Basic Calculus on Time Scales and some of its Applications

“…To understand the so-called dynamic equation and follow this paper easily, we present some preliminary definitions and notations of time scale which are very common in the literature ( see [1,2,7,11,12,21] and references therein). We generalize some definitions given in these references for the functions f :…”

Section: Preliminariesmentioning

confidence: 99%

“…Since the time Hilger formed the definitions of a derivative and integral on a time scale, several authors have extended on various aspects of the theory [1,2,7,9,11,12,16,19]. Time scale has been shown to be applicable to any field that can be described by means of discrete or continuous models.…”

Section: Introductionmentioning

confidence: 99%

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Dynamic equations x(Δm)(t) = f(t, x(t)) on time scales

Sikorska-Nowak

2011

Demonstratio Mathematica

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Abstract. In this paper we prove the existence of solutions and Carathéodory's type solutions of the dynamic Cauchy problem x (∆m) (t) = f (t, x(t)), t ∈ T,where x (∆m) denotes a mth order ∆ -derivative, T denotes an unbounded time scale (nonempty closed subset of R such that there exists a sequence (a n ) in T and a n → ∞), E -a Banach space and f is a continuous function or satisfies Carathéodory's conditions and some conditions expressed in terms of measures of noncompactness.The Sadovskii fixed point theorem and Ambrosetti's lemma are used to prove the main result.As dynamic equations are an unification of differential and difference equations our result is also valid for differential and difference equations. The results presented in this paper are new not only for Banach valued functions but also for real valued functions. IntroductionThe study of dynamic equations on time scales, which goes back to its founder Stefan Hilger [18], is an area of mathematics that has recently received a lot of attention. It has been created in order to unify the study of differential and difference equations. Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts. The study of dynamic equations on time scales reveals such discrepancies and helps avoid proving results twice-once for differential equations and once again for difference equations. The general idea is to prove a result for a dynamic equation, where the domain of the unknown function is a so-called time scale T , which may be an arbitrary 2000 Mathematics Subject Classification: 34G20, 34A40, 39A13.

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“…Much recent attention has been given to differential equations on time scales, and we refer the reader to [2,4,8] for some historical works as well as to the more recent papers [1,5,6] and the book [10] for excellent references on these types of equations. Before introducing the problems of interest for this paper, we present some definitions and notation which are common to the recent literature.…”

Section: Introductionmentioning

confidence: 99%