Electron. J. Differential Equations,
Vol. 2017 (2017), No. 263, pp. 113.
A simplified approach to Gronwall's inequality on time scales with
applications to new bounds for solutions to linear dynamic equations
Christopher C. Tisdell, Stephen Meagher
Abstract:
The purpose of this work is to advance and simplify our understanding of
some of the basic theory of linear dynamic equations and dynamic inequalities
on time scales.
Firstly, we revisit and simplify approaches to Gronwall's inequality on time scales.
We provide new, simple and direct proofs that are accessible to those with only
a basic understanding of calculus.
Secondly, we apply the ideas to second and higher order linear dynamic equations
on time scales. Part of the novelty herein involves a strategic choice of metric,
notably the taxicab metric, to produce {\em a priori} bounds on solutions.
This choice of metric significantly simplifies usual approaches and extends ideas
from the literature.
Thirdly, we examine mathematical applications of the aforementioned bounds.
We form results concerning the nonmultiplicity of solutions to linear problems;
and error estimates on solutions to initial value problems when the initial
conditions are imprecisely known.
Submitted April 5, 2017. Published October 19, 2017.
Math Subject Classifications: 34N05, 26E70, 97I99, 97D99.
Key Words: Gronwall inequality; linear dynamic equations on time scales;
uniqueness of solutions; a priori bounds; taxicab distance.
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Christopher C. Tisdell
School of Mathematics and Statistics
The University of New South Wales
UNSW, 2052, Australia
email: cct@unsw.edu.au

 Stephen Meagher
School of Mathematics and Statistics
The University of New South Wales
UNSW, 2052, Australia
email: s.meagher@unsw.edu.au

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