Oscillation and nonoscillation of advanced differential equations with variable coefficients ✩

The oscillation and nonoscillation of the advanced differential equations x (t ) − p(t)x(t + τ) = 0, t t0 (∗) and x (t ) − n i=1 pi (t)x(t + τi ) = 0, t t0 (∗∗) are investigated, where p(t),pi (t ) ∈ C([t0,∞), [0,∞)), τ and τi are positive constants. At first, a sharp sufficient condition for the oscillation of Eq. (∗) is obtained, then the result is generalized to Eq. (∗∗). These results improve the corresponding conclusions derived by Ladas and Stavroulakis (J. Differential Equations 44 (1982) 134–152). Next, two examples are given to illustrate the advantages of our results. Finally, the sufficient conditions for these two equations to be nonoscillatory are also obtained.  2002 Elsevier Science (USA). All rights reserved