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On the delay differential equation y'(x) = ay(τ(x)) + by(x)

Jan Čermák — 1999

Annales Polonici Mathematici

The paper discusses the asymptotic properties of solutions of the scalar functional differential equation y ' ( x ) = a y ( τ ( x ) ) + b y ( x ) , x [ x 0 , ] . Asymptotic formulas are given in terms of solutions of the appropriate scalar functional nondifferential equation.

Asymptotic properties of differential equations with advanced argument

Jan Čermák — 2000

Czechoslovak Mathematical Journal

The paper discusses the asymptotic properties of solutions of the scalar functional differential equation y ' ( x ) = a y ( τ ( x ) ) + b y ( x ) , x [ x 0 , ) of the advanced type. We show that, given a specific asymptotic behaviour, there is a (unique) solution y ( x ) which behaves in this way.

Asymptotic estimation for functional differential equations with several delays

Jan Čermák — 1999

Archivum Mathematicum

We discuss the asymptotic behaviour of all solutions of the functional differential equation y ' ( x ) = i = 1 m a i ( x ) y ( τ i ( x ) ) + b ( x ) y ( x ) , where b ( x ) < 0 . The asymptotic bounds are given in terms of a solution of the functional nondifferential equation i = 1 m | a i ( x ) | ω ( τ i ( x ) ) + b ( x ) ω ( x ) = 0 .

On transformations of functional-differential equations

Jan Čermák — 1993

Archivum Mathematicum

The paper contains applications of Schrőder’s equation to differential equations with a deviating argument. There are derived conditions under which a considered equation with a deviating argument intersecting the identity y = x can be transformed into an equation with a deviation of the form τ ( x ) = λ x . Moreover, if the investigated equation is linear and homogeneous, we introduce a special form for such an equation. This special form may serve as a canonical form suitable for the investigation of oscillatory...

Asymptotic behaviour of solutions of some linear delay differential equations

Jan Čermák — 2000

Mathematica Bohemica

In this paper we investigate the asymptotic properties of all solutions of the delay differential equation y’(x)=a(x)y((x))+b(x)y(x),      xI=[x0,). We set up conditions under which every solution of this equation can be represented in terms of a solution of the differential equation z’(x)=b(x)z(x),      xI and a solution of the functional equation |a(x)|((x))=|b(x)|(x),      xI.

Několik poznámek ke zlomkovému kalkulu

Jan ČermákTomáš KiselaLuděk Nechvátal — 2020

Pokroky matematiky, fyziky a astronomie

Článek přináší základní pohled na oblast tzv. zlomkového kalkulu, tedy partii matematické analýzy, která je věnována derivacím neceločíselných řádů a souvisejícím otázkám. Je zde popsán historický vývoj tohoto pojmu, včetně motivací a aplikací. Speciálně se pak text zaměřuje na oblast diferenciálních rovnic s neceločíselnými derivacemi, na základní otázky spojené s jejich vyšetřováním a také na některé nové výzvy, které tato disciplína přináší.

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