Dynamic Equations on Time Scales

Abstract

Abstract. In this paper we present a proof by construction that the hyperspace CL(R) of closed, nonemtpy subsets of R is simply connected under the Vietoris topology. This is useful in considering the convergence of time scales. We also present a construction of the (known) fact that this hyperspace is also path connected, as part of the proof. AMS Classification: 54B20, 54D05

Existence of positive periodic solutions for nonlinear neutral dynamic equations with variable coefficients on a time scale

“…We suppose that the reader is familiar with the basic concepts concerning the calculus on time scales for dynamic equations. Otherwise one can find in Bohner and Peterson books [8] and [9] most of the material needed to read this paper. We start by giving some definitions necessary for our work.…”

“…( 1.1) where x △ is the △-derivative on T (see [8]). Throughout this paper we assume that τ = mω if T has period ω and τ is fixed if T = R. Our purpose here is to use the Krasnosel'skiȋ's fixed point theorem to show the existence of positive periodic solutions on time scales for equation (1.1).…”

Multiple positive periodic solutions in shifts &delta;<SUB align="right">± for an impulsive nabla dynamic equation on time scales

“…For more details about the calculus on time scales, see Bohner and Peterson (2001). Let * T be a nonempty subset of the time scale T and * 0 t ∈ T be a fixed number; define operators * * 0 : [ , ) .…”

Eigenvalues for iterative systems of dynamic equations with integral boundary conditions

“…The time scales approach, not only uni…es di¤erential and di¤erence equations, but also provides accurate information of phenomena that manifest themselves partly in continuous time and partly in discrete time. We refer the books [5,6] which include basic de…nitions and theorems on time scales.…”