Nonlinear the first kind Fredholm integro-differential first-order equation with degenerate kernel and nonlinear maxima

In this note, the problems of solvability and construction of solutions for a nonlinear Fredholm one-order integro-differential equation with degenerate kernel and nonlinear maxima are considered.  Using the method of degenerate kernel combined with the method of regularization, we obtain an implicit the first-order functional-differential equation  with the nonlinear maxima.  Initial boundary conditions are used to ensure the solution uniqueness.  In order to use the method of a successive approximations and prove the one value solvability, the obtained implicit functional-differential equation is transformed to the nonlinear Volterra type integro-differential equation with the nonlinear maxima.

  1. Bakirova E. A., Assanova A. T., Kadirbayeva Z. M.  A problem with parameter for the integro-differential equations.  Mathematical Modelling and Analysis.  26 (1), 34–54 (2021).
  2. Efendiev M., Vougalter V.  Solvability of some integro-differential equations with drift.  Osaka Journal of Mathematics.  57 (2), 247–265 (2020).
  3. El-Sayeda A. M. A., Aahmed R. G.  Solvability of the functional integro-differential equation with self-reference and state-dependence.  Journal of Nonlinear Sciences and Applications.  13 (1), 1–8 (2020).
  4. Sidorov N., Sidorov D., Dreglea A.  Solvability and bifurcation of solutions of nonlinear equations with Fredholm operator.  Symmetry.  12 (6), 912 (2020).
  5. Rojas E. M., Sidorov N. A., Sinitsyn A. V.  A boundary value problem for noninsulated magnetic regime in a vacuum diode.  Symmetry.  12 (4), 617 (2020).
  6. Zhang Y.  Solvability of a class of integro-differential equations and connections to one-dimensional inverse problems.  Journal of Mathematical Analysis and Applications.  321 (1), 286–298 (2006).
  7. Falaleev M. V.  Fundamental operator-valued functions of singular integrodifferential operators in Banach spaces.  Journal of Mathematical Sciences.  230 (5), 782–785 (2018).
  8. Falaleev M. V., Orlov S. S.  Degenerate integro-differential operators in Banach spaces and their applications.  Russian Mathematics.  55 (10), 59–69 (2011).
  9. Assanova A. T., Bakirova E. A., Kadirbayeva Z. M.  Numerical solution to a control problem for integro-differential equations.  Computational Mathematics and Mathematical Physics.  60 (2), 203–221 (2020).
  10. Dzhumabaev D. S.  New general solutions to linear Fredholm integro-differential equations and their applications on solving the boundary value problems.  Journal of Computational and Applied Mathematics.  327 (1), 79–108 (2018).
  11. Dzhumabaev D. S., Mynbayeva S. T.  New general solution to a nonlinear Fredholm integro-differential equation.  Eurasian Mathematical Journal.  10 (4), 24–33 (2019).
  12. Dzhumabaev D. S., Mynbayeva S. T.  One approach to solve a nonlinear boundary value problem for the Fredholm integro-differential equation.  Bulletin of the Karaganda university-Mathematics. 97 (1), 27–36 (2020).
  13. Dzhumabaev D. S., Zharmagambetov A. S.  Numerical method for solving a linear boundary value problem for Fredholm integro-differential equations.  News of the National Academy of Sciences of the Republic of Kazakhstan-Series Physico-Mathematical.  2 (312), 5–11 (2017).
  14. Yuldashev T. K.  Nonlocal mixed-value problem for a Boussinesq-type integrodifferential equation with degenerate kernel.  Ukrainian Mathematical Journal.  68 (8), 1278–1296 (2016).
  15. Yuldashev T. K.  Mixed problem for pseudoparabolic integrodifferential equation with degenerate kernel.  Differential equations.  53 (1), 99–108 (2017).
  16. Yuldashev T. K.  Determination of the coefficient and boundary regime in boundary value problem for integro-differential equation with degenerate kernel.  Lobachevskii Journal of Mathematics.  38 (3), 547–553 (2017).
  17. Yuldashev T. K.  Nonlocal boundary value problem for a nonlinear Fredholm integro-differential equation with degenerate kernel.  Differential Equations.  54 (2), 1646–1653 (2018).
  18. Yuldashev T. K.  Spectral features of the solving of a Fredholm homogeneous integro-differential equation with integral conditions and reflecting deviation.  Lobachevskii Journal of Mathematics.  40 (12), 2116–2123 (2019).
  19. Yuldashev T. K.  On the solvability of a boundary value problem for the ordinary Fredholm integrodifferential equation with a degenerate kernel.  Computational Mathematics and Mathematical Physics.  59 (2), 241–252 (2019).
  20. Yuldashev T. K.  On a boundary-value problem for a fourth-order partial integro-differential equation with degenerate kernel.  Journal of Mathematical Sciences.  245 (4), 508–523 (2020).
  21. Imanaliev M. I., Asanov A.  Regularization, Uniqueness and Existence of Solution for Volterra Integral Equations of First Kind.  Studies by Integro-Diff. Equations. Frunze, Ilim.  21, 3–38 (1988), (in Russian).
  22. Lavrent'ev M. M., Romanov V. G. Shishatskii S. R.  Noncorrect problems of mathematical physics and analysis. Moscow, Nauka (1980), (in Russian).