Stability Analysis for Higher Order Adjacent Derivative in Parametrized Vector Optimization

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Volume 2010, Article ID 510838,15pages doi:10.1155/2010/510838

Research Article

Stability Analysis for Higher-Order Adjacent

Derivative in Parametrized Vector Optimization

X. K. Sun and S. J. Li

College of Mathematics and Science, Chongqing University, Chongqing 400030, China

Correspondence should be addressed to X. K. Sun,[email protected]

Received 29 March 2010; Accepted 3 August 2010

Academic Editor: Jong Kim

Copyrightq2010 X. K. Sun and S. J. Li. 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.

By virtue of order adjacent derivative of set-valued maps, relationships between higher-order adjacent derivative of a set-valued map and its profile map are discussed. Some results concerning stability analysis are obtained in parametrized vector optimization.

1. Introduction

Research on stability and sensitivity analysis is not only theoretically interesting but also practically important in optimization theory. A number of useful results have been obtained in scalar optimization see1,2. Usually, by stability, we mean the qualitative analysis, which is the study of various continuity properties of the perturbation or marginal function or mapof a family of parametrized optimization problems. On the other hand, by sensitivity, we mean the quantitative analysis, which is the study of derivatives of the perturbation function.

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On the other hand, some interesting results have been proved for stability analysis in vector optimization problems. In8, Tanino studied some qualitative results concerning the behavior of the perturbation map in convex vector optimization. In9, Li investigated the continuity and the closedness of contingent derivative of the marginal map in multiobjective optimization. In10, Xiang and Yin investigated some continuity properties of the mapping which associates the set of eﬃcient solutions to the objective function by virtue of the additive weight method of vector optimization problems and the method of essential solutions.

To the best of our knowledge, there is no paper to deal with the stability of higher-order adjacent derivative for weak perturbation maps in vector optimization problems. Motivated by the work reported in3–9, in this paper, by higher-order adjacent derivative of set-valued maps, we first discuss some relationships between higher-order adjacent derivative of a set-valued map and its profile map. Then, by virtue of the relationships, we investigate the stability of higher-order adjacent derivative of the perturbation maps.

The rest of this paper is organized as follows. In Section 2, we recall some basic definitions. InSection 3, after recalling the concept of higher-order adjacent derivative of set-valued maps, we provide some relationships between the higher-order adjacent derivative of a set-valued map and its profile map. In Section 4, we discuss some stability results of higher-order adjacent derivative for perturbation maps in parametrized vector optimization.

2. Preliminaries

Throughout this paper, let X and Y be two finite dimensional spaces, and letK ⊆ Y be a pointed closed convex cone with a nonempty interior intK, whereK is said to be pointed if K ∩−K {0}. Let F : X ⇒ Y be a set-valued map. The domain and the graph of

F are defined by DomF {x ∈ X : Fx/∅} and GraphF {x, y ∈ X ×Y : y ∈

Fx, x ∈ DomF}, respectively. The so-called profile map F K : X ⇒ Y is defined by

FKx:Fx K,for allx∈DomF.

At first, let us recall some important definitions.

Definition 2.1see11. LetQbe a nonempty subset ofY. An elementsy ∈ Qis said to be a minimal pointresp. weakly minimal pointofQifQ−y∩−K {0}resp.,Q−y∩

−intK ∅. The set of all minimal pointsresp., weakly minimal pointofQis denoted by MinKQresp., WMinKQ.

Definition 2.2see12. A base forKis a nonempty convex subsetBofKwith 0/∈Bsuch that everyk∈K,k /0 has a unique representationkαb, whereb∈Bandα >0.

Definition 2.3see13. The weak domination property is said to hold for a subsetHofYif

H⊆WMin_KHintK∪ {0}.

Definition 2.4see14. LetFbe a set-valued map fromXtoY.

iFis said to be lower semicontinuousl.s.catx∈Xif for any generalized sequence

{xn} withxn → x andy ∈ Fx, there exists a generalized sequence{yn}with yn∈Fxnsuch thatyn → y.

iiF is said to be upper semicontinuous u.s.c at x ∈ X if for any neighborhood

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iiiFis said to be closed atx∈Xif for any generalized sequencexn, yn∈GraphF,

xn, yn → x, y, it yieldsx, y∈GraphF.

We say thatFis l.s.cresp., u.s.c, closedonXif it is l.s.cresp., u.s.c, closedat eachx∈X.

Fis said to be continuous onXif it is both l.s.c and u.s.c onX.

Definition 2.5see14. Fis said to be Lipschitz aroundx∈ X if there exist a real number

M >0 and a neighborhoodNxofxsuch that

Fx1⊆Fx2 Mx1−x2BY, ∀x1, x2∈Nx, 2.1

whereBY denotes the closed unit ball of the origin inY.

Definition 2.6 see 14. F is said to be uniformly compact near x ∈ X if there exists a neighborhoodNxofxsuch that_x_∈_N_xFxis a compact set.

3. Higher-Order Adjacent Derivatives of Set-Valued Maps

In this section, we recall the concept of higher-order adjacent derivative of set-valued maps and provide some basic properties which are necessary in the following section. Throughout this paper, letmbe an integer number andm >1.

Definition 3.1 see 15. Let x ∈ C ⊆ X and u1, . . . , um−1 be elements of X. The set T_Cbmx, u1, . . . , um−1is called themth-order adjacent set ofCatx, u1, . . . , um−1, if and only

if, for anyx ∈ T_Cbmx, u1, . . . , um−1, for any sequence{hn} ⊆ R\ {0}with hn → 0, there

exists a sequence{xn} ⊆Xwithxn → xsuch that

xhnu1h2nu2· · ·hnm−1um−1hmnxn∈C, ∀n. 3.1

Definition 3.2 see 15. Letx, y ∈ GraphF and ui, vi ∈ X ×Y, i 1,2, . . . , m−1.

Themth-order adjacent derivativeDbmFx, y, u1, v1, . . . , um−1, vm−1ofFatx, yfor vectors

u1, v1, . . . ,um−1, vm−1is the set-valued map fromXtoYdefined by

GraphDbmFx, y, u1, v1, . . . , u_m−1, v_m₋₁T_Graphbm_Fx, y, u1, v1, . . . , u_m−1, v_m₋₁. 3.2

Proposition 3.3. Letx, y ∈ GraphFandu_i, v_i ∈ X×Y,i 1,2, . . . , m−1. Then, for any x∈DomDbmFx, y, u1, v1, . . . , um−1, vm−1,

Dbm_F_{x, y, u1, v1, . . . , u}

m−1, vm−1x K⊆DbmFKx, y, u1, v1, . . . , um−1, vm−1x.

3.3

Proof. The proof follows on the lines of Proposition 2.1 in 3 by replacing contingent derivative bymth-order adjacent derivative.

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Example 3.4. LetXY RandKR, letF :X ⇒Ybe defined by

Fx

⎧ ⎨ ⎩

{0} ifx≤0,

−1, x3 _if_{x >}₀_. 3.4

Letx, y 0,0∈GraphFandu1, v1 u2, v2 1,0. For anyx >0, we have

Db2_F_{x, y, u1, v1}_x _{₀_}_, _Db2_F_K_{x, y, u1, v1}_x _R,

Db3_F_{x, y, u1, v1, u2, v2}_x _{₁_}_, _Db3_F_K_{x, y, u1, v1, u2, v2}_x _R. 3.5

Thus, for anyx >0, we have

Db2_F_K_{x, y, u1, v1}_x_/_⊆_Db2_F_{x, y, u1, v1}_x _K,

Db3_F_K_{x, y, u1, v1, u2, v2}_x_/_⊆_Db3_F_{x, y, u1, v1, u2, v2}_x _K. 3.6

Proposition 3.5. Letx, y∈GraphFandu_i, v_i∈ X×Y,i 1,2, . . . , m−1. Assume thatK has a compact base. Then, for anyx∈DomDbmFx, y, u1, v1, . . . , um−1, vm−1,

WMin_KDbmFKx, y, u1, v1, . . . , u_m−1, v_m₋₁x⊆DbmFx, y, u1, v1, . . . , u_m−1, v_m₋₁x.

3.7

whereK is a closed convex cone contained inintK∪ {0}.

Proof. If WMinKDbmFKx, y, u1, v1, . . . , um−1, vm−1x ∅,the inclusion holds trivially.

Thus, we suppose that WMinKDbmF Kx, y, u1, v1, . . . , um−1, vm−1x/∅. Let y0 ∈

WMin_KDbmFKx, y, u1, v1, . . . , u_m−1, v_m₋₁x.Then,

y0∈Dbm_F_K_{x, y, u1, v1, . . . , u}

m−1, vm−1x. 3.8

SinceK ⊆intK∪ {0},

WMinKDbm

FKx, y, u1, v1, . . . , um−1, vm−1x

⊆MinKDbm

FKx, y, u1, v1, . . . , um−1, vm−1x,

3.9

then it follows that

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From3.8and the definition ofmth-order adjacent derivative, we have that for any sequence

{hn} ⊆ R \ {0}withhn → 0, there exist sequences{xn, yn}withxn, yn → x, y0and

{kn} ⊆K such that

yhnv1· · ·hmn−1vm−1hmnyn−kn∈F

xhnu1· · ·hnm−1um−1hmnxn

. 3.11

SinceK is a closed convex cone contained inintK∪ {0},K has a compact base. It is clear thatB∩K is a compact base forK, whereBis a compact base forK. In this proposition, we assume thatBis a compact base ofK. Sincekn ∈K, there existαn >0 andbn ∈Bsuch that

knαnbn. SinceBis compact, we may assume without loss of generality thatbn → b∈B.

Now, we show thatα_n/hm_n → 0.Suppose thatα_n/hm_n 0.Then, for someε >0, we may assume, without loss of generality, thatαn/hmn ≥ε, for alln. Letkn εhmn/αnkn ∈K.

Then, we have

kn−kn∈K. 3.12

By3.11and3.12, we obtain that

yhnv1· · ·hmn−1vm−1hmnyn−kn∈F

K. 3.13

From3.13andkn/hmn ε/αnknεbn → εb /0,we have

y0−εb∈Dbm_F_K_{x, y, u1, v1, . . . , u}

m−1, vm−1x, 3.14

which contradicts3.10. Therefore,αn/hmn → 0 andyn−kn/hmn → y0.Thus, it follows from

3.11thaty0∈DbmFx, y, u1, v1, . . . , u_m−1, v_m₋₁x,and the proof is complete.

Remark 3.6. The inclusion of

WMinKDbmFKx, y, u1, v1, . . . , um−1, vm−1x⊆DbmFx, y, u1, v1, . . . , um−1, vm−1x

3.15

may not hold under the assumptions ofProposition 3.5. The following example explains the case where we only takem2,3.

Example 3.7. LetXRandY R2_{, let}_K_R2

andF:X⇒Y be defined by

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Suppose thatx, y 0,0,0 ∈ GraphF,u1, v1 u2, v2 1,0,0. Then, for any

x∈X,

Db2_F_{x, y, u1, v1}_x _{₀_,₀_}_,

Db3_F_{x, y, u1, v1, u2, v2}_x _{₁_,₁_}_,

Db2_F_K_{x, y, u1, v1}_x _{y1, y2}_∈_R2_:_y1_≥₀_{, y2}_≥₀_,

Db3_F_K_{x, y, u1, v1, u2, v2}_x _{y1, y2}_∈_R2_:_y1_≥₁_{, y2}_≥₁_.

3.17

Naturally, we have

WMinKDb2FKx, y, u1, v1x y1, y2∈R2:y1y20, y1≥0, y2≥0

,

WMinKDb3FKx, y, u1, v1, u2, v2x y1, y2∈R2:y1 ≥1, y21

∪y1, y2∈R2_:_y1₁_{, y2}_≥₁_.

3.18

Thus, for anyx∈X,

WMinKDb2FKx, y, u1, v1x/⊆Db2Fx, y, u1, v1x,

WMinKDb3FKx, y, u1, v1, u2, v2x/⊆Db3Fx, y, u1, v1, u2, v2x.

3.19

Proposition 3.8. Let x, y ∈ GraphF, andui, vi ∈ X ×Y,i 1,2, . . . , m−1, and let K has a compact base. Suppose thatPx:DbmFKx, y, u1, v1, . . . , um−1, vm−1xfulfills the weak domination property for any x ∈ DomDbmFx, y, u1, v1, . . . , um−1, vm−1. Then, for any x∈DomDbmFx, y, u1, v1, . . . , u_m−1, v_m₋₁,

WMinKDbm

FKx, y, u1, v1, . . . , um−1, vm−1x

WMin_KDbmFx, y, u1, v1, . . . , u_m−1, v_m₋₁x,

3.20

whereK is a closed convex cone contained inintK∪ {0}.

Proof. Lety0∈WMin_KDbmFKx, y, u1, v1, . . . , u_m−1, v_m₋₁x.Then,

y0∈Dbm_F_K_{x, y, u1, v1, . . . , u}

m−1, vm−1x. 3.21

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Suppose thaty0/∈WMinKDbmFx, y, u1, v1, . . . , um−1, vm−1x.Then, there existsy∈ Dbm_F_{x, y, u1, v1, . . . , u}

m−1, vm−1xsuch that

y0−y_∈_int_K. _3.22

Fromy∈DbmFx, y, u1, v1, . . . , um−1, vm−1xandProposition 3.3, we have

y_∈_Dbm_F_K_{x, y, u1, v1, . . . , u}

m−1, vm−1x. 3.23

So, by3.21,3.22, and3.23,y0/∈WMin_KDbmFKx, y, u1, v1, . . . , u_m−1, v_m₋₁x,which leads to a contradiction. Thus,y0∈WMinKDbmFx, y, u1, v1, . . . , um−1, vm−1x.

Conversely, lety0∈WMinKDbmFx, y, u1, v1, . . . , um−1, vm−1x. Then,

y0∈Dbm_F_{x, y, u1, v1, . . . , u}

m−1, vm−1x⊆Dbm

FKx, y, u1, v1, . . . , um−1, vm−1x.

3.24

Suppose thaty0/∈WMinKDbmFKx, y, u1, v1, . . . , um−1, vm−1x. Then, there existsy∈ Dbm_F_K_{x, y, u1, v1, . . . , u}

m−1, vm−1xsuch that

y0−y_k_∈_int_K. _3.25

SincePx fulfills the weak domination property for any x ∈ DomDbmFx, y, u1, v1, . . . ,

um−1, vm−1, there existsk∈intK∪ {0}such that

y₋_k_∈_WMin KDbm

FKx, y, u1, v1, . . . , um−1, vm−1x. 3.26

From3.25and3.26, we have

y0−k−k_∈_WMin KDbm

FKx, y, u1, v1, . . . , um−1, vm−1x. 3.27

It follows fromProposition 3.5and3.27that

y0−k−k_∈_Dbm_F_{x, y, u1, v1, . . . , u}

m−1, vm−1x, 3.28

which contradicts y0 ∈ WMinKDbmFx, y, u1, v1, . . . , um−1, vm−1x. Thus, y0 ∈

WMinKDbmFKx, y, u1, v1, . . . , um−1, vm−1x, and the proof is complete.

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Remark 3.9. FromExample 3.7, the equality of

WMin_KDbmFKx, y, u1, v1, . . . , u_m−1, v_m₋₁x

WMin_KDbmFx, y, u1, v1, . . . , u_m−1, v_m₋₁x

3.29

may still not hold under the assumptions ofProposition 3.8.

Proposition 3.10. Let x, y ∈ GraphF and u_i, v_i ∈ X × Y, i 1,2, . . . , m − 1. Suppose that F is Lipschitz at x. Then, DbmFx, y, u1, v1, . . . , um−1, vm−1 is continuous on

DomDbmFx, y, u1, v1, . . . , um−1, vm−1.

Proof. SinceFis Lipschitz atx, there exist a real numberM >0 and a neighborhoodNxof

xsuch that

Fx1⊆Fx2 Mx1−x2BY, ∀x1, x2∈Nx. 3.30

First, we prove thatDbmFx, y, u1, v1, . . . , u_m−1, v_m₋₁is l.s.c. atx∈DomDbmFx, y,

u1, v1, . . . , um−1, vm−1. Indeed, for anyy ∈ DbmFx, y, u1, v1, . . . , um−1, vm−1x. From the

definition ofmth-order adjacent derivative, we have that for any sequence{hn} ⊆ R\ {0} withhn → 0, there exists a sequence{xn,yn}withxn,yn → x, ysuch that

yhnv1· · ·hmn−1vm−1hmnyn∈F

. 3.31

Take anyx∈ X andxn → x. Obviously,xhnu1· · ·hnm−1um−1hmnxn,xhnu1· · · hm−1

n um−1hmnxn∈Nx,for anynsuﬃciently large. Therefore, by3.30, we have

Fxhnu1· · ·hnm−1um−1hmnxn

⊆Fxhnu1· · ·hnm−1um−1hmnxn

Mhm

nxn−xnBY.

3.32

So, with3.31, there exists−bn∈BY such that

yhnv1· · ·hnm−1vm−1hmnynMxn−xnbn∈F

.

3.33

We may assume, without loss of generality, thatb_n → b∈B_Y. Thus, by3.33,

yMx−xb∈Dbm_F_{x, y, u1, v1, . . . , u}

m−1, vm−1x. 3.34

It follows from3.34that for any sequence {xk}with xk → x,y ∈ DbmFx, y, u1, v1, . . . , um−1, vm−1x, there exists a sequence{yk}with

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Obviously,yk → y. Hence,DbmFx, y, u1, v1, . . . , um−1, vm−1is l.s.c. on DomDbmFx, y, u1, v1, . . . , um−1, vm−1.

We will prove thatDbmFx, y, u1, v1, . . . , um−1, vm−1is u.s.c. onx∈DomDbmFx, y, u1, v1, . . . , um−1, vm−1.In fact, for anyε >0, we consider the neighborhoodx ε/MBX of

x. Letx ∈ x ε/MBX andy ∈ DbmFx, y, u1, v1, . . . , um−1, vm−1x. From the definition

ofDbmFx, y, u1, v1, . . . , u_m−1, v_m₋₁x, we have that for any sequence{h_n} ⊆R\ {0}with

hn → 0, there exists a sequence{xn, yn}withxn, yn → x, ysuch that

yhnv1· · ·hnm−1vm−1hmnyn∈F

. 3.36

Take anyxn → x. Obviously,xhnu1· · ·hmn−1um−1hnmxn,xhnu1· · ·hnm−1um−1hmnxn∈ Nx,for anynsuﬃciently large. Therefore, by3.30, we have

Mhm

nxn−xnBY.

3.37

Similar to the proof of l.s.c., there existsb∈B_Y such that

yMx−xb∈Dbm_F_{x, y, u1, v1, . . . , u}

m−1, vm−1x. 3.38

Thus, y ∈ DbmFx, y, u1, v1, . . . , um−1, vm−1x εBY. Hence, Dbm_F_{x, y, u1, v1, . . . , u}

m−1, vm−1 is u.s.c. on DomDbmFx, y, u1, v1, . . . , um−1, vm−1,

and the proof is complete.

4. Continuity of Higher-Order Adjacent Derivative for

Weak Perturbation Map

In this section, we consider a family of parametrized vector optimization problems. LetFbe a set-valued map fromUtoY, whereUis the Banach space of perturbation parameter vectors,

Y is the objective space, andF is considered as the feasible set map in the objective space. In the optimization problem corresponding to each parameter valuedx, our aim is to find the set of weakly minimal points of the feasible objective valued setFx. Hence, we define another set-valued mapSfromUtoY by

Sx WMinKFx, for anyx∈U. 4.1

The set-valued map S is called the weak perturbation map. Throughout this section, we suppose thatK is a closed convex cone contained inintK∪ {0}.

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Remark 4.2. SinceSx⊆Fx, theK-minicompleteness ofFbySnearximplies that

Sx KFx K, for any x∈Nx. 4.2

Hence, ifFisK-minicomplete bySnearx, then, for anyy∈Sx

Dbm_F_K_{x, y, u1, v1, . . . , u}

m−1, vm−1DbmSKx, y, u1, v1, . . . , um−1, vm−1.

4.3

The following lemma palys a crucial role in this paper.

Lemma 4.3. Letx, y ∈ GraphSand ui, vi ∈ U×Y,i 1,2, . . . , m−1, and let K have a compact base. Suppose that the following conditions are satisfied:

iPx : DbmF Kx, y, u1, v1, . . . , um−1, vm−1x fulfills the weak domination property for anyx∈DomDbmSx, y, u1, v1, . . . , um−1, vm−1;

iiFis Lipschitz atx;

iiiFisK-minicomplete by Snearx.

Then, for anyx∈DomDbmSx, y, u1, v1, . . . , um−1, vm−1,

Dbm_S_{x, y, u1, v1, . . . , u}

m−1, vm−1x WMinKDbmFx, y, u1, v1, . . . , um−1, vm−1x.

4.4

Proof. We first prove that

WMin_KDbmFx, y, u1, v1, . . . , u_m−1, v_m₋₁x⊆DbmSx, y, u1, v1, . . . , u_m−1, v_m₋₁x.

4.5

In fact, fromProposition 3.5,Proposition 3.8, and theK-minicompleteness ofFbySnearx, we have

WMinKDbmFx, y, u1, v1, . . . , um−1, vm−1x

WMin_KDbmFKx, y, u1, v1, . . . , u_m−1, v_m₋₁x

WMinKDbm

SKx, y, u1, v1, . . . , um−1, vm−1x

⊆Dbm_S_{x, y, u1, v1, . . . , u}

m−1, vm−1x.

4.6

Thus, result4.5holds. Now, we prove that

Dbm_S_{x, y, u1, v1, . . . , u}

m−1, vm−1x⊆WMinKDbmFx, y, u1, v1, . . . , um−1, vm−1x.

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In fact, assume thaty∈DbmSx, y, u1, v1, . . . , um−1, vm−1x. Then, for any sequence{hn} ⊆ R\ {0}withhn → 0, there exists a sequence{xn, yn}withxn, yn → x, ysuch that

yhnv1· · ·hnm−1vm−1hmnyn∈S

.

4.8

Suppose that y /∈WMinKDbmFx, y, u1, v1, . . . , um−1, vm−1x. Then, there exists y ∈ Dbm_F_{x, y, u1, v1, . . . , u}

m−1, vm−1xsuch thaty−y∈intK.Thus, for the preceding sequence

{hn},there exists a sequence{xn,yn}withxn,yn → x, ysuch that

yhnv1· · ·hmn−1vm−1hmnyn∈F

. 4.9

Obviously,xhnu1· · ·hnm−1um−1hmnxn, xhnu1· · ·hmn−1um−1hmnxn ∈Nx,for any nsuﬃciently large. Therefore, byii, we have

Mhm

nxn−xnBY.

4.10

So, with4.9, there exists−b_n∈B_Y such that

yhnv1· · ·hnm−1vm−1hmnynMxn−xnbn∈F

.

4.11

Sinceyn−ynMxn−xnbn → y−yandy−y∈intK,yn−ynMxn−xnbn∈intK,

fornsuﬃciently large. Then, we have

yhnv1· · ·hnm−1vm−1hmnyn−

yhnv1· · ·hnm−1vm−1hmnynMxn−xnbn∈intK,

4.12

which contradicts 4.8. Then, y ∈ WMinKDbmFx, y, u1, v1, . . . , um−1, vm−1x. This

completes the proof.

The following example shows that the K-minicompleteness of F is essential in Lemma 4.3, where we only takem2,3.

Example 4.4Fis notK-minicomplete bySnearx. LetU R,Y R2_and_K _R2

, and let

F:U⇒Y be defined by

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Then, for anyx∈U,

Fx K Fx, Sx y1, y2∈R2_:_y1_≥₀_{, y2}₀_. _4.14

Suppose thatx, y 0,0,0∈GraphS,u1, v1 u2, v2 1,0,0. Then,Fis Lipschitz atx, and for anyx∈U,

Db2_F_K_{x, y, u1, v1}_x _Db3_F_K_{x, y, u1, v1, u2, v2}_x

y1, y2∈R2_:_y1_≥₀_{, y2}_≥₀

∪y1, y2∈R2 _:_y2_≥_y1

4.15

fulfills the weak domination property. We also have

Db2_F_{x, y, u1, v1}_x _Db3_F_{x, y, u1, v1, u2, v2}_x

y1, y2∈R2_:_y1_≥₀_{, y2}_≥₀

∪y1, y2∈R2_:_y2_≥_y1_.

4.16

On the other hand,

Db2_S_{x, y, u1, v1}_x _Db3_S_{x, y, u1, v1, u2, v2}_x _S_x_,

WMin_KDb2Fx, y, u1, v1x WMin_KDb3Fx, y, u1, v1, u2, v2x

y1, y2∈R2_:_y1_≥₀_{, y2} ₀

∪y1, y2∈R2_:_y2₋_{y1, y1}_<₀_.

4.17

Thus, for anyx∈X,

Db2_S_{x, y, u1, v1}_x_/_WMin

KDb2Fx, y, u1, v1x, Db3_S_{x, y, u1, v1, u2, v2}_x_/_WMin

KDb3Fx, y, u1, v1, u2, v2x.

4.18

Theorem 4.5. Let x, y ∈ GraphS and u_i, v_i ∈ U × Y,i 1,2, . . . , m − 1. Then, Dbm_S_{x, y, u1, v1, . . . , u}_m_{−1, v}_m₋₁_x_{is closed on}_Dom_Dbm_S_{x, y, u1, v1, . . . , u}_m_{−1, v}_m₋₁_.

Proof. From the definition ofDbmSx, y, u1, v1, . . . , um−1, vm−1x, we have that

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Since T_Graphbm_Sx, y, u1, v1, . . . ,u_m−1, v_m₋₁ is closed set, DbmSx, y, u1, v1, . . . , u_m−1, v_m₋₁ is closed on DomDbmSx, y, u1, v1, . . . , u_m−1, v_m₋₁, and the proof is complete.

Theorem 4.6. Letx, y∈GraphSandui, vi∈U×Y, i1,2, . . . , m−1. IfYis a compact space, thenDbmSx, y, u1, v1, . . . , u_m−1, v_m₋₁is u.s.c. onDomDbmSx, y, u1, v1, . . . , um−1, vm−1.

Proof. Since Y is a compact space, it follows from Corollary 9 of Chapter 3 in 14 and Theorem 4.5 that DbmSx, y, u1, v1, . . . , um−1, vm−1 is u.s.c. on

DomDbmSx, y, u1, v1, . . . , um−1, vm−1.Thus, the proof is complete.

Theorem 4.7. Let x ∈ DomDbmSx, y, u1, v1, . . . , u_m−1, v_m₋₁. Suppose that DbmFx, y, u1, v1, . . . , um−1, vm−1xis a compact set and the assumptions of Lemma 4.3 are satisfied. Then, Dbm_S_{x, y, u1, v1, . . . , u}

m−1, vm−1is u.s.c. atx.

Proof. It follows from Lemma 4.3 and Theorem 4.5 that WMin_KDbmFx, y,

u1, v1, . . . , um−1, vm−1 is closed. By Proposition 3.10, we have that DbmFx, y, u1, v1, . . . , um−1, vm−1 is u.s.c. at x. Since DbmFx, y, u1, v1, . . . , um−1, vm−1x is a compact set, it

follows from Theorem 8 of Chapter 3 in14that

Dbm_S_{x, y, u1, v1, . . . , u}

m−1, vm−1WMinKDbmFx, y, u1, v1, . . . , um−1, vm−1

WMinKDbmFx, y, u1, v1, . . . , um−1, vm−1

∩Dbm_F_{x, y, u1, v1, . . . , u}

m−1, vm−1

4.20

is u.s.c. atx, and the proof is complete.

Now, we give an example to illustrateTheorem 4.7, where we also takem2,3.

Example 4.8. LetU 0,1,Y R2_{, and}_K_R2

, and letF:U⇒Ybe defined by

Fx y1, y2∈R2_{: 0}_≤_y1_≤_x3_,₀_≤_y2_≤_x3_. _4.21

Then, for anyx∈U,

Sx y1, y2∈R2_{: 0}_≤_y1_≤_x3_{, y2}₀_∪_{y1, y2}_∈_R2_:_y1 ₀_,₀_≤_y2_≤_x3_. _4.22

Suppose thatx, y 0,0,0∈GraphS,x1/3,u1, v1 u2, v2 1,0,0and

K {y1, y2∈R2

:1/4y2≤y1 ≤4y2}. Obviously,Khas a compact base,Fis Lipschitz at

x,andFisK-minicomplete bySnearx. By direct calculating, for anyx∈U,

Db2_F_K_{x, y, u1, v1}_x _K,

Db3_F_K_{x, y, u1, v1, u2, v2}_x _{y1, y2}_∈_R2_{: 0}_≤_y1_≤₄_y2₁_,₀_≤_y2_≤₄_y1₁

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fulfill the weak domination property, which is a strong property for a set-valued map. We also have

Db2_F_{x, y, u1, v1}_x _{₀_,₀_}_,

Db2_S_{x, y, u1, v1}_x _{₀_,₀_}_,

Db3_F_{x, y, u1, v1, u2, v2}_x _{y1, y2}_∈_R2_{: 0}_≤_y1_≤₁_,₀_≤_y2_≤₁_,

Db3_S_{x, y, u1, v1, u2, v2}_x _{y1, y2}_∈_R2_{: 0}_≤_y1_≤₁_{, y2}₀

∪y1, y2∈R2_:_y1₀_,₀_≤_y2_≤₁_.

4.24

Thus, the conditions ofTheorem 4.7are satisfied. Obviously, bothDb2_S_{x, y, u1, v1}_x_and

Db3_S_{x, y, u1, v1, u2, v2}_x_{are u.s.c at}_x_.

Theorem 4.9. Let x ∈ DomDbmSx, y, u1, v1, . . . , u_m−1, v_m₋₁. Suppose that DbmFx, y, u1, v1, . . . , um−1, vm−1is uniformly compact nearxand the assumptions ofLemma 4.3are satisfied. Then,DbmSx, y, u1, v1, . . . , u_m−1, v_m₋₁is l.s.c. atx.

Proof. ByLemma 4.3, it suﬃces to prove that WMin_KDbmFx, y, u1, v1, . . . , u_m−1, v_m₋₁is l.s.c. atx. Letxn → xand

y∈WMinKDbmFx, y, u1, v1, . . . , um−1, vm−1x. 4.25

ByProposition 3.10, we have thatDbmFx, y, u1, v1, . . . , um−1, vm−1is l.s.c. atx. Then, there

exists a sequence{y_n}withy_n ∈ DbmFx, y, u1, v1, . . . , u_m−1,v_m₋₁x_nsuch thaty_n → y. SinceK ⊆intK∪ {0},

WMinKDbmFx, y, u1, v1, . . . , um−1, vm−1x⊆Min_KDbmFx, y, u1, v1, . . . , um−1, vm−1x.

4.26

Then, for any sequence {y_n} with y_n ∈ WMin_KDbmFx, y, u1, v1, . . . , u_m−1, v_m₋₁x_n, we have

y

n∈Min_KDbmFx, y, u1, v1, . . . , um−1, vm−1xn, 4.27

then it follows that

yn−yn∈K. 4.28

Since DbmFx, y, u1, v1, . . . , um−1, vm−1 is uniformly compact near x, we may assume,

without loss of generality, that y_n → y. It follows from the closedness of DbmFx, y,

u1, v1, . . . , um−1, vm−1that

y∈Dbm_F_{x, y, u1, v1, . . . , u}

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From4.28andKis closed, we havey−y∈K ⊆intK∪ {0}. Then, it follows from4.25and

4.29thatyy. Thus, WMin_KDbmFx, y, u1, v1, . . . , u_m−1, v_m₋₁is l.s.c. atx, and the proof is complete.

It is easy to see thatExample 4.8can also illustrateTheorem 4.9.

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions which helped to improve the paper. This research was partially supported by the National Natural Science Foundation of China Grant no. 10871216 and Chongqing University Postgraduates Science and Innovation FundGrant no. 201005B1A0010338.

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