Page 110 - IJOCTA-15-1

D. Kasinathan et.al. / IJOCTA, Vol.15, No.1, pp.103-122 (2025)

are of great importance in the modeling of several system. The dynamical system contains a Pois-
physical phenomena. The model considered here son process, and the mild solution to the system
contains all above characteristics. is not a continuous stochastic process, but this
The concept of semigroups of bounded linear op- process has C˜adl˜ag path. Hence, for solvability,
erators is taken as an important concept to deal- the complete continuity condition has no meaning
ing with differential and integro-differential equa- in the presence of a jump term. To overcome this
tions in Banach spaces. On the other hand, in complexity, we have to decompose the solution
numerous mathematical models of real-world or operator to satisfy a suitable FPT, and an ob-
man-made phenomena, we are led to dynami- tained fixed point is a mild solution of a proposed
cal systems which involve some inherent random- problem.
ness. These systems are called stochastic systems.
Stochastic differential equations (SDEs) have at- The impact of stochastic integro-differential equa-
tracted much attention and have played an im- tions of the type (1) may be seen in. 17 This type
portant role in many ways such as option pricing, of equation can be found, for example;
forecast of the growth of population, etc. The (1) Stochastic formulation of reactor dynam-
modeling of most problems in real situations is de-
scribed by SDEs rather than deterministic equa- ics problems.
(2) The study of biological population
tions. Thus, it is of great importance to design
growth.
stochastic effects in the study of fractional-order
(3) Theory of automatic systems resulting in
dynamical systems. The focus on second-order
delay-differential equations.
equations is to study them directly rather than
(4) A variety of other problems in biology,
converting them to first-order equations. In many
physics, and engineering.
cases, it is advantageous to treat the second-order
SDEs directly rather than converting them to In 1990, Byszewski and Lakshmikantham 18 in-
first-order systems, refer to W.E. Fitzgibbon. 16 troduced and examined nonlocal problems. Dif-
A variety of problems arising in, mechanics, elas- ferential equations with the nonlocal conditions
ticity theory, molecular dynamics and quantum have been widely investigated for many years,
mechanics can be described in general by sec- since it has been proved that nonlocal problems
ond order nonlinear differential equations. The have greater impacts in applications than classi-
second-order differential equations involving ran- cal ones. The primary issue in working with the
domness are seem to be correct model in continu- nonlocal problem is to get the compactness of the
ous time to account for integrated processes that solution operator at zero, particularly whenever
can be made stationary. Due to this reason, re- the nonlocal problem is Lipschitz continuous or
searchers’ interest is focused on second order dif- continuous only. To overcome this challenge, nu-
ferential equations. For instance, engineers use merous authors developed various methods and
the second-order SDEs to describe mechanical vi- means. We suggest the readers to refer 19–21 for
brations or charges on a capacitor or condenser further information on this topic.
exposed to white noise stimulation.
Fixed point theory (FPT) is a fruitful tool in Furthermore, fBm is among the simplest Gauss-
present mathematics, and it plays a substantial ian stochastic processes with self–similarity and
role in nonlinear analysis. Two important ap- stationary increments. fBm is a generalization of
proaches to the solution of nonlinear equations conventional Brownian motion, which is defined
are described as topological and variational meth- by the Hurst index H ∈ (0, 1). When H = 1 ,
2
ods. Topological methods are obtained from FPT the stochastic process is a conventional Brownian
and are usually based on the idea of the topolog- motion; when H ̸= 1 , it acts quite differently;
2
ical degree. Variational methods represent the in particular, it is neither a semi-martingale nor
solutions as critical points of a suitable functional a Markov process. fBm is a natural choice as a
and find some ways of locating them. The FPT is model for noise in a wide range of physical phe-
regarded as a potent and essential tool to study nomena, including telecommunications networks,
the existence of a solution. Finding a solution of financial markets, and so on, due to these im-
major problems in abstract space, a fixed point portant qualities. 22 Li et al. 23 provided the com-
approach is a better way since it is advantageous, parative study of the classical stochastic model
and delivers suitable convergence theory. The for European option pricing, namely, the Black
fixed point techniques have broad application in Scholes model with the stochastic equation with
theoretical as well as numerical aspects and ex- fBm and stochastic equation with fractional time
tensively applied to the field of the stochastic derivative. It has been shown that the stochastic
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