Proof. Let the zero solution of Equation (
10) be uniformly Lipschitz stable. Let
be an arbitrary. Without loss of generality, we assume
. From Condition 3, there exist
such that for any
the inequality
holds, where
is a solution of Equation (
10) with the initial data
.
From the inclusions and , there exist a function and a positive constant . Without loss of generality, we can assume . Choose the constant such that and . Therefore, .
Let
. Choose the initial function
such that
. Therefore,
, i.e.,
for
. Consider the solution
of the system in Equation (
1) for the chosen initial data
.
Let
From the choice of
and the properties of the function
applying condition
we get
. Therefore, the function
satisfies Equation (
18) for
with
, where
is a solution of Equation (
10) with initial data
.
Let
be an arbitrary number. We prove
For
, we get
, i.e., the inequality in Equation (
19) holds.
Assume Equation (
19) is not true.
Case 1. There exists a point such that for , , i.e., for . Then, from Condition 2(i), we obtain the inequalities for , i.e., for and, according to Condition 2(ii) of Theorem 1 with , it follows that Condition 3(i) of Lemma 2 is satisfied for the solution on the interval and .
According to Lemma 2, we get
From the inequality in Equation (
20) and Condition
, we obtain
The contradiction proves the validity of Equation (
19). From the inequality in Equation (
19) and Condition 2(i), we have Theorem 1.
Case 2. There exists a point such that for , . Then, as in Case 1 we get for . Let for a natural number j. According to Condition 2(iii) of Theorem 1, we obtain . The contradiction proves this case is not possible.
Case 3. There exists a natural number k such that for and . Therefore, . The contradiction proves this case is not possible.
The proof of globally uniformly Lipschitz stability is analogous so we omit it. □
Theorem 2. Let the conditions of Theorem 1 be satisfied where Condition 2(i) is replaced by:
the inequalities holds, where and there exists positive constant such that , for , and .
If the zero solution of Equation (10) is uniformly Lipschitz stable (uniformly globally Lipschitz stable), then the zero solution of Equation (1) is uniformly Lipschitz stable (uniformly globally Lipschitz stable). Proof. The proof is similar to the one in Theorem 1 where .
Theorem 3. (Dini fractional derivative/ Caputo fractional Dini derivative) Let the following conditions be satisfied:
- 1.
Assumptions A1–A8 are fulfilled.
- 2.
There exist a function , and
- (i)
The inequalitiesholds, where , . - (ii)
For any function for such that for any , k is a non-negative integer, such that the inequalityholds where is one of the following two derivatives: the Dini fractional derivative defined by Equation (4) or the Caputo fractional Dini derivative defined by Equation (5) and . - (iii)
For any and the inequalityholds.
- 3.
The zero solution of Equation (10) is uniformly Lipschitz stable (uniformly globally Lipschitz stable).
Then, the zero solution of Equation (1) is uniformly Lipschitz stable (uniformly globally Lipschitz stable). The proof of Theorem 3 is similar to the one in Theorem 1 where Lemma 4 is applied instead of Lemma 2.
Example 3. Let and . Consider the non-instantaneous impulsive fractional differential equationswhere , , and . Let , .
Let
be a solution of Equation (
22). Let the point
,
k is a non-negative integer, be such that
and
. Using the notation
and Assumption A2, it follows that
and therefore
or
and
Then, for all
and
, we get the inequality
In addition, for any natural number i, and , we get with .
According to Example 2, Case 1 and Theorem 1, the zero solution of Equation (
22) is uniformly Lipschitz stable. □