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, x , x i =0, i= 1, n. Here, for any i = 1, n the function f i :[0, 1]× Rn × Rn ×R→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]).

implicit differential equation, boundary value problem, existence of solutions, estimates of solutions, Chaplygin’s theorem on differential inequality