Abstract:
By the dynamic equations on time scale, we mean an equation that keeps the equations of
continuous, discrete and quantum calculus within themselves in the same equation. Here,
the time scale is a non-empty closed subset of real numbers. Moreover, the significance of
this equation is very much evident in those circumstances where we need to deal with differential and difference equations together. By keeping the applications under observation,
a lot of studies of several orders of this equation have been done by many authors.
Through this thesis, we present several oscillatory results for the first and second
order dynamic equations on time scale. Our studies are more general to the studies given
in the literature. Furthermore, we establish the results by using less restrictive conditions
as compared to the existing conditions in the literature. These conditions are easy to verify
and implement. As an application, by means of coincidence degree theory, we establish
the existence of positive periodic solution of the general N-prey and M-predator model on
time scale. On the other hand, we also establish a few sufficient conditions for oscillation
of the p-Laplacian dynamic equation on time scale for which we use a relaxed technique
that compliments the existing techniques to prove oscillatory results. Further, we also use
the Riccati transformation technique to transform second-order dynamic equations into
the first-order dynamic equation. Furthermore, we derive some important inequalities and
directly utilize the use of a well-known Young’s inequality for some of oscillatory results.
In addition, we make the conditions of our findings such that we can easily demonstrate
the well- known Kamenev and Philos-type oscillation criteria for our dynamic equations
on time scale. Besides, our contribution is not limited to this, we provide a new trend
of finding a derivative of continuous, discrete and quantum calculus. This derivative is
denoted by the name “black-delta” (symbol N) -derivative on time scale. Through this
derivative, we put together the usual and discrete derivatives at the same time. Further, it
has several applications in various fields, for instance, engineering, population dynamics,
biology, economics, social-sciences and quantum physics, etc. Some fundamental results,
associated with this derivative, are presented. From the point of application purposes, a
necessary and sufficient condition for this derivative is also provided. Moreover, a drastic
connection between the well-known Hilger derivative and new derivative on time scale is
demonstrated which makes our outcome better in the comparison to Hilger-derivative on
some time scale. Furthermore, some crucial examples are presented so that we could shed
light on the practicability and effectiveness of our results.