Mathematica Eterna
Open Access

Abstract

On the convergence of sets and the approximation property for dynamic equations on time scales

Mieczys law Cicho´n and Ahmet Yantir

The main goal of the paper is to propose a new approach to the problem of approximation of solutions for differential problems. A standard approach is based on discrete approximations. We replace it by a sequence of dynamic equations. In this paper, we investigate the convergence of closed sets being domains of considered problems, i.e. time scales. Then we apply our results for the study of an approximation property of dynamic equations. Our results allow us to characterize a set of solutions for differential problems as a limit of a sequence of dynamic ones. We point out a kind of convergence of time scales which is applicable and most useful for the study of continuous dependence of solutions for dynamic equations on time scales. It forms an approximation for the differential equations by dynamic equations and allows us to extend the difference approach in numerical algorithms. Finally, we study some Cauchy problems without uniqueness of solutions, which are approximated by simple dynamic problems.