New Stability Results for Ekeland s Variational Principles for Vector-Valued and Set-Valued Maps

doi:10.1006rjmaa.2000.7357, available online at http:rrwww.idealibrary.com on

New Stability Results for Ekeland’s

⑀

Variational

Principles for Vector-Valued and Set-Valued Maps

X. X. Huang

Department of Mathematics and Computer Science, Chongqing Normal Uni¨ersity, Chongqing 400047, China

E-mail: [email protected] Submitted by Boris S. Mordukho¨ich

Received April 29, 1999

In this paper, we discuss the stability of Ekeland’s variational principles for vector-valued and set-valued maps when the dominating cone is a closed pointed convex cone whose interior may be empty. We provide a new approach to the study of the stability of Ekeland’s variational principles for vector-valued and set-valued maps without scalarization. 䊚2001 Academic Press

Key Words: Mosco convergence; Painleve᎐Kuratowski convergence; Ekeland’s variational principle; extreme points; stability.

1. INTRODUCTION

It is well known that Ekeland’s variational principle plays an important

Ž w x

role in nonlinear analysis and optimization theory see, e.g., 7 and the

.

references therein . Generalizing Ekeland’s variational principles to vec-tor-valued functions and even to set-valued maps has attracted a lot of

Ž w x .

attention see, e.g., 3, 4, 6 and the references therein . Just because of the importance of the variational principles, it is necessary to study the stability of the principles. The stability of the scalar Ekeland variational

w x

principle was established by Attouch and Riahi in 5 . Taking advantage of the so-called ␰ function, by scalarization, we established stability results for Ekeland’s variational principles for vector-valued and set-valued maps respectively under the assumption that the ordering cone is with nonempty

Ž w x.

interior see 1, 2 . However, this type of scalarization becomes invalid for the stability study when the underlying dominating cone is with empty interior. To deal with this situation we shall, in this paper, attempt a new

Ž .

approach without scalarizing the functions concerned to the stability of

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Copyright䊚2001 by Academic Press All rights of reproduction in any form reserved.

the variational principles for vector-valued and set-valued maps when the vector-valued or set-valued map sequence converges to a vector-valued or set-valued map in the sense of Mosco or Painleve and Kuratowski. This

Ž

approach directly defines a Phelps cone in a product Banach space of two

. w x

Banach spaces . Similar to the approach in 5 , some equivalence between

Ž

Ekeland’s principles and the extreme points of some sets in the product

.

space with respect to this type of Phelps cone is established. Conse-quently, the stability of the variational principles can be tackled by study-ing the convergence properties of sets of the extreme points of a sequence of relevant closed sets.

Throughout this paper, we assume X and Y are both Banach spaces. The dominating cone C;Y is a nontrivial pointed closed convex cone

Žwhose interior may be empty .. C induces a partial order in Y Ži.e.,

. U

᭙y₁,y₂gY, y₁F_c y₂ iff y₂yy₁gC. We shall denote by C the

U U _{Ž . 4}

positive polar cone of C, i.e., C s lgY :l c G0, ᭙cgC. Let eg

4

C_0 .

In the following, we introduce some concepts which will be used in the sequel.

DEFINITION1.1. Let f: XªY be a vector-valued map. f is said to be

Ž .

bounded below on X if᭚y₀gY such that f x yy₀gCfor all xgX. f

Ž .

is said to be lower semicontinuous l.s.c. on X if ᭙ygY, the set

xgX:f xŽ .FC y4 is closed.

DEFINITION1.2. Let F:Xª2Y _{be a set-valued map. We say that F is}

Ž .

bounded below on X if ᭚y₀gY such that F x yy₀;C for all xgX.

Ž .

Take x₀gX. We say that F is upper semicontinuous u.s.c. at x₀ if for

Ž .

any open set U with F x₀ ;U there exists an open set V with x₀gV

Ž .

such that F x ;U,᭙xgV.

If F is u.s.c. at every xgX then we say F is u.s.c. on X.

DEFINITION 1.3. Let f_n: XªY be a sequence of vector-valued maps.

4

We say f_n is uniformly bounded below on X if ᭚y₀gY such that

Ž .

f x_n yy₀gC,᭙xgX,᭙ngN.

Similarly, suppose that F : Xª2Y _{is a sequence of set-valued maps;} n

4 Ž .

we say F_n is uniformly bounded below if᭚y₀gY such that F x_n yy₀; C,᭙xgX,᭙ngN.

2. EKELAND’S VARIATIONAL PRINCIPLES FOR VECTOR-VALUED AND SET-VALUED MAPS

In this section, we present Ekeland’s variational principles for vector-valued and set-vector-valued maps, respectively.

w x

THEOREM2.1 3 . Let f: XªY be l.s.c. and bounded below. Gi¨en⑀)0, if x satisfies_⑀

f x

Ž .

yf x

Ž

_⑀

.

q⑀efyC, ᭙xgX, then᭙␭)0,᭚xgX such that

Ž .i f xŽ .FC f xŽ ⑀.;

Ž .ii 5xyx5F␭;

Žiii. f xŽ .yf xŽ .q⑀r␭5xyx e5 fyC,᭙xgX_ 4x.

w x Y

THEOREM 2.2 4 . Let F: Xª2 be u.s.c. nonempty compact-¨alued and bounded below.

Ž . Gi¨en⑀)0, x_⑀gX, y_⑀gF x_⑀ satisfying

4

F x

Ž

.

yy l y

Ž

C_ 0

.

s⭋ and

Ž

⑀ ⑀

.

F X

Ž

.

yy q⑀e l y

Ž

C

.

s⭋,

Ž

⑀

.

Ž .

then᭙␭)0,᭚xgX and ygF x such that

Ž .i ŽF xŽ .yy.l yŽ C_ 4.0 s⭋; Ž .ii yFC y⑀;

Žiii. 5x_⑀yx5F␭;

Živ. ŽF xŽ .yyq⑀r␭5xyx e5 .l yŽ C.s⭋,᭙xgX_ 4x .

Ž . Ž .

Remark 2.1. Since F x is nonempty compact ᭙xgX and C is a

Ž .

pointed closed convex cone, we deduce that F x is externally stable

Ž᭙xgX.. It follows from Proposition 3.2 or Proposition 3.3 of 4 andŽ . w x

w x

Corollary 4.1 of 4 that Theorem 2.2 holds.

3. STABILITY RESULTS

First, we recall the concepts of the convergence of set sequences. w x

DEFINITION 3.1 5 . Let X be a normed space. A sequence of sets

M

D_n;X:ngN4 is said to be Mosco convergent to D;X Ži.e., D_nªD.

if

wylim supDn;D; lim infDn nª⬁ nª⬁

with

4

lim infD_ns xslim_nªq⬁ x_n: x_ngD_n,᭙ngN

nª⬁

wy lim supD_ns

xswylim_kªq⬁ x_n :x_n gD_n ,᭙kgN,

k k k

nª⬁

4

where xswylim_kªq⬁ x_n stands for the weak convergence of x_n

k k

to x.

Ž . Ž

We say that D_n Painleve᎐Kuratowski P.K. converges to D i.e.,

P.K.6

.

D_n D if

lim supDn;D;lim infDn,

nª⬁ nª⬁

4

where lim sup_nª⬁D_ns xslim_kªq⬁ x_n gD_n,᭙kgN, n_k is a

subse-k k

4

quence of N .

Let f:XªY be a vector-valued map. Define the epigraph of f as epi f

Ž

.

s

Ž

x,y

.

:ygf x

Ž .

qC, xgX

4

.

Analogously, suppose that F: Xª2Y _{is a set-valued map. We define the}

epigraph of F as

epi F

Ž

.

s

Ž

x,y

.

: ygF x

Ž .

qC, xgX

4

.

DEFINITION 3.2. Let fn: XªY,᭙ngN be a sequence of vector-val-ued maps and f: XªY be a vector-valued map. We say that f_n Mosco

M P.K.6 M

ŽP.K.. converges to f Ži.e., fnªf. Žfn f. if epi fŽ n.ªepi fŽ .

P.K.6

Žepi fŽ _n. epi fŽ ...

Ž .

Similarly, we can define a sequence of set-valued maps F_n Mosco P.K.

M P.K.6 M

Ž . Ž . Ž .

converges to a set-valued map F i.e., FnªF Fn F if epi Fn ª P.K.6

Ž . Ž Ž . Ž ..

epi F epi Fn epi F .

IfC1 is a cone in a linear space Z, A;Z, we shall denote by ext AC₁ the

Ž .

set of efficient maximal points of A with respect to C1, i.e., zgext AC₁

Ž . 4

iff Al zqC1 s z.

DEFINITION 3.3. Let f:XªY be a vector-valued map, ␭)0, ⑀)0.

Ž . Ž . 5 5

We define ⑀r␭yextfs x:f x yf x q⑀r␭ xyx efyC, ᭙xg

44

X_ x .

Similarly, if F:Xª2Y is a set-valued map, we define ⑀r␭yextFs

x:᭚ygext_yCF xŽ .such thatŽF xŽ .yyq⑀r␭5xyx e5 .l yCs⭋,᭙x 44

gX_ x .

PROPOSITION 3.1. Let f:XªY be a ¨ector-¨alued map, which is bounded below.Then᭙⑀)0,᭚x_⑀gX such that

f x

Ž .

yf x

Ž

_⑀

.

q⑀efyC, ᭙xgX.

w x The proof of Proposition 3.1 is similar to that of Proposition 2.1 in 1 ;

w x for details, see 1 .

PROPOSITION 3.2. Let F: Xª2Y _{be a nonempty compact-¨alued map,}

Ž .

which is bounded below.Then᭙⑀)0,᭚x_⑀gX, y_⑀gext_yCF x_⑀ such that F X

Ž

.

yy q⑀e l yCs⭋.

Ž

⑀

.

w x Proposition 3.2 is an immediate consequence of Theorem 2.1 in 6 . Let

5 5

Ks

Ž

x,y

.

gX=Y:yyq⑀r␭ x eg yC

4

, ⑀)0, ␭)0. It is easy to check that K is a nonempty closed pointed convex cone. Note that this Phelps cone is directly defined in the product space X=Y,

w x

which is different from the Phelps cone in 1, 2, 5 , where it is defined in X=R1_.

PROPOSITION 3.3. Let f:XªY be a ¨ector-¨alued map. Gi¨en ⑀)0,

Ž Ž .. Ž .

␭)0, then xg⑀r␭yext f iff x,f x gext_yKepi f . Proof. Necessity. We prove by contradiction.

Ž . Ž .

Suppose that᭚ x,y gepi f such that

4

x,y y x,f x g yK_ 0 .

Ž

.

Ž

Ž .

.

Then 5 5

4

yyf x

Ž .

q⑀r␭ xyx eg yC_ 0 , implying 5 5

4

f x

Ž .

yf x

Ž .

q⑀r␭ xyx eg yC_ 0 . If xsx, then this expression cannot hold.

If x/x, then this expression contradicts xg⑀r␭yext f. Sufficiency. Once again, we prove by contradiction.

4

Suppose that᭚xgX_ x such that

5 5 f x

Ž .

yf x

Ž .

q⑀r␭ xyx eg yC, implying

4

x,f x

Ž .

y x,f x

Ž .

g yK_ 0

Ž

.

Ž

.

Ž Ž .. Ž .

since x/x, contradicting x,f x gext_yKepi f .

PROPOSITION3.4. Let F: Xª2Y _{be a nonempty set-¨alued map.}

Ž . Ž .

Gi¨en ⑀)0, ␭)0, then xg⑀r␭yext F iff x,y gext_yKepi F , for

Ž .

Ž .

Proof. Necessity. Since xg⑀r␭yext F, we have a ygext_yCF x such that

5 5

4

F x

Ž .

yyq⑀r␭ xyx e l y

Ž

K

.

s⭋, ᭙xgX_ x .

Ž .

1

Ž

.

We show by contradiction that

x,y gext epi F

Ž

.

.

Ž

.

yK

Ž . Ž .

Suppose that᭚ x,y gepi F such that

4

x,y y x,y g yK_ 0 ,

Ž

.

Ž

.

implying 5 5

4

yyyq⑀r␭ xyx eg yC_ 0 . X _{Ž .} So᭚y gF x such that X _{5 5}

4

y yyq⑀r␭ xyx eg yC_ 0 .

Ž .

2 X _{4 Ž .}

If xsx, then y yyg yC_0 , contradicting ygext_yCF x .

Ž . Ž .

If x/x, then 2 contradicts 1 .

Sufficiency. Once again, we prove by contradiction.

4 Ž .

Suppose that᭚xgX_ x and ygF x such that

5 5 yyyq⑀r␭ xyx eg yC implying

4

x,y y x,y g yK_ 0

Ž

.

Ž

.

Ž . Ž .

since x/x, contradicting x,y gext_yKepi F . The proof is complete.

LEMMA 3.1. Let f:XªY be a ¨ector-¨alued map. If ᭙l1gCU,

Ž . 1 Ž . Ž .

l f1 : XªR is l.s.c. this implies that f is l.s.c. , then epi f is nonempty and closed.

Ž . Ž . Ž . Ž .

Proof. Let xn,yn gepi f with xn,yn ª x,y . Then

ynyf x

Ž

n

.

gC, implying

Hence

l1

Ž .

y s lim l1

Ž

yn

.

G lim infl1

Ž

f x

Ž

n

.

.

Gl1

Ž

f x

Ž .

.

«yyf x

Ž .

gC,

nªq⬁ nª⬁

Ž . Ž .

i.e., x,y gepi f .

LEMMA3.2. Let F: Xª2Y be an u.s.c.nonempty compact-¨alued map.

Ž .

Then epi F is nonempty and closed.

Ž . Ž . Ž . Ž .

Proof. Suppose that x_n,y_n gepi F and x_n,y_n ª x,y . Then y_ng

Ž . X Ž . X

F xn qC. Thus᭚yngF xn and cngCsuch that ynsynqcn. As F is

X 4 4

compact-valued and u.s.c. at x, we obtain a sequence yn_k of yn and

U _{Ž .} X U X U

y gF x such that ynkªy . Consequently,cnksynkyynkªyyy gC

Ž . Ž . Ž .

since C is closed. Therefore, ygF x qC, i.e., x,y gepi F . The proof is complete.

w x

LEMMA 3.3 3 . If C is a pointed closed con_¨ex cone, then for any

4 U Ž .

egC_0 there exists l₁gC such that l e₁ )0. It is easy to check that the following lemma holds.

Ž . Ž . Ž .5 5 4

LEMMA 3.4. yK; x,y gX=Y:l x,y q⑀r␭l e₁ x F0 , where

Ž . Ž . U Ž .

l x,y sl y and l_{1 1}gC is such that l e₁ )0.

Ž

THEOREM3.1. Let X and Y be reflexi¨e Banach spaces hence, X=Y is 5Ž .5 5 5 5 54 Ž .

a reflexi¨e Banach space with norm x,y smax x , y ,᭙ x,y gX=

. U Ž .

Y . Let fn: XªY be uniformly bounded below, and ᭙l1gC , l f1 n is

Ž . U Ž .

l.s.c. ᭙ngN . Let f: XªY be such that ᭙l1gC , l f1 is l.s.c., and

M

f_nªf.Gi_¨en⑀)0, ␭)0, then

⑀r␭yext f; lim inf⑀r␭yext f_n.

nª⬁

M

Proof. Since fn is uniformly bounded below and fnªf, we deduce

Ž Ž .. Ž . Ž . Ž . Ž .

that for any x,f x gepi f , ᭚ x_n,y_n gepi f_n such that x_n,y_n ª

Žx,f xŽ .., hence y_nyf x_nŽ _n.gC and y_nªf xŽ .. So y_nyy₀gC. Letting

Ž .

nªq⬁, we have f x yy0gC, i.e., f is bounded below.

U _{Ž .}

In addition, ᭙l₁gC , l f₁ is l.s.c., and it follows that f is l.s.c. By Proposition 3.1 and Theorem 2.1, we conclude that ⑀r␭yext f/⭋.

Similarly, we have ⑀r␭yext f_n/⭋,᭙ngN.

Under the assumption of this theorem, by Lemma 3.1, we know that

Ž . Ž .

D_nsepi f_n ,Dsepi f are all nonempty closed subsets of X=Y such

M

that DnªD.

Ž .

Furthermore, it follows from Lemma 3.4 that yK; x,y gX=

Ž . 5 5 Ž . 4 U 4 Ž .

Y:l x,y q⑀r␭ x l e1 F0 , where ᭙l1gC _0 is such that l e1 )0

Ž . Ž .

and l x,y sl y1 . Besides, yK is also a pointed closed convex cone in X=Y.

Since f_n is uniformly bounded below, we know that ᭚y₀gY such that

Ž .

f x_n yy₀gC,᭙xgX,᭙ngN.

Ž . Ž . Ž .

For all x,y gDnsepi fn , yyf xn gC, hence yyy0gC, imply-ing l1

Ž .

y Gl1

Ž

y0

.

. Consequently, l x,

Ž

y

.

sl1

Ž .

y Gl1

Ž

y0

.

. So inf inf l x

Ž

,y

.

)y⬁. ngNŽx,y.gDn w x Ž

As a result, all the conditions of Theorem 3.5 in 5 are satisfied withC

Ž ..

replaced by ouryK,⑀ replaced by our⑀r␭l0,e . Applying Theorem 3.5 w x

in 5 , we obtain

ext_yKepi f

Ž

.

sextyKD; lim infextyKDns lim infextyKepi f

Ž

n

.

.

Ž .

3

nª⬁ nª⬁

Ž Ž .. Ž .

Finally, let xg⑀r␭yext f. Then x,f x gext_yKepi f .

Ž . Ž Ž .. Ž . Ž

By 3 ,᭚ xn,f xn gextyKepi fn hence xng⑀r␭yext fn by

Propo-.

sition 3.3 such that xnªx. The proof is complete.

Remark 3.1. Our assumption on C in Theorem 3.1 is weaker than the

w x w x

one in Theorem 3.1 in 1 while Theorem 3.1 in 1 only requires Y to be a normed space, which is weaker than the assumption onY in this Theorem 3.1. In addition, in this Theorem 3.1 our assumption on fn and f is

w x

stronger than the one in Theorem 3.1 in 1 , where we only require fn,f to be l.s.c.

THEOREM3.2. Let X,Y be reflexi_¨e Banach spaces. Let F : Xª2Y _be n

uniformly bounded below and u.s.c. nonempty compact-¨alued, F:Xª2Y M

be u.s.c. nonempty compact-¨alued, and FnªF.Gi¨en ␭)0, ⑀)0, then

⑀r␭yext F; lim inf⑀r␭yext Fn.

nª⬁ M

4

Proof. Since FnªF and Fn is uniformly bounded below, we deduce that F is bounded below. Applying Theorem 2.2 and Proposition 3.2, we know that

Ž . Ž . Ž Ž ..5 5 4

In addition, yK; x,y ;X=Y:l x,y q⑀r␭ l0,e x F0 is a pointed closed convex cone in X=Y, where l is as defined in the proof of Theorem 3.1.

Ž . Ž .

With the help of Lemma 3.2, we have that Dnsepi Fn ,Dsepi F are

M

nonempty closed subsets of X=Y such that DnªD.

4

Since Fn is uniformly bounded below, we deduce that inf inf l x,

Ž

y

.

)y⬁.

ngNŽx,y.gDn

w x Ž

Thus, all the conditions of Theorem 3.5 in 5 are satisfied with C

Ž ..

replaced by ouryK, ⑀ replaced by our cr␭l0,e . Applying Theorem 3.5 w x

in 5 , we obtain

ext_yKepi F

Ž

.

sextyKD; lim infextyKDnslim infextyKepi F

Ž

n

.

.

Ž .

4

nª⬁ nª⬁

Ž . Ž . Ž .

Let xg⑀r␭yext F. Then ᭚ygF x such that x,y gext_yKepi F .

Ž . Ž . Ž . Ž . Ž

By 4 , ᭚ x_n,y_n gext_yKepi F_n with y_ngF x_{n n} hence x_ng⑀r␭y

.

ext F_n by Proposition 3.4 such that x_nªx. The proof is complete.

Remark 3.2. Our assumption on C in Theorem 3.2 is weaker than the w x

one in Theorem 4.1 of 2 , whereC is assumed to have nonempty interior. But our assumption on Y in this Theorem 3.2 is stronger than the one in

w x

Theorem 4.1 in 2 , where we only require Y to be a Banach space. THEOREM3.3. Let X,Y be Banach spaces, fn :XªY be a sequence of ¨ector-¨alued functions, which is uniformly bounded below, and ᭙l₁gCU,

Ž .

᭙ngN, l f1 n is l.s.c. on X. Let f:XªY be a l.s.c. ¨ector-¨alued

P.K.6 function.Suppose that f_n f.

Let⑀)0, ␭)0.

Further assume that ᭙␳)0, there exists a compact subset K_␳;X=Y such that

Anl␳B;K␳,

where BsB₁=B is the unit ball of X₂ =Y, B is the unit ball of X,₁ B is₂ 5Ž .5 Ž5 5 5 5.

the unit ball of Y, X=Y is normed as x,y smax x , y , for any

Žx,y.gX=Y, and AnsŽxn,f xŽ n..: xng⑀r␭yext fn4. Then

⑀r␭yext f; lim inf⑀r␭yext fn.

nª⬁

P.K.6

4

Proof. Since fn is uniformly bounded below and fn f, we deduce

U _{Ž .}

Proposition 3.1 and Theorem 2.1, we know that ⑀r␭yext f/⭋. Similarly, we have ⑀r␭yext f_n/⭋,᭙ngN.

Ž . Ž .

Applying Lemma 3.1, we deduce that᭙ngN, Dnsepi fn , Dsepi f

P.K.6

are all nonempty closed subsets in X=Y. Furthermore, fn f means

P.K.6

Dn D.

Ž . Ž .

It follows from Lemma 3.4 that yK; x,y gX=Y:l x,y q

Ž .5 5 4 Ž . Ž . U Ž .

⑀r␭l e1 x F0 , where l x,y sl y1 and l1gC is such that l e1 )0. Note thatyK is also a pointed closed convex cone in X=Y. As shown in the proof of Theorem 3.1, we have

inf inf l x

Ž

,y

.

)y⬁.

ngNŽx,y.gDn

Ž . w x Ž

Up to now, we have verified a in Theorem 3.3 in 5 holds with X

.

replaced by our X=Y andC replaced by ouryK .

Ž . w x

Now we turn to checking that b in Theorem 3.3 in 5 also holds. In fact,᭙␳)0, by Proposition 3.3,

Anl␳BsextyKepi f

Ž

n

.

l␳BsextyKDnl␳B;K␳. w x Ž

Applying Theorem 3.3 in 5 with X replaced by our X=Y and C

.

replaced by ouryK , we have

ext_yKepi f

Ž

.

sextyKD; lim infextyKDnslim infextyKepi f

Ž

n

.

.

nª⬁ nª⬁

Ž Ž ..

Now᭙xg⑀r␭yextf, then x,f x gext_yKD by Proposition 3.3. By

Ž . Ž

the relation above, we obtain x_n,y_n gext_yKD_n hence x_ng⑀r␭yext f_n

. Ž . Ž Ž ..

by Proposition 3.3 such that xn,yn ª x,f x , implying xnªx. The proof is complete.

Remark 3.3. In Theorem 3.3, our assumption onC is weaker than the w x

one in Theorem 3.2 in 1 , where the additional condition that C is with nonempty interior is required. But our assumption

A_nl␳B;K_␳ is stronger than

⑀r␭yext fnl␳B1;K␳X,

where B1 is the unit ball of X and K␳X;X is a compact set. Besides, our assumption on fn and f in this theorem is stronger than the one in

w x

Theorem 3.2 in 1 , where f_n,f are assumed to be l.s.c. Similarly, we can prove

THEOREM 3.4. Let X,Y be Banach spaces, and F : Xª2Y _{be a} se-n

quence of u.s.c. nonempty compact ¨alued mappings, which is uniformly bounded below. Let F:Xª2Y _{be an u.s.c. nonempty compact ¨alued}

P.K.6 mapping. Suppose that F_n F.

Let⑀)0,␭)0.

Further assume that ᭙␳)0, there exists a compact subset K_␳;X=Y such that

A_nl␳B;K_␳,

where BsB₁=B is the unit ball of X₂ =Y, B is the unit ball of X,₁ B is₂ 5Ž .5 Ž5 5 5 5.

the unit ball of Y, X=Y is normed as x,y smax x , y , for any

Žx,y.gX=Y, and A_nsŽx_n,y_n.:x_ng⑀r␭yext F_n4, y_n is such that

Ž . Ž Ž . 5 5 . Ž .

yn gF xn and F x yyn q ⑀r␭ xyxn e l yC s ⭋, ᭙xg

44

X_ xn . Then

⑀r␭yext F; lim inf⑀r␭yext Fn.

nª⬁

Remark 3.4. In Theorem 3.4, our assumption onC is weaker than the w x

one in Theorem 4.2 in 2 , where C is additionally required to be with nonempty interior. But our assumption Anl␳B;K␳ is stronger than

⑀r␭yext Fnl␳B1;K␳X where B1 is the unit ball of X and K␳X;X is a compact set.

4. CONCLUSIONS

In this paper, we have presented a new approach to the stability of Ekeland’s variational principles for vector-valued and set-valued maps without scalarizing the maps concerned when the ordering cone may be

w x

with empty interior. Stability results similar to 1, 2 have been established under different assumptions.

ACKNOWLEDGMENT

This research was done under the supervision of Professor G. Y. Chen. The author thanks him for instruction.

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