On the reduction principle for differential equations with piecewise constant argument of generalized type


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Akhmet M. U.

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, vol.336, no.1, pp.646-663, 2007 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 336 Issue: 1
  • Publication Date: 2007
  • Doi Number: 10.1016/j.jmaa.2007.03.010
  • Title of Journal : JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
  • Page Numbers: pp.646-663
  • Keywords: integral manifolds, Reduction principle, piecewise constant argument of generalized type, continuation of solutions, stability, GREENS-FUNCTION, PERIODIC-SOLUTIONS, INTEGRAL MANIFOLD, LINEARIZATION, SYSTEM

Abstract

In this paper we introduce a new type of differential equations with piecewise constant argument (EPCAG), more general than EPCA [K.L. Cooke, J. Wiener, Retarded differential equations with piecewise constant delays, J. Math. Anal. Appl. 99 (1984) 265-297; J. Wiener, Generalized Solutions of Functional Differential Equations, World Scientific, Singapore, 1993]. The Reduction Principle [V.A. Pliss, The reduction principle in the theory of the stability of motion, Izv. Akad. Nauk SSSR Ser. Mat. 27 (1964) 1297-1324 (in Russian); V.A. Pliss, Integral Sets of Periodic Systems of Differential Equations, Nauka, Moskow, 1977 (in Russian)] is proved for EPCAG. The structure of the set of solutions is specified. We establish also the existence of global integral manifolds of quasilinear EPCAG in the so-called critical case and investigate the stability of the zero solution. (c) 2007 Elsevier Inc. All rights reserved.