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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 691401, 11 pages
http://dx.doi.org/10.1155/2012/691401
Research Article

Continuity of the Solution Maps for Generalized Parametric Set-Valued Ky Fan Inequality Problems

Z. Y. Peng1,2 and X. B. Li1,3

1College of Science, Chongqing Jiaotong University, Chongqing 400074, China
2Department of Mathematics, Inner Mongolia University, Hohhot 010021, China
3College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

Received 24 January 2012; Revised 27 March 2012; Accepted 10 April 2012

Academic Editor: Ya Ping Fang

Copyright © 2012 Z. Y. Peng and X. B. 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.

Abstract

Under new assumptions, we provide suffcient conditions for the (upper and lower) semicontinuity and continuity of the solution mappings to a class of generalized parametric set-valued Ky Fan inequality problems in linear metric space. These results extend and improve some known results in the literature (e.g., Gong, 2008; Gong and Yoa, 2008; Chen and Gong, 2010; Li and Fang, 2010). Some examples are given to illustrate our results.

1. Introduction

The Ky Fan inequality is a very general mathematical format, which embraces the formats of several disciplines, as those for equilibrium problems of mathematical physics, those from game theory, those from (vector) optimization and (vector) variational inequalities, and so on (see [1, 2]). Since Ky Fan inequality was introduced in [1, 2], it has been extended and generalized to vector or set-valued mappings. The Ky Fan Inequality for a set-/vector-valued mapping is known as the (weak) generalized Ky Fan inequality ((W)GKFI, in short). In the literature, existing results for various types of (generalized) Ky Fan inequalities have been investigated intensively, see [35] and the references therein.

It is well known that the stability analysis of solution maps for parametric Ky Fan inequality (PKFI, in short) is an important topic in optimization theory and applications. There are some papers to discuss the upper and/or lower semicontinuity of solution maps. Cheng and Zhu [6] discussed the upper semicontinuity and the lower semicontinuity of the solution map for a PKFI in finite-dimensional spaces. Anh and Khanh [7, 8] studied the stability of solution sets for two classes of parametric quasi-KFIs. Huang et al. [9] discussed the upper semicontinuity and lower semicontinuity of the solution map for a parametric implicit KFI. By virtue of a density result and scalarization technique, Gong [10] first discussed the lower semicontinuity of the set of efficient solutions for a parametric KFI with vector-valued maps. By using the ideas of Cheng and Zhu [6], Gong and Yao [11] studied the continuity of the solution map for a class of weak parametric KFI in topological vector spaces. Then, Kimura and Yao [12] discussed the semicontinuity of solution maps for parametric quasi-KFIs. Based on the work of [6, 10], the continuity of solution sets for PKFIs was discussed in [13] without the uniform compactness assumption. Recently, Li and Fang [14] obtained a new sufficient condition for the lower semicontinuity of the solution maps to a generalized PKFI with vector-valued mappings, where their key assumption is different from the ones in [11, 13].

Motivated by the work reported in [10, 11, 14, 15], this paper aims at studying the stability of the solution maps for a class of generalized PKFI with set-valued mappings. We obtain some new sufficient conditions for the semicontinuity of the solution sets to the generalized PKFI. Our results are new and different from the corresponding ones in [6, 10, 11, 1317].

The rest of the paper is organized as follows. In Section 2, we introduce a class of generalized set-valued PKFI and recall some concepts and their properties which are needed in the sequel. In Section 3, we discuss the upper semicontinuity and lower semicontinuity of the solution mappings for the class of generalized PKFI and compare our main results with the corresponding ones in the recent literature ([10, 11, 1315]). We also give two examples to illustrate that our main results are applicable.

2. Preliminaries

Throughout this paper, if not, otherwise, specified, 𝑑(,) denotes the metric in any metric space. Let 𝐵(0,𝛿) denote the closed ball with radius 𝛿0 and center 0 in any metric linear spaces. Let 𝑋 and 𝑌 be two real linear metric spaces. Let 𝑍 be a linear metric space and let Λ be a nonempty subset of 𝑍. Let 𝑌 be the topological dual space of 𝑌, and let 𝐶 be a closed, convex, and pointed cone in 𝑌 with int𝐶, where int𝐶 denotes the interior of 𝐶. Let 𝐶=𝑓𝑌𝑓(𝑦)0,𝑦𝐶(2.1) be the dual cone of 𝐶.

Let 𝐴 be a nonempty subset of 𝑋, and let 𝐹𝐴×𝐴𝑌{} be a set-valued mapping. We consider the following generalized KFI which consist in finding 𝑥𝐴(𝜆) such that 𝐹(𝑥,𝑦)(int𝐶)=,𝑦𝐴(𝜆).(KFI)

When the set 𝐴 and the function 𝐹 are perturbed by a parameter 𝜆 which varies over a set Λ of 𝑍, we consider the following weak generalized PKFI which consist in finding 𝑥𝐴(𝜆) such that𝐹(𝑥,𝑦,𝜆)(int𝐶)=,𝑦𝐴(𝜆),(PKFI) where 𝐴Λ𝑋{} is a set-valued mapping and 𝐹𝐵×𝐵×Λ𝑋×𝑋×𝑍𝑌{} is a set-valued mapping with 𝐴(Λ)=𝜆Λ𝐴(𝜆)𝐵.

For each 𝜆Λ, the solution set of (PKFI) is defined by 𝑉(𝐹,𝜆)={𝑥𝐴(𝜆)𝐹(𝑥,𝑦,𝜆)(int𝐶)=,𝑦𝐴(𝜆)}.(2.2) For each 𝑓𝐶{0} and for each 𝜆Λ, the 𝑓-solution set of (PKFI) is defined by𝑉𝑓(𝐹,𝜆)=𝑥𝐴(𝜆)inf𝑧𝐹(𝑥,𝑦,𝜆)𝑓(𝑧)0,𝑦𝐴(𝜆).(2.3)

Special Case
(i)If for any 𝜆Λ,𝑥,𝑦𝐴(𝜆), 𝐹(𝑥,𝑦,𝜆)=𝜑(𝑥,𝑦,𝜆)+𝜓(𝑦,𝜆)𝜓(𝑥,𝜆), where 𝜑𝐴(𝜇)×𝐴(𝜇)×Λ2𝑌 is a set-valued mapping and 𝜓𝐴(𝜇)×Λ𝑌 is a single-valued mapping, the (PKFI) reduces to the weak parametric vector equilibrium problem ((W)PVEP) considered in [15].(ii)When 𝐹 is a vector-valued mapping, that is, 𝐹𝐵×𝐵×Λ𝑋×𝑋×𝑍𝑌, the (PKFI) reduces to the parametric Ky Fan inequality in [14].(iii)If for any 𝜆Λ,𝑥,𝑦𝐴(𝜆), 𝐹(𝑥,𝑦,𝜆)=𝜑(𝑥,𝑦,𝜆)+𝜓(𝑦,𝜆)𝜓(𝑥,𝜆), where 𝜑𝐴(𝜇)×𝐴(𝜇)×Λ𝑌 and 𝜓𝐴(𝜇)×Λ𝑌 are two vector-valued maps, the (PKFI) reduces to the parametric (weak) vector equilibrium problem (PVEP) considered in [10, 11, 13, 16].

Throughout this paper, we always assume 𝑉(𝐹,𝜆) for all 𝜆Λ. This paper aims at investigating the semicontinuity and continuity of the solution mapping 𝑉(𝐹,𝜆) as set-valued map from the set Λ into 𝑋. Now, we recall some basic definitions and their properties which are needed in this paper.

Definition 2.1. Let 𝐹𝑋×𝑋×𝑍𝑌{} be a trifunction.(i)𝐹(𝑥,,𝜆) is called 𝐶-convex function on 𝐴(𝜆), if and only if for every 𝑥1,𝑥2𝐴(𝜆) and 𝑡[0,1], 𝑡𝐹(𝑥,𝑥1,𝜆)+(1𝑡)𝐹(𝑥,𝑥2,𝜆)𝐹(𝑥,𝑡𝑥1+(1𝑡)𝑥2,𝜆)+𝐶.(ii)𝐹(𝑥,,𝜆) is called 𝐶-like-convex function on 𝐴(𝜆), if and only if for any 𝑥1,𝑥2𝐴(𝜆) and any 𝑡[0,1], there exists 𝑥3𝐴(𝜆) such that 𝑡𝐹(𝑥,𝑥1,𝜆)+(1𝑡)𝐹(𝑥,𝑥2,𝜆)𝐹(𝑥,𝑥3,𝜆)+𝐶.(iii)𝐹(,,) is called 𝐶-monotone on 𝐴(Λ)×𝐴(Λ)×Λ, if and only if for any 𝜆Λ and 𝑥,𝑦𝐴(𝜆), 𝐹(𝑥,𝑦,𝜆)+𝐹(𝑦,𝑥,𝜆)𝐶. The mapping 𝐹 is called 𝐶-strictly monotone (or called 𝐶-strongly monotone in [10]) on 𝐴(Λ)×𝐴(Λ)×Λ if 𝐹 is 𝐶-monotone and if for any given 𝜆Λ, for all 𝑥,𝑦𝐴(𝜆) and 𝑥𝑦, s.t. 𝐹(𝑥,𝑦,𝜆)+𝐹(𝑦,𝑥,𝜆)int𝐶.

Definition 2.2 (see [18]). Let 𝑋 and 𝑌 be topological spaces, 𝑇𝑋𝑌{} be a set-valued mapping.(i)𝑇 is said to be upper semicontinuous (u.s.c., for short) at 𝑥0𝑋 if and only if for any open set 𝑉 containing 𝑇(𝑥0), there exists an open set 𝑈 containing 𝑥0 such that 𝑇(𝑥)𝑉 for all 𝑥𝑈.(ii)𝑇 is said to be lower semicontinuous (l.s.c., for short) at 𝑥0𝑋 if and only if for any open set 𝑉 with 𝑇(𝑥0)𝑉, there exists an open set 𝑈 containing 𝑥0 such that 𝑇(𝑥)𝑉 for all 𝑥𝑈.(iii)𝑇 is said to be continuous at 𝑥0𝑋, if it is both l.s.c. and u.s.c. at 𝑥0𝑋. 𝑇 is said to be l.s.c. (resp. u.s.c.) on 𝑋, if and only if it is l.s.c. (resp., u.s.c.) at each 𝑥𝑋.

From [19, 20], we have the following properties for Definition 2.2.

Proposition 2.3. Let 𝑋 and 𝑌 be topological spaces, let 𝑇𝑋𝑌{} be a set-valued mapping.(i)𝑇 is l.s.c. at 𝑥0𝑋 if and only if for any net {𝑥𝛼}𝑋 with 𝑥𝛼𝑥0 and any 𝑦0𝑇(𝑥0), there exists 𝑦𝛼𝑇(𝑥𝛼) such that 𝑦𝛼𝑦0.(ii)If 𝑇 has compact values (i.e., 𝑇(𝑥) is a compact set for each 𝑥𝑋), then 𝑇 is u.s.c. at 𝑥0 if and only if for any net {𝑥𝛼}𝑋 with 𝑥𝛼𝑥0 and for any 𝑦𝛼𝑇(𝑥𝛼), there exist 𝑦0𝑇(𝑥0) and a subnet {𝑦𝛽} of {𝑦𝛼}, such that 𝑦𝛽𝑦0.

3. Semicontinuity and Continuity of the Solution Map for (PKFI)

In this section, we obtain some new sufficient conditions for the semicontinuity and continuity of the solution maps to the (PKFI).

Firstly, we provide a new result of sufficient condition for the upper semicontinuity and closeness of the solution mapping to the (PKFI).

Theorem 3.1. For the problem (PKFI), suppose that the following conditions are satisfied:(i)𝐴() is continuous with nonempty compact value on Λ;(ii)𝐹(,,) is l.s.c. on 𝐵×𝐵×Λ.Then, 𝑉(𝐹,) is u.s.c. and closed on Λ.

Proof. (i) Firstly, we prove 𝑉(𝐹,) is u.s.c. on Λ. Suppose to the contrary, there exists some 𝜇0Λ such that 𝑉(𝐹,) is not u.s.c. at 𝜇0. Then, there exist an open set 𝑉 satisfying 𝑉(𝐹,𝜇0)𝑉 and sequences 𝜇𝑛𝜇0 and 𝑥𝑛𝑉(𝐹,𝜇𝑛), such that 𝑥𝑛𝑉,𝑛.(3.1) Since 𝑥𝑛𝐴(𝜇𝑛) and 𝐴() are u.s.c. at 𝜇0 with compact values by Proposition 2.3, there is an 𝑥0𝐴(𝜇0) such that 𝑥𝑛𝑥0 (here, we can take a subsequence {𝑥𝑛𝑘} of {𝑥𝑛} if necessary).
Now, we need to show that 𝑥0𝑉(𝐹,𝜇0). By contradiction, assume that 𝑥0𝑉(𝐹,𝜇0). Then, there exists 𝑦0𝐴(𝜇0) such that 𝐹𝑥0,𝑦0,𝜇0(int𝐶),(3.2) that is, 𝑧0𝑥𝐹0,𝑦0,𝜇0,s.t.𝑧0int𝐶.(3.3)
By the lower semicontinuity of 𝐴() at 𝜇0, for 𝑦0𝐴(𝜇0), there exists 𝑦𝑛𝐴(𝜇𝑛) such that 𝑦𝑛𝑦0.
It follows from 𝑥𝑛𝑉(𝐹,𝜇𝑛) and 𝑦𝑛𝐴(𝜇𝑛) that 𝐹𝑥𝑛,𝑦𝑛,𝜇𝑛(int𝐶)=.(3.4)
Since 𝐹(,,) is l.s.c. at (𝑥0,𝑦0,𝜆0), for 𝑧0𝐹(𝑥0,𝑦0,𝜇0), there exists 𝑧𝑛𝐹(𝑥𝑛,𝑦𝑛,𝜇𝑛) such that 𝑧𝑛𝑧0.(3.5) From (3.3), (3.5), and the openness of int𝐶, there exists a positive integer 𝑁 sufficiently large such that for all 𝑛𝑁, 𝑧𝑛int𝐶,for𝑧𝑛𝑥𝐹𝑛,𝑦𝑛,𝜇𝑛,(3.6) which contradicts (3.4). So, we have 𝑥0𝑉𝐹,𝜇0𝑉.(3.7) Since 𝑥𝑛𝑥0 (here we can take a subsequence {𝑥𝑛𝑘} of {𝑥𝑛} if necessary), we can find (3.7) contradicts (3.1). Consequently, 𝑉(𝐹,) is u.s.c. on Λ.
(ii) In a similar way to the proof of (i), we can easily obtain the closeness of 𝑉(𝐹,) on Λ. This completes the proof.

Remark 3.2. Theorem 3.1 generalizes and improves the corresponding results of Gong [10, Theorem 3.1] in the following four aspects:(i)the condition that 𝐴() is convex values is removed;(ii)the vector-valued mapping 𝐹(,,) is extended to set-valued mapping, and the condition that 𝐶-monotone of mapping is removed;(iii)the assumption (iii) of Theorem  3.1 in [10] is removed;(iv)the condition that 𝐴() is uniformly compact near 𝜇Λ is not required.Moreover, we also can see that the obtained result extends Theorem 2.1 of [15].

Now, we give an example to illustrate that Theorem 3.1 is applicable.

Example 3.3. Let 𝑋=𝑍=𝑌=, 𝐶=+, Λ=[0,21/2] be a subset of 𝑍. Let 𝐹𝑋×𝑋×Λ𝑌 be a set-valued mapping defined by 𝐹(𝑥,𝑦,𝜆)=[(𝑦+1)(𝜆2+1)(𝑥𝜆),10+𝜆2] and let 𝐴Λ𝑋 defined by 𝐴(𝜆)=[𝜆2,2].
It follows from direct computation that 𝐴[][](Λ)=0,2,𝑉(𝐹,𝜆)=𝜆,2,𝜆Λ=0,21/2.(3.8) Then, we can verify that all assumptions of Theorem 3.1 are satisfied. By Theorem 3.1, 𝑉(𝐹,) is u.s.c. and closed on Λ. Therefore, Theorem 3.1 is applicable.

When 𝐹𝑋×𝑋×𝑍𝑌 is a vector-valued mapping, one can get the following corollary.

Corollary 3.4. For the problem (PKFI), suppose that 𝐹𝑋×𝑋×𝑍𝑌 is a vector-valued mapping and the following conditions are satisfied:(i)𝐴() is continuous with nonempty compact value on Λ;(ii)𝐹(,,) is continuous on 𝐵×𝐵×Λ.Then, 𝑉(𝐹,) is u.s.c. and closed on Λ.

Now, we give a sufficient condition for the lower semicontinuity of the solution maps to the (PKFI).

Theorem 3.5. Let 𝑓𝐶{0}. Suppose that the following conditions are satisfied:(i)𝐴() is continuous with nonempty compact value on Λ;(ii)𝐹(,,) is u.s.c. with nonempty compact values on 𝐵×𝐵×Λ;(iii) for each 𝜆Λ, 𝑥𝐴(𝜆)𝑉𝑓(𝐹,𝜆), there exists 𝑦𝑉𝑓(𝐹,𝜆), such that 𝐹(𝑥,𝑦,𝜆)+𝐹(𝑦,𝑥,𝜆)+𝐵(0,𝑑𝛾(𝑥,𝑦))𝐶,(3.9) where 𝛾>0 is a positive constant.

Then, 𝑉𝑓(𝐹,) is l.s.c. on Λ.

Proof. By the contrary, assume that there exists 𝜆0Λ, such that 𝑉𝑓(𝐹,) is not l.s.c. at 𝜆0. Then, there exist 𝜆𝛼 with 𝜆𝛼𝜆0 and 𝑥0𝑉𝑓(𝐹,𝜆0), such that for any 𝑥𝛼𝑉𝑓(𝐹,𝜆𝛼) with 𝑥𝛼𝑥0.
Since 𝑥0𝐴(𝜆0) and 𝐴() are l.s.c. at 𝜆0, there exists ̂𝑥𝛼𝐴(𝜆𝛼) such that ̂𝑥𝛼𝑥0. We claim that ̂𝑥𝛼𝐴(𝜆)𝑉𝑓(𝐹,𝜆𝛼). If not, for ̂𝑥𝛼𝑉𝑓(𝐹,𝜆𝛼), it follows from above-mentioned assumption that ̂𝑥𝛼𝑥0, which is a contradiction.
By (iii), there exists 𝑦𝛼𝑉𝑓(𝐹,𝜆𝛼), such that 𝐹̂𝑥𝛼,𝑦𝛼,𝜆𝛼𝑦+𝐹𝛼,̂𝑥𝛼,𝜆𝛼+𝐵0,𝑑𝛾̂𝑥𝛼,𝑦𝛼𝐶.(3.10)
For 𝑦𝛼𝑉𝑓(𝐹,𝜆𝛼)𝐴(𝜆𝛼), because 𝐴() is u.s.c. at 𝜆0 with compact values by Proposition 2.3, there exist 𝑦0𝐴(𝜆0) and a subsequence {𝑦𝛼𝑘} of {𝑦𝛼} such that 𝑦𝛼𝑘𝑦0. In particular, for (3.10), we have 𝐹̂𝑥𝛼𝑘,𝑦𝛼𝑘,𝜆𝛼𝑘𝑦+𝐹𝛼𝑘,̂𝑥𝛼𝑘,𝜆𝛼𝑘+𝐵0,𝑑𝛾̂𝑥𝛼𝑘,𝑦𝛼𝑘𝐶.(3.11) Then, there exist ̂𝑧𝛼𝑘𝐹(̂𝑥𝛼𝑘,𝑦𝛼𝑘,𝜆𝛼𝑘) and ̃𝑧𝛼𝑘𝐹(𝑦𝛼𝑘,̂𝑥𝛼𝑘,𝜆𝛼𝑘) such that ̂𝑧𝛼𝑘+̃𝑧𝛼𝑘+𝐵0,𝑑𝛾̂𝑥𝛼𝑘,𝑦𝛼𝑘𝐶.(3.12) Since 𝐹(,,) is u.s.c. with compact values on 𝐵×𝐵×Λ by Proposition 2.3, there exist ̂𝑧0𝐹(𝑥0,𝑦0,𝜆0) and ̃𝑧0𝐹(𝑦0,𝑥0,𝜆0) such that ̂𝑧𝛼𝑘̂𝑧0, ̃𝑧𝛼𝑘̃𝑧0. From ̂𝑥𝛼𝑘𝑥0, 𝑦𝛼𝑘𝑦0, the continuity of 𝑑(,), and the closedness of 𝐶, we have ̂𝑧0+̃𝑧0+𝐵0,𝑑𝛾𝑥0,𝑦0𝐶.(3.13) It follows from 𝑥0𝑉𝑓(𝐹,𝜆0) and 𝑦0𝐴(𝜆0) that inf𝑧𝐹(𝑥0,𝑦0,𝜆0)𝑓(𝑧)0. Particularly, we have 𝑓̂𝑧00.(3.14) On the other hand, since 𝑦𝛼𝑘𝑉𝑓(𝐹,𝜆𝛼𝑘) and ̂𝑥𝛼𝑘𝐴(𝜆𝛼𝑘), we have inf𝑧𝐹(𝑦𝛼𝑘,̂𝑥𝛼𝑘,𝜆𝛼𝑘)𝑓(𝑧)0. Also, we have 𝑓(̃𝑧𝛼𝑘)0. It follows from the continuity of 𝑓 that we have 𝑓̃𝑧00.(3.15) By (3.14), (3.15), and the linearity of 𝑓, we get 𝑓̂𝑧0+̃𝑧0=𝑓̂𝑧0+𝑓̃𝑧00.(3.16)
For the above 𝑥0 and 𝑦0, we consider two cases:
Case i. If 𝑥0𝑦0, by (3.13), we can obtain that ̂𝑧0+̃𝑧0int𝐶.(3.17) Then, it follows from 𝑓𝐶{0} that 𝑓̂𝑧0+̃𝑧0<0,(3.18) which is a contradiction to (3.16).Case ii. If 𝑥0=𝑦0, since 𝑦𝛼𝑉𝑓(𝐹,𝜆𝛼), 𝑦𝛼𝑦0=𝑥0, this contradicts that for any 𝑥𝛼𝑉𝑓(𝐹,𝜆𝛼), 𝑥𝛼 do not converge to 𝑥0. Thus, 𝑉𝑓(𝐹,) is l.s.c. on Λ. The proof is completed.

Remark 3.6. Theorem 3.5 generalizes and improves the corresponding results of [14, Lemma 3.1] in the following three aspects:(i)the condition that 𝐴() is convex values is removed;(ii)the vector-valued mapping 𝐹(,,) is extended to set-valued map;(iii)the constant 𝛾 can be any positive constant (𝛾>0) in Theorem 3.5, while it should be strictly restricted to 𝛾=1 in Lemma 3.1 of [14]. Moreover, we also can see that the obtained result extends the ones of Gong and Yao [11, Theorem 2.1], where a strong assumption that 𝐶-strict/strong monotonicity of the mappings is required.

The following example illustrates that the assumption (iii) of Theorem 3.5 is essential.

Example 3.7. Let 𝑋=𝑌=,𝐶=+. Let Λ=[3,5] be a subset of 𝑍. For each 𝜆Λ,𝑥,𝑦𝑋, let 𝐴(𝜆)=[𝜆3,2] and 𝐹𝑋×𝑋×Λ𝑌{} be a set-valued mapping defined by 𝐹(𝑥,𝑦,𝜆)=39𝜆2𝜆𝜆+63𝑥(𝑥𝑦),68+(2𝜆1)2+𝜆3.(3.19) Obviously, assumptions (i) and (ii) of Theorem 3.5 are satisfied, and 𝐴(𝜆)=[0,2],forall𝜆Λ. For any given 𝜆Λ, let 𝑓(𝐹(𝑥,𝑦,𝜆))=𝑧/3,forall𝑧𝐹(𝑥,𝑦,𝜆). Then, it follows from a direct computation that 𝑉𝑓(𝐹,3)={0,2},𝑉𝑓](𝐹,𝜆)=2,𝜆(3,5.(3.20)
However, 𝑉𝑓(𝐹,𝜆) is even not l.c.s. at 𝜆=3. The reason is that the assumption (iii) is violated. Indeed, if 𝑥=0𝑉𝑓(𝐹,𝜆), for 𝜆=3 and for all 𝛾>0, there exist 𝑦=1/2𝐴(𝜆)𝑉𝑓(𝐹,𝜆)=(0,2), such that 𝐹(𝑥,𝑦,𝜆)+𝐹(𝑦,𝑥,𝜆)+𝐵(0,𝑑𝛾=(𝑥,𝑦))39𝜆2𝜆𝜆+63𝑥(𝑥𝑦),68+(2𝜆1)2+𝜆3+39𝜆2𝜆𝜆+63𝑦(𝑦𝑥),68+(2𝜆1)2+𝜆3+𝐵(0,𝑑𝛾=(𝑥,𝑦))39𝜆2𝜆𝜆+63(𝑥𝑦)2,136+2(2𝜆1)2+2𝜆3+𝐵(0,𝑑𝛾=(𝑥,𝑦))274|||102|||𝛾|||1,188+02|||𝛾̸𝐶;(3.21) if 𝑥=2𝑉𝑓(𝐹,𝜆), for 𝜆=3 and for all 𝛾>0, there exist 𝑦=1/2𝐴(𝜆)𝑉𝑓(𝐹,𝜆), using a similar method, we have 𝐹(𝑥,𝑦,𝜆)+𝐹(𝑦,𝑥,𝜆)+𝐵(0,𝑑𝛾̸(𝑥,𝑦))𝐶. Therefore, (iii) is violated.
Now, we show that 𝑉𝑓(𝐹,) is not l.s.c. at 𝜆=3. Indeed, there exists 0𝑉𝑓(𝐹,3) and there exists a neighborhood (2/9,2/9) of 0, for any neighborhood 𝑁(3) of 3, there exists ̃3<𝜆<5 such that ̃𝜆𝑁(3) and 𝑉𝑓̃𝜆2𝐹,=29,29.(3.22) Thus, 𝑉𝑓̃𝜆2𝐹,9,29=.(3.23) By Definition 2.2 (or page 108 in [18]), we know that 𝑉𝑓(𝐹,) is not l.c.s. at 𝜆=3. So, the assumption (iii) of Theorem 3.5 is essential.

By virtue of Theorem  1.1 in [15] (or Lemma  2.1 in [16]), we can get the following proposition.

Proposition 3.8. Suppose that for each 𝜆Λ and 𝑥𝐴(𝜆), 𝐹(𝑥,𝐴(𝜆),𝜆)+𝐶 is a convex set, then 𝑉(𝐹,𝜆)=𝑓𝐶{0}𝑉𝑓(𝐹,𝜆).(3.24)

Theorem 3.9. For the problem (PKFI), suppose that the following conditions are satisfied:(i)𝐴() is continuous with nonempty compact convex value on Λ;(ii)𝐹(,,) is continuous with nonempty compact values on 𝐵×𝐵×Λ;(iii)for each 𝜆Λ, 𝑥𝐴(𝜆)𝑉𝑓(𝐹,𝜆), there exists 𝑦𝑉𝑓(𝐹,𝜆), such that 𝐹(𝑥,𝑦,𝜆)+𝐹(𝑦,𝑥,𝜆)+𝐵(0,𝑑𝛾(𝑥,𝑦))𝐶,(3.25) where 𝛾>0 is a positive constant.(iv)for each 𝜆Λ and for each 𝑥𝐴(𝜆), 𝐹(𝑥,,𝜆) is 𝐶-like-convex on 𝐴(𝜆).

Then, 𝑉(𝐹,) is closed and continuous (i.e., both l.s.c. and u.s.c.) on Λ.

Proof. From Theorem 3.1, it is easy to see that 𝑉(𝐹,) is u.s.c. and closed on Λ. Now, we will only prove that 𝑉(𝐹,) is l.s.c. on Λ. For each 𝜆Λ and for each 𝑥𝐴(𝜆), since 𝐹(𝑥,,𝜆) is 𝐶-like-convex on 𝐴(𝜆), 𝐹(𝑥,𝐴(𝜆),𝜆)+𝐶 is convex. Thus, by virtue of Proposition 3.8, for each 𝜆Λ, it holds 𝑉(𝐹,𝜆)=𝑓𝐶{0}𝑉𝑓(𝐹,𝜆).(3.26) By Theorem 3.5, for each 𝑓𝐶{0}, 𝑉𝑓(𝐹,) is l.s.c. on Λ. Therefore, in view of Theorem 2 in [20, page 114], we have 𝑉(𝐹,) is l.s.c. on Λ. This completes the proof.

Remark 3.10. Theorem 3.9 generalizes and improves the work in [15, Theorems  3.4-3.5]. Our approach on the (semi)continuity of the solution mapping 𝑉(𝐹,) is totally different from the ones by Chen and Gong [15]. In [15], the 𝑉𝑓(𝐹,𝜆) is strictly to be a singleton, while it may be a set-valued one in our paper. In addition, the assumption that 𝐶-strictly monotonicity of the mapping 𝐹 is not required and the 𝐶-convexity of 𝐹 is generalized to the 𝐶-like-convexity.

When the mapping 𝐹 is vector-valued, we obtain the following corollary.

Corollary 3.11. For the problem (PKFI), suppose that 𝐹𝑋×𝑋×𝑍𝑌 is a vector-valued mapping and the following conditions are satisfied:(i)𝐴() is continuous with nonempty compact convex value on Λ;(ii)𝐹(,,) is continuous on 𝐵×𝐵×Λ;(iii)for each 𝜆Λ, 𝑥𝐴(𝜆)𝑉𝑓(𝐹,𝜆), there exists 𝑦𝑉𝑓(𝐹,𝜆), such that 𝐹(𝑥,𝑦,𝜆)+𝐹(𝑦,𝑥,𝜆)+𝐵(0,𝑑𝛾(𝑥,𝑦))𝐶,(3.27) where 𝛾>0 is a positive constant.(iv)for each 𝜆Λ and for each 𝑥𝐴(𝜆), 𝐹(𝑥,,𝜆) is 𝐶-like-convex on 𝐴(𝜆).Then, 𝑉(𝐹,) is closed and continuous (i.e., both l.s.c. and u.s.c.) on Λ.

Remark 3.12. Corollary 3.11 generalizes and improves [10, Theorem 4.2] and [13, Theorem 4.2], respectively, because the assumption that 𝐶-strict monotonicity of the mapping 𝐹 is not required.
Next,we give the following example to illustrate the case.

Example 3.13. Let 𝑋=𝑍=,𝑌=2,𝐶=2+, Λ=[1,1] be a subset of 𝑍. Let 𝐹𝑋×𝑋×Λ𝑌 be a mapping defined by 3𝐹(𝑥,𝑦,𝜆)=2𝜆2,𝜆2𝑥+1(3.28) and define 𝐴Λ2𝑌 by 𝐴(𝜆)=[1,1].
Obviously, 𝐴() is a continuous set-valued mapping from Λ to 𝑅 with nonempty compact convex values, and conditions (ii) and (iv) of Corollary 3.11 are satisfied.
Let 𝑓=(0,2)𝐶{0}, it follows from a direct computation that 𝑉𝑓(𝐹,𝜆)=[0,1] for any 𝜆Λ. Hence, for any 𝑥𝐴(𝜆)𝑉𝑓(𝐹,𝜆), there exists 𝑦=0𝑉𝑓(𝐹,𝜆), such that, 𝐹(𝑥,𝑦,𝜆)+𝐹(𝑦,𝑥,𝜆)+𝐵(0,𝑑𝛾=3(𝑥,𝑦))2𝜆2,𝜆2𝑥+3+12𝜆2,𝜆2𝑦+1+𝐵(0,𝑑𝛾(=𝑥,𝑦))32𝜆2,𝜆2𝑥||||+1+𝐵0,𝑥0𝛾𝐶.(3.29) Thus, the condition (iii) of Corollary 3.11 is also satisfied. By Corollary 3.11, 𝑉(𝐹,) is closed and continuous (i.e., both l.s.c. and u.s.c.) on Λ.
However, the condition that 𝐹 is a 𝐶-strictly monotone mapping is violated. Indeed, for any 𝜆Λ=[1,1] and 𝑥𝐴(𝜆)𝑉𝑓(𝐹,), there exist 𝑦=𝑥𝑉𝑓(𝐹,) with 𝑦=𝑥, such that 𝐹(𝑥,𝑦,𝜆)+𝐹(𝑦,𝑥,𝜆)=32𝜆2,0int𝐶,(3.30) which implies that 𝐹(,,) is not 2+-strictly monotone on 𝐴(Λ)×𝐴(Λ)×Λ. Then, Theorem  4.2 in [10] and Theorem  4.2 in [13] are not applicable, and the corresponding results in references (e.g., [11, Lemma 2.2, Theorem 2.1]) are also not applicable.

Acknowledgments

The authors would like to express their deep gratitude to the anonymous referees and the associate editor for their valuable comments and suggestions, which helped to improve the paper. This work was supported by the National Natural Science Foundation of China (10831009, 11001287), the Natural Science Foundation Project of Chongqing (CSTC2011BA0030, CSTC2011AC6104, CSTC2010BB9254), and the Education Committee Project Research Foundation of Chongqing nos. (KJ100711, KJ120401).

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