Uri Elias

Oscillation theory of two-term differential equations

Preface. 0. Introduction. 1. The Basic Lemma. 2. Boundary Value Functions. 3. Bases of Solutions. 4. Comparison of Boundary Value Problems. 5. Comparison Theorems for Two Equations. 6. Disfocality and Its Characterization. 7. Various Types of Disfocality. 8. Solutions on an Infinite Interval. 9. Disconjugacy and its Characterization. 10. Eigenvalue Problems. 11. More Extremal Points. 12. Minors of the Wronskian. 13. The Dominance Property of Solutions. References. Index.

NONOSCILLATION AND EVENTUAL DISCONJUGACY

If every solution of an nth order linear differential equation has only a finite number of zeros in (0, oo), it is not generally true that for sufficiently large c, c > 0, every solution has at most n — 1 zeros in (c, oo). Settling a known conjecture, we show that for any n, the above implication does hold for a special type of equation, L^y + p(x)y = 0, where Ln is an nth order disconjugate differential operator and p(x) is a continuous function of a fixed sign.

The Dynamics of a Piecewise Linear Map and its Smooth Approximation

The paper describes the dynamics of a piecewise linear area preserving map of the plane, F: (x, y) → (1 - y - |x|, x), as well as that portion of the dynamics that persists when the map is approximated by the real analytic map Fe: (x, y) → (1 - y - fe(x), x), where fe(x) is real analytic and close to |x| for small values of e. Our goal in this paper is to describe in detail the island structure and the chaotic behavior of the piecewise linear map F. Then we will show that these islands do indeed persist and contain infinitely many invariant curves for Fe, provided that e is small.

A classification of the solutions of a differential equation according to their behaviour at infinity, II

The solutions of the differential equation L n y + p(×)y = 0, where L n is a disconjugate operator, are classified according to their behaviour as × →∞ . The solution space is decomposed into disjoint sets. We study the dominance properties of the solutions which belong to different sets.

Discrete Sturm comparison theorems on finite and infinite intervals

The Sturm comparison theorem for second-order Sturm–Liouville difference equations on infinite intervals is established and discussed.

A Binomial Identity via Differential Equations

Abstract In the following we discuss a well-known binomial identity. Many proofs by different methods are known for this identity. Here we present another proof, which uses linear ordinary differential equations of the first order.

Rearrangement of a Conditionally Convergent Series

One of the surprising results in an elementary calculus course is that a rearrangement of a conditionaly convergent series may change its sum, even its very convergence. Observing typical textbook examples of this phenomenon, it turns out that during the rearrangement some of the terms are moved arbitrarily large distances from their original locations. Is this necessary? The answer is positive. Indeed, we can assert:

Nonlinear eigenvalue problems for a class of ordinary differential equations

(ani 1(x)(¢¢¢(a1(x)((a0(x)u p0¤ ) 0 ) p1¤ ) 0 ¢¢¢) pni 1¤ ) 0 = ¶ b(x)u r¤ ; where y p¤ = jyj p sgny, pi > 0 and p0p1 ::: pni 1 = r, with various boundary conditions. We prove the existence of eigenvalues and study the zero properties and structure of the corresponding eigenfunctions.

Comparison Theorems for Disfocality and Disconjugacy of Differential Equations

Pairs of ordinary differential equations are compared with respect to disfocality and disconjugacy.

Approximation of the Jacobi Polynomials and the Racah Coefficients

This is the second part of a project which provides asymptotic approximations to the Jacobi polynomials P (α,β) n (x) and to the Racah coefficients P (an+c,bn+d) n (x), as n→∞, where a, b, c, d are constants. The approximations to P (α,β) n (x) are generated by the construction of certain fundamental sets of solutions to a hypergeometric differential equation. In a first step we construct approximations to the Jacobi polynomials and the Racah coefficients on a closed interval [z1, 1] where the solutions are free of zeros. This poses a special challenge since the two endpoints of the interval are a regularsingular point and a turning point of the corresponding differential equation. In the second step we “connect” the approximations of the Jacobi polynomials on [1,∞) through the singular regular point x = 1 to yield a global approximation on [z1,∞). Our global approximation of the Jacobi polynomials on [z1,∞) is obtained without the intervention of “special functions”.