Semicontinuity of solution mappings for a class of parametric generalized vector equilibrium problems

Research Article

Semicontinuity of solution mappings for a class of parametric generalized vector equilibrium problems

Jue Lu^a,b, Yu Han^a, Nan-Jing Huang^a,∗

aDepartment of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China.

bSchool of Mathematics, Physics and Information Science, Shaoxing University, Shaoxing, Zhejiang 312000, China.

Communicated by Y. J. Cho

Abstract

In this paper, we discuss the upper and lower semicontinuity of the strong efficient solution mapping, the weakly efficient solution mapping and the efficient solution mapping to a class of parametric generalized vector equilibrium problems by using scalarization methods and a new density result. 2016 All rightsc reserved.

Keywords: Parametric generalized vector equilibrium problem, solution mapping, lower semicontinuity, upper semicontinuity.

2010 MSC: 49J40, 90C29.

1. Introduction

Vector equilibrium problem, as a generalization of the equilibrium problem [7] and the vector variational inequality [16], plays a very important role in many fields such as mathematical physics, economics theory, operations research, management science, engineering design and others. The existence theory concerned with solutions for the vector variational inequalities and the vector equilibrium problems has been extensively studied by many authors under quite different conditions (see, for example, [4, 5, 8, 12, 14, 15, 17, 18, 26, 28, 30, 32, 35] and the references therein).

On the other hand, the stability analysis in connection with the solution mappings to vector equilibrium problems is an important topic in vector optimization theory. Recently, the lower semicontinuity and the up- per semicontinuity of the solution mappings to parametric vector equilibrium problems have been intensively

∗Corresponding author

Email addresses: [email protected](Jue Lu),[email protected](Yu Han),[email protected] (Nan-Jing Huang)

Received 2016-06-03

studied in the literature, for instance, we refer the reader to [1–3, 9–11, 13, 19, 20, 22, 23, 27, 29, 31, 33, 34].

We note that, in order to get the semicontinuity of the solution mappings for the parametric vector equi- librium problems, the authors of [3, 9–11, 19, 20, 29, 31, 34] employed the monotonicity of mappings or the information about the solution mappings. It is worth mentioning that the monotonicity of mappings may yield that the set of solutions is a singleton and the assumptions involving information of solution mappings are not reasonable from the view of real problems. Therefore, it is important and interesting to discuss the semicontinuity of the solution mappings for a parametric generalized vector equilibrium problem (for short, PGVEP) under some new conditions.

The rest of the paper is organized as follows. Section 2 presents some necessary notations and lemmas. In Section 3, we obtain a new scalarization result and a new density result for a generalized vector equilibrium problem. Then we establish the lower semicontinuity of strong efficient solution mapping, weakly efficient solution mapping and efficient solution mapping to (PGVEP) by using the scalarization methods and the density result. In Section 4, we discuss the upper semicontinuity of strong efficient solution mapping and weakly efficient solution mapping to (PGVEP). Moreover, we establish the Hausdorff upper semicontinuity of efficient solution mapping to (PGVEP), which is a generalization of Theorem 5.4 of [24] from the finite dimensional space to the infinite dimensional space.

2. Preliminaries

Throughout this paper, unless otherwise specified, let Λ,W, ∆, X andY be five normed vector spaces.

Assume thatC ⊆Y is a closed, convex, pointed cone with nonempty interior,P ⊆∆ is a convex, pointed cone, andR+={x∈R:x≥0}. LetY^∗ be the topological dual space ofY and C^∗ be defined by

C^∗ ={f ∈Y^∗ :f(c)≥0, ∀c∈C}. Denote the quasi-interior ofC^∗ byC^#, i.e.,

C^#={f ∈Y^∗:f(c)>0, ∀c∈C\ {0}}. Let Dbe a nonempty subset ofY. The cone hull of Dis defined as

cone (D) ={td:t≥0, d∈D}.

Denote the closure of D by cl (D) and the interior of D by intD. A nonempty convex subset B of the convex coneC is called a base ofCifC= cone (B) and 0∈/cl (B). It is easy to see thatC^#6=∅if and only ifC has a base. Let ebe a fixed point in intC,

B^∗={f ∈C^∗ :f(e) = 1}, and

B^#= n

f ∈C^#:f(e) = 1 o

.

Then it is easy to see that B^∗ is a weak* compact base of C^∗,B^# is a base ofC^# and B^∗ = cl B^# with respect to the weak* topology.

LetK be a nonempty subset ofX andS :X⇒∆ andF :X×∆×X⇒Y be two set-valued mappings.

We consider the following generalized vector equilibrium problem consisting of findingx0 ∈K such that (GVEP) F(x0, u, y)∩(−Ω) =∅, ∀u∈S(x0),∀y∈K,

where Ω∪ {0} is a cone in Y.

Let W(F, S, K) denote the set of all weakly efficient solutions of (GVEP), i.e., W (F, S, K) ={x∈K :F(x, u, y)∩(−intC) =∅, ∀u∈S(x),∀y∈K}.

and E(F, S, K) denote the set of all efficient solutions of (GVEP), i.e.,

E(F, S, K) ={x∈K :F(x, u, y)∩(−C\ {0}) =∅, ∀u∈S(x),∀y∈K}. For anyf ∈C^∗, let Q(f) denote the set of allf-solutions of (GVEP), i.e.,

Q(f) ={x∈K :f(F(x, u, y))⊆R+,∀u∈S(x),∀y∈K}.

LetK be a nonempty subset ofX andS :X⇒∆ andF :X×∆×X⇒Y be two set-valued mappings.

Let F :X×∆×X×W ⇒ Y and K : Λ ⇒ X be two set-valued mappings. For any (α, λ) ∈W ×Λ, we consider the following parametric generalized vector equilibrium problem consisting of finding x0 ∈ K(λ) such that

(PGVEP) F(x₀, u, y, α)∩(−Ω) =∅, ∀u∈S(x₀),∀y∈K(λ), where Ω∪ {0} is a cone in Y.

For any (α, λ)∈W ×Λ, letM(α, λ) denote the set of all strong efficient solutions of (PGVEP), i.e., M(α, λ) ={x∈K(λ) :F(x, u, y, α)⊆C,∀u∈S(x),∀y∈K(λ)},

and W(α, λ) denote the set of all weakly efficient solutions of (PGVEP), i.e.,

W(α, λ) ={x∈K(λ) :F(x, u, y, α)∩(−intC) =∅,∀u∈S(x),∀y∈K(λ)}.

For anyf ∈C^∗ and (α, λ)∈W ×Λ, letS_f(α, λ) denote the set of all f-solutions of (PGVEP), i.e., S_f(α, λ) ={x∈K(λ) :f(F(x, u, y, α))⊆R+,∀u∈S(x),∀y ∈K(λ)}.

Definition 2.1. A set-valued mapping Φ : ∆⇒Y is said to be P-C-increasing, if for any u₁, u₂∈∆ with u1−u2∈P, one has

Φ (u1)⊆Φ (u2) +C.

Remark 2.2. The special case is as follows: a functionf :R→Ris said to be R+-R+-increasing, if for any u1, u2 ∈Rwith u1 ≥u2, one hasf(u1)≥f(u2).

Definition 2.3. LetD be a nonempty convex subset ofX. A set-valued mapping Φ :D⇒Y is said to be (i) C-concave, if for anyx₁, x₂ ∈D andt∈[0,1], one has

Φ (tx1+ (1−t)x2)⊆tΦ (x1) + (1−t) Φ (x2) +C;

(ii) strictlyC-concave, if for anyx₁, x₂ ∈D withx₁ 6=x₂ and, for anyt∈]0,1[, one has Φ (tx₁+ (1−t)x₂)⊆tΦ (x₁) + (1−t) Φ (x₂) + intC;

(iii) C-convexlike, if for anyx1, x2∈Dand, for any t∈[0,1], there existsx3∈Dsuch that tΦ (x1) + (1−t) Φ (x2)⊆Φ (x3) +C.

Now, we give the following example to illustrate that strictly C-concavity is easy to be verified.

Example 2.4. Let Y = R², C = R²+ =

(x₁, x₂)∈R²:x₁≥0, x₂ ≥0 , X = R and D = [−1,1]. We denote byB_Y the closed unit ball in Y. Let a set-valued mapping Φ :D⇒Y be defined as follows

Φ (x) = −x²,2 cosx +B_Y. Then it is easy to check that Φ is strictly C-concave.

Definition 2.5. A set-valued mapping G:T ⇒T₁ is said to be

(i) Hausdorff upper semicontinuous (H-u.s.c.) atu₀ ∈T, if for any neighborhoodV of 0∈T₁, there exists a neighborhoodU(u₀) ofu₀ such that for every u∈U(u₀),G(u)⊆G(u₀) +V;

(ii) upper semicontinuous (u.s.c.) atu₀ ∈T, if for any neighborhoodV of G(u₀), there exists a neighbor- hood U(u₀) of u₀ such that for every u∈U(u₀),G(u)⊆V;

(iii) lower semicontinuous (l.s.c.) at u₀ ∈ T, if for any x ∈ G(u₀) and any neighborhood V of x, there exists a neighborhoodU(u0) ofu0 such that for every u∈U(u0),G(u)∩V 6=∅.

We say that G is H-u.s.c., u.s.c. and l.s.c. on T if it is H-u.s.c., u.s.c. and l.s.c. at each point u ∈ T, respectively. We say thatGis continuous onT if it is both u.s.c. and l.s.c. on T.

Lemma 2.6 ([6]). A set-valued mapping Φ : T ⇒ T₁ is l.s.c. at u₀ ∈ T if and only if for any sequence {u_n} ⊆T with un→u0 and for any x0 ∈Φ (u0), there exists xn∈Φ (un) such that xn→x0.

Lemma 2.7 ([21]). Let Φ :T ⇒ T1 be a set-valued mapping. For any given u0 ∈T, if Φ (u0) is compact, thenΦ is u.s.c. atu0∈T if and only if for any sequence{u_n} ⊆T with un→u0 and for any xn∈Φ (un), there exist x₀ ∈Φ (u₀) and a subsequence{x_nk} of {x_n} such that x_nk →x₀.

Lemma 2.8 ([25]). A set-valued mapping G:T ⇒T1 is l.s.c. on T if and only if, for anyA⊆T, one has [

u∈cl(A)

G(u)⊆cl [

u∈A

G(u)

! .

3. Lower semicontinuity

In this section, we establish the lower semicontinuity of strong efficient solution mapping, weakly efficient solution mapping and efficient solution mapping to (PGVEP).

Lemma 3.1. Let K be a nonempty compact convex subset of X. Assume that (i) S(·) is l.s.c. and P-concave onK with nonempty compact values;

(ii) for any(x, y)∈K×K, F(x,·, y) is P-C-increasing;

(iii) for anyy∈K, F(·,·, y) is strictly C-concave onK×∆;

(iv) F(·,·,·) is continuous onK×∆×K with nonempty compact values.

ThenQ(·) is l.s.c. on C^∗\ {0_Y^∗}, where the topology onC^∗\ {0_Y^∗} is the weak* topology.

Proof. Suppose to the contrary that Q(·) is not l.s.c. at f₀ ∈C^∗\ {0_Y^∗}. Then there exist x₀ ∈ Q(f₀), a neighborhood W0 of 0∈X and a sequence{f_n}with

fn w^∗

−→f0, such that

(x0+W0)∩Q(fn) =∅, ∀n∈N. (3.1)

There are two cases to be considered.

Case 1. Q(f₀) is singleton. Let

xn∈Q(fn), ∀n∈N. (3.2)

It clear that x_n ∈K. SinceK is compact, without loss of generality, we can assume that x_n→ x¯∈K.

We claim that ¯x∈Q(f0). In fact, if not, then there existu0 ∈S(x0) and y0 ∈K such that f0(F(¯x, u0, y0))6⊂R+.

Then there exists z0∈F(¯x, u0, y0) such that

f0(z0)<0. (3.3)

Since S(·) is l.s.c. at x0, it follows from Lemma 2.6 that there exists un ∈S(xn) such that un → u0. Noting thatF(·,·, y₀) is l.s.c. at (x₀, u₀), by Lemma 2.6, there existsz_n∈F(x_n, u_n, y₀) such thatz_n→z₀. It follows from

fn w^∗

−→f0,

thatfn(zn)→f0(z0). By this together with (3.3), we havefn(zn)<0 fornlarge enough, which contradicts (3.2). Therefore, ¯x∈Q(f₀). It follows fromQ(f₀) is singleton that ¯x=x₀and sox_n→x₀. By this together with (3.2), we have

xn∈(x0+W0)∩Q(fn), fornlarge enough, which contradicts (3.1).

Case 2. Q(f0) is not singleton. Then there exists x⁰ ∈ Q(f0) such thatx⁰ 6=x0. Sincex⁰, x0 ∈Q(f0), we have

f₀ F x⁰, u, y

⊆R+, ∀u∈S x⁰

, ∀y∈K, (3.4)

and

f0(F(x0, u, y))⊆R+, ∀u∈S(x0), ∀y ∈K. (3.5) Since S(·) isP-concave onK, for anyt∈]0,1[, we have

S tx⁰+ (1−t)x₀

⊆tS x⁰

+ (1−t)S(x₀) +P.

For anyu_t∈S(tx⁰+ (1−t)x₀), there existu⁰ ∈S(x⁰),u₀∈S(x₀) and p₀ ∈P such that u_t=tu⁰+ (1−t)u₀+p₀.

By noting thatF(tx⁰+ (1−t)x₀,·, y) is P-C-increasing, we have F tx⁰+ (1−t)x₀, u_t, y

⊆F tx⁰+ (1−t)x₀, tu⁰+ (1−t)u₀, y

+C. (3.6)

Since F(·,·, y) is strictlyC-concave on K×∆, we have F tx⁰+ (1−t)x₀, tu⁰+ (1−t)u₀, y

⊆tF x⁰, u⁰, y

+ (1−t)F(x₀, u₀, y) + intC. (3.7) Let x(t) :=tx⁰+ (1−t)x₀. Then it is clear that x(t)∈K. It is easy to see that there existst₀ ∈]0,1[

such that x(t0) ∈x0+W0. It follows from (3.1) that x(t0)∈/ Q(fn). Then there exist un ∈S(x(t0)) and yn∈K such that

f_n(F(x(t₀), u_n, y_n))6⊂R+. Thus, there existsz_n∈F(x(t₀), u_n, y_n) such that

f_n(z_n)<0. (3.8)

Since S(x(t0)) andK are compact, without loss of generality, we can assume thatun→ u¯∈S(x(t0)) and yn → y0 ∈K. By Lemma 2.7, there exist z0 ∈F(x(t0),u, y¯ 0) and a subsequence {z_nk} of {z_n} such that z_nk → z₀. Without loss of generality, we can assume that z_n → z₀. It follows that f_n(z_n)→ f₀(z₀).

By (3.8), we have

f0(z0)≤0. (3.9)

On the other hand, from (3.4), (3.5), (3.6) and (3.7), we know that f0(z0)>0, which contradicts (3.9).

This completes the proof.

Lemma 3.2. Assume that, for each x∈K, F(x,·,·) is C-convexlike on S(x)×K. Then W(F, S, K) = [

f∈B^∗

Q(f).

Proof. For anyx∈S

f∈B^∗Q(f), there exists f₀∈B^∗ such that x∈Q(f₀). Thus,

f0(F(x, u, y))⊆R+, ∀u∈S(x), ∀y ∈K. (3.10) Suppose that x /∈W (F, S, K). Then there existu₀∈S(x) and y₀∈K such that

F(x, u₀, y₀)∩(−intC)6=∅,

and so there exists z0∈F(x, u0, y0) such that f0(z0) < 0, which contradicts (3.10). Therefore, we know thatx∈W(F, S, K). Next, we show that

W(F, S, K)⊆ [

f∈B^∗

Q(f).

Let x∈W(F, S, K). Then

F(x, u, y)∩(−intC) =∅, ∀u∈S(x), ∀y∈K.

It is easy to see that

(F(x, S(x), K) +C)∩(−intC) =∅.

For each x ∈ K, since F(x,·,·) is C-convexlike on S(x)×K, we can see that F(x, S(x), K) +C is a convex set. By the separation theorem of convex sets, there existsg∈Y^∗\ {0}such that

inf{g(z+c) :u∈S(x), y∈K, z ∈F(x, u, y), c∈C} ≥sup g c⁰

:c⁰ ∈ −C . It follows that g∈C^∗ and

g(F(x, u, y))⊆R+, ∀u∈S(x), ∀y∈K.

Since e∈intC and g∈C^∗\ {0}, it follows that g(e)>0. Letψ= _g(e)^g . We can see that ψ∈B^∗ and ψ(F(x, u, y))⊆R+, ∀u∈S(x), ∀y ∈K.

Thus,x∈Q(ψ) and so x∈S

f∈B^∗Q(f). This completes the proof.

Lemma 3.3. Let K be a nonempty compact convex subset of X. Assume that (i) S(·) is l.s.c. and P-concave onK with nonempty compact values;

(ii) for any(x, y)∈K×K, F(x,·, y) is P-C-increasing;

(iii) for anyy∈K, F(·,·, y) is strictly C-concave onK×∆;

(iv) F(·,·,·) is continuous onK×∆×K with nonempty compact values;

(v) for each x∈K, F(x,·,·) is C-convexlike on S(x)×K.

Then

[

f∈B^#

Q(f)⊆E(F, S, K)⊆W (F, S, K) = [

f∈B^∗

Q(f)⊆cl



 [

f∈B^#

Q(f)



.

Proof. It follows from Lemma 3.2 and the definitions that [

f∈B^#

Q(f)⊆E(F, S, K)⊆W (F, S, K) = [

f∈B^∗

Q(f).

By Lemma 3.1, we know thatQ(·) is l.s.c. onB^∗ = cl B^#

, by Lemma 2.8, one has

[

f∈B^∗

Q(f)⊆cl



 [

f∈B^#

Q(f)



,

and so

[

f∈B^#

Q(f)⊆E(Ω,Γ)⊆W (Ω,Γ) = [

f∈B^∗

Q(f)⊆cl



 [

f∈B^#

Q(f)



.

This completes the proof.

Theorem 3.4. Let (α0, λ0)∈W ×Λ. Assume that

(i) K(λ₀) is nonempty convex compact andK(·) is continuous at λ₀;

(ii) S(·) is continuous andP-concave on K(λ0) with nonempty compact values;

(iii) for any(x, y)∈K(λ0)×K(λ0), F(x,·, y, α₀) is P-C-increasing;

(iv) for anyy∈K(λ₀), F(·,·, y, α₀) is strictly C-concave on K(λ₀)×∆;

(v) F(·,·,·,·) is continuous onK(λ0)×∆×K(λ0)× {α₀} with nonempty compact values.

ThenM(·,·) is l.s.c. at (α₀, λ₀).

Proof. Suppose to the contrary thatM(·,·) is not l.s.c. at (α₀, λ0). Then there existx0∈M(α0, λ0) and a neighborhoodW₀ of 0∈X such that, for any neighborhoodU⁰×V⁰ of (α₀, λ₀), there exists (α⁰, λ⁰)∈U⁰×V⁰ satisfying

(x0+W0)∩M α⁰, λ⁰

=∅.

Hence, there exists a sequence {(α_n, λ_n)} with (α_n, λ_n)→(α₀, λ₀) such that

(x0+W0)∩M(αn, λn) =∅, ∀n∈N. (3.11) There are two cases to be considered.

Case 1. M(α₀, λ₀) is singleton. Let

xn∈M(αn, λn), ∀n∈N. (3.12)

It is clear that x_n ∈ K(λ_n) for all n ∈ N. By Lemma 2.7, there exist ¯x ∈ K(λ₀) and a subsequence {x_nk} of {x_n} such that xnk →x. Without loss of generality, we can assume that¯ xn →x. We claim that¯

¯

x ∈ M(α₀, λ₀). In fact, suppose to the contrary that ¯x /∈ M(α₀, λ₀). Then there exist u₀ ∈ S(¯x) and y₀∈K(λ₀) such that

F(¯x, u0, y0, α0)6⊂C.

It follows that there existsz₀∈F(¯x, u₀, y₀, α₀) such that

z0 ∈/ C. (3.13)

Since S(·) is l.s.c. at ¯x and K(·) is l.s.c. at λ0, it follows from Lemma 2.6 that there existsun∈S(xn) such that u_n → u₀ and there exists y_n ∈ K(λ_n) such that y_n → y₀. By noting that F(·,·,·,·) is l.s.c.

at (¯x, u₀, y₀, α₀), by Lemma 2.6, there exists z_n ∈ F(x_n, u_n, y_n, α_n) such that z_n → z₀. It follows from (3.13) that zn ∈/ C fornlarge enough, which contradicts (3.12). Therefore, ¯x∈M(α0, λ0). It follows from M(α0, λ0) is singleton that ¯x=x0 and so xn→x0. By this together with (3.12), we have

xn∈(x0+W0)∩M(αn, λn), fornlarge enough, which contradicts (3.11).

Case 2. M(α0, λ0) is not singleton. Then there exists x⁰ ∈ M(α0, λ0) such that x⁰ 6= x0. Since x⁰, x0 ∈ M(α0, λ0), one has

F x⁰, u, y, α₀

⊆C, ∀u∈S x⁰

, ∀y∈K(λ₀), (3.14)

and

F(x₀, u, y, α₀)⊆C, ∀u∈S(x₀), ∀y∈K(λ₀). (3.15) Since S(·) isP-concave onK(λ0), for anyt∈]0,1[, we have

S tx⁰+ (1−t)x0

⊆tS x⁰

+ (1−t)S(x0) +P.

For anyut∈S(tx⁰+ (1−t)x0), there existu⁰ ∈S(x⁰),u0∈S(x0) and p0 ∈P such that ut=tu⁰+ (1−t)u0+p0.

By noting thatF(tx⁰+ (1−t)x0,·, y, α₀) isP-C-increasing, we have F tx⁰+ (1−t)x0, ut, y, α0

⊆F tx⁰+ (1−t)x0, tu⁰+ (1−t)u0, y, α0

+C. (3.16)

Since F(·,·, y, α₀) is strictlyC-concave on K(λ0)×∆, we have F tx⁰+ (1−t)x0, tu⁰+ (1−t)u0, y, α0

⊆tF x⁰, u⁰, y, α0

+ (1−t)F(x0, u0, y, α0) + intC. (3.17) Let x(t) := tx⁰ + (1−t)x0. Then it is clear that x(t) ∈ K(λ0). For the above W0, there exists a neighborhood W1 of 0∈X such that

W₁+W₁ ⊆W₀.

Obviously, there existst0 ∈]0,1[ such that x(t0)∈x0+W1. Thus,

x(t0) +W1 ⊆x0+W1+W1⊆x0+W0. (3.18) Since x(t0) ∈ K(λ0), by Lemma 2.6, there exists x⁰_n ∈ K(λn) such that x⁰_n → x(t0) and so x⁰_n ∈ x(t₀) +W₁ for n large enough. By noting (3.11) and (3.18), we have x⁰_n ∈/ M(u_n, λ_n) and so there exist y⁰_n∈K(λ_n) and u⁰_n∈S(x⁰_n) such that

F x⁰_n, u⁰_n, y⁰_n, α_n 6⊂C.

Thus, there existsz⁰_n∈F(x⁰_n, u⁰_n, y⁰_n, α_n) satisfying

z⁰_n∈/ C. (3.19)

Since y⁰_n ∈ K(λ_n), it follows from Lemma 2.7 that there exist y⁰ ∈ K(λ₀) and a subsequence y_n⁰

k

of {y⁰_n} such that y⁰_nk → y⁰. Without loss of generality, we can assume that y⁰_n → y⁰. Since u⁰_n ∈ S(x⁰_n), it follows from Lemma 2.7 that there exist u⁰ ∈ S(x(t₀)) and a subsequence

u⁰_nk of {u⁰_n} such that u⁰_n

k → u⁰. Without loss of generality, we can assume that u⁰_n → u⁰. By noting the fact that F(·,·,·,·) is u.s.c. at (x(t0), u⁰, y⁰, α0), there exist z⁰ ∈ F(x(t0), u⁰, y⁰, α0) and a subsequence

z_n⁰_k of {z_n⁰} such that z_n⁰_k →z⁰. Without loss of generality, we can assume thatz⁰_n→z⁰. It follows from (3.19) that

z⁰ ∈/ intC. (3.20)

On the other hand, from (3.14), (3.15), (3.16) and (3.17), we know that z⁰ ∈ intC, which contradicts (3.20). This completes the proof.

Similar to the proof of Theorem 3.4, we can get the following lemma.

Lemma 3.5. Let f ∈C^∗\ {0} and(α0, λ0)∈W ×Λ. Assume that (i) K(λ₀) is nonempty convex compact andK(·) is continuous at λ₀;

(ii) S(·) is continuous andP-concave on K(λ₀) with nonempty compact values;

(iii) for any(x, y)∈K(λ0)×K(λ0), F(x,·, y, α₀) is P-C-increasing;

(iv) for anyy∈K(λ₀), F(·,·, y, α₀) is strictly C-concave on K(λ₀)×∆;

(v) F(·,·,·,·) is continuous onK(λ₀)×∆×K(λ₀)× {α₀} with nonempty compact values.

ThenSf(·,·) is l.s.c. at (α0, λ0).

Theorem 3.6. Let (α₀, λ₀)∈W ×Λ. Assume that

(i) K(λ0) is nonempty convex compact andK(·) is continuous at λ0;

(ii) S(·) is continuous andP-concave on K(λ₀) with nonempty compact values;

(iii) for any(x, y)∈K(λ₀)×K(λ₀), F(x,·, y, α₀) is P-C-increasing;

(iv) for anyy∈K(λ0), F(·,·, y, α₀) is strictly C-concave on K(λ0)×∆;

(v) F(·,·,·,·) is continuous onK(λ₀)×∆×K(λ₀)× {α₀} with nonempty compact values;

(vi) for anyx∈K(λ₀), F(x,·,·, α₀) isC-convexlike on S(x)×K(λ₀).

ThenW (·,·) is l.s.c. at (α0, λ0). Moreover, E(·,·) is l.s.c. at (α0, λ0).

Proof. It follows from Lemma 3.2 that

W(α₀, λ₀) = [

f∈B^∗

S_f(α₀, λ₀).

For anyx₀∈W(α₀, λ₀) and any neighborhood U of x₀, there existsf₀∈C^∗ such thatx₀ ∈S_f0(α₀, λ₀).

It follows from Lemma 3.5 thatS_f0(·,·) is l.s.c. at (α₀, λ₀) and so there exists a neighborhoodU(α₀)×U(λ₀) of (α0, λ0) such that

U∩S_f0(α, λ)6=∅, ∀(α, λ)∈U(α0)×U(λ0). It is easy to see that

S_f0(α, λ)⊆W (α, λ), and so

U ∩W (α, λ)6=∅, ∀(α, λ)∈U(α₀)×U(λ₀). Therefore, W(·,·) is l.s.c. at (α₀, λ₀). It follows from Lemma 3.3 that

[

f∈B^#

Sf(α0, λ0)⊆E(α0, λ0)⊆W(α0, λ0) = [

f∈B^∗

Sf(α0, λ0)⊆cl



 [

f∈B^#

Sf(α0, λ0)



.

For anyx∈E(α0, λ0) and any open neighborhood V of x, since

x∈E(α₀, λ₀)⊆cl



 [

f∈B^#

S_f(α₀, λ₀)



,

we have

V ∩



 [

f∈B^#

S_f(α₀, λ₀)



6=∅.

Then there exists f ∈B^# such that

V ∩S_f(α₀, λ₀)6=∅.

By Lemma 3.5,S_f(·,·) is l.s.c. at (α₀, λ0). Thus, there exists a neighborhoodU(α0)×U(λ0) of (α0, λ0) such that

V ∩Sf(α, λ)6=∅, ∀(α, λ)∈U(α0)×U(λ0). Since f ∈B^#, it is clear that

S_f(α, λ)⊆E(α, λ). Then,

V ∩E(α, λ)6=∅, ∀(α, λ)∈U(α₀)×U(λ₀). Therefore, E(·,·) is l.s.c. at (α₀, λ0). This completes the proof.

4. Upper semicontinuity

In this section, we establish the upper semicontinuity of strong efficient solution mapping and weakly efficient solution mapping to (PGVEP) and the Hausdorff upper semicontinuity of efficient solution mapping to (PGVEP).

Theorem 4.1. Let (α₀, λ₀)∈W×Λ. Assume that K(λ₀) is nonempty compact, K(·) is continuous atλ₀, S(·) is l.s.c. on K(λ0) and F(·,·,·,·) is l.s.c. on K(λ0)×∆×K(λ0)× {α₀}. Then M(·,·) is u.s.c. at (α₀, λ₀). Moreover, W(·,·) is u.s.c. at (α₀, λ₀).

Proof. Suppose to the contrary that M(·,·) is u.s.c. at (α₀, λ₀). Then there exist a neighborhood W₀ of M(α0, λ0) and a sequence{(α_n, λn)} with (αn, λn)→(α0, λ0) such that

M(α_n, λ_n)6⊂W₀. Then there exists

x_n∈M(α_n, λ_n), (4.1)

such that

x_n∈/ W₀, ∀n∈N. (4.2)

Since xn∈ K(λn), by Lemma 2.7, there exist x0 ∈K(λ0) and a subsequence {x_nk} of {x_n} such that xn_k →x0. Without loss of generality, we can assume that xn→x0.

We claim that x₀ ∈M(α₀, λ₀). In fact, suppose to the contrary that x₀ ∈/ M(α₀, λ₀). Then there exist u0 ∈S(x0) andy0 ∈K(λ0) such that

F(x0, u0, y0, α0)6⊂C.

Then, there exists z₀ ∈F(x₀, u₀, y₀, α₀) such that

z₀ ∈/ C. (4.3)

SinceS(·) is l.s.c. atx0 andK(·) is l.s.c. atλ0, it follows from Lemma 2.6 that there existsun∈S(xn) such that u_n → u₀ and there exists y_n ∈ K(λ_n) such that y_n → y₀. By noting that F(·,·,·,·) is l.s.c. at (x₀, u₀, y₀, α₀), by Lemma 2.6, there exists z_n ∈F(x_n, u_n, y_n, α_n) such thatz_n →z₀. It follows from (4.3) that zn ∈/ C for n large enough, which contradicts (4.1). Therefore, x0 ∈ M(α0, λ0). We can see that xn→x0 ∈W0, which contradicts (4.2).

By the similar arguments, we can prove that W(·,·) is u.s.c. at (α₀, λ₀). This completes the proof.

Lemma 4.2. Assume that K is a nonempty closed subset of X, S(·) is l.s.c. on K and for any y ∈ K, F(·,·, y) is l.s.c. on K×∆. Then Q(f) is closed.

Proof. Let{x_n} ⊆Q(f) withx_n→x₀. Then

f(F(x_n, u, y))⊆R+, ∀u∈S(x_n), ∀ ∈K. (4.4) It follows from the closedness ofK thatx0 ∈K. For any ¯u∈S(x0), sinceS(·) is l.s.c. atx0, by Lemma 2.6, there existsun∈S(xn) such thatun→u. For any¯ z∈F(x0,u, y), by noting that¯ F(·,·, y) is l.s.c. at (x₀,u), by Lemma 2.6, there exists¯ z_n ∈ F(x_n, u_n, y) such that z_n → z. By (4.4), we have f(z_n) ≥ 0. It follows fromf(zn)→f(z) thatf(z)≥0. Then

f(F(x0,u, y))¯ ⊆R+, ∀u¯∈S(x0), ∀y ∈K,

which means thatx₀ ∈Q(f). Therefore, Q(f) is closed. This completes the proof.

Lemma 4.3. Let(f0, α0, λ0)∈B^∗×W×Λ . Assume thatK(λ0)is nonempty compact,K(·) is continuous atλ0, S(·) is l.s.c. on K(λ0) and F(·,·,·,·) is l.s.c. onK(λ0)×∆×K(λ0)× {α₀}. Then S·(·,·) is u.s.c.

at(f₀, α₀, λ₀), where the topology onB^∗ is the weak* topology.

Proof. Suppose to the contrary thatS·(·,·) is u.s.c. at (f₀, α0, λ0). Then there exist a neighborhood W0 of S_f0(α₀, λ₀) and a sequence {(f_n, α_n, λ_n)} with (f_n, α_n, λ_n)→(f₀, α₀, λ₀) such that

S_fn(α_n, λ_n)6⊂W₀. Then there exists

xn∈S_fn(αn, λn), (4.5)

such that

xn∈/ W0, ∀n∈N. (4.6)

Since xn∈ K(λn), by Lemma 2.7, there exist x0 ∈K(λ0) and a subsequence {x_nk} of {x_n} such that x_nk →x₀. Without loss of generality, we can assume that x_n→x₀.

We claim thatx₀ ∈S_f0(α₀, λ₀). In fact, suppose to the contrary thatx₀ ∈/S_f0(α₀, λ₀). Then there exist u0 ∈S(x0) andy0 ∈K(λ0) such that

f₀(F(x₀, u₀, y₀, α₀))6⊂R+. Then, there exists z₀ ∈F(x₀, u₀, y₀, α₀) such that

f0(z0)<0. (4.7)

SinceS(·) is l.s.c. atx0 andK(·) is l.s.c. atλ0, it follows from Lemma 2.6 that there existsun∈S(xn) such that u_n → u₀ and there exists y_n ∈ K(λ_n) such that y_n → y₀. By noting that F(·,·,·,·) is l.s.c. at (x₀, u₀, y₀, α₀), by Lemma 2.6, there exists z_n ∈ F(x_n, u_n, y_n, α_n) such that z_n → z₀. By noting the fact that

f_n ^w

∗

−→f₀,

it is easy to see thatf_n(z_n)→f₀(z₀). By this together with (4.7), we havef_n(z_n)<0 forn large enough, which contradicts (4.5). Therefore, x0 ∈ Sf0(α0, λ0). We can see that xn → x0 ∈ W0, which contradicts (4.6). This completes the proof.

Theorem 4.4. Let (α0, λ0)∈W ×Λ. Assume that

(i) K(λ₀) is nonempty convex compact andK(·) is continuous at λ₀; (ii) S(·) is l.s.c. and P-concave onK(λ₀) with nonempty compact values;

(iii) for any(x, y)∈K(λ₀)×K(λ₀), F(x,·, y, α₀) is P-C-increasing;

(iv) for anyy∈K(λ0), F(·,·, y, α₀) is strictly C-concave on K(λ0)×∆;

(v) F(·,·,·,·) is continuous onK(λ0)×∆×K(λ0)× {α₀} with nonempty compact values;

(vi) for any(α, λ)∈W ×Λ and for any x∈K(λ), F(x,·,·, α) isC-convexlike on S(x)×K(λ).

Then,E(·,·) is H-u.s.c. at (α₀, λ₀).

Proof. Suppose to the contrary thatE(·,·) is not H-u.s.c. at (α₀, λ0). Then there exist a neighborhood W0

of 0∈X and a sequence{(α_n, λ_n)} with (α_n, λ_n)→(α₀, λ₀) such that E(αn, λn)6⊂E(α0, λ0) +W0, ∀n∈N.

Thus, there exists

xn∈E(αn, λn), (4.8)

satisfying

xn∈/ E(α0, λ0) +W0, ∀n∈N. (4.9) From Lemma 3.2, one has

W(αn, λn) = [

f∈B^∗

Sf(αn, λn).

It is clear that

E(α_n, λ_n)⊆W(α_n, λ_n), ∀n∈N. This together with (4.8) implies that

x_n∈ [

f∈B^∗

S_f(α_n, λ_n), ∀n∈N, and so there exists fn∈B^∗ such that

xn∈S_fn(αn, λn). (4.10)

Since B^∗ is weak* compact, without loss of generality, we can assume that f_n ^w

∗

−→f₀ ∈B^∗.

It follows from Lemma 4.2 thatSf0(α0, λ0) is closed. SinceSf0(α0, λ0)⊆K(λ0) andK(λ0) is compact, we can see that S_f0(α₀, λ₀) is compact. By Lemma 4.3, we can see that S·(·,·) is u.s.c. at (f₀, α₀, λ₀).

By noting (4.10) and Lemma 2.7, there exist a subsequence {x_nk} of {x_n} and x₀ ∈S_f0(α₀, λ₀) such that xnk →x0. It follows from Lemma 3.3 that

[

f∈B^#

Sf(α0, λ0)⊆E(α0, λ0)⊆W(α0, λ0) = [

f∈B^∗

Sf(α0, λ0)⊆cl



 [

f∈B^#

Sf(α0, λ0)



.

Thus, one has x0 ∈ [

f∈B^∗

Sf(α0, λ0)⊆cl



 [

f∈B^#

Sf(α0, λ0)



= cl (E(α0, λ0))⊆E(α0, λ0) +W0. This together withx_nk →x₀ shows that

xnk ∈E(α0, λ0) +W0, forklarge enough, which contradicts (4.9). This completes the proof.

Remark 4.5. Theorem 4.4 is a generalization of Theorem 5.4 of [24] from the finite dimensional space to the infinite dimensional space.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (11471230, 11671282) and the joint Foundation of the Ministry of Education of China and China Mobile Communication Corporation (MCM20150505).

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