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This work is concerned with the differential-difference equation x(t) = - f_α(X(t-1)), α > where f_α is an odd simple function. We use the concept of ejectivity to show the existence of nontrivial periodic solutions for any α > 0. By elementary arguments we also show the uniqueness of these solutions for each α < 2 and the existence of at least two such solutionis for each α ≥ 2.