## Fuzzy Linear and Nonlinear Integral Equations: Numerical Methods

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# On Properties of Pseudointegrals Based on Pseudoaddition Decomposable Measures

**Academic Editor:**Soheil Salahshour

#### Abstract

We mainly discussed pseudointegrals based on a pseudoaddition decomposable measure. Particularly, we give the definition of the pseudointegral for a measurable function based on a strict pseudoaddition decomposable measure by generalizing the definition of the pseudointegral of a bounded measurable function. Furthermore, we got several important properties of the pseudointegral of a measurable function based on a strict pseudoaddition decomposable measure.

#### 1. Introduction

The classical measure theory is one of the most important theories in mathematics [1, 2]. Although the additive measures are widely used, they do not allow modelling many phenomena involving interaction between criteria. For this reason, the fuzzy measure proposed by Sugeno is an extension of classical measure in which the additivity is replaced by a weaker condition, that is, monotonicity [3, 4]. Therefore, fuzzy measure and the corresponding integrals, for example, Choquet and Sugeno, are introduced [5–10].

So far, there have been many different fuzzy measures, such as the decomposable measure, the -additive measure, the belief measure, the possibility measure, and the plausibility measure. Among the fuzzy measures mentioned before, the decomposable measure was independently introduced by Dubois and Prade [11] and Weber [12]. Since the close relations with the classical measure theory, further developments of decomposable measures and related integrals have been extensive [13–18]. Decomposable measures include several well-known fuzzy measures such as the -additive measure and probability and possibility measures, and they provide a natural setting for relaxing probabilistic assumptions regarding the modeling of uncertainty [19, 20]. Decomposable measures and the corresponding integrals are very useful in decision theory and the theory of nonlinear differential and integral equations [21–24].

In many problems with uncertainty as in the theory of probabilistic metric spaces [20, 25, 26], multivalued logics [27, 28], and general measures [1, 4] often we work with many operations different from the usual addition and multiplication of reals. Some of them are triangular norms, triangular conorms, pseudoadditions, pseudomultiplications, and so forth [21, 29]. Based on the above-mentioned measures, pseudoanalysis as a generalization of the classical analysis is developed, where instead of the field of real numbers a semiring is taken on a real interval endowed with pseudoaddition and with pseudomultiplication (see [13, 19, 30–33]). The families of the pseudooperations generated by a function turn out to be solutions of well-known nonlinear functional equations [22–24].

In this paper, we will discuss pseudointegrals based on pseudoaddition decomposable measures. In Section 2, we recall the concepts of the pseudoaddition and the pseudomultiplication , which form a real semiring on the interval and the notion of the --decomposable measure. Then we will give the definition of the pseudointegral of a measurable function based on a strict pseudoaddition decomposable measure by generalizing the definition of the pseudointegral of a bounded measurable function. In Section 3, we will discuss several important properties of the pseudointegral of a measurable function based on the strict pseudoaddition decomposable measure.

#### 2. Preliminaries

Let be a closed subinterval of (in some cases we will also take semiclosed subintervals). The total order on will be denoted by . This can be the usual order of the real line, but it can also be another order. We will denote by maximum element on (usually is either or ) with respect to this total order.

*Definition 1 (see [34]). *Let be a sequence from .(1)If whenever , then we say that the sequence is an increasing sequence.(2) If whenever , then we say that the sequence is a strict increasing sequence.(3) If whenever , then we say that the sequence is a decreasing sequence.(4) If whenever , then we say that the sequence is a strict decreasing sequence.

Let be a nonempty set; we will denote by , , and algebra, -algebra, and Borel -algebra of subsets of a set , respectively.

Denote by the set of all functionals from to . For each the constant functional in with value will also be denoted by . It will be clear from the context which usage is intended. A functional is said to be finite if for all . The functional is said to be bounded if there exists , such that for all . Denote by the set of all bounded functionals.

Let and be two functions defined on and with values in and let be arbitrary binary operation on . Then, we define for any and for any , . Let be a subset of . If for all , then is -closed. The total order on induces a partial order on defined pointwise by stipulating that if and only if for all . Thus is a poset, and whenever we consider as a poset then it will always be with respect to this partial order. Let .

*Definition 2 (see [35]). *A binary operation is called a pseudoaddition, if it satisfies the following conditions, for all :(1), where is a zero element (usually is either or ) (boundary condition);(2) whenever and (monotonicity);(3) (commutativity);(4) (associativity).

A pseudoaddition is said to be continuous if it is a continuous function in ; a pseudoaddition is called strict if is continuous and strictly monotone. The following are examples of pseudoadditions: if and only if ; , where is a strictly monotone and continuous generator surjective function and if and only if . It is obvious that for all .

Let . In this paper, we assume .

*Definition 3 (see [35]). *A binary operation is called a pseudomultiplication, if it satisfies the following conditions, for all :(1), where is a unit element (boundary condition);(2) whenever and (monotonicity);(3) (commutativity);(4) (associativity).

A pseudomultiplication is said to be continuous if it is a continuous function in . The following are examples of pseudomultiplications: if and only if ; , where is a strictly monotone and continuous generator surjective function and if and only if . It is obvious that .

We assume also that and that is a distributive pseudomultiplication with respect to ; that is, The structure is called a real semiring.

Because of the associative property of the pseudoaddition , it can be extended by induction to -ary operation by setting Due to monotonicity, for each sequence of elements of , the following limit can be considered:

*Definition 4 (see [36]). *Let be a nonempty set and a pseudoaddition. A binary operation is called a pseudometric on , if it satisfies the following conditions, for all :(1) if and only if ;(2);(3)there exists such that
where is a distributive pseudomultiplication with respect to .

Let be a sequence from . The sequence is said to be convergent, if for any , there exists positive integer , such that for all , denoted by , and is said to be the limit of the sequence ; is said to be the lower limit of the sequence ; is said to be the upper limit of the sequence . It is obvious that . Let be a sequence from . The sequence is said to be convergent, if for any , and for each point , there exists positive integer , such that for all , denoted by , and is said to be the limit functional of the functionals sequence .

Let be a subset of . The poset is said to be upper complete if for each increasing sequence from ; the poset is said to be lower complete if for each decreasing sequence from ; the poset is said to be complete if for each sequence from , where the limit of the sequence of functionals is given by for all .

For any continuous pseudoaddition and with , there exists at least one point such that . If pseudoaddition is strict, then there exists only one point such that for all with . Thus we have the following concepts.

*Definition 5 (see [34]). *For any continuous pseudoaddition and with , the paracomplement set is a nonempty set of all points such that .

*Example 6. *Let the total order on be the usual order of the real line and let the pseudoaddition be the usual multiplication of the real numbers. It is obvious that zero element is . If , then and . If , then for any , we have .

*Definition 7 (see [34]). *For any continuous pseudoaddition , if , then define the paracomplement set as the set of all those functionals such that
for all .

*Definition 8 (see [34]). *For any strict pseudoaddition and with , the complement is defined as

*Definition 9 (see [34]). *For any strict pseudoaddition , if , then define the complement functional pointwise as
for all .

*Definition 10 (see [34]). *For any pseudoaddition , a nonempty subset of is said to be a functional space with respect to , denoted by , if for all and , where is a distributive pseudomultiplication with respect to .

It is clear that is the greatest functional space with respect to any pseudoaddition . Thus the functional space with is also called a subspace of . If is a functional space with respect to , then we just write instead of whenever can be determined from the context.

*Definition 11 (see [34]). *For each subset of the upper closure of , denoted by , is the set of all elements of having the form for some increasing sequence from .

It follows from Definition 11 that and if and only if is upper complete.

*Definition 12 (see [34]). *For any continuous pseudoaddition , a subspace will be called paracomplemented if for all ; for any strict pseudoaddition , a subspace will be called complemented if for all .

*Definition 13 (see [34]). *For any continuous pseudoaddition , a paracomplemented subspace is regular if it contains and is closed under ; for any strict pseudoaddition , a complemented subspace is normal if it contains and is closed under .

Note that for all and thus a paracomplemented subspace of is -closed if and only if it is -closed. It is obvious that regular and normal are closed under .

*Definition 14 (see [37]). *The pseudocharacteristic function of a set is defined with
where is zero element for and is unit element for .

*Definition 15 (see [21]). *A functional is said to be elementary if it has the following representation:
for each and pairwise disjoint and with , and the set of such elementary functionals will be denoted by . It is obvious that , for all .

*Definition 16 (see [21]). *A set function (or semiclosed interval) is called a --decomposable measure if it satisfies the following conditions:(1);
(2) for all with ;(3) for all and ;(4) for any sequence of pairwise disjoint sets from .

A pair consisting of a nonempty set and a -algebra of subsets of is called a measurable space. A functional is said to be a measurable functional if . Let be the set of all measurable mappings from to ; that is, Then will denote the set of those elements for which for each . In particular, this means that . Denote by the set of all bounded measurable functionals.

*Definition 17 (see [38]). *Let be a continuous pseudoaddition and a --decomposable measure. Let be a sequence of measurable functionals of a.e. pseudofinite on . If there exists a measurable functional of a.e. pseudofinite on , such that
for arbitrary , then the functionals sequence is said to be convergent to with respect to -measure, denoted by . If the functionals sequence does not converge to with respect to -measure, denote by .

*Definition 18 (see [35]). *Let be a continuous pseudoaddition and a --decomposable measure.(i)If , then the pseudointegral of an elementary measurable function is defined by
for and pairwise disjoint and with .(ii)If and is the sequence of elementary measurable functions such that, for each ,
where a sequence of elementary functions from the previous definition is constructed in [34], then the pseudointegral of a bounded measurable function is defined by

If there exists an increasing sequence of sets with , , such that , then we say that is -finite set of -measure and is a -measure finite and monotone cover of . The sequence of bounded measurable functionals is given by , and . It is obvious that is an increasing functionals sequence.

*Definition 19. *Let be a strict pseudoaddition and a --decomposable measure. If is -finite of -measure and is a -measure finite and monotone cover of , then the pseudointegral of a measurable function is defined by

#### 3. Main Results

Lemma 20 (see [21]). *Let be a continuous pseudoaddition and a --decomposable measure. If , then for all , we have*(1)*;*(2) *;*(3)*If , then
*(4)*;*(5)*, where with and ;*(6)* whenever with .*

Theorem 21. *Let be a strict pseudoaddition and a --decomposable measure. If is -finite of -measure and . Let be two different -measure finite and monotone covers of and let be two different positive integer sequences with . Then
*

*Proof. *Let . Since is an increasing sequence, we have
for every positive integer . Let with and is an arbitrary positive integer. If , then we have
Since is a decreasing sequence and
by Theorem 3.3 in [38], we have
which implies that
In particular, let and . Then we have
for every positive integer . Hence, we get that
On the contrary, using a similar argument, we can obtain

In Theorem 21, put and . Then we can easily see that the pseudointegral in Definition 19 has a unique value. In particular, we can get some elementary properties of the pseudointegral in the following theorem.

Theorem 22. *Let be a strict pseudoaddition and a --decomposable measure. If there exists an increasing sequence of sets with , , such that , then for all , we have*(1)*;*(2)*;*(3)*;
*(4)*;*(5)*, where with and ;*(6)* whenever with .*

*Proof. *For (1) and (2), we only prove (1) holds. By a similar proof, we can prove (2) holds. Since
, we get that
which implies that
Thus, by (1) of Lemma 20, we have

(3) Since
, we get that
Thus, we have
By (3) of Lemma 20, we have
which implies that
Hence, we get that
which implies that
that is,

(4) If , then , . Thus, by (4) of Lemma 20, we have
Hence, we get that
that is,

(5) Since with , we have with and with . By (5) of Lemma 20, we have
which implies that

(6) Since , we have . By the monotonicity of --decomposable measure , we get that if , then . By (6) of Theorem 22, we have
which implies that

Theorem 23. *Let be a strict pseudoaddition and a --decomposable measure.*(1)*If and is a -finite set of -measure, then if and only if a.e. on .*(2)*If , then for any , .*

*Proof. *(1) Suppose . For arbitrary , let . Then we get that
Thus, we have . Since is arbitrary, we have .

Suppose a.e. on , that is, . By (6) of Theorem 22, we have

(2) If there exists , such that for all , then
For any , we have
which implies that

Lemma 24 (see [38]). *Let be a strict pseudoaddition. The function given by
**
is a pseudometric on with .*

Theorem 25. *Let be a strict pseudoaddition and a -finite set of -measure. If is a --decomposable measure, then for any ,
*

*Proof. *Let and . Then and are two -measure -finite sets of . By (4) of Theorem 22, we have
Thus, by (3) of Theorem 22, we have
which implies that
If , then we have
which implies that
Similarly, if , we can also get this conclusion.

Theorem 26. *Let be a strict pseudoaddition, and let be a -finite set of -measure and a --decomposable measure. If*(1)*;*(2)* a.e. on , , and ;*(3)*,**then and
*

*Proof. *Since on , by Theorem 3.8 in [38], there exists a subsequence of that a.e. converges to on . By Theorem 3.5 in [38], we have .

(I) Suppose . By (2) of Theorem 23, for arbitrary , there exists such that if with , we have
Since , there exists a natural number , such that for all , where . Thus, we get that
Hence, by Theorem 25, we have
By Lemma 24, we obtain that

(II) Suppose . For arbitrary , there exists with , such that
Thus, we have
that is, . Since the measurable functionals sequence satisfies(i) a.e. on ;(ii) on ,by (I), we get that there exists a natural number , such that
for all . Hence, by Theorem 25, we have
Consequently, we obtain that

Corollary 27. *If the condition (3) of Theorem 26 is replaced by a.e. on , then the conclusion of Theorem 26 holds.*

*Proof. *Since a.e. on , by Theorem 3.5 in [38], we have .

(I) Suppose . By Theorem 3.9 in [38], if a.e. on , then . By Theorem 26 (I), we have

(II) Suppose . Since is -finite set of -measure, there exists an increasing sequence of sets with , , such that . For any , , the sequence of measurable functionals satisfies(i) a.e. on , ;(ii) a.e. on , .By Theorem 3.9 in [38], we have on , .By (I) and proof of Theorem 26 (II), we have

Lemma 28. *Let be a strict pseudoaddition, and let be a -finite set of -measure and a --decomposable measure. If is a monotone sequence, then the sequence is convergence.*

*Proof. *If is an increasing sequence, then

If is a decreasing sequence, then
Thus, we have
By Theorem 3.2 in [38], we get that the sequence is convergent.

Theorem 29. *Let be a strict pseudoaddition and let be a -finite set of -measure and a --decomposable measure. If is an increasing sequence of measurable functionals on , then
*

*Proof. *Let be an increasing sequence of measurable functionals on . By Lemma 28, we get that the sequence of measurable functionals is convergent. Let . By Theorem 3.5 in [38], we have with on . By (4) of Theorem 22, we get that
which implies that

On the contrary, since is -finite set of -measure, there exists an increasing sequence of sets with , , such that . For any given integer , is an increasing sequence of measurable functionals and on , for all . Now we show that
For arbitrary ,(i)if , that is, for all , then , that is, . Thus, we have
(ii)if there exists , such that , then ; that is, for all ; it follows that ; that is, . Thus, we have