Sturm Separation Theorem Research Articles

Sturm Separation Theorems for a Fourth-Order Equation on a Graph

Sturm Separation Theorems for a Fourth-Order Equation on a Graph

Sturm Separation Theorems for a Fourth-Order Equation on a Graph

Sturm Separation Theorems for a Fourth-Order Equation on a Graph

A converse of Sturm's separation theorem

We show that Sturm's classical separation theorem on the interlacing of the zeros of linearly independent solutions of real second order two-term ordinary differential equations necessarily fails in the presence of a turning point in the principal part of the equation. Related results are discussed.

The Sturm Separation Theorem for Impulsive Delay Differential Equations

Abstract Wronskian is one of the classical objects in the theory of ordinary differential equations. Properties of Wronskian lead to important conclusions on behaviour of solutions of delay equations. For instance, non-vanishing Wronskian ensures validity of the Sturm separation theorem (between two adjacent zeros of any solution there is one and only one zero of every other nontrivial linearly independent solution) for delay equations. We propose the Sturm separation theorem in the case of impulsive delay differential equations and obtain assertions about its validity.

On the relative equilibrium configurations in the planar five-body problem

The number of central configurations in the Grebenicov-Elmabsout model of the planar five-body problem is estimated. An appropriate rational parameterization is used to reduce the equations defining such configurations to some polynomial ones. For the restricted five-body problem a sharp estimation is given by using the Sturm separation theorem.

A transformation for symplectic systems and the definition of a focal point

We examine transformations for symplectic difference systems and Riccati difference operators connected with permutations of rows of a conjoined basis. The concept of an integration path for a conjoined basis is introduced to formulate the definition of a focal point and the disconjugacy criteria and state Sturm's separation theorems in terms of solutions of the transformed Riccati equation.

Interval oscillation criteria for second order neutral nonlinear differential equations

Some oscillation criteria for the second order neutral nonlinear differential equation [y(t)+p(t)y(σ(t))] ″+∑ i=1 nq i(t)f i(y(τ i(t)))=0, t⩾t 0 are established. New oscillation criteria are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [ t 0,∞), rather than on the whole half-line. Our results are more natural according to the Sturm Separation Theorem.

Interval Criteria for Oscillation of Second-Order Linear Ordinary Differential Equations

New oscillation criteria are established for the equation (py′)′+qy=0 that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [t0,∞), rather than on the whole half-line. Our results are more natural according to the Sturm Separation Theorem and sharper than some previous results, and can be applied to extreme cases such as ∫∞t0q(t)dt=−∞.

A sturm separation theorem for a linear 2nth order self‐adjoint differential equation

For the 2nth order equation, (−1) nv(2n) + qv = 0, with q continuous, we obtain a Sturm Separation theorem, involving n + 1 solutions of the equation, which is somewhat analogous to the classical result that the zeros of two linearly independent solutions of the second order equation separate each other.

A Simple Proof for Sturm's Separation Theorem

A Simple Proof for Sturm's Separation Theorem

Comparison theorems for oscillation of nonlinear, non-self-adjoint equations by use of quadratic forms

The purpose of this paper is to show that the theory of quadratic forms developed by M. R. Hestenes and the author may be used to obtain results for oscillation of nonlinear, non-self-adjoint integral-differential equations. Perhaps more important than the results is the philosophy of developing tools and techniques for a quantitative study of oscillation of differential equations on an interval a < t < b as opposed to the qualitative study on a .< t cz 03 usually developed. In this way our results are “meaningful” for real problems (eigenvalue problems, optimization problems, etc.) and oscillation points can be computed by numerical techniques developed by the author. Section 2 is devoted to explaining the relationships between quadratic forms and linear, self-adjoint, integral-differential equations. Section 3 gives the expected “classical” Sturm separation theorem and the elementary comparison results. In Section 4 we consider nonlinear, nonself-adjoint equations. In Section 5 we consider comparison problems for integral-differential equations. We note that our final example involves techniques that will give further results for linear but non-self-adjoint differential equations (including odd order).

On Sturm's Separation Theorem

The purpose of this note is to obtain an extension of the classical Sturm separation theorem for the second order, linear selfadjoint differential equation1to the case of a noncompact interval. The classical theorem (cf. [3, p. 209], [4, p. 224]) assumes that r and s are continuous with r positive on a compact interval I, and concludes that between each pair of zeros (on I) of one (nontrivial) solution of (1) there lies precisely one zero of any other linearly independent solution of (1). If I is not compact, a function y which is a solution of (1) on /may often be extended (continuously) to an endpoint, say a, of I.

A Genralization of Sturm's Separation Theorem

A Genralization of Sturm's Separation Theorem

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 equation (1) is nonoscillatory. As a consequence of Sturm's Separation Theorem [21], if one of the solutions of (1) is oscillatory, then all of them are. The same is true for the nonoscillation of (1). The literature on second order linear oscillation is voluminous. The first such result is of course the classical theorem of Sturm which asserts that

A Supplement to the Sturm Separation Theorem, with Applications

A Supplement to the Sturm Separation Theorem, with Applications

A Supplement to the Sturm Separation Theorem, with Applications

A Supplement to the Sturm Separation Theorem, with Applications

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 will always mean a non-trivial (#0) solution. This note will be concerned principally with solutions w = w(z) of the differential equation (5) under the assumption that 1(z) is analytic on the circle I zI <1. (Unless the contrary is stated below, it will be supposed that 1(z) satisfies this assumption.) The following terminology will be used: If no solution of a differential equation has two zeros (on a given z-set), then the differential equation will be said to be disconjugate (on that set) [11]. Similarly, if no solution has an infinite set of zeros, the differential equation will be called non-oscillatory. (In contrast to the situation on the real field, where Sturm's separation theorem is valid, it is possible that a solution of (5) can have a finite number of zeros on I zj <1 and that another solution has an infinite number of zeros there.) 2. Reduction to a real independent variable. The results on the zeros of solutions of (5) in the case that z is a complex variable will be deduced from cases where 1(z) is a complex-valued function of a real variable (for example, z = x +iy for fixed y). The transfer of these results from the real case will be possible because of the following comparison theorem (cf., e.g., [9, p. 319]):

A Priori Laplace Transformations of Linear Differential Equations

2. If f(t) is a real-valued, continuous function on an unspecified halfline (2), the differential equation (1) is called oscillatory or non-oscillatory according as every or no real-valued solution x (t) * 0 changes sign an infinity of times as t -* oo. In view of Sturm's separation theorem, (1) must be non-oscillatory if it is not oscillatory. As observed by Kneser [3], the differential equation (1) cannot be oscillatory if (3)~ ~~~~~~0 !o J im sup t2f (t) < 4, t-).00