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

model with derivative in time replaced by frac- in. 32 Our primary objective in this research is
tional derivative performs better than the model to provide a set of suitable constraints for the
with fBm. From the application viewpoint, lim- existence and uniqueness of mild solutions to a
itation on Gaussian noise is unsuitable to model general class of abstract fractional second-order
and consequently to deal these situations, one can nonlocal stochastic equations driven by fBm. Em-
exchange Gaussian noise by Poisson random mea- ploying fixed point approach and semigroup con-
sure. For instance, the model of river pollution is cepts, we establish the solutions for stochastic
studied by Hausenblas and Marchis, in which the system.
number of deposits on a bounded region of the
river at infinitesimal length is dη, where η is the
The succeeding points summarized the following
distance coordinate along the river that behaves
key ideas of this manuscript.
according to the Poisson process.
(1) FNSIDEs with impulsive noises driven by
In various disciplines of industrial research, SDEs
fBm is considered in infinite dimensional
are among the most useful tools for modeling sys-
space in stochastic settings.
tems with stochastic disturbances. It can be used
(2) The existence result is obtained by us-
in a variety of fields, including population dy-
ing the generalized fractional derivative
namics, ecology, biology, engineering, and finance. namely (Caputo fractional derivative),
For fundamental concepts of SDEs see. 9,10,13–15,24
Banach fixed point theorem (BFPT) and
Moreover, numerous authors employed stochastic
assumptions on non-linear terms.
techniques and fixed point approach to investigate (3) Exponential stability of FNSIDEs with
the qualitative behaviors of SDEs, see 20,25,26 and
impulsive noises driven by fBm is ana-
references therein.
lyzed.
These equations play a crucial role in character- (4) An example is demonstrated to validate
izing many physical, biological, and engineering the obtained theories.
problems. They are essential from the viewpoint
of applications since they incorporate randomness
The article is summarized follows. Preliminaries
into the mathematical description of the phenom-
are introduced in section 2. Section 3 presents
ena and provide a more accurate description of
a new integral inequality that improves on the
it. Therefore, the theory of SDEs has developed 33
impulsive integral inequality reported in. Then,
quickly, and the investigation for SDEs has at-
using the new integral inequality, we verify the
tracted considerable attention.
stability of second-order FNSIDEs with impulses
driven by fBm under nonlocal conditions. In sec-
The stability theory of SDEs has been popularly tion 4, we provide an example to demonstrate the
applied in various sciences and technology fields. use of our findings. The paper ends with conclu-
Using different techniques, many authors have sion section 5.
studied the stability results of mild solutions for
SDEs. However, in addition to stochastic effects, Difficulty:
impulsive effects likewise exist in natural systems.
Many dynamical systems have variable structures (1) The main difficulty in dealing with ex-
subject to abrupt stochastic changes, which may ponential stability of mild solutions for
result from sudden phenomena such as stochastic impulsive stochastic differential equations
failures and repairs of the components, changes with delays comes from impulsive effects
in the interconnections of subsystems, sudden en- on the system.
vironment changes, etc. Therefore, it is necessary (2) Besides it should be pointed out that
to consider the existence, uniqueness, and other many methods used frequently fail to con-
quantitative and qualitative properties of solu- sider the exponential stability of mild so-
tions to stochastic systems with impulsive effects. lution for impulsive stochastic differen-
In light of recent developments in the theory of tial equation with delays; for example the
SDEs, it is becoming challenging to ignore the comparison theorem, Gronwall inequality,
existence of impulsive effects. Therefore several analytic technique are ineffective in deal-
studies have documented the impact of impulses ing with this kind of problems which do
in studying the SDEs driven by fBm see. 27–31 not have stochastic differentials.
(3) Therefore, by using impulsive integral in-
The outcomes of this work was inspired by recent equality techniques, we explore the expo-
work 22 and the second-order FSDEs investigated nential stability of such problem.
105