ABSTRACT
In this chapter we obtain upper estimates of distances between zeros of nontrivial solutions to homogeneous equations. Combining upper and lower estimates of these distances, we develop a new approach to the study of unique solvability of the periodic problem in Paragraphs 14.2 and 14.3. This approach allows to make conclusions about existence and uniqueness of solutions to the periodic problem on intervals the total length of which exceeds the period of the coefficients. In Chapter 15 we use these results to obtain estimates of zones of positivity and oscillation of partial functional differential equations. The Floquet theory for delay differential equations is developed in Paragraph 14.4. We use the presentations of solutions obtained based on the Floquet theory and the estimates of distances between zeros in the study of asymptotic properties of delay equations. Results on existence of unbounded solutions obtained in Chapter 13 do not allow to make conclusions about unboundedness of all solutions of homogeneous delay differential equations of the second order. Distance between zeros in a corresponding example, where a periodic solution exists, is equal to the period of a solution. It is demonstrated that in the case of different distance between zeros and the period, existence of bounded solution is impossible. To formulate explicit tests of this, we use obtained in Paragraphs 14.2 and 14.3 results about upper and lower estimates of distances between zeros.
