On the extension of Chaplygin's theorem to the differential equations of neutral type

We consider functional-differential equation $\dot{x}((g(t))=f\big(t,x(h(t))\big), \ t\in [0,1],$ where function $f$ satisfies the Caratheodory conditions, but not necessarily guarantee the boundedness of the respective superposition operator from the space of the essentially bounded functions into the space of integrable functions. As a result, we cannot apply the standard analysis methods (in particular the fixed point theorems) to the integral equivalent of the respective Cauchy problem. Instead, to study the solvability of such integral equation we use the approach based not on the fixed point theorems but on the results received in [A.V. Arutyunov, E.S. Zhukovskiy, S.E. Zhukovskiy. Coincidence points principle for mappings in partially ordered spaces // Topology and its Applications, 2015, v. 179, № 1, 13–33] on the coincidence points of mappings in partially ordered spaces. As a result, we receive the conditions on the existence and estimates of the solutions of the Cauchy problem for the corresponding functional-differential equation similar to the well-known Chaplygin theorem. The main assumptions in the proof of this result are the non-decreasing function $f(t,\cdot)$ and the existence of two absolutely continuous functions $v,w,$ that for almost each $t\in [0,1]$ satisfy the inequalities $\dot{v}(g(t))\geq f\big(t,v(h(t))\big),$ $\dot{w}(g(t))\leq f\big(t,w(h(t))\big).$ The main result is illustrated by an example.