**MSC2010** 34K20, 34K25

### The tests of the stability of one class of autonomous differential ``pseudo-linear'' equations of the first order with autoregulated delay

#### M. B. Ermolaev^{1}, P. M. Simonov^{2}

Annotation | The article is devoted to obtaining effective tests of exponential stability of some classes of autonomous differential equations of first order with autoregulated delay. An overview of works from the city of Perm from the city of Ivanovo on this topic. Given the criterion, S.A. Gusarenko of the continuity of the operator with autoregulated delay. Given the condition V.P. Maksimov on the complete continuity of the operator with autoregulated delay. Sufficient conditions for the existence and continuability of solutions are given. These theorems are based on theorems on stability in the first approximation from the book and from the articles N.V. Azbelev and P.M. Simonov. Theorems on stability in the first approximation in appearance, although resembling the well-known Lyapunov's theorems on the first approximation, however, in reality differ significantly from the latter. Lyapunov's theorems for ordinary differential or functional differential equations give a technique for investigating stability: with the aid of the linearization of the nonlinear part of the equation, the question of the stability of a nonlinear equation reduces to the question of the stability of a linear equation for which effective signs of stability have already been proved. In our case, it is not possible to linearize the nonlinear parts of the equations, and therefore the above technique is not applicable here. In the article, replacing the process of linearization of the equation, so to speak, with ``pseudo-linearization'', and also using the results of the article by V.V. Malygina, we obtained some analogues of theorems on the first approximation for scalar, autonomous equations with autoregulated delay. The main conclusions obtained on the basis of this idea, freely interpreted, can be formalized by the following phrase: autonomous differential equations with autoregulated delay, generally speaking, have similar stability properties than the corresponding differential autonomous equations with concentrated delay. |
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Keywords | autonomous differential equations with autoregulated delay, stability, nonlinear operator of inner superposition, Lyapunov's theorem on stability in the first approximation, contraction operator, fixed point of the operator, admissibility of pairs of spaces |

^{1}**Mikhail B. Ermolaev**, Acting Head of the Department of Economics and Finance, Professor of the Department of Economics and Finance, Ivanovo State University of Chemical Technology (7 Sheremetevsky Avenue, Ivanovo 153000, Ivanovo region, Russia), DSc of Economics, ORCID: http://orcid.org/0000-0003-0217-3821, ermol-mb@mail.ru^{2}**Pyotr M. Simonov**, Professor of the Department of Information Systems and Mathematical Methods in Economics, Perm State National Research University (15 Bukirev Str., Perm 614990, Perm region, Russia), DSc (Physics and Mathematics), ORCID: http://orcid.org/0000-0001-6357-662Х, simpm@mail.ru

**Citation**: M. B. Ermolaev, P. M. Simonov, "[The tests of the stability of one class of autonomous differential ``pseudo-linear'' equations of the first order with autoregulated delay]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,19:2 (2017) 31–52 (In Russian)

**DOI** 10.15507/2079-6900.19.201701.031-052