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This paper deals with the oscillation of the first-order differential equation with several delay arguments x′t+∑i=1mpitxτit=0,t≥t0, where the functions pi,τi∈Ct0,∞,R+, for every i=1,2,…,m,τit≤t for t≥t0 and limt→∞τit=∞. In this paper, the state-of-the-art on the sharp oscillation conditions are presented. In particular, several sufficient oscillation conditions are presented and it is shown that, under additional hypotheses dealing with slowly varying at infinity functions, some of the “liminf” oscillation conditions can be essentially improved replacing “liminf” by “limsup”. The importance of the slowly varying hypothesis and the essential improvement of the sufficient oscillation conditions are illustrated by examples.

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Abstract. In this article, we study oscillatory properties of the fourth-order di¤erence equation with middle-term 4 xm am 2 x m+1 + bmf (x m+ ) = 0;in case when the corresponding second-order di¤erence equation 2 hm amh m+1 = 0 is nonoscillatory.