Ukrainian Mathematical Journal

, Volume 71, Issue 4, pp 619–642 | Cite as

Singular Integral Equation Equivalent in the Space of Smooth Functions to an Ordinary Differential Equation, Method of Successive Approximations for the Construction of Its Smooth Solutions and Its Nonsmooth Solutions

  • A. M. SamoilenkoEmail author

We propose a singular integral equation whose definition is extended to a singular point by additional conditions. In the space of smooth functions, this equation becomes equivalent, by the indicated extended definition, to an ordinary differential equation, whereas in the space of piecewise discontinuous functions, it becomes equivalent to an impulsive differential equation. For smooth solutions of the singular equation, we substantiate the method of successive approximations. For the ordinary differential equation, this method turns into a new algorithm for the construction of successive approximations. For the investigated equation, we specify a solution of new type, which is equivalent, for the impulsive differential equation, to a solution with discontinuity of the second kind (a “solution with needle”). We propose an algorithmic formula for the general solution of the initial-value problem for the impulsive differential equation.


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  1. 1.
    A. M. Samoilenko, M. O. Perestyuk, and I. O. Parasyuk, Differential Equations [in Ukrainian], Kyivs’kyi Universytet, Kyiv (2010).Google Scholar
  2. 2.
    V. I. Sobolev, Lectures on Additional Chapters of Mathematical Analysis [in Russian], Nauka, Moscow (1968).Google Scholar
  3. 3.
    A. M. Samoilenko and A. D. Myshkis, “Systems with pushes at given times,” Mat. Sb., 74, No. 2, 202–208 (1967).MathSciNetGoogle Scholar
  4. 4.
    A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations [in Russian], Vyshcha Shkola, Kiev (1987).Google Scholar
  5. 5.
    E. P. Trofimchuk and S. I. Trofimchuk, “Impulse systems with fixed shock times of general disposition: Existence, uniqueness of solution, and the well-posedness of the Cauchy problem,” Ukr. Mat. Zh., 42, No. 2, 230–237 (1990); English translation:Ukr. Math. J., 42, No. 2, 204–209 (1990).MathSciNetCrossRefGoogle Scholar
  6. 6.
    L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, Mathematical Theory of Optimal Processes [in Russian], Nauka, Moscow (1983).zbMATHGoogle Scholar
  7. 7.
    T. Vogel, Théorie des Systémes Evolutifis, Gautnier-Villous, Paris (1965).Google Scholar

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Authors and Affiliations

  1. 1.Institute of MathematicsUkrainian National Academy of SciencesKyivUkraine

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