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Abstract
The theory of dynamic equations on time scales provides an important bridge between the fields of differential and difference equations. It is particularly useful in describing phenomena that possess a hybrid continuous-discrete behaviour in their growth, like many temperate--2one insect populations and crops. A dynamic equation on a time scale is a generalised. 'two-in-one' model, it serves as a differential equation for purely continuous domains and as a difference equation for purely discrete ones.
The field of "dynamic equations on time scales" is about 20 years old. As such, much of the basic (yet very important) linear theory has been established, however the non-linear extensions are yet to be fully developed. This thesis aims to fill this gap by providing the foundational framework of non-linear results from which further lines of inquiry can be launched.
This thesis answers several important questions regarding the qualitative and quantitative properties of solutions to non-linear dynamic equations on time scales.
Namely,
(i) When do solutions exist?
(ii) If solutions exist, then are they unique?
(iii) How can such solutions be closely approximated?
(iv) How can we explicitly solve certain problems to extract their solutions?
The methods employed to address the above questions include: dynamic inequalities; iterative techniques and the method of successive approximations; and the fixed point approaches of Banach and Schauder.