Journal of Mathematical Sciences

, Volume 230, Issue 5, pp 688–694 | Cite as

On Lower Estimates of Solutions and Their Derivatives to a Fourth-Order Linear Integrodifferential Volterra Equation

  • S. IskandarovEmail author
  • G. T. Khalilova


We examine solutions of the problem on sufficient conditions that guarantee a lower estimate and tending to infinity of solutions and their derivatives up to the third order to a fourth-order linear integrodifferential Volterra equation. For this purpose, we develop a method based on the nonstandard reduction method (S. Iskandarov), the Volterra transformation method, the method of shearing functions (S. Iskandarov), the method of integral inequalities (Yu. A. Ved’ and Z. Pakhyrov), the method of a priori estimates (N. V. Azbelev, V. P. Maksimov, L. F. Rakhmatullina, and P. M. Simonov, 1991, 2001), the Lagrange method for integral representations of solutions to first-order linear inhomogeneous differential equations, and the method of lower estimate of solutions (Yu. A. Ved’ and L. N. Kitaeva).

Keywords and phrases

integrodifferential equation a priori estimate lower estimate initial data instability 

AMS Subject Classification

53A40 2015 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    N. V. Azbelev, V. P. Maksimov, and L. F. Rakhmatullina, Introduction to the Theory of Linear Functional Differential Equations, World Federation Publ., Atlanta (1995).zbMATHGoogle Scholar
  2. 2.
    N. V. Azbelev and P. M. Simonov, Stability of Solutions to Ordinary Differential Equations [in Russian], Perm State. Univ., Perm (2001).Google Scholar
  3. 3.
    S. Iskandarov, Method of Weight and Shearing Functions and Asymptotic Properties of Solutions to Integrodifferential and Integral Equations of Volterra Type [in Russian], Ilim, Bishkek (2002).Google Scholar
  4. 4.
    S. Iskandarov, “Method of nonstandard reduction and the exponential stability of a third-order linear ordinary differential equation,” Differ, Equ., 46, No. 6, 898–899 (2010).Google Scholar
  5. 5.
    S. Iskandarov and G. T. Khalilova, “Lower estimates for solutions of a third-order linear Volterra integrodifferential equation,” Vest. Kyrgyz. Univ., Special Issue, 61–65 (2011).Google Scholar
  6. 6.
    L. N. Kitaeva, “On the presence of nonvertical asymptotes of solutions to second-order differential equations with delayed arguments,” in: Research in Integrodifferintial Equations in Kyrgyzstan [in Russian], 3, Ilim, Frunze (1965), pp. 213–222.Google Scholar
  7. 7.
    N. N. Krasovsky, Stability of Motion, Stanford Univ. Press, Stanford (1963).Google Scholar
  8. 8.
    G. I. Marchuk, Splitting Methods [in Russian], Nauka, Moscow (1988).Google Scholar
  9. 9.
    Yu. A. Ved’, “Sufficient conditions of the absence of singular points of III equations,” in: Research in Integrodifferential Equations in Kyrgyzstan [in Russian], 3, Ilim, Frunze (1965), pp. 123–135.Google Scholar
  10. 10.
    V. Volterra, Leçons sur la théorie mathématique de la lutte pour la vie, Gauthier-Villars, Paris (1931).zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Theoretical and Applied Mathematics of the National Academy of Sciences of the Kyrgyz RepublicBishkekKyrgyzstan
  2. 2.Kyrgyz-Russian Academy of EducationBishkekKyrgyzstan

Personalised recommendations