J. Math. Anal. Appl. 382 (2011) 77–85 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa Averaging for retarded functional differential equations M. Federson ∗,1 , J.G. Mesquita 2 Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo-Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos SP, Brazil article info abstract Article history: Received 12 November 2010 Available online 20 April 2011 Submitted by J. Mawhin Keywords: Averaging principle Retarded functional differential equations Kurzweil–Henstock integral We consider retarded functional differential equations in the setting of Kurzweil–Henstock integrable functions and we state an averaging result for these equations. Our result generalizes previous ones.  2011 Elsevier Inc. All rights reserved. 1. Introduction In [6] and [7], the authors stated very nice averaging results for retarded functional differential equations employing the tools of non-standard analysis. Their results encompass, for instance, the results by J. Hale and S. Verduyn Lunel in [3]. In the present paper, we establish an averaging result for retarded functional differential equations, we write RFDE for short, by means of classical analysis. The conditions we assume on the right-hand sides of the RFDEs are more general than those considered in [3,6] or [7]. Indeed, we consider that the right-hand sides of the equations are Kurzweil–Henstock integrable functions. In the frame of the Kurzweil–Henstock integral, functions having not only many discontinuities but also being highly oscillating can be treated properly. It is known, for instance, that the Kurzweil–Henstock integral encompasses the integrals of Newton, Riemann and Lebesgue. In fact, the Kurzweil–Henstock integral coincides with the Perron and restricted Denjoy integrals and hence it can integrate functions as the well-known example f (t ) = d dt F (t ), where F (t ) = t 2 sin t −2 on [0, 1] when defined. Furthermore, the Kurzweil–Henstock integral is invariant by Cauchy and Harnack extensions and it has good convergence properties. See, for instance, [2,5,8–10] and the references therein. Let t 0 ∈ R, r > 0 and σ > 0. Given t ∈[t 0 , t 0 + σ ] and a function y :[t 0 − r , t 0 + σ ]→ R n , let y t : [−r , 0]→ R n be defined as usual by y t (θ) = y(t + θ), θ ∈ [−r , 0]. We consider the following initial value problem for an RFDE ⎧ ⎨ ⎩ ˙ y = f  y t , t ε  , y 0 = φ, * Corresponding author. E-mail addresses: federson@icmc.usp.br (M. Federson), jaquebg@icmc.usp.br (J.G. Mesquita). 1 Supported by FAPESP grant 2008/02879-1 and by CNPq grant 304646/2008-3. 2 Supported by FAPESP grant 2007/02731-1. 0022-247X/$ – see front matter  2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2011.04.034