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Russia Nizhni Novgorod
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.
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