On transformations of functional-differential equations

Jan Čermák

Archivum Mathematicum (1993)

  • Volume: 029, Issue: 3-4, page 227-234
  • ISSN: 0044-8753

Abstract

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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 and asymptotic properties of the considered equation.

How to cite

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Čermák, Jan. "On transformations of functional-differential equations." Archivum Mathematicum 029.3-4 (1993): 227-234. <http://eudml.org/doc/247427>.

@article{Čermák1993,
abstract = {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 $\tau (x)=\lambda x$. 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 and asymptotic properties of the considered equation.},
author = {Čermák, Jan},
journal = {Archivum Mathematicum},
keywords = {Functional-differential equation; singular case; transformation; canonical form; differential equations with deviating arguments; oscillatory and asymptotic properties},
language = {eng},
number = {3-4},
pages = {227-234},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On transformations of functional-differential equations},
url = {http://eudml.org/doc/247427},
volume = {029},
year = {1993},
}

TY - JOUR
AU - Čermák, Jan
TI - On transformations of functional-differential equations
JO - Archivum Mathematicum
PY - 1993
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 029
IS - 3-4
SP - 227
EP - 234
AB - 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 $\tau (x)=\lambda x$. 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 and asymptotic properties of the considered equation.
LA - eng
KW - Functional-differential equation; singular case; transformation; canonical form; differential equations with deviating arguments; oscillatory and asymptotic properties
UR - http://eudml.org/doc/247427
ER -

References

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  1. Vvedenije v teoriju differencialnych uravnenij s otklonjajuščimsa argumentom, Nauka, Moscow, 1964. (Russian) (1964) MR0170049
  2. Asymptotic behavior of solutions of the functional differential equation x ' ( t ) = a x ( t ) + b x ( t α ) , α > 1 , J.Math.Anal.Appl. 44 (1973), 745–757. (1973) Zbl0289.34115MR0333405
  3. The functional-differential equation y ' ( x ) = a y ( λ x ) + b y ( x ) , Bull. Amer. Math. Soc. 77 (1971), 891–937. (1971) MR0283338
  4. Functional Equations in a Single Variable, Polish Scient.Publ., Warszawa, 1968. (1968) Zbl0196.16403MR0228862
  5. Oscillation Theory of Differential Equations with Deviating Arguments, Marcel Dekker, New York, 1983. (1983) 
  6. Asymptotic behavior of solutions of the functional differential equation x ' ( t ) = A x ( λ t ) + B x ( t ) , λ > 0 , J.Math.Anal.Appl. 55 (1976), 794–806. (1976) Zbl0336.34060MR0447749
  7. On transformations of differential equations and systems with deviating argument, Czechoslovak Math.J. 31(106) (1981), 87-90. (1981) Zbl0463.34051MR0604115
  8. Transformation and canonical forms of functional-differential equation, Proc. Roy.Soc.Edinburgh 115A (1990), 349-357. (1990) MR1069527
  9. Some observations on the asymptotic behaviors of the solutions of the equation x ' ( t ) = A ( t ) x ( λ t ) + B ( t ) x ( t ) , λ > 0 , J.Math.Anal.Appl. 67 (1979), 483–489. (1979) MR0528702
  10. Regular iteration of real and complex functions, Acta Math. 100 (1958), 203–258. (1958) Zbl0145.07903MR0107016
  11. The most general transformation of homogeneous retarded linear differential equations of the n -th order, Math.Slovaka 33 (1983), 15–21. (1983) Zbl0514.34058MR0689272

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