Weak and strong singularities for second-order nonlinear differential equations with a linear difference operator

A class of second-order singular differential equations with a linear difference operator is investigated in this paper. The novelty of the present article is that for the first time we show that weak and strong singularities enable the achievement of a new existence criterion of positive periodic solutions through an applications of a fixed point theorem of Krasnoselskii's, i.e., our results of the existence of positive periodic solutions reveal a delicate relation between the value of external force e(t) and the velocity of nonlinear term f(t, x(t - tau(t))) approaching towards infinity when x(t - tau (t)) tending to zero.