On the existence and estimates of solutions to functional equations

We consider the issues of solvability of operator inclusions in partially ordered spaces. We use the notion of ordered covering of multivalued mappings proposed by A. V. Arutyunov, E. S. Zhukovskiy, and S. E. Zhukovskiy in their paper “Coincidence points principle for set-valued mappings in partially ordered spaces”, Topology Appl. 201, 330-343 (2016). A statement on the preservation of properties of an ordered covering under antitone perturbations is proved. Conditions for an ordered covering of the multivalued Nemytskii operator acting from the space of essentially bounded functions to the space of measurable functions are obtained. More exactly, it is established that, if the multivalued mapping $f(t,x)$ is orderly covering in the second argument (in the space $\mathbb{R}^n$), then the corresponding Nemytskii operator (defined as the set of measurable sections of the mapping $f(t,x(t))$) is also orderly covering. These results are used to study a functional inclusion with a deviating argument of the form $0\in g(t,x(h(t)),x(t))$. It is assumed that the multivalued mapping $g(t,x,y)$ is nonincreasing in the second argument and is orderly covering in the third argument. For this inclusion, a solution existence theorem is proved and estimates of solutions are obtained.