Periodic solutions of nonlinear equations generalizing logistic equations with delay (Q1688327)

The following generalization of the logistic equation with delay is considered \[ u^{\,\prime}(t)=\gamma u(t)\, F_\alpha(1-u(t-\tau)), \eqno{(1)} \] where \(\alpha\geq0, \gamma>0, \tau>0\) are constants and the nonlinearity \(F_\alpha\) is defined as \(F_\alpha(v)=v|v|^{\alpha-1}\) with \(F_0(v)=\text{sig}\,(v)\) and \(F_1(v)=v\). The main result of the paper is the existence, shape, and asymptotic stability with the asymptotic phase of slowly oscillating periodic solutions of equation (1) for large value of parameters \(\gamma\) (and for normalized delay \(\tau=1\)). As an auxiliary result of author's considerations the existence of periodic solutions of period \(4\) is shown for the delay differential equation \[ v^{\,\prime}(t)=-F_\alpha(v(t-1)). \] The latter is a well known result due to \textit{J. L. Kaplan} and \textit{J. A. Yorke} [J. Math. Anal. Appl. 48, 317--324 (1974; Zbl 0293.34102)] and its numerous extensions by others.