Minwf(xn,K)∩UY≠∅ for all n≥n0. From Lemma 12, we have
(27)(MinwF(K)-F(xn))∩WY≠∅
for all n≥n0. We can pick some points ξn∈MinwF(K), and ηn∈F(xn) such that
(28)ξn-ηn∈WY
for all n≥n0. Since WY is symmetric, we have
(29)ηn∈ξn+WY⊂MinwF(K)+WY=F(w-Eff(F,K))+WY=F(Ω)+WY
for all n≥n0. Since F(Ω) is bounded, so is F(Ω)+WY. Therefore, the sequence {ηn} is bounded.Now, let us present another criterion forM-well-posedness of (GSEP).Theorem 14.
LetX, Y, Z, C, K, f, T be given the same as in Theorem 10. Suppose that  (i)
the solution setΩ of (GSEP) is nonempty and compact;(ii)
for everyx,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{(sn,xn)}⊂T(K)×K, and for each (y,s)∈K×Z, there is a sequence {ζn} with ζn∈f(s,xn,y) for each n∈ℕ is a bounded sequence in Y;(vi)
for everyx∈K∖Ω and for every s∈T(x), Minwf(s,x,K)⊂f(s,x,Ω). Then (GSEP) is M-well-posedness.Proof.
We prove it by contradiction. Suppose that (GSEP) is notM-well-posedness. Hence there exists a minimizing sequence {(sn,xn)}⊂T(K)×K and ϵ>0 such that the relation (17) holds. If the sequence {xn} is unbounded, by a similar process in Case 2 of Theorem 10 we know that (GSEP) is M-well-posedness. If the sequence {xn} is bounded, then it has a convergent subsequence that converges to some point x⋆∈X with a corresponding subsequence {sn} with sn∈T(xn) for every n∈ℕ. We still denote it by {xn} if there is no confusion. The relation (17) tells us that
(30)x⋆∉Ω.
Since {(sn,xn)} is a minimizing sequence, we can choose a sequence {τn} with τn→0, where τn∈Φ(xn)=Minwf(sn,xn,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 {sn} (without any confusion, we still denote it by {sn}) that converges to some point s⋆∈T(x⋆). Since xn∈K∖Ω, by condition (vi), τn∈Minwf(sn,xn,K)⊂f(sn,xn,Ω). Then, for each n∈ℕ, there is a zn∈Ω such that τn∈f(sn,xn,zn). Since Ω is compact, there is a subsequence {znk} of {zn} 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 WY∈ℕY(0) such that
(31)(f(s⋆,x⋆,z⋆)+WY)∩(C+WY)=∅,
or
(32)(f(s⋆,x⋆,z⋆)+WY-C)∩WY=∅.
By the above C-upper Hausdorff continuity of f, for this WY, there is k0∈ℕ such that
(33)τnk∈f(snk,xnk,znk)⊂f(s⋆,x⋆,z⋆)+WY-C,
for all k≥k0. Thus, from (32), τnk∉WY which contradicts the fact that τnk→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 forM-well-posedness of (GSEP) as follows.Theorem 15.
LetX, Y, Z, C, K, T, f be given the same as in Theorem 10. Suppose that  (i)
the solution setΩ 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 mappingx→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 everyx∈K∖cl(Ω) and for every s∈T(x), Minwf(s,x,K)⊂f(s,x,cl(Ω)). Then (GSEP) is M-well-posedness.Proof.
Suppose that (GSEP) is notM-well-posedness. Then we have a minimizing sequence {(sn,xn)}⊂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 {xn} is unbounded, combining Lemmas 12 and 13, we have the sequences {xnk}, {vn}, {vnk}, {λn}, {tn}, {tnk}, {ζnk}, 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 ξnk∈f(tnk,x-,v⋆) such that ξnk→ξ. For this ξnk, by (21), there exists νnk∈f(tnk,vnk,v⋆) such that
(34)λnkξnk+(1-λnk)ζnk∈νnk-C.
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 (tnk,vnk)→(t⋆,x-), for any given WY∈𝔑Y(0),
(35)f(tnk,vnk,v⋆)⊂f(t⋆,v⋆,v⋆)+WY=WY
for k large enough. From (34) and (35) and the fact that {λnk}→1, we have
(36)ξ∈WY-C.
Next, we claim that
(37)f(t⋆,x-,v⋆)∩(-C)≠∅.
Indeed, if not, f(t⋆,x-,v⋆)∩(-C)=∅. By Lemma 9, there exists NY∈𝔑Y(0) such that (f(t⋆,x-,v⋆)+NY)∩(NY-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 {xn} is bounded, the sequences {xnk}, {sn}, {snk}, the points x⋆, s⋆, and the number ϵ are the same as in the Case 1 of Theorem 10, so that xnk∉cl(Ω)+ϵB for all k∈ℕ. Hence x⋆∉cl(Ω). Since {(sn,xn)} is minimizing, we can choose a sequence ηn∈Φ(sn,xn)=Minwf(sn,xn,K) with ηn→0. By condition (v), ηn∈Minwf(sn,xn,K)⊂f(sn,xn,cl(Ω)). For each n∈ℕ, there is zn∈cl(Ω) such that ηn∈f(sn,xn,zn). Since cl(Ω) is compact, there is a subsequence {znk} of {zn} that converges to some point z⋆∈cl(Ω). Since f:Z×K×K⇉Y is upper semi-continuous with nonempty compact values, we have
(38)0∈f(s⋆,x⋆,z⋆).
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.Example7 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 Theorem14 (resp., Theorem 15) the following condition (A) holds:  (A)
for everyWY∈𝔑Y(0), there is a UY∈𝔑(0) such that
(39)f(g(K∖cl(Ω)))∩(C+UY)⊂WY,where g(x)=h(x)×cl(Ω) and h is given in (7).
Then (GSEP) isB-well-posed.Proof.
From Theorem14 (resp., Theorem 15), (GSEP) is M-well-posed. By condition (vi) of Theorem 14 (resp., condition (iv) of Theorem 15), we have
(40)Φ(s,x)⊂f(s,x,cl(Ω))
for every s∈T(x) and x∈K∖cl(Ω). Hence,
(41)Φ(h(K∖cl(Ω)))⊂f(g(K∖cl(Ω))).
Combining this with condition (A), the condition (7) holds. Hence, by Proposition 8, (GSEP) is B-well-posed.



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*Source: 419053-2013-10-21.xml*