Explicit criteria for the existence of positive solutions for a scalar differential equation with variable delay in the critical case

A scalar linear differential equation with time-dependent delay x@?(t)=-a(t)x(t-@t(t)) is considered, where t@?I@?[t"0,~), t"0@?R, a:I->R^+@?(0,~) is a continuous function and @t:I->R^+ is a continuous function such that t-@t(t)>t"0-@t(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 @t. A generalization of one known final (in a certain sense) result is given for the case of @t 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.

[1]  L. Berezansky,et al.  On oscillation properties of delay differential equations with positive and negative coefficients , 2002 .

[2]  G. Ladas,et al.  Oscillation Theory of Delay Differential Equations: With Applications , 1992 .

[3]  V. Lakshmikantham,et al.  Theory of Differential Equations with Unbounded Delay , 1994 .

[4]  Wan-Tong Li,et al.  Nonoscillation and Oscillation Theory for Functional Differential Equations , 2004 .

[5]  J. Diblík,et al.  Exponential solutions of equation ẏ(t)=β(t)[y(t−δ)−y(t−τ)] , 2004 .

[6]  Elena Braverman,et al.  On exponential stability of linear differential equations with several delays , 2006 .

[7]  Qingkai Kong,et al.  Oscillation Theory for Functional Di erential Equations , 1994 .

[8]  A change of variables in the asymptotic theory of differential equations with unbounded delays , 2002 .

[9]  Jurang Yan,et al.  Existence of Multiple Positive Periodic Solutions of Delayed Predator-Prey Models with Functional Responses , 2006, Comput. Math. Appl..

[10]  Positive and oscillating solutions of differential equations with delay in critical case , 1998 .

[11]  I. Stavroulakis,et al.  Oscillations of first-order delay differential equations in a critical state , 1996 .

[12]  Josef Diblík,et al.  Positive solutions of p-type retarded functional differential equations , 2006 .

[13]  Xiaojie Xu,et al.  A new existence theory for positive periodic solutions to functional differential equations , 2004 .

[14]  N. Koksch,et al.  Positive Solutions of the Equation ẋ(t) = −c(t)x(t − τ) in the Critical Case☆ , 2000 .

[15]  Elena Braverman,et al.  Positive solutions for a scalar differential equation with several delays , 2008, Appl. Math. Lett..

[16]  Á. Elbert,et al.  Oscillation and nonoscillation criteria for delay differential equations , 1995 .

[17]  Josef Diblík,et al.  Positive solutions of retarded functional differential equations , 2005 .