Using the properties of , integrals, and differential inequalities, we will establish the comparison principles in this section.
Definition 4.1 Define by
for all .
Theorem 4.1 Let be continuous with respect to and of uniformly bounded variation on T. Suppose
for all and , where and is nondecreasing with respect to φ for all . Suppose the maximal solution of the scalar differential equation
(2)
exists on T. Then, if , are any two solutions of (1) through , on T, respectively, and we have
for all .
Proof Let . Then . Thus, we get
for all . By Theorem 1.9.2 in [46], we obtain for all . □
Example 4.1 Define by the level sets of the fuzzy mapping
where is the Mareš core of , for all . Thus, we have
It is obvious that F is continuous with respect to . Since
for all , we see that F is of uniformly bounded variation. Define by
where the multiplication in is defined by Definition 2.6. It is obvious that f is continuous with respect to and of uniformly bounded variation. By Definition 2.10, we have
for all and . Define the scalar differential equation
where the function and . It is obvious that for all , and is nondecreasing with respect to φ for all . Then
for all and . Hence, we obtain
for all and . Let and , such that . By Theorem 4.1, we conclude that
for all , where , are two solutions of fuzzy differential equation
through , on .
Corollary 4.1 Let be continuous with respect to and of uniformly bounded variation on T. Suppose
for all and , where g satisfies the assumptions of Theorem 4.1. Then, if , we have
for all , where is any solution of (1) through on T and is the maximal solution of the scalar differential equation (2) on T.
Proof In fact, we can show this corollary by a similar method to Theorem 4.1. □
Theorem 4.2 Let be continuous with respect to and of uniformly bounded variation on T. Suppose
for all and , where . Suppose the maximal solution of the scalar differential equation (2) exists on T. Then, if , are any two solutions of (1) through , on T, respectively, and we have
for all .
Proof Let . Then and for any fixed and with , we have
Since
and
we get
Thus, by Definition 4.1, we get
By Theorem 1.4.1 in [46], we obtain , for all . □
Corollary 4.2 Let be continuous with respect to and of uniformly bounded variation on T. If
for all and , where g satisfies the assumptions of Theorem 4.2, then, if , we have
for all , where is any solution of (1) through on T and is the maximal solution of the scalar differential equation (2) on T.
Proof In fact, we can show this corollary by a similar method to Theorem 4.2. □
Theorem 4.3 Let be continuous with respect to and of uniformly bounded variation on T. Suppose
for all and , where . Suppose the maximal solution of the scalar differential equation (2) exists on T. Then, if , are any two solutions of (1) through , on T, respectively, and , we have
for all .
Proof Let . Then and for any fixed and with , from the proof of Theorem 4.2, we have
which implies that
By Theorem 1.4.1 in [46], we obtain , for all . □
Corollary 4.3 Let be continuous with respect to and of uniformly bounded variation on T. Suppose
for all and , where g satisfies the assumptions of Theorem 4.3. Then, if , we have
for all , where is any solution of (1) through on T and is the maximal solution of the scalar differential equation (2) on T.
Proof In fact, we can show this corollary by a similar method to Theorem 4.3. □
Example 4.2 Define two fuzzy mappings by the level sets
and
for all , where and are the Mareš core of and , respectively, for all . Thus, we have
for all and . It is obvious that and are continuous from the right at 0 and continuous from the left on with respect to α. Since and are decreasing with respect to α and by Lemma 2.4, we get
and
Thus, we see that and are of uniformly bounded variation. Since and are uniformly continuous with respect to , we see that and are continuous with respect to . Define by
where the addition and multiplication in are defined by Definitions 2.7 and 2.8, respectively. It is obvious that f is continuous with respect to and of uniformly bounded variation. By Definition 2.10, we get
for all sufficiently small. Hence
for all and . Define the scalar differential equation
where the function and . It is obvious that for all , . Then
for all and . Hence, we obtain
for all and . Let and , such that . By Theorem 4.3, we conclude that
for all , where , are two solutions of the fuzzy differential equation
through , on .