Abstract
Based on level sets of fuzzy sets, we propose definitions of limits of sequences of fuzzy sets, and limits and derivatives of fuzzy set-valued mappings. Then, their properties are derived. Limits of sequences of fuzzy sets, and limits and derivatives of fuzzy set-valued mappings are fuzzified ones of them for crisp sets, where ‘crisp’ means ‘non-fuzzy’.
MSC:03E72, 90C70.
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1 Introduction and preliminaries
The usefulness and importance of limits of sequences of crisp sets, and limits (continuity) and derivatives of crisp set-valued mappings have been recognized in many areas, for example, variational analysis, set-valued optimization, stability theory, sensitivity analysis, etc. For details, see, for example, [1–6]. The concept of limits of sequences of crisp sets is interesting and important for itself, and it is necessary to introduce the concepts of limits and derivatives of crisp set-valued mappings. Typical and important applications of them are (i) set-valued optimization and (ii) stability theory and sensitivity analysis for mathematical models. For the case (ii), consider the following system. Some mathematical model outputs the set of optimal values and the set of optimal solutions for a given input parameter . Then and are crisp set-valued mappings. Stability theory deals with the continuity of and . Sensitivity analysis deals with the derivative of .
In this article, limits of sequences of fuzzy sets, and limits and derivatives of fuzzy set-valued mappings are considered. They are generalizations of them for crisp sets. The aim of this article is to propose those concepts and to investigate their properties systematically.
Some research works deal with limits of sequences of fuzzy numbers or fuzzy sets with bounded supports [7–9], while few research works deal with limits of sequences of fuzzy sets. In addition, some research works deal with limits (continuity) and derivatives of fuzzy number or fuzzy set with bounded support-valued mappings [7, 10, 11], while few research works deal with limits (continuity) and derivatives of fuzzy set-valued mappings. Furthermore, their approaches need some assumptions that level sets of fuzzy sets are nonempty and compact. Our new approach in this article, however, does not need those assumptions. Limits of sequences of fuzzy sets, and limits and derivatives of fuzzy set-valued mappings can be expected to be useful and important for (i) fuzzy set-valued optimization, (ii) stability theory and sensitivity analysis for fuzzy mathematical models, etc. For the case (ii), consider the following system. Some fuzzy mathematical model outputs the fuzzy set of optimal values on , , and the set of optimal solutions for a given input parameter . Then is a fuzzy set-valued mapping, and is a crisp set-valued mapping. Then we will be able to deal with the continuity of and , and the derivative of . This means that our proposing approaches give useful tools for stability theory and sensitivity analysis in fuzzy set theory.
We use the following notations.
For , we set , , , and .
For , , we define and , where and if .
Let ℕ be the set of all natural numbers, and we set
A subsequence of a sequence is represented as for some . We write , , or when in ℕ, but or in the case of the convergence of for some N in or .
For a set , let be the closure of C.
Let , , and be sets of all closed, convex, and closed convex subsets of , respectively.
1.1 Limits of sequences of sets
First, we slightly review the definitions of limits of sequences of sets and their properties.
Definition 1.1 (Definition 4.1 in [3])
For a sequence of subsets of , its lower limit is defined as the set
and its upper limit is defined as the set
The limit of is said to exist if , and its limit is defined as the set
Example 1.1 For with , set and . For each , let if k is odd, and let if k is even. In this case, we have and .
Proposition 1.1 (Exercise 4.2(b) in [3])
For a sequence of subsets of , and .
Proposition 1.2 (Exercise 4.3 in [3])
For sequences , , of subsets of and , the following statements hold.
-
(i)
If (which means that ), then .
-
(ii)
If (which means that ), then .
-
(iii)
If , and , , then .
Proposition 1.3 (Proposition 4.4 in [3])
For sequences , of subsets of and , the following statements hold.
-
(i)
.
-
(ii)
If , , then and .
-
(iii)
If , , then .
Definition 1.2 (Example 4.13 in [3])
For nonempty closed sets , the Hausdorff distance between C and D is defined as
where , and is the Euclidean norm. A sequence of nonempty closed subsets of is said to converge to a nonempty closed set with respect to ρ if .
Proposition 1.4 (Example 4.13 in [3])
Let be a bounded set, and let , and be nonempty closed sets. Then converges to C in the sense of Definition 1.1 if and only if converges to C with respect to ρ.
Proposition 1.5 (Proposition 4.15 in [3])
If , then .
From Definition 1.1, the following Proposition 1.6 can be obtained.
Proposition 1.6 Let , , and let . In addition, let . Then .
1.2 Limits of set-valued mappings
Next, we give the definitions of limits of set-valued mappings and their properties.
A mapping F such that for each is called a set-valued mapping from to , and it is denoted by . The set-valued mapping F is said to be closed-valued, convex-valued, or closed convex-valued if , , or for any , respectively.
Definition 1.3 (See p.152 in [3])
Let , and let . The lower limit of F when is defined as the set
and the upper limit of F when is defined as the set
where and mean the intersection and the union with respect to any sequence such that , respectively. The limit of F when is said to exist if , and its limit is defined as the set
Example 1.2 For with , set and . Let be set-valued mappings defined as
for each . In this case, we have and .
Definition 1.4 (Definition 5.4 in [3])
Let , and let . The set-valued mapping F is said to be lower semicontinuous at if
and F is said to be upper semicontinuous at if
The set-valued mapping F is said to be continuous at if F is both lower and upper semicontinuous at , that is,
Example 1.3 Consider the set-valued mappings F and G defined in Example 1.2. Since and , F is lower semicontinuous at 0 but not upper semicontinuous at 0. Since and , G is upper semicontinuous at 0 but not lower semicontinuous at 0.
From Definition 1.3, the following Proposition 1.7 can be obtained.
Proposition 1.7 Let , and let . Then if and only if for any sequence such that .
1.3 Derivatives of set-valued mappings
In this subsection, we give the definition of derivatives of set-valued mappings and their properties.
A set is called a cone if and for any and any .
Definition 1.5 (Definition 3.4 in [2])
Let , and let . A vector is called a tangent vector of S at if there exists a sequence , which converges to , and a sequence of positive real numbers such that
The set of all tangent vectors of S at is called the tangent cone of S at , and it is denoted by .
A tangent cone in Definition 1.5 is also called a contingent cone. For details of tangent cones, see, for example, [1, 12].
Example 1.4 Let , and let . In addition, let . In this case, we have .
Proposition 1.8 (Theorem 3.7 in [2])
Let , and let . Then is a closed cone. Furthermore, if S is convex, then is also convex.
Proposition 1.9 (Theorem 3.8 in [2])
Let with , and let . Then .
For a set-valued mapping , the set
is called the graph of F.
Definition 1.6 (Definition 4.8 in [2], Definition 5.1.1 in [1])
Let , and let . Then the set-valued mapping defined as
for each is called the contingent derivative of F at .
From Definition 1.6, it can be seen that
Example 1.5 Let , and let be a set-valued mapping defined as . Then . From Example 1.4, we have for each .
From Definition 1.6 and Proposition 1.8, the following Proposition 1.10 can be obtained.
Proposition 1.10 Let , and let . Then the following statements hold.
-
(i)
is closed-valued.
-
(ii)
If , then is closed convex-valued.
1.4 Fuzzy sets
We investigate properties of level sets of fuzzy sets, which are necessary to consider limits of sequences of fuzzy sets, and limits and derivatives of fuzzy set-valued mappings.
We identify each fuzzy set on with its membership function, and is interpreted as . Let be the set of all fuzzy sets on .
For and , the set
is called the α-level set of .
For a crisp set , the function defined as
for each is called the indicator function of S. Whenever we consider as a fuzzy set, is interpreted as .
It is known as the decomposition theorem that can be represented as
(see, for example, [13]).
A fuzzy set is said to be closed if is upper semicontinuous. A fuzzy set is closed if and only if , .
A fuzzy set is said to be convex if
for any and any . Namely, is said to be convex if is a quasiconcave function, and is convex if and only if , .
Let , , and be sets of all closed, convex, and closed convex fuzzy sets on , respectively.
A fuzzy set is called a fuzzy cone if , are cones.
For , the set
is called the fuzzy hypograph of .
In order to investigate relationships between fuzzy sets and their level sets, we set
and define as
for each . For and , it can be seen that
where . By using M, the decomposition theorem can be represented as
for .
Proposition 1.11 (Proposition 3 in [14])
Let . If , , then .
Proposition 1.12 (Proposition 4 in [14])
Let , and let . Then for .
Definition 1.7 For , the fuzzy set
is called the closure of .
For a crisp set , it can be seen that . Thus, the closure for fuzzy sets is a generalization of the crisp one.
Proposition 1.13 (See pp.13-14 in [3])
Let , and let be the smallest closed fuzzy set among closed fuzzy sets such that . Then .
Proposition 1.14 Let . Then the following statements hold.
-
(i)
for .
-
(ii)
.
-
(iii)
is the smallest closed fuzzy set among closed fuzzy sets such that .
Proof (i) follows from Proposition 1.12. (ii) follows from (i). In order to show (iii), let be the smallest closed fuzzy set among closed fuzzy sets such that , and we show that . It follows that from (ii), and that from the decomposition theorem and Proposition 1.11. Thus, we have . Assume that there exists such that . We set and , and fix any . Since , it follows that
Since , it follows that
from Proposition 1.13. From (1.1), it follows that there exists such that . Since , , we have , and , which contradicts (1.2). □
The following Example 1.6 shows that does not hold in general.
Example 1.6 Let be a fuzzy set defined as
for each . In this case, we have for each , and .
From Propositions 1.11, 1.12, and 1.14, the following Proposition 1.15 can be obtained.
Proposition 1.15 Let , and let . Then .
Proposition 1.16 Let , , and let , . In addition, let , for each , where , for each if . Then and .
Proof Fix any .
First, assume that . Then it follows that , for each , and we have and .
Next, assume that . We set and .
From the definition of β, it follows that , and , . For any and any , it follows that since . Thus, for any . For each , since , there exists such that , and it follows that . Thus, for any , there exists such that . Therefore, we have .
By the same way, we have . □
Proposition 1.17 Let . If for any except for at most countable many α, then .
Proof Assume that there exists such that . Without loss of generality, assume that . We set and . Since
and
we have , , and , . Therefore, it is not true that for any except for at most countable many α. □
2 Limits of sequences of fuzzy sets
In this section, we propose the definitions of limits of sequences of fuzzy sets based on their level sets, and investigate their properties.
The following Definition 2.1 is a fuzzified one of Definition 1.1.
Definition 2.1 Let , and let
for each . The lower limit of is defined as the fuzzy set
and the upper limit of is defined as the fuzzy set
The limit of is said to exist if , and its limit is defined as the fuzzy set
For crisp sets , , we set , , and set if the limit of exists. Then it can be seen that , , and that if the limit of exists. Thus, the lower limit, upper limit, and limit for sequences of fuzzy sets are generalizations of the crisp ones.
Example 2.1 Let be fuzzy sets defined as
for each , and let be sequences of fuzzy sets defined as
for each . In this case, we have for , and and . Thus, it follows that . Therefore, we obtain the following statements.
-
(i)
, while .
-
(ii)
, while .
-
(iii)
, while .
-
(iv)
, while and there does not exist .
From Definition 1.1 and Propositions 1.3, 1.5, and 1.11, the following Proposition 2.1 can be obtained.
Proposition 2.1 Let , and let , for each . Then the following statements hold.
-
(i)
.
-
(ii)
for .
-
(iii)
.
-
(iv)
for .
-
(v)
Let , and assume that , . Then .
The following Proposition 2.2(i) is a fuzzified one of Proposition 1.3(i), (iii). The following Proposition 2.2(ii) is a fuzzified one of Proposition 1.5. They can be derived from Propositions 1.3, 1.12, and 2.1.
Proposition 2.2
-
(i)
Let . Then . If , for some , then .
-
(ii)
Let . Then .
The following Proposition 2.3 is a fuzzified one of Proposition 1.1. It can be derived from the decomposition theorem and Propositions 1.1, 1.15, and 1.16.
Proposition 2.3 Let . Then
From Propositions 1.1, 1.15, 1.16, and 2.3, the following Proposition 2.4 can be obtained.
Proposition 2.4 Let , , and let , . In addition, let , for each . Then and .
The following Proposition 2.5 is a fuzzified one of Proposition 1.3(ii). It can be derived from Propositions 1.3 and 2.4.
Proposition 2.5 Let , and assume that , . Then and .
The following Proposition 2.6 is a fuzzified one of Proposition 1.2. It can be derived from Propositions 1.2, 1.11, 1.15, and 1.16.
Proposition 2.6 For , the following statements hold.
-
(i)
If (which means that ), then .
-
(ii)
If (which means that ), then .
-
(iii)
If , and , then .
-
(iv)
If , and , then .
-
(v)
If , and , then .
The following Proposition 2.7 is a fuzzified one of Proposition 1.6. It can be derived from Definition 1.1 and Proposition 1.11.
Proposition 2.7 Let , and let . In addition, let . Then .
Throughout the rest of this section, let be a compact set, and let
For and , we set
Let , and let . Then is said to converge to in the sense of Yoshida et al. [9] if for any except for at most countable many α, where is the Hausdorff distance between and defined in Definition 1.2. From Proposition 1.4, if and only if in the sense of Definition 1.1.
The following Proposition 2.8 shows that the concept of the convergence for sequences of fuzzy sets in the sense of Definition 2.1 is weaker than that in the sense of Yoshida et al. [9].
Proposition 2.8 Let , and let . If converges to in the sense of Yoshida et al. [9], then in the sense of Definition 2.1.
Proof converges to in the sense of Yoshida et al. [9] if and only if in the sense of Definition 1.1 for any except for at most countable many α. Then it follows that for any except for at most countable many α. From Proposition 1.17 and the decomposition theorem, we have in the sense of Definition 2.1. □
3 Limits of fuzzy set-valued mappings
In this section, we propose the definitions of limits of fuzzy set-valued mappings, and investigate their properties.
For a fuzzy set-valued mapping and , we define the crisp set-valued mapping as
for each .
A fuzzy set-valued mapping is said to be closed-valued, convex-valued, or closed convex-valued if , , or for any , respectively.
The following Definition 3.1 is a fuzzified one of Definition 1.3 (see Proposition 3.3 which is mentioned later).
Definition 3.1 Let , and let . In addition, let
for each . The lower limit of when is defined as the fuzzy set
and the upper limit of when is defined as the fuzzy set
The limit of when is said to exist if , and its limit is defined as the fuzzy set
For a crisp set-valued mapping and , let , , and let if the limit of F when exists. Then , , and if the limit of F when exists. Thus, the lower limit, upper limit, and limit for fuzzy set-valued mappings are generalizations of the crisp ones.
Example 3.1 Let be fuzzy sets defined as
for each , and let be a fuzzy set-valued mapping defined as
for each , where ℚ is the set of all rational numbers, and , and for . Since
for each when , and
for each , we have for , and and . Thus, it follows that . Therefore, we obtain the following statements.
-
(i)
, while .
-
(ii)
, while .
-
(iii)
, while .
-
(iv)
, while and there does not exist .
The following Definition 3.2 is a fuzzified one of Definition 1.4.
Definition 3.2 Let , and let . The fuzzy set-valued mapping is said to be lower semicontinuous at if
and is said to be upper semicontinuous at if
The fuzzy set-valued mapping is said to be continuous at if is both lower and upper semicontinuous at , that is,
Example 3.2 Let be fuzzy sets defined as and for each , and let be fuzzy set-valued mappings defined as
for each . Since and , is lower semicontinuous at 0 but not upper semicontinuous at 0. Since and , is upper semicontinuous at 0 but not lower semicontinuous at 0.
From Definition 1.3 and Propositions 1.3, 1.5, and 1.11, the following Proposition 3.1 can be obtained.
Proposition 3.1 Let , and let . In addition, let
for each . Then the following statements hold.
-
(i)
.
-
(ii)
for .
-
(iii)
.
-
(iv)
for .
-
(v)
Let , and assume that is convex-valued. Then .
From Propositions 1.12 and 3.1, the following Proposition 3.2 can be obtained.
Proposition 3.2 Let , and let . Then the following statements hold.
-
(i)
.
-
(ii)
If is convex-valued, then .
The following Proposition 3.3 shows that Definition 3.1 is a fuzzified one of Definition 1.3. It can be derived from Proposition 1.16.
Proposition 3.3 Let , and let . Then
From Propositions 1.16, 2.4, and 3.3, the following Proposition 3.4 can be obtained.
Proposition 3.4 Let , , and assume that is defined as for each . In addition, let , and let , for each . Then and .
The following Proposition 3.5 is a fuzzified one of Proposition 1.7. It can be derived from Proposition 3.3.
Proposition 3.5 Let , and let . Then if and only if for any sequence such that .
4 Derivatives of fuzzy set-valued mappings
In this section, we propose the definition of derivatives of fuzzy set-valued mappings and investigate their properties.
The following Definition 4.1 is a fuzzified one of Definition 1.5.
Definition 4.1 Let , and let . Then the fuzzy set
is called the fuzzy tangent cone or fuzzy contingent cone of at .
Let , and let . Then it can be seen that
Thus, the fuzzy tangent cone is a generalization of the crisp one.
Example 4.1 Let be a function defined as for each , and let be a fuzzy set defined as, for each ,
if , and
if . In addition, let . For each , since , it follows that from Example 1.4. Therefore, for each , we have
if , and
if .
The following Proposition 4.1 is a fuzzified one of Proposition 1.8. It can be derived from Propositions 1.8 and 1.12.
Proposition 4.1 Let , and let . Then the following statements hold.
-
(i)
is a closed fuzzy cone.
-
(ii)
If , then is a closed convex fuzzy cone.
The following Proposition 4.2 is a fuzzified one of Proposition 1.9. It can be derived from Propositions 1.9 and 1.11.
Proposition 4.2 Let with , and let . Then .
The following Definition 4.2 is a fuzzified one of the graph for crisp set-valued mappings.
Definition 4.2 Let . The fuzzy set defined as
for each is called the fuzzy graph of .
From Definition 4.2, it can be seen that
for .
Let , and assume that is defined as for each . Then it follows that
for each . Thus, the fuzzy graph for fuzzy set-valued mappings is a generalization of the crisp one.
The following Definition 4.3 is a fuzzified one of Definition 1.6.
Definition 4.3 Let , and let . Then the fuzzy set-valued mapping such that
is called the fuzzy contingent derivative of at .
From Definition 4.3, it can be seen that
for each and each .
Let , and let . Assume that is defined as for each . Then it follows that
that is, , and that
that is,
Thus, the fuzzy contingent derivative for fuzzy set-valued mappings is a generalization of the crisp one.
Example 4.2 Let be the function defined in Example 4.1, and let be a fuzzy set-valued mapping defined as, for each ,
for each if , and
for each if . In addition, let be the fuzzy set defined in Example 4.1. Then it follows that and . From Example 4.1, we have, for each and each ,
if , and
Proposition 4.3 Let , and let . Then for any .
Proof Fix any and any . Then we have
□
The following Proposition 4.4 is a fuzzified one of Proposition 1.10. It can be derived from Proposition 4.1.
Proposition 4.4 Let , and let . Then the following statements hold.
-
(i)
is closed-valued.
-
(ii)
If , then is closed convex-valued.
5 Conclusions
In this article, we proposed definitions of limits of sequences of fuzzy sets, and limits and derivatives of fuzzy set-valued mappings based on level sets of fuzzy sets, and investigated their properties. They are fuzzified ones of them for crisp ones.
Derived results are very general in the sense that they deal with all fuzzy sets, especially fuzzy sets which are not support bounded.
Consider some fuzzy mathematical model whose optimal value/solution output is a fuzzy set for an input parameter. The concepts of limits of sequences of fuzzy sets, limits and derivatives of fuzzy set-valued mappings are necessary and important for stability theory and sensitivity analysis for such fuzzy mathematical models. Then derived results can be expected to be useful for them.
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MK derived all results communicating with HK. All authors read and approved the final manuscript.
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Kon, M., Kuwano, H. On sequences of fuzzy sets and fuzzy set-valued mappings. Fixed Point Theory Appl 2013, 327 (2013). https://doi.org/10.1186/1687-1812-2013-327
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DOI: https://doi.org/10.1186/1687-1812-2013-327