On Chaplygin's theorem for an implicit differential equation of order $n$

S. Benarab

Vestnik rossijskih universitetov. Matematika, Tome 26 (2021) no. 135, pp. 225-233

Résumé

We consider the Cauchy problem for the implicit differential equation of order $n$ $$g(t, x, \dot{x}, \ldots, x^{(n)}) = 0, \ \ t \in [0, T ], \ \ x(0) = A.$$ It is assumed that $A=(A_0, \ldots, A_{n-1}) \in \mathbb{R}^{n},$ the function $g: [0, T] \times \mathbb{R}^{n+1} \to \mathbb{R}$ is measurable with respect to the first argument $t \in [0, T],$ and for a fixed $t$, the function $g(t,\cdot): \mathbb{R}^{n+1} \to \mathbb{R}$ is right continuous and monotone in each of the first $n$ arguments, and is continuous in the last $n+1 $-th argument. It is also assumed that for some sufficiently smooth functions $\eta, \nu$, there hold the inequalities \begin{align*} & \nu^{(i)}(0) \geq A_i \geq {\eta}^{(i)}(0), \ \, i=\overline{0,n-1}, \ {\nu}^{(n)}(t) \geq {\eta}^{(n)}(t), \ \, t\in [0,T];\\ & g\big(t,\nu(t),\dot{\nu}(t),\ldots,\nu^{(n)}(t)\big)\geq 0, \ g\big(t,\eta(t),\dot{\eta}(t),\ldots,\eta^{(n)}(t)\big)\leq 0, \ t\in [0,T]. \end{align*} Sufficient conditions for the solvability of the Cauchy problem are derived as well as estimates of its solutions. Moreover, it is shown that under the listed conditions, the set of solutions satisfying the inequalities $ {\nu}^{(n)} (t) \leq {x}^{(n)} (t ) \leq {\nu}^{(n)} (t)$ is not empty and contains solutions with the largest and the smallest $n$-th derivative. This statement is similar to the classical Chaplygin theorem on differential inequality. The proof method uses results on the solvability of equations in partially ordered spaces. Examples of applying the results obtained to the study of second-order implicit differential equations are given.

Détail

Keywords: implicit differential equation of order $n$, largest and smallest solutions, estimates of solutions, Chaplygin's theorem on differential inequality.

Bibliographie

[1] S. A. Chaplygin, “Foundations of a new method of approximate integration of differential equations”, Collected Works, v. I, Gostekhizdat, Moscow, 1948, 348–368 (In Russian)

[2] N. N. Luzin, “On the method of approximate integration of academician S. A. Chaplygin”, Uspekhi Mat. Nauk, 6:46 (1951), 3–27 (In Russian) | Zbl

[3] V. P. Maksimov, L. F. Rakhmatullina, Selected Works of N. V. Azbelev, Institute for Computer Research, Moscow–Izhevsk, 2012 (In Russian)

[4] 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 | Zbl

[5] A. V. Arutyunov , E. S. Zhukovskiy, S. E. Zhukovskiy, “On coincidence points of mappings in partially ordered spaces”, Doklady Mathematics, 88:3 (2013), 710–713 | DOI | Zbl

[6] A. V. Arutyunov , E. S. Zhukovskiy, S. E. Zhukovskiy, “Coincidence points of set-valued mappings in partially ordered spaces”, Doklady Mathematics, 88:3 (2013), 727–729 | DOI | Zbl

[7] A. V. Arutyunov , E. S. Zhukovskiy, S. E. Zhukovskiy, “Coincidence points principle for set-valued mappings in partially ordered spaces”, Topology and its Applications, 201 (2016), 330–343 | DOI | Zbl

[8] A. V. Arutyunov , E. S. Zhukovskiy, S. E. Zhukovskiy, “Coincidence points principle for mappings in partially ordered spaces”, Topology and its Applications, 179:1 (2015), 13-33 | DOI | Zbl

[9] T. V. Zhukovskaya, E. S. Zhukovskiy, I. D. Serova, “Some questions of the analysis of mappings of metricand partially ordered spaces”, Russian Universities Reports. Mathematics, 25:132 (2020), 345–358 (In Russian)

[10] E. S. Zhukovskiy, “On order covering maps in ordered spaces and Chaplygin-type inequalities”, St. Petersburg Math. J., 30:1 (2019), 73–94 | DOI | Zbl

[11] E. S. Zhukovskiy, “On Ordered-Covering Mappings and Implicit Differential Inequalities”, Differential Equations, 52:12 (2016), 1539–1556 | DOI | Zbl

[12] T. V. Zhukovskaya, I. D. Serova, “On estimates of solutions of boundary-value problems for implicit differential equations with deviating argument”, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 2020, no. 186, 38–44 (In Russian)

[13] S. Benarab, Z. T. Zhukovskaya, E. S. Zhukovskiy, S. E. Zhukovskiy, “Functional and differential inequalities and their applications to control problems”, Differential Equations, 56:11 (2021), 1440–1451 | DOI

[14] S. Benarab, “Two-sided estimates for solutions of boundary value problems for implicit differential equations”, Russian Universities Reports. Mathematics, 26:134 (2021), 216–220 (In Russian)

[15] I. V. Shragin, “Superpositional measurability under generalized caratheodory conditions”, Tambov University Reports. Series: Natural and Technical Sciences, 19:2 (2014), 476–478 (In Russian)

[16] N. V. Azbelev, “How it was (On the main stages of development of modern theory of functional differential equations)”, Problems of Nonlinear Analysis in Engineering Systems, 9:1(17) (2003), 1–22 (In Russian)

[17] E. Kamke, Ordinary Differential Equations Handbook, Nauka Publ., Moscow, 1976, 589 pp. (In Russian)