The paper under review continues the recent investigations of the authors [Proc. Symp. Pure Math. 45, Pt. 2, 155–167 (1986; Zbl 0618.34061), Ann. Mat. Pura Appl., IV. Ser. 145, 33–128 (1986; Zbl 0617.34071)] and is concerned with consideration of the differential-delay equation \[ \varepsilon x'(t)=-x(t)+f(x(t-1)). \tag{1} \] The authors suggest general principles that help in verifying the hypotheses given in the papers cited above guaranteeing existence and asymptotic properties as \(\varepsilon \to +0\) of periodic solutions of (1) which oscillate about a fixed point of \(f\). As an application the specific forms of \(f\) such as \(f_1(x) = \mu - x^2,\) \(f_2(x) = x^3 - \mu x,\) \(f_3(x) = -\mu [\sin (x+\theta)-\sin \theta]\), \(f_4(x) = \mu x^{\nu}e^{-x}\), \(f_5(x) = \mu x^{\nu}(x^{\lambda}+1)^{-1}\) are considered.