Differential equations on variable time scales


AKHMET M., Turan M.

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, vol.70, no.3, pp.1175-1192, 2009 (Peer-Reviewed Journal) identifier identifier

  • Publication Type: Article / Article
  • Volume: 70 Issue: 3
  • Publication Date: 2009
  • Doi Number: 10.1016/j.na.2008.02.020
  • Journal Name: NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
  • Journal Indexes: Science Citation Index Expanded, Scopus
  • Page Numbers: pp.1175-1192
  • Keywords: Variable time scales, Differential equations on time scales, Impulsive differential equations, Existence and uniqueness, Bounded solutions, Periodic solutions, Stability, IMPACT, BIFURCATIONS, OSCILLATOR

Abstract

We introduce a class of differential equations on variable time scales with a transition condition between two consecutive parts of the scale. Conditions for existence and uniqueness of solutions are obtained. Periodicity, boundedness and stability of solutions are considered. The method of investigation is by means of two successive reductions: B-equivalence of the system [E. Akalfn, M.U. Akhmet, The principles of B-smooth discontinuous flows, Computers and Mathematics with Applications 49 (2005) 981-995; M.U. Akhmet, Perturbations and Hopf bifurcation of the planar discontinuous dynamical system, Nonlinear Analysis 60 (2005) 163-178; M.U. Akhmet, N.A. Perestyuk, The comparison method for differential equations with impulse action, Differential Equations 26 (9) (1990) 1079-1096] on a variable time scale to a system on a time scale, a reduction to an impulsive differential equation [M.U. Akhmet, Perturbations and Hopf bifurcation of the planar discontinuous dynamical system, Nonlinear Analysis 60 (2005) 163-178; M.U. Akhmet, M. Turan, The differential equations on time scales through impulsive differential equations, Nonlinear Analysis 65 (2006) 2043-2060]. Appropriate examples are constructed to illustrate the theory. (C) 2008 Elsevier Ltd. All rights reserved.