ABSTRACT
Generally speaking, the Sturm separation theorem is not valid for delay differential equations. It is even possible that a second order delay equation has both oscillating and nonoscillating solutions.
Wronskian can vanish and its zeros do not depend on the chosen fundamental system. In this chapter we connect nonvanishing Wronskians and consequently the validity of the Sturm separation theorem with estimates of distances between adjacent zeros of nontrivial solutions of homogeneous functional differential equations and between distances between zeros of these solutions and zeros of their derivative. In Paragraph 12.3, assuming a corresponding smallness of delays, we obtain estimates of the distance between zeros of nontrivial solution of homogeneous delay differential equations and conditions on the validity of the Sturm separation theorem on this basis. In Paragraph 12.4 we obtain assertions about nondecreasing Wronskian. In Paragraph 12.5, assuming smallness of delays and existence of fix points in the deviation of the neutral term, we obtain estimates of distances between zeros and the validity of the Sturm separation theorem for neutral delay differential equations. We prove the validity of the Sturm theorem for the binomial (one-term) delay differential equation with non-decreasing deviation in Paragraph 12.2. In Paragraph 12.6, a smallness of differences between delays instead of a smallness of delays is assumed. We demonstrate that a delay equation with several "close to each other" delays inherits oscillation properties of a corresponding binomial (one term) equation and the validity of the Sturm separation theorem. The smallness of difference between delays can be explained, using distances between zeros of nontrivial solutions and zeros of their derivative. These distances are estimated through spectral radii of corresponding compact operators. In Paragraph 12.7, we obtain estimates of distances between zeros of nontrivial solutions of integro-differential equations based on a smallness of the difference between upper and lower limits of integration. We prove that also integro-differential equation in this case preserve the oscillation properties of binomial equations. In Paragraph 12.8 we set a general operator Q instead of delay term. We define properties on this operator under which oscillation properties of binomial equations are preserved. In Paragraphs 12.9 and 12.10 we obtain on this basis estimates of distances between zeros of homogeneous equations and the validity of Sturm separation theorems for neutral delay equations with constant deviation and for integro-differential equation with degenerate kernels and unbounded memory respectively.
