Maximum Principles and Nonoscillation Intervals

In the previous chapters, as well as in the known monographs on nonoscillation of functional differential equations, nonoscillation was only interpreted as existence of eventually positive solutions. In this and the following two chapters, nonoscillation on an interval is considered. We start with a definition of homogeneous equations in such a form that they preserve one dimensional fundamental systems. This allows us to define nonoscillation interval for first-order functional differential equations as an interval where nontrivial solutions of the homogeneous equation do not have zeros. Nonoscillation results developed in this chapter consist of two parts: first, to discover a nonoscillation interval for functional differential equations, and, second, to find various corollaries and applications of nonoscillation results in maximum principles, boundary value problems and stability; in certain cases we can prove equivalence of nonoscillation and corresponding maximum principles. One of the most important results obtained in this chapter for functional differential equations with a positive Volterra operator (in a particular case, for delay differential equations with positive coefficients) is the theorem on 8 equivalences which include nonoscillation, positivity of the Cauchy (fundamental) function, a differential inequality, an integral inequality, an estimate of the spectral radius of a corresponding auxiliary operator and, under the additional condition (for example, that this Volterra operator is nonzero in the case of the periodic problem), positivity of Green’s function of a wide class of boundary value problems. In particular, a generalized periodic problem and problems with integrals in boundary conditions can be considered; many explicit tests of nonoscillation and positivity of the Cauchy and Green’s functions are obtained. These tests require either a short of memory of the Volterra operator (a delay for equation with concentrated delay is small) or smallness of the interval where the functional differential equation is considered. Chapter 15 also proposes new ideas on the study of nonoscillation and positivity of the Cauchy and Green’s functions in the case of a sum of positive and negative operators in first-order functional differential equations. Previous results deduced nonoscillation only in the case when the equation with the positive operator is nonoscillatory; our approach is based on the idea that positive and negative operators can compensate each other. As a result, we deduce nonoscillation in the cases when the equation with only the positive operator is oscillatory, and in the case when their solutions oscillate with amplitudes tending to infinity, we get exponential stability on the basis of nonoscillation.