The author extends the method of J. L. Kaplan
and J. A. Yorke
[J. Differ. Equations 23, 293-314 (1977; Zbl 0307.34070
)] to prove the existence of periodic solutions with certain period in scalar delay differential equations of the type
satisfies the relation
depending on parameters, the paper gives conditions under which Hopf and saddle-node bifurcations of periodic solutions occur. Moreover, the author provides examples showing that Hopf and saddle-node bifurcations often occur infinitely many times.