INITIAL PROBLEM FOR A THIRD ORDER NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS OF CONVOLUTION TYPE

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详细

The article obtains two-sided a priori estimates for the solution of a homogeneous third-order Volterra integro-differential equation with power-law nonlinearity and a difference kernel. It is shown that the lower a priori estimate, which plays the role of a weight function when constructing a metric in the cone of the space of continuous functions, is unimprovable. Using these estimates, using the method of weight metrics (analogous to A. Bielecki’s method), a global theorem on the existence, uniqueness and method of finding a nontrivial solution to the initial problem for the specified integro-differential equation in the class of non-negative continuous functions on the positive half-axis is proved. It is shown that the solution can be found by the method of successive approximations and an estimate of the rate of their convergence to the exact solution is obtained. Examples are given to illustrate the results obtained.

作者简介

S. Askhabov

Kadyrov Chechen State University; Chechen State Pedagogical University; Moscow Institute of Physics and Technology

Email: askhabov@yandex.ru
Grozny, Russia; Grozny, Russia; Dolgoprudny, Russia

参考

  1. Okrasinski, W. Nonlinear Volterra equations and physical applications / W. Okrasinski // Extracta Math. — 1989. — V. 4, № 2. — P. 51–74.
  2. Askhabov S.N. Nonlinear convolution type equations / S.N. Askhabov, M.A. Betilgiriev // Semin. Anal., Oper. Equat. Numer. Anal. 1989/90. — Berlin : Karl-Weierstrass-Institut fu¨r Mathematik, 1990. — P. 1–30.
  3. Brunner, H. Volterra integral equations: an introduction to the theory and applications / H. Brunner. — Cambridge : Cambridge Univ. Press, 2017. — 402 p.
  4. Асхабов, С.Н. Интегро-дифференциальное уравнение типа свертки со степенной нелинейностью и неоднородностью в линейной части / С.Н. Асхабов // Дифференц. уравнения. — 2020. — Т. 56, № 6. — С. 786–795.
  5. Askhabov, S.N. On a second-order integro-differential equation with difference kernels and power nonlinearity / S.N. Askhabov // Bulletin of the Karaganda University. Math. Series. — 2022. — № 2 (106). — P. 38–48.
  6. Эдвардс, Р. Функциональный анализ: теория и приложения / Р. Эдвардс ; пер. с англ. Г.Х. Бермана, И.Б. Раскиной ; под ред. В.Я. Лина. — М. : Мир, 1969. — 1071 с.
  7. Okrasinski, W., Nonlinear Volterra equations and physical applications, Extracta Math., 1989, vol. 4, no. 2, pp. 51– 74.
  8. Askhabov, S.N. and Betilgiriev, M.A., Nonlinear convolution type equations, Semin. Anal., Oper. Equat. Numer. Anal., 1989/90, Berlin: Karl–Weierstrass–Institut fu¨r Mathematik, 1990, pp. 1–30.
  9. Brunner, H., Volterra Integral Equations: an Introduction to the Theory and Applications, Cambridge: Cambridge University Press, 2017.
  10. Askhabov, S.N., Integro-differential equation of the convolution type with a power nonlinearity and an inhomogeneity in the linear part, Differ. Equat., 2020, vol. 56, no. 6, pp. 775–784.
  11. Askhabov, S.N., On a second-order integro-differential equation with difference kernels and power nonlinearity, Bulletin of the Karaganda University. Math. Series, 2022, no. 2 (106), pp. 38–48.
  12. Edwards, R.E., Functional Analysis: Theory and Applications, New York: Holt, Rinehart, and Winston, 1965.

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