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Exponential stability for higher-order impulsive fractional neutral stochastic integro-delay . . .

(H 9 ) The cosine family of bounded linear oper- Remark 1. (1) We know that in the manu-
ators are {S α (ς)} ς≥0 and {T α (ς)} ς≥0 sat- script, 34 authors studied Ulam–Hyers sta-
isfy the further conditions: There exist bility for second-order non-instantaneous
a 1 > 0, a 2 > 0, α 1 ≥ 1, and α 2 ≥ 1 ∋ impulsive fractional neutral stochastic
differential equations. Also, 35 discussed
p
p
∥S α (ς)∥ ≤ a 1 e −α 1 ς ; ∥T α (ς)∥ ≤ a 2 e −α 2 ς , ς ≥ 0. the well posedness of second-order non-
instantaneous impulsive fractional neu-
tral stochastic differential equations. Here
they focused on Ulam-Hyers stability via
Lemma 5. 33 For any ω > 0, there exist α i > impulsive Gronwall’s inequality. We
study the exponential stability of the sys-
0 (i = 1, 2, 3, 4, 5, 6), b k , d k (k = 1, m), and a
tem (1) employing impulsive integral in-
function Π : [−r, ∞) → [0, ∞), r > 0 s.t
equality method.
Π(ς) ≤ α i e −ως , ς ∈ (−∞, 0].
(2) If the system (1) is uniformly exponen-
tially stable, then it is Ulam-Hyers stable.
 −σ 1 ς −σ 2 ς
 α 1 e + α 2 e , ς ∈ (−∞, 0] Means Ulam-Hyers is a weaker notion of
 −σ 1 ς −σ 2 ς
 α 1 e + α 2 e + α 3 sup Π(ς + θ) the stability than exponential. to make



 θ∈[−r,0] the converse of above statement is true,

 ς

R
 −σ 1 (ς−ϑ) on has to prove that the sequence of mild
 e sup Π(ϑ + θ)dϑ
 +α 4

0
 θ∈[−r,0] solution is ω-periodic.


 ς

R
 e −σ 2 (ς−ϑ) sup Π(ϑ + θ)dϑ
Π(ς) ≤ +α 5 (3) Exponential stability is a form of asymp-
0 θ∈[−r,0]
ς

 totic stability. Systems that are not lin-
 R
 −σ 1 (ς−ϑ)
 +α 6 e sup Π(ϑ + θ)dϑ

 ear time invariant are exponentially stable
 0 θ∈[−r,0]


P −σ 1 (ς−ς k ) − if an exponential decay bound their con-
 + b k e Π(ς )


 k vergence. Exponential stability is quickly
 0<ς k <ς

 P −σ 2 (ς−ς k ) −
 + d k e Π(ς ), ς ≥ 0. convergent than the other stability crite-

 k
0<ς k <ς rion. An asymptotically stable systems
(6) converge to the fixed point. Asymptotic
stability of mild solution, which are appro-
for σ 1 , σ 2 ∈ (0, r].
priate for continuous processes, cannot be
applied to our system (1).
If
∞
α 4 + α 6 α 5 α 6 X
+ + α 3 + + (b k + d k )
σ 1 − ω σ 2 − ω σ 1 − ω
k=1
< 1, (7) Theorem 2. Assume that the assumptions (H 2 )-
(H 7 ) and (H 9 ) are fulfilled and
then
ˆ −ως
Π(ς) ≤ Me for ς ∈ (−∞, ∞), (8) p−1 n p p 1−p p p
¯ p
8 M (1 + M ) + [a β [L (1 + M )]
g σ 1 2 2 f σ 2
p
˜
+ a C p β 1− p 2 ¯ p −p CHς pH−1 ˆ
L
L G ] + a β
2 2 2 2
where ω ∈ (0, σ 1 Λ σ 2 ) is a positive root of the ∞ p ∞ p o
p X 1 q p X 2 q
equation + a 1 L k + a 2 L k < 1,
k=1 k=1
∞
α 4 + α 6 α 5 α 6 X
+ +α 3 + + (b k +d k ) = 1.
σ 1 − ω σ 2 − ω σ 1 − ω then the mild solution of system (1) is exponen-
k=1
tially stable.
n (σ 1 − ω)α 1 (σ 1 − ω)α 1
ˆ
M = max α 1 + α 2 , − ,
e −ωr α 4 e −ωr α 6
o
(σ 2 − ω)α 2
> 0.
α 5 e −ωr
Proof. Consider the mild solution
115