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On the Solvability and Asymptotics of Solutions of One Functional Differential Equation with Singularity

Authors:

Zernov A E

Institution:

(1) Odessa Polytechnic University, Odessa

Abstract:

We prove the existence of continuously differentiable solutions with required asymptotic properties as t rarr

+0 and determine the number of solutions of the following Cauchy problem for a functional differential equation:

$\alpha \left( t \right)x\prime \left( t \right) = at + b_1 x\left( t \right) + b_2 x\left( {g\left( t \right)} \right) + \phi \left( {t,x\left( t \right),x\left( {g\left( t \right)} \right),x\prime \left( {h\left( t \right)} \right)} \right),\quad x\left( 0 \right) = 0,$

where

: (0,

)

(0, +

), g: (0,

)

(0, +

), and h: (0, tau

)

(0, +

) are continuous functions, 0 < g(t)

t, 0 < h(t)

t, t

(0,

), $\begin{gathered} \alpha \left( t \right)x\prime \left( t \right) = at + b_1 x\left( t \right) + b_2 x\left( {g\left( t \right)} \right) + \phi \left( {t,x\left( t \right),x\left( {g\left( t \right)} \right),x\prime \left( {h\left( t \right)} \right)} \right),\quad x\left( 0 \right) = 0, \hfill \\ \mathop {\lim }\limits_{t \to + 0} \alpha \left( t \right) = 0 \hfill \\ \end{gathered}$ , and the function phiv

is continuous in a certain domain.

Keywords:

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