Non-oscillation theorems

  title={Non-oscillation theorems},
  author={Einar Hille},
  journal={Transactions of the American Mathematical Society},
  • E. Hille
  • Published 1948
  • Mathematics
  • Transactions of the American Mathematical Society
where F(x) is a real-valued function defined for x> 0 and belonging to L(E, 1/E) for each E>0. A solution of (1.1) is a real-valued function y(x), absolutely continuous together with its first derivative, which satisfies the equation for almost all x, in particular at all points of continuity of F(x). We shall say that the equation is non-oscillatory in (a, oo), a>O, if no solution can change its sign more than once in the interval. Since the zeros of linearly independent solutions separate… Expand
Oscillation Criteria for Certain System of Non-Linear Ordinary Differential Equations
where α > 0 and p, g : R+ → R are locally Lebesgue integrable functions. By a solution of system (1) on the interval J ⊆ [0,+∞[ we understand a pair (u, v) of functions u, v : J → R, which areExpand
Integral Inequalities and Second Order Linear Oscillation
on the interval [0, a), u 0. Below, by a solution we mean a non-trivial solution. A solution is oscillatory if it has an infinite number of zeros. Numerous oscillation criteria are known andExpand
Classification of second order linear differential equations with respect to oscillation
where r E P[a, 03), r > 0, and q E C[a, co), are classified by the behavior of their real solutions, as oscillatory or nonoscillatory. In the first instance, one, and thereby every, solution vanishesExpand
Oscillation criteria for second-order damped nonlinear differential equations with p-Laplacian
Abstract This paper is concerned with the oscillation problem for the nonlinear differential equation with a damping term, ( ϕ p ( x ′ ) ) ′ + 2 ( p − 1 ) t ϕ p ( x ′ ) + a ( t ) g ( x ) = 0 , whereExpand
On a Comparison Theorem for Second Order Nonlinear Ordinary Differential Equations
with given functions p and R on [t0 , cc). A function defined on an interval [to, 0), /3 +cc, is said to be oscillatory at 3 if for every a E (to, 0) it has an infinite number of zeros on theExpand
where f : [0, +∞) × R → R is a continuous function and ξ ∈ (0,∞). More precisely, we are looking for conditions yielding existence of positive solutions of (1.1), defined on the whole interval [0,Expand
Oscillation and nonoscillation of solutions of second order linear differential equations with integrable coefficients
(1) x"+a(t)x = 0, t > 0, where a(t) is a locally integrable function of t. We call equation (1) oscillatory if all solutions of (1) have arbitrarily large zeros on [0, oo), otherwise, we say equationExpand
Oscillation constants for second-order nonlinear differential equations with $p$-Laplacian
and a suitable smoothness condition to ensure the uniqueness of solutions ofequation (1.1) to the initial value problem. Then each solution of equation (1.1) and its derivative exist in the future,Expand
Asymptotic linearity of solutions of self-adjoint systems of second order differential equations
A prepared solution U is oscillatory if U(t,) is singular on some sequence t,, n = 1, 2, . . . . t, + co. Otherwise, U is nonoscillatory. In view of the Sturmtype separation theorem of Morse [9] theExpand
On linear, second order differential equations in the unit circle
Hence, in considering zeros of solutions of (1), it can be assumed that (1) has the form (5). The term "solution" will always mean a non-trivial (#0) solution. This note will be concerned principallyExpand


Concerning the zeros of the solutions of certain differential equations
In volume 42 of the Mathematische Annalen Kneser has sfiown that the solutions of equations of the form yin) + qy = 0 oscillate an infinite number of times, provided that xm q > k > 0 forExpand
A treatise on the theory of Bessel functions
1. Bessel functions before 1826 2. The Bessel coefficients 3. Bessel functions 4. Differential equations 5. Miscellaneous properties of Bessel functions 6. Integral representations of BesselExpand
Sur les solutions asymptotiques des équations différentielles
© Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1911, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www.Expand
A General Type of Singular Point.
  • E. Hille
  • Computer Science, Medicine
  • Proceedings of the National Academy of Sciences of the United States of America
  • 1924
Über die reellen Lösungen der linearen Differentialgleichungen zweiter Ordnung, Arkiv för Matematik
  • Astronomi och Fysik vol. 12, no
  • 1917
Weber, Die partiellen Differentialgleichungen der Mathematischen Physik, vol
  • II, 5th ed., Braunschweig,
  • 1912
Untersuchungen über die reellen Nidlstellen der Integrale linearer Differentialgleichungen, Math
  • Ann vol
  • 1893