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Some properties of the interval-valued -integrals and a standard interval-valued -convolution
Journal of Inequalities and Applications volume 2014, Article number: 88 (2014)
Abstract
Pap and Stajner (Fuzzy Sets Syst. 102:393-415, 1999) investigated a generalized pseudo-convolution of functions based on pseudo-operations. Jang (Fuzzy Sets Syst. 222:45-57, 2013) studied the interval-valued generalized fuzzy integral by using an interval-representable pseudo-multiplication.
In this paper, by using the concepts of interval-representable pseudo-multiplication and g-integral, we define the interval-valued -integral represented by its interval-valued generator and a standard interval-valued -convolution by means of the corresponding interval-valued -integral. We also investigate some characterizations of the interval-valued -integral and a standard interval-valued -convolution.
MSC:28E10, 28E20, 03E72, 26E50, 11B68.
1 Introduction
Benvenuti and Mesiar [1], Daraby [2], Deschrijver [3], Grbic et al. [4], Klement et al. [5], Mesiar et al. [6], Stajner-Papuga et al. [7], Sugeno [8], Sugeno and Murofushi [9], Wu et al. [10, 11] have been studying pseudo-multiplications and various pseudo-integrals of measurable functions. Markova and Stupnanova [12], Maslov and Samborskij [13], and Pap and Stajner [14] introduced a general notion of pseudo-convolution of functions based on pseudo-mathematical operations and investigated the idempotent with respect to a pseudo-convolution.
Many researchers [3, 4, 15–29] have studied the pseudo-integral of measurable multi-valued function, for example, the Aumann integral, the fuzzy integral, and the Choquet integral of measurable interval-valued functions, in many different mathematical theories and their applications.
Recently, Jang [26] defined the interval-valued generalized fuzzy integral by using an interval-representable pseudo-multiplication and investigated their characterizations. The purpose of this study is to define the interval-valued -integral represented by its interval-valued generator and a standard interval-valued -convolution by means of the corresponding interval-valued -integral, and to investigate an interval-valued idempotent function with respect to a standard interval-valued -convolution.
This paper is organized in five sections. In Section 2, we list definitions and some properties of a pseudo-addition, a pseudo-multiplication, a g-integral, and a g-convolution of functions by means of the corresponding g-integral. In Section 3, we define an interval-representable pseudo-addition, an interval-representable pseudo-multiplication, the interval-valued -integral represented by its interval-valued generator , and investigate some characterizations of the interval-valued -integral. In Section 4, we define a standard interval-valued -convolution by means of the corresponding interval-valued -integral and investigate some basic characterizations of them. In Section 5, we give a brief summary of results and some conclusions.
2 Definitions and preliminaries
Let X be a set, be a σ-algebra of X, and be a set of all measurable functions . We introduce a pseudo-addition and a pseudo-multiplication (see [1–4, 6, 7, 12, 14, 26, 30]).
Definition 2.1 ([12])
-
(1)
A binary operation is called a pseudo-addition if it satisfies the following axioms:
-
(i)
for all ,
-
(ii)
for all ,
-
(iii)
for all ,
-
(iv)
such that for all ,
-
(v)
, .
-
(2)
A binary operation is called a pseudo-multiplication with respect to ⊕ if it satisfies the following axioms:
-
(i)
for all ,
-
(ii)
for all ,
-
(iii)
such that for all ,
-
(iv)
for all ,
-
(v)
for all ,
-
(vi)
for all .
If is a generating function for a semigroup , then the pseudo-operations are of the following forms:
and
Definition 2.2 ([6])
A set function is called a -measure if it satisfies the following axioms:
-
(i)
,
-
(ii)
for any sequence of pairwise disjoint sets from , where .
Let be the set of all measurable functions . We introduce the g-integral with respect to a fuzzy measure induced by a pseudo-addition ⊕ and a pseudo-multiplication ⊙ in Remark 2.1.
Definition 2.3 ([6])
-
(1)
Let be a continuous strictly monotone increasing surjection function such that and . The g-integral of f on A is defined by
(3)
where dx is related to the Lebesgue measure and the integral on the right-hand side is the Lebesgue integral.
-
(2)
f is said to be integrable if .
Let be the set of all integrable functions. Then we obtain some basic properties of the g-integral with respect to a fuzzy measure.
Theorem 2.2 (1) If , and , then we have
-
(2)
Let be a continuous strictly monotone increasing surjection function such that , ⊕, ⊙ are the same pseudo-operations as in Remark 2.1. If , , then we have
(5) -
(3)
Let be a continuous strictly monotone increasing surjection function such that , ⊕, ⊙ are the same pseudo-operations as in Remark 2.1, and for . If , , , then we have
(6)
Proof (1) Note that if and , then
By Definition 2.3(1), (7), and the monotonicity of the Lebesgue integral,
-
(2)
By Definition 2.3(1) and the additivity of the Lebesgue integral,
(9) -
(3)
By Definition 2.3(1) and the linearity of the Lebesgue integral,
(10)
By using the g-integral, we define the g-convolution of functions by means of the corresponding g-integral (see [2, 12–14]). □
Definition 2.4 ([14])
Let g be the same function as in Theorem 2.2, let ⊕, ⊙ be the same pseudo-operations as in Remark 2.1, for , and . The g-convolution of f and h by means of the g-integral is defined by
for all .
Finally, we introduce the following basic characterizations of the g-convolution in [14].
Theorem 2.3 ([14])
If g is the same function as in Theorem 2.2, ⊕, ⊙ are the same pseudo-operations as in Remark 2.1, for , and , then we have
for all .
Theorem 2.4 ([14])
If g is the same function as in Theorem 2.2, ⊕, ⊙ are the same pseudo-operations as in Remark 2.1, for , and , then we have
and
3 The interval-valued -integrals
In this section, we consider the intervals, a standard interval-valued pseudo-addition, and a standard interval-valued pseudo-multiplication. Let be the set of all closed intervals (for short, intervals) in Y as follows:
where Y is or . For any , we define . Obviously, (see [1, 21–29]).
Definition 3.1 ([26])
If for all and , and , then we define arithmetic, maximum, minimum, order, inclusion, superior, and inferior operations as follows:
-
(1)
,
-
(2)
,
-
(3)
,
-
(4)
,
-
(5)
,
-
(6)
if and only if and ,
-
(7)
if and only if and ,
-
(8)
if and only if and ,
-
(9)
,
-
(10)
,
-
(11)
, and
-
(12)
.
Definition 3.2 ([26])
-
(1)
A binary operation is called a standard interval-valued pseudo-addition if there exist pseudo-additions and such that for all , and such that for all ,
(16)
Then and are called the representants of ⨁.
-
(2)
A binary operation is called a standard interval-valued pseudo-multiplication if there exist pseudo-multiplications and such that for all , and such that for all ,
(17)
Then and are called the representants of ⨀.
Theorem 3.1 If two pseudo-additions and are representants of a standard interval-valued pseudo-addition ⨁, two pseudo-multiplications and are representants of a standard interval-valued pseudo-multiplication ⨀, then we have
-
(1)
for all ,
-
(2)
for all ,
-
(3)
for all ,
-
(4)
for all ,
-
(5)
for all .
Proof (1) By the commutativity of and , for any , we have
-
(2)
By the associativity of and , for any , we have
(19) -
(3)
By the commutativity of and , for any , we have
(20) -
(4)
By the associativity of and in Definition 2.1(2)(ii), for any , we have
(21) -
(5)
By the distributivity of and for in Definition 2.1(2)(iv), for any , we have
(22)
By using a standard interval-valued pseudo-addition and a standard interval-valued pseudo-multiplication, we define the interval-valued -integral represented by its interval-valued generator . □
Definition 3.3 Let X be a set, two pseudo-additions and be representants of a standard interval-valued pseudo-addition ⨁, and two pseudo-multiplications and be representants of a standard interval-valued pseudo-multiplication ⨀.
-
(1)
An interval-valued function is said to be measurable if for any open set ,
(23) -
(2)
Let be a continuous strictly increasing surjective function for such that , , and for . The interval-valued -integral with respect to a fuzzy measure μ of a measurable interval-valued function is defined by
(24)
for all .
-
(3)
is said to be integrable on if
(25)
Let be the set of all measurable interval-valued functions and be the set of all integrable interval-valued functions. Then, by Definition 3.3, we directly obtain the following theorem.
Theorem 3.2 If is a continuous strictly increasing surjective function for such that , , and for , two pseudo-additions and are representants of a standard interval-valued pseudo-addition ⨁, and two pseudo-multiplications and are representants of a standard interval-valued pseudo-multiplication ⨀, then we have
Proof By Definition 2.3(1),
for . By (27) and Definition 3.3, we have
By the definition of the interval-valued -integral, we directly obtain the following basic properties. □
Theorem 3.3 Let be a continuous strictly increasing surjective function for such that , , and for , two pseudo-additions and be representants of a standard interval-valued pseudo-addition ⨁, two pseudo-multiplications and be representants of a standard interval-valued pseudo-multiplication ⨀, and two pseudo-multiplications and be representants of a standard interval-valued pseudo-multiplication ⨂.
-
(1)
If and and , then we have
(29) -
(2)
If and , then we have
(30) -
(3)
If and , , then we have
(31)
Proof (1) Note that if and , then
for . Since and are strictly monotone increasing,
for . By (33) and Theorem 2.2(1),
for . By (34) and Theorem 3.2,
-
(2)
Note that if , then
(36)
By Theorem 2.2(2),
for . By (37) and Theorem 3.2,
-
(3)
Note that if and , then
(39)
By Theorem 2.2(3),
for . By (40) and Definition 3.3(2),
□
4 An interval-valued -convolution
In this section, by using the interval-valued -integral, we define the interval-valued -convolution of interval-valued functions in .
Definition 4.1 If , ⨁, ⨀, and ⨂ satisfy the hypotheses of Theorem 3.2, then the interval-valued -convolution is defined by
for all .
From Definition 4.1, we directly obtain some characterization of an interval-valued -convolution by means of the interval-valued -integrals.
Theorem 4.1 If , ⨁, ⨀, and ⨂ satisfy the hypotheses of Theorem 3.2, then we have
where for .
Proof By Definition 2.4, we have
for . By Theorem 3.2 and (44),
□
From Theorem 4.1, we investigate the commutativity and the associativity of a standard interval-valued -convolution.
Theorem 4.2 If , ⨁, ⨀, and ⨂ satisfy the hypotheses of Theorem 3.2 and , , and , then we have
-
(1)
,
-
(2)
.
Proof Let , , . By (16), we have
By Theorem 4.1 and (46), we have
By (17), we have
By Theorem 4.1 and (48),
□
Finally, we illustrate the following examples which are related with the interval-valued -integral and the interval-valued -convolution as follows.
Example 4.1 We give three examples of the interval-valued -integral.
-
(1)
If for all are the generators of , , , and , and for all , and for all , then we have
(50) -
(2)
If , for all are the generators of , , and for all are the generators of , , and for all , and for all , then we have
(51) -
(3)
If , for all are the generators of , , and for all are the generators of , , and for all , and for all , then we have
(52)
Example 4.2 We give an example of the interval-valued -convolution.
If , for all are the generators of , , and for all are the generators of , , , , and for all , for all , and for all , then we have
5 Conclusions
In this paper, we have considered the g-integral represented by its generating g, the pseudo-addition, the pseudo-multiplication (see Definition 2.3). This study was to define the g-convolution by means of the g-integral (see Definition 2.4) and to investigate some characterizations of the g-integral and the commutativity and the associativity of the g-convolution (see Theorems 2.2, 2.3, and 2.4).
We also defined the interval-valued -integral represented by its interval-valued generator . By using general notions of an interval-representable pseudo-multiplication (see Definition 3.2), we defined an interval-valued -integral (see Definition 3.3) and investigated some basic characterizations of them (see Theorems 3.2, 3.3).
From Definitions 2.3, 2.4, and Theorems 2.2, 2.3, we defined a standard interval-valued -convolution (see Definition 4.1). We also investigated some characterizations of a standard interval-valued -convolution of interval-valued functions by means of the interval-valued -integral including commutativity and associativity of an interval-representable convolution (see Theorems 4.1, 4.2).
In the future, we can study various inequalities of the interval-valued -integral and expect that the standard interval-valued -convolutions are used (i) to generalize the g-Laplace transform, Hamilton-Jacobi equation on the space of functions, such as in nonlinearity and optimization and such as in information theory (see [1, 14, 29]); (ii) to generalize the Stolasky-type inequality for the pseudo-integral of functions such as in economics, finance, decision making (see [2, 30]), etc.
References
Benvenuti P, Mesiar R: Pseudo-arithmetical operations as a basis for the general measure and integration theory. Inf. Sci. 2004, 160: 1-11. 10.1016/j.ins.2003.07.005
Daraby B: Generalization of the Stolarsky type inequality for the pseudo-integrals. Fuzzy Sets Syst. 2012, 194: 90-96.
Deschrijver G: Generalized arithmetic operators and their relationship to t -norms in interval-valued fuzzy set theory. Fuzzy Sets Syst. 2009, 160: 3080-3102. 10.1016/j.fss.2009.05.002
Grbic T, Stajner I, Strboja M: An approach to pseudo-integration of set-valued functions. Inf. Sci. 2011, 181: 2278-2292. 10.1016/j.ins.2011.01.038
Klement EP, Mesiar R, Pap E: Triangular Norms. Kluwer Academic, Dordrecht; 2000.
Mesiar R, Pap E: Idempotent integral as limit of g -integrals. Fuzzy Sets Syst. 1999, 102: 385-392. 10.1016/S0165-0114(98)00213-9
Stajner-Papuga I, Grbic T, Dankova M: Pseudo-Riemann Stieltjes integral. Inf. Sci. 2010, 180: 2923-2933.
Sugeno, M: Theory of fuzzy integrals and its applications. Doctorial Thesis, Tokyo Institute of Technology, Tokyo (1974)
Sugeno M, Murofushi T: Pseudo-additive measures and integrals. J. Math. Anal. Appl. 1987, 122: 197-222. 10.1016/0022-247X(87)90354-4
Wu C, Wang S, Ma M: Generalized fuzzy integrals: Part 1. Fundamental concept. Fuzzy Sets Syst. 1993, 57: 219-226. 10.1016/0165-0114(93)90162-B
Wu C, Ma M, Song S, Zhang S: Generalized fuzzy integrals: Part 3. Convergence theorems. Fuzzy Sets Syst. 1995, 70: 75-87. 10.1016/0165-0114(94)00232-V
Markova-Stupnanova A: A note on the idempotent function with respect to pseudo-convolution. Fuzzy Sets Syst. 1999, 102: 417-421. 10.1016/S0165-0114(98)00215-2
Maslov VP, Samborskij SN Advances in Soviet Mathematics 13. In Idempotent Analysis. Am. Math. Soc., Providence; 1992.
Pap E, Stajner I: Generalized pseudo-convolution in the theory of probabilistic metric spaces, information, fuzzy numbers, optimization, system theory. Fuzzy Sets Syst. 1999, 102: 393-415. 10.1016/S0165-0114(98)00214-0
Aubin JP: Set-Valued Analysis. Birkhäuser Boston, Boston; 1990.
Aumann RJ: Integrals of set-valued functions. J. Math. Anal. Appl. 1965, 12: 1-12. 10.1016/0022-247X(65)90049-1
Grabisch M: Fuzzy integral in multicriteria decision making. Fuzzy Sets Syst. 1995, 69: 279-298. 10.1016/0165-0114(94)00174-6
Guo C, Zhang D: On set-valued measures. Inf. Sci. 2004, 160: 13-25. 10.1016/j.ins.2003.07.006
Jang LC, Kil BM, Kim YK, Kwon JS: Some properties of Choquet integrals of set-valued functions. Fuzzy Sets Syst. 1997, 91: 61-67.
Jang LC, Kwon JS: On the representation of Choquet integrals of set-valued functions and null sets. Fuzzy Sets Syst. 2000, 112: 233-239. 10.1016/S0165-0114(98)00184-5
Jang LC, Kim T, Jeon JD: On the set-valued Choquet integrals and convergence theorems (II). Bull. Korean Math. Soc. 2003,40(1):139-147.
Jang LC: Interval-valued Choquet integrals and their applications. J. Appl. Math. Comput. 2004,16(1-2):429-445.
Jang LC: A note on the monotone interval-valued set function defined by the interval-valued Choquet integral. Commun. Korean Math. Soc. 2007, 22: 227-234. 10.4134/CKMS.2007.22.2.227
Jang LC: On properties of the Choquet integral of interval-valued functions. J. Appl. Math. 2011., 2011: Article ID 492149
Jang LC: A note on convergence properties of interval-valued capacity functionals and Choquet integrals. Inf. Sci. 2012, 183: 151-158. 10.1016/j.ins.2011.09.011
Jang LC: A note on the interval-valued generalized fuzzy integral by means of an interval-representable pseudo-multiplication and their convergence properties. Fuzzy Sets Syst. 2013, 222: 45-57.
Jang LC: Some characterizations of the Choquet integral with respect to a monotone interval-valued set function. Int. J. Fuzzy Log. Intell. Syst. 2013,13(1):75-81.
Wang Z, Li KW, Wang W: An approach to multiattribute decision making with interval-valued intuitionistic fuzzy assessments and in complete weights. Inf. Sci. 2009, 179: 3026-3040. 10.1016/j.ins.2009.05.001
Wechselberger K: The theory of interval-probability as a unifying concept for uncertainty. Int. J. Approx. Reason. 2000, 24: 149-170. 10.1016/S0888-613X(00)00032-3
Pap E, Ralevic N: Pseudo-Laplace transform. Nonlinear Anal. 1998, 33: 533-550. 10.1016/S0362-546X(97)00568-3
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This paper was supported by Konkuk University in 2014.
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Jang, LC. Some properties of the interval-valued -integrals and a standard interval-valued -convolution. J Inequal Appl 2014, 88 (2014). https://doi.org/10.1186/1029-242X-2014-88
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DOI: https://doi.org/10.1186/1029-242X-2014-88