On the Study of Topological Dynamical Systems
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Date
June 2012
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Abstract
The purpose of the study was to apply Topological Dynamics to Integral Equations. Topo-
logical Dynamical techniques were used to analyse it and con rmed the results. Sell de-
veloped methods which allowed one to apply the theory of topological dynamics to a very
general class of nonautonomous ordinary di erential equations. This was extended to non-
linear Volterra's Integral Equations. This research took o from there and applied the
techniques of topological dynamics to an integral equation. The usage of limiting equa-
tions which were used by Sell on his application to integral equations were extended to
recurrent motions and then studied the solution path. It thus con rmed the existence of
contraction and the stationary point in the said paper. The study of Dynamical Systems
of Shifts in the space of piece-wise continuous functions analogue to the known Bebutov
system was embarked upon. The stability in the sense of Poisson discontinuous function
was shown. It was proved that a xed discontinuous function, f, is discontinuous for all its
shifts, , whereas the trajectory of discontinuous function is not a compact set. The study
contributes to literature by providing notions of Topological Dynamic techniques which
were used to analyse and con rm the existence and contractions and the stationary points
of a special Integral Equation.
Description
A Thesis Presented to the Department of Mathematics, Kwame Nkrumah University of Science and Technology
in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Pure Mathematics