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.