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Explicit criteria for the existence of positive solutions for a scalar differential equation with variable delay in the critical case. (English) Zbl 1155.34337
Summary: A scalar linear differential equation with time-dependent delay x ˙(t)=-a(t)x(t-τ(t)) is considered, where tI:[t 0 ,),t 0 ,a:I + :=(0,) is a continuous function and τ:I + is a continuous function such that t-τ(t)>t 0 -τ(t 0 ) if t>t 0 . The goal of our investigation is to give sufficient conditions for the existence of positive solutions as t in the critical case in terms of inequalities on a and τ. A generalization of one known final (in a certain sense) result is given for the case of τ being not a constant. Analysing this generalization, we show, e.g., that it differs from the original statement with a constant delay since it does not give the best possible result. This is demonstrated on a suitable example.
MSC:
34K05General theory of functional-differential equations
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