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Inequalities for positive solutions of the equation \(\dot y(t)=-(a_0+\frac{a_1}t)y(t-\tau_1)-(b_0+\frac{b_1}t)y(t-\tau_2)\). (English) Zbl 1066.34066

As an application of a theorem on the general functional-differential equation \(\frac{dy(t)}{dt}=f(t,y_{t})\), the author considers the existence of a positive solution \(y^*(t)\) to the equation \[ \frac{dy(t)}{dt}=-\left( a_{0}+\frac{a_{1}}{t}\right) y(t-\tau _{1})-\left( b_{0}+\frac{b_{1}}{t}\right) y(t-\tau _{2}), \] with \(a_{0},b_{0},\tau _{1},\tau _{2}\in \mathbb{R}^{+},\) \(a_{1},b_{1}\in \mathbb{R}\), which satisfies the inequality \[ e^{-\lambda ^{\ast }t}t^{r^{\ast }}\left( 1+\frac{A^{\ast }-\varepsilon _{1} }{t}\right) \leq y^{\ast }(t)\leq e^{-\lambda ^{\ast }t}t^{r^{\ast }}\left( 1+\frac{A^{\ast }+\varepsilon _{1}}{t}\right) , \] where \(\varepsilon _{1}\in (0,1)\) and \(r^{\ast }\) , \(A^{\ast }\) are defined in this paper, and \(\lambda ^{\ast },\lambda ^{\ast \ast }\) are two real positive different roots of the transcendental equation \(\lambda =a_{0}e^{\lambda \tau _{1}}+b_{0}e^{\lambda \tau _{2}}\).

MSC:

34K06 Linear functional-differential equations
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