Nonoscillation Intervals for n-th-Order Equations

In Chap. 17, n-th-order functional differential equations are considered. In order to create a concept of nonoscillation for functional differential equations which will be an analogue of nonoscillation theory for ordinary differential equations, we define a nonoscillation interval for the n-th-order homogeneous functional differential equation as interval, where each nontrivial solution can have at most n−1 zeros, counting every zero with its multiplicity. The study of many classical questions of the qualitative theory of n-th-order equations such as existence and uniqueness of solutions of the interpolation boundary value problems, positivity or corresponding regular behavior of their Green’s functions, maximum principles and stability, was considered in the general framework based on the notion of nonoscillation intervals of corresponding linear functional differential equations. It is demonstrated that if the Wronskian of the fundamental system is not equal to zero at the initial point of the interval, then the nonoscillation interval exists for functional differential equations; this is true for a particular case of the n-th-order functional differential equations with Volterra operators (delay equations). For second-order functional differential equations, nonvanishing Wronskian is equivalent to the Sturm theorem on separation of zeros (between two adjacent zeros of any nontrivial solution there is a zero of every other solution). The analogue of the classical Mammana theorem, establishing necessary and sufficient nonoscillation conditions through nonvanishing normal chains of the Wronskians, is proven for n-th-order functional differential equations. For an n-th-order linear ordinary differential equation the classical Levin-Chichkin theorems claims that Green’s functions of all interpolation (De la Vallee-Poussin) problems behave regularly (preserve their signs in corresponding zones) on the nonoscillation interval. Under corresponding sign conditions on the relevant operators, this theorem is extended to n-th-order functional differential equations. Several possibilities to avoid these sign conditions are proposed, and various explicit tests for positivity and negativity of Green’s functions of two and multipoint boundary value problems are obtained.