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The paper deals with the existence of positive ω-periodic solutions for a class of nonlinear delay differential equations. For example, such equations represent the model for the survival of red blood cells in an animal. The sufficient conditions for the exponential stability of positive ω-periodic solution are also considered.

In this work we investigate some oscillatory properties of solutions of non-linear differential systems with retarded arguments. We consider the system of the form $$y_i^{\text{'}}\left(t\right)-p_i\left(t\right)y_{i+1}\left(t\right)=0,i=1,2,\cdots ,n-2,y_{n-1}^{\text{'}}\left(t\right)-p_{n-1}\left(t\right)|y_n\left(h_n\left(t\right)\right){|}^{\alpha }\mathrm{s}gn\left[y_n\left(h_n\left(t\right)\right)\right]=0,y_n^{\text{'}}\left(t\right)\mathrm{s}gn\left[y_1\left(h_1\left(t\right)\right)\right]+p_n\left(t\right){\left|y_1\left(h_1\left(t\right)\right)\right|}^{\beta }\le 0,$$ where $n\ge 3$ is odd, $\alpha >0$, $\beta >0$.