A simplified approach to Gronwall's inequality on time scales with applications to new bounds for solutions to linear dynamic equations
Texas State University, Department of Mathematics
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 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 non-multiplicity of solutions to linear problems; and error estimates on solutions to initial value problems when the initial conditions are imprecisely known.
Gronwall inequality, Linear dynamic equations on time scales, Uniqueness of solutions, A priori bounds, Taxicab distance
Tisdell, C. C., & Meagher, S. (2017). A simplified approach to Gronwall's inequality on time scales with applications to new bounds for solutions to linear dynamic equations. <i>Electronic Journal of Differential Equations, 2017</i>(263), pp. 1-13.
Attribution 4.0 International