On the Cauchy problem for implicit differential equations of higher orders

A. V. Arutyunov ; E. A. Pluzhnikova

Vestnik rossijskih universitetov. Matematika, Tome 26 (2021) no. 136, pp. 348-362

Résumé

The article is devoted to the study of implicit differential equations based on statements about covering mappings of products of metric spaces. First, we consider the system of equations \begin{equation*} \Phi_i(x_i,x_1,x_2,\ldots,x_n)=y_i, \ \ \ i=\overline{1,n}, \end{equation*} where $\Phi_i: X_i \times X_1 \times \ldots \times X_n \to Y_i,$ $y_i \in Y_i,$ $X_i$ and $Y_i$ are metric spaces, $i=\overline{1,n}.$ It is assumed that the mapping $\Phi_i$ is covering in the first argument and Lipschitz in each of the other arguments starting from the second one. Conditions for the solvability of this system and estimates for the distance from an arbitrary given element $x_0 \in X$ to the set of solutions are obtained. Next, we obtain an assertion about the action of the Nemytskii operator in spaces of summable functions and establish the relationship between the covering properties of the Nemytskii operator and the covering of the function that generates it. The listed results are applied to the study of a system of implicit differential equations, for which a statement about the local solvability of the Cauchy problem with constraints on the derivative of a solution is proved. Such problems arise, in particular, in models of controlled systems. In the final part of the article, a differential equation of the $n$-th order not resolved with respect to the highest derivative is studied by similar methods. Conditions for the existence of a solution to the Cauchy problem are obtained.

Détail

Keywords: implicit differential equations, differential equations of higher orders, the Cauchy problem, covering map, metric space, the Nemytsky operator, functional space.

Bibliographie

[1] E. R. Avakov, A. V. Arutyunov, E. S. Zhukovskii, “Covering mappings and their applications to differential equations not solved with respect to the derivative”, Differential Equations, 45:5 (2009), 627–649 | DOI | MR | Zbl

[2] A. V. Arutyunov, E. S. Zhukovskii, S. E. Zhukovskii, “On the well-posedness of differential equations unsolved for the derivative”, Differential Equations, 47:11 (2011), 1541–1555 | DOI | MR | Zbl

[3] A. V. Arutyunov, E. S. Zhukovskiy, S. E. Zhukovskiy, “Covering mappings and well-posedness of nonlinear Volterra equations”, Nonlinear Analysis: Theory, Methods and Applications, 75:3 (2012), 1026–1044 | DOI | MR | Zbl

[4] W. Merchela, “On stability of solutions of integral equations in the class of measurable functions”, Russian Universities Reports. Mathematics, 26:133 (2021), 44–54 (In Russian) | Zbl

[5] E. S. Zhukovskii, E. A. Pluzhnikova, “Covering Mappings in a Product of Metric Spaces and Boundary Value Problems for Differential Equations Unsolved for the Derivative”, Differential Equations, 49:4 (2013), 420–436 | DOI | MR | Zbl

[6] E. S. Zhukovskii, E. A. Pluzhnikova, “A theorem on operator covering in the product of metric spaces”, Tambov University Reports. Series: Natural and Technical Sciences, 16:1 (2011), 70–72 (In Russian)

[7] E. S. Zhukovskiy, “On Coincidence Points of Multivalued Vector Mappings of Metric Spaces”, Mathematical Notes, 100:3 (2016), 21–37 | MR | Zbl

[8] E. S. Zhukovskiy, “On Coincidence Points for Vector Mappings”, Russian Mathematics, 60:10 (2016), 10–22 | DOI | MR | Zbl

[9] T. V. Zhukovskaia, E. S. Zhukovskiy, “On iterative methods for solving equations with covering mappings”, Numerical Analysis and Applications, 9:4 (2016), 277–287 | DOI | MR | Zbl

[10] E. S. Zhukovskii, E. A. Pluzhnikova, “On controlling objects whose motion is defined by implicit nonlinear differential equations”, Automation and Remote Control, 76:1 (2015), 24–43 | DOI | MR | Zbl

[11] A. V. Arutyunov, E. S. Zhukovskiy, S. E. Zhukovskiy, “On the cardinality of the set of coincidence points of mappings in metric, normed and partially ordered spaces”, Sbornik: Mathematics, 209:8 (2018), 1107–1130 | DOI | MR | Zbl

[12] A. V. Arutyunov, “Nakryvayushchie otobrazheniya v metricheskikh prostranstvakh i nepodvizhnye tochki [Covering mappings in metric spaces and fixed points]”, Doklady Akademii nauk –- Proceedings of the Russian Academy of Sciences, 416:2 (2007), 151–155 (In Russian) | Zbl

[13] A. V. Arutyunov, “Coincidence Points of Two Maps”, Functional Analysis and Its Applications, 48:1 (2014), 72–75 | DOI | MR | Zbl

[14] B. S. Mordukhovich, B. Wang, “Restrictive metric regularity and generalized differential calculus in Banach spaces”, Mathematics and Mathematical Sciences, 2004, no. 50, 2650–2683

[15] A. D. Ioffe, “Metric regularity and subdifferential calculus”, Russian Mathematical Surveys, 55:3 (2000), 501–558 | DOI | MR | Zbl

[16] M. A. Krasnosel'skij, P. P. Zabrejko, E. I. Pustyl'nik, P. E. Sobolevskij, Integral'nye operatory v prostranstvah summiruemyh funkcij, Nauka, Moscow, 1966 (In Russian)