On the existence of solutions of dynamic equations on time scales in Banach spaces

TL;DR

This paper investigates the existence of solutions for dynamic equations on time scales within Banach spaces, extending classical differential equation results to a more general setting.

Contribution

It introduces a generalized existence theorem for dynamic equations on arbitrary time scales in Banach spaces, utilizing measures of noncompactness and a new Kamke Δ-function.

Findings

Extended classical differential equation solvability results to time scales in Banach spaces.

Developed a framework involving measures of noncompactness and Kamke Δ-function.

Analyzed countable systems from semi-discretized parabolic partial dynamic equations.

Abstract

In this paper we address the question of solvability of dynamic equations on time scales in Banach spaces. In particular, our main theorem extends the result for classical differential equations in Banach spaces of Bana\'s and Goebel (1980), to an arbitrary time scale. Central role is played by the axiomatic theory of measures of noncompactness and the newly introduced Kamke $\Delta$-function. Also, we study countable systems of dynamic equations on time scales arising from semi-discretisation of parabolic partial dynamic equations.