Volume 2008, Article ID 657329,14pages doi:10.1155/2008/657329
Research Article
Levitin-Polyak Well-Posedness for Equilibrium
Problems with Functional Constraints
Xian Jun Long,1 Nan-Jing Huang,1, 2 and Kok Lay Teo3
1Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
2State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Chengdu 610500, China 3Department of Mathematics and Statistics, Curtin University of Technology,
Perth W.A. 6102, Australia
Correspondence should be addressed to Nan-Jing Huang,[email protected]
Received 8 November 2007; Accepted 11 December 2007
Recommended by Simeon Reich
We generalize the notions of Levitin-Polyak well-posedness to an equilibrium problem with both abstract and functional constraints. We introduce several types of generalized Levitin-Polyak posedness. Some metric characterizations and sufficient conditions for these types of well-posedness are obtained. Some relations among these types of well-well-posedness are also established under some suitable conditions.
Copyrightq2008 Xian Jun Long et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Equilibrium problem was first introduced by Blum and Oettli1, which includes optimization problems, fixed point problems, variational inequality problems, and complementarity prob-lems as special cases. In the past ten years, equilibrium problem has been extensively studied and generalizedsee, e.g.,2,3.
It is well known that the well-posedness is very important for both optimization the-ory and numerical methods of optimization problems, which guarantees that, for approxi-mating solution sequences, there is a subsequence which converges to a solution. The well-posedness of unconstrained and constrained scalar optimization problems was first introduced and studied by Tykhonov4and Levitin and Polyak5, respectively. Since then, various con-cepts of well-posedness have been introduced and extensively studied for scalar optimiza-tion problems 6–13, best approximation problems 14–16, vector optimization problems
of Levitin-Polyak well-posedness for convex scalar optimization problems with functional constraints started by Konsulova and Revalski 32. Recently, Huang and Yang generalized those results to nonconvexvectoroptimization problems with both abstract and functional constraints 33, 34. Very recently, Huang and Yang 35 studied Levitin-Polyak-type well-posedness for generalized variational inequality problems with abstract and functional con-straints. They introduced several types of generalized Levitin-Polyak well-posednesses and obtained some criteria and characterizations for these types of well-posednesses.
Motivated and inspired by the numerical method introduced by Mastroeni36and the works mentioned above, the purpose of this paper is to generalize the results in35to equi-librium problems. We introduce several types of Levitin-Polyak well-posedness for equilib-rium problems with abstract and functional constraints. Necessary and sufficient conditions for these types of posedness are obtained. Some relations among these types of well-posedness are also established under some suitable conditions.
2. Preliminaries
Let X,· be a normed space, and let Y, d be a metric space. Let K ⊆ X and D ⊆ Y be nonempty and closed. Letf fromX×XtoR∪ {± ∞}be a bifunction satisfyingfx, x 0 for anyx∈Xand letgfromKtoY be a function. LetS{x∈K:gx∈D}.
In this paper, we consider the following explicit constrained equilibrium problem: find-ing a pointx∈Ssuch that
fx, y≥0, ∀y∈S. EP
Denote byΓthe solution set ofEP. Throughout this paper, we always assume thatS /∅and gis continuous onK.
LetW, dbe a metric space andW1⊂W. We denote bydW1p inf{dp, p: p∈W1}
the distance from the pointpto the setW1.
Definition 2.1. A sequence{xn} ⊂Kis said to be as follows:
itype I Levitin-PolyakLP in shortapproximating solution sequence if there exists a se-quenceεn>0 withεn→0 such that
dS
xn
≤εn, 2.1
fxn, y
εn≥0, ∀y∈S; 2.2
iitype II LP approximating solution sequence if there exists a sequenceεn >0 withεn →0 and{yn} ⊂Ssuch that2.1and2.2hold, and
fxn, yn
≤εn; 2.3
iiia generalized type I LP approximating solution sequence if there exists a sequenceεn >0 withεn→0 such that
dD
gxn
≤εn 2.4
iva generalized type II LP approximating solution sequence if there exists a sequenceεn>0 withεn→0 and{yn} ⊂Ssuch that2.2,2.3, and2.4hold.
Definition 2.2. The explicit constrained equilibrium problemEPis said to be of type Iresp., type II, generalized type I, generalized type II LP well-posed if the solution set Γ of EP
is nonempty, and for any type I resp., type II, generalized type I, generalized type II LP approximating solution sequence{xn}has a subsequence which converges to some point ofΓ.
Remark 2.3. iIffx, y Fx, y−xfor allx, y ∈K, whereF :K → X∗is a mapping and X∗denotes the topological dual ofX, then type Iresp., type II, generalized type I, generalized type IILP well-posedness forEPdefined inDefinition 2.2 reduces to type Iresp., type II, generalized type I, generalized type IILP well-posedness for the variational inequality with functional constraints.
iiIt is easy to see that anygeneralizedtype II LP approximating solution sequence is a generalized type I LP approximating solution sequence. Thus,generalizedtype I LP well-posedness impliesgeneralizedtype II LP well-posedness.
iiiEach type of LP well-posedness forEPimplies that the solution setΓis nonempty and compact.
ivLetgbe a uniformly continuous function on the set
Sδ0
x∈K:dSx≤δ0
2.5
for someδ0 >0. Then, generalized type Itype IILP well-posedness implies type Itype II
LP well-posedness.
It is well known that an equilibrium problem is closely related to a minimization problem
see, e.g.,36. Thus, we need to recall some notions of LP well-posedness for the following general constrained optimization problem:
minhx s.t.x∈K, gx∈D, P
whereh : K → R∪ { ∞} is lower semicontinuous. The feasible set ofPis still denoted by S. The optimal set and optimal value ofPare denoted byΓandv, respectively. If Domh∩ S /∅, thenv < ∞, where
Domh x∈K:hx< ∞. 2.6
In this paper, we always assume thatv >− ∞. In33, Huang and Yang introduced the follow-ing LP well-posed for generalized constrained optimization problemP.
Definition 2.4. A sequence{xn} ⊂Kis said to be
itype I LP minimizing sequence forPif
dS
xn
−→0, 2.7
lim sup n→ ∞ h
xn
iitype II LP minimizing sequence forPif
lim n→ ∞h
xn
v 2.9
and2.7holds;
iiia generalized type I LP minimizing sequence forPif2.8holds and
dD
gxn
−→0; 2.10
iva generalized type II LP minimizing sequence forPif2.9and2.10hold.
Definition 2.5. The generalized constrained optimization problemPis said to be type Iresp., type II, generalized type I, generalized type IILP well-posed ifv is finite,Γ/∅and for any type Iresp., type II, generalized type I, generalized type IILP minimizing sequence{xn}has a subsequence which converges to some point ofΓ.
Mastroeni36introduced the following gap function forEP:
hx sup y∈S
−fx, y, ∀x∈K. 2.11
It is clear thathis a function fromKto−∞, ∞. Moreover, ifΓ/∅, then Domh∩S /∅. Lemma 2.6see36. Lethbe defined by2.11. Then
ihx≥0for allx∈S;
iihx 0if and only ifx∈Γ.
Remark 2.7. ByLemma 2.6, it is easy to see thatx0 ∈Γif and only ifx0minimizeshxoverS
withhx0 0.
Now, we show the following lemmas.
Lemma 2.8. Lethbe defined by2.11. Suppose thatfis upper semicontinuous onK×Kwith respect to the first argument. Thenhis lower semicontinuous onK.
Proof. Letα∈Rand let the sequence{xn} ⊂Ksatisfyxn → x0 ∈Kandhxn≤ α. It follows that, for anyε >0 and eachn,−fxn, y≤α εfor ally∈S.By the upper semicontinuity off with respect to the first argument, we know that−fx0, y≤α ε.This implies thathx0≤α ε.
From the arbitrariness ofε >0, we havehx0≤αand sohis lower semicontinuous onK. This completes the proof.
Remark 2.9. Lemma 2.8implies thathis lower semicontinuous. Therefore, if Domh∩S /∅, then it is easy to see that Theorems 2.1 and 2.2 of33are true.
Proof. SinceΓ/∅, it follows fromLemma 2.6thatx0is a solution ofEPif and only ifx0is an
optimal solution ofPwithvhx0 0, wherehis defined by2.11. It is easy to check that a sequence{xn}is a type Iresp., type II, generalized type I, generalized type IILP approxi-mating solution sequence ofEPif and only if it is a type Iresp., type II, generalized type I, generalized type IILP minimizing sequence ofP. Thus, the conclusions ofLemma 2.10hold. This completes the proof.
Consider the following statement:
Γ/∅and, for any type I resp., type II, generalized type I, generalized type II LP approximating solution sequence{xn}, we havedΓxn−→0
.
2.12
It is easy to prove the following lemma byDefinition 2.2.
Lemma 2.11. If (EP) is type I (resp., type II, generalized type I, generalized type II) LP well-posed, then
2.12holds. Conversely, if 2.12holds andΓis compact, then (EP) is type I (resp., type II, generalized type I, generalized type II) LP well-posed.
3. Metric characterizations of LP well-posedness for (EP)
In this section, we give some metric characterizations of various types of LP well-posedness forEPdefined inSection 2.
Given two nonempty subsetsAandBofX, the Hausdorffdistance betweenAandBis defined by
HA, B maxeA, B, eB, A, 3.1
whereeA, B supa∈Ada, Bwithda, B infb∈Bda, b.
For anyε >0, two types of the approximating solution sets forEPare defined, respec-tively, by
M1ε x∈K:fx, y ε≥0, ∀y∈S, dSx≤ε
, M2ε x∈K:fx, y ε≥0, ∀y∈S, dD
gx≤ε. 3.2 Theorem 3.1. Let X,·be a Banach space. Then, (EP) is type I LP well-posed if and only if the solution setΓof (EP) is nonempty, compact, and
eM1ε,Γ−→0 asε−→0. 3.3
Proof. LetEPbe type I LP well-posed. ThenΓis nonempty and compact. Now, we prove that
3.3holds. Suppose to the contrary that there existγ > 0,{εn}withεn → 0, andxn ∈M1εn such that
dΓxn
≥γ. 3.4
Since{xn} ⊂ M1εn, we know that{xn}is a type I LP approximating solution sequence for
Conversely, suppose thatΓis nonempty, compact, and3.3holds. Let{xn}be a type I LP approximating solution sequence forEP. Then there exists a sequence{εn}with εn > 0 andεn → 0 such thatfxn, y εn ≥ 0 for ally ∈ SanddSxn≤ εn.Thus,{xn} ⊂ M1εn. It follows from3.3that there exists a sequence{zn} ⊂Γsuch that
xn−znd xn,Γ
≤eM1
εn
,Γ−→0. 3.5
SinceΓis compact, there exists a subsequence {znk}of{zn}converging to x0 ∈ Γ, and so the corresponding subsequence{xnk}of {xn}converges to x0. Therefore,EPis type I LP well-posed. This completes the proof.
Example 3.2. LetXY R,K 0,2, andD 0,1. Let
gx x, fx, y x−y2, ∀x, y∈X. 3.6
Then it is easy to compute that S 0,1, Γ 0,1, and M1ε 0,1 ε. It follows that eM1ε,Γ→0 asε→0. ByTheorem 3.1,EPis type I LP well-posed.
The following example illustrates that the compactness condition inTheorem 3.1is es-sential.
Example 3.3. Let X Y R, K 0, ∞, D 0, ∞, and let g andf be the same as in
Example 3.2. Then, it is easy to compute thatS 0, ∞,Γ 0, ∞,M1ε 0, ∞, and eM1ε,Γ→ 0 asε→ 0. Letxn nforn 1,2, . . . .Then,{xn}is an approximating solution sequence forEP, which has no convergent subsequence. This implies thatEPis not type I LP well-posed.
Furi and Vignoli8characterized well-posedness of the optimization problemdefined in a complete metric spaceS, d1by the use of the Kuratowski measure of noncompactness of a subsetAofXdefined as
μA inf
ε >0 :A⊆
n
i1
Ai, diamAi< ε, i1,2, . . . , n , 3.7
where diamAiis the diameter ofAidefined by diamAisup{d1x1, x2:x1, x2∈Ai}.
Now, we give a Furi-Vignoli-type characterization for the various LP well-posed.
Theorem 3.4. Let X,·be a Banach space andΓ/∅. Assume thatf is upper semicontinuous on K×Kwith respect to the first argument. Then, (EP) is type I LP well-posed if and only if
lim ε→0μ
Proof. Let EP be type I LP well-posed. It is obvious that Γ is nonempty and compact. As proved inTheorem 3.1,eM1ε,Γ→0 asε→0. SinceΓis compact,μΓ 0 and the follow-ing relation holdssee, e.g.,7:
μM1ε≤2HM1ε,Γ μΓ 2HM1ε,Γ2eM1ε,Γ. 3.9
Therefore,3.8holds.
In order to prove the converse, suppose that 3.8holds. We first show that M1ε is nonempty and closed for anyε > 0. In fact, the nonemptiness ofM1εfollows from the fact thatΓ/∅. Let{xn} ⊂M1εwithxn→x0. Then
dS
xn
≤ε, 3.10
fxn, y
ε≥0, ∀y∈S. 3.11
It follows from3.10that
dS
x0
≤ε. 3.12
By the upper semicontinuity of f with respect to the first argument and 3.11, we have fx0, y ε ≥ 0 for ally ∈ S, which together with3.12 yieldsx0 ∈ M1ε, and so M1ε
is closed. Now we prove thatΓis nonempty and compact. Observe thatΓ ε>0M1ε.Since limε→0μM1ε 0, by the Kuratowski theorem37,38, page 318, we have
HM1ε,Γ−→0 asε−→0 3.13
and soΓis nonempty and compact.
Let{xn} be a type I LP approximating solution sequence forEP. Then, there exists a sequence{εn}withεn>0 andεn →0 such thatfxn, y εn ≥0 for ally∈SanddSxn≤εn. Thus,{xn} ⊂ M1εn. This fact together with3.13shows thatdΓxn → 0. ByLemma 2.11,
EPis type I LP well-posed. This completes the proof.
In the similar way to Theorems3.1and 3.4, we can prove the following Theorems 3.5
and3.6, respectively.
Theorem 3.5. LetX,·be a Banach space. Then, (EP) is generalized type I LP well-posed if and only if the solution setΓof (EP) is nonempty, compact, andeM2ε,Γ→0asε→0.
Theorem 3.6. Let X,·be a Banach space andΓ/∅. Assume thatf is upper semicontinuous on K×Kwith respect to the first argument. Then, (EP) is generalized type I LP well-posed if and only if
limε→0μM2ε 0.
In the following we consider a real-valued functionc ct, sdefined fors, t ≥ 0 suffi -ciently small, such that
ct, s≥0, ∀t, s, c0,0 0, sn−→0, tn≥0, c
tn, sn
−→0, imply tn −→0.
3.14
Theorem 3.7. Let (EP) be type II LP well-posed. Then there exists a functioncsatisfying3.14such that
hx≥cdΓx, dSx
, ∀x∈K, 3.15
wherehxis defined by2.11. Conversely, suppose thatΓis nonempty and compact, and3.15holds for somecsatisfying3.14. Then, (EP) is type II LP well-posed.
Similarly, we have the next theorem by applying33, Theorem 2.2andLemma 2.10.
Theorem 3.8. Let (EP) be generalized type II LP well-posed. Then there exists a functioncsatisfying
3.14such that
hx≥c
dΓx, dD
gx, ∀x∈K, 3.16
wherehxis defined by2.11. Conversely, suppose thatΓis nonempty and compact, and3.16holds for somecsatisfying3.14. Then, (EP) is generalized type II LP well-posed.
4. Sufficient conditions of LP well-posedness for (EP)
In this section, we derive several sufficient conditions for various types of LP well-posedness forEP.
Definition 4.1. LetZbe a topological space and letZ1⊂Zbe a nonempty subset. Suppose that
G : Z → R∪ { ∞} is an extended real-valued function. The functionG is said to be level-compact onZ1if, for anys∈R, the subset{z∈Z1:Gz≤s}is compact.
Proposition 4.2. Suppose thatf is upper semicontinuous onK×Kwith respect to the first argument andΓ/∅. Then, (EP) is type I LP well-posed if one of the following conditions holds:
ithere existsδ1>0such thatSδ1is compact, where
Sδ1
x∈K:dSx≤δ1
; 4.1
iithe functionhdefined by2.11is level-compact onK;
iiiXis a finite-dimensional normed space and
lim
x∈K,x→ ∞max
hx, dSx
∞; 4.2
ivthere existsδ1>0such thathis level-compact onSδ1defined by4.1.
Proof. iLet{xn}be a type I LP approximating solution sequence forEP. Then, there exists a sequence{εn}withεn>0 andεn→0 such that
dS
xn
≤εn, 4.3
fxn, y
From4.3, without loss of generality, we can assume that{xn} ⊂Sδ1. SinceSδ1is compact, there exists a subsequence{xnj}of{xn}andx0 ∈Sδ1such thatxnj →x0. This fact combined with4.3yieldsx0 ∈S. Furthermore, it follows from4.4thatfxnj, y ≥ −εnj for ally ∈S. By the upper semicontinuity off with respect to the first argument, we havefx0, y ≥ 0 for
ally∈Sand sox0∈Γ. Thus,EPis type I LP well-posed.
It is easy to see that conditioniiimplies conditioniv. Now, we show that condition
iiiimplies condition iv. SinceXis a finite-dimensional space and the function his lower semicontinuous onSδ1, we need only to prove that, for anys∈Randδ1>0, the setB{x∈
Sδ1: hx≤ s}is bounded, and thus Bis closed. Suppose by contradiction that there exist s ∈Rand{xn} ⊂Sδ1such thatx → ∞andhxn ≤s. It follows from{xn} ⊂Sδ1that
dSxn≤δ1and so
maxhxn
, dS
xn
≤maxs, δ1
, 4.5
which contradicts4.2.
Therefore, we need only to prove that if conditionivholds, thenEPis type I LP well-posed. Suppose that conditionivholds. From4.3, without loss of generality, we can assume that{xn} ⊂ Sδ1. By4.4, we can assume without loss of generality that{xn} ⊂ {x ∈ K : hx≤m}for somem >0. Sincehis level-compact onSδ1, the subset{x∈Sδ1:hx≤m} is compact. It follows that there exist a subsequence {xnj} of {xn} andx0 ∈ Sδ1 such that xnj → x0. This together with4.3yieldsx0 ∈S. Furthermore, by the upper semicontinuity of fwith respect to the first argument and4.4, we obtainx0∈Γ. This completes the proof.
Similarly, we can prove the next proposition.
Proposition 4.3. Assume thatfis upper semicontinuous onK×Kwith respect to the first argument andΓ/∅. Then, (EP) is generalized type I LP well-posed if one of the following conditions holds:
ithere existsδ1>0such thatS1δ1is compact, where
S1
δ1
x∈K:dD
gx≤δ1
; 4.6
iithe functionhdefined by2.11is level-compact onK;
iiiXis a finite-dimensional normed space and
lim
x∈K,x→ ∞max
hx, dD
gx ∞; 4.7
ivthere existsδ1>0such thathis level-compact onS1δ1defined by4.6.
Proposition 4.4. LetXbe a finite-dimensional space,f an upper semicontinuous function onK×K with respect to the first argument, andΓ/∅. Suppose that there existsy0∈Ssuch that
lim
x→ ∞− f
x, y0
∞. 4.8
Proof. Let{xn}be a type I LP approximating solution sequence for EP. Then, there exists a sequence{εn}withεn>0 andεn→0 such that
dS
xn
≤εn, 4.9
fxn, y
εn≥0, ∀y∈S. 4.10
By4.9, without loss of generality, we can assume that{xn} ⊂ Sδ1, where Sδ1is defined by4.1with someδ1>0. Now, we claim that{xn}is bounded. Indeed, if{xn}is unbounded,
without loss of generality, we can suppose that xn → ∞. By 4.8, we obtain limn→ ∞ −
fxn, y0 ∞,which contradicts4.10whennis sufficiently large. Therefore, we can assume without loss of generality thatxn →x0 ∈K. This fact together with4.9yieldsx0 ∈S. By the
upper semicontinuity of f with respect to the first argument and4.10, we getx0 ∈ Γ. This
completes the proof.
Example 4.5. LetXY R,K 0,2, andD 0,1. Let
gx 1
2x, fx, y yy−x, ∀x, y ∈X. 4.11
Then it is easy to see thatS 0,2and condition4.8inProposition 4.4is satisfied.
In view of the generalized type I LP well-posedness, we can similarly prove the following proposition.
Proposition 4.6. LetXbe a finite-dimensional space,f an upper semicontinuous function onK×K with respect to the first argument, andΓ/∅. If there existsy0 ∈Ssuch thatlimx→ ∞ −fx, y0
∞, then (EP) is generalized type I LP well-posed.
Now, we consider the case whenY is a normed space, Dis a closed and convex cone with nonempty interior intD. Lete∈intD. For anyδ≥0, denote
S2δ x∈K:gx∈D−δe. 4.12
Proposition 4.7. LetY be a normal space, letDbe a closed convex cone with nonempty interiorintD ande∈intD. Assume thatfis upper semicontinuous onK×Kwith respect to the first argument and
Γ/∅. If there existsδ1 > 0such that the functionhxdefined by2.11is level-compact onS2δ1,
then (EP) is generalized type I LP well-posed.
Proof. Let {xn} be a generalized type I LP approximating solution sequence for EP. Then, there exists a sequence{εn}withεn>0 andεn→0 such that
dD
gxn
≤εn, 4.13
fxn, y
εn≥0, ∀y∈S. 4.14
It follows from4.13that there exists{sn} ⊂Dsuch thatgxn−sn ≤2εnand so
gxn
whereBis a closed unit ball ofY. Now, we show that there existsM0>0 such that
B⊂D−M0e. 4.16
Suppose by contradiction that there existbn∈Band 0< Mn→ ∞such thatbn Mne /∈Dfor
n1,2, . . . .Thenbn Mne /∈intDand so
bn
Mn e /∈intD, n1,2, . . . . 4.17
Taking the limit in 4.17, we obtain e /∈intD. This gives a contradiction to the assumption. Thus, the combination of 4.15and 4.16 yieldsgxn−sn ∈ D−2M0εne and so gxn ∈
D−2M0εne.By4.12, we can assume without loss of generality that
xn∈S2
δ1
4.18
for 2M0εn→0 asn→ ∞. It follows from4.14that
hxn
≤εn, n1,2, . . . . 4.19
By4.18,4.19, and the level-compactness ofhonS2δ1, we know that there exist a subse-quence {xnj}of {xn} andx0 ∈ S2δ1such that xnj → x0. Taking the limit in 4.13 with n replaced bynj, we obtainx0 ∈ S. Furthermore, we get fx0, y ≥ 0 for ally ∈ S. Therefore,
x0∈Γ. This completes the proof.
5. Relations among various type of LP well-posedness for (EP)
In this section, we will investigate further relationships among the various types of LP well-posedness forEP.
By definition, it is easy to see that the following result holds.
Theorem 5.1. Assume that there existδ1,α >0andc >0such that
dSx≤cdαD
gx, ∀x∈S1
δ1
, 5.1
whereS1δ1is defined by4.6. If (EP) is type I (type II) LP well-posed, then (EP) is generalized type I (type II) LP well-posed.
Definition 5.2see6. LetWbe a topological space. A set-valued mappingF :W →2Xis said to be upper Hausdorffsemicontinuous atw∈W if, for anyε >0, there exists a neighborhood Uofwsuch thatFU⊂BFw, ε, where, forZ⊂Xandr >0,BZ, r {x∈X:dZx≤r}.
Clearly,S1δ1given by4.6is a set-valued mapping fromR toX.
Proof. We prove only type I case, the other case can be proved similarly. Let{xn}be a general-ized type I LP approximating solution sequence forEP. Then there exists a sequenceεn →0 such that
dD
gxn
≤εn, 5.2
fxn, y
εn≥0, ∀y∈S. 5.3
Note that S1δ1is upper Hausdorffsemicontinuous at 0. This fact together with5.2yields thatdSxn≤εn, which combining5.3implies that{xn}is type I LP approximating solution sequence. SinceEPis type I LP well-posed, there exists a subsequence{xnj}of{xn} converg-ing to some point ofΓ. Therefore,EPis generalized type I LP well-posed. This completes the proof.
LetY be a normed space and set
S3y x∈K:gx∈D y, ∀y∈Y. 5.4
Clearly,S3yis a set-valued mapping fromYtoX. Similar to the proof ofTheorem 5.3, we can prove the following result.
Theorem 5.4. Assume that the set-valued mappingS3ydefined by5.4is upper Hausdorff semi-continuous at0 ∈Y. If (EP) is type I (type II) LP well-posed, then (EP) is generalized type I (type II) LP well-posed.
Corollary 5.5. LetDbe a closed and convex cone with nonempty interiorintDande∈intD. Suppose that the set-valued mappingS2δdefined by4.12is upper Hausdorffsemicontinuous at0 ∈R . If (EP) is type I (type II) LP well-posed, then (EP) is generalized type I (type II) LP well-posed.
Acknowledgments
The work of the second author was supported by the National Natural Science Foundation of China10671135, the Specialized Research Fund for the Doctoral Program of Higher Ed-ucation20060610005, and the open fundPLN0703of State Key Laboratory of Oil and Gas Reservoir Geology and ExploitationSouthwest Petroleum University, and the work of the third author was supported by the ARC grant from the Australian Research Council. The au-thors are grateful to Professor Simeon Reich and the referees for their valuable comments and suggestions.
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