Ukrainian Mathematical Journal

, Volume 54, Issue 12, pp 2060–2066 | Cite as

Asymptotic Behavior of Solutions of the Cauchy Problem x′ = f(t, x, x′), x(0) = 0

  • A. E. Zernov
  • Yu. V. Kuzina


We prove the existence of continuously differentiable solutions \(x:(0,{\rho ]} \to \mathbb{R}^n\) such that
$$\left\| {x\left( t \right) - {\xi }\left( t \right)} \right\| = O\left( {{\eta }\left( t \right)} \right),{ }\left\| {x'\left( t \right) - {\xi '}\left( t \right)} \right\| = O\left( {{\eta }\left( t \right)/t} \right),{ }t \to + 0$$
$$\left\| {x\left( t \right) - S_N \left( t \right)} \right\| = O\left( {t^{N + 1} } \right),{ }\left\| {x'\left( t \right) - S'_N \left( t \right)} \right\| = O\left( {t^N } \right),{ }t \to + 0,$$
$${\xi }:\left( {0,{\tau }} \right) \to \mathbb{R}^n ,{ \eta }:\left( {0,{\tau }} \right) \to \left( {0, + \infty } \right),{ }\left\| {{\xi }\left( t \right)} \right\| = o\left( 1 \right),$$
$${\eta }\left( t \right) = o\left( t \right),{ \eta }\left( t \right) = o\left( {\left\| {{\xi }\left( t \right)} \right\|} \right),{ }t \to + 0,{ }S_N \left( t \right) = \sum\limits_{k = 2}^N {c_k t^k ,}$$
$$c_k \in \mathbb{R}^n ,k \in \left\{ {2,...,N} \right\},{ }0 < {\rho } < {\tau },{ \rho is sufficiently small}{.}$$


Asymptotic Behavior Cauchy Problem Differentiable Solution 
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  1. 1.
    A. N. Vityuk, “Generalized Cauchy problem for system of differential equations that is unsolvable with respect to derivatives,” Differents. Uravn., 7, No. 9, 1575–1580 (1971).Google Scholar
  2. 2.
    N. P. Erugin, I. Z. Shtokalo, and P. S. Bondarenko, Course of Ordinary Differential Equations [in Russian], Vyshcha Shkola, Kiev (1974).Google Scholar
  3. 3.
    V. P. Rudakov, “On existence and uniqueness of solution of systems of differential equations of first order that are partially solvable with respect to derivatives,” Izv. Vuzov. Matematika, No. 9, 79–84 (1971).Google Scholar
  4. 4.
    G. Anichini and G. Conti, “Boundary value problems for implicit ODE's in a singular case,” Different. Equat. Dynam. Systems, 7, No. 4, 437–459 (1999).Google Scholar
  5. 5.
    R. Conti, “Sulla risoluzione dell'equazione F(t, x dx/dt)=0,” Ann. Mat. Pura ed Appl., No. 48, 97–102 (1959).Google Scholar
  6. 6.
    M. Frigon and T. Kaczynski, “Boundary value problems for systems of implicit differential equations,” J. Math. Anal. Appl., 179, No. 2, 317–326 (1993).Google Scholar
  7. 7.
    Z. Kowalski, “The polygonal method of solving the differential equation y′=(t, y, y, y′),” Ann. Pol. Math., 13, No. 2, 173–204 (1963).Google Scholar
  8. 8.
    B. P. Demidovich, Lectures on Mathematical Theory of Stability [in Russian], Nauka, Moscow (1967).Google Scholar
  9. 9.
    N. P. Erugin, The Book for Reading on General Course of Differential Equations [in Russian], Nauka i Tekhnika, Minsk (1972).Google Scholar
  10. 10.
    V. V. Nemytskii and V. V. Stepanov, Quality Theory of Differential Equations [in Russian], Gostekhteoretizdat, Moscow (1949).Google Scholar
  11. 11.
    A. E. Zernov, “On solvability and asymptotic properties of solutions of a singular Cauchy problem,” Differents. Uravn., 28, No. 5, 756–760 (1992).Google Scholar
  12. 12.
    A. E. Zernov, “Quality analysis of implicit singular Cauchy problem,” Ukr. Mat. Zh., 54, No. 3, 302–310 (2001).Google Scholar

Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  • A. E. Zernov
    • 1
  • Yu. V. Kuzina
    • 1
  1. 1.South-Ukrainian Pedagogic UniversityOdessa

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