J Syst Sci Complex (015) 8: 131 135 Lower Semicontinuity of the Efficient Solution Mapping in Semi-Infinite Vector Optimization GONG Xunhua DOI: 10.1007/s1144-015-3058-8 Received: 1 March 013 c The Editorial Office of JSSC & Springer-Verlag Berlin Heidelberg 015 Abstract This paper establishes some sufficient conditions for the lower semicontinuity of the efficient solution mapping for the semi-infinite vector optimization problem with perturbations of both the objective function and the constraint set in normed linear spaces. The constraint set is the set of weakly efficient solutions of vector equilibrium problem, and perturbed by the perturbation of the criterion mapping to the vector equilibrium problem. Keywords Efficient solution mapping, lower semicontinuity, semi-infinite vector optimization, vector equilibrium problem. 1 Introduction One important aspect of vector optimization problems is to study the stability of solution set of vector optimization problems with respect to perturbation of feasible solution sets and objective functions. A number of papers have been devoted to this subject. Naccache [1] and Tanino and Sawaragi [] proved the upper semicontinuity and lower semicontinuity of the efficient point multifunction in objective space and the efficient solution multifunction for the parametric multiobjective optimization problems in finite-dimensional spaces, respectively. Penot and Sterna-Karwat [3] generalized the results of [] to infinite-dimensional spaces. Using the so-called domination property and containment property, Bednarczuk [4,5] studied the Hausdorff upper semicontinuity, the K-Hausdorff upper semicontinuity and the lower (upper) semicontinuity of the efficient point multifunction for the parametrized vector optimization problems. Applying the approach of Bednarczuk [4,5] and introducing the new concepts of local containment property, K-local domination property, and uniformly local closedness of a multifunction around a given point, Chuong, Yao and Yen [6] further obtained some results on the lower semicontinuity GONG Xunhua Department of Mathematics, Nanchang University, Nanchang 330031, China. Email: xunhuagong@gmail.com. This research is partially supported by the National Natural Science Foundation of China under Grant Nos. 1106103, 110116, and 1147191. This paper was recommended for publication by Editor WANG Shouyang.
LOWER SEMICONTINUITY OF THE EFFICIENT SOLUTION MAPPING 1313 of the efficient point multifunction in Hausdorff topological vector spaces. Xiang and Zhou [7] derived necessary and sufficient conditions for lower semicontinuity and upper semicontinuity of the Pareto solution mapping of the multiobjective optimization with functional perturbation of objective function. Recently, several authors investigated the stability of solution set of parametric semi-infinite vector optimization problems. In parametric semi-infinite vector optimization problem, the objective function is a vector-valued function, and the constraint set is the solution set of some scalar or vector equilibrium problem and depends on the choice of the criterion mapping to the equilibrium problems. Todorov [8] studied the upper semicontinuity and lower semicontinuity of the efficient and weakly efficient solution mapping in the Berge sense and in the Kuratowski sense for the parametric linear semi-infinite vector optimization problem undergoing the continuous perturbation of the constraints and the linear perturbation of the objective function. Chuong, et al. [9], and Chuong and Yao [10] established sufficient conditions for the pseudo-lipschitz property of the Pareto solution mapping for parametric linear and convex semi-infinite vector optimization problem with linear and convex perturbation of the objective function and continuous perturbation of the criterion mapping to the scalar equilibrium problem. Chuong [11] derived sufficient conditions for the lower semicontinuity of the efficient solution mapping of quasiconvex semi-infinite vector optimization problem with perturbation of the objective functional and continuous perturbation of criterion mapping to the scalar equilibrium problem. Chuong, et al. [1] established necessary conditions for lower and upper semicontinuity of the Pareto solution mapping for semi-infinite vector optimization problems with perturbation of the objective functional and continuous perturbation of criterion mapping to the vector equilibrium problem. It is worth pointing that the results in the works [8 1] were obtained in finite-dimensional spaces. On the other hand, the lower semicontinuity of the constraint set mapping plays an important role in the study of lower semicontinuity of the Pareto solution mapping for semi-infinite vector optimization problems. In order to guarantee the lower semicontinuity of the constraint set mapping, in [9 1], the authors assumed that the criterion mapping to the scalar equilibrium problem or vector equilibrium problem satisfies the Slater condition, which is a relatively strict condition. Inspired and motivated by the work in [9 1], in this paper, we will present some sufficient conditions for the lower semicontinuity of the efficient solution mapping for the semi-infinite vector optimization problem with perturbations of both the objective function and the constraint set in infinite-dimentional normed linear spaces. The constraint set is the set of the weakly efficient solutions of vector equilibrium problem, and perturbed by the perturbation of the criterion mapping to the vector equilibrium problem. Our main methods are different from the corresponding ones in [9 1]. Without the Slater condition, we provide sufficient conditions for the lower semicontinuity of the constraint set mapping by scalarization method. The proof of the lower semicontinuity of efficient solution mapping of semi-infinite vector optimization problems with perturbation of the objective function and the constraint set is much simpler than those in [9 1]. The rest of the paper is organized as follows. In Section, we give some basic definitions
1314 GONG XUNHUA and lemma. In Section 3, we present some sufficient conditions for the lower semicontinuity and upper semicontinuity of the constraint set mapping, which will be used in Section 4. In Section 4, we discuss the lower semicontinuity of the efficient solution mapping of semi-infinite vector optimization problem with perturbations of both the objective function and the constraint set in normed linear spaces. Preliminaries Throughout this paper, let X, Y and Z be real normed linear spaces. Let Y be the topological dual space of Y, C be a closed, convex and pointed cone in Y with intc, and let K be a closed, convex and pointed cone in Z. Let C = {y Y : y (y) 0 for all y C} be the dual cone of C. Definition.1 Let A be a nonempty convex subset of X. A mapping ψ : A Y is called C-convex on A if, for every x 1,x A, t [0, 1], tψ(x 1 )+(1 t)ψ(x ) ψ(tx 1 +(1 t)x )+C. Definition. Let A be a nonempty convex subset of X. A mapping ψ : A Y is called C-strictly concave on A if, for every distinct x 1,x A, and t (0, 1), tψ(x 1 )+(1 t)ψ(x ) ψ(tx 1 +(1 t)x ) intc. Definition.3 (see [13]) Let A be a nonempty subset of X. A mapping ψ : A Y is called C-upper semicontinuous at x 0 A if, for any neighborhood V of 0 in Y,thereisa neighborhood U(x 0 )ofx 0 such that ψ(x) ψ(x 0 )+V C for all x U(x 0 ) A. Let A be a nonempty compact convex subset of X. The space C[A, Z] isthesetofall continuous vector-valued functions f : A Z. The norm of f C[A, Z] isdefinedasfollows: f =max f(x). x A It is obvious that C[A, Z] is a normed linear space. Let A be a nonempty compact convex subset of X. We define the space C 0 [A A, Y ]={g : A A Y : g satisfies the following conditions: (i) g(x, x) = 0 for all x A; (ii) for each y A, g(,y) g is C-upper continuous on A; (iii) for each y A, g(,y)isc-strictly concave on A; (iv) for each x A, g(x, )isc-convex on A; (v) g is bounded on A A}.
LOWER SEMICONTINUITY OF THE EFFICIENT SOLUTION MAPPING 1315 For any g 1,g C 0 [A A, Y ], we define ρ 0 (g 1,g )= sup g 1 (x, y) g (x, y). (x, y) A A It is clear that ρ 0 is a metric on C 0 [A A, Y ]. Consider parametric semi-infinite vector optimization problems or generalized parametric vector optimization problems under perturbations of both objective function and constraint set on the parameter space P 0 := C[A, Z] C 0 [A A, Y ] formulated as follows: For every (f,g) P 0, we have the semi-infinite vector optimization problem (SVOP) (f,g) : min K f(x) subject to x D(g), where K is a closed convex pointed cone in Z and D(g) ={x A : g(x, y) / intc for all y A}. (1) We call the set-valued mapping D : C 0 [A A, Y ] A the constraint set mapping of (SVOP). Remark.1 In [14], the set {x A : g(x, y) / intc for all y A} is called the set of weakly efficient solutions to the vector equilibrium problems. In [15], the mapping g : A A Y is called the criterion mapping to the vector equilibrium problem. Definition.4 Let (f,g) P 0. A vector x D(g) is said to be an efficient solution of (SVOP) (f,g) if {f(x)} = f(d(g)) (f(x) K), where K is a closed convex pointed cone in Z. For each (f,g) P 0, let S((f,g)) denote the set of efficient solutions of (SVOP) (f,g),thatis S((f,g)) = {x D(g) :{f(x)} = f(d(g)) (f(x) K)}. () We call the set-valued mapping S : P 0 A the efficient solution mapping of (SVOP). Let g : A A Y be a mapping. For each y C \{0}, weset V y (g) ={x A : y (g(x, y)) 0 for all y A}. (3) By Theorem.1 of [14], we have the following lemma. Lemma.1 Let g : A A Y be a mapping and intc. If for each x A, g(x, A)+C is a convex set, then D(g) = V y (g). y C \{0}
1316 GONG XUNHUA Definition.5 (see [16]) Let G be a set-valued map from a topological space W to another topological space Q. (i) We say that G is upper semicontinuous at x 0 W if, for any open set U G(x 0 ), there exists a neighborhood U(x 0 )ofx 0 such that G(x) U for all x U(x 0 ). We say that G is upper semicontinuous on W if it is upper semicontinuous at each x W. (ii) We say that G is lower semicontinuous at x 0 W if, for any y 0 G(x 0 )andany neighborhood U(y 0 )ofy 0, there exists a neighborhood U(x 0 )ofx 0 such that G(x) U(y 0 ) for all x U(x 0 ). We say that G is lower semicontinuous on W if it is lower semicontinuous at each x W. (iii) We say that G is closed if Graph(G) ={(x, y) :x W, y G(x)} is a closed set in W Q. If W and Q are metric spaces, by [16], we can see that G is lower semicontinuous at x 0 W if and only if, for any y 0 G(x 0 ) and for any sequence {x n } with x n x 0,thereexistsa sequence {y n } such that y n G(x n )andy n y 0. 3 The Continuity of D In this section, we show the lower semicontinuity, closedness, and upper semicontinuity of the constraint set mapping D of (SVOP), which will be used in Section 4. Definition 3.1 (see [17]) Let X be a Hausdorff topological vector space, let A X be a nonempty set. We call a set-valued mapping G : A X a KKM mapping, if for any finite set {x 1,x,,x n } A, n co{x 1,x,,x n } G(x i ), where co{x 1,x,,x n } denoted the convex hull of {x 1,x,,x n }. Lemma 3.1 (see [17]) Let X be a Hausdorff topological vector space. Let A be a nonempty convex subset of X, andletg : A A be a KKM map. If for each x A, G(x) is closed in X, and if there exists a point x 0 A such that G(x 0 ) is compact, then x A G(x). Lemma 3. Let A X be a nonempty compact convex set, and let g : A A Y be a mapping. Assume that g satisfies the following conditions: (i) g(x, x) C for all x A; (ii) for each x A, g(x, ) is C-convex on A; (iii) for each y A, g(,y) is C-upper semicontinuous on A. Then for each y C \{0}, V y (g) is a nonempty set, and D(g) is a nonempty compact set, where V y (g) is given in (3) and D(g) is given in (1). i=1
LOWER SEMICONTINUITY OF THE EFFICIENT SOLUTION MAPPING 1317 Proof For each y C \{0}, we define the set-valued mapping M : A A by M(y) ={x A : y (g(x, y)) 0}. By assumption, y M(y) for all y A. SoM(y). We claim that M is a KKM mapping. If M is not a KKM mapping, then there exist y 1,y,,y n A and x co{y 1,y,,y n } such that x/ n i=1 M(y i). That is there exist t i 0,i=1,,,n with n i=1 t i =1suchthat and x/ M(y i ),i=1,,,n.thus x = n t i y i, By assumption, g(x, ) is a C-convex mapping on A, we have i=1 y (g(x, y i )) < 0, i =1,,,n. (4) g(x, t 1 y 1 + t y + + t n y n ) t 1 g(x, y 1 )+t g(x, y )+ + t n g(x, y n ) C. Thus, there exists c C such that By assumption, g(x, x) C, thus By (4), we have g(x, x)+c = t 1 g(x, y 1 )+t g(x, y )+ + t n g(x, y n ). y (t 1 g(x, y 1 )+t g(x, y )+ + t n g(x, y n )) = y (g(x, x)+c) 0. y (t 1 g(x, y 1 )+t g(x, y )+ + t n g(x, y n )) < 0. This is a contradiction. Thus, M is a KKM mapping. We show that for each y A, y (g(,y)) is upper semicontinuous on A. By assumption, for each y A, andforeachx 0 A, g(,y)isc-upper semicontinuous at x 0. Then for any neighborhood V of 0 in Y, there is a neighborhood U(x 0 )ofx 0 such that By y C \{0}, for any ε>0, we set g(x, y) g(x 0,y)+V C for all x U(x 0 ) A. (5) V ε = {y Y : y (y) <ε}. For this V ε, by (5) and the fact that y C \{0}, wehave y (g(x, y)) y (g(x 0,y)) + ε for all x U(x 0 ) A. Thus, y (g(,y)) is upper semicontinuous on A. We show that M(y) is a closed set for each y A. Let {x n } M(y) andx n x 0. It is clear that x 0 A. Sincex n M(y), we have y (g(x n,y)) 0. (6)
1318 GONG XUNHUA Since y (g(,y)) is upper semicontinuous at x 0 for each y A, by(6),wehave that is 0 lim n y (g(x n,y)) y (g(x 0,y)), 0 y (g(x 0,y)). Thus x 0 M(y). Hence M(y) is closed. Since A is compact, M(y) is compact. By Lemma 3.1, we have y A M(y). Thus, there exists x y A M(y). This means that there exists x A such that y (g(x, y)) 0 for all y A. Therefore, V y (g). It follows from the inclusion V y (g) D(g) thatd(g). We show that D(g) isclosed. Let{x n } D(g) withx n x 0.SinceA is closed, x 0 A. Ifx 0 / D(g), then there exists y 0 A such that g(x 0,y 0 ) intc. Thus, there exists a neighborhood V of 0 in Y such that g(x 0,y 0 )+V intc. Hence, we have g(x 0,y 0 )+V C intc. By assumption, g(,y 0 )isc-upper semicontinuous at x 0,sothereexistsn 0 N such that g(x n0,y 0 ) g(x 0,y 0 )+V C intc. This contradicts the fact that x n0 D(g). Hence x 0 D(g). This means that D(g) isclosed. It is clear that D(g) is compact because that D(g) is closed subset of A. Lemma 3.3 Let A X be a nonempty compact convex set, and let g : A A Y be a mapping. Assume that the following conditions are satisfied: (i) g(x, x) =0for all x A; (ii) for each y A, g(,y) is C-upper semicontinuous and C-strictly concave on A. (iii) for each x A, g(x, ) is C-convex on A. Then for each y C \{0}, V y (g) is a singleton. Proof By Lemma 3., V y (g) for each y C \{0}. Suppose to the contrary that V y (g) is not a singleton. Then there exist x 1,x V y (g) withx 1 x,thenwehave y (g(x 1,y)) 0 for all y A (7) and y (g(x,y)) 0 for all y A. (8) In particular, we have y (g(x 1,tx 1 +(1 t)x )) 0, y (g(x,tx 1 +(1 t)x )) 0, (9)
LOWER SEMICONTINUITY OF THE EFFICIENT SOLUTION MAPPING 1319 by (7) and (8). For fixed t (0, 1), we have ty (g(x 1,tx 1 +(1 t)x )) + (1 t)y (g(x,tx 1 +(1 t)x )) 0, (10) by (9). On the other hand, by assumptions (i) and (ii), Since y C \{0}, tg(x 1,tx 1 +(1 t)x )+(1 t)g(x,tx 1 +(1 t)x ) g(tx 1 +(1 t)x,tx 1 +(1 t)x ) intc = intc. ty (g(x 1,tx 1 +(1 t)x )) + (1 t)y (g(x,tx 1 +(1 t)x )) < 0, which contradicts (10). Thus, V y (g) is a singleton. Lemma 3.4 For each y C \{0}, V y : C 0 [A A, Y ] A is a continuous mapping, where V y is given in (3). Proof For each y C \{0} and for each g C 0 [A A, Y ], V y (g) isasingltonby Lemma 3.3. Thus, V y : C 0 [A A, Y ] A is a single-valued mapping. ρ 0 Let {g n } C 0 [A A, Y ]andg n g C0 [A A, Y ]. We will show that V y (g n ) V y (g). Let {x n } = {V y (g n )} and {x} = {V y (g)}. For any n N, wehave For any subsequence {V y (g nk )} of {V y (g n )}, notingthat y (g n (x n,y)) 0 for all y A. (11) {x nk } = {V y (g nk )} for all k N, we have that the sequence {x nk } A. By the compactness of set A, there exists a subsequence {x nkj } of {x nk } with {x nkj } x A. For any fixed y A, andforanyε>0, we set V ε {y = Y : y (y) < ε }. V ε is a neighborhood of zero in Y. Since g(,y)isc-upper semicontinuous at x, thereisa neighborhood U(x) ofx such that g(x, y) g(x, y)+vε C for all x U(x) A. (1) Since {x nkj } x, by (1), there exists j 1 N such that j>j 1 implies ρ 0 Since g nkj g, thereexistsj N such that j>j implies g(x nkj,y) g(x, y)+vε C. (13) ρ 0 (g nkj,g) < ε y. (14)
130 GONG XUNHUA It follows from (14) and y (g nkj (x nkj,y) g(x nkj,y)) y g nkj (x nkj,y) g(x nkj,y) y ρ 0 (g nkj,g) < ε y y = ε, that g nkj (x nkj,y) g(x nkj,y) V ε. (15) Taking j 0 =max{j 1,j }, by (13) and (15), we have g nkj (x nkj,y)=g nkj (x nkj,y) g(x nkj,y)+g(x nkj,y) V ε + V ε + g(x, y) C for all j j 0. (16) By (11), (16) and y C \{0}, we get 0 y (g nkj (x nkj,y)) y (g(x, y)) + ε for all j j 0. Thus Since ε>0 is arbitrary, we get Since y A is arbitrary, 0 y (g(x, y)) + ε. 0 y (g(x, y)). y (g(x, y)) 0 for all y A, that is x V y (g). Thus, {x} = {V y (g)} = {x}, and hence V y (g nkj ) V y (g). Hence, for any subsequence {V y (g nk )} of {V y (g n )}, there exists a subsequence {V y (g nkj )} of {V y (g nk )} converging to V y (g). This implies that V y (g n ) V y (g). Thus, V y ( ) is continuous on C 0 [A A, Y ]. Theorem 3.1 D : C 0 [A A, Y ] A is lower semicontinuous. Proof In view of Lemma 3., D(g) for each g C 0 [A A, Y ]. We can see that for each x A, g(x, A)+C is a convex set. By Lemma.1, we have D(g) = y C \{0} V y (g), g C 0 [A A, Y ]. (17) By Lemma 3.3, V y (g) isasingletonforeachy C \{0}. Take g n ρ 0 g and x D(g). By (17), there exists y C \{0} such that {x} = {V y (g)}. By Lemma 3.4, V y ( ) is continuous at g. Hence, V y (g n ) V y (g). Set {x n } = {V y (g n )},n N. SinceV y (g n ) D(g n ), we have x n D(g n )andx n x. By [16], D(g) is lower semicontinuous at g. Sinceg C 0 [A A, Y ]is arbitrary, we see that D( ) is lower semicontinuous on C 0 [A A, Y ]. Theorem 3. The mapping D : C 0 [A A, Y ] A are closed and upper semicontinuous.
LOWER SEMICONTINUITY OF THE EFFICIENT SOLUTION MAPPING 131 Proof By Lemma 3., D(g) for every g C 0 [A A, Y ]. We show that D is a closed ρ 0 mapping. Let {(g n,x n )} Graph(D),g n g C0 [A A, Y ],x n x 0. It is clear that x 0 A. Since x n D(g n ), for all n N, g n (x n,y) / intc for all y A. (18) We claim that g(x 0,y) / intc for all y A. (19) Suppose to the contrary that there exists y 0 A such that g(x 0,y 0 ) intc, then there exists a neighborhood U of zero in Y such that For this U, thereexistsε>0 such that U ε + U ε U, where g(x 0,y 0 )+U intc. (0) U ε = {y Y : y ε Since g(,y 0 )isc-upper semicontinuous at x 0,forthisUε, there exists a neighborhood U(x 0) of x 0 such that g(x, y 0 ) g(x 0,y 0 )+Uε C for all x U(x 0) A. (1) Since x n x 0,thereexistsn 1 N such that for any n n 1 we have }. x n U(x 0 ) A. () ρ 0 Since g n g, for above ε>0, there exists n N such that n n implies ρ 0 (g n,g) < ε. It follows from g n (x n,y 0 ) g(x n,y 0 ) ρ 0 (g n,g) < ε, that g n (x n,y 0 ) g(x n,y 0 ) U ε for all n n. (3) Taking n 0 =max{n 1,n }, when n n 0, by (0) () and (3), we get g n (x n,y 0 )=g n (x n,y 0 ) g(x n,y 0 )+g(x n,y 0 ) U ε + g(x 0,y 0 )+Uε C U + g(x 0,y 0 ) C intc C intc. This contradicts (18). Thus, (19) holds. This mens that x 0 D(g), and hence D is a closed mapping. Furthermore, as A is compact, by Corollary 9 of [16, p.111], we can see that D is upper semicontinuous on C 0 [A A, Y ].
13 GONG XUNHUA Remark 3.1 In [18, 19], by the strict monotonicity condition, the authors discussed the lower semicontinuity of the weakly efficient solution mapping and efficient solution mapping to the parametric vector equilibrium problems. By the monotonicity condition, Li and Fang [1], Peng, Yang and Peng [] extended and improved the results of [18]. By the strict concavity condition and continuity condition, Chen and Gong [0] discussed the lower semicontinuity of the weakly efficient solution mapping to the parametric vector equilibrium. In Section 3, we discussed the lower semicontinuity and upper semicontinuity of the solution mapping to vector equilibrium problem with perturbation of criterion mapping rather than with perturbation of constraint set. Comparing [18 ], we no longer need the monotonicity condition or the continuity condition for criterion mapping. The results of Theorems 3.1 and 3. are different from the results of [18 ], and they are new. 4 The Lower Semicontinuity of S In this section, we show the lower semicontinuity of the efficient solution mapping S which is given in () with perturbations of both the objective function and the constraint set mapping in normed linear space. We endowed the parameter space P 0 := C[A, Z] C 0 [A A, Y ] with the metric d(p 1,p )= f 1 f + ρ 0 (g 1,g ), for p 1 =(f 1,g 1 ),p =(f,g ) P 0. Theorem 4.1 Let p =(f,g) P 0. Assume that f is injective, i.e., f(x 1 ) f(x ), whenever x 1 x. Then the efficient solution mapping S is lower semicontinuous at p P 0. Proof Suppose to the contrary that S is not lower semicontinuous at p. Then there exists an element x S(p), an open neighborhood U(x) ofx, and a sequence {p n } = {(f n,g n )} P 0 such that {p n } converges to p =(f,g) and S(p n ) U(x) = for all n N. (4) Pick an open ball B(x,λ) :={x X : x x <λ} such that clb(x, λ) U(x). By Theorem 3.1, D is lower semicontinuous at g, whered is the set-valued mapping given in by (1). Since ρ 0 x S(p) D(g) andg n g, by [16], there exists a sequence {vn } such that v n D(g n )and v n x as n. Without loss of generality, we may assume that v n B(x, λ) for all n N. We claim that there exists n 1 N, such that for any n n 1,thereexistsz n D(g n )\B(x, λ) such that f n (v n ) f n (z n ) K\{0}, (5) where K is the pointed convex cone given in Definition.3. Suppose to the contrary that the claim in (5) is false. Then for any n N, thereexistsn 0 n, such that for any z D(g n0 )\B(x, λ), we have f n0 (v n0 ) f n0 (z) / K\{0}. Thus, there exists a subsequence {n i } of {n}. For convenience, we may assume that for each n N, f n (v n ) f n (z) / K\{0} for all z D(g n )\B(x, λ) (6)
LOWER SEMICONTINUITY OF THE EFFICIENT SOLUTION MAPPING 133 holds. In view of Lemma 3., D(g n ) is a nonempty compact set for each n N. By the continuity of f n, we know that f n (D(g n ) clb(x, λ)) (f n (v n ) K) is a nonempty compact set. By Theorem 6.5 of [3, p. 14], f n (D(g n ) clb(x, λ)) (f n (v n ) K) has an efficient point y n, that is there exist z n D(g n ) clb(x, λ) such that y n = f n (z n ), and {f n (z n )} = f n (D(g n ) clb(x, λ)) (f n (v n ) K) (f n (z n ) K). (7) By (7), we have f n (z n ) f n (v n ) K. (8) Since v n D(g n ) clb(x, λ), f n (D(g n ) clb(x, λ)) (f n (v n ) K) is a section of the set f n (D(g n ) clb(x, λ)). By Lemma 6. of [3, p. 140], f n (z n ) is an efficient point of f n (D(g n ) clb(x, λ)), that is, {f n (z n )} = f n (D(g n ) clb(x, λ)) (f n (z n ) K). (9) Suppose z n / S(p n ), then there exists ź n D(g n ) such that f n (z n ) f n (ź n ) K\{0}. (30) By (9) and (30), we have ź n / B(x, λ). Thus, ź n D(g n )\B(x, λ). By (8) and (30), we get f n (v n ) f n (ź n )=f n (v n ) f n (z n )+f n (z n ) f n (ź n ) K + K\{0} K\{0}, which contradicts (6). Thus, z n S(p n ). Hence z n S(p n ) clb(x, λ) S(p n ) U(x). This contradicts (4). Thus, (5) holds. Since z n D(g n ) A for any n n 1 and A is a compact set, we can without loss of generality assume that z n z 0 A. By Theorem 3., D is a closed mapping. This implies that z 0 D(g). Since d(p n,p) 0,z n z 0,andf is continuous, we have f n (z n ) f(z 0 ) f n (z n ) f(z n ) + f(z n ) f(z 0 ) = (f n f)(z n ) + f(z n ) f(z 0 ) f n f + f(z n ) f(z 0 ) 0, as n. Thus, Similarly, by v n x, we have f n (z n ) f(z 0 ). f n (v n ) f(x).
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