Two-sided estimates for solutions of boundary value problems for implicit differential equations

We consider a two-point (including periodic) boundary value problem for the following system of differential equations that are not resolved with respect to the derivative of the desired function:
$$ f_i (t, x, \dot {x}, \dot {x}_i) = 0, \ \ i = \overline{1, n}. $$
Here, for any $i = \overline{1, n},$ the function $f_i: [0,1] \times \mathbb{R}^n \times \mathbb {R}^n \times \mathbb{R} \to \mathbb {R}$ is measurable in the first argument, continuous in the last argument, right-continuous, and satisfies the special condition of monotonicity in each component of the second and third arguments. Assertions about the existence and two-sided estimates of solutions (of the type of Chaplygin's theorem on differential inequality) are obtained. Conditions for the existence of the largest and the smallest (with respect to a special order) solution are also obtained. The study is based on results on abstract equations with mappings acting from a partially ordered space to an arbitrary set (see [S. Benarab, Z. T. Zhukovskaya, E. S. Zhukovskiy, S. E. Zhukovskiy. On functional and differential inequalities and their applications to control problems // Differential Equations, 2020, 56:11, 1440–1451]).