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

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 [3–5] 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, 13–17].

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, 13–15]). 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 𝛿≥0 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^∗∶=𝑓∈𝑌^∗∶𝑓(𝑦)≥0,∀𝑦∈𝐶(2.1) be the dual cone of 𝐶.

Let 𝐴 be a nonempty subset of 𝑋, and let 𝐹∶𝐴×𝐴⇉𝑌⧵{∅} be a set-valued mapping. We consider the following generalized KFI which consist in finding 𝑥∈𝐴(𝜆) such that 𝐹(𝑥,𝑦)∩(−int𝐶)=∅,∀𝑦∈𝐴(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 𝑥∈𝐴(𝜆) such that𝐹(𝑥,𝑦,𝜆)∩(−int𝐶)=∅,∀𝑦∈𝐴(𝜆),(PKFI) where 𝐴∶Λ⇉𝑋⧵{∅} is a set-valued mapping and 𝐹∶𝐵×𝐵×Λ⊂𝑋×𝑋×𝑍⇉𝑌⧵{∅} is a set-valued mapping with 𝐴(Λ)=⋃_𝜆∈Λ𝐴(𝜆)⊂𝐵.

For each 𝜆∈Λ, the solution set of (PKFI) is defined by 𝑉(𝐹,𝜆)∶={𝑥∈𝐴(𝜆)∣𝐹(𝑥,𝑦,𝜆)∩(−int𝐶)=∅,∀𝑦∈𝐴(𝜆)}.(2.2) For each 𝑓∈𝐶^∗⧵{0} and for each 𝜆∈Λ, the 𝑓-solution set of (PKFI) is defined by𝑉_𝑓(𝐹,𝜆)∶=𝑥∈𝐴(𝜆)∣inf_{𝑧∈𝐹(𝑥,𝑦,𝜆)}𝑓(𝑧)≥0,∀𝑦∈𝐴(𝜆).(2.3)

Special Case
(i)If for any 𝜆∈Λ,𝑥,𝑦∈𝐴(𝜆), 𝐹(𝑥,𝑦,𝜆)∶=𝜑(𝑥,𝑦,𝜆)+𝜓(𝑦,𝜆)−𝜓(𝑥,𝜆), where 𝜑∶𝐴(𝜇)×𝐴(𝜇)×Λ→2^𝑌 is a set-valued mapping and 𝜓∶𝐴(𝜇)×Λ→𝑌 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, 𝐹∶𝐵×𝐵×Λ⊂𝑋×𝑋×𝑍→𝑌, the (PKFI) reduces to the parametric Ky Fan inequality in [14].(iii)If for any 𝜆∈Λ,𝑥,𝑦∈𝐴(𝜆), 𝐹(𝑥,𝑦,𝜆)∶=𝜑(𝑥,𝑦,𝜆)+𝜓(𝑦,𝜆)−𝜓(𝑥,𝜆), where 𝜑∶𝐴(𝜇)×𝐴(𝜇)×Λ→𝑌 and 𝜓∶𝐴(𝜇)×Λ→𝑌 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 𝜆∈Λ. 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 𝐹∶𝑋×𝑋×𝑍⇉𝑌⧵{∅} be a trifunction.(i)𝐹(𝑥,⋅,𝜆) is called 𝐶-convex function on 𝐴(𝜆), if and only if for every 𝑥₁,𝑥₂∈𝐴(𝜆) and 𝑡∈[0,1], 𝑡𝐹(𝑥,𝑥₁,𝜆)+(1−𝑡)𝐹(𝑥,𝑥₂,𝜆)⊂𝐹(𝑥,𝑡𝑥₁+(1−𝑡)𝑥₂,𝜆)+𝐶.(ii)𝐹(𝑥,⋅,𝜆) is called 𝐶-like-convex function on 𝐴(𝜆), if and only if for any 𝑥₁,𝑥₂∈𝐴(𝜆) and any 𝑡∈[0,1], there exists 𝑥₃∈𝐴(𝜆) such that 𝑡𝐹(𝑥,𝑥₁,𝜆)+(1−𝑡)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 𝜆∈Λ and 𝑥,𝑦∈𝐴(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 𝜆∈Λ, for all 𝑥,𝑦∈𝐴(𝜆) and 𝑥≠𝑦, s.t. 𝐹(𝑥,𝑦,𝜆)+𝐹(𝑦,𝑥,𝜆)⊂−int𝐶.

Definition 2.2 (see [18]). Let 𝑋 and 𝑌 be topological spaces, 𝑇∶𝑋⇉𝑌⧵{∅} be a set-valued mapping.(i)𝑇 is said to be upper semicontinuous (u.s.c., for short) at 𝑥₀∈𝑋 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 𝑥∈𝑈.(ii)𝑇 is said to be lower semicontinuous (l.s.c., for short) at 𝑥₀∈𝑋 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 𝑥∈𝑈.(iii)𝑇 is said to be continuous at 𝑥₀∈𝑋, if it is both l.s.c. and u.s.c. at 𝑥₀∈𝑋. 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 𝑥∈𝑋.

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

Proposition 2.3. Let 𝑋 and 𝑌 be topological spaces, let 𝑇∶𝑋⇉𝑌⧵{∅} be a set-valued mapping.(i)𝑇 is l.s.c. at 𝑥₀∈𝑋 if and only if for any net {𝑥_𝛼}⊂𝑋 with 𝑥_𝛼→𝑥₀ and any 𝑦₀∈𝑇(𝑥₀), there exists 𝑦_𝛼∈𝑇(𝑥_𝛼) such that 𝑦_𝛼→𝑦₀.(ii)If 𝑇 has compact values (i.e., 𝑇(𝑥) is a compact set for each 𝑥∈𝑋), then 𝑇 is u.s.c. at 𝑥₀ if and only if for any net {𝑥_𝛼}⊂𝑋 with 𝑥_𝛼→𝑥₀ and for any 𝑦_𝛼∈𝑇(𝑥_𝛼), there exist 𝑦₀∈𝑇(𝑥₀) and a subnet {𝑦_𝛽} of {𝑦_𝛼}, such that 𝑦_𝛽→𝑦₀.

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 𝜇₀∈Λ 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 𝜇_𝑛→𝜇₀ and 𝑥_𝑛∈𝑉(𝐹,𝜇_𝑛), such that 𝑥_𝑛∉𝑉,∀𝑛.(3.1) Since 𝑥_𝑛∈𝐴(𝜇_𝑛) and 𝐴(⋅) are u.s.c. at 𝜇₀ with compact values by Proposition 2.3, there is an 𝑥₀∈𝐴(𝜇₀) such that 𝑥_𝑛→𝑥₀ (here, we can take a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} if necessary).
Now, we need to show that 𝑥₀∈𝑉(𝐹,𝜇₀). By contradiction, assume that 𝑥₀∉𝑉(𝐹,𝜇₀). Then, there exists 𝑦₀∈𝐴(𝜇₀) such that 𝐹𝑥₀,𝑦₀,𝜇₀∩(−int𝐶)≠∅,(3.2) that is, ∃𝑧₀∈𝐹𝑥₀,𝑦₀,𝜇₀,s.t.𝑧₀∈−int𝐶.(3.3)
By the lower semicontinuity of 𝐴(⋅) at 𝜇₀, for 𝑦₀∈𝐴(𝜇₀), there exists 𝑦_𝑛∈𝐴(𝜇_𝑛) such that 𝑦_𝑛→𝑦₀.
It follows from 𝑥_𝑛∈𝑉(𝐹,𝜇_𝑛) and 𝑦_𝑛∈𝐴(𝜇_𝑛) that 𝐹𝑥_𝑛,𝑦_𝑛,𝜇_𝑛∩(−int𝐶)=∅.(3.4)
Since 𝐹(⋅,⋅,⋅) is l.s.c. at (𝑥₀,𝑦₀,𝜆₀), for 𝑧₀∈𝐹(𝑥₀,𝑦₀,𝜇₀), there exists 𝑧_𝑛∈𝐹(𝑥_𝑛,𝑦_𝑛,𝜇_𝑛) such that 𝑧_𝑛⟶𝑧₀.(3.5) From (3.3), (3.5), and the openness of int𝐶, there exists a positive integer 𝑁 sufficiently large such that for all 𝑛≥𝑁, 𝑧_𝑛∈−int𝐶,for𝑧_𝑛∈𝐹𝑥_𝑛,𝑦_𝑛,𝜇_𝑛,(3.6) which contradicts (3.4). So, we have 𝑥₀∈𝑉𝐹,𝜇₀⊂𝑉.(3.7) Since 𝑥_𝑛→𝑥₀ (here we can take a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} 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 𝜇∈Λ 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=ℝ, 𝐶=ℝ₊, Λ=[0,2^1/2] be a subset of 𝑍. Let 𝐹∶𝑋×𝑋×Λ⇉𝑌 be a set-valued mapping defined by 𝐹(𝑥,𝑦,𝜆)=[(𝑦+1)(𝜆²+1)(𝑥−𝜆),10+𝜆²] and let 𝐴∶Λ⇉𝑋 defined by 𝐴(𝜆)=[𝜆²,2].
It follows from direct computation that 𝐴(Λ)=[0,2],𝑉(𝐹,𝜆)=[𝜆,2],∀𝜆∈Λ=0,2^1/2.(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 𝐹∶𝑋×𝑋×𝑍→𝑌 is a vector-valued mapping, one can get the following corollary.

Corollary 3.4. For the problem (PKFI), suppose that 𝐹∶𝑋×𝑋×𝑍→𝑌 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 𝑓∈𝐶^∗⧵{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 𝜆∈Λ, 𝑥∈𝐴(𝜆)⧵𝑉_𝑓(𝐹,𝜆), there exists 𝑦∈𝑉_𝑓(𝐹,𝜆), such that 𝐹(𝑥,𝑦,𝜆)+𝐹(𝑦,𝑥,𝜆)+𝐵(0,𝑑^𝛾(𝑥,𝑦))⊂−𝐶,(3.9) where 𝛾>0 is a positive constant.

Then, 𝑉_𝑓(𝐹,⋅) is l.s.c. on Λ.

Proof. By the contrary, assume that there exists 𝜆₀∈Λ, such that 𝑉_𝑓(𝐹,⋅) is not l.s.c. at 𝜆₀. Then, there exist 𝜆_𝛼 with 𝜆_𝛼→𝜆₀ and 𝑥₀∈𝑉_𝑓(𝐹,𝜆₀), such that for any 𝑥_𝛼∈𝑉_𝑓(𝐹,𝜆_𝛼) with 𝑥_𝛼↛𝑥₀.
Since 𝑥₀∈𝐴(𝜆₀) and 𝐴(⋅) are l.s.c. at 𝜆₀, there exists 𝑥̂_𝛼∈𝐴(𝜆_𝛼) such that 𝑥̂_𝛼→𝑥₀. We claim that 𝑥̂_𝛼∈𝐴(𝜆)⧵𝑉_𝑓(𝐹,𝜆_𝛼). If not, for 𝑥̂_𝛼∈𝑉_𝑓(𝐹,𝜆_𝛼), it follows from above-mentioned assumption that 𝑥̂_𝛼↛𝑥₀, which is a contradiction.
By (iii), there exists 𝑦_𝛼∈𝑉_𝑓(𝐹,𝜆_𝛼), such that 𝐹𝑥̂_𝛼,𝑦_𝛼,𝜆_𝛼+𝐹𝑦_𝛼,𝑥̂_𝛼,𝜆_𝛼+𝐵0,𝑑^𝛾𝑥̂_𝛼,𝑦_𝛼⊂−𝐶.(3.10)
For 𝑦_𝛼∈𝑉_𝑓(𝐹,𝜆_𝛼)⊂𝐴(𝜆_𝛼), because 𝐴(⋅) is u.s.c. at 𝜆₀ with compact values by Proposition 2.3, there exist 𝑦₀∈𝐴(𝜆₀) and a subsequence {𝑦_𝛼𝑘} of {𝑦_𝛼} such that 𝑦_𝛼𝑘→𝑦₀. In particular, for (3.10), we have 𝐹𝑥̂_𝛼𝑘,𝑦_𝛼𝑘,𝜆_𝛼𝑘+𝐹𝑦_𝛼𝑘,𝑥̂_𝛼𝑘,𝜆_𝛼𝑘+𝐵0,𝑑^𝛾𝑥̂_𝛼𝑘,𝑦_𝛼𝑘⊂−𝐶.(3.11) Then, there exist 𝑧̂_𝛼𝑘∈𝐹(𝑥̂_𝛼𝑘,𝑦_𝛼𝑘,𝜆_𝛼𝑘) and 𝑧̃_𝛼𝑘∈𝐹(𝑦_𝛼𝑘,𝑥̂_𝛼𝑘,𝜆_𝛼𝑘) such that 𝑧̂_𝛼𝑘+𝑧̃_𝛼𝑘+𝐵0,𝑑^𝛾𝑥̂_𝛼𝑘,𝑦_𝛼𝑘⊂−𝐶.(3.12) Since 𝐹(⋅,⋅,⋅) is u.s.c. with compact values on 𝐵×𝐵×Λ by Proposition 2.3, there exist 𝑧₀̂∈𝐹(𝑥₀,𝑦₀,𝜆₀) and 𝑧₀̃∈𝐹(𝑦₀,𝑥₀,𝜆₀) such that 𝑧̂_𝛼𝑘→𝑧₀̂, 𝑧̃_𝛼𝑘→𝑧₀̃. From 𝑥̂_𝛼𝑘→𝑥₀, 𝑦_𝛼𝑘→𝑦₀, the continuity of 𝑑(⋅,⋅), and the closedness of 𝐶, we have 𝑧₀̂+𝑧₀̃+𝐵0,𝑑^𝛾𝑥₀,𝑦₀⊂−𝐶.(3.13) It follows from 𝑥₀∈𝑉_𝑓(𝐹,𝜆₀) and 𝑦₀∈𝐴(𝜆₀) that inf_{𝑧∈𝐹(𝑥0,𝑦0,𝜆0)}𝑓(𝑧)≥0. Particularly, we have 𝑓𝑧₀̂≥0.(3.14) On the other hand, since 𝑦_𝛼𝑘∈𝑉_𝑓(𝐹,𝜆_𝛼𝑘) and 𝑥̂_𝛼𝑘∈𝐴(𝜆_𝛼𝑘), we have inf_{𝑧∈𝐹(𝑦𝛼𝑘,}𝑥̂_{𝛼𝑘,𝜆𝛼𝑘)}𝑓(𝑧)≥0. Also, we have 𝑓(𝑧̃_𝛼𝑘)≥0. It follows from the continuity of 𝑓 that we have 𝑓𝑧₀̃≥0.(3.15) By (3.14), (3.15), and the linearity of 𝑓, we get 𝑓𝑧₀̂+𝑧₀̃=𝑓𝑧₀̂+𝑓𝑧₀̃≥0.(3.16)
For the above 𝑥₀ and 𝑦₀, we consider two cases:
Case i. If 𝑥₀≠𝑦₀, by (3.13), we can obtain that 𝑧₀̂+𝑧₀̃∈−int𝐶.(3.17) Then, it follows from 𝑓∈𝐶^∗⧵{0} that 𝑓𝑧₀̂+𝑧₀̃<0,(3.18) which is a contradiction to (3.16).Case ii. If 𝑥₀=𝑦₀, since 𝑦_𝛼∈𝑉_𝑓(𝐹,𝜆_𝛼), 𝑦_𝛼→𝑦₀=𝑥₀, this contradicts that for any 𝑥_𝛼∈𝑉_𝑓(𝐹,𝜆_𝛼), 𝑥_𝛼 do not converge to 𝑥₀. Thus, 𝑉_𝑓(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=ℝ,𝐶=ℝ₊. Let Λ=[3,5] be a subset of 𝑍. For each 𝜆∈Λ,𝑥,𝑦∈𝑋, let 𝐴(𝜆)=[𝜆−3,2] and 𝐹∶𝑋×𝑋×Λ⇉𝑌⧵{∅} be a set-valued mapping defined by 𝐹(𝑥,𝑦,𝜆)=39−𝜆²−𝜆√𝜆+63𝑥(𝑥−𝑦),68+(2𝜆−1)²+𝜆3.(3.19) Obviously, assumptions (i) and (ii) of Theorem 3.5 are satisfied, and 𝐴(𝜆)=[0,2],forall𝜆∈Λ. For any given 𝜆∈Λ, let 𝑓(𝐹(𝑥,𝑦,𝜆))=𝑧/3,forall𝑧∈𝐹(𝑥,𝑦,𝜆). Then, it follows from a direct computation that 𝑉_𝑓(𝐹,3)={0,2},𝑉_𝑓(𝐹,𝜆)=2,∀𝜆∈(3,5].(3.20)
However, 𝑉_𝑓(𝐹,𝜆) is even not l.c.s. at 𝜆=3. The reason is that the assumption (iii) is violated. Indeed, if 𝑥=0∈𝑉_𝑓(𝐹,𝜆), for 𝜆=3 and for all 𝛾>0, there exist 𝑦=1/2∈𝐴(𝜆)⧵𝑉_𝑓(𝐹,𝜆)=(0,2), such that 𝐹(=𝑥,𝑦,𝜆39)+−𝐹𝜆₂(𝑦,𝑥,𝜆)+𝐵(0,𝑑^𝛾(𝑥,𝑦))−𝜆√𝜆+63𝑥(𝑥−𝑦),68+(2𝜆−1)²+𝜆+39−𝜆₂3−𝜆√𝜆+63𝑦(𝑦−𝑥),68+(2𝜆−1)²+𝜆3+𝐵(0,𝑑^𝛾=39−𝜆₂(𝑥,𝑦))−𝜆√𝜆+63(𝑥−𝑦)²,136+2(2𝜆−1)²+2𝜆3+𝐵(0,𝑑^𝛾=27(𝑥,𝑦))4−|||0−12|||^𝛾,188+|||0−12|||^𝛾⊈−𝐶;(3.21) if 𝑥=2∈𝑉_𝑓(𝐹,𝜆), for 𝜆=3 and for all 𝛾>0, there exist 𝑦=1/2∈𝐴(𝜆)⧵𝑉_𝑓(𝐹,𝜆), using a similar method, we have 𝐹(𝑥,𝑦,𝜆)+𝐹(𝑦,𝑥,𝜆)+𝐵(0,𝑑^𝛾(𝑥,𝑦))⊈−𝐶. Therefore, (iii) is violated.
Now, we show that 𝑉_𝑓(𝐹,⋅) is not l.s.c. at 𝜆=3. Indeed, there exists 0∈𝑉_𝑓(𝐹,3) and there exists a neighborhood (−2/9,2/9) of 0, for any neighborhood 𝑁(3) of 3, there exists 3<𝜆̃<5 such that 𝜆̃∈𝑁(3) and 𝑉_𝑓𝐹,𝜆̃=2∉−29,29.(3.22) Thus, 𝑉_𝑓𝐹,𝜆̃∩−29,29=∅.(3.23) By Definition 2.2 (or page 108 in [18]), we know that 𝑉_𝑓(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 𝜆∈Λ and 𝑥∈𝐴(𝜆), 𝐹(𝑥,𝐴(𝜆),𝜆)+𝐶 is a convex set, then 𝑉(𝐹,𝜆)=_{𝑓∈𝐶∗⧵{0}}𝑉_𝑓(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 𝜆∈Λ, 𝑥∈𝐴(𝜆)⧵𝑉_𝑓(𝐹,𝜆), there exists 𝑦∈𝑉_𝑓(𝐹,𝜆), such that 𝐹(𝑥,𝑦,𝜆)+𝐹(𝑦,𝑥,𝜆)+𝐵(0,𝑑^𝛾(𝑥,𝑦))⊂−𝐶,(3.25) where 𝛾>0 is a positive constant.(iv)for each 𝜆∈Λ and for each 𝑥∈𝐴(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 𝜆∈Λ and for each 𝑥∈𝐴(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 𝜆∈Λ, it holds 𝑉(𝐹,𝜆)=_{𝑓∈𝐶∗⧵{0}}𝑉_𝑓(𝐹,𝜆).(3.26) By Theorem 3.5, for each 𝑓∈𝐶^∗⧵{0}, 𝑉_𝑓(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 𝑉_𝑓(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 𝐹∶𝑋×𝑋×𝑍→𝑌 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 𝜆∈Λ, 𝑥∈𝐴(𝜆)⧵𝑉_𝑓(𝐹,𝜆), there exists 𝑦∈𝑉_𝑓(𝐹,𝜆), such that 𝐹(𝑥,𝑦,𝜆)+𝐹(𝑦,𝑥,𝜆)+𝐵(0,𝑑^𝛾(𝑥,𝑦))⊂−𝐶,(3.27) where 𝛾>0 is a positive constant.(iv)for each 𝜆∈Λ and for each 𝑥∈𝐴(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=ℝ,𝑌=ℝ²,𝐶=ℝ²⁺, Λ=[−1,1] be a subset of 𝑍. Let 𝐹∶𝑋×𝑋×Λ→𝑌 be a mapping defined by 𝐹(𝑥,𝑦,𝜆)=−32−𝜆²,𝜆²+1𝑥(3.28) and define 𝐴∶Λ→2^𝑌 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 𝑉_𝑓(𝐹,𝜆)=[0,1] for any 𝜆∈Λ. Hence, for any 𝑥∈𝐴(𝜆)⧵𝑉_𝑓(𝐹,𝜆), there exists 𝑦=0∈𝑉_𝑓(𝐹,𝜆), such that, 𝐹(=𝑥,𝑦−,𝜆3)+𝐹(𝑦,𝑥,𝜆)+𝐵(0,𝑑^𝛾(𝑥,𝑦))2−𝜆²,𝜆²+1𝑥+−32−𝜆²=−3−2𝜆₂,𝜆²+1𝑦+𝐵(0,𝑑^𝛾(𝑥,𝑦))∈−𝐶.,𝜆²+1𝑥+𝐵0,||𝑥−0||^𝛾(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 𝜆∈Λ=[−1,1] and 𝑥∈𝐴(𝜆)⧵𝑉_𝑓(𝐹,⋅), there exist 𝑦=−𝑥∈𝑉_𝑓(𝐹,⋅) with 𝑦=−𝑥, such that 𝐹(𝑥,𝑦,𝜆)+𝐹(𝑦,𝑥,𝜆)=−3−2𝜆²,0∉−int𝐶,(3.30) which implies that 𝐹(⋅,⋅,⋅) is not ℝ²⁺-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).