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The existence of positive periodic solutions for a kind of Rayleigh equation with a deviating argument $$x^{\text{'}\text{'}}\left(t\right)+f\left(x^{\text{'}}\left(t\right)\right)+g(t,x(t-\tau \left(t\right)))=p\left(t\right)$$ is studied. Using the coincidence degree theory, some sufficient conditions on the existence of positive periodic solutions are obtained.

Let $f:X\to X$ be a continuous map with the specification property on a compact metric space $X$. We introduce the notion of the maximal Birkhoff average oscillation, which is the “worst” divergence point for Birkhoff average. By constructing a kind of dynamical Moran subset, we prove that the set of points having maximal Birkhoff average oscillation is residual if it is not empty. As applications, we present the corresponding results for the Birkhoff averages for continuous functions on a repeller and locally...