Differential Equations with Functional Response on Piecewise Constant Argument

Abstract

We introduce a new class of functional differential equations with functional response on piecewise constant argument, FDEPCA. It contains functional differential equations with continuous time as well as differential equations with piecewise constant argument which were introduced in Akhmet (On the integral manifolds of the differential equations with piecewise constant argument of generalized type. In: Proceedings of the conference on differential and difference equations at the Florida institute of technology, August 1–5, 2005, Melbourne, Florida, ed. by R.P. Agarval, K. Perera (Hindawi Publishing Corporation, London, 2006), pp. 11–20; Nonlinear Anal 66:367–383, 2007; Discontinuity Nonlinearity Complexity 1:1–6,2012; J Math Anal Appl 336:646–663, 2007; Nonlinear Anal Hybrid Syst 2:456–467, 2008), and then developed in many papers and books. This chapter considers equations which are introduced and developed in our papers. We concentrate only on retarded equations, but one can easily extend the discussion to any type of piecewise constant argument and functional differential equations. Nonlinear and quasilinear systems are under consideration. At the end of the chapter, we suggest how one can apply the systems for solution of real-world problems, provided more general systems for future investigations.