Time-scale calculus

Carathéodory Theory of Dynamic Equations on Time Scales

Published in S. A. Mohiuddine, Bipan Hazarika, Sequence Space Theory with Applications, 2023

In the Carathéodory theory, we study the solutions of dynamic equations under weak conditions that the right-hand side of the dynamic equations is not necessarily continuous. This allows us to broaden the class of dynamic equations beyond those with continuous right-hand sides. We assume that the reader is familiar with the classical theory of Carathéodory. In this section, several concepts and results of the time scale calculus are briefly exposed that provide a framework for the study of dynamic equations in the sense of Carathéodory. The reader can find information on time scales calculus, for examples, in [2, 3].

Variational approach for a p-Laplacian boundary value problem on time scales

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Journal Information

Published in Applicable Analysis, 2018

You-Hui Su, Zhaosheng Feng

Time scale calculus is an exciting mathematical theory which unites the two approaches of dynamic modeling: difference and differential equations. In many scientific fields, such as mathematical biology, and ecology, it can be used to model dynamic processes whose time domains are more complex than the set of integers (difference equations) or real numbers (differential equations) [1]. For example, time scale calculus is often used to model insect populations that are continuous now and then, and many follow a difference scheme with variable step size, that is, they die out in winter, while their eggs are incubating or dormant, and then hatch in a new season, giving rise to a non-overlapping population [2]. Hence, it has become a crucial role in various equations and systems arising in biology, entomology, astronomy, medical sciences, and economics [2–6].

Time-scale calculus

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Carathéodory Theory of Dynamic Equations on Time Scales

Variational approach for a p-Laplacian boundary value problem on time scales