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Russia Nizhni Novgorod
Year
2024
Volume
34
Issue
1
Pages
109-136
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Section Mathematics
Title Investigation of conditions for preserving global solvability of operator equations by means of comparison systems in the form of functional-integral equations in $\mathbf{C}[0;T]$
Author(-s) Chernov A.V.a
Affiliations Nizhni Novgorod State Universitya
Abstract Let $U$ be the set of admissible controls, $T>0$ and it be given a scale of Banach spaces $W[0;\tau]$, $\tau\in(0;T]$, such that the set of constrictions of functions from $W=W[0;T]$ to a closed segment $[0;\tau]$ coincides with $W[0;\tau]$; $F[\cdot;u]\colon W\to W$ be a controlled Volterra operator, $u\in U$. For the operator equation $x=F[x;u]$, $x\in W$, we introduce a comparison system in the form of functional-integral equation in the space $\mathbf{C}[0;T]$. We establish that, under some natural hypotheses on the operator $F$, the preservation of the global solvability of the comparison system pointed above is sufficient to preserve (under small perturbations of the right-hand side) the global solvability of the operator equation. This fact itself is analogous to some results which were obtained by the author earlier. The central result of the paper consists in a set of signs for stable global solvability of functional-integral equations mentioned above which do not use hypotheses similar to local Lipschitz continuity of the right-hand side. As a pithy example of special interest, we consider a nonlinear nonstationary Navier–Stokes system in the space $\mathbb{R}^3$.
Keywords second kind evolutionary Volterra equation of general form, functional-integral equation, comparison system, preservation of global solvability, uniqueness of solution, nonlinear nonstationary Navier–Stokes system
UDC 517.957, 517.988, 517.977.56
MSC 47J05, 47J35, 47N10
DOI 10.35634/vm240108
Received 15 July 2023
Language Russian
Citation Chernov A.V. Investigation of conditions for preserving global solvability of operator equations by means of comparison systems in the form of functional-integral equations in $\mathbf{C}[0;T]$, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2024, vol. 34, issue 1, pp. 109-136.
References
  1. Chernov A.V. On totally global solvability of evolutionary Volterra equation of the second kind, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2022, vol. 32, issue 4, pp. 593–614 (in Russian). https://doi.org/10.35634/vm220407
  2. Chernov A.V. On preservation of global solvability of controlled second kind operator equation, Ufa Mathematical Journal, 2020, vol. 12, issue 1, pp. 56–81. https://doi.org/10.13108/2020-12-1-56
  3. Sumin V.I. Volterra functional-operator equations in the theory of optimal control of distributed systems, IFAC-PapersOnLine, 2018, vol. 51, issue 32, pp. 759–764. https://doi.org/10.1016/j.ifacol.2018.11.454
  4. Chernov A.V. Preservation of the solvability of a semilinear global electric circuit equation, Computational Mathematics and Mathematical Physics, 2018, vol. 58, issue 12, pp. 2018–2030. https://doi.org/10.1134/S0965542518120096
  5. Sumin V.I. Controlled Volterra functional equations and the contraction mapping principle, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2019, vol. 25, no. 1, pp. 262–278 (in Russian). https://doi.org/10.21538/0134-4889-2019-25-1-262-278
  6. Sumin V.I. The features of gradient methods for distributed optimal control problems, USSR Computational Mathematics and Mathematical Physics, 1990, vol. 30, issue 1, pp. 1–15. https://doi.org/10.1016/0041-5553(90)90002-A
  7. Kalantarov V.K., Ladyzhenskaya O.A. The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types, Journal of Soviet Mathematics, 1978, vol. 10, issue 1, pp. 53–70. https://doi.org/10.1007/BF01109723
  8. Sumin V.I. Funktsional'nye vol'terrovy uravneniya v teorii optimal'nogo upravleniya raspredelennymi sistemami. Chast' I. Vol'terrovy uravneniya i upravlyaemye nachal'no-kraevye zadachi (Functional Volterra equations in the theory of optimal control of distributed systems. Part I. Volterra equations and controlled initial boundary value problems), Nizhny Novgorod: Nizhny Novgorod State University, 1992.
  9. Korpusov M.O., Sveshnikov A.G. Blow-up of solutions to strongly nonlinear equations of the pseudoparabolic type, Sovremennaya Matematika i ee Prilozheniya, 2006, vol. 40, pp. 3–138 (in Russian).
  10. Sumin V.I., Chernov A.V. Volterra operator equations in Banach spaces: existence stability of global solutions, Nizhny Novgorod State University, Nizhny Novgorod, 2000, 75 p. Deposited in VINITI 25.04.2000, no. 1198-В00 (in Russian).
  11. Hartman P. Ordinary differential equations, New York–London–Sydney: John Wiley and Sons, 1964. https://zbmath.org/0125.32102
  12. Kantorovich L.V., Akilov G.P. Functional analysis, Oxford: Pergamon Press, 1982. https://zbmath.org/0484.46003
  13. Agarwal R.P., O'Regan D., Wong P.J.Y. Constant-sign solutions of systems of integral equations, Cham: Springer, 2013. https://doi.org/10.1007/978-3-319-01255-1
  14. Yang Zhilin. Positive solutions for a system of nonlinear Hammerstein integral equations and applications, Applied Mathematics and Computation, 2012, vol. 218, issue 22, pp. 11138–11150. https://doi.org/10.1016/j.amc.2012.05.006
  15. Bugajewska D., Bugajewski D., Hudzik H. $BV_\varphi$-solutions of nonlinear integral equations, Journal of Mathematical Analysis and Applications, 2003, vol. 287, issue 1, pp. 265–278. https://doi.org/10.1016/S0022-247X(03)00550-X
  16. Hernández-Verón M.A., Yadav N., Martínez E., Singh S. Kurchatov-type methods for non-differentiable Hammerstein-type integral equations, Numerical Algorithms, 2023, vol. 93, issue 1, pp. 131–155. https://doi.org/10.1007/s11075-022-01406-8
  17. Moroz V., Zabreiko P. On Hammerstein equations with natural growth conditions, Zeitschrift für Analysis und ihre Anwendungen, 1999, vol. 18, no. 3, pp. 625–638. https://doi.org/10.4171/ZAA/902
  18. Cabada A., Infante G., Fernández Tojo F.A. Nontrivial solutions of Hammerstein integral equations with reflections, Boundary Value Problems, 2013, vol. 2013, article number: 86. https://doi.org/10.1186/1687-2770-2013-86
  19. López-Somoza L., Minhós F. Existence and multiplicity results for some generalized Hammerstein equations with a parameter, Advances in Difference Equations, 2019, vol. 2019, article number: 423. https://doi.org/10.1186/s13662-019-2359-y
  20. Graef J., Kong Lingju, Minhós F. Generalized Hammerstein equations and applications, Results in Mathematics, 2017, vol. 72, nos. 1–2, pp. 369–383. https://doi.org/10.1007/s00025-016-0615-y
  21. Aziz W., Leiva H., Merentes N. Solutions of Hammerstein equations in the space $BV(I_a$$b$ $)$, Quaestiones Mathematicae, 2014, vol. 37, issue 3, pp. 359–370. https://doi.org/10.2989/16073606.2014.894675
  22. Bugajewski D. On BV-solutions of some nonlinear integral equations, Integral Equations and Operator Theory, 2003, vol. 46, issue 4, pp. 387–398. https://doi.org/10.1007/s00020-001-1146-8
  23. Bugajewska D., O'Regan D. On nonlinear integral equations and $\Lambda$-bounded variation, Acta Mathematica Hungarica, 2005, vol. 107, issue 4, pp. 295–306. https://doi.org/10.1007/s10474-005-0197-8
  24. Pachpatte B.G. On a generalized Hammerstein-type integral equation, Journal of Mathematical Analysis and Applications, 1985, vol. 106, issue 1, pp. 85–90. https://doi.org/10.1016/0022-247X(85)90132-5
  25. Polyanin A.D., Manzhirov A.V. Handbook of integral equations, Boca Raton: CRC Press, 2008. https://zbmath.org/1154.45001
  26. Trenogin V.A. Funktsional'nyi analiz (Functional analysis), Moscow: Nauka, 1980. https://zbmath.org/0517.46001
  27. Chernov A.V. Differentiation of the functional in a parametric optimization problem for the higher coefficient of an elliptic equation, Differential Equations, 2015, vol. 51, issue 4, pp. 548–557. https://doi.org/10.1134/S0012266115040114
  28. Seregin G.A., Shilkin T.N. Liouville-type theorems for the Navier–Stokes equations, Russian Mathematical Surveys, 2018, vol. 73, issue 4, pp. 661–724. https://doi.org/10.1070/RM9822
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