A partly sharp oscillation criterion for first-order delay differential equations *

摘要

In this paper, a partly sharp oscillation criterion is established for the first-order delay differential equation x'(t) + p(t)x(tau (t )) = 0 , t > T 0 , where p(t) is continuous and p(t) > 0 for all t > T0;tau (t) is continuous, non-decreasing, tau(t) < t for all t > T 0 , and limt ->. tau(t) = +.. By improving techniques from previous re-searches, we show that when alpha= lim inft ->. t tau (t) p(s)ds E (0 , 1 ], all solutions of the equa-e tion are oscillatory if lim sup t ->. t p(s)ds > 2 alpha + 2 1 tau(t) lambda 1 under no additional constraints, where lambda 1 is the smaller root of the equation lambda = e alpha lambda. This result is proved to be weaker than previous ones, and sharp when alpha E (0 , ln 2 by con-2 structing a specific example.

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