Asymptotic estimation of the convergence of solutions of the equation x ˙ ( t ) = b ( t ) x ( t - τ ( t ) )

Josef Diblík; Denis Khusainov

Archivum Mathematicum (2001)

  • Volume: 037, Issue: 4, page 279-287
  • ISSN: 0044-8753

Abstract

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The main result of the present paper is obtaining new inequalities for solutions of scalar equation x ˙ ( t ) = b ( t ) x ( t - τ ( t ) ) . Except this the interval of transient process is computed, i.e. the time is estimated, during which the given solution x ( t ) reaches an ε - neighbourhood of origin and remains in it.

How to cite

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Diblík, Josef, and Khusainov, Denis. "Asymptotic estimation of the convergence of solutions of the equation $\dot{x}(t)=b(t) x(t-\tau (t))$." Archivum Mathematicum 037.4 (2001): 279-287. <http://eudml.org/doc/248755>.

@article{Diblík2001,
abstract = {The main result of the present paper is obtaining new inequalities for solutions of scalar equation $\dot\{x\}(t)=b(t)x(t-\tau (t))$. Except this the interval of transient process is computed, i.e. the time is estimated, during which the given solution $x(t)$ reaches an $\varepsilon $ - neighbourhood of origin and remains in it.},
author = {Diblík, Josef, Khusainov, Denis},
journal = {Archivum Mathematicum},
keywords = {stability of trivial solution; estimation of convergence of nontrivial solutions; stability of trivial solution; estimation of convergence of nontrivial solutions},
language = {eng},
number = {4},
pages = {279-287},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Asymptotic estimation of the convergence of solutions of the equation $\dot\{x\}(t)=b(t) x(t-\tau (t))$},
url = {http://eudml.org/doc/248755},
volume = {037},
year = {2001},
}

TY - JOUR
AU - Diblík, Josef
AU - Khusainov, Denis
TI - Asymptotic estimation of the convergence of solutions of the equation $\dot{x}(t)=b(t) x(t-\tau (t))$
JO - Archivum Mathematicum
PY - 2001
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 037
IS - 4
SP - 279
EP - 287
AB - The main result of the present paper is obtaining new inequalities for solutions of scalar equation $\dot{x}(t)=b(t)x(t-\tau (t))$. Except this the interval of transient process is computed, i.e. the time is estimated, during which the given solution $x(t)$ reaches an $\varepsilon $ - neighbourhood of origin and remains in it.
LA - eng
KW - stability of trivial solution; estimation of convergence of nontrivial solutions; stability of trivial solution; estimation of convergence of nontrivial solutions
UR - http://eudml.org/doc/248755
ER -

References

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  8. Kolmanovskij V., Myshkis A., Introduction to the Theory and Applications of Functional Differential Equations, Kluwer Acad. Publ., 1999. (1999) Zbl0917.34001MR1680144
  9. Kolmanovskij V. B., Nosov V. R., Stability and Periodic Modes of Regulated Systems with Delay, Nauka, Moscow, 1981. (In Russian). (1981) 
  10. Krasovskii N. N., Stability of Motion, Stanford Univ. Press, 1963. (1963) Zbl0109.06001MR0147744
  11. Krisztin T., Asymptotic estimation for functional differential equations via Lyapunov functions, Colloq. Math. Soc. János Bolyai, Qualitative Theory of Differential Equations, Szeged, Hungary, 1988, 365–376. (1988) MR1062660
  12. Myshkis A. D., Linear Differential Equations with Delayed Argument, Nauka, Moscow, 1972. (In Russian). (1972) MR0352648
  13. Pituk M., Asymptotic behavior of solutions of differential equation with asymptotically constant delay, Nonlinear Anal. 30 (1997), 1111–1118. (1997) MR1487679
  14. Razumikhin B. S., Stability of Hereditary Systems, Nauka, Moscow, 1988. (In Russian). (1988) MR0984127

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