- Open Access
Topological aspects of circular metric spaces and some observations on the KKM property towards quasi-equilibrium problems
© Chaipunya and Kumam; licensee Springer 2013
- Received: 30 October 2012
- Accepted: 8 July 2013
- Published: 26 July 2013
The main purpose of this paper is to study some topological nature of circular metric spaces and deduce some fixed point theorems for maps satisfying the KKM property. We also investigate the solvability of a variant of a quasi-equilibrium problem as an application.
- circular metric space
- KKM property
- fixed point
The theory of modulars on linear spaces and the corresponding theory of modular linear spaces were founded by Nakano [1, 2]. The attempt to generalize the notion of a modular to avoid its restriction on a linear space or on a space with additional algebraic structure resulted in defining and developing a new modular which works on an arbitrary set. This new modular is called a metric modular and was introduced by Chistyakov in . He also pointed out the connection between a modular metric space, a metric space and a modular space. He even studied one of its applications in superposition operator theory. Recently, the concept of a metric modular has been involved in the development of fixed point theory (see [4–7]).
Amongst the flourishing growth of fixed point theory, it is the KKM maps that attracts the interests of most mathematicians. In [8, 9], the notion of an admissible hull is used to overcome the invalidity of a convex hull, and the class of KKM maps, generalized KKM maps and the KKM property were explored within metrizable spaces.
The purposes of this paper are to introduce and study the circular metric space, which generalizes the modular metric space, together with some elementary properties. Then, under the scope of circular metric space, we present a number of fixed point theorems for maps satisfying the KKM property. Examples of the contents are also provided alongside. In the final part of the paper, we employ the solvability of a variant of a quasi-equilibrium problem as an application of our fixed point results.
In this section, we adopt the concept of circular metric space, which extends the notion of modular metric space. Some properties of the space are also deduced, emphasizing on those related to the compactness.
Definition 2.1 Let X be a nonempty set. A function is said to be a circular metric if the following conditions are satisfied:
(C1) for all if and only if ;
(C2) for all ;
In this case, the pair is called a circular metric space.
Remark 2.2 If the inequality in (C3) holds for all , we say that is a modular metric space (see also ).
It is obvious that every modular metric space is in turn a circular metric space. It is natural to ask the converse, which is not true as we shall illustrate in the next example.
which makes fail to be a modular metric space.
Now, let us turn to some topological aspects of circular metric spaces.
to be a closed ball centered at of radius γ.
Let ℬ be the family of all open balls in a circular metric space . Then ℬ determines, as a base, a unique topology τ on X. In what follows, a circular metric space will always be equipped with this topology. One may observe that τ is -separable. Note also that a closed ball in X is closed.
Next, we give some characterizations of a compact subset of a circular metric space, where the concept of convergence is embodied.
Theorem 2.5 A nonempty subset D of a circular metric space is sequentially compact if and only if every infinite subset of D has an accumulated point.
Proof (⇒) Let D be an infinite subset of X which is sequentially compact, and let A be an infinite subset of D. We may construct a sequence of distinct points in A. Since is also in D, we can find a subsequence which converges to some . Clearly, a is an accumulated point of A.
(⇐) Let D be an infinite subset of X such that every infinite subset of D has an accumulated point a. Let be a sequence in D. Since is an infinite subset of D, there exists an accumulated point a of . Clearly, there is a sequence which converges to a. In fact, is a subsequence of . Since is arbitrarily defined, the conclusion consequently holds. □
Theorem 2.6 A nonempty subset D of a circular metric space is compact if and only if D is sequentially compact.
Observe that these open balls form an infinite open cover for D. Since D is compact, there exists a finite subcover. Since each of these open balls contains at most one point of A, A is a finite set. This contradicts our hypothesis. Therefore, every infinite subset of D has an accumulated point. That is, D is sequentially compact.
We claim that . Assume the contrary, so there exists a sequence in D such that . Since D is sequentially compact, has a convergent subsequence, namely , which converges to some point . Observe that for some , so there exists such that . For every large enough, we have so that . This is a contradiction. So, we have proved our claim.
Assume that there are infinitely many points satisfying (2.1). This will result in the lack of existence of a convergent subsequence of , which contradicts the sequential compactivity. So, the above process stops at some . Clearly, . Note that for each , there exists such that . Therefore, . That is, the family is a finite subcover of . In other words, D is compact. □
Unlike in a metric space, a finite subset in a circular metric space needs not be bounded. We will provide an easy example of such a situation in the following.
Example 2.8 Let and be a circular metric on X. Then any nonempty nonsingleton finite subset of X is not bounded.
Remark 2.9 Example 2.8 further implies that in a circular metric space, a compact set needs not be bounded.
Definition 2.10 Let D be a nonempty subset of a circular metric space , and let be given. A nonempty subset of X is said to be an ε-net for D if for any there exists such that . If for any there is a bounded finite ε-net for D, then D is said to be totally bounded.
Proposition 2.11 In any circular metric space, a totally bounded set is bounded.
Therefore, we can conclude that D is bounded. □
Note that the converse of this proposition is not generally true. In fact, the uniform boundedness is the necessary but not sufficient condition for the uniform total boundedness.
Theorem 2.12 Let D be a nonempty subset of a circular metric space . If D is compact, then for each we can find an ε-net in X for D.
Proof Let be arbitrary. Observe that the family is an open cover for D. Since D is compact, we can find a subcover . Therefore, V is a finite ε-net for D. □
Corollary 2.13 Let D be a nonempty subset of a circular metric space . If D is relatively compact, i.e., the closure is compact, then for each we can find a finite ε-net in X for D.
Proof According to the proof of the previous theorem and since , we have our conclusion. □
The above two assertions lead us to the following consequences.
Corollary 2.14 A bounded compact subset in a circular metric space is totally bounded.
Corollary 2.15 A bounded relatively compact subset in a circular metric space is totally bounded.
We shall discuss now some fixed point theorems for maps that obey the KKM property. To begin with, we shall give some brief recollection of notions in multivalued analysis which will be used sooner or later in this paper.
. If , we say that D is admissible;
D is said to be subadmissible if , where denotes the family of all nonempty finite subsets of D.
upper semicontinuous if for each nonempty closed set , is closed in X;
lower semicontinuous if for each nonempty open set , is open in X;
continuous if it is both upper and lower semicontinuous;
closed if its graph is closed;
firmly compact if for each nonempty bounded set , is bounded and relatively compact.
Remark 3.3 It is well known that if the multivalued map is upper semicontinuous, then F is closed. The converse holds when the space Y is compact.
Now, let us introduce the class of maps satisfying the KKM property.
Definition 3.4 Let M be a circular metric space and X be a subadmissible subset of M. A multivalued map is said to be a KKM map if for each we have .
Definition 3.5 Let M be a circular metric space, X be a subadmissible subset of M and Y be a topological space. Let be two multivalued maps. If for each we have , then G is said to be a generalized KKM map with respect to F.
Definition 3.6 Let M be a circular metric space, X be a subadmissible subset of M and Y be a topological space. A multivalued map is said to satisfy the KKM property if for any generalized KKM map with respect to F, the family has the finite intersection property.
Definition 3.7 Let be a circular metric space and X be a nonempty subset of M. A multivalued map is said to have the approximate fixed point property if for any , there exists such that . In other words, there exists such that .
Now, we are ready to give a fixed point theorem for such a class.
Theorem 3.8 Let be a circular metric space, X be a nonempty subadmissible subset of M and . If is totally bounded, then F has the approximate fixed point property.
Proof Let . By the uniform total boundedness of Y, for each , there exists such that . Now, define a multivalued map by for all . So, is closed for each and . Therefore, G is not a generalized KKM map with respect to F. This implies that there exists a finite subset of X such that . Thus, there exists such that . By the definition of G, it follows that . Therefore, for each . Hence, so that . Suppose that for some , then we have . This further implies that . Therefore, . That is, . □
Theorem 3.9 Let be a circular metric space, X be a nonempty subadmissible subset of M and . If F is closed and firmly compact, then F has a fixed point.
Since is arbitrary, we say that converges also to . Since , we have . Since is a sequence in which converges to and is closed, we may conclude that . Therefore, . □
Before we stride on further results, we would like to give a simple example to visualize and support Theorem 3.9.
Clearly, we have X being subadmissible and F being closed and firmly compact. Now, let be a given KKM map w.r.t. F. It is clear that for all . Since F has the finite intersection property, so does G. Therefore, we have . In view of Theorem 3.9, F has a fixed point. To be precise, every point in is a fixed point of F.
Now, we give the following lemma which enables us to obtain the Shauder-type fixed point.
Lemma 3.11 Let be a circular metric space, Y be a topological space and X be a nonempty subadmissible subset of M. Suppose that is continuous. If , then .
Proof Let be a generalized KKM map with respect to fF, and let . So, . Note also that . Since is arbitrary, we have that is a generalized KKM map with respect to F. Now, since , the family has the finite intersection property, so does the family . This shows that . □
Corollary 3.12 Let be a circular metric space, Y be a topological space and X be a nonempty subadmissible subset of M. Suppose that is continuous and is bounded and compact for all nonempty bounded subset A of X. If , then f has a fixed point.
Proof According to Lemma 3.11, we have . Since f is continuous, f is closed. So, we can apply Theorem 3.9 to obtain the desired result. □
We give some consequences of our results, some of which also appeared in , as follows.
Theorem 3.13 Let be a modular metric space, X be a nonempty subadmissible subset of M and . If is totally bounded, then F has the approximate fixed point property.
Theorem 3.14 Let be a modular metric space, X be a nonempty subadmissible subset of M and . If F is closed and firmly compact, then F has a fixed point.
Corollary 3.15 
Let be a metric space, X be a nonempty subadmissible subset of M and . If is totally bounded, then F has the approximate fixed point property.
Corollary 3.16 
Let be a metric space, X be a nonempty subadmissible subset of M and . If F is closed and compact, then F has a fixed point.
In fact, this problem (QEP1) is set in the context where the linear structure is absent. Hence, the convexity is as well not present. While most of the studies in optimization require the linearity, our next result will eventually overcome the situation when such notion is not valid.
- (a)the function defined by
- (b)the map defined by
is in the class .
Then problem (QEP1) has a solution.
Thus, it must be the case that . Since H is closed and for all , we conclude that . Consequently, we have , and so F is closed. Furthermore, since H is firmly compact, F is also firmly compact (by definition). Finally, Theorem 3.9 yields the existence of a fixed point of F. In turn, the point is actually a solution of problem (QEP1). □
The following solvability of (QEP2) resulted directly.
Corollary 4.2 According to Theorem 4.1, additionally assume that for all . Then problem (QEP2) has a solution.
The authors were supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRU-CSEC No. 56000508). The second author gratefully acknowledges the support provided by the Department of Mathematic and Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT) during his stay at the Department of Mathematical and Statistical Sciences, University of Alberta as a visitor for the short term research.
- Nakano H Tokyo Math. Book Ser. 1. In Modulared Semi-Ordered Linear Spaces. Maruzen, Tokyo; 1950.Google Scholar
- Nakano H Tokyo Math. Book Ser. 3. In Topology and Linear Topological Spaces. Maruzen, Tokyo; 1951.Google Scholar
- Chistyakov VV: Modular metric spaces, I: basic concepts. Nonlinear Anal. 2010, 72: 1–14. 10.1016/j.na.2009.04.057MathSciNetView ArticleGoogle Scholar
- Chaipunya P, Mongkolkeha C, Sintunavarat W, Kumam P: Fixed-point theorems for multivalued maps in modular metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 503504Google Scholar
- Chaipunya P, Cho YJ, Kumam P: Geraghty-type theorems in modular metric spaces with an application to partial differential equation. Adv. Differ. Equ. 2012., 2012: Article ID 83 10.1186/1687-1847-2012-83Google Scholar
- Cho YJ, Saadati R, Sadeghi G: Quasi-contractive maps in modular metric spaces. J. Appl. Math. 2012., 2012: Article ID 907951Google Scholar
- Mongkolkeha C, Sintunavarat W, Kumam P: Fixed point theorems for contraction maps in modular metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 93Google Scholar
- Chang TH, Yen CL: KKM property and fixed point theorems. J. Math. Anal. Appl. 1996, 203: 224–235. 10.1006/jmaa.1996.0376MathSciNetView ArticleGoogle Scholar
- Amini A, Fakhar M, Zafarani J: KKM maps in metric spaces. Nonlinear Anal. 2005, 60: 1045–1052. 10.1016/j.na.2004.10.003MathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.