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Ky Fan minimax inequalities for set-valued mappings
Fixed Point Theory and Applications volume 2012, Article number: 64 (2012)
Abstract
In this article, by virtue of the Kakutani-Fan-Glicksberg fixed point theorem, two types of Ky Fan minimax inequalities for set-valued mappings are obtained. Some examples are given to illustrate our results.
Mathematics Subject Classification (2010): 49J35; 49K35; 90C47.
1 Introduction
It is well known that Ky Fan minimax inequalities play a very important role in many fields, such as variational inequalities, game theory, mathematical economics, control theory, and fixed point theory. Because of its wide applications, Ky Fan minimax inequalities have been generalized in various ways. Since 1960s, Ky Fan minimax theorems of the real-valued functions have been discussed, such as [1–4] and references therein.
In recent years, based on the development of vector optimization, a great deal of articles have devoted to the study of the Ky Fan minimax theorems for vector-valued functions. In [5], Chen proved a Ky Fan minimax inequality for vector-valued mappings on H-spaces by using a generalized Fan's section theorem and a generalized Browder's fixed point theorem. Chang et al. [6] obtained a Ky Fan minimax inequality for vector-valued mappings on W-spaces by applying a generalized section theorem and a generalized fixed point theorem. Li and Wang [7] established the following Ky Fan minimax inequalities for vector-valued mappings:
Luo [8] also obtained some generalized Ky Fan minimax inequalities for vector-valued mappings by applying the classical Browder fixed point theorem and the Kakutani-Fan-Glicksberg fixed point theorem.
There are also many articles to study the minimax theorems for vector-valued mappings.
Nieuwenhuis [9] proved that
where the vector-valued function is f(x, y) = x + y. Tanaka [10–12] obtained minimax theorems of the separated vector-valued function of the type f(x, y) = u(x) + v(y) and investigated some existence results of cone saddle points for general vector-valued functions. Furthermore, by using the existence results of cone saddle points for vector-valued mappings, he obtained the following result:
such that
Shi and Ling [13] proved, respectively, a minimax theorem and a cone saddle point theorem for a class of vector-valued functions, which include the separated functions as its proper subset. Ferro [14, 15] studied minimax theorems for general vector-valued functions. Gong [16] obtained a strong minimax theorem and established an equivalent relationship between the strong minimax inequality and a strong cone saddle point theorem for vector-valued functions. Li et al. [17] investigated a minimax theorem and a saddle point theorem for vector-valued functions in the sense of lexicographic order, respectively.
To the best of authors' knowledge, there are few articles to investigate minimax problems for set-valued mappings. Li et al. [18] obtained some minimax inequalities for set-valued mappings by using a section theorem and a linear scalarization function. Li et al. [19] studied some generalized minimax theorems for set-valued mappings by using a nonlinear scalarization function. Zhang et al. [20] investigated some minimax problems for set-valued mappings by applying the Fan-Browder Fixed Point Theorem. Motivated by the study of [7, 10, 13, 18, 20], we obtain two types of Ky Fan minimax inequalities for set-valued mappings.
The rest of the article is organized as follows. In Section 2, we introduce notations and preliminary results. In Section 3, we obtain two types of Ky Fan minimax inequalities for set-valued mappings. We also give some examples to illustrate our results.
2 Preliminaries
Let X and V be real Hausdorff topological vector spaces. Assume that S is a pointed closed convex cone in V with its interior . Some fundamental terminologies are presented as follows.
Definition 2.1 [21] Let A ⊂ V be a nonempty subset.
(i) A point z ∈ A is said to be a minimal point of A iff A⋂(z-S) = {z}, and Min A denotes the set of all minimal points of A.
(ii) A point z ∈ A is said to be a weakly minimal point of A iff A⋂(z - intS) = Ø, and Min w A denotes the set of all weakly minimal points of A.
(iii) A point z ∈ A is said to be a maximal point of A iff A⋂(z + S) = {z}, and MaxA denotes the set of all maximal points of A.
(iv) A point z ∈ A is said to be a weakly maximal point of A iff A ⋂(z + intS) = Ø, and Max w denotes the set of all weakly maximal points of A.
It is easy to verity that
Definition 2.2 [22] Let F : X → 2Vbe a set-valued mapping with nonempty values.
(i) F is said to be upper semicontinuous (u.s.c.) at x0 ∈ X, iff for any neighborhood N(F(x0)) of F(x0), there exists a neighborhood N(x0) of x0 such that
(ii) F is said to be lower semicontinuous (l.s.c.) at x0 ∈ X, iff for any open neighborhood N in V satisfying F(x0) ⋂N ≠Ø, there exists a neighborhood N(x0) of x0 such that
(iii) F is said to be continuous at x0 ∈ X iff F is both u.s.c. and l.s.c. at x0.
Remark 2.1 [22] The nonempty compact-valued mapping F is said to be u.s.c. at x0 ∈ X0 if and only if for any net {x α } ⊂ X with x α → x0 and for any y α ∈ F(x α ), there exist y0 ∈ F(x0) and a subnet {y β } of {y α }, such that y β → y0.
Definition 2.3 Let X0 be a nonempty convex subset of X, and let F : X0 → 2Vbe a set-valued mapping with nonempty values.
(i) F is said to be properly S-quasiconvex on X0, iff for any x1, x2 ∈ X0 and l ∈ [0, 1], either
F is said to be properly S-quasiconcave on X0, iff - F is properly S-quasiconvex on X0.
(ii) F is said to be S-quasiconvex [23] on X0, iff for any point z ∈ V, the level set
is convex. F is said to be S-quasiconcave on X0, iff - F is S-quasiconvex on X0.
Remark 2.2 If F is a vector-valued mapping, then properly S-quasiconvex reduces to the ordinary properly S-quasiconvex in [14].
Lemma 2.1 Let X0 be a compact subset of X. Suppose that F : X0 × X0 → 2Vis a continuous set-valued mapping and for each (x, y) ∈ X0 × X0, F(x, y) is a compact set. Then and are u.s.c. and compact-valued on X0, respectively.
Proof. It follows from Lemma 2.2 in [18] that Γ and Φ are u.s.c. By the compactness of X0 and the closeness of weakly minimal (maximal) point sets, Γ and Φ are also compact-valued.
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Lemma 2.2 [22] Let X0 be a nonempty subset of X, and let F : X0 → 2Vbe a set-valued mapping with nonempty values. If X0 is compact and if F is u.s.c. and compact-valued, then is compact.
Lemma 2.3 [14] Let A ⊂ V be a nonempty compact subset. Then (i) Min A ≠Ø; (ii) A ⊂ Min A + S; (iii) A ⊂ Min w A + intS∪{0 V }; (iv) MaxA ≠Ø; (v) A ⊂ MaxA - S; and (vi) A ⊂ Max w A - intS∪{0 V }.
Lemma 2.4 [24] (Kakutani-Fan-Glicksberg fixed point theorem) Let X0 be a nonempty compact convex subset of X. If is u.s.c, and for any x ∈ X0, T(x) is a nonempty, closed and convex set, then T has a fixed point.
3 Ky Fan minimax inequalities for set-valued mappings
First, we prove the following interesting lemma.
Lemma 3.1 Let X0 be a nonempty compact convex subset of X, and let F : X0 × X0 → 2Vbe a continuous set-valued mapping with nonempty compact values.
(i) If for each x ∈ X0, F(x, ⋅) is properly S-quasiconcave on X0, then there exists such that
(ii) If for each y ∈ X0, F(⋅, y) is properly S-quasiconvex on X0, then there exists such that
Proof. (i) We define a multifunction by the formula
First, we show that T(x) ≠Ø, for each x ∈ X0. Since F(x, ⋅) is u.s.c. with compact values and X0 is compact, by Lemma 2.2, is a compact set for each x ∈ X0. By Lemma 2.3, . For each x ∈ X0, let . Then, there exists y x ∈ X0 such that z x ∈ F(x, y x ). Namely,
Hence, for each x ∈ X0, T(x) ≠Ø.
Second, we show that T(x) is a closed set, for each x ∈ X0. Let a net {y α : α ∈ I} ⊂ T(x), for each x ∈ X0 and y α → y0. By the definition of T, there exists {z α } such that z α ∈ F(x, y α ) and . Since F(x, ⋅) is u.s.c. with nonempty compact values, by Remark 2.1, there exist a subnet {z β } of {z α } and z0 ∈ F(x, y0) satisfying z β → z0. By the closeness of the weakly maximal point set, . Thus, we have that
and hence for each x ∈ X0, T(x) is a closed set.
Now, we show that T(x) is a convex set, for each x ∈ X0. For each x ∈ X0, let y1, y2 ∈ T(x) and l ∈ [0, 1]. Suppose that there exists l0 ∈ [0, 1] such that
Since F(x, l0y1 + (1 - l0)y2) ⊂ F(x, X0), by Lemma 2.3,
Then, by assumptions and (1), we have that either
or
Thus, we claim that either
In fact, if (2) does not hold, i.e., there exist z1, z2 ∈ V such that
Then, for z1, z2, there exist such that either
Clearly, this is a contradiction. Therefore, (2) holds, which also contradicts the assumption about y1 and y2. Hence, T(x) is a convex set, for each x ∈ X0.
Next, we show that T is u.s.c. on X0. Since X0 is compact, we only need to show that T is a closed map (see [22]). Let a net
and (x α , y α ) → (x0.y0). By the definition of T, there exists {z α } satisfying z α ∈ F(x α , y α ) and . By assumptions and Lemma 2.2, {z α } must have a convergence subnet. For convenience, let the convergence subnet be itself. Since F is u.s.c. with nonempty compact values, by Remark 2.1, there exist a subnet {z β } of {z α } and z0 ∈ F(x0, y0) satisfying
By Lemma 2.1, is u.s.c. and compact-valued. Then, by Remark 2.1, there exist a subnet {z γ } of {z α } and satisfying
Clearly, . That is (x0, y0) ∈ GraphT. Hence, T is u.s.c. on X0.
Therefore, by Lemma 2.4, there exists such that , i.e.,
-
(ii)
We also define a multifunction by the formula
Similar to the above proof, we can prove that all conditions of Lemma 2.4 are satisfied. By Lemma 2.4, there exists such that , i.e.,
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Remark 3.1 When F is a real-valued function, Lemma 3.1 (i) reduces to Lemma 6 in [7].
Theorem 3.1 Let X 0 be a nonempty compact convex subset of X. Suppose that the following conditions are satisfied:
(i) F : X0 × X0 → 2Vis a continuous set-valued mapping with nonempty compact values;
(ii) for each x ∈ X0, F(x, ⋅) is properly S-quasiconcave on X0.
Then,
such that
Proof. By assumptions and Lemmas 2.1-2.3,
Then, by Lemma 3.1, there exists such that
By Lemmas 2.1 and 2.2, and are two compact sets.
Thus, by Lemma 2.3, we have
and
Namely, for every and , there exist and such that
Particularly, taking u = v, we have z1 ∈ z2 + S. This completes the proof. □
Corollary 3.1 Let X 0 be a nonempty compact convex subset of X. Suppose that the following conditions are satisfied:
(i) f : X0 × X0 → V is a continuous vector-valued mapping;
(ii) for each x ∈ X0, f(x, ⋅) is properly S-quasiconcave on X0.
Then,
such that
Proof. Since , by the proof of Theorem 3.1, the conclusion follows readily. â–¡
Remark 3.2 Corollary 3.1 is different from Theorems 3 and 4 in [7] and Corollary 3.8 in [8]. The following example illustrates that when Theorem 3 in [7] and Corollary 3.8 in [8] are not applicable, Corollary 3.1 is applicable.
Example 3.1 Let X = R, V = R2, X0 = [0, 1] and S = {(u, v)|u ≥ 0, v ≥ 0}. Let f : [0, 1] × [0, 1] → R2
Obviously, f is continuous and f(x, ⋅) is properly S-quasiconcave for each x ∈ X0. All conditions of Corollary 3.1 are satisfied. So, inclusion (4) holds. Indeed, by the definition of f, we have
and for each x ∈ X0,
Then, by computing,
and
Thus,
and
Taking ,
However, taking x0 ≠1, we have
Namely, the condition (iii) of Theorem 3 in [7] and the condition (ii) of Corollary 3.8 in [8] do not hold. So, Theorem 3 in [7] and Corollary 3.8 in [8] are not applicable.
Theorem 3.2 Let X 0 be a nonempty compact convex subset of X. Suppose that the following conditions are satisfied:
(i) F : X0 × X0 → 2Vis a continuous set-valued mapping with nonempty compact values;
(ii) for each y ∈ X0, F(⋅,y) is properly S-quasiconvex on X0.
Then,
such that
Proof. By assumptions and Lemmas 2.1-2.3,
Then, by Lemma 3.1, there exists such that
By Lemmas 2.1 and 2.2, and are two compact sets. Thus, by Lemma 2.3, we have
and
Namely, for every and , there exist and such that
Particularly, taking u = v, we have z1 ∈ z2 - S. This completes the proof. □
Theorem 3.3 Let X 0 be compact convex subset of X. Suppose that the following conditions are satisfied:
(i) F : X0 × X0 → 2Vis a continuous set-valued mapping with nonempty compact values;
(ii) for each x ∈ X0, and any , the level set
is convex.
(iii) for any x ∈ X0,
Then,
Proof. By assumptions and Lemmas 2.1-2.3,
Let . We define a multifunction by the formula
Obviously, by the conditions (ii) and (iii), we have that W(x) is a nonempty convex set, for all x ∈ X0.
Now, we show that W(x) is a closed set, for any x ∈ X0. Let a net {y α : α ∈ I} ⊂ W(x), for each x ∈ X0 and y α → y0. By the definition of W, there exists {z α } such that z α ∈ F(x, y α ) and z α ∈ β + S. Since F(x, ⋅) is u.s.c. with compact values, by Remark 2.1, there exist a subnet {z β } of {z α } and z0 ∈ F(x, y0) satisfying z β → z0. By the closeness of S, z0 ∈ β + S. Thus, we have
and hence for each x ∈ X0, W(x) is a closed set.
Next, we show that W is upper semicontinuous on X0. Since X0 is compact, we only need to show W is a closed map (see [22]). Let a net
and (x α , y α ) → (x0.y0). By the definition of W, there exists {z α } satisfying z α ∈ F(x α , y α ) and z α ∈ β + S. Since F is u.s.c. with compact values, by Remark 2.1, there exist a subnet {z γ } of {z α } and z0 ∈ F(x0, y0) satisfying z γ → z0. By the closeness of S, z0 ∈ β + S. That is (x0, y0) ∈ GraphW. Namely, W is upper semicontinuous on X0.
Therefore, by Lemma 2.4, there exists such that , i.e.,
By (7) and Lemma 2.3, we have
Hence, inclusion (6) holds. This completes the proof. â–¡
Remark 3.3 (i) The condition (ii) of Theorem 3.3 can be replaced by "for any x ∈ X0, F(x, ⋅) is S-quasiconcave on X0".
(ii) If F is a scalar set-valued mapping, the condition (iii) of Theorem 3.3 always holds.
(iii) When F is a vector-valued mapping, Theorem 3.3 reduces to corresponding ones in [7, 8].
Theorem 3.4 Let X 0 be a compact convex subset of X. Suppose that the following conditions are satisfied:
(i) F : X0 × X0 → 2Vis a continuous set-valued mapping with nonempty compact values;
(ii) for each y ∈ X0, and any , the level set
is convex.
(iii) for any y ∈ X0,
Then,
Proof. By assumptions and Lemmas 2.1-2.3,
Let . We define a multifunction by the formula
From the proof process of Theorem 3.3, inclusion (8) holds. This completes the proof. â–¡
Remark 3.4 (i) The condition (ii) of Theorem 3.4 can be replaced by "for any y ∈ X0, F(⋅, y) is S-quasiconvex on X0".
(ii) If F is a scalar set-valued mapping, the condition (iii) of Theorem 3.4 always holds.
Next, we give an example for explaining Theorem 3.4.
Example 3.2 Let X = R, V = R2, X0 = [0, 1] ⊂ X, S = {(u, v)|u ≥ 0, v ≥ 0}, and M = {(u, v)|0 ≤ u ≤ 1,0 ≤ v ≤ 1}. Let f: [0, 1] × [0, 1] → R2 and ,
and
Obviously, F is continuous with nonempty compact values and F(⋅, y) is S-quasiconvex for every y ∈ X0. By the definition of F,
and for each y ∈ X0,
Moreover, by computing,
Then, for each y ∈ X0,
namely, the condition (iii) of Theorem 3.4 holds. Thus, all conditions of Theorem 3.4 are satisfied. So, inclusion (8) holds. Indeed, we have that
and
Then,
Hence, we have that
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Acknowledgements
The authors would like to thank the anonymous referees for their valuable comments and suggestions, which helped to improve the article. This study was supported by the National Natural Science Foundation of China No. 11171362.
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Zhang, Y., Li, SJ. Ky Fan minimax inequalities for set-valued mappings. Fixed Point Theory Appl 2012, 64 (2012). https://doi.org/10.1186/1687-1812-2012-64
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DOI: https://doi.org/10.1186/1687-1812-2012-64