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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 105617, 4 pages
The Relationship between Two Kinds of Generalized Convex Set-Valued Maps in Real Ordered Linear Spaces
College of Mathematics and Statistics, Chongqing University of Technology, Chongqing 400054, China
Received 12 March 2013; Accepted 14 May 2013
Academic Editor: Graziano Crasta
Copyright © 2013 Zhi-Ang Zhou. 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.
A new notion of the ic-cone convexlike set-valued map characterized by the algebraic interior and the vector closure is introduced in real ordered linear spaces. The relationship between the ic-cone convexlike set-valued map and the nearly cone subconvexlike set-valued map is established. The results in this paper generalize some known results in the literature from locally convex spaces to linear spaces.
In optimization theory, the generalized convexity of set-valued maps plays an important role. Corley  introduced the cone convexity of set-valued maps. To extend the cone convexity of set-valued maps, some authors [2–5] introduced new generalized convexity such as cone convexlikeness, cone subconvexlikeness, generalized cone subconvexlikeness, nearly cone subconvexlikeness, and ic-cone-convexlikeness. The above generalized convexity set-valued maps mentioned were defined in topological spaces. Recently, Li  has introduced the cone subconvexlike set-valued map based on the algebraic interior in linear spaces. Very recently, Hernández et al.  have defined the cone subconvexlikeness of the set-valued map characterized by the relative algebraic interior. Xu and Song  gave the relationship between ic-cone convexity and nearly cone subconvexlikeness in locally convex spaces. In this paper, we will extend the results obtained by Xu and Song  from locally convex spaces to linear spaces.
This paper is organized as follows. In Section 2, we give some preliminaries, including notations and lemmas. In Section 3, we obtain the relationship between ic-cone convexity and nearly cone subconvexlikeness in linear spaces. Our results generalize and improve the ones obtained by Xu and Song .
In this paper, we always suppose that is a nonempty set and is a real ordered linear space. Let denote the zero element for every space. Let be a nonempty subset in . The affine hull of is defined as , , , . The generated cone of is defined as . Write . Clearly, . is called a cone if and only if for any . Note that some authors defined the cone in the following way: is called a cone if and only if for any . It is possible that if is a cone in the sense of the latter definition. Moreover, if is a cone in the sense of the latter definition, then is a cone in the sense of the former definition. In this paper, if not specially specified, we suppose that all the cones mentioned are defined in the sense of the former definition. is called a convex set if and only if Clearly, a cone is convex if and only if . is said to be nontrivial if and only if and .
From now on, we suppose that is a nontrivial convex cone in and satisfies the condition . We recall the following well-known concepts.
Definition 1 (see ). Let be a nonempty subset in . The algebraic interior of is the set
Definition 2 (see ). Let be a nonempty subset in . The relative algebraic interior of is the set
Remark 3. Clearly, . Moreover, if , then .
Definition 4 (see ). Let be a nonempty subset in . The vector closure of is the set
Let be a set-valued map on . .
Definition 5 (see ). A set-valued map is called nearly -subconvexlike on if and only if is a convex set in .
Remark 6. When the set-valued map becomes a vector-valued map , Definition 5 reduces to Definition 4.1 in . When the linear spaces becomes a topological space, Definition 5 becomes Definition 2.2 in .
In locally convex spaces, Sach  introduced the ic--convexlikeness of the set-valued map. Now, we use the vector closure and the algebraic interior to introduce the ic--convexlikeness of the set-valued map in linear spaces.
Definition 7. A set-valued map is called ic--convexlike on if and only if is a convex set in and .
Lemma 8. Let and be two nonempty sets in . Then, .
Proof. Since and , . Now, we prove Suppose that . Then, and . For any , there exists such that For the above , there exists such that It follows from (6) and (7) that, for the above , there exists such that which implies that . Therefore, (5) holds. Thus, we obtain .
Lemma 9 (see ). If is a nonempty convex set in and , then(a) is a convex set in ;(b), namely, is vectorially closed;(c).
Lemma 10 (see ). Let be a nonempty subset of , and let be a nontrivial and convex cone with . Then, .
Remark 11. The conclusions of Lemma 10 are true when is replaced by .
3. The Relationship between Two Kinds of Generalized Convexity
In this section, we will give the relationship between two kinds of generalized convexity in real ordered linear spaces.
Theorem 12. Let be a set-valued map on and . If is ic--convexlike on , then is nearly -subconvexlike on .
Proof. Since is ic--convexlike on , is a convex set in and , which implies that Using the convexity of and (b) of Lemma 9, we have It follows from (9) and (10) that Clearly, By (11) and (12), we obtain Since is a convex set in , it follows from (13) and (a) of Lemma 9 that is a convex set in . Using Lemma 8, we have Now, we prove that Let . Then, , for all , , and we have Take in . By (16), , for all , , and we have which implies . Therefore, (15) holds. It follows from (14) and (15) that Since is a convex set in , it follows from (18) that is a convex set in . Therefore, is nearly -subconvexlike on .
Remark 13. If is a locally convex space or a finite dimensional linear space, then the condition can be dropped. Thus, Theorem 12 generalizes Theorem 3.2 in  from locally convex spaces to linear spaces.
The following example shows that the converse of Theorem 12 is not true.
Example 14. Let , , , and . The set-valued map is defined as follows: It is easy to check that . Moreover, is a convex set in . Therefore, is nearly -subconvexlike on . However, is not a convex set in . Therefore, is not ic--convexlike on .
In Theorem 12, we do not suppose that . If , we have the following result.
Theorem 15. Let be a set-valued map on . If , then is ic--convexlike on if and only if is nearly -subconvexlike on .
Proof. Necessity. Suppose that is ic--convexlike on . Clearly,
Since , . It follows from Lemma 10 that
which implies that
By (20) and (22), we have . Since is ic--convexlike on , it follows from Theorem 12 that is nearly -subconvexlike on .
Sufficiency. We suppose that is nearly -subconvexlike on . Since , it follows from Lemma 10 and (18) that Since is nearly -subconvexlike on , is a convex set in . Hence, is a convex set in .
Clearly, Since implies , . By the near -subconvexlikeness of , it is easy to check that is a convex set in . It follows from (c) of Lemma 9 that By Lemma 10, we have By (24), (25), and (26), we have . Therefore, is ic--convexlike on .
This work was supported by the National Nature Science Foundation of China (11271391 and 11171363), the Natural Science Foundation of Chongqing (CSTC 2011jjA00022 and CSTC 2011BA0030), and the Science and Technology Project of Chongqing Municipal Education Commission (KJ130830).
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