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Sun et al. Complex Eng Syst 2022;2:17 I http://dx.doi.org/10.20517/ces.2022.48 Page 9 of 15

Using Definition 1, we have that for ≥ 0 and ∈ Γ,

− 0
+ 0
Λ( , ( ), ( )) ≤ 1 J −

ln

≤ 1 0 J −

( )
ln
− −
= 2 J , (25)
where 2 = 1 . This implies
0

2
| ( )| ≤ . (26)
1
Therefore, for all ∈ N, we obtain

Λ( , ( ), ( )) ≤ 2 .
This means that the unique solution ( ) will not explode for ∈ [ , +1 ) and ∈ N. Hence, there exists a
unique global solution { ( ), ≥ 0} for SSS (1). Moreover, from (25), we obtain that

sup E| ( )| < ∞.
− ≤ ≤∞
The proof is completed. □

Remark 3 To deal with the time-varying delay , some new inequalities (e.g., see (15) and (16) for details) are
constructed in the proof for Theorem 1. Compared with the results reported in existing studies [21–25] , the time
delay in this paper is merely a Borel-measurable function, which invalidates these existing methods. By virtue
of Lemma 1, a more general form of time delay can be imposed on system (1).
We now refer to the equation (25) in the proof of Theorem 1. The following theorem provides sufficient con-
ditions for the th exponential stability of system (1) .
Theorem 2 UnderthesameconditionsasthoseconsideredinTheorem1, thesolutionofsystem(1)withtheinitial
value (2) is th moment exponentially stable. That is,

1
lim sup ln | ( )| < 0. (27)
→∞
Proof. Applying (25) yields
( )
ln
− −
Λ( , ( ), ( )) ≤ 2 J .
Recalling condition (7), we have
( )
ln
− −

1 | ( )| ≤ 2 J . (28)
Hence, from (12), we observe that
( )
ln ( )
1 1 − − J ln
lim sup ln | ( )| ≤ lim sup ln 3 = − − < 0,
→∞ →∞ J
2
where 3 = , which is the required assertion in (27). The proof is completed.
1