ABSTRACT
In Chapter 18, an operator-differential equation of quite general type is considered. For investigation of the behavior of the solutions of this equation in the neighborhood of infinity, in subsection 18.1.1 some new properties of monotone functions are studied. For the case of continuous mappings, the problem of existence of positive solutions of integral inequalities is studied in an infinite interval. For a general functional-differential equation, comparison theorems are proved using the Schauder-Tychonoff fixed point theorem. Quite general theorems are proved on the given operator-differential equation having Property A or B. In the case where the right-hand side of the equation is subject to two-sided estimates, necessary and sufficient conditions are proved for the given equation to have Property A or B. From the obtained general assertions, sufficient (necessary and sufficient) conditions are obtained for the given equation to have oscillatory, unbounded and vanishing solutions.
