Qualitative properties of Deterministic and Stochastic Dynamic Equations on Time Scales Theory
The theory of time scales, which has recently received a lot of attention, was introduced by Stefan Hilger in his PhD thesis in 1988 in order to unify continuous and discrete analysis. Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different in nature from their continuous counterparts. The study of dynamic equations on time scales reveals such discrepancies, and helps avoid proving results twice, once for differential equations and once for difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale, which is an arbitrary nonempty closed subset of the reals. By choosing the time scale to be the set of real numbers, the general result yields a result concerning an ordinary differential equation as studied in a first course in differential equations.
Stochastic differential equations give a connection between probability theory and a lot more established and progressively created fields of ordinary and partial differential equations. Stochastic differential equations are utilized to demonstrate different phenomena, for example, physical systems subject to thermal fluctuations or unstable stock prices. Commonly, SDEs contain a variable which represents random white noise calculated as the derivative of Brownian motion or the Wiener process.
An almost periodic function is a function of a real number that is periodic to within any ideal degree of precision, given reasonably long, well-distributed 'almost-periods’ Almost periodicity is a property of dynamical frameworks that seem to follow their ways through stage space, but not exactly. A model would be a planetary framework, with planets in circles moving with periods that are not commensurable (i.e., with a period vector that isn’t relative to a vector of integers). "Almost automorphic" functions are more general than almost periodic ones so we can say almost automorphic functions are a natural generalization of the classical idea of almost periodic functions. Bochner introduced the concept of almost automorphic functions in relation to some aspects of differential geometry.
Our work deals with the study of the existence, uniqueness and stability of the almost automorphic solution and its generalization for deterministic and stochastic dynamic equations on time scales. We introduce the concept of equipotentially square-mean almost automorphic sequence, square-mean piecewise almost automorphic functions on time scales, time scale version of the Stepanov-like square mean almost automorphic functions and some new results on composition theorem on time scales for the space of square-mean weighted Stepanov-like pseudo almost automorphic functions. As applications, Stochastic Cellular Neural Network, stochastic Nicholson's Blowflies model and non-autonomous Leslie-Gower prey-predator model are considered. We had derived some sufficient conditions for the existence of square mean almost auto-morphic solution for Stochastic Cellular Neural Network on time scales.
