Approximate properties of principal solutions of Volterra-type integrodifferential equations with infinite aftereffect

Yu. A. Ryabov

Mathematica Bohemica (1995)

  • Volume: 120, Issue: 3, page 265-282
  • ISSN: 0862-7959

Abstract

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The integrodifferential system with aftereffect (“heredity” or “prehistory”) dx/dt=Ax+-t R(t,s)x(s,)ds, is considered; here ε is a positive small parameter, A is a constant n × n matrix, R ( t , s ) is the kernel of this system exponentially decreasing in norm as t → ∞ . It is proved, if matrix A and kernel R ( t , s ) satisfy some restrictions and ε does not exceed some bound ε * , then the n -dimensional set of the so-called principal two-sided solutions x ˜ ( t , ε ) approximates in asymptotic sense the infinite-dimensional set of solutions x ( t , ε ) corresponding a sufficiently wide class of initial functions. For t growing to infinity an estimate of the difference between x ( t , ε ) and x ˜ ( t , ε ) is obtained.

How to cite

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Ryabov, Yu. A.. "Approximate properties of principal solutions of Volterra-type integrodifferential equations with infinite aftereffect." Mathematica Bohemica 120.3 (1995): 265-282. <http://eudml.org/doc/247794>.

@article{Ryabov1995,
abstract = {The integrodifferential system with aftereffect (“heredity” or “prehistory”) dx/dt=Ax+-t R(t,s)x(s,)ds, is considered; here $\varepsilon $ is a positive small parameter, $A$ is a constant $n\times n$ matrix, $R(t,s)$ is the kernel of this system exponentially decreasing in norm as $t\rightarrow \infty $. It is proved, if matrix $A$ and kernel $R(t,s)$ satisfy some restrictions and $\varepsilon $ does not exceed some bound $\varepsilon _\ast $, then the $n$-dimensional set of the so-called principal two-sided solutions $\tilde\{x\}(t,\varepsilon )$ approximates in asymptotic sense the infinite-dimensional set of solutions $x(t,\varepsilon )$ corresponding a sufficiently wide class of initial functions. For $t$ growing to infinity an estimate of the difference between $x(t,\varepsilon )$ and $\tilde\{x\}(t,\varepsilon )$ is obtained.},
author = {Ryabov, Yu. A.},
journal = {Mathematica Bohemica},
keywords = {integrodifferential system with after-effect; principal two-sided solutions; integrodifferential equations; principal solutions; small parameter; integrodifferential system with after-effect; principal two-sided solutions},
language = {eng},
number = {3},
pages = {265-282},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Approximate properties of principal solutions of Volterra-type integrodifferential equations with infinite aftereffect},
url = {http://eudml.org/doc/247794},
volume = {120},
year = {1995},
}

TY - JOUR
AU - Ryabov, Yu. A.
TI - Approximate properties of principal solutions of Volterra-type integrodifferential equations with infinite aftereffect
JO - Mathematica Bohemica
PY - 1995
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 120
IS - 3
SP - 265
EP - 282
AB - The integrodifferential system with aftereffect (“heredity” or “prehistory”) dx/dt=Ax+-t R(t,s)x(s,)ds, is considered; here $\varepsilon $ is a positive small parameter, $A$ is a constant $n\times n$ matrix, $R(t,s)$ is the kernel of this system exponentially decreasing in norm as $t\rightarrow \infty $. It is proved, if matrix $A$ and kernel $R(t,s)$ satisfy some restrictions and $\varepsilon $ does not exceed some bound $\varepsilon _\ast $, then the $n$-dimensional set of the so-called principal two-sided solutions $\tilde{x}(t,\varepsilon )$ approximates in asymptotic sense the infinite-dimensional set of solutions $x(t,\varepsilon )$ corresponding a sufficiently wide class of initial functions. For $t$ growing to infinity an estimate of the difference between $x(t,\varepsilon )$ and $\tilde{x}(t,\varepsilon )$ is obtained.
LA - eng
KW - integrodifferential system with after-effect; principal two-sided solutions; integrodifferential equations; principal solutions; small parameter; integrodifferential system with after-effect; principal two-sided solutions
UR - http://eudml.org/doc/247794
ER -

References

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  1. Yu. A. Ryabov, Two-sided solutions of linear integrodifferential equation of Volterra-type with infinite lower limit, The researches on the theory of differential equations, ed. MADI. 1986, pp. 3-16. (In Russian.) (1986) MR1012539
  2. Yu. A. Ryabov, The existence of two-sided solutions of linear integrodifferential equations of Volterra type with aftereffect, Čas. Pěstov. Mat. 111 (1986), 26-33. (InRussian.) (1986) MR0833154
  3. Yu. A. Ryabov, Principal two-sided solutions of Volterra-type linear integrodifferential equations with infinite aftereffect, Ukrain. Matemat. Zhurnal 39 (1987), no. 1, 92-97. (In Russian.) (1987) Zbl0654.45006MR0887728
  4. Ya. V. Bykov, Problems of the Theory of Integrodifferential Equations, Ilim, Fгunze, 1957, 320 p. (In Russian.) (1957) 

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