Non-oscillation domains of differential equations with two parameters
The second order differential equation (1) \(y''+(-\alpha +\beta B(x))y=0\) where \(\alpha\), \(\beta\) are real parameters and x varies over some real interval, arises in many branches of pure and applied mathematics. Hill's equation occurs if B(x) is periodic, and a particularly important case is Mathieu's equation in which \(B(x)=\cos 2x\). Apart from direct applications, these equations, which exhibit parametric excitation, also arise as variational equations from the perturbation of periodic solutions in the theory of nonlinear oscillators. There has been renewed interest in the properties of such equations in the chaotic output of certain forced oscillators. Equation (1) is said to be disconjugate on (-\(\infty,\}\) if and only if every one of its nontrivial solutions has at most one zero in (- \(\infty,\infty)\) It is said to be non-oscillatory on (-\(\infty,\infty)\) if and only if every one of its nontrivial solutions has at most a finite number of zeros in (-\(\infty,\infty)\). The collection of all \((\alpha,\beta)\in R^ 2\) for which (1) is disconjugate (non- oscillatory) on (-\(\infty,\infty)\) will be called the disconjugacy (non- oscillatory) domain of (1). Suppose that these domains are denoted by D and N respectively. \textit{R. A. Moore} [J. Analyse Math. 5, 183-196 (1957; Zbl 0077.087)] showed that \(D=N\) for (1) and that N is a closed, convex and unbounded subset of the \(\alpha\) \(\beta\)-plane. This monograph starts by presenting a review of the properties of the disconjugacy and non-oscillation domains for the more general equation (2) \(y''+(-\alpha A(x)+\beta B(x))y=0,\) on the closed half-line [0,\(\infty)\). The requirements of A and B are the minimal ones of A,B:[0,\(\infty)\to R\) such that each is Lebesgue integrable on every compact subset of [0,\(\infty)\). It is shown that D is always closed and convex. In general \(D\subset N\) and N is also convex though not always closed. In chapter 2 various special properties of D and N are discussed, including conditions on A and B which imply that \(D=N.\) Chapter 3 reviews the notion of disconjugacy for second-order vector differential equations of the form (2) where, in this case, \(y\in R^ n\) and A and B are now real \(n\times n\) matrices whose elements are functions which are Lebesgue integrable on certain compact subsets of \(R^ n\). The properties of D and N in relation to Volterra-Stieltjes integral equations are investigated in Chapter 4. The integral equation formulation includes certain differential and difference equations. Throughout the methods of proof are generally variational in form. The bibliography contains 52 references.
scientific article; zbMATH DE number 193255