Painlevé–Kuratowski convergences of the solution sets for generalized vector quasi-equilibrium problems

Acta Mathematicae Applicatae Sinica, English Series Vol. 30, No. 4 (2014) 845–858 DOI: 10.1007/s10255-014-0426-4 http://www.ApplMath.com.cn & www.SpringerLink.com Acta Mathemacae Applicatae Sinica, English Series © The Editorial Office of AMAS & Springer-Verlag Berlin Heidelberg 2014 Painlevé-Kuratowski Convergences of the Solution Sets for Perturbed Vector Equilibrium Problems without Monotonicity Zai-yun PENG1 , Xin-min YANG2 1 College of Science, Chongqing JiaoTong University, Chongqing 400074, China (E-mail: pengzaiyun@126.com) 2 Department of Mathematics, Chongqing Normal University, Chongqing 400047, China (E-mail: xmyang@cqnu.edu.cn) Abstract In this paper, we obtain some stability results for perturbed vector equilibrium problems. Under new assumptions, which are weaker than the assumption of C-strict monotonicity, we provide sufficient conditions for the Painlevé-Kuratowski Convergence of the weak efficient solution sets and efficient solution sets for the perturbed vector equilibrium problems with a sequence of mappings converging in real linear metric spaces. These results extend and improve some known results in the literature. Keywords stability; Painlevé-Kuratowski convergence; efficient solution; perturbed vector equilibrium prob- lem; scalarization 2000 MR Subject Classification 1 90C33; 49K40; 34D10 Introduction It is well known that the vector equilibrium problem (VEP) is a very general mathematical model, which embraces the formats of several disciplines, as those for equilibrium problems of mathematical physics, Game Theory, vector variational inequality problem, (vector) complementarity problem and (vector) saddle point problem, and so on (see [12,13]). In the literature, existence results for various types of vector equilibrium problems have been investigated intensively (see [9,12,13,14] and references therein). The stability analysis of the solution set maps for parametric (VEP) is of considerable interest. Some results on the semicontinuity of the solution set maps for the parametric (VEP) (or parametric variational inequalities) with the parameter perturbed in the space of parameters are now available in the literature. Cheng and Zhu[7] discussed the upper semicontinuity and the lower semicontinuity of the solution mapping for a class of parametric variational inequalities in finite-dimensional spaces. Gong and Yao[16] first obtained the lower semicontinuity of the efficient solutions of parametric generalized systems by virtue of a density result and scalarization technique. By using the ideas of Cheng and Zhu[7] , Gong[17] discussed the continuity of the solution set mapping for a class of parametric weak vector equilibrium problems in topological vector spaces. Huang et al.[19] discussed the parametric implicit vector equilibrium problems and established the lower semicontinuity of solution mapping using the local existence results of the solutions. Anh and Khanh[3] investigated the semicontinuity of solution mapping of paraManuscript received July 15, 2013. Revised October 9, 2013. Supported by the National Natural Science Foundation of China (No.11301571.11271389.11271391), the Natural Science Foundation Project of ChongQing (No.CSTC, 2012jjA00016.2011BA0030) and the Education Committee Research Foundation of ChongQing (KJ130428). Z.Y. PENG, X.M. YANG 846 metric vector quasi-equilibrium problems by virtue of the closedness or openness assumptions for some certain sets. Chen and Li[6] discussed the continuity of various efficient solution sets for a parametric generalized system and improved the results of [16,17]. Recently, Peng and Yang[27] obtained sufficient conditions for the lower semicontinuity of the solution mappings for two classes of parametric weak generalized equilibrium problems (they called weak generalized Ky Fan Inequalities) in Hausdorff topological vector spaces under weak C-strict monotonicity, which improved the results of [6,16]. As for the stability result investigated on the convergence of the sequence of mappings, there are some results for the vector optimization and vector variational inequality with a sequence of sets converging in the sense of Painlevé-Kuratowski (e.g., [10, 20-22]). Huang[20] discussed the convergence of the approximate efficient sets to the efficient sets of vector-valued and setvalued optimization problems in the sense of Painlevé-Kuratowski and Mosco. Lucchetti and Miglierina[22] investigated the Painlevé-Kuratowski set convergence of the solution sets of the perturbed problems both in the given space and its image space for a convex vector optimization problem. But, to the best of our knowledge, there are few stability results available for the perturbed vector equilibrium problem with the convergence of a sequence of mappings. It appears that two relevant paper are [9] and [11]. In [9], where Durea considered the vector equilibrium problems with the perturbations of the multifunction and obtained the PainlevéKuratowski upper convergence of the solution sets. In [11], under the C-strict monotonicity, Fang et al. obtained the Painlevé-Kuratowski Convergence of the efficient solution sets, the weak efficient solution sets and various proper efficient solution sets for a perturbed generalized system with a sequence of mappings converging in locally convex Hausdorff topological vector spaces. Since the perturbed vector equilibrium problem with a sequence of mappings converging is different from the parametric vector equilibrium problem with the parameter perturbed in a space of parameters, it is important to study the Painlevé-Kuratowski Convergence of the sequence of the solution sets. We also observe that the Painlevé-Kuratowski Convergence of the solution sets to perturbed vector equilibrium problem has been discussed under the restrictive assumption of C-strict monotonicity, which implies that the f -solution set of the perturbed vector equilibrium problem is a singleton for a linear continuous functional f (see recent literature[11] ). However, it is well known that the f -solution set of the perturbed vector equilibrium problem should be general, but not a singleton. So, in this paper, we aim at studying the Painlevé-Kuratowski Convergence of the solution sets to perturbed vector equilibrium problem when the f -solution set is a general set by removing the assumption of C-strict monotonicity. The obtained results extend and improve the recent ones in the literature[11] . The rest of the paper is organized as follows. In Section 2, we introduce the problems (VEP) and (VEP)n , recall some definitions and important properties. In Section 3 and Section 4, we discuss the Painlevé-Kuratowski convergences of the weak efficient solution sets and the efficient solution sets, respectively, and provide some examples to illustrate that our main results extend the corresponding ones in [11,17,27]. 2 Preliminaries Throughout this paper, unless otherwise specified, d(·, ·) denote the metric in any metric space. Let B(0, δ) denote the closed ball with radius δ ≥ 0 and center 0 in any metric linear spaces. Let X and Y be two real linear metric spaces. Let Y ∗ be the topological dual space of Y , and C be a closed convex pointed cone in Y with nonempty topological interior intC. Let   C ∗ := f ∈ Y ∗ : f (y) ≥ 0, ∀ y ∈ C Convergences of the Solution Sets for Perturbed Vector Equilibrium Problems without Monotonicity 847 be the dual cone of C. Denote the quasi-interior of C ∗ by C  , i.e,  C  := {f ∈ Y ∗ : f (y) > 0, ∀y ∈ C \ {0} . It is easy to see that C  = ∅ if and only if C has a base. Let A be a nonempty subset of X and F : A × A → Y be a bifunction. We consider the following vector equilibrium problem (VEP) Find x ∈ A such that F (x, y) ∈ −K, ∀ y ∈ A, where K ∪ {0} is a convex cone in Y . For a sequence of bifunctions Fn : A × A → Y, n = 1, 2, · · · , we define a sequence of vector equilibrium problems (VEP)n Find xn ∈ A such that Fn (xn , y) ∈ −K, ∀ y ∈ A, where K ∪ {0} is a convex cone in Y . Special Case. Let X = E and Y = E ∗ , where E is a reflexive Banach space with dual E ∗ . Let An be a single-valued operators from E to E ∗ . Let Fn (x, y) = An (x), y − x , where ·, · denotes the inner product. Then, (VEP)n reduces to (VI)n considered in [21]. For each f ∈ C ∗ \ {0}, let Vf and Vfn denote the set of f -efficient solutions to the (VEP) and (VEP)n , i.e.,     Vfn = x ∈ A : f (Fn (xn , y)) ≥ 0, ∀ y ∈ A . Vf = x ∈ A : f (F (x, y)) ≥ 0, ∀ y ∈ A , Throughout this paper, for all f ∈ C ∗ \ {0}, we always assume Vf = ∅ and Vfn = ∅. In this paper, under new assumptions, we discuss the Painlevé-Kuratowski convergence of the weak efficient solution sets and the efficient solution sets of (VEP)n . In the following, we recall some concepts of the convergence of set sequence and mapping sequence which will be used in the sequel. Definition 2.1[9,10] . Let X be a normed space. A sequence of sets {An ⊂ X : n ∈ N } is said P.K. to converge in the sense of Painlevé-Kuratowski (P.K.) to A(denoted as An −→ A) if lim sup An ⊂ A ⊂ lim inf An n→∞ n→∞ with   lim inf An := x ∈ X | ∃ (xn ), xn ∈ An , ∀ n ∈ N, xn → x , n→∞   lim sup An := x ∈ X | ∃ (nk ), ∃ (xnk ), xnk ∈ Ank , ∀ k ∈ N, xnk → x . n→∞ Definition 2.2[26] . Let Fn , F : X → Y (n ∈ N ) be a vector-valued mapping and let U (x) be Γ C F) the family of neighborhoods of x. We say that (Fn )n∈N ΓC -converges to F (denoted as Fn −→ if for every x ∈ X : (i) ∀ U ∈ U (x), ∀ ε ∈ int C, ∃nε,U ∈ N such that ∀ n ≥ nε,U , ∃ xn ∈ U such that Fn (xn ) ∈ F (x) + ε − C; (ii) ∀ ε ∈ int C, ∃ Uε ∈ U (x), kε ∈ N such that ∀ x ∈ Uε , ∀ n ≥ kε , Fn (x ) ∈ F (x) − ε + C. Z.Y. PENG, X.M. YANG 848 Definition 2.3[28] . Let Fn , F : X → Y (n ∈ N ) be vector-valued mapping. We say that Fn continuously converges to F if the fact that xn → x implies that Fn (xn ) → F (x). Let T : Λ → 2X be a set-valued mapping, and given λ ∈ Λ. Definition 2.4[1,2]. (i) T is called lower semicontinuous (l.s.c, in short) at λ iff for any open set V satisfying V T (λ) = ∅, there exists δ > 0, such that for every λ ∈ B(λ, δ), V ∩ T (λ) = ∅. (ii) We say T is l.s.c on Λ iff it is l.s.c at each λ ∈ Λ. Definition 2.5. Let F : X × X → Y be a vector-valued mapping. (i) F (x, ·) is called C-convex if, for each x1 , x2 ∈ A and t ∈ [0, 1], tF (x, x1 ) + (1 − t)F (x, x2 ) ∈ F (x, tx1 + (1 − t)x2 ) + C. (ii) F (x, ·) is called C-convexlike on A, iff for any x1 , x2 ∈ A and any t ∈ [0, 1], there exists x3 ∈ A such that tF (x, x1 ) + (1 − t)F (x, x2 ) ∈ F (x, x3 ) + C. (iii) A set D ⊂ Y is called a C-convex set, iff D + C is a convex set in Y. Remark 2.1. (i) F is C-convexlike on X if and only if F (X) + C is convex. (ii) From the definitions, we obtain immediately the following implications for the map F : C-convexity ⇒ C-convexlikeness. However, one simple example in [25] (Example 3.2) shows that the converse implication is generally not valid. Hence, the class of C-convexlike maps is larger than the class of C-convex maps. Proposition 2.1[28] . For any sequence of sets C ν , Dν ⊂ Rn , both the inner limit set lim inf C ν ν and the outer limit set lim sup C ν are closed. Furthermore, they depend only on the closures cl C ν in the sense that ν cl C ν = clDν =⇒ ⎧ ⎨ lim inf C ν = lim inf Dν , ν ν ⎩ lim sup C ν = lim sup Dν . ν ν Thus, whenever lim C ν exists, it is closed. (If C ν ≡ C, then lim C ν = cl C). ν ν [2,4] Proposition 2.2 . Let X and Y be topological spaces, T : X → 2Y be a set-valued mapping. T is l.s.c. at x0 ∈ X if and only if for any net {xα } ⊂ X with xα → x0 and any y0 ∈ T (x0 ), there exists yα ∈ T (xα ) such that yα → y0 . 3 Painlevé-Kuratowski Convergence of the Weak Efficient Solution Sets Denote by I W and InW the sets of the weak efficient sets to (VEP) and (VEP)n , respectively, i.e.,   I W = x ∈ A | F (x, y) ∈ Y \ −int C, ∀ y ∈ A ,   InW = xn ∈ A | Fn (xn , y) ∈ Y \ −int C, ∀ y ∈ A . In this section, without using the assumption of C-strict monotonicity, we mainly discuss the Painlevé-Kuratowski Convergence of InW . Lemma 3.1. Let A be a nonempty compact set. For n = 0, 1, 2 · · · , assume that the following conditions are satisfied: Convergences of the Solution Sets for Perturbed Vector Equilibrium Problems without Monotonicity 849 Γ C (i) For each y ∈ A, Fn (·, y) −→ F (·, y); Γ C −F (·, y). (ii) For each y ∈ A, −Fn (·, y) −→ Then, for each f ∈ C ∗ \ {0}, lim Vfn = Vf . n→∞ Proof. 1) “ lim Vfn n→∞ any xn ∈ Vfn with xn ⊃ Vf ”. On the contrary, assume that there exists x0 ∈ Vf , such that for  x0 . Since x0 ∈ A and A is compact, there exists a net x̂n ∈ A such that x̂n → x0 . Obviously, x̂n ∈ / Vfn . Then there exists yβ ∈ A such that f (Fn (x̂n , yβ )) < 0. Take an arbitrary subnet {x̂nk } ⊆ {x̂n } ⊆ A (taking a subnet of {x̂nk } if necessary), then x̂nk → x0 . Also, we have f (Fnk (x̂nk , yβ )) < 0. (3.1) It follows from x0 ∈ Vf and yβ ∈ A that f (F (x0 , yβ )) ≥ 0. (3.2) Γ C F (·, yβ ), it follows from Definition 2.2 that for any ε ∈ int C, ∃ nε ∈ N such Since Fn (·, yβ ) −→ that Fnk (xnk , yβ )) ∈ F (x0 , yβ )) − ε + C, ∀ nk ≥ n ε . Furthermore, we have f (Fnk (xnk , yβ ))) ≥ f (F (x0 , yβ )) − f (ε). From the arbitrariness of ε, we get f (Fnk (xnk , yβ ))) ≥ f (F (x0 , yβ )). (3.3) Combining with (3.2) and (3.3), we obtain f (Fnk (xnk , yβ ))) ≥ 0, which contradicts (3.1). Then for each x ∈ Vf , there exists xn ∈ Vfn such that xn → x (n → ∞), i.e., x ∈ lim Vfn . So we n→∞ conclude lim Vfn ⊃ Vf . n→∞ nq 2) “ lim Vfn ⊂ Vf ”. Take any x ∈ lim Vfn . Then there exists xnq ∈ Vf n→∞ n→∞ such that lim xnq = x, where {nq } is a subnet of N . Then, we have that for each y ∈ A n→∞ f (Fnq (xnq , y)) ≥ 0. (3.4) Γ C −F (·, y), we get that for any ε ∈ int C, ∃ lε ∈ N such that Because −Fn (·, y) −→ −Fnq (xnq , y)) ∈ −F (x, y) − ε + C, ∀ nq ≥ l ε . So we conclude f (Fnq (xnq , y)) − f (F (x, y)) − f (ε) ≤ 0. From the arbitrariness of ε, we get f (F (x, y)) ≥ f (Fnq (xnq , y)), ∀ y ∈ A. (3.5) By (3.4) and (3.5), we can obtain f (F (x, y)) ≥ 0 for each y ∈ A, i.e. x ∈ Vf . This completes the proof. 2 Remark 3.1. In [11], under the condition of C-strict monotonicity, the relation of Vfn and Vf to (VEP)n and (VEP) are obtained (Lemma 2.2 of [11]). However, this condition is so strict that the f -effcient solution set to (VEP)n and (VEP) is confined to be a singleton. In our paper, we discuss the relationships of Vfn and Vf to (VEP)n and (VEP) without using C-strict monotonicity. Furthermore, the f -efficient solution set may be a general set, but not a singleton. The following example is given to illustrate the case. Z.Y. PENG, X.M. YANG 850   2 Example 3.1. Let X = R, Y = R2 , C = R+ = x = (x1 , x2 ) : x1 ≥ 0, x2 ≥ 0 . Let A = [0, 1]. It is clear that A is a nonempty, compact set. Define the mapping Fn : A × A → R2 by  3 1 3 x− , n = 0, for all x, y ∈ A, F (x, y) = − (x2 + y 2 ), 2 4 8   3   1 2 1 1 1 2 3 Fn (x, y) = − x− x− − , , + y− 2 n n 4 n 8 n = 1, 2 · · · , for all x, y ∈ A. It is clear that A is compact. Let f = (0, 1) ∈ C ∗ \ {0}. Then, it is easy to get that = 18 + n1 , n1 + 1 and Vf = 18 , 1 . Obviously, they are general set, but not a singleton. Because the verification method for Condition (i) is similar to (ii), now we check Condition (ii) of Lemma 3.1 as follows: (a) ∀ ε ∈ int C, ∃ Uε = x− n1 , x+ n1 ∈ U (x), ∃N such that ∀ x1 ∈ Uε = x− n1 , x+ n1 , ∀ n ≥ N, we have  3  1 3  1 2  1 2 1 x1 − x1 − − ,− −Fn (x1 , y) = + y− 2 n n 4 n 8 3 2y 2 1 3 2x1 1 x21 + y 2 − − + 2 , − x1 − − , = 2 n n n 4 n 8 3 3 2 2 3 −Fn (x1 , y) + F (x, y) = (x21 − x2 ) − (x1 + y) + 2 , − (x1 − x) + . 2 n n 4 4n Vfn Then we obtain −Fn (x1 , y) ∈ −F (x, y) − ε + C. (b) ∀ U ∈ U (x), ∀ ε ∈ int C, ∃ nε,U ∈ N such that ∀ n ≥ nε,U , ∃ xn = x + −Fn (xn , y) + F (x, y) =  3y  1 2n n − 2y , − 1 n ∈ U such that 3 . 32 Thus, we also get −Fn (xn , y) ∈ −F (x, y) + ε − C. So, the conditions are fulfilled. It follows from Lemma 3.1 that, for each f ∈ C ∗ \{0}, lim Vfn = n→∞ Vf = 18 , 1 . However, the condition that Fn is a C-strictly monotone mapping is violated. Indeed, x = 12 , y = 1 ∈ A and x = y, we find that  F0 (x, y) + F0 (y, x) = 1 3 (x + y) − 4 4 ∈ −intC 2 1 3 (x+y)− − 4 n 4 ∈ −intC, − 3(x2 + y 2 ), and   1 Fn (x, y)+Fn (y, x) = −3 x− n 2  1 + y− n 2 , for n ≥ 2. Hence, the Lemma 2.2 of [11] is not applicable. In a similar way of Lemma 2.1 in [14], we obtain the following Lemma. Lemma 3.2. on A. Then, Suppose that int C = ∅ and for n = 0, 1, 2, · · · , ∀ x ∈ A, Fn (x, ·) is C-convexlike InW =  f ∈C ∗ \{0} Vfn . Convergences of the Solution Sets for Perturbed Vector Equilibrium Problems without Monotonicity 851 Theorem 3.3. Let A be a nonempty compact set. For n = 0, 1, 2 · · · , assume that the following conditions are satisfied: ΓC F (·, y); (i) For each y ∈ A, Fn (·, y) −→ Γ C −F (·, y). (ii) For each y ∈ A, −Fn (·, y) −→ (iii) For each x ∈ A, Fn (x, ·) is C-convexlike on A. Then, we have InW → I W in the sense of Painlevé-Kuratowski. Proof. i) First, we prove lim sup InW ⊆ I W . On the contrary, assume that there exists x ∈ n→∞ lim sup InW with x ∈ I W . n→∞ From x ∈ lim sup InW , we have x = lim xnk , where xnk ∈ InWk and {nk } is a subnet of N. Thus, we have nk →∞ n→∞ Fnk (xnk , y) ∈ Y \ −int C, ∀ y ∈ A. (3.6) W Noting that x ∈ I , there exists y0 ∈ A such that F (x, y0 ) ∈ −int C. (3.7) In particular, for (3.6), we have Fnk (xnk , y0 ) ∈ Y \ −int C. Γ C Since −Fn (·, y0 ) −→ −F (·, y0 ), we have ∀ ε ∈ int C, ∃ Uε ∈ U (x), kε ∈ N such that for xnk ∈ Uε , ∀ n ≥ kε , −Fn (xnk , y0 ) ∈ −F (x, y0 ) − ε + C. Then, F (x, y0 ) ∈ Y \ −int C, which contradicts (3.7). Hence, lim sup InW ⊆ I W . n→∞ ii) We prove I W ⊆ lim inf InW . By virtue of Lemma3.2, we need to prove that n→∞  f ∈C ∗ \{0} Vf ⊂ lim inf n→∞  Vfn . f ∈C ∗ \{0}  Vf , from Lemma 3.1, there exists f  ∈ C ∗ \ {0} such that x ∈  Vfn , such that xn → x. This implies Vf  = lim Vfn . Then, there exists xn ∈ Vfn ⊂ n→∞ ∗ f ∈C \{0}    Vfn . Therefore, we have Vf ⊂ lim inf Vfn . The proof is that x ∈ lim In fact, for any x ∈ f ∈C ∗ \{0} n→∞ f ∈C ∗ \{0} f ∈C ∗ \{0} complete. 2 Example 3.2. Let X = R, Y = R2 , C = R+ A = [0, 1]. It is clear that A is a compact set. Define the mapping Fn : A × A → R2 by n→∞ f ∈C ∗ \{0} 2   = x = (x1 , x2 ) : x1 ≥ 0, x2 ≥ 0 . Let   1 2 , n = 0, for all x, y ∈ A, F (x, y) = ey (siny − x2 + 1), sin(y − 1) + x − 2   1 2 1 Fn (xn , y) = ey (siny − x2n + 1), sin(y − 1) + xn − , n = 1, 2 · · · , where xn = x − . 2 n Using the same method as that in Example 3.1, it is easy to check that Conditions (i)–(ii) of Theorem 3.3 are satisfied. Z.Y. PENG, X.M. YANG 852 The condition (iii) of Theorem 3.3 can be checked as follows: Let g(y) = ey (siny − x2n + 1), h(y) = sin(y − 1) + (xn − 12 )2 , we have g  (y) = ey (siny − x2n + 1) + ey cos, g  (y) = ey (2cosy − x2n + 1) > 0, h (y) = cos(y − 1), h (y) = −sin(y − 1) ≥ 0. Obviously, for each x ∈ A, F (x, ·) is C-convexlike on A. So, by virtu of Theorem 3.3, InW −→ I W in the sense of Painlevé-Kuratowski. 4 Painlevé-Kuratowski Convergence of the Efficient Solution Sets Denote by I and In the sets of the efficient sets to (VEP) and (VEP)n , respectively, i.e.,   I = x ∈ A | F (x, y) ∈ / −C \ {0}, ∀ y ∈ A ,   In = xn ∈ A | Fn (xn , y) ∈ / −C \ {0}, ∀ y ∈ A . In this section, without using C-strict monotonicity, we will mainly discuss the PainlevéKuratowski Convergence of the efficient solution sets of (VEP)n . Definition 4.1[14] . A vector x ∈ A is called a positive proper efficient solution to the (VEP) if there exists f ∈ C  such that f (F (x, y)) ≥ 0, ∀ y ∈ A. First, we give a result of lower semicontinuous to Vfn for (VEP)n . Lemma 4.1. Let f ∈ C ∗ \ {0}. Suppose the following conditions are satisfied: (i) A is a nonempty, compact set; (ii) Fn (·, ·) continuously converges to F (·, ·) on X × X; (iii) For n = 0, 1, 2, · · · , each x ∈ A \ Vfn , there exists y ∈ Vfn , such that Fn (x, y) + Fn (y, x) + B(0, dr (x, y)) ⊂ −C, where γ > 0 is a positive constant. Let us define the set-valued mapping Hn : C ∗ \ {0} → 2A by Hn (f ) = Vfn , ∀ f ∈ C ∗ \ {0}. then we have Hn (·) is l.s.c on C ∗ \ {0}. Proof. We just give the proof of the case when n = 0, that is, H(·) is l.s.c on C ∗ \ {0} (set H0 (·) = H(·)). Suppose to the contrary that there exists f0 ∈ C ∗ \ {0}, such that H(·) be not l.s.c. at f0 . Then, there exist a sequence {fm } with fm → f0 with respect to the topology β(Y ∗ , Y ) and x0 ∈ H(f0 ) = Vf0 such that for any xm ∈ H(fm ) = Vfm , xm  x0 . Since x0 ∈ A and A is nonempty compact, then there exists xm ∈ A, such that xm → x0 . Obviously, xm ∈ A \ H(fm ). By (iii), there exists ym ∈ H(fm ) such that F (xm , ym ) + F (ym , xm ) + B(0, dr (xm , ym )) ⊂ −C, (4.1) where γ > 0 is a positive constant. For ym ∈ H(fm ) implies ym ∈ A, because A is nonempty compact, there exist y0 ∈ A and a subsequence {ymk } of {ym }, such that ymk → y0 . In particular, for (4.1), we have F (xmk , ymk ) + F (ymk , xmk ) + B(0, dr (xmk , ymk )) ⊂ −C. (4.2) Taking the limit as mk → +∞, it follows from the assumption (ii) that we have F (x0 , y0 ) + F (y0 , x0 ) + B(0, dr (x0 , y0 )) ⊂ −C. (4.3) Convergences of the Solution Sets for Perturbed Vector Equilibrium Problems without Monotonicity 853 Assume that x0 = y0 , by (4.3), we obtain F (x0 , y0 ) + F (y0 , x0 ) ∈ −int C. Thus, it follows from f0 ∈ C ∗ \ {0} that we have f0 (F (x0 , y0 ) + F (y0 , x0 )) < 0. (4.4) Noting that x0 ∈ H(f0 ) and y0 ∈ A, we have f0 (F (x0 , y0 )) ≥ 0. (4.5) Moreover, since ymk ∈ H(fmk ) and xmk ∈ A, it follows from the continuity of f0 and assumption (ii) that f0 (F (y0 , x0 )) ≥ 0. (4.6) By (4.5), (4.6) and the linearity of f0 , we have f0 (F (x0 , y0 ) + F (y0 , x0 )) ≥ 0. which contradicts (4.4). Therefore x0 = y0 , which also leads to a contradiction. Indeed, if x0 = y0 , for ymk ∈ H(fmk ), it has ymk → y0 = x0 , this contradicts that for any xmk ∈ H(fmk ), xmk does not converge to x0 . Thus, H(·) is l.s.c on C ∗ \ {0}. This completes the proof. 2 Lemma 4.2. Let f ∈ C ∗ \ {0} and C  = ∅. Suppose the following conditions are satisfied: (i) A is a nonempty, compact set; (ii) Fn (·, ·) continuously converges to F (·, ·) on X × X; (iii) For n = 0, 1, 2, · · · , each x ∈ A \ Vfn , there exists y ∈ Vfn , such that Fn (x, y) + Fn (y, x) + B(0, dr (x, y)) ⊂ −C, where γ > 0 is a positive constant; (iv) For n = 0, 1, 2, · · · , each x ∈ A, Fn (x, ·) is C-convexlike on A. Then,    Vfn ⊂ In ⊂ cl Vfn . f ∈C  Proof. f ∈C  We just give the proof of the case when n = 0, that is,    Vf ⊂ I ⊂ cl Vf . f ∈C  f ∈C  Since Vf = ∅ for each f ∈ C ∗ \ {0}. Then, by definition, we have  Vf ⊂ I ⊂ I w . (4.7) f ∈C  Since for any x ∈ A, F (x, ·) is C-convexlike, then F (x, A) + C is a convex set. From Lemma 3.2, we have  Iw = Vf . (4.8) f ∈C ∗ \{0} By (4.7) and (4.8), we get  f ∈C  Vf ⊂ I ⊂  f ∈C ∗ \{0} Vf . (4.9) Z.Y. PENG, X.M. YANG 854  We need to show that Vf ⊂ cl   f ∈C ∗ \{0} Vf . f ∈C  Let us define set-valued mapping Hn : C ∗ \ {0} → 2A by Hn (f ) = Vfn , ∀ f ∈ C ∗ \ {0}. ∗ From Lemma 4.1,  Hn (f ) is lower semicontinuous on∗ C \ {0}. Let x0 ∈ Vf . Then, there exists f0 ∈ C \ {0} such that f ∈C ∗ \{0} x0 ∈ Vf0 = H( f0 ). Since C  = ∅, let g ∈ C  and set fm = f0 + (1/m)g.  Then, fm ∈ C . We show that {fm } converges to f0 with respect to the weak∗ topology β(Y ∗ , Y ). For any neighborhood U of 0 with respect to β(Y ∗ , Y ), there exist bounded subsets Bi ⊂ Y (i = 1, 2, · · · , k) and ε > 0 such that k    f ∈ Y ∗ : sup |f (y)| < ε ⊂ U. y∈Bi i=1 Since Bi is bounded and g ∈ Y ∗ , |g(Bi )| is bounded for i = 1, · · · , k. Thus, there exists K such that sup |(1/m)g(y)| < ε, i = 1, · · · , k, k ≥ K. y∈Bi Hence (1/m)g ∈ U, that is, fm − f0 ∈ U. This means that {fm } converges to f0 with respect to β(Y ∗ , Y ). Since H(f ) is l.s.c at f0 (set H0 (f ) = H(f )), then  for sequence {fm } ⊂ C ∗ \ {0}, fm → f0 and x0 ∈ H(f0 ), there exists xm ∈ H(fm ) = Vfm ⊂ Vf , such that xm → x0 . This means that x0 ∈ cl By the arbitrariness of x0 ∈ Vf . f ∈C   f ∈C ∗ \{0} Vf , we have  f ∈C ∗ \{0} By (4.9) and (4.10), we obtain that  f ∈C  This completes the proof.   f ∈C    Vf ⊂ cl Vf . (4.10) f ∈C    Vf ⊂ I ⊂ cl Vf . f ∈C  2 Remark 4.1. Lemma 4.2 improves and extends Lemma 4.1 of [11] and Lemma 1.2 of [16]. In [11] and [16], the density results have been presented under the condition of C-strict monotonicity (called C-strong monotonicity in [16]), where the f -solution set for (VEP) and (VEP)n is confined to be a singleton (also can see Lemma 1.1 of [16], Lemma 4.1 of [17]). In our paper, Convergences of the Solution Sets for Perturbed Vector Equilibrium Problems without Monotonicity 855 we use Assumption (iii) of Lemma 4.2 to weaken this condition. Furthermore, the f -efficient (or positive proper efficient) solution set may be a general set, but not a singleton. The following example is given to illustrate the case. 2 Example 4.1. Let X = R, Y = R2 , C = R+ := [0, +∞) × [0, +∞), A = [−1, 1]. For each 2 x, y ∈ A, define the mapping Fn : A × A −→ R by   1 1 8 2 3 n = 0, for all x, y ∈ A, F (x, y) = − x2 + x + y − , 4x 3 y 2 + x2 , 5 2 3 2   1 8  1 13  2 3  1 2 1 1 2 2 x − 2 + y − ,4 x − 2 x− 2 y + , Fn (x, y) = − x − 2 + n 5 n 2 3 n 2 n n = 1, 2 · · · , for all x, y ∈ A. For any given μ > 0, let f ((x, y)) = μ1 y. It follows from a direct computation that = n12 , 1 , Vf (A, F ) = [0, 1]. Obviously, the f -solution set of (GS) and (GS)n is set-valued, but not a singleton. It is clear that conditions (i), (ii) and (iv) of Lemma 4.2 are satisfied. The assumption (iii) can be checked as follows (we just give the verification of the case when n = 0): For any x ∈ A \ Vf (A, F ) = [−1, 0), there exists y = 0 ∈ Vf (A, F ) and r = 73 > 0 such that Vfn (A, F ) F (x, y) + F (y, x) + B(0, dr (x, y))   1 1 8 2 3 = − x2 + x + y − , 4x 3 y 2 + x2 5 2 3 2   1 1 8 2 3 2 2 + − y + y + x − , 4y 3 x + y 2 + B(0, dr (x, 0)) 5 2 3 2  9 7 2 r = − x + x − 8, 6x 3 + B(0, d (x, 0)) ⊂ −C. 10   By Lemma 4.2, we obtain Vfn ⊂ In ⊂ cl Vfn . f ∈C  f ∈C  However, the condition of C-strict monotonicity in [17, 18] (or called C-strong monotonicity) does not hold. Indeed, for any x ∈ A \ Vf (A, F ) = [−1, 0), there exists y = −x ∈ Vf (A, F ) = [0, 1], such that F (x, y) + F (y, x)   1 1 8 2 3 = − x2 + x + y − , 4x 3 y 2 + x2 5 2 3 2 =(−2x2 − 8, 0) ∈ −∂C,  +  1 1 8 2 3 − y 2 + y + x − , 4y 3 x2 + y 2 5 2 3 2 where ∂C is the boundary of C. Obviously, F (x, y) + F (y, x) ∈ −int C, which implies F (·, ·) is not C-strictly monotone on A × A. Then, the density Lemma 4.1 in [11] and Lemma 1.2 of [16] are not applicable. Theorem 4.3. Suppose that all conditions of Lemma 4.2 are satisfied. Then, we have In → I in the sense of Painlevé-Kuratowski.   Proof. i) “lim sup In ⊂ I”. First, we prove lim sup Vfn ⊂ Vf . Take any x ∈  n→∞ n→∞ f ∈C  f ∈C   Vfn . Then, there exist xnk ∈ Vfnk such that lim sup n→∞ f ∈C  f ∈C  x = lim xnk , k→∞ where {nk } is a subset of N . Therefore, there exists f  ∈ C  such that xnk ∈ Vfn k . Using the same method of Lemma 3.1, we obtain x = lim xnk ∈ lim Vfn k = Vf  , nk →∞ nk →∞ Z.Y. PENG, X.M. YANG 856 then we have x ∈  f ∈L Vf  , i.e., lim sup n→∞   Vfn ⊂ f ∈C  Vf . By virtue of Lemma 4.2, we have    Vfn ⊂ In ⊂ cl Vfn , f ∈C  and (4.11) f ∈C  n = 1, 2, · · · (4.12) f ∈C   Vf ⊂ I ⊂ cl   f ∈C  Vf , n = 0. (4.13) f ∈C  From (4.12) and (4.13), we have cl (In ) = cl  f ∈C  Vfn ). Then, by Proposition 2.1, we get  lim sup In = lim sup n→∞ n→∞ (4.14) f ∈C  It follows from (4.11), (4.13), (4.14) that lim sup In ⊂ n→∞ Vfn .  f ∈C  Vf ⊂ I. ii) Using the same method of i), with suitable modification, we can get that I ⊂ lim inf In . P.K. n→∞ Hence, we obtain that lim sup In ⊂ I ⊂ lim inf In , i.e., In −→ I. This completes the proof. n→∞ n→∞ 2 Remark 4.2. Theorem 4.3 generalizes and improves the corresponding results of [11] (Theorems 4.1–4.2) in the following three aspects: (i) The condition that A(·) is convex valued is removed; (ii) The C-strict monotonicity is substituted by Assumption (iii), then the f -efficient solution may be general in our paper; (iii) The condition that C-convexity of Fn (A, A) is extended to C-convexlikeness; (iv) The boundness condition for Fn (x, x) ∈ C (∀ x ∈ A) and Fn (A, A) are removed. We give an example to illustrate that our results extend the corresponding results in the literature.   2 Example 4.2. Let X = R, Y = R2 , C = R+ = x = (x1 , x2 ) : x1 ≥ 0, x2 ≥ 0 . Let A = [0, 1]. It is clear that A is a compact set. Define the mapping Fn : A × A −→ R2 by  9 3 F (x, y) = ey (sin(x − 1) − 1) + (x + y) − , 2 2 11  2 1 x y + x+3 , n = 0, for all x, y ∈ A, 3 2     1 1 3 9 Fn (x, y) = ey sin x − 3 − 1 − 1 + x− 3 +y − , n 2 n 2 11  1  2 1 1 x− 3 y + x− 3 +3 , n = 1, 2 · · · , for all x, y ∈ A. 3 n 2 n By using the same method in the above examples, we easily get that all conditions of Theorem 4.3 are satisfied. By virtue of Theorem 4.3, we have In → I in the sense of Painlevé-Kuratowski. However, ∀ x ∈ A, there exists y = −x ∈ A with y = x, such that Fn (x, y) + Fn (y, x) ∈ −int C, which implies that Fn (·, ·) is not C-strictly monotone on A×A. 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